Abstract
Existence and uniqueness of fixed points of a general class of (\psi ,\varphi ) contractive mappings on complete rectangular metric spaces are discussed. One of the theorems is a generalization of a fixed point theorem recently introduced by Lakzian and Samet. Fixed points of (\psi ,\varphi ) contractions under conditions involving rational expressions are also investigated. Several particular cases and applications as well as an illustrative example are given.
MSC:46T99, 47H10, 54H25.
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1 Introduction and preliminaries
Fixed point theory has been one of the most rapidly developing fields in analysis during the last few decades. Wide application potential of this theory has accelerated the research activities which resulted in an enormous increase in publications [1–5]. In a large class of studies the classical concept of a metric space has been generalized in different directions by partly changing the conditions of the metric. Among these generalizations, one can mention the partial metric spaces introduced by Matthews [6, 7] (see also [1, 2, 8–10]), and rectangular metric spaces defined by Branciari [11].
Branciari defined a rectangular metric space (RMS) by replacing the sum at the righthand side of the triangle inequality by a threeterm expression. He also proved an analog of the Banach Contraction Principle. The intriguing nature of these spaces has attracted attention, and fixed points theorems for various contractions on rectangular metric spaces have been established (see, e.g., [12–17]).
In 1969 Boyd and Wong [18] defined a class of contractive mappings called ϕ contractions. In 1997, Alber and GuerreDelabriere [19] generalized this concept by introducing weak ϕ contraction. A selfmapping T on a metric space (X,d) is said to be weak ϕ contractive if there exists a map \varphi :[0,+\mathrm{\infty})\u27f6[0,+\mathrm{\infty}) with \varphi (0)=0 and \varphi (t)>0 for all t>0 such that
for all x,y\in X. Contractions of this type have been studied by many authors (see, e.g., [20, 21]). The larger class of (\psi ,\varphi ) weakly contractive mappings has also been a subject of interest (see, e.g., [10, 22–25]).
In a recent paper, Lakzian and Samet [26] stated and proved a fixed point theorem for the (\psi ,\varphi ) weakly contractive mappings on complete rectangular metric spaces. They also provided interesting examples as particular cases of such mappings. In this paper, we generalize the result of Lakzian and Samet [26] and, in addition, investigate maps satisfying rational type contractive conditions. We also give applications and an example.
We state some basic definitions and notations to be used throughout this paper. Rectangular metric spaces are defined as follows.
Definition 1 ([11])
Let X be a nonempty set and let d:X\times X\u27f6[0,\mathrm{\infty}] satisfy the following conditions for all x,y\in X and all distinct u,v\in X each of which is different from x and y.
Then the map d is called a rectangular metric and the pair (X,d) is called a rectangular metric space.
The concepts of convergence, Cauchy sequence and completeness in a RMS are defined below.
Definition 2

(1)
A sequence \{{x}_{n}\} in a RMS (X,d) is RMS convergent to a limit x if and only if d({x}_{n},x)\to 0 as n\to \mathrm{\infty}.

(2)
A sequence \{{x}_{n}\} in a RMS (X,d) is RMS Cauchy if and only if for every \epsilon >0 there exists a positive integer N(\epsilon ) such that d({x}_{n},{x}_{m})<\epsilon for all n>m>N(\epsilon ).

(3)
A RMS (X,d) is called complete if every RMS Cauchy sequence in X is RMS convergent.
We also use the following modified notations of Lakzian and Samet [26].
Let Ψ denote the set of all continuous functions \psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) for which \psi (t)=0 if and only if t=0. Nondecreasing functions which belong to the class Ψ are also known as altering distance functions (see [27]).
In their paper, Lakzian and Samet [26] stated the following fixed point theorem.
Theorem 3 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X and \psi ,\varphi \in \mathrm{\Psi}, where ψ is nondecreasing. Then T has a unique fixed point in X.
