Research  Open  Published:
Some new fixed point results in partial ordered metric spaces via admissible mappings
Fixed Point Theory and Applicationsvolume 2014, Article number: 117 (2014)
Abstract
The purpose of this paper is to discuss the existence of fixed points for new classes of mappings defined on an ordered metric space. The obtained results generalize and improve some fixed point results in the literature. Some examples show the usefulness of our results.
MSC:46N40, 47H10, 54H25, 46T99.
1 Introduction and preliminaries
Over the last decades, the fixed point theory has become increasingly useful in the study of nonlinear phenomena. In fact, the fixed point theorems and techniques have been developed in pure and applied analysis, topology and geometry. It is well known that a fundamental result of this theory is Banach’s contraction principle [1]. Consequently, in the last 50 years, it has been extensively studied and generalized to many settings; see for example [2–14].
In 2008, Dutta and Choudhury proved the following theorem.
Theorem 1.1 (See [15])
Let $(X,d)$ be a complete metric space and $f:X\to X$ be such that
where $\psi ,\varphi :[0,+\mathrm{\infty}[\phantom{\rule{0.2em}{0ex}}\to [0,+\mathrm{\infty}[$ are continuous, nondecreasing, and $\psi (t)=\varphi (t)=0$ if and only if $t=0$. Then f has a unique fixed point ${x}^{\ast}\in X$.
Note that the above theorem remains true if the hypothesis on ϕ is replaced by ϕ is lower semicontinuous and $\varphi (t)=0$ if and only if $t=0$ (see e.g. [16, 17]).
Eslamian and Abkar stated the following theorem as a generalization of Theorem 1.1.
Theorem 1.2 Let $(X,d)$ be a complete metric space and $f:X\to X$ be such that
where $\psi ,\alpha ,\beta :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ are such that ψ is continuous and nondecreasing, α is continuous and β is lower semicontinuous,
Then f has a unique fixed point ${x}^{\ast}\in X$.
Aydi et al. [18] proved that Theorem 1.2 is a consequence of Theorem 1.1
On the other hand, Ran and Reurings [19] initiate the fixed point theory in the metric spaces equipped with a partial order relation. Let X be a nonempty set equipped with a partial order relation ⪯ such that the function $d:X\times X\to [0,\mathrm{\infty})$ is a metric on X, then the triple $(X,d,\u2aaf)$ is called a partially ordered metric space. Two elements $x,y\in X$ are comparable if either $x\u2aafy$ or $y\u2aafx$. We write $x\prec y$ if $x\u2aafy$ but $x\ne y$. A sequence $\{{x}_{n}\}$ in X is said to be nondecreasing with respect to ⪯ if ${x}_{n}\u2aaf{x}_{n+1}$ for all $n\in \mathbb{N}$. A mapping $f:X\to X$ is said to be nondecreasing with respect to ⪯ if $x\u2aafy$ implies $fx\u2aaffy$. In further discussion, if there is no confusion, for the mappings on X and sequences in X, we use the phrase ‘nondecreasing’ instead ‘nondecreasing with respect to ⪯’.
Harjani and Sadarangani [20] extended Theorem 1.1 in the framework of partially ordered metric spaces in the following way.
Theorem 1.3 Let $(X,d,\u2aaf)$ be a partially ordered complete metric space. Let $f:X\to X$ be a continuous nondecreasing mapping such that
where $\psi ,\varphi :[0,+\mathrm{\infty}[\phantom{\rule{0.2em}{0ex}}\to [0,+\mathrm{\infty}[$ are continuous and nondecreasing and $\psi (t)=\varphi (t)=0$ if and only if $t=0$. If there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aaff{x}_{0}$, then f has a fixed point ${x}^{\ast}\in X$.
Choudhury and Kundu [21] generalized Theorems 1.2 and 1.3 as follows.
Theorem 1.4 Let $(X,d,\u2aaf)$ be a partially ordered complete metric space. Let $f:X\to X$ be a nondecreasing mapping such that
where $\psi ,\alpha ,\beta :[0,+\mathrm{\infty}[\phantom{\rule{0.2em}{0ex}}\to [0,+\mathrm{\infty}[$ are such that ψ is continuous and nondecreasing, α is continuous, β is lower semicontinuous,
If there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aaff{x}_{0}$, then f has a unique fixed point ${x}^{\ast}\in X$.
Aydi et al. [18] proved that Theorem 1.4 is a consequence of Theorem 1.3.
Karapinar and Salimi [22] proved the following theorem as a generalization of Theorems 1.2 and 1.3 where the approach of Aydi et al. [18] cannot be modified for it.
Theorem 1.5 Let $(X,d,\u2aaf)$ be an ordered metric space such that $(X,d)$ is complete and let $f:X\to X$ be a nondecreasing self mappings. Assume that there exist $\psi \in \mathrm{\Psi}$, $\alpha \in {\mathrm{\Phi}}_{\alpha}$, and $\beta \in {\mathrm{\Phi}}_{\beta}$ such that
and
for all comparable $x,y\in X$ where
and
Suppose that either

