一类非线性算子方程组的迭代算法及应用

Iterative solutions for a nonlinear system of operator equations

在无穷维Ｂａｎａｃｈ空间中研究了一类不具有单调性的算子方程组ｕ＝Ｆ（ｕ，ｖ），ｖ＝Ｇ（ｖ，ｕ），其中Ｆ，Ｇ可以表示成Ｆ＝Ｆ１＋Ｆ２，Ｇ＝Ｇ１＋Ｇ２，Ｆ１，Ｇ１是混合单调的，Ｆ２，Ｇ２是反向混合单调的（Ｆ２≠０，Ｇ２≠０），得到了可解性定理．当Ｐ是正规极小锥时，通过构造一系列的确界迭代生成列，建立了解的非单调迭代算法．最后，推广了最大－最小解的概念，定义了极大－极小解，并且研究了其存在的条件．主要特点是不要求算子具有混合单调性，可以说从本质上推广了许多已知的结论．

This paper deals with a class of systems of operator equations u=F(u,v), v=G(v,u), Where F and G have the decomposition F=F 1+F 2,G=G 1+G 2,F 1,G 1 are mixed monotone, F 2,G 2 are anti mixed monotone. Via constructing a series of iteratively bound generated sequences, We not only obtain some existence results, but also establish nonmonotone iteration processes. As application we study the existence of solutions for a nonlinear system of integro differential equations with nonmonotone terms. Our conclusions generalize many well known results.