Some Results on the Approximation of Solutions of Variational Inequalities for Multivalued Maps on Banach Spaces

Abstract

Multivalued $*$-nonexpansive mappings are studied in Banach spaces. The demiclosedness principle is established. Here we focus on the problem of solving a variational inequality which is defined on the set of fixed points of a multivalued $*$-nonexpansive mapping. For this purpose, we introduce two algorithms approximating the unique solution of the variational inequality.

1 Introduction

The notion of $*$-nonexpansive maps seems to be interesting because the $*$-nonexpansivity holds when given two sets that are images of two different points of the domain; it is possible to choose for each set (at least) a closest point to the corresponding point of the domain so that the distance between these two does not exceed the distance between the starting points. Therefore, it is an idea that immediately calls back the usual nonexpansivity of the single-valued case.

More precisely, let X be a Banach space and let C be a subset of X. Let K(C) be the family of compact subsets of C.

Definition 1.1

[10] A mapping $W:C\rightarrow K(C)$ is said to be $*$-nonexpansive if for all $x,y\in C$ and $x^W\in Wx$ such that $\Vert x-x^W\Vert =d(x,Wx)$, there exists $y^W\in Wy$ with $\Vert y-y^W\Vert =d(y,Wy)$ such that

$$\begin{aligned} \Vert x^W-y^W\Vert \le \Vert x-y\Vert \end{aligned}$$

Recall that a point $x\in C$ is said to be a fixed point for a multivalued mapping W if $x\in Wx$.

The concept of $*$-nonexpansive multivalued maps was introduced by Husain and Latif [10] in 1988; it is a generalisation of the known notion of nonexpansiveness for single-valued maps. In general, $*$-nonexpansive multivalued maps may neither be continuous (Example 1.1 in [9]) nor nonexpansive with respect to the definition obtained by means of Hausdorff metric(see also [26]).

However, $*$-nonexpansivity and multivalued nonexpansivity are not so far. In Theorem 3 of [15], it is proved that a multivalued map $W:C\rightarrow K(C)$ is $*$-nonexpansive if and only if the metric projection

$$\begin{aligned} P_W(x):=\left\{ u_x\in Wx:\Vert x-u_x\Vert =\inf _{y\in Wx}\Vert x-y\Vert \right\} \end{aligned}$$

is nonexpansive.

Existence results of fixed points for multivalued mappings are, in general, subtle and sometimes, surprising. For instance, the multivalued improvement of the classical Banach contraction principle, proved by Nadler in 1969 [18], guarantees the existence of a fixed point for a multivalued contraction, but not its uniqueness. Again, unlike to the single-valued case, the set of fixed points of a multivalued nonexpansive mapping $W:C\rightarrow K(C)$ on a strictly convex Banach space is not, in general, a convex set, see [12] Section 3, and the same holds for $*$-nonexpansive mappings too.

Xu, 1991 [26], has proved two existence results of fixed points for $*$-nonexpansive on strictly convex Banach spaces; Lopez-Acedo and Xu in [15] (1995) have obtained existence result in the setting of Banach space satisfying Opial condition.

Other surprising results, compared to the single-valued case, can be found in the literature about the approximation of fixed points of multivalued mappings. We refer to a well-known counterexample due to Pietramala, proved in [20] (1991): she proved that Browder approximation Theorem 1 in [2] cannot be extended to the genuine multivalued case even on a finite dimensional space ${\mathbb {R}}^2$.

The problem that we are concerned within this paper is the following: given a reflexive Banach space X and a closed subset $C\subset X$, to find $x^*\in C$ such that

$$\begin{aligned} \langle Ax^*,j(y-x^*)\rangle \ge 0, \qquad \forall y\in C\subset D(A), \end{aligned}$$

(1.1)

where

• $j(x)\in J(x)$ and $J:X\rightarrow X^*$ is the normalized duality mapping defined by

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

  (1.2)

• $A:D(A)\subset X\rightarrow X$ is a $\eta -$strongly accretive operator, i.e. it satisfies

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

The solution of (VIP)(1.1) is a singleton; indeed, given $x^*$ and ${\bar{x}}$ two different solutions, one immediately notes that

$$\begin{aligned} \langle Ax^*,j({\bar{x}}-x^*)\rangle \ge 0 \text{ and } \langle A{\bar{x}},j(x^*-{\bar{x}})\rangle \ge 0 \end{aligned}$$

hold jointly. Therefore, adding the inequalities

$$\begin{aligned} -\eta \Vert {\bar{x}}-x^*\Vert ^2\ge -\langle A{\bar{x}}-Ax^*,j({\bar{x}}-x^*)\rangle \ge 0, \end{aligned}$$

i.e. ${\bar{x}}=x^*$.

In a Hilbert space H, (VIP)(1.1) is equivalent to a variational inequality problem on the set of fixed points Fix(W) of a suitable nonexpansive mapping W; for instance, the metric projection on the subset C. In the setting of a general Banach space, since there exist closed and convex sets that are not fixed point sets of a nonexpansive mapping $W:X\rightarrow X$ (see page 25 in [6]), this is no longer true.

In this note, we will work on a feasible set C that is the fixed point set of a multivalued $*$-nonexpansive mapping, i.e. given a strongly accretive operator $A:X\rightarrow X$ and a multivalued $*$-nonexpansive mapping W with fixed points, we focus on some approximation algorithms of the unique solution of the variational inequality

$$\begin{aligned} \langle Ax^*,j(y-x^*)\rangle \ge 0, \qquad \forall y\in Fix(W). \end{aligned}$$

(1.3)

This problem encloses, as a particular case, viscosity problems

$$\begin{aligned} \langle (I-f)x^*,j(y-x^*)\rangle \ge 0, \qquad \forall y\in Fix(W), \end{aligned}$$

(1.4)

when $A=I-f$ and, if $A=I-u$, the problem

$$\begin{aligned} \langle x^*-u, j(y-x^*)\rangle \ge 0, \qquad \forall y\in Fix(W), \end{aligned}$$

(1.5)

that is equivalent to the minimum problem $\displaystyle \min _{x\in Fix(W)}\Vert x-u\Vert ^2$.

These problems are widely studied as for the single-valued as for the multivalued case; for details one should refer to [5, 17, 19, 25, 29].

The novelty of our work can be immediately recognised: the use of $*$-nonexpansive mapping is no longer developed with respect to multivalued nonexpansive although they can be of interest in view of Example 1.1 in [9] and Example 1-2 in [26] respectively.

In our approach, with respect to multivalued nonexpansive case, we do not use Banach limit.

Remark 1.2

We want to emphasise here that this last approach is not always fully correct. Indeed, taking into account [31], we note that in many papers, see for instance [8, 11, 24, 32], Banach limits are used to define a function $\phi$ by

$$\begin{aligned} \phi (x):=\mathrm{LIM}_{n\rightarrow +\infty }\Vert x_n-x\Vert ^2,\quad x\in X, \end{aligned}$$

where $(x_n)_{n\in {\mathbb {N}}}$ is a bounded sequence in X (which is generated by an iterative method). It is easily verified that $\phi$ is continuous, convex and coercive (i.e. $\phi (x)\rightarrow \infty$, as $\Vert x\Vert \rightarrow +\infty$). Hence, reflexivity of X ensures that $\phi$ attains its minimum on a closed convex set C. Let $p\in C$ be a minimiser of $\phi$ over C. If C is a nonexpansive retract of X, this minimum is a global minimum on X. The goal is to prove that p is a fixed point of W. Using compactness arguments, they proved that, given $(w_n)_{n\in {\mathbb {N}}}\subset Wp$, there exists a subsequence strongly convergent to $w\in Tp$ (wrongfully indicated by the same sequence $(w_n)_{n\in {\mathbb {N}}}$). Therefore, using the formula that defines the iteration, they proved that

$$\begin{aligned} \phi (w)=\mathrm{LIM}_{n\rightarrow \infty }\Vert x_n-w\Vert ^2 \le \ldots \le \mathrm{LIM}_{n\rightarrow \infty }\Vert x_n-p\Vert ^2=\phi (p)=\min _{X}\phi \nonumber \\ \end{aligned}$$

(1.6)

and then drew the conclusion that $w=p$ and thus $p\in Wp$.

