Hybrid Projection Algorithms for Asymptotically Strict Quasi-φ-Pseudocontractions

and Applied Analysis 3 It is well known that, in an infinite dimensional Hilbert space, the normal Mann iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Hybrid projection algorithms are popular tool to prove strong convergence of iterative sequences without compactness assumptions. Recently, hybrid projection algorithms have received rapid developments; see, for example, 3, 5–24 . In this paper, we will introduce a new mapping, asymptotically strict quasi-φ-pseudocontractions, and give a strong convergence theorem by a simple hybrid projection algorithm in a real Banach space. 2. Preliminaries Let E be a Banach space with the dual space E∗. We denote by J the normalized duality mapping from E to 2 ∗ defined by Jx { f∗ ∈ E∗ : 〈x, f∗〉 ‖x‖ ∥∥f∗∥∥2 } , ∀x ∈ E, 2.1 where 〈·, ·〉 denotes the generalized duality pairing of elements between E and E∗; see 25 . It is well known that if E∗ is strictly convex, then J is single valued, and if E∗ is uniformly convex, then J is uniformly continuous on bounded subsets of E. It is also well known that if C is a nonempty closed convex subset of a Hilbert space H and PC : H → C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber 26 recently introduced a generalized projection operator ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces. Recall that a Banach spaceE 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 said to be uniformly convex if limn→∞‖xn −yn‖ 0 for any two sequences {xn} and {yn} in E such that ‖xn‖ ‖yn‖ 1 and limn→∞‖ xn yn /2‖ 1. E is said to have Kadec-Klee property if a sequence {xn} of E satisfying that xn ⇀ x and ‖xn‖ → ‖x‖, then xn → x. It is known that if E is uniformly convex, then E enjoys KadecKlee property; see 25, 27 for more details. Let UE {x ∈ E : ‖x‖ 1} be the unit sphere of E then the Banach space E is said to be smooth provided lim t→ 0 ∥ ∥x ty ∥ ∥ − ‖x‖ t 2.2 exists for each x, y ∈ UE. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ UE. It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. Let E be a smooth Banach space. Consider the functional defined by φ ( x, y ) ‖x‖ − 2〈x, Jy〉 ∥∥y∥∥2, ∀x, y ∈ E. 2.3 Observe that, in a Hilbert space H, 2.3 is reduced to φ x, y ‖x − y‖2 for all x, y ∈ H. The generalized projection ΠC : E → C is a mapping that assigns to an arbitrary point x ∈ E the minimum point of the functional φ x, y , that is, ΠCx x, where x is the solution to the following minimization problem: φ x, x min y∈C φ ( y, x ) . 2.4 4 Abstract and Applied Analysis The existence and uniqueness of the operatorΠC follow from the properties of the functional φ x, y and the strict monotonicity of the mapping J ; see, for example, 26–29 . In Hilbert spaces, ΠC PC. It is obvious from the definition of the function φ that (∥ ∥y ∥ ∥ − ‖x‖) ≤ φ(y, x) ≤ (∥∥y∥∥ ‖x‖), ∀x, y ∈ E, 2.5 φ ( x, y ) φ x, z φ ( z, y ) 2 〈 x − z, Jz − Jy〉, ∀x, y, z ∈ E. 2.6 Remark 2.1. If E is a reflexive, strictly convex, and smooth Banach space, then, for all x, y ∈ E, φ x, y 0 if and only if x y. It is sufficient to show that if φ x, y 0, then x y. From 2.5 , we have ‖x‖ ‖y‖. This implies that 〈x, Jy〉 ‖x‖2 ‖Jy‖2. From the definition of J , we see that Jx Jy. It follows that x y; see 25, 27 for more details. Now, we give some definitions for our main results in this paper. Let C be a closed convex subset of a real Banach space E and T : C → C a mapping. 1 A point p in C is said to be an asymptotic fixed point of T 30 if C contains a sequence {xn} which converges weakly to p such that limn→∞‖xn − Txn‖ 0. The set of asymptotic fixed points of T will be denoted by F̃ T . 2 T is said to be relatively nonexpansive 15, 31, 32 if F̃ T F T / ∅, φ ( p, Tx ) ≤ φ(p, x), ∀x ∈ C, p ∈ F T . 2.7 The asymptotic behavior of a relatively nonexpansive mapping was studied in 30– 32 . 3 T is said to be relatively asymptotically nonexpansive 6, 11 if F̃ T F T / ∅, φ ( p, Tx ) ≤ (1 μn ) φ ( p, x ) , ∀x ∈ C, p ∈ F T , 2.8 where {μn} ⊂ 0,∞ is a sequence such that μn → 1 as n → ∞. 4 T is said to be φ-nonexpansive 14, 16, 17 if φ ( Tx, Ty ) ≤ φ(x, y), ∀x, y ∈ C. 2.9 5 T is said to be quasi-φ-nonexpansive 14, 16, 17 if F T / ∅, φ ( p, Tx ) ≤ φ(p, x), ∀x ∈ C, p ∈ F T . 