Explicit Iteration Methods for Solving Variational Inequalities in Banach Spaces

Abstract

The problem of finding a solution of a variational inequality over the set of common fixed points of a nonexpansive semigroup is considered in a real and uniformly convex Banach space without imposing the sequential weak continuity of the normalized duality mapping. Two new explicit iterative methods are introduced based on the steepest-descent method, and conditions are given to obtain their strong convergence. A numerical example is showed to illustrate the convergence analysis of the proposed methods.

1 Introduction

Let H be a real Hilbert space with inner product $\langle \cdot , \cdot \rangle$ and norm $\Vert \cdot \Vert$. Let C be a nonempty, closed, convex subset of H, and let $F:H\rightarrow H$ be a mapping. The classical variational inequality problem (VI(F, C) for short) is formulated as finding a point $p^* \in C$ such that

$$\begin{aligned} \langle Fp^*, p - p^* \rangle \ge 0 \quad \forall p \in C. \end{aligned}$$

(1.1)

This problem was introduced by Stampacchia [14]. Recently, it has been extended to study a large variety of methods arising in structural analysis, economics, optimization, operations research, and engineering sciences (see, for example, [6,7,8,9,10,11,12,13, 22] and the references therein).

In 2001, Yamada [19] proposed the hybrid steepest-descent method for solving problem VI(F, C) when the feasible set C is the set of common fixed points of a finite family of nonexpansive mappings to avoid the complexity caused by the use of the metric projection $P_C$. Besides, these research works are important because they contain many applications arising from the theory of signal recovery problems, power control problems, bandwidth allocation problems, and optimal control problems (see [8,9,10] and the references therein). Based on the hybrid steepest-descent method introduced by Yamada, many authors have been considering methods for solving variational inequality over the feasible set C with more complicated structure such as the common fixed point set of a countably infinite family of nonexpansive mappings (see for example, Yao et al. [21] and Wang [17]) or nonexpansive semigroup which is an uncountably infinite family of nonexpansive mappings [20].

There have been studies showing that solutions of an evolution equation with a m-accretive mapping $A: E \rightarrow E$ in a Banach space constitute a nonexpansive semigroup generated by operator A, and further, the set of common fixed points of $\{T (s): s \ge 0 \}$ is the set of zero points of A, that is $\mathscr {F}: =\cap _{s \ge 0}$Fix$(T(s))=A^{-1}(0)$ (see [4] for more details).

Along with the results achieved on different methods for solving the variational inequality (1.1) in a Hilbert space H (for example, Buong and Duong [2], Yang et al. [20]), many authors have recently studied solution methods for variational inequalities in Banach spaces (to name a few, Suzuki [15], Thuy and Hieu [16]). It is known that, among Banach spaces, Hilbert space H is a space with very nice geometrical properties such as the parallelogram identity, the existence of an inner product, or the uniqueness of the projection onto a nonempty, closed and convex subset of H which might be not valid in a general Banach space.

It should be added that a number of mathematical problems and research are set in Banach spaces related to variational inequalities such as differential equations and partial differential equations in Banach spaces or fixed point problems in Banach spaces. Therefore, methods for solving variational inequalities in Banach spaces or extensions to Banach spaces of methods developed in Hilbert spaces have been a topical issue attracting the attention of mathematicians. For some recent published results on solution methods for variational inequalities in Banach spaces, one needs to assume, in order to ensure their strong convergence, the weak continuity of the normalized duality mapping (such as, Ceng et al. [3], Chen and Song [5], and Suzuki [15]). Until now it has been shown that the $l^p, \ 1<p<\infty$, satisfies this weak continuity property, while the $L^p{[a, b]}, \ 1<p<\infty$, does not (see [4]). A natural question arising here is whether it is possible to develop methods for solving variational inequalities in Banach spaces without requiring the weak continuity of the normalized duality mapping. If the answer is affirmative, then the scope of applications of the algorithms in question can be expanded towards more general Banach spaces such as $L^p[a, b]$, rather than applicable only for $l^p, \ 1<p<\infty$.

Now, consider problem (1.1) in a Banach space E with $C=\mathscr {F}$, the set of common fixed points of a nonexpansive semigroup in E. Precisely, the problem has the form: Find an element $p^* \in \mathscr {F}$ such that

$$\begin{aligned} \langle Fp^*, j(p^* - p) \rangle \le 0 \quad \forall p \in \mathscr {F}, \end{aligned}$$

(1.2)

where $F: E \rightarrow E$ is a strongly accretive and strictly pseudocontractive mapping, $j(p^*-p) \in J(p^*-p)$, $J: E \rightarrow E^*$ being the normalized duality mapping of E, and $\langle x, x^* \rangle$ denotes the value of $x^* \in E^*$, the dual space of E, at the point $x \in E$.

In order to solve (1.2), Thuy and Hieu [16] proposed recently three implicit iterative methods, when E is a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. The first method is defined by

$$\begin{aligned} y_k =\frac{1}{t_k} \int _0^{t_k}T(s)(I-\lambda _kF)y_k\mathrm{d}s, \end{aligned}$$

(1.3)

the second one is in the form

$$\begin{aligned} y_k=\gamma _k (I-\lambda _k F)y_k+(1-\gamma _k) \frac{1}{t_k} \int _0^{t_k}T(s)y_k\mathrm{d}s, \end{aligned}$$

(1.4)

and finally the third one is formulated as

$$\begin{aligned} y_k=\gamma _k (I-\lambda _k F)y_k+(1-\gamma _k)T(v_k) y_k, \end{aligned}$$

(1.5)

for $k \ge 1$, where $0 <t_k \rightarrow \infty$ as $k \rightarrow \infty$ in (1.3) and (1.4), whereas $0< v_k \rightarrow 0$ as $k \rightarrow \infty$ in (1.5).

The following result has been proved when iteration (1.5) is used:

Theorem 1.1

  [16] Let F be an $\eta -$strongly accretive and $\gamma -$strictly pseudocontractive mapping with $\eta +\gamma >1$ and let $\{T(s): \ s\ge 0\}$ be a nonexpansive semigroup on E, which is a real uniformly convex Banach space with a uniformly Gâteaux differentiable norm, such that $\mathscr {F}=\cap _{s \ge 0}\mathrm{Fix}(T(s)) \ne \emptyset$. Then, the sequence $\{y_k\}$, defined by (1.5) where $\gamma _k \in (0,1), \ \lambda _k \in (0,1]$ and $v_k >0$ such that $\lim _{k \rightarrow \infty }v_k =\lim _{k \rightarrow \infty } \frac{\gamma _k}{v_k} = 0$, converges to a unique element $p^*$, solving (1.2).

