Multiple Positive Solutions for Second-Order -Laplacian Dynamic Equations with Integral Boundary Conditions
© Y. Li and T. Zhang. 2011
Received: 13 July 2010
Accepted: 25 November 2010
Published: 6 December 2010
We are concerned with the following second-order -Laplacian dynamic equations on time scales , , with integral boundary conditions , . By using Legget-Williams fixed point theorem, some criteria for the existence of at least three positive solutions are established. An example is presented to illustrate the main result.
Boundary value problems with -Laplacian have received a lot of attention in recent years. They often occur in the study of the -dimensional -Laplacian equation, non-Newtonian fluid theory, and the turbulent flow of gas in porous medium [1–7]. Many works have been carried out to discuss the existence of solutions or positive solutions and multiple solutions for the local or nonlocal boundary value problems.
On the other hand, the study of dynamic equations on time scales goes back to its founder Stefan Hilger  and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete equations. Further, the study of time scales has led to several important applications, for example, in the study of insect population models, neural networks, heat transfer, and epidemic models, we refer to [8–10]. In addition, the study of BVPs on time scales has received a lot of attention in the literature, with the pioneering existence results to be found in [11–16].
where is positive parameter, for with and , is the delta derivative, is the nabla derivative, is a time scale which is a nonempty closed subset of with the topology and ordering inherited from , 0 and are points in , an interval , with for all , , with , and where .
The main purpose of this paper is to establish some sufficient conditions for the existence of at least three positive solutions for BVPs (1.1)-(1.2) by using Legget-Williams fixed point theorem. This paper is organized as follows. In Section 2, some useful lemmas are established. In Section 3, by using Legget-Williams fixed point theorem, we establish sufficient conditions for the existence of at least three positive solutions for BVPs (1.1)-(1.2). An illustrative example is given in Section 4.
In this section, we will first recall some basic definitions and lemmas which are used in what follows.
Definition 2.1 (see ).
The point is called left-dense, left-scattered, right-dense, or right-scattered if , , and or , respectively. Points that are right-dense and left-dense at the same time are called dense. If has a left-scattered maximum , defined ; otherwise, set . If has a right-scattered minimum , defined ; otherwise, set .
Definition 2.2 (see ).
Definition 2.3 (see ).
A function is rd-continuous provided it is continuous at each right-dense point in and has a left-sided limit at each left-dense point in . The set of rd-continuous functions will be denoted by . A function is left-dense continuous (i.e., ld-continuous) if is continuous at each left-dense point in and its right-sided limit exists (finite) at each right-dense point in . The set of left-dense continuous functions will be denoted by .
Definition 2.4 (see ).
Lemma 2.5 (see ).
If , then for all .
So Lemma 2.6 is proved.
The proof of sufficiency is complete.
The proof of Lemma 2.7 is complete.
for all . Obviously, for all .
If , then .
It is easily obtained from the second part of the proof in Lemma 2.7. The proof is complete.
is complete continuous.
So is equicontinuous for any . Using Arzela-Ascoli theorem on time scales , we obtain that is relatively compact. In view of Lebesgue's dominated convergence theorem on time scales , it is easy to prove that is continuous. Hence, is complete continuous. The proof of this lemma is complete.
and introduce two assumptions with regard to the functionals , as follows:
(H1) there exists such that for all ;
(H2) for any and .
The following fixed point theorem duo to Bai and Ge is crucial in the arguments of our main result.
Lemma 2.10 (see ).
Let be Banach space, a cone, and , . Assume that and are nonnegative continuous convex functionals satisfying (H1) and (H2), is a nonnegative continuous concave functional on such that for all , and is a complete continuous operator. Suppose
(C1) , for ;
(C2) , for ;
(C3) for with .
3. Main Result
In this section, we will give sufficient conditions for the existence of at least three positive solutions to BVPs (1.1)-(1.2).
Suppose that there are positive numbers , , and with , and such that the following conditions are satisfied.
(H4) for all .
(H5) for all , where
Then BVPs (1.1)-(1.2) have at least three positive solutions.
By the definition of the operator and its properties, it suffices to show that the conditions of Lemma 2.10 hold with respect to the operator .
Then it is easy to see that and (H1)-(H2) hold.
As in the argument above, we can get that . Thus, condition (C2) of Lemma 2.10 holds.
Therefore, for . So condition (C1) in Lemma 2.10 is satisfied.
Thus by Lemma 2.10 and the assumption that on , BVPs (1.1)-(1.2) have at least three positive solutions. The proof is complete.
Suppose that there are positive numbers , , and with , , and such that (H3)-(H4) and the following condition are satisfied.
(H6) for all , where
Then BVPs (1.1)-(1.2) have at least three positive solutions.
Therefore, for . So condition (C1) in Lemma 2.10 is satisfied. Using a similar proof to Theorem 3.1, the other conditions in Lemma 2.10 are satisfied. By Lemma 2.10, BVPs (1.1)-(1.2) have at least three positive solutions. The proof is complete.
4. An Example
Then BVPs (4.1)-(4.2) have at least three positive solutions.
Moreover, we have
(H4) for all ,
(H5)for all ,
Therefore, conditions (H3)–(H5) in Theorem 3.1 are satisfied. Further, it is easy to verify that the other conditions in Theorem 3.1 hold. By Theorem 3.1, BVPs (4.1)-(4.2) have at least three positive solutions. The proof is complete.
This work is supported the by the National Natural Sciences Foundation of China under Grant no. 10971183.
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