 Research
 Open Access
dNeighborhood system and generalized Fcontraction in dislocated metric space
 P Sumati Kumari^{1}Email author,
 Kastriot Zoto^{2} and
 Dinesh Panthi^{3}
 Received: 7 January 2015
 Accepted: 10 June 2015
 Published: 22 July 2015
Abstract
This paper, gives an answer for the Question 1.1 posed by Hitzler (Generalized metrics and topology in logic programming semantics, 2001) by means of "Topological aspects of dmetric space with dneighborhood system". We have investigated the topological aspects of a dneighborhood system obtained from dislocated metric space (simply dmetric space) which has got useful applications in the semantic analysis of logic programming. Further more we have generalized the notion of Fcontraction in the view of dmetric spaces and investigated the uniqueness of fixed point and coincidence point of such mappings.
Keywords
 dMetric space
 dNeighborhood system
 \(\mathcal {G}F\)Contraction.
Mathematics Subject Classification
 47H10
 54H25
Background
Metrics appear everywhere in Mathematics: Geometry, Probability, statistics, coding theory, graph theory, pattern recognition, networks, computer graphics, molecular biology, theory of information and computer semantics are some of the fields in which metrics and/or their cousins play a significant role. The notion of metric spaces introduced by Frechet (1906), is one of the helpful topic in Analysis. Banach (1922) proved a fixed point theorem for contraction mapping in a complete metric space. The Banach contraction theorem is one of the primary result of functional analysis. After Banach contraction theorem, huge number of fixed point theorems have been established by various authors and they made different generalizations of this theorem.
Matthews (1985) generalized Banach contraction mapping theorem in dislocated metric space. Hitzler (2001) introduce the notion of dislocated metric (dmetric) space and presented variants of Banach contraction principle for various modified forms of a metric space including dislocated metric space and applied them to semantic analysis of logic programs. Hitzler (2001) has applied fixed point theorems for self maps on dislocated metric spaces, quasi dislocated metric spaces, generalized ultra metric spaces in his thesis “Generalized Metrics and Topology in Logic Programming Semantics”. In this context, Hitzler raised some related questions on the topological aspects of dislocated metrics.
Recently, Sarma and Kumari (2012) initiated the concept of dballs and established topological properties on dmetric space. In the context of dmetric space, many papers have been published concerning fixed point, coincidence point and common fixed point theorems satisfying certain contractive conditions in dislocated metric space (see Karapinar and Salimi 2013; Kumari et al. 2012a, b; Zoto et al. 2014; Ahamad et al. 2013; Ren et al. 2013) which become an interesting topic in nowadays.
Of late several weaker forms of metric are extensively used in various fields such as programming languages, qualitative domain theory and so on.
Motivated by above, we give an answer for the Question 1.1 posed by Hitzler, further more we discuss some topological properties in dneighborhood system obtained from dislocated metric space. Moreover, we generalize the notion of Fcontraction initiated by Wardowski (2012) and we prove fixed point theorem. Our established results generalize similar results in the framework of dislocated metric space. Further more, we provide coincidence theorem in the setting of dneighborhood systems.
Preliminaries and notations
First, we collect some fundamental definitions, notions and basic results which are used throughout this section. For more details, the reader can refer to Hitzler (2001).
Definition 2.1
Let X be a set. A relation \(<\!\!\!\!\!\circ \subseteq X\times \mathcal {P}(X)\) is called a dmembership relation (on X) if it satisfies the following property for all \(x\in X\) and \(A,B\subseteq X:\) \(x<\!\!\!\!\!\circ \, A\) and \(A\subseteq B\) implies \(x<\!\!\!\!\!\circ \, B\).
