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# A monotone projection algorithm for fixed points of nonlinear operators

- Changqun Wu
^{1}Email author and - Lijuan Sun
^{2}

**2013**:318

https://doi.org/10.1186/1687-1812-2013-318

© Wu and Sun; licensee Springer. 2013

**Received: **23 August 2013

**Accepted: **7 November 2013

**Published: **25 November 2013

## Abstract

In this paper, a monotone projection algorithm is investigated for equilibrium and fixed point problems. Strong convergence theorems for common solutions of the two problems are established in the framework of reflexive Banach spaces.

**MSC:**47H09, 47J25, 90C33.

## Keywords

*ϕ*-nonexpansive mappinggeneralized asymptotically quasi-

*ϕ*-nonexpansive mappingbifunctionequilibrium problemfixed point

## 1 Introduction and preliminaries

*E*be a real Banach space with the dual ${E}^{\ast}$. Recall that the normalized duality mapping

*J*from

*E*to ${2}^{{E}^{\ast}}$ is defined by

*E*. Recall that

*E*is said to be smooth iff ${lim}_{t\to 0}\frac{\parallel x+ty\parallel -\parallel x\parallel}{t}$ exists for each $x,y\in {B}_{E}$. It is also said to be uniformly smooth iff the above limit is attained uniformly for $x,y\in {B}_{E}$.

*E*is said to be strictly convex iff $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in E$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. It is said to be uniformly convex iff ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{y}_{n}\parallel =0$ for any two sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in

*E*such that $\parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1$ and ${lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =1$. It is well known that

*E*is uniformly smooth if and only if ${E}^{\ast}$ is uniformly convex. In what follows, we use ⇀ and → to stand for weak and strong convergence, respectively. Recall that

*E*enjoys the Kadec-Klee property iff for any sequence $\{{x}_{n}\}\subset E$, and $x\in E$ with ${x}_{n}\rightharpoonup x$, and $\parallel {x}_{n}\parallel \to \parallel x\parallel $, then $\parallel {x}_{n}-x\parallel \to 0$ as $n\to \mathrm{\infty}$. It is well known that if

*E*is a uniformly convex Banach space, then

*E*enjoys the Kadec-Klee property. Let

*E*be a smooth Banach space. Let us consider the functional defined by

Recently, Alber [1] introduced a generalized projection operator ${\mathrm{\Pi}}_{C}$ in a Banach space *E* which is an analogue of the metric projection ${P}_{C}$ in Hilbert spaces. Recall that the generalized projection ${\mathrm{\Pi}}_{C}:E\to C$ is a map that assigns to an arbitrary point $x\in E$ the minimum point of the functional $\varphi (x,y)$, that is, ${\mathrm{\Pi}}_{C}x=\overline{x}$, where $\overline{x}$ is the solution to the minimization problem $\varphi (\overline{x},x)={min}_{y\in C}\varphi (y,x)$. Existence and uniqueness of the operator ${\mathrm{\Pi}}_{C}$ follows from the properties of the functional $\varphi (x,y)$ and strict monotonicity of the mapping *J*. If *E* is a reflexive, strictly convex and smooth Banach space, then $\varphi (x,y)=0$ if and only if $x=y$. In Hilbert spaces, ${\mathrm{\Pi}}_{C}={P}_{C}$. It is obvious from the definition of function *ϕ* that ${(\parallel x\parallel -\parallel y\parallel )}^{2}\le \varphi (x,y)\le {(\parallel y\parallel +\parallel x\parallel )}^{2}$, $\mathrm{\forall}x,y\in E$.

*F*be a bifunction from $C\times C$ to ℝ, where ℝ denotes the set of real numbers. Let $\phi :C\to \mathbb{R}$ be a real-valued function and $A:C\to {E}^{\ast}$ be a mapping. The so-called generalized mixed equilibrium problem is to find $p\in C$ such that

Next, we give some special cases:

which is called the mixed equilibrium problem.

which is called the mixed variational inequality of Browder type.

which is called the generalized equilibrium problem.

which is called the equilibrium problem.

