Chatterjea-Type Mappings Overview

• Chatterjea-type mappings are contractive operators defined by cross terms (d(x,Ty) and d(y,Tx)) that guarantee a unique fixed point when the contraction constant is below 1/2.
• They extend classical fixed point theory to enriched, multivalued, cyclic, and generalized distance frameworks, accommodating mappings excluded by traditional Lipschitz conditions.
• Iterative methods such as Krasnoselskij averaging play a crucial role in ensuring convergence even when mappings are discontinuous or defined on nontraditional metric spaces.

Chatterjea-type mappings are contractive-type operators whose defining feature is the use of cross terms involving the distances from a point to the image of another point. In the standard metric formulation, a self-map $T:X\to X$ is of Chatterjea type if there exists $\gamma\in[0,\tfrac12)$ such that

$$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$

On a complete metric space, this condition yields a unique fixed point. Subsequent work has extended the scheme to enriched, multivalued, three-point, cyclic, iterated, and nonclassical-distance settings, while also clarifying limits of uniqueness, continuity, and the role of the chosen iterative process (Popescu et al., 2024, Sharma et al., 20 May 2026, Berinde et al., 2019).

1. Classical form and basic structural features

The classical Chatterjea condition differs from Banach and Kannan contractivity in that it does not control $d(Tx,Ty)$ directly by $d(x,y)$ or by $d(x,Tx)+d(y,Ty)$, but by the mixed distances $d(x,Ty)$ and $d(y,Tx)$. Recent expositions emphasize that Chatterjea-type mappings are non-equivalent to Banach and Kannan contractions and that they may be discontinuous. At the same time, the standard complete-metric fixed point theorem remains: if $T$ satisfies the Chatterjea inequality with constant in $[0,\tfrac12)$, then $\gamma\in[0,\tfrac12)$0 has a unique fixed point (Bisht et al., 2024, Popescu et al., 2024, Sharma et al., 20 May 2026).

This cross-term formulation has several consequences that recur throughout the literature. First, the class is flexible enough to accommodate mappings excluded by Banach-type Lipschitz conditions. Second, continuity is not intrinsic to the definition. Third, once the ambient geometry or the contractive inequality is altered, uniqueness can be lost, weakened, or replaced by “at most two fixed points,” as happens in some three-point analogues. A persistent theme is therefore not merely existence of fixed points, but the interaction among contractive geometry, admissible iterates, and the topology or generalized distance structure of the space (Popescu et al., 2024, Bisht et al., 2024).

2. Enriched Chatterjea mappings and Krasnoselskij averaging

A major normed-space development is the notion of a $\gamma\in[0,\tfrac12)$1-enriched Chatterjea mapping. For a normed linear space $\gamma\in[0,\tfrac12)$2, a map $\gamma\in[0,\tfrac12)$3 is called $\gamma\in[0,\tfrac12)$4-enriched Chatterjea if there exist $\gamma\in[0,\tfrac12)$5 and $\gamma\in[0,\tfrac12)$6 such that

$\gamma\in[0,\tfrac12)$7

for all $\gamma\in[0,\tfrac12)$8. The case $\gamma\in[0,\tfrac12)$9 recovers the classical Chatterjea pattern. In this framework, all Banach contractions with constant $$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$0 and all Kannan contractions with constant $$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$1 are included as enriched Chatterjea mappings with $$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$2, so the class is explicitly designed as a unifying extension (Berinde et al., 2019).

The associated fixed point theorem is formulated in Banach spaces. If $$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$3 is $$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$4-enriched Chatterjea, then $$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$5, and there exists $$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$6 such that the Krasnoselskij iteration

$$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$7

converges to $$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$8 for every starting point $$d(Tx,Ty)\le \gamma\big(d(x,Ty)+d(y,Tx)\big),\qquad x,y\in X.$$9. The paper also derives the estimate

$d(Tx,Ty)$0

which yields the Cauchy property and convergence. The role of Krasnoselskij averaging is essential: the paper exhibits $d(Tx,Ty)$1 on $d(Tx,Ty)$2, a map that is not Banach, Kannan, or Chatterjea, yet is enriched Chatterjea for suitable parameters, has unique fixed point $d(Tx,Ty)$3, and fails to have convergent Picard iteration except at the fixed point itself. The same work also notes discontinuous examples within the enriched class (Berinde et al., 2019).

