Báo cáo hóa học: Research Article On the Existence of Solutions for Dynamic Boundary Value Problems under Barrier Strips Condition

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Báo cáo hóa học: " Research Article On the Existence of Solutions for Dynamic Boundary Value Problems under Barrier Strips Condition"Hindawi Publishing CorporationAdvances in Difference EquationsVolume 2011, Article ID 378686, 9 pagesdoi:10.1155/2011/378686Research ArticleOn the Existence of Solutions forDynamic Boundary Value Problems underBarrier Strips Condition Hua Luo1 and Yulian An2 1 School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China 2 Department of Mathematics, Shanghai Institute of Technology, Shanghai 200235, China Correspondence should be addressed to Hua Luo, luohuanwnu@gmail.com Received 24 November 2010; Accepted 20 January 2011 Academic Editor: Jin Liang Copyright q 2011 H. Luo and Y. An. 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. 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. IntroductionCalculus on time scales, which unify continuous and discrete analysis, is now still an activearea of research. We refer the reader to 1–5 and the references therein for introductionon this theory. In recent years, there has been much attention focused on the existence andmultiplicity of solutions or positive solutions for dynamic boundary value problems on timescales. See 6–17 for some of them. Under various growth restrictions on nonlinear term ofdynamic equation, many authors have obtained many excellent results for the above problemby using Topological degree theory, fixed-point theorems on cone, bifurcation theory, and soon. In 2004, Ma and Luo 18 firstly obtained the existence of solutions for the dynamicboundary value problems on time scales xΔΔ t f t, x t , xΔ t , t ∈ 0, 1 Ì, 1.1 xΔ σ 1 x0 0, 0 Advances in Difference Equations2under a barrier strips condition. A barrier strip P is defined as follows. There are pairs twoor four of suitable constants such that nonlinear term f t, u, p does not change its sign onsets of the form 0, 1 Ì × −L, L × P , where L is a nonnegative constant, and P is a closedinterval bounded by some pairs of constants, mentioned above. The idea in 18 was from Kelevedjiev 19 , in which discussions were for boundaryvalue problems of ordinary differential equation. This paper studies the existence of solutionsfor the nonlinear two-point dynamic boundary value problem on time scales xΔΔ t f t, x σ t , x Δ t , t ∈ a, ρ2 b Ì, 1.2 xΔ a 0, xb 0,where Ì is a bounded time scale with a inf Ì, b sup Ì, and a < ρ2 b . We obtain theexistence of at least one solution to problem 1.2 without any growth restrictions on f butan 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 explainedspecifically, in almost all literature related to time scales. Here, in order to make this paperread easily, we recall some necessary definitions here. A time scale Ì is a nonempty closed subset of Ê; assume that Ì has the topology that itinherits from the standard topology on Ê. Define the forward and backward jump operatorsσ, ρ : Ì → Ì by inf{τ > t | τ ∈ Ì}, sup{τ < t | τ ∈ Ì}. σt ρt 1.3In this definition we put inf ∅ sup Ì, sup ∅ inf Ì. Set σ 2 t σ σ t , ρ2 t ρ ρ t . Thesets Ìk and Ìk which are derived from the time scale Ì are as follows: Ìk : t ∈ Ì : t is not maximal or ρ t t, 1.4 ...