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αψGeraghty contractions on generalized metric spaces
Journal of Inequalities and Applications volume 2014, Article number: 423 (2014)
Abstract
In this work, we introduce the class of αψGeraghty contraction as well as generalized αψGeraghty contraction mappings in the context of generalized metric spaces where ψ is an auxiliary function which does not require the subadditive property and set up some fixed point results for both classes individually. Our results will extend, improve and generalize several existing results in the literature.
MSC:46T99, 47H10, 54H25.
1 Introduction
Fixed point theory focuses on the strategies for solving nonlinear equations of kind Tx=x, where the function T is defined on some abstract space X. It is well known that the classical contraction mapping principal of Banach is one of the most useful and fundamental results in the theory of fixed point. It guarantees the existence and uniqueness of fixed points for certain selfmaps in a complete metric space and provides a constructive method to find those fixed points. Due to its practical implication, several authors studied and extended it in many directions; for example, see [1–50] and the references therein.
In 1973, Geraghty [33] introduced an interesting class of auxiliary functions to refine the Banach contraction mapping principle. Let ℱ denote all functions \beta :[0,\mathrm{\infty})\to [0,1) which satisfy the condition
By using the function \beta \in \mathcal{F}, Geraghty [33] proved the following remarkable theorem.
Theorem 1 (Geraghty [33])
Let (X,d) be a complete metric space and T:X\to X be an operator. Suppose that there exists \beta :[0,\mathrm{\infty})\to [0,1) satisfying the condition
If T satisfies the following inequality
then T has a unique fixed point.
On the other hand, Branciari [1] made an attempt to generalize the Banach contraction principle by bringing out the notion of generalized metric, as well known as rectangular metric, where he replaced the triangle inequality with the weaker assumption, namely, quadrilateral inequality. The space X equipped with generalized metric or rectangular metric became known as generalized metric space or rectangular space. Afterward, several authors studied these spaces and provided various fixed point results in such spaces (see, e.g., [2, 5–13, 46]).
Recently, Samet et al. [14] introduced the class of αψ contractive type mappings and obtained a fixed point result for this new class of mappings in the set up of a metric space which properly contains several wellknown fixed point theorems including the Banach contraction principle. The technique used in this paper has been studied and improved by a number of authors (see, e.g., [15–20, 42–47]).
In this paper, we introduce two notions viz. αψGeraghty contraction mappings and generalized αψGeraghty contraction mappings and investigate the existence and uniqueness of fixed points for both classes in the setting of a generalized metric space, where ψ is an auxiliary function which does not require the subadditive property.
2 Preliminaries
In this section, we recall some useful definitions and auxiliary results that will be needed in the sequel. Throughout this paper, ℕ and ℝ denote the set of natural numbers and the set of real numbers, respectively.
Definition 2 [1]
Let X be a nonempty set, and let d:X\times X\to [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 generalized metric and abbreviated as GM. Here, the pair (X,d) is called generalized metric space and abbreviated as GMS. Given a generalized metric d on X and \u03f5>0, we call {B}_{d}(x,\u03f5)=\{y\in Xd(x,y)<\u03f5\} ϵball centered at x.
In the above definition, the expression (GM3) is called quadrilateral inequality. Notice also that if d satisfies only (GM1) and (GM2), then it is called semimetric (see, e.g., [3]).
Remark 3

(1)
Any metric space is generalized metric space, but the converse is not true in general, as shown in [1, 8, 12].

(2)
In [1], it was taken for granted that a generalized metric space is a Hausdorff topological space and as in a metric space, the topology of a generalized metric space can be generated by the collection of all ϵballs {B}_{d}(x,\u03f5) for x\in X and \u03f5>0. But Das and Lahiri [7] showed that these assumptions are not true in an arbitrary generalized metric space (see [[7], Example 1 and Example 2]). Nevertheless, it is to be observed that the GMS (X,d) becomes a topological space when a subset U of X is said to be open if to each a\in U, there exists a positive number {\u03f5}_{a} such that {B}_{d}(a,{\u03f5}_{a})\subseteq U. For a useful discussion on the topological structure of GMS, one can refer to [28].
The concepts of convergence, Cauchy sequence, completeness and continuity on a GMS are defined below.
Definition 4

(1)
A sequence \{{x}_{n}\} in a GMS (X,d) is GMS 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 GMS (X,d) is GMS Cauchy if and only if for every \u03f5>0 there exists a positive integer N(\u03f5) such that d({x}_{n},{x}_{m})<\u03f5 for all n>m>N(\u03f5).

(3)
A GMS (X,d) is called complete if every GMS Cauchy sequence in X is GMS convergent.

