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Strong convergence of split equality variational inequality, variational inclusion, and multiple sets fixed point problems in Hilbert spaces with application

Abstract

This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem.

1 Introduction

The variational inequality problem (VIP) was developed by Stampacchia [1] in 1964, and it provides useful mathematical tools for investigating interesting issues like game theory, economic equilibrium mechanics, partial differential equations, and optimization theory; see [28]. Due to their active role and significance in nonlinear analysis, VIPs are currently growing in both theory and practice; for examples, see [8, 9]. The variational inequality problem (VIP), one of the significant problems, has drawn the interest of numerous mathematicians throughout the years. One can define VIP as finding \(\bar{r}^{*}\in Q\) such that

$$ \bigl\langle L\bigl(\bar{r}^{*}\bigr), p- \bar{r}^{*} \bigr\rangle \ge 0 \quad \text{for all } p \in Q, $$
(1.1)

where L is a nonlinear mapping. The set of solutions of the VIP is represented by \(\operatorname{VIP}(Q, L)\). One of the most often used methods for studying VIPs is the projection method, which can be applied to demonstrate the equivalence between fixed points and VIPs.

In 1994, Censor and Elfving [10] introduced the split feasibility problem (SFP) for modeling inverse problems. SFP identifies a point in a closed convex subset of a Hilbert space such that the image of that point under a particular bounded linear operator belongs to a closed convex subset of a different Hilbert space. Currently, SFPs are implemented in a number of fields, including signal processing [11], computer tomography [12], image restoration [13, 14], and intensity modulated radiation treatment (IMRT) [12, 15]. Recently, numerous authors have presented various split-type problems; see [11, 12, 15] for significant developments in this direction or related topics. In 2005, Censor et al. [12] proposed multiple-sets split feasibility problems (MSSFPs), which have applications in the inverse problem of intensity-modulated radiation therapy. In 2009, Censor and Segal [16] proposed split common fixed point problems (SCFPPs) to generalize the split feasibility problem, the convex feasibility problem, and the multiple-sets split feasibility problem. For more related problems, see [1723].

The split equality problem (SEP) was firstly proposed by Moudafi [24] in 2013. Partial and asymmetric relationships between the variables of two spaces are permitted by SEP. Moudafi [25] studied the split equality fixed point problem (SEFPP), which involves strongly nonexpansive mapping. Additionally, Moudafi and Al-Shemas [26] proved a weak convergence result and proposed the simultaneous iterative method. One can see the above-mentioned iterative approaches need the norm of the bounded linear operators \(\|A\|\) and \(\|B\|\). Calculating the norm of a bounded linear operator can be difficult in some situations. Lopez et al. [27] provided a valuable solution to this numerical challenge. In 2014, Zhao [28] improved the result of Moudafi [16, 24, 25]. By considering the step size that does not need previous knowledge of the operator norm and obtained the weak convergence result for quasi-nonexpansive mappings. A significant convergence result involving quasi-nonexpansive mappings for solving the split common fixed point problem was demonstrated by Shehu et al. [29] in 2017. The split equality variational inequality problem (SEVIP) has been extensively studied and used to solve many real-world issues, including modeling intensity-modulated radiation therapy treatment planning [30]. Mathematically, SEVIPs are very general since they include common solutions of the variational inequality problem [31], split equality zero point problem [32], split equality feasibility problem [33], and common zeros of mappings [34]. SEVIP is stated as a problem of finding \((\bar{r}^{*}, \bar{s}^{*})\in Q_{1}\times Q_{2}\) such that

$$\begin{aligned} \bigl(\bar{r}^{*}, \bar{s}^{*}\bigr)\in \operatorname{VIP} \bigl(Q_{1},L^{x}\bigr)\times \operatorname{VIP} \bigl(Q_{2},L^{y}\bigr) \quad \text{and} \quad A \bar{r}^{*}=B\bar{s}^{*}, \end{aligned}$$
(1.2)

where A, B are bounded linear operators, \(Q_{1}\), \(Q_{2}\) are nonempty closed convex subsets of \(H_{1}\) and \(H_{2}\), respectively, and \(L^{x}:H_{1}\to H_{1}\), \(L^{y}:H_{2}\to H_{2}\) are nonlinear operators.

In 2014, Wu et al. [35] proposed the multiple-sets split equality fixed point problem (MSEFPP) stated as a problem of finding \((\bar{r}^{*}, \bar{s}^{*})\in H_{1}\times H_{2}\) such that

$$\begin{aligned} \bar{r}^{*}\in \bigcap_{k _{1}=1}^{N} \operatorname{Fix}(S_{k_{1}}), \qquad \bar{s}^{*}\in \bigcap _{k _{2}=1}^{P}\operatorname{Fix}(U_{k_{2}}) \quad \text{and} \quad A\bar{r}^{*}=B\bar{s}^{*}, \end{aligned}$$
(1.3)

where \(S_{k_{1}}:H_{1}\to H_{1}\), \(U_{k_{2}}:H_{2}\to H_{2}\) are nonlinear operators with \(1\le k_{1}\le N \), \(1\le k_{2}\le P\) and A, B are bounded linear operators. MSEFPP includes the multiple-sets split equality problem (MSEP), the split equality fixed point problem (SEFPP), the multiple-sets split feasibility problem (MSFP), the split equality problem (SEP), and many others; see [36]. MSEFPP had a significant impact on the growth of various fields including signal processing and image restoration.

Variational inclusion problems had a significant impact on the growth of many fields including mathematical programming, optimal control, mathematical economics, variational inequalities, game theory, complementarity problems, etc. In 2011, Moudafi [33] studied the split monotone variational inclusion problem (SMVIP). SMVIP is to find \(\bar{r}^{*} \in H\) such that

$$\begin{aligned} 0\in K\bigl(\bar{r}^{*}\bigr) \quad \text{and} \quad 0\in \bigcap _{i=1}^{\infty}T\bigl(A\bar{r}^{*} \bigr), \end{aligned}$$
(1.4)

where A is a bounded linear operators, \(K:H_{1}\to H_{1}\) and \(T:H_{2}\to H_{2}\) are set-valued maximal monotone mappings. The general split equality variational inclusion problem (GSEMVIP) was studied by Chang et al. [37] in 2014, generalizing the SMVIP. GSEMVIP is stated as finding \((\bar{r}^{*},\bar{s}^{*})\in H_{1}\times H_{2}\) such that

$$\begin{aligned} 0\in \bigcap_{i=1}^{\infty}K_{i} \bigl(\bar{r}^{*}\bigr), \qquad 0\in \bigcap _{i=1}^{\infty}T_{i}\bigl(\bar{s}^{*} \bigr) \quad \text{and} \quad A\bar{r}^{*}=B\bar{s}^{*}. \end{aligned}$$
(1.5)

The set of solutions to the GSEMVIP is represented by Ω. Chang et al. [37] proved a strong convergence theorem for GSEMVIP. In 2017, Latif and Eslamian [38] introduced a method to find the common solution of three split equality problems, namely variational inequality, equilibrium, and fixed point problems of nonexpansive semigroups. In 2019, Kazmi et al. [39] proposed an iterative method to find the common solution of three split equality problems, namely variational inclusion, generalized general variational-like inequality, and fixed point problems for quasi-nonexpansive mapping.

The inertial extrapolation methods were extensively used as an acceleration technique. The researchers used inertial extrapolation to build numerous iterative algorithms. The critical characteristic of inertial extrapolation is that the following iteration is determined using the results of the previous two iterations. The efficiency of its original (non-inertial) algorithms, like [40, 41], has been significantly enhanced by inertial approaches, as demonstrated by numerical studies. As a result, a lot of research is currently focused on exploiting inertial extrapolation to improve current methods (see [42, 43] and the references therein).

Suppose that \(H_{1}\), \(H_{2}\), and \(H_{3}\) are real Hilbert spaces. Assume that \(A:H_{1}\rightarrow H_{3}\) and \(B:H_{2}\rightarrow H_{3}\) are two bounded linear operators. Suppose that \(\{K_{i}\}_{i=1}^{\infty}:H_{1} \to 2^{H_{1}}\) and \(\{T_{i}\}_{i=1}^{\infty}:H_{2} \to 2^{H_{2}}\) are maximal monotone mappings. Let \(S_{k_{1}}:H_{1}\to H_{1}\) and \(U_{k_{2}}:H_{2}\to H_{2}\) be demicontractive operators such that \(I-S_{k_{1}}\) and \(I-U_{k_{2}}\) are demiclosed at zero for \(1\le {k_{1}} \le N\), \(1\le{k_{2}} \le P\) with constants \(0\le \kappa _{k_{1}}<1\) and \(0\le \phi _{k_{2}}<1\), respectively, where \(\kappa =\max {\kappa _{k_{1}} }\) and \(\phi =\max {\phi _{k_{2}}}\). Define \(S_{k_{1}}=S_{k_{1}(\mod N)+1}\) and \(U_{k_{2}}=U_{k_{2}(\mod P)+1}\). Let \(L_{j}^{x}:H_{1} \to H_{1}\), \(L_{j}^{y}:H_{2} \to H_{2}\) be \(c_{j}^{x}\), \(c_{j}^{y}\)-inverse strongly monotone mappings, respectively, \(\bar{c}^{x}=\min_{j=1,2,\ldots,M}\{c_{j}^{x}\}\) and \(\bar{c}^{y}=\min_{j=1,2,\ldots,M}\{c_{j}^{y}\}\). We represent the required solution set by Γ, where

$$\begin{aligned} \Gamma={}& \Biggl\{ x^{*} \in \bigcap ^{N}_{{k_{1}}=1}\operatorname{Fix}(S_{k_{1}}) \cap \bigcap_{i=1}^{\infty}(K_{i})^{-1}(0) \cap \bigcap_{j=1}^{M}\operatorname{VIP} \bigl(L_{j}^{x}, Q_{1}\bigr), \\ &{} y^{*}\in \bigcap^{M}_{{k_{2}}=1} \operatorname{Fix}(U_{k_{2}}) \cap \bigcap_{i=1}^{ \infty}(T_{i})^{-1}(0) \cap \bigcap_{j=1}^{M} \operatorname{VIP} \bigl(L_{j}^{y}, Q_{2}\bigr) \text{ such that } Ax^{*}=By^{*} \Biggr\} . \end{aligned}$$
(1.6)

2 Preliminaries

In this study, we take H to be a real Hilbert space with the inner product \(\langle .,. \rangle \) and the corresponding norm \(\|.\|\). For weak and strong convergence of \(\{x_{n}\}\) to \(x^{*}\), we use the notations \(x_{n} \rightharpoonup x^{*}\) and \(x_{n} \rightarrow x^{*}\). Assume that Q is a nonempty, closed, and convex subset of a Hilbert space H. The collection of all fixed points for mapping Ū is denoted by \(\operatorname{Fix}(\bar{U})\).

Definition 2.1

Suppose that \(\bar{U}: H\rightarrow H\) is a mapping. Then Ū is called

  1. (i)

    contraction if \(\lambda \in [0,1)\) such that

    $$\Vert \bar{U}u-\bar{U}v \Vert \leq \lambda \Vert \bar{u}-\bar{v} \Vert \quad \text{for all } \bar{u},\bar{v} \in H; $$
  2. (ii)

    λ-demicontractive if there is \(\lambda \in [0,1)\) such that

    $$\Vert \bar{U}\bar{u}-\bar{v} \Vert ^{2}\leq \Vert \bar{u}- \bar{v} \Vert ^{2}+ \lambda \Vert \bar{u}-\bar{U}\bar{u} \Vert ^{2}\quad \text{for all }\bar{u} \in H\text{ and } \bar{v} \in \operatorname{Fix}(\bar{U}); $$
  3. (iii)

    ω̄- inverse strongly monotone (ism) if there is \(\bar{\omega}>0\) such that

    $$\langle \bar{U}\bar{u}-\bar{U}\bar{v},\bar{u}-\bar{v} \rangle \geq \bar{\omega} { \Vert \bar{U}\bar{u}-\bar{U}\bar{v} \Vert }^{2}\quad \text{for all } \bar{u}, \bar{v} \in H. $$

Definition 2.2

Suppose that \(\bar{U}:H \rightarrow 2^{H}\) is a mapping. Then Ū is called

  1. (i)

    monotone if

    $$\langle \bar{r}-\bar{t}, \bar{u}-\bar{v} \rangle \ge 0\quad \text{for all } \bar{r},\bar{t} \in H, \bar{u} \in \bar{U}\bar{r}, \bar{v} \in \bar{U} \bar{t}; $$
  2. (ii)

    maximal monotone if it is monotone and the graph \(G(\bar{U})\) is not properly contained in any other monotone mapping, where

    $$G(\bar{U})= \bigl\{ (\bar{x},\bar{y}) \in H \times H :\bar{y} \in \bar{U}( \bar{x})\bigr\} . $$

Lemma 2.3

[44] Let H be a real Hilbert space, then

  1. (i)

    \(2 \langle \bar{r},\bar{t} \rangle =\|\bar{r}\|^{2}+\| \bar{t}\|^{2}-\|\bar{r}-\bar{t}\|^{2}=\|\bar{r}+\bar{t}\|^{2}-\| \bar{r}\|^{2}-\|\bar{t}\|^{2}\) for all \(\bar{r},\bar{t} \in H\);

  2. (ii)

    \(\|\bar{r}+\bar{t}\|^{2} \leq \|\bar{r}\|^{2}+2 \langle \bar{t}, \bar{r}+\bar{t} \rangle \) for all \(\bar{r},\bar{t} \in H\);

  3. (iii)

    \(\|\alpha _{0} \bar{r}_{0}+\alpha _{1} \bar{r}_{1}+\alpha _{2} \bar{r}_{2}\|^{2}7= \alpha _{0}\|\bar{r}_{0}\|^{2}+\alpha _{1}\| \bar{r}_{1}\|^{2}+\alpha _{2} \|\bar{r}_{2}\|^{2}-\alpha _{0}\alpha _{1} \|\bar{r}_{0}-\bar{r}_{1}\|^{2} -\alpha _{0}\alpha _{2}\|\bar{r}_{0}-\bar{r}_{2}\|^{2}-\alpha _{2} \alpha _{1}\|\bar{r}_{1}-\bar{r}_{2}\|^{2}\) for \(\bar{r}_{i} \in H\), \(\alpha _{i} \in [0,1]\), \(i=0,1,2\), and \(\sum_{i=0}^{2}\alpha _{i}=1\).

Lemma 2.4

[45] The metric projection \(P_{Q}\) satisfies:

  1. (i)

    \(\bar{z}=P_{Q}\bar{u}\) iff \(\langle \bar{u}-\bar{z},\bar{z}-\bar{y} \rangle \leq 0 \) for all \(\bar{y},\bar{z}\in Q\) and \(\bar{u}\in H\);

  2. (ii)

    \(\|P_{Q}\bar{u}-\bar{z}\|^{2} \le \|\bar{u}-\bar{z}\|^{2}-\|P_{Q} \bar{u}-\bar{u}\|^{2} \) for all \(\bar{u}\in H\) and \(\bar{z} \in Q\).

Lemma 2.5

[46] Suppose that \(\bar{U}:H\rightarrow H\) is a κ-demicontractive operator and \(\bar{U}_{\lambda}:=(1-\lambda )I+\lambda \bar{U}\) for any \(\lambda \in (0,1-\kappa )\), where \(\kappa <1\), then for \(\bar{v}^{*} \in \operatorname{Fix}(\bar{U})\) and \(\bar{v}\in H\)

$$ { \bigl\Vert \bar{U}_{\lambda}(\bar{v})-\bar{v}^{*} \bigr\Vert }^{2} \leq { \bigl\Vert \bar{v}- \bar{v}^{*} \bigr\Vert }^{2}-\lambda (1-\kappa -\lambda ){ \Vert \bar{v}-\bar{U} \bar{v} \Vert }^{2}. $$
(2.1)

Lemma 2.6

[47] Suppose that \(\{s_{n} \}\) is a bounded sequence in a Hilbert space H, \(\lim_{n\rightarrow \infty}\|s_{n+1}-s_{n}\|=0\), and \(z^{*}\in \omega _{w}(s_{n})\). Then there is a subsequence \(\{s_{m_{k}}\}\) of \(\{s_{n}\}\) such that \(m_{k}(\mod N)+1=i\) and \(s_{m_{k}}\rightharpoonup z^{*}\), where \(i \in \{1,2,\ldots,N\}\).

