Positive Solutions for Resonant and Nonresonant Nonlinear Third-Order Multipoint Boundary Value Problems
Positive solutions for a kind of third-order multipoint boundary value problem under the nonresonant conditions and the resonant conditions are considered. In the nonresonant case, by using the Leggett-Williams fixed point theorem, the existence of at least three positive solutions is obtained. In the resonant case, by using the Leggett-Williams norm-type theorem due to O’Regan and Zima, the existence result of at least one positive solution is established. It is remarkable to point out that it is the first time that the positive solution is considered for the third-order boundary value problem at resonance. Some examples are given to demonstrate the main results of the paper.
We consider the existence of positive solutions for third-order -point boundary value problem: where , , , , and .
If condition holds, the problem is called resonant boundary value problem or boundary value problem at resonance. Otherwise, the associated problem is called nonresonant boundary value problem. Here, the condition is denoted by (1.2).
Third-order differential equations arise in a variety of different areas of applied mathematics and physics, as the deflection of a curved beam having a constant or varying cross section, three-layer beam, and so on . In recent years, the existence of positive solutions for nonresonant two-point or three-point boundary value problems (Bvp for short) for nonlinear third-order ordinary differential equations has been studied by several authors. For examples, Anderson  established the existence of at least three positive solutions to problem where is continuous and .
By using the well-known Guo-Krasnoselski fixed point theorem , Palamides and Smyrlis  proved that there exists at least one positive solution for third-order three-point problem: For more existence results of positive solutions for boundary value problems of third-order ordinary differential equations, one can see [5–12] and references therein.
For resonant problem of second-order or higher-order differential equations, many existence results of solutions have been established, see [13–25]. In , the authors considered the problem By using the Mawhin continuation theorem, the existence results of solutions are established under the resonant condition , , and , , , respectively.
It is well known that the problem of existence for positive solution to nonlinear Bvp is very difficult when the resonant case is considered. Only few works gave the approach in this area for first- and second-order differential equations [26–31]. To our best knowledge, no paper deal with the existence result of positive solution for resonant third-order boundary value problems. Motivated by the approach in [27–29], we study the existence of positive solution for problem (1) under the nonresonant condition and resonant condition , respectively. By using the Leggett-Williams fixed point theorem and its generalization [28, 30], we establish the existence results of positive solutions. The results obtained in this paper are interesting because(1)the results obtained in the nonresonant case are more general than those established before;(2)it is the first time that the positive solution is considered for third-order Bvp at resonance.
The rest of the paper is organized as follows. Some definitions and lemmas are given in Section 2. In Section 3, we consider the nonresonant case for problem (1). In Section 4, we discuss the existence of positive solution for problem (1) with resonant condition (1.2). Finally, in Section 5, we give some examples to illustrate the main results of the paper.
2. Background Definitions and Lemmas
For the convenience of the reader, we present here the necessary definitions and two-fixed point theorems.
Let , be real Banach spaces. A nonempty convex closed set is said to be a cone provided that(1), for all , ,(2) implies .
Definition 1. The map is said to be a nonnegative continuous concave functional on provided that is continuous and for all and .
Definition 2. Let be given and let be a nonnegative continuous concave functional on the cone . Define the convex sets and by
Lemma 3 (the Leggett-Williams fixed point theorem ). Let be a completely continuous operator and let be a nonnegative continuous concave functional on such that for all . Suppose that there exist such that and , for , for , for with .Then, has at least three-fixed points , , and such that
Operator is called a Fredholm operator with index zero, that is, is closed and , which implies that there exist continuous projections and such that and . Moreover, since , there exists an isomorphism . Denote by , the restriction of to to and its inverse by , so and the coincidence equation is equivalent to Denote to be a retraction, that is, a continuous mapping such that for all and
Lemma 4 (the Leggett-Williams norm-type theorem ). Let be a cone in , and let , be open bounded subsets of with , . Assume that is a Fredholm operator of index zero and is continuous and bounded, is compact on every bounded subset of , for all and , maps subsets of into bounded subsets of C,, where stands for the Brouwer degree, there exists such that for , where for some and such that for every ,,, then the equation has a solution in the set .
3. Positive Solution for the Nonresonant Problem
In this section, we suppose that and . We begin with some preliminary results. Consider the problem
Proof. Integrating both sides of (10) and considering the boundary condition , we have
Let be the Green function of problem
From (15), we can suppose that
For the definition and properties of the Green function together with (16), we have
for , .
Considering (14) together, we obtain that problems (10) and (11) have the unique solution
Lemma 6. The function established in Lemma 5 satisfies that , .
Proof. For , and , For , and , These ensures that , .
Proof. For , , we get that is decreasing on . Then, the condition ensures that have . This together with is concave and decreasing on . Thus, From the concavity of , we have Multiplying both sides with and considering , we have This completes the proof of Lemma 7.