Based on (1.3), (1.4), and (1.5), we construct two explicit iteration methods that are strongly convergent to a solution of (1.2) without imposing a sequentially weakly continuity property on the normalized duality mapping of Banach spaces. The methods are defined by

$$\begin{aligned} x_{n+1}= (1-\gamma _n) x_n + \gamma _n T_n F_n x_n, \end{aligned}$$

(1.6)

and

$$\begin{aligned} x_{n+1}= (1-\gamma _n) T_nx_n + \gamma _n F_n x_n, \quad n\ge 1, \ x_1 \in E, \end{aligned}$$

(1.7)

where $F_n$ and $T_n$ are, respectively, defined by

$$\begin{aligned} F_n x = (I-\lambda _n F) x \ \ \text {and} \ \ T_n x = \frac{1}{t_n} \int _0^{t_n} T(s) x \mathrm{d}s, \ x \in E, \end{aligned}$$

and $\{ \gamma _n \}$, $\{ \lambda _n \}$, $\{ t_n \}$ are three sequences satisfying the following conditions:

$$\begin{aligned}&\lambda _n \in (0,1), \ \lambda _n \rightarrow 0, \ \sum \limits _{n=1}^\infty \lambda _n = \infty ; \end{aligned}$$

(1.8)

$$\begin{aligned}&\lim _{n\rightarrow \infty } t_n = \infty , \ \{| t_{n+1} - t_n | \} \ \text{ is } \text{ bounded }; \end{aligned}$$

(1.9)

and

$$\begin{aligned} \gamma _n \in (0,1) \ \text{ such } \text{ that } \ 0< \liminf \limits _{n \rightarrow \infty } \gamma _n \le \limsup \limits _{n \rightarrow \infty } \gamma _n <1. \end{aligned}$$

(1.10)

The remainder of the paper is organized as follows. In the next Section, some preliminary results are recalled. In Sect. 3, two explicit iteration methods are studied for solving a variational inequality problem over the set of common fixed points of a nonexpansive semigroup in the framework of Banach spaces. The strong convergence of the sequences generated by each of these algorithms is obtained. It is related to the strong convergence of implicit methods given in [16]. Section 4 is devoted to show a numerical example for illustration purpose of the theoretical result. A final conclusion of the paper is given in Sect. 5.

2 Preliminaries

For simplicity, the norms of E and $E^*$ are denoted by the same symbol $\Vert \cdot \Vert$. A mapping J from E into $2^{E^*}$ satisfying the condition

$$\begin{aligned} J(x) = \{ x^* \in E^*: \ \langle x, x^* \rangle = \Vert x\Vert ^2 \ \text{ and } \ \Vert x^* \Vert = \Vert x \Vert \} \end{aligned}$$

is called a normalized duality mapping of E. It is well known that if $x \ne 0$, then $J(tx)=tJ(x)$ for all $t >0$ and $x \in E$, and $J(-x)=-J(x)$.

A mapping $T:E \rightarrow E$ is said to be a nonexpansive mapping on E if

$$\begin{aligned} \Vert Tx - Ty \Vert \le \Vert x-y \Vert \quad \forall \, x, y \in E, \end{aligned}$$

and

$$\begin{aligned} \text {Fix}(T) = \{x \in E: \ x = Tx \} \end{aligned}$$

denotes the fixed point set of T. Let $\{T(s): s\ge 0\}$ be a nonexpansive semigroup on E, that is,

• (1) for each $s>0$, T(s) is a nonexpansive mapping on E;

• (2) $T(0)x = x$ for all $x \in E$;

• (3) $T(s_1+s_2) = T(s_1) \circ T(s_2)$ for all $s_1, s_2\ge 0$;

• (4) for each $x \in C$, the mapping $T( \cdot )x$ from $[0,\infty )$ into E is continuous.

Denote $\mathscr {F}:=\cap _{s\ge 0}\text {Fix}(T(s))$ the set of common fixed points of the nonexpansive semigroup $\{T(s): s\ge 0\}$. Through this paper we assume that $\mathscr {F} \ne \emptyset$. Concerning the unemptiness of the set $\mathscr {F}$, some conditions posed on the nonexpansive semigroup $\{T(s): s\ge 0\}$ were mentioned in [4], such as if $\{T(s): s\ge 0\}$ is a bounded semigroup on a closed and convex subset C of a uniformly convex Banach space E, that is $\sup _{s \ge 0} \Vert T(s) x \Vert < +\infty , \ \forall x \in C$, then $\mathscr {F} \ne \emptyset$ and vice versa.

A mapping F with domain D(F) and range R(F) in E is called an $\eta$-strongly accretive and $\gamma$-strictly pseudocontractive mapping if it satisfies

$$\begin{aligned} \langle Fx - Fy, j(x-y) \rangle \ge \eta \Vert x - y \Vert ^2 \end{aligned}$$

and

$$\begin{aligned} \langle Fx - Fy, j(x-y) \rangle \le \Vert x - y \Vert ^2 - \gamma \Vert (I-F)x - (I-F)y \Vert ^2, \end{aligned}$$

(2.1)

respectively, for all x, $y \in D(F)$ and some element $j(x-y) \in J(x-y)$, where I denotes the identity mapping of E, and $\eta > 0$, $\gamma \in (0,1)$ are fixed constants. Clearly, from (2.1), it follows that $\Vert Fx - Fy \Vert \le L \Vert x - y\Vert$ with $L=1+1/ \gamma$. If $L \in [0,1)$, then F is called contractive, and if F satisfies (2.1) with $\gamma = 0$, then it is said to be pseudocontractive. Clearly, every nonexpansive mapping is continuous and pseudocontractive.

Let $S_1(0):=\{ x \in E: \Vert x \Vert =1\}$ be the unit sphere in E. The space E is called to have a Gâteaux differentiable norm (or to be smooth) if the limit

$$\begin{aligned} \lim _{t \rightarrow 0}\frac{\Vert x+ ty\Vert -\Vert x\Vert }{t} \end{aligned}$$

exists for all x, y in the sphere $S_1(0)$. The space E is said to have a uniformly Gâteaux differentiable norm if the limit is attained uniformly for $x \in S_1(0)$.

It is well known that if E is smooth, then the normalized duality mapping J is single valued. And if the norm of E is uniformly Gâteaux differentiable, then J is norm to weak-star uniformly continuous on every bounded subset of E (see [4]). In the sequel, we shall denote the single valued normalized duality mapping by j.

