# A New Iterative Algorithm for the Set of Fixed-Point Problems of Nonexpansive Mappings and the Set of Equilibrium Problem and Variational Inequality Problem

Authors: Atid Kangtunyakarn
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We introduce a new iterative scheme and a new mapping generated by infinite family of nonexpansive mappings and infinite real number. By using both of these ideas, we obtain strong convergence theorem for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of fixed-point problems of infinite family of nonexpansive mappings. Moreover, we apply our main result to obtain strong convergence theorems for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of common fixed point of pseudocontractive mappings.


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## Body

## 1. Introduction

LetH be a real Hilbert space and let C be a nonempty closed convex subset of H. Let A:C→H be a nonlinear mapping and let F:C×C→ℝ be a bifunction. A mapping T of H into itself is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥forallx,y∈H. We denote by F(T) the set of fixed points of T (i.e., F(T)={x∈H:Tx=x}). Goebel and Kirk [1] showed that F(T) is always closed convex, and also nonempty provided T has a bounded trajectory.A bounded linear operatorA on H is called strongly positive with coefficient γ¯ if there is a constant γ¯>0 with the property(1.1)〈Ax,x〉≥γ¯‖x‖2.The equilibrium problem forF is to find x∈C, such that(1.2)F(x,y)≥0,∀y∈C.
The set of solutions of (1.2) is denoted by EP(F). Many problems in physics, optimization, and economics are seeking some elements of EP(F), see [2, 3]. Several iterative methods have been proposed to solve the equilibrium problem, see, for instance, [2–4]. In 2005, Combettes and Hirstoaga [3] introduced an iterative scheme for finding the best approximation to the initial data when EP(F) is nonempty and proved a strong convergence theorem.The variational inequality problem is to find a pointu∈C, such that(1.3)〈v-u,Au〉≥0,∀v∈C.
The set of solutions of the variational inequality is denoted by VI(C,A). Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games reduce to find element of (1.2) and (1.3).A mappingA of C into H is called inverse-strongly monotone, see [5], if there exists a positive real number α, such that(1.4)〈x-y,Ax-Ay〉≥α‖Ax-Ay‖2
for all x,y∈C.The problem of finding a common fixed point of a family of nonexpansive mappings has been studied by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed-point sets of a family of nonexpansive mapping (see [6, 7]).The problem of finding a common element ofEP(F) and the set of all common fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and importance. Many iterative methods are purposed for finding a common element of the solutions of the equilibrium problem and fixed-point problem of nonexpansive mappings, see [8–10].In 2007, S. Takahashi and W. Takahashi [10] introduced a general iterative method for finding a common element of EP(F,A) and F(T). They defined {xn} in the following way:(1.5)u,x1∈C,arbitrarily;F(zn,y)+1λn〈y-zn,zn-xn〉≥0,∀y∈C,xn+1=βnf(xn)+(1-βn)Szn,∀n∈N,
where {βn}⊂[0,1], and proved strong convergence of the scheme (1.5) to z∈F(T)⋂EP(F), where z=PF(S)⋂EP(F)f(z) in the framework of a Hilbert space, under some suitable conditions on {βn}, {λn} and bifunction F.In this paper, by motive of (1.5), we prove strong convergence theorem for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of fixed-point problems by using a new mapping generated by infinite family of nonexpansive mapping and infinite real number. Moreover, we apply our main result to obtain strong convergence theorems for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of common fixed point of pseudocontractive mappings.


## 2. Preliminaries

In this section, we collect and give some useful lemmas that will be used for our main result in the next section.LetC be closed convex subset of a real Hilbert space H, and let PC be the metric projection of H onto C, that is, for x∈H, PCx satisfies the property(2.1)∥x-PCx∥=min⁡y∈C∥x-y∥.
The following characterizes the projection PC.Lemma 2.1 (see [11]).
Givenx∈H and y∈C, then PCx=y if and only if there holds the inequality
(2.2)〈x-y,y-z〉≥0,∀z∈C.Lemma 2.2 (see [12]).
LetE be a uniformly convex Banach space, let C be a nonempty closed convex subset of E, and let S:C→C be a nonexpansive mapping, then I-S is demiclosed at zero.Lemma 2.3 (see [13]).
Let{sn} be a sequence of nonnegative real numbers satisfying
(2.3)sn+1=(1-αn)sn+δn,∀n≥0,
where {αn} is a sequence in (0,1) and {δn} is a sequence, such that (1)
∑n=1∞αn=∞,(2)
limsupn→∞δn/αn≤0or∑n=1∞|δn|<∞ then limn→∞sn=0.For solving the equilibrium problem for a bifunctionF:C×C→ℝ, let us assume that F satisfies the following conditions: (A1)
F(x,x)=0,forallx∈C,(A2)
Fismonotone, that is, F(x,y)+F(y,x)≤0,forallx,y∈C,(A3)
for allx,y,z∈C,(2.4)lim⁡t→0+⁡F(tz+(1-t)x,y)≤F(x,y),(A4)
for allx∈C,y↦F(x,y) is convex and lower semicontinuous.The following lemma appears implicitly in [2].Lemma 2.4 (see [2]).
LetC be a nonempty closed convex subset of H, and let F be a bifunction of C×C into ℝ satisfying (A1)–(A4). Let r>0 and x∈H, then there exists z∈C, such that
(2.5)F(z,y)+1r〈y-z,z-x〉,
for all x∈C.Lemma 2.5 (see [3]).
