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A fixed point theorem for condensing operators and applications to Hammerstein integral equations in Banach spaces. (English) Zbl 0846.45006
The object of the paper is the study of the Hammerstein integral equation $x(t)= h(t)+ \int^1_0 k(t,s) f(s,x (s)) ds \qquad (t\in [0, 1]),$ where $$x$$ takes values in a real Banach space $$B$$. The author establishes first a fixed point theorem of Furi-Pera type for operators being condensing with respect to the Kuratowski measure of noncompactness. Next, using this theorem and assuming some growth or monotonicity restrictions on the nonlinear function $$f$$, a few existence theorems for the equation in question are proved.

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 45G10 Other nonlinear integral equations
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