# Inertial KM-type extragradient scheme for solving a variational inequality and a hierarchical fixed point problems

## Abstract

We propose an inertial KM-type extragradient scheme to approximate a common solution of a variational inequality problem and a hierarchical fixed point problem for nonexpansive mappings. This scheme generalizes and unifies a number of known iterative schemes. Furthermore, we discuss the weak convergence for the proposed scheme. We also discuss an example to illustrate the main theorem.

## Introduction

Let $${\mathcal{C}}$$ be a nonempty convex and closed set in a real Hilbert space $${\mathcal{H}}$$ and $$\langle \cdot ,\cdot \rangle$$ and $$\Vert \cdot \Vert$$ denote the inner product and induced norm on $${\mathcal{H}}$$. A mapping $$U:{\mathcal{C}} \to {\mathcal{C}}$$ is said to be nonexpansive if $$\Vert Uu-Uv\Vert \leq \Vert u-v\Vert$$, $$\forall u,v \in {\mathcal{C}}$$. Note that if $$\mathrm{F}(U):= \{ u \in {\mathcal{C}}: Uu=u\} \neq \emptyset$$ then set $$\mathrm{F}(U)$$ is convex and closed. Let $${\mathrm{F}}(U)\neq \emptyset$$. The subdifferential of a proper function $$g:{\mathcal{H}} \to (-\infty , +\infty ]$$ is the set-valued operator $$\partial g: {\mathcal{H}}\to 2^{\mathcal{H}}$$ defined by $$\partial g(u)=\{w \in {\mathcal{H}} : \langle y-u, w\rangle +g(u) \leq g(y), \forall y \in {\mathcal{H}} \}$$. Let $$u \in {\mathcal{H}}$$. Then g is subdifferential at u if $$\partial g(u) \neq \emptyset$$. The indicator function $$\psi _{\mathcal{C}}: {\mathcal{H}} \to (-\infty , +\infty ]$$ is given by

\begin{aligned}& \partial \psi _{\mathcal{C}}(u)= \textstyle\begin{cases} 0,&u\in {\mathcal{C}}, \\ \infty ,&\text{otherwise}. \end{cases}\displaystyle \end{aligned}

Note that $$\psi _{\mathcal{C}}$$ is a convex function when $${\mathcal{C}}$$ is a convex set.

In 2006, Moudafi et al.  discussed the convergence of a scheme for the following hierarchical fixed point problem (in short, H-FPP): Find $$\bar{u}\in {\mathrm{F}}(U)$$ such that

$$\langle \bar{u}-V\bar{u},\bar{u}-u\rangle \leq 0, \quad \forall u \in {\mathrm{F}}(U),$$
(1.1)

where the mappings $$U,V:{\mathcal{C}} \to {\mathcal{C}}$$ are nonexpansive. Let Φ denote the set of solutions of $$\operatorname{H\text{-}FPP}(\mbox{1.1})$$. If $$\bar{u}\in {\mathrm{F}}(U)$$ then $$(\mbox{1.1}) \Leftrightarrow \langle -(I-V)\bar{u}, u-\bar{u}\rangle + \psi _{{\mathrm{F}}(U)}(\bar{u})\leq \psi _{{\mathrm{F}}(U)}(u) \Leftrightarrow -(I-V) {\bar{u}} \in \partial \psi _{{\mathrm{F}}(U)} (\bar{u})$$. Hence $$\operatorname{H\text{-}FPP}(\mbox{1.1})$$ is equivalent to the variational inclusion: Find $$\bar{u}\in {{\mathrm{F}}(U)}$$ such that

$$0 \in (I-V)\bar{u}+ N_{{\mathrm{F}}(U)}(\bar{u}),$$
(1.2)

where the mapping I is identity on $${\mathcal{C}}$$ and $$N_{{\mathrm{F}}(U)}(\bar{u})$$ denotes the normal cone to $${\mathrm{F}}(U)$$ at ū given by

$$N_{{\mathrm{F}}(U)}(\bar{u})= \partial \psi _{{\mathrm{F}}(U)}(\bar{u})= \textstyle\begin{cases} \{w\in {\mathcal{H}}:\langle u-{\bar{u}},w\rangle \leq 0, \forall u\in { \mathrm{F}}(U)\},&\text{if }{\bar{u}}\in {\mathrm{F}}(U), \\ \emptyset ,&\text{otherwise}. \end{cases}$$

If we set $$V=I$$, then Φ is just $${\mathrm{F}}(U)$$. Furthermore, we mention that $$\operatorname{H\text{-}FPP}(\mbox{1.1})$$ is worth to study because it includes as special cases, the important problems such as the variational inequality on fixed point sets and hierarchical minimization problems; see Moudafi .

In 2007, Moudafi  proposed the following Krasnoselski–Mann (KM)-type scheme for solving $$\operatorname{H\text{-}FPP}(\mbox{1.1})$$: For given $$u_{0}\in {\mathcal{C}}$$,

$$u_{k+1}=(1-\alpha _{k})u_{k}+\alpha _{k}\bigl(\sigma _{k}Vu_{k}+(1-\sigma _{n})Uu_{k}\bigr), \quad \forall n \geq 0,$$
(1.3)

where $$\{\alpha _{k}\}\subset (0,1)$$ and $$\{\sigma _{k}\}\subset (0,1)$$. For further work related to scheme (1.3), see for example [1, 37].

In 2008, Mainge  introduced an inertial version of KM-type scheme by unifying the KM-type scheme and the inertial extrapolation, for approximating a fixed point of nonexpansive mappings and discussed the weak convergence. Recently, Bot et al.  derived some the convergence results of the following inertial KM-type scheme to approximate a fixed point of nonexpansive mapping U on $${\mathcal{H}}$$ which generalize the results of Mainge :

$$\left . \textstyle\begin{array}{l} t_{k} = u_{k}+\eta _{k}(u_{k}-u_{k-1}), \\ u_{k+1}=(1-\alpha _{k})t_{k}+\alpha _{k}Ut_{k}, \end{array}\displaystyle \right \}$$
(1.4)

for each $$k\geq 1$$, where $$\eta _{k}$$ is a damping-type term and $$\alpha _{k}$$ is a relaxation factor. Recently, the interest of studying inertial type algorithms has been increased due to their fast convergence. For further study of scheme (1.4) and its generalizations; see for example .

