Open Access

On the Existence of Solutions for Dynamic Boundary Value Problems under Barrier Strips Condition

Advances in Difference Equations20112011:378686

https://doi.org/10.1155/2011/378686

Received: 24 November 2010

Accepted: 20 January 2011

Published: 14 February 2011

Abstract

By defining a new terminology, scatter degree, as the supremum of graininess functional value, this paper studies the existence of solutions for a nonlinear two-point dynamic boundary value problem on time scales. We do not need any growth restrictions on nonlinear term of dynamic equation besides a barrier strips condition. The main tool in this paper is the induction principle on time scales.

1. Introduction

Calculus on time scales, which unify continuous and discrete analysis, is now still an active area of research. We refer the reader to [15] and the references therein for introduction on this theory. In recent years, there has been much attention focused on the existence and multiplicity of solutions or positive solutions for dynamic boundary value problems on time scales. See [617] for some of them. Under various growth restrictions on nonlinear term of dynamic equation, many authors have obtained many excellent results for the above problem by using Topological degree theory, fixed-point theorems on cone, bifurcation theory, and so on.

In 2004, Ma and Luo [18] firstly obtained the existence of solutions for the dynamic boundary value problems on time scales
(1.1)

under a barrier strips condition. A barrier strip is defined as follows. There are pairs (two or four) of suitable constants such that nonlinear term does not change its sign on sets of the form , where is a nonnegative constant, and is a closed interval bounded by some pairs of constants, mentioned above.

The idea in [18] was from Kelevedjiev [19], in which discussions were for boundary value problems of ordinary differential equation. This paper studies the existence of solutions for the nonlinear two-point dynamic boundary value problem on time scales
(1.2)

where is a bounded time scale with , , and . We obtain the existence of at least one solution to problem (1.2) without any growth restrictions on but an existence assumption of barrier strips. Our proof is based upon the well-known Leray-Schauder principle and the induction principle on time scales.

The time scale-related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to time scales. Here, in order to make this paper read easily, we recall some necessary definitions here.

A time scale is a nonempty closed subset of ; assume that has the topology that it inherits from the standard topology on . Define the forward and backward jump operators by
(1.3)
In this definition we put , . Set , . The sets and which are derived from the time scale are as follows:
(1.4)

Denote interval on by .

Definition 1.1.

If is a function and , then the delta derivative of at the point is defined to be the number (provided it exists) with the property that, for each , there is a neighborhood of such that
(1.5)

for all . The function is called -differentiable on if exists for all .

Definition 1.2.

If holds on , then we define the Cauchy -integral by
(1.6)

Lemma 1.3 (see [2, Theorem 1.16 (SUF)]).

If is -differentiable at , then
(1.7)

Lemma 1.4 (see [18, Lemma 3.2]).

Suppose that is -differentiable on , then

(i) is nondecreasing on if and only if ,

(ii) is nonincreasing on if and only if .

Lemma 1.5 (see [4, Theorem 1.4]).

Let be a time scale with . Then the induction principle holds.

Assume that, for a family of statements , , the following conditions are satisfied.

(1) holds true.

(2)For each with , one has .

(3)For each with , there is a neighborhood of such that for all , .

(4)For each with , one has for all .

Then is true for all .

Remark 1.6.

For , we replace with and with , substitute < for >, then the dual version of the above induction principle is also true.

By , we mean the Banach space of second-order continuous -differentiable functions equipped with the norm
(1.8)

where , , . According to the well-known Leray-Schauder degree theory, we can get the following theorem.

Lemma 1.7.

Suppose that is continuous, and there is a constant , independent of , such that for each solution to the boundary value problem
(1.9)

Then the boundary value problem (1.2) has at least one solution in .

Proof.

The proof is the same as [18, Theorem 4.1].

2. Existence Theorem

To state our main result, we introduce the definition of scatter degree.

Definition 2.1.

For a time scale , define the right direction scatter degree (RSD) and the left direction scatter degree (LSD) on by
(2.1)

respectively. If , then we call (or ) the scatter degree on .