Lemma 2.7

[48] Assume that \(\{s_{n}\}\) is a sequence of real numbers and there is a subsequence \(\{n_{k}\}\) of \(\{n\}\) satisfying \(s_{n_{k}}< s_{n_{k}+1} \) for all \(k \in \mathbb{N} \). Then there is a nondecreasing sequence \(\{m_{i}\} \subset \mathbb{N}\) such that \(m_{i}\rightarrow \infty \) and satisfies

$$ s_{m_{i}}\leq s_{m_{i}+1} \quad \textit{and} \quad s_{i}\leq s_{m_{i}+1}, $$
(2.2)

where \(i\in \mathbb{N}\) and \(m_{i}=\max \{k\leq i:s_{k} < s_{k+1}\}\).

Lemma 2.8

[49] Assume that \(\{s_{n}\}\subset [0,\infty )\), \(\{t_{n}\} \subset (-\infty , \infty )\), and \(\{\omega _{n}\}\subset [0,1]\) such that

$$ s_{n+1}\leq (1-\omega _{n})s_{n}+ \omega _{n}t_{n} \quad \forall n\in \mathbb{N}, $$
(2.3)

where \(\sum_{n=0}^{\infty}\omega _{n}=\infty \) and \(\limsup_{n\rightarrow \infty} t_{n}\leq 0\), then \(\lim_{n\rightarrow \infty}s_{n}=0\).

Lemma 2.9

[50] Assume that \(L : H \to 2^{H}\) is a maximal monotone mapping, \(\tau > 0\), and \(J^{L}_{\tau} \) is the resolvent mapping of L defined by \(J^{L}_{\tau}:= (I +{\tau} L)^{-1}\). Suppose that \(L^{-1}(0)\ne \phi \). Then \(\langle \bar{r}^{*}-J^{L}_{\tau}\bar{r}^{*}, J^{L}_{\tau} \bar{r}^{*}-q \rangle \ge 0\) for all \(\bar{r}^{*}\in H\), \(\tau >0\) and \(q\in L^{-1}(0)\).

Lemma 2.10

[51] Let \(L:Q\to H\) be a mapping and \(\bar{r}^{*}\in Q\). Then, for \(b>0\), \(\bar{r}^{*}\in \operatorname{VIP}(Q,L)\) iff \(\bar{r}^{*}= P_{Q}(I-bL) \bar{r}^{*} \), where \(P_{Q}\) is the metric projection of H onto Q.

Lemma 2.11

[52] Let \(L_{j} :Q\to H\) be a \(c_{j}\)-ism mapping with \(\bar{c}=\min_{j=1,2,\ldots,M}\{c_{j}\}\), where \(j=1,2,\ldots,M\) and \(\bigcap_{j=1}^{M} \operatorname{VIP}(Q,L_{j})\ne \phi \), then \(\bigcap_{j=1}^{M} \operatorname{VIP}(Q,L_{j}) =\operatorname{VIP}(Q, \sum_{j=1}^{M}c_{j}L_{j})\), where \(0< b_{j}<1\) for every \(j=1,2,\ldots,M\) and \(\sum_{j=1}^{M}c_{j}=1\).

Lemma 2.12

[53] Let \(\bar{U}:Q\rightarrow Q\) be a μ-demicontractive mapping and \(\operatorname{Fix}(\bar{U})\ne \phi \), then \(\operatorname{Fix}(\bar{U})\) is closed and convex.

Inspired and motivated by ongoing research in this direction, we study a new inertial simultaneous cyclic iterative algorithm to find the common solution of three split equality problems, namely the finite family of variational inequalities, infinite family of variational inclusion, and multiple-sets fixed point problems for demicontractive operators. The proposed algorithm does not require prior knowledge of the operator norm, and we demonstrate our algorithm’s strong convergence under some mild circumstances. We also discuss how our findings can be used to solve the problem of intensity-modulated radiation therapy (IMRT). In addition, we provide a numerical example to illustrate the numerical behavior of the suggested method and to compare it with different methods.

3 Main results

Let \(H_{1}\), \(H_{2}\), \(H_{3}\) be Hilbert spaces, \(Q_{1}\) and \(Q_{2}\) be closed convex subsets of \(H_{1}\) and \(H_{2}\), respectively. Assume that \(\psi _{1}:H_{1} \to H_{1}\), \(\psi _{2}:H_{2} \to H_{2}\) are \(\lambda _{1}\), \(\lambda _{2}\) contractions mappings with \(\lambda =\max \{\lambda _{1}, \lambda _{2}\}\) and \(A : H_{1} \to H_{3}\) and \(B : H_{2} \to H_{3}\) are two (nonzero) bounded linear operators. Let \(A^{*}\) and \(B^{*}\) denote the adjoint of A and B respectively. Let \(L_{j}^{x}:H_{1} \to H_{1}\), \(L_{j}^{y}:H_{2} \to H_{2}\) be \(c_{j}^{x}\), \(c_{j}^{y}\)-inverse strongly monotone mappings, respectively, \(\bar{c}^{x}=\min_{j=1,2,\ldots,M}\{c_{j}^{x}\}\) and \(\bar{c}^{y}=\min_{j=1,2,\ldots,M}\{c_{j}^{y}\}\). Suppose that \(\{K_{i}\}_{i=1}^{\infty}:H_{1} \to 2^{H_{1}}\) and \(\{T_{i}\}_{i=1}^{\infty}:H_{2} \to 2^{H_{2}}\) are maximal monotone mappings. Let \(S_{k_{1}}:H_{1}\to H_{1}\) and \(U_{k_{2}}:H_{2}\to H_{2}\) be demicontractive operators such that \(I-S_{k_{1}}\) and \(I-U_{k_{2}}\) are demiclosed at zero for \(1\le {k_{1}} \le N\), \(1\le{k_{2}} \le P\) with constants \(0\le \kappa _{k_{1}}<1\) and \(0\le \phi _{k_{2}}<1\) respectively, where \(\kappa =\max {\kappa _{k_{1}} }\) and \(\phi =\max {\phi _{k_{2}}}\). Define \(S_{k_{1}}=S_{k_{1}(\mod N)+1}\) and \(U_{k_{2}}=U_{k_{2}(\mod P)+1}\).

Algorithm 3.1

Consider \(0< a<\nu ^{x}_{n}\le 2\bar{c}^{x}\), \(0< b<\nu ^{y}_{n}\le 2\bar{c}^{y}\) for some \(a,b\in \mathbb{R} \), \(b_{j}^{x}, b_{j}^{y}\in (0,1)\) for \(j=1,2,\ldots,M\), \(\{\delta _{n}\}, \{\mu _{n}\}, \{\eta _{n,i}\}, \{\sigma _{n}\} \subset [\bar{d},\bar{e}]\subset (0,1)\), \(a_{n} \in (\delta ,1-\phi -\delta )\), \(\beta \in (0,1)\), \(\sum_{n=1}^{\infty}\tau _{n}<\infty \) and \(\{\kappa _{n}\}\subset [\kappa , \bar{\kappa}] \), where \(0<\kappa \le \bar{\kappa}\), \(\delta >0\), and \(\bar{e} \in \mathbb{R}\). Choose \(x_{0}, x_{1} \in Q\) and \(\gamma _{n}\) such that \(0 \le \gamma _{n}\le \bar{\gamma _{n}}\), where

$$\begin{aligned} \bar{\gamma _{n}}= \textstyle\begin{cases} \beta & \text{if } x_{n}=x_{n-1} \text{ and } y_{n}=y_{n-1} \\ \min \{{\frac{\tau _{n}}{\sqrt{ \Vert x_{n}-x_{n-1} \Vert + \Vert y_{n}-y_{n-1} \Vert }}},\beta \} & \text{if otherwise}. \end{cases}\displaystyle \end{aligned}$$
(3.1)

Compute

$$\begin{aligned} \textstyle\begin{cases} w_{n}=x_{n}+\gamma _{n}(x_{n}-x_{n-1}),\\ t_{n}=y_{n}+\gamma _{n}(y_{n}-y_{n-1}), \\ h_{n}=\sigma _{n} \psi _{1}(x_{n})+(1-\sigma _{n})P_{Q_{1}}(I-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x})w_{n},\\ x_{n+1}=\delta _{n} h_{n}+\mu _{n}((1-a_{n})h_{n}+a_{n}S_{n}(h_{n})) \\ \hphantom{x_{n+1}=}{} +\sum_{i=1}^{\infty}\eta _{n,i}(J_{\theta _{i}}^{K_{i}}(h_{n}-\rho _{n}(A^{*}(Ah_{n}-Bv_{n})))),\\ v_{n}=\sigma _{n} \psi _{2}(y_{n})+(1-\sigma _{n})P_{Q_{2}}(I-\nu _{n}^{y}\sum_{j=1}^{M} b_{j}^{y}L_{j}^{y})t_{n},\\ y_{n+1}=\delta _{n} v_{n}+\mu _{n} ((1-a_{n})v_{n}+a_{n}U_{n}(v_{n})) \\ \hphantom{y_{n+1}=}{}+\sum_{i=1}^{\infty}\eta _{n,i}(J_{\theta _{i}}^{T_{i}}(v_{n}+\rho _{n}(B^{*}(Ah_{n}-Bv_{n})))), \end{cases}\displaystyle \end{aligned}$$
(3.2)

where the stepsize \(\rho _{n}\) is given as

$$\begin{aligned}& \rho _{n}\in \biggl(\epsilon , \frac{2 \Vert Ah_{n}-Bv_{n} \Vert ^{2}}{ \Vert A^{*}(Ah_{n}-Bv_{n}) \Vert ^{2}+ \Vert B^{*}(Ah_{n}-Bv_{n}) \Vert ^{2}}-\epsilon \biggr), \\& n\in \Delta \text{ and }\epsilon >0\text{ otherwise }\rho _{n}=\rho , \text{and the index set }\Delta =\{n; Ah_{n}-Bv_{n}\ne 0\}. \end{aligned}$$

Remark 3.2

Assume that the solution set Γ is nonempty, then \(\{\rho _{n}\}\) in Algorithm 3.1 is well defined; see [54].

Lemma 3.3

Let the solution set Ω be nonempty, \(\{x_{n}\}\) and \(\{y_{n}\}\) be bounded sequences. Suppose that \(\|l_{n}-J_{\theta _{i}}^{(K_{i},T_{i})}(I-\rho _{{n}}(H^{*}H)l_{n}) \|\to 0\), \(\|h_{n}-w_{n}\|\to 0\), \(\|v_{n}-t_{n}\|\to 0\), \(\|x_{n}-w_{n}\|\to 0\), and \(\|y_{n}-t_{n}\|\to 0\). Then \(l^{*}=(x^{*},y^{*})\) is a solution of Problem (1.5) iff \(l^{*}= J_{\theta _{i}}^{(K_{i},T_{i})}(I-\rho _{n}(H^{*}H))l^{*}\) for every \(\theta _{i}>0\) and \(\rho _{n}>0\), where \(i\ge 1\), \(n\in \mathbb{N}\),

$$\begin{aligned}& H= \begin{bmatrix} A & -B \end{bmatrix} ,\qquad H^{*}= \begin{bmatrix} A^{*} \\ -B^{*} \end{bmatrix} , \\& H^{*}H= \begin{bmatrix} A^{*}A & -A^{*}B \\ -B^{*}A& B^{*}B \end{bmatrix} \quad \textit{and}\quad J_{\theta _{i}}^{(K_{i},T_{i})}= \begin{bmatrix} J_{\theta _{i}}^{K_{i}} \\ J_{\theta _{i}}^{T_{i}} \end{bmatrix} . \end{aligned}$$

Proof

Now, we have to show that \(l^{*}=(x^{*}, y^{*})\in \Omega \). For this we have to show for \(i\ge 1\), \(l^{*}=J_{\theta _{i}}^{(K_{i},T_{i})}(I-\rho _{n}(H^{*}H))l^{*}\). Suppose that \(l^{*}\ne J_{\theta _{i}}^{(K_{i},T_{i})}(I-\rho _{n}(H^{*}H))l^{*}\). As \(\{x_{n}\}\) and \(\{y_{n}\}\) are bounded, take subsequences \(\{x_{n_{q}}\}\) of \(\{x_{n}\}\) and \(\{y_{n_{q}}\}\) of \(\{y_{n}\}\) such that \(x_{n_{q}} \rightharpoonup x^{*} \in H_{1} \) and \(y_{n_{q}} \rightharpoonup y^{*} \in H_{2} \). Additionally, from given conditions, there are subsequences \(\{h_{n_{q}}\}\) \(\{w_{n_{q}}\}\), \(\{v_{n_{q}}\}\), and \(\{t_{n_{q}}\}\) respectively such that \(h_{n_{q}}\rightharpoonup x^{*} \), \(w_{n_{q}}\rightharpoonup x^{*} \), \(v_{n_{q}}\rightharpoonup y^{*} \), and \(t_{n_{q}}\rightharpoonup y^{*} \). Define \(l_{n}=(h_{n}, v_{n})\). Using the given condition, we have

$$\begin{aligned} \lim_{q\rightarrow \infty} \bigl\Vert l_{n_{q}}- \bigl(J_{\theta _{i}}^{(K_{i},T_{i})}\bigl(I- \rho _{{n_{q}}} \bigl(H^{*}H\bigr)l_{n_{q}}\bigr)\bigr) \bigr\Vert =0. \end{aligned}$$
(3.3)

Now, using Opial’s condition and equation (3.3), we get

$$\begin{aligned} \liminf_{q\to \infty} \bigl\Vert l_{n_{q}}-l^{*} \bigr\Vert < {}& \liminf_{q\to \infty} \bigl\Vert l_{n_{q}}-J_{ \theta _{i}}^{(K_{i},T_{i})}\bigl(I-\rho _{n} \bigl(H^{*}H\bigr)\bigr)l^{*} \bigr\Vert \\ \le{}& \liminf_{q\to \infty} \bigl\Vert l_{n_{q}}-J_{\theta _{i}}^{(K_{i},T_{i})} \bigl(I- \rho _{n}\bigl(H^{*}H\bigr)\bigr)l_{n_{q}} \bigr\Vert \\ &{}+\liminf_{q\to \infty} \bigl\Vert J_{\theta _{i}}^{(K_{i},T_{i})} \bigl(I-\rho _{n}\bigl(H^{*}H\bigr)\bigr)l_{n_{q}}-J_{ \theta _{i}}^{(K_{i},T_{i})} \bigl(I-\rho _{n}\bigl(H^{*}H\bigr)\bigr)l^{*} \bigr\Vert \\ \le{}& \liminf_{q\to \infty} \bigl\Vert l_{n_{q}}-l^{*} \bigr\Vert , \end{aligned}$$

which is a contradiction. Hence \(l^{*}= J_{\theta _{i}}^{(K_{i},T_{i})}(I-\rho _{n}(H^{*}H))l^{*}\) for \(i\ge 1\).

