Fixed point results of almost generalized $(\phi , \psi ,\theta )_s$-contractive mappings in ordered b-metric spaces

Abstract

The aim of this article is to establish some fixed point, coupled coincidence point and coupled common fixed point results for mappings satisfying an almost generalized $(\phi , \psi , \theta )_s$-contractive conditions in the frame work of partially ordered b-metric spaces. These results generalize, extend and unify several comparable results in the existing literature. Few examples are illustrated to support our results.

1 Introduction and preliminaries

The Banach contraction principle is one of the most important results in nonlinear analysis. It plays an important role in many branches of mathematical analysis, and it has many applications in solving nonlinear equations and scientific problems. Later, it has been generalized and improved in many different directions, one of the most influential generalization is a b-metric space, also called metric type space by some authors, introduced and studied by Bakhtin [11] and Czerwik [16]. There after, a large number of articles have been dedicated to the improvement of the fixed point theory for single valued and multivalued operators in b-metric spaces, the readers may refer to [1, 5, 6, 8, 10, 17, 18, 20, 21, 23, 28, 36] and the references therein. The concept of coupled fixed points of mixed monotone mappings in partially ordered metric spaces was introduced by Bhaskar and Lakshmikantham [13] and applied theiry results to first order differential equation with boundary condition. After that Lakshmikantham and Ćirić [30] have introduced the concept of coupled coincidence and coupled common fixed point for mappings with mixed monotone property and generalized the result of Bhaskar and Lakshmikantham [13]. Then, several authors have obtained fixed point, common fixed point, coupled fixed point and coupled common fixed point results for mappings under various contractive conditions in ordered b-metric spaces, some of which are in [2,3,4, 7, 9, 14, 15, 19, 22, 31, 32] and the references therein. Recently, some results on fixed point, coincidence point, coupled coincidence point for the self mappings satisfying generalized weak contractions have been discussed by Belay Mituku et al. [12], Seshagiri Rao et al. [33,34,35] and Kalyani et al. [24,25,26,27] in partially ordered b-metric space with necessary topological properties. Some important results of fixed points of distance spaces can be found from Todoŕcević [37] and William Kirk et al. [29].

In this paper, some fixed point, coincidence point, coupled coincidence point and coupled common fixed points for mappings satisfying an almost generalized $(\phi , \psi ,\theta )_s$-contraction conditions in complete partially ordered b-metric spaces are proved. These results generalize and extend the results of [13, 30, 33, 34] and several comparable results in the existing literature. Some examples are presented to support our results.

For the sake of convenience some definitions and suitable results are recalled from [2, 19, 30, 32] which will be needed in what follows.

Definition 1.1

[35] A map $d: P \times P \rightarrow [0, +\infty )$, where P is a non-empty set is said to be a b-metric, if it satisfies the properties given below for any $\upsilon ,\xi ,\mu \in P$ and for some real number $s \ge 1$:

• (a) $d(\upsilon ,\xi )=0$ if and only if $\upsilon =\xi$,

• (b) $d(\upsilon ,\xi )=d(\xi ,\upsilon )$,

• (c) $d(\upsilon ,\xi ) \le s \left( d(\upsilon ,\mu )+d(\mu ,\xi )\right)$.

Then (P, d, s) is known as a b-metric space. If $(P,\preceq )$ is still a partially ordered set, then $(P,d,s, \preceq )$ is called a partially ordered b-metric space.

Definition 1.2

[33] Let (P, d, s) be a b-metric space. Then

• (1) a sequence $\{\upsilon _n\}$ is said to converge to $\upsilon$, if $\lim \limits _{n \rightarrow +\infty }d(\upsilon _n,\upsilon )=0$ and written as $\lim \nolimits _{n \rightarrow +\infty }\upsilon _n=\upsilon$.

• (2) $\{\upsilon _n\}$ is said to be a Cauchy sequence in P, if $\lim \limits _{n,m \rightarrow +\infty }d(\upsilon _n,\upsilon _m)=0$.

• (3) (P, d, s) is said to be complete, if every Cauchy sequence in it is convergent.

Definition 1.3

If the metric d is complete then $(P,d,s,\preceq )$ is called complete partially ordered b-metric space.

Definition 1.4

[32] Let $(P,\preceq )$ be a partially ordered set and let $f,S:P \rightarrow P$ are two mappings. Then

• (1) S is called a monotone nondecreasing, if $S(\upsilon )\preceq S(\xi )$ for all $\upsilon ,\xi \in P$ with $\upsilon \preceq \xi$.

• (2) an element $\upsilon \in P$ is called a coincidence (common fixed) point of f and S, if $f\upsilon =S\upsilon ~(f\upsilon =S\upsilon =\upsilon )$.

• (3) f and S are called commuting, if $fS\upsilon =Sf\upsilon$, for all $\upsilon \in P$.

• (4) f and S are called compatible, if any sequence $\{\upsilon _n\}$ with $\lim \nolimits _{n \rightarrow +\infty }f\upsilon _n= \lim \nolimits _{n \rightarrow +\infty }S\upsilon _n =\mu ,~ \text {for}~ \mu \in P$ then $\lim \nolimits _{n \rightarrow +\infty } d(Sf\upsilon _n,fS\upsilon _n) =0$.

• (5) a pair of self maps (f, S) is called weakly compatible, if $fS\upsilon =Sf\upsilon$, when $S\upsilon =f\upsilon$ for some $\upsilon \in P$.

• (6) S is called monotone f-nondecreasing, if

  $$\begin{aligned} f\upsilon \preceq f\xi ~\text {implies}~ S\upsilon \preceq S\xi ,~\text {for any} ~\upsilon ,\xi \in P. \end{aligned}$$

• (7) a non empty set P is called well ordered set, if very two elements of it are comparable i.e., $\upsilon \preceq \xi$ or $\xi \preceq \upsilon$, for $\upsilon , \xi \in P$.

Definition 1.5

[2, 30] Suppose $(P, \preceq )$ be a partially ordered set and let $S: P \times P \rightarrow P$ and $f: P \rightarrow P$ be two mappings. Then

• (1) S has the mixed f-monotone property, if S is non-decreasing f-monotone in its first argument and is non-increasing f-monotone in its second argument, that is for any $\upsilon , \xi \in P$

  $$\begin{aligned} \begin{aligned}&\upsilon _1, \upsilon _2 \in P,~~f\upsilon _1 \preceq f\upsilon _2 ~\text {implies}~ S(\upsilon _1,\xi )\preceq S(\upsilon _2,\xi )~ \text {and} \\ {}&\xi _1, \xi _2 \in P,~~f\xi _1 \preceq f\xi _2 ~\text {implies}~ S(\upsilon ,\xi _1)\succeq S(\upsilon ,\xi _2). \end{aligned} \end{aligned}$$

  Suppose, if f is an identity mapping then S is said to have the mixed monotone property.

• (2) an element $(\upsilon ,\xi ) \in P \times P$ is called a coupled coincidence point of S and f, if $S(\upsilon ,\xi )=f\upsilon$ and $S(\xi ,\upsilon )=f\xi$. Note that, if f is an identity mapping then $(\upsilon ,\xi )$ is said to be a coupled fixed point of S.

• (3) an element $\upsilon \in P$ is called a common fixed point of S and f, if $S(\upsilon ,\upsilon )=f\upsilon =\upsilon$.

• (4) S and f are commutative, if for all $\upsilon , \xi \in P$, $S(f\upsilon ,f\xi )=f(S\upsilon ,S\xi )$.

• (5) S and f are said to be compatible, if

  $$\begin{aligned} \lim \limits _{n \rightarrow +\infty } d(f(S(\upsilon _n, \xi _n)), S(f\upsilon _n,f\xi _n))=0~\text {and}~\lim \limits _{n \rightarrow +\infty } d(f(S(\xi _n,\upsilon _n)), S(f\xi _n,f\upsilon _n))=0, \end{aligned}$$

  whenever $\{\upsilon _n\}$ and $\{\xi _n\}$ are any two sequences in P such that $\lim \nolimits _{n \rightarrow +\infty } S(\upsilon _n,\xi _n)=\lim \nolimits _{n \rightarrow +\infty } f\upsilon _n=\upsilon$ and $\lim \nolimits _{n \rightarrow +\infty } S(\xi _n,\upsilon _n)=\lim \nolimits _{n \rightarrow +\infty } f\xi _n=\xi$, for any $\upsilon , \xi \in P$.

