非线性差分方程初值问题

INITIAL VALUE PROBLEMS FOR NONLINEAR DIFFERENCE EQUATIONS

用上、下解方法研究非线性差分方程初值问题：△ｕｋ＋ｆ（ｋ，ｕｋ）＝０，ｋ＝１，…，ｎ；ｕ＿０＝０，其中△ｕｋ＝ｕｋ－ｕｋ－１，给出下解ｖ不超过上解ｗ的一个充分条件：当ｆ（ｋ，ｕ）关于ｕ不减时，有ｖ≤ｗ．用Ｂｒｏｕｗｅｒ不动点定理以及修正初值问题的技巧，建立了解的存在定理：在ｆ（ｋ，ｕ）关于ｕ连续的条件下，对给定的下解ｖ与上解ｗ；ｖ≤ｗ，初值问题存在解ｕ满足ｖ≤ｎ≤ｗ。

In this paper, using the method of upper and lower solutions,the following initialvalue problem(IVP) for nonlinear difference equation is studied :△u_k+f(k,u_k)=0,k=1,2,…,n;u_0=0,where △u_k=u_k-u_(k-1).In section 2, a sufficient condition is given which assuresthat the lower solution v doesn’t exceed the upper solution w,that is,if f(k,u) is nondecreasingin u,then v≤w,In section 3,employing the Brouwer fixed point theorem and the technique ofmodified IVP,an existence theorem of solution for IVP is established:under the assumptionsthat f(k,u)is continuous in u and there exist lower and upper solutions v,w,such that v≤w,the IVP has a solution u such that v ≤u ≤w.