(a)
f is continuous, or

(b)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
If there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aaff{x}_{0}$, then f has a fixed point.
On the other hand, in 2012, Samet et al. [23] introduced the concepts of αψcontractive and αadmissible mappings and established various fixed point theorems for such mappings in complete metric spaces. More recently, Salimi et al. [24] modified the notions of αψcontractive and αadmissible mappings and established fixed point theorems which are proper generalizations of the recent results in [22, 23]. For more on αadmissible mappings, see [25–27] and the references therein.
Samet et al. [23] defined the notion of αadmissible mappings as follows.
Definition 1.6 Let T be a selfmapping on X and $\alpha :X\times X\to [0,+\mathrm{\infty})$ be a function. We say that T is an αadmissible mapping if
In [23] the authors consider the family Ψ of nondecreasing functions $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ such that ${\sum}_{n=1}^{+\mathrm{\infty}}{\psi}^{n}(t)<+\mathrm{\infty}$ for each $t>0$, where ${\psi}^{n}$ is the n th iterate of ψ and give the following theorem.
Theorem 1.7 Let $(X,d)$ be a complete metric space and T be an αadmissible mapping. Assume that
for all $x,y\in X$, where $\psi \in \mathrm{\Psi}$. Also, suppose that the following assertions hold:

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

(ii)
either T is continuous or 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 T has a fixed point.
Recently, Hussain et al. [28] obtained the following Geraghty type [29] fixed point theorems via αadmissible mappings.
Theorem 1.8 Let $(X,d)$ be a complete metric space and $f:X\to X$ be an αadmissible mapping. Assume that there exists a function $\beta :[0,\mathrm{\infty})\to [0,1]$ such that for any bounded sequence $\{{t}_{n}\}$ of positive reals, $\beta ({t}_{n})\to 1$ implies ${t}_{n}\to 0$ and
for all $x,y\in X$ where $\ell \ge 1$. Suppose that either

(a)
f is continuous, or

(b)
if $\{{x}_{n}\}$ is a sequence in X such that ${x}_{n}\to x$ and $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all n, then $\alpha (x,fx)\ge 1$.
If there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},f{x}_{0})\ge 1$, then f has a fixed point.
Theorem 1.9 Let $(X,d)$ be a complete metric space and $f:X\to X$ be an αadmissible mapping. Assume that there exists a function $\beta :[0,\mathrm{\infty})\to [0,1]$ such that for any bounded sequence $\{{t}_{n}\}$ of positive reals, $\beta ({t}_{n})\to 1$ implies ${t}_{n}\to 0$ and
for all $x,y\in X$ where $0<\ell \le 1$. Suppose that either

(a)
f is continuous, or

(b)
if $\{{x}_{n}\}$ is a sequence in X such that ${x}_{n}\to x$ and $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all n, then $\alpha (x,fx)\ge 1$.
If there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},f{x}_{0})\ge 1$, then f has a fixed point.
Theorem 1.10 Let $(X,d)$ be a metric space such that $(X,d)$ is complete and $f:X\to X$ be an αadmissible mapping. Assume that there exists a function $\beta :[0,\mathrm{\infty})\to [0,1]$ such that for any bounded sequence $\{{t}_{n}\}$ of positive reals, $\beta ({t}_{n})\to 1$ implies ${t}_{n}\to 0$ and
for all $x,y\in X$. Suppose that either

(a)
f is continuous, or

(b)
if $\{{x}_{n}\}$ is a sequence in X such that ${x}_{n}\to x$ and $\alpha ({x}_{n},f{x}_{n})\ge 1$ for all n, then $\alpha (x,fx)\ge 1$.
If there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},f{x}_{0})\ge 1$, then f has a fixed point.
For more details on αadmissible mappings and related fixed point results we refer the reader to [30–32].
More recently, Salimi et al. [24] modified and generalized the notions of αψcontractive mappings and αadmissible mappings by the following ways.
Definition 1.11 [24]
Let T be a selfmapping on X and $\alpha ,\eta :X\times X\to [0,+\mathrm{\infty})$ be two functions. We say that T is an αadmissible mapping with respect to η if
Note that if we take $\eta (x,y)=1$ then this definition reduces to Definition 1.6. Also, if we take, $\alpha (x,y)=1$ then we say that T is an ηsubadmissible mapping.
The following result properly contains Theorem 1.7, and Theorems 2.3 and 2.4 of [22].
Theorem 1.12 [24]
Let $(X,d)$ be a complete metric space and T be an αadmissible mapping with respect to η. Assume that
where $\psi \in \mathrm{\Psi}$ and
Also, suppose that the following assertions hold:

(i)
there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},T{x}_{0})\ge \eta ({x}_{0},T{x}_{0})$,

(ii)
either T is continuous or for any sequence $\{{x}_{n}\}$ in X with $\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, we have $\alpha ({x}_{n},x)\ge \eta ({x}_{n},x)$ for all $n\in \mathbb{N}\cup \{0\}$.
Then T has a fixed point.
For more details on modified αψcontractive mappings and related fixed point results we refer the reader to [33, 34].
2 Main results
In this section, motivated by the work of Hussain et al. [28] and Salimi et al. [24] we state and prove the following fixed point results in the setting of partially ordered metric spaces.
Theorem 2.1 Let $(X,d,\u2aaf)$ be a partially ordered metric space such that $(X,d)$ is complete. Assume $f:X\to X$ and $\gamma :X\times X\to [0,\mathrm{\infty})$ be two mappings such that f is a nondecreasing and γadmissible mapping. Assume that there exist $\psi \in \mathrm{\Psi}$, $\alpha \in {\mathrm{\Phi}}_{\alpha}$, and $\beta \in {\mathrm{\Phi}}_{\beta}$ such that
and
for all comparable $x,y\in X$ where $\ell \ge 1$. Suppose that either

(i)
f is continuous, or

(ii)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, $\gamma ({x}_{n},f{x}_{n})\ge 1$, and $\gamma ({x}_{n},{x}_{n})\ge 1$ for all n, then $\gamma (x,x)\ge 1$, $\gamma (x,fx)\ge 1$, and ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
If there exists ${x}_{0}\in X$ such that $\gamma ({x}_{0},{x}_{0})\ge 1$, $\gamma ({x}_{0},f{x}_{0})\ge 1$, and ${x}_{0}\u2aaff{x}_{0}$, then f has a fixed point.
Proof Let ${x}_{0}\u2aaff{x}_{0}$. We define an iterative sequence $\{{x}_{n}\}$ in the following way:
Since f is nondecreasing and ${x}_{0}\u2aaff{x}_{0}$, we have
and hence $\{{x}_{n}\}$ is a nondecreasing sequence. Let $\gamma ({x}_{0},{x}_{0})\ge 1$. Since f is a γadmissible mapping and $\gamma ({x}_{0},{x}_{0})\ge 1$, we deduce that $\gamma ({x}_{1},{x}_{1})=\gamma (f{x}_{0},f{x}_{0})\ge 1$. By continuing this process, we get $\gamma ({x}_{n},{x}_{n})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$. Also, assume $\gamma ({x}_{0},f{x}_{0})\ge 1$. Similarly we get $\gamma ({x}_{n},f{x}_{n})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$. If ${x}_{{n}_{0}}={x}_{{n}_{0}+1}=f{x}_{{n}_{0}}$ for some ${n}_{0}\in \mathbb{N}$, then the point ${x}_{0}$ is the desired fixed point of f which completes the proof. Hence, we suppose that ${x}_{n}\ne {x}_{n+1}$, that is, $d({x}_{n1},{x}_{n})>0$ for all n. Hence, (2.3) implies
We want to show that the sequence $\{{d}_{n}:=d({x}_{n},{x}_{n+1})\}$ is nonincreasing sequence of reals. Suppose, to the contrary, that there exists some ${n}_{0}\in \mathbb{N}$ such that
Since ψ is nondecreasing, we obtain
Taking $x={x}_{n1}$ and $y={x}_{n}$ in (2.2) we derive
Hence
for all $n\in \mathbb{N}$. Now, by taking $x={x}_{{n}_{0}1}$ and $y={x}_{{n}_{0}}$ in (2.7) and applying (2.6) we have
which contradicts (2.12). Therefore, we conclude that ${d}_{n}<{d}_{n1}$ holds for all $n\in \mathbb{N}$. Hence $\{{d}_{n}\}$ is a nonincreasing sequence of positive real numbers. Thus, there exists $r\ge 0$ such that ${lim}_{n\to \mathrm{\infty}}{d}_{n}=r$. We shall show that $r=0$ by method of reductio ad absurdum. For this purpose, we assume that $r>0$. By (2.7) together with the properties of α, β, ψ we have
which is a contradiction. Hence
We shall show that the sequence $\{{x}_{n}\}$ is a Cauchy sequence. Suppose that it is not. Then there are $\epsilon >0$ and sequences $m(k)$ and $n(k)$ such that for all positive integers k with $n(k)>m(k)>k$
