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.

1 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. [1] 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 [2].

In 2007, Moudafi [2] 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, 3–7].

In 2008, Mainge [8] 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. [9] 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 [8]:

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

(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 [10–13].

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 [14] 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 [15] 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. [16–24] 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. [25] 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, 25–27].

2 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:

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

  $$ \langle hu-hv , u-v\rangle \geq 0; $$

• (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 [28] and demiclosed [29] on ${\mathcal{H}}$.

Lemma 2.2

([30])

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:

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

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

Lemma 2.3

([31])

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

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

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

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

3 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:

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

• (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 [3]. 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 [9].

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 [32] and hence $0 \in G v$ if and only if $v\in \operatorname{Sol}(\operatorname{VI}(\mbox{1.5}))$ [33]. 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:

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

• (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 [25].

• (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 [34].

4 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 $k \to +\infty$.

Figure 1

Convergence of $\{u_{k}\}$, $\{v_{k}\}$ and $\{w_{k}\}$ when $u_{0}=1$, $u_{1}=2$

Table 1 Values of $u_{k}$, $v_{k}$ and $w_{k}$

 □

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.