New Implicit and explicit approximation methods for solutions of integral equations of Hammerstein type

Let H   be a real Hilbert spae and F,K:H→HF,K:H→H be mappings such that D(K)=D(F)=HD(K)=D(F)=H. Suppose that Hammerstein equation of the type u+KFu=0u+KFu=0 has a solution in H, then we studied in this paper methods that contain an auxiliary mapping (defined on an appropriate real Hilbert space in terms of the mappings K and F) which is pseudocontractive whenever K and F   are monotone; and approximation of a fixed point of this pseudocontractive mapping induces approximation of a solution of the equation u+KFu=0u+KFu=0. Moreover, the mappings K and F need not be defined on compact subset of H or angle bounded on H  . Furthermore, our methods which do not involve K-1K-1 provide an implicit algorithm   for approximation of solutions of the equation u+KFu=0u+KFu=0 whenever K and F are assumed to be bounded and continuous; if K and F are assumed to be Lipschitz continuous, then an explicit iterative algorithm   for computation of solutions of the equation u+KFu=0u+KFu=0 is provided, still without involving K-1K-1.