Additionally, corresponding to $m(k)$, we may choose $n(k)$ such that it is the smallest integer satisfying (2.9) and $n(k)>m(k)\ge k$. Thus,
Now, for all $k\in \mathbb{N}$ we have
So
Again, we have
and
By taking the limit as $k\to +\mathrm{\infty}$ in the above inequalities and applying (2.8) and (2.10), we deduce
Now, from (2.2) with $x={x}_{m(k)}$ and $y={x}_{n(k)}$ we have
Then
Taking the lim inf as $k\to +\mathrm{\infty}$ in the above inequality, we have
So we have
which contradicts the fact that $\psi (t)\alpha (t)+\beta (t)>0$ for all $t>0$. Hence
that is, the sequence $\{{x}_{n}\}$ is a Cauchy sequence. Since $(X,d)$ is complete, then there exists ${x}^{\ast}\in X$ such that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. Suppose that (i) holds. Then
Hence, ${x}^{\ast}$ is a fixed point of f. Suppose that (ii) holds, that is, $\gamma ({x}^{\ast},{x}^{\ast})\ge 1$, $\gamma ({x}^{\ast},f{x}^{\ast})\ge 1$, and ${x}_{n}\u2aaf{x}^{\ast}$ for all $n\ge 0$. We claim that ${x}^{\ast}$ is a fixed point of f, that is, ${lim}_{n\to \mathrm{\infty}}d({x}_{n+1},f{x}^{\ast})=0$. Suppose, to the contrary, that ${lim}_{n\to \mathrm{\infty}}d({x}_{n+1},f{x}^{\ast})=d({x}^{\ast},f{x}^{\ast})>0$. Due to condition (2.2), we have
Taking the lim inf as $n\to \mathrm{\infty}$ in the above inequality, we obtain
which is a contradiction. Hence ${lim}_{n\to \mathrm{\infty}}d({x}_{n+1},f{x}^{\ast})=d({x}^{\ast},f{x}^{\ast})=0$ and so, ${x}^{\ast}=f{x}^{\ast}$. □
Example 2.2 Let $X=[0,\mathrm{\infty})$ be endowed with the usual metric $d(x,y)=xy$ for all $x,y\in X$ and $f:X\to X$ be defined by
Define also $\gamma :X\times X\to [0,+\mathrm{\infty})$, $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$, $\alpha :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$, and $\beta :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ by
We prove that Theorem 2.1 can be applied to f. But Theorem 1.5 cannot be applied.
Clearly, $(X,d)$ is a complete metric space. We show that f is a γadmissible mapping. Let $x,y\in X$. If $\gamma (x,y)\ge 1$ then $x,y\in [0,1]$. On the other hand, for all $x\in [0,1]$ we have $fx\le 1$. It follows that $\gamma (fx,fy)\ge 1$. Thus the assertion holds. Because of the above arguments, $\gamma (0,0)\ge 1$. Now, if $\{{x}_{n}\}$ is a sequence in X such that $\gamma ({x}_{n},{x}_{n})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\{{x}_{n}\}\subset [0,1]$ and hence $x\in [0,1]$. This implies that $\gamma (x,x)\ge 1$. Also $\psi (t)=t+1/2>t/2+1/2=\alpha (t)\beta (t)$ and $\psi (t)=t+1/2>1/2=\alpha (0)\beta (0)$ for all $t>0$. Let $\gamma (x,fx)\gamma (y,fy)\ge 1$. Then $x,y\in [0,1]$. Indeed, if $x\notin [0,1]$ or $y\notin [0,1]$. So, $\gamma (x,fx)=0$ or $\gamma (y,fy)$. That is, $\gamma (x,fx)\gamma (y,fy)=0<1$ which is a contradiction. Without any loss of generality we assume that $y\ge x$. We get
Then the condition of Theorem 2.1 holds and f has a fixed point. Let $x=2$ and $y=3$, then
That is, the contractive condition of Theorem 1.5 does not hold for this example.
Corollary 2.3 Let $(X,d,\u2aaf)$ be a partially ordered metric space such that $(X,d)$ is complete. Assume $f:X\to X$ and $\gamma :X\times X\to [0,\mathrm{\infty})$ be two mappings such that f is a nondecreasing γadmissible mapping. Assume that there exist $\psi \in \mathrm{\Psi}$, $\alpha \in {\mathrm{\Phi}}_{\alpha}$, and $\beta \in {\mathrm{\Phi}}_{\beta}$ such that
Suppose that either

