Open Access

Necessary and Sufficient Conditions for the Existence of Positive Solution for Singular Boundary Value Problems on Time Scales

Advances in Difference Equations20092009:737461

https://doi.org/10.1155/2009/737461

Received: 27 March 2009

Accepted: 15 September 2009

Published: 13 October 2009

Abstract

By constructing available upper and lower solutions and combining the Schauder's fixed point theorem with maximum principle, this paper establishes sufficient and necessary conditions to guarantee the existence of as well as positive solutions for a class of singular boundary value problems on time scales. The results significantly extend and improve many known results for both the continuous case and more general time scales. We illustrate our results by one example.

1. Introduction

Recently, there have been many papers working on the existence of positive solutions to boundary value problems for differential equations on time scales; see, for example, [122]. This has been mainly due to its unification of the theory of differential and difference equations. An introduction to this unification is given in [10, 14, 23, 24]. Now, this study is still a new area of fairly theoretical exploration in mathematics. However, it has led to several important applications, for example, in the study of insect population models, neural networks, heat transfer, and epidemic models; see, for example, [9, 10].

Motivated by works mentioned previously, we intend in this paper to establish sufficient and necessary conditions to guarantee the existence of positive solutions for the singular dynamic equation on time scales:

(11)

subject to one of the following boundary conditions:

(12)

or

(13)

where is a time scale, , where is right dense and is left dense. and is continuous. Suppose further that is nonincreasing with respect to , and there exists a function such that

(14)

A necessary and sufficient condition for the existence of as well as positive solutions is given by constructing upper and lower solutions and with the maximum principle. The nonlinearity may be singular at and/or . By singularity we mean that the functions in (1.1) is allowed to be unbounded at the points and/or A function is called a (positive) solution of (1.1) if it satisfies (1.1) ( ); if even exist, we call it is a solution.

To the best of our knowledge, there is very few literature giving sufficient and necessary conditions to guarantee the existence of positive solutions for singular boundary value problem on time scales. So it is interesting and important to discuss these problems. Many difficulties occur when we deal with them. For example, basic tools from calculus such as Fermat's theorem, Rolle's theorem, and the intermediate value theorem may not necessarily hold. So we need to introduce some new tools and methods to investigate the existence of positive solutions for problem (1.1) with one of the above boundary conditions.

The time scale related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to time scales. The readers who are unfamiliar with this area can consult, for example, [6, 1113, 25, 26] for details.

The organization of this paper is as follows. In Section 2, we provide some necessary background. In Section 3, the main results of problem (1.1)-(1.2) will be stated and proved. In Section 4, the main results of problem (1.1)–(1.3) will be investigated. Finally, in Section 5, one example is also included to illustrate the main results.

2. Preliminaries

In this section we will introduce several definitions on time scales and give some lemmas which are useful in proving our main results.

Definition 2.1.

A time scale is a nonempty closed subset of .

Definition 2.2.

Define the forward (backward) jump operator at for at for by
(21)

for all . We assume throughout that has the topology that it inherits from the standard topology on and say is right scattered, left scattered, right dense and left dense if and , respectively. Finally, we introduce the sets and which are derived from the time scale as follows. If has a left-scattered maximum , then , otherwise . If has a right-scattered minimum , then , otherwise .

Definition 2.3.

Fix and let . Define to be the number (if it exists) with the property that given there is a neighborhood of with
(22)
for all , where denotes the (delta) derivative of with respect to the first variable, then
(23)
implies
(24)

Definition 2.4.

Fix and let . Define to be the number (if it exists) with the property that given there is a neighborhood of with
(25)

for all . Call the (nabla) derivative of at the point .

If then . If then is the forward difference operator while is the backward difference operator.

Definition 2.5.

A function is called rd-continuous provided that it is continuous at all right-dense points of and its left-sided limit exists (finite) at left-dense points of . We let denote the set of rd-continuous functions .

Definition 2.6.

A function is called -continuous provided that it is continuous at all left-dense points of and its right-sided limit exists (finite) at right-dense points of . We let denote the set of -continuous functions .

Definition 2.7.

A function is called a delta-antiderivative of provided that holds for all . In this case we define the delta integral of by
(26)

for all .

Definition 2.8.

A function is called a nabla-antiderivative of provided that holds for all . In this case we define the delta integral of by
(27)

for all .

Throughout this paper, we assume that is a closed subset of with .

Let , equipped with the norm

(28)

It is clear that is a real Banach space with the norm.

Lemma 2.9 (Maximum Principle).

Let and . If , , and Then

3. Existence of Positive Solution to (1.1)-(1.2)

In this section, by constructing upper and lower solutions and with the maximum principle Lemma 2.9, we impose the growth conditions on which allow us to establish necessary and sufficient condition for the existence of (1.1)-(1.2).

We know that

(31)

is the Green's function of corresponding homogeneous BVP of (1.1)-(1.2).

