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On modified αϕfuzzy contractive mappings and an application to integral equations
Journal of Inequalities and Applications volume 2016, Article number: 67 (2016)
Abstract
We introduce the notion of a modified αϕfuzzy contractive mapping and prove some results in fuzzy metric spaces for such kind of mappings. The theorems presented provide a generalization of some interesting results in the literature. Two examples and an application to integral equations are given to illustrate the usability of our theory.
Introduction
A mathematical framework to work with ‘uncertainty’ or ‘vagueness’ was suggested by Zadeh in 1965; see [1]. In fact, the notion of a fuzzy set born as natural extension of the concept of set. Since then, this notion was used in mathematics and its applications (logic, topology, algebra, analysis, artificial intelligence, image processing and others). In particular many authors have expansively developed the theory of fuzzy metric spaces in different directions. Briefly, we say that this concept is strongly related to the concept of probabilistic metric space [2, 3]. Also, probabilistic metric spaces are generalizations of metric spaces in which the distances take values in the class of distribution functions. Fuzzy metric spaces were introduced by Kramosil and Michalek [4]; then George and Veeramani [5] modified the concept of fuzzy metric spaces in [4] and showed that every fuzzy metric induces Hausdorff topology.
On the other hand, fixed point theorems in fuzzy metric spaces have attracted much attention. Recently Gregori and Sapena [6] introduced a kind of contractive mapping for proving Banach’s contraction principle by using strong condition for completeness (Gcompleteness) in fuzzy metric spaces. Following this direction, in 2007, Mihet [7] proved fuzzy fixed point theorems for a Banach’s contraction in Mcomplete fuzzy metric spaces. As well as she introduced some new type of contractive conditions in the setting of fuzzy metric spaces (see [8, 9]). Very recently, motivated by Samet et al. [10], the concept of αϕfuzzy contractive mapping was discussed by Gopal and Vetro [11].
In view of above considerations, the fuzzy sets theory emerged as a potential area of interdisciplinary research and this is the motivation of our study. Then we introduce a new concept of modified αϕfuzzy contractive mapping and investigate the existence and uniqueness of fixed point of such kind of mappings. Finally, two examples and an application to integral equations are given to illustrate the usability of our theory.
Preliminaries
The following definitions and results will be needed in the sequel.
Definition 2.1
([12])
A binary operation \(*: [0, 1]\times[0, 1]\to[0, 1]\) is called a continuous triangular norm (tnorm) if the following conditions hold:

(i)
∗ is associative and commutative;

(ii)
∗ is continuous;

(iii)
\(a*1 = a\) for all \(a \in[0, 1]\);

(iv)
\(a*b \leq c*d\), whenever \(a \leq c\) and \(b \leq d\), for all \(a, b, c, d \in[0,1]\).
Four basic examples of continuous tnorms are: \(a*_{1} b = \min\{a, b\}\), \(a*_{2} b = \frac{ab}{\max\{a, b, \lambda\}}\) for \(\lambda\in(0, 1)\), \(a*_{3} b = ab\), \(a*_{4} b = \max\{a+b1, 0\}\).
Definition 2.2
([4])
A fuzzy metric space is a triple \((X, M, *)\), where X is a nonempty set, ∗ is a continuous tnorm and M is a fuzzy set on \(X\times X\times[0, +\infty)\) satisfying, for all \(x, y \in X\) and \(t>0\), the following properties:
 (K_{1}):

\(M(x, y, 0) = 0\);
 (K_{2}):

\(M(x, y, t) = 1\) iff \(x = y\);
 (K_{3}):

\(M(x, y, t) = M(y, x, t)\);
 (K_{4}):

\(M(x, y, \cdot) :[0, +\infty) \to[0, 1]\) is left continuous;
 (K_{5}):

\(M(x, z, t+s) \geq M(x, y, t) * M(y, z, s) \), for all \(z \in X\) and for all \(s > 0\).
George and Veeramani [5] modified the above definition and introduced a Hausdorff topology on the fuzzy metric space as follows.
Definition 2.3
([5])
A fuzzy metric space is a triple \((X, M, *)\), where X is a nonempty set, ∗ is a continuous tnorm and M is a fuzzy set on \(X\times X\times(0, +\infty)\), satisfying, for all \(x, y \in X\), the following properties:
 (G_{1}):