Assume thatF:C×C→ℝ satisfies (A1)–(A4). For r>0 and x∈H, define a mapping Tr:H→C as follows:
(2.6)Tr(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C},
for all z∈H, then the following hold: (1)
Tr is single valued,(2)
Tr is firmly nonexpansive, that is,(2.7)‖Tr(x)-Tr(y)‖2≤〈Tr(x)-Tr(y),x-y〉∀x,y∈H,(3)
F(Tr)=EP(F),(4)
EP(F) is closed and convex.Lemma 2.6 (see [14]).
LetH be a Hibert space, let C be a nonempty closed convex subset of H, and let A be a mapping of C into H. Let u∈C, then for λ>0,
(2.8)u=PC(I-λA)u⟺u∈VI(C,A),
where PC is the metric projection of H onto C.Definition 2.7.
LetC be a nonempty convex subset of a real Hilbert space. Let Ti,i=1,2,… be mappings of C into itself. For each j=1,2,…, let αj=(α1j,α2j,α3j)∈I×I×I where I∈[0,1] and α1j+α2j+α3j=1. For every n∈ℕ, we define the mapping Sn:C→C as follows:
(2.9)Un,n+1=I,Un,n=α1nTnUn,n+1+α2nUn,n+1+α3nI,Un,n-1=α1n-1Tn-1Un,n+α2n-1Un,n+α3n-1I,⋮Un,k+1=α1k+1Tk+1Un,k+2+α2k+1Un,k+2+α3k+1I,Un,k=α1kTkUn,k+1+α2kUn,k+1+α3kI,⋮Un,2=α12T2Un,3+α22Un,3+α32I,Sn=Un,1=α11T1Un,2+α21Un,2+α31I.
This mapping is called S-mapping generated by Tn,…,T1 and αn,αn-1,…,α1.Lemma 2.8.
LetC be a nonempty closed convex subset of a real Hilbert space. Let {Ti}i=1∞ be nonexpansive mappings of C into itself with ⋂i=1∞F(Ti)≠∅, and let αj=(α1j,α2j,α3j)∈I×I×I, where I=[0,1],α1j+α2j+α3j=1,α1j+α2j≤b<1, and α1j,α2j,α3j∈(0,1),forallj=1,2,…. For every n∈ℕ, let Sn be S-mapping generated by Tn,…,T1 and αn,αn-1,…,α1, then for every x∈C and k∈ℕ,limn→∞Un,kx exists.Proof.
Letx∈C and y∈⋂i=1∞F(Ti). Fix k∈ℕ, then for every n∈ℕ with n≥k, we have
(2.10)‖Un+1,kx-Un,kx‖2=‖α1kTkUn+1,k+1x+α2kUn+1,k+1x+α3kx-α1kTkUn,k+1x-α2kUn,k+1x-α3kx‖2=‖α1k(TkUn+1,k+1x-TkUn,k+1x)+α2k(Un+1,k+1x-Un,k+1x)‖2≤α1k‖TkUn+1,k+1x-TkUn,k+1x‖2+α2k‖Un+1,k+1x-Un,k+1x‖2≤α1k‖Un+1,k+1x-Un,k+1x‖2+α2k‖Un+1,k+1x-Un,k+1x‖2≤(1-α3k)‖Un+1,k+1x-Un,k+1x‖2⋮≤∏j=kn(1-α3j)‖Un+1,n+1x-Un,n+1x‖2=∏j=kn(1-α3j)‖α1n+1Tn+1Un+1,n+2x+α2n+1Un+1,n+2x+α3n+1x-x‖2=∏j=kn(1-α3j)‖α1n+1Tn+1x+(1-α1n+1)x-x‖2=∏j=kn(1-α3j)‖α1n+1(Tn+1x-x)‖2≤∏j=kn(1-α3j)(∥Tn+1x-y∥+∥y-x∥)2≤∏j=kn(1-α3j)(∥x-y∥+∥y-x∥)2≤∏j=kn(1-α3j)(2∥x-y∥)2≤bn-(k-1)(2∥x-y∥)2.
It follows that
(2.11)∥Un+1,kx-Un,kx∥≤b(n-(k-1))/2(2∥x-y∥)=bn/2b(k-1)/2(2∥x-y∥)=anak-1M,
where a=b1/2∈(0,1) and M=2∥x-y∥.
For anyk,n,p∈ℕ,p>0,n≥k, we have
(2.12)∥Un+p,kx-Un,kx∥≤∥Un+p,kx-Un+p-1,kx∥+∥Un+p-1,kx-Un+p-2,kx∥+⋯+∥Un+1,kx-Un,kx∥=∑j=nn+p-1∥Uj+1,kx-Uj,kx∥≤∑j=nn+p-1ajak-1M≤an(1-a)ak-1M.
Since a∈(0,1), we have limn→∞an=0. From (2.12), we have that {Un,kx} is a Cauchy sequence. Hence, limn→∞Un,kx exists.For everyk∈ℕ and x∈C, we define mapping U∞,k and S:C→C as follows:(2.13)lim⁡n→∞Un,kx=U∞,kx,lim⁡n→∞Snx=lim⁡n→∞Un,1x=Sx.
Such a mapping S is called S-mapping generated by Tn,Tn-1,… and αn,αn-1,….Remark 2.9.
