Abstract
In this paper, we establish some fixed point results for fuzzy mappings in a complete dislocated b-metric space. Our results generalize and extend the results of Joseph et al. (SpringerPlus 5:Article ID 217, 2016). We also give examples to support our results, and applications relating the results to a fixed point for multivalued mappings and fuzzy mappings are studied.
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1 Introduction and preliminaries
Fixed point theory plays an important role in various fields of mathematics. It provides very important tools for finding the existence and uniqueness of solutions. The Banach contraction theorem has an important role in fixed point theory, and it has become very popular due to iterations which can be easily implemented on the computers. The idea of a fuzzy set was first laid down by Zadeh [2]. Later on Weiss [3] and Butnariu [4] gave the idea of a fuzzy mapping and obtained many fixed point results. Afterward, Heilpern [5] initiated the idea of fuzzy contraction mappings and proved a fixed point theorem for fuzzy contraction mappings which is a fuzzy analogue of Nadler’s [6] fixed point theorem for multivalued mappings.
Recently, Beg et al. [7] proved the result concerning the existence of fixed points of a mapping satisfying locally contractive conditions on a closed ball (see also [8–16]). It is also possible that the mapping satisfies locally contractive conditions on a sequence contained in a closed ball in M. One can obtain fixed point results for such a mapping by using the suitable conditions.
The notion of dislocated topologies has useful applications in the context of logic programming semantics (see [17]). A dislocated metric space (metric-like space) (see [18, 19]) is a generalization of partial metric space (see [20]). Aydi et al. [21] established a fixed point theorem for set-valued quasi contraction in b-metric spaces. Nawab et al. [22] introduced the new concept of dislocated b-metric space as a generalization of metric space and established to prove some common fixed point results for four mappings satisfying the generalized weak contractive conditions in a partially ordered dislocated b-metric space.
In this paper, we obtain a fixed point and a common fixed point for fuzzy mappings for a generalized contraction on a closed ball in a complete b-metric space. An example which supports the proved results is also given. We give the following definitions and results which will be needed in the sequel.
Definition 1.1
([22])
Let X be a nonempty set. A function \(d_{lb}:X\times X\rightarrow[0,\infty )\) is called dislocated b-metric (or simply \(d_{lb}\)-metric) if, for any \(x,y,z\in X\) and \(b\geq1\), the following conditions hold:
-
(i)
If \(d_{lb}(x,y)=0\), then \(x=y\);
-
(ii)
\(d_{lb}(x,y)=d_{lb}(y,x)\);
-
(iii)
\(d_{lb}(x,y)\leq b[d_{lb}(x,z)+d_{lb}(z,y)]\).
The pair \((X,d_{lb})\) is called a dislocated b-metric space. It should be noted that the class of \(d_{lb}\) metric spaces is effectively larger than that of \(d_{l}\) metric spaces, since \(d_{lb}\) is a \(d_{l}\) metric when \(b=1\).
It is clear that if \(d_{lb}(x,y)=0\), then from (i), \(x=y\). But if \(x=y\), \(d_{lb}(x,y)\) may not be 0. For \(x\in X\) and \(\varepsilon>0\), \(\overline{B(x,\varepsilon)}=\{y\in X:d_{lb}(x,y)\leq\varepsilon\}\) is a closed ball in \((X,d_{lb})\).
Example 1.2
If \(X=\mathbb{R} ^{+}\cup\{0\}\), then \(d_{lb}(x,y)=(x+y)^{2}\) defines a dislocated b-metric \(d_{lb}\) on X.
Definition 1.3
([22])
Let \((X,d_{lb})\) be a dislocated b-metric space.
-
(i)
A sequence \(\{x_{n}\}\) in \((X,d_{lb})\) is called Cauchy sequence if, given \(\varepsilon>0\), there corresponds \(n_{0}\in N\) such that, for all \(n,m\geq n_{0}\), we have \(d_{lb}(x_{m},x_{n})<\varepsilon\) or \(\lim_{n,m\rightarrow\infty}d_{lb}(x_{n},x_{m})=0\).
