Strong Convergence Theorems for a Generalized Mixed Equilibrium Problem and a Family of Total Quasi-φ-Asymptotically Nonexpansive Multivalued Mappings in Banach Spaces

and Applied Analysis 3 Let X be a smooth Banach space. We always use φ : X × X → R to denote the Lyapunov functional defined by φ ( x, y ) ‖x‖ − 2〈x, Jy〉 ∥∥y∥∥2, ∀x, y ∈ X. 1.7 It is obvious from the definition of the function φ that (‖x‖ − ∥∥y∥∥)2 ≤ φ(x, y) ≤ (‖x‖ ∥∥y∥∥)2, ∀x, y ∈ X. 1.8 Following Alber 4 , the generalized projection ΠC : X → C is defined by ΠC x arg inf y∈C φ ( y, x ) , ∀x ∈ X. 1.9 Lemma 1.2 see 4 . Let X be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of X. Then, the following conclusions hold: a φ x,ΠCy φ ΠCy, y ≤ φ x, y for all x ∈ C and y ∈ X, b if x ∈ X and z ∈ C, then z ΠCx iff 〈 z − y, Jx − Jz〉 ≥ 0, ∀y ∈ C, 1.10 c for x, y ∈ X, φ x, y 0 if and only if x y. Let X be a smooth, strictly convex, and reflexive Banach space, C a nonempty closed convex subset of X, and T : C → C a mapping. A point p ∈ C is said to be an asymptotic fixed point of T if there exists a sequence {xn} ⊂ C such that xn ⇀ p and ‖xn − Txn‖ → 0. We denoted the set of all asymptotic fixed points of T by F̃ T . Definition 1.3. 1 A mapping T : C → C is said to be relatively nonexpansive 5 if F T / ∅, F T F T̃ and φ ( p, Tx ) ≤ φ(p, x), ∀x ∈ C, p ∈ F T . 1.11 2 A mapping T : C → C is said to be closed if, for any sequence {xn} ⊂ C with xn → x and Txn → y, Tx y. LetC be a nonempty closed convex subset of a Banach spaceX. LetN C be the family of nonempty subsets of C. 4 Abstract and Applied Analysis Definition 1.4. 1 Let T : C → N C be a multivalued mapping and q a point in C. The definitions of Tq, T2q, T3q, . . . , Tq, n ≥ 1 are as follows: Tq : { q1 : q1 ∈ T ( q )} , T2q T ( T ( q )) : ⋃ q1∈T q T ( q1 ) , T3q T ( T2 ( q )) : ⋃ q2∈T2 q T ( q2 ) , .. Tq T ( Tn−1 ( q )) : ⋃ qn−1∈Tn−1 q T ( qn−1 ) , n ≥ 1. 1.12 2 Let T : C → N C be a multivalued mapping. A point p ∈ C is said to be an asymptotic fixed point of T if there exists a sequence {xn} ⊂ C such that xn ⇀ p and limn→∞d xn, T xn 0. We denoted the set of all asymptotic fixed points of T by F̃ T . 3 A multivalued mapping T : C → N C is said to be relatively nonexpansive 5 if F T / ∅, F T F̃ T and φ ( p,w ) ≤ φ(p, x), ∀x ∈ C, w ∈ Tx, p ∈ F T . 1.13 4 A multivalued mapping T : C → N C is said to be closed if, for any sequence {xn} ⊂ C with xn → x and wn ∈ T xn with wn → y, then y ∈ Tx. Definition 1.5. 1 Amultivalued mapping T : C → N C is said to be quasi-φ-nonexpansive if F T / ∅ and φ ( p,w ) ≤ φ(p, x), ∀x ∈ C, w ∈ Tx, p ∈ F T . 1.14 2 A multivalued mapping T : C → N C is said to be quasi-φ-asymptotically nonexpansive if F T / ∅ and there exists a real sequence {kn} ⊂ 1,∞ with kn → 1 such that φ ( p,wn ) ≤ knφ ( p, x ) , ∀n ≥ 1, x ∈ C, wn ∈ Tx, p ∈ F T . 1.15 3 A multivalued mapping T : C → N C is said to be total quasi-φ-asymptotically nonexpansive if F T / ∅ and there exist nonnegative real sequences {νn}, {μn} with νn → 0, μn → 0 as n → ∞ and a strictly increasing continuous function ζ : R → R with ζ 0 0 such that for all x ∈ C, p ∈ F T φ ( p,wn ) ≤ φ(p, x) νnζ ( φ ( p, x )) μn, ∀n ≥ 1, wn ∈ Tx. 1.16 Abstract and Applied Analysis 5 Definition 1.6. 