(i)
f is continuous, or

(ii)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, $\gamma ({x}_{n},f{x}_{n})\ge 1$, and $\gamma ({x}_{n},{x}_{n})\ge 1$ for all n, then $\gamma (x,x)\ge 1$, $\gamma (x,fx)\ge 1$, and ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$,

(iii)
$$\gamma (x,fx)\gamma (y,fy){(\psi (d(fx,fy))+\ell )}^{\gamma (x,x)\gamma (y,y)}\le \alpha (d(x,y))\beta (d(x,y))+\ell $$
for all comparable $x,y\in X$ where $\ell \ge 1$.
If there exists ${x}_{0}\in X$ such that $\gamma ({x}_{0},{x}_{0})\ge 1$, $\gamma ({x}_{0},f{x}_{0})\ge 1$, and ${x}_{0}\u2aaff{x}_{0}$, then f has a fixed point.
Proof Let $\gamma (x,fx)\gamma (y,fy)\ge 1$. Then from (iii) we have
That is,
Hence, all conditions of Theorem 2.1 hold and f has a fixed point. □
Now, we prove our second main result as follows.
Theorem 2.4 Let $(X,d,\u2aaf)$ be a partially ordered metric space such that $(X,d)$ is complete. Assume $f:X\to X$ and $\gamma :X\times X\to [0,\mathrm{\infty})$ are two mappings such that f is a nondecreasing γadmissible mapping. Assume that there exist $\psi \in \mathrm{\Psi}$, $\alpha \in {\mathrm{\Phi}}_{\alpha}$, and $\beta \in {\mathrm{\Phi}}_{\beta}$ such that
and
for all comparable $x,y\in X$. Suppose that either

