An algorithm for approximating a common solution of some nonlinear problems in Banach spaces with an application

Abdulmalik U Bello,Monday O Nnakwe

doi:10.1186/s13662-021-03268-1

Abstract

In this paper, we construct a new Halpern-type subgradient extragradient iterative algorithm. The sequence generated by this algorithm converges strongly to a common solution of a variational inequality, an equilibrium problem, and a J-fixed point of a continuous J-pseudo-contractive map in a uniformly smooth and two-uniformly convex real Banach space. Also, the theorem is applied to approximate a common solution of a variational inequality, an equilibrium problem, and a convex minimization problem. Moreover, a numerical example is given to illustrate the implementability of our algorithm. Finally, the theorem proved complements, improves, and unifies some related recent results in the literature.

Highlights

Let Q∗ be the dual space of a real normed linear space Q and D be a nonempty, closed, and convex subset of Q
We study the classical variational inequality of Fichera [1] and Stampacchia [2], the equilibrium problem of Blum and Otelli [3], and some fixed point problems
In 2014, Zegeye and Shazard [49] studied the problem of finding a common solution in the set of fixed points of a Lipschitz pseudo-contractive map S and solution sets of a variational inequality for a γ -inverse strongly monotone map A in a real Hilbert space H by considering the following iterative algorithm:

Summary

Introduction

Let Q∗ be the dual space of a real normed linear space Q and D be a nonempty, closed, and convex subset of Q. It is obvious that the fixed point technique introduced by Browder in the year 1967 for approximating zeros of accretive maps is not applicable in this case, where A from a real Banach space to its dual space is monotone. In 2014, Zegeye and Shazard [49] studied the problem of finding a common solution in the set of fixed points of a Lipschitz pseudo-contractive map S and solution sets of a variational inequality for a γ -inverse strongly monotone map A in a real Hilbert space H by considering the following iterative algorithm:

Results

Conclusion