THE SET OF SOLUTIONS OFVOLTERRA AND URYSOHN INTEGRAL EQUATIONS IN BANACH SPACES

摘要

We prove two existence theorems of solu-tions of nonlinear integral equations of Urysohn type x(t) = ψ(t) + λ ∫_0~α f (t, s, x(s)) ds and Volterra type x(t) = ψ(t) ∫_0~α f(t,s,x(s))ds, t ∈ I_α = [0, α], α, λ ∈ R_+, with the Henstock-Kurzweil-Pettis integral. Moreover, we show that the set S of all solutions of the Volterra integral equation is compact and connected. The assumptions about the function f are really weak: scalar measurability and weak sequential continuity with respect to the third variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of the measure of weak noncompactness.