期刊
ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS
卷 -, 期 121, 页码 1-28出版社
UNIV SZEGED, BOLYAI INSTITUTE
DOI: 10.14232/ejqtde.2016.1.121
关键词
second order ODE; time singularity; existence and uniqueness; phi-Laplacian; damped solution; half-line
资金
- Grant Agency of the Czech Republic [14-06958S]
We study analytical properties of a singular nonlinear ordinary differential equation with a phi-Laplacian. In particular we investigate solutions of the initial value problem (p(t)phi(u'(t)))' + p(t)f(phi(u(t))) = 0, u(0) = u(0) is an element of [L-0, L], u'(0) = 0 on the half-line [0, infinity). Here, f is a continuous function with three zeros phi(L0) < 0 < phi(L), function p is positive on (0, infinity) and the problem is singular in the sense that p(0) - 0 and 1/p(t) may not be integrable on [0,1]. The main goal of the paper is to prove the existence of damped solutions defined as solutions u satisfying sup{u(t), t is an element of [0, infinity)} < L. Moreover, we study the uniqueness of damped solutions. Since the standard approach based on the Lipschitz property is not applicable here in general, the problem is more challenging. We also discuss the uniqueness of other types of solutions.
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