Abstract
In this paper, we define a multivalued contraction mapping which is a hybrid construction of rational Kannan type and Pata type inequalities and show that such mappings have fixed points in complete metric spaces. We utilize both Hausdorff and \(\delta\)-distances. The results are illustrated with examples. The methodology of the proof is the newly introduced one for Pata type. The work is in the domain of set valued analysis.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In this paper we investigate the fixed point property of a new class of multivalued mappings defined by putting together the ideas of contraction satisfying rational types inequalities and Pata type inequalities, respectively. The former class was first introduced in the work of Dass et al. [13] in which the inequality to be satisfied by the contraction contains rational terms. These contractions are known as rational contractions. Their considerations occupy a large area of fixed point theory. Some of the fixed point results of rational type contractions are in [1, 4, 6,7,8, 10, 11, 14, 15, 21, 22]. Pata type inequalities are recent introductions in fixed point theory. It was initiated in a paper of Pata [24] in which a generalization of the Banach contraction mapping principle was established. The inequality in this case is not a single one, instead it is a group of inequalities which is obtained by varying a parameter over a certain range. A new methodology was introduced in [24] to ensure the existence of the fixed points of such mappings. Several works have appeared by following the idea of Pata-type contraction. Some works along this line of research are noted in [5, 12, 16, 17]. The idea was extended to the domain of setvalued analysis through the work of Kolagar et al. [20] where a multivalued version of the result due to Pata was established in metric spaces.
In this paper, by combining a rational term containing Kannan type expressions with Pata type terms, we define a hybrid multivalued contraction and establish that such mappings have fixed points. It may be mentioned that the class of Kannan type mappings [18, 19, 25, 26] is considered to be an important class of contractive mappings with discontinuities and has been considered quite extensively in fixed point theory. Our function satisfies two types of inequalities of the above type with Hausdorff and \(\delta\)-distances, respectively, in two different situations. The main result has a corollary and is illustrated with examples. One of which shows that the corollary is properly contained in main theorem. We establish our theorem without assuming the continuity of the function.
2 Mathematical preliminaries
The following are the concepts from setvalued analysis which we use in this paper. Let \((X,\ d)\) be a metric space. Then
\(N(X) = \{ A : A\ \text {is a non-empty subset of}\ X\}\),
\(CB(X) = \{A : A\ \text {is a non-empty closed and bounded subset of}\ X\}\) and
\(C(X) = \{ A : A\ \text {is a non-empty compact subset of}\ X\}\).
For \(x\in X\) and \(A,\ B\in CB(X)\), the functions \(D(x,\ B)\), \(\delta (A,\ B)\) and \(H(A,\ B)\) are defined as follows:
and
The \(\delta\)-distance has all the properties of a metric except one. It has been used in works like [2, 3, 9]. H is known as the Hausdorff metric induced by the metric d on CB(X) [23]. Further, if \((X,\ d)\) is complete then \((CB(X),\ H)\) is also complete.
We use the following result in our subsequent discussion.
Lemma 2.1
Let\((X,\ d)\)be a metric space and\(B \in C(X)\). Then for every\(x\in X\)there exists\(y\in B\)such that\(d(x,\ y)= D(x,\ B)\).
Definition 2.1
Let \(T: X \longrightarrow CB(Y)\) be a multivalued mapping, where \((X,\ \rho )\), \((Y,\ d)\) are two metric spaces and H is the Hausdorff metric on CB(Y). The mapping T is said to be continuous at \(x\in X\) if for any sequence \(\{x_{n}\}\) in X, \(H(Tx_n,\ Tx)\longrightarrow 0\) whenever \(\rho (x_n,\ x)\longrightarrow 0\) as \(n \longrightarrow \infty\).
Definition 2.2
Let X be a non-empty set, \(f : X\longrightarrow X\) be a single valued mapping and \(T : X\longrightarrow N(X)\) be a multivalued mapping. A point \(x\in X\) is a fixed point of f (resp. T ) if and only if \(x = fx\) (resp. \(x \in Tx\)).
