Iterative Schemes for Fixed Points of Relatively Nonexpansive Mappings and Their Applications

and Applied Analysis 3 for all n ∈ N, where x1 x ∈ C and {αn} is a sequence of 0, 1 . Strong convergence of this type of iterative sequence has beenwidely studied: for instance, see 13, 14 and the references therein. In 2006, Martinez-Yanes and Xu 15 have adapted Nakajo and Takahashi’s 10 idea to modify the process 1.5 for a nonexpansive mapping T in a Hilbert space: x0 x ∈ C, yn αnx0 1 − αn Txn, Cn { z ∈ C : ∥ ∥yn − z ∥ ∥2 ≤ ‖xn − z‖ αn ( ‖x0‖ 2〈xn − x0, z〉 )} , Qn {z ∈ C : 〈xn − z, x0 − xn〉 ≥ 0}, xn 1 PCn∩Qnx0, n 0, 1, 2, . . . , 1.6 where PCn∩Qn is the metric projection from C onto Cn ∩ Qn. They proved that if {αn} ⊂ 0, 1 and limn→∞ αn 0, then the sequence {xn} generated by 1.6 converges strongly to PF T x. The ideas to generalize the process 1.3 and 1.6 from Hilbert space to Banach space have been studied by many authors. Matsushita and Takahashi 6 , Qin and Su 16 , and Plubtieng and Ungchittrakool 17 generalized the process 1.3 and 1.6 and proved the strong convergence theorems for relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach spacey; see, for instance, 2, 6, 16, 18–22 and the references therein. Recently, Nakajo et al. 18 introduced the following condition. Let C be a nonempty closed convex subset of a Hilbert space H, let {Tn} be a family of mappings of C into itself with F : ⋂∞ n 1 F Tn / ∅, and let ωw zn denote the set of all weak subsequential limits of a bounded sequence {zn} in C. {Tn} is said to satisfy the NST-condition II if, for every bounded sequence {zn} in C, lim n→∞ ‖zn − Tnzn‖ 0 implies ωw zn ⊂ F. 1.7 Very recently, Nakajo et al. 19 introduced the more general condition so-called the NST∗condition, {Tn} is said to satisfy the NST∗-condition if, for every bounded sequence {zn} in C, lim n→∞ ‖zn − Tnzn‖ lim n→∞ ‖zn − zn 1‖ 0 implies ωw zn ⊂ F. 1.8 It follows directly from the definitions above that if {Tn} satisfies the NST-condition I , then {Tn} satisfies the NST∗-condition. Motivated and inspired by Wei and Cho 2 , in this paper, we introduce two iterative schemes 3.1 and 3.14 and use the NST∗-condition for a countable family of relatively nonexpansive mappings to obtain the strong convergence theorems for finding a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space. Using this result, we also discuss the problem of strong convergence concerning variational inequality, equilibrium, and nonexpansive mappings in Hilbert spaces. Moreover, we also apply our convergence 4 Abstract and Applied Analysis theorems to the maximal monotone operators in Banach spaces. The results obtained in this paper improve and extend the corresponding result of Matsushita and Takahashi 6 , Qin and Su 16 , Wei and Cho 2 , Wei and Zhou 22 , and many others. 