Higher-Order Dynamic Delay Differential Equations on Time Scales
Hua Su,1Lishan Liu,2and Xinjun Wang3
Academic Editor: Yansheng Liu
Received11 Oct 2011
Accepted12 Feb 2012
Published24 Apr 2012
Abstract
We study the existence of positive solutions for the nonlinear four-point singular boundary value problem with higher-order -Laplacian dynamic delay differential equations on time scales, subject to some boundary conditions. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with -Laplacian operator are obtained.
1. Introduction
The study of dynamic equations on time scales goes back to its founder Stefan Hilger [1] and is a new area of still fairly theoretical exploration in mathematics. Boundary value problems for delay differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory (see [2, 3]). In recent years, many authors have begun to pay attention to the study of boundary value problems or with -Laplacian equations or with -Laplacian dynamic equations on time scales (see [4β19] and the references therein).
In [7], Sun and Li considered the existence of positive solution of the following dynamic equations on time scales:
where , , , . They obtained the existence of single and multiple positive solutions of the problem (1.1) by using fixed point theorem and Leggett-Williams fixed point theorem, respectively.
In [10], Avery and Anderson discussed the following dynamic equation on time scales:
He obtained some results for the existence of one positive solution of the problem (1.2) based on the limits and .
In [11], Wang et al. discussed the following dynamic equation by using Avery-Peterson fixed theorem (see [10]):
They obtained some results for the existence, three positive solutions of the problem (1.3), (1.4) and (1.3), (1.5), respectively.
However, there are not many concerning the -Laplacian problems on time scales. Especially, for the singular multi point boundary value problems for higher-order -Laplacian dynamic delay differential equations on time scales, with the authorβs acknowledg, no one has studied the existence of positive solutions in this case.
Recently, in [16], we study the existence of positive solutions for the following nonlinear two-point singular boundary value problem with -Laplacian operator
by using the fixed point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for nonlinear singular boundary value problem (1.6) with -Laplacian operator are obtained.
Now, motivated by the results mentioned above, in this paper, we study the existence of positive solutions for the following nonlinear four-point singular boundary value problem with higher-order -Laplacian dynamic delay differential equations operator on time scales (SBVP):
where is -Laplacian operator, that is, , , , ; , is prescribed and , , , , , .
In this paper, by constructing one integral equation which is equivalent to the problem (1.7), (1.8), we research the existence of positive solutions for nonlinear singular boundary value problem (1.7), (1.8) when and satisfy some suitable conditions.
Our main tool of this paper is the following fixed point index theory.
Theorem 1.1 (see [18]). Suppose is a real Banach space, is a cone, let . Let operator be completely continuous and satisfy , for all . Then(i)if , for all , then ; (ii)if , for all , then .
This paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. In Section 3, we study the existence of at least two solutions of the systems (1.7), (1.8). In Section 4, we give an examples as the application.
2. Preliminaries and Lemmas
A time scale is an arbitrary nonempty closed subset of real numbers . In [1, 14, 20], we can find some basic definitions about time scale. The operators and from to :
are called the forward jump operator and the backward jump operator, respectively.
If , then . If , then is the forward difference operator, while is the backward difference operator.
A function is left-dense continuous (i.e., -continuous), if is continuous at each left-dense point in and its right-sided limit exists at each right-dense point in . It is well known that is -continuous.
If , then we define the nabla integral by
If , then we define the delta integral by
In the rest of this paper, is closed subset of with , . And let
Here,
Then is a Banach space with the norm . And let
Obviously, is a cone in . Set .
Definition 2.1. is called a solution of SBVP (1.7) and (1.8) if it satisfies the following:(1);(2) for all and satisfy conditions (1.8);(3) hold for . In the rest of the paper, we also make the following assumptions: ();() and there exists , such that(), on and .It is easy to check that condition () implies that
We can easily get the following lemmas.
Lemma 2.2. Suppose condition holds. Then there exists a constant satisfing
Furthermore, the function
is positive continuous functions on ; therefore, has minimum on . Hence we suppose, that there exists such that , .
Proof. At first, it is easily seen that is continuous on . Next, let
Then, from condition , we have that the function is strictly monotone nondecreasing on and , the function is strictly monotone nonincreasing on and , which implies . The proof is complete.
Lemma 2.3. Let and of Lemma 2.2, then
The proof of the above lemma is similar to the proof in [17, Lemma 2.2], so we omit it.
