Existence of multiple solutions for some functional boundary value problems

Staněk, Svatoslav

Archivum mathematicum, Tome 28 (1992) no. 1-2, pp. 57-65

Résumé

Let $X$ be the Banach space of $C^0$-functions on $\langle 0,1\rangle $ with the sup norm and $\alpha ,\beta \in X \rightarrow {R}$ be continuous increasing functionals, $\alpha (0)= \beta (0)=0$. This paper deals with the functional differential equation (1) $x^{\prime \prime \prime } (t) = Q [ x, x^\prime , x^{\prime \prime }(t)] (t)$, where $Q:{X}^2 \times {R} \rightarrow {X}$ is locally bounded continuous operator. Some theorems about the existence of two different solutions of (1) satisfying the functional boundary conditions $\alpha (x)=0=\beta (x^\prime )$, $x^{\prime \prime }(1)-x^{\prime \prime }(0)=0$ are given. The method of proof makes use of Schauder linearizatin technique and the Schauder fixed point theorem. The results are modified for 2nd order functional differential equations.

Détail

Classification : 34B10, 34B15
Keywords: Schauder linearization technique; Schauder differential equation; functional boundary conditions; boundary value problem