Abstract
In this paper, we obtain a Suzuki type unique common fixed point theorem using C-condition in partial metric spaces. In addition, we give an example which supports our main theorem.
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Introduction
The notion of a partial metric space was introduced by Matthews [12] as a part of the study of denotational semantics of data flow networks. In fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation and domain theory in computer science (see [6]).
Matthews [12] and Romaguera [16] and Altun et al. [2] proved some fixed point theorems in partial metric spaces for a single map. For more works on fixed, common fixed point theorems in partial metric spaces, we refer [1, 3–5, 7–11, 13–15, 17–19]).
The aim of this paper is to prove a Suzuki type unique common fixed point theorem for four maps using (C)-condition in partial metric spaces.
First, we give the following theorem of Suzuki [18].
Theorem 1.1
(See [18]) Let (X, d) be a complete metric space and let T be a mapping on X. Define a non-increasing function \(\theta : \left[ 0,1\right) \rightarrow (\frac{1}{2},1]\) by
Assume that there exists \(r\in \left[ 0,1\right),\) such that
for all \(x,y\in X.\) Then, there exists a unique fixed point z of T. Moreover, \(\lim _{n}T^{n}x=z\) for all \(x\in X.\)
Definition 1.2
(See [11]) A mapping T on a metric space (X, d) is called a non-expensive mapping if
Definition 1.3
(See [11]) A mapping T on a metric space (X, d) satisfies the C-condition if
First, we recall some basic definitions and lemmas which play crucial role in the theory of partial metric spaces.
Definition 1.4
(See [12]) A partial metric on a nonempty set X is a function \(p:X\times X \rightarrow R^+,\) such that for all \(x,y, z \in X\):
-
\((p_1)\) \(x = y \Leftrightarrow p(x, x) = p(x, y) = p(y, y)\),
-
\((p_2)\) \(p(x, x) \le p(x, y), p(y, y)\le p(x, y),\)
-
\((p_3)\) \(p(x, y) = p(y, x),\)
-
\((p_4)\) \(p(x, y) \le p(x, z) + p(z, y) - p(z, z).\)
The pair (X, p) is called a partial metric space (PMS).
If p is a partial metric on X, then the function \(p^s : X\times X \rightarrow \mathbb {R}^+\) given by
is a metric on X.
Example 1.5
(See [1, 9, 12]) Consider \(X=[0,\infty )\) with \(p(x,y)=\max \{x,y\}.\) Then, (X, p) is a partial metric space. It is clear that p is not a (usual) metric. Note that in this case, \(p^s(x,y)=\left| x-y\right|.\)
Example 1.6
(See [7]) Let \(X=\{[a,b]: a,b,\in \mathbb R, \ a\le b\}\) and define \(p([a,b],[c,d])=\max \{b,d\}-\min \{a,c\}.\) Then, (X, p) is a partial metric space.
We now state some basic topological notions (such as convergence, completeness, and continuity) on partial metric spaces (see [1, 2, 9, 10, 12].)
Definition 1.7
-
(i)
A sequence \(\{x_n \}\) in the PMS (X, p) converges to the limit x if and only if \(\displaystyle p(x,x)=\lim \nolimits _{n\rightarrow \infty } p(x,x_n).\)
-
(ii)
A sequence \(\{x_n \}\) in the PMS (X, p) is called a Cauchy sequence if \(\displaystyle \lim \nolimits _{n,m\rightarrow \infty } p(x_n,x_m)\) exists and is finite.
-
(iii)
A PMS (X, p) is called complete if every Cauchy sequence \(\{x_n \}\) in X converges with respect to \(\tau _p,\) to a point \(x\in X,\) such that \(\displaystyle p(x,x)=\lim \nolimits _{n,m\rightarrow \infty } p(x_n,x_m).\)
The following lemma is one of the basic results in PMS ([1, 2, 9, 10, 12]).