On the other hand, we consider the classical variational inequality (in short, VI): Find $$\bar{u}\in {\mathcal{C}}$$ such that

$$\bigl\langle h(\bar{u}), v-\bar{u}\bigr\rangle \geq 0, \quad \forall v \in {\mathcal{C}},$$
(1.5)

introduced in  where $$h: {\mathcal{H}} \to {\mathcal{H}}$$. The set of solutions of $$\operatorname{VI}(\mbox{1.5})$$ is denoted by $$\operatorname{Sol}(\operatorname{VI}(\mbox{1.5}))$$. Note that the projected gradient scheme for solving $$\operatorname{VI}(\mbox{1.5})$$ is

$$u_{k+1}={\mathcal{P}}_{{\mathcal{C}}}\bigl(u_{k}- \mu h(u_{k})\bigr),$$
(1.6)

where $$\mu >0$$ and $${\mathcal{P}}_{{\mathcal{C}}}$$ is the metric projection onto $${\mathcal{C}}$$. In order to converge, this scheme requires the restrictive condition that h is inverse strongly (or strongly) monotone. To overcome this difficulty, Korpelevich  proposed an extragradient iterative scheme by

$$\left . \textstyle\begin{array}{l} v_{k}={\mathcal{P}}_{{\mathcal{C}}}(u_{k}-\mu h(u_{k})), \\ u_{k+1}={\mathcal{P}}_{{\mathcal{C}}}(u_{k}-\mu h(v_{k})), \end{array}\displaystyle \right \}$$
(1.7)

where $$\mu \in (0, \frac{1}{L})$$, where $$L>0$$ is Lipschitz constant of h. Since then many researchers improved scheme (1.7) in various directions; see, e.g.  and the references therein. Note that the calculation of two projections onto $${\mathcal{C}}$$ might affect the efficiency of such scheme. Therefore, Dong et al.  proposed the following inertial KM-type extragradient scheme for $$\operatorname{VI}(\mbox{1.5})$$:

$$\left . \textstyle\begin{array}{l} t_{k}= u_{k}+\eta _{k}(u_{k}-u_{k-1}), \\ v_{k}={\mathcal{P}}_{{\mathcal{C}}}(t_{k}-\mu h(t_{k})), \\ u_{k+1}=(1-\alpha _{k})t_{k}+\alpha _{k}{\mathcal{P}}_{{\mathcal{C}}}(t_{k}- \mu h(v_{k})), \end{array}\displaystyle \right \}$$
(1.8)

where $$\{\eta _{k}\}\subset [0, \eta ]$$, k is nondecreasing with $$\eta _{1}=0$$ and $$0\leq \eta _{k} \leq \eta < 1$$, for every $$k\geq 1$$ such that

$$\delta > \frac{\eta [(1+\mu L)^{2}\eta (1+\eta )+(1-\mu ^{2}L^{2})\eta \sigma +\sigma (1+\mu L)^{2}]}{1-\mu ^{2}L^{2}}$$

and

\begin{aligned}& 0< \alpha \leq \alpha _{k}\leq \frac{\delta (1-\mu ^{2}L^{2})-\eta [(1+\mu L)^{2}\eta (1+\eta )+(1-\mu ^{2}L^{2})\eta \sigma +\sigma (1+\mu L)^{2}]}{\delta [(1+\mu L)^{2}\eta (1+\eta )+(1-\mu ^{2}L^{2})\eta \sigma +\sigma (1+\mu L)^{2}]}, \\& \quad \text{where } \alpha , \sigma , \delta >0. \end{aligned}

They proved the weak convergence theorem for scheme (1.8).

In this paper, we propose an inertial version of KM-type extragradient scheme by combining iterative schemes (1.3) and (1.8) to approximate a common solution of $$\operatorname{H\text{-}FPP}(\mbox{1.1})$$ and $$\operatorname{VI}(\mbox{1.5})$$. We prove a weak convergence theorem for the proposed scheme. Furthermore, we discuss an example to illustrate the main theorem. The theorems of the paper unify and generalize previously known corresponding theorems; see for example [2, 8, 9, 2527].

## Preliminaries

We give some definitions and results of convex and nonlinear analysis, which will be used in the proof of the weak convergence theorem.

A mapping $${\mathcal{P}}_{{\mathcal{C}}}$$ is called the metric projection of $${\mathcal{H}}$$ onto $${\mathcal{C}}$$ if for every point $$u \in {\mathcal{H}}$$, there exists a unique point in $${\mathcal{C}}$$ denoted by $${\mathcal{P}}_{{\mathcal{C}}} u$$ such that

$$\Vert u-{\mathcal{P}}_{{\mathcal{C}}}u \Vert \leq \Vert u-v \Vert , \quad \forall v \in {\mathcal{C}}.$$

Note that $${\mathcal{P}}_{{\mathcal{C}}}$$ is nonexpansive and satisfies

$$\langle u-v ,{\mathcal{P}}_{{\mathcal{C}}}u-{\mathcal{P}}_{{\mathcal{C}}}v \rangle \geq \Vert {\mathcal{P}}_{{\mathcal{C}}}u-{\mathcal{P}}_{{\mathcal{C}}}v \Vert ^{2}, \quad \forall u \in {\mathcal{H}}.$$

Moreover, $${\mathcal{P}}_{{\mathcal{C}}}u$$ is characterized by the fact $${\mathcal{P}}_{{\mathcal{C}}}u\in {\mathcal{C}}$$ and

$$\langle u-{\mathcal{P}}_{{\mathcal{C}}}u,v-{\mathcal{P}}_{{\mathcal{C}}}u \rangle \leq 0, \quad \forall v\in {\mathcal{C}},$$

which implies that

$$\Vert u-v \Vert ^{2}\geq \Vert u-{ \mathcal{P}}_{{\mathcal{C}}}u \Vert ^{2} + \Vert v-{ \mathcal{P}}_{{ \mathcal{C}}}u \Vert ^{2}, \quad \forall u\in { \mathcal{H}}, v\in {\mathcal{C}}.$$