In this paper, we give a generalization of Theorem 3 for a larger class of (\psi ,\varphi ) weakly contractive mappings and improve the results obtained by Lakzian and Samet. Moreover, Theorem 3 can be considered as a particular case of our generalized theorem.
2 Main results and applications
We present our main results in this section. First, we state the following fixed point theorem.
Theorem 4 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X and \psi ,\varphi \in \mathrm{\Psi}, where L>0, the function ψ is nondecreasing and
Then T has a unique fixed point in X.
Proof First, we prove the existence part. Let {x}_{0}\in X be an arbitrary point. Define the sequence \{{x}_{n}\}\subset X as
Assume that {x}_{n}\ne {x}_{n+1}=T{x}_{n} for all n\ge 1. Substitute x={x}_{n1} and y={x}_{n} in (2.1) and note that
Then we obtain,
where
If M({x}_{n1},{x}_{n})=d({x}_{n},{x}_{n+1}), then we have
which implies \varphi (d({x}_{n},{x}_{n+1}))=0, and hence d({x}_{n},{x}_{n+1})=0. Then {x}_{n}={x}_{n+1}=T{x}_{n}, which contradicts the initial assumption. Therefore, we must have M({x}_{n1},{x}_{n})=d({x}_{n1},{x}_{n}), that is,
Since ψ is nondecreasing, then d({x}_{n},{x}_{n+1})\le d({x}_{n1},{x}_{n}) for all n\ge 1, that is, the sequence \{d({x}_{n},{x}_{n+1})\} is decreasing and positive. Hence, it converges to a positive number, say s>0. Taking limit as n\to \mathrm{\infty} in (2.4), we obtain
which leads to \varphi (s)=0 and hence to s=0. Thus,
Next, we will show that T has a periodic point, that is, there exist a positive integer p and a point z\in X such that z={T}^{p}z. Assume the contrary, that is, T has no periodic point. Then, all elements of the sequence \{{x}_{n}\} are distinct, i.e., {x}_{n}\ne {x}_{m} for all n\ne m. Suppose also that \{{x}_{n}\} is not a RMS Cauchy sequence. Therefore, there exists \epsilon >0 for which one can find subsequences \{{x}_{n(i)}\} and \{{x}_{m(i)}\} of \{{x}_{n}\} with n(i)>m(i)>i such that
where n(i) is the smallest integer satisfying (2.6), that is,
We apply the rectangular inequality (RM3) and use (2.6) and (2.7) to obtain
Taking limit as i\to \mathrm{\infty} in (2.8) and using (2.5), we get
Employing the rectangular inequality (RM3) once again, we write the following inequalities:
from which we obtain
using (2.5) and (2.9); and therefore,
Now we substitute x={x}_{n(i)1} and y={x}_{m(i)1} in (2.1), which yields
where
Clearly, as i\to \mathrm{\infty} we have M({x}_{n(i)1},{x}_{m(i)1})\to max\{\epsilon ,0,0\}=\epsilon and m({x}_{n(i)1},{x}_{m(i)1})\to 0 due to (2.5) and (2.12). Then letting i\to \mathrm{\infty} in (2.13), we get
which implies \varphi (\epsilon )=0, and hence \epsilon =0. This contradicts the assumption that \{{x}_{n}\} is not RMS Cauchy, thus, \{{x}_{n}\} must be RMS Cauchy. Since (X,d) is complete, then \{{x}_{n}\} converges to a limit, say u\in X. Let x={x}_{n} and y=u in (2.1). This gives
with
Note that m({x}_{n},u)\to 0 as n\to \mathrm{\infty}. If M({x}_{n},u)=d({x}_{n},u) or M({x}_{n},u)=d({x}_{n},{x}_{n+1}), then we have M({x}_{n},u)\to 0 as n\to \mathrm{\infty}, due to (2.5) and the fact that the sequence \{{x}_{n}\} converges to u. Regarding the continuity of ψ and ϕ, we have
Hence, {lim}_{n\to \mathrm{\infty}}d(T{x}_{n},Tu)=0, that is, in either case, we end up with {x}_{n+1}=T{x}_{n}\to Tu. Since X is Hausdorff, we deduce that u=Tu. If, on the other hand, M({x}_{n},u)=d(u,Tu) passing to limit as n\to \mathrm{\infty} in (2.15), we get \varphi (M(u,Tu))=0, hence d(u,Tu)=0, that is u=Tu. This result contradicts the assumption that T has no periodic points. Therefore, T has a periodic point, that is, z={T}^{p}z for some z\in X and a positive integer p.