We can prove that has the following properties.

Proposition 3.1.

For , one has
(32)

To obtain positive solutions of problem (1.1)-(1.2), the following results of Lemma 3.2 are fundamental.

Lemma 3.2.

Assume that holds. If and exist and are finite, then one has
(33)

Proof.

Without loss of generality, we suppose that there is only one right-scattered point . Then we have
(34)
that is,
(35)
Similarly, we can prove
(36)

The proof is complete.

Theorem 3.3.

Suppose that holds. Then problem (1.1)-(1.2) has a positive solution if and only if the following integral condition holds:
(37)

Proof.

  1. (1)

    Necessity

     

By (H), there exists such that . Without loss of generality, we assume that is nonincreasing on with .

Suppose that is a positive solution of problem (1.1)-(1.2), then

(38)

which implies that is concave on . Combining this with the boundary conditions, we have Therefore . So by [10, Theorem  1.115], there exists satisfying or . And for , , for . Denote , then .

First we prove .

By ( ), for any fixed , we have

(39)
It follows that
(310)
If , then we have by (3.10)
(311)

This means , then , which is a contradiction with being positive solution. Thus , then .

Second, we prove .

If then

(312)
If , then , , and
(313)
It follows that
(314)
By (3.14) we have
(315)
(316)
Combining this with (3.10) we obtain
(317)
Similarly
(318)
Then we can obtain
(319)
  1. (2)

    Sufficiency

     
Let
(320)
Then
(321)

Let

(322)

then

Let then

(323)
So, we have
(324)

and . Hence are lower and upper solutions of problem (1.1)-(1.2), respectively. Obviously for .

Now we prove that problem (1.1)-(1.2) has a positive solution with

Define a function

(325)
Then is continuous. Consider BVP
(326)

Define mapping by

(327)

Then problem (1.1)-(1.2) has a positive solution if and only if has a fixed point with

Obviously is continuous. Let By (3.7) and (3.16), for all , we have

(328)

Then is bounded. By the continuity of we can easily found that are equicontinuous. Thus is completely continuous. By Schauder fixed point theorem we found that has at least one fixed point .

We prove . If there exists such that

(329)

Let then for Thus . By (3.24) we know that And . By Lemma 2.9 we have , which is a contradiction. Then . Similarly we can prove . The proof is complete.

Theorem 3.4.

Suppose that ( ) holds. Then problem (1.1)-(1.2) has a positive solution if and only if the following integral condition holds:
(330)

Proof.

  1. (1)

    Necessity

     
Let be a positive solution of problem (1.1)-(1.2). Then is decreasing on . Hence is integrable and
(331)
By simple computation and using [10, Theorem  1.119], we obtain . So there exist such that . By we obtain
(332)
By
(333)

we have

  1. (2)

    Sufficiency

     
Let , then
(334)
Similar to Theorem 3.3, let , there exists satisfying , and
(335)

then is integral and exist, hence is a positive solution in . The proof is complete.

4. Existence of Positive Solution to (1.1)–(1.3)

Now we deal with problem (1.1)–(1.3). The method is just similar to what we have done in Section 3, so we omit the proof of main result of this section.

Let

(41)

be the Green's function of corresponding homogeneous BVP of (1.1)–(1.3).

We can prove that has the following properties.

Similar to (3.2), we have

(42)

Theorem 4.1.

Suppose that holds, then problem (1.1)–(1.3) has a positive solution if and only if the following integral condition holds:
(43)

Theorem 4.2.

Suppose that holds, then problem (1.1)–(1.3) has a positive solution if and only if the following integral condition holds:
(44)

5. Example

To illustrate how our main results can be used in practice we present an example.

Example 5.1.

We have
(51)
where Select , then we have Moreover, we have
(52)

By Theorem 3.3, problem (5.1) has a positive solution in .

Remark 5.2.

Example 5.1 implies that there is a large number of functions that satisfy the conditions of Theorem 3.3. In addition, the conditions of Theorem 3.3 are also easy to check.

Declarations

Acknowledgments

This work is sponsored by the National Natural Science Foundation of China (10671012, 10671023) and the Scientific Creative Platform Foundation of Beijing Municipal Commission of Education (PXM2008-014224-067420).

