Abstract
We are concerned with the following nonlinear second-order three-point boundary value problem on time scales , , , , where with and . A new representation of Green's function for the corresponding linear boundary value problem is obtained and some existence criteria of at least one positive solution for the above nonlinear boundary value problem are established by using the iterative method.
Highlights
Let T be a time scale, that is, T is an arbitrary nonempty closed subset of R
It is interesting that the method used in this paper is completely different from that in 6, 7, 9, 10, that is, a new representation of Green’s function for the corresponding linear BVP is obtained and some existence criteria of at least one positive solution to the BVP 1.2 are established by using the iterative method
We impose the following hypotheses: H1 f : a, b T × R → R is continuous; H2 for fixed t ∈ a, b T, f t, u is monotone increasing on u; H3 there exists q ∈ 0, 1 such that f t, ru ≥ rqf t, u for r ∈ 0, 1, t, u ∈ a, b T × R
Summary
Let T be a time scale, that is, T is an arbitrary nonempty closed subset of R. Three-point boundary value problems BVPs for short for second-order dynamic equations on time scales have received much attention. In 2002, Anderson 6 studied the following second-order three-point BVP on time scales: uΔ∇ tatfut 0, t ∈ 0, T T, 1.1 u 0 0, u T αu η , where 0, T ∈ T, η ∈ 0, ρ T T and 0 < α < T/η. Some existence results of at least one positive solution and of at least three positive solutions were established by using the well-known Krasnoselskii and Leggett-Williams fixed point theorems. In 2003, Kaufmann 7 applied the Krasnoselskii fixed point theorem to obtain the existence of multiple positive solutions to the BVP 1.1. For some other related results, one can refer to 8–10 and references therein
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