# Approximation of a Common Element of the Fixed Point Sets of Multivalued Strictly Pseudocontractive-Type Mappings and the Set of Solutions of an Equilibrium Problem in Hilbert Spaces

Authors: F. O. Isiogugu
Journal: Abstract and Applied Analysis
 (2016)

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## Abstract

The strong convergence of a hybrid algorithm to a common element of the fixed point sets of multivalued strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces is obtained using a strict fixed point set condition. The obtained results improve, complement, and extend the results on multivalued and single-valued mappings in the contemporary literature.


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## Body

## 1. Introduction

LetX be a nonempty set and let T:X→X be a map. A point x∈X is called a fixed point of T if x=Tx. If T:X→2X is a multivalued map then x is a fixed point of T if x∈Tx. If Tx={x} then x is called a strict fixed point of T. The set F(T)={x∈D(T):x∈Tx} (resp., F(T)={x∈D(T):x=Tx}) is called the fixed point set of multivalued (resp., single-valued) map T, while the set Fs(T)={x∈D(T):Tx={x}} is called the strict fixed point set of T.LetX be a normed space. A subset K of X is called proximinal if for each x∈X there exists k∈K such that (1)x-k=inf⁡x-y:y∈K=dx,K.It is known that every closed convex subset of a uniformly convex Banach space is proximinal. We will denote the family of all nonempty closed and bounded subsets of X by CB(X), the family of all nonempty subsets of X by 2X, and the family of all proximinal subsets of X by P(X), for a nonempty set X.LetH denote the Hausdorff metric induced by the metric d on X; that is, for every A,B∈CB(X), (2)HA,B=max⁡supa∈A⁡da,B,supb∈B⁡db,A.Let X be a normed space. Let T:D(T)⊆X→2X be a multivalued mapping on X. A multivalued mapping T:D(T)⊆X→2X is called L-Lipschitzian if there exists L≥0 such that for all x,y∈D(T)(3)HTx,Ty≤Lx-y.In (3) if L∈[0,1)T is said to be a contraction while T is nonexpansive if L=1. T is called quasi-nonexpansive if F(T)={x∈D(T):x∈Tx}≠∅ and for all p∈F(T), (4)HTx,Tp≤x-p.Clearly every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive. T is said to be k-strictly pseudocontractive-type of Isiogugu [1] if there exists k∈(0,1) such that, given any pair x,y∈D(T) and u∈Tx, there exists v∈Ty satisfying u-v≤H(Tx,Ty) and (5)H2Tx,Ty≤x-y2+kx-u-y-v2.If k=1 in (5), T is said to be pseudocontractive-type, while T is nonexpansive-type if k=0. Every multivalued nonexpansive mapping T:D(T)⊆X→P(X) is nonexpansive-type. In a real Hilbert space H, T:D(T)⊆H→CB(H) is said to be k-strictly pseudocontractive of Chidume et al. [2] if there exists k∈(0,1) such that for all x,y∈D(T)(6)H2Tx,Ty≤x-y2+kx-u-y-v2,∀u∈Tx,v∈Ty.If k=1, T is said to be pseudocontractive. It is easy to see that every k-strictly pseudocontractive mapping T:D(T)⊆H→P(H) is k-strictly pseudocontractive-type.LetH be a real Hilbert space with inner product 〈·,·〉 and norm ·, respectively, and let C be a nonempty closed convex subset of H. Given an operator A:H→H and a closed convex set C, the variational inequality problem is the problem of finding x∗∈C such that 〈x-x∗,A(x∗)〉≥0, for all x∈C. This variational inequality problem is usually denoted as VIP(A,C).LetF:C×C→R be a bifunction, where R is the set of real numbers. The equilibrium problem for F:C×C→R is to find x∈C such that (7)Fx,y≥0∀y∈C.The set of solutions of (7) is denoted by EP(F). Several algorithms were introduced by authors for approximating solutions of equilibrium problems for a bifunction (or finite family of bifunctions) (see, e.g., [3] and references therein). Given a mapping A:C→H, let F(x,y)=〈Ax,y-x〉 for all x,y∈C; then z∈EP(F) if and only if 〈Az,y-z〉≥0 for all y∈C; that is, z is a solution of the variational inequality VIP(A,C). Numerous problems in physics, optimization, and economics are reduced to the problem of finding the solutions of (7) (see, e.g., [4–6] and the references therein).The purpose of this work is to first establish closed and convexity property for a strict fixed point set of a multivalued strictly pseudocontractive-type mappings. Second, establish with a strict fixed point set condition a strong convergence of a hybrid algorithm to a common element of the fixed point sets of two multivalued strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces. The obtained results extend, complement, and improve the results on equilibrium problems as well as multivalued and single-valued mappings in the contemporary literature.


