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Common fixed point results for αψcontractions on a metric space endowed with graph
Journal of Inequalities and Applications volume 2014, Article number: 136 (2014)
Abstract
Abdeljawad (Fixed Point Theory Appl., 2013:19) introduced the concept of αadmissible for a pair of mappings. More recently Salimi et al. [Fixed Point Theory Appl., 2013:151] modified the notion of αψcontractive mappings. In this paper we introduce the concept of an αadmissible map with respect to η and modify the αψcontractive condition for a pair of mappings and establish common fixed point results for two, three, and four mappings in a closed ball in complete dislocated metric spaces. As an application, we derive some new common fixed point theorems for ψgraphic contractions defined on dislocated metric space endowed with a graph as well as preordered dislocated metric space. Some comparative examples are constructed which illustrate the superiority of our results to the existing ones in the literature.
MSC: 46S40, 47H10, 54H25.
1 Introduction and preliminaries
Fixed point results of mappings satisfying certain contractive condition on the entire domain has been at the center of rigorous research activities, for example, see [1–33]. From application point of view the situation is not yet completely satisfactory because it frequently happens that a mapping T is a contraction not on the entire space X but merely on a subset Y of X. Recently Arshad et al. [8] proved a result concerning the existence of fixed points of a mapping satisfying a contractive condition on closed ball in a complete dislocated metric space (see also [9, 14, 15, 25, 33]). The notion of dislocated topologies has useful applications in the context of logic programming semantics (see [5, 17, 29]).
The existence of fixed points of αψcontractive and αadmissible mappings in complete metric spaces has been studied by several researchers (see [18–20] and references therein). In this paper we discuss common fixed point results for αψcontractive type mappings in a closed ball in complete dislocated metric space. Our results improve several well known recent conventional results in [2, 8, 31]. We also derive some new common fixed point theorems for ψgraphic contractions as well as ordered contractions on preordered metric space. We give examples which show how these results can be used when the corresponding results cannot.
Consistent with [2, 7, 8, 17, 31], the following definitions and results will be needed in the sequel.
Definition 1.1 [17]
Let X be a nonempty set and let {d}_{l}:X\times X\to [0,\mathrm{\infty}) be a function, called a dislocated metric (or simply {d}_{l}metric) if the following conditions hold for any x,y,z\in X:

(i)
if {d}_{l}(x,y)=0, then x=y;

(ii)
{d}_{l}(x,y)={d}_{l}(y,x);

(iii)
{d}_{l}(x,y)\le {d}_{l}(x,z)+{d}_{l}(z,y).
The pair (X,{d}_{l}) is then called a dislocated metric space. It is clear that if {d}_{l}(x,y)=0, then from (i), x=y. But if x=y, {d}_{l}(x,y) may not be 0.
Definition 1.2 [17]
A sequence \{{x}_{n}\} in a {d}_{l}metric space (X,{d}_{l}) is called a Cauchy sequence if given \epsilon >0, there corresponds {n}_{0}\in N such that for all n,m\ge {n}_{0} we have {d}_{l}({x}_{m},{x}_{n})<\epsilon.
Definition 1.3 [17]
A sequence \{{x}_{n}\} in {d}_{l}metric space converges with respect to {d}_{l} if there exists x\in X such that {d}_{l}({x}_{n},x)\to 0 as n\to \mathrm{\infty}. In this case, x is called a limit of \{{x}_{n}\} and we write {x}_{n}\to x.
Definition 1.4 [17]
A {d}_{l}metric space (X,{d}_{l}) is called complete if every Cauchy sequence in X converges to a point in X.
Definition 1.5 Let X be a nonempty set and T,f:X\to X. A point y\in X is called point of coincidence of T and f if there exists a point x\in X such that y=Tx=fx, here x is called coincidence point of T and f. The mappings T, f are said to be weakly compatible if they commute at their coincidence point (i.e., Tfx=fTx whenever Tx=fx).
We require the following lemmas for subsequent use.
Lemma 1.6 [8]
Let X be a nonempty set and f:X\to X be a function. Then there exists E\subset X such that fE=fX and f:E\to X is onetoone.
Lemma 1.7 [7]
Let X be a nonempty set and the mappings S,T,f:X\to X have a unique point of coincidence v in X. If (S,f) and (T,f) are weakly compatible, then S, T, f have a unique common fixed point.
Let Ψ denote the family of all nondecreasing functions \psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) such that {\sum}_{n=1}^{+\mathrm{\infty}}{\psi}^{n}(t)<+\mathrm{\infty} for all t>0, where {\psi}^{n} is the n th iterate of ψ.
Lemma 1.8 [31]
If \psi \in \mathrm{\Psi}, then \psi (t)<t for all t>0.
Definition 1.9 [2]
Let S,T:X\to X and \alpha :X\times X\to [0,+\mathrm{\infty}). We say that the pair (S,T) is αadmissible if x,y\in X such that \alpha (x,y)\ge 1, then we have \alpha (Sx,Ty)\ge 1 and \alpha (Tx,Sy)\ge 1.
Definition 1.10 [31]
Let T:X\to X and \alpha ,\eta :X\times X\to [0,+\mathrm{\infty}) two functions. We say that T is αadmissible mapping with respect to η if x,y\in X such that \alpha (x,y)\ge \eta (x,y), then we have \alpha (Tx,Ty)\ge \eta (Tx,Ty). Note that if we take \eta (x,y)=1, then T is called an αadmissible mapping [32].
2 Common fixed point results in dislocated metric space
We first extend the concept of αηadmissibility for the pair of mappings.
Definition 2.1 Let S,T:X\to X and \alpha ,\eta :X\times X\to [0,+\mathrm{\infty}) two functions. We say that the pair (S,T) is αadmissible with respect to η if x,y\in X such that \alpha (x,y)\ge \eta (x,y), then we have \alpha (Sx,Ty)\ge \eta (Sx,Ty) and \alpha (Tx,Sy)\ge \eta (Tx,Sy). Also, if we take \eta (x,y)=1, then the pair (S,T) is called αadmissible, if we take, \alpha (x,y)=1, then we say that the pair (S,T) is ηsubadmissible mapping. If we take \eta (x,y)=1, then we obtain Definition 1 of Abdeljawad [2]. Also, if we take S=T, we obtain Definition 1.10.
Theorem 2.2 Let (X,{d}_{l}) be a complete dislocated metric space and S,T:X\to X be two mappings. Suppose there exist two functions, \alpha ,\eta :X\times X\to [0,+\mathrm{\infty}) such that the pair (S,T) is αadmissible with respect to η. For r>0, {x}_{0}\in \overline{B({x}_{0},r)}, and \psi \in \mathrm{\Psi}, assume that
and
Suppose that the following assertions hold:

(i)
\alpha ({x}_{0},S{x}_{0})\ge \eta ({x}_{0},S{x}_{0});

(ii)
for any sequence \{{x}_{n}\} in \overline{B({x}_{0},r)} such that \alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1}) for all n\in N\cup \{0\} and {x}_{n}\to u\in \overline{B({x}_{0},r)} as n\to +\mathrm{\infty} then \alpha ({x}_{n},u)\ge \eta ({x}_{n},u) for all n\in \mathbb{N}\cup \{0\}.
Then there exists a point {x}^{\ast} in \overline{B({x}_{0},r)} such that {x}^{\ast}=S{x}^{\ast}=T{x}^{\ast}.
Proof Let {x}_{1} in X be such that {x}_{1}=S{x}_{0} and {x}_{2}=T{x}_{1}. Continuing this process, we construct a sequence {x}_{n} of points in X such that
By assumption \alpha ({x}_{0},{x}_{1})\ge \eta ({x}_{0},{x}_{1}) and the pair (S,T) is αadmissible with respect to η, we have, \alpha (S{x}_{0},T{x}_{1})\ge \eta (S{x}_{0},T{x}_{1}) from which we deduce that \alpha ({x}_{1},{x}_{2})\ge \eta ({x}_{1},{x}_{2}) which also implies that \alpha (T{x}_{1},S{x}_{2})\ge \eta (T{x}_{1},S{x}_{2}). Continuing in this way we obtain \alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1}) for all n\in \mathbb{N}\cup \{0\}. First, we show that {x}_{n}\in \overline{B({x}_{0},r)} for all n\in \mathbb{N}. Using inequality (2), we have
It follows that
Let {x}_{2},\dots ,{x}_{j}\in \overline{B({x}_{0},r)} for some j\in \mathbb{N}. If j=2i+1, where i=0,1,2,\dots \frac{j1}{2} then using inequality (1), we obtain
Thus we have
If j=2i+2, then as {x}_{1},{x}_{2},\dots ,{x}_{j}\in \overline{B({x}_{0},r)} where (i=0,1,2,\dots ,\frac{j2}{2}), we obtain,
Thus from inequality (3) and (4), we have
Now,
Thus {x}_{j+1}\in \overline{B({x}_{0},r)}. Hence {x}_{n}\in \overline{B({x}_{0},r)} for all n\in \mathbb{N}. Now inequality (5) can be written as
Fix \epsilon >0 and let n(\epsilon )\in \mathbb{N} such that \sum {\psi}^{n}({d}_{l}({x}_{0},{x}_{1}))<\epsilon. Let n,m\in \mathbb{N} with m>n>k(\epsilon ), then by using the triangle inequality, we obtain
Thus we proved that \{{x}_{n}\} is a Cauchy sequence in (\overline{B({x}_{0},r)},{d}_{l}). As every closed ball in a complete dislocated metric space is complete, so there exists {x}^{\ast}\in \overline{B({x}_{0},r)} such that {x}_{n}\to {x}^{\ast}. Also
On the other hand, from (ii), we have
Now using the triangle inequality, together with (1) and (8), we get
Letting i\to \mathrm{\infty} and by using inequality (7), we obtain {d}_{l}(S{x}^{\ast},{x}^{\ast})<0. Hence S{x}^{\ast}={x}^{\ast}. Similarly by using
we obtain {d}_{l}(T{x}^{\ast},{x}^{\ast})=0, that is, T{x}^{\ast}={x}^{\ast}. Hence S and T have a common fixed point in \overline{B({x}_{0},r)}. □
If \eta (x,y)=1 for all x,y\in X in Theorem 2.2, we obtain the following result.
Corollary 2.3 Let (X,{d}_{l}) be a complete dislocated metric space and S,T:X\to X, r>0 and {x}_{0} be an arbitrary point in \overline{B({x}_{0},r)}. Suppose there exists \alpha :X\times X\to [0,+\mathrm{\infty}) such that the pair (S,T) is αadmissible. For \psi \in \mathrm{\Psi}, assume that
and
Suppose that the following assertions hold:

(i)
\alpha ({x}_{0},S{x}_{0})\ge 1;

(ii)
for any sequence \{{x}_{n}\} in \overline{B({x}_{0},r)} such that \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n\in \mathbb{N}\cup \{0\} and {x}_{n}\to u\in \overline{B({x}_{0},r)} as n\to +\mathrm{\infty} then \alpha ({x}_{n},u)\ge 1 for all n\in \mathbb{N}\cup \{0\}.
Then there exists a point {x}^{\ast} in \overline{B({x}_{0},r)} such that {x}^{\ast}=S{x}^{\ast}=T{x}^{\ast}.
If \alpha (x,y)=1 for all x,y\in X in Theorem 2.2, we obtain following result.
Corollary 2.4 Let (X,{d}_{l}) be a complete dislocated metric space and S,T:X\to X be two mappings. Suppose there exists \eta :X\times X\to [0,+\mathrm{\infty}) such that the pair (S,T) is ηsubadmissible. For \psi \in \mathrm{\Psi} and {x}_{0}\in \overline{B({x}_{0},r)}, assume that
and
Suppose that the following assertions hold:

(i)
\eta ({x}_{0},S{x}_{0})\le 1;

(ii)
for any sequence \{{x}_{n}\} in \overline{B({x}_{0},r)} such that \eta ({x}_{n},{x}_{n+1})\le 1 for all n\in \mathbb{N}\cup \{0\} and {x}_{n}\to u\in \overline{B({x}_{0},r)} as n\to +\mathrm{\infty} then \eta ({x}_{n},u)\le 1 for all n\in \mathbb{N}\cup \{0\}.
Then there exists a point {x}^{\ast} in \overline{B({x}_{0},r)} such that {x}^{\ast}=S{x}^{\ast}=T{x}^{\ast}.
Corollary 2.5 (Theorem 2.2 of [32])
Let (X,d) be a complete metric space and S:X\to X be an αadmissible mapping. Assume that for \psi \in \mathrm{\Psi},
holds for all x,y\in X. Also, suppose that the following assertions hold:

(i)
there exists {x}_{0}\in X such that \alpha ({x}_{0},S{x}_{0})\ge 1;