We know that b-metric is not continuous, so the following lemma is used frequently in our results for the convergence of sequences in a b-metric spaces.

Lemma 1.6

[2] Let $(P,d,s,\preceq )$ be a b-metric space with $s>1$ and suppose that $\{\upsilon _n \}$ and $\{\xi _n\}$ are b-convergent to $\upsilon$ and $\xi$ respectively. Then

$$\begin{aligned} \frac{1}{s^2}d(\upsilon ,\xi )\le \lim \limits _{n \rightarrow +\infty } \inf d(\upsilon _n,\xi _n) \le \lim \limits _{n \rightarrow +\infty } \sup d(\upsilon _n,\xi _n)\le s^2 d(\upsilon ,\xi ). \end{aligned}$$

In particular, if $\upsilon =\xi$, then $\lim \nolimits _{n \rightarrow +\infty } d(\upsilon _n,\xi _n)=0$. Moreover, for each $\tau \in P$, we have

$$\begin{aligned} \frac{1}{s}d(\upsilon ,\tau )\le \lim \limits _{n \rightarrow +\infty } \inf d(\upsilon _n,\tau ) \le \lim \limits _{n \rightarrow +\infty } \sup d(\upsilon _n,\tau )\le s d(\upsilon ,\tau ). \end{aligned}$$

2 Main results

Throughout this paper, we use the following denotations of the distances functions.

A self mapping $\phi$ defined on $[0, +\infty )$ is said to be an altering distance function, if it satisfies the following conditions:

• (i) $\phi$ is continuous,

• (ii) $\phi$ is nondecreasing,

• (iii) $\phi (t)=0$ if and only if $t=0$.

Let us denote the set of all altering distance functions on $[0, +\infty )$ by $\Phi$.

Similarly, $\Psi$ denote the set of all functions $\psi :[0, +\infty )\rightarrow [0, +\infty )$ satisfying the following conditions:

• (i) $\psi$ is lower semi-continuous,

• (ii) $\psi (t)=0$ if and only if $t=0$.

and $\Theta$ denote the set of all continuous functions $\theta :[0,+\infty )\rightarrow [0,+\infty )$ with $\theta (t)=0$ if and only if $t=0$.

Let $(P,d,s,\preceq )$ be a partially ordered b-metric space with parameter $s > 1$ and, let $S:P \rightarrow P$ be a mapping. Set

$$\begin{aligned} M(\upsilon ,\xi )=\max \left\{ \frac{d(\xi ,S\xi ) \left[ 1+d(\upsilon ,S\upsilon )\right] }{1+d(\upsilon ,\xi )},\frac{d(\upsilon ,S\xi )+ d(\xi ,S\upsilon )}{2s}, d(\upsilon ,S\upsilon ),d(\xi ,S\xi ), d(\upsilon ,\xi )\right\} , \end{aligned}$$

(1)

and

$$\begin{aligned} N(\upsilon ,\xi )=\min \{d(\upsilon ,S\upsilon ),d(\xi ,S\xi ),d(\xi ,S\upsilon ), d(\upsilon ,S\xi )\}. \end{aligned}$$

(2)

Let $\phi \in \Phi$, $\psi \in \Psi$ and $\theta \in \Theta$. The mapping S is called an almost generalized $(\phi ,\psi ,\theta )_s$-contraction mapping if it satisfies the following condition

$$\begin{aligned} \phi (sd(S\upsilon ,S\xi ))\le \phi (M(\upsilon ,\xi ))-\psi (M(\upsilon ,\xi ))+L \theta (N(\upsilon ,\xi )), \end{aligned}$$

(3)

for all $\upsilon ,\xi \in P$ with $\upsilon \preceq \xi$ and $L\ge 0$.

Now in this paper, we start with the following fixed point theorem in the context of partially ordered b-metric space.

Theorem 2.1

Suppose that $(P,d,s,\preceq )$ be a complete partially ordered b-metric space with parameter $s > 1$. Let $S:P \rightarrow P$ be an almost generalized $(\phi ,\psi ,\theta )_s$-contractive mapping, and be continuous, nondecreasing mapping with regards to $\preceq$. If there exists $\upsilon _0 \in P$ with $\upsilon _0 \preceq S\upsilon _0$, then S has a fixed point in P.

Proof

For some $\upsilon _0 \in P$ such that $S\upsilon _0=\upsilon _0$, then the proof is finished. Assume that $\upsilon _0 \prec S\upsilon _0$, then construct a sequence $\{\upsilon _n\} \subset P$ by $\upsilon _{n+1}=S\upsilon _n$ for $n\ge 0$. Since S is nondecreasing, then by induction we obtain that

$$\begin{aligned} \upsilon _0 \prec S\upsilon _0=\upsilon _1\preceq \cdots \preceq \upsilon _n \preceq S\upsilon _n=\upsilon _{n+1}\preceq \cdots . \end{aligned}$$

(4)

If for some $n_0\in {\mathbb {N}}$ such that $\upsilon _{n_0}=\upsilon _{n_0+1}$ then from (4), $\upsilon _{n_0}$ is a fixed point of S and we have nothing to prove. Suppose that $\upsilon _n \ne \upsilon _{n+1}$ for all $n \ge 1$. Since $\upsilon _n>\upsilon _{n-1}$ for any $n \ge 1$ and then by contraction condition (3), we have

$$\begin{aligned} \begin{aligned} \phi (d(\upsilon _n,\upsilon _{n+1}))&= \phi (d(S\upsilon _{n-1},S\upsilon _n))\le \phi (sd(S\upsilon _{n-1},S\upsilon _n)) \\ {}&\le \phi (M(\upsilon _{n-1},\upsilon _n))-\psi (M(\upsilon _{n-1},\upsilon _n))+L\theta (N(\upsilon _{n-1},\upsilon _n)), \end{aligned} \end{aligned}$$

(5)

where

$$\begin{aligned} \begin{aligned} M(\upsilon _{n-1},\upsilon _n)&=\max \Bigg \{\frac{d(\upsilon _n,S\upsilon _n) \left[ 1+d(\upsilon _{n-1},S\upsilon _{n-1})\right] }{1+d(\upsilon _{n-1},\upsilon _n)}, \frac{d(\upsilon _{n-1},S\upsilon _n)+ d(\upsilon _n,S\upsilon _{n-1})}{2s},\\ {}&\quad \times d(\upsilon _{n-1},S\upsilon _{n-1}),d(\upsilon _n,S\upsilon _n), d(\upsilon _{n-1},\upsilon _n)\Bigg \} \\ {}&= \max \Bigg \{d(\upsilon _n,\upsilon _{n+1}), \frac{d(\upsilon _{n-1},\upsilon _{n+1})+ d(\upsilon _n,\upsilon _n)}{2s}, d(\upsilon _{n-1},\upsilon _n)\Bigg \} \\ {}&\le \max \Bigg \{d(\upsilon _n,\upsilon _{n+1}),\frac{d(\upsilon _{n-1},\upsilon _n)+ d(\upsilon _n,\upsilon _{n+1})}{2}, d(\upsilon _{n-1},\upsilon _n)\Bigg \} \\ {}&\le \max \{d(\upsilon _n,\upsilon _{n+1}),d(\upsilon _{n-1},\upsilon _n)\} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} N(\upsilon _{n-1},\upsilon _n)&=\min \{d(\upsilon _{n-1},S\upsilon _{n-1}),d(\upsilon _n,S\upsilon _n),d(\upsilon _n,S\upsilon _{n-1}), d(\upsilon _{n-1},S\upsilon _n)\} \\ {}&=\min \{d(\upsilon _{n-1},\upsilon _n),d(\upsilon _n,\upsilon _{n+1}),d(\upsilon _n,\upsilon _n), d(\upsilon _{n-1},\upsilon _{n+1})\}=0. \end{aligned} \end{aligned}$$