Conversely, assume that \(l^{*}= J_{\theta _{i}}^{(K_{i},T_{i})}(I-\rho _{n}(H^{*}H))l^{*}\). Subsequently,

$$\begin{aligned} \textstyle\begin{cases} x^{*}= J_{\theta _{i}}^{K_{i}}( x^{*}-\rho _{n}(A^{*}(A x^{*}-By^{*}))) \\ y^{*}= J_{\theta _{i}}^{T_{i}}(y^{*}+\rho _{n}(B^{*}(A x^{*}-By^{*}))). \end{cases}\displaystyle \end{aligned}$$
(3.4)

As the solution set is nonempty, \(K_{i}^{-1}(0)\) and \(T_{i}^{-1}(0)\) are nonempty. Using Lemma 2.9 and equation (3.4), we have

$$\begin{aligned} \bigl\langle x^{*}-\bigl( x^{*}-\rho _{n} \bigl(A^{*}\bigl(A x^{*}-By^{*}\bigr)\bigr)\bigr), x_{1}-x^{*} \bigr\rangle \ge 0 \quad \text{for all } x_{1}\in K_{i}^{-1}(0), \end{aligned}$$

which implies

$$\begin{aligned} \bigl\langle A x^{*}-By^{*}, Ax_{1}-Ax^{*} \bigr\rangle \ge 0 \quad \text{for all } x_{1}\in K_{i}^{-1}(0). \end{aligned}$$
(3.5)

Similarly,

$$\begin{aligned} \bigl\langle A x^{*}-By^{*}, By^{*}-By_{1} \bigr\rangle \ge 0 \quad \text{for all } y_{1}\in T_{i}^{-1}(0). \end{aligned}$$
(3.6)

Adding equations (3.5) and (3.6), we have

$$\begin{aligned} \bigl\langle A x^{*}-By^{*}, Ax_{1}-Ax^{*}+By^{*}-By_{1} \bigr\rangle \ge 0 \quad \text{for all } x_{1}\in K_{i}^{-1}(0) \text{ and } y_{1}\in T_{i}^{-1}(0). \end{aligned}$$
(3.7)

On simplifying

$$\begin{aligned} \bigl\Vert A x^{*}-By^{*} \bigr\Vert ^{2}\le \bigl\langle A x^{*}-By^{*}, Ax_{1}-By_{1} \bigr\rangle \quad \text{for all } x_{1}\in K_{i}^{-1}(0)\text{ and } y_{1}\in T_{i}^{-1}(0). \end{aligned}$$

As the solution set is nonempty, let \(l'=(x',y')\in \Omega \) for each \(i\ge 1\). Therefore, \(x' \in K_{i}^{-1}(0)\) and \(y'\in T_{i}^{-1}(0)\) and \(Ax'=By'\). Taking \(x_{1}=x'\) and \(y_{1}=y'\) in equation (3.7), we have

$$\begin{aligned} \bigl\Vert A x^{*}-By^{*} \bigr\Vert =0, \end{aligned}$$

which implies \(A x^{*}=By^{*}\), and using \(A x^{*}=By^{*}\) in equation (3.4), we have

$$\begin{aligned} \textstyle\begin{cases} x^{*}= J_{\theta _{i}}^{K_{i}}( x^{*}) \\ y^{*}= J_{\theta _{i}}^{T_{i}}(y^{*}), \end{cases}\displaystyle \end{aligned}$$
(3.8)

which implies \(0\in K_{i}(x^{*})\) and \(0\in T_{i}(y^{*})\) for all \(i\ge 1\). Hence, \(l^{*}\) is the solution of Problem (1.5). □

Theorem 3.4

Let the sequence \(\{(x_{n}, y_{n})\}\) be generated by the iterative Algorithm 3.1and a solution set Γ be nonempty. Suppose that the following conditions are satisfied:

  1. (i)

    \(\sum_{j=1}^{M}b_{j}^{x}=\sum_{j=1}^{M}b_{j}^{y}=1\), \(\sum_{n=1}^{\infty}\nu ^{x}_{n}<\infty \), \(\sum_{n=1}^{\infty}\nu ^{y}_{n}< \infty \),

    $$0< a< \nu ^{x}_{n}\le 2\bar{c}^{x},\qquad 0< b< \nu ^{y}_{n}\le 2\bar{c}^{y}\quad \textit{for some }a,b \in \mathbb{R} ; $$
  2. (ii)

    \(\tau _{n}=o(\sigma _{n})\), \(\lim_{n\to \infty}\frac{\tau _{n}}{\sigma _{n}}=0\), \(\lim_{n\to \infty}\sigma _{n}=0\), \(\sum_{n=1}^{\infty}\sigma _{n}= \infty \);

  3. (iii)

    \(\delta _{n}+\mu _{n}+\sum_{i=1}^{\infty}\eta _{n,i}=1\) for \(n\ge 0\),

then the sequence generated by the iterative Algorithm 3.1converges strongly to \((\bar{r}^{*}, \bar{s}^{*})\in \Gamma \).

Proof

Firstly, we will show that the sequence \(\{(x_{n}, y_{n})\}\) is bounded. Take \((\bar{r}^{*}, \bar{s}^{*})\in \Gamma \). Let

$$\begin{aligned} u_{n}^{1}=h_{n}-\rho _{n}\bigl(A^{*}(Ah_{n}-Bv_{n})\bigr) \quad \text{and} \quad u_{n}^{2}=v_{n}+\rho _{n}\bigl(B^{*}(Ah_{n}-Bv_{n})\bigr), \end{aligned}$$
(3.9)

then using Lemma 2.3, we estimate

$$\begin{aligned} \bigl\Vert u_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2}={}& \bigl\Vert h_{n}-\rho _{n}\bigl(A^{*}(Ah_{n}-Bv_{n}) \bigr)- \bar{r}^{*} \bigr\Vert ^{2} \\ ={}& \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+\rho _{n}^{2} \bigl\Vert A^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2}-2 \rho _{n} \bigl\langle h_{n}- \bar{r}^{*}, A^{*}(Ah_{n}-Bv_{n}) \bigr\rangle \\ ={}& \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+\rho _{n}^{2} \bigl\Vert A^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2}-2 \rho _{n} \bigl\langle Ah_{n}-A \bar{r}^{*}, Ah_{n}-Bv_{n} \bigr\rangle \\ ={}& \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+\rho _{n}^{2} \bigl\Vert A^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2}- \rho _{n} \bigl\Vert Ah_{n}-A \bar{r}^{*} \bigr\Vert ^{2} \\ &{}-\rho _{n} \Vert Ah_{n}-Bv_{n} \Vert ^{2}+ \rho _{n} \bigl\Vert Bv_{n}-A \bar{r}^{*} \bigr\Vert ^{2} \end{aligned}$$
(3.10)

and

$$\begin{aligned} \bigl\Vert u_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2}={}& \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}+\rho _{n}^{2} \bigl\Vert B^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2}-\rho _{n} \bigl\Vert Bv_{n}-B\bar{s}^{*} \bigr\Vert ^{2} \\ &{} - \rho _{n} \Vert Ah_{n}-Bv_{n} \Vert ^{2}+\rho _{n} \bigl\Vert Ah_{n}-B \bar{s}^{*} \bigr\Vert ^{2}. \end{aligned}$$
(3.11)

As \(A\bar{r}^{*}=B\bar{s}^{*}\) and adding equations (3.10) and (3.11), we have

$$\begin{aligned} & \bigl\Vert u_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert u_{n}^{2}-\bar{s}^{*} \bigr\Vert ^{2} \\ &\quad = \bigl\Vert h_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}+\rho _{n}^{2} \bigl( \bigl\Vert A^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2}+ \bigl\Vert B^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2} \bigr) \\ &\qquad {}-2\rho _{n} \Vert Ah_{n}-Bv_{n} \Vert ^{2} \\ &\quad = \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}-\rho _{n} \bigl[2 \Vert Ah_{n}-Bv_{n} \Vert ^{2}-\rho _{n} \bigl( \bigl\Vert A^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2} \\ &\qquad {}+ \bigl\Vert B^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2} \bigr) \bigr] \\ &\quad \le \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}-\rho _{n} \epsilon \bigl( \bigl\Vert A^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2}+ \bigl\Vert B^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2} \bigr) \end{aligned}$$
(3.12)
$$\begin{aligned} &\quad \le \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2} . \end{aligned}$$
(3.13)

Let \(z_{n}^{1}=P_{Q_{1}}(I-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x})w_{n}\) and \(z_{n}^{2}=P_{Q_{2}}(I-\nu _{n}^{y}\sum_{j=1}^{M} b_{j}^{y}L_{j}^{y})t_{n}\).

As \(\bar{r}^{*} \in \operatorname{VIP}(L_{j}^{x},Q_{1})\). Also, from the definition of inverse strongly monotone and from condition (i), we have

$$\begin{aligned} \bigl\Vert z_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2}={}& \Biggl\Vert P_{Q_{1}}\Biggl(I-\nu _{n}^{x}\sum _{j=1}^{M} b_{j}^{x}L_{j}^{x} \Biggr)w_{n}-P_{Q_{1}}\Biggl(I-\nu _{n}^{x} \sum_{j=1}^{M} b_{j}^{x}L_{j}^{x} \Biggr) \bar{r}^{*} \Biggr\Vert ^{2} \\ \le{}& \Biggl\Vert \Biggl(I-\nu _{n}^{x}\sum _{j=1}^{M} b_{j}^{x}L_{j}^{x} \Biggr)w_{n}-\Biggl(I- \nu _{n}^{x}\sum _{j=1}^{M} b_{j}^{x}L_{j}^{x} \Biggr)\bar{r}^{*} \Biggr\Vert ^{2} \\ ={}& \Biggl\Vert \bigl(w_{n}-\bar{r}^{*}\bigr)-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}\bigl(L_{j}^{x}w_{n}-L_{j}^{x} \bar{r}^{*}\bigr) \Biggr\Vert ^{2} \\ \le{}& \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}-2\nu _{n}^{x}\sum _{j=1}^{M} b_{j}^{x} \bigl\langle w_{n}-\bar{r}^{*}, L_{j}^{x}w_{n}-L_{j}^{x} \bar{r}^{*} \bigr\rangle \\ &{} +\sum_{j=1}^{M} b_{j}^{x} \bigl(\nu _{n}^{x}\bigr)^{2} \bigl\Vert L_{j}^{x}w_{n}-L_{j}^{x} \bar{r}^{*} \bigr\Vert ^{2} \\ \le{}& \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}-2\nu _{n}^{x}\sum _{j=1}^{M} b_{j}^{x}c_{j}^{x} \bigl\Vert L_{j}^{x}w_{n}-L_{j}^{x} \bar{r}^{*} \bigr\Vert ^{2}+\sum _{j=1}^{M} b_{j}^{x}\bigl( \nu _{n}^{x}\bigr)^{2} \bigl\Vert L_{j}^{x}w_{n}-L_{j}^{x} \bar{r}^{*} \bigr\Vert ^{2} \\ \le{}& \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}-\nu _{n}^{x}\sum _{j=1}^{M} b_{j}^{x}\bigl(2 \bar{c}^{x}-\nu _{n}^{x}\bigr) \bigl\Vert L_{j}^{x}w_{n}-L_{j}^{x} \bar{r}^{*} \bigr\Vert ^{2} \end{aligned}$$
(3.14)
$$\begin{aligned} \le{}& \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}. \end{aligned}$$
(3.15)

Similarly,

$$\begin{aligned} \bigl\Vert z_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2}&= \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2}-\nu _{n}^{y} \sum_{j=1}^{M} b_{j}^{y}\bigl(2\bar{c}^{y}-\nu _{n}^{y}\bigr) \bigl\Vert L_{j}^{y}t_{n}-L_{j}^{y} \bar{s}^{*} \bigr\Vert ^{2} \end{aligned}$$
(3.16)
$$\begin{aligned} & \le \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2}. \end{aligned}$$
(3.17)

Adding equations (3.14) and (3.16), we have

$$\begin{aligned} \bigl\Vert z_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert z_{n}^{2}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2}-\nu _{n}^{x}\sum _{j=1}^{M} b_{j}^{x}\bigl(2 \bar{c}^{x}-\nu _{n}^{x}\bigr) \bigl\Vert L_{j}^{x}w_{n}-L_{j}^{x} \bar{r}^{*} \bigr\Vert ^{2} \\ &{}-\nu _{n}^{y}\sum_{j=1}^{M} b_{j}^{y}\bigl(2\bar{c}^{y}-\nu _{n}^{y}\bigr) \bigl\Vert L_{j}^{y}t_{n}-L_{j}^{y} \bar{s}^{*} \bigr\Vert ^{2} \end{aligned}$$
(3.18)
$$\begin{aligned} \le{}& \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2} . \end{aligned}$$
(3.19)

As \(\bar{r}^{*} \in \operatorname{VIP}(L_{j}^{x},Q_{1})\), we have

$$\begin{aligned} \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2}={}& \Biggl\Vert P_{Q_{1}}\Biggl(I-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}\Biggr)w_{n}-P_{Q_{1}} \Biggl(I-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}\Biggr) \bar{r}^{*} \Biggr\Vert ^{2} \\ \le{}& \Biggl\langle z_{n}^{1}-\bar{r}^{*}, \Biggl(I-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}\Biggr)w_{n}- \Biggl(I-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}\Biggr) \bar{r}^{*} \Biggr\rangle \\ \le{}& \frac{1}{2}\{ \bigl\Vert z_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2}+ \Biggl\Vert \Biggl(I-\nu _{n}^{x} \sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}\Biggr)w_{n}- \Biggl(I-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}\Biggr) \bar{r}^{*} \Biggr\Vert ^{2} \\ &{}- \Biggl\Vert \bigl(z_{n}^{1}-w_{n}\bigr)+ \nu _{n}^{x} \sum_{j=1}^{M} b_{j}^{x}\bigl(L_{j}^{x}(w_{n})-L_{j}^{x} \bigl(\bar{r}^{*}\bigr)\bigr) \Biggr\Vert ^{2}. \end{aligned}$$
(3.20)

Using \(I-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}\) is nonexpansive and on rearranging the terms, we have

$$\begin{aligned} \bigl\Vert z_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2}\le{}& \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}- \bigl\Vert z_{n}^{1}-w_{n} \bigr\Vert ^{2}- \Biggl(\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}\Biggr)^{2} \bigl\Vert L_{j}^{x}(w_{n})-L_{j}^{x} \bigl( \bar{r}^{*}\bigr) \bigr\Vert ^{2} \\ &{}+2\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x} \bigl\langle w_{n}-z_{n}^{1},L_{j}^{x}(w_{n})-L_{j}^{x} \bigl(\bar{r}^{*}\bigr) \bigr\rangle \\ \le{}& \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}- \bigl\Vert z_{n}^{1}-w_{n} \bigr\Vert ^{2}+2\nu _{n}^{x} \sum _{j=1}^{M} b_{j}^{x} \bigl\Vert w_{n}-z_{n}^{1} \bigr\Vert \bigl\Vert L_{j}^{x}(w_{n})-L_{j}^{x} \bigl( \bar{r}^{*}\bigr) \bigr\Vert . \end{aligned}$$
(3.21)

Similarly,

$$\begin{aligned} \bigl\Vert z_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2} &\le \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2}- \bigl\Vert z_{n}^{2}-t_{n} \bigr\Vert ^{2}+2 \nu _{n}^{y}\sum_{j=1}^{M} b_{j}^{y} \bigl\Vert t_{n}-z_{n}^{2} \bigr\Vert \bigl\Vert L_{j}^{y}(t_{n})-L_{j}^{y} \bigl( \bar{s}^{*}\bigr) \bigr\Vert . \end{aligned}$$
(3.22)

Adding equations (3.21) and (3.22), we have

$$\begin{aligned} \bigl\Vert z_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2} + \bigl\Vert z_{n}^{2}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2}- \bigl\Vert z_{n}^{1}-w_{n} \bigr\Vert ^{2}- \bigl\Vert z_{n}^{2}-t_{n} \bigr\Vert ^{2} \\ &{}+2\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x} \bigl\Vert w_{n}-z_{n}^{1} \bigr\Vert \bigl\Vert L_{j}^{x}(w_{n})-L_{j}^{x} \bigl(\bar{r}^{*}\bigr) \bigr\Vert \\ &{}+2\nu _{n}^{y}\sum_{j=1}^{M} b_{j}^{y} \bigl\Vert t_{n}-z_{n}^{2} \bigr\Vert \bigl\Vert L_{j}^{y}(t_{n})-L_{j}^{y} \bigl( \bar{s}^{*}\bigr) \bigr\Vert . \end{aligned}$$
(3.23)

Using equation (3.2), we get

$$\begin{aligned} \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert &= \bigl\Vert x_{n}+\gamma _{n}(x_{n}-x_{n-1})- \bar{r}^{*} \bigr\Vert \\ &\le \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert +\gamma _{n} \Vert x_{n}-x_{n-1} \Vert . \end{aligned}$$
(3.24)

Similarly,

$$\begin{aligned} \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert &\le \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert +\gamma _{n} \Vert y_{n}-y_{n-1} \Vert . \end{aligned}$$
(3.25)