1 Let {Ti}i 1 : C → N C be a sequence of mappings. {Ti}i 1 is said to be a family of uniformly total quasi-φ-asymptotically nonexpansive multivalued mappings if ∩i 1F Ti / ∅ and there exist nonnegative real sequences {νn}, {μn} with νn → 0, μn → 0 as n → ∞ and a strictly increasing continuous function ζ : R → R with ζ 0 0 such that for all i ≥ 1, x ∈ C, p ∈ ∩i 1F Tiand Applied Analysis 5 Definition 1.6. 1 Let {Ti}i 1 : C → N C be a sequence of mappings. {Ti}i 1 is said to be a family of uniformly total quasi-φ-asymptotically nonexpansive multivalued mappings if ∩i 1F Ti / ∅ and there exist nonnegative real sequences {νn}, {μn} with νn → 0, μn → 0 as n → ∞ and a strictly increasing continuous function ζ : R → R with ζ 0 0 such that for all i ≥ 1, x ∈ C, p ∈ ∩i 1F Ti φ ( p,wn,i ) ≤ φ(p, x) νnζ ( φ ( p, x )) μn, ∀wn,i ∈ T i x, ∀n ≥ 1. 1.17 2 A total quasi-φ-asymptotically nonexpansivemultivaluedmapping T : C → N C is said to be uniformly L-Lipschitz continuous if there exists a constant L > 0 such that ‖wn − sn‖ ≤ L ∥∥x − y∥∥, ∀x, y ∈ C, wn ∈ Tx, sn ∈ Ty, n ≥ 1. 1.18 In 2005, Matsushita and Takahashi 5 proved weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space X. In 2008, Plubtieng and Ungchittrakool 6 proved the strong convergence theorems to approximate a fixed point of two relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space X. In 2010, Chang et al. 7 obtained the strong convergence theorem for an infinite family of quasi-φ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space X with Kadec-Klee property. In 2011, Chang et al. 8 proved some approximation theorems of common fixed points for countable families of total quasi-φasymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space X with Kadec-Klee property. In 2011, Homaeipour and Razani 9 proved weak and strong convergence theorems for a single relatively nonexpansive multivalued mapping in a uniformly convex and uniformly smooth Banach space X. On the other hand, In 2009, Zhang 10 proved the strong convergence theorem for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a finite family of quasiφ-asymptotically nonexpansive mappings in a uniformly smooth and uniformly convex Banach space. Recently, Tang 11 , Cho et al. 12–21 , and Noor et al. 22–26 extended the finite family of quasi-φ-asymptotically nonexpansive mappings to infinite family of quasi-φasymptotically nonexpansive mappings. Motivated and inspired by the researches going on in this direction, the purpose of this paper is by using the hybrid iterative algorithm to find a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a infinite family of total quasiφ-asymptotically nonexpansive multivalued mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. In order to get the strong convergence theorems, the hybrid algorithms are presented and used to approximate the fixed point. The results presented in the paper improve and extend some recent results announced by some authors. 