For eachn∈ℕ,Sn is nonexpansive and limn→∞supx∈D∥Snx-Sx∥=0 for every bounded subset D of C. To show this, let x,y∈C and D be a bounded subset of C, then we have
(2.14)‖Snx-Sny‖2=‖α11(T1Un,2x-T1Un,2y)+α21(Un,2x-Un,2y)+α31(x-y)‖2≤α11‖T1Un,2x-T1Un,2y‖2+α21‖Un,2x-Un,2y‖2+α31‖x-y‖2≤α11‖Un,2x-Un,2y‖2+α21‖Un,2x-Un,2y‖2+α31‖x-y‖2=(1-α31)‖Un,2x-Un,2y‖2+α31‖x-y‖2≤(1-α31)((1-α32)‖Un,3x-Un,3y‖2+α32‖x-y‖2)+α31‖x-y‖2=(1-α31)(1-α32)‖Un,3x-Un,3y‖2+α32(1-α31)‖x-y‖2+α31‖x-y‖2=∏j=12(1-α3j)‖Un,3x-Un,3y‖2+(1-∏j=12(1-α3j))‖x-y‖2⋮≤∏j=12(1-α3j)‖Un,n+1x-Un,n+1y‖2+(1-∏j=12(1-α3j))‖x-y‖2=‖x-y‖2.
Then, we have that S:C→C is also nonexpansive, Indeed, observe that for each x,y∈C,
(2.15)∥Sx-Sy∥=lim⁡n→∞∥Snx-Sny∥≤∥x-y∥.
By (2.11), we have
(2.16)∥Sn+1x-Snx∥=∥Un+1,1x-Un,1x∥≤anM.
This implies that for m>n and x∈D,
(2.17)∥Smx-Snx∥≤∑j=nm-1∥Sj+1x-Sjx∥≤∑j=nm-1ajM≤an1-aM.
By letting m→∞, for any x∈D, we have
(2.18)∥Sx-Snx∥≤an1-aM.
It follows that
(2.19)lim⁡n→∞sup⁡x∈D⁡∥Snx-Sx∥=0.Lemma 2.10.
LetC be a nonempty closed convex subset of a real Hilbert space. Let {Ti}i=1∞ be nonexpansive mappings of C into itself with ⋂i=1∞F(Ti)≠∅, and let αj=(α1j,α2j,α3j)∈I×I×I, where I=[0,1],α1j+α2j+α3j=1,α1j+α2j≤b<1, and α1j,α2j,α3j∈(0,1)forallj=1,2,…. For every n∈ℕ, let Sn and S be S-mappings generated by Tn,…,T1 and αn,αn-1,…,α1 and Tn,Tn-1,…, and αn,αn-1,…, respectively, then F(S)=⋂i=1∞F(Ti).Proof.
It is easy to see that⋂i=1∞F(Ti)⊆F(S). For every n,k∈ℕ,withn≥k, let x0∈F(S) and x*∈⋂i=1∞F(Ti), then we have
(2.20)‖Snx0-x*‖2=‖α11(T1Un,2x0-x*)+α21(Un,2x0-x*)+α31(x0-x*)‖2≤α11‖T1Un,2x0-x*‖2+α21‖Un,2x0-x*‖2+α31‖x0-x*‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2≤α11‖Un,2x0-x*‖2+α21‖Un,2x0-x*‖2+α31‖x0-x*‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2=(1-α31)‖Un,2x0-x*‖2+α31‖x0-x*‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2≤(1-α31)((1-α32)‖Un,3x0-x*‖2+α32‖x0-x*‖2-α12α22‖T2Un,3x0-Un,3x0‖2-α22α32‖Un,3x0-x0‖2)+α31‖x0-x*‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2=(1-α31)(1-α32)‖Un,3x0-x*‖2+α32(1-α31)‖x0-x*‖2+α31‖x0-x*‖2-α12α22(1-α31)‖T2Un,3x0-Un,3x0‖2-α22α32(1-α31)‖Un,3x0-x0‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2=∏j=12(1-α3j)‖Un,3x0-x*‖2+(1-∏j=12(1-α3j))‖x0-x*‖2-α12α22(1-α31)‖T2Un,3x0-Un,3x0‖2-α22α32(1-α31)‖Un,3x0-x0‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2≤∏j=12(1-α3j)((1-α33)‖Un,4x0-x*‖2+α33‖x0-x*‖2-α13α23‖T3Un,4x0-Un,4x0‖2-α23α33‖Un,4x0-x0‖2)+(1-∏j=12(1-α3j))‖x0-x*‖2-α12α22(1-α31)‖T2Un,3x0-Un,3x0‖2-α22α32(1-α31)‖Un,3x0-x0‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