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 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.
Highlights
Calculus on time scales, which unify continuous and discrete analysis, is still an active area of research
There has been much attention focused on the existence and multiplicity of solutions or positive solutions for dynamic boundary value problems on time scales
In 2004, Ma and Luo 18 firstly obtained the existence of solutions for the dynamic boundary value problems on time scales xΔΔ t f t, x t, xΔ t, t ∈ 0, 1 Ì, x 0 0, xΔ σ 1 0
Summary
Calculus on time scales, which unify continuous and discrete analysis, is still an active area of research. There has been much attention focused on the existence and multiplicity of solutions or positive solutions for dynamic boundary value problems on time scales. In 2004, Ma and Luo 18 firstly obtained the existence of solutions for the dynamic boundary value problems on time scales xΔΔ t f t, x t , xΔ t , t ∈ 0, 1 Ì, x 0 0, xΔ σ 1 0. This paper studies the existence of solutions for the nonlinear two-point dynamic boundary value problem on time scales xΔΔ t f t, xσ t , xΔ t , t ∈ a, ρ2 b Ì, xΔ a 0, x b 0, 1.2 where Ì is a bounded time scale with a inf Ì, b sup Ì, and a < ρ2 b. The proof is the same as 18, Theorem 4.1
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