Lemma 1.8
-
(i)
A sequence \(\{x_n\}\) is a Cauchy sequence in the PMS (X, p) if and only if it is a Cauchy sequence in the metric space \((X,p^s).\)
-
(ii)
A PMS (X, p) is complete if and only if the metric space \((X,p^s)\) is complete. Moreover
$$\begin{aligned} \lim _{n\rightarrow \infty }p^s(x,x_n)=0 \Leftrightarrow p(x,x) =\lim _{n\rightarrow \infty } p(x,x_n)=\lim _{n,m\rightarrow \infty } p(x_n,x_m). \end{aligned}$$
Next, we give two simple lemmas which will be used in the proof of our main result. For the proofs, we refer to [1].
Lemma 1.9
Assume \(x_n\rightarrow z\) as \(n\rightarrow \infty\) in a PMS (X, p), such that \(p(z,z)=0.\) Then, \(\displaystyle \lim _{n\rightarrow \infty } p(x_n,y)=p(z,y)\) for every \(y \in X.\)
Lemma 1.10
Let (X, p) be a PMS. Then
-
(A)
If \(p(x,y)=0,\) then \(x=y.\)
-
(B)
If \(x\ne y,\) then \(p(x,y)>0.\)
Remark 1.11
If \(x=y,\) p(x, y) may not be 0.
Definition 1.12
A pair (T, g) is called weakly compatible pair if they commute at coincidence points.
Now, we prove our main result.
Main result
Theorem 2.1
Let (X, p) be a partial metric space and let \(S, T, f,g : X \rightarrow X\) be mappings satisfying
-
(2.1.1) \(\frac{1}{2} \min \{p(fx,Sx),p(gy,Ty)\} \le p(fx,gy)\) implies that \(\psi \left( {p\left( {Sx,Ty} \right) } \right) \le \alpha \left( {M\left( {x,y} \right) } \right) - \beta \left( {M\left( {x,y} \right) } \right) ,\) for all x, y in X, where \(\psi ,\alpha ,\beta :[0,\infty ) \rightarrow [0,\infty )\) are such that \(\psi\) is an altering distance function, \(\alpha\) is continuous, and \(\beta\) is lower semi continuous, \(\alpha (0)=\beta (0)=0\) and \(\psi (t)-\alpha (t)+\beta (t)>0,\) for all \(t>0\) and
$$\begin{aligned} \begin{array}{rl} M\left( {x,y} \right) &{}= \max \left\{ {\begin{array}{l} {p(fx,gy),p(fx,Sx),p(gy,Ty),} \\ {\frac{1}{2}\left[ {p(fx,Ty) + p(gy, Sx)} \right] } \\ \end{array}} \right\} , \end{array} \end{aligned}$$ -
(2.1.2) \(S(X)\subseteq g(X), T(X)\subseteq f(X),\)
-
(2.1.3) either f(X) or g(X) is a complete subspace of X,
-
(2.1.4) the pairs (f, S) and (g, T) are weakly compatible.
Then, S, T, f and g have a unique common fixed point in X.
Proof
Let \(x_0 \in X\) be arbitrary point in X. From (2.1.2), there exist sequences of \(\{x_n\}\) and \(\{y_{n}\}\) in X, such that \(Sx_{2n} = gx_{2n+1} = y_{2n},\) \(Tx_{2n+1} = fx_{2n+2} = y_{2n+1}, \quad n = 0,1,2,\ldots.\)
Case (i): Assume that \(y_{n} \ne y_{n + 1}\) for all n.
Denote \(p_n = p(y_n, y_{n+1}).\)
We show that \(p_{n} \le p_{n-1}, \quad n = 1, 2, 3,\ldots\)
Now
From (2.1.1), we get
Hence, \(\psi (p_{2n}) \le \alpha (\max \left\{ p_{2n-1},p_{2n}\right\} ) - \beta (\max \left\{ p_{2n-1},p_{2n}\right\} ).\)
If \(p_{2n}\) is maximum, then we have \(\psi (p_{2n}) \le \alpha (p_{2n}) - \beta (p_{2n}),\) thus \(\psi (p_{2n}) - \alpha (p_{2n}) + \beta (p_{2n})\le 0\), which is a contradiction.