-
(ii)
A sequence \(\{x_{n}\}\) dislocated b-converges (for short \(d_{lb}\)-converges) to x if \(\lim_{n\rightarrow\infty}d_{lb}(x_{n},x)=0\). In this case x is called a \(d_{lb}\)-limit of \(\{x_{n}\}\).
Definition 1.4
Let K be a nonempty subset of dislocated b-metric space X, and let \(x\in X\). An element \(y_{0}\in K\) is called a best approximation in K if
If each \(x\in X\) has at least one best approximation in K, then K is called a proximinal set. We denote by \(P(X)\) the set of all proximinal subsets of X.
Definition 1.5
The function \(H_{d_{lb}}:P(X)\times P(X)\rightarrow R^{+}\), defined by
is called dislocated Hausdorff b-metric on \(P(X)\).
A fuzzy set in X is a function with domain X and values in \([0,1]\), \(F(X) \) is the collection of all fuzzy sets in X. If A is a fuzzy set and \(x\in X\), then the function value \(A(x)\) is called the grade of membership of x in A. The α-level set of a fuzzy set A is denoted by \([A]_{\alpha}\) and defined as follows:
Let X be any nonempty set and Y be a metric space. A mapping T is called fuzzy mapping if T is a mapping from X into \(F(Y)\). A fuzzy mapping T is a fuzzy subset on \(X\times Y\) with membership function \(T(x)(y)\). The function \(T(x)(y)\) is the grade of membership of y in \(T(x)\). For convenience, we denote the α-level set of \(T(x)\) by \([Tx]_{\alpha}\) instead of \([T(x)]_{\alpha}\) [23].
Definition 1.6
([23])
A point \(x\in X\) is called a fuzzy fixed point of a fuzzy mapping \(T:X\rightarrow F(X)\) if there exists \(\alpha\in(0,1]\) such that \(x\in[ Tx]_{\alpha}\).
Lemma 1.7
Let A and B be nonempty proximal subsets of a dislocated b-metric space \((X,d_{lb})\). If \(a\in A\), then
Lemma 1.8
Let \((X,d_{lb})\) be a dislocated metric space. Let \((P(X),H_{d_{lb}})\) be a dislocated Hausdorff b-metric space. Then, for all \(A,B\in P(X)\) and for each \(a\in A\), there exists \(b_{a}\in B\) satisfying
then
2 Main results
Theorem 2.1
Let \((X,d_{lb})\) be a complete dislocated b-metric space with constant \(b\geq1\). Let \(T:X\rightarrow F(X)\) be a fuzzy mapping, and let \(x_{0}\) be any arbitrary point in X. Suppose there exists \(\alpha (x)\in(0,1]\) for all \(x\in X\) satisfying the following conditions:
and
for all \(x,y\in\overline{B_{d_{lb}}(x_{0},r)}\), \(r>0\) and \(b\mu<1\), where \(\mu=\frac{(a_{1}+ba_{3}+a_{5}+a_{6})}{1-(a_{2}+ba_{3})}\). Also, \(a_{i}\geq0 \), where \(i=1,2,\ldots,6\) with \(ba_{1}+a_{2}+b(1+b)a_{3}+b(a_{5}+a_{6})<1\) and \(\sum_{i=1}^{6} a_{i}<1\). Then there exists \(x^{\ast }\) in \(\overline{B_{d_{lb}}(x_{0},r)}\) such that \(x^{\ast}\in[ Tx^{\ast}]_{\alpha(x^{\ast})}\).
