Hybrid approximation of solutions of integral equations of the Hammerstein type

Abstract

Let X be a uniformly convex and uniformly smooth real Banach space with dual X * . Let F : X → X * and K : X * → X be continuous monotone operators. Suppose that the Hammerstein equation u + K Fu = 0 has a solution in X. It is proved that a hybrid-type approximation sequence converges strongly to u * , where u * is a solution of the equation u + K Fu = 0. In our theorems, the operator K or F need not be defined on a compact subset of X and no invertibility assumption is imposed on K .

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

“…Consequently, methods for approximating solutions of such equations are of interest. For earlier and more recent works on approximation of solutions of equations of Hammerstein type, the reader may consult any of the following: Brezis and Browder[5,6], Chidume and Shehu[27], Chidume and Ofoedu[25], Chidume and Zegeye[29], Chidume and Djitte[22], Ofoedu and Onyi[45], Ofoedu and Malonza[44], Zegeye and Malonza[58], Chidume and Bello[20],Minjibir and Mohammed…”