Positive solutions of third-order nonlocal boundary value problems at resonance
© Zhang and Sun; licensee Springer 2012
Received: 21 April 2012
Accepted: 3 September 2012
Published: 18 September 2012
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.
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 .
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.
KerL has a finite dimension, and
ImL is closed and has a finite codimension.
Throughout the paper, we will assume that
1∘L is a Fredholm operator of index zero, that is, ImL is closed and .
for all and , and
Note that every cone induces a partial order ≤ in X by defining if and only if . The following property is valid for every cone in a Banach space.
Lemma 2.3 ()
Let P be a cone in X. Then for every, there exists a positive numbersuch thatfor all.
Our main results are based on the following theorem due to O’Regan and Zima.
Theorem 2.4 ()
Let C be a cone in X and let, be open bounded subsets of X withand. Assume that 1∘is satisfied and if the following assumptions hold:
(H1) is continuous and bounded, andis compact on every bounded subset of X;
(H2) for alland;
(H3) γ maps subsets ofinto bounded subsets of C;
(H4) , wherestands for the Brouwer degree;
(H5) there existssuch thatfor, whereandis such thatfor all;
then the equationhas a solution in the set.
3 Main results
for every . We also let .
We can now state our result on the existence of a positive solution for the BVP (1.1).
there exists a constant such that for all ;
- (2)there exist , , , and continuous functions , such that for , is non-increasing on with
Then the resonant BVP (1.1) has at least one positive solution on.
Proof Consider the Banach spaces with .
which shows that , which together with implies that . Note that and thus . Therefore, L is a Fredholm operator of index zero.
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.
Moreover, . Let and for . Then γ is a retraction and maps subsets of into bounded subsets of C, which means that (H3) of Theorem 2.4 holds.
Let , where and will be defined in the following proof.
Let . Now, we verify that and .
First, we show . Suppose, on the contrary, that achieves maximum value M only at . Then in combination with yields that , which is a contradiction.
Suppose, on the contrary, that . The step is divided into two cases:
which implies is increasing close to 1. This together with induces (t close to 1), that is, is decreasing close to 1, which contradicts .
Case 2. Assume that has zero points on ; we may choose nearest to 1 with . Then there is a constant such that . Similar to the above arguments, we easily get a contradiction too.
Hence, we can choose so that . This gives and . By , we know and . Similarly, we also divide the part of the proof into two cases.
which is a contradiction.
which is a contradiction. Therefore, (H2) of Theorem 2.4 holds.
which shows that (H4) of Theorem 2.4 holds.
which shows that . These ensure that (H6), (H7) of Theorem 2.4 hold. It remains to show that (H5) is satisfied.
That is, for all , which shows that (H5) of Theorem 2.4 holds.
Summing up, all the hypotheses of Theorem 2.4 are satisfied. Therefore, the equation has a solution . And so, the resonant BVP (1.1) has at least one positive solution on . □
4 An example
for all ;
- (2)for , is non-increasing on with
Thus, all the conditions of Theorem 3.1 are satisfied. Then the resonant problem (4.1) has at least one positive solution on .
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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