Existence and uniqueness of a common fixed point under a limit contractive condition
Journal of Inequalities and Applications
Existence and uniqueness of a common fixed point under a limit contractive condition
Mohammad Imdad 2
Sunny Chauhan 0
Muhammad Alamgir Khan 1
0 Near Nehru Training Centre , H. No. 274, Nai Basti B-14, Bijnor, Uttar Pradesh 246701 , India
1 Department of Natural Resources Engineering and Management, University of Kurdistan , Hewler , Iraq
2 Department of Mathematics, Aligarh Muslim University , Aligarh, 202 002 , India
In this paper, utilizing the notion of common limit range property for two pairs of self mappings, we prove common fixed point theorems in fuzzy metric spaces under a limit contractive condition, which improve and extend the results of Zhu et al. [Common fixed point theorems of new contractive conditions in fuzzy metric spaces, J. Appl. Math. 2013:145190, 2013]. We also give some examples to demonstrate the validity of the hypotheses of our results. As an application to our main result, we obtain a fixed point theorem for four finite families of self mappings in fuzzy metric space. MSC: 47H10; 54H25
fuzzy metric space; weakly compatible mappings; common limit in the range property; fixed point theorems
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Motivated by the results of Imdad et al. [], Chauhan [] extended the notion of
common limit range property to two pairs of self mappings in fuzzy metric spaces and proved
several results on the existence and uniqueness of common fixed points in fuzzy
metric spaces. For the sake of completeness, we refer the readers to [–]. Most recently,
Zhu et al. [] proved common fixed point theorems by using the notions of the property
(E.A) and the common property (E.A) in George and Veeramani type fuzzy metric spaces
(briefly, GV-type) under a new limit contraction condition.
In the present paper, we prove some common fixed point theorems for weakly
compatible mappings in fuzzy metric spaces employing the common limit range property. As an
application to our main result, we also derive a common fixed point theorem for four finite
families of mappings in fuzzy metric spaces using the notion of pairwise commuting
families (due to Imdad et al.[]) and utilize the same to derive common fixed point theorems
for six mappings. In the process, many known results (especially the ones contained in
Zhu et al. []) are enriched and improved. Some related results are also derived besides
furnishing illustrative examples.
2 Preliminaries
In this section, we present background material required in our subsequent discussion.
Definition . [] A binary operation ∗ : [, ] × [, ] → [, ] is a continuous t-norm if
it satisfies the following conditions:
() ∗ is associative and commutative,
() ∗ is continuous,
() a ∗ = a for all a ∈ [, ],
() a ∗ b ≤ c ∗ d, whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [, ].
Examples of continuous t-norms are a ∗ b = min{a, b}, a ∗ b = ab and a ∗ b = max{a + b –
, }.
Definition . [] A -tuple (X, M, ∗) is said to be a fuzzy metric space if X is an arbitrary
set, ∗ is a continuous t-norm, and M is a fuzzy set on X × (, ∞) satisfying the following
conditions: for all x, y, z ∈ X, t, s > ,
(GV-) M(x, y, t) > ,
(GV-) M(x, y, t) = if and only if x = y,
(GV-) M(x, y, t) = M(y, x, t),
(GV-) M(x, y, t) ∗ M(y, z, s) ≤ M(x, z, t + s),
(GV-) M(x, y, ·) : [, ∞) → [, ] is continuous.
In view of (GV-) and (GV-), it is worth pointing out that < M(x, y, t) < (for all t > )
provided x = y (see []). In what follows, fuzzy metric spaces in the sense of George and
Veeramani [] will be referred to as GV-fuzzy metric spaces.
Every metric space is a fuzzy metric space and is called a standard fuzzy metric space
(see []). Here, we give some new and interesting examples of fuzzy metric spaces due to
Gregori et al. []. For further details and related examples, one can see [].
Example . [] Let f : X → R+ be a one-one function, and let g : R+ → [, ∞) be an
increasing continuous function. For fixed α, β > , define M as
M(x, y, t) =
Example . [] Define a function M as
M(x, y, t) = e– dg(x(t,y)) ,
M(x, y, t) = – dg(x(t,)y) ,
then (X, M, ∗) is a fuzzy metric space on X, wherein ∗ is the product t-norm and g : R+ →
[, ∞) is an increasing continuous function.
Example . [] Let (X, d) be a bounded metric space with d(x, y) < k (for all x, y ∈ X).
Let g : R+ → [k, ∞) be an increasing continuous function. Define a function M as
then (X, M, ∗) is a fuzzy metric space on X, wherein ∗ is a Lukasiewicz t-norm, i.e., a ∗ b =
max{a + b – , }.
Definition . [] A sequence {xn} in a GV-fuzzy metric space (X, M, ∗) is said to be
convergent to some x ∈ X if for all t > , there is some n ∈ N such that
for all n ≥ n.
Definition . [] A pair (A, S) of self mappings of a GV-fuzzy metric space (X, M, ∗) is
said to be compatible if and only if M(ASxn, SAxn, t) → for all t > , whenever {xn} is a
sequence in X such that Axn, Sxn → z for some z ∈ X as n → ∞.
Definition . [] A pair (A, S) of self (...truncated)