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

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.


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 [2][3][4][5][6][7][8].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 r * ∈ Q such that L r * , p -r * ≥ 0 for all p ∈ Q, (1.1) where L is a nonlinear mapping.The set of solutions of the VIP is represented by 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 [17][18][19][20][21][22][23].
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 (r * , s * In 2014, Wu et al. [35] proposed the multiple-sets split equality fixed point problem (MSEFPP) stated as a problem of finding (r * , s * ) ∈ H 1 × H 2 such that Fix(U k 2 ) and Ar * = Bs * , ( where S k 1 : 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 r * ∈ H such that where A is a bounded linear operators, K : H The set of solutions to the GSEMVIP is represented by .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).

Preliminaries
In this study, we take H to be a real Hilbert space with the inner product ., .and the corresponding norm . .For weak and strong convergence of {x n } to x * , we use the notations x n x * and x n → 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 Fix( Ū).
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.

Let
where the stepsize ρ n is given as n ∈ and > 0 otherwise ρ n = ρ, and the index set = {n; Ah n -Bv n = 0}.
Remark 3.2 Assume that the solution set is nonempty, then {ρ n } in Algorithm 3.1 is well defined; see [54].

Lemma 3.3
Let the solution set be nonempty, {x n } and {y n } be bounded sequences.Suppose that l n -J .
Proof Now, we have to show that l * = (x * , y * ) ∈ .For this we have to show for i ≥ 1, 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 x * ∈ H 1 and y n q y * ∈ 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 x * , w n q x * , v n q y * , and t n q y * .Define l n = (h n , v n ).Using the given condition, we have Now, using Opial's condition and equation (3.3), we get which is a contradiction.Hence l * = J As the solution set is nonempty, K -1 i (0) and T -1 i (0) are nonempty.Using Lemma 2.9 and equation (3.4), we have Similarly, Adding equations (3.5) and (3.6), we have On simplifying As the solution set is nonempty, let l = (x , y ) ∈ for each i ≥ 1.Therefore,