Recall that a Banach space E is said to be strictly convex if, for $x, y \in S_1(0)$ with $x \ne y$, $\Vert (1-\lambda )x+\lambda y\Vert <1$ for all $\lambda \in (0,1)$, and uniformly convex if, for any $\varepsilon$, $0< \varepsilon \le 2$, the inequalities $\Vert x\Vert \le 1$, $\Vert y\Vert \le 1$ and $\Vert x-y\Vert \ge \varepsilon$ imply that there exists a $\delta =\delta (\varepsilon ) \ge 0$ such that $\Vert (x+y)/2 \Vert \le 1- \delta$. It is well known that every uniformly convex Banach space is reflexive and strictly convex.

Now we give some facts that will be used in the proof of our results.

Lemma 2.1

[1] Let E be a real normed linear space and let J be the normalized duality mapping on E. Then, the following inequality holds

$$\begin{aligned} \Vert x+y\Vert ^2 \le \Vert x\Vert ^2 + 2 \langle y, j(x+y)\rangle \quad \forall \, x, y \in E, \quad \forall \, j(x+y) \in J(x+y). \end{aligned}$$

Lemma 2.2

[3] Let E be a real smooth Banach space and let $F: E \rightarrow E$ be an $\eta$-strongly accretive and $\gamma$-strictly pseudocontractive mapping with $\eta +\gamma >1$. Then, for any $\lambda \in (0,1)$, $I-\lambda F$ is a contraction with contractive coefficient $1- \lambda \tau$, where $\tau = 1-\sqrt{(1-\eta )/ \gamma } \in (0,1)$.

Lemma 2.3

[5] Let C be a nonempty, bounded, closed, convex subset of a uniformly convex Banach space E and let $\{T(s): s \ge 0\}$ be a nonexpansive semigroup on C. Then, for any $r >0$ and $h > 0$,

$$\begin{aligned} \lim _{t \rightarrow \infty } \sup _{x \in C\cap B_r} \biggl \Vert \, T(h)\biggl ( \frac{1}{t}\int _0^tT(s)x \mathrm{d}s\biggl ) - \frac{1}{t}\int _0^tT(s)x \mathrm{d}s \, \biggl \Vert =0, \end{aligned}$$

where $B_r=\{x \in E: \Vert x\Vert \le r\}$.

Lemma 2.4

[15] Let $\{x_n\}$ and $\{z_n\}$ be bounded sequences in a Banach space E such that $x_{n+1} = (1- \gamma _n) x_n + \gamma _n z_n$ for $n \ge 1$, where $\{\gamma _n\} \subset (0,1)$ such that $0<\liminf _{n \rightarrow \infty } \gamma _n \le \limsup _{n \rightarrow \infty } \gamma _n <1$. Assume that

$$\begin{aligned} \limsup _{n \rightarrow \infty } \big [ \Vert z_{n+1}- z_n \Vert - \Vert x_{n+1} - x_n \Vert \big ] \le 0. \end{aligned}$$

Then $\lim _{n\rightarrow \infty } \Vert x_n - z_n \Vert =0$.

Lemma 2.5

[18] Let $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ be three sequences of nonnegative numbers such that

$$\begin{aligned} a_{n+1} \le (1-b_n)a_n +b_n c_n, \end{aligned}$$

where $b_n \in (0,1), \ \sum _{n=1}^\infty b_n = \infty$, $\lim \limits _{n \rightarrow \infty } b_n = 0$, and $\limsup \limits _{n \rightarrow \infty } c_n \le 0$. Then $\lim \limits _{n \rightarrow \infty } a_n = 0$.

Proposition 2.1

Let E be a real uniformly convex Banach space with a uniformly Gâteaux differentiable norm and $F: E \rightarrow E$ be an L-Lipschitz continuous mapping. Let $\{T(t): t \ge 0\}$ be a nonexpansive semigroup on E. If there exists a bounded sequence $\{x_n\}$ satisfying $\lim _{n\rightarrow \infty }\Vert x_n-T(t)x_n\Vert = 0$ for all $t \ge 0$ and if $\lim _{k \rightarrow \infty }y_k = p^*$, where $\{y_k\}$ is a sequence defined by (1.5), that is

$$\begin{aligned} y_k=\gamma _k(I-\lambda _kF)y_k + (1-\gamma _k) T(v_k) y_k,\ v_k>0, \ k \ge 1, \end{aligned}$$

then,

$$\begin{aligned} \limsup _{n \rightarrow \infty } \left\langle Fp^*, j(p^* - x_n)\right\rangle \le 0. \end{aligned}$$

(2.2)

Proof

Let k and n be fixed integers. Then

$$\begin{aligned} y_k-x_n = \gamma _k [(I-\lambda _kF)y_k - x_n] + (1-\gamma _k) [T(v_k) y_k - x_n] \end{aligned}$$

and

$$\begin{aligned} \langle T(v_k)x - T(v_k)y, j(x-y) \rangle \le \Vert x-y\Vert ^2 \quad \text{ for } \text{ all } x,y \in E. \end{aligned}$$

So

$$\begin{aligned} \Vert y_k - x_n\Vert ^2= & {} (1-\gamma _k) \langle T(v_k)y_k-x_n, j(y_k-x_n)\rangle \\&+\, \gamma _k\langle (I-\lambda _kF)y_k - x_n, j(y_k-x_n) \rangle \\= & {} (1-\gamma _k) [\langle T(v_k)y_k-T(v_k)x_n, j(y_k-x_n)\rangle \\&+\, \langle T(v_k)x_n-x_n, j(y_k-x_n) \rangle ]\\&+ \,\gamma _k \langle (I-\lambda _kF)y_k-y_k, j(y_k-x_n)\rangle + \gamma _k \Vert y_k-x_n\Vert ^2\\\le & {} (1-\gamma _k) [\Vert y_k-x_n\Vert ^2 + \Vert T(v_k)x_n-x_n\Vert \,\Vert y_k-x_n\Vert ]\\&+\, \gamma _k \langle (I-\lambda _kF)y_k-y_k, j(y_k-x_n)\rangle + \gamma _k \Vert y_k-x_n\Vert ^2\\\le & {} \Vert y_k-x_n\Vert ^2 + \Vert T(v_k)x_n-x_n\Vert \,\Vert y_k-x_n\Vert - \gamma _k \lambda _k \langle Fy_k, j(y_k-x_n)\rangle . \end{aligned}$$

