Bifurcation from Interval and Positive Solutions of a Nonlinear Second-Order Dynamic Boundary Value Problem on Time Scales

and Applied Analysis 3 A2 f t, u > 0 for t, u ∈ 0, σ T T × 0,∞ . A3 There exists a function c ∈ C 0, σ T T , 0,∞ such that f t, u ≥ c t u, t, u ∈ 0, σ T T × 0,∞ . 1.8 Obviously, A1 means that f is not necessarily linearizable at 0 and ∞. We consider the existence of positive solutions of problem 1.1 in this paper by using bifurcation techniques. The difference from 6 is that the branches of positive solutions under consideration now bifurcate from not one point, but an interval. Our main idea is from 7 , in which they considered positive solutions of fourth-order boundary value problems for differential equations. The main tool we will use is the following global bifurcation theorem for problems which is not necessarily linearizable. Theorem A Rabinowitz, 8 . Let V be a real reflexive Banach space. Let F : R × V → V be completely continuous such that F λ, 0 0, for all λ ∈ R. Let a, b ∈ R a < b be such that u 0 is an isolated solution of the equation u − F λ, u 0, u ∈ V, 1.9 for λ a, and λ b, where a, 0 , b, 0 are not bifurcation points of 1.9 . Furthermore, assuming that deg I − F a, · , Br 0 , 0 / deg I − F b, · , Br 0 , 0 , 1.10 where Br 0 is an isolating neighborhood of the nontrivial solution, and deg I − F, Br 0 , 0 denote the degree of I − F on Br 0 with respect to 0. Let S { λ, u : λ, u is a solution of 1.9 with u/ 0} ∪ a, b × {0} . 1.11 Then there exists a connected component C of S containing a, b × {0}, and either i C is unbounded, or ii C ∩ R \ a, b × {0} / ∅. The rest of the paper is organized as follows. In Section 2, we firstly introduce the time scales concepts and notations that we will use in this paper. Next, Section 3 states some notations and proves some necessary preliminary results, and Section 4 studies the bifurcation from the trivial solution for a nonlinear problem which is not necessarily linearizable and then establishes our main result. 4 Abstract and Applied Analysis 2. Introduction for Time Scales A time scale T is a nonempty closed subset of R, assuming that T has the topology that it inherits from the standard topology on R. Define the forward and backward jump operators σ, ρ : T → T by σ t inf{τ > t | τ ∈ T}, ρ t sup{τ < t | τ ∈ T}. 2.1 Here, we put inf ∅ supT, sup ∅ infT. Let T which is derived from the time scale T be T k : { t ∈ T : t is nonmaximal or ρ t t}, 2.2 and T 2 : Tk. Define interval I on T by IT I ∩ T. Definition 2.1. If u : T → R is a function and t ∈ T, then the Δ-derivative of u at the point t is defined to be the number uΔ t provided that it exists with the property that for each ε > 0, there is a neighborhood U of t such that ∣∣u σ t − u s − uΔ t σ t − s ∣∣ ε|σ t − s| 2.3 for all s ∈ U. The function u is called Δ-differentiable on T if uΔ t exists for all t ∈ T. The second Δ-derivative of u at t ∈ Tk2 , if it exists, is defined to be uΔ2 t uΔΔ t : uΔ Δ t . We also define the function u : u ◦ σ and u : u ◦ ρ. Definition 2.2. IfUΔ u holds on T, we define the Cauchy Δ-integral by ∫ t s u τ Δτ U t −U s , s, t ∈ T. 2.4 Lemma 2.3 see 2, Theorems 2.7 and 2.8 . Assume a, b ∈ T, then ∫b a fΔ t g t Δt f t g t ∣∣b a − ∫b