A Strong Convergence Theorem for Relatively Nonexpansive Mappings and Equilibrium Problems in Banach Spaces

and Applied Analysis 3 xn ⇀ x ∈ X and ‖xn‖ → ‖x‖, then xn → x. It is known that if X is uniformly convex, then X has the Kadec-Klee property. The normalized duality mapping J from X to X∗ is defined by Jx { x∗ ∈ X∗ : 〈x, x∗〉 ‖x‖ ‖x∗‖2 } 2.3 for any x ∈ X. We list some properties of mapping J as follows. i If X is a smooth Banach space with Gâteaux differential norm , then J is singlevalued and demicontinuous. If X is a smooth reflexive Banach space, then J is single-valued and hemicontinuous. If X is a strongly smooth Banach space with Fréchet differential norm , then J is single-valued and continuous. ii J is uniformly continuous on every bounded set of a uniformly smooth Banach space. iii If X is a reflexive, smooth and strictly convex Banach space, J∗ : X∗ → X is the duality mapping of X∗, then J−1 J∗, JJ∗ IX∗ , J∗J IX . Let X be a smooth Banach space andK be a nonempty, closed and convex subset of X. The function φ : X ×X → R is defined by φ ( y, x ) ∥∥y∥∥2 − 2〈y, Jx〉 ‖x‖ 2.4 for all x, y ∈ X. Next, we recall the concept of the generalized f-projector operator, together with its properties. Let G : K ×X∗ → R ∪ { ∞} be a functional defined as follows: G ( ξ, φ ) ‖ξ‖ − 2〈ξ, φ〉 ∥∥φ∥∥2 2ρf ξ , 2.5 where ξ ∈ K, φ ∈ X∗, ρ is a positive number and f : K → R ∪ { ∞} is proper, convex, and lower semicontinuous. From the definitions of G and f , it is easy to have the following properties: i G ξ, φ is convex and continuous with respect to φwhen ξ is fixed; ii G ξ, φ is convex and lower semicontinuous with respect to ξ when φ is fixed. Definition 2.1. Let X be a real smooth Banach space andK be a nonempty, closed and convex subset of X. We say that ΠfK : X → 2 is a generalized f-projection operator if ΠfKx { u ∈ K : G u, Jx inf ξ∈K G ξ, Jx } , ∀x ∈ X. 2.6 In order to obtain our results, the following lemmas are crucial to us. Lemma 2.2 see 15 . Let X be a real Banach space and f : X → R ∪ { ∞} be a lower semicontinuous convex functional. Then there exist x∗ ∈ X∗ and α ∈ R such that f x ≥ 〈x, x∗〉 α, ∀x ∈ X. 2.7 4 Abstract and Applied Analysis Lemma 2.3 see 16 . Let X be a uniformly convex and smooth Banach space and let {yn}, {zn} be two sequences of X. If φ yn, zn → 0 and either {yn} or {zn} is bounded, then yn − zn → 0. Let K be a closed subset of a real Banach space X, and let T be a mapping from K to K. We denote by F T the set of all fixed points of T . A point p in K is said to be an asymptotic fixed point of T , ifK contains a sequence {xn}which converges weakly to p such that limn→∞ xn−Txn 0. The set of all asymptotic fixed points of T will be denoted by F̂ T . T is called nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ K, and relatively nonexpansive if F̂ T F T and φ p, Tx ≤ φ p, x for all x ∈ K and p ∈ F T . Obviously, the definition of relatively nonexpansive mapping T is equivalent to F̂ T F T and G p, JTx ≤ G p, Jx for all x ∈ K and p ∈ F T . Lemma 2.4 see 17 . Let X be a strictly convex and smooth Banach space, let K be a closed, and convex subset of X, and let T be a relatively nonexpansive mapping from K into itself. Then F T is closed, and convex. Lemma 2.5 see 10 . Let X be a real reflexive and smooth Banach space and let K be a nonempty, closed, and convex subset of X. The following statements hold: i ΠfKx is a nonempty, closed, and convex subset of K for all x ∈ X; ii for all x ∈ X, x̂ ∈ ΠfKx if and only if 〈 x̂ − y, Jx − Jx̂〉 ρf(y) − ρf x̂ ≥ 0, ∀y ∈ K; 2.8 iii if X is strictly convex, then ΠfK is a single-valued mapping. Lemma 2.6 see 10 . Let X be a real reflexive and smooth Banach space, let K be a nonempty, closed, and convex subset of X, and let x ∈ X, x̂ ∈ ΠfKx. Then φ ( y, x̂ ) G x̂, Jx ≤ G(y, Jx), ∀y ∈ K. 2.9 Lemma 2.7 see 10 . Let X be a Banach space and y ∈ X. Let f : X → R ∪ { ∞} be a proper, convex and lower semicontinuous functional with convex domainD f . If {xn} is a sequence inD f such that xn ⇀ x̂ ∈ int D f and limn→∞G xn, Jy G x̂, Jy , then