6 Abstract and Applied Analysis 2. Preliminaries Lemma 2.1 see 8 . LetX be a real uniformly smooth and strictly convex Banach space with KadecKlee property and C a nonempty closed convex set ofX. Let {xn} and {yn} be two sequences in C such that xn → p and φ xn, yn → 0, where φ is the function defined by 1.7 , and then yn → p. Lemma 2.2. Let X and C be as in Lemma 2.1. Let T : C → N C be a closed and total quasi-φasymptotically nonexpansive multivalued mapping with nonnegative real sequences {νn}, {μn} and a strictly increasing continuous function ζ : R → R such that νn → 0, μn → 0 (as n → ∞), and ζ 0 0. If μ1 0, then the fixed point set F T is a closed and convex subset of C. Proof. Letting {xn} be a sequence in F T with xn → p as n → ∞ , we prove that p ∈ F T . In fact, by the assumption that T is a total quasi-φ-asymptotically nonexpansive multivalued mapping and μ1 0, we have φ xn, u ≤ φ ( xn, p ) ν1ζ ( φ ( xn, p )) , ∀u ∈ Tp. 2.1 Furthermore, we have φ ( p, u ) lim n→∞ φ xn, u ≤ lim n→∞ ( φ ( xn, p ) ν1ζ ( φ ( xn, p ))) 0, ∀u ∈ Tp. 2.2 By Lemma 1.2 c , p u. Hence, p ∈ Tp. This implies that p ∈ F T , that is, F T is closed. Next, we prove that F T is convex. For any x, y ∈ F T , t ∈ 0, 1 , putting q tx 1 − t y, we prove that q ∈ F T . Indeed, let {un} be a sequence generated by u1 ∈ Tq, u2 ∈ Tu1 ⊂ T2q, u3 ∈ Tu2 ⊂ T3q, .. un ∈ Tun−1 ⊂ Tq, .. 2.3 In view of the definition of φ x, y , for all un ∈ Tun−1 ⊂ Tq, we have φ ( q, un ) ∥∥q∥∥2 − 2〈q, Jun 〉 ‖un‖ ∥∥q∥∥2 − 2t〈x, Jun〉 − 2 1 − t 〈 y, Jun 〉 ‖un‖ ∥∥q∥∥2 tφ x, un 1 − t φ ( y, un ) − t‖x‖ − 1 − t ∥∥y∥∥2 2.4 Abstract and Applied Analysis 7 since tφ x, un 1 − t φ ( y, un ) ≤ t(φ(x, q) νnζ ( φ ( x, q )) μn ) 1 − t (φ(y, q) νnζ ( φ ( y, q )) μn )and Applied Analysis 7 since tφ x, un 1 − t φ ( y, un ) ≤ t(φ(x, q) νnζ ( φ ( x, q )) μn ) 1 − t (φ(y, q) νnζ ( φ ( y, q )) μn ) t ( ‖x‖ − 2〈x, Jq〉 ∥∥q∥∥2 νnζ ( φ ( x, q )) μn ) 1 − t ∥∥y∥∥2 − 2〈y, Jq〉 ∥∥q∥∥2 νnζ ( φ ( y, q )) μn ) t‖x‖ 1 − t ∥∥y∥∥2 − ∥∥q∥∥2 tνnζ ( φ ( x, q )) 1 − t νnζ ( φ ( y, q )) μn. 2.5 Substituting 2.5 into 2.4 and simplifying it, we have φ ( q, un ) ≤ tνnζ ( φ ( x, q )) 1 − t νnζ ( φ ( y, q )) μn −→ 0 n −→ ∞ . 2.6 By Lemma 2.1, we have un → q as n → ∞ . This implies that un 1 → q as n → ∞ . Since T is closed, we have q ∈ Tq, that is, q ∈ F T . This completes the proof of Lemma 2.2. Lemma 2.3 see 7 . LetX be a uniformly convex Banach space, r > 0, a positive number, and Br 0 a closed ball ofX. Then, for any given sequence {xn}n 1 ⊂ Br 0 and for any given sequence {λn}n 1 of positive numbers with Σn 1λn 1, there exists a continuous, strictly increasing, and convex function g : 0, 2r → 0,∞ with g 0 0 such that for any positive integers i, j with i < j, ∥∥∥∥ ∞ ∑ n 1 λnxn ∥∥∥∥ 2 ≤ ∞ ∑ n 1 λn‖xn‖ − λiλjg ∥xi − xj ∥∥). 