Hence

$$\begin{aligned} \langle Fy_k, j(y_k-x_n)\rangle \le \frac{\Vert x_n-T(v_k)x_n\Vert }{\gamma _k \lambda _k}\,\Vert y_k-x_n\Vert . \end{aligned}$$

(2.3)

By assumption, since $v_k \ge 0$, we have

$$\begin{aligned} \lim _{n \rightarrow \infty }\Vert x_n-T(v_k)x_n\Vert = 0 \quad \text{ for } \text{ all } k. \end{aligned}$$

(2.4)

Consequently, the sequence $\{x_n\}$ being bounded, it follows from (2.3) and (2.4) that

$$\begin{aligned} \limsup _{n \rightarrow \infty }\,\langle Fy_k, j(y_k-x_n)\rangle \le 0 \quad \text{ for } \text{ all } k. \end{aligned}$$

(2.5)

On the other hand, for all k and n, we have

$$\begin{aligned} \begin{array}{l} |\langle Fy_k, j(y_k-x_n)\rangle - \langle Fp^*, j(p^*-x_n)\rangle |\\ = |\langle Fy_k-Fp^*, j(y_k-x_n)\rangle + \langle Fp^*, j(y_k-x_n) - j(p^*-x_n)\rangle |\\ \le \Vert Fy_k-Fp^*\Vert \,\Vert y_k-x_n\Vert + |\langle Fp^*, j(y_k-x_n) - j(p^*-x_n)\rangle |. \end{array} \end{aligned}$$

Since F is Lipschitz continuous with constant L, since the sequences $\{y_k\}$ and $\{x_n\}$ are bounded, and $\Vert y_k-p^*\Vert \rightarrow 0$, we obtain that

$$\begin{aligned} \lim _{k \rightarrow \infty } \Vert Fy_k-Fp^*\Vert = 0 \text{ and } \lim _{k \rightarrow \infty } j(y_k-x_n) - j(p^*-x_n) = 0 \text{ uniformly } \text{ in } n. \end{aligned}$$

Then we can write

$$\begin{aligned} \forall \varepsilon >0, \exists K \text{ such } \text{ that } \forall k \ge K,\ \forall n\textstyle {:} \ |\langle Fy_k, j(y_k-x_n)\rangle - \langle Fp^*, j(p^*-x_n)\rangle | < \varepsilon , \end{aligned}$$

and

$$\begin{aligned} \forall \varepsilon >0, \exists K \text{ such } \text{ that } \forall k \ge K,\ \forall n\textstyle {:} \ \langle Fp^*, j(p^*-x_n)\rangle < \langle Fy_k, j(y_k-x_n)\rangle + \varepsilon . \end{aligned}$$

Hence, thanks to (2.5), we can deduce that

$$\begin{aligned} \forall \varepsilon >0, \exists K \text{ such } \text{ that } \forall k \ge K\textstyle {:} \ \limsup _{n \rightarrow \infty }\langle Fp^*, j(p^*-x_n)\rangle \le \varepsilon . \end{aligned}$$

Since $\varepsilon >0$ is arbitrary, we finally obtain that

$$\begin{aligned} \limsup _{n \rightarrow \infty } \left\langle Fp^*, j(p^* - x_n)\right\rangle \le 0. \end{aligned}$$

This completes the proof. $\square$

3 Main Results

Theorem 3.1

Let E be a real uniformly convex Banach space with a uniformly Gâteaux differentiable norm and $F: E \rightarrow E$ be an $\eta$-strongly accretive and $\gamma$-strictly pseudocontractive mapping with $\eta + \gamma >1$. Let also $\{T(t): t \ge 0\}$ be a nonexpansive semigroup on E such that $\mathscr {F} = \cap _{t \ge 0}\mathrm{Fix} (T(t)) \ne \emptyset$. Then the sequence $\{x_n\}$ defined by (1.6) converges strongly to the unique solution $p^*$ of (1.2).

Proof

The proof consists of three steps.

Step 1. We show that there exists a positive constant $M_1$ such that $\Vert x_n \Vert , \Vert T_n x_n \Vert ,$$\Vert F_n x_n \Vert , \Vert F_{n+1} x_n - p\Vert , \Vert F x_n \Vert \le M_1$ for all $n \ge 1$ and all $p \in \mathscr {F}$.

Let $n \ge 1$. For a fixed point $p \in \mathscr {F}$, we have $T_n p = p,$ and hence, by Lemma 2.2,

$$\begin{aligned} \Vert x_{n+1} - p \Vert= & {} \Vert (1-\gamma _n) x_n + \gamma _n T_n F_nx_n - p \Vert \\\le & {} (1-\gamma _n) \Vert x_n-p \Vert + \gamma _n \Vert T_n F_nx_n - T_n p \Vert \\\le & {} (1-\gamma _n) \Vert x_n-p \Vert + \gamma _n \Vert F_nx_n - p \Vert \\= & {} (1-\gamma _n) \Vert x_n-p \Vert \\&+\, \gamma _n \Vert (I-\lambda _nF)x_n -(I-\lambda _nF)p - \lambda _nFp \Vert \\\le & {} (1-\gamma _n) \Vert x_n-p \Vert + \gamma _n \big [ (1-\lambda _n \tau ) \Vert x_n - p\Vert + \lambda _n \Vert Fp \Vert \big ] \\= & {} (1-\gamma _n \lambda _n \tau ) \Vert x_n-p \Vert + \gamma _n \lambda _n \tau \frac{\Vert Fp \Vert }{\tau } \\\le & {} \max \left\{ \Vert x_n - p\Vert , \frac{1}{\tau }\Vert Fp \Vert \right\} \\\le & {} \cdots \le \max \left\{ \Vert x_1 - p\Vert , \frac{1}{\tau } \Vert Fp \Vert \right\} . \end{aligned}$$

Therefore, the sequence $\{x_n\}$ is bounded. Since $T_n$ is nonexpansive and $I- \lambda _n F$ is contractive, we have

$$\begin{aligned} \Vert T_nx_n -T_np \Vert \le \Vert x_n-p\Vert \le \max \left\{ \Vert x_1 - p\Vert , \frac{1}{\tau } \Vert Fp \Vert \right\} \end{aligned}$$

and

$$\begin{aligned} \Vert F_nx_n-F_np\Vert= & {} \Vert (I-\lambda _nF) x_n - (I-\lambda _nF)p\Vert \le (1-\lambda _n\tau ) \Vert x_n-p\Vert \\\le & {} (1-\lambda _n\tau ) \max \left\{ \Vert x_1 - p\Vert , \frac{1}{\tau } \Vert Fp \Vert \right\} . \end{aligned}$$

Consequently the sequences $\{T_nx_n\}$ and $\{F_nx_n\}$ are bounded and thus also the sequence $\{F_{n+1}x_n - p \}$. Since F is a $\gamma$-strictly pseudocontractive mapping the sequence $\{Fx_n\}$ is also bounded. So, the existence of $M_1$ is proved.