2=∏j=12(1-α3j)(1-α33)‖Un,4x0-x*‖2+α33∏j=12(1-α3j)‖x0-x*‖2-α13α23∏j=12(1-α3j)‖T3Un,4x0-Un,4x0‖2-α23α33∏j=12(1-α3j)‖Un,4x0-x0‖2+(1-∏j=12(1-α3j))‖x0-x*‖2-α12α22(1-α31)‖T2Un,3x0-Un,3x0‖2-α22α32(1-α31)‖Un,3x0-x0‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2=∏j=13(1-α3j)‖Un,4x0-x*‖2+(1-∏j=13(1-α3j))‖x0-x*‖2-α13α23∏j=12(1-α3j)‖T3Un,4x0-Un,4x0‖2-α23α33∏j=12(1-α3j)‖Un,4x0-x0‖2-α12α22(1-α31)‖T2Un,3x0-Un,3x0‖2-α22α32(1-α31)‖Un,3x0-x0‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2⋮≤∏j=1k+1(1-α3j)‖Un,k+2x0-x*‖2+(1-∏j=1k+1(1-α3j))‖x0-x*‖2-α1k+1α2k+1∏j=1k(1-α3j)‖Tk+1Un,k+2x0-Un,k+2x0‖2-α2k+1α3k+1∏j=1k(1-α3j)‖Un,k+2x0-x0‖2-α1kα2k∏j=1k-1(1-α3j)‖TkUn,k+1x0-Un,k+1x0‖2-α2kα3k∏j=1k-1(1-α3j)‖Un,k+1x0-x0‖2-α1k-1α2k-1∏j=1k-2(1-α3j)‖Tk-1Un,kx0-Un,kx0‖2-α2k-1α3k-1∏j=1k-2(1-α3j)‖Un,kx0-x0‖2⋮-α13α23∏j=12(1-α3j)‖T3Un,4x0-Un,4x0‖2-α23α33∏j=12(1-α3j)‖Un,4x0-x0‖2-α12α22(1-α31)‖T2Un,3x0-Un,3x0‖2-α22α32(1-α31)‖Un,3x0-x0‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2⋮≤∏j=1n(1-α3j)‖Un,n+1x0-x*‖2+(1-∏j=1n(1-α3j))‖x0-x*‖2-α1nα2n∏j=1n-1(1-α3j)‖TnUn,n+1x0-Un,n+1x0‖2-α2nα3n∏j=1n-1(1-α3j)‖Un,n+1x0-x0‖2⋮-α1k+1α2k+1∏j=1k(1-α3j)‖Tk+1Un,k+2x0-Un,k+2x0‖2-α2k+1α3k+1∏j=1k(1-α3j)‖Un,k+2x0-x0‖2-α1kα2k∏j=1k-1(1-α3j)‖TkUn,k+1x0-Un,k+1x0‖2-α2kα3k∏j=1k-1(1-α3j)‖Un,k+1x0-x0‖2-α1k-1α2k-1∏j=1k-2(1-α3j)‖Tk-1Un,kx0-Un,kx0‖2-α2k-1α3k-1∏j=1k-2(1-α3j)‖Un,kx0-x0‖2⋮-α13α23∏j=12(1-α3j)‖T3Un,4x0-Un,4x0‖2-α23α33∏j=12(1-α3j)‖Un,4x0-x0‖2-α12α22(1-α31)‖T2Un,3x0-Un,3x0‖2-α22α32(1-α31)‖Un,3x0-x0‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2=‖x0-x*‖2-α1nα2n∏j=1n-1(1-α3j)‖TnUn,n+1x0-Un,n+1x0‖2⋮-α1k+1α2k+1∏j=1k(1-α3j)‖Tk+1Un,k+2x0-Un,k+2x0‖2-α2k+1α3k+1∏j=1k(1-α3j)‖Un,k+2x0-x0‖2-α1kα2k∏j=1k-1(1-α3j)‖TkUn,k+1x0-Un,k+1x0‖2-α2kα3k∏j=1k-1(1-α3j)‖Un,k+1x0-x0‖2-α1k-1α2k-1∏j=1k-2(1-α3j)‖Tk-1Un,kx0-Un,kx0‖2-α2k-1α3k-1∏j=1k-2(1-α3j)‖Un,kx0-x0‖2⋮-α13α23∏j=12(1-α3j)‖T3Un,4x0-Un,4x0‖2-α23α33∏j=12(1-α3j)‖Un,4x0-x0‖2-α12α22(1-α31)‖T2Un,3x0-Un,3x0‖2-α22α32(1-α31)‖Un,3x0-x0‖2-α11α21‖T1Un,2x0-Un,2x0‖2-α21α31‖Un,2x0-x0‖2.
For k∈ℕ and (2.20), we have
(2.21)α2k-1α3k-1∏j=1k-2(1-α3j)‖Un,kx0-x0‖2≤‖x0-x*‖2-‖Snx0-x*‖2,
as n→∞. This implies that U∞,kx0=x0,forallk∈ℕ.
Again by (2.20), we have
(2.22)α1kα2k∏j=1k-1(1-α3j)‖TkUn,k+1x0-Un,k+1x0‖2≤‖x0-x*‖2-‖Snx0-x*‖2,
as n→∞. Hence,
(2.23)α1kα2k∏j=1k-1(1-α3j)‖TkU∞,k+1x0-U∞,k+1x0‖2≤0.
From U∞,kx0=x0,forallk∈ℕ, and (2.23), we obtain that Tkx0=x0,forallk∈ℕ. This implies that x0∈⋂i=1∞F(Ti).


## 3. Main Result

Theorem 3.1.