(i)
f is continuous, or

(ii)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, $\gamma ({x}_{n},{x}_{n})\ge 1$, and $\gamma ({x}_{n},f{x}_{n})\ge 1$ for all n, then $\gamma (x,x)\ge 1$, $\gamma (x,fx)\ge 1$, and ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
If there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},{x}_{0})\ge 1$, $\alpha ({x}_{0},f{x}_{0})\ge 1$, and ${x}_{0}\u2aaff{x}_{0}$, then f has a fixed point.
Proof Let ${x}_{0}\u2aaff{x}_{0}$. We define an iterative sequence $\{{x}_{n}\}$ in the following way:
From Theorem 2.1 we know that $\{{x}_{n}\}$ is a nondecreasing sequence, $\gamma ({x}_{n},{x}_{n})\ge 1$ and $\gamma ({x}_{n},f{x}_{n})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$. Also, similarly, we suppose that $d({x}_{n1},{x}_{n})>0$ for all n. We shall show that the sequence $\{{d}_{n}:=d({x}_{n},{x}_{n+1})\}$ is nonincreasing sequence of reals. Assume that there exists some ${n}_{0}\in \mathbb{N}$ such that
Hence
Taking $x={x}_{n1}$ and $y={x}_{n}$ in (2.13) and applying (2.6) we get
Hence
for all $n\in \mathbb{N}$. Now, by taking $x={x}_{{n}_{0}1}$ and $y={x}_{{n}_{0}}$ in (2.15) and using (2.14) we deduce
which is a contradiction. Then ${d}_{n}<{d}_{n1}$ holds for all $n\in \mathbb{N}$ and so there exists $r\ge 0$ such that ${lim}_{n\to \mathrm{\infty}}{d}_{n}=r$. Reviewing the proof of Theorem 2.1 we can show that $r=0$. Now, suppose, to the contrary, that $\{{x}_{n}\}$ is not a Cauchy sequence. Then there exist $\epsilon >0$ and sequences $m(k)$ and $n(k)$ such that for all positive integers k with $n(k)>m(k)>k$
and
By (2.13) with $x={x}_{m(k)}$ and $y={x}_{n(k)}$ we have
and so
By taking the lim inf as $k\to +\mathrm{\infty}$ in the above inequality, we have
Therefore
which is a contradiction. Hence,
Then $\{{x}_{n}\}$ is a Cauchy sequence. Since $(X,d)$ is complete, there exists ${x}^{\ast}\in X$ such that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. Let (i) hold. Then
So, ${x}^{\ast}$ is a fixed point of f. Now, we assume that (ii) holds, that is, $\gamma ({x}^{\ast},{x}^{\ast})\ge 1$, $\gamma ({x}^{\ast},f{x}^{\ast})\ge 1$, and ${x}_{n}\u2aaf{x}^{\ast}$ for all $n\ge 0$. We claim that ${x}^{\ast}$ is a fixed point of f, equivalently, ${lim}_{n\to \mathrm{\infty}}d({x}_{n+1},f{x}^{\ast})=0$. Suppose, to the contrary, that ${lim}_{n\to \mathrm{\infty}}d({x}_{n+1},f{x}^{\ast})=d({x}^{\ast},f{x}^{\ast})>0$. From (2.13), we have
Taking the lim inf as $n\to \mathrm{\infty}$ in the above inequality, we obtain
which is a contradiction. Then ${lim}_{n\to \mathrm{\infty}}d({x}_{n+1},f{x}^{\ast})=d({x}^{\ast},f{x}^{\ast})=0$ and hence, ${x}^{\ast}=f{x}^{\ast}$. □
Example 2.5 Let X and d be as in Example 2.2. Define $f:X\to X$ by
Define also γ, ψ, α, and β as in Example 2.2. We shall show that Theorem 2.4 can be applied to f, but Theorem 1.5 cannot be applied. Proceeding as in the proof of Example 2.2 f is a γadmissible mapping, $\alpha (0,0)\ge 1$, and if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\gamma (x,x)\ge 1$. Let $\gamma (x,fx)\gamma (y,fy)\ge 1$. Then $x,y\in [0,1]$. Assume $y\ge x$. We get
Then the condition of Theorem 2.4 holds and so f has a fixed point. Let $x=ln2$ and $y=ln4$, then
Hence, the condition of Theorem 1.5 does not hold for this example.
Corollary 2.6 Let $(X,d,\u2aaf)$ be a partially ordered metric space such that $(X,d)$ is complete. Assume $f:X\to X$ and $\gamma :X\times X\to [0,\mathrm{\infty})$ are two mappings such that f is a nondecreasing γadmissible mapping. Assume that there exist $\psi \in \mathrm{\Psi}$, $\alpha \in {\mathrm{\Phi}}_{\alpha}$, and $\beta \in {\mathrm{\Phi}}_{\beta}$ such that
and
for all comparable $x,y\in X$. Suppose that either

(i)
f is continuous, or

(ii)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, $\gamma ({x}_{n},{x}_{n})\ge 1$, and $\gamma ({x}_{n},f{x}_{n})\ge 1$ for all n, then $\gamma (x,x)\ge 1$, $\gamma (x,fx)\ge 1$, and ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
If there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},{x}_{0})\ge 1$, $\alpha ({x}_{0},f{x}_{0})\ge 1$, and ${x}_{0}\u2aaff{x}_{0}$, then f has a fixed point.
Theorem 2.7 Let $(X,d,\u2aaf)$ be a partially ordered metric space such that $(X,d)$ is complete. Assume that $f:X\to X$ and $\gamma :X\times X\to [0,\mathrm{\infty})$ are two mappings such that f is a nondecreasing γadmissible mapping. Assume that there exist $\psi \in \mathrm{\Psi}$, $\alpha \in {\mathrm{\Phi}}_{\alpha}$, and $\beta \in {\mathrm{\Phi}}_{\beta}$ such that
and
for all comparable $x,y\in X$. Suppose that either

