Positive solutions for a system of third-order differential equation with multi-point and integral conditions

Jebari, Rochdi, and Boukricha, Abderrahman. "Positive solutions for a system of third-order differential equation with multi-point and integral conditions." Commentationes Mathematicae Universitatis Carolinae 56.2 (2015): 187-207. <http://eudml.org/doc/270108>.

@article{Jebari2015,
abstract = {This paper concerns the following system of nonlinear third-order boundary value problem: \begin\{equation*\} u\_\{i\}^\{\prime \prime \prime \}(t)+f\_\{i\}(t,u\_\{1\}(t),\dots ,u\_\{n\}(t),u^\{\prime \}\_\{1\}(t),\dots ,u^\{\prime \}\_\{n\}(t))= 0, 0<t<1, i\in \lbrace 1,\dots ,n\rbrace \end\{equation*\} with the following multi-point and integral boundary conditions: \[ \{\left\lbrace \begin\{array\}\{ll\} u\_\{i\}(0)=0 u\_\{i\}^\{\prime \}(0)=0 u\_\{i\}^\{\prime \}(1)= \sum ^\{p\}\_\{j=1\}\beta \_\{j,i\}u\_\{i\}^\{\prime \}(\eta \_\{j,i\}) + \int ^\{1\}\_\{0\}h\_\{i\}(u\_\{1\}(s),\dots ,u\_\{n\}(s))\,ds \end\{array\}\right.\} \] where $\beta _\{j,i\}>0$, $0< \eta _\{1,i\}<\dots <\eta _\{p,i\}<\frac\{1\}\{2\}$, $f_\{i\}:[0,1]\times \mathbb \{R\}^\{n\}\times \mathbb \{R\}^\{n\}\rightarrow \mathbb \{R\}$ and $h_\{i\}:[0,1]\times \mathbb \{R\}^\{n\}\rightarrow \mathbb \{R\}$ are continuous functions for all $i\in \lbrace 1,\dots ,n\rbrace $ and $j\in \lbrace 1,\dots ,p\rbrace $. Using Guo-Krasnosel’skii fixed point theorem in cone, we discuss the existence of positive solutions of this problem. We also prove nonexistence of positive solutions and we give some examples to illustrate our results.},
author = {Jebari, Rochdi, Boukricha, Abderrahman},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {third-order differential equation; multi-point and integral boundary conditions; Guo-Krasnosel'skii fixed point theorem in cone; positive solutions; third-order differential equation; multi-point and integral boundary conditions; Guo-Krasnosel'skii fixed point theorem in cone; positive solutions},
language = {eng},
number = {2},
pages = {187-207},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Positive solutions for a system of third-order differential equation with multi-point and integral conditions},
url = {http://eudml.org/doc/270108},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Jebari, Rochdi
AU - Boukricha, Abderrahman
TI - Positive solutions for a system of third-order differential equation with multi-point and integral conditions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 2
SP - 187
EP - 207
AB - This paper concerns the following system of nonlinear third-order boundary value problem: \begin{equation*} u_{i}^{\prime \prime \prime }(t)+f_{i}(t,u_{1}(t),\dots ,u_{n}(t),u^{\prime }_{1}(t),\dots ,u^{\prime }_{n}(t))= 0, 0<t<1, i\in \lbrace 1,\dots ,n\rbrace \end{equation*} with the following multi-point and integral boundary conditions: \[ {\left\lbrace \begin{array}{ll} u_{i}(0)=0 u_{i}^{\prime }(0)=0 u_{i}^{\prime }(1)= \sum ^{p}_{j=1}\beta _{j,i}u_{i}^{\prime }(\eta _{j,i}) + \int ^{1}_{0}h_{i}(u_{1}(s),\dots ,u_{n}(s))\,ds \end{array}\right.} \] where $\beta _{j,i}>0$, $0< \eta _{1,i}<\dots <\eta _{p,i}<\frac{1}{2}$, $f_{i}:[0,1]\times \mathbb {R}^{n}\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ and $h_{i}:[0,1]\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ are continuous functions for all $i\in \lbrace 1,\dots ,n\rbrace $ and $j\in \lbrace 1,\dots ,p\rbrace $. Using Guo-Krasnosel’skii fixed point theorem in cone, we discuss the existence of positive solutions of this problem. We also prove nonexistence of positive solutions and we give some examples to illustrate our results.
LA - eng
KW - third-order differential equation; multi-point and integral boundary conditions; Guo-Krasnosel'skii fixed point theorem in cone; positive solutions; third-order differential equation; multi-point and integral boundary conditions; Guo-Krasnosel'skii fixed point theorem in cone; positive solutions
UR - http://eudml.org/doc/270108
ER -