Advances in Difference Equations

, 2009:209707 | Cite as

Existence Results for Higher-Order Boundary Value Problems on Time Scales

Open Access
Research Article
Part of the following topical collections:
  1. Boundary Value Problems on Time Scales

Abstract

By using the fixed-point index theorem, we consider the existence of positive solutions for the following nonlinear higher-order four-point singular boundary value problem on time scales Open image in new window , Open image in new window ; Open image in new window , Open image in new window ; Open image in new window , Open image in new window ; Open image in new window , Open image in new window , where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window is rd-continuous.

Keywords

Boundary Value Problem Small Positive Number Singular Boundary Jump Operator Nonempty Closed Subset 

1. Introduction

Time scales and time-scale notationare introduced well in the fundamental texts by Bohner and Peterson [1, 2], respectively, as important corollaries. In, the recent years, many authors have paid much attention to the study of boundary value problems on time scales (see, e.g., [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]). In particular, we would like to mention some results of Anderson et al. [3, 5, 6, 14, 16], DaCunha et al. [4], and Agarwal and O'Regan [7], which motivate us to consider our problem.

In [3], Anderson and Karaca discussed the dynamic equation on time scales
and the eigenvalue problem

with the same boundary conditions where Open image in new window is a positive parameter. They obtained some results for the existence of positive solutions by using the Krasnoselskii, the Schauder, and the Avery-Henderson fixed-point theorem.

In [4], by using the Gatica-Oliker-Waltman fixed-point theorem, DaCunha, Davis, and Singh proved the existence of a positive solution for the three-point boundary value problem on a time scale Open image in new window given by

where Open image in new window is fixed, and Open image in new window is singular at Open image in new window and possibly at Open image in new window .

Anderson et al. [5] gave a detailed presentation for the following higher-order self-adjoint boundary value problem on time scales:

and got many excellent results.

In related papers, Sun [11] considered the following third-order two-point boundary value problem on time scales:

where Open image in new window and Open image in new window . Some existence criteria of solution and positive solution are established by using the Leray-Schauder fixed point theorem.

In this paper, we consider the existence of positive solutions for the following higher-order four-point singular boundary value problem (BVP) on time scales

where Open image in new window , Open image in new window Open image in new window , and Open image in new window is rd-continuous. In the rest of the paper, we make the following assumptions:

() Open image in new window

() Open image in new window .

In this paper, by constructing one integral equation which is equivalent to the BVP (1.6) and (1.7), we study the existence of positive solutions. Our main tool of this paper is the following fixed-point index theorem.

Theorem 1.1 ([18]).

Suppose Open image in new window is a real Banach space, Open image in new window is a cone, let Open image in new window . Let operator Open image in new window Open image in new window be completely continuous and satisfy Open image in new window . Then

(i)if Open image in new window , then Open image in new window

(ii)if Open image in new window , then Open image in new window .

The outline of the paper is as follows. In Section 2, for the convenience of the reader we give some definitions and theorems which can be found in the references, and we present some lemmas in order to prove our main results. Section 3 is developed in order to present and prove our main results. In Section 4 we present some examples to illustrate our results.

2. Preliminaries and Lemmas

For convenience, we list the following definitions which can be found in [1, 2, 9, 14, 17]. A time scale Open image in new window is a nonempty closed subset of real numbers Open image in new window . For Open image in new window and Open image in new window , define the forward jump operator Open image in new window and backward jump operator Open image in new window , respectively, by
for all Open image in new window . If Open image in new window , Open image in new window is said to be right scattered, and if Open image in new window , Open image in new window is said to be left scattered; if Open image in new window , Open image in new window is said to be right dense, and if Open image in new window , Open image in new window is said to be left dense. If Open image in new window has a right scattered minimum Open image in new window , define Open image in new window ; otherwise set Open image in new window . If Open image in new window has a left scattered maximum Open image in new window , define Open image in new window ; otherwise set Open image in new window . In this general time-scale setting, Open image in new window represents the delta (or Hilger) derivative [13, Definition 1.10],
A function Open image in new window is right-dense continuous provided that it is continuous at each right-dense point Open image in new window (a point where Open image in new window ) and has a left-sided limit at each left-dense point Open image in new window . The set of right-dense continuous functions on Open image in new window is denoted by Open image in new window . It can be shown that any right-dense continuous function Open image in new window has an antiderivative (a function Open image in new window with the property Open image in new window for all Open image in new window ). Then the Cauchy delta integral of Open image in new window is defined by

Throughout we assume that Open image in new window are points in Open image in new window , and define the time-scale interval Open image in new window . In this paper, we also need the the following theorem which can be found in [1].

Theorem 2.1.

In this paper, let
Then Open image in new window is a Banach space with the norm Open image in new window . Define a cone Open image in new window by

Obviously, Open image in new window is a cone in Open image in new window . Set Open image in new window . If Open image in new window on Open image in new window then we say Open image in new window is concave on Open image in new window We can get the following.

Lemma 2.2.

Suppose condition Open image in new window holds. Then there exists a constant Open image in new window satisfies
Furthermore, the function

is a positive continuous function on Open image in new window , therefore Open image in new window has minimum on Open image in new window . Then there exists Open image in new window such that Open image in new window .