\(M(x, y, t) > 0\), for all \(t > 0\);
 (G_{2}):

\(M(x, y, t) = 1\) for all \(t > 0\) iff \(x = y\);
 (G_{3}):

\(M(x, y, t) = M(y, x, t)\) for all \(t > 0\);
 (G_{4}):

\(M(x, y, \cdot) :(0, +\infty) \to[0, 1]\) is continuous;
 (G_{5}):

\(M(x, z, t+s) \geq M(x, y, t) * M(y, z, s) \), for all \(z\in X\) and for all \(t, s > 0\).
For more properties and examples of fuzzy metric spaces, the reader can refer to [4, 13–21].
Definition 2.4
Let \((X, M, *)\) be a fuzzy metric space. Then:

(i)
A sequence \(\{x_{n}\}\) converges to \(x\in X\), that is, \(\lim_{n \to+\infty} x_{n} = x \), if \(\lim_{n\to+\infty} M(x_{n},x, t) = 1\) for all \(t > 0\).

(ii)
A sequence \(\{x_{n}\}\) is said to be MCauchy, if for each \(\epsilon\in(0, 1)\) and \(t > 0\) there exists \(n_{0} \in \mathbb{N}\) such that \(M(x_{n},x_{m}, t)> 1  \epsilon\) for all \(m, n \geq n_{0}\).

(iii)
A sequence \(\{x_{n}\}\) is said to be GCauchy, if \(\lim_{n \to+\infty}M(x_{n},x_{n+m}, t)= 1\) for each \(m \in\mathbb{N}\) and \(t > 0\).
Now, a fuzzy metric space \((X, M, *)\) is said to be Mcomplete (Gcomplete) if every MCauchy (GCauchy) sequence is convergent in X.
Definition 2.5
Let \((X, M, *)\) be a fuzzy metric space in the sense of George and Veeramani. The fuzzy metric M is said to be triangular if the following condition holds:
for all \(x, y, z \in X\) and for all \(t > 0\).
Some fixed point theorems
In this section, we introduce the new notion of a modified αϕfuzzy contractive mapping and αadmissible mapping with respect to η in fuzzy metric spaces.
Denote by Φ the family of all right continuous functions \(\phi: [0, +\infty) \to[0, +\infty)\), with \(\phi(c) < c\) for all \(c >0\).
Remark 3.1
Note that for every function \(\phi\in\Phi\), \(\lim_{n \to+\infty} \phi^{n}(c) = 0\), where \(\phi^{n}(c)\) denotes the nth iterate of ϕ.
According to [24] (see also [25]) we use the concept of αadmissible mapping in the following form.
Definition 3.1
Let \((X, M, *)\) be a fuzzy metric space in the sense of George and Veeramani and let \(\alpha, \eta: X \times X \times(0, +\infty) \to[0, +\infty)\) be two functions. We say that \(T:X \to X\) is αadmissible with respect to η if
Note that if we take \(\eta(x, y, t) = 1\), then this definition reduces to the definition of αadmissible mapping (Definition 3.4 of Gopal and Vetro [11]). Also, if we take \(\alpha(x, y, t) = 1\), then we say that T is an ηsubadmissible mapping.
In [11], Gopal and Vetro introduced the concept of an αϕfuzzy contractive mapping in fuzzy metric spaces as follows.
Definition 3.2
([11])
Let \((X, M, *)\) be a fuzzy metric space in the sense of George and Veeramani. We say that \(T: X\to X\) is an αϕfuzzy contractive mapping if there exist two functions \(\alpha: X \times X \times(0, +\infty) \to[0, +\infty)\) and \(\phi\in\Phi\) such that, for all \(x, y \in X\) and for all \(t > 0\), we have
Motivated by the above definition we give the following generalization.
Definition 3.3
Let \((X, M, *)\) be a fuzzy metric space in the sense of George and Veeramani. We say that \(T: X\to X\) is a modified αϕfuzzy contractive mapping if there exist three functions \(\alpha, \eta: X \times X \times(0, +\infty) \to[0, +\infty)\) and \(\phi\in\Phi\) such that, for all \(x, y \in X\) and for all \(t > 0\), we have
where \(N(x, y, t) = \min\{M(x, y, t), M(x, Tx, t), M(y, Ty, t) \}\).
Remark 3.2
If \(\eta(x, y,t) = 1\) and \(N(x, y, t) = M(x, y, t)\), then this definition reduces to Definition 3.2 in [11], thus it will imply the definition of the fuzzy contractive mapping given by Gregori and Sapena [6]. It follows that a fuzzy contractive mapping is a modified αϕfuzzy contractive mapping; in general the converse is not true.
Our first main result is the following theorem.
Theorem 3.1
Let \((X, M, *)\) be a Gcomplete fuzzy metric space in the sense of George and Veeramani. Let \(T: X\to X\) be a modified αϕfuzzy contractive mapping satisfying the following conditions:

(i)
T is αadmissible with respect to η;

(ii)
there exists \(x_{0} \in X\) such that \(\alpha(x_{0}, Tx_{0}, t)\geq\eta(x_{0}, Tx_{0}, t)\) for each \(t > 0\);

(iii)
T is continuous.
Then T has a fixed point, that is, there exists \(x^{*} \in X\) such that \(Tx^{*} = x^{*}\).
Proof
Let \(x_{0} \in X\) such that \(\alpha(x_{0}, Tx_{0}, t)\geq \eta(x_{0}, Tx_{0}, t)\) for all \(t > 0\). Define a sequence \(\{x_{n}\}\) in X by \(x_{n} = Tx_{n1} = T^{n}x_{0}\) for all \(n \in \mathbb{N}\). Clearly, if \(x_{n} = x_{n+1}\) for some \(n \in\mathbb{N}\) then \(x = x_{n}\) is a fixed point of T. Hence we suppose that \(x_{n+1}\neq x_{n}\) for all \(n \in\mathbb{N}\). Since T is an αadmissible mapping with respect to η and \(\alpha(x_{0}, Tx_{0}, t)\geq\eta(x_{0}, Tx_{0}, t)\), we deduce that
Continuing this process, we get \(\alpha(x_{n}, x_{n+1}, t) \geq\eta(x_{n}, x_{n+1}, t)\) for all \(n \in\mathbb{N}\). Now by (2) with \(x = x_{n1}\), \(y = x_{n}\), we get
where
It follows
Now, if \(\min\{M(x_{n1},x_{n}, t), M(x_{n}, x_{n+1}, t)\} = M(x_{n}, x_{n+1}, t)\) for some \(n \in\mathbb{N}\), then
which is a contradiction. Hence for all \(n\in\mathbb{N}\) we have
Consequently, \(M(x_{n}, x_{n+1}, t) > M(x_{n1}, x_{n}, t)\) for all \(n \in\mathbb{N}\), thus \(\{M(x_{n1}, x_{n}, t)\}\) is an increasing sequence of positive reals in \([0, 1]\).
Let \(s(t) = \lim_{n\to+\infty} M(x_{n1}, x_{n}, t)\); we show that \(s(t) = 1\) for all \(t > 0\). Indeed, we assume that there exists \(t_{0} > 0\) such that \(s(t_{0}) < 1\). From
by using the right continuity of the function ϕ, letting \(n \to +\infty\), we obtain the contradiction
This implies that \(\lim_{n\to+\infty} M(x_{n1}, x_{n}, t) = 1\) for all \(t > 0\). Then, for a fixed \(p \in\mathbb{N}\), we get
as \(n \to+\infty\) and hence the sequence \(\{x_{n}\}\) is GCauchy. Since \((X,M,\ast)\) is Gcomplete, then \(\{x_{n}\}\) converges to some \(x^{*} \in X\). Also, the continuity of T leads to \(Tx_{n} \to Tx^{*}\) and hence
for all \(t > 0\), that is, the sequence \(x_{n}\) converges to \(Tx^{*}\). By uniqueness of the limit, we deduce that \(x^{*} = Tx^{*}\); this completes the proof. □
In the second theorem, we replace the continuity hypothesis with another regularity hypothesis.
Theorem 3.2
Let \((X, M, *)\) be a Gcomplete fuzzy metric space in the sense of George and Veeramani with M triangular. Let \(T: X\to X\) be a modified αϕfuzzy contractive mapping satisfying the following conditions:

(i)
T is αadmissible with respect to η;

(ii)
there exists \(x_{0} \in X\) such that \(\alpha(x_{0}, Tx_{0}, t)\geq\eta(x_{0}, Tx_{0}, t)\) for each \(t > 0\);