If p=1, then z=Tz, so z is a fixed point of T. Let p>1. We claim that the fixed point of T is {T}^{p1}z. Suppose the contrary, that is, {T}^{p1}z\ne T({T}^{p1}z). Then d({T}^{p1}z,{T}^{p}x)>0 and so is \varphi (d({T}^{p1}z,{T}^{p}x)). Letting x={T}^{p1}z and y={T}^{p}z in (2.1), we have
where
For M({T}^{p1}z,{T}^{p}z)=d({T}^{p}z,{T}^{p+1}z), (2.16) becomes
Thus, we get \varphi (d({T}^{p}z,{T}^{p+1}z))=0 and hence d({T}^{p}z,{T}^{p+1}z)=d(z,Tz)=0, which is not possible since p>1. If, on the other hand, M({T}^{p1}z,{T}^{p}z)=d({T}^{p1}z,{T}^{p}z), then (2.16) turns into
and taking into account the fact that ψ is nondecreasing, we deduce
Now we write x={T}^{p2}z and y={T}^{p1}z in (2.1) and get
where
For M({T}^{p2}z,{T}^{p1}z)=d({T}^{p1}z,{T}^{p}z), we obtain
which is possible only if \varphi (d({T}^{p1}z,{T}^{p}z))=0 and hence, d({T}^{p1}z,{T}^{p}z)=0. However, we assumed that d({T}^{p1}z,{T}^{p}z)>0. Thus, we must have M({T}^{p2}z,{T}^{p1}z)=d({T}^{p2}z,{T}^{p1}z), so that
which implies d({T}^{p2}z,{T}^{p1}z)\ge d({T}^{p1}z,{T}^{p}z) since ψ is nondecreasing. This leads to
We continue in this way and end up with the inequality
which yields d(z,Tz)<d(z,Tz). Therefore, the assumption d({T}^{p1}z,{T}^{p}x)>0 is wrong, that is, d({T}^{p1}z,{T}^{p}x)=0 and {T}^{p1}z is the fixed point of T.
Finally, to prove the uniqueness, we assume that T has two distinct fixed points, say z and w. Then letting x=z and y=w in (2.1), we have
where
Thus, we have
implying \varphi (d(z,w))=0, and hence d(z,w)=0, which completes the proof of the uniqueness. □
It is worth mentioning that the Theorem 3.1 given in [26] is a particular case of Theorem 4. We next give some consequences of Theorem 4.
Corollary 5 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X and \psi ,\varphi \in \mathrm{\Psi}, where the function ψ is nondecreasing and
Then T has a unique fixed point in X.
Proof Observe that
for some L>0. Then by Theorem 4, T has a unique fixed point in X. □
Corollary 6 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X and some 0\le k<1. Then T has a unique fixed point in X.
Proof Let \psi (t)=t and \varphi (t)=(1k)t. Then by Corollary 5, T has a unique fixed point. □
Corollary 7 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X and some 0\le k<\frac{1}{3} and L>0. Then T has a unique fixed point in X.
Proof Obviously,
Let \psi (t)=t and \varphi (t)=(13k)t. Then by Theorem 4, T has a unique fixed point. □
Our next corollary is concerned with weak ϕ contractions.
Corollary 8 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X, where
Then T has a unique fixed point in X.