Hence \(p_{2n - 1}\) is maximum. Thus
\(< \psi (p_{2n-1}).\)
Since \(\psi\) is increasing, we have \(p_{2n} \le p_{2n-1}.\)
Similarly, we can show that \(p_{2n-1} \le p_{2n-2}.\)
Thus, \(p_{n} \le p_{n-1}, \quad n = 1, 2, 3,\ldots\)
Thus, \(\{p_n\}\) is a non-increasing sequence of non- negative real numbers and must converge to a real number, say, \(l \ge 0.\) Suppose \(l>0.\)
Letting \(n\rightarrow \infty\) in (2), we get \(\psi (l) \le \alpha (l) - \beta (l).\)
Thus, \(\psi (l) - \alpha (l) + \beta (l)\le 0,\) which is a contradiction. Hence, \(l = 0.\)
Thus
Hence, from \((p_2),\) we get
By definition of \(p^s,\) (3), and (4), we get
Now, we prove that \(\{y_{2n} \}\) is a Cauchy sequence in \((X,p^s).\) On contrary, suppose that \(\{y_{2n}\}\) is not Cauchy.
There exist \(\epsilon > 0\) and monotone increasing sequences of natural numbers \(\{2m_k\}\) and \(\{2n_k\},\) such that \(n_k > m_k,\)
and
Letting \(k \rightarrow \infty\) and then using (5), we get
Hence, from definition of \(p^s\) and (4), we have
Letting \(k \rightarrow \infty\) and then using (8) and (5) in \(|p^s(y_{2n_k+1}, y_{2m_k}) - p^s(y_{2n_k}, y_{2m_k})| \le p^s(y_{2n_k+1}, y_{2n_k})\) we obtain
Hence, we have
Letting \(k \rightarrow \infty\) and then using (8) and (5) in \(|p^s(y_{2n_k}, y_{2m_k-1}) - p^s(y_{2n_k}, y_{2m_k})| \le p^s(y_{2m_k-1}, y_{2m_k}),\) we get
Hence, we have
Letting \(k \rightarrow \infty\) and then using (12) and (5) in \(|p^s(y_{2m_k-1}, y_{2n_k+1}) - p^s(y_{2m_k-1}, y_{2n_k})| \le p^s(y_{2n_k+1}, y_{2n_k})\) we obtain
Hence, we get
If \(\frac{1}{2}\min \{p(y_{2m_k-1},y_{2m_k}),p(y_{2n_k},y_{2n_k+1})\} > p(y_{2m_k-1},y_{2n_k}),\) then letting \(k\rightarrow \infty,\) we get \(0 \ge \frac{\epsilon }{2}\) from (3) and (13).
It is a contradiction. Hence \(\frac{1}{2}\min \{p(y_{2m_k-1},y_{2m_k}),p(y_{2n_k},y_{2n_k+1})\} \le p(y_{2m_k-1},y_{2n_k}) = p(fx_{2m_k},gx_{2n_k+1}).\)
From (2.1.1), we have
Letting \(k \rightarrow \infty\) and then using (11), (13), (3), (15), and (9), we have
which is a contradiction. Hence, \(\{y_{2n}\}\) is Cauchy.
In addition, \(|p^s(y_{2n+1}, y_{2m+1}) - p^s(y_{2n}, y_{2m})| \le p^s(y_{2n+1}, y_{2n})+p^s(y_{2m}, y_{2m+1}).\)
Letting \(n ,m \rightarrow \infty,\) we have
Hence, \(\{y_{2n+1}\}\) is Cauchy. Thus \(\{y_n\}\) is a Cauchy sequence in \((X, p^s)\).
Hence, we have \(\mathop {\lim }\limits _{n ,~ m \rightarrow \infty }p^s(y_n, y_{m}) = 0.\)
Now, from the definition of \(p^s\) and from (4), we obtain
Therefore, \(\{y_n\}\) is Cauchy sequence in X.
Suppose g(X) is complete.