As \(\lim_{n\rightarrow \infty}\frac{\tau _{n}}{\sigma _{n}}=0\), there is \(M_{1}>0\) such that \(\tau _{n}\le \sigma _{n} M_{1}\). Using equations (3.24) and (3.25) and using Minkowski’s inequality, we get

$$\begin{aligned} &\bigl( \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)^{\frac{1}{2}} \\ &\quad \le \bigl(\bigl( \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert +\gamma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr)^{2}+\bigl( \bigl\Vert y_{n}- \bar{s}^{*} \bigr\Vert +\gamma _{n} \Vert y_{n}-y_{n-1} \Vert \bigr)^{2} \bigr)^{ \frac{1}{2}} \\ &\quad \le \bigl( \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)^{ \frac{1}{2}}+\gamma _{n}\bigl( \Vert x_{n}-x_{n-1} \Vert ^{2}+ \Vert y_{n}-y_{n-1} \Vert ^{2}\bigr)^{ \frac{1}{2}} \\ &\quad \le \bigl( \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)^{ \frac{1}{2}}+\tau _{n} \\ &\quad \le \bigl( \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)^{ \frac{1}{2}}+\sigma _{n} M_{1}. \end{aligned}$$
(3.26)

Now, using Lemma 2.5, we have

$$\begin{aligned} \bigl\Vert (S_{n})_{a_{n}}(h_{n})- \bar{r}^{*} \bigr\Vert ^{2}&= \bigl\Vert \bigl((1-a_{n})h_{n}+a_{n} S_{n}(h_{n}) \bigr)- \bar{r}^{*} \bigr\Vert ^{2} \\ &\le \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}-a_{n}(1-a_{n}-\kappa _{{k_{1}}}) \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert ^{2} \\ &\le \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}-a_{n}(1-a_{n}-\kappa ) \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert ^{2} \end{aligned}$$
(3.27)
$$\begin{aligned} &\le \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2} \end{aligned}$$
(3.28)

and

$$\begin{aligned} \bigl\Vert (U_{n})_{a_{n}}(v_{n})- \bar{s}^{*} \bigr\Vert ^{2} &\le \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}-a_{n}(1-a_{n}- \phi ) \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert ^{2}. \end{aligned}$$
(3.29)

Using Lemma 2.3, condition (iii), and equation (3.27), we have

$$\begin{aligned} \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}={}& \Biggl\Vert \delta _{n} h_{n}+\mu _{n}\bigl((1-a_{n})h_{n}+a_{n}S_{n}(h_{n}) \bigr)+ \sum_{i=1}^{\infty}\eta _{n,i}J_{\theta _{i}}^{K_{i}}\bigl(u_{n}^{1} \bigr)- \bar{r}^{*} \Biggr\Vert ^{2} \\ \le{}& \delta _{n} \bigl\Vert h_{n}- \bar{r}^{*} \bigr\Vert ^{2}+\mu _{n}\bigl( \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}-a_{n}(1-a_{n}- \kappa ) \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert ^{2}\bigr) \\ &{}+\sum_{i=1}^{ \infty}\eta _{n,i} \bigl\Vert J_{\theta _{i}}^{K_{i}}\bigl(u_{n}^{1} \bigr)-\bar{r}^{*} \bigr\Vert ^{2} \\ \le{}& \Biggl(1-\sum_{i=1}^{\infty}\eta _{n,i}\Biggr) \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \sum_{i=1}^{\infty}\eta _{n,i} \bigl\Vert u_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2} \\ &{}-\mu _{n}a_{n}(1-a_{n}- \kappa ) \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert ^{2}. \end{aligned}$$
(3.30)

Similarly,

$$\begin{aligned} \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \Biggl(1-\sum_{i=1}^{\infty} \eta _{n,i}\Biggr) \bigl\Vert v_{n}- \bar{s}^{*} \bigr\Vert ^{2}+\sum_{i=1}^{\infty} \eta _{n,i} \bigl\Vert u_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2} \\ &{}-\mu _{n}a_{n}(1-a_{n}-\phi ) \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert ^{2}. \end{aligned}$$
(3.31)

Adding equations (3.30), (3.31) and using equation (3.13), we have

$$\begin{aligned} & \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2} \\ &\quad \le \Biggl(1-\sum_{i=1}^{ \infty}\eta _{n,i}\Biggr) \bigl( \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)+ \sum_{i=1}^{\infty} \eta _{n,i}\bigl[ \bigl\Vert u_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert u_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2}\bigr] \\ &\qquad {}-\mu _{n}a_{n}(1-a_{n}-\phi ) \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert ^{2}-\mu _{n}a_{n}(1-a_{n}- \kappa ) \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert ^{2} \\ &\quad \le \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}-\mu _{n}a_{n}(1-a_{n}- \phi ) \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert ^{2} \\ &\qquad {}-\mu _{n}a_{n}(1-a_{n}-\kappa ) \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert ^{2} \end{aligned}$$
(3.32)
$$\begin{aligned} &\quad \le \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2} . \end{aligned}$$
(3.33)

Using equations (3.2), (3.18), and (3.33), we estimate

$$\begin{aligned} & \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2} \\ &\quad \le \bigl\Vert h_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2} \\ &\quad = \bigl\Vert \sigma _{n} \psi _{1}(x_{n})+(1- \sigma _{n})z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \sigma _{n} \psi _{2}(y_{n})+(1-\sigma _{n})z_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2} \\ &\quad \le \sigma _{n} \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+(1-\sigma _{n}) \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2}+\sigma _{n} \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2} \\ &\qquad {}+(1-\sigma _{n}) \bigl\Vert z_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2} \end{aligned}$$
(3.34)
$$\begin{aligned} &\quad \le \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}\bigr)+(1-\sigma _{n}) \Biggl( \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert ^{2} \\ &\qquad {}-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}\bigl(2\bar{c}^{x}-\nu _{n}^{x}\bigr) \bigl\Vert L_{j}^{x}w_{n}-L_{j}^{x} \bar{r}^{*} \bigr\Vert ^{2}-\nu _{n}^{y} \sum_{j=1}^{M} b_{j}^{y} \bigl(2\bar{c}^{y}- \nu _{n}^{y}\bigr) \bigl\Vert L_{j}^{y}t_{n}-L_{j}^{y} \bar{s}^{*} \bigr\Vert ^{2}\Biggr) \end{aligned}$$
(3.35)
$$\begin{aligned} &\quad \le \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}\bigr)+ \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2} . \end{aligned}$$
(3.36)

As \(\lambda =\max \{\lambda _{1}, \lambda _{2}\}\), using equations (3.2), (3.19), (3.26), and Minkowski’s inequality, we have

$$\begin{aligned} &\bigl( \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)^{\frac{1}{2}} \\ &\quad = \bigl( \bigl\Vert \sigma _{n} \psi _{1}(x_{n})+(1- \sigma _{n})z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \sigma _{n} \psi _{2}(y_{n})+(1-\sigma _{n})z_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2} \bigr)^{\frac{1}{2}} \\ &\quad \le \bigl(\bigl(\sigma _{n} \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert +(1-\sigma _{n}) \bigl\Vert z_{n}^{1}- \bar{r}^{*} \bigr\Vert \bigr)^{2}+\bigl(\sigma _{n} \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert \\ &\qquad {}+(1-\sigma _{n}) \bigl\Vert z_{n}^{2}- \bar{s}^{*} \bigr\Vert \bigr)^{2} \bigr)^{ \frac{1}{2}} \\ &\quad \le \bigl(\bigl(\sigma _{n} \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert +(1-\sigma _{n}) \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert \bigr)^{2}+\bigl(\sigma _{n} \bigl\Vert \psi _{2}(y_{n})-\bar{s}^{*} \bigr\Vert \\ &\qquad {}+(1-\sigma _{n}) \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert \bigr)^{2} \bigr)^{\frac{1}{2}} \\ &\quad \le \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{\frac{1}{2}}+(1-\sigma _{n}) \bigl( \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2} \bigr)^{\frac{1}{2}} \\ &\quad \le \sigma _{n} \bigl(\bigl( \bigl\Vert \psi _{1}(x_{n})-\psi _{1}\bigl(\bar{r}^{*} \bigr) \bigr\Vert + \bigl\Vert \psi _{1}\bigl(\bar{r}^{*} \bigr)-\bar{r}^{*} \bigr\Vert \bigr)^{2}+\bigl( \bigl\Vert \psi _{2}(y_{n})- \psi _{2}\bigl( \bar{s}^{*}\bigr) \bigr\Vert \\ &\qquad {}+ \bigl\Vert \psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*} \bigr\Vert \bigr)^{2} \bigr)^{\frac{1}{2}} \\ &\qquad {}+(1-\sigma _{n}) \bigl(\bigl( \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2} \bigr)^{ \frac{1}{2}}+\sigma _{n} M_{1}\bigr) \\ &\quad \le \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})- \psi _{1}\bigl(\bar{r}^{*}\bigr) \bigr\Vert ^{2}+\| \psi _{2}({y_{n}})-\psi _{2} \bigl(\bar{s}^{*}\bigr)\|^{2}\bigr)^{\frac{1}{2}}+\bigl( \bigl\Vert \psi _{1}\bigl(\bar{r}^{*}\bigr)- \bar{r}^{*} \bigr\Vert ^{2} \\ &\qquad {}+ \bigl\Vert \psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{\frac{1}{2}} \\ &\qquad {}+(1-\sigma _{n}) \bigl( \bigl\Vert x_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{ \frac{1}{2}}+\sigma _{n} M_{1} \\ &\quad \le \sigma _{n} \bigl( \bigl( \lambda _{1}^{2} \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \lambda _{2}^{2} \bigl\Vert y_{n}- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{\frac{1}{2}} \\ &\qquad {}+\bigl( \bigl\Vert \psi _{1}\bigl( \bar{r}^{*}\bigr)- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}\bigl(\bar{s}^{*}\bigr)-\bar{s}^{*} \bigr\Vert ^{2}\bigr)^{\frac{1}{2}} \bigr) \\ &\qquad {}+(1-\sigma _{n}) \bigl( \bigl\Vert x_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{ \frac{1}{2}}+\sigma _{n} M_{1} \\ &\quad \le \bigl(1-\sigma _{n}(1-\lambda )\bigr) \bigl( \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}- \bar{s}^{*} \bigr\Vert ^{2} \bigr)^{\frac{1}{2}} \\ &\qquad {}+\sigma _{n}\bigl( \bigl\Vert \psi _{1}\bigl( \bar{r}^{*}\bigr)- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{ \frac{1}{2}}+\sigma _{n} M_{1}. \end{aligned}$$
(3.37)

Using equations (3.23), (3.33), and (3.34), we estimate

$$\begin{aligned} \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \bigl\Vert h_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2} \\ \le{}& \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &{}+(1-\sigma _{n}) \Biggl( \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert ^{2}- \bigl\Vert z_{n}^{1}-w_{n} \bigr\Vert ^{2} \\ &{}- \bigl\Vert z_{n}^{2}-t_{n} \bigr\Vert ^{2}+2\nu _{n}^{x}\sum _{j=1}^{M} b_{j}^{x} \bigl\Vert w_{n}-z_{n}^{1} \bigr\Vert \bigl\Vert L_{j}^{x}(w_{n})-L_{j}^{x} \bigl(\bar{r}^{*}\bigr) \bigr\Vert \\ &{}+2\nu _{n}^{y}\sum_{j=1}^{M} b_{j}^{y} \bigl\Vert t_{n}-z_{n}^{2} \bigr\Vert \bigl\Vert L_{j}^{y}(t_{n})-L_{j}^{y} \bigl( \bar{s}^{*}\bigr) \bigr\Vert \Biggr) \\ \le {}&\sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}\bigr)+ \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2} \\ &{}+ \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2}- \bigl\Vert z_{n}^{1}-w_{n} \bigr\Vert ^{2}- \bigl\Vert z_{n}^{2}-t_{n} \bigr\Vert ^{2} \\ &{}+2\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x} \bigl\Vert w_{n}-z_{n}^{1} \bigr\Vert \bigl\Vert L_{j}^{x}(w_{n})-L_{j}^{x} \bigl( \bar{r}^{*}\bigr) \bigr\Vert \\ &{}+2\nu _{n}^{y}\sum_{j=1}^{M} b_{j}^{y} \bigl\Vert t_{n}-z_{n}^{2} \bigr\Vert \bigl\Vert L_{j}^{y}(t_{n})-L_{j}^{y} \bigl( \bar{s}^{*}\bigr) \bigr\Vert . \end{aligned}$$
(3.38)

From Lemma 2.3 and equation (3.28), we have

$$\begin{aligned} \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}={}& \Biggl\Vert \delta _{n} h_{n}+\mu _{n}\bigl((1-a_{n})h_{n}+a_{n}S_{n}(h_{n}) \bigr)+ \sum_{i=1}^{\infty}\eta _{n,i}J_{\theta _{i}}^{K_{i}}\bigl(u_{n}^{1} \bigr)- \bar{r}^{*} \Biggr\Vert ^{2} \\ \le{}& \delta _{n} \bigl\Vert h_{n}- \bar{r}^{*} \bigr\Vert ^{2}+\mu _{n} \bigl\Vert (1-a_{n})h_{n}+a_{n}S_{n}(h_{n})- \bar{r}^{*} \bigr\Vert ^{2} \\ &{}+\sum_{i=1}^{\infty}\eta _{n,i} \bigl\Vert J_{\theta _{i}}^{K_{i}}\bigl(u_{n}^{1} \bigr)- \bar{r}^{*} \bigr\Vert ^{2}-\delta _{n} \mu _{n} \bigl\Vert h_{n}-(1-a_{n})h_{n}-a_{n}S_{n}(h_{n}) \bigr\Vert ^{2} \\ &{}-\delta _{n}\sum_{i=1}^{\infty} \eta _{n,i} \bigl\Vert h_{n}-J_{\theta _{i}}^{K_{i}} \bigl(u_{n}^{1}\bigr) \bigr\Vert ^{2} \\ \le{}& \delta _{n} \bigl\Vert h_{n}- \bar{r}^{*} \bigr\Vert ^{2}+\mu _{n} \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+\sum _{i=1}^{\infty}\eta _{n,i} \bigl\Vert J_{\theta _{i}}^{K_{i}}\bigl(u_{n}^{1}\bigr)- \bar{r}^{*} \bigr\Vert ^{2} \\ &{}-\delta _{n}\mu _{n}a_{n} \bigl\Vert h_{n}-S_{n}(h_{n}) \bigr\Vert ^{2}- \delta _{n}\sum_{i=1}^{ \infty}\eta _{n,i} \bigl\Vert h_{n}-J_{\theta _{i}}^{K_{i}} \bigl(u_{n}^{1}\bigr) \bigr\Vert ^{2} \\ \le{}& \Biggl(1-\sum_{i=1}^{\infty}\eta _{n,i}\Biggr) \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \sum_{i=1}^{\infty}\eta _{n,i} \bigl\Vert u_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2} \\ &{}-\delta _{n}\mu _{n}a_{n} \bigl\Vert h_{n}-S_{n}(h_{n}) \bigr\Vert ^{2}- \delta _{n}\sum_{i=1}^{ \infty}\eta _{n,i} \bigl\Vert h_{n}-J_{\theta _{i}}^{K_{i}} \bigl(u_{n}^{1}\bigr) \bigr\Vert ^{2} \end{aligned}$$
(3.39)
$$\begin{aligned} \le{}& \Biggl(1-\sum_{i=1}^{\infty}\eta _{n,i}\Biggr) \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \sum_{i=1}^{\infty}\eta _{n,i} \bigl\Vert u_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2} . \end{aligned}$$
(3.40)

Similarly,

$$\begin{aligned} \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \Biggl(1-\sum_{i=1}^{\infty} \eta _{n,i}\Biggr) \bigl\Vert v_{n}- \bar{s}^{*} \bigr\Vert ^{2}+\sum_{i=1}^{\infty} \eta _{n,i} \bigl\Vert u_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2} \\ &{}-\delta _{n}\mu _{n}a_{n} \bigl\Vert v_{n}-U_{n}(v_{n}) \bigr\Vert ^{2}- \delta _{n}\sum_{i=1}^{ \infty}\eta _{n,i} \bigl\Vert v_{n}-J_{\theta _{i}}^{T_{i}} \bigl(u_{n}^{2}\bigr) \bigr\Vert ^{2} \end{aligned}$$
(3.41)
$$\begin{aligned} \le{}& \Biggl(1-\sum_{i=1}^{\infty}\eta _{n,i}\Biggr) \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}+ \sum_{i=1}^{\infty}\eta _{n,i} \bigl\Vert u_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2} . \end{aligned}$$
(3.42)

