# Fixed point results via *α*-admissible mappings and cyclic contractive mappings in partial *b*-metric spaces

- Abdul Latif
^{1}Email author, - Jamal Rezaei Roshan
^{2}, - Vahid Parvaneh
^{3}and - Nawab Hussain
^{1}

**2014**:345

https://doi.org/10.1186/1029-242X-2014-345

© Latif et al.; licensee Springer 2014

**Received: **7 March 2014

**Accepted: **8 August 2014

**Published: **3 September 2014

## Abstract

Considering *α*-admissible mappings in the setup of partial *b*-metric spaces, we establish some fixed and common fixed point results for ordered cyclic weakly $(\psi ,\phi ,L,A,B)$-contractive mappings in complete ordered partial *b*-metric spaces. Our results extend several known results in the literature. Examples are also provided in support of our results.

**MSC:**47H10, 54H25.

## Keywords

## 1 Introduction

There are a lot of generalizations of the concept of metric space. The concepts of *b*-metric space and partial metric space were introduced by Czerwik [1] and Matthews [2], respectively. Combining these two notions, Shukla [3] introduced another generalization which is called a partial *b*-metric space. Also, in [4], Mustafa *et al.* introduced a modified version of partial *b*-metric spaces. In fact, the advantage of their definition of partial *b*-metric is that by using it one can define a dependent *b*-metric which is called the *b*-metric associated with the partial *b*-metric.

**Definition 1.1** [4]

Let *X* be a (nonempty) set and $s\ge 1$ be a given real number. A function ${p}_{b}:X\times X\to {\mathbb{R}}^{+}$ is a partial *b*-metric if, for all $x,y,z\in X$, the following conditions are satisfied:

$({p}_{b1})$ $x=y\u27fa{p}_{b}(x,x)={p}_{b}(x,y)={p}_{b}(y,y)$,

$({p}_{b2})$ ${p}_{b}(x,x)\le {p}_{b}(x,y)$,

$({p}_{b3})$ ${p}_{b}(x,y)={p}_{b}(y,x)$,

$({p}_{b4})$ ${p}_{b}(x,y)\le s({p}_{b}(x,z)+{p}_{b}(z,y)-{p}_{b}(z,z))+(\frac{1-s}{2})({p}_{b}(x,x)+{p}_{b}(y,y))$.

The pair $(X,{p}_{b})$ is called a partial *b*-metric space.

**Example 1.2** [3]

Then $(X,{p}_{b})$ is a partial *b*-metric space with the coefficient $s={2}^{q-1}>1$, but it is neither a *b*-metric nor a partial metric space.

Some more examples of partial *b*-metrics can be constructed with the help of following propositions.

**Proposition 1.3** [3]

*Let* *X* *be a nonempty set and let* *p* *be a partial metric and* *d* *be a* *b*-*metric with the coefficient* $s\ge 1$ *on* *X*. *Then the function* ${p}_{b}:X\times X\to {\mathbb{R}}^{+}$ *defined by* ${p}_{b}(x,y)=p(x,y)+d(x,y)$, *for all* $x,y\in X$, *is a partial* *b*-*metric on* *X* *with the coefficient* *s*.

**Proposition 1.4** [3]

*Let* $(X,p)$ *be a partial metric space and* $q\ge 1$. *Then* $(X,{p}_{b})$ *is a partial* *b*-*metric space with the coefficient* $s={2}^{q-1}$, *where* ${p}_{b}$ *is defined by* ${p}_{b}(x,y)={[p(x,y)]}^{q}$.

**Proposition 1.5** [4]

*Every partial*

*b*-

*metric*${p}_{b}$

*defines a*

*b*-

*metric*${d}_{{p}_{b}}$,

*where*

*for all* $x,y\in X$.

Now, we recall some definitions and propositions in a partial *b*-metric space.

**Definition 1.6** [4]

*b*-metric space. Then for an $x\in X$ and an $\u03f5>0$, the ${p}_{b}$-ball with center

*x*and radius

*ϵ*is

**Proposition 1.7** [4]

*Let* $(X,{p}_{b})$ *be a partial* *b*-*metric space*, $x\in X$, *and* $r>0$. *If* $y\in {B}_{{p}_{b}}(x,r)$ *then there exists a* $\delta >0$ *such that* ${B}_{{p}_{b}}(y,\delta )\subseteq {B}_{{p}_{b}}(x,r)$.

is a base of a ${T}_{0}$ topology ${\tau}_{{p}_{b}}$ on *X* which we call the ${p}_{b}$-metric topology.

The topological space $(X,{p}_{b})$ is ${T}_{0}$, but it does not need to be ${T}_{1}$.

**Definition 1.8** [4]

*b*-metric space $(X,{p}_{b})$ is said to be:

- (i)
${p}_{b}$-convergent to a point $x\in X$ if ${lim}_{n\to \mathrm{\infty}}{p}_{b}(x,{x}_{n})={p}_{b}(x,x)$.

- (ii)
A ${p}_{b}$-Cauchy sequence if ${lim}_{n,m\to \mathrm{\infty}}{p}_{b}({x}_{n},{x}_{m})$ exists (and is finite).

- (iii)
A partial

*b*-metric space $(X,{p}_{b})$ is said to be ${p}_{b}$-complete if every ${p}_{b}$-Cauchy sequence $\{{x}_{n}\}$ in*X*${p}_{b}$-converges to a point $x\in X$ such that ${lim}_{n,m\to \mathrm{\infty}}{p}_{b}({x}_{n},{x}_{m})={lim}_{n,m\to \mathrm{\infty}}{p}_{b}({x}_{n},x)={p}_{b}(x,x)$.