### Definition 2.1

A mapping $$h:{\mathcal{H}} \to {\mathcal{H}}$$ is called:

1. (i)

monotone, if for all $$u,v \in {\mathcal{H}}$$, we have

$$\langle hu-hv , u-v\rangle \geq 0;$$
2. (ii)

L-Lipschitz continuous, if there exists a constant $$L >0$$ such that, for all $$u,v \in {\mathcal{H}}$$, we have

$$\Vert hu-hv \Vert \leq L \Vert u-v \Vert .$$

### Lemma 2.1

If a mapping U is nonexpansive on $${\mathcal{H}}$$ then $$I-U$$ is maximal monotone  and demiclosed  on $${\mathcal{H}}$$.

### Lemma 2.2

()

Let $$\{\psi _{k}\}$$, $$\{\delta _{k}\}$$ and $$\{\eta _{k}\}$$ be the sequences in $$[0, \infty )$$ such that $$\psi _{k+1}\leq \psi _{k}+\eta _{k}(\psi _{k}-\psi _{k-1})+\gamma _{k}$$, $$\forall k\geq 1$$, $$\sum_{k=1}^{\infty }\gamma _{k} < +\infty$$ and there is a number η with $$0\leq \eta _{k}\leq \eta <1$$, $$\forall k\geq 1$$. Then the following hold:

1. (a)

$$\sum_{k= 1}^{\infty }[\psi _{k}-\psi _{k-1}]_{+}< +\infty$$, where $$[r]_{+} := \max \{r, 0\}$$;

2. (b)

there is a $$\psi ^{*}\in [0, \infty )$$ such that $$\lim_{k\to \infty } \psi _{k}=\psi ^{*}$$.

### Lemma 2.3

()

Let $${\mathcal{C}}$$ be a nonempty subset of $${\mathcal{H}}$$ and the sequence $$\{u_{k}\}$$ in $${\mathcal{H}}$$ satisfy the conditions:

1. (a)

$$\lim_{k \to \infty } \Vert u_{k} -u\Vert$$ exists for every $$u \in {\mathcal{C}}$$;

2. (b)

any weak cluster point of $$\{u_{k}\}$$ is in $${\mathcal{C}}$$.

Then $$\{u_{k}\}$$ is weak convergent to a point in $${\mathcal{C}}$$.

## Weak convergence theorem

We propose the following inertial KM-type extragradient scheme for solving $$\operatorname{H\text{-}FPP}(\mbox{1.1})$$ and $$\operatorname{VI}(\mbox{1.5})$$.

### Scheme

Choose initial values $$u_{0}, u_{1}\in {\mathcal{H}}$$ arbitrarily. The sequence $$\{u_{k}\}$$ be generated by the scheme:

$$\left . \textstyle\begin{array}{l} t_{k}=u_{k}+\eta _{k}(u_{k}-u_{k-1}), \\ v_{k}= {\mathcal{P}}_{{\mathcal{C}}}(t_{k}-\mu h(t_{k})), \\ w_{k}= {\mathcal{P}}_{{\mathcal{C}}}(t_{k}-\mu h(v_{k})), \\ u_{k+1}= (1-\alpha _{k})t_{k}+\alpha _{n}(\sigma _{k}Vw_{k}+(1- \sigma _{n})Uw_{k}), \end{array}\displaystyle \right \}$$
(3.1)

where $$\{\eta _{k}\}\subset [0, \eta ]$$, k, is nondecreasing with $$\eta _{1}=0$$ and $$0\leq \eta _{k} \leq \eta < 1$$, $$\{\sigma _{k}\}\subseteq [c,d]$$, $$c,d\in (0,1)$$, $$\mu \in (0,\frac{1}{L})$$, $$L>0$$ and $$\{\alpha _{k}\}$$ is a real sequence with conditions:

$$\delta > \frac{\eta ^{2}(1+\eta )+\eta \sigma }{1-\eta ^{2}} \quad \text{and} \quad 0< \alpha \leq \alpha _{k}\leq \frac{\delta -\eta [\eta (1+\eta )+\eta \delta +\sigma ]}{\delta [1+\eta (1+\eta )+\eta \delta +\sigma ]}, \quad \text{where } \alpha , \sigma , \delta >0.$$

Now, we discuss the weak convergence for scheme (3.1).

### Theorem 3.1

Let $${\mathcal{H}}$$ be a real Hilbert space and $${\mathcal{C}}\subset {\mathcal{H}}$$ be a nonempty, convex and closed set; let the mappings $$U,V:{\mathcal{C}}\to {\mathcal{C}}$$ be nonexpansive and $$h:{\mathcal{H}}\to {\mathcal{H}}$$ be L-Lipschitz continuous and monotone. Assume that $$\Gamma =\operatorname{Sol}(\operatorname{VI}(\mbox{1.5}))\cap \Phi \cap {\mathrm{F}(V)} \neq \emptyset$$. Let the sequence $$\{u_{k}\}$$ be defined by scheme (3.1). Then the following results hold:

1. (a)

$$\sum_{k=1}^{\infty }\Vert u_{k+1}-u_{k}\Vert ^{2}< +\infty$$;

2. (b)

the sequence $$\{u_{k}\}$$ converges weakly to $$\bar{u} \in \Gamma$$.