3. Multivalued variants and average-operator methods

Chatterjea-type fixed point theory has also been extended to multivalued mappings. In complete partial cone metric spaces $d(Tx,Ty)$4, with $d(Tx,Ty)$5 a normal cone and $d(Tx,Ty)$6 denoting the nonempty, closed, bounded subsets of $d(Tx,Ty)$7, a multivalued map $d(Tx,Ty)$8 is assumed to satisfy

$d(Tx,Ty)$9

where $d(x,y)$0 is the Hausdorff-type partial metric built from $d(x,y)$1. Under this Chatterjea-type condition, $d(x,y)$2 has a fixed point $d(x,y)$3 with $d(x,y)$4. The proof proceeds through iterative selections $d(x,y)$5, geometric control of successive partial distances, and completeness of the partial cone metric space. An explicit finite example on $d(x,y)$6 shows the theorem in a setting where self-distances may be nonzero and the distance values are vectors in $d(x,y)$7 (Shateri et al., 2022).

A second multivalued branch uses normed spaces and enrichment. For $d(x,y)$8, a multivalued enriched Chatterjea mapping is defined by the existence of $d(x,y)$9 and $d(x,Tx)+d(y,Ty)$0 such that

$d(x,Tx)+d(y,Ty)$1

with $d(x,Tx)+d(y,Ty)$2 the Hausdorff metric and $d(x,Tx)+d(y,Ty)$3 the corresponding Minkowski sum. The central tool is the average operator

$d(x,Tx)+d(y,Ty)$4

for which $d(x,Tx)+d(y,Ty)$5. This reduces the enriched condition to a standard multivalued Chatterjea condition and yields existence of fixed points. The same framework provides data dependence: if $d(x,Tx)+d(y,Ty)$6 satisfies $d(x,Tx)+d(y,Ty)$7 for all $d(x,Tx)+d(y,Ty)$8, then for every $d(x,Tx)+d(y,Ty)$9 there exists $d(x,Ty)$0 such that

$d(x,Ty)$1

If $d(x,Ty)$2 is itself multivalued enriched Chatterjea, then

$d(x,Ty)$3

These results place stability of fixed point sets on the same footing as existence (Hacıoğlu et al., 2021).

4. Three-point, generalized, and cyclic formulations

A substantial recent line of work replaces the two-point Chatterjea inequality by a genuinely three-point condition. One formulation defines a generalized Chatterjea-type mapping on a metric space $d(x,Ty)$4, $d(x,Ty)$5, by requiring

$d(x,Ty)$6

for all pairwise distinct $d(x,Ty)$7, with $d(x,Ty)$8. Under the additional assumption $d(x,Ty)$9 whenever $d(y,Tx)$0, the map has at least one fixed point and at most two fixed points. The paper gives examples of generalized Chatterjea maps that are not standard Chatterjea maps, including discontinuous examples and examples outside Kannan and triangle-perimeter classes (Popescu et al., 2024).

A related paper works with the constant range $d(y,Tx)$1 and the assumption that $d(y,Tx)$2 has no periodic points of prime period $d(y,Tx)$3. It again concludes existence of a fixed point and that the number of fixed points is at most two. It further proves continuity at fixed points for generalized Chatterjea-type mappings and derives fixed point theorems even when completeness is not mandatory, provided appropriate orbit-limit or density-and-continuity hypotheses are satisfied. It also identifies parameter regimes under which standard Chatterjea mappings, generalized Kannan mappings, and mappings contracting perimeters of triangles are contained in the generalized Chatterjea class (Bisht et al., 2024).

A different generalization appears in $d(y,Tx)$4-metric spaces. There, a cyclical operator $d(y,Tx)$5 is a $d(y,Tx)$6-cyclic $d(y,Tx)$7-Chatterjea contraction if for suitable $d(y,Tx)$8 with $d(y,Tx)$9 and $T$0,

$T$1

for $T$2, $T$3. In a complete $T$4-metric space with closed subsets $T$5, such a cyclical operator has a unique fixed point in $T$6. This formulation introduces altering-distance and control functions while preserving the essential Chatterjea cross-term structure (Al-khaleel et al., 2018).

5. Iterated forms, $T$7-continuity, and minimal assumptions

Another direction studies Chatterjea behavior on iterates rather than on $T$8 itself. The 2026 paper “Another Perspective on Chatterjea Contraction” introduces $T$9-Chatterjea contractions and proves existence and uniqueness of fixed points on complete metric spaces under an additional $[0,\tfrac12)$0-continuity assumption, meaning that some iterate $[0,\tfrac12)$1 is continuous. For $[0,\tfrac12)$2 and for general $[0,\tfrac12)$3, the iterative sequence $[0,\tfrac12)$4 converges to the unique fixed point. At the same time, the paper gives counterexamples showing that without $[0,\tfrac12)$5-continuity the iterates may converge while no fixed point exists, and it provides examples demonstrating that the $[0,\tfrac12)$6-Chatterjea class strictly contains the classical Chatterjea class (Sharma et al., 20 May 2026).

