Higher order multi-point boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions

J. R. Graef, University of Tennessee at Chattanooga, Chattanooga, TN, U.S.A.
Lingju Kong, University of Tennessee at Chattanooga, Chattanooga, TN, U.S.A.
Qingkai Kong, Northern Illinois University, DeKalb, IL, U.S.A.
James S. W. Wong, The University of Hong Kong, City University of Hong Kong and Chinney Investment Ltd., Hong Kong

E. J. Qualitative Theory of Diff. Equ., No. 28. (2010), pp. 1-40.

Abstract: We study classes of $n$th order boundary value problems consisting of an equation having a sign-changing nonlinearity $f(t,x)$ together with several different sets of nonhomogeneous multi-point boundary conditions. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. Conditions are determined by the behavior of $f(t,x)/x$ near $0$ and $\pm\infty$ when compared to the smallest positive characteristic values of some associated linear integral operators. This work improves and extends a number of recent results in the literature on this topic. The results are illustrated with examples.

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