Lemma 2.3.

Proof.

Suppose Open image in new window .

We will discuss it from three perspectives.

(i) Open image in new window . It follows from the concavity of Open image in new window that

which means Open image in new window .

(ii) Open image in new window . If Open image in new window , we have

If Open image in new window , we have

and this means Open image in new window .

(iii) Open image in new window . Similarly, we have

which means Open image in new window .

From the above, we know Open image in new window . The proof is complete.

Lemma 2.4.

Suppose that conditions Open image in new window hold, then Open image in new window is a solution of boundary value problem (1.6), (1.7) if and only if Open image in new window is a solution of the following integral equation:

Proof.

Necessity. By the equation of the boundary condition, we see that Open image in new window , then there exists a constant Open image in new window such that Open image in new window . Firstly, by delta integrating the equation of the problems (1.6) on Open image in new window , we have

By Open image in new window and the boundary condition (1.7), let Open image in new window on (2.23), we have

By the equation of the boundary condition (1.7), we get
Secondly, by (2.24) and let Open image in new window on (2.24), we have
Then by delta integrating (2.29) for Open image in new window times on Open image in new window , we have
Similarly, for Open image in new window , by delta integrating the equation of problems (1.6) on Open image in new window , we have
Therefore, for any Open image in new window , Open image in new window can be expressed as the equation

where Open image in new window is expressed as (2.22).

Sufficiency. Suppose that

then by (2.22), we have

which imply that (1.6) holds. Furthermore, by letting Open image in new window and Open image in new window on (2.22) and (2.34), we can obtain the boundary value equations of (1.7). The proof is complete.

Now, we define a mapping Open image in new window given by

where Open image in new window is given by (2.22).

Lemma 2.5.

Suppose that conditions Open image in new window hold, the solution Open image in new window of problem (1.6), (1.7) satisfies
and for Open image in new window in Lemma 2.2, one has

Proof.

If Open image in new window is the solution of (1.6), (1.7), then Open image in new window is a concave function, and Open image in new window , thus we have
that is,

By Lemma 2.3, for Open image in new window , we have

then Open image in new window . The proof is complete.

Lemma 2.6.

Open image in new window is completely continuous.

Proof.

is continuous, decreasing on Open image in new window and satisfies Open image in new window . Then, Open image in new window for each Open image in new window and Open image in new window . This shows that Open image in new window . Furthermore, it is easy to check that Open image in new window is completely continuous by Arzela-ascoli Theorem.

For convenience, we set

where Open image in new window is the constant from Lemma 2.2. By Lemma 2.5, we can also set

3. The Existence of Positive Solution

Theorem 3.1.

Suppose that conditions ( Open image in new window ), ( Open image in new window ) hold. Assume that Open image in new window also satisfies

Open image in new window

Open image in new window

where Open image in new window

Then, the boundary value problem (1.6), (1.7) has a solution Open image in new window such that Open image in new window lies between Open image in new window and Open image in new window .

Theorem 3.2.

Suppose that conditions ( Open image in new window ), ( Open image in new window ) hold. Assume that Open image in new window also satisfies

Open image in new window

Open image in new window

Then, the boundary value problem (1.6), (1.7) has a solution Open image in new window such that Open image in new window lies between Open image in new window and Open image in new window .

Theorem 3.3.

Suppose that conditions ( Open image in new window ), ( Open image in new window ) hold. Assume that Open image in new window also satisfies

() Open image in new window

() Open image in new window

Then, the boundary value problem (1.6), (1.7) has a solution Open image in new window such that Open image in new window lies between Open image in new window and Open image in new window .

Proof of Theorem 3.1.

Without loss of generality, we suppose that Open image in new window . For any Open image in new window , by Lemma 2.3, we have
For any Open image in new window , by (3.1) we have

For Open image in new window and Open image in new window , we will discuss it from three perspectives.

(i)If Open image in new window , thus for Open image in new window , by ( Open image in new window ) and Lemma 2.4, we have

  1. (ii)

    If Open image in new window , thus for Open image in new window , by ( Open image in new window ) and Lemma 2.4, we have

     
  1. (iii)

    If Open image in new window , thus for Open image in new window , by ( Open image in new window ) and Lemma 2.4, we have

     
Therefore, no matter under which condition, we all have
Then by Theorem 2.1, we have

On the other hand, for Open image in new window , we have Open image in new window ; by ( Open image in new window ) we know

Then, by Theorem 2.1, we get

Therefore, by (3.8), (3.11), Open image in new window , we have

Then operator Open image in new window has a fixed point Open image in new window , and Open image in new window . Then the proof of Theorem 3.1 is complete .

Proof of Theorem 3.2.

First, by Open image in new window , for Open image in new window , there exists an adequately small positive number Open image in new window , as Open image in new window , we have
Then let Open image in new window , thus by (3.13)
So condition ( Open image in new window ) holds. Next, by condition ( Open image in new window ), Open image in new window , then for Open image in new window , there exists an appropriately big positive number Open image in new window , as Open image in new window , we have

Let Open image in new window , thus by (3.15), condition ( Open image in new window ) holds. Therefore by Theorem 3.1 we know that the results of Theorem 3.2 hold. The proof of Theorem 3.2 is complete.