Unfortunately, the above argument holds for a subsequence $(w_{n_k})_{k\in {\mathbb {N}}}$ of $(w_n)_{n\in {\mathbb {N}}}$ only, and so (1.6) holds for a subsequence $(x_{n_k})$ only; that is, the correct statement of (1.6) should be

$$\begin{aligned} \mathrm{LIM}_{k\rightarrow \infty }\Vert x_{n_k}-w\Vert ^2 \le \ldots \le \mathrm{LIM}_{k\rightarrow \infty }\Vert x_{n_k}-p\Vert ^2. \end{aligned}$$

(1.7)

Consequently, the conclusion $w=p$ cannot be drawn from (1.7). Notice that Banach limits are sensitive to subsequences, as the following simple example shows: consider the real sequence $a_n=1+(-1)^n$; then we have

$$\begin{aligned} \mathrm{LIM}_{n\rightarrow \infty } a_n=1,\ \mathrm{LIM}_{n\rightarrow \infty } a_{2n+1}=0, \ \mathrm{LIM}_{n\rightarrow \infty } a_{2n}=2. \end{aligned}$$

Therefore, the claim $w=p$ in the above proof is not convincing.

The paper is organised as follows: in the next section, we introduce some definitions and tools which are used in our proofs. In Sect. 3, we prove our results and raise some open problems.

2 Preliminaries

Let $(X,\Vert \cdot \Vert )$ be a Banach space. Denote by K(X) the family of compact subset of X.

In (1.2), we have quickly introduced the normalised duality mapping; indeed it is the special case of the following.

A function $\varphi :{\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$ is said to be a gauge if:

• 1. $\varphi (0)=0$;

• 2. $\varphi$ is continuous and strictly increasing;

• 3. $\varphi (t)\rightarrow +\infty$, as $t\rightarrow +\infty$.

Associated with a gauge $\varphi$ is the duality map

$$\begin{aligned} J_\varphi (x)=\{x^*\in X^*:\langle x,x^*\rangle = \Vert x\Vert \Vert x^*\Vert ,\varphi (\Vert x\Vert )=\Vert x^*\Vert \}. \end{aligned}$$

Choosing $\varphi (t)=t^{p-1}$, for some $p\in (1,+\infty )$, the duality map is referred to as the generalised duality map of order p; for $p=2$, we get J(x). It is well known that the Asplund’s result¹ proved that $J_\varphi$ is the sub-differential of the convex functional $\Phi (\Vert \cdot \Vert )$ defined as

$$\begin{aligned} \Phi (t)=\int _0^t\varphi (s) \mathrm{d}s. \end{aligned}$$

Since the relationship

$$\begin{aligned} J(x)\varphi (\Vert x\Vert )=\Vert x\Vert ^2J_\varphi (x) \end{aligned}$$

(2.1)

holds, it is easy to notice that the (VIP)(1.1) is equivalent to

$$\begin{aligned} \langle Ax^*,j_\varphi (y-x^*)\rangle \ge 0, \qquad \forall y\in C. \end{aligned}$$

Following Browder [1], recall that a Banach space X has a weakly sequentially continuous duality map $J_{\varphi }$ for some gauge $\varphi$ if $J_{\varphi }x_n\rightarrow J_{\varphi }x$ in the $\hbox {weak}^*$ topology of $X^*$ whenever $x_n\rightarrow x$ in the weak topology of X. The following result is useful in fixed point theory and geometry of Banach spaces [14].

Lemma 2.1

Let X be a Banach space with a weakly sequentially continuous duality map $J_{\varphi }$ for some gauge $\varphi$. Assume $(x_n)$ is a sequence in X weakly converging to $x^*$. Then

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Phi (\Vert x_n-x\Vert )=\limsup _{n\rightarrow \infty }\Phi (\Vert x_n-x^*\Vert )+\Phi (\Vert x-x^*\Vert ) \end{aligned}$$

(2.2)

for all $x\in X$. In particular, X satisfies Opial’s condition, i.e.

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert x_n-x^*\Vert <\limsup _{n\rightarrow \infty }\Vert x_n-x\Vert , \qquad \forall x\in X \end{aligned}$$

(2.3)

(but not vice versa [7, 28]).

Definition 2.2

[3] An operator $A:X\rightarrow X$ is said to be $\lambda -$strict pseudocontractive ($\lambda \in (0,1)$) if for every $x,y\in X$ there exists $j(x-y)\in J(x-y)$ such that

$$\begin{aligned} \langle Ax-Ay,j(x-y)\rangle \le \Vert x-y\Vert ^2-\lambda \Vert (I-A)x-(I-A)y\Vert ^2. \end{aligned}$$

Proposition 2.3

[3] Let X be a smooth Banach space, $A:X\rightarrow X$ be an operator.

• (i) If A is $\lambda -$strict pseudocontractive then A is $L-$Lipschitzian, where $L=1+\lambda ^{-1}$.

• (ii) If A is $\eta -$strongly accretive and $\lambda -$ strict pseudocontractive with $\eta +\lambda >1$ then $(I-\tau A)$ is a $(1-\tau \rho )$-contraction, for all $\tau \in (0,1)$, where $\displaystyle \rho :=\left( 1-\sqrt{\frac{1-\eta }{\lambda }}\right)$.

Remark 2.4

Each linear operator A defined by $\displaystyle Ax=k x$, $k>1$, is not a strict pseudocontractive mapping therefore vice versa of statement (i) does not hold.

In [28] it is proved that if A is a $\eta$-strongly monotone and $L-$Lipschitzian operator, then $(I-\tau A)$ is a contraction if $\displaystyle \tau \le \frac{2\eta }{L^2}$ in the setting of a Hilbert space.

A similar result is proved on q-uniformly smooth Banach spaces (see [16]).

Next Lemma, proved in [27], will be a useful tool for our proof.

Lemma 2.5

Assume $(b_{n})_{n\in {\mathbb {N}}}$ is a sequence of nonnegative numbers for which,

$$\begin{aligned} b_{n+1}\le (1-a_{n})b_{n}+\delta _{n},\quad n\ge 0, \end{aligned}$$

where $(a_{n})_{n\in {\mathbb {N}}}$ is a sequence in (0, 1) and $(\delta _{n})_{n\in {\mathbb {N}}}$ is a sequence in ${\mathbb {R}}$ such that,

• 1. $\sum _{n=1}^{\infty }a_{n}=\infty ;$

• 2. $\lim \sup _{n\rightarrow \infty }\frac{\delta _{n}}{a_{n}}\le 0$ or $\sum _{n=1}^{\infty }|\delta _{n}|<\infty .$

Then $\lim _{n\rightarrow \infty }b_{n}=0.$

3 Results

Let $W:X{\rightarrow } K(X)$ be a $*$-nonexpansive multivalued mapping with nonempty Fix(W).