**Lemma 1.9** [4]

- (1)
*A sequence*$\{{x}_{n}\}$*is a*${p}_{b}$-*Cauchy sequence in a partial**b*-*metric space*$(X,{p}_{b})$*if and only if it is a**b*-*Cauchy sequence in the**b*-*metric space*$(X,{d}_{{p}_{b}})$. - (2)
*A partial**b*-*metric space*$(X,{p}_{b})$*is*${p}_{b}$-*complete if and only if the**b*-*metric space*$(X,{d}_{{p}_{b}})$*is**b*-*complete*.*Moreover*, ${lim}_{n\to \mathrm{\infty}}{d}_{{p}_{b}}(x,{x}_{n})=0$*if and only if*$\underset{n\to \mathrm{\infty}}{lim}{p}_{b}(x,{x}_{n})=\underset{n,m\to \mathrm{\infty}}{lim}{p}_{b}({x}_{n},{x}_{m})={p}_{b}(x,x).$

**Definition 1.10** [4]

Let $(X,{p}_{b})$ and $({X}^{\prime},{p}_{b}^{\prime})$ be two partial *b*-metric spaces and let $f:(X,{p}_{b})\to ({X}^{\prime},{p}_{b}^{\prime})$ be a mapping. Then *f* is said to be ${p}_{b}$-continuous at a point $a\in X$ if for a given $\epsilon >0$, there exists $\delta >0$ such that $x\in X$ and ${p}_{b}(a,x)<\delta +{p}_{b}(a,a)$ imply that ${p}_{b}^{\prime}(f(a),f(x))<\epsilon +{p}_{b}^{\prime}(f(a),f(a))$. The mapping *f* is ${p}_{b}$-continuous on *X* if it is ${p}_{b}$-continuous at all $a\in X$.

**Proposition 1.11** [4]

*Let* $(X,{p}_{b})$ *and* $({X}^{\prime},{p}_{b}^{\prime})$ *be two partial* *b*-*metric spaces*. *Then a mapping* $f:X\to {X}^{\prime}$ *is* ${p}_{b}$-*continuous at a point* $x\in X$ *if and only if it is* ${p}_{b}$-*sequentially continuous at* *x*; *that is*, *whenever* $\{{x}_{n}\}$ *is* ${p}_{b}$-*convergent to* *x*, $\{f({x}_{n})\}$ *is* ${p}_{b}^{\prime}$-*convergent to* $f(x)$.

**Definition 1.12** A triple $(X,\u2aaf,{p}_{b})$ is called an ordered partial *b*-metric space if $(X,\u2aaf)$ is a partially ordered set and ${p}_{b}$ is a partial *b*-metric on *X*.

The following crucial lemma is useful in proving our main results.

**Lemma 1.13** [4]

*Let*$(X,{p}_{b})$

*be a partial*

*b*-

*metric space with the coefficient*$s>1$

*and suppose that*$\{{x}_{n}\}$

*and*$\{{y}_{n}\}$

*are convergent to*

*x*

*and*

*y*,

*respectively*.

*Then we have*

*In particular*, *if* ${p}_{b}(x,y)=0$, *then we have* ${lim}_{n\to \mathrm{\infty}}{p}_{b}({x}_{n},{y}_{n})=0$.

*Moreover*,

*for each*$z\in X$

*we have*

*In particular*,

*if*${p}_{b}(x,x)=0$,

*then we have*

One of the interesting generalizations of the Banach contraction principle was given by Kirk *et al.* [5] in 2003 by introducing the notion of cyclic representation.

**Definition 1.14** [5]

Let *A* and *B* be nonempty subsets of a metric space $(X,d)$ and $T:A\cup B\to A\cup B$. Then *T* is called a cyclic map if $T(A)\subseteq B$ and $T(B)\subseteq A$.

The following interesting theorem for a cyclic map was given in [5].

**Theorem 1.15** [5]

*Let*

*A*

*and*

*B*

*be nonempty closed subsets of a complete metric space*$(X,d)$.

*Suppose that*$T:A\cup B\to A\cup B$

*is a cyclic map such that*

*for all* $x\in A$ *and* $y\in B$, *where* $k\in [0,1)$ *is a constant*. *Then* *T* *has a unique fixed point* *u* *and* $u\in A\cap B$.

Berinde initiated in [6, 7] the concept of almost contractions and obtained several interesting fixed point theorems for Ćirić strong almost contractions. Babu *et al.* introduced in [8] the class of mappings which satisfy ‘condition (*B*)’. Moreover, they proved the existence of fixed points for such mappings on complete metric spaces. Finally, Ćirić *et al.* in [9], and Aghajani *et al.* in [10] introduced the concept of almost generalized contractive conditions (for two, resp. four mappings) and proved some important results in ordered metric spaces. Let us recall one of these definitions.

**Definition 1.16** [9]

*f*and

*g*be two self-mappings on a metric space $(X,d)$. They are said to satisfy almost generalized contractive condition, if there exist a constant $\delta \in (0,1)$ and some $L\ge 0$ such that

for all $x,y\in X$.

**Definition 1.17** [11]

- (1)
*φ*is continuous and nondecreasing. - (2)
$\phi (t)=0$ if and only if $t=0$.

