If $Y$ is a subset of the space $\mathbb{R}^{n}\times {\mathbb{R}^{n}}$, we call a pair of continuous functions $U$, $V$$Y$-compatible, if they map the space $\mathbb{R}^{n}$ into itself and satisfy $Ux\cdot Vy\ge 0$, for all $(x,y)\in Y$ with $x\cdot y\ge {0}$. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential $n$-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer’s fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points $P_{\delta }$ is obtained. Then passing to the limits as $\delta $ tends to zero the so-obtained accumulation points are solutions of the problem.
If $Y$ is a subset of the space $\mathbb{R}^{n}\times {\mathbb{R}^{n}}$, we call a pair of continuous functions $U$, $V$$Y$-compatible, if they map the space $\mathbb{R}^{n}$ into itself and satisfy $Ux\cdot Vy\ge 0$, for all $(x,y)\in Y$ with $x\cdot y\ge {0}$. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential $n$-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer’s fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points $P_{\delta }$ is obtained. Then passing to the limits as $\delta $ tends to zero the so-obtained accumulation points are solutions of the problem.
@article{CMJ_2005_55_3_a1,
author = {Karakostas, G. L. and Palamides, P. K.},
title = {Boundary value problems with compatible boundary conditions},
journal = {Czechoslovak Mathematical Journal},
pages = {581--592},
year = {2005},
volume = {55},
number = {3},
mrnumber = {2153084},
zbl = {1081.34039},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a1/}
}
TY - JOUR
AU - Karakostas, G. L.
AU - Palamides, P. K.
TI - Boundary value problems with compatible boundary conditions
JO - Czechoslovak Mathematical Journal
PY - 2005
SP - 581
EP - 592
VL - 55
IS - 3
UR - http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a1/
LA - en
ID - CMJ_2005_55_3_a1
ER -
%0 Journal Article
%A Karakostas, G. L.
%A Palamides, P. K.
%T Boundary value problems with compatible boundary conditions
%J Czechoslovak Mathematical Journal
%D 2005
%P 581-592
%V 55
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a1/
%G en
%F CMJ_2005_55_3_a1
Karakostas, G. L.; Palamides, P. K. Boundary value problems with compatible boundary conditions. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 581-592. http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a1/
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