# The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

Authors: Rabian Wangkeeree; Panatda Boonman
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We first introduce the iterative procedure to approximate a common element of the fixed-point
set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real
Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally,
we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading
mapping and asymptoticTJ mapping and the solution set of SMEP in a real Hilbert space.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, let C be a nonempty closed convex subset of H, and let T be a mapping of C into H, then T:C→H is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C. A mapping T:C→H is said to be quasinonexpansive if ∥Tx-y∥≤∥x-y∥ for all x∈C and y∈F(T):={x∈C:Tx=x}. It is well known that the set F(T) of fixed points of a quasi-nonexpansive mapping T is a closed and convex set [1]. A mapping T:C→C is said to be firmly nonexpansive [2] if
(1.1)‖Tx-Ty‖2≤〈x-y,Tx-Ty〉,
for all x,y∈C, and it is an important example of nonexpansive mappings in a Hilbert space.Letφ:C→ℝ be a real-valued function, and let F:C×C→ℝ be an equilibrium bifunction, that is, F(u,u)=0 for each u∈C. The mixed equilibrium problem is to find x∈C such that(1.2)F(x,y)+φ(y)-φ(x)≥0∀y∈C.
Denote the set of solution of (1.2) by MEP(F,φ). In particular, if φ=0, this problem reduces to the equilibrium problem, which is to find x∈C such that(1.3)F(x,y)≥0∀y∈C.
The set of solution of (1.3) is denoted by EP(F).The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, Min-Max problems, the Nash equilibrium problems in noncooperative games, and others; see, for example, Blum and Oettli [3] and Moudafi [4]. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.3).LetF1,F2:C×C→ℝ be two monotone bifunctions and μ>0 is constant. In 2009, Moudafi [5] introduced an alternating algorithm for approximating a solution of the system of equilibrium problems, finding (x′,y′)∈C×C such that(1.4)(SEP){F1(x′,z)+1μ〈y′-x′,x′-z〉≥0,∀z∈C,F2(y′,z)+1μ〈x′-y′,y′-z〉≥0,∀z∈C.
For such mappings F1 and F2 and two given positive constants λ,μ>0, Plubtieng and Sombut [6] considered the following system of mixed equilibrium problem, finding (x′,y′)∈C×C such that(1.5)(SMEP){F1(x′,z)+φ(z)-φ(x′)+1λ〈y′-x′,x′-z〉≥0,∀z∈C,F2(y′,z)+φ(z)-φ(y′)+1μ〈x′-y′,y′-z〉≥0,∀z∈C.
In particular, if λ=μ and φ≡0, then problem (SMEP) reduces to (SEP). Furthermore, Plubtieng and Sombut [6] introduced the following iterative procedure to approximate a common element of the fixed-point set of a quasi-nonexpansive mapping T and the solution set of (SMEP) in a Hilbert space H. Let {xn},{yn}, and {un} be given by
(1.6)x1∈Cchosenarbitrary,un∈C,F2(un,z)+φ(z)-φ(un)+1μ〈z-un,un-xn〉≥0,∀z∈C,yn∈C,F1(yn,z)+φ(z)-φ(yn)+1λ〈z-yn,yn-un〉≥0,∀z∈C,xn+1=αnxn+(1-αn)Tyn,∀n∈N,
where {αn}⊂[a,b] for some a,b∈(0,1) and satisfying appropriate conditions. The weak convergence theorems are obtained in a real Hilbert space.On the other hand, in 1953, Mann [7] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H:(1.7)xn+1=αnxn+(1-αn)Txn,∀n∈N,
where the initial point x0 is taken in C arbitrarily, and {αn} is a sequence in [0,1].For two nonexpansive mappingsT1,T2 of C into itself, Moudafi [4] studied weak convergence theorems in the following iterative process:(1.8)x0∈Cchosenarbitrary,xn+1=αnxn+(1-αn)(βnT1xn+(1-βn)T2xn),
for all n∈ℕ, where {αn} and {βn} are appropriate sequences in [0,1] and F(T1)∩F(T2)≠∅. Recently, Iemoto and Takahashi [8] also considered this iterative procedure for T1 is a nonexpansive mapping and T2:C→C is a nonspreading mapping. Very recently, Kim [9] studied the weak and strong convergence for the Moudafi’s iterative scheme (1.8) of two quasi-nonexpansive mappings.In this paper, inspired and motivated by Plubtieng and Sombut [6], Moudafi [4], Iemoto and Takahashi [8], and Kim [9], we first introduce the iterative procedure to approximate a common element of the common fixed point set of two quasi-nonexpansive mappings and the solution set of SMEP in a real Hilbert space. Next, we prove the weak convergence theorem for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed point of asymptotic nonspreading mapping and asymptotic TJ mapping and the solution set of SMEP in a real Hilbert space.