Theorem 3.4
Let the sequence {(x n , y n )} be generated by the iterative Algorithm 3.1 and a solution set be nonempty.Suppose that the following conditions are satisfied: (i) for n ≥ 0, then the sequence generated by the iterative Algorithm 3.1 converges strongly to (r * , s * ) ∈ .
From equation (3.2), we have Adding equations (3.50) and (3.51) and using equations (3.18) and (3.49), we get where Hence, where Case 1. Assume that there exists N ∈ N such that {p n } is decreasing for n ≥ N .As {p n } is bounded and monotonic and subsequently convergent, as lim n→∞ σ n = 0 and τ n = o(σ n ), using equations (3.35), (3.43), (3.49), we have Taking limit n → ∞ in equation (3.58) and using condition (ii), we get From equations (3.32), (3.35), and (3.49), we get which implies Taking limit n → ∞ in equation (3.60) and using condition (ii), we get From equations (3.35), (3.45), and (3.49), we have Taking limit n → ∞ in equation (3.63) and using condition (ii), we get   From equation (3.2), we have From equation (3.2), we have Similarly, Also, Similarly, Since τ n = o(σ n ) and lim n→∞ σ n = 0, we have  As A and B are bounded linear operators, therefore, Ax n q Ax * and By n q By * .Clearly, it follows from weak semicontinuity of norm that Ax * -By * ≤ lim inf q→∞ Ax n q -By n q = 0. (3.82) This implies Ax * = By * .Subsequently, (x * , y * ) ∈ .Fix an index k 1 ∈ {1, 2, . . ., N}.As the pool of index is finite and lim n→∞ x nx n-1 = 0, from Lemma 2.6, one can find a subsequence {x m q } of {x n } such that x m q x * and m q ( mod N) + 1 = k 1 .Using equations (3.72) and (3.78), we get h m q x * .Additionally, and I -S k 1 is demiclosed at zero for each k 1 ∈ {1, 2, . . ., N}, which implies x * ∈ N k 1 =1 Fix(S k 1 ).Similarly, and I -U k 2 is demiclosed at zero for each k 2 ∈ {1, 2, . . ., P}, which implies y * ∈ P k 2 =1 Fix(U k 2 ).As {x n } is bounded, take a subsequence {x n q } of {x n } such that x n q x * ∈ 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 x * and w n q x * .Suppose that x * / ∈ M j=1 VIP(Q 1 , L x j ).Then, from Lemmas (2.10) and (2.11), we have ).Using Opial's condition and equation (3.69), we estimate which is a contradiction.Thus x * ∈ Fix(P Q 1 (Iν x M j=1 b x j L x j )).Similarly, {y n } is bounded, take a subsequence {y n q } of {y n } such that y n q y * ∈ 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 y * and t n q y * .Suppose that y * / ∈ M j=1 VIP(Q 2 , L y j ).Then, from Lemmas (2.10) and (2.11), we have . Using Opial's condition and equation (3.69), we estimate Now, using Lemma 3.3, l * ∈ .As w n q x * and t n q y * , from equation (3.69), we estimate z 1 n q x * and z 2 n q y * .Using Lemma 2.4, we get As lim n→∞ σ n = 0 and lim n→∞ τ n σ n = 0, using equation (3.88), we get lim sup n→∞ sn ≤ 0, and using Lemma 2.8, we have which implies lim n→∞ x n -r * 2 = lim n→∞ y n -s * 2 = 0. Thus x n → r * and y n → s * .Case 2. Suppose that there is a subsequence {p n q } of {p n } such that pn q +1 ≥ pn q for all q ∈ N.
Hence, by Lemma 2.7, there is a nondecreasing sequence of natural numbers {n l } ⊂ N such that n l → ∞ as l → ∞, we have + pn l -pn l +1 + 2 n l w n l -r * + t n l -s * . (3.92) Taking limit l → ∞, using lim l→∞ ρ n l = 0 and n l = o(ρ n l ), we get As d n l > 0, we have pn l ≤ sn l .Subsequently, As lim l→∞ σ n l = 0, lim l→∞ n l σ n l = 0 and from equation (3.101), we obtain pn l → 0 as l → ∞.Also using equation (3.102), we get pn l +1 → 0 as l → ∞.Subsequently, from equation (3.90), we get pn l +1 ≥ pl .This implies lim l→∞ pl = 0 i.e. x lx * 2 + y ly * 2 → 0 as l → ∞, which implies x l → r * and y l → s * .Remark 3.5 (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.(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.(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.

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, . . ., J, beam into K-beamlets, k = 1, 2, . . ., K .Assume that a jk ≥ 0 is the dose absorbed at jth as a result of unit intensity from kth beamlet, x k ≥ 0 is the intensity of k beamlet, and c j denotes the dose absorbed by jth voxel given as

.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, . . ., Q and A s are OARs for s = Q + 1, Q + 2, . . ., R. Let l s and u s be lower and upper bound doses for jth voxel, and some upper bound constraints are given as the lower bound constraint the equivalent uniform dose (EUD) constraint U s : R J → R given as Y k 2 such that Ar = Bs.(4.5) Thus, the main objective of IMRT is to find an intensity r that is closest to the entire radiation intensity space in the Varian machine and an intensity s 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 r * ∈ Fix(P Q ) if and only if x ∈ 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 + ∩X j (k 1 = 1, 2, . . ., N ) and U k 2 = P Y + ∩Y k (k 2 = 1, 2, . . ., P) in Algorithm 3.1.For the multiple-sets split equality problem, we obtain the following cyclic iterative algorithm.
where the stepsize ρ n is given as n ∈ and > 0 otherwise ρ n = ρ, and the index set = {n; Ah n -Bv n = 0}.
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: Ar -Bs 2 .(4.8) The multiple-sets split equality problem (4.1), on the other hand, can be written as the minimization problem min r∈R N ,s∈R P f (r, s).(4.9)

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.

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.

Table 2
Numerical analysis of Alg 3.1 for different cases

Table 3
Numerical analysis of Alg 3.1 for different values of ρ 1