Global Structure Of Solutions For Two Dirichlet Problems Of Nonlinear Difference Equations

In this paper,we study the unilateral global bifurcation structure of solutions for two Dirichlet boundary value problems of nonlinear difference equations by using bifurcation theory and topological degree.The main results are described as follows:1.We study the unilateral global bifurcation structure of solutions for Dirichlet boundary value problem of second-order difference equation by using the bifurcation theorems from an interval,where T>1 is an integer,T={1,2.…,T},T={0,1,…,T+1},??G[0,?)is a parameter,,p:{0,1,2,…,T}?(0,?),q? C(T,[0,?)),a:T?(0,?),F ? C(T ×R3,R)and F(t,u,s,?)are not neccssarily linearizable near u=0 or infinity.The maini results not only are the discretization of the results of Ma.Dai[10],but also provide theory basis for numerical calculation of these class of problems.Based on the above unilateral interval bifurcation results,we study the existence and nonexistence of solutions for problem with nondifferentiable nonlinearities where ? is a parameter,a(t)is a positive continuous function in T,f ?C(R,R).2.We show the existence of the principal eigenvalues and determine the sign of the corresponding eigenfunctions for the eigenvalue problem where A is a parameter.q(t)?0,the weight function a(t)>0,?p(s)is p-Laplacian operator,i.e.?p(s)=|s|p-2s.Then,we apply topological degree theory and global bifurcation theorem,the u-nilateral global bifurcation theorem of Dancer type for the following one-dimensional p-Laplacian difference equation is established where g:T×R2?R satisfies Caratheodory condition.The main results in this section not only apply the spectral results directly to the one-dimensional p-Laplacian difference equation,but also generalize the bifurcation theory of this quasi-linear problem.Finally,we prove the existence of definite solutions for the p-Laplacian difference equation based on the above results where f? C(T ×R.R).According to the behavior of nonlinearity.f at 0 or ?,The existence of nodal solutions of p-Laplacian difference equation with signum condition is discussed separately.