## 2. Preliminaries

Throughout this paper, letℕ be the set of positive integers, and let ℝ be the set of real numbers. Let H be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively, and let C be a closed convex subset of H. We denote the strong convergence and the weak convergence of {xn} to x∈H by xn→x and xn⇀x, respectively.From [10], for each x,y∈H and λ∈[0,1], we have(2.1)‖λx+(1-λ)y‖2=λ‖x‖2+(1-λ)‖y‖2-λ(1-λ)‖x-y‖2.
For every point x∈H, there exists a unique nearest point in C, denoted by PCx, such that(2.2)‖x-PCx‖≤‖x-y‖∀y∈C.PC is called the metric projection of H onto C. It is well know that PC is a nonexpansive mapping of H onto C and satisfies(2.3)‖PCx-PCy‖2≤〈PCx-PCy,x-y〉∀x,y∈H.
Moreover, PCx is characterized by the following properties: PCx∈C, (2.4)〈x-PCx,PCy-y〉≥0,‖x-y‖2≥‖x-PCx‖2+‖y-PCy‖2∀x∈H,y∈C.
Further, for all x∈H and y∈C,y=PCx if and only if 〈x-y,y-z〉≥0, for all z∈C.Lemma 2.1 (see [11]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let P be the metric projection of H onto C, and let {xn}n∈ℕ be in H. If
(2.5)‖xn+1-u‖≤‖xn-u‖,
for all u∈C and n∈ℕ. Then, {PCxn} converges strongly to an element of C.Theorem 2.2 (Opial’s theorem, [10]).
LetH be a real Hilbert space, and suppose that xn⇀x, then
(2.6)liminf⁡n→∞‖xn-x‖<liminf⁡n→∞‖xn-y‖,
for all y∈H with x≠y.All Hilbert space andlp(1<p<∞) satisfy Opial’s condition, while Lp with 1<p≠2<∞ do not.For solving the mixed equilibrium problem for an equilibrium bifunctionF:C×C→ℝ, let us assume that F satisfies the following conditions:(A1)
F(x,x)=0 for all x∈C,(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0 for all x,y∈C,(A3)
for eachy∈C,x↦F(x,y) is weakly upper semicontinuous,(A4)
for eachx∈C,y↦F(x,y) is convex and semicontinuous.The following lemma appears implicitly in [3, 12].Lemma 2.3 (see [3]).
LetC be a nonempty closed convex subset of H, and let F:C×C→ℝ be a bifunction satisfying (A1)–(A4). Let r>0 and x∈H, then there exists z∈C such that
(2.7)F(z,y)+1r〈y-z,z-x〉≥0∀y∈C.The following lemma was also given in [12].Lemma 2.4 (see [12]).
LetC be a nonempty closed closed convex subset of H and let F:C×C→ℝ be a bifunction satisfying (A1)–(A4), then, for any r>0 and x∈H, define a mapping Trx:H→C as follows:
(2.8)Tr(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C},
for allz∈H,r∈ℝ. Then the following hold:(i)
Tr is single valued,(ii)
Tr is firmly nonexpansive, that is,
(2.9)‖Trx-Try‖2≤〈Trx-Try,x-y〉,∀x,y∈H;(iii)
F(Tr)=EP(F),(iv)
EP(F) is closed and convex.We note that Lemma2.4 is equivalent to the following lemma.Lemma 2.5 (see [6]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C×C→ℝ be an equilibrium bifunction satisfying (A1)–(A4) and let φ:C→ℝ be a lower semicontinuous and convex functional. For each r>0 and x∈H, define a mapping
(2.10)Sr(x)={y∈C:F(y,z)+φ(z)-φ(y)+1r〈y-x,z-y〉≥0,∀z∈C},∀x∈H.
Then, the following results hold:(i)
for eachx∈C, Sr(x)≠∅,(ii)
Sr is single valued,(iii)
Sr is firmly nonexpansive, that is, for any x,y∈H,
(2.11)‖Sr(x)-Sr(y)‖2≤〈Sr(x)-Sr(y),x-y〉,(iv)
F(Sr)=MEP(F,φ),(v)
MEP(F,φ) is closed and convex.