In view of Definition 1.1, for a given $x\in X$ and $p\in Fix(W)$, for any $x^W\in Wx$ such that $\Vert x-x^W\Vert =d(x,Wx)$, i.e $x^W\in P_Wx$, there exists $p^W$ such that $\Vert p-p^W\Vert =d(p,Wp)=0$, i.e. $p=p^W\in P_Wp$, and

$$\begin{aligned} \Vert x^W-p\Vert \le \Vert x-p\Vert . \end{aligned}$$

(3.1)

Let us start with two easy examples showing that the convergence of a classical viscosity method ([17, 29]) is not guaranteed without any attention on the choice of $x^W\in Wx$.

Counter-Example 3.1

Let $W:{\mathbb {R}}\rightarrow K({\mathbb {R}})$ be defined as $Wx=\{x,x+1\}$.

Note that W is $*$-nonexpansive mapping, $Fix(W)={\mathbb {R}}$ and $Wx\ne \{x\}$ if $x\in Fix(W)$.

Let $\displaystyle f(x)=\frac{x}{2}$ and let us consider the implicit iteration:

$$\begin{aligned} x_n=\tau _nf(x_{n})+(1-\tau _n)x^W_n, \end{aligned}$$

(3.2)

where $x^W_n\in Wx_n$.

Since $Wx=\{x,x+1\}$, we choose $W(x_n)\ni x^W_n=x_n+1$ and our iteration becomes:

$$\begin{aligned} x_n=\frac{2(1-\tau _n)}{\tau _n}. \end{aligned}$$

(3.3)

It is clear that $(x_n)_{n\in {\mathbb {N}}}$ does not converge for any null sequence $(\tau _n)_{n\in {\mathbb {N}}}$ (i.e. $\tau _n\rightarrow 0$ as $n\rightarrow +\infty$).

Lemma 3.2

Let $W: X \rightarrow K(X)$ and let $P_W$ be defined as

$$\begin{aligned} P_Wx:=\{u_x\in Wx:\Vert x-u_x\Vert =d(x,Wx)\}, \end{aligned}$$

i.e. the projection of x on the set Wx. Then the following hold:

• (i) If $P_W$ is the identity mapping then W is $*$-nonexpansive.

• (ii) W is $*$-nonexpansive if and only if $P_W$ is nonexpansive.

• (iii) If $Fix(W)\ne \emptyset$ then $P_W|_{Fix(W)}$ is single-valued, i.e. $P_Wx=\{x\}$ for each $x\in Fix(W)$ and $Fix(W)=Fix(P_W)$.

Proof

(i) follows by definitions and (ii) is proved in Theorem 3 in [15].

To prove (iii), if ${\tilde{x}}\in P_W({\tilde{x}})$ then ${\tilde{x}}\in \{u_{{\tilde{x}}}\in W{{\tilde{x}}}:\Vert {\tilde{x}}-u_{{\tilde{x}}}\Vert =d({{\tilde{x}}},W{{\tilde{x}}})\}$; hence, $d({\tilde{x}},W{\tilde{x}})=0$. This implies that ${\tilde{x}}\in Fix(W)$, and

$$\begin{aligned} Fix(P_W)\subset Fix(W) \end{aligned}$$

is proved. On the other hand, if $x\in Fix(W)$, $d(x,Wx)=0$ then $P_Wx=\{u_x:\Vert u_x-x\Vert =0\}$, i.e. $x=u_x$.

Therefore, $P_W(x)=\{x\}$, i.e $P_W$ is single-valued on the set of fixed points of W and $x\in Fix(P_W)$ that conclude our proof. $\square$

The next counter-example shows that the convergence of a classical viscosity method is not certain even under the strong condition that the metric projection $P_W$ is single-valued on Fix(W).

Counter-Example 3.3

Let $W:[0,1]\rightarrow K([0,1])$ defined as:

$$\begin{aligned} Wx=\left\{ \begin{array}{l} \displaystyle \left[ x,x+\frac{1}{2}\right] , \qquad 0\le x< \frac{1}{2}\\ \\ \displaystyle \left[ x-\frac{1}{2},x\right] , \qquad \frac{1}{2}\le x\le 1.\\ \end{array}\right. \end{aligned}$$

Since $P_W$ is the identity mapping, W is $*$-nonexpansive by Lemma 3.2 (i).

Let us consider iteration

$$\begin{aligned} x_n=\tau _nf(x_{n-1})+(1-\tau _n)x^W_n, \end{aligned}$$

where $f:[0,1]\rightarrow [0,1]$ is a contraction such that $f(x)< x$, $x>0$ (so the unique fixed point is $x=0$) and the following choice for $x^W_n$ is done:

$$\begin{aligned} x^W_n=\left\{ \begin{array}{l} \displaystyle x_n+\frac{1}{2}, \qquad 0\le x_n< \frac{1}{2}\\ \\ \displaystyle x_n-\frac{1}{2}, \qquad \frac{1}{2}\le x_n\le 1.\\ \end{array}\right. \end{aligned}$$

By induction we prove that if $\displaystyle x_0\le \frac{1}{2}$ then $\displaystyle x_n\le \frac{1}{2}$ for all $n\in {\mathbb {N}}$.

Let us suppose, by contradiction, that $\displaystyle x_n\le \frac{1}{2}$ and $\displaystyle x_{n+1}>\frac{1}{2}$.

Thus, $\displaystyle x^W_{n+1}=x_{n+1}-\frac{1}{2}$ and

$$\begin{aligned}&x_{n+1}=\tau _{n+1}f(x_{n})+(1-\tau _{n+1})x_{n+1}-\frac{(1-\tau _{n+1})}{2}\\&\quad \Rightarrow x_{n+1}=f(x_{n})-\frac{1}{2}\frac{(1-\tau _{n+1})}{\tau _{n+1}}\le f(x_{n})\le x_n\le \frac{1}{2} \end{aligned}$$

by inductive hypothesis. This is a contradiction; therefore, $\displaystyle x_0\le \frac{1}{2}$ implies $\displaystyle x_n\le \frac{1}{2}$ for all $n\in {\mathbb {N}}$.²

Let $\displaystyle x_0\in \left[ 0,\frac{1}{2}\right]$; then the entire sequence lies in the same interval. This is not possible because it can be written as

$$\begin{aligned} x_n=f(x_{n-1})+\frac{(1-\tau _n)}{2\tau _n}. \end{aligned}$$

Since f is positive $\displaystyle x_n>\frac{(1-\tau _n)}{2\tau _n}$ therefore our sequence $(x_n)_{n\in {\mathbb {N}}}$ can not be found in $\displaystyle \left[ 0,\frac{1}{2}\right]$ for any null sequence $(\tau _n)_{n\in {\mathbb {N}}}$. This contradiction shows that by such a choice of $x_n^W$, the algorithm does not work.

3.1 Iterative Approach

To prove our convergence results, the following demiclosedness type-principle is necessary; for multivalued nonexpansive mapping demiclosedness principle is well known by Theorem 3.1- [13]; for multivalued $*$-nonexpansive mapping demiclosedness principle given in the next Lemma 3.4 seems to be new.

Lemma 3.4

Let X be a reflexive space satisfying Opial condition (2.3).

Let $W:X\rightarrow K(X)$ be a $*$-nonexpansive multivalued mapping with fixed points.

Let $(y_n)_{n\in {\mathbb {N}}}$ be a bounded sequence such that

$$\begin{aligned} d(y_n,Wy_n)\rightarrow 0, \text{ as } n\rightarrow \infty . \end{aligned}$$

Then the weak cluster points of $(y_n)_{n\in {\mathbb {N}}}$ belong to Fix(W), (i.e. $\omega _w(y_n)\subset Fix(W)$).

Proof

Since X is reflexive, let $(y_{n_k})_{k\in {\mathbb {N}}}\subset (y_n)_{n\in {\mathbb {N}}}$ weak convergent to z.