**Definition 1.18** [12]

Let $(X,\u2aaf)$ be a partially ordered set and *A* and *B* be closed subsets of *X* with $X=A\cup B$. Let $f,g:X\to X$ be two mappings. The pair $(f,g)$ is said to be $(A,B)$-weakly increasing if $fx\u2aafgfx$, for all $x\in A$ and $gy\u2aaffgy$, for all $y\in B$.

In [13], Hussain *et al.* introduced the notion of ordered cyclic weakly $(\psi ,\phi ,L,A,B)$-contractive pair of self-mappings as follows.

**Definition 1.19** [13]

*b*-metric space, let $f,g:X\to X$ be two mappings, and let

*A*and

*B*be nonempty closed subsets of

*X*. The pair $(f,g)$ is called an ordered cyclic weakly $(\psi ,\phi ,L,A,B)$-contraction if

- (1)
$X=A\cup B$ is a cyclic representation of

*X*w.r.t. the pair $(f,g)$; that is, $fA\subseteq B$ and $gB\subseteq A$; - (2)there exist two altering distance functions
*ψ*,*φ*and a constant $L\ge 0$, such that for arbitrary comparable elements $x,y\in X$ with $x\in A$ and $y\in B$, we have$\psi ({s}^{2}d(fx,gy))\le \psi ({M}_{s}(x,y))-\phi ({M}_{s}(x,y))+L\psi (N(x,y)),$

Also, in [13] the authors proved the following results.

**Theorem 1.20** [13]

*Let*$(X,\u2aaf,d)$

*be a complete ordered*

*b*-

*metric space and*

*A*

*and*

*B*

*be closed subsets of*

*X*.

*Let*$f,g:X\to X$

*be two*$(A,B)$-

*weakly increasing mappings with respect to*⪯.

*Suppose that*:

- (a)
*the pair*$(f,g)$*is an ordered cyclic weakly*$(\psi ,\phi ,L,A,B)$-*contraction*; - (b)
*f**or**g**is continuous*.

*Then* *f* *and* *g* *have a common fixed point* $u\in A\cap B$.

An ordered *b*-metric space $(X,\u2aaf,d)$ is called *regular* if for any nondecreasing sequence $\{{x}_{n}\}$ in *X* such that ${x}_{n}\to x\in X$, as $n\to \mathrm{\infty}$, one has ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.

**Theorem 1.21** [13]

*Let the hypotheses of Theorem * 1.20 *be satisfied*, *except that condition* (b) *is replaced by the assumption*

(b′) *the space* $(X,\u2aaf,d)$ *is regular*.

*Then* *f* *and* *g* *have a common fixed point in* *X*.

In this paper, first we prove some fixed point results for *α*-admissible mappings in the context of partial *b*-metric spaces. Then we express some common fixed point results for cyclic generalized almost contractive mappings. Our results extend and generalize some recent results in [4] and [13]. In fact, they are cyclic variants of the results in [4].

## 2 Fixed point results via *α*-admissible mappings in partial *b*-metric spaces

Samet *et al.* [14] defined the notion of *α*-admissible mappings and proved the following result.

**Definition 2.1** [14]

*T*be a self-mapping on

*X*and $\alpha :X\times X\to [0,\mathrm{\infty})$ be a function. We say that

*T*is an

*α*-admissible mapping if

Denote by ${\mathrm{\Psi}}^{\prime}$ 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*.

**Theorem 2.2** [14]

*Let*$(X,d)$

*be a complete metric space and*

*T*

*be an*

*α*-

*admissible mapping*.

*Assume that*

*where*$\psi \in {\mathrm{\Psi}}^{\prime}$.

*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*$n\in \mathbb{N}\cup \{0\}$*such that*${x}_{n}\to x$*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*.

We now recall the concept of $(c)$*-comparison function* which was introduced by Berinde [15].

**Definition 2.3** (Berinde [15])

A function $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is said to be a $(c)$-comparison function if

(${c}_{1}$) *φ* is increasing,

(${c}_{2}$) there exist ${k}_{0}\in \mathbb{N}$, $a\in (0,1)$, and a convergent series of nonnegative terms ${\sum}_{k=1}^{\mathrm{\infty}}{v}_{k}$ such that ${\phi}^{k+1}(t)\le a{\phi}^{k}(t)+{v}_{k}$, for $k\ge {k}_{0}$ and any $t\in [0,\mathrm{\infty})$.

Later, Berinde [16] introduced the notion of $(b)$-comparison function as a generalization of a $(c)$-comparison function.

**Definition 2.4** (Berinde [16])

- (1)
*φ*is monotone increasing; - (2)
there exist ${k}_{0}\in \mathbb{N}$, $a\in (0,1)$, and a convergent series of nonnegative terms ${\sum}_{k=1}^{\mathrm{\infty}}{v}_{k}$ such that ${s}^{k+1}{\phi}^{k+1}(t)\le a{s}^{k}{\phi}^{k}(t)+{v}_{k}$, for $k\ge {k}_{0}$ and any $t\in [0,\mathrm{\infty})$.

Let ${\mathrm{\Psi}}_{b}$ be the class of $(b)$-comparison functions $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$. It is clear that the notion of $(b)$-comparison function coincides with $(c)$-comparison function for $s=1$.

We now recall the following lemma, which will simplify the proofs.

**Lemma 2.5** (Berinde [17])

*If*$\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*is a*$(b)$-

*comparison function*,

*then we have the following*.