Lemma 2.4. Suppose that conditions hold, is a solution of the following boundary value problems:
where
Then, , is a positive solution to the SBVP (1.7) and (1.8).
Proof. It is easy to check that satisfies (1.7) and (1.8).
So in the rest of the sections of this paper, we focus on SBVP (2.13) and (2.14).
Lemma 2.5. Suppose that conditions hold, is a solution of boundary value problems (2.13), (2.14) if and only if is a solution of the following integral equation:
where
Proof. Necessity. Obviously, for , we have . If , by the equation of the boundary condition, we have , , then there exists a constant such that . Firstly, by integrating the equation of the problems (2.13) on , we have
then
thus
By and condition (2.18), let on (2.18), we have
By the equation of the boundary condition (2.14), we have
then
Then, by (2.20) and leting on (2.20), we have
Then
Then, by integrating (2.25) for times on , we have
Similarly, for , by integrating the equation of problems (2.13) on , we have
Therefore, for any , can be expressed as equation
where is expressed as (2.17). Then the results of Lemma 2.3 hold. Sufficiency. Suppose that , . Then by (2.17), we have
So, , . These imply that (2.13) holds. Furthermore, by letting and on (2.17) and (2.29), we can obtain the boundary value equations of (2.14). The proof is complete.
Now, we define an operator equation given by
where is given by (2.17).
From the definition of and the previous discussion, we deduce that, for each , . Moreover, we have the following lemmas.
Lemma 2.6. is completely continuous.
Proof. Because
is continuous, decreasing on and satisfies , then, for each and . This shows that . Furthermore, it is easy to check by Arzela-ascoli Theorem that is completely continuous.
Lemma 2.7. Suppose that conditions hold, the solution of problem (2.13), (2.14) satisfies
and for in Lemma 2.2, one has
Proof. Firstly, we can have
Next, if is the solution of problem (2.13), (2.14), then is concave function, and , . Thus, we have
that is, , . Finally, by Lemma 2.3, for , we have . By , we have
The proof is complete.
For convenience, we set
where is the constant from Lemma 2.2. By Lemma 2.5, we can also set
where .
3. The Existence of Multiple Positive Solutions
In this section, we also make the following conditions:(), for , ;(), for , .
Next, we will discuss the existence of multiple positive solutions.
Theorem 3.1. Suppose that conditions (), (), (), and () hold. Assume that also satisfies();().Then, the SBVP (2.13), (2.14) hase at last two solutions , such that
Proof. For any , by Lemma 2.3, we have
First, by condition (), for any , there exists a constant such that
Set . For any , by (3.2) we have
For any , by (3.3) and Lemmas 2.3β2.6, we will discuss it from three perspectives.(i)If , we have
(ii)If , we have(iii)If , we have
Therefore, no matter under which condition, we all have
Then, by Theorem 1.1, we have
Next, by condition (), for any , there exists a constant such that
We choose a constant , obviously . Set . For any , by Lemma 2.3, we have
Then, by (3.10), Lemmas 2.3β2.6 and also similar to the previous proof, we can also have from three perspectives that
Then, by Theorem 1.1, we have
Finally, set . For any , we have , by () we know
Thus,
Then, by Theorem 1.1, we have
Therefore, by (3.9), (3.13), (3.16), we have
Then has fixed point and fixed point . Obviously, are all positive solutions of problem (2.13), (2.14) and . Proof of Theorem 3.1 is complete.
Theorem 3.2. Suppose that conditions (), (), (), () hold. Assume that also satisfies();().Then, the SBVP (2.13), (2.14) has at last two solutions such that .
Proof. First, by , for , there exists a constant such that
Set , for any , by (3.18), we have
that is,
Then, by Theorem 1.1, we have
Next, let ; note that is monotone increasing with respect to . Then, from , it is easy to see that
Therefore, for any , there exists a constant such that
Set , for any , by (3.23), we have
that is,
Then, by Theorem 1.1, we have
Finally, set . For any , by , Lemma 2.3 and also similar to the previous proof of Theorem 3.1, we can also have
Then, by Theorem 1.1, we have
Therefore, by (3.21), (3.28), (3.26), , we have
Then has fixed point and fixed point . Obviously, are all positive solutions of problem (2.13), (2.14) and . The proof of Theorem 3.2 is complete.