(i)
f is continuous, or

(ii)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, $\gamma ({x}_{n},{x}_{n})\ge 1$, and $\gamma ({x}_{n},f{x}_{n})\ge 1$ for all n, then $\gamma (x,x)\ge 1$, $\gamma (x,fx)\ge 1$, and ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
If there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},{x}_{0})\ge 1$, $\alpha ({x}_{0},f{x}_{0})\ge 1$, and ${x}_{0}\u2aaff{x}_{0}$, then f has a fixed point.
Proof Let ${x}_{0}\u2aaff{x}_{0}$. We define an iterative sequence $\{{x}_{n}\}$ in the following way:
From Theorem 2.1 we know that $\{{x}_{n}\}$ is a nondecreasing sequence, $\gamma ({x}_{n},{x}_{n})\ge 1$, and $\gamma ({x}_{n},f{x}_{n})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$. Also, similarly, we suppose that $d({x}_{n1},{x}_{n})>0$ for all n. We shall show that the sequence $\{{d}_{n}:=d({x}_{n},{x}_{n+1})\}$ is nonincreasing. Assume that there exists some ${n}_{0}\in \mathbb{N}$ such that
Hence
Taking $x={x}_{n1}$ and $y={x}_{n}$ in (2.18) and applying (2.19) we get
Hence
for all $n\in \mathbb{N}$. Now, by taking $x={x}_{{n}_{0}1}$ and $y={x}_{{n}_{0}}$ in (2.20) and using (2.19) we deduce
which is a contradiction. Then ${d}_{n}<{d}_{n1}$ holds for all $n\in \mathbb{N}$ and so there exists $r\ge 0$ such that ${lim}_{n\to \mathrm{\infty}}{d}_{n}=r$. Proceeding as in the proof of Theorem 2.1 we conclude that $r=0$. Now, suppose, to the contrary that $\{{x}_{n}\}$ is not a Cauchy sequence. Then there exist $\epsilon >0$ and sequences $m(k)$ and $n(k)$ such that for all positive integers k with $n(k)>m(k)>k$
and
By (2.18) with $x={x}_{m(k)}$ and $y={x}_{n(k)}$ we have
and so
Taking the lim inf as $k\to +\mathrm{\infty}$ in the above inequality, we have
Therefore
which is a contradiction. Hence
that is, $\{{x}_{n}\}$ is a Cauchy sequence. Since $(X,d)$ is complete, there exists ${x}^{\ast}\in X$ such that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. Let (i) hold. Then
So, ${x}^{\ast}$ is a fixed point of f. Now, we assume that (ii) holds, that is, $\gamma ({x}^{\ast},{x}^{\ast})\ge 1$, $\gamma ({x}^{\ast},f{x}^{\ast})\ge 1$, and ${x}_{n}\u2aaf{x}^{\ast}$ for all $n\ge 0$. We claim that ${x}^{\ast}$ is a fixed point of f, or equivalently, ${lim}_{n\to \mathrm{\infty}}d({x}_{n+1},f{x}^{\ast})=0$. Suppose, to the contrary, that ${lim}_{n\to \mathrm{\infty}}d({x}_{n+1},f{x}^{\ast})=d({x}^{\ast},f{x}^{\ast})>0$. From (2.18), we have
Taking the lim inf as $n\to \mathrm{\infty}$ in the above inequality, we obtain
which is a contradiction. Then ${lim}_{n\to \mathrm{\infty}}d({x}_{n+1},f{x}^{\ast})=d({x}^{\ast},f{x}^{\ast})=0$, and hence ${x}^{\ast}=f{x}^{\ast}$. □
Example 2.8 Let X and d be as in Example 2.2. Define $f:X\to X$ by
Define also γ, ψ, α, and β as in Example 2.2. We shall show that Theorem 2.7 can be applied for f, but Theorem 1.5 cannot be applied. Reviewing the proof of Example 2.2, f is a γadmissible mapping, $\alpha (0,0)\ge 1$ and if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\gamma (x,x)\ge 1$. Let $\gamma (x,fx)\gamma (y,fy)\ge 1$. Then $x,y\in [0,1]$. Assume $y\ge x$. We get
Then the condition of Theorem 2.7 holds and f has a fixed point. Clearly, the condition of Theorem 1.5 does not hold for this example.
Corollary 2.9 Let $(X,d,\u2aaf)$ be a partially ordered metric space such that $(X,d)$ is complete. Assume $f:X\to X$ and $\gamma :X\times X\to [0,\mathrm{\infty})$ are two mappings such that f is a nondecreasing γadmissible mapping. Assume that there exist $\psi \in \mathrm{\Psi}$, $\alpha \in {\mathrm{\Phi}}_{\alpha}$, and $\beta \in {\mathrm{\Phi}}_{\beta}$ such that
and
for all comparable $x,y\in X$. Suppose that either