- (1)
*the series*${\sum}_{k=0}^{\mathrm{\infty}}{s}^{k}{\phi}^{k}(t)$*converges for any*$t\in {\mathbb{R}}_{+}$; - (2)
*the function*${b}_{s}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$,*defined by*${b}_{s}(t)={\sum}_{k=0}^{\mathrm{\infty}}{s}^{k}{\phi}^{k}(t)$, $t\in [0,\mathrm{\infty})$,*is increasing and continuous at*0.

**Theorem 2.6**

*Let*$(X,{p}_{b})$

*be a*${p}_{b}$-

*complete partial*

*b*-

*metric space*,

*f*

*be a continuous*

*α*-

*admissible mapping on*

*X*,

*there exists*${x}_{0}\in X$

*such that*$\alpha ({x}_{0},f{x}_{0})\ge 1$

*and if any sequence*$\{{x}_{n}\}$

*in*

*X*${p}_{b}$-

*converges to a point*

*x*,

*where*$\alpha ({x}_{n},{x}_{n+1})\ge 1$

*for all*

*n*,

*then we have*$\alpha (x,x)\ge 1$.

*Assume that*

*for all*$x,y\in X$,

*where*$\psi \in {\mathrm{\Psi}}_{b}$

*and*

Then *f* has a fixed point.

*Proof* Let ${x}_{0}\in X$ be such that $\alpha ({x}_{0},f{x}_{0})\ge 1$. Define a sequence $\{{x}_{n}\}$ by ${x}_{n}={f}^{n}{x}_{0}$ for all $n\in \mathbb{N}$. Since *f* is an *α*-admissible mapping and $\alpha ({x}_{0},{x}_{1})=\alpha ({x}_{0},f{x}_{0})\ge 1$, we deduce that $\alpha ({x}_{1},{x}_{2})=\alpha (f{x}_{0},f{x}_{1})\ge 1$. Continuing this process, we get that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$.

Now, we will finish the proof in the following steps.

for each $n=1,2,3,\dots $ .

If ${x}_{n}={x}_{n+1}$, for some $n\in \mathbb{N}$, then ${x}_{n}=f{x}_{n}$. Thus, ${x}_{n}$ is a fixed point of *f*. Therefore, we assume that ${x}_{n}\ne {x}_{n+1}$, for all $n\in \mathbb{N}$.

a contradiction.

So (2.3) holds.

as $n\u27f6\mathrm{\infty}$.

*b*-metric space

*X*, from Lemma 1.9, $\{{x}_{n}\}$ is a

*b*-Cauchy sequence in the

*b*-metric space $(X,{d}_{{p}_{b}})$. ${p}_{b}$-Completeness of $(X,{p}_{b})$ shows that $(X,{d}_{{p}_{b}})$ is also

*b*-complete. Then there exists $z\in X$ such that

*f*we have

Hence, ${p}_{b}(z,fz)\le \psi ({p}_{b}(z,fz))$. Thus, ${p}_{b}(z,fz)=0$, that is, $z=fz$. □

In Theorem 2.6, we omit the continuity of the mapping *f* and we replace $\alpha ({x}_{n},x)\ge 1$ instead of $\alpha (x,x)\ge 1$ and rearrange it as follows.

**Theorem 2.7**

*Let*$(X,{p}_{b})$

*be a*${p}_{b}$-

*complete partial*

*b*-

*metric space and*

*f*

*be an*

*α*-

*admissible mapping on*

*X*

*such that*

*for all*$x,y\in X$,

*where*$\psi \in {\mathrm{\Psi}}_{b}$.

*Assume that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},f{x}_{0})\ge 1$; - (ii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all**n**and*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$,*then*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *f* *has a fixed point*.

*Proof*Let ${x}_{0}\in X$ be such that $\alpha ({x}_{0},f{x}_{0})\ge 1$ and define a sequence $\{{x}_{n}\}$ in

*X*by ${x}_{n}={f}^{n}{x}_{0}=f{x}_{n-1}$ for all $n\in \mathbb{N}$. Following the proof of Theorem 2.6, we have $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and there exists $z\in X$ such that ${x}_{n}\to z$ as $n\to \mathrm{\infty}$ which ${p}_{b}(z,z)=0$. Hence, from (ii) we deduce that $\alpha ({x}_{n},z)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$. Therefore, by (2.7), we obtain

which implies that $z=fz$. □

**Definition 2.8** [18]

Let $f:X\to X$ and $\alpha :X\times X\to \mathbb{R}$. We say that *f* is a triangular *α*-admissible mapping if

(T1) $\alpha (x,y)\ge 1$ implies $\alpha (fx,fy)\ge 1$, $x,y\in X$,

(T2) $\{\begin{array}{l}\alpha (x,z)\ge 1\\ \alpha (z,y)\ge 1\end{array}$ implies $\alpha (x,y)\ge 1$, $x,y,z\in X$.

**Example 2.9** [18]

Let $X=\mathbb{R}$, $fx=\sqrt[3]{x}$, and $\alpha (x,y)={e}^{x-y}$, then *f* is a triangular *α*-admissible mapping. Indeed, if $\alpha (x,y)={e}^{x-y}\ge 1$, then $x\ge y$ which implies that $fx\ge fy$, that is, $\alpha (fx,fy)={e}^{fx-fy}\ge 1$. Also, if $\{\begin{array}{l}\alpha (x,z)\ge 1\\ \alpha (z,y)\ge 1\end{array}$, then $\{\begin{array}{l}x-z\ge 0\\ z-y\ge 0\end{array}$, that is, $x-y\ge 0$ and therefore $\alpha (x,y)={e}^{x-y}\ge 1$.

