Abstract
In this paper, we investigate the existence of positive solutions for a class of third-order nonlocal boundary value problems at resonance. Our results are based on the Leggett-Williams norm-type theorem, which is due to O’Regan and Zima. An example is also included to illustrate the main results.
Highlights
1 Introduction This paper is devoted to the existence of positive solutions for the following third-order nonlocal boundary value problem (BVP for short):
It is well known that the problem of the existence of positive solutions to BVPs is very difficult when the resonant case is considered
Third-order or higher-order derivatives do not have the convexity; to the best of our knowledge, no results are available for the existence of positive solutions for thirdorder or higher- order BVPs at resonance
Summary
This paper is devoted to the existence of positive solutions for the following third-order nonlocal boundary value problem (BVP for short):. All of the papers on third-order BVPs focused their attention on the positive solutions with non-resonance cases. It is well known that the problem of the existence of positive solutions to BVPs is very difficult when the resonant case is considered. Few papers deal with the existence of positive solutions to BVPs at resonance, and just to second-order BVPs [ – ]. Third-order or higher-order derivatives do not have the convexity; to the best of our knowledge, no results are available for the existence of positive solutions for thirdorder or higher- order BVPs at resonance. Some new existence results of at least one positive solution are established by applying the Leggett-Williams norm-type theorem due to O’Regan and Zima [ ]. Our main results are based on the following theorem due to O’Regan and Zima
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