Singular higher order boundary value problems for ordinary differential equations

C. Kunkel, University of Tennessee, at Martin, Martin, TN, U.S.A.

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 19., pp. 1-11.

Abstract: We study singular boundary value problems for differential equations with left focal boundary conditions of the form,
$$(-1)^{n} u^{(n)}(t) + f(t,u(t), \ldots, u^{(n-2)}(t)) = 0, \quad t \in (0,1),$$
$$u^{(n-1)}(0) = u^{(n-2)}(1) = \cdots = u(1) = 0.$$
We assume that $f(t,x_{0},\ldots,x_{n-2})$ is continuous on $[0,1] \times (0,\infty) \times \R^{n-2}$ and $f$ has a singularity at $x_{0}=0.$ We prove the existence of a positive solution by means of the lower and upper solutions method, the Brouwer fixed point theorem, and by perturbation methods to approximate regular problems.

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