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On the existence of weak solutions of integral equations in Banach spaces. (English) Zbl 0816.45012
The author proves an existence theorem for weakly continuous solutions of the Hammerstein integral equation $x(t) = g(t) + \lambda \int_ I K(t,s)f \bigl( s,x(s) \bigr) ds$ and a Kneser-type theorem (about the existence, weak compactness etc.) for weakly continuous solutions of the Volterra integral equation $x(t) = g(t) + \lambda \int_ 0^ t K(t,s)f \bigl( s,x(s) \bigr) ds.$ Conditions are found in terms of the measure of weak non-compactness $$\omega (X)$$ for a nonvoid bounded subset $$X$$ of a Banach space $$E$$, where $$\omega (X) = \inf \{t > 0 : \exists C,X \subset C + tB,C$$ is weakly compact, $$B$$ is the unit ball}.

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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