Step 2. We prove that $\lim _{n\rightarrow \infty } \Vert T(t)x_n - x_n \Vert =0$, for all $t \ge 0$.

Let $n \ge 1$, and define $z_n = T_n F_n x_n$. From (1.6), it follows that

$$\begin{aligned} x_{n+1} = (1-\gamma _n)x_n + \gamma _n z_n, \end{aligned}$$

(3.1)

and

$$\begin{aligned} \Vert z_{n+1} - z_n \Vert= & {} \Vert T_{n+1} F_{n+1}x_{n+1} - T_n F_n x_n \Vert \\\le & {} \Vert T_{n+1} F_{n+1} x_{n+1} - T_{n+1} F_{n+1} x_{n} \Vert \\&+ \Vert T_{n+1} F_{n+1} x_{n} - T_{n} F_{n+1} x_{n} \Vert + \Vert T_{n} F_{n+1} x_{n} - T_n F_n x_n \Vert . \end{aligned}$$

Since

$$\begin{aligned} \Vert T_{n+1} F_{n+1} x_{n} - T_{n} F_{n+1} x_{n} \Vert= & {} \biggl \Vert \frac{1}{t_{n+1}} \int _0^{t_{n+1}} \big [T(s) F_{n+1} x_n - T(s) p \big ] \mathrm{d}s \\&-\frac{1}{t_{n}} \int _0^{t_{n}} \big [T(s) F_{n+1} x_n - T(s) p \big ] \mathrm{d}s \biggl \Vert \\= & {} \biggl \Vert \biggl ( \frac{1}{t_{n+1}} - \frac{1}{t_n}\biggl ) \int _0^{t_n} \big [T(s) F_{n+1}x_n - T(s) p \big ] \mathrm{d}s \\&+ \frac{1}{t_{n+1}} \int _{t_n}^{t_{n+1}} \big [T(s) F_{n+1} x_n - T(s) p \big ] \mathrm{d}s \biggl \Vert \\\le & {} \biggl \vert \frac{1}{t_{n+1}} - \frac{1}{t_n}\biggl \vert t_n M_1 + \frac{\vert t_{n+1} - t_n \vert }{t_{n+1}} M_1\\= & {} 2 \frac{\vert t_{n+1} - t_n \vert }{t_{n+1}} M_1, \end{aligned}$$

we obtain that

$$\begin{aligned} \Vert z_{n+1} - z_n \Vert\le & {} \Vert F_{n+1} x_{n+1} - F_{n+1} x_n \Vert \\&+\, 2\frac{\vert t_{n+1} - t_n \vert }{t_{n+1}} M_1 + \Vert F_{n+1} x_{n} - F_{n} x_n \Vert \\\le & {} \Vert x_{n+1} - x_n \Vert + 2\lambda _{n+1} M_1 + 2\frac{\vert t_{n+1} - t_n \vert }{t_{n+1}} M_1 + \vert \lambda _{n+1} - \lambda _n \vert M_1. \end{aligned}$$

This together with (1.8) and (1.9) implies that

$$\begin{aligned} \limsup _{n \rightarrow \infty } \big [ \Vert z_{n+1}-z_n \Vert -\Vert x_{n+1} -x_n \Vert \big ] \le 0. \end{aligned}$$

Hence, we can deduce from (1.10) and Lemma 2.4 that

$$\begin{aligned} \lim \limits _{n \rightarrow \infty } \Vert x_n - z_n \Vert = 0. \end{aligned}$$

(3.2)

Since $\Vert F_nx_n - x_n \Vert \le \lambda _n M_1$ and $\lambda _n \rightarrow 0$ as $n \rightarrow \infty$, we have also

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert F_nx_n - x_n \Vert = 0. \end{aligned}$$

(3.3)

On the other hand, from (3.2), (3.3), and

$$\begin{aligned} \Vert T_nx_n-x_n\Vert\le & {} \Vert T_nx_n-z_n\Vert + \Vert z_n-x_n\Vert \\= & {} \Vert T_nx_n - T_nF_nx_n\Vert + \Vert z_n-x_n\Vert \le \Vert x_n-F_nx_n\Vert +\Vert z_n-x_n\Vert , \end{aligned}$$

we get

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert T_nx_n - x_n \Vert = 0. \end{aligned}$$

(3.4)

Now, for any $t>0$, observe that

$$\begin{aligned} \Vert T(t) x_n - x_n \Vert\le & {} \Vert T(t) x_n - T(t) T_nx_n \Vert \\&+\, \Vert T(t) T_nx_n - T_n x_n \Vert + \Vert T_n x_n - x_n \Vert \\\le & {} 2 \Vert T_n x_n - x_n \Vert + \Vert T(t)T_n x_n - T_n x_n \Vert . \end{aligned}$$

This together with (3.4) and Lemma 2.3 implies that

$$\begin{aligned} \lim \limits _{n \rightarrow \infty } \Vert T(t) x_n - x_n \Vert = 0. \end{aligned}$$

(3.5)

Step 3. We claim that $\lim _{n\rightarrow \infty } \Vert x_n-p^*\Vert =0$, where $p^*$ is the unique solution of the variational inequality (1.2).

