Approximation of Solutions of Nonlinear Equations of Monotone and Hammerstein Type

Abstract

Let X be a real uniformly smooth and uniformly convex Banach space with dual X ^*. Let A: X → X ^* be a bounded uniformly submonotone map. It is proved that a Mann-type approximation sequence converges strongly to Jx ^* where x ^* ∈ N(A). Furthermore, as an application of this result an iterative sequence which converges strongly to a solution of the Hammerstein equation u+KFu = 0 is constructed where, F:X→X^* and K:X^*→X are monotone-type mappings. No invertibility assumption is imposed on K. Moreover, neither K nor F need be compact. Finally, our method is of independent interest.

Keywords:

• Monotone
• Uniformly monotone
• Uniformly submonotone maps
• Uniformly convex
• Uniformly smooth spaces

1991 Mathematics Subject Classifications:

• 47H05
• 47H06
• 47H30
• 47J05
• 47J25