 Research
 Open Access
On modified αϕfuzzy contractive mappings and an application to integral equations
 Urmila Mishra^{1},
 Calogero Vetro^{2} and
 Poom Kumam^{3, 4}Email author
https://doi.org/10.1186/s1366001610072
© Mishra et al. 2016
 Received: 11 September 2015
 Accepted: 3 February 2016
 Published: 18 February 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.
Keywords
 αadmissible mapping with respect to η
 fixed point
 modified αϕfuzzy contractive mapping
 integral equations
MSC
 47H10
 45D05
1 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.
2 Preliminaries
The following definitions and results will be needed in the sequel.
Definition 2.1
([12])
 (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])
 (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])
 (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
 (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
3 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
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])
Motivated by the above definition we give the following generalization.
Definition 3.3
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
 (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.
Proof
In the second theorem, we replace the continuity hypothesis with another regularity hypothesis.
Theorem 3.2
 (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 \(t > 0\).
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
 (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 \(t > 0\).
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
 (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 \(t > 0\).
Now, we give two simple illustrative examples.
Example 3.1
Proof
By a slight modification of Example 3.1, one can obtain, for instance, an example covered by Corollary 3.2.
Example 3.2
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. □
 (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
Theorem 3.4
 (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 \(t > 0\).
Proof
By taking \(\eta(x, y, t) = 1\) in Theorem 3.4, we deduce the following corollary.
Corollary 3.3
 (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 \(t > 0\).
Also, by taking \(\alpha(x, y, t) = 1\) in Theorem 3.4 we deduce the following corollary.
Corollary 3.4
 (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 \(t > 0\).
4 Application to integral equation
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
 (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 getwhere \(\int^{r}_{0} f(r,s)\, ds\) is bounded on \([0,I]\) and \(\sup_{r \in [0,I]} \int^{r}_{0} f(r,s)\, ds \leq\lambda<1\).$$\begin{aligned}& \bigl\vert K\bigl(r,s,x(s)\bigr)K\bigl(r,s,y(s)\bigr)\bigr\vert \\& \quad \leq f(r,s) \max\bigl\{ \bigl\vert x(s)y(s)\bigr\vert ,\bigl\vert x(s)Tx(s)\bigr\vert ,\bigl\vert y(s)Ty(s)\bigr\vert \bigr\} , \end{aligned}$$
Proof
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). □
Declarations
Acknowledgements
This research project was supported by the Theoretical and Computational Science (TaCS) Center.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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