New semi-implicit midpoint rule for zero of monotone mappings in Banach spaces

Abstract

Let E be a nonempty closed uniformly convex and 2-uniformly smooth Banach space with dual E∗ and A : E∗ → E be Lipschitz continuous monotone mapping with A− 1(0) ≠ ∅. A new semi-implicit midpoint rule (SIMR) with the general contraction for monotone mappings in Banach spaces is established and proved to converge strongly to x∗ ∈ E, where Jx∗ ∈ A− 1(0). Moreover, applications to convex minimization problems, solution of Hammerstein integral equations, and semi-fixed point of a cluster of semi-pseudo mappings are included.