- (A1)
$F(x,x)=0$, $\mathrm{\forall}x\in C$;

- (A2)
*F*is monotone,*i.e.*, $F(x,y)+F(y,x)\le 0$, $\mathrm{\forall}x,y\in C$; - (A3)$\underset{t\downarrow 0}{lim\hspace{0.17em}sup}F(tz+(1-t)x,y)\le F(x,y),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y,z\in C;$
- (A4)
for each $x\in C$, $y\mapsto F(x,y)$ is convex and weakly lower semi-continuous.

Iterative algorithms have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization; see [2–27] and the references therein. The computation of solutions of nonlinear operator equations (inequalities) is important in the study of many real world problems. Recently, the study of the convergence of various iterative algorithms for solving various nonlinear mathematical models forms the major part of numerical mathematics.

*C*be a nonempty subset of

*E*, and let $T:C\to C$ be a mapping. In this paper, we use $F(T)$ to stand for the fixed point set of

*T*. Recall that

*T*is said to be asymptotically regular on

*C*iff for any bounded subset

*K*of

*C*, ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\{\parallel {T}^{n+1}x-{T}^{n}x\parallel :x\in K\}=0$. Recall that

*T*is said to be closed iff for any sequence $\{{x}_{n}\}\subset C$ such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}={x}_{0}$ and ${lim}_{n\to \mathrm{\infty}}T{x}_{n}={y}_{0}$, then $T{x}_{0}={y}_{0}$. Recall that a point

*p*in

*C*is said to be an asymptotic fixed point of

*T*iff

*C*contains a sequence $\{{x}_{n}\}$ which converges weakly to

*p*such that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-T{x}_{n}\parallel =0$. The set of asymptotic fixed points of

*T*will be denoted by $\tilde{F}(T)$.

*T*is said to be relatively nonexpansive iff $\tilde{F}(T)=F(T)\ne \mathrm{\varnothing}$ and

*T*is said to be relatively asymptotically nonexpansive iff $\tilde{F}(T)=F(T)\ne \mathrm{\varnothing}$ and

where $\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$ is a sequence such that ${\mu}_{n}\to 0$ as $n\to \mathrm{\infty}$.

*T*is said to be quasi-

*ϕ*-nonexpansive iff $F(T)\ne \mathrm{\varnothing}$ and

*T*is said to be asymptotically quasi-

*ϕ*-nonexpansive iff there exists a sequence $\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$ with ${\mu}_{n}\to 0$ as $n\to \mathrm{\infty}$ such that

**Remark 1.1** The class of relatively asymptotically nonexpansive mappings, which is an extension of the class of relatively nonexpansive mappings, was first introduced in [28].

**Remark 1.2** The class of asymptotically quasi-*ϕ*-nonexpansive mappings, which is an extension of the class of quasi-*ϕ*-nonexpansive mappings, was considered in [29–31]. The class of quasi-*ϕ*-nonexpansive mappings and the class of asymptotically quasi-*ϕ*-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-*ϕ*-nonexpansive mappings and asymptotically quasi-*ϕ*-nonexpansive mappings do not require the restriction $F(T)=\tilde{F}(T)$.

*T*is said to be generalized asymptotically quasi-

*ϕ*-nonexpansive iff $F(T)\ne \mathrm{\varnothing}$, and there exist two nonnegative sequences $\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$ with ${\mu}_{n}\to 0$ and $\{{\xi}_{n}\}\subset [0,\mathrm{\infty})$ with ${\xi}_{n}\to 0$ as $n\to \mathrm{\infty}$ such that

**Remark 1.3** The class of generalized asymptotically quasi-*ϕ*-nonexpansive mappings [32] is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings in the framework of Banach spaces which was studied by Agarwal *et al.* [33].

In this paper, we consider a projection algorithm for a common solution of a family of generalized asymptotically quasi-*ϕ*-nonexpansive mappings and generalized mixed equilibrium problems. A strong convergence theorem is established in a Banach space. In order to prove our main results, we need the following lemmas.

**Lemma 1.4** [21]

*E*be a uniformly convex Banach space, and let $r>0$. Then there exists a strictly increasing, continuous and convex function $g:[0,2r]\to R$ such that $g(0)=0$ and

for all ${x}_{1},{x}_{2},\dots ,\in {B}_{r}=\{x\in E:\parallel x\parallel \le r\}$ and ${\alpha}_{1},{\alpha}_{2},\dots ,\in [0,1]$ such that ${\sum}_{i=1}^{\mathrm{\infty}}{\alpha}_{i}=1$.

