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

Abstract

Let T be a time scale such that 0 , T ∈ T . Consider the following three-point boundary value problem on time scales: u Δ ∇ ( t ) + a ( t ) f ( t , u ( t ) ) = 0 , t ∈ ( 0 , T ) , β u ( 0 ) − γ u Δ ( 0 ) = 0 , α u ( η ) = u ( T ) , where β , γ ⩾ 0 , β + γ > 0 , η ∈ ( 0 , ρ ( T ) ) , 0 < α < T / η , and d = β ( T − α η ) + γ ( 1 − α ) > 0 . By using fixed point theorems in cones, some new and general results are obtained for the existence of single and multiple positive solutions of the above problem. In particular, our criteria generalize and improve some known results.