(iii)
for any sequence \(\{x_{n}\}\) in X with \(\alpha(x_{n}, x_{n+1}, t) \geq\eta(x_{n}, x_{n+1}, t)\) for all \(n \in\mathbb{N}\), \(t >0\) and \(x_{n} \to x\) as \(n \to+\infty\), then \(\alpha(x_{n}, x, t) \geq\eta(x_{n}, x, t)\), for all \(n \in \mathbb{N}\) and \(t > 0\).
Then T has a fixed point, that is, there exists \(x^{*} \in X\) such that \(Tx^{*} = x^{*}\).
Proof
By the proof of Theorem 3.1, we see that \(\{x_{n}\}\) is a GCauchy sequence in the Gcomplete fuzzy metric space \((X, M, *)\) such that \(\alpha(x_{n}, x_{n+1},t) \geq\eta(x_{n}, x_{n+1},t)\) for all \(n \in \mathbb{N}\). Then there exists a point \(x^{*}\) in X such that \(x_{n} \to x^{*}\) as \(n \to+\infty\). Now, hypothesis (iii) of the theorem implies that
for all \(n \in\mathbb{N}\) and for all \(t >0\).
If \(x^{*} \neq Tx^{*}\), that is, \(M(x^{*},Tx^{*},t) <1\) for some \(t>0\), then, from (1), (3), and (2), respectively, since M is triangular we write
Letting \(n \to+\infty\), we get
and so, to avoid contradiction with \(\phi(c)< c\) for \(c>0\), we conclude that
It follows that \(Tx^{*}= x^{*}\); this completes the proof. □
Some of the following corollaries can be deduced from the above results. In particular, by taking \(\eta(x, y, t) = 1\) in Theorem 3.1 and Theorem 3.2 (with M triangular), we have the following corollary.
Corollary 3.1
Let \((X, M, *)\) be a Gcomplete fuzzy metric space in the sense of George and Veeramani (with M triangular). Let \(T: X\to X\) be an αadmissible mapping. Assume that there exists \(\phi\in\Phi\) such that, for \(x, y \in X\) and \(t>0\),
where \(N(x, y, t) = \min\{M(x, y, t), M(x, Tx, t), M(y, Ty, t) \}\). Also suppose that the following assertions hold:

(i)
there exists \(x_{0} \in X\) such that \(\alpha(x_{0}, Tx_{0}, t)\geq1\) for all \(t > 0\);

(ii)
either T is continuous or for any sequence \(\{x_{n}\}\) in X with \(\alpha(x_{n}, x_{n+1}, t) \geq1\) for all \(n \in\mathbb{N}\), \(t >0\), and \(x_{n} \to x\) as \(n \to+\infty\), then \(\alpha(x_{n}, x, t) \geq1\), for all \(n \in\mathbb{N}\) and \(t > 0\).
Then T has a fixed point.
Also, by taking \(\alpha(x, y, t) = 1\) in Theorem 3.1 and Theorem 3.2 (with M triangular), we have the following corollary.
Corollary 3.2
Let \((X, M, *)\) be a Gcomplete fuzzy metric space in the sense of George and Veeramani (with M triangular). Let \(T: X\to X\) be an ηsubadmissible mapping. Assume that there exists \(\phi\in\Phi\) such that, for \(x, y \in X\) and \(t>0\),
where \(N(x, y, t) = \min\{M(x, y, t), M(x, Tx, t), M(y, Ty, t) \}\). Also suppose that the following assertions hold:

(i)
there exists \(x_{0} \in X\) such that \(\eta(x_{0}, Tx_{0}, t)\leq1\) for all \(t > 0\);