LetC be a nonempty closed convex subset of a Hilbert space H. Let F be bifunctions from C×C into ℝ satisfying (A1)–(A4). Let A:C→H be a α-inverse-strongly monotone mapping. Let {Ti}i=1∞ be infinite family of nonexpansive mappings with 𝔉=⋂i=1∞F(Ti)⋂EP(F)⋂VI(C,A), and let ρj=(α1j,α2j,α3j)∈I×I×I, where I=[0,1],α1j+α2j+α3j=1,α1j+α2j≤b<1, and α1j,α2j,α3j∈(0,1)forallj=1,2,…. For every n∈ℕ, let Sn and S be S-mappings generated by Tn,…,T1 and ρn,ρn-1,…,ρ1 and Tn,Tn-1,…, and ρn,ρn-1,…, respectively. Let {xn},{un} be sequences generated by x1,u∈C and
(3.1)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnu+βnPC(I-λA)xn+γnSnPC(I-λA)un,∀n≥1,
where {αn},{βn},{γn}∈(0,1), such that αn+βn+γn=1,βn∈[c,d]⊂(0,1)rn∈[a,b]⊂(0,2α),λ⊂(0,2α). Assume that(i)
limn→∞αn=0and∑n=0∞αn=∞,(ii)
∑n=1∞α1n<∞,(iii)
∑n=1∞|rn+1-rn|,∑n=1∞|γn+1-γn|,∑n=1∞|αn+1-αn|,∑n=1∞|βn+1-βn|<∞,then the sequence {xn},{un} converge strongly to z=P𝔉u.Proof.
First, we show that(I-λA) is nonexpansive. Let x,y∈C. Since A is α-inverse-strongly monotone and λ<2α, we have
(3.2)‖(I-λA)x-(I-λA)y‖2=‖x-y-λ(Ax-Ay)‖2=‖x-y‖2-2λ〈x-y,Ax-Ay〉+λ2‖Ax-Ay‖2≤‖x-y‖2-2αλ‖Ax-Ay‖2+λ2‖Ax-Ay‖2=‖x-y‖2+λ(λ-2α)‖Ax-Ay‖2≤‖x-y‖2.
Thus, (I-λA) is nonexpansive. We will divide our proof into 5 steps.Step 1.
We shall show that the sequence{xn} is bounded. Since
(3.3)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C.
By Lemma 2.5, we have un=Trnxn and EP(F)=F(Trn).
Letz∈𝔉. By nonexpansiveness of I-λA and Trn, we have
(3.4)∥xn+1-z∥=∥αnu+βnPC(I-λA)xn+γnSnPC(I-λA)un-z∥=∥αn(u-z)+βn(PC(I-λA)xn-z)+γn(SnPC(I-λA)un-z)∥≤αn∥u-z∥+βn∥PC(I-λA)xn-z∥+γn∥SnPC(I-λA)un-z∥≤αn∥u-z∥+βn∥xn-z∥+γn∥un-z∥=αn∥u-z∥+βn∥xn-z∥+γn∥Trnxn-z∥≤αn∥u-z∥+(1-αn)∥xn-z∥≤max⁡{∥u-z∥,∥xn-z∥}.
By induction, we can prove that {xn} is bounded and so is {un}.Step 2.
We will show thatlimn→∞∥xn+1-xn∥=0. By definition of xn, we have
(3.5)∥xn+1-xn∥=‖αnu+βnPC(I-λA)xn+γnSnPC(I-λA)un-αn-1u-βn-1PC(I-λA)xn-1-γn-1Sn-1PC(I-λA)un-1‖=‖(αn-αn-1)u+βnPC(I-λA)xn-βn-1PC(I-λA)xn+βn-1PC(I-λA)xn+γnSnPC(I-λA)un-γn-1SnPC(I-λA)un+γn-1SnPC(I-λA)un-βn-1PC(I-λA)xn-1-γn-1Sn-1PC(I-λA)un-1‖=‖(αn-αn-1)u+(βn-βn-1)PC(I-λA)xn+βn-1(PC(I-λA)xn-PC(I-λA)xn-1)+(γn-γn-1)SnPC(I-λA)un+γn-1(SnPC(I-λA)un-Sn-1PC(I-λA)un-1)‖≤|αn-αn-1|∥u∥+|βn-βn-1|∥PC(I-λA)xn∥+βn-1∥PC(I-λA)xn-PC(I-λA)xn-1∥+|γn-γn-1|∥SnPC(I-λA)un∥+γn-1∥SnPC(I-λA)un-Sn-1PC(I-λA)un-1∥≤|αn-αn-1|∥u∥+|βn-βn-1|∥PC(I-λA)xn∥+βn-1∥xn-xn-1∥+|γn-γn-1|∥SnPC(I-λA)un∥+γn-1(∥SnPC(I-λA)un-SnPC(I-λA)un-1∥+∥SnPC(I-λA)un-1-Sn-1PC(I-λA)un-1∥)≤|αn-αn-1|∥u∥+|βn-βn-1|∥PC(I-λA)xn∥+βn-1∥xn-xn-1∥+|γn-γn-1|∥SnPC(I-λA)un∥+γn-1∥un-un-1∥+γn-1∥SnPC(I-λA)un-1-Sn-1PC(I-λA)un-1∥.
Since un=Trnxn, by definition of Trn, we have
(3.6)F(Trnxn,y)+1rn〈y-Trnxn,Trnxn-xn〉≥0,∀y∈C.
Similarly,
(3.7)F(Trn+1xn+1,y)+1rn+1〈y-Trn+1xn+1,Trn+1xn+1-xn+1〉≥0,∀y∈C.