Since Wz is compact, it is closed and there exists $(z_{n_k})_{k\in {\mathbb {N}}}\subset Wz$ such that

$$\begin{aligned} \Vert y_{n_k}-z_{n_k}\Vert =d(y_{n_k},Wz). \end{aligned}$$

Still because of the compactness of Wz, there exists a subsequence $(z_{n_{k_j}})_j\subset (z_{n_k})_k\subset Wz$ strongly convergent to ${\tilde{z}}\in Wz$.

By definition of $*-$nonexpansivity, for any $j\in {\mathbb {N}}$ and $y_{n_{k_j}}$ there exists $u_{y(j)}\in Wy_{n_{k_j}}$ with $\Vert y_{n_{k_j}}-u_{y(j)}\Vert =d(y_{n_{k_j}},Wy_{n_{k_j}})$ and $u_{z(j)}\in Wz$ with $\Vert u_{z(j)}-z\Vert =d(z, Wz)$ such that

$$\begin{aligned} \Vert u_{y(j)}-u_{z(j)}\Vert \le \Vert y_{n_{k_j}}-z\Vert . \end{aligned}$$

We now prove that $z={\tilde{z}}$, so will be $z\in Wz$. If not, since $d(y_n, Wy_n)\rightarrow 0$ and using the Opial’s inequality

$$\begin{aligned} \limsup _{j\rightarrow \infty }\Vert y_{n_{k_j}}-{\tilde{z}}\Vert\le & {} \limsup _{j\rightarrow \infty }[\Vert y_{n_{k_j}}-z_{n_{k_j}}\Vert +\Vert z_{n_{k_j}}-{\tilde{z}}\Vert ]\\= & {} \limsup _{j\rightarrow \infty }d(y_{n_{k_j}},Tz)\le \limsup _{j\rightarrow \infty }\Vert y_{n_{k_j}}-u_{z(j)}\Vert \\\le & {} \limsup _{j\rightarrow \infty }[\Vert y_{n_{k_j}}-u_{y(j)}\Vert +\Vert u_{y(j)}-u_{z(j)}\Vert ]\\= & {} \limsup _{j\rightarrow \infty }[d(y_{n_{k_j}},Ty_{n_{k_j}})+\Vert u_{y(j)}-u_{z(j)}\Vert ]\\\le & {} \limsup _{j\rightarrow \infty }\Vert y_{n_{k_j}}-z\Vert \\< & {} \limsup _{j\rightarrow \infty }\Vert y_{n_{k_j}}-{\tilde{z}}\Vert , \end{aligned}$$

and this is absurd. Therefore $z\in Wz$, i.e. $z\in Fix(W)$. $\square$

Our first result concerns the existence of a unique solution of our (VIP) on the set of fixed point of a $*$-nonexpansive mapping.

Proposition 3.5

Let X be a reflexive Banach space with duality mapping $J_{\varphi }$ that is weakly sequentially continuous, for some gauge $\varphi$.

Let $W:X\rightarrow K(X)$ a $*$-nonexpansive multivalued mapping such that Fix(W) is nonempty.

Let $A:X\rightarrow X$ an $\eta$-strongly accretive and $k-$strict pseudocontractive such that $\eta +k>1$.

Then

$$\begin{aligned} \langle Ax^*,j(y-x^*)\rangle \ge 0, \qquad \forall y\in Fix(W) \end{aligned}$$

(3.4)

has a unique solution.

Proof

Uniqueness of the solution is already noted by means of the strong accretivity of A. Let us prove the existence.

Since W is $*$-nonexpansive, $P_W$ is nonexpansive by Lemma 3.2 (ii). Let $(\alpha _n)_{n\in {\mathbb {N}}}\subset (0,1)$ be a sequence such that $\alpha _n\rightarrow 0$ as $n\rightarrow +\infty$ and let $\mu \in (0,1)$. For any $n\in {\mathbb {N}}$, consider the multivalued mapping

$$\begin{aligned} \Gamma _n:=\alpha _n(I-\mu A)+(1-\alpha _n)P_W. \end{aligned}$$

It is easy to verify that each $\Gamma _n$ is a contraction. Indeed, if $x,y\in X$, $w=(I-\mu _nA)x\in X$, $v=(I-\mu _nA)y\in X$ we obtain that

$$\begin{aligned} H(\Gamma _nx,\Gamma _ny)\le & {} \alpha _n\Vert w-v\Vert +(1-\alpha _n)H(P_Wx,P_Wy)\\\le & {} \alpha _n\Vert (I{-}\mu A)x{-}(I{-}\mu A)y\Vert {+}(1{-}\alpha _n)H(P_Wx,P_Wy)\\ (\hbox {by Proposition }2.3(\hbox {ii}))\le & {} \alpha _n(1-\mu \rho )\Vert x-y\Vert +(1-\alpha _n)\Vert x-y\Vert \\= & {} (1-\alpha _n\mu \rho )\Vert x-y\Vert . \end{aligned}$$

Then by Nadler fixed point principle, $\Gamma _n$ has fixed point and

$$\begin{aligned} x_n=\alpha _n(I-\mu _nA)x_n+(1-\alpha _n)x^P_n \end{aligned}$$

is well defined for an opportune $x^P_n\in P_Wx_n$. Let $p\in Fix(W)$; then $p=Fix(P_W)$ by Lemma 3.2 (iii) and $P_Wp=\{p\}$ i.e. $P_W$ is single-valued on Fix(W). Thus

$$\begin{aligned} \Vert x_n-p\Vert= & {} \Vert \alpha _n(I-\mu A)x_n+(1-\alpha _n)x^P_n-p\Vert \\\le & {} \alpha _n\Vert (I-\mu A)x_n-(I-\mu A)p\Vert +\alpha _n\mu \Vert Ap\Vert +(1-\alpha _n)\Vert x^P_n-p\Vert \\ (\hbox {by } (3.1))\le & {} \alpha _n(1-\mu \rho )\Vert x_n-p\Vert +\alpha _n\mu \Vert Ap\Vert +(1-\alpha _n)\Vert x_n-p\Vert \\= & {} (1-\alpha _n\mu \rho )\Vert x_n-p\Vert +\alpha _n\mu \rho \frac{\Vert Ap\Vert }{\rho }; \end{aligned}$$

therefore,

$$\begin{aligned} \Vert x_n-p\Vert \le \frac{\Vert Ap\Vert }{\rho }, \end{aligned}$$

i.e. our sequence is bounded.

Moreover, for each $w\in Fix(W)$,

$$\begin{aligned} \Vert x_n-w\Vert ^2= & {} \langle x_n-w,j(x_n-w)\rangle \nonumber \\\le & {} \alpha _n\langle x_n-\mu Ax_n-w,j(x_n-w)\rangle +(1-\alpha _n)\Vert x_n-w\Vert ^2\nonumber \\= & {} \Vert x_n-w\Vert ^2-\alpha _n\mu \langle Ax_n,j(x_n-w)\rangle ; \end{aligned}$$

(3.5)

hence,

$$\begin{aligned} \langle Ax_n,j(x_n-w)\rangle \le 0, \qquad \forall w\in Fix(T). \end{aligned}$$

(3.6)

On the other hand, since A is $\eta$-strongly accretive, it follows from (3.6) that

$$\begin{aligned} 0&\ge \langle Ax_n-Aw,J(x_n-w)\rangle +\langle Aw,J(x_n-w)\rangle \\&\ge \eta \Vert x_n-w\Vert ^2+\langle Aw,j(x_n-w)\rangle . \end{aligned}$$

This implies that

$$\begin{aligned} \Vert x_n-w\Vert ^2\le -\frac{1}{\eta }\langle Aw,j(x_n-w)\rangle . \end{aligned}$$

(3.7)

Since $(x_n)_{n\in {\mathbb {N}}}$ is bounded and $(\alpha _n)_{n\in {\mathbb {N}}}$ is a null sequence it holds

$$\begin{aligned} \Vert x_n-x_n^P\Vert =\alpha _n\Vert (I-\mu A)x_n-x_n^P\Vert \rightarrow 0, \text{ as } n\rightarrow \infty ; \end{aligned}$$

therefore, $d(x_n,P_Wx_n)\rightarrow 0$ as $n\rightarrow \infty$ and, as a rule, $d(x_n,Wx_n)\rightarrow 0$ as $n\rightarrow \infty$ . By Lemma 3.4, the weak limit of $(x_n)_{n\in {\mathbb {N}}}$ are fixed points for W.