(4)
A mapping T:(X,d)\to (X,d) is continuous if for any sequence \{{x}_{n}\} in X such that d({x}_{n},x)\to 0 as n\to \mathrm{\infty}, we have d(T{x}_{n},Tx)\to 0 as n\to \mathrm{\infty}.
Lemma 5 [[46], Lemma 1]
Let (X,d) be a generalized metric space, and let \{{x}_{n}\} be a Cauchy sequence in X such that {x}_{m}\ne {x}_{n} whenever m\ne n. Then \{{x}_{n}\} can converge to at most one point.
Lemma 6 [46]
Let (X,d) be a generalized metric space, and let \{{x}_{n}\} be a sequence in X with distinct elements ({x}_{n}\ne {x}_{m} for n\ne m). Suppose that d({x}_{n},{x}_{n+1}) and d({x}_{n},{x}_{n+2}) tend to 0 as n\to \mathrm{\infty} and that \{{x}_{n}\} is not a Cauchy sequence. Then there exist \u03f5>0 and two sequences \{{m}_{k}\} and \{{n}_{k}\} of positive integers such that {n}_{k}>{m}_{k}>k and the following four sequences
tend to ϵ as k\to \mathrm{\infty}.
Proposition 7 [5]
Suppose that \{{x}_{n}\} is a Cauchy sequence in a GMS (X,d) with
where u\in X. Then {lim}_{n\to \mathrm{\infty}}d({x}_{n},z)=d(u,z) for all z\in X.
Samet et al. [14] introduced the notion of αadmissible mappings as follows.
Definition 8 Let X be a nonempty set, and let T:X\to X and \alpha :X\times X\to [0,\mathrm{\infty}) be mappings. Then T is called αadmissible if for all x,y\in X, we have
Some interesting examples of such mappings are given in [14]. Afterward, several authors (see, e.g., [25, 26, 38–41]) studied such mappings and used them to prove some interesting results in fixed point theory.
Recently, Karapınar et al. [42] defined the notion of triangular αadmissible mappings as follows.
Definition 9 Let X be a nonempty set, and let T:X\to X and \alpha :X\times X\to \mathbb{R} be mappings. Then T is called triangular αadmissible if

(1)
x,y\in X, \alpha (x,y)\ge 1\Rightarrow \alpha (Tx,Ty)\ge 1;

(2)
x,y,z\in X, \alpha (x,z)\ge 1 and \alpha (y,z)\ge 1\Rightarrow \alpha (x,y)\ge 1.
Lemma 10 [42]
Let T:X\to X be a triangular αadmissible map. Assume that there exists {x}_{1}\in X such that \alpha ({x}_{1},T{x}_{1})\ge 1. Define a sequence \{{x}_{n}\} by {x}_{n+1}=T{x}_{n}. Then we have \alpha ({x}_{n},{x}_{m})\ge 1 for all m,n\in \mathbb{N} with n<m.
Now, we define the following class of auxiliary functions which will be used densely in the sequel. Let Ψ denote the class of functions \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) which satisfy the following conditions:

(a)
ψ is nondecreasing;

(b)
ψ is continuous;

(c)
\psi (t)=0\iff t=0.
It is important to note that this work contains the fixed point results for αψGeraghty contraction and generalized αψGeraghty contraction mappings which are proved by keeping in view the fact that a generalized metric space need not be continuous, neither the respective topology needs to be Hausdorff and also the auxiliary function ψ defined above omits the assumption of subadditive property used in [47].
3 Main results
We start this section with the following definition.
Definition 11 Let (X,d) be a generalized metric space, and let \alpha :X\times X\to \mathbb{R} be a function. A map T:X\to X is called αψGeraghty contraction mapping if there exists \beta \in \mathcal{F} such that for all x,y\in X,
where \psi \in \mathrm{\Psi}.
Note that if we take \psi (t)=t in Definition 11, then T is called αGeraghty contraction mapping. Again, if we take \alpha (x,y)=1 for all x,y\in X in Definition 11, then T is called ψGeraghty contraction mapping.
Theorem 12 Let (X,d) be a complete generalized metric space, \alpha :X\times X\to \mathbb{R} be a function, and let T:X\to X be a map. Suppose that the following conditions are satisfied:

(1)
T is an αψGeraghty contraction mapping;

(2)
T is triangular αadmissible;

(3)
there exists {x}_{0}\in X such that \alpha ({x}_{0},T{x}_{0})\ge 1 and \alpha ({x}_{0},{T}^{2}{x}_{0})\ge 1;