### Proof

(a). Let for any $$q\in \Gamma$$. Since h is L-Lipschitz continuous and monotone then we can easily get

$$\Vert w_{k}-q \Vert ^{2}\leq \Vert t_{k}-q \Vert ^{2}-\bigl(1-\mu ^{2}L^{2} \bigr) \Vert t_{k}-v_{k} \Vert ^{2};$$
(3.2)

see . From the nonexpansivity of $${\mathcal{P}}_{{\mathcal{C}}}$$ and Lipschitz continuity of h, it follows that

\begin{aligned} \Vert v_{k}-w_{k} \Vert = \bigl\Vert { \mathcal{P}}_{{\mathcal{C}}}\bigl(t_{k}-\mu h(t_{k})\bigr)-{ \mathcal{P}}_{{ \mathcal{C}}}\bigl(t_{k}-\mu h(v_{k})\bigr) \bigr\Vert \leq & \mu \bigl\Vert h(t_{k})-h(v_{k}) \bigr\Vert \\ \leq & \mu L \Vert t_{k}-v_{k} \Vert , \end{aligned}
(3.3)

which yields

$$\Vert t_{k}-w_{k} \Vert \leq \Vert t_{k}-v_{k} \Vert + \Vert v_{k}-w_{k} \Vert \leq (1+\mu L) \Vert t_{k}-v_{k} \Vert .$$
(3.4)

As follows from (3.2), (3.4) and $$\mu L\in (0,1)$$, we have

$$\Vert w_{k}-q \Vert ^{2}\leq \Vert t_{k}-q \Vert ^{2}- \frac{1-\mu ^{2}L^{2}}{(1+\mu L)^{2}} \Vert t_{k}-w_{k} \Vert ^{2}.$$
(3.5)

Let for any $$q\in \Gamma$$ and $$T_{\sigma _{k}}:=\sigma _{k}V+(1-\sigma _{k})U$$. Now, by using (3.5), we estimate

\begin{aligned} \Vert u_{k+1}-q \Vert ^{2} =& \bigl\Vert (1-\alpha _{k})t_{k}+\alpha _{k}T_{\sigma _{k}}w_{k}-q \bigr\Vert ^{2} \\ \leq &(1-\alpha _{k}) \Vert t_{k}-q \Vert ^{2}+\alpha _{k} \Vert T_{\sigma _{n}}w_{k}-q \Vert ^{2}-\alpha _{k}(1-\alpha _{k}) \Vert T_{\sigma _{k}}w_{k}-t_{k} \Vert ^{2} \\ \leq &(1-\alpha _{k}) \Vert t_{k}-q \Vert ^{2}+\alpha _{k}\bigl(\sigma _{k} \Vert Vw_{k}-q \Vert ^{2}+(1-\sigma _{k}) \Vert Uw_{k}-q \Vert ^{2} \\ & {} -\sigma _{k}(1-\sigma _{k}) \Vert Vw_{k}-Uw_{k} \Vert ^{2}\bigr)-\alpha _{k}(1- \alpha _{k}) \Vert T_{\sigma _{k}}w_{k}-t_{k} \Vert ^{2} \\ \leq & \Vert t_{k}-q \Vert ^{2}-\alpha _{k} \sigma _{k}(1-\sigma _{k}) \Vert Vw_{k}-Uw_{k} \Vert ^{2}-\frac{1-\mu ^{2}L^{2}}{(1+\mu L)^{2}} \Vert t_{k}-v_{k} \Vert ^{2} \\ & {} -\alpha _{k}(1-\alpha _{k}) \Vert T_{\sigma _{k}}w_{k}-t_{k} \Vert ^{2} \end{aligned}
(3.6)
\begin{aligned} \leq & \Vert t_{k}-q \Vert ^{2}-\alpha _{k}(1-\alpha _{k}) \Vert T_{\sigma _{k}}w_{k}-t_{k} \Vert ^{2}. \end{aligned}
(3.7)

Next, we estimate

\begin{aligned} \Vert t_{k}-q \Vert ^{2} =& \bigl\Vert u_{k}+\eta _{k}(u_{k}-u_{k-1})-q \bigr\Vert ^{2} \\ =& (1+\eta _{k}) \Vert u_{k}-q \Vert ^{2}- \eta _{k} \Vert u_{k-1}-q \Vert ^{2} \\ & {} +\eta _{k}(1+\eta _{k}) \Vert u_{k}-u_{k-1} \Vert ^{2}. \end{aligned}
(3.8)

From (3.7) and (3.8), we have

\begin{aligned} \Vert u_{k+1}-q \Vert ^{2}-(1+\eta _{k}) \Vert u_{k}-q \Vert ^{2}+\eta _{k} \Vert u_{k-1}-q \Vert ^{2} \leq & -\alpha _{k}(1-\alpha _{k}) \Vert T_{\sigma _{k}}u_{k}-t_{k} \Vert ^{2} \\ & {} +\eta _{k}(1+\eta _{k}) \Vert u_{k}-u_{k-1} \Vert ^{2}. \end{aligned}
(3.9)

Furthermore, from scheme (3.1), we have

\begin{aligned} \Vert T_{\sigma _{k}}w_{k}-t_{k} \Vert ^{2} =& \biggl\Vert \frac{1}{\alpha _{k}}(u_{k+1}-u_{k})+ \frac{\eta _{k}}{\alpha _{k}}(u_{k-1}-u_{k}) \biggr\Vert ^{2} \\ \geq & \frac{1}{\alpha _{k}^{2}} \Vert u_{k+1}-u_{k} \Vert ^{2}+ \frac{\eta _{k}^{2}}{\alpha _{k}^{2}} \Vert u_{k}-u_{k-1} \Vert ^{2} \\ & {} +\frac{\eta _{k}}{\alpha _{k}^{2}} \biggl(-\rho _{k} \Vert u_{k+1}-u_{k} \Vert ^{2}- \frac{1}{\rho _{k}} \Vert u_{k}-u_{k-1} \Vert ^{2}\biggr), \end{aligned}
(3.10)

where $$\rho _{k}:=\frac{1}{\eta _{k}+\delta \alpha _{k}}$$. Thus, it follows from (3.9) and (3.10) that

\begin{aligned} \Vert u_{k+1}-q \Vert ^{2}-(1+\eta _{k}) \Vert u_{k}-q \Vert ^{2}+\eta _{k} \Vert u_{k-1}-q \Vert ^{2} \leq & \frac{(1-\alpha _{k})(\eta _{k}\rho _{k}-1)}{\alpha _{k}} \Vert u_{k+1}-u_{k} \Vert ^{2} \\ & {} +\gamma _{k} \Vert u_{k}-u_{k-1} \Vert ^{2}, \end{aligned}
(3.11)

where

$$\gamma _{k}:= \eta _{k}(1+\eta _{k})+\eta _{k}(1-\alpha _{k}) \frac{(1-\eta _{k}\rho _{k})}{\alpha _{k}\rho _{k}}>0,$$
(3.12)

since $$\eta _{k}\rho _{k} <1$$ and $$\alpha _{k}\in (0,1)$$. It follows from $$\delta =\frac{(1-\eta _{k}\rho _{k})}{\alpha _{k}\rho _{k}}$$ and (3.12) that

$$\gamma _{k}:= \eta _{k}(1+\eta _{k})+\eta _{k}(1-\alpha _{k})\delta \leq \eta (1+ \eta )+\eta \delta , \quad \forall k\geq 1.$$
(3.13)

