- Research Article
- Open Access
On the Existence of Solutions for Dynamic Boundary Value Problems under Barrier Strips Condition
© Hua Luo and Yulian An. 2011
- Received: 24 November 2010
- Accepted: 20 January 2011
- Published: 14 February 2011
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.
- Nonlinear Term
- Growth Restriction
- Bifurcation Theory
- Dynamic Boundary
- Degree Theory
Calculus on time scales, which unify continuous and discrete analysis, is now still an active area of research. We refer the reader to [1–5] 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 [6–17] 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, fixed-point theorems on cone, bifurcation theory, and so on.
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.
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.
Denote interval on by .
for all . The function is called -differentiable on if exists for all .
Lemma 1.3 (see [2, Theorem 1.16 (SUF)]).
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 .
For , we replace with and with , substitute < for >, then the dual version of the above induction principle is also true.
where , , . According to the well-known Leray-Schauder degree theory, we can get the following theorem.
Then the boundary value problem (1.2) has at least one solution in .
The proof is the same as [18, Theorem 4.1].
To state our main result, we introduce the definition of scatter degree.
respectively. If , then we call (or ) the scatter degree on .
If , then . If , then . If and , then . (2) If is bounded, then both and are finite numbers.
Let be continuous. Suppose that there are constants , , with , satisfying
for , for ,
Then problem (1.2) has at least one solution in .
where is bounded everywhere and continuous.
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.
We firstly prove that there exists , independent of and , such that .
Let , . We employ the induction principle on time scales (Lemma 1.5) to show that holds step by step.
From the boundary condition and the assumption of , holds.
For each with , suppose that holds, that is, . Note that ; we divide this discussion into three cases to prove that holds.
which contradicts (H2). So .
If , similar to Case 2, then holds.
Therefore, is true.
For each , with , and holds, then there is a neighborhood of such that holds for all , by virtue of the continuity of .
This contradiction shows that . In the same way, we claim that .
According to (2.22) and Lemma 1.7, the dynamic boundary value problem (1.2) has at least one solution in .
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 . Applying to the dual version of the induction principle on time scales (Remark 1.6), we can obtain the following result.
Let be continuous. Suppose that there are constants , , with , satisfying
for , for ,
has at least one solution.
has at least one solution. Here is bounded everywhere and continuous.
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).
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