A note on the second order boundary value problem on a half-line

S. McArthur, University of Arkansas at Little Rock, Little Rock, AR, U.S.A.
N. Kosmatov, University of Arkansas at Little Rock, Little Rock, AR, U.S.A.

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 21., pp. 1-8.

Abstract: We consider the existence of a solution to the second order nonlinear differential equation
\begin{displaymath}
(p(t)u'(t))'=f(t,u(t),u'(t)), \ a.\, e. \ \mathrm{in} \ (0,\infty),
\end{displaymath}
that satisfies the boundary conditions \begin{displaymath} u'(0) = 0, \lim_{t \to \infty} u(t) = 0, \end{displaymath} where $f: [0,\infty) \times \mathbb{R}^2 \to \mathbb{R}$ is Carath\'{e}odory with respect to $L_r[0,\infty)$, $r > 1$. The main technique used in this note is the Leray-Schauder Continuation Principle.

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