一类泛函方程解的存在唯一性及其性质

Existence and properties of solutions for a class of functional equations

讨论了一类源于动态规划的泛函方程,主要研究在某些条件下该泛函方程解的存在性、唯一性、迭代逼近及解的某些性质等问题.研究过程中,首先定义了巴拿赫空间中一个有界闭凸子集上的映射,其次证明该映射为非扩张的自映射,最后再证明该空间的任意收敛点列都是柯西列,从而由巴拿赫空间的完备性、不动点定理和非扩张映射证明了解的存在性.得到的结果拓宽、深化了刘泽庆等人的一些已有结论,并在一定程度上统一和归纳了由Bellman,Bhakta,Mitra等学者得到的早期研究结果.

In this paper we consider a class of functional equations arising in dynamic programming.The existence, uniqueness, iterative approximation and some properties of the solutions for the func- tional equations are discussed. During the process of the study, we firstly define a mapping on a closed bounded and convex subset of Banach space, and then prove that this mapping is a non-expan- sive mapping into itself. Furthermore, we show that every convergent sequence on this subset is a Cauchy sequence. Finally, by using non-expansive mapping, fixed point theorem and the complete- ness of Banach space, the existence of the solution is obtained. The results presented in this paper not only generalize the results in Liu, but also unify the previous results due to Bellman, Bhakta, Mitra and others.