Proof
Let \(x_{0}\) be any arbitrary point in X such that \(x_{1}\in[ Tx_{0}]_{\alpha(x_{0})}\). Consider the sequence \(\{ x_{n}\}\) of points in X such that \(x_{n}\in[ Tx_{n-1}]_{\alpha (x_{n-1})}\). First we show that \(x_{n}\in\overline{B_{d_{lb}}(x_{0},r)}\) for all \(n\in \mathbb{N} \). Using (2.2), we get
which implies \(x_{1}\in\overline{B_{d_{lb}}(x_{0},r)}\). Let \(x_{2},x_{3},\ldots,x_{j}\in\overline{B_{d_{lb}}(x_{0},r)}\), \(j\in \mathbb{N} \). Now, by using Lemma 1.8, we get
Then we have
Continuing in this way and by using (2.3), we have
Now,
which implies \(x_{j+1}\in\overline{B_{d_{lb}}(x_{0},r)}\). Hence, by induction \(x_{n}\in\overline{B_{d_{lb}}(x_{0},r)}\) for all \(n\in N\). Now inequality (2.4) can be written as
Now, for any positive integers m, n (\(n>m\)), we have
Hence \(\{x_{n}\}\) is a Cauchy sequence in \(\overline{B_{d_{lb}}(x_{0},r)}\). As \(\overline{B_{d_{lb}}(x_{0},r)}\) is complete, there exists \(z\in X\) such that \(x_{n}\rightarrow z\) as \(n\rightarrow\infty\). Now, by Lemma 1.7 and (2.1), we get
Taking limit \(n\rightarrow\infty\), we get
So, we get
Hence, \(z\in X\) is a fixed point. □
Theorem 2.2
Let \((X,d_{lb})\) be a complete dislocated b-metric space with constant \(b\geq1\). Let \(S,T:X\rightarrow F(X)\) be two fuzzy mappings, and let \(x_{0}\) be any arbitrary point in X. Suppose there exist \(\alpha_{S}(x),\alpha_{T}(x)\in(0,1]\) for all \(x\in X\) satisfying the following conditions:
and
for all \(x,y\in\overline{B_{d_{lb}}(x_{0},r)}\), \(r>0\) and \(b\mu<1\), where \(\mu=\frac {a_{1}+a_{2}+ba_{3}+ba_{4}+2a_{5}}{2-(a_{1}+a_{2}+ba_{3}+ba_{4})}\). Also, \(a_{i}\geq0\), where \(i=1,2,\ldots,6\) with \((a_{1}+a_{2})(b+1)+b(a_{3}+a_{4})(b+1)+2ba_{5}<2\) and \(\sum_{i=1}^{5} a_{i}<1\). Then there exists \(x^{\ast}\) in \(\overline{B_{d_{lb}}(x_{0},r)}\) such that \(x^{\ast}\) is a common fixed point of S and T.
Proof
Let \(x_{0}\) be any arbitrary point in X such that \(x_{1}\in[ Tx_{0}]_{\alpha_{T}(x_{0})}\). Consider the sequence \(\{x_{n}\}\) of points in X such that \(x_{2i+1}\in[ Tx_{2i}]_{\alpha (x_{2i})}\), \(x_{2i+2}\in[ Sx_{2i+1}]_{\alpha(x_{2i+1})}\) for \(i=0,1,2,\ldots\) . First we show that \(x_{n}\in\overline {B_{d_{lb}}(x_{0},r)}\) for all \(n\in \mathbb{N} \). Using (2.7), we get
which implies \(x_{1}\in\overline{B_{d_{lb}}(x_{0},r)}\). Let \(x_{2},x_{3},\ldots,x_{j}\in\overline{B_{d_{lb}}(x_{0},r)}\), \(j\in \mathbb{N} \). If \(j=2i+1\), where \(i=0,1,2,\ldots,\frac{j-1}{2}\). Now, by using Lemma 1.8, we get
Now, we have
Similarly, by symmetry, we have
So, we have
Adding (2.8) and (2.9), we get
As
then, by (2.10), we have
Similarly, if \(j=2i+2\), where \(i=0,1,2,\ldots,\frac{j-2}{2}\), we have
Now, by (2.11)
Also, by (2.12)
By combining (2.13) and (2.14), we get
Now,
which implies \(x_{j+1}\in\overline{B_{d_{lb}}(x_{0},r)}\). Hence, by induction \(x_{n}\in\overline{B_{d_{lb}}(x_{0},r)}\) for all \(n\in N\). Now inequality (2.15) can be written as
Now, for any positive integers m, n (\(n>m\)), we have
Hence, \(\{x_{n}\}\) is a Cauchy sequence in \(\overline{B_{d_{lb}}(x_{0},r)}\). As \(\overline{B_{d_{lb}}(x_{0},r)}\) is complete, there exists \(z\in X\) such that \(x_{n}\rightarrow z\) as \(n\rightarrow\infty\). Now, by Lemma 1.7 and (2.1), we prove \(z\in X\) to be the common fixed point of S and T.