## 3. Main Results

In this section, we prove the weak convergence for approximating a common element of the common fixed point set of two quasi-nonexpansive mappings and the solution set of the system of mixed equilibrium problems in a Hilbert space.To begin with, let us state and proof the following characterizations of the solution set of GMEP.Lemma 3.1.
LetC be a closed convex subset of a real Hilbert space H. Let F1 and F2 be two mappings from C×C→ℝ satisfying (A1)–(A4), and let S1,λ and S2,μ be defined as in Lemma 2.5 associated to F1 and F2, respectively. For given x′,y′∈C,(x′,y′) is a solution of problem (1.5) if and only if x′ is a fixed point of the mapping G:C→C defined by
(3.1)G(x)=S1,λ(S2,μx),∀x∈C,
where y′=S2,μx′.Proof.
For givenx′,y′∈C, we observe the following equivalency:
(3.2)(x′,y′)isasolutionofproblem(1.5)⟺{F1(x′,z)+φ(z)-φ(x′)+1λ〈y′-x′,x′-z〉≥0∀z∈C,F2(y′,z)+φ(z)-φ(y′)+1μ〈x′-y′,y′-z〉≥0∀z∈C,⟺x′=S1,λy′y′=S2,μx′,⟺x′=S1,λ(S2,μx′),⟺x′=Gx′.
This completes the proof.We note from Lemma3.1 that the mapping G is nonexpansive. Moreover, if C is a closed bounded convex subset of H, then the solution of problem (1.5) always exists. Throughout this paper, we denote the set of fixed points of G by Ω.Theorem 3.2.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F1 and F2 be two bifunctions from C×C→ℝ satisfying (A1)–(A4). Let r,λ>0 and S1,r and S2,λ be defined as in Lemma 2.5 associated to F1 and F2, respectively. Let Ti:C→C, i=1,2, be two quasi-nonexpansive mappings such that I-Ti are demiclosed at zero, that is, if {wn}⊂C, wn⇀w, and (I-Ti)wn→0, then w∈F(Ti), with F(T1)∩F(T2)∩Ω≠∅. Let the sequences {xn},{yn}, and {un}, be given by
(3.3)x1∈Cchosenarbitrary,un∈C,F2(un,z)+φ(z)-φ(un)+1λ〈z-un,un-xn〉≥0,∀z∈C,yn∈C,F1(yn,z)+φ(z)-φ(yn)+1r〈z-yn,yn-un〉≥0,∀z∈C,xn+1=anxn+(1-an)(bnT1yn+(1-bn)T2yn),∀n∈N,
where {an}, {bn}⊂[a,b] for some a,b∈(0,1), and satisfy
(3.4)liminf⁡n→∞an(1-an)>0,liminf⁡n→∞bn(1-bn)>0,
then xn⇀x¯:=limn→∞PF(T1)∩F(T2)∩Ωxn and (x¯,y¯) is a solution of problem (1.5), where y¯=S2,λx¯.Proof.
Letx*∈F(T1)∩F(T2)∩Ω, then x*=T1x*=T2x* and x*=S1,r(S2,λx*).
Puttingy*=S2,λx*,yn=S1,run, and un=S2,λxn, we have
(3.5)‖yn-x*‖=‖S1,run-S1,ry*‖≤‖un-y*‖=‖S2,λxn-S2,λx*‖≤‖xn-x*‖.
Next, we prove that
(3.6)lim⁡n→∞‖xn-x*‖exists.
Since T1 and T2 are quasi-nonexpansive, we obtain that
(3.7)‖bnT1yn+(1-bn)T2yn-x*‖2=‖bn(T1yn-x*)+(1-bn)(T2yn-x*)‖2=bn‖T1yn-x*‖2+(1-bn)‖T2yn-x*‖2-bn(1-bn)‖T1yn-T2yn‖2≤bn‖T1yn-x*‖2+(1-bn)‖T2yn-x*‖2≤bn‖yn-x*‖2+(1-bn)‖yn-x*‖2=‖yn-x*‖2≤‖xn-x*‖2,
which gives that
(3.8)‖xn+1-x*‖2=‖anxn+(1-an)(bnT1yn+(1-bn)T2yn)-x*‖2=‖an(xn-x*)+(1-an)(bnT1yn+(1-bn)T2yn-x*)‖2=an‖xn-x*‖2+(1-an)‖bnT1yn+(1-bn)T2yn-x*‖2-an(1-an)‖xn-(bnT1yn+(1-bn)T2yn)‖2≤an‖xn-x*‖2+(1-an)‖xn-x*‖2-an(1-an)‖xn-(bnT1yn+(1-bn)T2yn)‖2=‖xn-x*‖2-an(1-an)‖xn-(bnT1yn+(1-bn)T2yn)‖2≤‖xn-x*‖2.