Remark 2.2.
  1. (1)

    If , then . If , then . If and , then . (2) If is bounded, then both and are finite numbers.

     

Theorem 2.3.

Let be continuous. Suppose that there are constants , , with , satisfying

  1. (H1)

    , ,

     
  2. (H2)

    for , for ,

     
where
(2.2)

Then problem (1.2) has at least one solution in .

Remark 2.4.

Theorem 2.3 extends [19, Theorem 3.2] even in the special case . Moreover, our method to prove Theorem 2.3 is different from that of [19].

Remark 2.5.

We can find some elementary functions which satisfy the conditions in Theorem 2.3. Consider the dynamic boundary value problem
(2.3)

where is bounded everywhere and continuous.

Suppose that , then for
(2.4)

It implies that there exist constants , , satisfying (H1) and (H2) in Theorem 2.3. Thus, problem (2.3) has at least one solution in .

Proof of Theorem 2.3.

Define as follows:
(2.5)
For all , suppose that is an arbitrary solution of problem
(2.6)

We firstly prove that there exists , independent of and , such that .

We show at first that
(2.7)

Let , . We employ the induction principle on time scales (Lemma 1.5) to show that holds step by step.

  1. (1)

    From the boundary condition and the assumption of , holds.

     
  2. (2)

    For each with , suppose that holds, that is, . Note that ; we divide this discussion into three cases to prove that holds.

     

Case 1.

If , then from Lemma 1.3, Definition 2.1, and (H1) there is
(2.8)

Similarly, .

Case 2.

If , then similar to Case 1 we have
(2.9)
Suppose to the contrary that , then
(2.10)

which contradicts (H2). So .

Case 3.

If , similar to Case 2, then holds.

Therefore, is true.

  1. (3)

    For each , with , and holds, then there is a neighborhood of such that holds for all , by virtue of the continuity of .

     
  2. (4)
    (4)For each , with , and is true for all , since implies that
    (2.11)

    we only show that and .

     
Suppose to the contrary that . From
(2.12)
, and the continuity of , there is a neighborhood of such that
(2.13)
So , . Combining with , , we have from (H2), , , . So from Lemma 1.4
(2.14)

This contradiction shows that . In the same way, we claim that .

Hence, , , holds. So
(2.15)
From Definition 1.2 and Lemma 1.3, we have for
(2.16)
There are, from and (2.7),
(2.17)
for . In addition,
(2.18)
Thus,
(2.19)
that is,
(2.20)
Moreover, by the continuity of , the equation in (2.6), (2.7) and the definition of
(2.21)
where is defined in (2.2). Now let . Then, from (2.15), (2.20), and (2.21),
(2.22)
Note that from (2.19) we have
(2.23)
that is, , . So is also an arbitrary solution of problem
(2.24)

According to (2.22) and Lemma 1.7, the dynamic boundary value problem (1.2) has at least one solution in .

3. An Additional Result

Parallel to the definition of delta derivative, the notion of nabla derivative was introduced, and the main relations between the two operations were studied in [7]. Applying to the dual version of the induction principle on time scales (Remark 1.6), we can obtain the following result.

Theorem 3.1.

Let be continuous. Suppose that there are constants , , with , satisfying

  1. (S1)

    , ,

     
  2. (S2)

    for , for ,

     
where
(3.1)
Then dynamic boundary value problem
(3.2)

has at least one solution.

Remark 3.2.

According to Theorem 3.1, the dynamic boundary value problem related to the nabla derivative
(3.3)

has at least one solution. Here is bounded everywhere and continuous.

Declarations

Acknowledgments

H. Luo was supported by China Postdoctoral Fund (no. 20100481239), the NSFC Young Item (no. 70901016), HSSF of Ministry of Education of China (no. 09YJA790028), Program for Innovative Research Team of Liaoning Educational Committee (no. 2008T054), and Innovation Method Fund of China (no. 2009IM010400-1-39). Y. An was supported by 11YZ225 and YJ2009-16 (A06/1020K096019).

