Boundary value problems with compatible boundary conditions

George L. Karakostas; P. K. Palamides

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 3, page 581-592
  • ISSN: 0011-4642

Abstract

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If Y is a subset of the space ℝ n × ℝ n , we call a pair of continuous functions U , V Y -compatible, if they map the space ℝ n into itself and satisfy U x · V y ≥ 0 , for all ( x , y ) ∈ Y with x · y ≥ 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 δ is obtained. Then passing to the limits as δ tends to zero the so-obtained accumulation points are solutions of the problem.

How to cite

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Karakostas, George L., and Palamides, P. K.. "Boundary value problems with compatible boundary conditions." Czechoslovak Mathematical Journal 55.3 (2005): 581-592. <http://eudml.org/doc/30970>.

@article{Karakostas2005,
abstract = {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.},
author = {Karakostas, George L., Palamides, P. K.},
journal = {Czechoslovak Mathematical Journal},
keywords = {differential equations of second order; two-point boundary value problems; differential equations of second order; two-point boundary value problems},
language = {eng},
number = {3},
pages = {581-592},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Boundary value problems with compatible boundary conditions},
url = {http://eudml.org/doc/30970},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Karakostas, George L.
AU - Palamides, P. K.
TI - Boundary value problems with compatible boundary conditions
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 3
SP - 581
EP - 592
AB - 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.
LA - eng
KW - differential equations of second order; two-point boundary value problems; differential equations of second order; two-point boundary value problems
UR - http://eudml.org/doc/30970
ER -

References

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  9. 10.1090/S0002-9939-98-04212-9, Proc. Amer. Math. Soc. 126 (1998), 145–152. (1998) MR1443173DOI10.1090/S0002-9939-98-04212-9

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