Existence Of Solutions For Three Classes Of Discrete Delta-nabla Fractional Boundary Value Problems With P-Laplacian Operators

In recent years,the mathematical model of fractional difference equations has attracted the attention of many scholars,and its related research results have been gradually applied to the fields of electrical engineering,chemistry,and biomedicine.The existence,uniqueness and multiplicity of solutions to value problems have become the research hotspots of scholars.this paper studies three types of discrete delta-nabla fractional boundary value problem with p-Laplacian operators:The first category considers the following discrete delta-nabla fractional boundary value problems with p-Laplacian operators:(?)whereb?Z+,t?T=[?-?-1,b+?-?-1]N?-?-1,1<?,??2,3??+??4,0<?<1,??(0,+?),??-2,b???,are left and right fractional difference operators,respectively,?p(s)=|s|p-2s,p>1,?p=?q-1,1/p+1/q=1.Through transformation,the above problem is transformed into the following frac-tional difference boundary value problem:(?)The existence of positive solutions for the differential boundary value problem under sublinear boundary conditions is obtained by using the upper and lower solutions method and Schauder fixed point theorem for the transformed equation,and then the argument that a positive solution exists for the original equation is established.The second category considers discrete delta-nabla fractional boundary value prob-lems with p-Laplacian operators under non-homogeneous boundary value conditions:(?)whereb?? Z+,T=[?-1,b+?-2]Na-1,0<??1<??2,2<?+?<3,??-2,b???,are left and right fractional difference operators,respectively,?p(s)=|s|p-2s,p>1,?p=?q-1,1/p+1/q=1.In response to this problem,the author constructs the corresponding Green's function,and according to the properties of the Green's function,obtains the suffi-cient conditions for the existence of a positive solution to the equation.In addition,corresponding examples are given to illustrate the validity of the results.The third category considers the delta-nabla discrete ordered fractional boundary value problem with p-Laplacian operator:(?)whereb?+N,h:[?-1,b+?]N?-1?(0,?),?=?1+?2+?3,?1?(0,1),?2,?3?(1,2),2<?2+?3<3,2<?1+?2+?3<3,v=v1+v2+v3,v1,v2,v3?(0,1),0<v1+v2+v3<1.This equation is to promote the study of delta-nabla fractional difference equations in ordered problems,and to use Krasnosel'skii fixed point theorem to establish the existence of understanding.