Fixed point theorems for Kannantype maps
 Jeong Sheok Ume^{1}Email author
https://doi.org/10.1186/s1366301502865
© Ume; licensee Springer. 2015
Received: 16 December 2014
Accepted: 23 February 2015
Published: 11 March 2015
Abstract
We introduce the new classes of Kannantype maps with respect to udistance and prove some fixed point theorems for these mappings. Then we present several examples to illustrate the main theorems.
Keywords
1 Introduction
Using the concept of Hausdorff metric, Nadler [4] proved the fixed point theorem for multivalued contraction maps, which is a generalization of the Banach contraction principle [2]. Since then various fixed point results concerning multivalued contractions have appeared; for example, see [5–7] and the references cited there.
Without using the concept of Hausdorff metric, most recently Dehaish and Latif [8] generalized fixed point theorems of Latif and Abdou [9], Suzuki [10], Suzuki and Takahashi [11].
In 1996, Kada et al. [12] introduced the notion of wdistance and improved several classical results including Caristi’s fixed point theorem. Suzuki and Takahashi [11] introduced singlevalued and multivalued weakly contractive maps with respect to wdistance and proved fixed point results for such maps. Generalizing the concept of wdistance, in 2001, Suzuki [10] introduced the notion of τdistance on a metric space and improved several classical results including the corresponding results of Suzuki and Takahashi [11]. In 2010, Ume [13] introduced the new concept of a distance called udistance, which generalizes wdistance, Tataru’s distance and τdistance. Then he proved a new minimization theorem and a new fixed point theorem by using udistance on a complete metric space.
Distances in uniform spaces were given by Vályi [14]. More general concepts of distances were given by Wlodarczyk and Plebaniak [15–18] and Wlodarczyk [19].
In this paper, we introduce the new classes of Kannantype multivalued maps without using the concept of Hausdorff metric and Kannantype singlevalued maps with respect to udistance and prove some fixed point theorems for these mappings. Then we present several examples to illustrate the main theorems.
2 Preliminaries
Throughout this paper we denote by N the set of all positive integers, by R the set of all real numbers and by \(R_{+}\) the set of all nonnegative real numbers.
 (\(u_{1}\)):

\(p(x, z) \leq p(x, y) + p(y, z)\).
 (\(u_{2}\)):

\(\theta(x, y, 0, 0) = 0\) and \(\theta(x, y, s, t) \geq\min\{s,t\}\) for all \(x,y\in X\) and \(s, t \in R_{+} \), for any \(x \in X\) and for every \(\varepsilon> 0\), there exists \(\delta> 0\) such that \( s  s_{0} < \delta\), \( t  t_{0} < \delta\), \(s, s_{0}, t, t_{0} \in R_{+}\) and \(y \in X\) imply \(\theta(x,y,s,t) \theta(x,y,s_{0} , t_{0} ) < \varepsilon\).
 (\(u_{3}\)):

\(\lim_{n \rightarrow\infty} x_{n} =x\) and \(\lim_{n \rightarrow \infty} \sup\{ \theta(w_{n} , z_{n} , p(w_{n} , x_{m}), p(z_{n} ,x_{m} )) : m\geq n \} = 0\) imply \(p(y,x) \leq\lim_{n \rightarrow\infty} \inf p(y, x_{n} )\) for all \(y\in X\).
 (\(u_{4}\)):

\(\lim_{n \rightarrow\infty} \sup\{p(x_{n} , w_{m}):m \geq n \} =0\), \(\lim_{n \rightarrow\infty} \sup\{p(y_{n} , z_{m}):m \geq n \} =0\), \(\lim_{n \rightarrow\infty} \theta(x_{n} , w_{n} , s_{n} ,t_{n} )=0\) and \(\lim_{n \rightarrow\infty} \theta(y_{n} , z_{n} , s_{n} ,t_{n} )=0\) imply \(\lim_{n \rightarrow\infty} \theta(w_{n} , z_{n} , s_{n} ,t_{n} )=0\) or \(\lim_{n \rightarrow\infty} \sup\{p(w_{m} , x_{n} ): m \geq n \} =0\), \(\lim_{n \rightarrow\infty} \sup\{p(z_{m} , y_{n}): m \geq n \} =0\), \(\lim_{n \rightarrow\infty} \theta(x_{n} , w_{n} , s_{n} ,t_{n} )=0\) and \(\lim_{n \rightarrow\infty} \theta(y_{n} , z_{n} , s_{n} ,t_{n} )=0\) imply \(\lim_{n \rightarrow\infty} \theta(w_{n} , z_{n} , s_{n} ,t_{n} )=0\).
 (\(u_{5}\)):

