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INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue III, March 2026
Generalized Theorems of Fixed Point for Fuzzy Contractions in
Fuzzy Metric Space
Atul Kumar Agnihotri
1
,S.K. Pandey
2
1
Department of mathematical sciences A.P.S. University Reewa (M.P.), 485001, India
2
Prof. of mathematics department of mathematics, P.M. college of excellence govt. Vivekanand P.G.
college Maihar (M.P.) 485001, India.
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.150300039
Received: 16 March 2026; Accepted: 25 March 2026; Published: 09 April 2026
ABSTRACT
In this paper, we establish a generalized fixed-point theorem for fuzzy contractions in fuzzy metric spaces. The
result extends the well-known Banach contraction principle into the setting of fuzzy metric spaces by employing
a generalized fuzzy contraction. Examples are provided to demonstrate the applicability and generalization of
classical fixed-point results. Fuzzy fixed-point techniques are used in mathematical modelling to solve problems
where traditional methods fail due to imprecise or uncertain data. To obtain fuzzy fixed points, different
contraction conditions are implemented in a fuzzy context.
Keywords: Fuzzy metric space, Fixed point theorem, Fuzzy contraction, Generalized fuzzy metric, Banach
contraction
INTRODUCTION
The concept of fuzzy metric space was first introduced by Kramosil and Michalek (1975) as a generalization of
the classical metric space to handle uncertainty and imprecision. Later, George and Veeramani (1994) modified
the definition to make it suitable for topological analysis.
Fixed point theory, originally studied by Banach (1922), plays an essential role in nonlinear analysis and its
applications. Extending fixed point theorems to fuzzy metric spaces has become an active area of research, as it
integrates uncertainty with analytical rigor.
In this paper, we propose a generalized fuzzy contraction mapping and prove a fixed point theorem for such
mappings in fuzzy metric spaces.
Preliminaries
Definition
A fuzzy metric space is a triple
where:
1. is a non-empty set
2. is a continuous t-norm
3.
satisfying the following conditions:
1.
if and only if
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2.
3.
4. is non-decreasing in
5.
Generalized Fuzzy Contraction
Definition
Let
be a fuzzy metric space. A mapping is called a generalized fuzzy contraction if there exists
such that for all and for all ,
This condition generalizes the classical fuzzy contraction and allows a broader class of mappings to satisfy fixed
point results.
Main Theorem (Results)
Generalized Fixed Point Theorem
Let
be a complete fuzzy metric space and be a generalized fuzzy contraction.
Then has a unique fixed point
, i.e.,
Proof
Let
be arbitrary and define a sequence
by:
Using the generalized contraction property,
Similarly,
By induction,
Since ,
as
Hence,
Thus,
is a Cauchy sequence in () .
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Now we have show completences of X
Because is complete, the sequence
converges to some
, i.e.,
for all
Now we show that
is a fixed point of
We have
.
Now consider:
As ,
.
, and by
By continuity of ,
Uniqueness of Fixed Point
Let
and
be two fixed points of .
Let (X,P,*) be fuzzy metric stage and let T:X X be a self-mapping satisfying the generalized fuzzy contraction
condition
where is continuous and strictly increasing such that for all . If T has a fixed
point in X the it is unique.
Proof
Let
and
be two fixed points of T then
Since
for all , the above inequality is possible only if
By the properties of fuzzy metric space,
Hence, the fixed point of is unique.
Applications
Differential Equations
Many fuzzy differential and integral equations can be converted into fixed point problems in fuzzy metric spaces,
ensuring the existence of solutions.
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Computer Science
Fuzzy metric spaces are used in pattern recognition and image processing fixed point results guarantee
convergence of iterative algorithms.
CONCLUSION
In this paper, we proved a generalized fixed point theorem for contraction mappings in fuzzy metric spaces. The
result provides a unifying framework for several well-known fixed point theorems and demonstrates that under
suitable fuzzy contractive conditions, the mapping admits a unique fixed point.
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