Fixed-Point Theorems in Fuzzy Normed Linear Spaces for Contractive Mappings with Applications to Dynamic-Programming

Abstract

The aim of this paper is to provide new ways of dealing with dynamic programming using a context of newly proven results about fixed-point problems in linear spaces endowed with a fuzzy norm. In our paper, the general framework is set to fuzzy normed linear spaces as they are defined by Nădăban and Dzitac. When completeness is required, we will use the George and Veeramani (G-V) setup, which, for our purposes, we consider to be more suitable than Grabiec-completeness. As an important result of our work, we give an original proof for a version of Banach’s fixed-point principle on this particular setup of fuzzy normed spaces, a variant of Jungck’s fixed-point theorem in the same setup, and they are proved in G-V-complete fuzzy normed spaces, paving the way for future developments in various fields within this framework, where our application of dynamic programming makes a proper example. As the uniqueness of almost every dynamic programming problem is necessary, the fixed-point theorems represent an important tool in achieving that goal.

Keywords

• dynamic programming
• fixed-point theorems
• fuzzy normed linear space
• G-V-completeness
• fuzzy continuous