Recalling (2.1) we can write (3.7) as

$$\begin{aligned} \varphi (\Vert w-x_n\Vert )\Vert x_n-w\Vert ^2\le \frac{1}{\eta } \Vert w-x_n\Vert ^2\langle Aw,j_\varphi (w-x_n)\rangle ) \end{aligned}$$

(3.8)

and so

$$\begin{aligned} \varphi (\Vert w-x_n\Vert )\le \frac{1}{\eta }\langle Aw,j_\varphi (w-x_n)\rangle ). \end{aligned}$$

Let $w\in \omega _w(x_n)$; there exists $x_{n_k}\rightharpoonup w$ and $w\in Fix(T)$. Since the duality map $J_\varphi$ is weakly sequentially continuous

$$\begin{aligned} \varphi (\Vert x_{n_k}-w\Vert )\le \frac{1}{\eta } \langle Aw,j_\varphi (w-x_{n_k})\rangle \rightarrow 0, \end{aligned}$$

as $k\rightarrow \infty$; hence, $x_{n_k}\rightarrow w$, by properties of $\varphi$. Rewriting (3.6) with respect to $J_\varphi$ we get

$$\begin{aligned} 0\ge \langle Ax_{n_k}, j_\varphi (x_{n_k}-p)\rangle \rightarrow \langle Aw, j_\varphi (w-p)\rangle , \qquad \forall p\in Fix(T). \end{aligned}$$

Then w is a solution of (VIP) and, by the uniqueness of the solution, $\omega _w(x_n)=\omega _s(x_n)=\{w\}$ and the thesis follows.$\square$

Next we can define our first iteration.

Let $A : D(A)\subset X \rightarrow X$ be a strongly accretive operator and strict pseudocontractive.

Let $W:X\rightarrow 2^X$ be a multivalued $*$-nonexpansive mapping; let $x_0$ and $x_0^W\in Wx_0$ such that $\Vert x_0-x_0^W\Vert =d(x_0,Wx_0)$, i.e. $x_0^W\in P_Wx_0$. Let

$$\begin{aligned} x_1=\lambda _0(x_0-\mu _0Ax_0)+(1-\lambda _0)x_0^W. \end{aligned}$$

Using definition of $*-$nonexpansivity, there exists $x_1^W\in Wx_1$ such that $\Vert x_1-x_1^W\Vert =d(x_1,Wx_1)$, i.e. $x_1^W=P_Wx_1$ and

$$\begin{aligned} \Vert x_1^W-x_0^W\Vert \le \Vert x_1-x_0\Vert . \end{aligned}$$

In a same manner, let

$$\begin{aligned} x_2=\lambda _1(x_1-\mu _1Ax_1)+(1-\lambda _0)x_1^W, \end{aligned}$$

and choose $x_2^W\in P_Wx_2$ and

$$\begin{aligned} \Vert x_2^W-x_1^W\Vert \le \Vert x_2-x_1\Vert . \end{aligned}$$

Iterating this process we get a sequence

$$\begin{aligned} x_{n+1}=\lambda _n(x_n-\mu _nAx_n)+(1-\lambda _n)x_n^W \end{aligned}$$

(3.9)

such that

$$\begin{aligned} \Vert x_{n+1}^W-x_n^W\Vert \le \Vert x_{n+1}-x_n\Vert . \end{aligned}$$

(3.10)

Theorem 3.6

Let X be a reflexive Banach space with duality mapping $J_\varphi$ that is weakly sequentially continuous .

Let $W:X\rightarrow K(X)$ a $*$-nonexpansive multivalued mapping such that Fix(W) is nonempty.

Let $A:X\rightarrow X$ an $\eta$-strongly accretive and $k-$strict pseudocontractive such that $\eta +k>1$.

Let $(\mu _n)_{n\in {\mathbb {N}}}\subset (0,1)$ and $(\lambda _n)_{n\in {\mathbb {N}}}\subset [0,a]\subset [0,1)$ such that

• $\lambda _n\mu _n\rightarrow 0$, as $n\rightarrow \infty$ and $\displaystyle \sum _{n\in {\mathbb {N}}}\lambda _n\mu _n=\infty$.

• $\displaystyle \lim _{n\rightarrow \infty }\frac{|\lambda _n-\lambda _{n-1}|}{\lambda _n\mu _n}=0$.

• $\displaystyle \lim _{n\rightarrow \infty }\frac{|\mu _n-\mu _{n-1}|}{\mu _n}=0$.

Then, for any choice $x_0$ as a starting point, the explicit process

$$\begin{aligned} x_{n+1}=\lambda _n (I-\mu _nA)x_n+(1-\lambda _n)x^W_n, \end{aligned}$$

(3.11)

defined choosing $x_n^W$ in such a way that (3.10) is satisfied, strongly converges, as $n\rightarrow \infty$, to the unique solution of (VIP)

$$\begin{aligned} \langle Ax^*,j(y-x^*)\rangle \ge 0, \qquad \forall y\in Fix(W). \end{aligned}$$

(3.12)

Proof

Defining $B_n:=(I-\mu _nA)$, our iteration can be described by

$$\begin{aligned} x_{n+1}=\lambda _n B_nx_n+(1-\lambda _n)x^W_n. \end{aligned}$$

By hypotheses, every $B_n$ is a contraction using Proposition 2.3 (ii).

Let $p\in Fix(W)$ be a given fixed point of W; then by (3.1),

$$\begin{aligned} \Vert x_{n+1}-p\Vert\le & {} \lambda _n\Vert B_nx_n-p\Vert +(1-\lambda _n)\Vert x^W_n-p\Vert \\\le & {} \lambda _n\Vert B_nx_n-B_np\Vert +\lambda _n\Vert B_np-p\Vert +(1-\lambda _n)\Vert x_n-p\Vert \\\le & {} \lambda _n(1-\mu _n\rho )\Vert x_n-p\Vert +(1-\lambda _n)\Vert x_n-p\Vert +\lambda _n\mu _n\Vert Ap\Vert \\= & {} \left( 1-\lambda _n\mu _n\rho \right) \Vert x_n-p\Vert +\lambda _n\mu _n\rho \frac{\Vert Ap\Vert }{\rho }\\\le & {} \max \left\{ \Vert x_n-p\Vert ,\frac{\Vert Ap\Vert }{\rho }\right\} \le \ldots \le \max \left\{ \Vert x_1-p\Vert ,\frac{\Vert Ap\Vert }{\rho }\right\} , \end{aligned}$$

and the boundedness of our sequence immediately holds.

Recalling that X is reflexive and since it satisfies Opial condition because it has a weakly sequentially continuous duality mapping $J_\varphi$ (Lemma 2.1), our next step is: to show that $\omega _w(x_n)\subset Fix(W)$.