Taking limit \(n\rightarrow\infty\), we get
So, we get
This implies that \(z\in X\) is a fixed point of S. Similarly, we can prove that z is a fixed point of T. Hence, z is a common fixed point of S and T. □
Example 2.3
Let \(X=\mathbb{Q} ^{+}\cup\{0\}\) and \(d_{lb}(x,y)=(x+y)^{2}\), whenever \(x,y\in X\), then \((X,d_{lb})\) is a complete dislocated b-metric space with \(b>1\). Define a fuzzy mapping \(T:X\rightarrow F(X)\) by
For all \(x\in X\), there exists \(\alpha(x)=1\) such that
Consider \(x_{0}=1\) and \(r=4\), then \(\overline{B_{d_{lb}}(x_{0},r)}=[0,1]\). Let \(a_{1}=\frac{1}{10}\), \(a_{2}=\frac{1}{20}\), \(a_{3}=\frac {1}{30}\), \(a_{4}=\frac{1}{40}\), \(a_{5}=\frac{1}{50}\), \(a_{6}=\frac{1}{60}\).Then
and
where
Since all the conditions of Theorem 2.1 are satisfied, there exists \(0\in\overline{B_{d_{lb}}(x_{0},r)}\) which is the fixed point of T.
Example 2.4
Consider \(X= \{0,1,2 \}\). Let \(d_{lb} : X \times X \rightarrow [0,\infty)\) be the mapping defined by
It is clear that d is a complete dislocated b-metric space with the constant \(b = \frac{4}{3} \). Note that \(d(2,2) \neq0\), so d is not a b-metric and also d is not a metric. Consider \(x_{0}=1\) and \(r=1\), then \(\overline{B_{d_{lb}}(x_{0},r)}=0\). Define the fuzzy mapping \(S,T:X\rightarrow F(X)\) by
and
Define \(\alpha_{S}(x) =\alpha_{T}(x) = \alpha\), where \(\alpha\in (0,\frac{3}{4}]\). Now we have
and
For \(x,y\in X\), we get
Let \(a_{1}=a_{2}=\frac{1}{4}\), \(a_{3}=0\), \(a_{4}=\frac{1}{4}\), \(a_{5}=0\), we can see that \(b\mu= \frac{20}{21}<1\), where \(\mu=\frac {a_{1}+a_{2}+ba_{3}+ba_{4}+2a_{5}}{2-(a_{1}+a_{2}+ba_{3}+ba_{4})} = \frac{5}{7}\). Also, \(a_{i}\geq0\), where \(i=1,2,\ldots,6\) with \((a_{1}+a_{2})(b+1)+b(a_{3}+a_{4}) (b+1)+2ba_{5} = \frac{35}{18}<2\) and \(\sum_{i=1}^{5} a_{i}<1\). It easy to prove that condition (2.6) in Theorem 2.2 holds. Then there exists \(0 \in[Sx]_{\alpha_{S}(x)} \cap[Ty]_{\alpha_{T}(y)}\).
3 Application
In this section, we indicate that Theorem 2.1 and Theorem 2.2 can be utilized to derive a common fixed point for a multivalued mapping in a dislocated b-metric space.