(ii)
either T is continuous or for any sequence \(\{x_{n}\}\) in X with \(\alpha(x_{n}, x_{n+1}, t) \geq1\) for all \(n \in\mathbb{N}\), \(t >0\) and \(x_{n} \to x\) as \(n \to+\infty\), then \(\alpha(x_{n}, x, t) \geq1\), for all \(n \in\mathbb{N}\) and \(t > 0\).
Then T has a fixed point.
Now, we give two simple illustrative examples.
Example 3.1
Let \(X = [0, +\infty)\) endowed with the fuzzy metric \(M(x, y, t) = \frac {t}{t+xy}\) for all \(x,y \in X\) and \(t>0\), and the tnorm ∗_{3}. Clearly M is triangular. Define \(T: X \to X\) by
and \(\alpha,\eta: X\times X\times(0, +\infty) \to[0, +\infty)\) by
and \(\eta(x,y,t)= 1/3\) for all \(x,y \in X\) and \(t>0\). Then the existence of a fixed point for T follows by an application of Theorem 3.2 with \(\phi(c) = \frac{c}{2}\) for all \(c \geq0\).
Proof
By completeness of \((X, d)\), where \(d(x,y)=xy\) for all \(x,y \in X\), one deduces easily that \((X,M,\ast_{3})\) is a Gcomplete metric space with M triangular. We show that T is an αadmissible mapping with respect to η. Indeed, let \(x, y \in X\); if \(\alpha(x, y,t) \geq\eta(x, y,t)\) for all \(t>0\), then \(x, y \in[0,1]\). On the other hand, for all \(x,y \in[0,1]\), we have \(Tx,Ty \in[0,1]\), which implies that \(\alpha(Tx, Ty,t) \geq\eta (Tx, Ty,t)\) for all \(t>0\). Trivially, for every \(x_{0} \in[0,1]\) we get \(\alpha(x_{0},Tx_{0},t) \geq\eta(x_{0}, Tx_{0},t)\) for all \(t>0\). Now, if \(\{x_{n}\}\) is a sequence in X such that \(\alpha(x_{n}, x_{n+1},t) \geq\eta(x_{n}, x_{n+1},t)\) for all \(n \in\mathbb{N}\) and \(x_{n} \to x\) as \(n \to+ \infty\), then \(\{x_{n}\} \subset[0, 1]\) and hence \(x \in[0, 1]\). This implies that \(\alpha(x_{n}, x,t) \geq\eta(x_{n}, x,t)\) for all \(n \in\mathbb{N}\) and \(t>0\). Next, let \(\alpha(x,y,t) \geq\eta(x,y,t)\) for all \(t>0\), that is, \(x,y \in[0,1]\). Then the contractive condition in Theorem 3.2, that is,
is always true since \(x+y \leq2\). Thus, all the hypotheses of Theorem 3.2 are satisfied and T has at least a fixed point; here 0 and 2 are two fixed points of T.
On the other hand, T is not a fuzzy contractive mapping of Gregori and Sapena [6]. In fact, for \(x=1\) and \(y=2\), there does not exist \(k \in(0,1)\) such that
□
By a slight modification of Example 3.1, one can obtain, for instance, an example covered by Corollary 3.2.
Example 3.2
Let \(X= \{\frac{1}{n}, n \in\mathbb{N} \} \cup\{0,2\}\) endowed with the fuzzy metric \(M(x, y, t) = \frac{t}{t+xy}\) for all \(x,y \in X\) and \(t>0\), and the tnorm ∗_{3}. Define \(T: X \to X\) by
and \(\eta: X\times X\times(0, +\infty) \to[0, +\infty)\) by
Then the existence of a fixed point of T follows by an application of Corollary 3.2 with \(\phi(c) = \frac{c}{2}\) for all \(c \geq0\).
Proof
Clearly, T is an ηsubadmissible continuous mapping. Shortly, let \(x,y\in X\) such that \(\eta(x,y,t) \leq1\) for all \(t>0\), this implies that \(x,y \in X \setminus\{2\}\) and, by the definitions of T and η, we have \(Tx, Ty \in X \setminus\{2\} \) and \(\eta(Tx,Ty,t) =1 \) for all \(t>0\), that is, T is ηsubadmissible. Further, for every \(x_{0} \in X \setminus\{2\}\) we get \(\eta(x_{0},Tx_{0},t) = 1\) for all \(t>0\).
Next, let \(\eta(x,y,t) \leq1\) for all \(t>0\), that is, \(x,y \in X \setminus\{2\}\). By using the same argument as in Example 3.1, the contractive condition in Corollary 3.2 is always true since \(x+y \leq2\). Thus, all the hypotheses of Corollary 3.2 are satisfied; again 0 and 2 are two fixed points of T. □
Next, for establishing the uniqueness of fixed point of modified αϕfuzzy contractive mapping, we will consider the following hypothesis:

(H)
for all \(x, y \in X\) and for all \(t>0\) there exists \(z \in X\) such that \(\alpha(x, z, t) \geq\eta(x, z, t)\), \(\alpha(y, z, t) \geq\eta(y, z, t)\) and \(\lim_{n \to+\infty}M(T^{n1}z,T^{n}z,t)=1\).
Theorem 3.3
Adding hypothesis (H) in Theorem 3.1 (Theorem 3.2), one obtains that \(x^{*}\) is the unique fixed point of T provided that \(\phi\in\Phi \) is nondecreasing.
Proof
Assume that \(x^{*}\) and \(y^{*}\) are two fixed points of T. If \(\alpha (x^{*}, y^{*}, t) \geq\eta(x^{*}, y^{*}, t)\), then by condition (2), we get \(x^{*} = y^{*}\). Suppose \(\alpha (x^{*}, y^{*}, t) < \eta(x^{*}, y^{*}, t)\), then from hypothesis (H) there exists \(z \in X\) such that
Since T is αadmissible with respect to η, then we have
for all \(n \in\mathbb{N}\) and for all \(t>0\). Now we prove that \(M(x^{*},T^{n}z,t) \to1\), as \(n \to+ \infty\), for all \(t>0\). Reasoning by absurd, we suppose that there exists \(t>0\) such that \(\lim_{n \to+\infty}M(x^{*},T^{n}z,t)<1\). Then, from (2) and (5), we get
where
Let \(n_{0} \in\mathbb{N}\) such that \(M(x^{*},T^{n}z,t) \leq M(T^{n}z,T^{n+1}z,t)\) for all \(n \geq n_{0}\). Therefore, for all \(n > n_{0}\), we obtain
This implies that
Then, letting \(n \to+\infty\), we get \(T^{n}z \to x^{*}\). Similarly one can obtain \(T^{n}z \to y^{*}\), as \(n \to+\infty\). Consequently, we deduce that \(x^{*} = y^{*}\); the uniqueness is proved. □
The following classes of functions are used in our subsequent results. Let
and
where \(\psi(r) = \phi(r) = 0\) if and only if \(r = 0\).
Theorem 3.4
Let \((X, M, *)\) be a Gcomplete fuzzy metric space and let \(T:X \to X\) be an αadmissible mapping with respect to η. Assume that there exist \(\psi\in\Psi\) and \(\phi\in\Phi_{1}\) such that, for \(x, y \in X\) and \(t>0\),
Also suppose that the following conditions hold:

(i)
there exists \(x_{0} \in X\) such that \(\alpha(x_{0}, Tx_{0}, t)\geq\eta(x_{0}, Tx_{0}, t)\) for all \(t > 0\);