(4)
T is continuous.
Then T has a fixed point {x}^{\ast}\in X, and \{{T}^{n}{x}_{0}\} converges to {x}^{\ast}.
Proof By (3) let {x}_{0}\in X, construct the sequence \{{x}_{n}\} as {x}_{n+1}=T{x}_{n}, n\in \mathbb{N}. If {x}_{n}={x}_{n+1} for some n\in \mathbb{N}\cup \{0\}, then {x}^{\ast}={x}_{n} is a fixed point of T. Assume further that {x}_{n}\ne {x}_{n+1} for each n\in \mathbb{N}\cup \{0\}. Since T is triangular αadmissible, it follows from (3) that
Due to Lemma 10, we have
for all n\in \mathbb{N}. Notice that we also find \alpha ({x}_{n},{x}_{n+m})\ge 1 for each m,n\in \mathbb{N}.
Now, we shall prove that {lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=0. By taking x={x}_{n1} and y={x}_{n} in (3.1) and regarding (3.2), we get that
for each n\in \mathbb{N}.
Since ψ is nondecreasing, we conclude from (3.3) that
for each n\in \mathbb{N}. Thus, we conclude that the sequence \{d({x}_{n},{x}_{n+1})\} is nonnegative and nonincreasing. As a result, there exists r\ge 0 such that {lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=r. We claim that r=0. Suppose, on the contrary, that r>0. Then, on account of (3.3), we get that
which yields that {lim}_{n\to \mathrm{\infty}}\beta (\psi (d({x}_{n},{x}_{n+1})))=1. We derive
due to the fact that \beta \in \mathcal{F}. On the other hand, the continuity of ψ together with (3.4) yields that
Analogously, we shall prove that {lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+2})=0. By substituting x={x}_{n1} and y={x}_{n+1} in (3.1) and taking (3.2) into account, we find that
for each n\in \mathbb{N}. Since ψ is nondecreasing, we derive from (3.6) that
for each n\in \mathbb{N}. Thus, we observe that the sequence \{d({x}_{n1},{x}_{n+1})\} is nonnegative and nonincreasing. Consequently, there exists r\ge 0 such that {lim}_{n\to \mathrm{\infty}}d({x}_{n1},{x}_{n+1})=r. We assert that r=0. Suppose, on the contrary, that r>0. Then, by regarding (3.6), we get that
which implies that {lim}_{n\to \mathrm{\infty}}\beta (\psi (d({x}_{n1},{x}_{n+1})))=1. We derive
due to the fact that \beta \in \mathcal{F}. On the other hand, the continuity of ψ together with (3.7) yields that
Suppose that {x}_{n}={x}_{m} for some m,n\in \mathbb{N}, m<n. Then
a contradiction. Hence, all elements of the sequence \{{x}_{n}\} are distinct.
We are ready to prove that \{{x}_{n}\} is a Cauchy sequence in (X,d). Suppose, on the contrary, that we have
Regarding the quadrilateral inequality, we need to examine two possible cases as follows.
Case 1. Suppose that k=nm is odd, where k\ge 1. Then we have
which is equivalent to
Since T is triangular αadmissible, by applying ψ, we get that
Letting m,n\to \mathrm{\infty}, we deduce that
So, by using (3.5), (3.9) and the continuity of ψ, we get
which implies {lim}_{m,n\to \mathrm{\infty}}\beta (\psi (d({x}_{n},{x}_{m})))=1. Consequently, we get {lim}_{m,n\to \mathrm{\infty}}d({x}_{n},{x}_{m})=0, which is a contradiction.
Case 2. Suppose that k=nm is even, where k\ge 1. So, we have
that can be written as
Due to the fact that T is triangular αadmissible, by applying ψ, we obtain that
Letting m,n\to \mathrm{\infty}, we find that
So, by using (3.8), (3.9) and the continuity of ψ, we observe
which yields {lim}_{m,n\to \mathrm{\infty}}\beta (\psi (d({x}_{n+1},{x}_{m+1})))=1. Thus, we conclude that {lim}_{m,n\to \mathrm{\infty}}d({x}_{n+1},{x}_{m+1})=0, which is a contradiction.
From Case 1 and Case 2 we concluded that \{{x}_{n}\} is a Cauchy sequence. Since (X,d) is a complete generalized metric space, there exists {x}^{\ast}\in X such that {lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}^{\ast})=0. Since T is continuous, we have
By Lemma 5, we have that T{x}^{\ast}={x}^{\ast}. □
If we let \psi (t)=t in Theorem 12, we get the following result.
Corollary 13 Let (X,d) be a complete generalized metric space, \alpha :X\times X\to \mathbb{R} be a function, and let T:X\to X be a map. Suppose that the following conditions are satisfied:

(1)
T is an αGeraghty contraction mapping;

(2)
T is triangular αadmissible;

(3)
there exists {x}_{0}\in X such that \alpha ({x}_{0},T{x}_{0})\ge 1 and \alpha ({x}_{0},{T}^{2}{x}_{0})\ge 1;

