Nonlinear integral equations in Banach spaces and Henstock-Kurzweil-Pettis integrals. (English) Zbl 1154.45011

Sikorska-Nowak, Aneta

Dyn. Syst. Appl. 17, No. 1, 97-107 (2008).

The following nonlinear integral equation is considered:
\[ x(t) = f(t) + \int_0^a k_1(t,s) x(s)\,ds + \int_0^a k_2(t,s)g(x(s),s)\,ds \]
with \(a\in (0, \infty)\) and the Henstock-Kurzweil-Pettis integrals. That integral equation is explored as a nonlinear Fredholm equation expressed as a perturbed linear equation. The author proves an existence theorem for that equation under the following assumptions: The function \(g\) is scalarly measurable and weakly sequential continuous with respect to the second variable. Moreover, she supposes that the function \(g\) satisfies some conditions expressed in terms of the measure of weak noncompactness.

Reviewer: Michael I. Gil’ (Beer-Sheva)

MSC:

45N05	Abstract integral equations, integral equations in abstract spaces
45G10	Other nonlinear integral equations
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

nonlinear Fredholm integral equation; existence of solution; Pettis integral; Henstock-Kurzweil-Pettis integral; measure of weak noncompactness