Nonlinear eigenvalue problems for higher order Lidstone boundary value problems

P. Eloe, University of Dayton, Dayton, Ohio, U.S.A.

E. J. Qualitative Theory of Diff. Equ., No. 2. (2000), pp. 1-8.

Abstract: In this paper, we consider the Lidstone boundary value problem $y^{(2m)}(t) = \lambda a(t)f(y(t), ..., y^{(2j)}(t), ... y^{(2(m-1))}(t), 0 < t < 1, y^{(2i)}(0) = 0 = y^{(2i)}(1), i = 0, ..., m - 1$, where $(-1)^m f > 0$ and $a$ is nonnegative. Growth conditions are imposed on $f$ and inequalities involving an associated Green's function are employed which enable us to apply a well-known cone theoretic fixed point theorem. This in turn yields a $\lambda$ interval on which there exists a nontrivial solution in a cone for each $\lambda$ in that interval. The methods of the paper are known. The emphasis here is that $f$ depends upon higher order derivatives. Applications are made to problems that exhibit superlinear or sublinear type growth.

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