(4)
T is continuous.
Then T has a fixed point {x}^{\ast}\in X, and \{{T}^{n}{x}_{0}\} converges to {x}^{\ast}.
It is also possible to remove the continuity of the mapping T by replacing a weaker condition.
Definition 14 Let (X,d) be a complete generalized metric space, \alpha :X\times X\to \mathbb{R} be a function, and let T:X\to X be a map. We say that the sequence \{{x}_{n}\} is αregular, the following condition is satisfied:
If \{{x}_{n}\} is a sequence in X such that \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n and {x}_{n}\to x\in X as n\to +\mathrm{\infty}, then there exists a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} such that \alpha ({x}_{{n}_{k}},x)\ge 1 for all k.
Theorem 15 Let (X,d) be a complete generalized metric space, \alpha :X\times X\to \mathbb{R} be a function, and let T:X\to X be a map. Suppose that the following conditions are satisfied:

(1)
T is an αψGeraghty contraction mapping;

(2)
T is triangular αadmissible;

(3)
there exists {x}_{0}\in X such that \alpha ({x}_{0},T{x}_{0})\ge 1 and \alpha ({x}_{0},{T}^{2}{x}_{0})\ge 1;

(4)
either T is continuous or \{{x}_{n}\} is αregular.
Then T has a fixed point {x}^{\ast}\in X, and \{{T}^{n}{x}_{0}\} converges to {x}^{\ast}.
Proof Following the proof of Theorem 12, we know that the sequence \{{x}_{n}\}, defined by {x}_{n+1}=T{x}_{n} for all n\ge 0, converges to some {x}^{\ast}\in X. From (3.2) and assumption (4) of the theorem, there exists a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} such that \alpha ({x}_{{n}_{k}},{x}^{\ast})\ge 1 Applying (3.1), for all k, we get that
After letting k\to \mathrm{\infty} in (3.16), we have
Therefore, in view of Proposition 7, we can say {x}^{\ast}=T{x}^{\ast}. □
The following result can be derived from Theorem 15 by letting \psi (t)=t.
Corollary 16 Let (X,d) be a complete generalized metric space, \alpha :X\times X\to \mathbb{R} be a function, and let T:X\to X be a map. Suppose that the following conditions are satisfied:

(1)
T is an αGeraghty contraction mapping;

(2)
T is triangular αadmissible;

(3)
there exists {x}_{0}\in X such that \alpha ({x}_{0},T{x}_{0})\ge 1 and \alpha ({x}_{0},{T}^{2}{x}_{0})\ge 1;

(4)
either T is continuous or \{{x}_{n}\} is αregular.
Then T has a fixed point {x}^{\ast}\in X, and \{{T}^{n}{x}_{0}\} converges to {x}^{\ast}.
Now we introduce the notion of generalized αψGeraghty contraction.
Definition 17 Let (X,d) be a generalized metric space, and let \alpha :X\times X\to \mathbb{R} be a function. A map T:X\to X is called generalized αψGeraghty contraction mapping if there exists \beta \in \mathcal{F} such that for all x,y\in X,
where M(x,y)=max\{d(x,y),d(x,Tx),d(y,Ty)\} and \psi \in \mathrm{\Psi}.
Note that if we take \psi (t)=t in the above definition, then T is called generalized αGeraghty contraction mapping. Again, if we take \alpha (x,y)=1 for all x,y\in X, then T is called generalized ψGeraghty contraction mapping.
Theorem 18 Let (X,d) be a complete generalized metric space, \alpha :X\times X\to \mathbb{R} be a function, and let T:X\to X be a map. Suppose that the following conditions are satisfied:

(1)
T is a generalized αψGeraghty contraction mapping;

(2)
T is triangular αadmissible;

(3)
there exists {x}_{0}\in X such that \alpha ({x}_{0},T{x}_{0})\ge 1 and \alpha ({x}_{0},{T}^{2}{x}_{0})\ge 1;