From (3.6) and (3.7), we obtain
(3.8)F(Trnxn,Trn+1xn+1)+1rn〈Trn+1xn+1-Trnxn,Trnxn-xn〉≥0,F(Trn+1xn+1,Trnxn)+1rn+1〈Trnxn-Trn+1xn+1,Trn+1xn+1-xn+1〉≥0.
By (3.8), we have
(3.9)1rn〈Trn+1xn+1-Trnxn,Trnxn-xn〉+1rn+1〈Trnxn-Trn+1xn+1,Trn+1xn+1-xn+1〉≥0.
It follows that
(3.10)〈Trnxn-Trn+1xn+1,Trn+1xn+1-xn+1rn+1-Trnxn-xnrn〉≥0.
This implies that
(3.11)0≤〈Trn+1xn+1-Trnxn,Trnxn-Trn+1xn+1+Trn+1xn+1-xn-rnrn+1(Trn+1xn+1-xn+1)〉.
It follows that
(3.12)‖Trn+1xn+1-Trnxn‖2≤〈Trn+1xn+1-Trnxn,Trn+1xn+1-xn-rnrn+1(Trn+1xn+1-xn+1)〉=〈Trn+1xn+1-Trnxn,xn+1-xn+(1-rnrn+1)(Trn+1xn+1-xn+1)〉≤∥Trn+1xn+1-Trnxn∥‖xn+1-xn+(1-rnrn+1)(Trn+1xn+1-xn+1)‖≤∥Trn+1xn+1-Trnxn∥(∥xn+1-xn∥+|1-rnrn+1|∥Trn+1xn+1-xn+1∥)=∥Trn+1xn+1-Trnxn∥(∥xn+1-xn∥+1rn+1|rn+1-rn|∥Trn+1xn+1-xn+1∥)≤∥Trn+1xn+1-Trnxn∥(∥xn+1-xn∥+1a|rn+1-rn|∥Trn+1xn+1-xn+1∥).
It follows that
(3.13)∥un+1-un∥≤∥xn+1-xn∥+1a|rn+1-rn|∥un+1-xn+1∥.
Putting yn=PC(I-λA)un, then {yn} is bounded. By definition of Sn,foralln∈ℕ, we have
(3.14)∥Snyn-1-Sn-1yn-1∥=∥Un,1yn-1-Un-1,1yn-1∥=∥α11T1Un,2yn-1+α21Un,2yn-1+α31yn-1-α11T1Un-1,2yn-1-α21Un-1,2yn-1-α31yn-1∥≤(1-α31)∥Un,2yn-1-Un-1,2yn-1∥≤(1-α31)(1-α32)∥Un,3yn-1-Un-1,3yn-1∥=∏j=12(1-α3j)∥Un,3yn-1-Un-1,3yn-1∥⋮≤∏j=1n-1(1-α3j)∥Un,nyn-1-Un-1,nyn-1∥≤∥Un,nyn-1-yn-1∥=∥α1nTnyn-1+(1-α1n)yn-1-yn-1∥=α1n∥Tnyn-1-yn-1∥≤α1n2∥yn-1-z∥.
Substituting (3.13) and (3.14) into (3.5), we have
(3.15)∥xn+1-xn∥≤|αn-αn-1|∥u∥+|βn-βn-1|∥PC(I-λA)xn∥+βn-1∥xn-xn-1∥+|γn-γn-1|∥SnPC(I-λA)un∥+γn-1∥un-un-1∥+γn-1∥SnPC(I-λA)un-1-Sn-1PC(I-λA)un-1∥≤|αn-αn-1|∥u∥+|βn-βn-1|∥PC(I-λA)xn∥+βn-1∥xn-xn-1∥+|γn-γn-1|∥SnPC(I-λA)un∥+γn-1(∥xn+1-xn∥+1a|rn+1-rn|∥un+1-xn+1∥)+2γn-1α1n∥yn-1-z∥≤|αn-αn-1|∥u∥+|βn-βn-1|∥PC(I-λA)xn∥+(1-αn-1)∥xn-xn-1∥+|γn-γn-1|∥SnPC(I-λA)un∥+1a|rn+1-rn|∥un+1-xn+1∥+2γn-1α1n∥yn-1-z∥≤(1-αn-1)∥xn-xn-1∥+|βn-βn-1|M1+|αn-αn-1|M1+|γn-γn-1|M1+1a|rn+1-rn|M1+2α1nM1,
where M1=maxn∈ℕ{∥u∥,∥PC(I-λA)xn∥,∥SnPC(I-λA)un∥,∥un-xn∥,∥yn-z∥}. By (3.15), Lemma 2.3, and conditions (i)–(iii), we obtain
(3.16)lim⁡n→∞∥xn+1-xn∥=0.Step 3.
We shall show thatlimn→∞∥xn-un∥=0.
Letv∈𝔉. Since un=Trnxn and Trn is firmly nonexpansive, we have
(3.17)‖v-Trnxn‖2=‖Trnv-Trnxn‖2≤〈Trnv-Trnxn,v-xn〉=12(‖Trnxn-v‖2+‖xn-v‖2-‖Trnxn-xn‖2).
Hence,
(3.18)‖un-v‖2≤‖xn-v‖2-‖un-xn‖2.
