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

We study the existence of nontrivial solutions to the boundary value problem

$-u^{\prime \prime}+cu\prime+\lambda u$ $=f(x,u),\ \ -\infty<x<+\infty$

$u(-\infty)=u(+\infty)=0$

and to the system

$-u^{\prime \prime}+c_{1}u^{\prime}+\lambda_{1}u$ $=f\left( x,u,v\right),\ \ \ \ -\infty<x<+\infty,$

$v^{\prime \prime}+c_{2}v^{\prime}+\lambda_{2}v$ $=g\left( x,u,v\right),\ \ \ -\infty<x<+\infty,$

$u\left( -\infty \right)$ $=u\left( +\infty \right) =0,\ \ v\left(-\infty \right) =v\left( +\infty \right) =0,$

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.

Toufik Moussaoui
Department of Mathematics, E.N.S., P.O. Box 92, 16050 Kouba, Algiers, Algeria

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Boundary value problem; fixed point theorem.

T. Moussaoui, R. Precup, Existence of solutions for second-order differential equations and systems on infinite intervals, Electronic Journal of Differential Equations 2009 (2009) no. 94, 1-13.

https://ictp.acad.ro/wp-content/uploads/2023/03/2009c-Precup-Existence-of-solutions-for-second-order-differential-equations-and-systems-on-infinite-intervals.pdf

https://www.emis.de/journals/EJDE/2009/94/moussaoui.pdf

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