Positive solutions of third-order nonlocal boundary value problems at resonance

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.

This paper is devoted to the existence of positive solutions for the following third-order nonlocal boundary value problem (BVP for short):

where , () and . The problem (1.1) happens to be at resonance in the sense that the associated linear homogeneous BVP

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 [1].

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 [13] 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 [16], 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 [16]. An example is also included to illustrate the main results.

For the convenience of the reader, we present here the necessary definitions and a new fixed point theorem due to O’Regan and Zima.

Definition 2.1 Let X and Z be real Banach spaces. A linear operator is called a Fredholm operator if the following two conditions hold:

(i) KerL has a finite dimension, and

(ii) 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 .

From Definition 2.1, it follows that there exist continuous projectors and such 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

(1) for all and , and

(2) implies .

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 ([19])

LetPbe a cone inX. Then for every, there exists a positive numbersuch thatfor all.

Let be a retraction, that is, a continuous mapping such that for all . Set

and

Our main results are based on the following theorem due to O’Regan and Zima.

Theorem 2.4 ([16])

LetCbe a cone inXand let, be open bounded subsets ofXwithand. Assume that 1^∘is satisfied and if the following assumptions hold:

(H1) is continuous and bounded, andis compact on every bounded subset ofX;

(H2) for alland;

(H3) γmaps subsets ofinto bounded subsets ofC;

(H4) , wherestands for the Brouwer degree;

(H5) there existssuch thatfor, whereandis such thatfor all;

(H6) ;

(H7) ,

then the equationhas a solution in the set.

For simplicity of notation, we set

where , and

It is easy to check that , , and since , we get

for every . We also let .

We can now state our result on the existence of a positive solution for the BVP (1.1).

Theorem 3.1Assume thatis continuous and

(1) there exists a constantsuch thatfor all;

(2) there exist, , , and continuous functions, such thatfor, is non-increasing onwith

(3) .

Then the resonant BVP (1.1) has at least one positive solution on.

Proof Consider the Banach spaces with .

Let and with

be given by and for . Then

and

Clearly, and ImL is closed. It follows from that

In fact, for each , we have

which shows that , which together with implies that . Note that and thus . Therefore, L is a Fredholm operator of index zero.

Next, define the projections by

and by

Clearly, , and . Note that for , the inverse of is given by

where

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

Let

and

Clearly, and are bounded and open sets and

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.

Suppose that there exist and such that . Then

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.

Next, we show . It follows from that there is a constant such that , and thus . By the condition we have, for (), there exists such that

Suppose, on the contrary, that . The step is divided into two cases:

Case 1. Assume that on . Let . Then (3.1) yields

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.

Case 1. If on , then there is a constant such that . Thus we have

Let . Then it follows from (3.1) that

which is a contradiction.

Case 2. If has zero points on , we may choose nearest to with . Then there is a constant such that . Thus we have

Let . Then it follows from (3.1) that

which is a contradiction. Therefore, (H2) of Theorem 2.4 holds.

Consider . Then on . Similar to [13], we define

Suppose . Then in view of (1), we obtain

Hence, implies . Furthermore, if , then we have

contradicting (3.1). Thus for and . Therefore,

However,

This gives

which shows that (H4) of Theorem 2.4 holds.

Let and . Then

From (1), we know that

Hence . Moreover, for , we have

which shows that . These ensure that (H6), (H7) of Theorem 2.4 hold. It remains to show that (H5) is satisfied.

Taking on and , we confirm that

Let . Then we have , , and , . Therefore, in view of (2), for all , we obtain

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 . □

Consider the BVP

Here , , , and

By a simple computation, we get

We may choose , , , , . It is easy to check

(1) for all ;

(2) for , is non-increasing on with

(3) .

Thus, all the conditions of Theorem 3.1 are satisfied. Then the resonant problem (4.1) has at least one positive solution on .

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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