From (5), we get

$$\begin{aligned} d(\upsilon _n,\upsilon _{n+1})= d(S\upsilon _{n-1},S\upsilon _n)\le \frac{1}{s} M(\upsilon _{n-1},\upsilon _n). \end{aligned}$$

(6)

If $\max \{d(\upsilon _n,\upsilon _{n+1}),d(\upsilon _{n-1}, \upsilon _n)\}= d(\upsilon _n,\upsilon _{n+1})$ for some $n \ge 1$, then from (6) follows

$$\begin{aligned} d(\upsilon _n,\upsilon _{n+1})\le \frac{1}{s} d(\upsilon _n,\upsilon _{n+1}), \end{aligned}$$

(7)

which is a contradiction. This means that $\max \{d(\upsilon _n,\upsilon _{n+1}),d(\upsilon _{n-1},\upsilon _n)\}= d(\upsilon _{n-1},\upsilon _n)$ for $n \ge 1$. Hence, we obtain from (6) that

$$\begin{aligned} d(\upsilon _n,\upsilon _{n+1})\le \frac{1}{s} d(\upsilon _{n-1},\upsilon _n). \end{aligned}$$

(8)

Since, $\frac{1}{s}\in (0,1)$ then the sequence $\{\upsilon _n\}$ is a Cauchy sequence by [1, 5, 10, 18]. But P is complete, then there exists some $\mu \in P$ such that $\upsilon _n \rightarrow \mu$.

From the continuity of S implies that

$$\begin{aligned} S\mu =S\left( \lim \limits _{n\rightarrow +\infty }\upsilon _n\right) =\lim \limits _{n\rightarrow +\infty }S\upsilon _n=\lim \limits _{n\rightarrow +\infty }\upsilon _{n+1}=\mu . \end{aligned}$$

(9)

Therefore, $\mu$ is a fixed point of S in P. $\square$

By relaxing the continuity criteria of a map S in Theorem 2.1, we have the following result.

Theorem 2.2

In Theorem 2.1, assume that P satisfies

$$\begin{aligned} \text {if a nondecreasing sequence}~ \{\upsilon _n\} \rightarrow \mu ~ \text {in}~ P,~ \text {then}~ \upsilon _n \preceq \mu ~ \text {for all}~ n \in {\mathbb {N}}, i.e., \mu =\sup \upsilon _n. \end{aligned}$$

Then a nondecreasing mapping S has a fixed point in P.

Proof

From Theorem 2.1, we construct a nondecreasing Cauchy sequence $\{\upsilon _n\}$ in P such that $\upsilon _n \rightarrow \mu \in P$. Therefore from the hypotheses, we have $\upsilon _n \preceq \mu$ for any $n \in {\mathbb {N}}$, implies that $\mu =\sup \upsilon _n$.

Now, we prove that $\mu$ is a fixed point of S. Suppose that $S\mu \ne \mu$. Let

$$\begin{aligned} \begin{aligned} M(\upsilon _n,\mu )&=\max \Bigg \{\frac{d(\mu ,S\mu ) \left[ 1+d(\upsilon _n,S\upsilon _n)\right] }{1+d(\upsilon _n,\mu )},\frac{d(\upsilon _n,S\mu )+ d(\mu ,S\upsilon _n)}{2s}, d(\upsilon _n,S\upsilon _n),\\ {}&\quad \times d(\mu ,S\mu ), d(\upsilon _n,\mu )\Bigg \}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} N(\upsilon _n,\mu )=\min \{d(\upsilon _n,S\upsilon _n),d(\mu ,S\mu ),d(\mu ,S\upsilon _n), d(\upsilon _n,S\mu )\}. \end{aligned}$$

Letting $n\rightarrow +\infty$ and use of $\lim \nolimits _{n\rightarrow +\infty }\upsilon _n=\mu$, we get

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }M(\upsilon _n, \mu )= \max \left\{ d(\mu ,S\mu ),\frac{d(\mu ,S\mu )}{2s},0\right\} =d(\mu ,S\mu ), \end{aligned}$$

(10)

and

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }N(\upsilon _n, \mu )= \min \{0,d(\mu ,S\mu )\}=0. \end{aligned}$$

(11)

We know that $\upsilon _n \preceq \mu$, for all n then from contraction condition (3), we get

$$\begin{aligned} \phi (d(\upsilon _{n+1}, S\mu ))=\phi (d(S\upsilon _n, S\mu ))\le \phi (s d(S\upsilon _n, S\mu ))\le \phi (M(\upsilon _n, \mu ))-\psi (M(\upsilon _n, \mu )).\nonumber \\ \end{aligned}$$

(12)

Letting $n \rightarrow +\infty$ and use of (10) and (11), we get

$$\begin{aligned} \phi (d(\mu ,S\mu )) \le \phi (d(\mu ,S\mu ))-\psi (d(\mu ,S\mu ))< \phi (d(\mu ,S\mu )), \end{aligned}$$

(13)

which is a contradiction under (13). Thus, $S\mu =\mu$, that is S has a fixed point $\mu$ in P. $\square$

Now we give the sufficient condition for the uniqueness of the fixed point exists in Theorem 2.1 and Theorem 2.2.

$$\begin{aligned} \text {every pair of elements has a lower bound or an upper bound.} \end{aligned}$$

(14)

This condition is equivalent to,

$$\begin{aligned} \text {for every}~ \upsilon , \xi \in P, ~\text {there exists}~ w \in P ~\text {which is comparable to}~ \upsilon ~ \text {and}~ \xi . \end{aligned}$$

Theorem 2.3

In addition to the hypotheses of Theorem 2.1 (or Theorem 2.2), condition (14) provides uniqueness of a fixed point of S in P.

Proof

From Theorem 2.1 (or Theorem 2.2), we conclude that S has a nonempty set of fixed points. Suppose that $\upsilon ^*$ and $\xi ^*$ be two fixed points of S then, we claim that $\upsilon ^*=\xi ^*$. Suppose that $\upsilon ^*\ne \xi ^*$, then from the hypotheses we have

$$\begin{aligned} \begin{aligned} \phi (d(S\upsilon ^*, S\xi ^*))&\le \phi (sd(S\upsilon ^*, S\xi ^*)) \le \phi (M(\upsilon ^*, \xi ^*))-\psi (M(\upsilon ^*, \xi ^*))+L \theta (N(\upsilon ^*, \xi ^*)), \end{aligned}\nonumber \\ \end{aligned}$$

(15)

where

$$\begin{aligned} \begin{aligned} M(\upsilon ^*,\xi ^*)&=\max \Bigg \{\frac{d(\xi ^*,S\xi ^*) \left[ 1+d(\upsilon ^*,S\upsilon ^*)\right] }{1+d(\upsilon ^*,\xi ^*)},\frac{d(\upsilon ^*,S\xi ^*)+ d(\xi ^*,S\upsilon ^*)}{2s}, d(\upsilon ^*,S\upsilon ^*),\\ {}&\quad \times d(\xi ^*,S\xi ^*), d(\upsilon ^*,\xi ^*)\Bigg \} \\ {}&=\max \Bigg \{\frac{d(\xi ^*,\xi ^*) \left[ 1+d(\upsilon ^*,\upsilon ^*)\right] }{1+d(\upsilon ^*,\xi ^*)},\frac{d(\upsilon ^*,\xi ^*)+ d(\xi ^*,\upsilon ^*)}{2s}, d(\upsilon ^*,\upsilon ^*),\\ {}&\quad \times d(\xi ^*,\xi ^*), d(\upsilon ^*,\xi ^*)\Bigg \} \\ {}&= \max \Bigg \{0,\frac{d(\upsilon ^*,\xi ^*)}{s}, d(\upsilon ^*,\xi ^*) \Bigg \} \\ {}&=d(\upsilon ^*,\xi ^*), \end{aligned} \end{aligned}$$

and

$$\begin{aligned} N(\upsilon ^*,\xi ^*)=\min \{d(\upsilon ^*,S\upsilon ^*),d(\xi ^*,S\xi ^*),d(\xi ^*,S\upsilon ^*), d(\upsilon ^*,S\xi ^*)\}=0. \end{aligned}$$