Hence, {∥xn-x*∥} is a nonincreasing sequence, and hence, limn→∞∥xn-x*∥ exists. This implies that {xn},{yn},{un},{T1yn}, and {T2yn} are bounded. From (3.8), we have
(3.9)an(1-an)‖xn-(bnT1yn+(1-bn)T2yn)‖2≤‖xn-x*‖2-‖xn+1-x*‖2.
Since liminfn→∞an(1-an)>0, this implies that
(3.10)lim⁡n→∞‖xn-(bnT1yn+(1-bn)T2yn)‖=0.
Furthermore, since 0<a≤an≤b<1, we have
(3.11)‖xn+1-xn‖=‖anxn+(1-an)(bnT1yn+(1-bn)T2yn)-xn‖=‖(1-an)(bnT1yn+(1-bn)T2yn)-(1-an)xn‖=(1-an)‖bnT1yn+(1-bn)T2yn-xn‖≤(1-a)‖bnT1yn+(1-bn)T2yn-xn‖.
From (3.10), we conclude that
(3.12)lim⁡n→∞‖xn+1-xn‖=0.
From (3.7), we have
(3.13)bn(1-bn)‖T1yn-T2yn‖2=bn‖T1yn-x*‖2+(1-bn)‖T2yn-x*‖2-‖bnT1yn+(1-bn)T2yn-x*‖2≤bn‖yn-x*‖2+(1-bn)‖yn-x*‖2-‖bnT1yn+(1-bn)T2yn-x*‖2=‖yn-x*‖2-‖bnT1yn+(1-bn)T2yn-x*‖2≤‖xn-x*‖2-‖bnT1yn+(1-bn)T2yn-x*‖2=(‖xn-x*‖-‖bnT1yn+(1-bn)T2yn-x*‖)×(‖xn-x*‖+‖bnT1yn+(1-bn)T2yn-x*‖)≤M(‖xn-x*‖-‖bnT1yn+(1-bn)T2yn-x*‖)≤M(‖xn-(bnT1yn+(1-bn)T2yn)‖),
where M is a constant satisfying M≥supn≥1[∥xn-x*∥+∥bnT1yn+(1-bn)T2yn-x*∥]. Again from (3.10), we conclude that
(3.14)lim⁡n→∞bn(1-bn)‖T1yn-T2yn‖=0.
Using liminfn→∞bn(1-bn)>0, we have
(3.15)lim⁡n→∞‖T1yn-T2yn‖=0.
Now, we prove that
(3.16)lim⁡n→∞‖xn+1-T1yn‖=0,lim⁡n→∞‖xn+1-T2yn‖=0.
We observe that
(3.17)‖xn+1-T1yn‖=‖anxn+(1-an)(bnT1yn+(1-bn)T2yn)-T1yn‖=‖anxn+(1-an)(bnT1yn+(1-bn)T2yn)-(an+(1-an))T1yn)‖=‖an(xn-T1yn)+(1-an)(bnT1yn+(1-bn)T2yn-T1yn)‖=‖an(xn-T1yn)+(1-an)(bnT1yn+T2yn-bnT2yn-T1yn)‖=‖an(xn-T1yn)+(1-an)((1-bn)T2yn-(1-bn)T1yn)‖=‖an(xn-T1yn)+(1-an)(1-bn)(T2yn-T1yn)‖≤an‖xn-T1yn‖+(1-an)(1-bn)‖T2yn-T1yn‖=an‖xn-xn+1+xn+1-T1yn‖+(1-an)(1-bn)‖T2yn-T1yn‖≤an‖xn-xn+1‖+an‖xn+1-T1yn‖+(1-an)(1-bn)‖T2yn-T1yn‖,
which gives that
(3.18)(1-an)‖xn+1-T1yn‖≤an‖xn-xn+1‖+(1-an)(1-bn)‖T2yn-T1yn‖.
Since 0<a≤an≤b<1, we have
(3.19)an(1-an)‖xn+1-T1yn‖≤(1-an)‖xn+1-T1yn‖≤an‖xn-xn+1‖+(1-an)(1-bn)‖T2yn-T1yn‖.
