EXPLICIT ALGORITHM FOR HAMMERSTEIN EQUATIONS WITH BOUNDED, HEMI-CONTINUOUS AND MONOTONE MAPPINGS J. T. MENDY, M. SENE, AND N. DJITTE Abstract . Let E be a reflexive smooth and strictly convex real Banach space and E * its dual. Let F : E → E * and K : E * → E be bounded hemi-continuous mappings such that D(F) = E and R(F) = D(K) = E * . Suppose that the Hammerstein equation u + KFu = 0 has a solution in E. We present in this paper a method containing an auxiliary mapping, defined on an appropriate Banach space in terms of F and K and which is maximal montone. The solutions of the Hammerstein equation are derived from the zeros of this map. Our method provides an implicit algorithm and explicit one that converge strongly to a solution of the equation u + KFu = 0. No invertibility assumption is imposed on K and the operator F need not be defined on a compact subset of E. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings. Finally, illustration of the proposed method is given in L p spaces. 1. Introduction Let E be a normed linear space with dual E * . Consider the functional equations of the form: (1.1) u + KFu = 0, where F : E → E * and K : E * → E are maps such that D(K) = R(F ) = E * . This class of equations includes nonlinear integral equations of Hammerstein type: (1.2) u( x) + Z Ω κ( x, y) f (y, u(y))dy = 0, where dy is a σ-finite measure on the measure space Ω; the real kernel κ is defined on Ω × Ω, f is a real- valued function defined on Ω × R. The mappings K and F are given by (1.3) Kv( x) = Z Ω κ( x, y)v(y)dy, a.e. x ∈ Ω, Fu(y) = f (y, u(y)) a.e. y ∈ Ω. There exist various motivations for studying equations of type (1.1). For illustration, let us mention one of them. The study of Hammerstein equations is related to nonsmooth calculus of variations (see e.g., the monograph [35]). Suppose that we are interested in minimizing the energy functional (1.4) J (u) = Z Ω  h(u(t)) - f ( s, u(t))  ds, where h denotes the kinetic energy of the system, and f is a potential energy generating a superposition operator. Assume further that the functional J is not differentiable in the usual sense, but admits a generalized 2010 Mathematics Subject Classification. 47H04, 47H06, 47H15, 47H17, 47J25. Key words and phrases. Maximal Monotone Mapping, Hammerstein Equations, Iterative Methods. 1