Consequently, we get

$$\begin{aligned} d(\upsilon ^*, \xi ^*)= d(S\upsilon ^*, S\xi ^*) \le \frac{1}{s} M(\upsilon ^*, \xi ^*). \end{aligned}$$

(16)

Therefore from (16), we obtain that

$$\begin{aligned} d(\upsilon ^*, \xi ^*) \le \frac{1}{s} d(\upsilon ^*, \xi ^*)<d(\upsilon ^*, \xi ^*), \end{aligned}$$

(17)

which is a contradiction. Hence, $\upsilon ^*= \xi ^*$. This completes the proof.$\square$

Let $(P,d,s,\preceq )$ be a partially ordered b-metric space with parameter $s > 1$ and let $S,f:P \rightarrow P$ be two mappings. Set

$$\begin{aligned} \begin{aligned} M_f(\upsilon ,\xi )&=\max \{\frac{d(f\xi ,S\xi ) \left[ 1+d(f\upsilon ,S\upsilon )\right] }{1+d(f\upsilon ,f\xi )},\frac{d(f\upsilon ,S\xi )+ d(f\xi ,S\upsilon )}{2s},\\ {}&\quad \times d(f\upsilon ,S\upsilon ),d(f\xi ,S\xi ), d(f\upsilon ,f\xi )\}, \end{aligned} \end{aligned}$$

(18)

and

$$\begin{aligned} N_f(\upsilon ,\xi )=\min \{d(f\upsilon ,S\upsilon ),d(f\xi ,S\xi ),d(f\xi ,S\upsilon ), d(f\upsilon ,S\xi )\}. \end{aligned}$$

(19)

Now, we introduce the following definition.

Definition 2.4

Let $(P,d,s,\preceq )$ be a partially ordered b-metric space with $s > 1$. The mapping $S:P \rightarrow P$ is called an almost generalized $(\phi ,\psi ,\theta )_s$-contraction mapping with respect to $f:P \rightarrow P$ for some $\phi \in \Phi$, $\psi \in \Psi$ and $\theta \in \Theta$, if

$$\begin{aligned} \phi (sd(S\upsilon ,S\xi ))\le \phi (M_f(\upsilon ,\xi ))-\psi (M_f(\upsilon ,\xi ))+L\theta (N_f(\upsilon ,\xi )), \end{aligned}$$

(20)

for any $\upsilon ,\xi \in P$ with $f\upsilon \preceq f\xi$, $L \ge 0$ and where $M_f(\upsilon ,\xi )$ and $N_f(\upsilon ,\xi )$ are given by (18) and (19) respectively.

Theorem 2.5

Suppose that $(P,d,s,\preceq )$ be a complete partially ordered b-metric space with $s> 1$. Let $S: P \rightarrow P$ be an almost generalized $(\phi ,\psi ,\theta )_s$-contractive mapping with respect to $f: P \rightarrow P$ and, S and f are continuous such that S is a monotone f-non decreasing mapping, compatible with f and $SP \subseteq fP$. If for some $\upsilon _0 \in P$ such that $f\upsilon _0 \preceq S\upsilon _0$, then S and f have a coincidence point in P.

Proof

By following the proof of a Theorem 2.2 in [7], we construct two sequences $\{\upsilon _n\}$ and $\{\xi _n\}$ in P such that

$$\begin{aligned} \xi _n=S\upsilon _n=f\upsilon _{n+1} ~\text {for all}~n\ge 0, \end{aligned}$$

(21)

for which

$$\begin{aligned} f\upsilon _0 \preceq f\upsilon _1 \preceq \cdots \preceq f\upsilon _n \preceq f\upsilon _{n+1} \preceq \cdots . \end{aligned}$$

(22)

Again from [7], we have to show that

$$\begin{aligned} d(\xi _n,\xi _{n+1})\le \lambda d(\xi _{n-1},\xi _n), \end{aligned}$$

(23)

for all $n \ge 1$ and where $\lambda \in [0, \frac{1}{s})$. Now from (20) and use of (21) and (22), we have

$$\begin{aligned} \begin{aligned} \phi (sd(\xi _n,\xi _{n+1}))&=\phi (sd(S\upsilon _n,S\upsilon _{n+1})) \\ {}&\le \phi (M_f(\upsilon _n,\upsilon _{n+1}))-\psi (M_f(\upsilon _n,\upsilon _{n+1})) +L\theta (N_f(\upsilon _n,\upsilon _{n+1})), \end{aligned} \end{aligned}$$

(24)

where

$$\begin{aligned} \begin{aligned} M_f(\upsilon _n,\upsilon _{n+1})&=\max \Bigg \{\frac{d(f\upsilon _{n+1},S\upsilon _{n+1}) \left[ 1+d(f\upsilon _n,S\upsilon _n)\right] }{1+d(f\upsilon _n,f\upsilon _{n+1})},\frac{d(f\upsilon _n,S\upsilon _{n+1})+ d(f\upsilon _{n+1},S\upsilon _n)}{2s},\\ {}&\quad \times d(f\upsilon _n,S\upsilon _n),d(f\upsilon _{n+1},S\upsilon _{n+1}), d(f\upsilon _n,f\upsilon _{n+1})\Bigg \} \\&=\max \Bigg \{\frac{d(\xi _n,\xi _{n+1}) \left[ 1+d(\xi _{n-1},\xi _n)\right] }{1+d(\xi _{n-1},\xi _n)}, \frac{d(\xi _{n-1},\xi _{n+1})+d(\xi _n,\xi _n)}{2s},d(\xi _{n-1},\xi _n),\\ {}&\quad \times d(\xi _n,\xi _{n+1}), d(\xi _{n-1},\xi _n)\Bigg \} \\&=\max \Bigg \{d(\xi _n,\xi _{n+1}),\frac{d(\xi _{n-1},\xi _n)+d(\xi _n,\xi _{n+1})}{2s},d(\xi _{n-1},\xi _n)\Bigg \} \\ {}&\le \max \{d(\xi _n,\xi _{n+1}), d(\xi _{n-1},\xi _n)\} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} N_f(\upsilon _n,\upsilon _{n+1})&=\min \{d(f\upsilon _n,S\upsilon _n),d(f\upsilon _{n+1},S\upsilon _{n+1}),d(f\upsilon _{n+1},S\upsilon _n), d(f\upsilon _n,S\upsilon _{n+1})\} \\ {}&=\min \{d(\xi _{n-1},\xi _n),d(\xi _n,\xi _{n+1}),d(\xi _n,\xi _n),d(\xi _{n-1},\xi _{n+1})\}=0. \end{aligned} \end{aligned}$$

Therefore from Eq. (24), we get

$$\begin{aligned} \phi (sd(\xi _n,\xi _{n+1}))\le \phi (\max \{d(\xi _{n-1},\xi _n),d(\xi _n,\xi _{n+1})\}) -\psi (\max \{d(\xi _{n-1},\xi _n),d(\xi _n,\xi _{n+1})\}).\nonumber \\ \end{aligned}$$

(25)

If $0<d(\xi _{n-1},\xi _n)\le d(\xi _n,\xi _{n+1})$ for some $n \in {\mathbb {N}}$, then from (25) we get

$$\begin{aligned} \phi (sd(\xi _n,\xi _{n+1}))\le \phi (d(\xi _n,\xi _{n+1}))-\psi (d(\xi _n,\xi _{n+1}))<\phi (d(\xi _n,\xi _{n+1})), \end{aligned}$$

