On the existence of solutions of nonlinear integral equations in Banach spaces and Henstock-Kurzweil integrals. (English) Zbl 1101.45006

Sikorska-Nowak, Aneta

Ann. Pol. Math. 83, No. 3, 257-267 (2004).

Summary: We prove some existence theorems for nonlinear integral equations of the Urysohn type \(x(t)=\varphi(t)+\lambda\int_0^a f(t,s,x(s))\,ds\) and Volterra type \(x(t)=\varphi(t)+\int_0^tf(t,s,x(s))\,ds\), \(t\in I_a=[0,a]\), where \(f\) and \(\varphi\) are functions with values in Banach spaces. Our fundamental tools are: measures of noncompactness and properties of the Henstock-Kurzweil integral.

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:

existence; measure of noncompactness; nonlinear Volterra integral equation; nonlinear Urysohn integral equation; Henstock-Kurzweil integral; Banach spaces