\(\lim_{n \rightarrow\infty} \theta(w_{n} , z_{n} , p(w_{n} , x_{n}), p(z_{n} ,x_{n} ))=0\) and \(\lim_{n \rightarrow\infty} \theta(w_{n} , z_{n} , p(w_{n} , y_{n}), p(z_{n} ,y_{n} ))= 0\) imply \(\lim_{n \rightarrow\infty} d(x_{n}, y_{n})=0\) or \(\lim_{n \rightarrow\infty} \theta(a_{n} , b_{n} , p(x_{n} , a_{n}), p(x_{n} ,b_{n} ))=0\) and \(\lim_{n \rightarrow\infty} \theta(a_{n} , b_{n} , p(y_{n} , a_{n}), p(y_{n} ,b_{n} ))=0\) imply \(\lim_{n \rightarrow\infty} d(x_{n},y_{n})=0\).
We recall remark, examples, definition and lemmas which will be useful in what follows.
Remark 2.1
([13])
(a) Suppose that θ from \(X \times X \times R_{+} \times R_{+} \) into \(R_{+} \) is a mapping satisfying \((u_{2}) {\sim}(u_{5})\). Then there exists a mapping η from \(X \times X \times R_{+} \times R_{+} \) into \(R_{+} \) such that η is nondecreasing in its third and fourth variable, satisfying \((u_{2})_{\eta}{\sim}(u_{5})_{\eta}\), where \((u_{2})_{\eta}{\sim}(u_{5})_{\eta}\) stand for substituting η for θ in \((u_{2}) {\sim}(u_{5})\), respectively.
(b) On account of (a), we may assume that θ is nondecreasing in its third and fourth variables, respectively, for a function θ from \(X \times X \times R_{+} \times R_{+} \) into \(R_{+} \) satisfying \((u_{2}) {\sim}(u_{5})\).
(c) Each τdistance p on a metric space \((X,d)\) is also a udistance on X.
We present some examples of udistance which are not τdistance (for details, see [13]).
Example 2.2
Let \(X=R_{+}\) with the usual metric. Define \(p: X \times X \rightarrow R_{+} \) by \(p(x,y) = (\frac{1}{4})x^{2}\). Then p is a udistance on X but not a τdistance on X.
Example 2.3
Let X be a normed space with \(\\cdot \\), then a function \(p:X \times X \rightarrow R_{+} \) defined by \(p(x,y)=\ x \\) for every \(x, y \in X \) is a udistance on X but not a τdistance.
It follows from the above examples and (c) of Remark 2.1 that udistance is a proper extension of τdistance.
Definition 2.4
([13])
Lemma 2.5
([13])
Let X be a metric space with a metric d and let p be a udistance on X. If \(\{x_{n}\}\) is a pCauchy sequence, then \(\{x_{n} \}\) is a Cauchy sequence.
Lemma 2.6
([13])
 (1)
If sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) of X satisfy \(\lim_{n \rightarrow\infty} p(z,x_{n}) =0 \) and \(\lim_{n \rightarrow \infty} p(z,y_{n}) =0\) for some \(z \in X\), then \(\lim_{n\rightarrow\infty} d(x_{n}, y_{n}) =0\).
 (2)
If \(p(z,x)=0\) and \(p(z,y)=0\), then \(x=y\).
 (3)
Suppose that sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) of X satisfy \(\lim_{n \rightarrow\infty} p(x_{n},z) =0 \) and \(\lim_{n \rightarrow\infty} p(y_{n} ,z) =0\) for some \(z \in X\), then \(\lim_{n\rightarrow\infty} d(x_{n}, y_{n}) =0\).
 (4)
If \(p(x,z)=0\) and \(p(y,z)=0\), then \(x=y\).
Lemma 2.7
([13])
3 Main result
The following lemma plays an important role in proving our theorems.
Lemma 3.1
Proof
Definition 3.2
Let \((X,d)\) be a metric space, \(2^{X}\) be a set of all nonempty subsets of X and \(\operatorname{Cl}(X)\) be a set of all nonempty closed subsets of X. Let \(T: X\rightarrow2^{X} \). Then an element \(z \in X\) is a fixed point of T if \(z \in Tz\).