Using (3.12) and (3.15), we conclude that
(3.20)lim⁡n→∞an(1-an)‖xn+1-T1yn‖=0,
which gives that
(3.21)lim⁡n→∞‖xn+1-T1yn‖=0,
since liminfn→∞an(1-an)>0. Similarly, we have
(3.22)‖xn+1-T2yn‖=‖anxn+(1-an)(bnT1yn+(1-bn)T2yn)-T2yn‖=‖an(xn-T2yn)+(1-an)(bnT1yn+(1-bn)T2yn-T2yn)‖=‖an(xn-T2yn)+(1-an)bn(T1yn-T2yn)‖≤an‖xn-T2yn‖+(1-an)bn‖T1yn-T2yn‖=an‖xn-xn+1+xn+1-T2yn‖+(1-an)bn‖T2yn-T1yn‖≤an‖xn-xn+1‖+an‖xn+1-T2yn‖+(1-an)bn‖T2yn-T1yn‖,
which implies that
(3.23)(1-an)‖xn+1-T2yn‖≤an‖xn-xn+1‖+(1-an)bn‖T2yn-T1yn‖.
Thus, we have
(3.24)an(1-an)‖xn+1-T2yn‖≤(1-an)‖xn+1-T2yn‖≤an‖xn-xn+1‖+(1-an)bn‖T2yn-T1yn‖≤b‖xn-xn+1‖+(1-an)‖T2yn-T1yn‖≤b‖xn-xn+1‖+(1-a)‖T2yn-T1yn‖.
Hence, limn→∞an(1-an)∥xn+1-T2yn∥=0. Since liminfn→∞an(1-an)>0, we have
(3.25)lim⁡n→∞‖xn+1-T2yn‖=0.
Next, we prove that
(3.26)lim⁡n→∞‖yn-T1yn‖=0,lim⁡n→∞‖yn-T2yn‖=0.
Since S1,r and S2,λ are firmly nonexpansive, it follows that
(3.27)‖un-y*‖2=‖S2,λxn-S2,λx*‖2≤〈S2,λxn-S2,λx*,xn-x*〉=〈un-y*,xn-x*〉,
which gives that
(3.28)‖(un-y*)-(xn-x*)‖2=‖un-y*‖2-2〈un-y*,xn-x*〉+‖xn-x*‖2≤‖un-y*‖2-2‖un-y*‖2+‖xn-x*‖2=-‖un-y*‖2+‖xn-x*‖2.
This implies that
(3.29)‖un-y*‖2≤‖xn-x*‖2-‖(un-y*)-(xn-x*)‖2.
By the convexity of ∥·∥2, we have
(3.30)‖xn+1-x*‖2=‖anxn+(1-an)(bnT1yn+(1-bn)T2yn)-x*‖2=‖an(xn-x*)+(1-an)(bnT1yn+(1-bn)T2yn-x*)‖2≤an‖xn-x*‖2+(1-an)‖bnT1yn+(1-bn)T2yn-x*‖2≤an‖xn-x*‖2+(1-an)‖yn-x*‖2≤an‖xn-x*‖2+(1-an)‖un-y*‖2≤an‖xn-x*‖2+(1-an)(‖xn-x*‖2-‖(un-y*)-(xn-x*)‖2)=‖xn-x*‖2-(1-an)‖(un-y*)-(xn-x*)‖2.
Thus,
(3.31)(1-an)‖(un-y*)-(xn-x*)‖2≤‖xn-x*‖2-‖xn+1-x*‖2.
Since 0<a≤an≤b<1, we have
(3.32)(1-b)‖(un-y*)-(xn-x*)‖2≤(1-an)‖(un-y*)-(xn-x*)‖2≤‖xn-x*‖2-‖xn+1-x*‖2.
Since limn→∞∥xn-x*∥ exists, we have
(3.33)lim⁡n→∞‖un-xn+x*-y*‖=0.
Similarly, we have
(3.34)‖yn-x*‖2=‖S1,rnun-S1,rny*‖2≤〈S1,rnun-S1,rny*,un-y*〉=〈yn-x*,un-x*〉,
which gives that
(3.35)‖(yn-x*)-(un-y*)‖2=‖yn-x*‖2-2〈yn-x*,un-x*〉+‖un-y*‖2≤‖yn-x*‖2-2‖yn-x*‖2+‖un-y*‖2=-‖yn-x*‖2+‖un-y*‖2.