2. Preliminaries Let E be a real Banach space with norm ‖ · ‖ and let E∗ be the dual of E. For all x ∈ E and x∗ ∈ E∗, we denote the value of x∗ at x by 〈x, x∗〉. We denote the strong convergence and the weak convergence of a sequence {xn} to x in E by xn → x and xn ⇀ x, respectively. We also denote the weak∗ convergence of a sequence {x∗ n} to x∗ in E∗ by x∗ n⇀x. An operator T ⊂ E × E∗ is said to be monotone if 〈x − y, x∗ − y∗〉 ≥ 0 whenever x, x∗ , y, y∗ ∈ T . We denote the set {x ∈ E : 0 ∈ Tx} by T−10. A monotone T is said to be maximal if its graph G T { x, y : y ∈ Tx} is not properly contained in the graph of any other monotone operator. If T is maximal monotone, then the solution set T−10 is closed and convex. The normalized duality mapping J from E to E∗ is defined by Jx { x∗ ∈ E∗ : 〈x, x∗〉 ‖x‖ ‖x∗‖ } 2.1 for x ∈ E. By Hahn-Banach theorem, J x is nonempty; see 23 for more details. A Banach space E is said to be strictly convex if ‖ x y /2‖ < 1 for all x, y ∈ E with ‖x‖ ‖y‖ 1 and x / y. It is also said to be uniformly convex if, for each ε ∈ 0, 2 , there exists δ > 0 such that ‖x y‖/2 ≤ δ for x, y ∈ E with ‖x‖ ‖y‖ 1 and ‖x − y‖ ≥ ε. Let S E {x ∈ E : ‖x‖ 1} be the unit sphere of E. Then the Banach space E is said to be smooth provided that lim t→ 0 ∥x ty ∥ − ‖x‖ t 2.2 exists for each x, y ∈ S E . It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ S E . It is well known that if E is smooth, strictly convex, and reflexive, then the duality mapping J is single valued, one-to-one, and onto. Let E be a smooth, strictly convex, and reflexive Banach space and letC be a nonempty closed convex subset of E. Throughout this paper, we denote by φ the function defined by φ ( y, x ) ∥y ∥2 − 2 〈 y, Jx 〉 ‖x‖, ∀x, y ∈ C. 2.3 It is obvious from the definition of the function φ that ‖x‖ − ‖y‖ 2 φ y, x ‖y‖2 ‖x‖2 for all x, y ∈ E. Abstract and Applied Analysis 5and Applied Analysis 5 Following Alber 24 , the generalized projection PC from E onto C is defined by PC x arg min y∈C φ ( y, x ) , ∀x ∈ E. 2.4 If E is a Hilbert space, then φ y, x ‖y − x‖2 and PC is the metric projection of H onto C. We need the following lemmas for the proof of our main results. Lemma 2.1 Kamimura and Takahashi 25 . Let E be a uniformly convex and smooth Banach space and let {xn} and {yn} be sequences in E such that either {xn} or {yn} is bounded. If limn→∞ φ xn, yn 0, then limn→∞‖xn − yn‖ 0. Lemma 2.2 Alber 24 , Kamimura and Takahashi 25 . Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Then φ ( x, PCy ) φ ( PCy, y ) ≤ φ ( x, y ) , ∀x ∈ C, y ∈ E. 2.5 Lemma 2.3 Alber 24 , Kamimura and Takahashi 25 . Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let x ∈ E, and let x0 ∈ C. Then x0 PCx ⇐⇒ 〈z − x0, Jx0 − Jx〉 ≥ 0, ∀z ∈ C. 2.6 Lemma 2.4 Matsushita and Takahashi 6 . LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, and let T be a relatively nonexpansive mapping from C into itself. Then F T is closed and convex. Lemma 2.5 see 2 . Let E be a real smooth and uniformly convex Banach space. If S : E → E is a mapping which is nonexpansive with respect to the Lyapunov functional, then F S is convex and closed subset of E. 3. Main Results In this section, by