Since \(y_{2n} = Sx_{2n} = gx_{2n+1},\) it follows \(\left\{ {y_{2n } } \right\} \subseteq g(X)\) is a Cauchy sequence in the complete metric space \((g(X), p^s),\) it follows that \(\left\{ {y_{2n} } \right\}\) converges in \((g(X), p^s).\)
Thus, \(\mathop {\lim }\limits _{n \rightarrow \infty } p^s(y_{2n}, u) = 0\) for some \(u \in g(X).\)
That is, \(y_{2n} \rightarrow u = gt \in g(X)\) for some \(t \in X.\)
Since \(\left\{ y_n\right\}\) is Cauchy in X and \(\{y_{2n}\}\rightarrow u,\) it follows that \(\{y_{2n+1}\}\rightarrow u.\)
From Lemma (1.2.5), we get
Now, we claim that, for each \(n \ge 1,\) at least, one of the following assertions holds:
On the contrary, suppose that
for some \(n \ge 1\).
Then we have
which is a contradiction, and so, the claim holds.
Sub case(a) : Suppose \(\frac{1}{2}p(y_{2n-1},y_{2n})\le p(y_{2n-1},u).\)
Suppose \(Tt \ne u.\)
We have
From (2.1.1), we get
Letting \(n \rightarrow \infty\) and using (17), (18), we get
It is a contradiction. Hence, \(Tt = u = gt.\)
Since the pair (g, T) is weakly compatible, we have \(gu = Tu.\)
Suppose \(Tu \ne u.\)
Since \(\frac{1}{2} \min \{p(fx_{2n},Sx_{2n}),p(gu,Tu)\} \le p(fx_{2n},gu),\) from (2.1.1), we get
Letting \(n \rightarrow \infty ,\) we have
which is a contradiction.
Hence, \(Tu=u.\)
Therefore, \(u = Tu = gu.\)
Since \(T(X)\subseteq f(X),\) then there exists \(v \in X,\) such that \(Tu = fv = u.\)
Suppose \(Sv \ne fv.\)
Since \(\frac{1}{2}\min \{p(fv,Sv),p(gu,Tu)\} \le p(fv,gu),\) from (2.1.1), we have
Hence, \(Sv = fv = u.\)
Since the pair (f, S) is weakly compatible, we have \(fu = Su.\)
Suppose \(Su \ne u.\)
Since \(\frac{1}{2}\min \{p(fu,Su),p(gt,Tt)\} \le p(fu,gt),\) from (2.1.1), we have
Is a contradiction. Hence, \(u = Su = fu.\)
Thus, \(Tu = gu = Su =fu =u.\)
Hence, u is a common fixed point of S, T, f and g.
Let w be another common fixed point of S, T, f and g.
Since \(\frac{1}{2} \min \{p(fu,Su),p(gw,Tw)\} \le p(fu,gw),\) from (2.1.1), we obtain
which is a contradiction. Hence, \(u=w.\)
Thus, u is the unique common fixed point of S, T, f and g.
Sub case(b) : Suppose \(\frac{1}{2}p(y_{2n},y_{2n+1})\le p(y_{2n},u).\)
In this case, also, we can prove that u is the unique common fixed point of S, T, f and g by proceeding as in Subcase(a).
Case(ii): Suppose \(y_{2m} = y_{2m+1}\) for some m.
Assume that \(y_{2m+1} \ne y_{2m+2}.\)
However, \(\begin{array}{l} p(y_{2m + 1} ,y_{2m} )= p(y_{2m + 1} ,y_{2m+1} ) \le p(y_{2m + 1} ,y_{2m+2} ),~~ \text{ from } (p_2) \\ \end{array}\) and
Hence, \(M(x_{2m+2}, x_{2m+1}) = p(y_{2m+1} ,y_{2m + 2} ).\)
from (2.1.1), we get
It is a contradiction. Hence, \(y_{2m+2} = y_{2m + 1}.\)
Continuing in this way, we can conclude that \(y_{n} = y_{n + k}\) for all \(k > 0.\)
Thus, \(\{y_n\}\) is a Cauchy sequence.