2.10 6 T is said to be asymptotically φ-nonexpansive 14 if there exists a real sequence {μn} ⊂ 0,∞ with μn → 0 as n → ∞ such that φ ( Tx, Ty ) ≤ (1 μn ) φ ( x, y ) , ∀x, y ∈ C. 2.11 7 T is said to be asymptotically quasi-φ-nonexpansive 14 if there exists a real sequence {μn} ⊂ 0,∞ with μn → 0 as n → ∞ such that F T / ∅, φ ( p, Tx ) ≤ (1 μn ) φ ( p, x ) , ∀x ∈ C, p ∈ F T . 2.12 Abstract and Applied Analysis 5 8 T is said to be a strict quasi-φ-pseudocontraction if F T / ∅, and there exists a constant κ ∈ 0, 1 such thatand Applied Analysis 5 8 T is said to be a strict quasi-φ-pseudocontraction if F T / ∅, and there exists a constant κ ∈ 0, 1 such that φ ( p, Tx ) ≤ φ(p, x) κφ x, Tx , ∀x ∈ C, p ∈ F T . 2.13 We remark that T is said to be a quasistrict pseudocontraction in 13 . 9 T is said to be asymptotically regular on C if, for any bounded subset K of C, lim n→∞ sup x∈K {∥ ∥ ∥T 1x − Tx ∥ ∥ ∥ } 0. 2.14 Remark 2.2. The class of quasi-φ-nonexpansive mappings and the class of asymptotically quasi-φ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-φnonexpansive mappings and asymptotically quasi-φ-nonexpansive mappings do not require F T F̃ T , where F̃ T denotes the asymptotic fixed-point set of T . Remark 2.3. In the framework of Hilbert spaces, quasi-φ-nonexpansive mappings and asymptotically quasi-φ-nonexpansive mappings are reduced to quasinonexpansive mappings and asymptotically quasinonexpansive mappings. In this paper, we introduce a new nonlinear mapping: asymptotically strict quasi-φpseudocontractions. Definition 2.4. Recall that a mapping T : C → C is said to be an asymptotically strict quasiφ-pseudocontraction if F T / ∅, and there exists a sequence {μn} ⊂ 0,∞ with μn → 0 as n → ∞ and a constant κ ∈ 0, 1 such that φ ( p, Tx ) ≤ (1 μn ) φ ( p, x ) κφ x, Tx , ∀x ∈ C, p ∈ F T . 2.15 Remark 2.5. In the framework of Hilbert spaces, asymptotically strict quasi-φ-pseudocontractions are asymptotically strict quasipseudocontractions. Next, we give an examplewhich is an asymptotically strict quasi-φ-pseudocontraction. Let E l2 : {x {x1, x2, . . .} : ∑∞ n 1 |xn| < ∞}, and let BE be the closed unit ball in E. Define a mapping T : BE → BE by T x1, x2, . . . ( 0, x2 1, a2x2, a3x3, . . . ) , 2.16 where {ai} is a sequence of real numbers such that a2 > 0, 0 < aj < 1, where i / 2, and Πi 2aj 1/2. Then φ ( p, Tx ) ∥ ∥p − Tnx∥∥2 ≤ 2(Πi 2aj )∥ ∥p − x∥∥2 κ‖x − Tnx‖ 2 ( Πni 2aj ) φ ( p, x ) κφ x, Tx , ∀x ∈ BE, n ≥ 2, 2.17 6 Abstract and Applied Analysis where p 0, 0, . . . is a fixed point of T and κ ∈ 0, 1 is a real number. In view of limn→∞2 Πni 2aj 1, we see that T is an asymptotically strict quasi-φ-pseudocontraction. In order to prove our main results, we also need the following lemmas. Lemma 2.6 see 29 . Let E be a uniformly convex and smooth Banach space, and let {xn}, {yn} be two sequences of E. If φ xn, yn → 0 and either {xn} or {yn} is bounded, then xn − yn → 0. Lemma 2.7 see 26 . Let C be a nonempty closed convex subset of a smooth Banach space E and x ∈ E then x0 ΠCx if and only if 〈 x0 − y, Jx − Jx0 〉 ≥ 0, ∀y ∈ C. 2.18 Lemma 2.8 see 26 . Let E be a reflexive, strictly convex, and smooth Banach space, C a nonempty closed convex subset of E, and x ∈ E then φ ( y,ΠCx ) φ ΠCx, x ≤ φ ( y, x ) , ∀y ∈ C. 2.19 3. Main Results Theorem 3.1. Let C be a nonempty closed and convex subset of a uniformly convex and smooth Banach space E. Let T : C → C be a closed and asymptotically strict quasi-φ-pseudocontraction with a sequence {μn} ⊂ 0,∞ such that μn → 0 as n → ∞. Assume that T is uniformly asymptotically regular on C and F T is nonempty and bounded. Let {xn} be a sequence generated in the following manner: x0 ∈ E chosen arbitrarily, C1 C, x1 ΠC1x0, Cn 1 { u ∈ Cn : φ xn, Txn ≤ 2 1 − κ 〈 xn − u, Jxn − JTxn 〉 μn Mn 1 − κ } , xn 1 ΠCn 1x0, ∀n ≥ 0, Υ whereMn sup{φ p, xn : p ∈ F T } then the sequence {xn} converges strongly to x ΠF T x0. Proof. The proof is split into five steps. Step 1. Show that F T is closed and convex. Let {pn} be a sequence in F T such that pn → p as n → ∞. We see that p ∈ F T . Indeed, we obtain from the definition of T that φ ( pn, T p ) ≤ (1 μn ) φ ( pn, p ) κφ ( p, Tp ) . 3.1 In view of 2.6 , we see that φ ( pn, T p ) φ ( pn, p ) φ ( p, Tp ) 2 〈 pn − p, Jp − JTp 〉 . 3.2 Abstract and Applied Analysis 7 It follows thatand Applied Analysis 7 It follows that φ ( pn, p ) φ ( p, Tp ) 2 〈 pn − p, Jp − JTp 〉 ≤ (1 μn ) φ ( pn, p ) κφ ( p, Tp ) , 3.3