Next, we define the sequences $$\{\phi _{k}\}$$ and $$\{\psi _{k}\}$$ by

$$\phi _{k}:= \Vert x_{k}-q \Vert ^{2}, \quad\quad \psi _{k}:= \phi _{k}-\eta _{k}\phi _{k-1}+ \gamma _{k} \Vert u_{k}-u_{k-1} \Vert ^{2}, \quad \forall k\geq 1.$$
(3.14)

Now, using the monotonicity of $$\{\eta _{k}\}$$ and the fact that $$\phi _{k}\geq 0$$ for all $$k\in \mathbb{N}$$, we have

$$\psi _{k+1}-\psi _{k}\leq \phi _{k+1}-(1+\eta _{k})\phi _{k}+\eta _{k} \phi _{k-1}+\gamma _{k+1} \Vert u_{k+1}-u_{k} \Vert ^{2}-\gamma _{k} \Vert u_{k}-u_{k-1} \Vert ^{2}.$$
(3.15)

Hence, it follows from (3.11) and (3.15) that

\begin{aligned} \psi _{k+1}-\psi _{k} \leq & \frac{(1-\alpha _{k})(\eta _{k}\rho _{k}-1)}{\alpha _{k}} \Vert u_{k+1}-u_{k} \Vert ^{2}+\gamma _{k+1} \Vert u_{k+1}-u_{k} \Vert ^{2} \\ =& \biggl(\frac{(1-\alpha _{k})(\eta _{k}\rho _{k}-1)}{\alpha _{k}}+ \gamma _{k+1} \biggr) \Vert u_{k+1}-u_{k} \Vert ^{2}. \end{aligned}
(3.16)

Now, we note that

$$\frac{(1-\alpha _{k})(\eta _{k}\rho _{k}-1)}{\alpha _{k}}+\gamma _{k+1} \leq -\sigma , \quad \forall k\geq 1;$$
(3.17)

see .

Therefore, it follows from (3.16) and (3.17) that

\begin{aligned} \psi _{k+1}-\psi _{k}\leq -\sigma \Vert u_{k+1}-u_{k} \Vert ^{2}. \end{aligned}
(3.18)

Since $$\eta _{1}=0$$, it follows from (3.14) that $$\psi _{1}=\phi _{1}\geq 0$$ and hence (3.18) shows that $$\{\psi _{k}\}$$ is bounded. Furthermore, (3.14) and the boundedness of $$\{\eta _{k}\}$$ yield

\begin{aligned} -\eta \phi _{k-1}\leq \phi _{k}-\eta \phi _{k-1}\leq \psi _{k}\leq \psi _{1}. \end{aligned}
(3.19)

Thus, we obtain

\begin{aligned} \phi _{k}\leq \eta ^{k}\phi _{0}+ \psi _{1}\sum_{j=1}^{k-1} \eta ^{j}\leq \eta ^{k}\phi _{0}+\frac{1}{1-\eta }\psi _{1}. \end{aligned}
(3.20)

Now, it follows from (3.18), (3.19), (3.20) and the boundedness of $$\{\psi _{k}\}$$ that

\begin{aligned} \sigma \sum_{j=1}^{k} \Vert u_{j+1}-u_{j} \Vert ^{2}\leq \psi _{1}- \psi _{k+1}\leq \psi _{1}+\eta \phi _{k}\leq \psi _{1}+\eta ^{k} \phi _{0}+ \frac{1}{1-\eta }\psi _{1}, \end{aligned}
(3.21)

which implies that $$\sum_{k=1}^{\infty }\Vert u_{k+1}-u_{k}\Vert ^{2}<+\infty$$.

Proof of (b). Since $$\eta _{k}\rho _{k} <1$$, it follows from (3.11), (3.13), $$\sum_{k=1}^{\infty }\Vert u_{k+1}-u_{k}\Vert ^{2}<+\infty$$, and Lemma 2.2 that

$$\lim_{k\to \infty } \Vert u_{k}-q \Vert \quad \text{exists and finite},$$
(3.22)

and hence $$\{u_{k}\}$$ is bounded. It follows furthermore from $$\sum_{k=1}^{\infty }\Vert u_{k+1}-u_{k}\Vert ^{2}<+\infty$$ that

$$\lim_{k\to \infty } \Vert u_{k+1}-u_{k} \Vert =0.$$
(3.23)

Next, by the definition of $$t_{k}$$ in (3.1) and $$\eta _{k}\leq \eta$$, k, we have

$$\Vert t_{k}-u_{k} \Vert =\eta _{k} \Vert u_{k}-u_{k-1} \Vert \leq \eta \Vert u_{k}-u_{k-1} \Vert ,$$

which implies that

$$\lim_{k\to \infty } \Vert t_{k}-u_{k} \Vert =0,$$
(3.24)

and hence $$\{t_{k}\}$$ is bounded. Since

$$\Vert t_{k}-u_{k+1} \Vert \leq \Vert t_{k}-u_{k} \Vert + \Vert u_{k}-u_{k+1} \Vert ,$$
(3.25)

it follows from (3.23), (3.24) and (3.25) that

$$\lim_{k\to \infty } \Vert t_{k}-u_{k+1} \Vert =0.$$
(3.26)

