Strong convergence of path for continuous pseudo-contractive mappings

Abstract

Abstract. The purpose of this paper is to study the convergence of a path that begins at the unique fixed point of a strongly pseudo-contractive operator defined on a closed and convex subset of a reflexive Banach space and converges to a fixed point of a pseudo-contractive mapping. Primarily, it is proven that a convex combination of these two operators is indeed strongly pseudo-contractive under the weakly inward condition. This fact generalizes a result of Barbu for accretive operators.

Iterative Methods for Pseudocontractive Mappings in Banach Spaces

“…Within the past 40 years or so, many authors have been devoting their study to the existence of zeros of accretive mappings or fixed points of pseudocontractive mappings and iterative construction of zeros of accretive mappings and of fixed points of pseudocontractive mappings (see [5,[7][8][9][10]). Also, several iterative methods for approximating fixed points (zeros) of nonexpansive and pseudocontractive mappings (accretive mappings) in Hilbert spaces and Banach spaces have been introduced and studied by many authors.…”

“…where : [0, ∞) → [0, ∞) is a nondecreasing continuous function, they proved that the sequence { } generated by (10) converges strongly to a fixed point of . In particular, in 2007, by using the viscosity iterative method studied by [18,19], Song and Chen [17] introduced a modified implicit iterative method (12) below for a continuous pseudocontractive mapping without compactness assumption on its domain in a real reflexive and strictly convex Banach space having a uniformly Gâteaux differentiable norm: for arbitrary initial value 0 ∈ ,…”

Strong convergence for the modified Mann’s iteration of λ-strict pseudocontraction

“…Morales and Jung [13], in 2000, proved the following behavior for pseudocontractive mappings. Also see Song and Chen [17,18] for more details.…”

“…The same result of nonexpansive mapping was shown by Reich [15] in 1980. [13,17,18]) Let C be a nonempty, closed and convex subset of a uniformly smooth Banach space E, and let T : C → C be a continuous pseudocontractive mapping with F (T ) = ∅. Suppose that for t ∈ (0, 1) and u ∈ C, x t defined by…”

Weak Convergence Theorems for Generalized Mixed Equilibrium Problems, Monotone Mappings and Pseudocontractive Mappings

“…Moreover, this inclusion is strict due to an example in [8] (see, also Example 5.7.1 and Example 5.7.2 in [2]). Fixed point problems for pseudocontractive mappings and strictly pseudocontractive mappings were studied by many authors, see, for example, [1,9,14,16,19,22,23,24,35] and the references therein. Recently, many authors have introduced some iterative algorithms for finding a common element of the set of the solutions of the GMEP, the GEP, the MEP, the EP, and the VIP and the set of fixed points of a countable family of nonexpansive mappings, and have proved strong convergence of the sequences generated by the proposed iterative algorithms; see [6,13,15,17,25,26,27,29,30,31,32] and the references therein.…”