- Research
- Open Access
- Published:
Common fixed point theorems for generalized \(( \phi, \psi )\)-weak contraction condition in complete metric spaces
Journal of Inequalities and Applications volume 2015, Article number: 139 (2015)
Abstract
The intent of this manuscript is to establish some common fixed point theorems in a complete metric space under weak contraction condition for two pairs of discontinuous weak compatible maps. The results proved herein are the generalization of some recent results in literature. We give an example to support our results.
1 Introduction
Let \((X, d)\) be a metric space. A function \(T : (X,d) \rightarrow (X,d) \) is said to satisfy the contraction condition if there exists a real number \(k \in(0,1)\) and for all \(x , y \in X\) such that
It is remarkable that a self-map T satisfying condition (1.1) implies that T is continuous (see [1]). One can see in the literature on metric fixed point theory that condition (1.1) has been extended and generalized by fixed point theorists in many ways for obtaining fixed points, common fixed points and, very recently, proximal points in different spaces.
The title of this paper implies that the pair of maps in a complete metric space satisfies certain inequality by generalizing the contraction condition of Banach, but the inequality itself does not force the mapping to be continuous. It was an open question for almost four decades after Banach’s fixed point theorem [1]. In 1968, Kannan [2] answered this question affirmatively by introducing the following inequality:
Rakotch [3] and later Boyd and Wond [4] generalized the inequality of Banach by introducing a control function as follows:
and
and it satisfies certain conditions. In a Hilbert space, Alber and Guerre-Delabriere [5] introduced a weak contraction condition, which was later extended by Rhoades [6] in a complete metric space, and they argued that the result of Alber and Guerre-Delabriere [5] still holds in a complete metric space.
A mapping \(T:X\rightarrow X\) satisfies the following condition:
for all \(x, y \in X\), and \(\varphi: [0,\infty) \rightarrow [0,\infty) \) is a continuous and nondecreasing function such that \(\varphi(t)=0\) if and only if \(t=0\).
Remark
If \(\varphi(t) = (1 - k)t\), where \(k\in(0, 1)\) in (1.5), then we shall have the condition (1.1) of Banach [1].
So, in view of (1.1), condition (1.5) is a weaker condition, and we call this condition a weak contraction condition.
Definition 1.1
(Altering distance function [7])
A function \(\psi: [0,\infty)\rightarrow[0,\infty)\) is called an altering distance function if it satisfies the following conditions:
-
(1)
ψ is monotone increasing and continuous,
-
(2)
\(\psi(t)= 0\) if and only if \(t= 0\).
Definition 1.2
([8])
A pair of self-mappings A and B of a metric space \((X,d)\) is said to be weakly compatible if they commute at their coincidence points. In other words, if \(Ax = Bx\) for some \(x\in X\), then \(ABx = BAx\).
The first and second author of this paper with Choudhury [9] proved a common fixed point theorem in Saks spaces. The main purpose of this note is to establish a few common fixed point theorems by generalizing the results of Murthy et al. [9] for two pairs of discontinuous functions in a complete metric space by using a weaker condition than condition (1.5) called \((\varphi, \psi)\)-weak contraction condition in metric spaces. For example, refer to Rhoades [6], Dutta and Choudhury [10], Zhang and Song [11], Doric [12], Hosseini [13], Abkar and Choudhury [14], Ahmad et al. [15], Hussain et al. [16], etc.
2 Fixed point theorems in a complete metric space
Theorem 2.1
Let \((X,d) \) be a complete metric space, and let A, B, S and \(T : X\rightarrow X \) be mappings satisfying
for all \(x , y \in X\), with \(x \neq y\) and
and
which is lower semi-continuous for all \(t > 0\) and ϕ is discontinuous at \(t = 0\) with \(\phi(0) = 0\),
Then A, B, S and T have a unique common fixed point in X.
Proof
Let \(x_{0} \in X\) be an arbitrary point. Since \(A(X)\subset T(X)\) and \(B(X)\subset S(X)\), then there exists a point \(x_{1} \in X\) such that \(Ax_{0} = Tx_{1}\), and for \(x_{1}\in X\), there exists a point \(x_{2}\in X\) such that \(Bx_{1}= Sx_{2}\). Inductively, we can construct a sequence
We assume, for all \(n \in N \cup\{0\}\),
At first, we shall show that \(d(y_{2n},y_{2n+1})\rightarrow0 \) as \(n\rightarrow\infty\) for all \(n \in N \cup\{0\}\), where N is a set of natural numbers.