From (3.6) and (3.26), and $$\{\alpha _{k}\}\subseteq (0,1)$$, $$\{\sigma _{k}\}\subseteq [c,d]$$, $$c,d \in (0,1)$$, we have

\begin{aligned} \alpha _{k}\sigma _{k}(1-\sigma _{k}) \Vert Vw_{k}-Uw_{k} \Vert ^{2} =& \Vert t_{k}-q \Vert ^{2}- \Vert u_{k+1}-q \Vert ^{2} \\ \leq & \Vert t_{k}-u_{k+1} \Vert \bigl( \Vert t_{k}-q \Vert + \Vert u_{k+1}-q \Vert \bigr) \\ =& \Vert t_{k}-u_{k+1} \Vert M_{1}, \end{aligned}

where $$M_{1}:=\sup_{k}\{\Vert t_{k}-q\Vert +\Vert u_{k+1}-q\Vert \}$$. Hence, it follows

$$\lim_{k\to \infty } \Vert Vw_{k}-Uw_{k} \Vert =0.$$
(3.27)

From (3.6) and (3.26), and $$\mu L\in (0,1)$$, we have

\begin{aligned} \frac{1-\mu ^{2}L^{2}}{(1+\mu L)^{2}} \Vert t_{k}-w_{k} \Vert ^{2} \leq & \Vert t_{k}-q \Vert ^{2}- \Vert u_{k+1}-q \Vert ^{2} \\ =& \Vert t_{k}-u_{k+1} \Vert M_{1}, \end{aligned}

it follows that

$$\lim_{k\to \infty } \Vert t_{k}-w_{k} \Vert =0.$$
(3.28)

It follows from (3.26) and (3.28) that

$$\lim_{k\to \infty } \bigl\Vert t_{k}-u_{k+1}- \alpha _{k}(t_{k}-w_{k}) \bigr\Vert =0.$$
(3.29)

Furthermore, we have

\begin{aligned}& \alpha _{k} \Vert Uw_{k}-w_{k} \Vert \leq \Vert u_{k+1}-t_{k} \Vert +\alpha _{k} \Vert t_{k}-w_{k} \Vert +\alpha _{k}\sigma _{k} \Vert Uw_{k}-Vw_{k} \Vert , \\& \Vert Uw_{k}-w_{k} \Vert \leq \frac{1}{\alpha _{k}} \Vert u_{k+1}-t_{k} \Vert + \Vert t_{k}-w_{k} \Vert +\sigma _{k} \Vert Uw_{k}-Vw_{k} \Vert . \end{aligned}
(3.30)

Since $$\alpha _{k}>\alpha >0$$, k, it follows from (3.26), (3.27), (3.28) and (3.30) that

$$\lim_{k\to \infty } \Vert Uw_{k}-w_{k} \Vert =0.$$
(3.31)

From (3.27) and (3.31), we have

$$\lim_{k\to \infty } \Vert Vw_{k}-w_{k} \Vert =0.$$
(3.32)

Now, let ū be a sequential weak cluster point of $$\{u_{k}\}$$, that is, there exists a subsequence $$\{u_{k_{i}}\}$$ of $$\{u_{k}\}$$ such that $$\{u_{k_{i}}\}$$ converges weakly to ū, say, in $${\mathcal{H}}$$. Furthermore, (3.24) and (3.28) imply that $$\{u_{k}\}$$, $$\{t_{k}\}$$ and $$\{w_{k}\}$$ all have the same asymptotic behavior and hence there exist subsequences $$\{t_{k_{i}}\}$$ of $$\{t_{k}\}$$ and $$\{w_{k_{i}}\}$$ of $$\{w_{k}\}$$ and such that $$t_{k_{i}}$$ and $$w_{k_{i}}$$ both converge weakly to ū. Now, Lemma 2.1, (3.31) and (3.32) imply that $$\bar{u}\in {\mathrm{F}}(U)$$ and $$\bar{u}\in {\mathrm{F}}(V)$$.

Next, we prove that $$\bar{u}\in \Phi$$. Since

$$u_{k+1}-t_{k}=\alpha _{k}(w_{k}-t_{k})+ \alpha _{k}\bigl(\sigma _{k}(Vw_{k}-w_{k})+(1- \sigma _{k}) (Uw_{k}-w_{k})\bigr),$$
(3.33)

and hence

$$\frac{1}{\alpha _{k}\sigma _{k}} \bigl(t_{k}-u_{k+1}-\alpha _{k}(t_{k}-w_{k}) \bigr)=(I-V)w_{k}+ \biggl(\frac{1-\sigma _{k}}{\sigma _{k}} \biggr) (I-U)w_{k},$$
(3.34)

and therefore for all $$z\in {\mathrm{F}}(U)$$ and by making use of the monotonicity of $$I-V$$, we have

\begin{aligned} \biggl\langle \frac{1}{\alpha _{k}\sigma _{k}} \bigl(t_{k}-u_{k+1}- \alpha _{k}(t_{k}-w_{k}) \bigr), w_{k}-z\biggr\rangle =&\bigl\langle (I-V)w_{k}-(I-V)z,w_{k}-z \bigr\rangle \\ & {} +\bigl\langle (I-V)z,w_{k}-z\bigr\rangle \\ & {} +\frac{1-\sigma _{k}}{\sigma _{k}}\langle w_{k}-Uw_{k},w_{k}-z \rangle \\ \geq &\bigl\langle (I-V)z,w_{k}-z\bigr\rangle \\ & {} +\frac{1-\sigma _{k}}{\sigma _{k}}\langle w_{k}-Uw_{k},w_{k}-z \rangle . \end{aligned}
(3.35)

Hence,

\begin{aligned}& \biggl\langle \frac{1}{\alpha _{k_{i}}\sigma _{k_{i}}} \bigl(t_{k_{i}}-u_{{k_{i}}+1} - \alpha _{k_{i}}(t_{k_{i}}-w_{k_{i}}) \bigr), w_{k_{i}}-z \biggr\rangle \\& \quad \geq \bigl\langle (I-V)z,w_{k_{i}}-z\bigr\rangle \\& \quad\quad{} +\frac{1-\sigma _{k_{i}}}{\sigma _{k_{i}}}\langle w_{k_{i}}-Uw_{k_{i}},w_{k_{i}}-z \rangle . \end{aligned}
(3.36)