using the NST∗-condition, we proved two strong convergence theorems for finding a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space. Theorem 3.1. Let E be a real uniformly smooth and uniformly convex Banach space, let C be a nonempty closed convex subset of E, and let S : C → C be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let {Tn} be a family of relatively nonexpansive 6 Abstract and Applied Analysis mappings of C into itself such that F : ⋂∞ n 1 F Tn ∩ F S / ∅ and satisfy the NST∗-condition. Then the sequence {xn} generated by x0 ∈ C, yn Tn xn en , zn J−1 ( αnJxn 1 − αn Jyn ) , un Szn, Hn { z ∈ C : φ z, un φ z, zn αnφ z, xn 1 − αn φ z, xn en } , Wn {z ∈ C : 〈z − xn, Jx0 − Jxn〉 ≤ 0}, xn 1 PHn∩Wnx0, n 0, 1, 2, . . . 3.1 converges strongly to PFx0 provided that i {αn} ⊂ 0, 1 with αn ≤ 1 − β for some β ∈ 0, 1 ; ii the error sequence is {en} ⊂ C such that ‖en‖ → 0 as n → ∞. Proof. We split the proof into five steps. Step 1 BothHn and Wn are closed and convex subset of C . Noting the facts that φ z, zn αnφ z, xn 1 − αn φ z, xn en ⇐⇒ ‖zn‖ − αn‖xn‖ − 1 − αn ‖xn en‖ ≤ 2〈z, Jzn − αnJxn − 1 − αn J xn en 〉, φ z, un φ z, zn ⇐⇒ ‖zn‖ − ‖un‖ ≥ 2〈z, Jzn − Jun〉, 3.2 we can easily know that Hn is closed and convex subset of E. It is obvious that Wn is also a closed and convex subset of E. Step 2 F : ⋂∞ n 1 F Tn ∩ F S ⊆ Hn ∩Wn for all n ∈ N ∪ {0} . To observe this, take p ∈ F. Then it follows from the convexity of ‖ · ‖2 that φ ( p, un ) φ ( Sp, Szn ) ≤ φ ( p, zn ) ∥p ∥∥2 − 2 〈 p, αnJxn 1 − αn Jyn 〉 ∥αnJxn 1 − αn Jyn ∥2 ≤ ∥p ∥2 − 2αn 〈 p, Jxn 〉 − 2 1 − αn 〈 p, Jyn 〉 αn‖xn‖ 1 − αn ∥yn ∥2 αnφ ( p, xn ) 1 − αn φ ( p, yn ) αnφ ( p, xn ) 1 − αn φ ( p, Tn xn en ) ≤ αnφ ( p, xn ) 1 − αn φ ( p, xn en ) 3.3 Abstract and Applied Analysis 7and Applied Analysis 7 for all n ∈ N ∪ {0}. Hence F ⊂ Hn for all n ∈ N ∪ {0}. On the other hand, it is clear that p ∈ W0 C. Then p ∈ H0 ∩ W0 and x1 PH0∩W0x0 is well defined. Suppose that p ∈ Wn−1 for some n ≥ 1. Then p ∈ Hn−1 ∩ Wn−1 and xn PHn−1∩Wn−1x0 is well defined. It follows from Lemma 2.3 that 〈 p − xn, Jx0 − Jxn 〉 〈 p − PHn−1∩Wn−1x0, Jx0 − JPHn−1∩Wn−1x0 〉 ≤ 0, 3.4 which implies that p ∈ Wn. Therefore p ∈ Hn ∩ Wn, and hence xn 1 PHn∩Wnx0 is well defined. Then by induction, the sequence {xn} generated by 3.1 is well defined, for each n ≥ 0. Moreover F ⊂ Hn ∩Wn for each nonnegative integer n. Step 3 {xn} is bounded sequence of C . In fact, for all p ∈ F ⊂ Hn ∩Wn ⊂ Wn, it follows from Lemma 2.2 that φ ( p, PWnx0 ) φ PWnx0, x0 ≤ φ ( p, x0 ) . 3.5 By the definition of Wn and Lemma 2.2, we note that xn PWnx0 and hence φ ( p, xn ) φ xn, x0 ≤ φ ( p, x0 ) . 3.6 Therefore {xn} is bounded. Step 4 ωw xn ⊂ F : ⋂∞ n 1 F Tn ∩ F S . From the facts xn PWnx0, xn 1 ∈ Wn, and Lemma 2.2, we have φ xn 1, xn φ xn, x0 ≤ φ xn 1, x0 . 3.7 Therefore, limn→∞ φ xn, x0 exists. Then φ xn 1, xn → 0, which implies from Lemma 2.1 that ‖xn 1 − xn‖ → 0 as n → ∞. Since xn 1 ∈ Hn, we have φ xn 1, un φ xn 1, zn αnφ xn 1, xn 1 − αn φ xn 1, xn en . 3.8