Definition 2.2
 (i)
If \(U\in \mathcal {U}_{x},\) then \(x<\!\!\!\!\!\circ \, U\)
 (ii)
If \(U,V\in \mathcal {U}_{x},\) then \(U\cap V\in \mathcal {U}_{x}\)
 (iii)
If \(U\in \mathcal {U}_{x},\) then there is a \(V\subseteq U\) with \(V\in \mathcal {U}_{x}\) such that for all \(y<\!\!\!\!\!\circ \, V\) we have \(U\in \mathcal {U}_{y}\)
 (iv)
If \(U\in \mathcal {U}_{x},\) and \(U\subseteq V,\) then \(V\in \mathcal {U}_{x}.\)
Each \(U\in \mathcal {U}_{x}\) is called a dneighborhood of x. Finally, let X be a set and \(<\!\!\!\!\!\circ \) be a dmembership relation on X and, for each \(x\in X\), let \((\mathcal {U}_{x},<\!\!\!\!\!\circ \, )\) be a dneighborhood system for x. Then \((X,\mathcal {U},<\!\!\!\!\!\circ \, ) \, (or \ simply \ X)\) is called a dtopological space, where \(\mathcal {U}=\{\mathcal {U}_{x}/x\in X\}.\)
Proposition 2.3

(\(d_1\)) \(d(x, y) \) = 0 \(\implies x= y\)

(\(d_2\)) \(d(x, y)= d(y, x)\)

(\(d_3\)) \(d(x, z) \le d(x, y) + d (y, z) \) for all x, y, z in X
If \(x\in X\) and \(\epsilon >0,\) the set \(\mathcal {B}_{\epsilon }(x)=\{y/y\in X\) and \(d(x,y)<\epsilon \}\) is called the ball with center at x and radius \(\epsilon .\)
Proposition 2.4
Let \((X,\varrho )\) be a d metric space. Define the d membership relation \(<\!\!\!\!\!\circ \) as the relation \(\{(x,A)/\) there exists \(\epsilon >0\) for which \(\mathcal {B}_{\epsilon }(x)\subseteq A\}.\) For each \(x\in X,\) let \(\mathcal {U}_{x}\) be the collection of all subsets A of X such that \(x<\!\!\!\!\!\circ \, A\) . Then \((\mathcal {U}_{x},<\!\!\!\!\!\circ \, )\) is a d neighborhood system for x ; for each \(x\in X.\)
Definition 2.5
Let \((X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be a dtopological spaces and let \(x\in X.\) A net \((x_{\gamma })\) dconverges to \(x\in X\) if for each dneighborhood U of x we have that \(x_{\gamma }\) is eventually in U, that is, there exists some \(\gamma _{0}\) such that \(x_{\gamma }\in U\) for each \(\gamma >\gamma _{0}.\)
Definition 2.6
Let \((X,\varrho )\) be a dmetric space and let \((X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be a dtopological spaces as in Proposition 2.4. Let \((x_{n})\) be a sequence in X. Then \((x_{n})\) converges in \((X,\varrho )\) if and only if \((x_{n})\) dconverges in \((X,\mathcal {U},<\!\!\!\!\!\circ \, ).\)
Definition 2.7
Let X and Y be dtopological spaces and let \(f:X\rightarrow Y\) be a function. Then f is dcontinuous at \(x_{0}\in X\) if for each dneighborhood V of \(f(x_{0})\) in Y there is a dneighborhood U of \(x_{0}\) in X such that \(f(U)\subseteq V.\) We Say f is dcontinuous on X if f is dcontinuous at each \(x_{0} \in X.\)
Theorem 2.8
Let X and Y be d topological spaces and let \(f:X\rightarrow Y\) be a function. Then f is a d continuous if and only if for each net \((x_{\gamma })\) in X which d converges to some \(x_{0} \in X,\) \((f(x_{\gamma }))\) is a net in Y which d converges to \(f(x_{0})\in Y.\)
Proposition 2.9
Let \((X,\varrho )\) and \((Y,\varrho ^{\prime })\) be d metric spaces, let \(f:X\rightarrow Y\) be a function and let \((X,\mathcal {U},<\!\!\!\!\!\circ \, )\) and \((Y,\mathcal {V},<\!\!\!\!\!\circ ^{\prime })\) be the d topological spaces obtained from \((X,\varrho )\) , respectively \((Y,\varrho ^{\prime })\) as in Proposition 2.4. Then f is d continuous at \(x_{0} \in X\) if and only if for each \(\epsilon >0\) there exists a \(\delta >0\) such that \(f(\mathcal {B}_{\delta }(x_{0}))\subseteq \mathcal {B}_{\epsilon }(f(x_{0})).\)
Definition 2.10
Let \((X,\varrho )\) be a dmetric space, let \(f:X\rightarrow X\) be a contraction with contractivity factor \(\gamma \) and let \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be the d topological space obtained from dmetric \((X,\varrho )\) as in Proposition 2.4. Then f is dcontinuous.