Using (3.29), (3.31), and the conditions on the parameters $$\alpha _{k}$$ and $$\sigma _{k}$$ in (3.36), we have

$$\limsup_{i\to \infty }\langle z-Vz,w_{k_{i}}-z \rangle \leq 0 \quad \forall z\in {\mathrm{F}}(U).$$
(3.37)

Since $$w_{k_{i}}$$ converges weakly to ū, we get

$$\bigl\langle (I-V)z,\bar{u}-z\bigr\rangle \leq 0, \quad \forall z \in {\mathrm{F}}(U).$$
(3.38)

Since $${\mathrm{F}}(U)$$ is convex, $$\beta z+(1-\beta )\hat{u}\in {\mathrm{F}}(U)$$ for $$\beta \in (0,1)$$ and hence

\begin{aligned}& \bigl\langle (I-V) \bigl(\beta z+(1-\beta )\bar{u}\bigr),\bar{u}- \bigl(\beta z+(1-\beta ) \bar{u}\bigr)\bigr\rangle \end{aligned}
(3.39)
\begin{aligned}& \quad =\beta \bigl\langle (I-V) \bigl(\beta z+(1-\beta )\bar{u}\bigr),\bar{u}-z \bigr\rangle \end{aligned}
(3.40)
\begin{aligned}& \quad \leq 0, \quad \forall z\in {\mathrm{F}}(U), \end{aligned}
(3.41)

which implies

$$\bigl\langle (I-V) \bigl(\beta z+(1-\beta )\bar{u}\bigr),\bar{u}-z\bigr\rangle \leq 0, \quad \forall z\in {\mathrm{F}}(U).$$

On taking the limit $$\beta \to 0_{+}$$, we have

$$\bigl\langle (I-V)\bar{u},\bar{u}-z\bigr\rangle \leq 0, \quad \forall z\in {\mathrm{F}}(U),$$
(3.42)

which implies $$\bar{u}\in \Phi$$.

Now, we show that $$\bar{u}\in \operatorname{Sol}(\operatorname{VI}(\mbox{1.5}))$$. Since $$\lim_{k\to \infty }\Vert v_{k}-t_{k}\Vert =0$$ and $$\lim_{k\to \infty }\Vert t_{k}-u_{k}\Vert =0$$, there exist subsequences $$\{t_{k_{i}}\}$$ of $$\{t_{k}\}$$ and $$\{v_{k_{i}}\}$$ of $$\{v_{k}\}$$, respectively, such that $$\{t_{k_{i}}\}$$, $$\{v_{k_{i}}\}$$ both converge weakly to ū. Let

\begin{aligned} G v = \textstyle\begin{cases} h v + N_{{\mathcal{C}}}(v),& \text{if } v \in {\mathcal{C}}; \\ \emptyset ,& \text{if } v \notin {\mathcal{C}}, \end{cases}\displaystyle \end{aligned}

then the monotone mapping G is maximal  and hence $$0 \in G v$$ if and only if $$v\in \operatorname{Sol}(\operatorname{VI}(\mbox{1.5}))$$ . Let $$(v,w)\in \operatorname{graph}(G)$$, then $$w\in Gv=h v + N_{{\mathcal{C}}}(v)$$ and hence $$w-h v\in N_{{\mathcal{C}}}(v)$$, i.e., $$\langle v-u, w-hv \rangle \geq 0$$, for all $$u \in {\mathcal{C}}$$.

On the other hand, from $$v_{k}={\mathcal{P}}_{{\mathcal{C}}}(I-\mu h)t_{k}$$ and $$v \in {\mathcal{C}}$$, we get

\begin{aligned} \bigl\langle (I-\mu h)t_{k}-v_{k}, v_{k}-v \bigr\rangle \geq & 0. \end{aligned}

This implies that

\begin{aligned} \biggl\langle v^{*}-v_{k}, \frac{v_{k}-t_{k}}{\mu } + h t_{k} \biggr\rangle \geq & 0. \end{aligned}

Since $$\langle v-u, w-hv \rangle \geq 0$$, for all $$u \in {\mathcal{C}}$$ and $$v_{k_{i}} \in {\mathcal{C}}$$, using the monotonicity of h, we have

\begin{aligned} \langle v-v_{k_{i}}, w \rangle \geq & \langle v-v_{k_{i}}, hv \rangle \\ \geq & \langle v-v_{k_{i}}, hv \rangle - \biggl\langle v-v_{k_{i}}, \frac{v_{k_{i}}-t_{k_{i}}}{\mu } + h t_{k_{i}} \biggr\rangle \\ =& \langle v-v_{k_{i}}, hv-hv_{k_{i}} \rangle +\langle v-v_{k_{i}}, hv_{k_{i}}-ht_{k_{i}} \rangle - \biggl\langle v-y_{k_{i}}, \frac{v_{k_{i}}-t_{k_{i}}}{\mu } \biggr\rangle \\ \geq &\langle v-v_{k_{i}}, hv_{k_{i}}-ht_{k_{i}} \rangle - \biggl\langle v-v_{k_{i}}, \frac{v_{k_{i}}-t_{k_{i}}}{\mu } \biggr\rangle . \end{aligned}

Since h is continuous, on taking the limit $$i\to \infty$$ we have $$\langle v-\bar{u}, w \rangle \geq 0$$. Since G is maximal monotone, we have $$\bar{u} \in G^{-1}0$$ and hence $$\bar{u}\in \operatorname{Sol}(\operatorname{VI}(\mbox{1.5}))$$ and thus $$\bar{u}\in \Gamma$$.

Finally, it follows from (3.22) and Lemma 2.3 that the sequence $$\{u_{k}\}$$ converges weakly to $$\bar{u}\in \Gamma$$. □

### Remark 3.2

One can derive a number of schemes from scheme (3.1); some special cases are as follows:

1. (i)

Setting $$\eta _{k}=0$$, k then scheme (3.1) reduces to extragradient scheme for solving $$\operatorname{VI}(\mbox{1.5})$$ and $$\operatorname{H\text{-}FPP}(\mbox{1.1})$$.