This implies that
(3.36)‖yn-x*‖2≤‖un-y*‖2-‖yn-un-x*+y*‖2.
By the convexity of ∥·∥2, we have
(3.37)‖xn+1-x*‖2=‖anxn+(1-an)(bnT1yn+(1-bn)T2yn)-x*‖2≤an‖xn-x*‖2+(1-an)‖yn-x*‖2≤an‖xn-x*‖2+(1-an)(‖un-y*‖2-‖yn-un-x*+y*‖2)≤an‖xn-x*‖2+(1-an)‖xn-x*‖2-(1-an)‖yn-un-x*+y*‖2=‖xn-x*‖2-(1-an)‖(yn-x*)-(un-y*)‖2.
Thus,
(3.38)(1-an)‖yn-un-x*+y*‖2≤‖xn-x*‖2-‖xn+1-x*‖2.
Since 0<a≤an≤b<1, we have
(3.39)(1-b)‖yn-un-x*+y*‖2≤(1-an)‖yn-un-x*+y*‖2≤‖xn-x*‖2-‖xn+1-x*‖2.
Since limn→∞∥xn-x*∥ exists, we have
(3.40)lim⁡n→∞‖yn-un-x*+y*‖=0.
Hence,
(3.41)‖(bnT1yn+(1-bn)T2yn)-yn‖=‖(bnT1yn+(1-bn)T2yn)-xn+xn-un+un-y*+y*-x*+x*-yn‖≤‖(bnT1yn+(1-bn)T2yn)-xn‖+‖xn-un-x*+y*‖+‖un-yn+x*-y*‖.
It follow from (3.10), (3.33), and (3.40) that
(3.42)lim⁡n→∞‖(bnT1yn+(1-bn)T2yn)-yn‖=0,
from which it follows that
(3.43)‖yn-xn‖=‖yn-(bnT1yn+(1-bn)T2yn)+(bnT1yn+(1-bn)T2yn)-xn‖≤‖yn-(bnT1yn+(1-bn)T2yn)‖+‖(bnT1yn+(1-bn)T2yn)-xn‖⟶0asn⟶∞,
that is,
(3.44)lim⁡n→∞‖yn-xn‖=0.
Thus,
(3.45)‖yn-T1yn‖≤‖yn-xn‖+‖xn-xn+1‖+‖xn+1-T1yn‖⟶0.
Similarly, we have ∥yn-T2yn∥→0. Since {yn} is bounded sequence, there exists a subsequence {yni} of {yn} such that yni⇀x¯ as i→∞. Since T1 and T2 are demiclosed at 0, we conclude that x¯∈F(T1)∩F(T2). Let G be a mapping which is defined as in Lemma 3.1. Thus, we have
(3.46)‖yn-G(yn)‖=‖S1,rS2,λxn-G(yn)‖=‖G(xn)-G(yn)‖≤‖xn-yn‖,
and hence,
(3.47)‖xn-G(xn)‖=‖xn-yn+yn-G(yn)+G(yn)-G(xn)‖≤‖xn-yn‖+‖yn-G(yn)‖+‖G(yn)-G(xn)‖≤‖xn-yn‖+‖xn-yn‖+‖yn-xn‖=3‖xn-yn‖⟶0.
This together with xni⇀x¯ implies that x¯∈F(G):=Ω, if {ynj} is another subsequence of {yn} such that ynj⇀x̂ as i→∞. Since T1 and T1 are demiclosed at 0, we conclude that x̂∈F(T1)∩F(T2)∩Ω. From xni⇀x¯ and xnj⇀x̂, we will show that x¯=x̂. Assume that x¯≠x̂. Since limn→∞∥xn-x*∥ exists for all x*∈F(T1)∩F(T2)∩Ω, by Opial’s Theorem 2.2, we have
(3.48)lim⁡n→∞‖xn-x¯‖=liminf⁡i→∞‖xni-x¯‖<liminf⁡i→∞‖xni-x̂‖=lim⁡n→∞‖xn-x̂‖=liminf⁡j→∞‖xnj-x̂‖<liminf⁡j→∞‖xnj-x¯‖=lim⁡n→∞‖xn-x¯‖.
This is a contradiction. Thus, we have x¯=x̂. This implies that yn⇀x¯∈F(T1)∩F(T2)∩Ω. Since ∥xn-yn∥→0, we have xn⇀x¯. Put zn=PF(T1)∩F(T2)∩Ωxn. Finally, we show that x¯=limn→∞zn. Now from (2.4) and x¯∈F(T1)∩F(T2)∩Ω, we have
(3.49)〈x¯-zn,zn-xn〉≥0.
