Extremal solutions of nonlinear Fredholm integral equations in ordered Banach spaces.

*(English)*Zbl 0755.45024Let $E$ 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) $u\left(t\right)={\int}_{I}F(t,s,u\left(s\right))ds$, 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}.

Reviewer: J.Banaś (Rzeszów)

##### MSC:

45N05 | Abstract integral equations, integral equations in abstract spaces |

45G10 | Nonsingular nonlinear integral equations |