## 2. Preliminaries

In the sequel, we will need the following definitions and lemmas.Definition 1.
LetT:X→2X be a multivalued mapping; for each x∈X, PTx is defined by (8)PTx=y∈Tx:x-y=dx,Tx.For solving the equilibrium problems for a bifunction F: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 eachx,y,z∈C, limt↓0⁡F(tz+(1-t)x,y)≤F(x,y).(A4)
For eachx∈C, y↦F(x,y) is convex and lower semicontinuous.Lemma 2 (see [4]).
LetC be a nonempty closed convex subset of a real Hilbert space H and F:C×C→R a bifunction satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that (9)Fz,y+1ry-z,z-x≥0,∀y∈C.Lemma 3 (see [6]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Assume that F:C×C→R satisfying (A1)–(A4). Let r>0 and x∈H. Define Tr:H→2C by (10)Trx=z∈C:Fz,y+1ry-z,z-x≥0,∀y∈C.Then the following hold:(1)
Tr is single valued.(2)
Tr is firmly nonexpansive; that is, for any x,y∈H, Trx-Try2≤〈Trx-Try,x-y〉.(3)
F(Tr)=EP(F).(4)
EP(F) is closed and convex.Lemma 4 (see [7]).
LetC be a nonempty closed convex subset of a real Hilbert space H and F:C×C→R a bifunction satisfying (A1)–(A4). Let r>0 and x∈H. Then for all x∈H and p∈F(Tr)(11)p-Trx2+Trx-x2≤p-x2.Lemma 5.
LetH be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let PC be the convex projection onto C. Then, convex projection is characterized by the following relations:(i)
x∗=PC(x)⇔〈x-x∗,y-x∗〉≤0, for all y∈C.(ii)
x-PCx2≤x-y2-y-PCx2.(iii)
x-PCy2≤x-y2-PCy-y2.


