One of the generalizations of Krasnoselskii’s theorem on cone expansion and compression was obtained in [R. W. Leggett, L. R. Williams, J. Math. Anal. Appl., Vol. 76, 91–97 (1980; Zbl 0448.47044)]. In the present paper, the author proves the following Leggett-Williams type theorem: Theorem. Let be a normal cone in a Banach space and let be the normal constant of . Assume that and are bounded open sets in such that and . Let be a completely continuous operator and . If either for and for , or for and for is satisfied, then has a fixed point in the set . The author applies this theorem to prove the existence of positive solutions to the following second order three-point boundary value problem:
where , is a continuous function, , , and .