(ii)
either T is continuous or for any sequence \(\{x_{n}\}\) in X with \(\alpha(x_{n}, x_{n+1}, t) \geq\eta(x_{n}, x_{n+1}, t)\) for all \(n \in\mathbb{N}\), \(t >0\), and \(x_{n} \to x\) as \(n \to+\infty\), then \(\alpha(x_{n}, x, t) \geq\eta(x_{n}, x, t)\) and \(\alpha(x, Tx, t) \geq\eta(x, Tx, t)\), for all \(n \in\mathbb{N}\) and \(t > 0\).
Then T has a fixed point.
Proof
Let \(x_{0}\) in X such that \(\alpha(x_{0}, Tx_{0}, t)\geq \eta(x_{0}, Tx_{0}, t)\). Define a sequence \(\{x_{n}\}\) in X such that \(x_{n} = Tx_{n1} = T^{n}x_{0}\) for all \(n \in\mathbb{N}\). If \(x_{n+1} = x_{n}\) for some \(n \in X\), then \(x = x_{n}\) is a fixed point for T and the result is proved in this case. Now, we assume that \(x_{n} \neq x_{n+1}\) for all \(n \in \mathbb{N}\). Since T is αadmissible with respect to η and \(\alpha(x_{0}, Tx_{0}, t) \geq\eta(x_{0}, Tx_{0}, t)\), we deduce that
Continuing this process we get
for all \(n \in\mathbb{N}\). Clearly,
Now by (6) with \(x=x_{n1}\), \(y = x_{n}\), we have
which implies that
If \(M(x_{n1},x_{n},t)=1\), then \(M(x_{n},x_{n+1},t)=1\) too. Otherwise, if \(M(x_{n1},x_{n},t)<1\), then
and, since ψ is nondecreasing, we get \(M(x_{n}, x_{n+1}, t) \geq M(x_{n1}, x_{n}, t)\), for all \(n \in\mathbb{N}\). Thus, \(\{M(x_{n1}, x_{n}, t)\}\) is a nondecreasing sequence of positive reals in \([0, 1]\). Let \(s(t) = \lim_{n\to+\infty} M(x_{n1}, x_{n}, t)\). Now we show that \(s(t) = 1\) for all \(t > 0\). If not, then there exists \(t > 0\) such that \(s(t) <1\). Therefore, from the above inequality, on taking \(n \to+\infty\), we obtain
that is a contradiction and so we get \(\lim_{n\to+\infty} M(x_{n1}, x_{n}, t) = 1\), for all \(t > 0\). Next, for a fixed \(p \in\mathbb{N}\), we have
as \(n \to+\infty\) and thus \(\{x_{n}\}\) is a GCauchy sequence. Since \((X,M,\ast)\) is Gcomplete, therefore \(\{x_{n}\}\) converges to \(x^{*}\) for some \(x^{*} \in X\). Now, we distinguish two cases.
Case 1. T is continuous. Then this implies that \(Tx_{n} \to Tx^{*}\) and so
for all \(t > 0\), that is, \(x_{n} \to Tx^{*}\). By the uniqueness of the limit, we get \(x^{*} = Tx^{*}\), that is, \(x^{*}\) is a fixed point of T.
Case 2. For any sequence \(\{x_{n}\}\) in X with \(\alpha(x_{n}, x_{n+1}, t) \geq\eta(x_{n}, x_{n+1}, t)\) for all \(n \in\mathbb{N}\), \(t >0\), and \(x_{n} \to x^{*}\) as \(n \to+\infty\), then \(\alpha(x_{n}, x^{*}, t) \geq\eta(x_{n}, x^{*}, t)\) and \(\alpha (x^{*}, Tx^{*}, t) \geq\eta(x^{*}, Tx^{*}, t)\), for all \(n \in\mathbb{N}\) and \(t > 0\). In this case, we get easily
Now by (6), we have
If \(M(x_{n},x^{*},t)=1\), then \(M(x_{n+1},Tx^{*},t)=1\) too. Otherwise, if \(M(x_{n},x^{*},t)<1\), then
This implies that \(M(x_{n+1}, Tx^{*}, t) \geq M(x_{n}, x^{*}, t) \to1\) as \(n \to+ \infty\) for all \(t>0\) and so \(x^{*} = Tx^{*}\). □
By taking \(\eta(x, y, t) = 1\) in Theorem 3.4, we deduce the following corollary.
Corollary 3.3
Let \((X, M, *)\) be a Gcomplete fuzzy metric space and let \(T:X \to X\) be an αadmissible mapping. Assume that there exist \(\psi\in\Psi\) and \(\phi\in\Phi _{1}\) such that, for \(x, y \in X\) and \(t>0\),
Also suppose that the following conditions hold:

(i)
there exists \(x_{0} \in X\) such that \(\alpha(x_{0}, Tx_{0}, t)\geq1\) for all \(t > 0\);

(ii)
either T is continuous or for any sequence \(\{x_{n}\}\) in X with \(\alpha(x_{n}, x_{n+1}, t) \geq1\) for all \(n \in\mathbb{N}\), \(t >0\) and \(x_{n} \to x\) as \(n \to+\infty\), then \(\alpha(x_{n}, x, t) \geq1\) and \(\alpha(x, Tx, t) \geq1\), for all \(n \in\mathbb{N}\) and \(t > 0\).
Then T has a fixed point.
Also, by taking \(\alpha(x, y, t) = 1\) in Theorem 3.4 we deduce the following corollary.
Corollary 3.4
Let \((X, M, *)\) be a Gcomplete fuzzy metric space and let \(T: X \to X\) be an ηsubadmissible mapping. Assume that there exist \(\psi\in\Psi\) and \(\phi\in\Phi_{1}\) such that, for \(x, y \in X\) and \(t>0\),
Also suppose that the following conditions hold:

(i)
there exists \(x_{0} \in X\) such that \(\eta(x_{0}, Tx_{0}, t)\leq1\) for all \(t > 0\);

