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

##### 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.