Abstract
In this paper, we investigate and analyze a proximal point algorithm via viscosity approximation method with error. This algorithm is introduced for finding a common zero point for a countable family of inverse strongly accretive operators and a countable family of nonexpansive mappings in Banach spaces. Our result can be extended to some well known results from a Hilbert space to a uniformly convex and 2−uniformly smooth Banach space. Finally, we establish the strong convergence theorems for the proximal point algorithm. Also, some illustrative numerical examples are presented.
Highlights
The problems of finding a zero point for monotone operators play an important role in modern optimization and analysis
In 2000, Moudafi [16] introduced the viscosity approximation method for finding fixed point of a nonexpansive mapping S in a Hilbert space; for given x0 ∈ C, the sequence is defined by the following algorithm: xn+1 = αnf + (1 − αn)Sxn, ∀n ≥ 0, where f : C → C is a contraction mapping and {αn} ⊆ (0, 1) satisfies some condition
In this paper, motivated by [8, 16], we are interested in the problem for finding a common solution of two different fixed point problems which are a common element of a zero point for an infinite countable family of β-inverse strongly accretive operators and a common fixed point of an infinite countable nonexpansive mappings in uniformly convex and 2−uniformly smooth Banach space
Summary
The problems of finding a zero point for monotone operators play an important role in modern optimization and analysis. In 2000, Moudafi [16] introduced the viscosity approximation method for finding fixed point of a nonexpansive mapping S in a Hilbert space; for given x0 ∈ C, the sequence is defined by the following algorithm: xn+1 = αnf (xn) + (1 − αn)Sxn, ∀n ≥ 0, where f : C → C is a contraction mapping and {αn} ⊆ (0, 1) satisfies some condition This sequence converges strongly to a fixed point of S. In this paper, motivated by [8, 16], we are interested in the problem for finding a common solution of two different fixed point problems which are a common element of a zero point for an infinite countable family of β-inverse strongly accretive operators and a common fixed point of an infinite countable nonexpansive mappings in uniformly convex and 2−uniformly smooth Banach space. Some illustrative numerical examples (using Matlab software) are presented
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