Existence and Uniqueness of Solutions of Integral Equations of Hammerstein Type

Abstract

Let X be a real Banach space, X ∗ its conjugate dual space. Let A be a monotone angle-bounded continuous linear mapping of X into X ∗ with constant of angle-boundedness c ≥ 0. Let N be a hemicontinuous (possibly non-linear) mapping of X ∗ into X such that for a given constant k ≥ 0, hv 1 − v 2 , N v 1 − N v 2 i ≥ −kkv 1 − v 2 k 2 X ∗ for all v 1 and v 2 in X ∗ . Suppose finally that there exists a constant R with k(1 + c 2 )R < 1 such that for u ∈ X hAu, ui ≤ Rkuk 2 X . Then, there exists exactly one solution w in X ∗ of the nonlinear equation w + AN w = 0. Existence and uniqueness is also proved using variational methods.