Let Banach space be endowed with the maximum norm. We define the cone by
Define the nonnegative continuous concave functional by Define the constants , by
Theorem 8. Suppose that there exist constants such that , ,, ,, ,then problem (1) has at least three positive solutions , , and satisfying
Proof. We define operator by
It is easy to check that and is completely continuous.
Next, the conditions of Lemma 3 are checked. If , then , and condition implies that Thus, . In the same way, we see that . Hence, condition of Lemma 3 is satisfied.
The fact that constant function implies that . If , from the assumption (A2), . Thus, which ensures that condition (H1) of Lemma 3 is satisfied. Finally, we show that condition of Lemma 3 also holds. Suppose that with , So, condition of Lemma 3 is satisfied. Thus, an application of Lemma 3 implies that the nonresonant third-order boundary value problem (1) has at least three positive solutions , , and satisfying (31).
4. Positive Solution for Resonant Problem
In this section, the condition is considered. Obviously, problem (1) is at resonance under this condition. The norm-type Leggett-Williams fixed point theorem will be used to establish the existence results of positive solution. We define the Banach spaces endowed with the maximum norm.
Define linear operator , , , where and with It is obvious that . Denote the function as follows: Note that , .
Denote the function and positive number as follows:
Theorem 9. Assume that there exists positive constant such that is continuous and satisfies the following conditions:, for , for , there exists , and continuous functions , such that , and is nonincreasing on with then problem (1) with resonant condition (1.2) has at least one positive solution.
Proof. Firstly, we claim that
In fact, for each , we take
It is easy to check that , and , which means . Thus,
On the other hand, for each , there exists ,
Integrating both sides on , we have
Considering condition , and , we conclude that
which equivalents to the conclusion that . So, we have
Clearly, and are closed. Next, we see that , where
In fact, for each , we have
This shows that . Since , we have . Thus, is a Fredholm operator with index zero.
Then, define the projections , by Clearly, , and . Note that for , the inverse of is given by where In fact, it is easy to check that Considering that can be extended continuously on ; condition (C1) of Lemma 4 is fulfilled.
Define the cone of nonnegative functions and subsets of by Clearly, and are bounded and open sets, furthermore
Let the isomorphism and for . Then, it is easy to check that is a retraction and maps subsets of into bounded subsets of , which means that condition (C3) of Lemma 4 is satisfied.
Next, we confirm that (C2) of Lemma 4 holds. For this purpose, suppose that there exists and such that . Then, for all . Thus, Let . We verify that and . The step is divided into three cases:(1)we show that . Suppose, on the contrary, that achieves maximum value only at . Then, in combination with yields that , which is a contradiction,(2)we show that . Suppose, on the contrary, that achieves maximum value at . Then, which together with the condition yield that is increasing near the point . This contradicts to the fact that achieves maximum value at .
Thus, there exists such that . We may choose nearest to with . From the mean value theory, we claim that there exists such that Here, Thus, Then, which contradict to condition (S2). Thus, (C2) holds.
Remark 10. The sign of third-order derivative of a function at point cannot be confirmed even if is a maximal value of . Thus, the method in  is not applicable directly to problem (1). In our opinion, it is the key that the conditions in this paper are stronger than that in .
For , define where and . Suppose that . In view of , we obtain Hence, implies . Furthermore, if , we get contradicting to . Thus, for and . Therefore, This ensures Let and . From condition , we see Hence, . Moreover, for , we have which means . These ensure that (C6) and (C7) of Lemma 4 hold.
At last, we confirm that (C5) is satisfied. Taking on , we see and we can take . Let , we have Therefore, in view of (S3), we obtain, for all , So, , for all , which means (C5) of Lemma 4 holds.
Thus, by Lemma 4, we confirm that the equation has a solution , which implies that nonlinear third-order multipoint boundary value problem (1) with resonance condition (1.2) has at least one positive solution.
In this section, we give two examples to illustrate the main results of the paper. First, we consider the nonresonant four-point boundary value problem where Here, , , , , and By a simply computation, we can get that
We choose , , and . It is easy to check that (1), ,(2), ,(3),. Thus, all the conditions of Theorem 8 are satisfied. This ensures that problem (75) has at least three positive solutions , , and satisfying Next, we consider the resonant third-order four-point boundary value problem where , and Here, By a simple computation, we have Choose , , , , and .
We take Then, It is easy to check that (1), for all ,(2), for all ,(3) and is nonincreasing on with Then, all conditions of Theorem 9 are satisfied. This ensures that the resonant problem has at least one solution, positive on .
The paper is supported by the Natural Science Foundation of China (no. 11201109), the Natural Science Foundation of Anhui Educational Department (KJ2012Z335 and KJ2012B144), the Excellent Talents Foundation of University of Anhui Province (2012SQRL165), and the NSF of Hefei Normal University (2012Kj09).
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