(4)
T is continuous.
Then T has a fixed point {x}^{\ast}\in X, and \{{T}^{n}{x}_{0}\} converges to {x}^{\ast}.
Proof By (3) let {x}_{0}\in X, construct the sequence \{{x}_{n}\} as {x}_{n+1}=T{x}_{n}, n\in \mathbb{N}. If {x}_{n}={x}_{n+1} for some n\in \mathbb{N}\cup \{0\}, then {x}^{\ast}={x}_{n} is a fixed point of T. Assume further that {x}_{n}\ne {x}_{n+1} for each n\in \mathbb{N}\cup \{0\}. Since T is triangular αadmissible, it follows from (3) that
So by induction we get
for n\in \mathbb{N}. And we also find \alpha ({x}_{n},{x}_{n+m})\ge 1 for each m,n\in \mathbb{N}.
Therefore, by (3.17)
for each n\ge 1, where
If max\{d({x}_{n1},{x}_{n}),d({x}_{n},{x}_{n+1})\}=d({x}_{n},{x}_{n+1}), then by (3.19) we get
which is a contradiction. Hence max\{d({x}_{n1},{x}_{n}),d({x}_{n},{x}_{n+1})\}=d({x}_{n1},{x}_{n}), therefore (3.19) gives
This yields that for each n\in \mathbb{N},
Thus, we conclude that the sequence \{d({x}_{n},{x}_{n+1})\} is nonnegative and nonincreasing. As a result, there exists t\ge 0 such that {lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=t. We claim that t=0. Suppose, on the contrary, that t>0. Then, on account of (3.19), we get that
which yields that {lim}_{n\to \mathrm{\infty}}\beta (\psi (d({x}_{n},{x}_{n+1})))=1. We derive
due to the fact that \beta \in \mathcal{F}. On the other hand, the continuity of ψ together with (3.23) yields that
Now, we shall show
Regarding (3.17) and (3.18), we find that
for all n\in \mathbb{N}, where
In view of (3.22), we obtain
Define {a}_{n}=d({x}_{n},{x}_{n+2}) and {b}_{n}=d({x}_{n},{x}_{n+1}). Then, taking (3.26) into account, we get
This yields that for each n\in \mathbb{N},
In the light of (3.22), we have
Therefore
Thus, the sequence max\{{a}_{n},{b}_{n}\} is nonnegative and nonincreasing, so it converges to some r\ge 0. Clearly, by (3.24)
Now we have to show that r=0. If to the contrary r>0, then in view of (3.26), we have
which yields that {lim}_{n\to \mathrm{\infty}}\beta (\psi (M({x}_{n1},{x}_{n+1})))=1. We derive
due to the fact that \beta \in \mathcal{F}. On the other hand, the continuity of ψ together with (3.30) yields that
which is a contradiction and hence r=0.
Suppose that {x}_{n}={x}_{m} for some m,n\in \mathbb{N}, m<n. Then
a contradiction. Hence, all elements of the sequence \{{x}_{n}\} are distinct.
In order to prove that \{{x}_{n}\} is a Cauchy sequence in (X,d), suppose that it is not. Then by Lemma 6, using (3.24) and (3.25), we assert that there exist \u03f5>0 and two sequences \{{m}_{k}\} and \{{n}_{k}\} of positive integers such that {n}_{k}>{m}_{k}>k and sequences given in (2.2) tends to ϵ as k\to \mathrm{\infty}. By substituting x={x}_{{m}_{k}} and y={x}_{{n}_{k+1}} in (3.17) and taking (3.18) into account, we obtain
On the other hand, we have
and hence
From (3.31) we have
Letting k\to \mathrm{\infty}, it follows that
Thus, {lim}_{k\to \mathrm{\infty}}\psi (M({x}_{{m}_{k1}},{x}_{{n}_{k}}))=0 and hence (3.33) gives \psi (\u03f5)=0, which is a contradiction. Therefore, \{{x}_{n}\} is a Cauchy sequence. Since (X,d) is a complete generalized metric space, there exists {x}^{\ast}\in X such that {lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}^{\ast})=0. As T is continuous, therefore we have
By Lemma 5, we get that T{x}^{\ast}={x}^{\ast}. □
If we take \psi (t)=t in Theorem 18, we get the following.
Corollary 19 Let (X,d) be a complete generalized metric space, \alpha :X\times X\to \mathbb{R} be a function, and let T:X\to X be a map. Suppose that the following conditions are satisfied:

(1)
T is a generalized αGeraghty contraction mapping;

(2)
T is triangular αadmissible;

(3)
there exists {x}_{0}\in X such that \alpha ({x}_{0},T{x}_{0})\ge 1 and \alpha ({x}_{0},{T}^{2}{x}_{0})\ge 1;

(4)
T is continuous.
Then T has a fixed point {x}^{\ast}\in X, and \{{T}^{n}{x}_{0}\} converges to {x}^{\ast}.
Theorem 20 Let (X,d) be a complete generalized metric space, \alpha :X\times X\to \mathbb{R} be a function, and let T:X\to X be a map. Suppose that the following conditions are satisfied:

(1)
T is a generalized αψGeraghty contraction mapping;

(2)
T is triangular αadmissible;

(3)
there exists {x}_{0}\in X such that \alpha ({x}_{0},T{x}_{0})\ge 1 and \alpha ({x}_{0},{T}^{2}{x}_{0})\ge 1;

