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

• Main Authors: Imran Talib, Naseer Ahmad Asif, Cemil Tunc
• Format: Article
• Language: English
• Published: Texas State University 2015-12-01
• Series: Electronic Journal of Differential Equations
• Subjects: Lower and upper solutions; coupled system; coupled boundary conditions; Arzela-Ascoli theorem; Schauder's fixed point theorem
• Online Access: http://ejde.math.txstate.edu/Volumes/2015/313/abstr.html

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.