A New Definition of Limit of Periodic Function and Periodic g-Contractive Mapping at Infinity

Tian-Quan Yun

Tian-Quan Yun School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, 510641, P.R. China.

Keywords: Limit, periodic function, fixed point theorem, Banach contractive mapping theorem, g-contractive mapping theorem

Abstract

Limit is a basic concept of calculus. However, according to the updated definition, the limit of periodic function at infinity is not in existence. This conclusion of description does not suit with the periodic phenomenon. For example, the temperature on earth is changed periodically every year since the birth of the earth (viewed as t=0). Today (viewed as t ??) the temperature on earth is continuing. Continuation means that the limit exists. In this paper, a new definition of limit of periodic function and periodic g-contractive mapping at infinity is defined by the value of its initial point based on transformation of variables. Similar definition is made for g- contractive ratio of periodic g-contractive mapping with k-related fixed points. These definitions can be used to describe the k-polar problems and calculation the limit of combinations of periodic functions at infinity. Furthermore, the new definition on contractive ratio of periodic iterative g-contractive mapping at infinity can help us to find the constant G and improves the application of the periodic iterative g-contractive mapping theorem.