Topological aspects of dmetric space with dneighborhood system
The following question was put forth in Hitzler Thesis.
(Question 1.1). Question: Is there a reasonable notion of dopen set corresponding to the notions of dneighborhood, dconvergence and dcontinuity.
We provided an answer for the above open question by constructing below theorems.
Theorem 3.1
Let \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be a d topological space. Define \(\mathfrak {J}=\{V/ \) for each \(x\in V \, {{there \, exists}} \, A\in \mathcal {U}_{x} \, {{such \, that}}\, A\subset V \}\) . Then \(\mathfrak {J}\) is a topology on X.
Proof
Clearly \(\mathfrak {J}\) contains X and \(\emptyset .\)
Let \(\{V_{\alpha }\}\) be an indexed family of nonempty elements of \(\mathfrak {J}\).
Let \(x\in \cup V_{\alpha }\) which implies that \(x\in V_{\alpha } \) for some \(\alpha. \)
Thus there exists \(A\in \mathcal {U}_{x}\) such that \(A\subset V_{\alpha }\subset \cup V_{\alpha }\). Which implies that \(\cup V_{\alpha }\in \mathfrak {J}.\)
Let \(\{ V_{\alpha _{i}} \}^{n}_{i=1}\) be any finite intersection of elements of \(\mathfrak {J}.\)
We have to prove that \(\cap ^{n}_{i=1}V_{\alpha _{i}}\in \mathfrak {J}.\) To obtain this, first we prove that if \(G_{1},G_{2}\in \mathfrak {J}\) then \(G_{1}\cap G_{2}\in \mathfrak {J}.\) Let \(x\in G_{1}\cap G_{2}\).
Which implies that \( x\in G_{1}\) and \(x\in G_{2}\) then there exists \(A_{1}\in \mathcal {U}_{x} \, {\text{such that}}\, A_{1}\subset G_{1}\) and there exists \(A_{2}\in \mathcal {U}_{x} \, {\text{such that}}\,\, A_{2}\subset G_{2}.\)
Which implies \( A_{1}\cap A_{2}\in \mathcal {U}_{x}\) and \(A_{1}\cap A_{2}\subset G_{1}\cap G_{2}\).
Thus \( G_{1}\cap G_{2}\in \mathfrak {J}.\) Hence by induction, we get \(\cap ^{n}_{i=1}V_{\alpha _{i}}\in \mathfrak {J}.\) \(\square \)
Definition 3.2
Let \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be a dtopological space and \(A \subseteq X\) be a dopen if for every \(x\in A\) there exists \(\mathcal {U}\in \mathcal {U}_{x}\ni \mathcal {U}\subset A.\)
Definition 3.3
Let \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be a dtopological space and \(A \subseteq X\) is dopen then \(A^{c}\) is dclosed.
Definition 3.4
Let \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be a dtopological space and \(A\subseteq X.\) A point x in A is called an interior point of A if \(x<\!\!\!\!\!\circ \, A.\)
Remark
Interior point of A is an open set.
Definition 3.5
Let \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be a dtopological space and \(A\subseteq X.\) A point x in X is said to be limit point of A if for every \(U\in \mathcal {U}_{x}\) there exist \(y\ne x\) in A such that \(y<\!\!\!\!\!\circ \, U.\)
Definition 3.6
Let (X, d) be a dmetric space and \(f : X \rightarrow X\). If there is a number \(0 < \alpha < 1\) such that \(d(f(x), f(y)) \le \alpha d(x, y) \forall x, y \in X\) then f is called a contraction.
