
Minimax Theory and its Applications 02 (2017), No. 2, 319343 Copyright Heldermann Verlag 2017 Explicit Algorithm for Hammerstein Equations with Bounded, HemiContinuous and Monotone Mappings J. T. Mendy Gaston Berger University, Saint Louis, Senegal jt.mendy@utg.edu.gm Moustapha Sene Gaston Berger University, Saint Louis, Senegal msene84@yahoo.com Ngalla Djitte Gaston Berger University, Saint Louis, Senegal ngalla.djitte@ugb.edu.sn [Abstractpdf] Let $E$ be a reflexive smooth and strictly convex real Banach space and $E^*$ its dual. Let $F\colon E\rightarrow E^*$ and $K\colon E^*\rightarrow E$ be bounded hemicontinuous 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 monotone. 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. Keywords: Maximal monotone mapping, Hammerstein equations, iterative methods. MSC: 47H04, 47H06, 47H15, 47H17, 47J25. [ Fulltextpdf (166 KB)] for subscribers only. 