The claim will follow by Lemma 3.4 and by the asymptotic regularity of $(x_n)_{n\in {\mathbb {N}}}$. Computing:

$$\begin{aligned} \Vert x_{n+1}{-}x_n\Vert= & {} \Vert \lambda _nB_nx_n+(1-\lambda _n)x^W_n-\lambda _{n-1}B_{n-1}x_{n-1}-(1-\lambda _{n-1})x^W_{n-1}\Vert \ \\\le & {} \lambda _n\Vert B_nx_n-B_{n-1}x_{n-1}\Vert +|\lambda _n-\lambda _{n-1}|\Vert B_{n-1}x_{n-1}-x^W_{n-1}\Vert \\&+(1-\lambda _n)\Vert x^W_n-x^W_{n-1}\Vert \\\le & {} \lambda _n\Vert B_nx_n-B_{n}x_{n-1}\Vert +\lambda _n\Vert B_{n}x_{n-1}-B_{n-1}x_{n-1}\Vert \\&+|\lambda _n-\lambda _{n-1}|\Vert B_{n-1}x_{n-1}-x^W_{n-1}\Vert \\&+(1-\lambda _n)\Vert x_{n}-x_{n-1}\Vert \\\le & {} \lambda _n(1-\mu _n\rho )\Vert x_n-x_{n-1}\Vert +\lambda _n\Vert B_{n}x_{n-1}-B_{n-1}x_{n-1}\Vert \\&+|\lambda _n-\lambda _{n-1}|\Vert B_{n-1}x_{n-1}-x^W_{n-1}\Vert +(1-\lambda _n)\Vert x_n-x_{n-1}\Vert \\\le & {} (1-\lambda _n\mu _n\rho )\Vert x_n-x_{n-1}\Vert +\lambda _n|\mu _n-\mu _{n-1}|\Vert Ax_{n-1}\Vert \\&+|\lambda _n-\lambda _{n-1}|\Vert B_{n-1}x_{n-1}-x^W_{n-1}\Vert . \end{aligned}$$

The boundedness of $(x_n)_{n\in {\mathbb {N}}}$ guarantees that there exists a constant M such that

$$\begin{aligned} \Vert x_{n+1}-x_n\Vert\le & {} (1-\lambda _n\mu _n\rho )\Vert x_n-x_{n-1}\Vert +M\left[ \lambda _n|\mu _n-\mu _{n-1}|+|\lambda _n-\lambda _{n-1}|\right] \\= & {} (1-a_n)\Vert x_n-x_{n-1}\Vert +M\delta _n, \end{aligned}$$

where

$$\begin{aligned} a_n:=\lambda _n\mu _n\rho ;\qquad \delta _n=\left[ \lambda _n|\mu _n-\mu _{n-1}|+|\lambda _n-\lambda _{n-1}|\right] ; \end{aligned}$$

hence, asymptotic regularity for $(x_n)_{n\in {\mathbb {N}}}$ follows by Lemma 2.5. Moreover,

$$\begin{aligned} \Vert x_n-x^W_n\Vert\le & {} \Vert x_n-x_{n+1}\Vert +\Vert x_{n+1}-x^W_n\Vert \\\le & {} \Vert x_n-x_{n+1}\Vert +\lambda _n\Vert B_nx_n-x^W_n\Vert ; \end{aligned}$$

thus,

$$\begin{aligned} (1-\lambda _n)\Vert x_n-x^W_n\Vert \le \Vert x_n-x_{n+1}\Vert +\lambda _n\mu _n\Vert Ax^W_n\Vert . \end{aligned}$$

Since $(\lambda _n\mu _n)_{n\in {\mathbb {N}}}$ is a null sequence and by asymptotic regularity, $\Vert x_n-x^W_n\Vert \rightarrow 0$.

Observing that

$$\begin{aligned} d(x_n,Wx_n)\le \Vert x_n-x^W_n\Vert \rightarrow 0, \end{aligned}$$

as $n\rightarrow \infty$, by Lemma 3.4, the weak limits of $(x_n)_{n\in {\mathbb {N}}}$ are fixed points for W.

To prove the strong convergence, let $w\in Fix(W)$ the unique solution for (1.1). Such a (unique) solution exists by Proposition 3.5.

Since $J_{\varphi }$ is the sub-differential of $\Phi$, we have

$$\begin{aligned} \Phi (\Vert x_{n+1}-w\Vert )= & {} \Phi (\Vert \lambda _n(B_nx_n-w)+(1-\lambda _n)(x^W_n-w)\Vert ) \nonumber \\= & {} \Phi ( \Vert \lambda _n(B_nx_n{-}B_nw){+}\lambda _n(B_nw{-}w){+}(1{-}\lambda _n)(x^W_n{-}w)\Vert ) \nonumber \\\le & {} \Phi (\Vert \lambda _n(B_nx_n-B_nw)+(1-\lambda _n)x^W_n-w)\Vert ) \nonumber \\&-\lambda _n\mu _n\langle Aw, j_\varphi (x_{n+1}-w)\rangle \nonumber \\\le & {} \lambda _n(1-\mu _n\rho )\Phi (\Vert x_n-w\Vert )+(1-\lambda _n)\Phi (\Vert x^W_n-w\Vert ) \nonumber \\&-\lambda _n\mu _n\langle Aw, j_\varphi (x_{n+1}-w)\rangle \nonumber \\\le & {} \left[ 1-\lambda _n\mu _n\rho \right] \Phi (\Vert x_n-w\Vert )-\lambda _n\mu _n\langle Aw, j_\varphi (x_{n+1}-w)\rangle . \nonumber \\ \end{aligned}$$

(3.13)

Then

$$\begin{aligned} \Phi (\Vert x_{n+1}-w\Vert )\le & {} \left( 1-a_n\right) \Phi (\Vert x_n-w\Vert )+a_n\frac{\langle -Aw, j_\varphi (x_{n+1}-w)\rangle }{,}{\rho }, \end{aligned}$$

where $a_n=\lambda _n\mu _n\rho$ satisfies Lemma 2.5 (1). To apply Lemma 2.5, note that there exists a subsequence of $(x_n)_{n\in {\mathbb {N}}}$ for which

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle -Aw, j_\varphi (x_{n+1}-w)\rangle =\lim _{k\rightarrow \infty }\langle -Aw, j_\varphi (x_{n_k}-w)\rangle . \end{aligned}$$

Since $(x_n)_{n\in {\mathbb {N}}}$ is bounded and X is reflexive, there exists a subsequence $(x_{n_{k_j}})_{j\in {\mathbb {N}}}\subset (x_{n_k})_{k\in {\mathbb {N}}}$ weak convergence to p; moreover, $p\in Fix(W)$. By the weak sequential continuity of the duality map we, therefore, conclude that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle Aw,j_\varphi (w-x_{n+1})\rangle =\lim _{j\rightarrow \infty }\langle Aw,j_\varphi (w-x_{n_{k_j}})\rangle =\langle Aw,j_\varphi (w-p)\rangle \le 0 \end{aligned}$$

since w is the unique solution of (1.1). Lemma 2.5 gives that $\Vert x_{n}-w\Vert \rightarrow 0$, as $n\rightarrow \infty$, being $\Phi (\Vert x_n-w\Vert )\rightarrow 0$, and the thesis follows. $\square$

By means of Theorem 3.6, we obtain viscosity iteration and Halpern approach to minimisation problem

Corollary 3.7

Let X be a reflexive Banach space with duality mapping that is weakly sequentially continuous, $J_\varphi$.

Let $W:X\rightarrow K(X)$ a $*$-nonexpansive multivalued mapping such that Fix(W) is nonempty. Let $f:X\rightarrow X$ a $\eta$-contraction.