Authors’ Affiliations

(1)
School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics
(2)
Department of Mathematics, Shanghai Institute of Technology

References

  1. Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results in Mathematics 1999,35(1-2):3-22.MathSciNetView ArticleMATHGoogle Scholar
  2. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar
  3. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar
  4. 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
  5. Kaymakcalan B, Lakshmikantham V, Sivasundaram S: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.MATHGoogle Scholar
  6. Agarwal RP, O'Regan D: Triple solutions to boundary value problems on time scales. Applied Mathematics Letters 2000,13(4):7-11. 10.1016/S0893-9659(99)00200-1MathSciNetView ArticleMATHGoogle Scholar
  7. Atici FM, Guseinov GS: 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. Special issue on "Dynamic equations on time scales", edited by R. P. Agarwal, M. Bohner and D. O'Regan 10.1016/S0377-0427(01)00437-XMathSciNetView ArticleMATHGoogle Scholar
  8. Bohner M, Luo H: Singular second-order multipoint dynamic boundary value problems with mixed derivatives. Advances in Difference Equations 2006, 2006:-15.Google Scholar
  9. Chyan CJ, Henderson J: Twin solutions of boundary value problems for differential equations on measure chains. Journal of Computational and Applied Mathematics 2002,141(1-2):123-131. Special issue on "Dynamic equations on time scales", edited by R. P. Agarwal, M. Bohner and D. O'Regan 10.1016/S0377-0427(01)00440-XMathSciNetView ArticleMATHGoogle Scholar
  10. Erbe L, Peterson A, Mathsen R: Existence, multiplicity, and nonexistence of positive solutions to a differential equation on a measure chain. Journal of Computational and Applied Mathematics 2000,113(1-2):365-380. 10.1016/S0377-0427(99)00267-8MathSciNetView ArticleMATHGoogle Scholar
  11. Gao C, Luo H: Positive solutions to nonlinear first-order nonlocal BVPs with parameter on time scales. Boundary Value Problems 2011, 2011:-15.Google Scholar
  12. Henderson J:Multiple solutions for 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
  13. Li W-T, Sun H-R: Multiple positive solutions for nonlinear dynamical systems on a measure chain. Journal of Computational and Applied Mathematics 2004,162(2):421-430. 10.1016/j.cam.2003.08.032MathSciNetView ArticleMATHGoogle Scholar
  14. Luo H, Ma R: Nodal solutions to nonlinear eigenvalue problems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2006,65(4):773-784. 10.1016/j.na.2005.09.043MathSciNetView ArticleMATHGoogle Scholar
  15. Sun H-R:Triple positive solutions for -Laplacian -point boundary value problem on time scales. Computers & Mathematics with Applications 2009,58(9):1736-1741. 10.1016/j.camwa.2009.07.083MathSciNetView ArticleMATHGoogle Scholar
  16. Sun J-P, Li W-T: Existence and nonexistence of positive solutions for second-order time scale systems. Nonlinear Analysis: Theory, Methods & Applications 2008,68(10):3107-3114. 10.1016/j.na.2007.03.003MathSciNetView ArticleMATHGoogle Scholar
  17. Wang D-B, Sun J-P, Guan W: Multiple positive solutions for functional dynamic equations on time scales. Computers & Mathematics with Applications 2010,59(4):1433-1440. 10.1016/j.camwa.2009.12.019MathSciNetView ArticleMATHGoogle Scholar
  18. Ma R, Luo H: Existence of solutions for a two-point boundary value problem on time scales. Applied Mathematics and Computation 2004,150(1):139-147. 10.1016/S0096-3003(03)00204-2MathSciNetView ArticleMATHGoogle Scholar
  19. Kelevedjiev P: Existence of solutions for two-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 1994,22(2):217-224. 10.1016/0362-546X(94)90035-3MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Hua Luo and Yulian An. 2011

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.