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On the Existence of Solutions for Dynamic Boundary Value Problems under Barrier Strips Condition
Advances in Difference Equations volumeÂ 2011, ArticleÂ number:Â 378686 (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 twopoint 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 [1â€“5] 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 [6â€“17] 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, fixedpoint 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
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 twopoint dynamic boundary value problem on time scales
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 wellknown LeraySchauder principle and the induction principle on time scales.
The time scalerelated 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
In this definition we put ,. Set ,. The sets and which are derived from the time scale are as follows:
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
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
Lemma 1.3 (see [2, Theorem 1.16 (SUF)]).
If is differentiable at , then
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 secondorder continuous differentiable functions equipped with the norm
where , , . According to the wellknown LeraySchauder 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
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
respectively. If , then we call (or ) the scatter degree on .
Remark 2.2.

(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

(H1)
,,

(H2)
for , for ,
where
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
where is bounded everywhere and continuous.
Suppose that , then for
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:
For all , suppose that is an arbitrary solution of problem
We firstly prove that there exists , independent of and , such that .
We show at first that
Let ,. We employ the induction principle on time scales (Lemma 1.5) to show that holds step by step.

(1)
From the boundary condition and the assumption of , holds.

(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
Similarly, .
Case 2.
If , then similar to Case 1 we have
Suppose to the contrary that , then
which contradicts (H2). So .
Case 3.
If , similar to Case 2, then holds.
Therefore, is true.

(3)
For each , with , and holds, then there is a neighborhood of such that holds for all , by virtue of the continuity of .

(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
, and the continuity of , there is a neighborhood of such that
So , . Combining with , , we have from (H2), , , . So from Lemma 1.4
This contradiction shows that . In the same way, we claim that .
Hence, , , holds. So
From Definition 1.2 and Lemma 1.3, we have for
There are, from and (2.7),
for . In addition,
Thus,
that is,
Moreover, by the continuity of , the equation in (2.6), (2.7) and the definition of
where is defined in (2.2). Now let . Then, from (2.15), (2.20), and (2.21),
Note that from (2.19) we have
that is, , . So is also an arbitrary solution of problem
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

(S1)
, ,

(S2)
for , for ,
where
Then dynamic boundary value problem
has at least one solution.
Remark 3.2.
According to Theorem 3.1, the dynamic boundary value problem related to the nabla derivative
has at least one solution. Here is bounded everywhere and continuous.
References
Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results in Mathematics 1999,35(12):322.
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. BirkhÃ¤user, Boston, Mass, USA; 2001:x+358.
Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. BirkhÃ¤user, Boston, Mass, USA; 2003:xii+348.
Hilger S: Analysis on measure chainsâ€”a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(12):1856.
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.
Agarwal RP, O'Regan D: Triple solutions to boundary value problems on time scales. Applied Mathematics Letters 2000,13(4):711. 10.1016/S08939659(99)002001
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(12):7599. Special issue on "Dynamic equations on time scales", edited by R. P. Agarwal, M. Bohner and D. O'Regan 10.1016/S03770427(01)00437X
Bohner M, Luo H: Singular secondorder multipoint dynamic boundary value problems with mixed derivatives. Advances in Difference Equations 2006, 2006:15.
Chyan CJ, Henderson J: Twin solutions of boundary value problems for differential equations on measure chains. Journal of Computational and Applied Mathematics 2002,141(12):123131. Special issue on "Dynamic equations on time scales", edited by R. P. Agarwal, M. Bohner and D. O'Regan 10.1016/S03770427(01)00440X
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(12):365380. 10.1016/S03770427(99)002678
Gao C, Luo H: Positive solutions to nonlinear firstorder nonlocal BVPs with parameter on time scales. Boundary Value Problems 2011, 2011:15.
Henderson J:Multiple solutions for order SturmLiouville boundary value problems on a measure chain. Journal of Difference Equations and Applications 2000,6(4):417429. 10.1080/10236190008808238
Li WT, Sun HR: Multiple positive solutions for nonlinear dynamical systems on a measure chain. Journal of Computational and Applied Mathematics 2004,162(2):421430. 10.1016/j.cam.2003.08.032
Luo H, Ma R: Nodal solutions to nonlinear eigenvalue problems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2006,65(4):773784. 10.1016/j.na.2005.09.043
Sun HR:Triple positive solutions for Laplacian point boundary value problem on time scales. Computers & Mathematics with Applications 2009,58(9):17361741. 10.1016/j.camwa.2009.07.083
Sun JP, Li WT: Existence and nonexistence of positive solutions for secondorder time scale systems. Nonlinear Analysis: Theory, Methods & Applications 2008,68(10):31073114. 10.1016/j.na.2007.03.003
Wang DB, Sun JP, Guan W: Multiple positive solutions for functional dynamic equations on time scales. Computers & Mathematics with Applications 2010,59(4):14331440. 10.1016/j.camwa.2009.12.019
Ma R, Luo H: Existence of solutions for a twopoint boundary value problem on time scales. Applied Mathematics and Computation 2004,150(1):139147. 10.1016/S00963003(03)002042
Kelevedjiev P: Existence of solutions for twopoint boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 1994,22(2):217224. 10.1016/0362546X(94)900353
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. 2009IM010400139). Y. An was supported by 11YZ225 and YJ200916 (A06/1020K096019).
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Luo, H., An, Y. On the Existence of Solutions for Dynamic Boundary Value Problems under Barrier Strips Condition. Adv Differ Equ 2011, 378686 (2011). https://doi.org/10.1155/2011/378686
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DOI: https://doi.org/10.1155/2011/378686
Keywords
 Nonlinear Term
 Growth Restriction
 Bifurcation Theory
 Dynamic Boundary
 Degree Theory