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.
@article{107436,
author = {Stan\v ek, Svatoslav},
title = {Existence of multiple solutions for some functional boundary value problems},
journal = {Archivum Mathematicum},
volume = {028},
year = {1992},
pages = {57-65},
zbl = {0782.34074},
mrnumber = {1201866},
language = {en},
url = {http://dml.mathdoc.fr/item/107436}
}
Staněk, Svatoslav. Existence of multiple solutions for some functional boundary value problems. Archivum Mathematicum, Tome 028 (1992) pp. 57-65. http://gdmltest.u-ga.fr/item/107436/
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