For this, suppose that \(x = x_{2n}\), \(y = x_{2n+1}\) in (2.1), we have
where
and
Then, by the triangular inequality, we have
If
then we get
and (2.7) implies the following:
Using the monotonically increasing property of ψ function, we have
From (2.9) and (2.10), we have
Since
we have \(N(x_{2n},x_{2n+1})>0\), then from (2.7), (2.11) and the property of ϕ and ψ functions, we have
which is a contradiction. Thus we have
So, we obtain the following:
Now putting (2.14) and (2.15) in (2.7), we have
Therefore \(d(y_{2n} , y_{2n+1})\) is a monotonically decreasing sequence of nonnegative real numbers, then there exists a number \(r > 0 \) such that
By virtue of (2.6) and (2.12), we have \(N(x_{2n},x_{2n+1})>0\). Taking \(n\rightarrow\infty\) in (2.16) and using (2.17), we get
This implies that
We observe that the last term on the right-hand side of the above inequality is non-zero. We get a contradiction with ϕ function. Therefore, we have
Putting \(x= x_{2n+1}\) and \(y = x_{2n+2}\) in (2.1) and arguing as above, we obtain
Therefore, for all \(n \in N \cup\{0\}\), we have
Next, we prove that \(\{y_{n}\}\) is a Cauchy sequence. For this, it is enough to show that the subsequence \(\{y_{2n}\}\) is a Cauchy sequence. To the contrary, suppose that \(\{y_{2n}\}\) is not a Cauchy sequence, then there exist \(\epsilon> 0 \) and the sequence of natural numbers \(\{2n(k)\}\) and \(\{2m(k)\}\) such that \(2n(k) > 2m(k) >2k \) for \(k \in N \) and
corresponding to \({2m(k)}\). We can choose \({2n(k)}\) to be the smallest such that (2.19) is satisfied. Then we have
Putting \(x = x_{2m(k)-1}\) and \(y = x_{2n(k)-1}\) in (2.1), where for all \(k \in N\),
where
and
Using the triangle inequality, we have
Taking the limit \(k\rightarrow\infty\), we get
Again for all k, we have
On letting limit \(k\rightarrow\infty\) and using (2.18)-(2.22), we get
Again for all positive integers k, we have
Letting limit \(k\rightarrow\infty\) and using (2.18)-(2.23), we get
Again for all positive integers k, we have
Letting limit \(k\rightarrow\infty\) and using (2.18)-(2.24), we get
and
Letting \(k\rightarrow\infty\) in (2.21), we get
Using discontinuity of ϕ at \(t=0\) and \(\phi(t)>0\) for \(t>0\), we observe that the last term on the right-hand side of the above inequality is non-zero. Thus we arrive at a contradiction.
Hence \(\{y_{n} \}\) is a Cauchy sequence.
Therefore, the Cauchy sequence \(\{y_{n}\}\) is a convergent sequence, and it converges to a point z (say) in X.
Consequently, the subsequences also converge to z in X:
Now we shall prove that z is a common fixed point of A, B, S and T.
Since \(B(X)\subset S(X)\), there exists \(v \in X \) such that \(z=Sv\). Let \(d(z,Av) \neq0\). Putting \(x = v \) and \(y = x_{2n+1}\) in (2.1), we get
where
and
Taking \(n\rightarrow\infty\) and using \(z=Sv\), we have
Also we have
Using discontinuity of ϕ at \(t=0\) and \(\phi(t) > 0\) for \(t > 0\), we observe that the last term on the right-hand side of the above inequality is non-zero. Therefore we obtain
Hence we arrive at a contradiction with the ψ function.
Therefore \(d(z,Av)=0\Rightarrow Av = z \Rightarrow Av = z = Sv\).
Since \((A,S)\) is a weakly compatible pair of maps, so it commutes at their coincidence point v, i.e., \(ASv=SAv \Rightarrow Az= Sz\).
Now we shall show that \(Az= Sz= z\).
For this, putting \(x = z \) and \(y = x_{2n+1}\) in (2.1), we get
where
and
Taking \(n\rightarrow\infty\) and using \(Az=Sz\), we get
Now (2.27) implies that
Using discontinuity of ϕ at \(t=0\) and \(\phi(t)>0\) for \(t>0\), we observe that the last term on the right-hand side of the above inequality is non-zero. Therefore we obtain
which is contradiction. Therefore \(d(Sz,z)=0\Rightarrow Sz = z \Rightarrow Sz = Az=z\). Similarly, we can show that \(Tz = Bz=z \). Hence \(Sz = Az=Tz = Bz= z\).
Now we shall show that z is the unique common fixed point of A, B, S and T.
