A strong convergence theorem for generalized-Φ-strongly monotone maps, with applications

Abstract

Abstract Let X be a uniformly convex and uniformly smooth real Banach space with dual space X∗ $X^{*}$. In this paper, a Mann-type iterative algorithm that approximates the zero of a generalized-Φ-strongly monotone map is constructed. A strong convergence theorem for a sequence generated by the algorithm is proved. Furthermore, the theorem is applied to approximate the solution of a convex optimization problem, a Hammerstein integral equation, and a variational inequality problem. This theorem generalizes, improves, and complements some recent results. Finally, examples of generalized-Φ-strongly monotone maps are constructed and numerical experiments which illustrate the convergence of the sequence generated by our algorithm are presented.

Keywords

• Generalized-Φ-strongly monotone map
• Optimization problem
• Hammerstein integral equation
• Variational inequality problem
• Strong convergence