By Theorem 1.1 (see also Theorem 3.2 in [16]), we have that $p^*$ exists and is unique. Furthermore, we also know that the sequence $\{y_k\}$ defined, for each k, by (1.5) converges to $p^*$. So, by Proposition 2.1, we have that

$$\begin{aligned} \limsup _{n \rightarrow \infty }\ \langle Fp^*, j(p^*-x_n \rangle \le 0. \end{aligned}$$

(3.6)

Now, using the convexity of $\Vert \cdot \Vert ^2$, Lemmas 2.1, 2.2, and the property $j(-x)=-j(x)$ for all $x \in E$, we can estimate the value of $\Vert x_{n+1}-p^* \Vert ^2$ as follows:

$$\begin{aligned} \Vert x_{n+1}-p^* \Vert ^2\le & {} (1-\gamma _n) \Vert x_n - p^* \Vert ^2 + \gamma _n \Vert T_n F_nx_n - p^* \Vert ^2 \\= & {} (1-\gamma _n) \Vert x_n - p^* \Vert ^2 + \gamma _n \Vert T_n F_nx_n - T_n p^* \Vert ^2 \\\le & {} (1-\gamma _n) \Vert x_n - p^* \Vert ^2 + \gamma _n \Vert F_nx_n - p^* \Vert ^2 \\= & {} (1-\gamma _n) \Vert x_n - p^* \Vert ^2 + \gamma _n \Vert F_nx_n -F_n p^* -\lambda _n Fp^* \Vert ^2 \\\le & {} (1-\gamma _n) \Vert x_n - p^* \Vert ^2 + \gamma _n \big [(1-\lambda _n \tau ) \Vert x_n - p^* \Vert ^2 \\&-\, 2\lambda _n \langle Fp^*, j(x_n - p^* - \lambda _n Fx_n ) \rangle \big ] \\= & {} (1-\gamma _n \lambda _n \tau ) \Vert x_n - p^* \Vert ^2 + 2\gamma _n \lambda _n \big [ \langle Fp^*, j(p^* - x_n) \rangle \\&+\, \langle Fp^*, j(p^*-x_n + \lambda _n Fx_n) - j(p^* - x_n) \rangle \big ]\\\le & {} (1-\gamma _n \lambda _n \tau ) \Vert x_n - p^* \Vert ^2 + \gamma _n \lambda _n \tau 2 \big [ \langle Fp^*, j(p^* - x_n) \rangle \\&+\, \langle Fp^*, j(p^* - x_n + \lambda _n Fx_n) - j(p^* - x_n) \rangle \big ] / \tau , \end{aligned}$$

or

$$\begin{aligned} \Vert x_{n+1}-p^* \Vert ^2 \le (1- b_n) \Vert x_n - p^* \Vert ^2 + b_n c_n, \end{aligned}$$

(3.7)

where

$$\begin{aligned} b_n= & {} \gamma _n \lambda _n \tau ,\\ c_n= & {} 2 \big [ \langle Fp^*, j(p^*-x_n) \rangle + \langle Fp^*, j(p^* - x_n + \lambda _n Fx_n) - j(p^* - x_n) \rangle \big ] / \tau . \end{aligned}$$

Since $\{ \lambda _n \}$ and $\{ \gamma _n \}$ satisfy conditions (1.8) and (1.10), respectively, we have that

$$\begin{aligned} b_n \rightarrow 0 \quad \text{ and } \quad \sum _{n=1}^\infty b_n = \infty . \end{aligned}$$

Using (3.6) combined with $\lambda _n \rightarrow 0$ as $n \rightarrow \infty$, we obtain that

$$\begin{aligned} \limsup _{n \rightarrow \infty }\ c_n \le 0. \end{aligned}$$

Then, from Lemma 2.5 and (3.7) we can conclude that $\lim _{n \rightarrow \infty } \Vert x_n - p^* \Vert ^2 = 0$. This completes the proof. $\square$

Theorem 3.2

Assume the conditions in Theorem 3.1 are satisfied. Then the sequence $\{x_n\}$ defined by (1.7) converges strongly to the unique solution $p^*$ of (1.2).

Proof

We prove the result in three steps.

Step 1. We show that the sequence $\{x_n\}$ is bounded.

For a fixed point $p \in \mathscr {F}$, we have $T_n p = p$ and hence, by Lemma 2.2,

$$\begin{aligned} \Vert x_{n+1}-p \Vert= & {} \Vert \gamma _n(I-\lambda _n F)x_n + (1-\gamma _n) T_n x_n - p \Vert \\\le & {} \gamma _n \Vert (I-\lambda _n F) x_n - p \Vert + (1-\gamma _n ) \Vert T_n x_n - T_n p \Vert \\\le & {} \gamma _n [ (1-\lambda _n \tau ) \Vert x_n -p \Vert + \lambda _n \Vert Fp \Vert ] + (1-\gamma _n) \Vert x_n - p \Vert \\= & {} (1-\gamma _n \lambda _n \tau ) \Vert x_n - p \Vert + \gamma _n \lambda _n \tau \frac{\Vert Fp \Vert }{\tau } \\\le & {} \max \{\Vert x_1 - p \Vert , \Vert Fp \Vert /\tau \}. \end{aligned}$$

Therefore, $\{x_n\}$ is bounded, and so are the sequences $\{T_nx_n\}$, $\{F_nx_n\}$, $\{Fx_n\}$, and $\{x_n - p \}$. We assume that they are bounded by a positive constant $M_2$.

Step 2. We prove that $\lim _{n\rightarrow \infty } \Vert T(t)x_n - x_n \Vert =0$ for all $t \ge 0$.

Put $z_n = T_nx_n-\gamma _n\lambda _nFx_n/(1-\gamma _n)$. From (1.7) it follows that

$$\begin{aligned} x_{n+1} = \gamma _n x_n +(1- \gamma _n) z_n, \end{aligned}$$

and

$$\begin{aligned} \Vert z_{n+1} - z_n \Vert\le & {} \Vert T_{n+1} x_{n+1} - T_n x_n \Vert + \Vert \gamma _n \lambda _n Fx_n / (1-\gamma _n) \\&-\, \gamma _{n+1} \lambda _{n+1} Fx_{n+1} /(1-\gamma _{n+1}) \Vert \\\le & {} \Vert T_{n+1} x_{n+1} - T_{n+1} x_n \Vert + \Vert T_{n+1} x_{n} - T_n x_n \Vert \\&+\, \Vert \gamma _n \lambda _n Fx_n / (1-\gamma _n) - \gamma _{n+1} \lambda _{n+1} Fx_{n+1} /(1-\gamma _{n+1}) \Vert \\\le & {} \Vert x_{n+1} - x_n \Vert + 2 \frac{\vert t_{n+1} - t_n \vert }{t_{n+1}} M_2 \\&+\, (\lambda _{n} + \lambda _{n+1}) \beta M_2 /(1-\beta ), \end{aligned}$$

where $\beta$ is a positive number in (0, 1) such that $\gamma _n \le \beta$. This together with $\lambda _n \rightarrow 0$ and $\vert t_{n+1}-t_n \vert / t_{n+1} \rightarrow 0$ as $n \rightarrow \infty$ implies that

$$\begin{aligned} \limsup \limits _{n \rightarrow \infty } \big [ \Vert z_{n+1}-z_n \Vert -\Vert x_{n+1} -x_n \Vert \big ] \le 0. \end{aligned}$$

Hence, we deduce from (1.10) and Lemma 2.4 that

$$\begin{aligned} \lim \limits _{n \rightarrow \infty } \Vert x_n - z_n \Vert = 0. \end{aligned}$$

On the other hand, since

$$\begin{aligned} \Vert z_n - T_n x_n \Vert = \Vert \gamma _n \lambda _n Fx_n / (1-\gamma _n) \Vert \le \lambda _n \beta M_2 / (1-\beta ) \end{aligned}$$

and $\lambda _n \rightarrow 0$ as $n \rightarrow \infty$, we have that $\Vert z_n - T_n x_n \Vert \rightarrow 0$.