2. (ii)

Setting $$\sigma _{k}=0$$, k, and $$V=I$$, $$U=I$$ then scheme (3.1) reduces to $$\operatorname{scheme}{(\mbox{1.8})}$$ for solving $$\operatorname{VI}(\mbox{1.5})$$ and hence we recover Theorem 3.1 .

3. (iii)

Setting $$V=I$$, $$\sigma _{k}=0$$, $$U=J_{\lambda _{k}}^{B}:=(I+\lambda _{k} B)^{-1}$$ (where $$B:{\mathcal{H}}\to 2^{{\mathcal{H}}}$$ is maximal monotone and $$\lambda _{k}\in (0, \infty )$$), and $$\alpha _{k}=\alpha$$ k, scheme (3.1) takes the following form:

$$\left . \textstyle\begin{array}{l} t_{k}=u_{k}+\eta _{k}(u_{k}-u_{k-1}), \\ v_{k}= {\mathcal{P}}_{{\mathcal{C}}}(t_{k}-\mu h(t_{k})), \\ w_{k}= {\mathcal{P}}_{{\mathcal{C}}}(t_{k}-\mu h(v_{k})), \\ u_{k+1}=(1-\alpha )t_{k}+\alpha J_{\lambda _{k}}^{B}w_{k}, \end{array}\displaystyle \right \}$$
(3.43)

which was considered with an additional error tolerance strategy in .

## Numerical example

We discuss an example to illustrate Theorem 3.1.

### Example 4.1

Let $${\mathcal{H}}=\mathbb{R}$$. Let $${\mathcal{C}}= (-\infty , +\infty )$$, the mappings $$h:{\mathcal{H}}\to {\mathcal{H}}$$ be defined by $$h(u)=3u-2$$, $$\forall u \in {\mathcal{C}}$$; and $$U, V: {\mathcal{C}}\to {\mathcal{C}}$$ be defined by $$Uu=\frac{u+4}{7}$$, $$Vu=\frac{u+6}{10}$$, $$\forall u \in {\mathcal{C}}$$, respectively. Setting $$\{\alpha _{k}\}=0.8$$, $$\{\eta _{k}\}=0.4$$ and $$\{\sigma _{k}\}=\{\frac{1}{1000}+\frac{0.9}{k^{2}}\}$$, $$\forall k \geq 1$$. Then there are unique sequences $$\{u_{k}\}$$, $$\{v_{k}\}$$ and $$\{w_{k}\}$$ obtained by scheme (3.1) converging to $$\bar{u}=\frac{2}{3}\in \Gamma$$.

### Proof

Since h is Lipschitz continuous with $$L=3$$ and monotone and hence $$\mu \in (0,\frac{1}{3})$$, we take $$\mu =\frac{1}{4}$$. Observe that the mappings U, V are nonexpansive with $${\mathrm{F}}(U)= \{ \frac{2}{3} \}$$, $${\mathrm{F}}(V)= \{ \frac{2}{3} \}$$, and hence $$\Phi =\operatorname{Sol}(\operatorname{H\text{-}FPP}(\mbox{1.1}))= \{ \frac{2}{3} \}$$. One can also obtain $$\operatorname{Sol}(\operatorname{VI}(\mbox{1.5}))= \{ \frac{2}{3} \}$$. Hence, $$\Gamma =\operatorname{Sol}(\operatorname{VI}(\mbox{1.5}))\cap \Phi \cap {\mathrm{F}(S)}= \{ \frac{2}{3} \} \neq \emptyset$$. Furthermore, scheme (3.1) reduces to the following scheme: Given initial values $$u_{0}$$, $$u_{1}$$,

$$\left . \textstyle\begin{array}{l} t_{k}=u_{k}+\eta _{k}(u_{k}-u_{k-1}), \\ v_{k}= {\mathcal{P}}_{{\mathcal{C}}}(t_{k}-\mu h(t_{k}))= \textstyle\begin{cases} 0,&\text{if } u< 0, \\ 1,&\text{if } u> 1, \\ \frac{1}{4}t_{k}+\frac{1}{2},&\text{otherwise}, \end{cases}\displaystyle \\ w_{k}= {\mathcal{P}}_{{\mathcal{C}}}(t_{k}-\mu h(v_{k}))= \textstyle\begin{cases} t_{k}+\frac{1}{2},&\text{if } u< 0, \\ t_{k}+\frac{1}{4},&\text{if } u> 1, \\ t_{k}-\frac{1}{4}(3y_{k}-2),&\text{otherwise}, \end{cases}\displaystyle \\ u_{k+1}= (1-\alpha _{k})t_{k}+\alpha _{k} (\sigma _{k} \frac{w_{k}+6}{10}+(1-\sigma _{k})\frac{w_{k}+7}{4} ). \end{array}\displaystyle \right \}$$
(4.1)

Finally, using MATLAB, we have Fig. 1 and Table 1, which show that $$\{u_{k}\}$$, $$\{v_{k}\}$$ and $$\{w_{k}\}$$ converge to $$\bar{u}=\frac{2}{3}$$ as $$k \to +\infty$$.

□

### Concluding remark 4.1

In this paper, we considered a variational inequality problem (VI) and a hierarchical fixed point problem (H-FPP) in Hilbert space. We proposed an inertial version of Krasnoselski–Mann (KM)-type extragradient scheme (3.1) by combining the KM-type scheme (1.3) and an inertial version of the extragradient scheme (1.8) to approximate a common solution of $$\operatorname{H\text{-}FPP}(\mbox{1.1})$$ and $$\operatorname{VI}(\mbox{1.5})$$. Furthermore, we proved a weak convergence theorem for the proposed scheme (3.1). Finally, we discussed an example to illustrate Theorem 3.1.

Not applicable.

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## Acknowledgements

This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program. The authors sincerely thank the anonymous referees for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.

## Funding

This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

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All authors contributed equally and study the final manuscript. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to K. R. Kazmi.

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