Definition 3.7
( Sarma and Kumari 2012) Let (X, d) be a dmetric space and \(f:X\rightarrow X\) be a mapping. Write \(V(x)=d(x,f(x))\) and \(Z(f)=\{x/V(x)=0\}\). We call points of Z(f) as coincidence point of f. Clearly every point of Z(f) is a fixed point of f but the converse is not necessarily true.
Theorem 3.8
A subset \(F\subseteq X\) is said to be dclosed iff a net \((x_{\gamma })\) in F dconverges to x then \(x\in F.\)
Proof
Suppose \(F\subseteq X\) is dclosed.
Let \((x_{\gamma })\) be a net in F such that lim \(d(x_{\gamma },x)=0.\)
We shall prove that \(x\in F.\)
Let us suppose \(x\notin F\) which implies that \( x\in XF,\) which is open.
Thus there exists \(A\in \mathcal {U}_{x}\) such that \( A\subset XF.\)
As \(A\in \mathcal {U}_{x}\) there exists \( \ \epsilon >0\) such that \( \mathcal {B}_{\epsilon }(x)\subset A\).
Since lim \(d(x_{\gamma },x)=0\) there exists \(\gamma _{0}\) such that \(d(x_{\gamma },x)< \epsilon \) for \(\gamma \ge \gamma _{0}\).
Hence \( x_{\gamma }\in \mathcal {B}_{\epsilon }(x)\subset A\subset XF.\) A contradiction.
It follows that \( x\in F.\)
Conversely, assume that if a net \((x_{\gamma })\) in F dconverges to x then \(x\in F.\)
We shall prove that \(F\subseteq X\) is dclosed.
\({\rm i.e} \ XF\) is dopen.
For this we have to prove that for every \(x\in XF\) there exists \(A\in \mathcal {U}_{x}\) such that
\(A\subseteq XF.\)
Suppose for some \(x\in XF\) there exists \(A\in \mathcal {U}_{x}\) such that \(A\nsubseteq XF.\)
Let \( x_{A}\in A(XF).\)
As \(\mathcal {U}_{x}\) is a direct set under set inclusion \(A\le B \ if \ B\subseteq A.\)
Thus \(\{x_{A}/A\in \mathcal {U}_{x}\}\) is a net.
Let \(\epsilon >0, \ A_{0}=\mathcal {B}_{\epsilon }(x)\in \mathcal {U}_{x}\).
If \(A\ge A_{0} \ ,A\subseteq A_{0}, \)
Thus \( x_{A}\in A_{0}\) implies that \(d(x_{A},x)<\epsilon \).
It follows that lim\(d(x_{A},x)=0\).
Which implies that \( x\in F.\) A Contradiction.
So for all \(x\in XF\) there exists \(A\in \mathcal {U}_{x}\) such that \(A\subseteq XF.\)
Which completes the proof. \(\square \)
Remark
For each \(\delta >0,\mathcal {B}_{\delta }(x)\) is a dneighborhood of x.
Theorem 3.9
 (i)
For each \(x\in X,\) there exists \(U\in \mathcal {U}_{x}\) such that \(x<\!\!\!\!\!\circ \, U. \)
 (ii)
If \(x<\!\!\!\!\!\circ \, U_{1}\cap U_{2}\) there exists \(U_{3}\in \mathcal {U}_{x} \,{{such \, that}}\, x<\!\!\!\!\!\circ \, U_{3}\) and \(U_{3}\subseteq U_{1}\cap U_{2}.\)
Proof
(i) is clear.
