Generalization of Chatterjea Fixed Point Theorem in

b-Metric Spaces

Suham M. Ali1,*

1 Department of Mathematics, College of Science, University of Kirkuk, Kirkuk, Iraq

Email: suhammuhammedali@uokirkuk.edu.iq

Received: October 09, 2025 Revised: December 16, 2025 Accepted: January 23, 2026 ⋆ Corresponding author

ABSTRACT

In this paper, we generalize Chatterjea’s fixed point theorem in b-metric spaces by using a subadditive altering

distance function. We also present corollaries and examples as applications of the theorem.

Keywords: Fixed point theory b-metric space Subadditive altering distance function

1. INTRODUCTION

Banach’s fixed point theorem is one of the principal tools

used to study nonlinear equations. Fixed point theorems have

applications in economics, engineering, biology, chemistry,

physics, computer science, and other fields. They are also

used to solve nonlinear and differential equations.

Chatterjea [1] introduced the following type of contraction.

Definition 1.1. A mapping T : X → X on a metric space

(X,d) is called a Chatterjea-type contraction if there exists

α ∈ (0, 12

) such that, for all x,y ∈ X,

d(Tx,Ty) ≤ α

􀀀

d(x,Ty)+d(y,Tx)

. (1.1)

Chatterjea proved that every Chatterjea-type contraction on a

complete metric space has a unique fixed point.

The notion of a b-metric space was developed in [2, 3]. A

function d : X ×X →[0,∞), where X ̸= ∅, is a b-metric with

parameter b ≥ 1 if, for all x,y, z ∈ X,

d(x,y) = 0 ⇐⇒ x = y, d(x, y) = d(y, x),

and

d(x, z) ≤ b

􀀀

d(x,y)+d(y, z)

.

The pair (X,d) is then called a b-metric space with parameter

b.

Many authors have generalized Banach’s fixed point theorem

in b-metric spaces [3, 4, 5, 6, 7, 8, 9, 10]. Boriceanu et al. [11]

defined convergence, Cauchy sequences, and completeness

in b-metric spaces. Other authors have extended Chatterjea’s

fixed point theorem [12, 13]. In 2025, Ampadu [14] established

a common fixed point theorem for four self-mappings

under Chatterjea contractive conditions in a cone pentagonal

metric space. In 2025, Anghelina [15] established Matkowski-

, Kannan-, and Chatterjea-type fixed point theorems in f -

metric spaces, which generalize b-metric spaces.

This paper extends Chatterjea’s result by using a subadditive

altering distance function. The following definition is used

throughout.

Definition 1.2 ([16]). A subadditive altering distance function

is a mapping ϕ : [0,∞) → [0,∞) satisfying, for all

x,y ∈ [0,∞):

(i) ϕ is continuous and strictly increasing;

(ii) ϕ(x) = 0 if and only if x = 0;

(iii) ϕ(σx+λy) ≤ σϕ(x)+λϕ(y) for σ,λ ∈ R.

2. MAIN RESULTS

Let ϕ be a subadditive altering distance function, and consider

a b-metric space (X,d) with the topology generated by d. We

first present the following fixed point theorem.

Theorem 2.1. Let (X,d) be a complete b-metric space with

parameter b ≥ 1, and let T : X → X satisfy, for some p ∈

(0, 1

b+2 ),

ϕ

􀀀

d(Tx,Ty)

≤ p

ϕ

􀀀

d(x,y)

+ϕ

􀀀

d(x,Ty)

(2.1)