The rest of the proof follows as in Case(i). \(\square\)
The following example illustrates our Theorem 2.1
Example 2.2
Let \(X = [0,1]\) and \(p(x, y) = \max \{x, y\}\) for all \(x,y \in X.\) Let \(f,g,S,T:X \rightarrow X,~f(x)= \frac{x}{2},~g(x)= \frac{x}{3},~~S(x)= \frac{x}{4}\) and \(T(x)= \frac{x}{6},\) Let \(\psi ,\alpha ,\beta : [0, \infty ) \rightarrow [0, \infty )\) be defined by \(\psi \left( t \right) ~= ~4t,\) \(\alpha \left( t \right) ~= ~7t\) and \(\beta \left( t \right) ~= ~\frac{7t}{2}.\) Clearly, \(\psi\) is an altering distance function and \(\alpha\) is continuous and \(\beta\) is lower semi continuous, \(\alpha (0) = \beta (0) = 0\) and \(\psi (t) -\alpha (t)+\beta (t) = \frac{t}{2} > 0,\) for all \(t >0.\)
Now
So
Therefore, all of the conditions of Theorem 2.1 are satisfied and 0 is the unique common fixed point of S, T, f and g.
References
Abdeljawad, T., Karapınar, E., Tas, K.: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett. 24(11), 1900–1904 (2011)
Altun, I., Sola, F., Simsek, H.: Generalized contractions on partial metric spaces. Topol. Appl. 157(18), 2778–2785 (2010)
Altun, I., Erduran, A.: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory and Applications Article ID 508730, p. 10 (2011). 10.1155/2011/508730
Aydi, H.: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 4(2), 01–12 (2011)
Fadail, Z.M., Ahmed, A.G.B., Ansar, A.H., Radenovic, S., Rajovic, M.: Some common fixed point results of mappings in 0-s-complete metric-like spaces via new functions. Appl. Math. Sci. 9(83), 5009–5027 (2015)
Heckmann, R.: Approximation of metric spaces by partial metric spaces. Appl. Categ. Struct. 7(1–2), 71–83 (1999)
Ilić, D., Pavlović, V., Rakočević, V.: Some new extensions of Banach’s contraction principle to partial metric spaces. Appl. Math. Lett. 24(8), 1326–1330 (2011)
Kadelburg, Z., Radenovic, S.: Fixed points under \(\alpha - \beta\) conditions in ordered partial metric spaces. Int. J. Anal. Appl. 5(1), 91–101 (2014)
Karapınar, E., Erhan, I.M.: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. 24(11), 1894–1899 (2011)
Karapınar, E.: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011, 4 (2011)
Karapinar, E., Erhan, I.M., Aksoy, U.: Weak \(\psi\)-contractions on partially ordered metric spaces and applications to boundary value problems. Bound. Value Probl. 2014, 149 (2014). doi:10.1186/s13661-014-0149-8
Matthews, S.G.: Partial metric topology. In: Proceedings of the 8th Summer Conference on General Topology and Applications 1994. vol. 728, pp. 183–197. Annals of the New York Academy of Sciences (1994)
Mustafa, Z., Huang, H., Radenovic, S.: Some remarks on the paper “Some fixed point generalizations are not real generalizations”. J. Adv. Math. Stud. 05/11/2015 (to appear)
Radenovic, S.: Classical fixed point results in 0-complete partial metric spaces via cyclic-type extension. Allahabad Math. Soc. 31(1), 39–55 (2016)
Rao, K.P.R., Kishore, G.N.V.: A unique common fixed point theorem for four maps under \(\psi - \phi\) contractive condition in partial metric spaces. Bull. Math. Anal. Appl. 3(3), 56–63 (2011)
Romaguera, S.: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. Article ID 493298, p. 6 (2010)
Shukla, S., Radenovic, S., Vetro, C.: Set-valued Hardy–Rogers type ontraction in 0-complete partial metric spaces. Int. J. Math. Math. Sci. Article ID 652925, p. 9 (2014)
Suzuki, T.: A generalized Banach contraction principle which characterizes metric completeness. Proc. Am. Math. Soc. 136, 1861–1869 (2008)
Valero, O.: On Banach fixed point theorems for partial metric spaces. Appl. Gen. Topol. 6(2), 229–240 (2005)
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Bindu, V.M.L.H., Kishore, G.N.V., Rao, K.P.R. et al. Suzuki type unique common fixed point theorem in partial metric spaces using (C)-condition. Math Sci 11, 39–45 (2017). https://doi.org/10.1007/s40096-016-0202-0
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DOI: https://doi.org/10.1007/s40096-016-0202-0