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Fixed point theory in partial metric spaces via φfixed point’s concept in metric spaces
Journal of Inequalities and Applications volume 2014, Article number: 426 (2014)
Abstract
Let X be a nonempty set. We say that an element x\in X is a φfixed point of T, where \phi :X\to [0,\mathrm{\infty}) and T:X\to X, if x is a fixed point of T and \phi (x)=0. In this paper, we establish some existence results of φfixed points for various classes of operators in the case, where X is endowed with a metric d. The obtained results are used to deduce some fixed point theorems in the case where X is endowed with a partial metric p.
MSC:54H25, 47H10.
1 Introduction and preliminaries
In 1994, Matthews [1] introduced the concept of partial metric spaces as a part of the study of denotational semantics of dataflow networks and showed that the Banach contraction principle can be generalized to the partial metric context for applications in program verification.
We start by recalling some basic definitions and properties of partial metric spaces (see [1, 2] for more details).
A partial metric on a nonempty set X is a function p:X\to X\to [0,\mathrm{\infty}) such that for all x,y,z\in X, we have

(P1) p(x,x)=p(y,y)=p(x,y)?x=y;

(P2) p(x,x)=p(x,y);

(P3) p(x,y)=p(y,x);

(P4) p(x,y)=p(x,z)+p(z,y)p(z,z).
A partial metric space is a pair (X,p) such that X is a nonempty set and p is a partial metric on X. It is clear that, if p(x,y)=0, then from (P1) and (P2), x=y; but if x=y, p(x,y) may not be 0. A basic example of a partial metric space is the pair ([0,\mathrm{\infty}),p), where p(x,y)=max\{x,y\}. Other examples of partial metric spaces which are interesting from a computational point of view may be found in [1].
Each partial metric p on X generates a {T}_{0} topology {\tau}_{p} on X which has as a base the family of open pballs \{{B}_{p}(x,\epsilon ):x\in X,\epsilon >0\}, where
Let (X,p) be a partial metric space. A sequence \{{x}_{n}\}\subset X converges to some x\in X with respect to p if and only if
A sequence \{{x}_{n}\}\subset X is said to be a Cauchy sequence if and only if {lim}_{m,n\to \mathrm{\infty}}p({x}_{n},{x}_{m}) exists and is finite. The partial metric space (X,p) is said to be complete if and only if every Cauchy sequence \{{x}_{n}\} in X converges to some x\in X such that {lim}_{n,m\to \mathrm{\infty}}p({x}_{n},{x}_{m})=p(x,x).
If p is a partial metric on X, then the function {d}_{p}:X\to X\to [0,\mathrm{\infty}) defined by
is a metric on X.
Lemma 1.1 Let (X,p) be a partial metric space. Then

(i)
\{{x}_{n}\} is a Cauchy sequence in (X,p) if and only if \{{x}_{n}\} is a Cauchy sequence in the metric space (X,{d}_{p});

(ii)
the partial metric space (X,p) is complete if and only if the metric space (X,{d}_{p}) is complete.
Recently, many works on fixed point theory in the partial metric context have been published. For more details, we refer to [2–22]. On the other hand, Haghi et al. [10] observed that some fixed point theorems for certain classes of operators can be deuced easily from the same theorems in metric spaces. The idea presented in [10] is interesting, however it cannot be applied for a large class of operators, as, for example, in the case of an implicit contraction.
In [23], Rus presented three interesting open problems in the context of a complete partial metric space (X,p).
Problem 1. If T:(X,p)\to (X,p) is an operator satisfying a certain contractive condition with respect to p, which condition satisfies T with respect to the metric {d}_{p} defined by (1)?
Problem 2. The problem is to give fixed point theorems for these new classes of operators on a metric space.
Problem 3. Use the results for the above problems to give fixed point theorems in a partial metric space.
In [18], Samet answered to the above problems by considering BoydWong contraction mappings. Other types of contractions were considered in [11, 19].
In this paper, we introduce the concept of a φfixed point, and we establish some φfixed point results for various classes of operators defined on a metric space (X,d). The obtained results are then used to obtain some fixed point theorems, in the case where X is endowed with a partial metric p.
2 φFixed point results
Let (X,d) be a metric space, \phi :X\to [0,\mathrm{\infty}) be a given function, and T:X\to X be an operator.
We denote by
the iterate operators of T. The set of all fixed points of the operator T will be denoted by
The set of all zeros of the function φ will be denoted by
We introduce the notion of φfixed point as follows.
Definition 2.1 An element z\in X is said to be a φfixed point of the operator T if and only if z\in {F}_{T}\cap {Z}_{\phi}.
Definition 2.2 We say that the operator T is a φPicard operator if and only if