We will require the following class of functions in our results. Let \(\Phi\) denote the family of all functions \(\varphi : [0,1] \longrightarrow [0, \infty )\) such that \(\varphi\) is continuous at zero with \(\varphi (0)= 0.\)
3 Main results
Theorem 3.1
Let\((X,\ d)\)be a complete metric space and\(T: X \longrightarrow C(X)\)be a multivalued mapping. Suppose there exist\(\Lambda \ge 0\), \(L \ge 0\), \(\eta > 0\), \(\alpha \ge 1\), \(\beta \in [0,\alpha ]\)and\(\varphi \in \Phi\)such that for every\(\varepsilon \in [0,1]\)and for all\(x,\ y \in X\),
whenever\(d(x,\ y)\ge \eta\)and
whenever\(d(x,\ y)< \eta\),
where
\(M(x,y)= \max \ \Big \{d(x,\ y), \dfrac{D(x,\ Tx)D(y,\ Ty)}{1+d(x,\ y)}\Big \}\),
\(N(x,y)=\min \ \{D(x,\ Tx),\ D(y,\ Ty),\ D(x,\ Ty),\ D(y,\ Tx)\}\) and
\(||x|| = d(x,z)\)and\(||Tx|| = D(z, Tx)\)for an arbitrary but fixed\(z \in X\). Also, suppose that there exists\(x_0\in X\)such that\(Tx_{0}\)is singleton. ThenThas a fixed point.
Proof
Suppose that \(x_{0}\in X\) be such that \(Tx_{0}\) is singleton. Let \(Tx_{0} = \{x_{1}\}\). Then clearly \(d(x_{0},\ x_{1})= D(x_{0},\ Tx_{0})\). By Lemma 2.1, there exists \(x_{2}\in Tx_{1}\) such that \(d(x_{1},\ x_{2})= D(x_{1},\ Tx_{1})\). Again, by Lemma 2.1, we can find \(x_{3}\in Tx_{2}\) such that \(d(x_{2},\ x_{3})= D(x_{2},\ Tx_{2})\). Continuing this process we construct a sequence \(\{x_{n}\}\) such that
Let
Since \(Tx_{0}\) is singleton and \(Tx_{0} = \{x_{1}\}\), we have
If \(d(x_{n},\ x_{n+1})\ge \eta\), applying the contraction condition (1) for \(0 <\varepsilon \le 1\), we have
If \(d(x_{n},\ x_{n+1})< \eta\), applying the contraction condition (2) for \(0<\varepsilon \le 1\), we have
Combining (6) and (7), we have
Now,
and
Suppose that \(d(x_{n+1},\ x_{n+2}) > d(x_{n},\ x_{n+1})\). Then \(d(x_{n+1},\ x_{n+2}) > 0\). From (8)–(10), we have
which implies that
that is,
Since \(\alpha \ge 1\), taking \(\varepsilon \longrightarrow 0\) in the above inequality and using the property of \(\varphi\), we have \(d(x_{n+1},\ x_{n+2})\le 0\), which contradicts the assumption that \(d(x_{n+1},\ x_{n+2}) > 0\). So we have
that is, \(\{d(x_{n},\ x_{n+1})\}\) is a decreasing sequence of nonnegative real numbers. So
and also there exists a nonnegative real number l such that
Now, \(c_{n}=d(x_{n},\ x_{0})\le d(x_{n},\ x_{n+1})+d(x_{n+1},\ x_{0})\le d(x_{n},\ x_{n+1})+d(x_{n+1},\ x_{1})\,+\,d(x_{1},\ x_{0})\).
For both the cases where \(d(x_{n},\ x_{0})\ge \eta\) and \(d(x_{n},\ x_{0})< \eta\), we have from (1), (2), (12) and the above inequality that
where
and
Therefore, we have
If possible, suppose that the sequence \(\{c_{n}\}\) is unbounded. Then there exists a subsequence \(\{c_{n_{k}}\}\) with \(c_{n_{k}}\longrightarrow \infty\) as \(k \longrightarrow \infty\). So there exists a natural number N such that
So, for all \(k \ge N\), we have
Now
Then, for all \(k \ge N\), we have from (17) and (18) that
that is,
Let \(a = \Lambda \ 2^{2\alpha }\ (1+2c_{1})^\alpha\) and \(b = (L+2)c_{1}\). So, we have
Choose \(\varepsilon = \varepsilon _{k} = \dfrac{1+b}{c_{n_{k}}}=\dfrac{1+(L+2)c_{1}}{c_{n_{k}}}\), where \(k \ge N\). Then by (16), we have \(0 < \varepsilon \le 1\). Now we have
which is a contradiction. Hence \(\{c_{n}\}\) is bounded. Therefore, there exists a real number \(P>0\) such that
Now from (14) and (19), we get
Taking limit \(n \longrightarrow \infty\) in the above inequality, using (13), we get
that is,
that is,
Since \(\alpha \ge 1\), taking \(\varepsilon \longrightarrow 0\) in the above inequality and using the property of \(\varphi\), we have \(l \le 0\), which is a contradiction unless \(l = 0\). So, we have
