Electron. J. Diff. Equ., Vol. 2015 (2015), No. 313, pp. 1-11.

Existence of solutions to second-order nonlinear coupled systems with nonlinear coupled boundary conditions

Imran Talib, Naseer Ahmad Asif, Cemil Tunc

Abstract:
In this article, study the existence of solutions for the second-order nonlinear coupled system of ordinary differential equations
$$\displaylines{
 u''(t)=f(t,v(t)),\quad t\in [0,1],\cr
 v''(t)=g(t,u(t)),\quad t\in [0,1],
 }$$
with nonlinear coupled boundary conditions
$$\displaylines{
 \phi(u(0),v(0),u(1),v(1),u'(0),v'(0))=(0,0), \cr
 \psi(u(0),v(0),u(1),v(1),u'(1),v'(1))=(0,0),
 }$$
where $f,g:[0,1]\times \mathbb{R}\to \mathbb{R}$ and $\phi,\psi:\mathbb{R}^6\to \mathbb{R}^2$ are continuous functions. Our main tools are coupled lower and upper solutions, Arzela-Ascoli theorem, and Schauder's fixed point theorem.

Submitted July 14, 2015. Published December 24, 2015.
Math Subject Classifications: 34B10, 34B15.
Key Words: Lower and upper solutions; coupled system; coupled boundary conditions; Arzela-Ascoli theorem; Schauder's fixed point theorem.

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Imran Talib
Department of Mathematics, School of Science
University of Management and Technology
CII Johar Town, Lahore, Pakistan
email: imrantaalib@gmail.com
Naseer Ahmad Asif
Department of Mathematics, School of Science
University of Management and Technology
CII Johar Town, Lahore, Pakistan
email: naseerasif@yahoo.com
Cemil Tunc
Department of Mathematics Faculty of Sciences
Yuzuncu Yil University, Van - Turkey
email: cemtunc@yahoo.com

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