(ii)
for any sequence \{{x}_{n}\} in X with \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n\in \mathbb{N}\cup \{0\} and {x}_{n}\to x as n\to +\mathrm{\infty}, we have \alpha ({x}_{n},x)\ge 1 for all n\in \mathbb{N}\cup \{0\}.
Then S has a fixed point.
Theorem 2.6 On adding the condition ‘if {x}^{\ast} is any common fixed point in \overline{B({x}_{0},r)} of S and T, x be any fixed point of S or T in \overline{B({x}_{0},r)}, then \alpha (x,{x}^{\ast})\ge \eta (x,{x}^{\ast})’ to the hypotheses of Theorem 2.2, S and T have a unique common fixed point {x}^{\ast} and {d}_{l}({x}^{\ast},{x}^{\ast})=0.
Proof Assume that {y}^{\ast} be another fixed point of T in \overline{B({x}_{0},r)}, then, by assumption, \alpha ({x}^{\ast},{y}^{\ast})\ge \eta ({x}^{\ast},{y}^{\ast}),
A contradiction to the fact that for each t>0, \psi (t)<t. So {x}^{\ast}={y}^{\ast}. Hence T has no fixed point other than {x}^{\ast}. Similarly, S has no fixed point other than {x}^{\ast}. Now, \alpha ({x}^{\ast},{x}^{\ast})\ge \eta ({x}^{\ast},{x}^{\ast}), then
This implies that
□
Example 2.7 Let X={Q}^{+}\cup \{0\} and {d}_{l}:X\times X\to X be defined by {d}_{l}(x,y)=x+y. Then (X,{d}_{l}) is complete dislocated metric space (see [8]). Let S,T:X\to X be defined by
and
Considering, {x}_{0}=1, r=2, \psi (t)=\frac{t}{3} and \alpha (x,y)=2. Now \overline{B({x}_{0},r)}=[0,1]\cap X. Also,
Also if x,y\in (1,\mathrm{\infty})\cap X, then
Then the contractive condition does not hold on X. Also, if x,y\in \overline{B({x}_{0},r)}, then
Therefore, all the conditions of Corollary 2.3 are satisfied and S and T have a common fixed point 0.
Now we apply our Theorem 2.6 to obtain unique common fixed point of three mappings on a closed ball in complete dislocated metric space.
Theorem 2.8 Let (X,{d}_{l}) be a dislocated metric space, S,T,f:X\to X such that SX\cup TX\subset fX, r>0 and {x}_{0}\in \overline{B({x}_{0},r)}. Suppose there exist two functions, \alpha ,\eta :X\times X\to [0,+\mathrm{\infty}) αadmissible with respect to η and \psi \in \mathrm{\Psi} such that
and
Suppose that

(i)
the pair (S,T) and f are αadmissible with respect to η;

(ii)
\alpha (f{x}_{0},S{x}_{0})\ge \eta (f{x}_{0},S{x}_{0});

(iii)
if \{{x}_{n}\} is a sequence in \overline{B(f{x}_{0},r)} such that \alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1}) for all n and {x}_{n}\to u\in \overline{B(f{x}_{0},r)} as n\to +\mathrm{\infty} then \alpha ({x}_{n},u)\ge \eta ({x}_{n},u) for all n\in \mathbb{N}\cup \{0\};

(iv)
if fx is any point in \overline{B(f{x}_{0},r)} such that Sx=Tx=fx and fy be any point in \overline{B(f{x}_{0},r)} such that Sy=fy or Ty=fy, then \alpha (fx,fy)\ge \eta (fx,fy);

(v)
fX is complete subspace of X and (S,f) and (T,f) are weakly compatible.
Then S, T, and f have a unique common fixed point fz in \overline{B(f{x}_{0},r)}. Moreover, {d}_{l}(fz,fz)=0.
Proof By Lemma 1.6, there exists E\subset X such that fE=fX and f:E\to X is onetoone. Now since SX\cup TX\subset fX, we define two mappings g,h:fE\to fE by g(fx)=Sx and h(fx)=Tx, respectively. Since f is onetoone on E, then g, h are well defined. Now f{x}_{0}\in \overline{B(f{x}_{0},r)}\subseteq fX. Then f{x}_{0}\in fX. Let {y}_{0}=f{x}_{0}, choose a point {y}_{1} in fX such that {y}_{1}=g({y}_{0}) and let {y}_{2}=h({y}_{1}). Continuing this process and having chosen {y}_{n} in fX such that
As f is αadmissible then \alpha (x,y)\ge \eta (x,y) implies
Also if (S,T) is αadmissible then \alpha (x,y)\ge \eta (x,y) implies
This implies that the pair (g,h) is αadmissible. As \alpha ({y}_{0},{y}_{1})\ge \eta ({y}_{0},{y}_{1})\u27f9\alpha (g{y}_{0},h{y}_{1})\ge \eta (g{y}_{0},h{y}_{1})\u27f9\alpha (h{y}_{1},g{y}_{2})\ge \eta (h{y}_{1},g{y}_{2}). Continuing this process, we have \alpha ({y}_{n},{y}_{n+1})\ge \eta ({y}_{n},{y}_{n+1}). Following similar arguments to those of Theorem 2.2, {y}_{n}\in \overline{B(f{x}_{0},r)}. Also by inequality (10).
Note that for fx,fy\in \overline{B(f{x}_{0},r)} and \alpha (fx,fy)\le \eta (fx,fy). Then by using inequality (9), we have
As fX is a complete space, all conditions of Theorem 2.6 are satisfied, we deduce that there exists a unique common fixed point fz\in \overline{B(f{x}_{0},r)} of g and h. Now fz=g(fz)=h(fz) or fz=Sz=Tz=fz. Thus fz is the point of coincidence of S, T and f. Let v\in \overline{B(f{x}_{0},r)} be another point of coincidence of f, S and T then there exists u\in \overline{B(f{x}_{0},r)} such that v=fu=Su=Tu, which implies that fu=g(fu)=h(fu). A contradiction as fz\in \overline{B(f{x}_{0},r)} is a unique common fixed point of g and h. Hence v=fz. Thus S, T and f have a unique point of coincidence fz\in \overline{B(f{x}_{0},r)}. Now since (S,f) and (T,f) are weakly compatible, by Lemma 1.7 fz is a unique common fixed point of S, T, and f. □
Similarly, we can apply our Theorem 2.6 to obtain unique common fixed point and point of coincidence of four mappings in complete dislocated metric space. One can easily obtain conclusion by using the technique given in the proof of Theorem 2.8 [8].
Theorem 2.9 Let (X,{d}_{l}) be a dislocated metric space and S, T, g and f be selfmappings on X such that SX,TX\subset fX=gX, r>0 and {x}_{0}\in \overline{B({x}_{0},r)}. Suppose there exist two functions \alpha ,\eta :X\times X\to [0,+\mathrm{\infty}) is αadmissible with respect to η and \psi \in \mathrm{\Psi} such that
and
Suppose that