Since \(x<\!\!\!\!\!\circ \, U_{1}\cap U_{2}\) implies \(U_{1}\cap U_{2}\in \mathcal {U}_{x}.\)
So there exists \(\epsilon >0\) such that \( \mathcal {B}_{\epsilon }(x)\subset U_{1}\cap U_{2}.\)
Since balls are dneighborhood, choose \(U_{3}=\mathcal {B}_{\frac{\epsilon }{2}}(x)\in \mathcal {U}_{x}.\)
Then \(x<\!\!\!\!\!\circ \, U_{3}\) and \(\mathcal {B}_{\frac{\epsilon }{2}}(x)\subset U_{1}\cap U_{2}.\) \(\square \)
Lemma 3.10
 (i)
\(\mathfrak {J}^{\prime }\) finer than \(\mathfrak {J}\) \((\mathfrak {J}\subseteq \mathfrak {J}^{\prime })\)
 (ii)
\({For \, each}\, x\in X\) and each basis element \(B\in \mathcal {B}\) with \(x\in B\) there exists a basis element \(B^{\prime }\in \mathcal {B}^{\prime }\) such that \(x\in B^{\prime }\) and \(B^{\prime }\subseteq B.\)
Theorem 3.11
Let \((X,d,\mathfrak {J})\) be the topology induced from the d topological space \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) obtained from d metric as in Proposition 2.4, \(\mathfrak {J}_{d}\) be the topology induced by the d metric then \(\mathfrak {J}=\mathfrak {J}_{d}.\)
Proof
Let \(\mathcal {V}_{\epsilon }(x)=\mathcal {B}_{\epsilon }(x)\cup \{x\}.\) Then the collection \(\mathfrak {B}=\{\mathcal {V}_{\epsilon }(x)/x\in X\}\) is a basis for \(\mathfrak {J}_{d},\) and \(\mathcal {U}_{x}=\{U\subset X /x <\!\!\!\!\!\circ \, U\}\) is a basis for \(\mathfrak {J}.\) Clearly \(\mathfrak {J}_{d}\subset \mathfrak {J},\) since \(\mathcal {V}_{\epsilon }(x)\) is a dneighborhood.
Let \(x\in X\) and \(U\in \mathcal {U}_{x}\) such that \(x\in U.\)
Since \(x<\!\!\!\!\!\circ \, U\) there exists \(\epsilon >0\) such that \(\mathcal {B}_{\epsilon }(x)\subseteq U.\)
Which implies \(\{x\}\cup \mathcal {B}_{\epsilon }(x)\subseteq U\).
So \(\mathcal {V}_{\epsilon }(x)\subseteq U.\) Hence \(\mathfrak {J}\subset \mathfrak {J}_{d}.\) \(\square \)
Theorem 3.12
 (1)
There exists \((x_{n})\in A \) such that lim \(d(x_{n},x)=0\)
 (2)
For every \(U\in \mathcal {U}_{x}\) there exists \(\ y\ne x\) in A such that \(y<\!\!\!\!\!\circ \, U.\)
Proof
Let \(U\in \mathcal {U}_{x}\) there exists \(\epsilon >0\) such that \(\mathcal {B}_{\epsilon }(x)\subseteq U.\)
Since (1) holds, lim\(d(x_{n},x)=0\).
Which implies that, there exists N such that \(d(x_{n},x)< \epsilon \) \(\forall n\ge N.\)
Let \(y=x_{N}\) and \(r=\epsilon d(x_{N},x)\) then \(\mathcal {B}_{r}(y)\subset \mathcal {B}_{\epsilon }(x)\subset U\).
It follows that \( \mathcal {B}_{r}(y)\subset U.\)
So \( y<\!\!\!\!\!\circ \, U.\) Hence (2) holds.
Assume that (2) holds. Let \(U=\mathcal {B}_{\frac{1}{n}}(x),\) there exists \(x_{n}\ne x\) in A such that \(x_{n}<\!\!\!\!\!\circ \, U=\mathcal {B}_{\frac{1}{n}}(x).\)
i.e there exists \(\epsilon _{n}<\frac{1}{n}\) such that \(\mathcal {B}_{\epsilon _{n}}(x_{n})\subset \mathcal {B}_{\frac{1}{n}}(x).\)
Let \(y_{n}\in \mathcal {B}_{\epsilon _{n}}(x_{n})\).
Which implies that \( d(x_{n},y_{n})<\frac{1}{n}\) and \(d(x,y_{n})<\frac{1}{n}\).
Hence \(d(x_{n},x)<d(x_{n},y_{n})+d(y_{n},x)<\frac{2}{n}\).
Which yields lim \(d(x_{n},x)=0.\) Hence (1) holds. \(\square \)
Theorem 3.13
Let \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be the d topological space obtained from d metric \((X,\varrho )\) as in Proposition 2.4 .Then balls are d open.
