1. Introduction
The notion of fuzzy metric spaces was first introduced by Kramosil et al. [
1]. George et al. [
2] modified the notion of Kramosil et al. [
1] by obtaining Hausdorff topology in fuzzy metric spaces and said that every metric induces a fuzzy metric. Later, fixed point theory was first introduced by Grabic [
3] in fuzzy metric spaces by extending Banach contraction conditions and Edelstein contraction conditions [
4] in terms of fuzzy, in the sense of Kramosil et al. [
1]. Grabic’s [
3] fixed point results were based on strong conditions associated with completeness of the fuzzy metric spaces called
G-completeness. After that, George and Veeramani weakened Grabic [
3] conditions and introduced M-completeness of fuzzy metric spaces.
Later, Tirado [
5], Gregori et al. [
6] and Mihet [
7] defined different classes of fuzzy contractive conditions. In 2012, Wardowski [
8] introduced a new contraction called
F-contraction in a metric space and proved some fixed point theorems in complete metric spaces. Recently, H. Hung et al. [
9] introduced a new type of condition called fuzzy
F-contraction in a fuzzy metric space. As compared to the
F-contraction, this is much simpler and more straightforward as it contains only one condition—that is, the function
F is strictly increasing and proves some fixed point theorems for fuzzy
F-contraction conditions. On the other hand, Hussain et al. [
10] introduced the concept of
-
-contraction conditions in a metric space as a generalization of
F-contraction and obtained some interesting fixed point results.
In this paper, we first introduce a family of function
, such as an implicit function, and give
-
-fuzzy contractive conditions depending on the class of functions (for instance,
-function in fuzzy metric spaces). We give here some fixed point theorems using the concepts of
-admissible and weaker conditions of continuity of the function. Finally, we obtain the fuzzy
F-contraction theorem given by H. Hung et al. [
9] as in the form of corollary of our main result, which proves that our generalization is fruitful. At the end, as an application of our result, we produce the existence of a solution of nonlinear fractional differential equations via the introduced fuzzy contractive conditions.
3. Fixed Point Theorems for α-ΓF-Fuzzy Contractions
In this section, we first introduce a new class of function called -function and the concept of --fuzzy contractions and prove some fixed point theorems in a fuzzy metric space. We begin with the following definition:
Let denote the set of all continuous functions satisfying:
1. For all with , there exists such that
We have the following examples:
, where .
.
.
Here, , and then .
Now, we define a new class of fuzzy contractive conditions depending on the class of functions.
Definition 9. Let be a fuzzy metric space and a mapping . Furthermore, suppose that be two functions. T is said to be an α-η--fuzzy contractive mapping on X, if for with and , we havewhere and . Next, we give the concept of --continuous mapping on a fuzzy metric space.
Definition 10. Let be a fuzzy metric space and and be a function. We say T is an α-η-continuous mapping on a fuzzy metric space. If, for a given and sequence withfor all . This implies that . Example 1. Let and t-norm be defined by for all , define a fuzzy set such that for all and is a fuzzy metric space. Let and be defined byand . Clearly, T is not continuous, but T is α-η-continuous mapping on . We need the following lemma to prove our main results.
Lemma 1 ([
9])
. Let be a fuzzy metric space and be a sequence in X such that for each ,
and for any ,If is not a Cauchy sequence in X, then there exists , , and two sequences of positive integers , , , , such that the following sequencestend to as . Now, we are ready to prove our main results.
Theorem 1. Let be a complete fuzzy metric space. Let be a self mapping satisfying the following assertions:
- 1.
T is α-admissible mapping with respect to η;
- 2.
T is an α-η--fuzzy contractive mapping;
- 3.
there exists such that ;
- 4.
T is an α-η-continuous map.
Then, T has a fixed point. Moreover, T has a unique fixed point whenever for all .
Proof. Let
such that
. For
, we define the sequence
by
for all
. Now, since
T is an
-admissible mapping with respect to
, then
by continuing this process, we have
for all
.
