Positive Solutions Of Second-order Difference Equation Boundary Value Problems With Indefinite Weight

This dissertation studies the explicit expression and properties of the Green function for the linear difference equation with indefinite weight, then by using the fixed-point theorem in cones, fixed point index theory, topological degree and bifurcation theory, we obtain the existence of positive solutions of some nonlinear second-order discrete boundary value problems with indefinite weight, respectively. We describe them in detail as follows.In Section1, we obtain the explicit expression and properties of the Green function of the discrete linear boundary value problems where λ is a parameter, T={1,2,...,T}, T>1be an integer, m(t):Tâ†'R. changes its sign and m(t)≠0on T; Furthermore, combining with the properties of the Green function, by using the fixed-point theorem in cones and fixed point index theory, we prove the existence and multiplicity of positive solutions of the nonlinear second-order discrete boundary value problems with indefinite weight, where f:T×IR+â†'IR+is continuous. The main results not only extend and improve the corresponding results of Agarwal and O’Regan [Appl. Math. Lett.,1997], Chu and O’Regan [Com. Appl. Anal.,2008]; but also when λ=0or m=0, they provide theory basis for numerical calculation of the results of Wang etc.[Pro. Amer. Math. Soc.,1994],[J. Math. Anal. Appl.,1994]. 2. Combining with the properties of the Green function of problem (P), by using the fixed point theorem, we show the existence of positive solutions of the nonlinear second-order discrete semipositone boundary value problems with indefinite weight, when nonlinearity/satisfy superlinear growth condition at infinity, where μ>0is a parameter. The result not only generalize the results of Agarwal and O’Regan [Nonlinear Anal.,2000], but also are the discretization of the results of Anuradha, D. D. Hai and Shivaji [Pro. Amer. Math. Soc.,1996].3. By using the Krein-Rutman theorem, topological degree theory and bifur-cation theorem, we still consider the existence of positive solutions of nonlinear second-order discrete semipositone boundary value problems with indefinite weight when f satisfy asymptotically linear condition at infinity, where μ>0is a parameter, we give bifurcation structure and parameter interval of the positive solutions. The main result not only generalize the existing discrete results when m=0, but also supplement the results of Zhang and Liu [J. Math. Anal. Appl.,2007], moreover, it can clearly describe the trend component of positive so-lutions.