Now we prove that \(\{x_{n}\}\) is a Cauchy sequence. Suppose \(\{x_{n}\}\) is not a Cauchy sequence. Then there exists a \(\xi\) satisfying \(0<\xi <\eta\) for which we can find two sequences of positive integers \(\{m(k)\}\) and \(\{n(k)\}\) such that for all positive integers k, \(n(k)> m(k) > k\) and \(d(x_{m(k)},\ x_{n(k)})\ge \xi\). Assuming that n(k) is the smallest such positive integer, we get
Now,
Taking the limit as \(k\longrightarrow \infty\) in the above inequality and using (20), we have
Again,
and
Taking the limit as \(k\longrightarrow \infty\) in the above inequalities, using (20) and (21), we have
Again,
and
Taking the limit as \(k\longrightarrow \infty\) in the above inequalities, using (20) and (21), we have
Similarly, we have
Since \(\xi < \eta\), (21) implies that there exists a positive integer \(k_0\) such that \(d(x_{m(k)},\ x_{n(k)})<\eta\) for all \(k>k_0\). When \(k>k_0\), applying (2) for \(0 < \varepsilon \le 1\), we have
Now,
So, we have
Taking limit \(k\longrightarrow \infty\) in the above inequality, using (20) and (21), we have
Taking limit \(k\longrightarrow \infty\) in the above inequality, using (20), (23) and (24), we have
Taking limit as \(k\longrightarrow \infty\) in (25), using (22), (26) and (27), we have
that is,
that is,
Since \(\alpha \ge 1\), taking limit as \(\varepsilon \longrightarrow 0\) in the above inequality and using the property of \(\varphi\), we have \(\xi \le 0\), which is a contradiction. Therefore, \(\{x_{n}\}\) is a Cauchy sequence in X.
As X is a complete, there exists \(z \in X\) such that
As \(\eta > 0\), there exists a positive integer \(n_{0}\) such that \(d(x_{n},\ z) < \eta\) for all \(n\ge n_{0}\). When \(n\ge n_{0}\), applying (2) for \(0 < \varepsilon \le 1\), we have
Now
Taking limit as \(n\longrightarrow \infty\) in the above inequality and using (28), we get
Taking limit as \(n\longrightarrow \infty\) in the above inequality and using (28), we get
Letting \(n\longrightarrow \infty\) in (29), we obtain
Since \(\alpha \ge 1\), taking limit as \(\varepsilon \longrightarrow 0\) in the above inequality and using the property of \(\varphi\), we have \(D(z,\ Tz)\le 0\), which implies that \(D(z,\ Tz) = 0\). Now \(D(z,\ Tz) = 0\) implies \(z \in \overline{Tz}\), where \(\overline{Tz}\) is the closure of Tz. Since \(Tz \in C(X)\), \(\overline{Tz} = Tz\). So \(z \in Tz\), that is, z is a fixed point of T.
Corollary 3.1
Let\((X,\ d)\)be a complete metric space and\(T: X \longrightarrow C(X)\)be a multivalued mapping. Suppose there exist\(\Lambda \ge 0\), \(L \ge 0\), \(\alpha \ge 1\), \(\beta \in [0,\ \alpha ]\)and\(\varphi \in \Phi\)such that the inequality (2) is satisfied, that is, for every\(\varepsilon \in [0,\ 1]\)and for all\(x,\ y \in X\),
where\(M(x,\ y)\), \(N(x,\ y)\), ||x|| and ||Tx|| are same as defined in Theorem3.1. Also, suppose that there exists\(x_0\in X\) such that \(Tx_{0}\)is singleton. ThenThas a fixed point.
Proof
Since \(H(Tx,\ Ty)\le \delta (Tx,\ Ty)\) for all \(x,\ y\in X\), for any \(\eta > 0\) the inequality of the corollary can be trivially decomposed into (1) and (2). Then all the conditions of Theorem 3.1 are satisfied and hence by an application of Theorem 3.1 we conclude that T has a fixed point.
Example 3.1
Let \(X= [0,\ 1]\bigcup \{3,\ 6\}\) and \(`` d ''\) be the usual metric on X. Let \(T : X\longrightarrow C(X)\) be defined as follows:
Let \(\varphi :[0,\ 1] \longrightarrow [0,\ \infty )\) be defined as follows:
Let \(\eta = \frac{1}{1000}\), \(\Lambda = 100\), \(\alpha = 2\), \(\beta = 1\) and \(L\ge 0\) be arbitrary. Then all the conditions of the Theorem 3.1 are satisfied and 0 is a fixed point of T.
