Existence of solutions for second-order differential equations and systems on infinite intervals

Abstract

We study the existence of nontrivial solutions to the boundary-value problem $$displaylines{ -u''+cu'+lambda u = f(x,u),quad -infty <x<+infty , cr u(-infty )=u(+infty ) =0 }$$ and to the system $$displaylines{ -u''+c_1u'+lambda _1u = f(x,u,v),quad -infty <x<+infty , cr -v''+c_2v'+lambda _2v = g(x,u,v),quad -infty <x<+infty , cr u(-infty )=u(+infty ) =0, quad v(-infty )=v(+infty ) =0, }$$ where $c,c_1,c_2,lambda ,lambda _1,lambda _2$ are real positive constants and the nonlinearities $f$ and $g$ satisfy suitable conditions. The proofs are based on fixed point theorems.

Keywords

• Boundary value problem
• fixed point theorem