(ii)
either T is continuous or for any sequence \(\{x_{n}\}\) in X with \(\eta(x_{n}, x_{n+1}, t) \leq1\) for all \(n \in\mathbb{N}\), \(t >0\), and \(x_{n} \to x\) as \(n \to+\infty\), then \(\eta(x_{n}, x, t) \leq1\) and \(\eta (x, Tx, t) \leq1\), for all \(n \in\mathbb{N}\) and \(t > 0\).
Then T has a fixed point.
Application to integral equation
Inspired by [3, 26], we present an application of fixed point techniques to integral equations, by using our theory. Precisely, we discuss the existence of solution for the following general integral equation:
where \(I>0\).
Let \(C([0, I],\mathbb{R})\) be the Banach space of all continuous functions defined on \([0, I]\) endowed with the supremum norm
and the induced metric
First of all, according to [5], let \(a \ast b=ab\) for all \(a,b \in [0,1]\) (in short ∗_{3}), and consider the fuzzy metric
This \(M_{d}\) is called the standard fuzzy metric induced by d. The topologies induced by the standard fuzzy metric and the corresponding metric are the same; see [5]. In view of (8), the Banach space \(C([0,I],\mathbb{R})\) may be regarded as a fuzzy Banach space. Indeed the space \((C([0,I],\mathbb {R}),M_{d},\ast_{3})\) is the Gcomplete fuzzy metric space induced by the Banach space \(C([0,I],\mathbb{R})\); see [2, 4] for more details.
Thus, Banach spaces and fuzzy Banach spaces are strongly related to each other; see for instance [27]. Without going in details, there are examples of fuzzy Banach spaces which are not Banach spaces. Therefore, fuzzy Banach spaces cover a broad spectrum of categories than the corresponding Banach spaces. This is a sufficient motivation to develop an application in a fuzzy setting.
Now, we discuss the existence of a solution for the integral equation (7).
Theorem 4.1
Let \(T: C([0,I],\mathbb{R}) \to C([0,I],\mathbb{R})\) be the integral operator given by
where \(K \in C([0, I]\times[0,I] \times\mathbb{R},\mathbb{R})\) satisfies the following condition:

(i)
there exists \(f:[0, I]\times[0,I] \to[0,+\infty)\) such that, for all \(r \in[0,I]\), \(f(r,\cdot) \in L^{1}([0,I],\mathbb{R})\), and, for all \(x,y \in C([0,I],\mathbb{R})\) and for all \(r,s \in[0,I]\), we get
$$\begin{aligned}& \bigl\vert K\bigl(r,s,x(s)\bigr)K\bigl(r,s,y(s)\bigr)\bigr\vert \\& \quad \leq f(r,s) \max\bigl\{ \bigl\vert x(s)y(s)\bigr\vert ,\bigl\vert x(s)Tx(s)\bigr\vert ,\bigl\vert y(s)Ty(s)\bigr\vert \bigr\} , \end{aligned}$$where \(\int^{r}_{0} f(r,s)\, ds\) is bounded on \([0,I]\) and \(\sup_{r \in [0,I]} \int^{r}_{0} f(r,s)\, ds \leq\lambda<1\).
Then the integral equation (7) has a solution \(x^{*}\in C([0,I],\mathbb{R})\).
Proof
For all \(x,y \in C([0,I],\mathbb{R})\), we get
It follows that
In view of (8), for all \(x,y \in C([0,I],\mathbb{R})\) and \(t> 0\), we can write
and so
Consequently, after routine calculations, we obtain
This implies that (4) holds true with \(\phi(c)=\lambda c\) for all \(c\geq0\) and \(\alpha(x,y,t)=1\) for all \(x,y \in C([0,I],\mathbb {R})\) and \(t > 0\).
As mentioned above, since \(C([0,I],\mathbb{R})\) is complete, then the fuzzy metric space \((C([0,I], \mathbb{R}),M_{d}, \ast_{3})\) is Gcomplete with M triangular. The other hypotheses of Corollary 3.1 are immediately satisfied and hence we deduce that the operator T has a fixed point \(x^{*}\in C([0,I],\mathbb{R})\), which is a solution of the integral equation (7). □
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Acknowledgements
This research project was supported by the Theoretical and Computational Science (TaCS) Center.
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Mishra, U., Vetro, C. & Kumam, P. On modified αϕfuzzy contractive mappings and an application to integral equations. J Inequal Appl 2016, 67 (2016). https://doi.org/10.1186/s1366001610072
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MSC
 47H10
 45D05
Keywords
 αadmissible mapping with respect to η
 fixed point
 modified αϕfuzzy contractive mapping
 integral equations