Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

Abstract

Abstract Let B be a uniformly convex and uniformly smooth real Banach space with dual space B ∗ $B^{*}$ . Let F : B → B ∗ $F:B\to B^{*}$ , K : B ∗ → B $K:B^{*} \to B$ be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation u + K F u = 0 $u+KFu=0$ . This theorem is a significant improvement of some important recent results which were proved in real Hilbert spaces under the assumption that F and K are maximal monotone continuous and bounded. The continuity and boundedness restrictions on K and F have been dispensed with, using our new method, even in the more general setting considered in our theorems. Finally, numerical experiments are presented to illustrate the convergence of the sequence of our algorithm.

Keywords

• Fixed point
• Maximal monotone
• Uniformly smooth
• Uniformly convex
• Hammerstein
• Quasi-bounded