Example 3.2
Let \(X= [0,\ \frac{1}{4}]\bigcup \{\frac{1}{2},\ 1\}\bigcup \{n\in \mathbb {N} : n\ge 2\}\) and \(`` d ''\) be the usual metric on X. Let \(T : X\longrightarrow C(X)\) be defined as follows:
Let \(\varphi :[0,\ 1] \longrightarrow [0,\ \infty )\) be defined as follows:
Let \(\Lambda = 2000\), \(\alpha = 2\), \(\beta = 1\) and \(L\ge 0\) be arbitrary. Then all the conditions of the Corollary 3.1 are satisfied and \(\frac{1}{20}\) is a fixed point of T.
Remark
In Example 3.1, when \(x=6\), \(y=3\) and \(\varepsilon = \dfrac{1}{10^{6}}\), the inequality of the Corollary 3.1, that is, (2) is not satisfied. So the Example 3.1 is not applicable to Corollary 3.1 and hence Theorem 3.1 properly contains Corollary 3.1. Further, it is observed that for any \(\varepsilon > 0\), the example does not satisfy the inequality \(H(Tx,\ Ty)\le (1-\varepsilon ) \Big [M(x,\ y)+ LN(x,\ y)\Big ]\) or \(\delta (Tx,\ Ty)\le (1-\varepsilon ) \Big [M(x,\ y)+ LN(x,\ y)\Big ]\) for all \(x,\ y \in X\). This indicates the second term \(\Lambda \varepsilon ^\alpha \ \varphi (\varepsilon )\Big [1+||x||+||y||+||Tx||+||Ty||\Big ]^\beta\) is essential for the theorem which is the spirit of Pata type results. The theorem is proved without any continuity assumption on the function.
The following theorem is the special case of Theorem 3.1 when we treat \(T : X \longrightarrow X\) as a multivalued mapping in which case Tx can be treated as a singleton set for every \(x \in X\).
Theorem 3.2
Let\((X,\ d)\)be a complete metric space and\(T: X \longrightarrow X\). Suppose there exist\(\Lambda \ge 0\), \(L \ge 0\), \(\alpha \ge 1\), \(\beta \in [0,\ \alpha ]\)and\(\varphi \in \Phi\)such that for every\(\varepsilon \in [0,\ 1]\)and for all\(x,\ y \in X\),
where
\(M^*(x,\ y)= \max \ \Big \{d(x,\ y), \dfrac{d(x,\ Tx)\ d(y,\ Ty)}{1+d(x,\ y)}\Big \}\),
\(N^*(x,\ y)=\min \ \{d(x,\ Tx),\ d(y,\ Ty),\ d(x,\ Ty),\ d(y,\ Tx)\}\) and
\(||x|| = d(x,\ z)\)for an arbitrary but fixed\(z \in X\).
ThenThas a fixed point.
Proof
We define \(S: X \longrightarrow C(X)\) by \(Sx=\{Tx\}\). Then, for all \(x,\ y\in X\), we have
Then for any \(\eta > 0\), (30) can be expressed in the form of (1) and (2). Then all the conditions of Theorem 3.1 are satisfied and hence by an application of Theorem 3.1, there exists \(z \in X\) such that \(z \in Sz = \{Tz\}\), which implies that \(z= Tz\), that is, z is a fixed point of T.
References
Abbas, M., V.Ć. Rajić, T. Nazir, and S. Radenović. 2015. Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces. Afrika Matematika 26: 17–30.
Ahmed, M.A. 2003. Common fixed point theorems for weakly compatible mappings. Rocky Mountain Journal of Mathematics 33: 1189–1203.
Altun, I., and D. Turkoglu. 2008. Some fixed point theorems for weakly compatible multivalued mappings satisfying an implicit relation. Filomat 22: 13–21.
Arshad, M., E. Karapinar, and J. Ahmad. 2013. Some unique fixed point theorems for rational contractions in partially ordered metric spaces. Journal of Inequalities and Applications 2013: 248.
Balasubramanian, S. 2014. A Pata-type fixed point theorem. Mathematical Sciences 8: 65–69.
Cabrera, I., J. Harjani, and K. Sadarangani. 2013. A fixed point theorem for contractions of rational type in partially ordered metric spaces. Annali Dell’Universita’Di Ferrara 59: 251–258.
