Multiple positive solutions of nonlinear third-order BVP for a class of [formula omitted]-Laplacian dynamic equations on time scales

Abstract

In this paper, by using fixed-point theorems in cones, the existence of multiple positive solutions is considered for singular nonlinear boundary value problem for the following third-order p -Laplacian dynamic equations on time scales ( Φ p ( u Δ Δ ( t ) ) ) ∇ + f ( t , u ( t ) ) = 0 , t ∈ [ a , b ] , α u ( ρ ( a ) ) − β u Δ ( ρ ( a ) ) = 0 , γ u ( b ) + δ u Δ ( b ) = 0 , u Δ Δ ( ρ ( a ) ) = 0 , where Φ p ( s ) is p -Laplacian operator, i.e., Φ p ( s ) = | s | p − 2 s , p > 1 , Φ p − 1 = Φ q , 1 p + 1 q = 1 . In particular, the conditions we used in the paper are different from those in [R.Y. Ma, Existence of solutions of nonlinear m -point boundary value problem, J. Math. Anal. Appl. 256 (2001) 556–567; A.M. Mao, S.X. Luan, Y.H. Ding, On the existence of positive solutions for a class of singular boundary value problems, J. Math. Appl. 298 (2004) 57–72].