**Example 2.10** [18]

Let $X=\mathbb{R}$, $fx={e}^{{x}^{7}}$, and $\alpha (x,y)=\sqrt[5]{x-y}+1$. Hence, *f* is a triangular *α*-admissible mapping. Indeed, if $\alpha (x,y)=\sqrt[5]{x-y}+1\ge 1$ then $x\ge y$ which implies that $fx\ge fy$, that is, $\alpha (fx,fy)\ge 1$.

Moreover, if $\{\begin{array}{l}\alpha (x,z)\ge 1\\ \alpha (z,y)\ge 1\end{array}$, then $x-y\ge 0$ and hence $\alpha (x,y)\ge 1$.

**Example 2.11** [18]

*f*is a triangular

*α*-admissible mapping. In fact, if

Thus, $\alpha (x,z)+\alpha (z,y)\le 2\alpha (x,y)$. Now, if $\{\begin{array}{l}\alpha (x,z)\ge 1\\ \alpha (z,y)\ge 1\end{array}$, then $\alpha (x,y)\ge 1$.

**Example 2.12** [18]

Let $X=\mathbb{R}$, $fx={x}^{3}+\sqrt[7]{x}$, and $\alpha (x,y)={x}^{5}-{y}^{5}+1$. Then *f* is a triangular *α*-admissible mapping.

**Lemma 2.13** [18]

*Let*

*f*

*be a triangular*

*α*-

*admissible mapping*.

*Assume that there exists*${x}_{0}\in X$

*such that*$\alpha ({x}_{0},f{x}_{0})\ge 1$.

*Define the sequence*$\{{x}_{n}\}$

*by*${x}_{n}={f}^{n}{x}_{0}$.

*Then*

A mapping $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is called a *comparison function* if it is increasing and ${\psi}^{n}(t)\to 0$, as $n\to \mathrm{\infty}$ for any $t\in [0,\mathrm{\infty})$.

**Lemma 2.14** (Berinde [15], Rus [19])

*If*$\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*is a comparison function*,

*then*:

- (1)
*each iterate*${\psi}^{k}$*of**ψ*, $k\ge 1$,*is also a comparison function*; - (2)
*ψ**is continuous at*0; - (3)
$\psi (t)<t$,

*for any*$t>0$.

Denote by Ψ the family of all continuous comparison functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$.

**Theorem 2.15**

*Let*$(X,{p}_{b})$

*be a*${p}_{b}$-

*complete partial*

*b*-

*metric space*,

*f*

*be a continuous triangular*

*α*-

*admissible mapping on*

*X*,

*there exists*${x}_{0}\in X$

*such that*$\alpha ({x}_{0},f{x}_{0})\ge 1$

*and if any sequence*$\{{x}_{n}\}$

*in*

*X*${p}_{b}$-

*converges to a point*

*x*,

*where*$\alpha ({x}_{n},{x}_{n+1})\ge 1$

*for all*

*n*,

*then we have*$\alpha (x,x)\ge 1$.

*Assume that*

*for all* $x,y\in X$. *Then* *f* *has a fixed point*.

*Proof* Let ${x}_{0}\in X$ be such that $\alpha ({x}_{0},f{x}_{0})\ge 1$. Define a sequence $\{{x}_{n}\}$ by ${x}_{n}={f}^{n}{x}_{0}$ for all $n\in \mathbb{N}$. Since *f* is an *α*-admissible mapping and $\alpha ({x}_{0},{x}_{1})=\alpha ({x}_{0},f{x}_{0})\ge 1$, we deduce that $\alpha ({x}_{1},{x}_{2})=\alpha (f{x}_{0},f{x}_{1})\ge 1$. Continuing this process, we get $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$.

Now, we will finish the proof in the following steps.

for each $n=1,2,3,\dots $ .

If ${x}_{n}={x}_{n+1}$, for some $n\in \mathbb{N}$, then ${x}_{n}=f{x}_{n}$. Thus, ${x}_{n}$ is a fixed point of *f*. Therefore, we assume that ${x}_{n}\ne {x}_{n+1}$, for all $n\in \mathbb{N}$.

a contradiction.

So (2.9) holds.

*X*. For this, we have to show that $\{{x}_{n}\}$ is a

*b*-Cauchy sequence in $(X,{d}_{{p}_{b}})$ (see Lemma 1.9). Suppose the contrary; that is, $\{{x}_{n}\}$ is not a

*b*-Cauchy sequence. Then there exists $\epsilon >0$ for which we can find two subsequences $\{{x}_{{m}_{i}}\}$ and $\{{x}_{{n}_{i}}\}$ of $\{{x}_{n}\}$ such that ${n}_{i}$ is the smallest index for which

a contradiction.

Step III. There exists *z* such that $fz=z$.

*b*-metric space

*X*, from Lemma 1.9, $\{{x}_{n}\}$ is a

*b*-Cauchy sequence in the

*b*-metric space $(X,{d}_{{p}_{b}})$. ${p}_{b}$-Completeness of $(X,{p}_{b})$ shows that $(X,{d}_{{p}_{b}})$ is also

*b*-complete. Then there exists $z\in X$ such that

*f*we have

Hence, ${p}_{b}(z,fz)\le \psi ({p}_{b}(z,fz))$. Thus, ${p}_{b}(z,fz)=0$, that is, $z=fz$. □

If in Theorem 2.15 we take $\alpha (x,y)=1$ then we deduce the following corollary.