Using equation (3.13) and adding equations (3.39) and (3.41), we get

$$\begin{aligned} \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \Biggl(1-\sum_{i=1}^{ \infty} \eta _{n,i}\Biggr) \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+\sum_{i=1}^{\infty} \eta _{n,i} \bigl\Vert u_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2} \\ &{}+\Biggl(1-\sum_{i=1}^{\infty}\eta _{n,i}\Biggr) \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}+\sum_{i=1}^{ \infty}\eta _{n,i} \bigl\Vert u_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2} \\ &{}-\delta _{n}\mu _{n}a_{n}\bigl( \bigl\Vert h_{n}-S_{n}(h_{n}) \bigr\Vert ^{2}+ \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert ^{2}\bigr) \\ &{}-\delta _{n}\sum_{i=1}^{\infty} \eta _{n,i}\bigl( \bigl\Vert h_{n}-J_{\theta _{i}}^{K_{i}} \bigl(u_{n}^{1}\bigr) \bigr\Vert ^{2}+ \bigl\Vert J_{\theta _{i}}^{T_{i}}\bigl(u_{n}^{2} \bigr)-v_{n} \bigr\Vert ^{2}\bigr) \\ \le {}&\Biggl(1-\sum_{i=1}^{\infty}\eta _{n,i}\Biggr) \bigl( \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}- \bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &{}+\sum_{i=1}^{\infty}\eta _{n,i} \bigl( \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}- \bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &{}-\delta _{n}\mu _{n}a_{n}\bigl( \bigl\Vert h_{n}-S_{n}(h_{n}) \bigr\Vert ^{2}+ \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert ^{2}\bigr) \\ &{}-\delta _{n}\sum_{i=1}^{\infty} \eta _{n,i}\bigl(\|h_{n}-J_{\theta _{i}}^{K_{i}} \bigl(u_{n}^{1}\bigr)\bigr) \Vert ^{2}+ \bigl\Vert J_{\theta _{i}}^{T_{i}}\bigl(u_{n}^{2} \bigr)-v_{n} \bigr\Vert ^{2}) \\ \le{}& \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2} \\ &{}-\delta _{n}\mu _{n}a_{n}\bigl( \bigl\Vert h_{n}-S_{n}(h_{n}) \bigr\Vert ^{2}+ \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert ^{2}\bigr) \\ &{}-\delta _{n}\sum_{i=1}^{\infty} \eta _{n,i}( \bigl\Vert h_{n}-J_{\theta _{i}}^{K_{i}} \bigl(u_{n}^{1}\bigr) \bigr\Vert ^{2}+\| \bigl(J_{\theta _{i}}^{T_{i}}\bigl(u_{n}^{2} \bigr)-v_{n}\|^{2}\bigr) \end{aligned}$$
(3.43)
$$\begin{aligned} \le{}& \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2} . \end{aligned}$$
(3.44)

Using equation (3.12) and adding equations (3.40) and (3.42), we get

$$\begin{aligned} \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \Biggl(1-\sum_{i=1}^{ \infty} \eta _{n,i}\Biggr) \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+\sum_{i=1}^{\infty} \eta _{n,i} \bigl\Vert u_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2} \\ &{}+\Biggl(1-\sum_{i=1}^{\infty}\eta _{n,i}\Biggr) \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}+\sum_{i=1}^{ \infty}\eta _{n,i} \bigl\Vert u_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2} \\ \le{}& \Biggl(1-\sum_{i=1}^{\infty}\eta _{n,i}\Biggr) \bigl( \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}- \bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &{}+\sum_{i=1}^{\infty}\eta _{n,i} \bigl( \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}- \bar{s}^{*} \bigr\Vert ^{2} \\ &{}-\rho _{n}\epsilon \bigl( \bigl\Vert A^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2}+ \bigl\Vert B^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2} \bigr) \bigr) \\ \le{}& \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2} \\ &{}-\sum_{i=1}^{\infty}\eta _{n,i}\rho _{n}\epsilon \bigl( \bigl\Vert A^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2} \\ &{}+ \bigl\Vert B^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2} \bigr). \end{aligned}$$
(3.45)

From equations (3.37) and (3.44), we have

$$\begin{aligned} &\bigl( \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)^{ \frac{1}{2}} \\ &\quad \le \bigl( \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)^{ \frac{1}{2}} \\ &\quad \le \bigl(1-\sigma _{n}(1-\lambda )\bigr) \bigl( \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}- \bar{s}^{*} \bigr\Vert ^{2} \bigr)^{\frac{1}{2}} \\ &\qquad {}+\sigma _{n}\bigl( \bigl\Vert \psi _{1}\bigl( \bar{r}^{*}\bigr)-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}\bigl( \bar{s}^{*}\bigr)- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{\frac{1}{2}}+\sigma _{n} M_{1} \\ &\quad \le \max \biggl\{ \bigl( \bigl\Vert x_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{ \frac{1}{2}}, \\ &\qquad \frac{1}{1-\lambda} \bigl[\bigl( \bigl\Vert \psi _{1}\bigl( \bar{r}^{*}\bigr)-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{\frac{1}{2}}+M_{1} \bigr] \biggr\} \\ &\quad \le \max \biggl\{ \bigl( \bigl\Vert x_{1}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{1}- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{ \frac{1}{2}}, \\ & \qquad \frac{1}{1-\lambda} \bigl[\bigl( \bigl\Vert \psi _{1}\bigl( \bar{r}^{*}\bigr)-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*} \bigr\Vert ^{2}\bigr)^{\frac{1}{2}}+M_{1} \bigr] \biggr\} . \end{aligned}$$
(3.46)

Hence \(\{x_{n}\}\) and \(\{y_{n}\}\) are bounded. Subsequently, \(\{w_{n}\}\), \(\{t_{n}\}\), \(\{h_{n}\}\), and \(\{v_{n}\}\) are also bounded.

$$\begin{aligned} \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}&= \bigl\Vert x_{n}+\gamma _{n}(x_{n}-x_{n-1})- \bar{r}^{*} \bigr\Vert ^{2} \\ &\le \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+2\gamma _{n} \bigl\langle x_{n}-x_{n-1}, w_{n}-\bar{r}^{*} \bigr\rangle \\ &\le \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+2\gamma _{n} \Vert x_{n}-x_{n-1} \Vert \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert \\ &\le \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+2\tau _{n} \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert . \end{aligned}$$
(3.47)

Similarly,

$$\begin{aligned} \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2}&\le \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}+2\tau _{n} \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert . \end{aligned}$$
(3.48)

Adding equations (3.47) and (3.48), we have

$$\begin{aligned} \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert ^{2}&\le \bigl\Vert x_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}+2\tau _{n}\bigl[ \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert \bigr]. \end{aligned}$$
(3.49)

From equation (3.2), we have

$$\begin{aligned} \bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}={}& \bigl\Vert \sigma _{n} \psi _{1}(x_{n})+(1- \sigma _{n})z_{n}^{1}- \bar{r}^{*} \bigr\Vert ^{2} \\ ={}&\sigma _{n}^{2} \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+(1-\sigma _{n})^{2} \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2} \\ &{}+2\sigma _{n}(1-\sigma _{n}) \bigl\langle \psi _{1}(x_{n})-\bar{r}^{*}, z_{n}^{1}- \bar{r}^{*} \bigr\rangle \\ ={}&\sigma _{n}^{2} \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+(1-\sigma _{n})^{2} \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2}+2\sigma _{n} \bigl\langle \psi _{1}(x_{n})- \bar{r}^{*}, z_{n}^{1}-\bar{r}^{*} \bigr\rangle \\ &{}-2\sigma _{n}^{2} \bigl\langle \psi _{1}(x_{n})-\bar{r}^{*}, z_{n}^{1}- \bar{r}^{*} \bigr\rangle \\ \le{}& \sigma _{n}^{2} \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert ^{2}+(1-\sigma _{n})^{2} \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2} \\ &{}+2\sigma _{n} \bigl\langle \psi _{1}(x_{n})- \psi _{1}\bigl(\bar{r}^{*}\bigr), z_{n}^{1}- \bar{r}^{*} \bigr\rangle +2\sigma _{n} \bigl\langle \psi _{1}\bigl( \bar{r}^{*}\bigr)-\bar{r}^{*}, z_{n}^{1}-\bar{r}^{*} \bigr\rangle \\ &{}+2\sigma _{n}^{2} \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert \bigl\Vert z_{n}^{1}- \bar{r}^{*} \bigr\Vert \\ \le{}& \sigma _{n}^{2} \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert ^{2}+(1-\sigma _{n})^{2} \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2} \\ &{}+2\sigma _{n} \bigl\Vert \psi _{1}(x_{n})- \psi _{1}\bigl(\bar{r}^{*}\bigr) \bigr\Vert \bigl\Vert z_{n}^{1}- \bar{r}^{*} \bigr\Vert \\ &{}+2\sigma _{n} \bigl\langle \psi _{1}\bigl( \bar{r}^{*}\bigr)-\bar{r}^{*}, z_{n}^{1}- \bar{r}^{*} \bigr\rangle +2\sigma _{n}^{2} \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert \\ \le{} &\sigma _{n}^{2} \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert ^{2}+(1-\sigma _{n})^{2} \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2}+2\sigma _{n}\lambda _{1} \bigl\Vert x_{n}- \bar{r}^{*} \bigr\Vert \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert \\ &{}+2\sigma _{n} \bigl\langle \psi _{1}\bigl( \bar{r}^{*}\bigr)-\bar{r}^{*}, z_{n}^{1}- \bar{r}^{*} \bigr\rangle +2\sigma _{n}^{2} \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert \\ \le {}&\sigma _{n}^{2} \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert ^{2}+(1-\sigma _{n})^{2} \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2}+\sigma _{n}\lambda _{1} \bigl( \bigl\Vert x_{n}- \bar{r}^{*} \bigr\Vert ^{2} \\ &{}+ \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2}\bigr)+2\sigma _{n} \bigl\langle \psi _{1}\bigl( \bar{r}^{*}\bigr)-\bar{r}^{*}, z_{n}^{1}-\bar{r}^{*} \bigr\rangle \\ &{} +2 \sigma _{n}^{2} \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert . \end{aligned}$$
(3.50)

Similarly,

$$\begin{aligned} \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \sigma _{n}^{2} \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}+(1-\sigma _{n})^{2} \bigl\Vert z_{n}^{2}-\bar{s}^{*} \bigr\Vert ^{2}+ \sigma _{n}\lambda _{2} \bigl( \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2} \\ &{}+ \bigl\Vert z_{n}^{2}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)+2\sigma _{n} \bigl\langle \psi _{2}\bigl( \bar{s}^{*}\bigr)-\bar{s}^{*}, z_{n}^{2}-\bar{s}^{*} \bigr\rangle \\ &{} +2 \sigma _{n}^{2} \bigl\Vert \psi _{2}(y_{n})-\bar{s}^{*} \bigr\Vert \bigl\Vert z_{n}^{2}-\bar{s}^{*} \bigr\Vert . \end{aligned}$$
(3.51)

Adding equations (3.50) and (3.51) and using equations (3.18) and (3.49), we get

$$\begin{aligned} &\bigl\Vert h_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert v_{n}-\bar{s}^{*} \bigr\Vert ^{2} \\ &\quad \le{} \sigma _{n}^{2} \bigl( \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &\qquad {}+(1-\sigma _{n})^{2}\bigl( \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert z_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2}\bigr)+\sigma _{n} \lambda \bigl( \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2} \\ &\qquad {}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)+\sigma _{n}\lambda \bigl( \bigl\Vert z_{n}^{1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert z_{n}^{2}- \bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &\qquad {}+2\sigma _{n}\bigl( \bigl\langle \psi _{1}\bigl( \bar{r}^{*}\bigr)-\bar{r}^{*}, z_{n}^{1}- \bar{r}^{*} \bigr\rangle + \bigl\langle \psi _{2}\bigl( \bar{s}^{*}\bigr)- \bar{s}^{*}, z_{n}^{2}- \bar{s}^{*} \bigr\rangle \bigr) \\ &\qquad {}+2\sigma _{n}^{2}\bigl( \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert . \bigl\Vert z_{n}^{1}- \bar{r}^{*} \bigr\Vert + \bigl\Vert \psi _{2}(y_{n})-\bar{s}^{*} \bigr\Vert . \bigl\Vert z_{n}^{2}-\bar{s}^{*} \bigr\Vert \bigr) \\ &\quad \le{} \bigl(1-2(1-\lambda )\sigma _{n}\bigr) \bigl( \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}- \bar{s}^{*} \bigr\Vert ^{2}\bigr)+\sigma _{n}^{2} M_{2} \\ &\qquad {}+2\sigma _{n}\bigl( \bigl\langle \psi _{1}\bigl( \bar{r}^{*}\bigr)-\bar{r}^{*}, z_{n}^{1}- \bar{r}^{*} \bigr\rangle + \bigl\langle \psi _{2}\bigl( \bar{s}^{*}\bigr)- \bar{s}^{*}, z_{n}^{2}- \bar{s}^{*} \bigr\rangle \bigr) \\ &\qquad {}+2\tau _{n}\bigl((1-\sigma _{n})^{2}+\sigma _{n}\lambda \bigr) \bigl( \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert \bigr), \end{aligned}$$
(3.52)

where

$$\begin{aligned} M_{2}={}&\sup_{n\in \mathbb{N}}\bigl\{ \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}- \bar{s}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert ^{2} \\ &{}+ \bigl\Vert \psi _{2}(y_{n})-\bar{s}^{*} \bigr\Vert ^{2}+2 \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert . \bigl\Vert z_{n}^{1}- \bar{r}^{*} \bigr\Vert \end{aligned}$$
(3.53)
$$\begin{aligned} &{} +2 \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert . \bigl\Vert z_{n}^{2}- \bar{s}^{*} \bigr\Vert \bigr\} . \end{aligned}$$
(3.54)

Hence,

$$\begin{aligned} \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \bigl(1-2(1- \lambda )\sigma _{n}\bigr) \bigl( \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2} \bigr) \\ &{}+\frac{2\sigma _{n}(1-\lambda )}{2(1-\lambda )} \biggl(2 \bigl( \bigl\langle \psi _{1}\bigl( \bar{r}^{*}\bigr)-\bar{r}^{*}, z_{n}^{1}- \bar{r}^{*} \bigr\rangle \\ &{}+ \bigl\langle \psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*}, z_{n}^{2}-\bar{s}^{*} \bigr\rangle \bigr)+4\frac{\tau _{n}}{\sigma _{n}} \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert \\ &{}+\sigma _{n} M_{2} \biggr), \end{aligned}$$
(3.55)

which gives

$$\begin{aligned} \bar{p}_{n+1}\le (1-d_{n}) \bar{p}_{n}+d_{n}\bar{s}_{n}, \end{aligned}$$
(3.56)

where \(\bar{p}_{n}=\|x_{n}-\bar{r}^{*}\|^{2}+\|y_{n}-\bar{s}^{*}\|^{2}\), \(d_{n}=2\sigma _{n}(1-\lambda )\) and

$$\begin{aligned}& \bar{s}_{n}=\frac{1}{2(1-\lambda )} \biggl[ \bigl( \bigl\langle \psi _{1}\bigl( \bar{r}^{*}\bigr)- \bar{r}^{*}, z_{n}^{1}-\bar{r}^{*} \bigr\rangle + \bigl\langle \psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*}, z_{n}^{2}-\bar{s}^{*} \bigr\rangle \bigr) \\& \hphantom{\bar{s}_{n}=}{}+4\frac{\tau _{n}}{\sigma _{n}} \bigl( \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert \bigr)+\sigma _{n}M_{2} \biggr]. \end{aligned}$$

