Abstract
The purpose of this paper is to consider that a modified Halpern's iterative sequence converges strongly to a fixed point of nonexpansive mappings in Banach spaces which have a uniformly Gâteaux differentiable norm. Our result is an extension of the corresponding results.
Highlights
Let E be a real Banach space and C a nonempty closed convex subset of E
In the last ten years, many papers have been written on the approximation of fixed point for nonlinear mappings by using some iterative processes see, e.g., 1–18
The purpose of this paper is to present a significant answer to the above open question. we will show that the sequence {αn} satisfying the conditions C1 and C2 is sufficient to guarantee the strong convergence of the modified Halpern’s iterative sequence for nonexpansive mappings
Summary
Let E be a real Banach space and C a nonempty closed convex subset of E. In 1992, Wittmann 16 proved, still in Hilbert spaces, the strong convergence of the sequence 1.2 to a fixed point of T , where {αn} satisfies the following conditions: C1 lim n→∞ The strong convergence of Halpern’s iteration to a fixed point of T has been proved in Banach spaces see, e.g., 2, 6, 10–12, 14, 15, 17, 18 .
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