(26)

or equivalently

$$\begin{aligned} sd(\xi _n,\xi _{n+1})\le d(\xi _n,\xi _{n+1}). \end{aligned}$$

(27)

This is a contradiction. Hence from (25) we obtain that

$$\begin{aligned} sd(\xi _n,\xi _{n+1})\le d(\xi _{n-1},\xi _n). \end{aligned}$$

(28)

Thus Eq. (23) holds, where $\lambda \in [0,\frac{1}{s})$. Therefore from (23) and Lemma 3.1 of [23], we conclude that $\{\xi _n\}=\{S\upsilon _n\}=\{f\upsilon _{n+1}\}$ is a Cauchy sequence in P and then converges to some $\mu \in P$ as P is complete such that

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }S\upsilon _{n}=\lim \limits _{n \rightarrow +\infty }f\upsilon _{n+1}=\mu . \end{aligned}$$

Thus by the compatibility of S and f, we obtain that

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }d(f(S\upsilon _n), S(f\upsilon _n))=0, \end{aligned}$$

(29)

and from the continuity of S and f, we have

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }f(S\upsilon _n)=f\mu , \quad \lim \limits _{n \rightarrow +\infty } S(f\upsilon _n)=S\mu . \end{aligned}$$

(30)

Further by use of triangular inequality and from Eqs. (29) and (30), we get

$$\begin{aligned} \frac{1}{s}d(S\mu ,f\mu )\le d(S\mu ,S(f\upsilon _n))+s d(S(f\upsilon _n), f(S\upsilon _n))+sd(f(S\upsilon _n), f\mu ). \end{aligned}$$

(31)

Finally, we arrive at $d(Sv,fv)=0$ as $n \rightarrow +\infty$ in (31). Therefore, v is a coincidence point of S and f in P. $\square$

Relaxing the continuity criteria of f and S in Theorem 2.5, we obtain the following result.

Theorem 2.6

In Theorem 2.5, assume that P satisfies

$$\begin{aligned} \begin{aligned} ~~~~~&\text { for any nondecreasing sequence}~ \{f\upsilon _n\}\subset P~\text {with}~ \lim \limits _{n \rightarrow +\infty } f\upsilon _n=f\upsilon ~ \text {in}~ fP,~ \text {where}~fP \\&\text {is a closed subset of}~ P~\text {implies that}~ f\upsilon _n \preceq f\upsilon , f\upsilon \preceq f(f\upsilon )~\text {for}~n \in {\mathbb {N}}. \end{aligned} \end{aligned}$$

If there exists $\upsilon _0 \in P$ such that $f\upsilon _0 \preceq S\upsilon _0$, then the weakly compatible mappings S and f have a coincidence point in P. Moreover, S and f have a common fixed point, if S and f commute at their coincidence points.

Proof

The sequence, $\{\xi _n\}=\{S\upsilon _n\}=\{f\upsilon _{n+1}\}$ is a Cauchy sequence from the proof of Theorem 2.5. Since fP is closed, then there is some $\mu \in P$ such that

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }S\upsilon _{n}=\lim \limits _{n \rightarrow +\infty }f\upsilon _{n+1}=f\mu . \end{aligned}$$

Thus from the hypotheses, we have $f\upsilon _n\preceq f\mu$ for all $n \in {\mathbb {N}}$. Now, we have to prove that $\mu$ is a coincidence point of S and f.

From equation (20), we have

$$\begin{aligned} \begin{aligned} \phi (sd(S\upsilon _n,S\upsilon ))\le \phi (M_f(\upsilon _n,\upsilon )) -\psi (M_f(\upsilon _n,\upsilon ))+L\theta (N_f(\upsilon _n,\upsilon )), \end{aligned} \end{aligned}$$

(32)

where

$$\begin{aligned} \begin{aligned} M_f(\upsilon _n,\mu )&=\max \Bigg \{\frac{d(f\mu ,S\mu ) \left[ 1+d(f\upsilon _n,S\upsilon _n)\right] }{1+d(f\upsilon _n,f\mu )},\frac{d(f\upsilon _n,S\mu )+ d(f\mu ,S\upsilon _n)}{2s},\\ {}&\quad \times d(f\upsilon _n,S\upsilon _n),d(f\mu ,S\mu ), d(f\upsilon _n,f\mu )\Bigg \} \\ {}&\rightarrow \max \Bigg \{d(f\mu ,S\mu ),\frac{d(f\mu ,S\mu )}{2s},0,d(f\mu ,S\mu ),0\Bigg \} \\ {}&= d(f\mu ,S\mu ) ~\text {as}~n \rightarrow +\infty , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} N_f(\upsilon _n,\mu )&=\min \{d(f\upsilon _n,S\upsilon _n),d(f\mu ,S\mu ),d(f\mu ,S\upsilon _n), d(f\upsilon _n,S\mu )\} \\ {}&\rightarrow \min \{0,d(f\mu ,S\mu ),0,d(f\mu ,S\mu )\} \\ {}&= 0 ~\text {as}~n \rightarrow +\infty . \end{aligned} \end{aligned}$$

Therefore Eq. (32) becomes

$$\begin{aligned} \phi \left( s\lim \limits _{n \rightarrow +\infty } d(S\upsilon _n,S\upsilon )\right) \le \phi (d(f\mu ,S\mu ))-\psi (d(f\mu ,S\mu ))< \phi (d(f\mu ,S\mu )). \end{aligned}$$

Consequently, we get

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }d(S\upsilon _n,S\upsilon ) < \frac{1}{s}d(f\mu ,S\mu ). \end{aligned}$$

(33)

Further by triangular inequality, we have

$$\begin{aligned} \frac{1}{s}d(f\mu ,S\mu )\le d(f\mu ,S\upsilon _n)+d(S\upsilon _n,S\mu ), \end{aligned}$$

(34)

then (33) and (34) lead to contradiction, if $f\mu \ne S\mu$. Hence, $f\mu =S\mu$.

Let $f\mu =S\mu =\rho$, that is S and f commute at $\rho$, then $S\rho = S(f\mu )=f(S\mu )=f\rho$. Since $f\mu =f(f\mu )=f\rho$, then by Eq. (32) with $f\mu =S\mu$ and $f\rho =S\rho$, we get

$$\begin{aligned} \begin{aligned} \phi (sd(S\mu ,S\rho ))\le \phi (M_f(\mu ,\rho ))-\psi (M_f(\mu ,\rho ))<\phi (d(S\mu ,S\rho )), \end{aligned} \end{aligned}$$

(35)

or equivalently,

$$\begin{aligned} sd(S\mu ,S\rho ) \le d(S\mu ,S\rho ), \end{aligned}$$

which is a contradiction, if $S\mu \ne S\rho$. Thus, $S\mu = S\rho = \rho$. Hence, $S\mu = f\rho =\rho$, that is $\rho$ is a common fixed point of S and f. $\square$

Definition 2.7

Let $(P,d,s,\preceq )$ be a partially ordered b-metric space with $s > 1$, $\phi \in \Phi$, $\psi \in \Psi$ and $\theta \in \Theta$. A mapping $S:P \times P \rightarrow P$ is said to be an almost generalized $(\phi ,\psi ,\theta )_s$-contractive mapping with respect to $f:P \rightarrow P$ such that

$$\begin{aligned} \phi (s^kd(S(\upsilon ,\xi ),S(\rho ,\tau )))\le \phi (M_f(\upsilon ,\xi ,\rho ,\tau )) -\psi (M_f(\upsilon ,\xi ,\rho ,\tau ))+L\theta (N_f(\upsilon ,\xi ,\rho ,\tau )),\nonumber \\ \end{aligned}$$