Case 1. Assume that there exists \(N\in \mathbb{N}\) such that \(\{\bar{p}_{n}\}\) is decreasing for \(n\geq N\). As \(\{\bar{p}_{n}\}\) is bounded and monotonic and subsequently convergent, as \(\lim_{n\rightarrow \infty}\sigma _{n}=0\) and \(\tau _{n}=o(\sigma _{n})\), using equations (3.35), (3.43), (3.49), we have

$$\begin{aligned} \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}\bigr)+ \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2} \\ &{}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}+2\tau _{n}\bigl[ \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert \bigr] \\ &{}-\delta _{n}\mu _{n}a_{n}\bigl( \bigl\Vert h_{n}-S_{n}(h_{n}) \bigr\Vert ^{2}+ \bigl\Vert v_{n}-U_{n}(v_{n}) \bigr\Vert ^{2}\bigr) \\ &{}-\delta _{n}\sum_{i=1}^{\infty} \eta _{n,i}\bigl( \bigl\Vert h_{n}-J_{\theta _{i}}^{K_{i}} \bigl(u_{n}^{1}\bigr) \bigr\Vert ^{2}+ \bigl\Vert J_{\theta _{i}}^{T_{i}}\bigl(u_{n}^{2} \bigr)-v_{n} \bigr\Vert ^{2}\bigr), \end{aligned}$$
(3.57)

which implies

$$\begin{aligned} \delta _{n}\sum_{i=1}^{\infty} \eta _{n,i}\bigl( \bigl\Vert h_{n}-J_{\theta _{i}}^{K_{i}} \bigl(u_{n}^{1}\bigr) \bigr\Vert ^{2}+ \bigl\Vert J_{\theta _{i}}^{T_{i}}\bigl(u_{n}^{2} \bigr)-v_{n} \bigr\Vert ^{2}\bigr)\le{}& \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})-\bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &{}+ \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2} \\ &{}-\bigl( \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &{}+2\tau _{n}[ \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert . \end{aligned}$$
(3.58)

Taking limit \(n \to \infty \) in equation (3.58) and using condition (ii), we get

$$\begin{aligned} \lim_{n\rightarrow \infty} \bigl\Vert h_{n}-J_{\theta _{i}}^{K_{i}} \bigl(u_{n}^{1}\bigr) \bigr\Vert =\lim _{n\rightarrow \infty} \bigl\Vert J_{\theta _{i}}^{T_{i}} \bigl(u_{n}^{2}\bigr)-v_{n} \bigr\Vert =0. \end{aligned}$$
(3.59)

From equations (3.32), (3.35), and (3.49), we get

$$\begin{aligned} \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &{}+ \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2} +2\tau _{n}\bigl[ \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert \\ &{}+ \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert \bigr]-\mu _{n}a_{n}(1-a_{n}-\phi ) \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert ^{2} \\ &{}-\mu _{n}a_{n}(1-a_{n}-\kappa ) \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert ^{2}, \end{aligned}$$

which implies

$$\begin{aligned} &\mu _{n}a_{n}(1-a_{n}-\phi _{k_{2}}) \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert ^{2} \end{aligned}$$
(3.60)
$$\begin{aligned} &\qquad {}+\mu _{n}a_{n}(1-a_{n}-\kappa _{n_{k_{1}}}) \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert ^{2} \\ &\quad \le \sigma _{(} \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}) \\ &\qquad {}+ \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}-\bigl( \bigl\Vert x_{n+1}- \bar{r}^{*} \bigr\Vert ^{2} \\ &\qquad {}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)+2\tau _{n}\bigl[ \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert \bigr]. \end{aligned}$$
(3.61)

Taking limit \(n \to \infty \) in equation (3.60) and using condition (ii), we get

$$\begin{aligned} \lim_{n\rightarrow \infty} \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert =\lim_{n \rightarrow \infty} \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert =0. \end{aligned}$$
(3.62)

From equations (3.35), (3.45), and (3.49), we have

$$\begin{aligned}& \sum_{i=1}^{\infty}\eta _{n,i}\rho _{n}\epsilon \bigl( \bigl\Vert A^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2}+ \bigl\Vert B^{*}(Ah_{n}-Bv_{n}) \bigr\Vert ^{2} \bigr) \\& \quad {} \le \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}\bigr)+ \bigl\Vert x_{n}- \bar{r}^{*} \bigr\Vert ^{2} \\& \qquad {}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}-\bigl( \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}- \bar{s}^{*} \bigr\Vert ^{2}\bigr) \\& \qquad {}+2\tau _{n}\bigl[ \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert \bigr]. \end{aligned}$$
(3.63)

Taking limit \(n \to \infty \) in equation (3.63) and using condition (ii), we get

$$\begin{aligned} \lim_{n\rightarrow \infty} \bigl\Vert A^{*}(Ah_{n}-Bv_{n}) \bigr\Vert =\lim_{n \rightarrow \infty} \bigl\Vert B^{*}(Ah_{n}-Bv_{n}) \bigr\Vert =0. \end{aligned}$$
(3.64)

As A and B are bounded linear operators and hence

$$\begin{aligned} \lim_{n\rightarrow \infty} \Vert Ah_{n}-Bv_{n} \Vert =0. \end{aligned}$$
(3.65)

From equations (3.34), (3.35), and (3.49), we get

$$\begin{aligned} &\nu _{n}^{x}\sum _{j=1}^{M} b_{j}^{x}\bigl(2 \bar{c}^{x}-\nu _{n}^{x}\bigr) \bigl\Vert L_{j}^{x}w_{n}-L_{j}^{x} \bar{r}^{*} \bigr\Vert ^{2}+\nu _{n}^{y} \sum_{j=1}^{M} b_{j}^{y} \bigl(2\bar{c}^{y}- \nu _{n}^{y}\bigr) \bigl\Vert L_{j}^{y}t_{n}-L_{j}^{y} \bar{s}^{*} \bigr\Vert ^{2} \\ &\quad \le \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})-\bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &\qquad {}+ \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}-\bigl( \bigl\Vert x_{n+1}- \bar{r}^{*} \bigr\Vert ^{2} \\ &\qquad {}+ \bigl\Vert y_{n+1}-\bar{s}^{*} \bigr\Vert ^{2}\bigr)+2\tau _{n}\bigl[ \bigl\Vert w_{n}- \bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n}- \bar{s}^{*} \bigr\Vert \bigr]. \end{aligned}$$
(3.66)

Taking limit \(n \to \infty \) in equation (3.66) and using conditions \((i)-(ii) \), we get

$$\begin{aligned} \lim_{n\rightarrow \infty} \bigl\Vert L_{j}^{x}w_{n}-L_{j}^{x} \bar{r}^{*} \bigr\Vert = \lim_{n\rightarrow \infty} \bigl\Vert L_{j}^{y}t_{n}-L_{j}^{y} \bar{s}^{*} \bigr\Vert =0, \quad \text{where } 1\le j\le M. \end{aligned}$$
(3.67)

From equations (3.35), (3.38), and (3.49), we get

$$\begin{aligned} &\bigl\Vert z_{n}^{1}-w_{n} \bigr\Vert ^{2}+ \bigl\Vert z_{n}^{2}-t_{n} \bigr\Vert ^{2} \\ &\quad \le{} \sigma _{n} \bigl( \bigl\Vert \psi _{1}(x_{n})-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n})- \bar{s}^{*} \bigr\Vert ^{2}\bigr)+ \bigl\Vert x_{n}-\bar{r}^{*} \bigr\Vert ^{2} \\ &\qquad {}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}-\bigl( \bigl\Vert x_{n+1}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}- \bar{s}^{*} \bigr\Vert ^{2}\bigr)+2\tau _{n}\bigl[ \bigl\Vert w_{n}-\bar{r}^{*} \bigr\Vert \\ &\qquad {}+ \bigl\Vert t_{n}-\bar{s}^{*} \bigr\Vert \bigr]+2 \nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x} \bigl\Vert w_{n}-z_{n}^{1} \bigr\Vert \bigl\Vert L_{j}^{x}(w_{n})-L_{j}^{x} \bigl(\bar{r}^{*}\bigr) \bigr\Vert \\ &\qquad {}+2\nu _{n}^{y}\sum_{j=1}^{M} b_{j}^{y} \bigl\Vert t_{n}-z_{n}^{2} \bigr\Vert \bigl\Vert L_{j}^{y}(t_{n})-L_{j}^{y} \bigl( \bar{s}^{*}\bigr) \bigr\Vert . \end{aligned}$$
(3.68)

Taking limit \(n \to \infty \) in equation (3.68) and using equation (3.67) and using condition (ii), we get

$$\begin{aligned} \lim_{n\rightarrow \infty} \bigl\Vert z_{n}^{1}-w_{n} \bigr\Vert = \lim_{n\rightarrow \infty} \bigl\Vert z_{n}^{2}-t_{n} \bigr\Vert =0. \end{aligned}$$
(3.69)

From equation (3.2), we have

$$\begin{aligned} \Vert h_{n}-w_{n} \Vert &\le \sigma _{n} \bigl\Vert \psi _{1}(x_{n})-w_{n} \bigr\Vert +(1-\sigma _{n}) \bigl\Vert z_{n}^{1}-w_{n} \bigr\Vert . \end{aligned}$$
(3.70)

Similarly,

$$\begin{aligned} \Vert v_{n}-t_{n} \Vert &\le \sigma _{n} \bigl\Vert \psi _{2}(y_{n})-t_{n} \bigr\Vert +(1-\sigma _{n}) \bigl\Vert z_{n}^{2}-t_{n} \bigr\Vert . \end{aligned}$$
(3.71)

Taking limit \(n \to \infty \) in equations (3.70) and (3.71), using equation (3.69) and \(\lim_{n\rightarrow \infty}\sigma _{n}=0\), we get

$$\begin{aligned} \lim_{n\rightarrow \infty} \Vert h_{n}-w_{n} \Vert =\lim_{n\rightarrow \infty} \Vert v_{n}-t_{n} \Vert =0. \end{aligned}$$
(3.72)

From equation (3.2), we have

$$\begin{aligned} \Vert x_{n+1}-h_{n} \Vert \le{}& \delta _{n} \Vert h_{n}-h_{n} \Vert +\mu _{n}\bigl((1-a_{n}) \Vert h_{n}-h_{n} \Vert +a_{n} \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert \bigr) \\ &{}+\sum_{i=1}^{\infty} \eta _{n,i} \bigl\Vert J_{\theta _{i}}^{K_{i}}\bigl(u_{n}^{1} \bigr)-h_{n} \bigr\Vert \\ \le{}& \mu _{n}a_{n} \bigl\Vert S_{n}(h_{n})-h_{n} \bigr\Vert +\sum_{i=1}^{\infty}\eta _{n,i} \bigl\Vert J_{\theta _{i}}^{K_{i}}\bigl(u_{n}^{1} \bigr)-h_{n} \bigr\Vert . \end{aligned}$$
(3.73)

Similarly,

$$\begin{aligned} \Vert y_{n+1}-v_{n} \Vert &\le \mu _{n}a_{n} \bigl\Vert U_{n}(v_{n})-v_{n} \bigr\Vert +\sum_{i=1}^{ \infty}\eta _{n,i} \bigl\Vert J_{\theta _{i}}^{K_{i}}\bigl(u_{n}^{2} \bigr)-v_{n} \bigr\Vert . \end{aligned}$$
(3.74)

Taking limit \(n \to \infty \) in equation (3.73) and (3.74), then using equations (3.59) and (3.62), we get

$$\begin{aligned} \lim_{n\rightarrow \infty} \Vert x_{n+1}-h_{n} \Vert =\lim_{n\rightarrow \infty} \Vert y_{n+1}-v_{n} \Vert =0. \end{aligned}$$
(3.75)

Also,

$$\begin{aligned} \Vert w_{n}-x_{n} \Vert =\gamma _{n} \Vert x_{n}-x_{n-1} \Vert \le \tau _{n}. \end{aligned}$$
(3.76)

Similarly,

$$\begin{aligned} \Vert t_{n}-y_{n} \Vert \le \tau _{n}. \end{aligned}$$
(3.77)

Since \(\tau _{n}=o(\sigma _{n})\) and \(\lim_{n\rightarrow \infty}\sigma _{n}=0\), we have

$$\begin{aligned} \lim_{n\rightarrow \infty} \Vert w_{n}-x_{n} \Vert =\lim_{n\rightarrow \infty} \Vert t_{n}-y_{n} \Vert =0. \end{aligned}$$
(3.78)

Using equations (3.72), (3.75), and (3.78), we get

$$\begin{aligned} \lim_{n\rightarrow \infty} \Vert x_{n}-x_{n+1} \Vert = \lim_{n\rightarrow \infty} \Vert y_{n}-y_{n+1} \Vert =0. \end{aligned}$$
(3.79)

Moreover,

$$\begin{aligned} \Vert Ax_{n}-By_{n} \Vert &\le \Vert Ax_{n}-Ah_{n} \Vert + \Vert Ah_{n}-Bv_{n} \Vert + \Vert Bv_{n}-By_{n} \Vert \\ &\le \Vert A \Vert \Vert x_{n}-h_{n} \Vert + \Vert Ah_{n}-Bv_{n} \Vert + \Vert B \Vert \Vert v_{n}-y_{n} \Vert . \end{aligned}$$
(3.80)

By equations (3.65), (3.72), and (3.78), we get

$$\begin{aligned} \lim_{n\rightarrow \infty} \Vert Ax_{n}-By_{n} \Vert =0. \end{aligned}$$
(3.81)

As A and B are bounded linear operators, therefore, \(Ax_{n_{q}}\rightharpoonup Ax^{*}\) and \(By_{n_{q}}\rightharpoonup By^{*}\). Clearly, it follows from weak semicontinuity of norm that

$$\begin{aligned} \bigl\Vert Ax^{*}-By^{*} \bigr\Vert \le \liminf _{q\to \infty} \Vert Ax_{n_{q}}-By_{n_{q}} \Vert =0. \end{aligned}$$
(3.82)

This implies \(Ax^{*}=By^{*}\). Subsequently, \((x^{*},y^{*})\in \Gamma \). Fix an index \(k_{1} \in \{1,2,\ldots, N\}\). As the pool of index is finite and \(\lim_{n\rightarrow \infty} \|x_{n}-x_{n-1}\|=0\), from Lemma 2.6, one can find a subsequence \(\{x_{m_{q}}\}\) of \(\{x_{n}\}\) such that \(x_{m_{q}}\rightharpoonup x^{*}\) and \(m_{q}(\mod N)+1=k_{1}\). Using equations (3.72) and (3.78), we get \(h_{m_{q}}\rightharpoonup x^{*}\). Additionally,

$$\begin{aligned} \lim_{q\rightarrow \infty} \bigl\Vert S_{k_{1}}(h_{m_{q}})-h_{m_{q}} \bigr\Vert =\lim_{q \rightarrow \infty} \bigl\Vert S_{m_{q}}(h_{m_{q}})-h_{m_{q}} \bigr\Vert =0 \end{aligned}$$
(3.83)

and \(I-S_{k_{1}}\) is demiclosed at zero for each \(k_{1}\in \{1,2,\ldots, N\}\), which implies \(x^{*}\in \bigcap_{k_{1}=1}^{N} \operatorname{Fix}(S_{k_{1}})\). Similarly,

$$\begin{aligned} \lim_{q\rightarrow \infty} \bigl\Vert U_{k_{2}}(v_{m_{q}})-v_{m_{q}} \bigr\Vert =\lim_{q \rightarrow \infty} \bigl\Vert U_{m_{q}}(v_{m_{q}})-v_{m_{q}} \bigr\Vert =0, \end{aligned}$$
(3.84)

and \(I-U_{k_{2}}\) is demiclosed at zero for each \(k_{2}\in \{1,2,\ldots, P\}\), which implies \(y^{*}\in \bigcap_{k_{2}=1}^{P} \operatorname{Fix}(U_{k_{2}})\). As \(\{x_{n}\}\) is bounded, take a subsequence \(\{x_{n_{q}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{q}} \rightharpoonup x^{*} \in H_{1} \). Additionally, from the boundedness of \(\{h_{n}\}\), \(\{w_{n}\}\) and from equations (3.72) and (3.78), there are subsequences \(\{h_{n_{q}}\}\), \(\{w_{n_{q}}\}\) respectively such that \(h_{n_{q}}\rightharpoonup x^{*} \) and \(w_{n_{q}}\rightharpoonup x^{*} \). Suppose that \(x^{*}\notin \bigcap_{j=1}^{M} \operatorname{VIP}(Q_{1}, L_{j}^{x})\). Then, from Lemmas (2.10) and (2.11), we have \(x^{*}\notin \operatorname{Fix}(P_{Q_{1}}(I-\nu _{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}))\). Using Opial’s condition and equation (3.69), we estimate

