Fixed point results for contractions of polynomial type

TL;DR

This paper introduces new classes of polynomial-type contractions on metric spaces and proves fixed point theorems for these classes, generalizing classical results like Banach's fixed point theorem.

Contribution

It defines polynomial and almost polynomial contractions and establishes fixed point theorems for these classes, extending existing contraction principles.

Findings

Fixed point theorems for polynomial contractions.

Fixed point theorems for almost polynomial contractions.

Generalizations of Banach's fixed point theorem.

Abstract

We introduce two new classes of single-valued contractions of polynomial type defined on a metric space. For the first one, called the class of polynomial contractions, we establish two fixed point theorems. Namely, we first consider the case when the mapping is continuous. Next, we weaken the continuity condition. In particular, we recover Banach's fixed point theorem. The second class, called the class of almost polynomial contractions, includes the class of almost contractions introduced by Berinde [Nonlinear Analysis Forum. 9(1) (2004) 43--53]. A fixed point theorem is established for almost polynomial contractions. The obtained result generalizes that derived by Berinde in the above reference. Several examples showing that our generalizations are significant, are provided.