Proof Let \psi (t)=t. Then by Theorem 4, T has a unique fixed point. □
As the second result, we state the following existence and uniqueness theorem under conditions involving rational expressions.
Theorem 9 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X and \psi ,\varphi \in \mathrm{\Psi}, where ψ is nondecreasing and
Then T has a unique fixed point in X.
Proof Let {x}_{0}\in X be an arbitrary point. Define the sequence \{{x}_{n}\}\subset X as
Assume that {x}_{n}\ne {x}_{n+1}=T{x}_{n} for all n\ge 1. Substituting x={x}_{n1} and y={x}_{n} in (2.27), we obtain
where
For the rest of the proof, one can follow the same steps as in the proof of Theorem 4. □
Setting \psi (t)=t and \varphi (t)=(1k)t, we obtain the following particular result.
Corollary 10 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X and some k\in [0,1). Then T has a unique fixed point in X.
We also generalize the applications of Theorem 3.1 in [26] given by Lakzian and Samet. Let Λ be the set of functions f:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) such that

(1)
f is Lebesgue integrable on each compact subset of [0,+\mathrm{\infty});

(2)
{\int}_{0}^{\epsilon}f(t)\phantom{\rule{0.2em}{0ex}}dt>0 for every \epsilon >0.
For this class of functions, we can state the following results.
Theorem 11 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X and f,g\in \mathrm{\Lambda}, where L>0 and
Then T has a unique fixed point in X.
Proof Let \psi (t)={\int}_{0}^{t}f(u)\phantom{\rule{0.2em}{0ex}}du and \varphi (t)={\int}_{0}^{t}g(u)\phantom{\rule{0.2em}{0ex}}du. Then ψ and ϕ are functions in Ψ, and moreover, the function ψ is nondecreasing. By Theorem 4, T has a unique fixed point. □
Corollary 12 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X and f\in \mathrm{\Lambda} and some 0\le k<1, L>0, where
Then T has a unique fixed point in X.
Proof Let g(t)=(1k)f(t). Then by Theorem 11, T has a unique fixed point. □
Theorem 13 Let (X,d) be a Hausdorff and complete RMS and let T:X\to X be a selfmap satisfying
for all x,y\in X and f,g\in \mathrm{\Lambda}, where
Then T has a unique fixed point in X.
Proof Let \psi (t)={\int}_{0}^{t}f(u)\phantom{\rule{0.2em}{0ex}}du and \varphi (t)={\int}_{0}^{t}g(u)\phantom{\rule{0.2em}{0ex}}du. Then ψ and ϕ are functions in Ψ, and moreover, the function ψ is nondecreasing. By Theorem 9, T has a unique fixed point. □
Finally, we give an illustrative example of (\psi ,\varphi ) contraction defined on a generalized metric space.
Example 14 Let X=A\cup B, where A=\{\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}\} and B=[1,2]. Define the generalized metric d on X as follows:
and
It is clear that d does not satisfy the triangle inequality on A. Indeed,
Notice that (RM3) holds, so d is a rectangular metric.
Let T:X\to X be defined as
Define \psi (t)=t and \varphi (t)=\frac{t}{5}. Then T satisfies the conditions of Theorem 4 and has a unique fixed point on X, i.e., x=\frac{1}{4}.
References
Abedelljawad T, Karapınar E, Taş K: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett. 2011, 24: 1900–1904. 10.1016/j.aml.2011.05.014
Aydi H, Karapınar E, Shatanawi W:Coupled fixed point results for (\psi ,\phi )weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl. 2011, 62: 4449–4460. 10.1016/j.camwa.2011.10.021
Radenović S, Rakočević V, Resapour S:Common fixed points for (g,f) type maps in cone metric spaces. Appl. Math. Comput. 2011, 218: 480–491. 10.1016/j.amc.2011.05.088
Berinde V: A common fixed point theorem for compatible quasi contractive self mappings in metric spaces. Appl. Math. Comput. 2009, 213: 348–354. 10.1016/j.amc.2009.03.027
Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 2011, 217: 5784–5789. 10.1016/j.amc.2010.12.060
Matthews, SG: Partial metric topology. Research report 212, Department of Computer Science, University of Warwick (1992)
Matthews SG: Partial metric topology. Annals of the New York Academi of Sciences 728. Proc. 8th Summer Conference on General Topology and Applications 1994, 183–197.