A related iterative generalization is the Singh-Chatterjea contraction. Here one assumes that for some $[0,\tfrac12)$7 and $[0,\tfrac12)$8,

$[0,\tfrac12)$9

On a complete metric space this implies a unique fixed point, and the entire orbit $\gamma\in[0,\tfrac12)$00 converges to it for every initial point. The paper states that this framework recovers classical Chatterjea when $\gamma\in[0,\tfrac12)$01, relates to Singh’s iterate-based extension of Kannan mappings, and strictly enlarges the class of admissible operators. It also shows that every Banach contraction becomes Singh-Chatterjea after sufficiently many iterates; the minimum $\gamma\in[0,\tfrac12)$02 is characterized by the inequality $\gamma\in[0,\tfrac12)$03 when $\gamma\in[0,\tfrac12)$04 is the Banach constant (Bekri et al., 13 Oct 2025).

At the opposite end of the spectrum, the 2025 CJM-based analysis identifies an essentially weakest Chatterjea-type condition on a complete metric space: $\gamma\in[0,\tfrac12)$05 This is the condition labeled $\gamma\in[0,\tfrac12)$06. The paper proves that, together with an orbit-wise CJM implication, $\gamma\in[0,\tfrac12)$07 is equivalent to the statement that $\gamma\in[0,\tfrac12)$08 has a unique fixed point and every Picard sequence converges to it. The significance is that no uniform contraction constant below $\gamma\in[0,\tfrac12)$09 is required; strict pointwise cross-term dominance suffices within the CJM framework (Hashimoto et al., 3 May 2025).

6. Nonclassical distance structures and broader analytical scope

Chatterjea-type ideas have been transplanted into several generalized distance frameworks. In semimetric spaces with triangle functions $\gamma\in[0,\tfrac12)$10, one studies both the classical Chatterjea inequality

$\gamma\in[0,\tfrac12)$11

and the Chatterjea-Bianchini condition

$\gamma\in[0,\tfrac12)$12

For complete semimetric spaces with continuous $\gamma\in[0,\tfrac12)$13 and triangle functions satisfying the paper’s conditions $\gamma\in[0,\tfrac12)$14, $\gamma\in[0,\tfrac12)$15, (i), and (ii), fixed point results are established for the Chatterjea-Bianchini case. The framework specializes to metric, $\gamma\in[0,\tfrac12)$16-metric, ultrametric, and power-type triangle functions, with explicit parameter ranges such as $\gamma\in[0,\tfrac12)$17 for $\gamma\in[0,\tfrac12)$18-metrics and $\gamma\in[0,\tfrac12)$19 for power triangle functions $\gamma\in[0,\tfrac12)$20 (Bisht et al., 2024).

In partial $\gamma\in[0,\tfrac12)$21-metric spaces $\gamma\in[0,\tfrac12)$22, Chatterjea-type mappings are treated through

$\gamma\in[0,\tfrac12)$23

For complete partial $\gamma\in[0,\tfrac12)$24-metric spaces with coefficient $\gamma\in[0,\tfrac12)$25 and $\gamma\in[0,\tfrac12)$26, the fixed point is unique and satisfies $\gamma\in[0,\tfrac12)$27. The same paper develops a max-type Chatterjea condition, a joint Chatterjea-Kannan theorem with coefficient constraints, and $\gamma\in[0,\tfrac12)$28-stability of Picard iteration. It also conjectures the $\gamma\in[0,\tfrac12)$29-property for these broader classes (Gaba et al., 2019).

In probabilistic cone metric spaces, the contractive condition becomes distributional: $\gamma\in[0,\tfrac12)$30 On complete probabilistic cone metric spaces, this guarantees a unique fixed point. The formulation combines uncertainty, via distribution-valued distances, with cone-induced order. The cited work presents this as relevant to stochastic processes, random operators, and uncertainty modeling in functional equations (Rada, 18 Aug 2025).

Taken together, these developments show that Chatterjea-type mappings are no longer confined to complete metric spaces and single-valued self-maps. They now encompass enriched averaging schemes in Banach spaces, set-valued dynamics, three-point and cyclic geometries, iterate-based contractions, semimetric and partial metric structures, and probabilistic cone settings. A plausible implication is that the class functions less as a single theorem and more as a contractive template adaptable to different ambient geometries. The cited works explicitly connect these extensions to nonlinear equations, variational inequalities, optimization problems, differential inclusions, economics, computer science, stochastic processes, and random operators (Berinde et al., 2019, Shateri et al., 2022, Rada, 18 Aug 2025).