Let \(z_{1}\) be also a fixed point of A, B, S and T. Putting \(x = z\) and \(y = z_{1}\) in (2.1), we have
a contradiction. Hence \(d(z , z_{1})= 0 \Rightarrow z = z_{1} \). Hence A, B, S and T have a unique common fixed point in X. □
When we take \(S = T = I\) identity map, we get the following theorem.
Theorem 2.2
Let \((X,d) \) be a complete metric space. Let \(A, B : X\rightarrow X \) be two self-mappings which satisfy the following inequality:
where \(x , y \in X\), \(x \neq y\),
and
-
(1)
\(\phi: [0,\infty)\rightarrow[0,\infty)\) is such that \(\phi (t)>0\) which is lower semi-continuous for all \(t>0\), and ϕ is discontinuous at \(t=0\) with \(\phi(0)=0\),
-
(2)
\(\psi: [0,\infty)\rightarrow[0,\infty)\) is an altering distance function.
Then there exists a unique fixed point of A, B in X.
When we take \(\psi(t) = t\) in Theorem 2.1 and Theorem 2.2, we get the following corollaries.
Corollary 2.3
Let \((X,d) \) be a complete metric space. Let A, B, S and \(T : X\rightarrow X \) be self-mappings which satisfy the following inequality:
where \(x , y \in X\), \(x \neq y\) and
-
(1)
\(A(X)\subset T(X)\) and \(B(X)\subset S(X)\),
-
(2)
\((A,S) \) and \((B,T)\) are weak compatible pairs,
-
(3)
\(\phi: [0,\infty)\rightarrow[0,\infty)\) is such that \(\phi (t)>0\) which is lower semi-continuous for all \(t>0\), and ϕ is discontinuous at \(t=0\) with \(\phi(0)=0\).
Then A, B, S and T have a unique common fixed point in X.
Corollary 2.4
Let \((X,d) \) be a complete metric space. Let \(A, B : X\rightarrow X \) be two self-mappings which satisfy the following inequality:
where \(x , y \in X\), \(x \neq y\),
-
\(\phi: [0,\infty)\rightarrow[0,\infty)\) is such that \(\phi (t)>0\) which is lower semi-continuous for all \(t>0\), and ϕ is discontinuous at \(t=0\) with \(\phi(0)=0\).
Then there exists a unique fixed point of A, B in X.
Now, we are ready to establish the following theorem in which we are going to replace
with
Theorem 2.5
Let \((X,d) \) be a complete metric space. Let A, B, S and \(T: X\rightarrow X \) be self-mappings which satisfy the following inequality:
for all \(x , y \in X\) with \(x \neq y\) and
and
-
(1)
\(A(X)\subset T(X)\) and \(B(X)\subset S(X)\),
-
(2)
\((A,S) \) and \((B,T)\) are weak compatible pairs,
-
(3)
\(\phi: [0,\infty)\rightarrow[0,\infty)\) is a lower semi-continuous function with \(\phi(t)>0\) for all \(t\in(0,\infty)\) and \(\phi(0)=0\),
-
(4)
\(\psi: [0,\infty)\rightarrow[0,\infty)\) is an altering distance function which in addition is strictly monotone increasing.
Then A, B, S and T have a unique common fixed point in X.
One can easily prove this theorem by using the technique given in the proof of Theorem 2.1.
When we take \(S = T = I\) identity map, we get the following theorem.
Theorem 2.6
Let \((X,d) \) be a complete metric space. Let A and \(B : X\rightarrow X \) be self-mappings which satisfy the following inequality:
for all \(x , y \in X\) with \(x \neq y\),
and
-
(1)
\(\phi: [0,\infty)\rightarrow[0,\infty)\) is a lower semi-continuous function with \(\phi(t)>0\) for all \(t\in(0,\infty)\) and \(\phi(0)=0\),
-
(2)
\(\psi: [0,\infty)\rightarrow[0,\infty)\) is an altering distance function which in addition is strictly monotone increasing.
Then there exists a unique fixed point of A, B in X.
When we take \(\psi(t) = t\) in Theorem 2.5 and Theorem 2.6, we get the following corollaries.
Corollary 2.7
Let \((X,d) \) be a complete metric space. Let A, B, S and \(T: X\rightarrow X \) be self-mappings which satisfy the following inequality:
where \(x , y \in X\), \(x \neq y\) and
-
(1)
\(A(X)\subset T(X)\) and \(B(X)\subset S(X)\),
-
(2)
\((A,S) \) and \((B,T)\) are weak compatible pairs,
-
(3)
\(\phi: [0,\infty)\rightarrow[0,\infty)\) is a lower semi-continuous function with \(\phi(t)>0\) for all \(t\in(0,\infty)\) and \(\phi(0)=0\).
Then A, B, S and T have a unique common fixed point in X.