 (i)
\(p(y_{1}, y_{2})\leq r[p(x_{1}, y_{1})+p(x_{2}, y_{2})]\) for any \(x_{1}, x_{2} \in X\), \(y_{1} \in Tx_{1}\) and \(y_{2} \in Tx_{2}\),
 (ii)
\(Ty \subseteq Tx\) for all \(x,y \in X\) with \(y \in Tx\).
In the next example we shall show that if \((X,d)\) is a complete metric space with a udistance p and a mapping \(T:X\rightarrow \operatorname{Cl}(X)\) is not Kannantype multivalued pcontractive, in general, T may have no fixed point in X.
Example 3.3
From (3.18) and (3.20) easily we can obtain that p is a udistance on X.
But there exist \(x=0 \in X\) and \(y=\frac{1}{4} \in X\) with \(y \in Tx\) such that \(Ty=T \frac{1}{4}= [ \frac{1}{40}, \frac{1}{20} ] \nsubseteq\{ \frac{1}{4} \} =T0\). Therefore T is not Kannantype multivalued pcontractive and T does not have a fixed point.
Using Lemma 3.1, we have the following main theorem.
Theorem 3.4
Let \((X,d)\) be a complete metric space and let \(T: X \rightarrow \operatorname{Cl}(X)\) be a Kannantype multivalued pcontractive mapping. Then T has a unique fixed point in X.
Proof
Now we shall know that \(\{a_{n}\}\) is a Cauchy sequence.
Now we give an example to support Theorem 3.4.
Example 3.5
Definition 3.6
 (iii)
\(p(Tx_{1},Tx_{2}) \leq r[p(x_{1},Tx_{1})+p(x_{2},Tx_{2})]\) for any \(x_{1}, x_{2} \in X\),
 (iv)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n+1}=Tx_{n}\) for each \(n \in N\) and \(\lim_{n \to\infty} x_{n} =c \in X\), then \(p(Tc,c) \leq r [p(Tc, Tc)+p(c, Tc) ]\) and \(p(c,Tc) \leq r [p(Tc, Tc)+p(Tc, c) ]\).
In the following example we show that if \((X,d)\) is a complete metric space and a mapping \(T:X \rightarrow X\) is not Kannantype singlevalued pcontractive, in general, T may have no fixed point in X.
Example 3.7
Theorem 3.8
Let \((X,d)\) be a metric space with a udistance p on X.
Let \(T: X \rightarrow X\) be a Kannantype singlevalued pcontractive mapping such that there exist a sequence \(\{x_{n}\}\) of X and \(c \in X\) satisfying \(x_{n+1}=Tx_{n}\) for each \(n \in N\) and \(\lim_{n \to\infty} x_{n} =c \in X\). Then c is a fixed point of T, i.e., \(Tc=c\).
Proof
Theorem 3.9
Let \((X,d)\) be a complete metric space and let \(T: X \to X \) be a Kannantype singlevalued pcontractive mapping.
Then T has a unique fixed point in X.
Proof
From Theorem 3.9, we have the following corollary.
Corollary 3.10
([1])
Then T has a unique fixed point in X.
Proof
Therefore T has a unique fixed point. □
Finally we shall present an example to show that all conditions of Theorem 3.9 are satisfied, but all conditions of Corollary 3.10 are not satisfied.
Example 3.11
Suppose that \(\{x_{n} \} \) is a sequence of X such that \(x_{n+1}=Tx_{n}\) for all \(x \in N\).
Declarations
Acknowledgements
The author would like to express his gratitude to the referees for giving valuable comments and suggestions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2013R1A1A2057665).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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