$$\begin{aligned} \liminf_{q\to \infty} \bigl\Vert w_{n_{q}}-x^{*} \bigr\Vert < {}&\liminf_{q\to \infty} \Biggl\Vert w_{n_{q}}-P_{Q_{1}}\Biggl(I- \nu _{x}\sum _{j=1}^{M} b_{j}^{x}L_{j}^{x} \Biggr)x^{*} \Biggr\Vert \\ \le{}& \liminf_{q\to \infty} \Biggl\Vert w_{n_{q}}-P_{Q_{1}} \Biggl(I-\nu _{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}\Biggr)w_{n_{q}} \Biggr\Vert \\ &{}+\liminf_{q\to \infty} \Biggl\Vert P_{Q_{1}}\Biggl(I-\nu _{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}\Biggr)w_{n_{q}}-P_{Q_{1}} \Biggl(I- \nu _{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}\Biggr)x^{*} \Biggr\Vert \\ \le{}& \liminf_{q\to \infty} \bigl\Vert w_{n_{q}}-x^{*} \bigr\Vert , \end{aligned}$$
(3.85)

which is a contradiction. Thus \(x^{*}\in \operatorname{Fix}(P_{Q_{1}}(I-\nu _{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x}))\). Similarly, \(\{y_{n}\}\) is bounded, take a subsequence \(\{y_{n_{q}}\}\) of \(\{y_{n}\}\) such that \(y_{n_{q}} \rightharpoonup y^{*} \in H_{2} \). Additionally, from the boundedness of \(\{v_{n}\}\), \(\{t_{n}\}\) and from equations (3.72) and (3.78), there are subsequences \(\{v_{n_{q}}\}\) \(\{t_{n_{q}}\}\) respectively such that \(v_{n_{q}}\rightharpoonup y^{*} \) and \(t_{n_{q}}\rightharpoonup y^{*} \). Suppose that \(y^{*}\notin \bigcap_{j=1}^{M} \operatorname{VIP}(Q_{2}, L_{j}^{y})\). Then, from Lemmas (2.10) and (2.11), we have \(y^{*}\notin \operatorname{Fix}(P_{Q_{2}}(I-\nu _{y}\sum_{j=1}^{M} b_{j}^{y}L_{j}^{y}))\). Using Opial’s condition and equation (3.69), we estimate

$$\begin{aligned} \liminf_{q\to \infty} \bigl\Vert t_{n_{q}}-y^{*} \bigr\Vert < {}&\liminf_{q\to \infty} \Biggl\Vert t_{n_{q}}-P_{Q_{2}}\Biggl(I- \nu _{y}\sum _{j=1}^{M} b_{j}^{y}L_{j}^{y} \Biggr)y^{*} \Biggr\Vert \\ \le{}& \liminf_{q\to \infty} \Biggl\Vert t_{n_{q}}-P_{Q_{2}} \Biggl(I-\nu _{y}\sum_{j=1}^{M} b_{j}^{y}L_{j}^{y}\Biggr)t_{n_{q}} \Biggr\Vert \\ &{}+\liminf_{q\to \infty} \Biggl\Vert P_{Q_{2}}\Biggl(I-\nu _{y}\sum_{j=1}^{M} b_{j}^{y}L_{j}^{y}\Biggr)t_{n_{q}}-P_{Q_{2}} \Biggl(I- \nu _{y}\sum_{j=1}^{M} b_{j}^{y}L_{j}^{y}\Biggr)y^{*} \Biggr\Vert \\ \le{}& \liminf_{q\to \infty} \bigl\Vert t_{n_{q}}-y^{*} \bigr\Vert , \end{aligned}$$
(3.86)

which is a contradiction. Thus \(y^{*}\in \operatorname{Fix}(P_{Q_{2}}(I-\nu _{y}\sum_{j=1}^{M} b_{j}^{y}L_{j}^{y}))\). Using equation (3.59), we get

$$\begin{aligned} \lim_{q\rightarrow \infty} \bigl\Vert l_{n_{q}}- \bigl(J_{\theta _{i}}^{(K_{i},T_{i})}\bigl(I- \rho _{{n_{q}}} \bigl(H^{*}H\bigr)l_{n_{q}}\bigr)\bigr) \bigr\Vert =0. \end{aligned}$$
(3.87)

Now, using Lemma 3.3, \(l^{*}\in \Omega \). As \(w_{n_{q}}\rightharpoonup x^{*} \) and \(t_{n_{q}}\rightharpoonup y^{*} \), from equation (3.69), we estimate \(z_{n_{q}}^{1}\rightharpoonup x^{*} \) and \(z_{n_{q}}^{2}\rightharpoonup y^{*} \). Using Lemma 2.4, we get

$$\begin{aligned}& \limsup_{n\rightarrow \infty} \bigl\langle z_{n}^{1}- \bar{r}^{*}, \psi _{1}\bigl(\bar{r}^{*}\bigr)- \bar{r}^{*} \bigr\rangle + \bigl\langle z_{n}^{2}- \bar{s}^{*},\psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*} \bigr\rangle \\& \quad =\limsup_{q\rightarrow \infty} \bigl\langle z_{n_{q}}^{1}- \bar{r}^{*},\psi _{1}\bigl(\bar{r}^{*}\bigr)- \bar{r}^{*} \bigr\rangle + \bigl\langle z_{n_{q}}^{2}- \bar{s}^{*},\psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*} \bigr\rangle \\& \quad = \limsup_{q\rightarrow \infty} \bigl\langle x^{*}- \bar{r}^{*}, \psi _{1}\bigl(\bar{r}^{*}\bigr)- \bar{r}^{*} \bigr\rangle + \bigl\langle y^{*}- \bar{s}^{*},\psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*} \bigr\rangle \\& \quad \le 0. \end{aligned}$$
(3.88)

As \(\lim_{n\rightarrow \infty}\sigma _{n}=0\) and \(\lim_{n\rightarrow \infty}\frac{\tau _{n}}{\sigma _{n}}=0\), using equation (3.88), we get \(\limsup_{n\to \infty} \bar{s}_{n}\le 0 \), and using Lemma 2.8, we have

$$\begin{aligned} \lim_{n\rightarrow \infty}\bar{p}_{n}= \lim_{n\rightarrow \infty} \bigl\Vert x_{n}- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-\bar{s}^{*} \bigr\Vert ^{2}=0, \end{aligned}$$
(3.89)

which implies \(\lim_{n\rightarrow \infty}\|x_{n}-\bar{r}^{*}\|^{2}=\lim_{n \rightarrow \infty}\|y_{n}-\bar{s}^{*}\|^{2}=0\). Thus \(x_{n}\to \bar{r}^{*}\) and \(y_{n}\to \bar{s}^{*}\).

Case 2. Suppose that there is a subsequence \(\{\bar{p}_{n_{q}}\}\) of \(\{\bar{p}_{n}\}\) such that

$$\begin{aligned} \bar{p}_{n_{q}+1}\ge \bar{p}_{n_{q}} \quad \text{for all } q\in \mathbb{N}. \end{aligned}$$

Hence, by Lemma 2.7, there is a nondecreasing sequence of natural numbers \(\{n_{l}\}\subset \mathbb{N}\) such that \(n_{l}\rightarrow \infty \) as \(l\rightarrow \infty \), we have

$$\begin{aligned} \bar{p}_{n_{l}+1}\ge \bar{p}_{n_{l}} \end{aligned}$$

and

$$\begin{aligned} \bar{p}_{n_{l}+1}\ge \bar{p}_{l}. \end{aligned}$$
(3.90)

This with equation (3.63) gives

$$\begin{aligned} &\sum_{i=1}^{\infty}\eta _{n_{l},i}\rho _{n_{l}}\epsilon \bigl( \bigl\Vert A^{*}(Ah_{n_{l}}-Bv_{n_{l}}) \bigr\Vert ^{2}+ \bigl\Vert B^{*}(Ah_{n_{l}}-Bv_{n_{l}}) \bigr\Vert ^{2} \bigr) \end{aligned}$$
(3.91)
$$\begin{aligned} &\quad \le \sigma _{n_{l}} \bigl( \bigl\Vert \psi _{1}(x_{n_{l}})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n_{l}})-\bar{s}^{*} \bigr\Vert ^{2}\bigr) \\ &\qquad {}+\bar{p}_{{n_{l}}}-\bar{p}_{n_{l}+1}+2\epsilon _{n_{l}}\bigl[ \bigl\Vert w_{n_{l}}- \bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n_{l}}-\bar{s}^{*} \bigr\Vert \bigr]. \end{aligned}$$
(3.92)

Taking limit \(l \to \infty \), using \(\lim_{l\to \infty}\rho _{n_{l}}=0\) and \(\epsilon _{n_{l}}=o(\rho _{n_{l}})\), we get

$$\begin{aligned} \bigl\Vert A^{*}(Ah_{n_{l}}-Bv_{n_{l}}) \bigr\Vert = \bigl\Vert B^{*}(Ah_{n_{l}}-Bv_{n_{l}}) \bigr\Vert =0. \end{aligned}$$
(3.93)

Similarly,

$$\begin{aligned}& \nu _{n_{l}}^{x}\sum _{j=1}^{M} b_{j}^{x}\bigl(2 \bar{c}^{x}-\nu _{n_{l}}^{x}\bigr) \bigl\Vert L_{j}^{x}w_{n_{l}}-L_{j}^{x} \bar{r}^{*} \bigr\Vert ^{2}+\nu _{n_{l}}^{y} \sum_{j=1}^{M} b_{j}^{y} \bigl(2\bar{c}^{y}-\nu _{n_{l}}^{y}\bigr) \bigl\Vert L_{j}^{y}t_{n_{l}}-L_{j}^{y} \bar{s}^{*} \bigr\Vert ^{2} \\& \quad \le \sigma _{n_{l}} \bigl( \bigl\Vert \psi _{1}(x_{n_{l}})- \bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi _{2}(y_{n_{l}})- \bar{s}^{*} \bigr\Vert ^{2}\bigr) \\& \qquad {}+\bar{p}_{{n_{l}}}-\bar{p}_{n_{l}+1}+2\epsilon _{n_{l}}\bigl[ \bigl\Vert w_{n_{l}}- \bar{r}^{*} \bigr\Vert \\& \qquad {}+ \bigl\Vert t_{n_{l}}-\bar{s}^{*} \bigr\Vert \bigr]. \end{aligned}$$
(3.94)

Taking limit \(l\to \infty \), using \(\lim_{l\to \infty}\rho _{n_{l}}=0\), condition (i), and \(\epsilon _{n_{l}}=o(\rho _{n_{l}})\), we get

$$\begin{aligned} \lim_{l\to \infty} \bigl\Vert L_{j}^{x}w_{n_{l}}-L_{j}^{x} \bar{r}^{*} \bigr\Vert =\lim_{l \to \infty} \bigl\Vert L_{j}^{y}t_{n_{l}}-L_{j}^{y} \bar{s}^{*} \bigr\Vert =0. \end{aligned}$$
(3.95)

Similarly, we can prove

$$\begin{aligned} &\lim_{l\rightarrow \infty} \bigl\Vert z_{n_{l}}^{1}-w_{n_{l}} \bigr\Vert = \lim_{l \rightarrow \infty} \bigl\Vert z_{n_{l}}^{2}-t_{n_{l}} \bigr\Vert = \lim_{l\rightarrow \infty} \Vert w_{n_{l}}-x_{n_{l}} \Vert =0\quad \text{and} \end{aligned}$$
(3.96)
$$\begin{aligned} &\lim_{l\rightarrow \infty} \Vert t_{n_{l}}-y_{n_{l}} \Vert =\lim_{l \rightarrow \infty} \Vert x_{n_{l}}-x_{{n_{l}}+1} \Vert = \lim_{l\rightarrow \infty} \Vert y_{n_{l}}-y_{{n_{l}}+1} \Vert =0. \end{aligned}$$
(3.97)

Using the same justification as in Case 1, we can prove

$$\begin{aligned} &\lim_{l\rightarrow \infty} \Vert Ax_{n_{l}}-By_{n_{l}} \Vert =\lim_{l \rightarrow \infty} \bigl\Vert A^{*}(Ah_{n_{l}}-Bv_{n_{l}}) \bigr\Vert =0 \end{aligned}$$
(3.98)
$$\begin{aligned} &\text{and}\quad \lim_{l\rightarrow \infty} \bigl\Vert U_{n_{l}}(v_{n_{l}})-v_{n_{l}} \bigr\Vert =\lim_{l \rightarrow \infty} \bigl\Vert S_{n_{l}}(h_{n_{l}})-h_{n_{l}} \bigr\Vert =\lim_{l \rightarrow \infty} \Vert h_{n_{l}}-w_{n_{l}} \Vert =0 \end{aligned}$$
(3.99)
$$\begin{aligned} &\text{and}\quad =\lim_{l\rightarrow \infty} \bigl\Vert B^{*}(Ah_{n_{l}}-Bv_{n_{l}}) \bigr\Vert =\lim_{l \rightarrow \infty} \Vert v_{n_{l}}-t_{n_{l}} \Vert =0. \end{aligned}$$
(3.100)

Again, following the same steps as in Case 1, we obtain

$$\begin{aligned} \limsup_{l\rightarrow \infty} \bigl\langle z_{n_{l}}^{1}- \bar{r}^{*}, \psi _{1}\bigl(\bar{r}^{*}\bigr)- \bar{r}^{*} \bigr\rangle + \bigl\langle z_{n_{l}}^{2}- \bar{s}^{*},\psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*} \bigr\rangle &\le 0. \end{aligned}$$
(3.101)

Additionally, from equation (3.56), we obtain

$$\begin{aligned} \bar{p}_{{n_{l}}+1}\le (1-d_{n_{l}}) \bar{p}_{{n_{l}}}+d_{n_{l}} \bar{s}_{n_{l}}, \end{aligned}$$
(3.102)

which implies

$$\begin{aligned} d_{n_{l}}\bar{p}_{{n_{l}}}&\le \bar{p}_{{n_{l}}}- \bar{p}_{{n_{l}}+1}+d_{n_{l}} \bar{s}_{n_{l}} \\ &\le d_{n_{l}}\bar{s}_{n_{l}}. \end{aligned}$$
(3.103)

As \(d_{n_{l}}>0\), we have \(\bar{p}_{{n_{l}}}\le \bar{s}_{n_{l}}\). Subsequently,

$$\begin{aligned} \bigl\Vert x_{n_{l}}-\bar{r}^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n_{l}}-\bar{s}^{*} \bigr\Vert ^{2}\le{}& \frac{1}{2(1-\lambda )} \biggl[ \bigl( \bigl\langle \psi _{1}\bigl(\bar{r}^{*}\bigr)- \bar{r}^{*}, z_{n_{l}}^{1}-\bar{r}^{*} \bigr\rangle \\ &{}+ \bigl\langle \psi _{2}\bigl(\bar{s}^{*}\bigr)- \bar{s}^{*}, z_{n_{l}}^{2}- \bar{s}^{*} \bigr\rangle \bigr) \\ &{}+4\frac{\epsilon _{n_{l}}}{\sigma _{n_{l}}} \bigl( \bigl\Vert w_{n_{l}}- \bar{r}^{*} \bigr\Vert + \bigl\Vert t_{n_{l}}- \bar{s}^{*} \bigr\Vert \bigr)+\sigma _{n_{l}}M_{2} \biggr]. \end{aligned}$$
(3.104)

As \(\lim_{l\rightarrow \infty}\sigma _{n_{l}}=0\), \(\lim_{l\rightarrow \infty}\frac{\epsilon _{n_{l}}}{\sigma _{n_{l}}}= 0\) and from equation (3.101), we obtain \(\bar{p}_{{n_{l}}}\to 0\) as \(l\to \infty \). Also using equation (3.102), we get \(\bar{p}_{n_{l}+1}\rightarrow 0\) as \(l \rightarrow \infty \). Subsequently, from equation (3.90), we get \(\bar{p}_{n_{l}+1}\ge \bar{p}_{l} \). This implies \(\lim_{l\rightarrow \infty}\bar{p}_{l}=0\) i.e. \(\|x_{l}-x^{*}\|^{2}+\|y_{l}-y^{*}\|^{2}\rightarrow 0\) as \(l \rightarrow \infty \), which implies \(x_{l}\rightarrow \bar{r}^{*}\) and \(y_{l}\rightarrow \bar{s}^{*}\). □

Remark 3.5

  1. (i)

    Izuchukwu et al. [55] established a strong convergence theorem for finding a common solution to two types of split equality problems, namely fixed point problem and finite families of variational inequalities problems. In this paper we proved a strong convergence theorem for finding a common solution to three types of split equality problems, namely infinite families of variational inclusion problems, finite families of variational inequalities, and a multiple-sets fixed point problem. Hence, our result is more desirable than Izuchukwu et al. [55] results.