(36)

for all $\upsilon ,\xi ,\rho ,\tau \in P$ with $f\upsilon \preceq f \rho$ and $f\xi \succeq f \tau$, $k>2$, $L \ge 0$ and where

$$\begin{aligned} \begin{aligned} M_f(\upsilon ,\xi ,\rho ,\tau )&=\max \Bigg \{\frac{d(f\rho ,S(\rho ,\tau )) \left[ 1+d(f\upsilon ,S(\upsilon ,\xi ))\right] }{1+d(f\upsilon ,f\rho )},\\ {}&\qquad \frac{d(f\upsilon ,S(\rho ,\tau ))+ d(f\rho ,S(\upsilon ,\xi ))}{2s},\\ {}&\quad \times d(f\upsilon ,S(\upsilon ,\xi )),d(f\rho ,S(\rho ,\tau )), d(f\upsilon ,f\rho )\Bigg \}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} N_f(\upsilon ,\xi ,\rho ,\tau )=\min \{d(f\upsilon ,S(\upsilon ,\xi )),d(f\rho ,S(\rho ,\tau )),d(f\rho ,S(\upsilon ,\xi )), d(f\upsilon ,S(\rho ,\tau ))\}. \end{aligned}$$

Theorem 2.8

Let $(P,d,s,\preceq )$ be a complete partially ordered b-metric space with $s > 1$. Suppose that $S:P \times P \rightarrow P$ be an almost generalized $(\phi ,\psi ,\theta )_s$-contractive mapping with respect to $f:P \rightarrow P$ and, S and f are continuous functions such that S has the mixed f-monotone property and commutes with f. Also assume that $S(P \times P) \subseteq f(P)$. Then S and f have a coupled coincidence point in P, if there exists $(\upsilon _0,\xi _0) \in P \times P$ such that $f\upsilon _0 \preceq S(\upsilon _0,\xi _0)$ and $f\xi _0 \succeq S(\xi _0,\upsilon _0)$.

Proof

From the hypotheses and following the proof of Theorem 2.2 of [7], we construct two sequences $\{\upsilon _n\}$ and $\{\xi _n\}$ in P such that

$$\begin{aligned} f\upsilon _{n+1}=S(\upsilon _n,\xi _n), ~~~~f\xi _{n+1}=S(\xi _n,\upsilon _n),~~\text {for all}~n\ge 0. \end{aligned}$$

In particular, $\{f\upsilon _n\}$ is nondecreasing and $\{f\xi _n\}$ is nonincreasing sequences in P. Now from (36) by replacing $\upsilon =\upsilon _n, \xi =\xi _n, \rho =\upsilon _{n+1}, \tau =\xi _{n+1}$, we get

$$\begin{aligned} \phi (s^kd(f\upsilon _{n+1},f\upsilon _{n+2}))= & {} \phi (s^kd(S(\upsilon _n,\xi _n),S(\upsilon _{n+1},\xi _{n+1}))) \nonumber \\\le & {} \phi (M_f(\upsilon _n,\xi _n,\upsilon _{n+1},\xi _{n+1}))-\psi (M_f(\upsilon _n,\xi _n,\upsilon _{n+1},\xi _{n+1})) \nonumber \\&+L\theta (N_f(\upsilon _n,\xi _n,\upsilon _{n+1},\xi _{n+1})), \end{aligned}$$

(37)

where

$$\begin{aligned} M_f(\upsilon _n,\xi _n,\upsilon _{n+1},\xi _{n+1})\le \max \{d(f\upsilon _n,f\upsilon _{n+1}),d(f\upsilon _{n+1},f\upsilon _{n+2})\}, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} N_f(\upsilon _n,\xi _n,\upsilon _{n+1},\xi _{n+1})&= \min \{d(f\upsilon _n,S(\upsilon _n,\xi _n)),d(f\upsilon _{n+1},S(\upsilon _{n+1},\xi _{n+1})),\\&\quad \times d(f\upsilon _n,S(\upsilon _{n+1},\xi _{n+1})),d(f\upsilon _{n+1},S(\upsilon _n,\xi _n))\} =0. \end{aligned} \end{aligned}$$

Therefore from (37), we have

$$\begin{aligned} \begin{aligned} \phi (s^kd(f\upsilon _{n+1},f\upsilon _{n+2}))&\le \phi (\max \{d(f\upsilon _n,f\upsilon _{n+1}),d(f\upsilon _{n+1},f\upsilon _{n+2})\})\\ {}&\quad -\psi (\max \{d(f\upsilon _n,f\upsilon _{n+1}),d(f\upsilon _{n+1},f\upsilon _{n+2})\}). \end{aligned} \end{aligned}$$

(38)

Similarly by taking $\upsilon =\xi _{n+1}, \xi =\upsilon _{n+1}, \rho =\upsilon _n, \tau =\upsilon _n$ in (36), we get

$$\begin{aligned} \begin{aligned} \phi (s^kd(f\xi _{n+1},f\xi _{n+2}))&\le \phi (\max \{d(f\xi _n,f\xi _{n+1}),d(f\xi _{n+1},f\xi _{n+2})\})\\ {}&\quad -\psi (\max \{d(f\xi _n,f\xi _{n+1}),d(f\xi _{n+1},f\xi _{n+2})\}). \end{aligned} \end{aligned}$$

(39)

From the fact that $\max \{\phi (c),\phi (d)\}=\phi \{\max \{c,d\}\}$ for all $c,d \in [0,+\infty )$. Then combining (38) and (39), we get

$$\begin{aligned} \begin{aligned} \phi (s^k \delta _n)&\le \phi (\max \{d(f\upsilon _n,f\upsilon _{n+1}),d(f\upsilon _{n+1},f\upsilon _{n+2}),d(f\xi _n,f\xi _{n+1}),d(f\xi _{n+1},f\xi _{n+2})\})\\ {}&\quad -\psi (\max \{d(f\upsilon _n,f\upsilon _{n+1}),d(f\upsilon _{n+1},f\upsilon _{n+2}),d(f\xi _n,f\xi _{n+1}),d(f\xi _{n+1},f\xi _{n+2})\}) \end{aligned}\nonumber \\ \end{aligned}$$

(40)

where

$$\begin{aligned} \delta _n=\max \{d(f\upsilon _{n+1},f\upsilon _{n+2}), d(f\xi _{n+1},f\xi _{n+2})\}. \end{aligned}$$

(41)

Let us denote,

$$\begin{aligned} \Delta _n=\max \{d(f\upsilon _n,f\upsilon _{n+1}),d(f\upsilon _{n+1},f\upsilon _{n+2}),d(f\xi _n,f\xi _{n+1}),d(f\xi _{n+1},f\xi _{n+2})\}. \end{aligned}$$

(42)

Hence from Eqs. (38)–(41), we obtain

$$\begin{aligned} s^k\delta _n\le \Delta _n. \end{aligned}$$

(43)

Next, we prove that

$$\begin{aligned} \delta _n\le \lambda \delta _{n-1}, \end{aligned}$$

(44)

for all $n \ge 1$ and where $\lambda =\frac{1}{s^k} \in [0,1)$.

Suppose that if $\Delta _n=\delta _n$ then from (43), we get $s^k\delta _n\le \delta _n$ which leads to $\delta _n=0$ as $s>1$ and hence (44) holds. If $\Delta _n=\max \{d(f\upsilon _n,f\upsilon _{n+1}), d(f\xi _n,f\xi _{n+1})\}$, i.e., $\Delta _n=\delta _{n-1}$ then (43) follows (44).