(4)
\{{x}_{n}\} is αregular.
Then T has a fixed point {x}^{\ast}\in X, and \{{T}^{n}{x}_{0}\} converges to {x}^{\ast}.
Proof Following the proof of Theorem 18, we know that the sequence \{{x}_{n}\} defined by {x}_{n+1}=T{x}_{n} for all n\ge 0, converges to some {x}^{\ast}\in X. Now, we shall show that T{x}^{\ast}={x}^{\ast}. Suppose, on the contrary, that T{x}^{\ast}\ne {x}^{\ast}, i.e., d({x}^{\ast},T{x}^{\ast})>0. Since {x}_{n} is αregular, then from (3.18) there exists a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} such that \alpha ({x}_{{n}_{k}},{x}^{\ast})\ge 1. Applying (3.17), for all k, we get that
where M({x}_{{n}_{k}},{x}^{\ast})=max\{d({x}_{{n}_{k}},{x}^{\ast}),d({x}_{{n}_{k}},T{x}_{{n}_{k}}),d({x}^{\ast},T{x}^{\ast})\}.
After letting k\to \mathrm{\infty} in (3.34), we have
In view of Proposition 7, we get a contradiction and hence {x}^{\ast}=T{x}^{\ast}. □
Corollary 21 Let (X,d) be a complete generalized metric space, \alpha :X\times X\to \mathbb{R} be a function, and let T:X\to X be a map. Suppose that the following conditions are satisfied:

(1)
T is a generalized αGeraghty contraction mapping;

(2)
T is triangular αadmissible;

(3)
there exists {x}_{0}\in X such that \alpha ({x}_{0},T{x}_{0})\ge 1 and \alpha ({x}_{0},{T}^{2}{x}_{0})\ge 1;

(4)
\{{x}_{n}\} is αregular.
Then T has a fixed point {x}^{\ast}\in X, and \{{T}^{n}{x}_{0}\} converges to {x}^{\ast}.
For the uniqueness of a fixed point of αψGeraghty contraction and generalized αψGeraghty contraction mapping, we will consider the following condition.
(U): For all x,y\in F(T), we have \alpha (x,y)\ge 1, where F(T) denotes the set of fixed points of T.
Theorem 22 Adding condition (U) to the hypothesis of Theorem 12 (respectively, Theorem 15), we obtain that u is the unique fixed point of T.
Proof We have to show that u is a unique fixed point of T. Let v be another fixed point of T with v\ne u. By hypothesis (U),
Now, using (3.1), we have
which is a contradiction. Hence, u=v. □
Theorem 23 Adding condition (U) to the hypothesis of Theorem 18 (respectively, Theorem 20), we obtain that u is the unique fixed point of T.
Proof As we have done in the proof of Theorem 22. Let v be another fixed point of T with v\ne u. Then, by assumption (U), we get
Now, using (3.17), we have
where
Therefore,
which is a contradiction. Hence, u=v. □
Theorem 24 Adding condition (U) to the hypothesis of Corollary 13 (respectively, Corollary 16, Corollary 19 and Corollary 21), we obtain that u is the unique fixed point of T.
4 Consequences
If we define the mapping \alpha :X\times X\to [0,\mathrm{\infty}) by \alpha (x,y)=1 for all x,y\in X, then, by Theorem 22 and Theorem 23, we obtain the following corollary as an extension of several known results in the literature.
Corollary 25 Let (X,d) be a complete generalized metric space and T:X\to X be a map. Assume that there exist two functions \beta \in \mathcal{F} and \psi \in \mathrm{\Psi} such that
for all x,y\in X. Then T has a unique fixed point.
Corollary 26 Let (X,d) be a complete generalized metric space and T:X\to X be a map. Assume that there exist two functions \beta \in \mathcal{F} and \psi \in \mathrm{\Psi} such that, for all x,y\in X,
where M(x,y)=max\{d(x,y),d(x,Tx),d(y,Ty)\}. Then T has a unique fixed point.
If we let \psi (t)=t, we get the following two corollaries.
Corollary 27 Let (X,d) be a complete generalized metric space and T:X\to X be a map. Assume that there exists a function \beta \in \mathcal{F} such that
for all x,y\in X. Then T has a unique fixed point.
Corollary 28 Let (X,d) be a complete generalized metric space and T:X\to X be a map. Assume that there exists a function \beta \in \mathcal{F} such that, for all x,y\in X,
where M(x,y)=max\{d(x,y),d(x,Tx),d(y,Ty)\}. Then T has a unique fixed point.
4.1 Fixed point theorems on metric spaces endowed with a partial order
In this section, we state some consequences of our main results in the context of a partially ordered metric space.
Definition 29 Let (X,\u2aaf) be a partially ordered set and T:X\to X be a given mapping. We say that T is nondecreasing with respect to ⪯ if
Definition 30 Let (X,\u2aaf) be a partially ordered set. A sequence \{{x}_{n}\}\subset X is said to be nondecreasing with respect to ⪯ if {x}_{n}\u2aaf{x}_{n+1} for all n.
Definition 31 Let (X,\u2aaf) be a partially ordered set and d be a metric on X. We say that (X,\u2aaf,d) is regular if for every nondecreasing sequence \{{x}_{n}\}\subset X such that {x}_{n}\to x\in X as n\to \mathrm{\infty}, there exists a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} such that {x}_{{n}_{k}}\u2aafx for all k.
We have the following result.
Corollary 32 Let (X,\u2aaf) be a partially ordered set and d be a generalized metric on X such that (X,d) is complete. Let T:X\to X be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function \psi \in \mathrm{\Psi} such that
for all x,y\in X with x\u2ab0y. Suppose also that the following conditions hold:

(i)
there exists {x}_{0}\in X such that {x}_{0}\u2aafT{x}_{0};

(ii)
T is continuous or (X,\u2aaf,d) is regular.
Then T has a fixed point. Moreover, if for all x,y\in X there exists z\in X such that x\u2aafz and y\u2aafz, we have uniqueness of the fixed point.
Proof Let \alpha :X\times X\to [0,\mathrm{\infty}) be a mapping defined by
Notice that T is a generalized αψGeraghty contraction mapping, that is,
for all x,y\in X. Due to (i), we have \alpha ({x}_{0},T{x}_{0})\ge 1. Owing to the monotone property of T, we observe that
for all x,y\in X. Hence, T is αadmissible. It is evident from Theorem 12 that T has a fixed point if it is continuous. Let us discuss the case that (X,\u2aaf,d) is regular. Let \{{x}_{n}\} be a sequence in X such that \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n and {x}_{n}\to x\in X as n\to \mathrm{\infty}. From the regularity hypothesis, there exists a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} such that {x}_{{n}_{k}}\u2aafx for all k. The definition of α yields that \alpha ({x}_{{n}_{k}},x)\ge 1 for all k. In this case, the existence of a fixed point follows from Theorem 15. The uniqueness follows from Theorem 18. □
The following results are immediate consequences of Corollary 32.
Corollary 33 Let (X,\u2aaf) be a partially ordered set and d be a generalized metric on X such that (X,d) is complete. Let T:X\to X be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function \psi \in \mathrm{\Psi} such that
for all x,y\in X with x\u2ab0y. Suppose also that the following conditions hold:

(i)
there exists {x}_{0}\in X such that {x}_{0}\u2aafT{x}_{0};

(ii)
T is continuous or (X,\u2aaf,d) is regular.
Then T has a fixed point. Moreover, if for all x,y\in X there exists z\in X such that x\u2aafz and y\u2aafz, we have uniqueness of the fixed point.
The following two corollaries can be concluded from the above results by taking \psi (t)=t.
Corollary 34 Let (X,\u2aaf) be a partially ordered set and d be a generalized metric on X such that (X,d) is complete. Let T:X\to X be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function \psi \in \mathrm{\Psi} such that
for all x,y\in X with x\u2ab0y. Suppose also that the following conditions hold:

(i)
there exists {x}_{0}\in X such that {x}_{0}\u2aafT{x}_{0};

(ii)
T is continuous or (X,\u2aaf,d) is regular.
Then T has a fixed point. Moreover, if for all x,y\in X there exists z\in X such that x\u2aafz and y\u2aafz, we have uniqueness of the fixed point.
Corollary 35 Let (X,\u2aaf) be a partially ordered set and d be a generalized metric on X such that (X,d) is complete. Let T:X\to X be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function \psi \in \mathrm{\Psi} such that
for all x,y\in X with x\u2ab0y. Suppose also that the following conditions hold:

(i)
there exists {x}_{0}\in X such that {x}_{0}\u2aafT{x}_{0};

(ii)
T is continuous or (X,\u2aaf,d) is regular.
Then T has a fixed point. Moreover, if for all x,y\in X there exists z\in X such that x\u2aafz and y\u2aafz, we have uniqueness of the fixed point.
In [49] Haghi et al. proved the following lemma.
Lemma 36 Let X be a nonempty set and f:X\to X be a function. Then there exists a subset E\subseteq X such that f(E)=f(X) and f:E\to X is one to one.
Theorem 37 Let (X,d) be a generalized metric space, and let T,f:X\to X be two selfmaps such that T(X)\subseteq f(X) and f(X) is a complete subset of X. If there exist two functions \beta \in \mathcal{F} and \psi \in \mathrm{\Psi} such that
holds for all x,y\in X, then T and f have a unique point of coincidence in X. Moreover, if T and f are weakly compatible, then T and f have a unique common fixed point.
Proof By Lemma 36, there exists E\subseteq X such that f(E)=f(X) and f:E\to X is one to one. We define a map g:f(E)\to f(E) by g(fx)=Tx. Clearly, g is well defined, since f is one to one. Now, using (4.5), we have
for all fx,fy\in f(E). Since f(E)=f(X) is complete, therefore by Corollary 25 there exists z\in X such that g(fz)=fz, which implies Tz=fz. Hence, T and f have a coincidence point. Again, if w is another coincidence point of T and f such that z\ne w, then by (4.5)
which is a contradiction. Hence z is a unique coincidence point of T and f. It is clear that T and f have a unique common fixed point whenever T and f are weakly compatible. □
Theorem 38 Let (X,d) be a generalized metric space, and let T,f:X\to X be two selfmaps such that T(X)\subseteq f(X) and f(X) is a complete subset of X. If there exist two functions \beta \in \mathcal{F} and \psi \in \mathrm{\Psi} such that, for all x,y\in X,
where M(x,y)=max\{d(fx,fy),d(fx,Tx),d(fy,Ty)\}. Then T and f have a unique point of coincidence in X. Moreover, if T and f are weakly compatible, then T and f have a unique common fixed point.
Proof Here, we utilize Corollary 26 instead of Corollary 25 in the proof of Theorem 37 and the proof completely follows the lines of the proof of Theorem 37 and hence it is omitted. □
Now, in support of the proved results, we present an example which is inspired by Example 4.11 of [50].
Example 39 Let X=\{\alpha ,\beta ,\gamma ,\delta \} and define d:X\times X\to [0,\mathrm{\infty}) by
Then it can be easily checked that (X,d) is a generalized metric space which is not a metric space since the inequality
is not true. Define T:X\to X
and \alpha :X\times X\to [0,\mathrm{\infty}) by
Firstly, we will prove that