limn→∞‖xn‖ ‖x̂‖. LetM be a closed and convex subset of a real Banach space X and g : M ×M → R be a bifunction. The equilibrium problem for g is as follows. Find x̂ ∈ M such that g ( x̂, y ) ≥ 0, ∀y ∈ M. 2.10 The set of all solutions for the above equilibrium problem is denoted by EP g . For solving the equilibrium problem, one always assumes that the bifunction g satisfies the following conditions: A1 g x, x 0, for all x ∈ M; A2 g is monotone, that is, g x, y g y, x ≤ 0, for all x, y ∈ M; Abstract and Applied Analysis 5 A3 for all x, y, z ∈ M, lim supt↓0g tz 1 − t x, y g x, y ; A4 for all x ∈ M, g x, · is convex and lower semicontinuous.and Applied Analysis 5 A3 for all x, y, z ∈ M, lim supt↓0g tz 1 − t x, y g x, y ; A4 for all x ∈ M, g x, · is convex and lower semicontinuous. In order to prove our results, we present several necessary lemmas. Lemma 2.8 see 14 . Let M be a closed and convex subset of a uniformly smooth, strictly convex and reflexive Banach space X, and g ·, · be a bifunction fromM ×M → R satisfying the conditions (A1)–(A4). For all r > 0 and x ∈ X, define the mappingas follows. Trx { z ∈ M : g(z, y) 1 r 〈 Jz − Jx, y − z〉 ≥ 0, ∀y ∈ M}. 2.11 Then, the following statements hold: B1 Tr is single-valued; B2 Tr is a firmly nonexpansive-type mapping, that is, for all x, y ∈ X, 〈 JTrx − JTry, Trx − Try 〉 ≤ 〈Jx − Jy, Trx − Try〉; 2.12 B3 F Tr F̂ Tr EP g ; B4 EP g is closed and convex. Lemma 2.9 see 14 . LetM be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space X, g be a bifunction from M ×M to R satisfying the conditions (A1)–(A4), and r > 0. Then, for any x ∈ X and q ∈ F Tr , φ ( q, Trx ) φ Trx, x ≤ φ ( q, x ) . 2.13 3. The Main Result In this section, we prove a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems in Banach spaces. Theorem 3.1. Let X be a uniformly convex and uniformly smooth Banach space, K and M be two nonempty, closed and convex subsets of X such that K ∩ M/ ∅. Let T : K → K be a relatively nonexpansive mapping and f : X → R a convex and lower semicontinuous mapping with K ⊂ int D f . Let g ·, · be a bifunction from M × M → R, which satisfies the conditions (A1)–(A4). 6 Abstract and Applied Analysis Assume that {αn}n 0 is a sequence in 0, 1 such that lim supn→∞αn < 1, and {rn} ⊂ a,∞ for some a > 0. Define a sequence {xn} in K ∩M by the following algorithm: x0 x ∈ K ∩M, H0 K ∩M, yn J−1 αnJxn 1 − αn JTxn , un ∈ M such that g ( un, y ) 1 rn 〈 Jun − Jyn, y − un 〉 ≥ 0, ∀y ∈ M, Hn 1 { z ∈ Hn : G z, Jun ≤ G ( z, Jyn ) ≤ G z, Jxn }, xn 1 Π f Hn 1 x, n 0, 1, 2, . . . . 3.1 If F F T ∩ EP g is nonempty, then {xn} converges strongly toΠfFx. Proof . The proof is divided into the following four steps. I First, we prove the following conclusion: Hn is a closed convex set and F ⊂ Hn for all n ≥ 0. It is obvious that H0 is a closed convex set and F ⊂ H0. Thus, we only need to show that Hn is a closed convex set and F ⊂ Hn for all n ≥ 1. Since G z, Jun ≤ G z, Jyn and G z, Jyn ≤ G z, Jxn are respectively equivalent to 2 〈 z, Jyn − Jun 〉 ‖un‖ − ∥∥yn∥∥2 ≤ 0, 2 〈 z, Jxn − Jyn 〉 ∥∥yn∥∥2 − ‖xn‖ ≤ 0, 3.2 it follows thatHn 1 is closed and convex for all n ≥ 0. Thus, we know that {xn} is well defined. Further, for any u ∈ F and n ≥ 0, we have G ( u, Jyn ) ‖u‖ − 2〈u, αnJxn 1 − αn JTxn〉 ‖αnJxn 1 − αn JTxn‖ 2ρf u ≤ ‖u‖ − 2αn〈u, Jxn〉 − 2 1 − αn 〈u, JTxn〉 αn‖xn‖ 1 − αn ‖Txn‖ 2ρf u αn ( ‖u‖ − 2〈u, Jxn〉 ‖xn‖ 2ρf u ) 1 − αn ( ‖u‖ − 2〈u, JTxn〉 ‖Txn‖ 2ρf u ) αnG u, Jxn 1 − αn G u, JTxn ≤ G u, Jxn . 3.3 On the other hand, it follows from the definition of {un} and Lemma 2.8 that un Trnyn. From Lemma 2.9, we obtain φ u, un φ ( u, Trnyn ) ≤ φ(u, yn), 3.4 Abstract and Applied Analysis 7 which implies that G u, Jun ≤ G ( u, Jyn ) . 3.5and Applied Analysis 7 which implies that G u, Jun ≤ G ( u, Jyn ) . 3.5 Therefore, u ∈ Hn 1 for all n ≥ 0. II Second, we show that {xn} is bounded and limn→∞G xn, Jx exists. Since f : X → R is a convex and lower semicontinuous mapping, a direct application of Lemma 2.2 yields that there exist x∗ ∈ X∗ and α ∈ R such that f ( y ) ≥ 〈y, x∗〉 α, ∀y ∈ X. 3.6