Positive solutions of singular p-Laplacian dynamic equations with sign changing nonlinearity

摘要

Let T be a time scale such that 0,T∈T. By the Schauder fixed-point theorem and the upper and lower solution method, we present some existence criteria of the positive solution of m-point singular p-Laplacian dynamic equation (φp(u△(t)))▽+q(t)f(t,u(t))=0,t∈(0,T)T with boundary conditions u(0)=0,∑i=1m-1ψi(u(ξi))+u△(T)=0,m⩾2, where φp(s)=|s|p-2s with p>1, ψi:R→R is continuous for i=1,2,…,m-1 and nonincreasing if m⩾3,0<ξ1<ξ2<⋯<ξm-2<ξm-1=T. The nonlinear term may be singular in its dependent variable and is allowed to change sign. Our results are new even for the corresponding differential (T=R) and difference equations (T=Z). As an application, an example is given to illustrate our result.