Proof
Let \(\mathcal {B}_{\epsilon }(x)\) be a ball with center at x and radius \(\epsilon \).
It sufficies to prove that \(\mathcal {B}_{\epsilon }(x)\) is dopen.
i.e we shall prove for every \(y \in \mathcal {B}_{\epsilon }(x)\) there exists \(U\in \mathcal {U}_{y}\) such that \( U\subset \mathcal {B}_{\epsilon }(x).\)
Since \(y\in \mathcal {B}_{\epsilon }(x)\) implies \( d(x,y)<\epsilon \).
Choose \(\delta =\epsilon d(x,y)\).
As \(\mathcal {B}_{\delta }(y)\) is a dneighborhood, now let \(U=\mathcal {B}_{\delta }(y)\).
So it is sufficient to prove that \(U\subset \mathcal {B}_{\epsilon }(x)\).
Let \(z\in U\).
This implies that \( d(y,z)<\delta <\epsilon d(x,y)\).
Then \( d(x,z)<\epsilon \).
It follows that \( z\in \mathcal {B}_{\epsilon }(x)\).
Hence \(U\subset \mathcal {B}_{\epsilon }(x).\) \(\square \)
Theorem 3.14
Let \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be a d topological space obtained from d metric \((X,\varrho )\) as in Proposition 2.4. Then \((X,d,\mathfrak {J})\) is a Haussdorff space.
Proof
Suppose \(x\ne y\Rightarrow d(x,y)>0\).
Let us choose \(\delta =d(x,y).\)
Let \(\mathcal {B}_{\frac{\delta }{2}}(x),\mathcal {B}_{\frac{\delta }{2}}(y)\) be the dneighborhoods of x and y respectively.
It sufficies to prove \(\mathcal {B}_{\frac{\delta }{2}}(x)\cap \mathcal {B}_{\frac{\delta }{2}}(y)=\phi \).
Let \(z\in \mathcal {B}_{\frac{\delta }{2}}(x)\cap \mathcal {B}_{\frac{\delta }{2}}(y)\).
Which implies that \( z\in \mathcal {B}_{\frac{\delta }{2}}(x)\) and \(z\in \mathcal {B}_{\frac{\delta }{2}}(y)\).
So \( d(x,z)< \frac{\delta }{2}\) and \(d(y,z)< \frac{\delta }{2}\).
It follows \( d(x,y)< \delta =d(x,y)\).
Which is a contradiction. \(\square \)
Theorem 3.15
Let \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be a d topological space obtained form d metric \((X,\varrho )\) as in Proposition 2.4. Then singleton sets are d closed in \((X,d,\mathfrak {J}).\)
Proof
Let \(x\in X\), we have to prove that \(\{x\}\) is dclosed or it is sufficies to prove \(X\{x\}\) is dopen.
i.e for each \(y\in X\{x\}\) there exists \(U\in \mathcal {U}_{y}\) such that \(U\subseteq X\{x\}.\)
Since \(y\ne x,\) implies \( d(x,y)>0\).
Which yields \(x\notin \mathcal {B}_{\epsilon }(y)\).
Thus, there is a dneighborhood,\(\mathcal {B}_{\epsilon }(y)\in \mathcal {U}_{y}\) such that \(\mathcal {B}_{\epsilon }(y)\subseteq X\{x\}\).
Hence \(\{x\}\) is dclosed. \(\square \)
Corollary 3.16
Let \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be a d topological space obtained form d metric \((X,\varrho )\) . Then \((X,d,\mathfrak {J})\) is a \(T_{1}\) space.
Corollary 3.17
Let \((X,d,\mathfrak {J})\) be a d topological space. Then the collection \(\{\mathcal {B}_{\epsilon }(x)/x\in X\}\) is an open base at x for X.
Main theorems
Wardowski (2012) introduced a new type of contraction called Fcontraction and proved a new fixed point theorem concerning Fcontraction and supported by computational data illustrate the nature of Fcontractions. In this section, we present a theorem which generalizes the Wardowski’s theorem.
