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.
MSC: 34B10, 34B15.
Keywords:third-order; nonlocal; at resonance; positive solution
This paper is devoted to the existence of positive solutions for the following third-order nonlocal boundary value problem (BVP for short):
has nontrivial solutions. Clearly, the resonant condition is . Third-order differential equations arise in a variety of different areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity-driven flows and so on .
Recently, the existence of positive solutions for third-order two-point or multi-point BVPs has received considerable attention; we mention a few works: [2-11] and the references therein. However, 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. Only few papers deal with the existence of positive solutions to BVPs at resonance, and just to second-order BVPs [12-15]. It is worth mentioning that Infante and Zima  studied the existence of positive solutions for the second-order m-point BVP
by means of the Leggett-Williams norm-type theorem due to O’Regan and Zima , where , () and .
However, 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 third-order or higher- order BVPs at resonance. The main purpose of this paper is to fill the gap in this area. Motivated greatly by the above-mentioned excellent works, in this paper we will investigate the third-order nonlocal BVP (1.1) at resonance, where , () and . 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 . An example is also included to illustrate the main results.
2 Some definitions and a fixed point theorem
For the convenience of the reader, we present here the necessary definitions and a new fixed point theorem due to O’Regan and Zima.
(i) KerL has a finite dimension, and
(ii) ImL is closed and has a finite codimension.
Throughout the paper, we will assume that
and that the isomorphism
is invertible. We denote the inverse of by . The generalized inverse of L denoted by is defined by . Moreover, since , there exists an isomorphism . Consider a nonlinear operator . It is known (see [17,18]) that the coincidence equation is equivalent to
Definition 2.2 Let X be a real Banach space. A nonempty closed convex set P is said to be a cone provided that
Lemma 2.3 ()
Our main results are based on the following theorem due to O’Regan and Zima.
Theorem 2.4 ()
3 Main results
For simplicity of notation, we set
We can now state our result on the existence of a positive solution for the BVP (1.1).
Considering that f can be extended continuously to , it is easy to check that is continuous and bounded, and is compact on every bounded subset of X, which ensures that (H1) of Theorem 2.4 is fulfilled.
Define the cone of nonnegative functions
which is a contradiction.
which is a contradiction. Therefore, (H2) of Theorem 2.4 holds.
Consider . Then on . Similar to , we define
which shows that (H4) of Theorem 2.4 holds.
From (1), we know that
4 An example
The authors declare that they have no competing interests.
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
This work is supported by the National Natural Science Foundation of China (10801068) and the Education Scientific Research Foundation of Tangshan College (120186). The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.
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