On a criterion for the existence of at least four solutions of functional boundary value problems

Staněk, Svatoslav

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Existence of multiple solutions for some functional boundary value problems

Staněk, Svatoslav (1992)

Archivum Mathematicum

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Let $X$ be the Banach space of $C^{0}$ -functions on $〈 0, 1 〉$ with the sup norm and $α, β \in X \to R$ be continuous increasing functionals, $α (0) = β (0) = 0$ . This paper deals with the functional differential equation (1) $x^{'''} (t) = Q [x, x^{'}, x^{''} (t)] (t)$ , where $Q : X^{2} \times R \to X$ is locally bounded continuous operator. Some theorems about the existence of two different solutions of (1) satisfying the functional boundary conditions $α (x) = 0 = β (x^{'})$ , $x^{''} (1) - x^{''} (0) = 0$ are given. The method of proof makes use of Schauder linearizatin technique and the Schauder fixed point theorem. The results are modified for 2nd...