Strong Convergence of Generalized Projection Algorithms for Nonlinear Operators

and Applied Analysis 3 nonexpansive mapping such that F S ∩ A−10/ ∅. Let {xn} be a sequence generated by x0 x ∈ C and un J−1 ( βnJxn ( 1 − βn ) JSJrnxn ) , Cn { z ∈ C : φ z, un ≤ φ z, xn } , Qn {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0}, xn 1 ΠCn∩Qnx0 1.6 for all n ∈ N ∪ {0}, where J is the duality mapping on E, {βn} ⊂ 0, 1 , and {rn} ⊂ a,∞ for some a > 0. If lim infn→∞ 1 − βn > 0, then {xn} converges strongly to ΠF S ∩A−10x0, where ΠF S ∩A−10 is the generalized projection of E onto F S ∩A−10. The purpose of this paper is to employ the idea of Inoue et al. 17 and Das and Debata 1 to introduce a new hybrid method for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings. We prove a strong convergence theorem of the new hybrid method. Moreover we apply our main results to obtain strong convergence for a maximal monotone operator and two nonexpansive mappings in a Hilbert space. 2. Preliminaries Throughout this paper, all linear spaces are real. LetN andR be the sets of all positive integers and real numbers, respectively. Let E be a Banach space and let E∗ be the dual space of E. For a sequence {xn} of E and a point x ∈ E, the weak convergence of {xn} to x and the strong convergence of {xn} to x are denoted by xn ⇀ x and xn → x, respectively. Let S E be the unit sphere centered at the origin of E. Then the space E is said to be smooth if the limit lim t→ 0 ∥x ty ∥ − ‖x‖ t 2.1 exists for all x, y ∈ S E . It is also said to be uniformly smooth if the limit exists uniformly in x, y ∈ S E . A Banach space E is said to be strictly convex if ‖ x y /2‖ < 1 whenever x, y ∈ S E and x / y. It is said to be uniformly convex if for each ∈ 0, 2 , there exists δ > 0 such that ‖ x y /2‖ < 1 − δ whenever x, y ∈ S E and ‖x − y‖ ≥ . We know the following 18 : i if E is smooth, then J is single-valued; ii if E is reflexive, then J is onto; iii if E is strictly convex, then J is one to one; iv if E is strictly convex, then J is strictly monotone; v if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. 4 Abstract and Applied Analysis A Banach space E is said to have the Kadec-Klee property if for a sequence {xn} of E satisfying that xn ⇀ x and ‖xn‖ → ‖x‖, xn → x. It is known that if E is uniformly convex, then E has the Kadec-Klee property; see 18, 19 for more details. Let E be a smooth, strictly convex, and reflexive Banach space and let C be a closed convex subset of E. Throughout this paper, define the function φ : E × E → R by φ ( y, x ) ∥ ∥y ∥ ∥2 − 2y, Jx ‖x‖, ∀y, x ∈ E. 2.2 Observe that, in a Hilbert space H, 2.2 reduces to φ x, y ‖x − y‖2, for all x, y ∈ H. It is obvious from the definition of the function φ that, for all x, y ∈ E, 1 ‖x‖ − ‖y‖ 2 ≤ φ x, y ≤ ‖x‖ ‖y‖ , 2 φ x, y φ x, z φ z, y 2〈x − z, Jz − Jy〉, 3 φ x, y 〈x, Jx − Jy〉 〈y − x, Jy〉 ≤ ‖x‖‖Jx − Jy‖ ‖y − x‖‖y‖. Following Alber 20 , the generalized projection ΠC from E onto C is a map that assigns to an arbitrary point x ∈ E the minimum point of the functional φ y, x ; that is, ΠCx x, where x is the solution to the minimization problem φ x, x min y∈C φ ( y, x ) . 2.3 Existence and uniqueness of the operator ΠC follows from the properties of the functional φ y, x and strict monotonicity of the mapping J . In a Hilbert space, ΠC 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 6 . Let E be a uniformly convex and smooth Banach space and let {xn} and {yn} be two 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 Matsushita and Takahashi 15 . Let C be a 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 2.3 Alber 20 and Kamimura and Takahashi 6 . Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space, x ∈ E and let z ∈ C. Then, z ΠCx if and only if 〈y − z, Jx − Jz〉 ≤ 0 for all y ∈ C. Lemma 2.4 Alber 20 and Kamimura and Takahashi 6 . Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space. Then φ ( x,ΠCy ) φ ( ΠCy, y ) ≤ φx, y, ∀x ∈ C, y ∈ E. 2.4 Let E be a smooth, strictly convex, and reflexive Banach space, and let A be a setvalued mapping from E to E∗ with graph G A { x, x∗ : x∗ ∈ Ax}, domain D A {z ∈ E : Az/ ∅}, and range R A ∪{Az : z ∈ D A }. We denote a set-valued operator A from E to E∗ by A ⊂ E × E∗. A is said to be monotone if 〈x − y, x∗ − y∗〉 ≥ 0, for all x, x∗ , y, y∗ ∈ A. A monotone operator A ⊂ E × E∗ is said to be maximal monotone if its graph is not properly Abstract and Applied Analysis 5 contained in the graph of any other monotone operator. We know that if A is a maximal monotone operator, then A−10 {z ∈ D A : 0 ∈ Az} is closed and convex. The following theorem is well known. Lemma 2.5 Rockafellar 21 . Let E be a smooth, strictly convex, and reflexive Banach space and letA ⊂ E ×E∗ be a monotone operator. ThenA is maximal if and only if R J rA E∗ for all r > 0. Let E be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty closed convex subset of E and let A ⊂ E × E∗ be a monotone operator satisfyingand Applied Analysis 5 contained in the graph of any other monotone operator. We know that if A is a maximal monotone operator, then A−10 {z ∈ D A : 0 ∈ Az} is closed and convex. The following theorem is well known. Lemma 2.5 Rockafellar 21 . Let E be a smooth, strictly convex, and reflexive Banach space and letA ⊂ E ×E∗ be a monotone operator. ThenA is maximal if and only if R J rA E∗ for all r > 0. Let E be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty closed convex subset of E and let A ⊂ E × E∗ be a monotone operator satisfying