Finally, using the same argument as in the proof of Theorem 3.1, we obtain that $\Vert x_n - T_n x_n \Vert \rightarrow 0$ and $\Vert T(t)x_n - x_n \Vert \rightarrow 0$ for all $t \ge 0$ as $n \rightarrow \infty$.

Step 3. We claim that $\lim _{n\rightarrow \infty } \Vert x_n-p^*\Vert =0$, where $p^*$ is the unique solution of the variational inequality (1.2).

By Theorem 1.1 (see also Theorem 3.2 in [16]), we have that $p^*$ exists and is unique. Furthermore, we also know that the sequence $\{y_k\}$ defined, for each k, by (1.5) converges to $p^*$. So, by Proposition 2.1, we have that

$$\begin{aligned} \limsup _{n \rightarrow \infty }\ \langle Fp^*, j(p^*-x_n \rangle \le 0. \end{aligned}$$

(3.8)

Now we estimate the value of $\Vert x_{n+1}-p^* \Vert ^2$ as follows:

$$\begin{aligned} \Vert x_{n+1}-p^* \Vert ^2= & {} (1-\gamma _n ) \langle T_n x_n - p^*, j(x_{n+1} - p^*) \rangle \\&+\, \gamma _n \langle (I-\lambda _n F)x_n - p^*, j(x_{n+1} - p^*) \rangle \\= & {} (1-\gamma _n) \langle T_n x_n - T_n p^*, j(x_{n+1} - p^*) \rangle \\&+\, \gamma _n \langle \lambda _n [(I- F)x_n - p^*] + (1-\lambda _n ) (x_n - p^*), j(x_{n+1} - p^*) \rangle \\\le & {} (1-\gamma _n ) \Vert x_n - p^* \Vert \Vert x_{n+1} - p^* \Vert \\&+\, \gamma _n (1-\lambda _n) \Vert x_n -p^* \Vert \Vert x_{n+1} - p^* \Vert \\&+\, \gamma _n \lambda _n \langle (I-F)x_n - p^*, j(x_{n+1} - p^*) \rangle . \end{aligned}$$

On the other hand, we can write

$$\begin{aligned} \begin{array}{llll} \langle (I-F)x_n - p^*, j(x_{n+1}-p^*)\rangle \\ = \langle x_n-Fx_n-p^* + Fp^*, j(x_{n+1}-p^*)\rangle - \langle Fp^*, j(x_{n+1}-p^*)\rangle \\ \le \Vert x_n-Fx_n-p^* + Fp^*\Vert \,\Vert x_{n+1}-p^*\Vert - \langle Fp^*, j(x_{n+1}-p^*)\rangle . \end{array} \end{aligned}$$

Since the operator F is $\gamma$-strictly pseudomonotone and $\eta$-strongly accretive, we have successively

$$\begin{aligned} \begin{array}{llll} \Vert x_n-Fx_n-p^* + Fp^*\Vert ^2 &{} \le \frac{1}{\gamma }\Vert x_n-p^*\Vert ^2 - \frac{1}{\gamma } \langle Fx_n-Fp^*, j(x_n-p^*)\rangle \\ &{} \le \frac{1-\eta }{\gamma }\Vert x_n-p^*\Vert ^2. \end{array} \end{aligned}$$

Since $\eta + \gamma >1$, we have that $\frac{1-\eta }{\gamma }<1$. Hence, with $\tau = 1-\sqrt{\frac{1-\eta }{\gamma }}$, we get

$$\begin{aligned} \Vert x_n-Fx_n-p^* + Fp^*\Vert \le (1-\tau ) \Vert x_n-p^*\Vert . \end{aligned}$$

Consequently,

$$\begin{aligned} \begin{array}{llll} \Vert x_{n+1}-p^*\Vert ^2 &{} \le (1-\gamma _n \lambda _n) \Vert x_n -p^*\Vert \,\Vert x_{n+1}-p^*\Vert \\ &+\, \gamma _n \lambda _n (1-\tau ) \Vert x_n -p^*\Vert \,\Vert x_{n+1}-p^*\Vert \\ &+\, \gamma _n \lambda _n \langle -Fp^*, j(x_{n+1}-p^*)\rangle \\ &{} \le (1-\gamma _n \lambda _n \tau ) \frac{\Vert x_n-p^*\Vert ^2 \, + \, \Vert x_{n+1}-p^*\Vert ^2}{2}\\ &+\, \gamma _n \lambda _n \tau \langle -Fp^*, j(x_{n+1}-p^*)\rangle / \tau . \end{array} \end{aligned}$$

Hence,

$$\begin{aligned} \begin{array}{llll} 2 \Vert x_{n+1}-p^*\Vert ^2 &{} \le &{} (1-\gamma _n \lambda _n \tau ) \Vert x_n-p^*\Vert ^2 + (1-\gamma _n \lambda _n \tau ) \Vert x_{n+1}-p^*\Vert ^2\\ &{}&{}+\, 2 \gamma _n \lambda _n \tau \langle -Fp^*, j(x_{n+1}-p^*)\rangle / \tau . \end{array} \end{aligned}$$

That is, we have

$$\begin{aligned} (1+\gamma _n \lambda _n \tau ) \Vert x_{n+1}-p^*\Vert ^2\le & {} (1-\gamma _n \lambda _n \tau ) \Vert x_n-p^*\Vert ^2\\&+\, 2 \gamma _n \lambda _n \tau \langle -Fp^*, j(x_{n+1}-p^*)\rangle / \tau . \end{aligned}$$

