The Study On Existence, Uniqueness And Properties Of Solutions For Some Classes Of Functional Equations

The concept, dynamic programming, was proposed by Richard Bellman for the first timein the middle of the last century. It is a method in the solution for finding the solutions ofoptimization of multistage decision, and its cote idea is principle of optimization. During thepast fifty years, dynamic programming plays a vital role in operation researches, controltheory, industrial engineering, economics, management and so on. It is worthy of noting thatthe embedding of functional equation is one of the most important features of dynamicprogramming. Combining dynamic programming with functional equation together providesnew ideas for future studies.Before the nineteenth century, most of the mathematics master paid more attention tosolving functional equations, until the19th century, it has started to shift its focus on the studyof the existence and uniqueness of solutions for functional equations. Generally, there arethree tools for proving the existence and uniqueness of solutions. Firstly, Ascoli-ArzelaLemma and Schauder fixed point theorem; Secondly, contraction mapping principle orsuccessive approximation method; Finally, the method of majorant series. Using thecontraction mapping principle and the method of majorant series not only can prove theexistence of solutions, but also can guarantee the uniqueness of the solutions.Today, international mathematics masters focus their efforts on how to improve theconditions and the methods for finding the solutions of dynamic programming. This paperstudies mainly the existence and uniqueness of solutions to functional equations arising fromdynamic programming, but we do not solve the functional equations. This paper is organizedas follows:In the second chapter, we study the existence and uniqueness of solutions of four types offunctional equations with single process variable involving the supremum and infimum. Fromthe establishment of the functional equations to the conditions imposed on the equations allpossess some kinds of innovation. The problem of solving the functional equations isconverted into solving the fixed-point problem. This paper mainly adopts two methods ofproof：One way is to define a continuous mapping in Banach space during the process ofproof, by repeating the iterations of mapping shows that the mapping is non-expansive andself-mapping, and then by the fixed point theorem we prove the existence of the solution.Another approach is to use the monotone iterative technique. Construct a sequence of functions at first, and then prove that the sequence of function is monotone and bounded, soit’s convergent. At last, we prove the limit of convergent sequence is the fixed point.In the third chapter, we focus our study on the properties of solutions to functionalequations with multi-process variables involving any nested bounds. The functional equationswe discussed in this chapter is more complicated, with the introduction of more processvariables, the amount of calculation will also be increased. At the same time, the processingmethod also need to be adjusted accordingly, such as, in order to overcome these difficultiesadding, cutting and changing inequality conditions which ensure the theorem hold are addedinto the theorem, some comparatively ideal result are obtained.