Electron. J. Diff. Eqns., Vol. 2009(2009), No. 94, pp. 1-13.

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

Toufik Moussaoui, Radu Precup

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.

Submitted October 7, 2008. Published August 6, 2009.
Math Subject Classifications: 34B40.
Key Words: Boundary value problem; fixed point theorem.

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Toufik Moussaoui
Department of Mathematics, E.N.S.
P.O. Box 92, 16050 Kouba, Algiers, Algeria
email: moussaoui@ens-kouba.dz
Radu Precup
Department of Applied Mathematics
Babes-Bolyai University
400084 Cluj, Romania

e-mail: r.precup@math.ubbcluj.ro

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