Chandok, S., and J.K. Kim. 2012. Fixed point theorem in ordered metric spaces for generalized contractions mappings satisfying rational type expressions. Journal of Nonlinear Functional Analysis 17: 301–306.
Chandok, S., B.S. Choudhury, and N. Metiya. 2015. Fixed point results in ordered metric spaces for rational type expressions with auxiliary functions. Journal of the Egyptian Mathematical Society 23 (1): 95–101.
Choudhury, B.S., N. Metiya, and P. Maity. 2013. Coincidence point results of multivalued weak C-contractions on metric spaces with a partial order. Journal of Nonlinear Sciences & Applications 6: 7–17.
Choudhury, B.S., N. Metiya, T. Som, and C. Bandyopadhyay. 2015. Multivalued fixed point results and stability of fixed point sets in metric spaces. Facta Universitatis, Series: Mathematics and Informatics 30: 501–512.
Choudhury, B.S., N. Metiya, and C. Bandyopadhyay. 2015. Fixed points of multivalued \(\alpha\)-admissible mappings and stability of fixed point sets in metric spaces. Rendiconti del Circolo Matematico di Palermo 64: 43–55.
Choudhury, B.S., N. Metiya, and S. Kundu. 2018. End point theorems of multivalued operators without continuity satisfying hybrid inequality under two different sets of conditions. Rendiconti del Circolo Matematico di Palermo Series. https://doi.org/10.1007/s12215-018-0344-z.
Dass, B.K., and S. Gupta. 1975. An extension of Banach contraction principle through rational expressions. Indian Journal of Pure and Applied Mathematics 6: 1455–1458.
Harjani, J., B. López, and K. Sadarangani. 2010. A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space. Abstract and Applied Analysis 2010: 190701. https://doi.org/10.1155/2010/190701.
Jaggi, D.S., and B.K. Das. 1980. An extension of Banach’s fixed point theorem through rational expression. Bulletin of the Calcutta Mathematical Society 72: 261–264.
Kadelburg, Z., and S. Radenović. 2015. A note on Pata-type cyclic contractions. Sarajevo Journal of Mathematics 11 (2): 235–245.
Kadelburg, Z., and S. Radenović. 2016. Fixed point theorems under Pata-type conditions in metric spaces. Journal of the Egyptian Mathematical Society 24: 77–82.
Kannan, R. 1968. Some results on fixed points. Bulletin of the Calcutta Mathematical Society 60: 71–76.
Kannan, R. 1969. Some results of fixed points-II. American Mathematical Monthly 76: 405–408.
Kolagar, S.M., M. Ramezani, and M. Eshaghi. 2016. Pata type fixed point theorems of multivalued operators type fixed point theorems of multivalued operators in ordered metric spaces with applications to hyperbolic differential inclusions. UPB Scientific Bulletin, Series A 78 (4): 21–34.
Kumam, P., F. Rouzkard, Md Imdad, and D. Gopal. 2013. Fixed point theorems on ordered metric spaces through a rational contraction. Abstract and Applied Analysis 2013: 206515. https://doi.org/10.1155/2013/206515.
Luong, N.V., and N.X. Thuan. 2011. Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces. Fixed Point Theory and Applications 46: 1–10.
Nadler Jr., S.B. 1969. Multivalued contraction mapping. Pacific Journal of Mathematics 30: 475–488.
Pata, V. 2011. A fixed point theorem in metric spaces. Journal of Fixed Point Theory and Applications 10: 299–305.
Reich, S. 1971. Kannan’s fixed point theorem. Bollettino dell’Unione Matematica Italiana 4: 1–11.
Reich, S. 1972. Fixed points of contractive functions. Bollettino dell’Unione Matematica Italiana 5: 26–42.
Acknowledgements
The work has been partially supported by the Science and Engineering Research Board, Government of India, under Research Project no. PDF/2016/000353. The support is gratefully acknowledged. The authors gratefully acknowledge the suggestions made by the learned referee.
Author information
Authors and Affiliations
Contributions
The authors contributed equally to this work. All authors have read and approved the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
All the authors declare that there is no conflict of interests.
Rights and permissions
About this article
Cite this article
Choudhury, B.S., Metiya, N., Bandyopadhyay, C. et al. Fixed points of multivalued mappings satisfying hybrid rational Pata-type inequalities. J Anal 27, 813–828 (2019). https://doi.org/10.1007/s41478-018-0131-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41478-018-0131-4