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Extremal solutions of nonlinear Fredholm integral equations in ordered Banach spaces. (English) Zbl 0755.45024
Let $E$ be a real Banach space ordered with help of a normal cone. The object of the paper is the study of the nonlinear integral equation (1) $u(t)=\int\sb IF(t,s,u(s))ds$, where $F$ is assumed to be uniformly continuous on the set $I\times I\times E$ $(I$ is an interval). Moreover, $F$ satisfies a Lipschitz condition expressed in terms of the measure of noncompactness. Requiring additionally certain monotonicity assumptions the author proves a few results on the existence of extremal solutions of (1). Some interesting and nontrivial examples are given. {Reviewer’s remark: Lemma 1 is well-known. Therefore, its proof in the paper is superfluous}.

45N05Abstract integral equations, integral equations in abstract spaces
45G10Nonsingular nonlinear integral equations