Furthermore, let such that then is fixed point of T and nothing to prove.
Let us assume
or
for all
. Since,
T is an
-
-
-contractive mapping, and thus we use
,
in (
2), we obtain
which implies
Since,
, by definition of
-function, there exists
such that
Since
F is a strictly increasing function
Thus, the sequence
is a strictly increasing bounded from above, and thus sequence
is convergent. In other words, there exists
such that
for any
and
. It follows that
by (
4) and (
5), for any
, we have
We have to show that
. Assume that
for some
and by taking limit as
in (
3) and using (
6), we obtain
This is a contradiction with
. Therefore,
Next, we have to prove that
is a Cauchy sequence. Suppose that
is not a Cauchy sequence. By using the Lemma 1, then there exists
,
and sequence
and
such that
Again, with
and
in (
2), we have
Letting limit as
, we have
this implies
Since
, there exists
such that
Using (
8) and (
9), implies that
This is a contraction with
. Thus, the sequence
is a Cauchy sequence in
X. Since fuzzy metric space
is complete, then there exists
such that
Let us prove that is a fixed point of T. Since T is an --continuous and for all . Then, implies — that is, . □
Let
such that
, by Equation (
2),
Since
, there exists
such that
. Thus, we can deduce above
This implies that
which is a contradiction. Thus,
T has a unique fixed point.
We can deduce the following Corollary.
Corollary 1. Let be a complete fuzzy metric space. Let be a self mapping satisfying the following assertions:
- 1.
T is α-admissible mapping with respect to η;
- 2.
if, for with and , we havewhere , and ; - 3.
there exists such that ;
- 4.
T is an α-η-continuous map.
Then, T has a fixed point. Moreover, T has a unique fixed point whenever for all .
Example 2. Suppose and t-norm is defined by for all . Define a fuzzy set such thatfor all and for all . Thus, is a complete fuzzy metric space. Define
such that
Let
defined by
for all
and
Furthermore, let be any strictly increasing function and consider function defined by , where .
Let then , on the other hand for all , then (or ). This means that T is an -admissible mapping with respect to .
There exist such that .
Let , for all . This implies . Thus, T is an - continuous map.
Thus,
T is an
-
-
-fuzzy contractive mapping. Thus,
is a fixed point for self map
T. Now, consider
,
and
Hence, this example can not hold the Theorem 1 proved in [
9], such as
does not hold.
When we use in Definition 9, Theorem 1 and Corollary 1, we obtain the following.
Definition 11. Let be a fuzzy metric space and a mapping . Furthermore, suppose that be a function. We say T is said to be an α--fuzzy contractive mapping on X if, for with and , we havewhere and . Theorem 2. Let be a complete fuzzy metric space. Let be a self mapping satisfying the following assertions:
- 1.
T is α-admissible mapping;
- 2.
T is an α--fuzzy contractive mapping;
- 3.
there exists such that ;
- 4.
T is α-continuous mapping.
Then, T has a fixed point. Moreover, T has a unique fixed point in X whenever for all .
Proof. Similar to the Proof of Theorem 1. □
Corollary 2. Let be a complete fuzzy metric space. Let be a self mapping satisfying the following assertions:
- 1.
T is α-admissible mapping;
- 2.
if, for with and , we havewhere , and ; - 3.
there exists such that ;
- 4.
T is α-continuous map.
Then, T has a fixed point. Moreover, T has a unique fixed point in X whenever for all .
Taking in Corollary 2 for all . We deduce the following fixed point result.
Corollary 3 ([
9])
. Let be a complete fuzzy metric space such thatfor all . If is a continuous fuzzy F-contraction, then T has a unique fixed-point in X. In the next theorem, we omit the continuity hypothesis of T.
Theorem 3. Let be a complete fuzzy metric space. Let be a self mapping satisfying the following assertions:
- 1.
T is α-admissible mapping with respect to η;
- 2.
T is an α-η--fuzzy contractive mapping;
- 3.
there exists such that ;
- 4.
if is a sequence in X such that with as , thenholds for all .
