Multiple positive solutions of nonlinear third-order BVP for a class of p-Laplacian dynamic equations on time scales

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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 (Φp(uΔΔ(t)))+f(t,u(t))=0,t[a,b],αu(ρ(a))βuΔ(ρ(a))=0,γu(b)+δuΔ(b)=0,uΔΔ(ρ(a))=0, where Φp(s) is p-Laplacian operator, i.e., Φp(s)=|s|p2s,p>1,Φp1=Φq,1p+1q=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].

Keywords

Time scales
p-Laplacian operator
Third-order BVPs
Fixed-point index
Positive solutions

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Project supported by Natural Science Foundation of Shanxi Province (2008011002-1) and by Development Foundation of Higher Education Department of Shanxi Province.