Corollary 2.8
Let \((X,d) \) be a complete metric space. Let \(A, B : X\rightarrow X \) be two self-mappings which satisfy the following inequality:
for all \(x , y \in X\) with \(x \neq y\),
and
-
\(\phi: [0,\infty)\rightarrow[0,\infty)\) is a lower semi-continuous function with \(\phi(t)>0\) for all \(t\in(0,\infty)\) and \(\phi(0)=0\).
Then there exists a unique fixed point of A, B in X.
Example 2.9
Let \(X = [0,2]\) be endowed with the Euclidean metric \(d(x,y) = |x-y|\), and let A, B, S and \(T \rightarrow X \) be defined by
where \(x,y \in X\). \(A(X) = [0 , \frac{7}{5}]\), \(S(X) = [0 , \frac{11}{5}]\), \(B(X) = [0 , \frac{9}{5}]\), \(T(X) = [0 , \frac{13}{5}]\). Here \(A(X)\subset T(X)\) and \(B(X)\subset S(X)\), and \((A,S)\) and \((B,T)\) are weakly compatible maps at \(x = 0\) but not compatible maps.
Take
Now we have to check the inequality of Theorem 2.1 for the following cases.
Case 1: If \(x = 0\) and \(y = 0\):
hence \(\psi( d(Ax , By )) = \psi(M(x,y))- \phi( N(x,y))\).
Case 2: If \(x = 0\) and \(y \neq0\):
and
Case 3: If \(x \neq0\) and \(y = 0\):
and
Case 4: If \(x \neq0\) and \(y \neq0\):
and
So the inequalities hold in each of the cases. Hence all the conditions of Theorem 2.1 hold and A, B, S and T have the unique common fixed point at \(x=0 \) in X.
References
Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133-181 (1922)
Kannan, R: Some results on fixed points. Bull. Calcutta Math. Soc. 6, 71-78 (1968)
Rakotch, A: A note on contraction mappings. Proc. Am. Math. Soc. 13, 459-462 (1962)
Boyd, DW, Wong, TSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458-464 (1969)
Alber, YI, Guerre-Delabriere, S: Principles of weakly contractive maps in Hilbert spaces. In: Gohberg, I, Lyubich, Y (eds.) New Results in Operator Theory and Its Applications, vol. 98, pp. 7-22. Birkhäuser, Basel (1997)
Rhoades, BE: Some theorems on weakly contractive maps. Nonlinear Anal. 47, 2683-2693 (2001)
Khan, MS, Swalesh, M, Sessa, S: Fixed points theorems by altering distances between the points. Bull. Aust. Math. Soc. 30, 1-9 (1984)
Jungck, G, Rhoades, BE: Fixed points for set valued functions without continuity. Indian J. Pure Appl. Math. 29, 227-238 (1998)
Murthy, PP, Tas, K, Choudhury, BS: Weak contraction mappings in Saks spaces. Fasc. Math. 48, 83-95 (2012)
Dutta, PN, Choudhury, BS: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl. 2008, Article ID 406368 (2008)
Zhang, Q, Song, Y: Fixed point theory for generalized ψ-weak contractions. Appl. Math. Lett. 22, 75-78 (2009)
Doric, D: Common fixed point for generalized \((\Psi, \varphi)\)-weak contractions. Appl. Math. Lett. 22, 1896-1900 (2009)
Hosseini, VR: Common fixed for generalized \((\varphi\mbox{-}\psi)\)-weak contractions mappings condition of integral type. Int. J. Math. Anal. 4(31), 1535-1543 (2010)
Abkar, A, Choudhury, BS: Fixed point result in partially ordered metric spaces using weak contractive inequalities. Facta Univ., Ser. Math. Inform. 27(1), 1-11 (2012)
Ahmad, J, Arshad, M, Vetro, P: Coupled coincidence point results for \((\Psi, \varphi)\)-contractive mappings in partially ordered metric spaces. Georgian Math. J. 21(2), 113-124 (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, 136 (2014)
Acknowledgements
The first author (PP Murthy) wishes to thank University Grants Commission, New Delhi, India for the MRP (File number 42-32/2013 (SR)).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Rights and permissions
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.
About this article
Cite this article
Murthy, P.P., Tas, K. & Patel, U.D. Common fixed point theorems for generalized \(( \phi, \psi )\)-weak contraction condition in complete metric spaces. J Inequal Appl 2015, 139 (2015). https://doi.org/10.1186/s13660-015-0647-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0647-y
MSC
- 47H10
- 54H25
Keywords
- fixed points
- \((\phi,\psi)\)-weak contraction condition
- altering distance function
- weak compatible maps
- complete metric spaces