Authors’ Affiliations

(1)
School of Science, Beijing Information Science & Technology University
(2)
Department of Mathematics and Physics, North China Electric Power University
(3)
Department of Applied Mathematics, Beijing Institute of Technology

References

  1. Anderson DR: Eigenvalue intervals for a two-point boundary value problem on a measure chain. Journal of Computational and Applied Mathematics 2002,141(1-2):57-64. 10.1016/S0377-0427(01)00435-6MathSciNetView ArticleMATHGoogle Scholar
  2. Avery RI, Anderson DR: Existence of three positive solutions to a second-order boundary value problem on a measure chain. Journal of Computational and Applied Mathematics 2002,141(1-2):65-73. 10.1016/S0377-0427(01)00436-8MathSciNetView ArticleMATHGoogle Scholar
  3. Atici FM, Guseinov GSh: On Green's functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics 2002,141(1-2):75-99. 10.1016/S0377-0427(01)00437-XMathSciNetView ArticleMATHGoogle Scholar
  4. Anderson D, Avery R, Henderson J:Existence of solutions for a one dimensional -Laplacian on time-scales. Journal of Difference Equations and Applications 2004,10(10):889-896. 10.1080/10236190410001731416MathSciNetView ArticleMATHGoogle Scholar
  5. Bohner M, Luo H: Singular second-order multipoint dynamic boundary value problems with mixed derivatives. Advances in Difference Equations 2006, 2006:-15.Google Scholar
  6. Luo H, Ma Q: Positive solutions to a generalized second-order three-point boundary-value problem on time scales. Electronic Journal of Differential Equations 2005,2005(17):1-14.MathSciNetGoogle Scholar
  7. Henderson J:Multiple solutions for 2 th order Sturm-Liouville boundary value problems on a measure chain. Journal of Difference Equations and Applications 2000,6(4):417-429. 10.1080/10236190008808238MathSciNetView ArticleMATHGoogle Scholar
  8. Henderson J, Peterson A, Tisdell CC: On the existence and uniqueness of solutions to boundary value problems on time scales. Advances in Difference Equations 2004,2004(2):93-109. 10.1155/S1687183904308071MathSciNetView ArticleMATHGoogle Scholar
  9. Agarwal RP, Bohner M, Li W-T: Nonoscillation and Oscillation: Theory for Functional Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 267. Marcel Dekker, New York, NY, USA; 2004:viii+376.Google Scholar
  10. Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar
  11. He Z:Double positive solutions of three-point boundary value problems for -Laplacian dynamic equations on time scales. Journal of Computational and Applied Mathematics 2005,182(2):304-315. 10.1016/j.cam.2004.12.012MathSciNetView ArticleMATHGoogle Scholar
  12. He Z, Jiang X:Triple positive solutions of boundary value problems for -Laplacian dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2006,321(2):911-920. 10.1016/j.jmaa.2005.08.090MathSciNetView ArticleMATHGoogle Scholar
  13. Tian Y, Ge W: Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008,69(9):2833-2842. 10.1016/j.na.2007.08.054MathSciNetView ArticleMATHGoogle Scholar
  14. Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.View ArticleGoogle Scholar
  15. Erbe L, Peterson A, Saker SH: Hille-Kneser-type criteria for second-order dynamic equations on time scales. Advances in Difference Equations 2006, 2006:-18.Google Scholar
  16. Khan RA, Nieto JJ, Otero-Espinar V: Existence and approximation of solution of three-point boundary value problems on time scales. Journal of Difference Equations and Applications 2008,14(7):723-736. 10.1080/10236190701840906MathSciNetView ArticleMATHGoogle Scholar
  17. Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations. Advances in Difference Equations 2008, 2008:-13.Google Scholar
  18. Wang D-B:Three positive solutions of three-point boundary value problems for -Laplacian dynamic equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008,68(8):2172-2180. 10.1016/j.na.2007.01.037MathSciNetView ArticleMATHGoogle Scholar
  19. Sun J-P: A new existence theorem for right focal boundary value problems on a measure chain. Applied Mathematics Letters 2005,18(1):41-47. 10.1016/j.aml.2003.04.008MathSciNetView ArticleMATHGoogle Scholar
  20. Feng M, Zhang X, Ge W: Positive solutions for a class of boundary value problems on time scales. Computers & Mathematics with Applications 2007,54(4):467-475. 10.1016/j.camwa.2007.01.031MathSciNetView ArticleMATHGoogle Scholar
  21. Feng M-Q, Li X-G, Ge W-G:Triple positive solutions of fourth-order four-point boundary value problems of -Laplacian dynamic equations on time scales. Advances in Difference Equations 2008, 2008:-9.Google Scholar
  22. Feng M, Feng H, Zhang X, Ge W:Triple positive solutions for a class of -point dynamic equations on time scales with -Laplacian. Mathematical and Computer Modelling 2008,48(7-8):1213-1226. 10.1016/j.mcm.2007.12.016MathSciNetView ArticleMATHGoogle Scholar
  23. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar
  24. Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1-2):18-56.MathSciNetView ArticleMATHGoogle Scholar
  25. Henderson J, Tisdell CC: Topological transversality and boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004,289(1):110-125. 10.1016/j.jmaa.2003.08.030MathSciNetView ArticleMATHGoogle Scholar
  26. He Z:Existence of two solutions of -point boundary value problem for second order dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2004,296(1):97-109. 10.1016/j.jmaa.2004.03.051MathSciNetView ArticleMATHGoogle Scholar

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© Meiqiang Feng et al. 2009

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