- Research Article
- Open Access
Positive Solutions for Boundary Value Problems of Second-Order Functional Dynamic Equations on Time Scales
Advances in Difference Equations volume 2009, Article number: 829735 (2009)
Criteria are established for existence of least one or three positive solutions for boundary value problems of second-order functional dynamic equations on time scales. By this purpose, we use a fixed-point index theorem in cones and Leggett-Williams fixed-point theorem.
In a recent paper , by applying a fixed-point index theorem in cones, Jiang and Weng studied the existence of positive solutions for the boundary value problems described by second-order functional differential equations of the form
Aykut  applied a cone fixed-point index theorem in cones and obtained sufficient conditions for the existence of positive solutions of the boundary value problems of functional difference equations of the form
In this article, we are interested in proving the existence and multiplicity of positive solutions for the boundary value problems of a second-order functional dynamic equation of the form
Throughout this paper we let be any time scale (nonempty closed subset of ) and be a subset of such that and for is not right scattered and left dense at the same time.
We will assume that the following conditions are satisfied.
(H2) is continuous with respect to and for , where denotes the set of nonnegative real numbers.
(H3) defined on satisfies
Let be nonempty subset of
if then ; for , where denotes the set of all positively regressive and rd-continuous functions.
(H5) and are defined on and , respectively, where
; furthermore, ;
There have been a number of works concerning of at least one and multiple positive solutions for boundary value problems recent years. Some authors have studied the problem for ordinary differential equations, while others have studied the problem for difference equations, while still others have considered the problem for dynamic equations on a time scale [5–10]. However there are fewer research for the existence of positive solutions of the boundary value problems of functional differential, difference, and dynamic equations [1, 2, 11–13].
Our problem is a dynamic analog of the BVPs (1.1) and (1.2). But it is more general than them. Moreover, conditions for the existence of at least one positive solution were studied for the BVPs (1.1) and (1.2). In this paper, we investigate the conditions for the existence of at least one or three positive solutions for the BVP (1.3). The key tools in our approach are the following fixed-point index theorem , and Leggett-Williams fixed-point theorem .
Theorem 1.1 (see ).
Let be Banach space and be a cone in . Let , and define . Assume is a completely continuous operator such that for
If for , then
If for , then
Theorem 1.2 (see ).
Let be a cone in the real Banach space . Set
Suppose that is a completely continuous operator and is a nonnegative continuous concave functional on with for all . If there exists such that the following conditions hold:
and for all
Then has at least three fixed points in satisfying
First, we give the following definitions of solution and positive solution of BVP (1.3).
We say a function is a solution of BVP (1.3) if it satisfies the following.
is nonnegative on .
as , where is defined as(21)
as , where is defined as(22)
is -differentiable, is -differentiable on and is continuous.
Furthermore, a solution of (1.3) is called a positive solution if for
Denote by and the solutions of the corresponding homogeneous equation
under the initial conditions
Since the Wronskian of any two solutions of (2.3) is independent of , evaluating at and using the initial conditions (2.4) yield
Using the initial conditions (2.4), we can deduce from (2.3) for and , the following equations:
Lemma 2.2 (see ).
Under the conditions (H1) and the first part of (H4) the solutions and possess the following properties:
Let be the Green function for the boundary value problem:
where and are given in (2.7) and (2.8), respectively. It is obvious from (2.6), (H1) and (H4), that holds.
Assume the conditions (H1) and (H4) are satisfied. Then
for , and , for . Besides, is nondecreasing and is nonincreasing, for . Therefore, we have
So statement (i) of the lemma is proved. If for a given then statement (ii) of the lemma is obvious for such values. Now, and . Consequently, , for all such Let us take any . Then we have for ,
and we have for ,
Let be endowed with maximum norm for , and let be a cone defined by
where is as in (2.12).
Suppose that is a solution of (1.3), then it can be written as
Throughout this paper we assume that is the solution of (1.3) with . Clearly, can be expressed as follows:
Let be a solution of (1.3) and . Noting that for , we have
Define an operator as follows:
It is easy to derive that is a positive solution of BVP (1.3) if and only if is a nontrivial fixed point of , where be defined as before.
For , we have . Moreover, we have from definition of that and , for and , respectively. Thus, where . It follows from the definition and Lemma 2.3 that
is completely continuous.
for all , then there exist such that , for and , for .
Choose such that
By using the first equality of (2.27), we can choose such that
If , then for , we have
Therefore we get
Thus, we have from Theorem 1.1, , for . On the other hand, the second equality of (2.27) implies for every , there is an , such that
Here we choose satisfying (2.28). For , we have definition of that
It follows from (2.32) that
This shows that
Thus, by Theorem 1.1, we conclude that for . The proof is therefore complete.