Now from (43), we obtain that $\delta _n\le \lambda ^n \delta _0$ and hence,

$$\begin{aligned} d(f\upsilon _{n+1},f\upsilon _{n+2})\le \lambda ^n \delta _0 ~~\text {and}~~d(f\xi _{n+1},f\xi _{n+2})\le \lambda ^n \delta _0. \end{aligned}$$

(45)

Therefore from Lemma 3.1 of [23], the sequences $\{f\upsilon _n\}$ and $\{f\xi _n\}$ are Cauchy sequences in P. Hence, by following the remaining proof of Theorem 2.2 of [3], we can show that S and f have a coincidence point in P. $\square$

Corollary 2.9

Let $(P,d,s,\preceq )$ be a complete partially ordered b-metric space with $s > 1$, and $S:P \times P \rightarrow P$ be a continuous mapping such that S has a mixed monotone property. Suppose there exists $\phi \in \Phi$, $\psi \in \Psi$ and $\theta \in \Theta$ such that

$$\begin{aligned} \phi (s^kd(S(\upsilon ,\xi ),S(\rho ,\tau )))\le \phi (M_f(\upsilon ,\xi ,\rho ,\tau )) -\psi (M_f(\upsilon ,\xi ,\rho ,\tau ))+L\theta (N_f(\upsilon ,\xi ,\rho ,\tau )), \end{aligned}$$

for all $\upsilon ,\xi ,\rho ,\tau \in P$ with $\upsilon \preceq \rho$ and $\xi \succeq \tau$, $k>2$, $L\ge 0$ and where

$$\begin{aligned} \begin{aligned} M_f(\upsilon ,\xi ,\rho ,\tau )&=\max \Bigg \{\frac{d(\rho ,S(\rho ,\tau )) \left[ 1+d(\upsilon ,S(\upsilon ,\xi ))\right] }{1+d(\upsilon ,\rho )},\frac{d(\upsilon ,S(\rho ,\tau ))+ d(\rho ,S(\upsilon ,\xi ))}{2s},\\ {}&\quad \times d(\upsilon ,S(\upsilon ,\xi )),d(\rho ,S(\rho ,\tau )), d(\upsilon ,\rho )\Bigg \}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} N_f(\upsilon ,\xi ,\rho ,\tau )=\min \{d(\upsilon ,S(\upsilon ,\xi )),d(\rho ,S(\rho ,\tau )),d(\rho ,S(\upsilon ,\xi )), d(\upsilon ,S(\rho ,\tau ))\}. \end{aligned}$$

Then S has a coupled fixed point in P, if there exists $(\upsilon _0,\xi _0) \in P \times P$ such that $\upsilon _0 \preceq S(\upsilon _0,\xi _0)$ and $\xi _0 \succeq S(\xi _0,\upsilon _0)$.

Proof

Set $f=I_P$ in Theorem  2.8. $\square$

Corollary 2.10

Let $(P,d,s,\preceq )$ be a complete partially ordered b-metric space with $s > 1$, and $S:P \times P \rightarrow P$ be a continuous mapping such that S has a mixed monotone property. Suppose there exists $\psi \in \Psi$ such that

$$\begin{aligned} d(S(\upsilon ,\xi ),S(\rho ,\tau ))\le \frac{1}{s^k}M_f(\upsilon ,\xi ,\rho ,\tau )-\frac{1}{s^k}\psi (M_f(\upsilon ,\xi ,\rho ,\tau )), \end{aligned}$$

for all $\upsilon ,\xi ,\rho ,\tau \in P$ with $\upsilon \preceq \rho$ and $\xi \succeq \tau$, $k>2$ where

$$\begin{aligned} \begin{aligned} M_f(\upsilon ,\xi ,\rho ,\tau )&=\max \Bigg \{\frac{d(\rho ,S(\rho ,\tau )) \left[ 1+d(\upsilon ,S(\upsilon ,\xi ))\right] }{1+d(\upsilon ,\rho )},\frac{d(\upsilon ,S(\rho ,\tau ))+ d(\rho ,S(\upsilon ,\xi ))}{2s},\\ {}&\quad \times d(\upsilon ,S(\upsilon ,\xi )),d(\rho ,S(\rho ,\tau )), d(\upsilon ,\rho )\Bigg \}. \end{aligned} \end{aligned}$$

If there exists $(\upsilon _0,\xi _0) \in P \times P$ such that $\upsilon _0 \preceq S(\upsilon _0,\xi _0)$ and $\xi _0 \succeq S(\xi _0,\upsilon _0)$, then S has a coupled fixed point in P.

Theorem 2.11

In addition to Theorem 2.8, if for all $(\upsilon ,\xi ),(r,s) \in P \times P$, there exists $(c^*,d^*)\in P \times P$ such that $(S(c^*,d^*), S(d^*,c^*))$ is comparable to $(S(\upsilon ,\xi ), S(\xi ,\upsilon ))$ and to (S(r, s), S(s, r)), then S and f have a unique coupled common fixed point in $P \times P$.

Proof

From Theorem 2.8, we know that there exists atleast one coupled coincidence point in P for S and f. Assume that $(\upsilon , \xi )$ and (r, s) are two coupled coincidence points of S and f, i.e., $S(\upsilon , \xi )=f\upsilon$, $S(\xi ,\upsilon , )=f\xi$ and $S(r,s)=fr$, $S(s,r)=fs$. Now, we have to prove that $f\upsilon =fr$ and $f\xi =fs$.

From the hypotheses, there exists $(c^*,d^*)\in P \times P$ such that $(S(c^*,d^*), S(d^*,c^*))$ is comparable to $(S(\upsilon ,\xi ), S(\xi ,\upsilon ))$ and to (S(r, s), S(s, r)). Suppose that

$$\begin{aligned} (S(\upsilon ,\xi ), S(\xi ,\upsilon )) \le (S(c^*,d^*), S(d^*,c^*)) ~\text {and}~ (S(r,s),S(s,r))\le (S(c^*,d^*), S(d^*,c^*)). \end{aligned}$$

Let $c^*_0=c^*$ and $d^*_0=d^*$ and then choose $(c^*_1,d^*_1) \in P \times P$ as

$$\begin{aligned} fc^*_1=S(c^*_0,d^*_0),~~ fd^*_1=S(d^*_0,c^*_0)~~(n \ge 1). \end{aligned}$$

By repeating the same procedure above, we can obtain two sequences $\{f c^*_{n}\}$ and $\{f d^*_{n}\}$ in P such that

$$\begin{aligned} fc^*_{n+1}=S(c^*_n,d^*_n),~~ fd^*_{n+1}=S(d^*_n,c^*_n)~~(n \ge 0). \end{aligned}$$

Similarly, define the sequences $\{f \upsilon _{n}\}$, $\{f \xi _{n}\}$ and $\{f r_{n}\}$, $\{f s_{n}\}$ as above in P by setting $\upsilon _0=\upsilon$, $\xi _0=\xi$ and $r_0=r$, $s_0=s$. Further, we have that

$$\begin{aligned} f\upsilon _{n} \rightarrow S(\upsilon ,\xi ),~f\xi _{n} \rightarrow S(\xi ,\upsilon ),~ fr_{n} \rightarrow S(r,s),~fs_n \rightarrow S(s,r)~~(n \ge 1). \end{aligned}$$

(46)

Since, $(S(\upsilon ,\xi ), S(\xi ,\upsilon ))=(f\upsilon ,f\xi )=(f\upsilon _1,f\xi _1)$ is comparable to $(S(c^*,d^*), S(d^*,c^*))=(fc^*,fd^*)=(fc^*_1,fd^*_1)$ and hence we get $(f\upsilon _1,f\xi _1) \le (fc^*_1,fd^*_1)$. Thus, by induction we obtain that

$$\begin{aligned} (f\upsilon _{n},f\xi _{n}) \le (fc^*_n,fd^*_n)~~(n \ge 0). \end{aligned}$$

(47)

Therefore from (36), we have

$$\begin{aligned} \begin{aligned} \phi (d(f\upsilon ,fc^*_{n+1}))&\le \phi (s^3d(f\upsilon ,fc^*_{n+1}))= \phi (d(S(\upsilon ,\xi ),S(c^*_n,d^*_n))) \\ {}&\le \phi (M_f(\upsilon ,\xi ,c^*_n,d^*_n))-\psi (M_f(\upsilon ,\xi ,c^*_n,d^*_n))+L\theta (N_f(\upsilon ,\xi ,c^*_n,d^*_n)), \end{aligned}\nonumber \\ \end{aligned}$$

