NONLINEAR INTEGRAL EQUATIONS IN BANACH SPACES AND HENSTOCK-KURZWEIL-PETTIS INTEGRALS

We prove an existence theorem for the nonlinear integral equation : x(t) = f(t) + Zα 0 k1(t, s)x(s)ds + Zα 0 k2(t, s)g(s, x(s))ds, t ∈ Iα = [0, α], α ∈ R+, with the Henstock-Kurzweil-Pettis integrals. This integral equation can be considered as a nonlinear Fredholm equation expressed as a perturbed linear equation. The assumptions about the function g are really-weak: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function g satisfies some conditions expressed in terms of the measure of weak noncompactness.