# Existence of multiple solutions for some functional boundary value problems

Archivum Mathematicum (1992)

• Volume: 028, Issue: 1-2, page 57-65
• ISSN: 0044-8753

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## Abstract

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Let $X$ be the Banach space of ${C}^{0}$-functions on $〈0,1〉$ with the sup norm and $\alpha ,\beta \in X\to R$ be continuous increasing functionals, $\alpha \left(0\right)=\beta \left(0\right)=0$. This paper deals with the functional differential equation (1) ${x}^{\text{'}\text{'}\text{'}}\left(t\right)=Q\left[x,{x}^{\text{'}},{x}^{\text{'}\text{'}}\left(t\right)\right]\left(t\right)$, where $Q:{X}^{2}×R\to X$ is locally bounded continuous operator. Some theorems about the existence of two different solutions of (1) satisfying the functional boundary conditions $\alpha \left(x\right)=0=\beta \left({x}^{\text{'}}\right)$, ${x}^{\text{'}\text{'}}\left(1\right)-{x}^{\text{'}\text{'}}\left(0\right)=0$ 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.

## How to cite

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Staněk, Svatoslav. "Existence of multiple solutions for some functional boundary value problems." Archivum Mathematicum 028.1-2 (1992): 57-65. <http://eudml.org/doc/247344>.

@article{Staněk1992,
abstract = {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.},
author = {Staněk, Svatoslav},
journal = {Archivum Mathematicum},
keywords = {Schauder linearization technique; Schauder differential equation; functional boundary conditions; boundary value problem; third order functional-differential equation; functional boundary conditions; Schauder's fixed point theorem; a priori estimates; degree theory; lower and upper solutions},
language = {eng},
number = {1-2},
pages = {57-65},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Existence of multiple solutions for some functional boundary value problems},
url = {http://eudml.org/doc/247344},
volume = {028},
year = {1992},
}

TY - JOUR
AU - Staněk, Svatoslav
TI - Existence of multiple solutions for some functional boundary value problems
JO - Archivum Mathematicum
PY - 1992
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 028
IS - 1-2
SP - 57
EP - 65
AB - 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.
LA - eng
KW - Schauder linearization technique; Schauder differential equation; functional boundary conditions; boundary value problem; third order functional-differential equation; functional boundary conditions; Schauder's fixed point theorem; a priori estimates; degree theory; lower and upper solutions
UR - http://eudml.org/doc/247344
ER -

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