, Volume 22, Issue 6, pp 1797-1804
Date: 12 Apr 2006

Positive Solutions for Second–Order m–Point Boundary Value Problems on Time Scales

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Abstract

Let \({\Bbb T}\) be a time scale such that 0, T \({\Bbb T}\) . By means of the Schauder fixed point theorem and analysis method, we establish some existence criteria for positive solutions of the m–point boundary value problem on time scales $$ \begin{array}{*{20}c} {{u^{{\Delta \nabla }} {\left( t \right)} + a{\left( t \right)}f{\left( {u{\left( t \right)}} \right)} = 0,t \in {\left( {0,T} \right)},}} & {{\beta u{\left( 0 \right)} - \gamma u^{\Delta } {\left( 0 \right)},u{\left( T \right)} - {\sum\limits_{i = 1}^{m - 2} {a_{i} u{\left( {\xi _{i} } \right)}} } = b,m \geqslant 3,}} \\ \end{array} $$ where aC ld ((0, T), [0,∞)), fC ld ([0,∞) × [0,∞), [0,∞)), β, γ ∈ [0,∞), ξ i ∈ (0, ρ(T)), b, a i ∈ (0,∞) (for i = 1, . . . ,m− 2) are given constants satisfying some suitable hypotheses. We show that this problem has at least one positive solution for sufficiently small b > 0 and no solution for sufficiently large b. Our results are new even for the corresponding differential equation ( \({\Bbb T}\) = ℝ) and difference equation ( \({\Bbb T}\) = ℤ).

Supported by the NNSF of China (10571078) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China, the Fundamental Research Fund for Physics and Mathematics of Lanzhou University (LZU05003)