Existence Of Positive Solutions For Two Classes Of Nonlinear Difference Equations With Discrete Singular ?-Laplacian

This paper studies the existence and the global structure of positive solutions for the quasilinear difference equation with diverse boundary conditions by using topological method and the unilateral global bifurcation theory respectively.The main results are described as follows:1.We use topological method to study existence and multiplicity of positive solutions for the quasilinear difference boundary value problem where ?,??0,T:={2,…,T-1},T>3 is an integer,?u(t)=u(t+1)-u(t)is the forward difference operator,?u(t)=u(t)-u(t-1)is the backward difference operator,?:(-1,1)?R is given by ?(s)=s/(?),a,b:T×R?R are continuous functions.The function f:=?a(t,s)+?b(t,s)is either sublinear,or superlinear,or sub-superlinear near s=0.The main results are the discretization of the results of Corsato,Obersnel,Omari et al.in[J.Math.Anal.Appl.,2013].2.We apply the unilateral global bifurcation theorem to investigate the global structure of positive solutions for nonlinear discrete Robin boundary value problem where ?>0,[1,T]Z:={1,…,T-1},T>2 is an integer,?:(-1,1)?R is given by ?(s)=s/(?),f:[1,T]Z×[0,?)?[0,?)is continuous for some ?>T and f(t,s)is either,or sublinear,or superlinear near s=0.In this part,the problem considered is the N-dimensional difference form of the problem studied by Ma et al.in[J.Funct.Anal.,2016].