**Lemma 1.5** [1]

*E*be a reflexive, strictly convex and smooth Banach space, let

*C*be a nonempty closed convex subset of

*E*and $x\in E$. Then

**Lemma 1.6** [1]

*C*be a nonempty closed convex subset of a smooth Banach space

*E*and $x\in E$. Then ${x}_{0}={\mathrm{\Pi}}_{C}x$ if and only if

**Lemma 1.7** [32]

Let *E* be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and let *C* be a nonempty closed and convex subset of *E*. Let $T:C\to C$ be a generalized asymptotically quasi-*ϕ*-nonexpansive mapping. Then $F(T)$ is closed and convex.

**Lemma 1.8** [34]

*C*be a closed convex subset of a smooth, strictly convex and reflexive Banach space

*E*. Let $A:C\to {E}^{\ast}$ be a continuous and monotone mapping, let $\phi :C\to \mathbb{R}$ be convex and lower semi-continuous, and let

*F*be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4). Let $r>0$ and $x\in E$. Then there exists $z\in C$ such that

- (1)
${T}_{r}$ is a single-valued firmly nonexpansive-type mapping,

*i.e.*, for all $x,y\in E$, $\u3008{T}_{r}x-{T}_{r}y,J{T}_{r}x-J{T}_{r}y\u3009\le \u3008{T}_{r}x-{T}_{r}y,Jx-Jy\u3009$; - (2)
$F({T}_{r})=\mathit{GMEP}(F,A,\phi )$ is closed and convex;

- (3)
${T}_{r}$ is quasi-

*ϕ*-nonexpansive; - (4)
$\varphi (q,{T}_{r}x)+\varphi ({T}_{r}x,x)\le \varphi (q,x)$, $\mathrm{\forall}q\in F({T}_{r})$.

## 2 Main results

**Theorem 2.1**

*Let*

*E*

*be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec*-

*Klee property*,

*and let*

*C*

*be a nonempty closed and convex subset of*

*E*.

*Let*Δ

*be an index set and*

*N*

*be an integer*.

*Let*${A}_{j}:C\to {E}^{\ast}$

*be a continuous and monotone mapping and*${\phi}_{j}:C\to \mathbb{R}$

*be a lower semi*-

*continuous and convex function*.

*Let*${F}_{j}$

*be a bifunction from*$C\times C$

*to*ℝ

*satisfying*(A1)-(A4)

*for every*$j\in \mathrm{\Delta}$.

*Let*${T}_{0}$

*be an identity mapping*,

*and let*${T}_{i}:C\to C$

*be a generalized asymptotically quasi*-

*ϕ*-

*nonexpansive mapping for every*$1\le i\le N$.

*Assume that*${T}_{i}$

*is closed asymptotically regular on*

*C*

*and*$\mathrm{\Psi}:={\bigcap}_{i=1}^{N}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}\mathit{GMEP}({F}_{j},{A}_{j},{\phi}_{j})$

*is nonempty and bounded*.

*Let*$\{{x}_{n}\}$

*be a sequence generated in the following manner*:

*where* $\{{\alpha}_{n,i}\}$ *is a real number sequence in* $(0,1)$ *for every* $i\le 1$, $\{{r}_{n,j}\}$ *is a real number sequence in* $[r,\mathrm{\infty})$, *where* *r* *is some positive real number*, *and* ${M}_{n}=sup\{\varphi (z,{x}_{n}):z\in \mathrm{\Psi}\}$. *Assume that* ${\sum}_{i=0}^{N}{\alpha}_{n,i}=1$ *and* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,i}>0$ *for every* $1\le i\le N$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{\mathrm{\Psi}}{x}_{0}$, *where* ${\mathrm{\Pi}}_{\mathrm{\Psi}}$ *is the generalized projection from* *E* *onto* Ψ.

*Proof* The proof is split into five steps.

Step 1. Show that the common solution set Ψ is convex and closed.

This step is clear in view of Lemma 1.7 and Lemma 1.8.

Step 2. Show that the set ${C}_{n}$ is convex and closed.

This proves that ${C}_{m+1,j}$ is convex. This proves that ${C}_{n}$ is closed and convex. This completes Step 2.

Step 3. Show that $\mathrm{\Psi}\subset {C}_{n}$.