By (3.18), we have
(3.19)‖xn+1-v‖2=‖αn(u-v)+βn(PC(I-λA)xn-v)+γn(SnPC(I-λA)un-v)‖2≤αn‖u-v‖2+βn‖xn-v‖2+γn‖un-v‖2≤αn‖u-v‖2+βn‖xn-v‖2+γn(‖xn-v‖2-‖un-xn‖2)≤αn‖u-v‖2+‖xn-v‖2-γn‖un-xn‖2.
it implies that
(3.20)γn‖un-xn‖2≤αn‖u-v‖2+‖xn-v‖2-‖xn+1-v‖2≤αn‖u-v‖2+(∥xn-v∥-∥xn+1-v∥)(∥xn-v∥+∥xn+1-v∥)≤αn‖u-v‖2+∥xn-xn+1∥(∥xn-v∥+∥xn+1-v∥).
By (3.16) and condition (i), we have
(3.21)lim⁡n→∞∥xn-un∥=0.
Let z∈𝔉 and by nonexpansiveness of I-λA, we have
(3.22)‖xn+1-z‖2≤αn‖u-z‖2+βn‖PC(I-λA)xn-z‖2+γn‖SnPC(I-λA)un-z‖2-βnγn∥PC(I-λA)xn-SnPC(I-λA)un∥≤αn‖u-z‖2+(1-αn)‖xn-z‖2-βnγn∥PC(I-λA)xn-SnPC(I-λA)un∥≤αn‖u-z‖2+‖xn-z‖2-βnγn∥PC(I-λA)xn-SnPC(I-λA)un∥.
It implies that
(3.23)βnγn∥PC(I-λA)xn-SnPC(I-λA)un∥≤αn‖u-z‖2+‖xn-z‖2-‖xn+1-z‖2=αn‖u-z‖2+(∥xn-z∥-∥xn+1-z∥)(∥xn-z∥+∥xn+1-z∥)=αn‖u-z‖2+∥xn-xn+1∥(∥xn-z∥+∥xn+1-z∥).
By (3.16) and condition (i), we have
(3.24)lim⁡n→∞∥PC(I-λA)xn-SnPC(I-λA)un∥=0.
Since
(3.25)∥PC(I-λA)xn-SnPC(I-λA)xn∥≤∥PC(I-λA)xn-SnPC(I-λA)un∥+∥SnPC(I-λA)un-SnPC(I-λA)xn∥≤∥PC(I-λA)xn-SnPC(I-λA)un∥+∥un-xn∥,
by (3.24) and (3.21), we have
(3.26)lim⁡n→∞∥PC(I-λA)xn-SnPC(I-λA)xn∥=0.
Since
(3.27)∥xn-PC(I-λA)xn∥≤∥xn-xn+1∥+∥xn+1-PC(I-λA)xn∥≤∥xn-xn+1∥+αn∥u-PC(I-λA)xn∥+γn∥SnPC(I-λA)un-PC(I-λA)xn∥,
by (3.24), (3.16), and condition (i), we have
(3.28)lim⁡n→∞∥xn-PC(I-λA)xn∥=0.
Since
(3.29)∥xn-SnPC(I-λA)un∥≤∥xn-xn+1∥+∥xn+1-SnPC(I-λA)un∥≤∥xn-xn+1∥+αn∥u-SnPC(I-λA)un∥+βn∥PC(I-λA)xn-SnPC(I-λA)un∥,
again by (3.24), (3.16), and condition (i), we have
(3.30)lim⁡n→∞∥xn-SnPC(I-λA)un∥=0.
Since
(3.31)∥xn-SnPC(I-λA)xn∥≤∥xn-SnPC(I-λA)un∥+∥SnPC(I-λA)un-SnPC(I-λA)xn∥≤∥xn-SnPC(I-λA)un∥+∥un-xn∥,
by (3.21) and (3.30), we have
(3.32)lim⁡n→∞∥xn-SnPC(I-λA)xn∥=0.Step 4.
Puttingz0=P𝔉u, we will show that
(3.33)limsup⁡n→∞〈u-z0,xn-z0〉≤0.
To show this inequality, take a subsequence {xnm} of {xn}, such that
(3.34)limsup⁡n→∞〈u-z0,xn-z0〉=limsup⁡m→∞〈u-z0,xnm-z0〉.
Without loss of generality, we may assume that xnm⇀ωasm→∞ where ω∈C. By nonexpansiveness of PC(I-λA), (3.28), and Lemma 2.2, we have ω∈F(PC(I-λA)). By Lemma 2.6, we obtain that ω∈VI(C,A). Since ∥unm-xnm∥→0 as m→∞, we have unm⇀ωasm→∞. Since
(3.35)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C.
By (A2), we have
(3.36)1rn〈y-un,un-xn〉≥F(y,un),∀y∈C.
In particular,
(3.37)〈y-unm,1rnm(unm-xnm)〉≥F(y,unm).
By condition (A4), F(y,·) is lower semicontinuous and convex, and thus weakly semicontinuous. By (3.21) imply that (1/rnm)(unm-xnm)→0 in norm. Therefore, letting m→∞ in (3.37), we have
(3.38)F(y,ω)≤lim⁡m→∞F(y,unm)≤0,∀y∈C.
Replacing y with yt:=ty+(1-t)ω, t∈(0,1], we have yt∈C, and using (A1), (A4), and (3.38), we obtain
(3.39)0=F(yt,yt)≤tF(yt,y)+(1-t)F(yt,ω)≤tF(yt,y).