Since {∥xn-x*∥} is nonnegative and nonincreasing for all x*∈F(T1)∩F(T2)∩Ω, it follows by Lemma 2.1 that {zn} converges strongly to some x̂∈F(T1)∩F(T2)∩Ω. By (3.49), we have
(3.50)〈x¯-x̂,x̂-x¯〉≥0.
Therefore, x¯=x̂.SettingT:=T1=T2 in Theorem 3.2, we have the following result.Corollary 3.3 (see [6]).
LetC be a closed convex subset of a real Hilbert space H. Let F1 and F2 be two bifunctions from C×C→ℝ satisfying (A1)–(A4). Let λ,μ>0, and let S1,λ and S2,μ be defined as in Lemma 2.5 associated to F1 and F2, respectively. Let T:C→C be a quasi-nonexpansive mapping such that I-T is demiclosed at zero and F(T)∩Ω≠∅. Suppose that x0=x∈C and {xn},{yn},and{zn} are given by
(3.51)zn∈C,F2(zn,z)+φ(z)-φ(zn)+1μ〈z-zn,zn-xn〉≥0,∀z∈C,yn∈C,F1(yn,z)+φ(z)-φ(yn)+1λ〈z-yn,yn-zn〉≥0,∀z∈C,xn+1=anxn+(1-an)Tyn,
for all n∈ℕ, where {an}⊂[a,b] for some a,b∈(0,1), and satisfy liminfn→∞an(1-an)>0, then {xn} converges weakly to x¯=limn→∞PF(T)∩Ωxn and (x¯,y¯) is a solution of problem (1.5), where y¯=S2,μx¯.SettingF1=F2≡0, φ≡0 in Theorem 3.2, we have the following result.Corollary 3.4 (see [9]).
LetH be a Hilbert space, let C be a nonempty, closed, and convex subset of H, and let T1,T2 be two quasi-nonexpansive mappings of C into itself such that I-T1, I-T2 are demiclosed at zero with F(T1)∩F(T2)≠∅. For any x1 in C, let {xn} be defined by
(3.52)xn+1=(1-an)xn+an(bnT1xn+(1-bn)T2xn),
where {an} and {bn} are chosen so that
(3.53)liminf⁡n→∞an(1-an)>0liminf⁡n→∞bn(1-bn)>0,
then xn⇀p∈F(T1)∩F(T2).


## 4. Applications

In this section, we apply our results to approximate a common element of the set of common fixed points of an asymptotic nonspreading mapping and an asymptoticTJ mapping and the solution set of SMEP in a real Hilbert space. We recall the following definitions. A mapping T:C→C is called nonspreading [13] if(4.1)2‖Tx-Ty‖2≤‖Tx-y‖2+‖Ty-x‖2∀x,y∈C.
Furthermore, Takahashi and Yao [14] also introduced two nonlinear mappings in Hilbert spaces. A mapping T:C→C is called a TJ-1 mapping [14] if(4.2)2‖Tx-Ty‖2=‖x-y‖2+‖Tx-y‖2,
for all x,y∈C. A mapping T:C→C is called a TJ-2 [14] mapping if(4.3)3‖Tx-Ty‖2=2‖Tx-y‖2+‖Ty-x‖2,
for all x,y∈C. For these two nonlinear mappings, TJ-1 and TJ-2 mappings, Takahashi and Yao [14] studied the existence results of fixed points in Hilbert spaces. Very recently, Lin et al. [15] introduced the following definitions of new mappings.Definition 4.1.
LetC be a nonempty closed convex subset of a Hilbert space H. We say that T:C→C is an asymptotic nonspreading mapping if there exist two functions α:C→[0,2) and β:C→[0,k],k<2, such that(A1)
2∥Tx-Ty∥2≤α(x)∥Tx-y∥2+β(x)∥Ty-x∥2 for all x,y∈C,(A2)
0<α(x)+β(x)≤2 for all x∈C.Remark 4.2.
The class of asymptotic nonspreading mappings contains the class of nonspreading mappings and the class ofTJ-2 mappings in a Hilbert space. Indeed, in Definition 4.1, we know that(i)
ifα(x)=β(x)=1 for all x∈C, then T is a nonspreading mapping,(ii)
ifα(x)=4/3 and β(x)=2/3 for all x∈C, then T is a TJ-2 mapping.Definition 4.3.
