Arbitrary positive solutions to a multi-point [formula omitted]-Laplacian boundary value problem involving the derivative on time scales

Abstract

This paper is concerned with the existence of positive solutions to the p -Laplacian dynamic equation ( φ p ( u Δ ( t ) ) ) ∇ + h ( t ) f ( t , u ( t ) , u Δ ( t ) ) = 0 , t ∈ [ 0 , T ] T subject to boundary conditions u ( 0 ) − B 0 ( ∑ i = 1 m − 2 a i u Δ ( ξ i ) ) = 0 , u Δ ( T ) = 0 , where φ p ( u ) = | u | p − 2 u with p > 1 . By using the fixed-point theorem due to Avery and Peterson, we prove that the boundary value problem has at least triple or arbitrary positive solutions. Our results are new for the special cases of difference equations and differential equations as well as in the general time scale setting. As an application, an example is given to illustrate the result. The interesting point in this paper is that the nonlinear term f is involved with the first-order derivative explicitly.