## 3. Main Results

Proposition 6.
LetK be a nonempty subset of a real Hilbert space H. And let T:K→P(K) be a k-strictly pseudocontractive-type mapping such that Fs(T) is nonempty. Then Fs(T) is closed and convex.Proof.
Let{xn}n=1∞⊆Fs(T) such that {xn}n=1∞ converges to x∈K. We show that x∈Fs(T). Let u∈Tx be arbitrary: (12)x-u≤x-xn+xn-u≤x-xn+HTxn,Tx≤x-xn+x-xn+kx-u.Taking limits as n→∞, we have that x-u≤kx-u. Hence, x=u∈Tx. Since u was arbitrary, we have that Tx={x}.
We now prove thatF(T) is convex. Let p1,p2∈F(T) and z=αp1+(1-α)p2 and then z-p1=(1-α)(p2-p1) and z-p2=α(p1-p2):(13)d2z,Tz≤z-u2,∀u∈Tz=αp1+1-αp2-u2=αp1-u2+1-αp2-u2-α1-αp2-p12.Now, k-strictly pseudocontractive-type condition on T and a strict fixed point condition on p1 and p2 imply that, for all u∈Tz, u-p1≤H(Tz,Tp1) and H2(Tz,Tp1)≤[z-p12+kz-u2]. and u-p2≤H(Tz,Tp2) and H2(Tz,Tp2)≤[z-p22+kz-u2]. It then follows that (14)d2z,Tz≤u-z2=αp1-u2+1-αp2-u2-α1-αp2-p12≤αH2Tz,Tp1+1-αH2Tz,Tp2-α1-αp1-p22≤αz-p12+kz-u2+1-αz-p22+kz-u2-α1-αp1-p22.In particular, for each u∈PTz, (15)d2z,Tz≤αz-p12+kd2z,Tz+1-αz-p22+kd2z,Tz-α1-αp1-p22=αp1+1-αp2-z2+kd2z,Tz=kd2z,Tz.Hence, d(z,Tz)=0. Since Tz is proximinal, there exists w∈Tz such that w-z=0; consequently, z∈Tz. Also, if v∈Tz, then (16)v-z2=v-αp1+1-αp22≤αz-p12+kz-v2+1-αz-p22+kz-v2-α1-αp1-p22=kz-v2which shows that z=v. Thus, Tz={z}.We now prove a strong convergence of multivalued version of the hybrid algorithm considered in [8] to a common element of the set of fixed points of two k-strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces. As a corollary, we obtain a hybrid algorithm for finding common elements of the set of fixed points of two multivalued strictly pseudocontractive mappings of [2] and the set of solutions of an equilibrium problem, with a strict fixed point set condition.Theorem 7.
LetC be a nonempty closed convex subset of a real Hilbert space H, let f:C×C→R be a bifunction satisfying (A1)–(A4), and let S,T:C→P(C) be two strictly pseudocontractive-type mappings with contractive coefficients λ1 and λ2, respectively, such that F=Fs(T)∩Fs(S)∩EP(f)≠∅. Let {xn} be a sequence generated from an arbitrary x0∈C as follows:(17)x0∈H,C1=C,x1=PCx0,yn=αnxn+1-αnβnvn+1-βnzn,un∈CsuchthatFun,y+1rny-un,un-yn≥0,∀y∈C,Cn+1=z∈Cn:z-un2≤z-xn2,xn+1=PCn+1x0,where vn∈Txn and zn∈Sxn. {αn}n=1∞ and {βn}n=1∞ are sequences in [0,1] satisfying (i)
αn≥max⁡{λ1,λ2},(ii)
liminfn→∞⁡(1-αn)(1-βn)(αn-λ1)>0 and liminfn→∞⁡(1-αn)(αn-λ2)βn>0,(iii)
{rn}⊂[a,∞) for some a>0.Then {xn} converges strongly to p∈PFx0.Proof.
Observe thatCn is closed and convex for all n≥1; therefore PCxn+1x0 is well defined and note that un=Trnyn. Next we show that F⊂Cn, for all n≥1. F⊂C1=C is obvious. Suppose F⊂Ck, set wn=βnvn+(1-βn)zn, and then using Lemma 3, for all q∈F, we have (18)q-uk2=q-Trkyk2≤q-yk2=q-αkxk+1-αkβkvk+1-βkzk2=q-αkxk+1-αkwk2=αkxk-q2+1-αkwk-q2-αk1-αkxk-wk2.Also, (19)wk-q2=βkvk+1-βkzk-q2=βkvk-q2+1-βkzk-q2-βk1-βkvk-zk2.Using (19) we obtain from (18) that (20)q-uk2≤αkxk-q2+1-αkβkvk-q2+1-αk1-βkzk-q2-1-αkβk1-βkvk-zk2-αk1-αkxk-wk2≤αkxk-q2+1-αkβkH2Txk,Tq+1-αk1-βkH2Sxk,Tq-1-αkβk1-βkvk-zk2-αk1-αkxk-wk2≤αkxk-q2+1-αkβkxk-q2+λ2xk-vk2+1-αk1-βkxk-q2+λ1xk-zk2-1-αkβk1-βkvk-zk2-αk1-αkxk-wk2.Also, (21)xk-wk2=xk-βkvk+1-βkzk2=βkxk-vk2+1-βkxk-zk2-βk1-βkvk-zk2.Using (21) we obtain from (20) that (22)q-uk2≤αk+1-αkβk+1-αk1-βkxk-q2+1-αkβkλ2-αk1-αkβkxk-vk2+1-αk1-βkλ1-αk1-αk1-βkxk-zk2+1-αk1-βkβkαk-1-αk1-βkβkvk-zk2=xk-q2-βk1-αkαk-λ2xk-vk2-1-αk1-βkαk-λ1xk-zk2-1-αk21-βkβkvk-zk2≤xk-q2.This