Hence, F(ty+(1-t)ω,y)≥0,forallt∈(0,1] and for all y∈C. Letting t→0+ and using assumption (A3), we can conclude that
(3.40)F(ω,y)≥0,y∈C.
Therefore, ω∈EP(F).
We will show thatω∈⋂i=1∞F(Ti). By Lemma 2.10, we have F(S)=⋂i=1∞F(Ti).  Assume that ω≠Sω. Using Opial s′ property, (3.32), ω∈F(PC(I-λA)), and Remark 2.9, we have
(3.41)liminf⁡m→∞∥xnm-ω∥<liminf⁡m→∞∥xnm-Sω∥≤liminf⁡m→∞(∥xnm-SnmPC(I-λA)xnm∥+∥SnmPC(I-λA)xnm-SnmPC(I-λA)ω∥+∥SnmPC(I-λA)ω-Sω∥)=liminf⁡m→∞(∥xnm-SnmPC(I-λA)xnm∥+∥SnmPC(I-λA)xnm-SnmPC(I-λA)ω∥+∥Snmω-Sω∥)≤liminf⁡m→∞∥xnm-ω∥.
This is a contradiction, then ω∈⋂i=1∞F(Ti). Hence, ω∈𝔉.
Sincexnm⇀ω and ω∈𝔉, we have
(3.42)limsup⁡n→∞〈u-z0,xn-z0〉=limsup⁡m→∞〈u-z0,xnm-z0〉=〈u-z0,ω-z0〉≤0.Step 5.
Finally, we show that{xn} and {un} converse strongly to z0=P𝔉u. Putting z0=P𝔉u, by nonexpansiveness of PC(I-λA), Sn, and Trn, we have
(3.43)‖xn+1-z0‖2=‖αn(u-z0)+βn(PC(I-λA)xn-z0)+γn(SnPC(I-λA)un-z0)‖2≤‖βn(PC(I-λA)xn-z0)+γn(SnPC(I-λA)un-z0)‖2+2αn〈u-z0,xn+1-z0〉≤βn‖PC(I-λA)xn-z0‖2+γn‖SnPC(I-λA)un-z0‖2+2αn〈u-z0,xn+1-z0〉≤βn‖xn-z0‖2+γn‖Trnxn-z0‖2+2αn〈u-z0,xn+1-z0〉≤(1-αn)∥xn-z0∥2+2αn〈u-z0,xn+1-z0〉.From  Step4 and Lemma 2.3, we obtain that {xn} converse strongly to z0=P𝔉u. By using (3.21), we have {un} converse strongly to z0=P𝔉u.


## 4. Application

Using our main theorem (Theorem3.1), we obtain the following strong convergence theorems involving infinite family of κ-strict pseudocontractions.To prove strong convergence theorem in this section, we need definition and lemma as follows.Definition 4.1.
A mappingT:C→C is said to be a κ-strongly pseudocontraction mapping, if there exists κ∈[0,1), such that
(4.1)‖Tx-Ty‖2≤‖x-y‖2+κ‖(I-T)x-(I-T)y‖2,∀x,y∈C.Lemma 4.2 (see [15]).
LetC be a nonempty closed convex subset of a real Hilbert space H and T:C→C a κ-strict pseudocontraction. Define S:C→C by Sx=αx+(1-α)Tx for each x∈C. Then, as α∈[κ,1)S is nonexpansive, such that F(S)=F(T).Theorem 4.3.
LetC be a nonempty closed convex subset of a Hilbert space H. Let F be bifunctions from C×C into ℝ satisfying (A1)–(A4). Let A:C→H be a α-inverse-strongly monotone mapping. Let {Ti}i=0∞ be infinite family of κi-pseudocontractions mappings with 𝔉=⋂i=1∞F(Ti)⋂EP(F)⋂VI(C,A). Define a mapping Tκi by Tκi=κix+(1-κi)Tix, forallx∈C,i∈ℕ, and let ρj=(α1j,α2j,α3j)∈I×I×I, where I=[0,1],α1j+α2j+α3j=1,α1j+α2j≤b<1, and α1j,α2j,α3j∈(0,1)forallj=1,2,…. For every n∈ℕ, let Sn and S be S-mappings generated by Tκn,…,Tκ1 and ρn,ρn-1,…,ρ1 and Tκn,Tκn-1,…, and ρn,ρn-1,…, respectively. Let {xn},{un} be sequences generated by x1,u∈C and
(4.2)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnu+βnPC(I-λA)xn+γnSnPC(I-λA)un,∀n≥1,
where {αn},{βn},{γn}∈(0,1), such that αn+βn+γn=1,βn∈[c,d]⊂(0,1)rn∈[a,b]⊂(0,2α),λ⊂(0,2α). Assume that(i)
limn→∞αn=0and∑n=0∞αn=∞,(ii)
∑n=1∞α1n<∞,(iii)
∑n=1∞|rn+1-rn|,∑n=1∞|γn+1-γn|,∑n=1∞|αn+1-αn|,∑n=1∞|βn+1-βn|<∞,
then the sequence{xn},{un} converges strongly to z=P𝔉u.Proof.
For everyi∈ℕ, by Lemma 4.2, we have that Tκi is nonexpansive mappings. From Theorem 3.1, we could have the desired conclusion.



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*Source: 562689-2011-04-12.xml*