Proof of Theorem 3.3.

Firstly, by condition ( Open image in new window ), Open image in new window , then for Open image in new window , there exists an adequately small positive number Open image in new window , as Open image in new window , we have

Let Open image in new window , so by (3.17), condition ( Open image in new window ) holds.

Secondly, by condition ( Open image in new window ), Open image in new window , then for Open image in new window , there exists a suitably big positive number Open image in new window , as Open image in new window , we have

If Open image in new window is unbounded, by the continuity of Open image in new window on Open image in new window , then there exist a constant Open image in new window , and a point Open image in new window such that
If Open image in new window is bounded, we suppose Open image in new window , there exists an appropriately big positive number Open image in new window , then choose Open image in new window , we have

Therefore, condition ( Open image in new window ) holds. Thus, by Theorem 3.1, we know that the result of Theorem 3.3 holds. The proof of Theorem 3.3 is complete.

4. Application

In this section, in order to illustrate our results, we consider the following examples.

Example 4.1.

Consider the following boundary value problem on the specific time scale Open image in new window :
and Open image in new window is the constant defined in Lemma 2.2,
Then obviously

By Theorem 2.1, we have

so conditions ( Open image in new window ), Open image in new window hold.

By simple calculations, we have

then Open image in new window , that is, Open image in new window , so condition Open image in new window holds.

For Open image in new window , it is easy to see that

so condition Open image in new window holds. Then by Theorem 3.2, BVP (4.1) has at least one positive solution.

Example 4.2.

Consider the following boundary value problem on the specific time scale Open image in new window .
and Open image in new window is the constant from Lemma 2.2,
Then obviously
By Theorem 2.1, we have
so conditions ( Open image in new window ), Open image in new window hold. By simple calculations, we have

then Open image in new window , that is, Open image in new window , so condition Open image in new window holds.

For Open image in new window , it is easy to see that

then condition Open image in new window holds. Thus by Theorem 3.3, BVP (4.8) has at least one positive solution.

Notes

Acknowledgment

The authors would like to thank the anonymous referee for his/her valuable suggestions, which have greatly improved this paper.

References

  1. [1]
    Bohner M, Peterson A: Dynamic Equations on Time Scales, An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.MATHCrossRefGoogle Scholar
  2. [2]
    Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar
  3. [3]
    Anderson DR, Karaca IY: Higher-order three-point boundary value problem on time scales. Computers & Mathematics with Applications 2008,56(9):2429–2443. 10.1016/j.camwa.2008.05.018MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    DaCunha JJ, Davis JM, Singh PK: Existence results for singular three point boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004,295(2):378–391. 10.1016/j.jmaa.2004.02.049MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Anderson DR, Guseinov GSh, Hoffacker J: Higher-order self-adjoint boundary-value problems on time scales. Journal of Computational and Applied Mathematics 2006,194(2):309–342. 10.1016/j.cam.2005.07.020MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Anderson DR, Avery R, Davis J, Henderson J, Yin W: Positive solutions of boundary value problems. In Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:189–249.CrossRefGoogle Scholar
  7. [7]
    Agarwal RP, O'Regan D: Nonlinear boundary value problems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2001,44(4):527–535. 10.1016/S0362-546X(99)00290-4MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Kaufmann ER: Positive solutions of a three-point boundary-value problem on a time scale. Electronic Journal of Differential Equations 2003, (82):1–11.Google Scholar
  9. [9]
    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-XMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    Boey KL, Wong PJY: Existence of triple positive solutions of two-point right focal boundary value problems on time scales. Computers & Mathematics with Applications 2005,50(10–12):1603–1620.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Sun J-P: Existence of solution and positive solution of BVP for nonlinear third-order dynamic equation. Nonlinear Analysis: Theory, Methods & Applications 2006,64(3):629–636. 10.1016/j.na.2005.04.046MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    Benchohra M, Henderson J, Ntouyas SK: Eigenvalue problems for systems of nonlinear boundary value problems on time scales. Advances in Difference Equations 2007, 2007:-10.Google Scholar
  13. [13]
    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
  14. [14]
    Anderson DR: Oscillation and nonoscillation criteria for two-dimensional time-scale systems of first-order nonlinear dynamic equations. Electronic Journal of Differential Equations 2009,2009(24):13. Article ID 796851.Google Scholar
  15. [15]
    Bohner M, Luo H: Singular second-order multipoint dynamic boundary value problems with mixed derivatives. Advances in Difference Equations 2006, 2006:-15.Google Scholar
  16. [16]
    Anderson DR, Ma R: Second-order -point eigenvalue problems on time scales. Advances in Difference Equations 2006, 2006:-17.Google Scholar
  17. [17]
    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/S1687183904308071MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar

Copyright information

© J. Liu and Y. Sang. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShandong Economics UniversityJinanChina
  2. 2.Department of MathematicsNorth University of ChinaTaiyuanChina

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