Theorem 3.1
Let \((X,d_{lb})\) be a complete dislocated b-metric space with constant \(b\geq1\). Suppose that \(R : X \rightarrow P(X)\) are two multivalued mappings satisfying the following conditions:
and
for all \(x,y\in\overline{B_{d_{lb}}(x_{0},r)}\), \(r>0\) and \(b\mu<1\), where \(\mu=\frac{(a_{1}+ba_{3}+a_{5}+a_{6})}{1-(a_{2}+ba_{3})}\). Also, \(a_{i}\geq0 \), where \(i=1,2,\ldots,6\) with \(ba_{1}+a_{2}+b(1+b)a_{3}+b(a_{5}+a_{6})<1\) and \(\sum_{i=1}^{6} a_{i}<1\). Then there exists \(x^{\ast }\) in \(\overline{B_{d_{lb}}(x_{0},r)}\) such that \(x^{\ast}\in Rx^{\ast}\).
Proof
Let \(\alpha: X \rightarrow(0, 1]\) be an arbitrary mapping. Consider two fuzzy mappings \(T : X \rightarrow F(X)\) defined by
We obtain that
Hence, condition (3.1) becomes condition (2.1) in Theorem 2.1. This implies that there exists \(z \in X\) such that \(z \in[Tz]_{\alpha (z)} = Rz\). □
Theorem 3.2
Let \((X,d_{lb})\) be a complete dislocated b-metric space with constant \(b\geq1\). Suppose that \(R, G : X \rightarrow P(X)\) are two multivalued mappings satisfying the following conditions:
and
for all \(x,y\in\overline{B_{d_{lb}}(x_{0},r)}\), \(r>0\) and \(b\mu<1\), where \(\mu=\frac {a_{1}+a_{2}+ba_{3}+ba_{4}+2a_{5}}{2-(a_{1}+a_{2}+ba_{3}+ba_{4})}\). Also, \(a_{i}\geq0\), where \(i=1,2,\ldots,6\) with \((a_{1}+a_{2})(b+1)+b(a_{3}+a_{4})(b+1)+2ba_{5}<2\) and \(\sum_{i=1}^{5} a_{i}<1\). Then there exists \(x^{\ast}\) in \(\overline{B_{d_{lb}}(x_{0},r)}\) such that \(x^{\ast}\) is a common fixed point of R and G.
Proof
Let \(\alpha: X \rightarrow(0, 1]\) be an arbitrary mapping. Consider two fuzzy mappings \(S, T : X \rightarrow F(X)\) defined by
We obtain that
and
Hence, condition (3.3) becomes condition (2.6) of Theorem 2.2. This implies that there exists \(z \in[Sz]_{\alpha(z)} \cap [Tz]_{\alpha(z)} = Rz \cap Gz\). □
4 Conclusion
In the present work we have shown the new concept of fuzzy mappings in a complete dislocated b-metric space. We have also obtained fixed point and common fixed point results for fuzzy mappings in a complete dislocated b-metric space. Our results generalize and extend the concept of Joseph et al. [1] and references therein. We have also given examples to support our results, showing that d is a complete dislocated b-metric space but is not b-metric and metric space. Finally, we related the results to a fixed point for multivalued mappings and fuzzy mappings.