*m*and for every $j\in \mathrm{\Delta}$. Then, for $\mathrm{\forall}z\in \mathrm{\Psi}\subset {C}_{m,j}$, we have

which proves that $z\in {C}_{m+1,j}$. This completes Step 3.

Step 4. Show that ${x}_{n}\to p$, where $p\in \mathrm{\Psi}$.

This gives that ${lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{0})=\varphi (p,{x}_{0})$. Hence, we have ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}\parallel =\parallel p\parallel $. Since the space *E* enjoys the Kadec-Klee property, we find that ${x}_{n}\to p$ as $n\to \mathrm{\infty}$.

*E*is reflexive and ${E}^{\ast}$ is also reflexive. We may assume that $J{u}_{n,j}\rightharpoonup {u}^{\ast ,j}\in {E}^{\ast}$. In view of the reflexivity of

*E*, we see that $J(E)={E}^{\ast}$. This shows that there exists ${u}^{j}\in E$ such that $J{u}^{j}={u}^{\ast ,j}$. It follows that $\varphi ({x}_{n+1},{u}_{n})={\parallel {x}_{n+1}\parallel}^{2}-2\u3008{x}_{n+1},J{u}_{n}\u3009+{\parallel J{u}_{n}\parallel}^{2}$. Taking ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}$ on the both sides of the equality above yields that

*E*enjoys the Kadec-Klee property, we obtain that ${u}_{n,j}\to p$ as $n\to \mathrm{\infty}$. Note that $\parallel {x}_{n}-{u}_{n,j}\parallel \le \parallel {x}_{n}-p\parallel +\parallel p-{u}_{n,j}\parallel $. This gives that

*J*is uniformly norm-to-norm continuous on any bounded sets, we have

*E*enjoys the Kadec-Klee property, we obtain that ${y}_{n}\to p$ as $n\to \mathrm{\infty}$. Since $\parallel {u}_{n,j}-{y}_{n}\parallel \le \parallel {u}_{n,j}-p\parallel +\parallel p-{y}_{n}\parallel $, we find that ${lim}_{n\to \mathrm{\infty}}\parallel {u}_{n,i}-{y}_{n}\parallel =0$. Since

*J*is uniformly norm-to-norm continuous on any bounded sets, we have ${lim}_{n\to \mathrm{\infty}}\parallel J{u}_{n,j}-J{y}_{n}\parallel =0$. From the assumption ${r}_{n,i}\ge r$, we see that ${lim}_{n\to \mathrm{\infty}}\frac{\parallel J{u}_{n,j}-J{y}_{n}\parallel}{{r}_{n,j}}=0$. Notice that

Taking the limit as $n\to \mathrm{\infty}$, we find that ${f}_{j}(y,p)\le 0$, $\mathrm{\forall}y\in C$. For $0<{t}_{j}<1$ and $y\in C$, define ${y}_{{t}_{j}}={t}_{j}y+(1-{t}_{j})p$. It follows that ${y}_{t,j}\in C$, which yields that ${f}_{j}({y}_{t,j},p)\le 0$. It follows from conditions (A1) and (A4) that $0={f}_{j}({y}_{t,j},{y}_{t,j})\le {t}_{j}{f}_{j}({y}_{t,j},y)+(1-{t}_{j}){f}_{j}({y}_{t,j},p)\le {t}_{j}{f}_{j}({y}_{t,j},y)$. This yields that ${f}_{j}({y}_{t,j},y)\ge 0$. Letting ${t}_{j}\downarrow 0$, we find from condition (A3) that ${f}_{j}(p,y)\ge 0$, $\mathrm{\forall}y\in C$. This implies that $p\in \mathit{EP}({f}_{j})=\mathit{GMEP}({F}_{j},{A}_{j},{\phi}_{j})$ for every $j\in \mathrm{\Delta}$.

