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.
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.
Lemma 1.3 (see [2, Theorem 1.16 (SUF)]).
Lemma 1.4 (see [18, Lemma 3.2]).
Lemma 1.5 (see [4, Theorem 1.4]).
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.
Proof of Theorem 2.3.
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 . Applying to the dual version of the induction principle on time scales (Remark 1.6), we can obtain the following result.
has at least one solution.
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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