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