Definition 4.1
 (i)
F is strictly increasing, i.e for all \(\alpha ,\beta \in R^{+}\) such that \(\alpha <\beta ,F(\alpha )<F(\beta )\)
 (ii)
For each sequence \(\{\alpha _{n}\}_{n\in \mathbb {N}}\) of positive numbers \(lim_{n\rightarrow \infty }\alpha _{n}=0\) iff \(lim_{n\rightarrow \infty }F(\alpha _{n})= \infty \)
 (iii)
There exists \(k\in (0,1)\) such that \(lim_{\alpha \rightarrow 0^{+}}\alpha ^{k}F(\alpha )=0\)
A mapping \(T:X\rightarrow X\) is said to be an Fcontraction if there exists \(\tau >0\) such that for all \(x,y\in X, d(Tx,Ty)>0 \Rightarrow \) \(\tau +F(d(Tx,Ty))\le F(d(x,y)).\)
Theorem 4.2
(Sgroi and Vetro 2013) Let (X, d) be a complete metric space and let \(T:X\rightarrow X\) be an Fcontraction then T has a unique fixed point \(x^{*}\in X\) and for every \(x_{0}\in X\) a sequence \(\{T^{n}x_{0}\}_{n\in \mathbb {N}}\) is convergent to \(x^{*}.\)
In the literature one can find some interesting papers concerning Fcontractions; (see for example Cosentino and Vetro 2014; Sgroi and Vetro 2013; Secelean 2013; Paesano and Vetro 2014; Hussain and Salimi 2014).
Definition 4.3
By \(\mathcal {G}\) we denote the set of all monotone decreasing real functions \(g:[0,\infty )\rightarrow [0,\infty ),\) such that \(g(x)=0\) iff \(x=0\) and \(lim_{t\rightarrow 0^{+}}\ g(t)=0.\)
Lemma 4.4
Let \(g\in \mathcal {G}\) and \(\{\epsilon _{n}\}\subseteq [0,\infty )\) , then from \(g(\epsilon _{n})\rightarrow 0\) it follows that \(\epsilon _{n}\rightarrow 0.\)
Proof
Routine. \(\square \)
Theorem 4.5
 (1)
\((X,d^{*})\) is a d metric space.
 (2)
lim\(d(x_{n},x)=0\) iff lim\(d^{*}(x_{n},x)=0.\)
 (3)
(X, d) is complete iff \((X,d^{*})\) is complete.
Proof
Let \(d^{*}(x,y)=0\).
Which yields \( g(d(x,y))=0\) implies \( d(x,y)=0.\)
So \( x=y.\) \(d^{*}(x,y)=d^{*}(y,x)\) follows from \(g(d(x,y))=g(d(y,x)).\)
Now consider \(d^{*}(x,z)=g(d(x,z))\)
\(\le g(d(x,y)+d(y,z))\)
\(\le g(d(x,y)+g(d(y,z))\) since g is subadditive.
\(=d^{*}(x,y)+d^{*}(y,z)\).
It follows that \( d^{*}(x,z)\le d^{*}(x,y)+d^{*}(y,z)\).
Hence \(d^{*}\) is a dmetric. This completes the proof of (1).
Let lim\(d(x_{n},x)=0\). It follows that lim\(gd(x_{n},x)=g(0)=0.\)
Which implies lim\(d^{*}(x_{n},x)=0.\)
Suppose lim\(d^{*}(x_{n},x)=0.\) By above lemma, it follows that lim\(d(x_{n},x)=0.\) Which completes the proof of (2).
Let us suppose that (X, d) is complete. Thus for every \(\epsilon >0\) there exist \(n_{1}\in N\) such that \(d(x_{n},x_{m})<\epsilon \) for all \(m,n\ge n_{1}\).
Which yields lim\(d(x_{n},x_{m})=0\).
Which implies lim\(d^{*}(x_{n},x_{m})={\text{lim}}gd(x_{n},x_{m})=g(0)=0\); because g is continuous at 0. So \(\{x_{n}\}\) is a Cauchy sequence in \((X,d^{*}).\) By using (2), we get \((X,d^{*})\) is complete.
