Fixed point results for contractions of polynomial type

TL;DR

The paper extends fixed point theory by introducing polynomial contractions and almost polynomial contractions on complete metric spaces, capturing nonlinear dependencies via a polynomial-inequality framework with coefficients \(a_i\) and powers \(d^i(\cdot,\cdot)\). It proves fixed point results under continuity (and a relaxed Picard-continuity) and shows how Banach's contraction principle is recovered as a special case; it also generalizes Berinde's almost contractions through the almost polynomial framework. The main contributions are existence (and in some settings uniqueness) of fixed points and convergence of Picard iterations for these new classes, together with illustrative examples and corollaries. These results broaden the applicability of contraction-type methods to nonlinear problems by accommodating polynomial-type and cross-term dependencies. The findings have potential implications for nonlinear integral and differential equations where standard contractions are too restrictive.

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.

Theorems & Definitions (27)

• Definition 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• proof
• Proposition 2.3
• proof
• Corollary 2.4
• Corollary 2.5
• Remark 2.6