Let $(\lambda _n)_{n\in {\mathbb {N}}}\subset [0,a]\subset [0,1)$ such that

$$\begin{aligned} \displaystyle \lim _{n\rightarrow +\infty }\lambda _n= 0, \displaystyle \sum _{n\in {\mathbb {N}}}\lambda _n=\infty \hbox { and }\displaystyle \lim _{n\rightarrow \infty }\frac{|\lambda _n-\lambda _{n-1}|}{\lambda _n}=0. \end{aligned}$$

Then, for any choice $x_0$ as a starting point, the explicit process

$$\begin{aligned} x_{n+1}=\lambda _n f(x_n)+(1-\lambda _n)x^W_n \end{aligned}$$

strongly converges, as $n\rightarrow \infty$, to the unique solution of (VIP)

$$\begin{aligned} \langle (I-f)x^*,j(y-x^*)\rangle \ge 0, \qquad \forall y\in Fix(W). \end{aligned}$$

(3.14)

Corollary 3.8

Let X be a reflexive Banach space with duality mapping that is weakly sequentially continuous, $J_\varphi$.

Let $W:X\rightarrow K(X)$ a $*$-nonexpansive multivalued mapping such that Fix(W) is nonempty. Let $(\lambda _n)_{n\in {\mathbb {N}}}\subset [0,a]\subset [0,1)$ such that

$$\begin{aligned} \displaystyle \lim _{n\rightarrow +\infty }\lambda _n= 0, \displaystyle \sum _{n\in {\mathbb {N}}}\lambda _n=\infty \hbox { and }\displaystyle \lim _{n\rightarrow \infty }\frac{|\lambda _n-\lambda _{n-1}|}{\lambda _n}=0. \end{aligned}$$

Then, for any choice $x_0$ as a starting point, the explicit process

$$\begin{aligned} x_{n+1}=\lambda _n u+(1-\lambda _n)x^W_n \end{aligned}$$

strongly converges, as $n\rightarrow \infty$, to the unique solution of minimisation problem

$$\begin{aligned} \min _{x\in Fix(W)}\Vert x-u\Vert . \end{aligned}$$

(3.15)

Example

Let us apply Corollary 3.8 to W defined as in counter-example 3.3. Let $W:[0,1]\rightarrow K([0,1])$ defined as

$$\begin{aligned} Wx=\left\{ \begin{array}{l} \displaystyle \left[ x,x+\frac{1}{2}\right] , \qquad 0\le x< \frac{1}{2}\\ \\ \displaystyle \left[ x-\frac{1}{2},x\right] , \qquad \frac{1}{2}\le x\le 1.\\ \end{array}\right. \end{aligned}$$

We have already noted that $P_W=I$ and $Fix(W)={\mathbb {R}}$. For any $(\lambda _n)_{n\in {\mathbb {N}}}$ satisfying the assumption of Corollary 3.8, following construction (3.9) our iteration becomes

$$\begin{aligned} x_{n+1}=\lambda _n u+(1-\lambda _n)x_n. \end{aligned}$$

Then a simple realisation of Lemma 2.5 applied to the unknown sequence $(x_n-u)$ gives the convergence of $(x_n)_{n\in {\mathbb {N}}}$ to u that solves the minimization problem (3.15).

By a suitable modification of the idea that defines (3.11), we can define another iteration; the proof of the following is based on the idea of Theorem 3.6.

Theorem 3.9

Let X, W and A as in Theorem 3.6. Let $(\mu _n)_{n\in {\mathbb {N}}}\subset (0,1)$ and $(\lambda _n)_{n\in {\mathbb {N}}}\subset [0,a]\subset [0,1)$ such that

• $\mu _n\rightarrow 0$, as $n\rightarrow \infty$ and $\displaystyle \sum _{n\in {\mathbb {N}}}\mu _n=\infty$.

• $\displaystyle \lim _{n\rightarrow \infty }\frac{|\lambda _n-\lambda _{n-1}|}{\mu _n}=0$.

• $\displaystyle \lim _{n\rightarrow \infty }\frac{|\mu _n-\mu _{n-1}|}{\mu _n}=0$.

Then choosing $x_n^W$ as in the previous theorem, i.e. in such a way that (3.10) is satisfied, the explicit iteration

$$\begin{aligned} x_{n+1}=\lambda _n x_n+(1-\lambda _n)x^W_n-(1-\lambda _n)\mu _nAx^W_n \end{aligned}$$

(3.16)

strongly converges, as $n\rightarrow \infty$, to the unique solution of (VIP)

$$\begin{aligned} \langle Ax^*,j(y-x^*)\rangle \ge 0, \qquad \forall y\in Fix(W). \end{aligned}$$

(3.17)

Proof

Again define $B_n:=(I-\mu _nA)$ in such a way that our iteration becomes:

$$\begin{aligned} x_{n+1}=\lambda _n x_n+(1-\lambda _n)B_nx^W_n. \end{aligned}$$

Let $p\in Fix(W)$ be a given fixed point of W. Then

$$\begin{aligned} \Vert x_{n+1}-p\Vert\le & {} \lambda _n\Vert x_n-p\Vert +(1-\lambda _n)\Vert B_nx^W_n-B_np\Vert +(1-\lambda _n)\mu _n\Vert Ap\Vert \\\le & {} \lambda _n\Vert x_n-p\Vert +(1-\lambda _n)(1-\mu _n\rho )\Vert x_n^W-p\Vert +(1-\lambda _n)\mu _n\Vert Ap\Vert \\ \hbox {by }(3.1)\le & {} \lambda _n\Vert x_n-p\Vert +(1-\lambda _n)(1-\mu _n\rho )\Vert x_n-p\Vert +(1-\lambda _n)\mu _n\Vert Ap\Vert \\= & {} \left( 1-(1-\lambda _n)\mu _n\rho \right) \Vert x_n-p\Vert +(1-\lambda _n)\mu _n\rho \frac{\Vert Ap\Vert }{\rho }\\\le & {} \max \left\{ \Vert x_n-p\Vert ,\frac{\Vert Ap\Vert }{\rho }\right\} \le \ldots \le \max \left\{ \Vert x_1-p\Vert ,\frac{\Vert Ap\Vert }{\rho }\right\} , \end{aligned}$$

and the boundedness of our sequence immediately holds. To show that $\omega _w(x_n)\subset Fix(W)$ asymptotic regularity is needed; therefore, let us compute

$$\begin{aligned} \Vert x_{n+1}-x_n\Vert\le & {} \lambda _n\Vert x_n-x_{n-1}\Vert +|\lambda _n-\lambda _{n-1}|\Vert x_{n-1}-B_{n-1}x^W_{n-1}\Vert +\\&(1-\lambda _n)\Vert B_nx^W_{n}-B_{n-1}x^W_{n-1}\Vert \\\le & {} \lambda _n\Vert x_n-x_{n-1}\Vert +|\lambda _n-\lambda _{n-1}|\Vert x_{n-1}-B_{n-1}x^W_{n-1}\Vert +\\&(1-\lambda _n)\Vert B_nx^T_{n}-B_{n}x^W_{n-1}\Vert +\\&(1-\lambda _n)\Vert B_nx^W_{n-1}-B_{n-1}x^W_{n-1}\Vert \\\le & {} \lambda _n\Vert x_n-x_{n-1}\Vert +|\lambda _n-\lambda _{n-1}|\Vert x_{n-1}-B_{n-1}x^W_{n-1}\Vert +\\&(1-\lambda _n)(1-\mu _n\rho )\Vert x^W_{n}-x^W_{n-1}\Vert +\\&(1-\lambda _n)|\mu _n-\mu _{n-1}|\Vert Ax^W_{n-1}\Vert \\ \hbox {by }(3.10)\le & {} \lambda _n\Vert x_n-x_{n-1}\Vert +|\lambda _n-\lambda _{n-1}|\Vert x_{n-1}-B_{n-1}x^W_{n-1}\Vert +\\&(1-\lambda _n)(1-\mu _n\rho )\Vert x_{n}-x_{n-1}\Vert +\\&(1-\lambda _n)|\mu _n-\mu _{n-1}|\Vert Ax^W_{n-1}\Vert . \end{aligned}$$