Similar to Theorems 3.1 and 3.2, we also obtain the following theorems.
Theorem 3.3. Suppose that conditions (), (), (), and () hold and(),().Then, the SBVP (2.13), (2.14) has at last two solutions such that .
Theorem 3.4. Suppose that conditions (), (), (), and () hold and();().Then, the SBVP (2.13), (2.14) has at last two solutions such that .
4. An Example
Example 4.1. Consider the following 3-order singular boundary value problem (SBVP) with -Laplacian:
where
So, by Lemma 2.4, we discuss the following SBVP:
where
Then, obviously,
so conditions (), , , (), and hold. Next,
we choose , and for , because of the monotone increasing of on , then
Therefore, by
we know
so condition holds. Then, by Theorem 3.1, SBVP (4.3) has at least two positive solutions and . Then, by Lemma 2.4, , , are the positive solutions of the SBVP (4.1).
Acknowldgments
The first and second authors were supported financially by Shandong Province Natural Science Foundation (ZR2009AQ004), and National Natural Science Foundation of China (11071141), and the third author was supported by Shandong Province planning Foundation of Social Science (09BJGJ14).
References
S. Hilger, βAnalysis on measure chainsβaunified approach to continuous and discrete calculus,β Results in Mathematics, vol. 18, no. 1-2, pp. 18β56, 1990.
G. B. Gustafson and K. Schmitt, βNonzero solutions of boundary value problems for second order ordinary and delay-differential equations,β Journal of Differential Equations, vol. 12, pp. 129β147, 1972.
L. H. Erbe and Q. Kong, βBoundary value problems for singular second-order functional-differential equations,β Journal of Computational and Applied Mathematics, vol. 53, no. 3, pp. 377β388, 1994.
J. Henderson and H. B. Thompson, βMultiple symmetric positive solutions for a second order boundary value problem,β Proceedings of the American Mathematical Society, vol. 128, no. 8, pp. 2373β2379, 2000.
F. M. Atici and G. Sh. Guseinov, βOn Green's functions and positive solutions for boundary value problems on time scales,β Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 75β99, 2002.
H. R. Sun and W. T. Li, βPositive solutions for nonlinear three-point boundary value problems on time scales,β Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 508β524, 2004.
J. W. Lee and D. O'Regan, βExistence results for differential delay equations-I,β Journal of Differential Equations, vol. 102, no. 2, pp. 342β359, 1993.
D. R. Anderson, βSolutions to second-order three-point problems on time scales,β Journal of Difference Equations and Applications, vol. 8, no. 8, pp. 673β688, 2002.
R. I. Avery and D. R. Anderson, βExistence of three positive solutions to a second-order boundary value problem on a measure chain,β Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 65β73, 2002.
Y. Wang, W. Zhao, and W. Ge, βMultiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional -Laplacian,β Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 641β654, 2007.
R. I. Avery and A. C. Peterson, βThree positive fixed points of nonlinear operators on ordered Banach spaces,β Computers & Mathematics with Applications, vol. 42, no. 3β5, pp. 313β322, 2001.
J. P. Sun, βExistence of solution and positive solution of BVP for nonlinear third-order dynamic equation,β Nonlinear Analysis, vol. 64, no. 3, pp. 629β636, 2006.
Z. M. He, βDouble positive solutions of three-point boundary value problems for -Laplacian dynamic equations on time scales,β Journal of Computational and Applied Mathematics, vol. 182, no. 2, pp. 304β315, 2005.
H. Su, Z. Wei, and F. Xu, βThe existence of positive solutions for nonlinear singular boundary value system with -Laplacian,β Journal of Applied Mathematics and Computation, vol. 181, no. 2, pp. 826β836, 2006.
H. Su, Z. Wei, and F. Xu, βThe existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p-Laplacian operator,β Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 319β332, 2007.
H. Su, B. Wang, and Z. Wei, βPositive solutions of four-point boundary-value problems for four-order -Laplacian dynamic equations on time scales,β Electronic Journal of Differential Equations, vol. 78, pp. 1β13, 2006.
D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego, Calif, USA, 1988.
X. Zhang and L. Liu, βPositive solutions for -point boundary-value problems with one-dimensional -Laplacian,β Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 523β531, 2011.
V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, vol. 370 of Mathematics and its Applications, Kluwer Academic, Boston, Mass, USA, 1996.