LetC be a nonempty closed convex subset of a Hilbert space H. We say T:C→C is an asymptotic TJ mapping if there exists two functions α:C→[0,2] and β:C→[0,k],k<2, such that(B1)
2∥Tx-Ty∥2≤α(x)∥x-y∥2+β(x)∥Tx-y∥2 for all x,y∈C,(B2)
α(x)+β(x)≤2 for all x∈C.Remark 4.4.
The class of asymptoticTJ mappings contains the class of TJ-1 mappings and the class of nonexpansive mappings in a Hilbert space. Indeed, in Definition 4.3, we know that(i)
ifα(x)=2 and β(x)=0 for each x∈C, then T is a nonexpansive mapping,(ii)
ifα(x)=β(x)=1 for each x∈C, then T is a TJ-1 mapping.It is well known that the setF(T) of fixed points of a quasi-nonexpansive mapping T is a closed and convex set [1]. Hence, if T:C→C is an asymptotic nonspreading mapping (resp., asymptotic TJ mapping) with F(T)≠∅, then T is a quasi-nonexpansive mapping, and this implies that F(T) is a nonempty closed convex subset of C.Theorem 4.5 (see [15]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let T:C→C be an asymptotic nonspreading mapping, then I-T is demiclosed at 0.Theorem 4.6 (see [15]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let T:C→C be an asymptotic TJ mapping, then I-T is demiclosed at 0.Applying the above results, we have the following theorem.Theorem 4.7.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F1 and F2 be two bifunctions from C×C→ℝ satisfying (A1)–(A4). Let r,λ>0 and S1,r and S2,λ be defined as in Lemma 2.5 associated to F1 and F2, respectively. Let Ti:C→C, i=1,2, be any one of asymptotic nonspreading mapping and asymptotic TJ mapping such that F(T1)∩F(T2)∩Ω≠∅. Let {xn},{yn}, and {un} be given by
(4.4)x1∈Cchosenarbitrary,un∈C,F2(un,z)+φ(z)-φ(un)+1λ〈z-un,un-xn〉≥0,∀z∈C,yn∈C,2F1(yn,z)+φ(z)-φ(yn)+1r〈z-yn,yn-un〉≥0,∀z∈C,xn+1=anxn+(1-an)(bnT1yn+(1-bn)T2yn),∀n∈N,
where {an},{bn}⊂[a,b] for some a,b∈(0,1) and satisfying
(4.5)liminf⁡n→∞an(1-an)>0,liminf⁡n→∞bn(1-bn)>0,
then xn⇀x¯:=limn→∞PF(T1)∩F(T2)∩Ωxn, and (x¯,y¯) is a solution of problem (1.5), where y¯=S2,λnx¯.SettingF1=F2:=F and φ≡0 in the above theorem, we have the following result.Corollary 4.8.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F be the bifunctions from C×C→ℝ satisfying (A1)–(A4). Let Ti:C→C, i=1,2, be any one of asymptotic nonspreading mapping and asymptotic TJ mapping such that ℱ:=F(T1)∩F(T2)∩EP(F)≠∅. For given u∈C and r>0, let the sequences {xn} and {un} be defined by
(4.6)x1∈Cchosenarbitrary,un∈C,F(un,y)+1r〈y-un,un-xn〉≥0∀y∈C,xn+1=anxn+(1-an)(bnT1un+(1-bn)T2un)∀n∈N,
where {an}, {bn} are two sequences in (0,1) satisfying
(4.7)liminf⁡n→∞an(1-an)>0,liminf⁡n→∞bn(1-bn)>0,
then xn⇀w for some w∈ℱ.SettingF1=F2≡0 and φ≡0 in Theorem 4.7, we have the following result.Corollary 4.9 (see [15]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let Ti:C→C, i=1,2, be any one of asymptotic nonspreading mapping and asymptotic TJ mapping. Let ℱ:=F(T1)∩F(T2)≠∅. Let {an} and {bn} be two sequences in (0,1). Let {xn} be defined by
(4.8)x1∈Cchosenarbitrary,xn+1=anxn+(1-an)(bnT1xn+(1-bn)T2xn).
Assume that liminfn→∞an(1-an)>0 and liminfn→∞bn(1-bn)>0, then xn⇀w for some w∈ℱ.



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*Source: 282094-2012-05-22.xml*