Thus, using the property $j(-x)=-j(x)$ for all $x \in E$, we can deduce

$$\begin{aligned} \Vert x_{n+1} - p^* \Vert ^2\le & {} \frac{1-\gamma _n \lambda _n \tau }{1+\gamma _n \lambda _n \tau } \Vert x_n - p^* \Vert ^2 \\&+\, \frac{2 \gamma _n \lambda _n \tau }{1+\gamma _n \lambda _n \tau } \frac{\langle -F(p^*), j(x_{n+1} - p^*) \rangle }{\tau } \\= & {} \biggl ( 1-\frac{2\gamma _n \lambda _n \tau }{1+\gamma _n \lambda _n \tau }\biggl ) \Vert x_n -p^* \Vert ^2 \\&+\, \frac{2\gamma _n \lambda _n \tau }{1+\gamma _n \lambda _n \tau } \frac{\langle F(p^*), j(p^* - x_{n+1}) \rangle }{\tau }, \end{aligned}$$

or

$$\begin{aligned} \Vert x_{n+1} - p^* \Vert ^2 \le (1-d_n) \Vert x_n - p^* \Vert ^2 + d_n e_n \end{aligned}$$

(3.9)

where

$$\begin{aligned} d_n = 2 \gamma _n \lambda _n \tau /(1+\gamma _n \lambda _n \tau ) \ \text{ and } \ e_n = \langle F(p^*), j(p^* - x_{n+1}) \rangle / \tau . \end{aligned}$$

Since $\sum _{n=0}^\infty \lambda _n = \infty$, we have $\sum _{k=0}^\infty d_n = \infty$. Using (3.8), we obtain $\limsup _{n \rightarrow \infty } e_n \le 0$.

Then, from Lemma 2.5 and (3.9) we have that $\lim _{n \rightarrow \infty } \Vert x_n - p^* \Vert ^2 = 0$. This completes the proof. $\square$

4 A Numerical Illustration

It is easy to see that our results are applicable for solving the following optimization problem: Find a point $p^*\in C$ such that

$$\begin{aligned} \varphi (p^*) = \min _{x \in C} \varphi (x). \end{aligned}$$

(4.1)

Here the function $\varphi : \mathbb {R}^N \rightarrow \mathbb {R}$ is assumed to have a strongly monotone and Lipschitz continuous derivative $\varphi '(x)$ on the Euclidean space $\mathbb {R}^N$, and C is the set of common fixed points of a nonexpansive semigroup $\{T(t): t\ge 0\}$ on $\mathbb {R}^N$.

For our computational purpose, we consider the case when $N=500$, $\varphi (x)=\Vert x-1\Vert ^2$, and $C:=\mathscr {F}$, the fixed point set of a nonexpansive semigroup $\{T(t): \mathbb {R}^{500} \rightarrow \mathbb {R}^{500}, \ t\ge 0\}, \ x \longmapsto T(t) x$ defined by the multiplication of the following matrix

$$\begin{aligned} \left( \begin{array}{cccccccccc} \cos (\alpha t) &{}\quad -\sin (\alpha t) &{}\quad 0 &{}\quad 0 &{}\quad 0&{}\quad \ldots &{}\quad 0&{}\quad 0&{}\quad 0 \\ \sin (\alpha t) &{}\cos (\alpha t) &{}\quad 0 &{} \quad 0 &{}\quad 0&{}\quad \ldots &{}\quad 0&{}\quad 0&{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \cos (\alpha t) &{}\quad -\sin (\alpha t) &{}\quad 0 &{}\quad \ldots &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad \sin (\alpha t) &{}\quad \cos (\alpha t) &{}\quad 0 &{}\quad \ldots &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0&{}\quad 1 &{}\quad \ldots &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ \vdots &{} \quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{} \quad \vdots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0&{}\quad 0 &{}\quad \ldots &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0&{}\quad 0 &{}\quad \ldots &{}\quad 0 &{}\quad \cos (\beta t) &{}\quad -\sin (\beta t) \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0&{}\quad 0 &{}\quad \ldots &{}\quad 0 &{}\quad \sin (\beta t) &{}\quad \cos (\beta t) \end{array}\right) \end{aligned}$$

by $(x_1, x_2, \ldots , x_{500})^\mathrm{T}$, where $x =(x_1,x_2,\ldots , x_{500}) \in \mathbb {R}^{500}$ and $\alpha , \beta \in \mathbb {R}$ are fixed constants. In this case, $C:=\mathscr {F} = \{ x \in \mathbb {R}^{500}: \ x = (0,\ldots ,0,x_5,\ldots , x_{498},0,0)^\mathrm{T}\}$ is a closed and convex subset of $\mathbb {R}^{500}$ and the optimal condition for (4.1) is the variational inequality (1.2) with $F(x)=\nabla \varphi (x)=2(x-1)$. It is easy to see that

$$\begin{aligned} p^* = (0,0,0,0,1,\ldots ,1,0,0)^\mathrm{T} \in \mathscr {F}\subset \mathbb {R}^{500} \end{aligned}$$

is the unique solution of (4.1).

We apply our methods (1.6) and (1.7) for solving (4.1) with $F(x)=2(x-1)$, the derivative of $\varphi$ at $x \in \mathbb {R}^{500}$. Clearly, F is 2-Lipschitz continuous and 1-strongly monotone on $\mathbb {R}^{500}$. For both methods, the initial guess is $x_0=(5, 5,\ldots , 5)\in \mathbb {R}^{500}$. We choose $\alpha = \pi /5, \beta =\pi /7$, and the other parameters are as follows: $t_n =(n+1)^{2}$, $\gamma _n=(n+1)^{-1/7}$, and $\lambda _n =(n+1)^{-1/ 2 }$ in the computation of (1.6) and $\gamma _n=(2n+1)^{-1/2}$, and $\lambda _n =(5n+1)^{-1/ 3 }$ in (1.7).

The numerical results obtained with our example are given in Tables 1 and 2.

Table 1 Results for method (1.6)

Table 2 Results for method (1.7)

The above numerical results show that the methods considered in this paper, based on (1.6) and (1.7), are able to solve efficiently some class of optimization problems and variational inequalities with control conditions. However, it seems that method (1.7) is more efficient than method (1.6).

5 Conclusion

We studied a class of accretive variational inequalities, where the feasible set is the set of common fixed points of a nonexpansive semigroup in a real uniformly convex Banach space with a uniformly Gâteaux differentiable norm. We introduced two explicit iterative methods to solve the considered problem. Our convergence analyses guarantee that the methods converge to a solution under certain mild assumptions. In particular, we do not impose the use of a sequentially weakly continuity property on the normalized duality mapping of Banach spaces. We presented a numerical example to illustrate the convergence analyses of the proposed methods.