Karapınar E, Erhan IM: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. 2011, 24: 1894–1899. 10.1016/j.aml.2011.05.013
Karapınar E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 4. doi:10.1186/1687–1812–2011–4
Karapınar E, Erhan IM, Yıldız Ulus A: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci. 2012, 6: 239–244.
Branciari A: A fixed point theorem of BanachCaccioppoli type on a class of generalized metric spaces. Publ. Math. (Debr.) 2000, 57: 31–37.
Das P: A fixed point theorem on a class of generalized metric spaces. Korean J. Math. Sci. 2002, 9: 29–33.
Das P: A fixed point theorem in a generalized metric spaces. Soochow J. Math. 2007, 33: 33–39.
Das P, Lahiri BK: Fixed point of a Ljubomir Ćirić’s quasicontraction mapping in a generalized metric space. Publ. Math. (Debr.) 2002, 61: 589–594.
Das P, Lahiri BK: Fixed point of contractive mappings in generalized metric space. Math. Slovaca 2009, 59: 499–504. 10.2478/s1217500901432
Azam A, Arshad M: Kannan fixed point theorems on generalized metric spaces. J. Nonlinear Sci. Appl. 2008, 1: 45–48.
Azam A, Arshad M, Beg I: Banach contraction principle on cone rectangular metric spaces. Appl. Anal. Discrete Math. 2009, 3: 236–241. 10.2298/AADM0902236A
Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S00029939196902395599
Alber YI, GuerreDelabriere S: Principles of weakly contractive maps in Hilbert spaces. Advances and Appl. 98. In New Results in Operator Theory. Edited by: Gohberg I, Lyubich Y. Birkhäuser, Basel; 1997:7–22.
Karapınar E: Fixed point theory for cyclic weak ϕ contraction. Appl. Math. Lett. 2011, 24: 822–825. 10.1016/j.aml.2010.12.016
Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal. TMA 2001, 47: 2683–2693. Proceedings of the Third World Congress of Nonlinear Analysis, Part 4, Catania (2000) 10.1016/S0362546X(01)003881
Karapınar E, Sadarangani K:Fixed point theory for cyclic (\psi ,\varphi )contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 69. doi:10.1186/1687–1812–2011–69
Karapınar, E: Best proximity points of Kannan type cyclic weak phicontractions in ordered metric spaces. An. Univ. Ovidius Constanta (in press)
Karapınar E: Best proximity points of cyclic mappings. Appl. Math. Lett. 2012, 25: 1761–1766. 10.1016/j.aml.2012.02.008
Gordji M, Baghani H, Kim G:Common fixed point theorems for (\psi ,\varphi )weak nonlinear contraction in partially ordered sets. Fixed Point Theory Appl. 2012., 2012: Article ID 62. doi:10.1186/1687–1812–2012–62
Lakzian H, Samet B:Fixed points for (\psi ,\phi )weakly contractive mapping in generalized metric spaces. Appl. Math. Lett. 2012, 25: 902–906. 10.1016/j.aml.2011.10.047
Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 1984, 30: 1–9. 10.1017/S0004972700001659
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Tanja Sekulić is thankful to the Ministry of Science and Technological Development of Republic Serbia.
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Erhan, İ.M., Karapınar, E. & Sekulić, T. Fixed points of (\psi ,\varphi ) contractions on rectangular metric spaces. Fixed Point Theory Appl 2012, 138 (2012). https://doi.org/10.1186/168718122012138
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DOI: https://doi.org/10.1186/168718122012138