On Asymptotically Quasi-φ-Nonexpansive Mappings in the Intermediate Sense

and Applied Analysis 3 It is easy to see that a quasi-nonexpansive mapping is an asymptotically quasi-nonexpansive mapping with the sequence {1}. T is said to be asymptotically nonexpansive in the intermediate sense if and only if it is continuous, and the following inequality holds: lim sup n→∞ sup x,y∈C (∥ ∥Tx − Tny∥∥ − ∥∥x − y∥∥) ≤ 0. 2.6 T is said to be asymptotically quasi-nonexpansive in the intermediate sense if and only if F T / ∅ and the following inequality holds: lim sup n→∞ sup p∈F T ,y∈C (∥ ∥p − Tny∥∥ − ∥∥p − y∥∥) ≤ 0. 2.7 The class of mappings which are asymptotically nonexpansive in the intermediate sense was considered by Bruck et al. 18 . It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense may not be Lipschitz continuous. However, asymptotically nonexpansive mappings are Lipschitz continuous. In what follows, we always assume that E is a Banach space with the dual space E∗. Let C be a nonempty, closed, and convex subset of E. We use the symbol J to stand for the normalized duality mapping from E to 2 ∗ defined by Jx { f∗ ∈ E∗ : 〈x, f∗〉 ‖x‖ ∥∥f∗∥∥2 } , ∀x ∈ E, 2.8 where 〈·, ·〉 denotes the generalized duality pairing of elements between E and E∗. It is well known that if E∗ is strictly convex, then J is single valued; if E∗ is reflexive and smooth, then J is single valued and demicontinuous; see 19 for more details and the references therein. It is also well known that if D is a nonempty, closed, and convex subset of a Hilbert space H, and PC : H → D is the metric projection from H onto D, then PD is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber 20 introduced a generalized projection operator in Banach spaces which is an analogue of the metric projection in Hilbert spaces. Recall that 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 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. Let UE {x ∈ E : ‖x‖ 1} be the unit sphere of E. Then the Banach space E is said to be smooth provided limt→ 0 ‖x ty‖−‖x‖ /t exists for all x, y ∈ UE. It is also said to be uniformly smooth if the limit is attained uniformly for all x, y ∈ UE. Recall that a Banach space E enjoys Kadec-Klee property if for any sequence {xn} ⊂ E and x ∈ E with xn ⇀ x, and ‖xn‖ → ‖x‖, then ‖xn − x‖ → 0 as n → ∞. For more details on Kadec-Klee property, the readers can refer to 19, 21 and the references therein. It is well known that if E is a uniformly convex Banach spaces, then E enjoys Kadec-Klee property. Let E be a smooth Banach space. Consider the functional defined by φ ( x, y ) ‖x‖ − 2〈x, Jy〉 ∥∥y∥∥2, ∀x, y ∈ E. 2.9 4 Abstract and Applied Analysis Notice that, in a Hilbert space H, 2.9 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 ) . 2.10 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, 19, 20 . 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, 2.11 φ ( x, y ) φ x, z φ ( z, y ) 2 〈 x − z, Jz − Jy〉, ∀x, y, z ∈ E. 2.12 Remark 2.2. 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 2.11 , 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 20 for more details. Next, we recall the following. 1 A point p in C is said to be an asymptotic fixed point of T 22 if and only 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 if and only if F̃ T F T / ∅, φ ( p, Tx ) ≤ φ(p, x), ∀x ∈ C, ∀p ∈ F T . 2.13 The asymptotic behavior of relatively nonexpansive mappings was studied in 23, 24 . 3 T is said to be relatively asymptotically nonexpansive if and only if F̃ T F T / ∅, φ ( p, Tx ) ≤ (1 μn ) φ ( p, x ) , ∀x ∈ C, ∀p ∈ F T , ∀n ≥ 1, 2.14 where {μn} ⊂ 0,∞ is a sequence such that μn → 1 as n → ∞. Remark 2.3. The class of relatively asymptotically nonexpansive mappings was first considered in Su and Qin 25 ; see also, Agarwal et al. 26 , and Qin et al. 27 . 4 T is said to be quasi-φ-nonexpansive if and only if F T / ∅, φ ( p, Tx ) ≤ φ(p, x), ∀x ∈ C, ∀p ∈ F T . 2.15 5 T is said to be asymptotically quasi-φ-nonexpansive if and only if there exists a 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 , ∀n ≥ 1. 2.16 Abstract and Applied Analysis 5 Remark 2.4. The class of quasi-φ-nonexpansive mappings and the class of asymptotically quasi-φ-nonexpansive mappings were first considered in Zhou et al. 28 ; see also Qin et al. 29 , Qin, and Agarwal 30 , Qin et al. 31 , Qin et al. 32 , and Qin et al. 33 . Remark 2.5. 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 do not require F T F̃ T . Remark 2.6. The class of quasi-φ-nonexpansive mappings and the class of asymptotically quasi-φ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.and Applied Analysis 5 Remark 2.4. The class of quasi-φ-nonexpansive mappings and the class of asymptotically quasi-φ-nonexpansive mappings were first considered in Zhou et al. 28 ; see also Qin et al. 29 , Qin, and Agarwal 30 , Qin et al. 31 , Qin et al. 32 , and Qin et al. 33 . Remark 2.5. 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 do not require F T F̃ T . Remark 2.6. The class of quasi-φ-nonexpansive mappings and the class of asymptotically quasi-φ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces. In this paper, based on asymptotically quasinonexpansive mappings in the intermediate sense which was first considered by Bruck et al. 18 , we introduce and consider the following new nonlinear mapping: asymptotically quasiφ-nonexpansive mappings in the intermediate sense. 6 T is said to be an asymptotically φ-nonexpansive mapping in the intermediate sense if and only if lim sup n→∞ sup x,y∈C ( φ ( Tx, Ty ) − φ(x, y)) ≤ 0. 2.17 7 T is said to be an asymptotically quasi-φ-nonexpansive mapping in the intermediate sense if and only if F T / ∅, and lim sup n→∞ sup p∈F T ,x∈C ( φ ( p, Tx ) − φ(p, x)) ≤ 0. 2.18 Remark 2.7. The class of asymptotically quasiφ-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasinonexpansive mappings in the intermediate sense in the framework of Banach spaces. Let E R1 and C 0, 1 . Define the following mapping T : C → C by