On the Convergence of Mann and Ishikawa Iterative Processes for Asymptotically φ-Strongly Pseudocontractive Mappings

and Applied Analysis 3 proving: 1 a fixed point theorem for an asymptotically pseudocontractive mapping that is also uniformly L-Lipschitzian and uniformly asymptotically regular, 2 that the set of fixed points of T is closed and convex, and 3 the strong convergence of a CQ-iterative method. The literature on asymptotical-type mappings is very wide see, 7–15 . In 1967, Browder 16 and Kato 17 , independently, introduced the accretive operators see, for details, Chidume 18 . Their interest was connected with the existence of results in the theory of nonlinear equation of evolution in Banach spaces. In 1974 Deimling 19 , studying the zeros of accretive operators, introduced the class of φ-strongly accretive operators. Definition 1.5. An operator A defined on a subset K of a Banach space X is said to be φstrongly accretive if 〈 Ax −Ay, j(x − y)〉 ≥ φ(∥∥x − y∥∥)∥∥x − y∥∥, 1.6 where φ : R → R is a strictly increasing function such that φ 0 0 and j x−y ∈ J x−y . Note that in the special case inwhich φ t kt, k ∈ 0, 1 , we obtain a strongly accretive operator. However, it is not difficult to prove see Osilike 20 that Ax x − x/ x 1 in R is φ-strongly accretive with φ s s2/ 1 s but not strongly accretive. Since an operator A is a strongly accretive operator if and only if I −A is a strongly pseudocontractive mapping i.e., 〈 I − A x − I − A y, j x − y 〉 ≤ k‖x − y‖2, k < 1 , taking into account Definition 1.5, it is natural to study the class of φ-pseudocontractive mappings, that is, the mappings such that 〈 Tx − Ty, j(x − y)〉 ≤ ∥∥x − y∥∥2 − φ(∥∥x − y∥∥)∥∥x − y∥∥, 1.7 where φ : R → R is a strictly increasing function such that φ 0 0. Of course the set of fixed points for this mapping contains, at most, only one point. Recently, in same papers the following has been introduced. Definition 1.6. A mapping T is a φ-strongly pseudocontractive mapping if 〈 Tx − Ty, j(x − y)〉 ≤ ∥∥x − y∥∥2 − φ(∥∥x − y∥∥), 1.8 where j x−y ∈ J x−y and φ : R → R is a strictly increasing function such that φ 0 0. In the literature this class is also known as generalized φ-strongly pseudocontractive according to 1.7 . We prefer to not use the term generalized because this class is narrower than pseudocontractive mappings. Choosing φ t φ t t, we obtain 1.7 . In Xiang’s paper 21 , it was remarked that it is an open problem if every φ-strongly pseudocontractive mapping is φ-pseudocontractive mapping. In the same paper, Xiang obtained a fixed point theorem for continuous and φstrongly pseudocontractive mappings in the setting of the Banach spaces. In this paper our attention is on the class of the asymptotically φ-strongly pseudocontractive mappings defined as follows. 4 Abstract and Applied Analysis Definition 1.7. If X is a Banach space and K is a subset of X, a mapping T : K → K is said to be asymptotically φ-strongly pseudocontractive if 〈 Tx − Ty, j(x − y)〉 ≤ kn ∥ ∥x − y∥∥2 − φ(∥∥x − y∥∥), 1.9 where j x − y ∈ J x − y , {kn}n ⊂ 1,∞ is converging to one, and φ : 0,∞ → 0,∞ is strictly increasing such that φ 0 0. One can note that if T has fixed points, then it is unique. In fact if x, z are fixed points for T , then, for every n ∈ N, ‖x − z‖ 〈Tnx − Tz, j x, z 〉 ≤ kn ∥ ∥x − y∥∥2 − φ(∥∥x − y∥∥), 1.10 so, passing n to ∞ results in ‖x − z‖ ≤ ‖x − z‖ − φ(∥∥x − y∥∥) ⇒ −φ(∥∥x − y∥∥) ≥ 0. 1.11 Since φ : 0,∞ → 0,∞ is strictly increasing and φ 0 0 then, x z. We now give two examples. Example 1.8. The mapping Tx x/ x 1 , where x ∈ 0, 1 , is asymptotically φ-strongly pseudocontractive with kn 1, for all n ∈ N and φ t t3/ 1 t . However, T is not strongly pseudocontractive, see 20 . Example 1.9. The mapping Tx x/ 1 0.01x , where x ∈ 0, 1 , is asymptotically φ-strongly pseudocontractive with kn n 100 /n, for all n ∈ N and φ x x3/ 1 x . However T is not strongly pseudocontractive, nor φ-strongly Pseudocontractive. Proof. First we prove that T is not strongly pseudocontractive. For arbitrary k < 1, there exist x, y ∈ 0, 1 , such that 1 1 0.01x ( 1 0.01y ) > k. 1.12