**Corollary 2.16**

*Let*$(X,{p}_{b})$

*be a*${p}_{b}$-

*complete partial*

*b*-

*metric space and*

*f*

*be a continuous mapping on*

*X*.

*Assume that*

*for all* $x,y\in X$. *Then* *f* *has a fixed point*.

In Theorem 2.15, we omit the continuity of the mapping *f* and we replace $\alpha ({x}_{n},x)\ge 1$ instead of $\alpha (x,x)\ge 1$ and rearrange it as follows.

**Theorem 2.17**

*Let*$(X,{p}_{b})$

*be a*${p}_{b}$-

*complete partial*

*b*-

*metric space and*

*f*

*be a triangular*

*α*-

*admissible mapping on*

*X*

*such that*

*for all*$x,y\in X$,

*where*$\psi \in \mathrm{\Psi}$.

*Assume that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},f{x}_{0})\ge 1$; - (ii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all**n**and*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$,*then*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *f* *has a fixed point*.

*Proof*Let ${x}_{0}\in X$ be such that $\alpha ({x}_{0},f{x}_{0})\ge 1$ and define a sequence $\{{x}_{n}\}$ in

*X*by ${x}_{n}={f}^{n}{x}_{0}=f{x}_{n-1}$ for all $n\in \mathbb{N}$. Following the proof of Theorem 2.15, we have $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and there exists $z\in X$ such that ${x}_{n}\to z$ as $n\to \mathrm{\infty}$ which ${p}_{b}(z,z)=0$. Hence, from (ii) we deduce that $\alpha ({x}_{n},z)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$. Therefore, by (2.29), we obtain

which implies that $z=fz$. □

**Example 2.18**Let $X=[0,1]$ and ${p}_{b}(x,y)=|x-y{|}^{2}$ be a ${p}_{b}$-metric on

*X*. Define $f:X\to X$ by $fx=ln(\frac{x}{4}+1)$ and $\alpha :X\times X\to [0,\mathrm{\infty})$ by

and $\psi (t)=\frac{t}{8}$ for all $t\in [0,\mathrm{\infty})$. Now, we prove that all the hypotheses of Theorem 2.17 are satisfied and hence *f* has a fixed point.

First, we see that $(X,{p}_{b})$ is a ${p}_{b}$-complete partial *b*-metric space. Let $x,y\in X$. If $\alpha (x,y)\ge 1$, then $x,y\in [0,\frac{1}{4}]$. On the other hand, for all $x\in [0,1]$, we have $fx\le \frac{x}{4}\le \frac{1}{4}$ and hence $\alpha (fx,fy)=1$. This implies that *f* is a triangular *α*-admissible mapping on *X*. Obviously, $\alpha (0,f0)=1$.

Now, if $\{{x}_{n}\}$ is a sequence in *X* such that $\alpha ({x}_{n},{x}_{n+1})=1$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, it is easy to see that $\alpha ({x}_{n},x)=1$.

Thus, all the conditions of Theorem 2.17 are satisfied and therefore *f* has a fixed point $(z=0)$.

## 3 Common fixed points of generalized almost cyclic weakly $(\psi ,\phi ,L,A,B)$-contractive mappings

In this section, we consider the notion of ordered cyclic weakly $(\psi ,\phi ,L,A,B)$-contractions in the setup of ordered partial *b*-metric spaces and then obtain some common fixed point theorems for these cyclic contractions in the setup of complete ordered partial *b*-metric spaces. Our results extend some fixed point theorems from the framework of ordered metric spaces and ordered *b*-metric spaces, in particular Theorems 1.20 and 1.21.

We shall call an ordered partial *b*-metric space $(X,\u2aaf,{p}_{b})$ *regular* if for any nondecreasing sequence $\{{x}_{n}\}$ in *X* such that ${x}_{n}\to x\in X$, as $n\to \mathrm{\infty}$, one has ${x}_{n}\u2aafx$, for all $n\in \mathbb{N}$.

**Definition 3.1**Let $(X,\u2aaf,{p}_{b})$ be an ordered partial

*b*-metric space, let $f,g:X\to X$ be two mappings, and let

*A*and

*B*be nonempty closed subsets of

*X*. The pair $(f,g)$ is called an ordered cyclic almost generalized weakly $(\psi ,\phi ,L,A,B)$-contraction if

- (1)
$X=A\cup B$ is a cyclic representation of

*X*w.r.t. the pair $(f,g)$; that is, $fA\subseteq B$ and $gB\subseteq A$; - (2)there exist two altering distance functions
*ψ*,*φ*and a constant $L\ge 0$, such that for arbitrary comparable elements $x,y\in X$ with $x\in A$ and $y\in B$, we have$\psi ({s}^{2}{p}_{b}(fx,gy))\le \psi ({M}_{s}(x,y))-\phi ({M}_{s}(x,y))+L\psi (N(x,y)),$(3.1)

**Theorem 3.2** *Let* $(X,\u2aaf,{p}_{b})$ *be a* ${p}_{b}$-*complete ordered partial* *b*-*metric space and* *A* *and* *B* *be two nonempty closed subsets of* *X*. *Let* $f,g:X\to X$ *be two* $(A,B)$-*weakly increasing mappings with respect to* ⪯. *Suppose that the pair* $(f,g)$ *is an ordered cyclic almost generalized weakly* $(\psi ,\phi ,L,A,B)$-*contraction*. *Then* *f* *and* *g* *have a common fixed point* $z\in A\cap B$.