(i)
{F}_{T}\cap {Z}_{\phi}=\{z\};

(ii)
{T}^{n}x\to z as n\to \mathrm{\infty}, for each x\in X.
Definition 2.3 We say that the operator T is a weakly φPicard operator if and only if

(i)
{F}_{T}\cap {Z}_{\phi}\ne \mathrm{\varnothing};

(ii)
the sequence \{{T}^{n}x\} converges for each x\in X, and the limit is a φfixed point of T.
We denote by ℱ the set of functions F:{[0,\mathrm{\infty})}^{3}\to [0,\mathrm{\infty}) satisfying the following conditions:

max\{a,b\}\le F(a,b,c), for all a,b,c\in [0,\mathrm{\infty});

F(0,0,0)=0;

F is continuous.
As examples, the following functions belong to ℱ:

(i)
F(a,b,c)=a+b+c;

(ii)
F(a,b,c)=max\{a,b\}+c;

(iii)
F(a,b,c)=a+{a}^{2}+b+c.
In this section, we study the existence and uniqueness of φfixed points for various classes of operators.
2.1 (F,\phi )Contraction mappings
Definition 2.4 Let (X,d) be a metric space, \phi :X\to [0,\mathrm{\infty}) be a given function, and F\in \mathcal{F}. We say that the operator T:X\to X is an (F,\phi )contraction with respect to the metric d if and only if
for some constant k\in (0,1).
Our first main result is the following.
Theorem 2.1 Let (X,d) be a complete metric space, \phi :X\to [0,\mathrm{\infty}) be a given function, and F\in \mathcal{F}. Suppose that the following conditions hold:
(H1) φ is lower semicontinuous;
(H2) T:X\to X is an (F,\phi )contraction with respect to the metric d.
Then

(i)
{F}_{T}\subseteq {Z}_{\phi};

(ii)
T is a φPicard operator;

(iii)
for all x\in X, for all n\in \mathbb{N}, we have
d({T}^{n}x,z)\le \frac{{k}^{n}}{1k}F(d(Tx,x),\phi (Tx),\phi (x)),
where \{z\}={F}_{T}\cap {Z}_{\phi}={F}_{T}.
Proof Suppose that \xi \in X is a fixed point of T. Applying (2) with x=y=\xi, we obtain
which implies (since k\in (0,1)) that
On the other hand, from (F1), we have
Using (3) and (4), we obtain \phi (\xi )=0, which proves (i).
Let x\in X be an arbitrary point. Using (2), we have
By induction, we obtain easily
which implies by property (F1) that
From (5), we have
which implies (since k\in (0,1)) that \{{T}^{n}x\} is a Cauchy sequence. Since (X,d) is complete, there is some z\in X such that
Now, we shall prove that z is a φfixed point of T. Observe that from (5), we have
Since φ is lower semicontinuous, from (6) and (7), we obtain
Using (2), we have
Letting n\to \mathrm{\infty} in the above inequality, using (6), (7), (8), (F2), and the continuity of F, we get
which implies from condition (F1) that
It follows from (8) and (9) that z is a φfixed point of T.
Suppose now that {z}^{\prime}\in X is another φfixed point of T. Applying (2) with x=z and y={z}^{\prime}, we obtain
which implies that d(z,{z}^{\prime})=0, that is, z={z}^{\prime}. So we get (ii).
Finally, using (5) and the triangle inequality, we get
Letting m\to \mathrm{\infty} in the above inequality, from (6), we obtain
which proves (iii). □
2.2 Graphic (F,\phi )contraction mappings
Definition 2.5 Let (X,d) be a metric space, \phi :X\to [0,\mathrm{\infty}) be a given function, and F\in \mathcal{F}. We say that the operator T:X\to X is a graphic (F,\phi )contraction with respect to the metric d if and only if
for some constant k\in (0,1).
Theorem 2.2 Let (X,d) be a complete metric space, \phi :X\to [0,\mathrm{\infty}) be a given function, and F\in \mathcal{F}. Suppose that the following conditions hold:
(H1) φ is lower semicontinuous;
(H2) T:X\to X is a graphic (F,\phi )contraction with respect to the metric d;
(H3) T is continuous.
Then

(i)
{F}_{T}\subseteq {Z}_{\phi};

(ii)
T is a weakly φPicard operator;