*E*is uniformly smooth, we know that ${E}^{\ast}$ is uniformly convex. It follows from Lemma 1.4 that

*g*that

On the other hand, we have $\parallel J{T}_{i}^{n}{x}_{n}-Jp\parallel \le \parallel J{T}_{i}^{n}{x}_{n}-J{x}_{n}\parallel +\parallel J{x}_{n}-Jp\parallel $. Combining (2.5) with (2.6), one obtains that ${lim}_{n\to \mathrm{\infty}}\parallel J{T}_{i}^{n}{x}_{n}-Jp\parallel =0$. Since ${J}^{-1}:{E}^{\ast}\to E$ is demicontinuous, one sees that ${T}_{i}^{n}{x}_{n}\rightharpoonup p$. Notice that $|\parallel {T}_{i}^{n}{x}_{n}\parallel -\parallel p\parallel |\le \parallel J{T}_{i}^{n}{x}_{n}-Jp\parallel $. This yields that ${lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}^{n}{x}_{n}\parallel =\parallel p\parallel $. Since the space *E* enjoys the Kadec-Klee property, we obtain that ${lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}^{n}{x}_{n}-p\parallel =0$. Note that $\parallel {T}^{n+1}{x}_{n}-p\parallel \le \parallel {T}^{n+1}{x}_{n}-{T}^{n}{x}_{n}\parallel +\parallel {T}^{n}{x}_{n}-p\parallel $. Since *T* is asymptotically regular, we find that ${lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}^{n+1}{x}_{n}-p\parallel =0$. That is, ${T}_{i}{T}_{i}^{n}{x}_{n}-p\to 0$ as $n\to \mathrm{\infty}$. It follows from the closedness of ${T}_{i}$ that ${T}_{i}p=p$ for every $i\in \{1,2,\dots ,N\}$. This completes Step 4.

Step 5. Show that $p={\mathrm{\Pi}}_{\mathrm{\Psi}}{x}_{0}$.

From Lemma 1.6, we can immediately obtain that $p={\mathrm{\Pi}}_{\mathrm{\Psi}}{x}_{0}$. This completes the proof. □

**Remark 2.2** Theorem 2.1 mainly improves the corresponding results in Kim [20], Yang *et al.* [21], Hao [23], Qin *et al.* [31], Qin *et al.* [35].

**Remark 2.3** The framework of the space in Theorem 2.1 can be applicable to ${L}^{p}$, $p\ge 1$.

If $N=2$ and $\mathrm{\Delta}=\{1\}$, then Theorem 2.1 is reduced to the following.

**Corollary 2.4**

*Let*

*E*

*be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec*-

*Klee property*,

*and let*

*C*

*be a nonempty closed and convex subset of*

*E*.

*Let*

*F*

*be a bifunction from*$C\times C$

*to*ℝ

*satisfying*(A1)-(A4).

*Let*${T}_{i}:C\to C$

*be a generalized asymptotically quasi*-

*ϕ*-

*nonexpansive mapping for every*$i\in \{1,2\}$.

*Assume that each*${T}_{i}$

*is closed asymptotically regular on*

*C*

*and*$F({T}_{1})\cap F({T}_{2})\cap \mathit{EP}(F)$

*is nonempty and bounded*.

*Let*$\{{x}_{n}\}$

*be a sequence generated in the following manner*:

*where* $\{{\alpha}_{n,0}\}$, $\{{\alpha}_{n,1}\}$, *and* $\{{\alpha}_{n,2}\}$ *are real number sequences in* $(0,1)$, $\{{r}_{n}\}$ *is a real number sequence in* $[r,\mathrm{\infty})$, *where* *r* *is some positive real number*, *and* ${M}_{n}=sup\{\varphi (z,{x}_{n}):z\in F({T}_{1})\cap F({T}_{2})\cap EF(F)\}$. *Assume that* ${\sum}_{i=0}^{2}{\alpha}_{n,i}=1$ *and* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,i}>0$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F({T}_{1})\cap F({T}_{2})\cap \mathit{EP}(F)}{x}_{0}$, *where* ${\mathrm{\Pi}}_{F({T}_{1})\cap F({T}_{2})\bigcap EF(F)}$ *is the generalized projection from* *E* *onto* $F({T}_{1})\cap F({T}_{2})\cap \mathit{EP}(F)$.

**Remark 2.5** Corollary 2.4 mainly improves the corresponding results in Qin *et al.* [31]. To be more clear, the mapping is extended from quasi-*ϕ*-nonexpansive mappings to generalized asymptotically quasi-*ϕ*-nonexpansive mappings and the framework of spaces is extended from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space.

## Declarations

### Acknowledgements

The authors are grateful to the reviewers’ useful suggestions which improved the contents of the article.

## Authors’ Affiliations

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