The boundedness of $(x_n)_{n\in {\mathbb {N}}}$ guarantees that there exists a constant M such that

$$\begin{aligned} \Vert x_{n+1}-x_n\Vert\le & {} [\lambda _n+(1-\lambda _n)(1-\mu _n\rho )]\Vert x_n-x_{n-1}\\&+M\left[ |\lambda _n-\lambda _{n-1}|+|\mu _n-\mu _{n-1}|\right] \\= & {} [1-(1-\lambda _n)\mu _n\rho )]\Vert x_n-x_{n-1}\Vert \\&+M\left[ |\lambda _n-\lambda _{n-1}|+|\mu _n-\mu _{n-1}|\right] \\= & {} (1-a_n)\Vert x_n-x_{n-1}\Vert +M\delta _n, \end{aligned}$$

where $a_n:=(1-\lambda _n)\mu _n\rho$ and $\delta _n=\left[ |\lambda _n-\lambda _{n-1}|+|\mu _n-\mu _{n-1}|\right]$.

Applying Lemma 2.5, we obtain asymptotic regularity for $(x_n)_{n\in {\mathbb {N}}}$. Moreover,

$$\begin{aligned} \Vert x_n-x^W_n\Vert\le & {} \Vert x_n-x_{n+1}\Vert +\Vert x_{n+1}-x^W_n\Vert \\\le & {} \Vert x_n-x_{n+1}\Vert +\Vert \lambda _n(x_n-x^W_n)+(1-\lambda _n)\mu _nAx^W_n\Vert \\\le & {} \Vert x_n-x_{n+1}\Vert +\lambda _n\Vert x_n-x^W_n\Vert +(1-\lambda _n)\mu _n\Vert Ax^W_n\Vert ; \end{aligned}$$

thus,

$$\begin{aligned} (1-\lambda _n)\Vert x_n-x^W_n\Vert \le \Vert x_n-x_{n+1}\Vert +(1-\lambda _n)\mu _n\Vert Ax^W_n\Vert . \end{aligned}$$

Since $(\mu _n)_{n\in {\mathbb {N}}}$ is a null sequence and by asymptotic regularity, $\Vert x_n-x^W_n\Vert \rightarrow 0$. Observing that

$$\begin{aligned} d(x_n,Wx_n)\le \Vert x_n-x^W_n\Vert \rightarrow 0, \end{aligned}$$

as $n\rightarrow \infty$, by Lemma 3.4, the weak limits of $(x_n)_{n\in {\mathbb {N}}}$ are fixed points for W.

To conclude, let $w\in Fix(W)$ the unique solution for (1.1) that there exists by Proposition 3.5. Thus,

$$\begin{aligned} \Phi (\Vert x_{n+1}-w\Vert )= & {} \Phi (\Vert \lambda _n(x_n-w)+(1-\lambda _n)(B_nx^W_n-w)\Vert ) \\= & {} \Phi ( \Vert \lambda _n(x_n-w)+(1-\lambda _n)(B_nx^W_n-B_nw)\\&-(1-\lambda _n)\mu _nAw)\Vert ) \\\le & {} \Phi (\Vert \lambda _n(x_n-w)+(1-\lambda _n)(B_nx^W_n-B_nw)\Vert ) \\&-(1-\lambda _n)\mu _n\langle Aw, j_\varphi (x_{n+1}-w)\rangle \\\le & {} \lambda _n\Phi (\Vert x_n-w\Vert )+(1-\lambda _n)(1-\mu _n\rho )\Phi (\Vert x^W_n-w\Vert ) \\&-(1-\lambda _n)\mu _n\langle Aw, j_\varphi (x_{n+1}-w)\rangle \\\le & {} \left[ 1-(1-\lambda _n)\mu _n\rho \right] \Phi (\Vert x_n-w\Vert ) \\&-(1-\lambda _n)\mu _n\langle Aw, j_\varphi (x_{n+1}-w)\rangle , \end{aligned}$$

i.e.

$$\begin{aligned} \Phi (\Vert x_{n+1}-w\Vert )\le & {} \left( 1-a_n\right) \Phi (\Vert x_n-w\Vert )+a_n\frac{\langle -Aw, j_\varphi (x_{n+1}-w)\rangle }{\rho }, \end{aligned}$$

where $a_n=(1-\lambda _n)\mu _n\rho .$ To apply Lemma 2.5, following proof of Theorem 3.6, we get that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle Aw,j_\varphi (w-x_{n+1})\rangle =\lim _{j\rightarrow \infty }\langle Aw,j_\varphi (w-x_{n_{k_j}})\rangle =\langle Aw,j_\varphi (w-p)\rangle \le 0 \end{aligned}$$

since w is the unique solution of (1.1). Lemma 2.5 gives that $\Vert x_{n}-w\Vert \rightarrow 0$, as $n\rightarrow \infty$, being $\Phi (\Vert x_n-w\Vert )\rightarrow 0$, and the thesis follows. $\square$

Example

Take $X=l^p$, $p>1$, that is reflexive and it has a weakly sequentially continuous duality mapping $J_\varphi$ with gauge $\varphi (t)=t^{p-1}$.

Take $Wx:=\{x,2x\}$, that is $*$-nonexpansive by Lemma 3.2(i).

Take $A=I-u$, u be fixed.

Consider the iterative process (3.16) with $\lambda _n=\frac{1}{2}$ and $\mu _n=\frac{1}{\sqrt{n}}$. Then all the hypotheses of Theorem 3.9 are satisfied and the iteration process (3.16) becomes

$$\begin{aligned} x_{n+1}-u=\left( 1-\frac{1}{2\sqrt{n}}\right) (x_n-u). \end{aligned}$$

So Lemma 2.5 still yields that $x_n\rightarrow u$, solution of the variational inequality (1.5).

3.2 Open Questions

In what follows, we include some open problems that we think that they may be of interest:

• Does the conclusion of our Theorems hold under weaker conditions on underlying Banach spaces?

• Is it possible to replace the strict pseudocontractivity of A with Lipschitzianity as in the setting of Hilbert spaces?

3.3 Conclusions

We studied multivalued $*$-nonexpansive mappings in Banach spaces. The demiclosedness principle is established in reflexive Banach spaces satisfying Opial’s condition (Lemma 3.4). Thus, the demiclosedness principle holds also in reflexive Banach spaces with duality mapping that is weakly sequentially continuous since these satisfy Opial’s condition.

We proved the existence of a unique solution of our Variational Inequality Problem (VIP) on the set of fixed point of a $*$-nonexpansive mapping (Proposition 3.5) when X is a reflexive Banach space with duality mapping that is weakly sequentially continuous $J_\varphi$ for some gauge $\varphi$. We constructed two iterative schemes (3.9) and (3.16) that converge to the solution of the (VIP) in reflexive Banach spaces with duality mapping that is weakly sequentially continuous (Theorems 3.6 and Theorem 3.9, respectively).

Some examples and counter-examples are given.

Some open questions whose answer could be interesting are pointed out.