(i)
f is continuous, or

(ii)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, $\gamma ({x}_{n},{x}_{n})\ge 1$, and $\gamma ({x}_{n},f{x}_{n})\ge 1$ for all n, then $\gamma (x,x)\ge 1$, $\gamma (x,fx)\ge 1$, and ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
If there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},{x}_{0})\ge 1$, $\alpha ({x}_{0},f{x}_{0})\ge 1$, and ${x}_{0}\u2aaff{x}_{0}$, then f has a fixed point.
References
 1.
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.
 2.
Berinde V, Vetro F: Common fixed points of mappings satisfying implicit contractive conditions. Fixed Point Theory Appl. 2012., 2012: Article ID 105
 3.
Chatterjea SK: Fixed point theorem. C. R. Acad. Bulgare Sci. 1972, 25: 727–730.
 4.
Ćirić LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 1974, 45: 267–273.
 5.
Damjanovic B, Samet B, Vetro C: Common fixed point theorems for multivalued maps. Acta Math. Sci. Ser. B, Engl. Ed. 2012, 32: 818–824.
 6.
Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, London; 2001.
 7.
Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 60: 71–76.
 8.
Kannan R: Some results on fixed points  II. Am. Math. Mon. 1969, 76: 405–408. 10.2307/2316437
 9.
Matthews SG: Partial metric topology. Ann. New York Acad. Sci. 728. Proc. 8th Summer Conference on General Topology and Applications 1994, 183–197.
 10.
Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7(2):289–297.
 11.
Nadler SB Jr.: Multivalued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475
 12.
Reich S: Kannan’s fixed point theorem. Boll. Unione Mat. Ital. 1971, 4: 1–11.
 13.
Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136: 1861–1869.
 14.
Vetro F: On approximating curves associated with nonexpansive mappings. Carpath. J. Math. 2011, 27: 142–147.
 15.
Dutta PN, Choudhury BS: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 406368
 16.
Abbas M, Dorić D: Common fixed point theorem for four mappings satisfying generalized weak contractive condition. Filomat 2010, 24(2):1–10. 10.2298/FIL1002001A
 17.
Dorić D:Common fixed point for generalized $(\psi ,\varphi )$weak contractions. Appl. Math. Lett. 2009, 22: 1896–1900. 10.1016/j.aml.2009.08.001
 18.
Aydi H, Karapinar E, Samet B: Remarks on some recent fixed point theorems. Fixed Point Theory Appl. 2012., 2012: Article ID 76
 19.
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002993903072204
 20.
Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003
 21.
Choudhury BS, Kundu A:$(\psi ,\alpha ,\beta )$Weak contractions in partially ordered metric spaces. Appl. Math. Lett. 2012, 25: 6–10. 10.1016/j.aml.2011.06.028
 22.
Karapinar E, Salimi P: Fixed point theorems via auxiliary functions. J. Appl. Math. 2012., 2012: Article ID 792174
 23.
Samet B, Vetro C, Vetro P: Fixed point theorems for α  ψ contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014
 24.
Salimi P, Latif A, Hussain N: Modified α  ψ contractive mappings with applications. Fixed Point Theory Appl. 2013., 2013: Article ID 151
 25.
Mohammadi B, Rezapour S, Shahzad N: Some results on fixed points of α  ψ Ciric generalized multifunctions. Fixed Point Theory Appl. 2013., 2013: Article ID 24
 26.
Asl JH, Rezapour S, Shahzad N: On fixed points of α  ψ contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212
 27.
Alikhani H, Rezapour S, Shahzad N: Fixed points of a new type of contractive mappings and multifunctions. Filomat 2013, 27: 1315–1319. 10.2298/FIL1307315A
 28.
Hussain N, Karapinar E, Salimi P, Akbar F: α Admissible mappings and related fixed point theorems. J. Inequal. Appl. 2013., 2013: Article ID 114
 29.
Geraghty M: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S00029939197303341765
 30.
Hussain N, Karapinar E, Salimi P, Vetro P: Fixed point results for ${G}^{m}$ MeirKeeler contractive and G  $(\alpha ,\psi )$ MeirKeeler contractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 34
 31.
Salimi P, Vetro C, Vetro P: Fixed point theorems for twisted $(\alpha ,\beta )$  ψ contractive type mappings and applications. Filomat 2013, 27(4):605–615. 10.2298/FIL1304605S
 32.
Salimi P, Vetro C, Vetro P: Some new fixed point results in nonArchimedean fuzzy metric spaces. Nonlinear Anal., Model. Control 2013, 18(3):344–358.
 33.
Hussain N, Kutbi MA, Salimi P: Best proximity point results for modified α  ψ proximal rational contractions. Abstr. Appl. Anal. 2013., 2013: Article ID 927457
 34.
Hussain N, Salimi P, Latif A: Fixed point results for single and setvalued α  η  ψ contractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 212
Acknowledgements
The authors are thankful to the referees for their valuable comments on this paper. Wei Long acknowledges support from the Research Project of Jiangxi Normal University (2012114).
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
About this article
Received
Accepted
Published
DOI
Keywords
 partially ordered set
 admissible mappings
 fixed point