(a)
T is triangular αadmissible;

(b)
there exists {x}_{0}\in X such that \alpha ({x}_{0},T{x}_{0})\ge 1 and \alpha ({x}_{0},{T}^{2}{x}_{0})\ge 1;

(c)
\{{x}_{n}\} is αregular;

(d)
hypothesis (U) is satisfied.
Proof (a) (i) Let x,y\in X such that \alpha (x,y)\ge 1. Then, by the definition of α, we have x,y\in X\{\gamma \}, therefore \alpha (Tx,Ty)=1.

(ii)
Let x,y,z\in X such that \alpha (x,z)\ge 1 and \alpha (y,z)\ge 1. Again the definition of α gives x,y,z\in X\{\gamma \} and hence \alpha (x,y)=1.
So, (i) and (ii) imply that T is triangular αadmissible.

(b)
Taking {x}_{0}=\delta, we have \alpha ({x}_{0},T{x}_{0})=\alpha (\delta ,\beta )=1 and \alpha ({x}_{0},{T}^{2}{x}_{0})=\alpha (\delta ,\beta )=1.

(c)
Let \{{x}_{n}\} be a sequence in X such that \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n and {x}_{n}\to x as n\to \mathrm{\infty}. By the definition of α, for each n, {x}_{n}\in X\{\gamma \}. Since X\{\gamma \}=\{\alpha ,\beta ,\delta \} is closed, we get that x\in X\{\gamma \}. Therefore, the definition of α gives \alpha ({x}_{n},x)=1 for each n\in \mathbb{N}.

(d)
Let x,y\in F(T). Clearly x=y=\beta, therefore, by the definition of α, we have \alpha (x,y)=\alpha (\beta ,\beta )=1. □
Also, we define \psi \in \mathrm{\Psi} by
and \beta :[0,\mathrm{\infty})\to [0,1) by
Clearly, \beta \in \mathcal{F}. Now, we will show that T is an αψGeraghty contraction mapping. Let x,y\in X with x\ne y and consider the following possible cases.
Case 1. If x,y\in \{\alpha ,\beta ,\delta \}, then d(Tx,Ty)=d(\beta ,\beta )=0 and hence (3.1) trivially holds.
Case 2. If x=\gamma ,y\in \{\alpha ,\beta ,\delta \}, then d(Tx,Ty)=d(\delta ,\beta )=1.2.
If y=\alpha, then
If y=\beta, then
If y=\delta, then
Hence inequality (3.1) is satisfied.
Case 3. If x\in \{\alpha ,\beta ,\delta \}, y=\gamma. Since d is symmetric, therefore (3.1) holds obviously from Case 2.
So, T is an αψGeraghty contraction mapping. Hence all the conditions of Theorem 22 are satisfied, and consequently T has a unique fixed point. Here β is such a point.
Remark 40

(1)
It can be easily checked that the map T defined in Example 39 also satisfies inequality (3.17), that is, T is a generalized αψGeraghty contraction mapping. Hence, Example 39 satisfied all the conditions of Theorem 23. Therefore, Theorem 23 also guarantees the existence and uniqueness of the fixed point of T.

(2)
It is to be noted that T defined in Example 39 does not satisfy (4.1) at x=\gamma, y=\alpha for any \psi \in \mathrm{\Psi} and \beta \in \mathcal{F}.
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Asadi, M., Karapınar, E. & Kumar, A. αψGeraghty contractions on generalized metric spaces. J Inequal Appl 2014, 423 (2014). https://doi.org/10.1186/1029242X2014423
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DOI: https://doi.org/10.1186/1029242X2014423
Keywords
 generalized metric spaces
 αψ contractions