Existence of multiple solutions for some functional boundary value problems

Staněk, Svatoslav

Archivum Mathematicum (1992)

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

Abstract

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Let X be the Banach space of C 0 -functions on 〈 0 , 1 〉 with the sup norm and α , β ∈ X → 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 × R → 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 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 -

References

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