(iii)
for all x\in X, if {T}^{n}x\to z as n\to \mathrm{\infty}, then
d({T}^{n}x,z)\le \frac{{k}^{n}}{1k}F(d(Tx,x),\phi (Tx),\phi (x)),\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.
Proof Suppose that \xi \in X is a fixed point of T. Applying (10) with x=\xi, we obtain
which implies (since k\in (0,1)) that
On the other hand, from (F1), we have
Using (11) and (12), we obtain \phi (\xi )=0, which proves (i).
Let x\in X be an arbitrary point. Using (10), we have
By induction, we obtain easily
which implies by property (F1) that
From (13), we have
which implies that \{{T}^{n}x\} is a Cauchy sequence. Since (X,d) is complete, there is some z\in X such that
Now, we shall prove that z is a φfixed point of T. Observe that from (13), we have
Since φ is lower semicontinuous, from (14) and (15), we obtain \phi (z)=0. On the other hand, using the continuity of T and (14), we get z=Tz. Then z is a φfixed point of T. So T is a weakly φPicard operator.
Finally, the proof of (iii) follows by using similar arguments as in the proof of (iii), Theorem 2.1. □
2.3 (F,\phi )Weak contraction mappings
Definition 2.6 Let (X,d) be a metric space, \phi :X\to [0,\mathrm{\infty}) be a given function, and F\in \mathcal{F}. We say that the operator T:X\to X is an (F,\phi )weak contraction with respect to the metric d if and only if
for all (x,y)\in {X}^{2}, for some constants k\in (0,1) and L\ge 0.
For this class of operators, we have the following result.
Theorem 2.3 Let (X,d) be a complete metric space, \phi :X\to [0,\mathrm{\infty}) be a given function, and F\in \mathcal{F}. Suppose that the following conditions hold:
(H1) φ is lower semicontinuous;
(H2) T:X\to X is an (F,\phi )weak contraction with respect to the metric d.
Then

(i)
{F}_{T}\subseteq {Z}_{\phi};

(ii)
T is a weakly φPicard operator;

(iii)
for all x\in X, if {T}^{n}x\to z as n\to \mathrm{\infty}, then
d({T}^{n}x,z)\le \frac{{k}^{n}}{1k}F(d(x,Tx),\phi (x),\phi (Tx)),\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.
Proof Let \xi \in X be a fixed point of T. Applying (16) with x=y=\xi, we get
which implies that F(0,\phi (\xi ),\phi (\xi ))=0. Using property (F1), we obtain \phi (\xi )=0, that is, \xi \in {Z}_{\phi}. Then (i) is proved.
Let x\in X be an arbitrary point. Applying (16), we obtain
By induction, we get
The rest of the proof follows using similar arguments to the proof of Theorem 2.2. □
3 Links with partial metric spaces
From the previous obtained results in metric spaces, we deduce in this section some fixed point theorems in partial metric spaces; see also [21].
We start by the Matthews fixed point theorem [1].
Corollary 3.1 Let (X,p) be a complete partial metric space and let T:X\to X be a mapping such that
for some constant k\in (0,1). Then T has a unique fixed point z\in X. Moreover, we have p(z,z)=0.
Proof Consider the metric {d}_{p} on X defined by (1) and the function \phi :X\to [0,\mathrm{\infty}) defined by \phi (x)=p(x,x). Applying Theorem 2.1 with F(a,b,c)=a+b+c, and using Lemma 1.1, we obtain the desired result. □
Similarly, from Theorem 2.2, we obtain the following result.
Corollary 3.2 Let (X,p) be a complete partial metric space and let T:X\to X be a mapping such that
for some constant k\in (0,1). Then T has a fixed point z\in X. Moreover, we have p(z,z)=0.
Finally, from Theorem 2.3, we obtain the following result.
Corollary 3.3 Let (X,p) be a complete partial metric space and let T:X\to X be a mapping such that
for some constants k\in (0,1) and L\ge 0. Then T has a fixed point z\in X. Moreover, we have p(z,z)=0.
Observe that if p is a metric on X, we obtain from Corollary 3.3 the Berinde fixed point theorem for (k,L)weak contractions [24].
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Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the International Research Group Project No. IRG1404.
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Jleli, M., Samet, B. & Vetro, C. Fixed point theory in partial metric spaces via φfixed point’s concept in metric spaces. J Inequal Appl 2014, 426 (2014). https://doi.org/10.1186/1029242X2014426
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DOI: https://doi.org/10.1186/1029242X2014426