Electron. J. Diff. Eqns., Vol. 2002(2002), No. 80, pp. 1-10.

A note on the singular Sturm-Liouville problem with infinitely many solutions

Nickolai Kosmatov

Abstract:
We consider the Sturm-Liouville nonlinear boundary-value problem
$$ \displaylines{ -u''(t) = a(t)f(u(t)), \quad 0 less than t less than 1, \cr \alpha u(0) - \beta u'(0) =0, \quad \gamma u(1) + \delta u'(1) = 0, } $$
where $\alpha$, $\beta$, $\gamma$, $\delta \geq 0$, $\alpha \gamma + \alpha \delta + \beta \gamma greater than 0$ and $a(t)$ is in a class of singular functions. Using a fixed point theorem we show that under certain growth conditions imposed on $f(u)$ the problem admits infinitely many solutions.

Submitted November 13, 2001. Published September 27, 2002.
Math Subject Classifications: 34B16, 34B18.
Key Words: Sturm-Liouville problem, Green's function, fixed point theorem, Holder's inequality, multiple solutions.

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Nickolai Kosmatov
Department of Mathematics and Statistics
University of Arkansas at Little Rock
Little Rock, Arkansas 72204-1099, USA
e-mail: nxkosmatov@ualr.edu
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