shows that q∈Ck+1. It then follows that F⊆Cn for all n≥1. From xn=PCnx0 we have from Lemma 5(i) that (23)xn-y,x0-xn≥0,∀y∈Cn.Since F⊆Cn for all n≥1, we have (24)xn-q,x0-xn≥0,∀q∈F.Using Lemma 5(ii) we obtain (25)xn-x02=PCnx0-x02≤x0-q2-q-xn2≤x0-q2,for each q∈F⊂Cn and for all n≥1. Consequently the sequence {xn} is bounded, and so are {zn} and {vn}. Furthermore, since xn=PCnx0 and xn+1=PCn+1x0∈Cn+1⊂Cn then from definition of PC we have xn-x0≤xn+1-x0 for all n≥1. Therefore the sequence {xn-x0} is nondecreasing. It then follows that limn→∞⁡xn-x0 exists. From the construction of Cn we have that Cm⊂Cn and xm=PCmx0∈Cn for any integer m≥n. It also follows from Lemma 5(iii) that (26)xm-xn2=xm-PCnx02≤xm-x02-PCnx0-x02=xm-x02-xn-x02.Letting m,n→∞ in (26), we have xm-xn→0. Hence {xn} is a Cauchy sequence. Since H is Hilbert and C is closed and convex we can assume that xn→p∈C as n→∞; that is, limn→∞⁡xn-p=0. We now show that p∈F(S). In particular when m=n+1 in (26) we obtain (27)limn→∞⁡xn+1-xn=0.Also, since xn+1∈Cn+1, we obtain (28)xn+1-un≤xn+1-xn.It then follows from (27) that (29)limn→∞⁡xn+1-un=0.Combining (27) and (29) we obtain (30)limn→∞⁡xn-un=0.It follows from limn→∞⁡xn-p=0 and (30) that (31)limn→∞⁡un-p=0.Setting n=k in (22) we have (32)un-q2≤xn-q2-βn1-αnαn-λ2xn-vn2-1-αn1-βnαn-λ1xn-zn2-1-αn21-βnβnvn-zn2.Observe that (33)q-xn2-q-un2=xn2-un2-2q,xn-un≤xn-unxn+un+qxn-un.It then follows from (30) that (34)limn→∞⁡q-xn-q-un=0.Using liminfn→∞⁡(1-αn)(1-βn)(αn-λ1)>0 and liminfn→∞⁡(1-αn)(αn-λ2)βn>0 we obtain from (32) that limn→∞⁡xn-vn=0 and limn→∞⁡xn-zn=0. Hence p∈F(S)∩F(T). It remains to show that p is in EP(f). Now from (32) (35)q-yn≤q-xn.Also, using un=Trnyn, Lemma 4, and (35) we have (36)un-yn2=Trnyn-yn2≤q-yn2-q-Trnyn2≤q-xn2-q-Trnyn2=q-yn2-q-un2.It then follows from (34) and (36) that (37)limn→∞⁡un-yn=0.Consequently, we obtain from (31) and (37) that (38)limn→∞⁡yn-p=0.From the assumption that rn≥a>0, (39)limn→∞⁡un-ynrn=0.Since un=Trnyn implies (40)fun,y+1rny-un,un-yn≥0,we have from (A2) that (41)un-yn2rn≥1rny-un,un-yn≥-fun,y≥fy,un,∀y∈C.By taking limit as n→∞ of the above inequality and from (A4), (31), and (38) we have f(y,p)≤0, for all y∈C. Let t∈(0,1) and for all y∈C, since p∈C, we have that yt=ty+(1-t)p∈C. Hence f(yt,p)≤0. It follows from (A1) that (42)0=fyt,yt≤tfyt,y+1-tfyt,p≤tfyt,y;that is, f(yt,y)≥0. Letting t↓0, from (A3) we obtain f(p,y)≥0 for all y∈C so that p∈EP(f). Hence y∈F.
Finally we show thatP=PFx0. By taking the limits as n→∞ in (23) we have (43)p-q,x0-p≥0,∀q∈F.It then follows from Lemma 5(i) that p=PFx0. This completes the proof.Corollary 8.
LetC be a nonempty closed convex subset of a real Hilbert space H, let f:C×C→R be a bifunction satisfying (A1)–(A4), and let S,T:C→P(C) be two strictly pseudocontractive mappings with contractive coefficients λ1 and λ2, respectively, such that F=Fs(T)∩Fs(S)∩EP(f)≠∅. Let {xn} be a sequence generated from an arbitrary x0∈C as follows: (44)x0∈H,C1=C,x1=PCx0,yn=αnxn+1-αnβnvn+1-βnzn,un∈CsuchthatFun,y+1rny-un,un-yn≥0,∀y∈C,Cn+1=z∈Cn:z-un2≤z-xn2,xn+1=PCn+1x0,where vn∈PTxn and zn∈PSxn. {αn}n=1∞ and {βn}n=1∞ are sequences in [0,1] satisfying (i)
αn≥max⁡{λ1,λ2},(ii)
liminfn→∞⁡(1-αn)(1-βn)(αn-λ1)>0 and liminfn→∞⁡(1-αn)(αn-λ2)βn>0,(iii)
{rn}⊂[a,∞) for some a>0.Then {xn} converges strongly to p∈PFx0.Proof.
The proof follows easily from Theorem7.



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*Source: 3094838-2016-03-16.xml*