2.7 For solving the generalized mixed equilibrium problem, let us assume that the function ψ : C → R is convex and lower semicontinuous, the nonlinear mapping A : C → X∗ is continuous and monotone, and the bifunction Θ : C × C → R satisfies the following conditions: A1 Θ x, x 0, for all x ∈ C, A2 Θ is monotone, that is, Θ x, y Θ y, x ≤ 0, for all x, y ∈ C, A3 lim supt↓0Θ x t z − x , y ≤ Θ x, y , for all x, y, z ∈ C, A4 the function y → Θ x, y is convex and lower semicontinuous. Lemma 2.4. Let X be a smooth, strictly convex and reflexive Banach space and C a nonempty closed convex subset of X. Let Θ : C × C → R be a bifunction satisfying conditions (A1)–(A4). Let r > 0 and x ∈ X. Then, the following hold. i [27] There exists z ∈ C such that Θ ( z, y ) 1 r 〈 y − z, Jz − Jx〉 ≥ 0, ∀y ∈ C. 2.8 8 Abstract and Applied Analysis ii [28] Define a mapping Tr : X → C by Trx { z ∈ C : Θ(z, y) 1 r 〈 y − z, Jz − Jx〉 ≥ 0, ∀y ∈ C } , x ∈ X. 2.9 Then, the following conclusions hold: a Tr is single-valued, b Tr is a firmly nonexpansive-type mapping, that is, for all z, y ∈ X, 〈 Tr z − Tr ( y ) , JTr z − JTr ( y )〉 ≤ Tr z − Tr ( y ) , Jz − Jy〉, 2.10 c F Tr EP Θ ̂ F Tr , d EP Θ is closed and convex, e φ q, Tr x φ Tr x , x ≤ φ q, x , for all q ∈ F Tr . Lemma 2.5 see 10 . Let X be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of X. Let A : C → X∗ be a continuous and monotone mapping, ψ : C → R a lower semicontinuous and convex function, andΘ : C×C → R a bifunction satisfying conditions (A1)–(A4). Let r > 0 be any given number and x ∈ X any given point. Then, the following hold. i There exists u ∈ C such that for all y ∈ C Θ ( u, y ) 〈 Au, y − u〉 ψ(y) − ψ u 1 r 〈 y − u, Ju − Jx〉 ≥ 0. 2.11 ii If one defines a mapping Kr : C → C by Kr x { u ∈ C : Θ(u, y) 〈Au, y − u〉 ψ(y) − ψ u 1 r 〈 y − u, Ju − Jx〉 ≥ 0, ∀y ∈ C } , x ∈ C, 2.12 then, the mapping Kr has the following properties: a Kr is single-valued, b Kr is a firmly nonexpansive-type mapping, that is, for all z, y ∈ X 〈 Kr z −Kr ( y ) , JKr z − JKr ( y )〉 ≤ Kr z −Kr ( y ) , Jz − Jy〉, 2.13 c F Kr Ω ̂ F Kr , d Ω is a closed convex set of C, e φ p,Kr z φ Kr z , z ≤ φ p, z , for all p ∈ F Kr , z ∈ X. Abstract and Applied Analysis 9 Remark 2.6. It follows from Lemma 2.4 that the mapping Kr : C → C defined by 2.12 is a relatively nonexpansive mapping. Thus, it is quasi-φ-nonexpansive.and Applied Analysis 9 Remark 2.6. It follows from Lemma 2.4 that the mapping Kr : C → C defined by 2.12 is a relatively nonexpansive mapping. Thus, it is quasi-φ-nonexpansive. 3. Main Results In this section, we will use the hybrid iterative algorithm to find a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a infinite family of total quasiφ-asymptotically nonexpansive multivalued mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. Theorem 3.1. Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and C a nonempty closed and convex subset of X. Let Θ : C × C → R be a bifunction satisfying conditions (A1)–(A4), A : C → X∗ a continuous and monotone mapping, and ψ : C → R a lower semicontinuous and convex function. Let {Ti}i 1 : C → N C be an infinite family of closed and uniformly total quasi-φ-asymptotically nonexpansive multivalued mappings with nonnegative real sequences {νn}, {μn} and a strictly increasing continuous function ζ : R → R such that μ1 0, νn → 0, μn → 0 (as n → ∞) and ζ 0 0 and for each i ≥ 1, Ti is uniformly Li-Lipschitz continuous. Let x0 ∈ C, C0 C, and let {xn} be a sequence generated by xn 1 ∏ Cn 1 x0, Cn 1 { ν ∈ Cn : φ ν, un ≤ φ ν, xn ξn } , ∀n ≥ 0, yn J−1 αnJxn 1 − αn Jzn , zn J−1 ( βn,0Jxn ∞ ∑