*Proof*First, note that $u\in A\cap B$ is a fixed point of

*f*if and only if

*u*is a fixed point of

*g*. Indeed, suppose that

*u*is a fixed point of

*f*. As $u\u2aafu$ and $u\in A\cap B$, by (3.1), we have

It follows that $\phi ({p}_{b}(u,gu))=0$. Therefore, ${p}_{b}(u,gu)=0$ and hence $gu=u$. Similarly, we can show that if *u* is a fixed point of *g*, then *u* is a fixed point of *f*.

*f*and

*g*are $(A,B)$-weakly increasing, we have

If ${x}_{2n}={x}_{2n+1}$, for some $n\in \mathbb{N}$, then ${x}_{2n}=f{x}_{2n}$. Thus ${x}_{2n}$ is a fixed point of *f*. By the first part of the proof, we conclude that ${x}_{2n}$ is also a fixed point of *g*. Similarly, if ${x}_{2n+1}={x}_{2n+2}$, for some $n\in \mathbb{N}$, then ${x}_{2n+1}=g{x}_{2n+1}$. Thus, ${x}_{2n+1}$ is a fixed point of *g*. By the first part of the proof, we conclude that ${x}_{2n+1}$ is also a fixed point of *f*. Therefore, we assume that ${x}_{n}\ne {x}_{n+1}$, for all $n\in \mathbb{N}$. Now, we complete the proof in the following steps.

*Step 1*. We will prove that

*Step 2*. We will prove that $\{{x}_{n}\}$ is a ${p}_{b}$-Cauchy sequence. Because of (3.7), it is sufficient to show that $\{{x}_{2n}\}$ is a ${p}_{b}$-Cauchy sequence. By Lemma 1.9, we should show that $\{{x}_{2n}\}$ is

*b*-Cauchy in $(X,{d}_{{p}_{b}})$. Suppose the contrary,

*i.e.*, that $\{{x}_{2n}\}$ is not a

*b*-Cauchy sequence in $(X,{d}_{{p}_{b}})$. Then there exists $\epsilon >0$ for which we can find two subsequences $\{{x}_{2{m}_{i}}\}$ and $\{{x}_{2{n}_{i}}\}$ of $\{{x}_{2n}\}$ such that ${n}_{i}$ is the smallest index for which

*b*-Cauchy sequence in the metric space $(X,{d}_{{p}_{b}})$. Since $(X,{p}_{b})$ is ${p}_{b}$-complete, from Lemma 1.9, $(X,{d}_{{p}_{b}})$ is a

*b*-complete

*b*-metric space. Therefore, the sequence $\{{x}_{n}\}$ converges to some $z\in X$, that is, ${lim}_{n\to \mathrm{\infty}}{d}_{{p}_{b}}({x}_{n},z)=0$. Since ${lim}_{m,n\to \mathrm{\infty}}{d}_{{p}_{b}}({x}_{n},{x}_{m})=0$, from the definition of ${d}_{{p}_{b}}$ and (3.7), we get

*Step 3*. In the above steps, we constructed an increasing sequence $\{{x}_{n}\}$ in

*X*such that ${x}_{n}\to z$, for some $z\in X$. As

*A*and

*B*are closed subsets of

*X*, we have $z\in A\cap B$. Using the regularity assumption on

*X*, we have ${x}_{n}\u2aafz$, for all $n\in \mathbb{N}$. Now, we show that $fz=gz=z$. By (3.1), we have

It follows that $\phi ({lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{M}_{s}({x}_{2n},z))=0$, and hence, by (3.24), that ${p}_{b}(z,gz)=0$. Thus, *z* is a fixed point of *g*. On the other hand, from the first part of the proof, $fz=z$. Hence, *z* is a common fixed point of *f* and *g*. □

**Theorem 3.3**

*Let*$(X,\u2aaf,{p}_{b})$

*be a*${p}_{b}$-

*complete ordered partial*

*b*-

*metric space and*

*A*

*and*

*B*

*be nonempty closed subsets of*

*X*.

*Let*$f,g:X\to X$

*be two*$(A,B)$-

*weakly increasing mappings with respect to*⪯.

*Suppose that*

Also, let *f* and *g* be continuous. Then *f* and *g* have a common fixed point $z\in A\cap B$.

*Proof* Repeating the proof of Theorem 3.2, we construct an increasing sequence $\{{x}_{n}\}$ in *X* such that ${x}_{n}\to z$, for some $z\in X$. As *A* and *B* are closed subsets of *X*, we have $z\in A\cap B$. Now, we show that $fz=gz=z$.

*f*and

*g*, we get

As *ψ* is nondecreasing, we have ${s}^{2}{p}_{b}(fz,gz)\le max\{{p}_{b}(z,fz),{p}_{b}(z,gz)\}$. Hence, by (3.28) we obtain ${s}^{2}{p}_{b}(fz,gz)=max\{{p}_{b}(z,fz),{p}_{b}(z,gz)\}$. But then, using (3.29), we get $\phi ({M}_{s}(z,z))=0$. Thus, we have $fz=gz=z$ and *z* is a common fixed point of *f* and *g*. □

As consequences, we have the following results.

By putting $A=B=X$ in Theorems 3.2 and 3.3 and $L=0$ in Theorem 3.2, we obtain the main results (Theorems 3 and 4) of Mustafa *et al.* [4].