(i)
the pairs (S,T) and (f,g) are αadmissible with respect to η;

(ii)
\alpha (f{x}_{0},S{x}_{0})\ge \eta (f{x}_{0},S{x}_{0});

(iii)
if \{{x}_{n}\} is a sequence in \overline{B(f{x}_{0},r)} such that \alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1}) for all n and {x}_{n}\to u\in \overline{B(f{x}_{0},r)} as n\to +\mathrm{\infty} then \alpha ({x}_{n},u)\ge \eta ({x}_{n},u) for all n;

(iv)
if fx=gx is any point in \overline{B(f{x}_{0},r)} such that Sx=Tx=fx and fy=gy be any point in \overline{B(f{x}_{0},r)} such that Sy=fy or Ty=fy, then \alpha (fx,fy)\ge \eta (fx,fy);

(v)
fX is complete subspace of Xand (S,f) and (T,g) are weakly compatible.
Then S, T, f, and g have a unique common fixed point fz in \overline{B(f{x}_{0},r)}.
A partial metric version of Theorem 2.2 is given below.
Theorem 2.10 Let (X,p) be a complete partial metric space, S,T:X\to X be two maps, r>0 and {x}_{0}\in \overline{B({x}_{0},r)}. Suppose there exist two functions, \alpha ,\eta :X\times X\to [0,+\mathrm{\infty}) such that (S,T) be αadmissible with respect to η and \psi \in \mathrm{\Psi}. Assume that
and
Suppose that the following assertions hold:

(i)
\alpha ({x}_{0},S{x}_{0})\ge \eta ({x}_{0},S{x}_{0});

(ii)
for any sequence \{{x}_{n}\} in \overline{B({x}_{0},r)} such that \alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1}) for all n\in N\cup \{0\} and {x}_{n}\to u\in \overline{B({x}_{0},r)} as n\to +\mathrm{\infty} then \alpha ({x}_{n},u)\ge \eta ({x}_{n},u) for all n\in \mathbb{N}\cup \{0\}.
Then there exists a point {x}^{\ast} in \overline{B({x}_{0},r)} such that {x}^{\ast}=S{x}^{\ast}=T{x}^{\ast}.
3 Fixed point results for graphic contractions in dislocated metric spaces
Consistent with Jachymski [24], let (X,{d}_{l}) be a dislocated metric space and Δ denotes the diagonal of the Cartesian product X\times X. Consider a directed graph G such that the set V(G) of its vertices coincides with X, and the set E(G) of its edges contains all loops, i.e., E(G)\supseteq \mathrm{\Delta}. We assume G has no parallel edges, so we can identify G with the pair (V(G),E(G)). Moreover, we may treat G as a weighted graph (see [24]) by assigning to each edge the distance between its vertices. If x and y are vertices in a graph G, then a path in G from x to y of length m (m\in \mathbb{N}) is a sequence {\{{x}_{i}\}}_{i=0}^{m} of m+1 vertices such that {x}_{0}=x, {x}_{m}=y and ({x}_{n1},{x}_{n})\in E(G) for i=1,\dots ,m. A graph G is connected if there is a path between any two vertices. G is weakly connected if \tilde{G} is connected (see for details [1, 11, 21, 24]).
Definition 3.1 [24]
We say that a mapping T:X\to X is a Banach Gcontraction or simply Gcontraction if T preserves the edges of G, i.e.,
and T decreases the weights of the edges of G in the following way:
Now we extend the concept of Gcontraction for the pair of maps as follows.
Definition 3.2 Let (X,{d}_{l}) be a dislocated metric space endowed with a graph G and S,T:X\to X be selfmappings. Assume that for r>0, {x}_{0}\in \overline{B({x}_{0},r)} and \psi \in \mathrm{\Psi} following conditions hold:
Then the mappings (S,T) are called a ψgraphic contractive mappings. If \psi (t)=kt for some k\in [0,1), then we say (S,T) are Gcontractive mappings.
Theorem 3.3 Let (X,{d}_{l}) be a complete dislocated metric space endowed with a graph G and S,T:X\to X be ψgraphic contractive mappings and {x}_{0}\in \overline{B({x}_{0},r)}. Suppose that the following assertions hold:

(i)
({x}_{0},S{x}_{0})\in E(G) and {\sum}_{i=0}^{j}{\psi}^{i}({d}_{l}({x}_{0},S{x}_{0}))\le r for all j\in \mathbb{N};

(ii)
if \{{x}_{n}\} is a sequence in \overline{B({x}_{0},r)} such that ({x}_{n},{x}_{n+1})\in E(G) for all n\in \mathbb{N} and {x}_{n}\to x as n\to +\mathrm{\infty}, then ({x}_{n},x)\in E(G) for all n\in \mathbb{N}.
Then S and T have a common fixed point.
Proof Define, \alpha :{X}^{2}\to (\mathrm{\infty},+\mathrm{\infty}) by \alpha (x,y)=\{\begin{array}{ll}1,& \text{if}(x,y)\in E(G),\\ 0,& \text{otherwise.}\end{array} At first we prove that the mappings (S,T) are αadmissible. Let x,y\in \overline{B({x}_{0},r)} with \alpha (x,y)\ge 1, then (x,y)\in E(G). As (S,T) are ψgraphic contractive mappings, we have (Sx,Ty)\in E(G) and (Tx,Sy)\in E(G). That is, \alpha (Sx,Ty)\ge 1 and \alpha (Tx,Sy)\ge 1. Thus S, T are αadmissible mappings. From (i) there exists {x}_{0} such that ({x}_{0},S{x}_{0})\in E(G). That is, \alpha ({x}_{0},S{x}_{0})\ge 1.
If x,y\in \overline{B({x}_{0},r)} with \alpha (x,y)\ge 1, then (x,y)\in E(G). Now, since S, T are ψgraphic contractive mappings, {d}_{l}(Sx,Ty)\le \psi ({d}_{l}(x,y)). That is,
Let \{{x}_{n}\}\subset \overline{B({x}_{0},r)} with {x}_{n}\to x as n\to \mathrm{\infty} and \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n\in \mathbb{N}. Then ({x}_{n},{x}_{n+1})\in E(G) for all n\in \mathbb{N} and {x}_{n}\to x as n\to +\mathrm{\infty}. So by (ii) we have ({x}_{n},x)\in E(G) for all n\in \mathbb{N}. That is, \alpha ({x}_{n},x)\ge 1. Hence, all conditions of Corollary 2.3 are satisfied and S and T have a common fixed point.
Theorem 3.2(2^{o}) [24] and Theorem 2.3(2) [12] are extended to ψgraphic contractive pair defined on a dislocated metric space as follows. □
Theorem 3.4 Let (X,{d}_{l}) be a complete dislocated metric space endowed with a graph G and S,T:X\to X be ψgraphic contractive mappings and {x}_{0}\in \overline{B({x}_{0},r)}. Suppose that the following assertions hold:

(i)
({x}_{0},S{x}_{0})\in E(G) and {\sum}_{i=0}^{j}{\psi}^{i}({d}_{l}({x}_{0},S{x}_{0}))\le r for all j\in \mathbb{N};
(iis) (x,z)\in E(G) and (z,y)\in E(G) imply (x,y)\in E(G) for all x,y,z\in X, that is, E(G) is a quasiorder [24]and if \{{x}_{n}\} is a sequence in \overline{B({x}_{0},r)} such that ({x}_{n},{x}_{n+1})\in E(G) for all n\in \mathbb{N} and {x}_{n}\to x as n\to +\mathrm{\infty}, then there is a subsequence \{{x}_{{k}_{n}}\} with ({x}_{{k}_{n}},x)\in E(G) for all n\in \mathbb{N}.
Then S, T have a common fixed point.
Proof Condition (iis) implies that of (ii) in Theorem 3.3 (see Remark 3.1 [24]). Now the conclusion follows from Theorem 3.3. □
Corollary 3.5 Let (X,{d}_{l}) be a complete dislocated metric space endowed with a graph G and S,T:X\to X be two mappings and {x}_{0}\in \overline{B({x}_{0},r)}. Suppose that the following assertions hold:

(i)
(S,T) are Gcontractive mappings;

(ii)
({x}_{0},S{x}_{0})\in E(G) and {d}_{l}({x}_{0},S{x}_{0})\le (1k)r;

(iii)
if \{{x}_{n}\} is a sequence in \overline{B({x}_{0},r)} such that ({x}_{n},{x}_{n+1})\in E(G) for all n\in \mathbb{N} and {x}_{n}\to x as n\to +\mathrm{\infty}, then ({x}_{n},x)\in E(G) for all n\in \mathbb{N}.
Then S and T have a common fixed point.
Corollary 3.6 Let (X,{d}_{l}) be a complete dislocated metric space endowed with a graph G and S:X\to X be a mapping and {x}_{0}\in \overline{B({x}_{0},r)}. Suppose that the following assertions hold:

(i)
S is Banach Gcontraction on \overline{B({x}_{0},r)};

(ii)
({x}_{0},S{x}_{0})\in E(G) and {d}_{l}({x}_{0},S{x}_{0})\le (1k)r;

(iii)
if \{{x}_{n}\} is a sequence in \overline{B({x}_{0},r)} such that ({x}_{n},{x}_{n+1})\in E(G) for all n\in \mathbb{N} and {x}_{n}\to x as n\to +\mathrm{\infty}, then ({x}_{n},x)\in E(G) for all n\in \mathbb{N}.
Then S has a fixed point.
Corollary 3.7 Let (X,{d}_{l}) be a complete dislocated metric space endowed with a graph G and S:X\to X be a mapping. Suppose that the following assertions hold:

(i)
S is Banach Gcontraction on X and there is {x}_{0}\in X such that ({x}_{0},S{x}_{0})\in E(G);

(ii)
if \{{x}_{n}\} is a sequence in X such that ({x}_{n},{x}_{n+1})\in E(G) for all n\in \mathbb{N} and {x}_{n}\to x as n\to +\mathrm{\infty}, then ({x}_{n},x)\in E(G) for all n\in \mathbb{N}.
Then S has a fixed point.
The study of existence of fixed points in partially ordered sets has been initiated by Ran and Reurings [28] with applications to matrix equations. Agarwal, et al. [3, 4], Bhaskar and Lakshmikantham [10], Ciric et al. [13] and Hussain et al. [22, 23] presented some new results for nonlinear contractions in partially ordered metric spaces and noted that their theorems can be used to investigate a large class of problems. Roldán et al. [30] and Harandi et al. [6] proved some results in preordered metric spaces which is a generalization of partially ordered metric spaces. Here as an application of our results we deduce some new common fixed point results in preordered dislocated metric spaces.
Recall that if (X,\u2aaf) is a preordered set and T:X\to X is such that for x,y\in X, with x\u2aafy implies Tx\u2aafTy, then the mapping T is said to be nondecreasing. If for x,y\in X, with x\u2aafy implies Sx\u2aafTy and Tx\u2aafSy, then the pair (S,T) is called jointly nondecreasing.
Let X be a nonempty set. Then (X,{d}_{l},\u2aaf) is called a preordered dislocated metric space if {d}_{l} is a dislocated metric on X and ⪯ is a preorder on X. Let (X,{d}_{l},\u2aaf) be a preordered dislocated metric space. Define the graph G by
For this graph, the first condition in Definition 3.2 means S, T are jointly nondecreasing with respect to this order. From Theorems 3.3Corollary 3.7 we derive the following important results in preordered dislocated metric spaces.
Theorem 3.8 Let (X,{d}_{l},\u2aaf) be a preordered complete dislocated metric space and let the pair (S,T) of selfmaps of X be jointly nondecreasing and {x}_{0}\in \overline{B({x}_{0},r)}. Suppose that the following assertions hold:

(i)
for all x,y\in \overline{B({x}_{0},r)}, with x\u2aafy\u27f9{d}_{l}(Sx,Ty)\le \psi ({d}_{l}(x,y));

(ii)
{x}_{0}\u2aafS{x}_{0} and {\sum}_{i=0}^{j}{\psi}^{i}({d}_{l}({x}_{0},S{x}_{0}))\le r for all j\in \mathbb{N};