References
Joseph, JM, Roselin, DD, Marudai, M: Fixed point theorem on multi-valued mappings in b-metric spaces. SpringerPlus 5, Article ID 217 (2016)
Zadeh, LA: Fuzzy sets. Inf. Control 8(3), 338-353 (1965)
Weiss, MD: Fixed points and induced fuzzy topologies for fuzzy sets. J. Math. Anal. Appl. 50, 142-150 (1975)
Butnariu, D: Fixed point for fuzzy mapping. Fuzzy Sets Syst. 7, 191-207 (1982)
Heilpern, S: Fuzzy mappings and fixed point theorem. J. Math. Anal. Appl. 83(2), 566-569 (1981)
Nadler, BS: Multivalued contraction mappings. Pac. J. Math. 30, 475-488 (1969)
Beg, I, Arshad, M, Shoaib, A: Fixed point on a closed ball in ordered dislocated metric space. Fixed Point Theory 16(2), 195-206 (2015)
Arshad, M, Shoaib, A, Vetro, P: Common fixed points of a pair of Hardy Rogers type mappings on a closed ball in ordered dislocated metric spaces. J. Funct. Spaces 2013, Article ID 638181 (2013)
Arshad, M, Shoaib, A, Abbas, M, Azam, A: Fixed points of a pair of Kannan type mappings on a closed ball in ordered partial metric spaces. Miskolc Math. Notes 14(3), 769-784 (2013)
Arshad, M, Azam, A, Abbas, M, Shoaib, A: Fixed points results of dominated mappings on a closed ball in ordered partial metric spaces without continuity. UPB Sci. Bull., Ser. A 76(2), 123-134 (2014)
Hussain, N, Arshad, M, Shoaib, A, Fahimuddin: Common fixed point results for α-ψ-contractions on a metric space endowed with graph. J. Inequal. Appl. 2014, Article ID 136 (2014)
Shoaib, A, Arshad, M, Ahmad, J: Fixed point results of locally contractive mappings in ordered quasi-partial metric spaces. Sci. World J. 2013, Article ID 194897 (2013)
Shoaib, A, Arshad, M, Kutbi, MA: Common fixed points of a pair of Hardy Rogers type mappings on a closed ball in ordered partial metric spaces. J. Comput. Anal. Appl. 17, 255-264 (2014)
Shoaib, A: α-η dominated mappings and related common fixed point results in closed ball. J. Concr. Appl. Math. 13(1-2), 152-170 (2015)
Phiangsungnoen, S, Sintunavarat, W, Kumam, P: Common α-fuzzy fixed point theorems for fuzzy mappings via \(\beta_{\mathcal{F}}\)-admissible pair. J. Intell. Fuzzy Syst. 27(5), 2463-2472 (2014)
Phiangsungnoena, S, Kumam, P: Fuzzy fixed point theorems for multivalued fuzzy contractions in b-metric spaces. J. Nonlinear Sci. Appl. 8, 55-63 (2015)
Hitzler, P, Seda, AK: Dislocated topologies. J. Electr. Eng. 51(12), 3-7 (2000)
Karapınar, E, Piri, H, Alsulami, HH: Fixed points of modified F-contractive mappings in complete metric-like spaces. J. Funct. Spaces 2015, Article ID 270971 (2015)
Ren, Y, Li, J, Yu, Y: Common fixed point theorems for nonlinear contractive mappings in dislocated metric spaces. Abstr. Appl. Anal. 2013, Article ID 483059 (2013)
Matthews, SG: Partial metric topology. Ann. N.Y. Acad. Sci. 728, 183-197 (1994). Proc. 8th Summer Conference on General Topology and Applications
Aydi, H, Bota, M-F, Karapınar, E, Mitrović, S: A fixed point theorem for set-valued quasi-contractions in b-metric spaces. Fixed Point Theory Appl. 2012, Article ID 88 (2012)
Hussain, N, Roshan, JR, Paravench, V, Abbas, M: Common fixed point results for weak contractive mappings in ordered dislocated b-metric space with applications. J. Inequal. Appl. 2013, Article ID 486 (2013)
Azam, A: Fuzzy fixed points of fuzzy mappings via a rational inequality. Hacet. J. Math. Stat. 40(3), 421-431 (2011)
Acknowledgements
This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Cluster (CLASSIC), Faculty of Science, KMUTT. PK acknowledges the financial support provided by King Mongkut’s University of Technology Thonburi through the “KMUTT 55th Anniversary Commemorative Fund”. SP supported by Rajamangala University of Technology Rattanakosin Research and Development Institute, Rajamangala University of Technology Rattanakosin (RMUTR).
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Shoaib, A., Kumam, P., Shahzad, A. et al. Fixed point results for fuzzy mappings in a b-metric space. Fixed Point Theory Appl 2018, 2 (2018). https://doi.org/10.1186/s13663-017-0626-8
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DOI: https://doi.org/10.1186/s13663-017-0626-8