Iterations of strongly pseudocontractive maps in Banach spaces.

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[section]1. Introduction

We denote by J the normalized duality mapping from X into [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by

J (x) = {f [member of] [X.sup.*] : = [[parallel]x[parallel].sup.2] = [[parallel]f[parallel].sup.2]},

where [X.sup.*] denotes the dual space of X and <.> denotes the generalized duality pairing.

Definition 1.1. [15] A mapping T : X [right arrow] X with domain D(T) and R(T) in X is called strongly pseudocontractive if for all x, y [member of] D(T), there exist j(x - y) [member of] J(x - y) and a constant k [member of] (0, 1) such that

[less than or equal to] k [[parallel]x - y[parallel].sup.2].

Closely related to the class of strongly pseudocontractive operators is the important class of strongly accretive operators. It is well known that T is strongly pseudo-contractive if and only if (I - T) is strongly accretive, where I denotes the identity map. Therefore, an operator T : X [right arrow] X is called strongly accretive if there exists a constant k [member of] (0, 1) such that

[greater than or equal to] k [[parallel]x - y[parallel].sup.2],

holds for all x, y [member of] X and some j( x - y) [member of] J( x - y). These operators have been studied and used by several authors (see, for example [13-20]).

The Mann iteration scheme [9], introduced in 1953, was used to prove the convergence of the sequence to the fixed points of mappings of which the Banach principle is not applicable. In 1974, Ishikawa [7] devised a new iteration scheme to establish the convergence of a Lipschitzian pseudocontractive map when Mann iteration process failed to converge. Noor et al. [13]. gave the following three-step iteration process for solving non-linear operator equations in real Banach spaces.

Let K be a nonempty closed convex subset of X and T : K [right arrow] K be a mapping. For an arbitrary [x.sub.0] [member of] K, the sequence [{[x.sub.n]}.sup.[infinity].sub.n=0] [subset] K, defined by

[x.sub.n+1] = (1 - [[alpha].sub.n])[x.sub.n] + [[alpha].sub.n]T[y.sub.n], [y.sub.n] = (1 - [[beta].sub.n])[x.sub.n] + [[beta].sub.n]T[z.sub.n], [Z.sub.n] = (1 - [[gamma].sub.n])[x.sub.n] + [[gamma].sub.n]T[x.sub.n], n [less than or equal to] 0, (1)

where [{[alpha].sub.n]}.sup.[infinity].sub.n=0], [{[beta].sub.n]}.sup.[infinity].sub.n=0] and [{[gamma].sub.n]}.sup.[infinity].sub.n=0] are three sequences in [0,1] for each n, is called the three-step iteration (or the Noor iteration). When [[gamma].sub.n] = 0, then the three-step iteration reduces to the Ishikawa iterative sequence [{[x.sub.n]}.sup.[infinity].sub.n=0] [subset] K defined by

[x.sub.n+1] = (1 - [[alpha].sub.n])[x.sub.n] + [[alpha].sub.n]T[y.sub.n], [y.sub.n] = (1 - [[gamma].sub.n])[x.sub.n] + [[gamma].sub.n]T[x.sub.n], n [less than or equal to] 0. (2)

If [[beta].sub.n] = [[gamma].sub.n] = 0, then (1) becomes the Mann iteration. It is the sequence [{[x.sub.n]}.sup.[infinity].sub.n=0] [subset] K defined by

[x.sub.n+1] = (1 - [[alpha].sub.n])[x.sub.n] + [[alpha].sub.n]T[x.sub.n], n [greater than or equal to] 0. (3)

Rafiq [15], recently studied the following of iterative scheme which he called the modified three-step iteration process, to approximate the unique common fixed points of a three strongly pseudocontractive mappings in Banach spaces.

Let [T.sub.1], [T.sub.2], [T.sub.3] : K [right arrow] K be three mappings. For any given [x.sub.0] [member of] K, the modified three-step iteration [{[x.sub.n]}.sup.[infinity].sub.n=0] [subset] K is defined by

[x.sub.n+1] = (1 -...