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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}.

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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