Then, T has a fixed point. Moreover, T has a unique fixed point whenever for all .
Proof. Let
such that
. Similar to the proof of the Theorem 1, we can conclude that
where,
. By assumption 4, either
holds for all
. This implies that
holds for all
. Equivalently, there exists a subsequence
of
such that
and by (
2), we obtain
which implies for any
,
Since
F is a strictly increasing function,
Taking limit as in the above inequality, we obtain —that is, . The uniqueness of the fixed point is similar to Theorem 1. □
Corollary 4. Let be a complete fuzzy metric space. Let be a self mapping satisfying the following assertions:
- 1.
T is α-admissible mapping with respect to η;
- 2.
if, for with and , we havewhere , and ; - 3.
there exists such that ;
- 4.
if is a sequence in X such that with as , thenholds for all .
Then, T has a fixed point. Moreover, T has a unique fixed point whenever for all .
When we consider in Theorem 3 and Corollary 4, we obtain the following.
Theorem 4. Let be a complete fuzzy metric space. Let be a self mapping satisfying the following assertions:
- 1.
T is an α-admissible mapping;
- 2.
T is an α--fuzzy contractive mapping;
- 3.
there exists such that ;
- 4.
if is a sequence in X such that with as , then or holds for all .
Then, T has a fixed point. Moreover, T has a unique fixed point whenever for all .
Proof. Let
such that
. Similarly to Theorem 3, we can conclude that
where
. By assumption 4,
holds for all
.
Equivalently, there exists a subsequence
of
such that
and by definition of
-
-fuzzy contractive mapping, we deduce that
Since
F is strictly increasing function,
Taking limit
in above inequality, we find
.
Uniqueness follows from the above Theorem 1. □
Corollary 5. Let be a complete fuzzy metric space. Let be a self mapping satisfying the following assertions:
- 1.
T is α-admissible;
- 2.
if, for with and , we havewhere , and ; - 3.
there exists such that ;
- 4.
if is a sequence in X such that with as , then either or holds for all .
Then, T has a fixed point Moreover, T has a unique fixed point, whenever , for all .
4. Application
As an application of the Corollary 5, we established here the existence theorem of solutions for a nonlinear fractional differential equation.
We study the problem considered in [
14] for the existence of solutions for the nonlinear fractional differential equation.
where
via the integral boundary conditions
where
denote the Caputo fractional derivative of order
and
is a continuous function. Here,
, where
, is the Banach space of continuous function from
into
R endow with the supremum norm
Let
be any complete fuzzy metric space. The triplet
is a fuzzy metric space, where the set
M is defined by
for all
and
. For a continuous function
, the Caputo derivative of fractional order
is defined as
, where
denote the integer part of the real number
.
Now, for continuous function
, the Reimann–Liouville fractional derivatives of order
is defined by
, the right hand side is point-wise defined on
.
Now, we give the following existence theorem.
Theorem 5. Suppose that
- 1.
there exists a function and such thatfor all and with ; - 2.
there exists such that for all , where the operater is defined by - 3.
for each and , implies ;
- 4.
If is a sequence in X such that in X and for all , then for all
Then, (12) has at least one solution. Proof. It is well-known that
is a solution of (
12) if and only if
is a solution of the integral equation
Then, problem (
12) is equivalent to find
, which is a fixed point of
T.
Now, let
such that
for all
By (i), we find
Thus, for each
with
for all
, we have
Now, consider the function
defined by
for each
such that
. The above inequality implies that
for all
with
Therefore,
T is an
-
-contractive mapping.
Next, by using assumption 3 of Theorem 5, implies , which implies , which implies for all . Hence, T is -admissible.
From assumption 2 of Theorem 5, there exists such that .
Finally, from assumption 4 of Theorem 5, if be a sequence in X such that for all implies for all , then for all implies for all . Therefore, condition 4 of Corollary 5 holds true.
With this as an application of our Corollary 5, we deduce that the existence of
such that
and
is a solution of the problem (
12). □