Taking $\phi =(1-\delta )\psi $, $0<\delta <1$ in Theorem 3.2, we get the following.

**Corollary 3.4**

*Let*$(X,\u2aaf,{p}_{b})$

*be a*${p}_{b}$-

*complete ordered partial*

*b*-

*metric space and*

*A*

*and*

*B*

*be closed subsets of*

*X*.

*Let*$f,g:X\to X$

*be two*$(A,B)$-

*weakly increasing mappings with respect to*⪯.

*Suppose that*:

- (a)
$X=A\cup B$

*is a cyclic representation of**X**w*.*r*.*t*.*the pair*$(f,g)$; - (b)
*there exist*$0<\delta <1$, $L\ge 0$,*and an altering distance function**ψ**such that for any comparable elements*$x,y\in X$*with*$x\in A$*and*$y\in B$,*we have*$\psi ({s}^{2}{p}_{b}(fx,gy))\le \delta \psi ({M}_{s}(x,y))+L\psi (N(x,y)),$(3.30)

*where*${M}_{s}(x,y)$

*and*$N(x,y)$

*are given by*(3.2)

*and*(3.3),

*respectively*;

- (c)
*f**and**g**are continuous*,*or*

(c′) *the space* $(X,\u2aaf,{p}_{b})$ *is regular*.

*Then* *f* *and* *g* *have a common fixed point* $z\in A\cap B$.

Taking $s=1$ and $L=0$ in Corollary 3.4, we obtain the partial version of Theorems 2.1 and 2.2 of Shatanawi and Postolache [12].

In Definitions 1.18 and 3.1 and Theorems 3.2 and 3.3, if we take $f=g$, then we have the following definitions and results.

**Definition 3.5** Let $(X,\u2aaf)$ be a partially ordered set and *A* and *B* be closed subsets of *X* with $X=A\cup B$. The mapping $f:X\to X$ is said to be $(A,B)$-weakly increasing if $fx\u2aaf{f}^{2}x$, for all $x\in A$ and $fy\u2aaf{f}^{2}y$, for all $y\in B$.

**Definition 3.6**Let $(X,\u2aaf,{p}_{b})$ be an ordered partial

*b*-metric space, let $f:X\to X$ be a mapping, and let

*A*and

*B*be nonempty closed subsets of

*X*. The mapping

*f*is called an ordered cyclic almost generalized weakly $(\psi ,\phi ,L,A,B)$-contraction if

- (1)
$X=A\cup B$ is a cyclic representation of

*X*w.r.t.*f*; that is, $fA\subseteq B$ and $fB\subseteq A$; - (2)there exist two altering distance functions
*ψ*,*φ*and a constant $L\ge 0$, such that for arbitrary comparable elements $x,y\in X$ with $x\in A$ and $y\in B$, we have$\psi ({s}^{2}{p}_{b}(fx,fy))\le \psi ({M}_{s}(x,y))-\phi ({M}_{s}(x,y))+L\psi (N(x,y)),$

**Corollary 3.7** *Let* $(X,\u2aaf,{p}_{b})$ *be a* ${p}_{b}$-*complete ordered partial* *b*-*metric space and* *A* *and* *B* *be two nonempty closed subsets of* *X*. *Let* $f:X\to X$ *be a* $(A,B)$-*weakly increasing mapping with respect to* ⪯. *Suppose that the mapping* *f* *is an ordered cyclic almost generalized weakly* $(\psi ,\phi ,L,A,B)$-*contraction*. *Then* *f* *has a fixed point* $z\in A\cap B$.

**Corollary 3.8**

*Let*$(X,\u2aaf,{p}_{b})$

*be a*${p}_{b}$-

*complete ordered partial*

*b*-

*metric space and*

*A*

*and*

*B*

*be nonempty closed subsets of*

*X*.

*Let*$f:X\to X$

*be a*$(A,B)$-

*weakly increasing mapping with respect to*⪯.

*Suppose that*

*Also*, *let* *f* *be continuous*. *Then* *f* *has a fixed point* $z\in A\cap B$.

We illustrate our results with the following example.

**Example 3.9**Consider the partial

*b*-metric space $X=[0,6]$ by ${p}_{b}(x,y)={[max\{x,y\}]}^{2}$. Define an order ⪯ on

*X*by

So, we have ${lim}_{n\to \mathrm{\infty}}{x}_{n}=\sqrt{u}$, which convergence holds in the case of the usual metric in *X*. Now, it is easy to see that ${lim}_{n,m\to \mathrm{\infty}}{p}_{b}({x}_{n},{x}_{m})={lim}_{n\to \mathrm{\infty}}{p}_{b}({x}_{n},\sqrt{u})={p}_{b}(\sqrt{u},\sqrt{u})=u$.

$\psi (t)=t$ and $\phi (t)=\frac{8}{9}t$ for all $t\in [0,\mathrm{\infty})$. Also, let $A=[0,1]$ and $B=[0,6]$. In order to check the conditions of Corollary 3.8, take $x,y\in X$ such that $x\u2aafy$ and consider the following two possible cases.

^{∘}$x\le 1$. Then obviously also $y\le 1$ and $x\ge y$. It is easy to check that

^{∘}$x>1$. Then $x=y>1$ and

Hence, all the conditions of Corollary 3.8 are satisfied and *f* has a fixed point (which is $z=0$).

## Declarations

### Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

## Authors’ Affiliations

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