Conversely suppose that \((X,d^{*})\) is complete.
Let \(\{x_{n}\}\) is a Cauchy sequence in \((X,d^{*}).\)
Then for every \(\epsilon >0\) there exist \(n_{1}\in N\) such that \(d^{*}(x_{n},x_{m})<\epsilon \) for all \(m,n\ge n_{1}.\)
Thus lim\(d^{*}(x_{n},x_{m})=0\).
It follows that lim\(gd(x_{n},x_{m})=0\).
By above Lemma, lim\(d(x_{n},x_{m})=0\).
Which implies that \(\{x_{n}\}\) is a Cauchy sequence in (X, d). By using (2) we conclude that (X, d) is complete. \(\square \)
Definition 4.6
 (i)
F is strictly increasing, i.e for all \(\alpha ,\beta \in R^{+}\) such that \(\alpha <\beta ,\ F(\alpha )<F(\beta ).\)
 (ii)
For each sequence \(\{\alpha _{n}\}_{n\in \mathbb {N}}\) of positive numbers \(lim_{n\rightarrow \infty }\alpha _{n}=0\) iff \(lim_{n\rightarrow \infty }F(\alpha _{n})=\infty \)
 (iii)
There exists \(k\in (0,1)\) such that \(lim_{\alpha \rightarrow 0^{+}}\alpha ^{k}F(\alpha )=0\). A mapping \(T:X\rightarrow X\) is said to be an \(\mathcal {G}F\)contraction if there exists \(\tau >0\) such that for all \(x,y\in X\), \(g(d(Tx,Ty))>0 \Rightarrow \) \(\tau +F(g(d(Tx,Ty)))\le F(g(d(x,y))).\)
Theorem 4.7
Let (X, d) be a complete d metric space and let \(T:X\rightarrow X\) be an \(\mathcal {G}F\) contraction. Then T has a unique fixed point.
Proof
Define \(d^{*}:X^{2}\rightarrow [0,\infty ]\) by \(d^{*}(x,y)=g(d(x,y))\) for any \(x,y\in X\) and \(g\in \mathcal {G}.\) By lemma 4.4 and theorem 4.5 it follows that \((X,d^{*})\) is a dmetric space.
We have, when \(d^{*}(Tx,Ty))>0\) implies \( \tau +F(d^{*}(Tx,Ty))\le F(d^{*}(x,y)).\)
Then by using same proof as in Theorem 4.2, we can conclude that T has a unique fixed point. \(\square \)
Theorem 4.8
Let \((X,\rho )\) be a complete d metric space and let \(f:X\rightarrow X\) be a contraction and \(( X,\mathcal {U},<\!\!\!\!\!\circ \, )\) be the d topological space obtained from \((X,\varrho )\) . Then f has a unique coincidence point for f.
Proof
Let \(x_{0}\in X\). Choose \(x_{n+1}=f(x_{n})=f^{n}(x_{0})\).
Then \(f^{n}(x_{0})\) is a cauchy sequence and converges in \((X,\varrho )\) to some point u.
i.e u = lim \(f^{n}(x_{0}).\) Since f is a contraction it is also dcontinuous, by Proposition 2.10, \(f(u)=lim f^{n+1}(x_{0}).\)
Hence \( d(u,f(u))=lim_{n\rightarrow \infty }d(f^{n}(x_{0}),f^{n+1}(x_{0}))<lim_{n\rightarrow \infty }\alpha ^{n}d(x_{0},f(x_{0}))\) = 0. Since \(0<\alpha <1\), \( d(u,f(u))=0\).
Thus u is a coincidence point of f. \(\square \)
Uniqueness
Let us suppose that v be the another coincidence point such that \(d(v ,f(v ))=0\).
Thus \(f(v )=v\) and \(f(u )=u\). By using triangle inequality, \(d(v ,u )\leq d(v ,f(v ))+d(f(v ),f(u ))+d(f(u ),u ) \leq \alpha d(v ,u )\) which implies \(d(v ,u )=0\).
Hence \(u=v \).
Declarations
Authors’ contributions
All authors participated in the design of this work and performed equally. All authors read and approved the final manuscript.
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