3. Existence of One Positive Solution
In this section, we investigate the conditions for the existence of at least one positive solution of the BVP (1.3).
In the next theorem, we will also assume that the following condition on .
where is large enough such that
and is small enough such that
where is the eigenfunction related to the smallest eigenvalue of the eigenvalue problem:
If (H1)–(H6) are satisfied, then the BVP (1.3) has at least one positive solution.
Fix and let for . Then, satisfies (2.27). Define by
Then is a completely continuous operator. One has from Lemma 2.6 that there exist such that
Define by then is a completely continuous operator. By the first equality in (H6) and the definition of , there are and such that
We now prove that for all and . In fact, if there exists and such that , then satisfies the equation
and the boundary conditions
Multiplying both sides of (3.10) by , then integrating from to , and using integration by parts in the left-hand side two times, we obtain
Combining (3.9) and (3.12), we get
We also have
Equations (3.13) and (3.14) lead to
This is impossible. Thus for and . By (3.7) and the homotopy invariance of the fixed-point index (see ), we get that
On the other hand, according to the second inequality of (H6), there exist and such that
then it follows that
Define by , then is a completely continuous operator. We claim that there exists such that
In fact, if for some and , then
where Combining (3.21) with (3.22), we have
Let Then we get
Consequently, by the homotopy invariance of the fixed-point index, we have
where is zero operator. Use (3.16) and (3.25) to conclude that
Hence, has a fixed point in .
Let . Since for and .
If (H1)–(H5) and (H7) are satisfied, then the BVP (1.3) has at least one positive solution.
Define by , then is a completely continuous operator. By the first inequality in (H7), there exist and such that
We claim that for and . In fact, if there exist and such that , then satisfies the boundary condition (3.11). Since , we have . Then we have
Multiplying the last equation by integrating from to , by (3.28), we obtain
then we have
Equations (3.30) and (3.31) lead to
This is impossible. By homotopy invariance of the fixed-point index, we get that
Define by , then is a completely continuous operator. By the second inequality in (H7), and definition of , there exist and such that
then, it is obvious that
We claim that there exists such that
In fact, if for some and , then using (3.36), it is analogous to the argument of (3.13) and (3.14) that
Equation (3.38) leads to Let . Then we get
Consequently, by (3.8) and the homotopy invariance of the fixed-point index, we have
In view of (3.33) and (3.40), we obtain
Therefore, has a fixed point in . The proof is completed.
Using the following (H8) or (H9) instead of (H6) or (H7), the conclusions of Theorems 3.1 and 3.2 are true. For ,
4. Existence of Three Positive Solutions
In this section, using Theorem 1.2 (the Leggett-Williams fixed-point theorem) we prove the existence of at least three positive solutions to the BVP (1.3).
Define the continuous concave functional to be , and the constants
Suppose there exists constants such that
(D3) one of the following is satisfied:
there exists a constant such that for and ,
where , , and are as defined in (2.12), (4.1), (4.2), respectively. Then the boundary value problem (1.3) has at least three positive solutions and satisfying
The technique here similar to that used in  Again the cone , the operator is the same as in the previous sections. For all we have If , then and the condition (a) of (D3) imply that
Thus there exist a and such that if , then . For , we have for all , for all Pick any
Then implies that
The condition (b) of (D3) implies that there exists a positive number such that for and . If , then
Thus Consequently, the assumption (D3) holds, then there exist a number such that and
The remaining conditions of Theorem 1.2 will now be shown to be satisfied.
By (D1) and argument above, we can get that Hence, condition (ii) of Theorem 1.2 is satisfied.
We now consider condition (i) of Theorem 1.2. Choose for , where . Then and so that . For , we have , . Combining with (D2), we get
for . Thus, we have
As a result, yields
Lastly, we consider Theorem 1.2(iii). Recall that . If and , then
Thus, all conditions of Theorem 1.2 are satisfied. It implies that the TPBVP (1.3) has at least three positive solutions with
Let Consider the BVP:
Since , . It is clear that (H1)–(H5) and (H8) are satisfied. Thus, by Corollary 3.3, the BVP (5.1) has at least one positive solution.
Let us introduce an example to illustrate the usage of Theorem 4.1. Let
Consider the TPBVP:
The Green function of the BVP (5.4) has the form
Clearly, is continuous and increasing . We can also see that . By (2.12), (4.1), and (4.2), we get , and .
Now we check that (D1), (D2), and (b) of (D3) are satisfied. To verify (D1), as , we take , then
and (D1) holds. Note that , when we set ,
holds. It means that (D2) is satisfied. Let , we have
from , so that (b) of (D3) is met. Summing up, there exist constants and satisfying
Thus, by Theorem 4.1, the TPBVP (5.4) has at least three positive solutions with
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