Bifurcation Of Solutions On Two Classes Of Boundary Value Problems Of Difference Equations

In this paper,we are concerned with the existence of solutions on two classes of nonlinear difference equations boundary value problems and obtain the local and global bifurcation structures of solution sets via bifurcation theory and topological degree theory.The main results are as follows:1.We use the Rabinowitz global bifurcation theorem to study the local and global structure of solution sets of periodic boundary value problems for secondorder difference equation (?) where T>1 is an integer,T={1,2,…,T},λ∈[0,∞)is a parameter,Δu(t)=u(t+1)-u(t),Δ2u(t)=u(t+2)-2u(t+1)+u(t),g:[1,T]×R→R is a continuous function and it grows sublinearly at infinity.This section mainly considers the difference forms of the equations studied by Gámez and Ruiz in[19]when periodic boundary conditions are satisfied.2.We study the existence of solutions of second order difference equation Dirichlet boundary value problem (?) where T>1 is an integer,T={1,2,…,T},λ∈[0,∞)is a parameter,0<q<1<p,b:T→R is sign changing.The problem considered in this part is the difference form of the problem studied by Willian Cintra et al.[41]in one-dimensional case.The global structure of solution sets of difference equations with concave-convex nonlinearity is obtained under certain conditions by using bifurcation theory and topological degree theory,it provides a theoretical basis for numerical calculation of this kind of problem.