Multiple positive solutions for nonlinear three-point boundary value problems on time scales

Abstract

In this paper, we consider the nonlinear second-order three-point boundary value problem u Δ ∇ ( t ) + h ( t ) f ( t , u ( t ) ) = 0 , t ∈ [ t 1 , t 3 ] ⊂ T , u Δ ( t 3 ) = 0 , α u ( t 1 ) − β u Δ ( t 1 ) = u Δ ( t 2 ) , where T is a time scale, 0 ≤ t 1 < t 2 < t 3 , α > 0 and β ≥ 0 are given constants. By using fixed-point theorems in cones, some new results are obtained for the existence of at least one,two and three positive solutions of the above problem.