Abstract
The aim of this paper is to introduce a new iterative scheme for finding common solutions of the variational inequalities for an inverse strongly accretive mapping and the solutions of fixed point problems for nonexpansive semigroups by using the modified viscosity approximation method associate with Meir‐Keeler type mappings and obtain some strong convergence theorem in a Banach spaces under some parameters controlling conditions. Our results extend and improve the recent results of Li and Gu (2010), Wangkeeree and Preechasilp (2012), Yao and Maruster (2011), and many others.
Highlights
The theory of variational inequalities and variational inclusions are among the most interesting and important mathematical problems and have been studied intensively in the past years since they have wide applications in the optimization and control, economics, engineering science, physical sciences, and applied sciences
The aim of this paper is to introduce a new iterative scheme for finding common solutions of the variational inequalities for an inverse strongly accretive mapping and the solutions of fixed point problems for nonexpansive semigroups by using the modified viscosity approximation method associate with Meir-Keeler type mappings and obtain some strong convergence theorem in a Banach spaces under some parameters controlling conditions
Let C be a nonempty closed convex subset of a real Banach space E and E∗ be the dual space of E with norm · and ·, · pairing between E and E∗
Summary
The theory of variational inequalities and variational inclusions are among the most interesting and important mathematical problems and have been studied intensively in the past years since they have wide applications in the optimization and control, economics, engineering science, physical sciences, and applied sciences. In 2011, Yao and Maruster 8 proved some strong convergence theorems for finding a solution of variational inequality problem 1.6 in Banach spaces They defined a sequence {xn} iteratively by given arbitrarily x0 ∈ C and xn 1 βnxn 1 − βn QC 1 − αn xn − λAxn , ∀n ≥ 0, 1.9 where QC is a sunny nonexpansive retraction from a uniformly convex and 2-uniformly smooth Banach space E, and A is an α-inverse strongly accretive operator of C into E. Motivated and inspired by the idea of Li and Gu , Wangkeeree and Preechasilp , and Yao and Maruster 8 , in this paper, we introduce a new iterative scheme for finding common solutions of the variational inequalities for an inverse strongly accretive mapping and the solutions of fixed point problems for a nonexpansive semigroup by using the modified viscosity approximation method associated with Meir-Keeler type mapping. Our results extend and improve the recent results of Li and Gu , Wangkeeree and Preechasilp , Yao and Maruster 8 , and many others
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