(iii)
if \{{x}_{n}\} is a nondecreasing sequence in \overline{B({x}_{0},r)} such that {x}_{n}\to x\in \overline{B({x}_{0},r)} as n\to +\mathrm{\infty}, then {x}_{n}\u2aafx for all n\in \mathbb{N}.
Then S and T have a common fixed point.
Corollary 3.9 Let (X,{d}_{l},\u2aaf) be a preordered complete dislocated metric space and let the pair (S,T) of selfmaps of X be jointly nondecreasing and {x}_{0}\in \overline{B({x}_{0},r)}. Suppose that the following assertions hold:

(i)
there exists k\in [0,1) such that {d}_{l}(Sx,Ty)\le k{d}_{l}(x,y) for all x,y\in \overline{B({x}_{0},r)} with x\u2aafy;

(ii)
{x}_{0}\u2aafS{x}_{0} and {d}_{l}({x}_{0},S{x}_{0})\le (1k)r;

(iii)
if \{{x}_{n}\} is a nondecreasing sequence in \overline{B({x}_{0},r)} such that {x}_{n}\to x\in \overline{B({x}_{0},r)} as n\to +\mathrm{\infty}, then {x}_{n}\u2aafx for all n\in \mathbb{N}.
Then S and T have a common fixed point.
Corollary 3.10 Let (X,{d}_{l},\u2aaf) be a preordered complete dislocated metric space and let the pair (S,T) of selfmaps of X be jointly nondecreasing. Suppose that the following assertions hold:

(i)
there exists k\in [0,1) such that {d}_{l}(Sx,Ty)\le k{d}_{l}(x,y) for all x,y\in X with x\u2aafy;

(ii)
{x}_{0}\u2aafS{x}_{0};

(iii)
if \{{x}_{n}\} is a nondecreasing sequence in X such that {x}_{n}\to x\in X as n\to +\mathrm{\infty}, then {x}_{n}\u2aafx for all n\in \mathbb{N}.
Then S and T have a common fixed point.
Corollary 3.11 Let (X,{d}_{l},\u2aaf) be a preordered complete dislocated metric space and S:X\to X be a nondecreasing map and {x}_{0}\in \overline{B({x}_{0},r)}. Suppose that the following assertions hold:

(i)
there exists k\in [0,1) such that {d}_{l}(Sx,Sy)\le k{d}_{l}(x,y) for all x,y\in \overline{B({x}_{0},r)} with x\u2aafy;

(ii)
{x}_{0}\u2aafS{x}_{0} and {d}_{l}({x}_{0},S{x}_{0})\le (1k)r;

(iii)
if \{{x}_{n}\} is a nondecreasing sequence in \overline{B({x}_{0},r)} such that {x}_{n}\to x\in \overline{B({x}_{0},r)} as n\to +\mathrm{\infty}, then {x}_{n}\u2aafx for all n\in \mathbb{N}.
Then S has a fixed point.
Corollary 3.12 Let (X,{d}_{l},\u2aaf) be a preordered complete dislocated metric space and S:X\to X be a nondecreasing map. Suppose that the following assertions hold:

(i)
there exists k\in [0,1) such that {d}_{l}(Sx,Sy)\le k{d}_{l}(x,y) for all x,y\in X with x\u2aafy;

(ii)
there exists {x}_{0}\in X such that {x}_{0}\u2aafS{x}_{0};

(iii)
if \{{x}_{n}\} is a nondecreasing sequence in X such that {x}_{n}\to x\in X as n\to +\mathrm{\infty}, then {x}_{n}\u2aafx for all n\in \mathbb{N}.
Then S has a fixed point.
Corollary 3.13 [27]
Let (X,d,\u2aaf) be a preordered complete metric space and S:X\to X be a nondecreasing mapping such that
for all x,y\in X with x\u2aafy where 0\le k<1. Suppose that the following assertions hold:

(i)
there exists {x}_{0}\in X such that {x}_{0}\u2aafS{x}_{0};

(ii)
if \{{x}_{n}\} is a sequence in X such that {x}_{n}\u2aaf{x}_{n+1} for all n\in \mathbb{N} and {x}_{n}\to x as n\to +\mathrm{\infty}, then {x}_{n}\u2aafx for all n\in \mathbb{N}.
Then S has a fixed point.
Remark 3.14 We can similarly obtain partial metric and preordered partial metric versions of all results proved here which provide new results in the literature.
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This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.
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Hussain, N., Arshad, M., Shoaib, A. et al. Common fixed point results for αψcontractions on a metric space endowed with graph. J Inequal Appl 2014, 136 (2014). https://doi.org/10.1186/1029242X2014136
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DOI: https://doi.org/10.1186/1029242X2014136
Keywords
 common fixed point
 complete dislocated metric space
 αψcontractive mappings
 ψgraphic contractions
 closed ball