  2. (ii)

    Theorem 3.4 generalizes the findings of Kazmi et al. [56] from the common solution of a multiple-sets split equality fixed point problem and a split equality variational inequality problem to a infinite family of split equality variational inclusion problems, a multiple-sets split equality fixed point problem, and a finite family of split equality variational inequality problems.

  3. (iii)

    Theorem 3.4 improves and generalizes the result of Guo et al. [57] from the common solution of a split equality variational inclusion problem and a finite family of fixed point problems to an infinite family of split equality variational inclusion problems, a finite family of split equality variational inequality problems, and a multiple-sets split equality fixed point problem. Further, our result generalize the Guo et al. [57] result from nonexpansive mapping to more generalized demicontractive mappings.

4 Applications

Intensity-modulated radiation therapy

Intensity-modulated radiation therapy (IMRT) is a cutting-edge radiotherapy technology that treats cancer while minimizing the amount of normal tissue exposed to radiation in the treatment area. IMRT has recently garnered much attention, as evidenced by [15, 58]. We often utilize an optimization technique to minimize the objective function of radiation beam weights, and radiations with varying intensities are transmitted into the body of patients while building IMRT.

The primary goal of IMRT is to deliver a sufficient dosage to the areas that require radiation therapy, known as planned target volumes (PTV), while limiting the amount given to other areas, known as organs at risk (OAR). To achieve a clinically appropriate dose distribution, we primarily evaluate the radiation dosage absorbed by irradiated tissue based on the distribution of beamlet intensities and the radiation intensity given by all beamlets when planning IMRT. Furthermore, we consider the irradiated object’s physical and biological features. While the restrictions on the deliverable radiation intensities of the beamlets are represented in the intensity space, the space whose elements are the deliverable radiation intensities of the beamlets, and the limits on the dose received by each voxel of the body are represented in the dose space. The intensity space and the dosage space are Euclidean spaces for vectors.

We employ volumetric modulated arc therapy (VMAT) and study two external sources/treatment equipment, the Varian and the Elekta. Here, we compare one dose space and two intensity spaces. We split the entire patient volume into J-voxels \(j = 1,2,\ldots,J\), beam into K-beamlets, \(k = 1,2,\ldots,K\). Assume that \(a_{jk}\ge 0\) is the dose absorbed at jth as a result of unit intensity from kth beamlet, \(x_{k}\ge 0\) is the intensity of k beamlet, and \(c_{j}\) denotes the dose absorbed by jth voxel given as

$$\begin{aligned} c_{j}=\sum_{k=1}^{K}a_{jk}x_{k}. \end{aligned}$$
(4.1)

We represent A as a dose influence matrix with nonnegative entries \(a_{jk}\). In the dose space, we will set a lower bound on the dose delivered to PTV and an upper bound on the dose delivered to OAR. Suppose that \(A_{s}\) denotes anatomical structures that are \(Q+R\) in number, where \(A_{s}\) are PTVs for \(s = 1,2,\ldots,Q\) and \(A_{s}\) are OARs for \(s= Q+1,Q+2,\ldots,R\). Let \(l_{s}\) and \(u_{s}\) be lower and upper bound doses for jth voxel, and some upper bound constraints are given as

$$\begin{aligned} J_{\max ,s}=\bigl\{ t \in R^{J} : t_{j}\le u_{s}, j\in A_{s}\bigr\} , \end{aligned}$$
(4.2)

the lower bound constraint

$$\begin{aligned} J_{\min ,s}=\bigl\{ t \in R^{J} : l_{s}\le t_{j}, j\in A_{s}\bigr\} , \end{aligned}$$
(4.3)

the equivalent uniform dose (EUD) constraint \(U_{s}:R^{J} \to R\) given as

$$\begin{aligned} U_{s}(t)= \biggl[\frac{1}{N_{s}}\sum _{j\in A_{s}}(t_{j})^{\alpha _{s}} \biggr]^{\frac{1}{\alpha _{s}}}. \end{aligned}$$
(4.4)

Assume that A and B are influenced matrices, \(X_{+}\) and \(Y_{+}\) are nonnegative constraints, \(X_{k_{1}}\), \({k_{1}} = 1,2,\ldots, N\), and \(Y_{k_{2}}, {k_{2}} = 1,2,\ldots, P\), are the beamlet intensity vectors that satisfy the nth constraint for Varian (V) and Elekta (E) machine respectively. Consider the multiple-sets split equality problem of finding

$$\begin{aligned} \text{Find } \bar{r}\in X_{+}\cap \Biggl(\bigcap _{{k_{1}}=1}^{N}X_{k_{1}}\Biggr), \qquad \bar{s}\in Y_{+}\cap \Biggl(\bigcap_{{k_{2}}=1}^{M}Y_{k_{2}} \Biggr) \quad \text{such that } A\bar{r}=B\bar{s}. \end{aligned}$$
(4.5)

Thus, the main objective of IMRT is to find an intensity that is closest to the entire radiation intensity space in the Varian machine and an intensity that is closest to the entire radiation intensity space in the Elekta machine, these both machines transmit an equal dose to the tumor. As we know that \(\bar{r^{*}}\in \operatorname{Fix}(P_{Q})\) if and only if \(x \in Q\), where \(P_{Q}\) is the metric projection defined on a closed convex subset Q of Hilbert space H.

Take \(S_{k_{1}}=P_{X_{+}\cap X_{j}}\) (\({k_{1}}=1,2,\ldots,N\)) and \(U_{k_{2}}=P_{Y_{+}\cap Y_{k}}\) (\({k_{2}}=1,2,\ldots,P\)) in Algorithm 3.1. For the multiple-sets split equality problem, we obtain the following cyclic iterative algorithm.

Algorithm 4.1

Consider \(0< a<\nu ^{x}_{n}\le 2\bar{c}^{x}\), \(0< b<\nu ^{y}_{n}\le 2\bar{c}^{y}\) for some \(a,b\in \mathbb{R} \), \(b_{j}^{x}, b_{j}^{y}\in (0,1)\) for \(j=1,2,\ldots,M\), \(\{\delta _{n}\}, \{\mu _{n}\}, \{\eta _{n,i}\}, \{\sigma _{n}\} \subset [\bar{d},\bar{e}]\subset (0,1)\), \(a_{n} \in (\delta ,1-\phi -\delta )\), \(\beta \in (0,1)\), \(\sum_{n=1}^{\infty}\tau _{n}<\infty \), and \(\{\kappa _{n}\}\subset [\kappa , \bar{\kappa}] \), where \(0<\kappa \le \bar{\kappa}\), \(\delta >0\), , and \(\bar{e} \in \mathbb{R}\). Choose \(x_{0}, x_{1} \in Q\) and \(\gamma _{n}\) such that \(0 \le \gamma _{n}\le \bar{\gamma _{n}}\), where

$$\begin{aligned} \bar{\gamma _{n}}= \textstyle\begin{cases} \beta & \text{if } x_{n}=x_{n-1} \text{ and } y_{n}=y_{n-1} \\ \min \{{\frac{\tau _{n}}{\sqrt{ \Vert x_{n}-x_{n-1} \Vert + \Vert y_{n}-y_{n-1} \Vert }}},\beta \} & \text{if otherwise}. \end{cases}\displaystyle \end{aligned}$$
(4.6)

Compute

$$\begin{aligned} \textstyle\begin{cases} w_{n}=x_{n}+\gamma _{n}(x_{n}-x_{n-1}),\\ t_{n}=y_{n}+\gamma _{n}(y_{n}-y_{n-1}), \\ h_{n}=\sigma _{n} \psi _{1}(x_{n})+(1-\sigma _{n})P_{Q_{1}}(I-\nu _{n}^{x}\sum_{j=1}^{M} b_{j}^{x}L_{j}^{x})w_{n},\\ x_{n+1}=\delta _{n} h_{n}+\mu _{n}((1-a_{n})h_{n}+a_{n}P_{X_{+} \cap X_{n}}(h_{n})) \\ \hphantom{x_{n+1}=}{}+\sum_{i=1}^{\infty}\eta _{n,i}(J_{\theta _{i}}^{K_{i}}(h_{n}-\rho _{n}(A^{*}(Ah_{n}-Bv_{n})))),\\ v_{n}=\sigma _{n} \psi _{2}(y_{n})+(1-\sigma _{n})P_{Q_{2}}(I-\nu _{n}^{y}\sum_{j=1}^{M} b_{j}^{y}L_{j}^{y})t_{n},\\ y_{n+1}=\delta _{n} v_{n}+\mu _{n} ((1-a_{n})v_{n}+a_{n}P_{Y_{+}\cap Y_{n}}(v_{n})) \\ \hphantom{y_{n+1}=}{}+\sum_{i=1}^{\infty}\eta _{n,i}(J_{\theta _{i}}^{T_{i}}(v_{n}+\rho _{n}(B^{*}(Ah_{n}-Bv_{n})))), \end{cases}\displaystyle \end{aligned}$$
(4.7)

where the stepsize \(\rho _{n}\) is given as

$$\begin{aligned}& \rho _{n}\in \biggl(\epsilon , \frac{2 \Vert Ah_{n}-Bv_{n} \Vert ^{2}}{ \Vert A^{*}(Ah_{n}-Bv_{n}) \Vert ^{2}+ \Vert B^{*}(Ah_{n}-Bv_{n}) \Vert ^{2}}-\epsilon \biggr), \\& n\in \Delta \text{ and }\epsilon >0\text{ otherwise }\rho _{n}=\rho , \text{and the index set }\Delta =\{n; Ah_{n}-Bv_{n}\ne 0\}. \end{aligned}$$

As we know, metric projection is firmly nonexpansive, and thus 0-demicontractive. As a result, using Algorithm 4.1 and the proof described in Theorem 3.4, we can obtain the strong convergence result for approximating the solution of the multiple-sets split equality problem (4.1).

It is not always possible to find an intensity that meets all of the constraints. In this case, we find a solution that is as close to all of the constraints as possible. To do so, we consider the following proximity function:

$$\begin{aligned} f(r,s)=\frac{a_{n}}{2}\sum_{{k_{1}}=1}^{N} \Vert P_{X_{+}\cap (X_{k_{1}})}r-r \Vert ^{2}+\frac{a_{n}}{2}\sum _{{k_{2}}=1}^{P} \Vert P_{Y_{+}\cap (Y_{k_{2}})}s-s \Vert ^{2}+\frac{1}{2} \Vert Ar-Bs \Vert ^{2}. \end{aligned}$$
(4.8)

The multiple-sets split equality problem (4.1), on the other hand, can be written as the minimization problem

$$\begin{aligned} \min_{r\in \mathbb{R}^{N}, s\in \mathbb{R}^{P}} f(r,s). \end{aligned}$$
(4.9)

5 Numerical example

In this section, we give a numerical example to compare the convergence of the algorithms given in [32, 37] to Algorithms 3.1.

Example 5.1

Let \(H_{1}=H_{2}=H_{3}=\mathbb{R}^{4}\) be the set of real numbers. Assume that

$$\begin{aligned} Q_{1}=Q_{2}=\bigl\{ (\bar{t_{1}}, \bar{t_{2}}, \bar{t_{3}}, \bar{t_{4}})^{T} \in \mathbb{R}^{4}: \bar{t_{1}}+ \bar{t_{2}}-3 \bar{t_{3}}+ \bar{t_{4}} \ge 0\bigr\} . \end{aligned}$$

Let \(A, B:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}\) be two bounded linear operators created from a normal distribution with unit variance and mean zero. Consider \(\psi _{1}:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}\) and \(\psi _{2}:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}\) as two contraction mappings such that \(\psi _{1}(x)=\frac{x}{24}\) and \(\psi _{2}(x)=\frac{x}{24}\) for all \(x\in \mathbb{R}^{4}\). Further, we take two demicontractive mappings \(S_{k_{1}}\) and \(U_{k_{1}}:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}\) such that \(S_{k_{1}}(x)=\frac{-3{k_{1}}}{{k_{1}}+1}x\), \(U_{k_{2}}(x)= \frac{-2{k_{2}}}{{k_{2}}+1}\), where \(k_{1}=k_{2}=1\), \(x=(x_{1}, x_{2}, x_{3}, x_{4})^{T}\). Take \(i=1\), and \({K_{1}}:H_{1} \to 2^{H_{1}}\) and \({T_{1}}:H_{2} \to 2^{H_{2}}\) are defined as \(K_{1}(x)=A^{*}A(x)\) and \(T_{1}(x)=B^{*}B(x)\). Take \(J=20\). Let \(L_{j}^{x}\) and \(L_{j}^{y}\) be defined by \(L_{j}^{y}(\bar{t})=\frac{\bar{t}+2}{5j}\) and \(L_{j}^{x}(\bar{t})=\frac{\bar{t}-1}{3j}\) with \(b_{j}^{x}=\frac{2}{3^{j}}+\frac{1}{N3^{N}}\), \(b_{j}^{y}=\frac{4}{5^{j}}+\frac{1}{N5^{N}}\) and \(\nu _{j}^{x}=\frac{1}{7j^{3}}\), \(\nu _{j}^{y}=\frac{1}{5j^{3}}\). Obviously, the solution set Γ is nonempty as \(0\in \Gamma \). Let \(\delta _{n}=\frac{1}{n+1}\), \(\mu _{n}=\frac{1+n}{3n}\), \(\eta _{n,1}= \frac{2n^{2}-2n-1}{3n (n+1)}\), \(\sigma _{n}=\frac{1}{2n+1}\), \(a_{n}=\frac{1}{n+1}\), and the step size \(\tau _{n}=\frac{1}{n^{5}}\). Choose \(\beta =0.5\) and initial values \(x=(0.1,0.1, 0.1, 0.1)^{T}\), \(y=(0.2, 0.2, 0.2, 0.2)^{T}\). We set \(E_{n} =\|x_{n}-x_{n-1}\|+\|y_{n}-y_{n-1}\|< 10^{-4}\) as a stopping criterion and plot the graphs of errors \(E_{n}\) over the number of iterations n. Furthermore, we provide numerical data for various values of \(\eta _{n}\) and the following values of \(x_{0}\) and \(x_{1}\).

Case 1: \(x_{0}=(0.01,0.01, 0.01, 0.01)\), \(x_{1}=(0.02,0.02, 0.02, 0.02)\);

Case 2: \(x_{0}=(0.1,0.1, 0.1, 0.1)\), \(x_{1}=(0.2,0.2, 0.2, 0.2)\);

Case 3: \(x_{0}=(1, 1, 1, 1)\), \(x_{1}=(2, 2, 2, 2)\).

Tables 13 and Figs. 13 present the numerical results.

Figure 1
figure 1

Comparison of Alg 3.1 with Eslamian Alg [32] and Chang Alg [37]

Figure 2
figure 2

Numerical analysis of Alg 3.1 for different cases

Figure 3
figure 3

Numerical analysis of Alg 3.1 for different values of \(\rho _{1}\)

Table 1 Comparison of Alg 3.1 with Eslamian Alg [32] and Chang Alg [37]
Table 2 Numerical analysis of Alg 3.1 for different cases
Table 3 Numerical analysis of Alg 3.1 for different values of \(\rho _{1}\)

6 Conclusion

In this paper, we have proposed a new inertial simultaneous cyclic iterative algorithm with a method for a finite family of split equality variational inequality problems, an infinite family of split equality variational inclusion problems, and a multiple-sets split equality fixed point problem for demicontractive operators in infinite dimensional Hilbert spaces. The proposed algorithm includes several well-known methods such as the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We prove strong convergence of the proposed algorithm. This result extends and unifies various known results in the literature. Finally, we give a numerical example to justify the main convergence theorem.

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Acknowledgements

The authors extend their appreciation to the Arab Open University, Saudi Arabia, for funding this work through AOU research fund No. (AOURG-2023-009).

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C.B., R.C., R.K., K.S., S.A. and N.M. wrote the main manuscript text and prepared Figs. 1-3. All authors reviewed the manuscript.

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Batra, C., Chugh, R., Kumar, R. et al. Strong convergence of split equality variational inequality, variational inclusion, and multiple sets fixed point problems in Hilbert spaces with application. J Inequal Appl 2024, 40 (2024). https://doi.org/10.1186/s13660-024-03118-0

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