(48)

where

$$\begin{aligned} \begin{aligned} M_f(\upsilon ,\xi ,c^*_n,d^*_n)&=\max \Bigg \{\frac{d(fc^*_n,S(c^*_n,d^*_n)) \left[ 1+d(f\upsilon ,S(\upsilon ,\xi ))\right] }{1+d(f\upsilon ,fc^*_n)},\\ {}&\qquad \frac{d(f\upsilon ,S(c^*_n,d^*_n))+ d(fc^*_n,S(\upsilon ,\xi ))}{2s},\\ {}&\quad \times d(f\upsilon ,S(\upsilon ,\xi )),d(fc^*_n,S(c^*_n,d^*_n)), d(f\upsilon ,fc^*_n)\Bigg \} \\ {}&= \max \Bigg \{0,\frac{d(f\upsilon ,fc^*_n)}{s},0,0,d(f\upsilon ,fc^*_n)\Bigg \} \\ {}&=d(f\upsilon ,fc^*_n) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} N_f(\upsilon ,\xi ,c^*_n,d^*_n)&=\min \{d(f\upsilon ,S(\upsilon ,\xi )),d(fc^*_n,S(c^*_n,d^*_n)), \\&\quad d(fc^*_n,S(\upsilon ,\xi )), d(f\upsilon ,S(c^*_n,d^*_n))\}=0. \end{aligned} \end{aligned}$$

Thus from (48),

$$\begin{aligned} \phi (d(f\upsilon ,fc^*_{n+1}))\le \phi (d(f\upsilon ,fc^*_n))-\psi (d(f\upsilon ,fc^*_n)). \end{aligned}$$

(49)

As by the similar process, we can prove that

$$\begin{aligned} \phi (d(f\xi ,fd^*_{n+1}))\le \phi (d(f\xi ,fd^*_n))-\psi (d(f\xi ,fd^*_n)). \end{aligned}$$

(50)

From (49) and (50), we have

$$\begin{aligned} \begin{aligned} \phi (\max \{d(f\upsilon ,fc^*_{n+1}),d(f\xi ,fd^*_{n+1})\})&\le \phi (\max \{d(f\upsilon ,fc^*_n),d(f\xi ,fd^*_n)\})\\ {}&\quad -\psi (\max \{d(f\upsilon ,fc^*_n),d(f\xi ,fd^*_n)\}) \\ {}&<\phi (\max \{d(f\upsilon ,fc^*_n),d(f\xi ,fd^*_n)\}). \end{aligned} \end{aligned}$$

(51)

Hence by the property of $\phi$, we get

$$\begin{aligned} \max \{d(f\upsilon ,fc^*_{n+1}),d(f\xi ,fd^*_{n+1})\} <\max \{d(f\upsilon ,fc^*_n),d(f\xi ,fd^*_n)\}, \end{aligned}$$

which shows that $\max \{d(f\upsilon ,fc^*_n),d(f\xi ,fd^*_n)\}$ is a decreasing sequence and by a result there exists $\gamma \ge 0$ such that

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\max \{d(f\upsilon ,fc^*_n),d(f\xi ,fd^*_n)\} =\gamma . \end{aligned}$$

From (51) taking upper limit as $n \rightarrow +\infty$, we get

$$\begin{aligned} \phi (\gamma )\le \phi (\gamma )-\psi (\gamma ), \end{aligned}$$

(52)

from which we get $\psi (\gamma )=0$, implies that $\gamma =0$. Thus,

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\max \{d(f\upsilon ,fc^*_n),d(f\xi ,fd^*_n)\} =0. \end{aligned}$$

Consequently, we get

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }d(f\upsilon ,fc^*_n) =0 \quad \text {and} \quad \lim \limits _{n \rightarrow +\infty }d(f\xi ,fd^*_n) =0. \end{aligned}$$

(53)

By similar argument, we get

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }d(fr,fc^*_n) =0 \quad \text {and} \quad \lim \limits _{n \rightarrow +\infty }d(fs,fd^*_n) =0. \end{aligned}$$

(54)

Therefore from (53) and (54), we get $f\upsilon =fr$ and $f\xi =fs$. Since $f\upsilon =S(\upsilon ,\xi )$ and $f\xi =S(\xi ,\upsilon )$, then by the commutativity of S and f, we have

$$\begin{aligned} f(f\upsilon )= f(S(\upsilon ,\xi ))=S(f\upsilon ,f\xi ) \quad \text {and} \quad f(f\xi )= f(S(\xi ,\upsilon ))=S(f\xi ,f\upsilon ). \end{aligned}$$

(55)

Let $f\upsilon =a^*$ and $f\xi =b^*$ then (55) becomes

$$\begin{aligned} f(a^*)= S(a^*,b^*) \quad \text {and} \quad f(b^*)= S(b^*,a^*), \end{aligned}$$

(56)

which shows that $(a^*,b^*)$ is a coupled coincidence point of S and f. It follows that $f(a^*)=fr$ and $f(b^*)=fs$ that is $f(a^*)=a^*$ and $f(b^*)=b^*$. Thus from (56), we get $a^*=f(a^*)= S(a^*,b^*)$ and $b^*=f(b^*)= S(b^*,a^*)$. Therefore, $(a^*,b^*)$ is a coupled common fixed point of S and f.

For the uniqueness let $(u^*,v^*)$ be another coupled common fixed point of S and f, then we have $u^*=fu^*= S(u^*,v^*)$ and $v^*=fv^*= S(v^*,u^*)$. Since $(u^*,v^*)$ is a coupled common fixed point of S and f, then we obtain that $fu^*=f\upsilon =a^*$ and $fv^*=f\xi =b^*$. Thus, $u^*=fu^*=fa^*=a^*$ and $v^*=fv^*=fb^*=b^*$. Hence the result. $\square$

Theorem 2.12

In addition to the hypotheses of Theorem 2.11, if $f\upsilon _0$ and $f\xi _0$ are comparable, then S and f have a unique common fixed point in P.

Proof

From Theorem 2.11, S and f have a unique coupled common fixed point $(\upsilon ,\xi ) \in P$. Now, it is enough to prove that $\upsilon =\xi$. From the hypotheses, we have $f\upsilon _0$ and $f\xi _0$ are comparable then we assume that $f\upsilon _0 \preceq f\xi _0$. Hence by induction we get $f\upsilon _n \preceq f\xi _n$ for all $n \ge 0$, where $\{f\upsilon _n\}$ and $\{f\xi _n\}$ are from Theorem 2.8.

Now by use of Lemma 1.6, we get

$$\begin{aligned} \begin{aligned} \phi (s^{k-2}d(\upsilon ,\xi ))&=\phi (s^k \frac{1}{s^2}d(\upsilon ,\xi )) \le \lim \limits _{n \rightarrow +\infty } \sup \phi (s^k d(\upsilon _{n+1}, \xi _{n+1})) \\ {}&= \lim \limits _{n \rightarrow +\infty }\sup \phi (s^k d(S(\upsilon _n, \xi _n),S(\xi _n,\upsilon _n))) \\ {}&\le \lim \limits _{n \rightarrow +\infty }\sup \phi (M_f(\upsilon _n, \xi _n,\xi _n,\upsilon _n))-\lim \limits _{n \rightarrow +\infty }\inf \psi (M_f(\upsilon _n, \xi _n,\xi _n,\upsilon _n))\\ {}&~~~~~~+\lim \limits _{n \rightarrow +\infty } \sup L\theta (N_f(\upsilon _n, \xi _n,\xi _n,\upsilon _n)) \\ {}&\le \phi (d(\upsilon ,\xi ))-\lim \limits _{n \rightarrow +\infty }\inf \psi (N_f(\upsilon _n, \xi _n,\xi _n,\upsilon _n)) \\ {}&<\phi (d(\upsilon ,\xi )), \end{aligned} \end{aligned}$$

which is a contradiction. Thus, $\upsilon =\xi$, i.e., S and f have a common fixed point in P. $\square$

Remark 2.13

It is well known that b-metric space is a metric space when $s=1$. So, from the result of Jachymski [22], the condition

$$\begin{aligned} \phi (d(S(\upsilon ,\xi ),S(\rho ,\tau )))&\le \phi (\max \{d(f\upsilon ,f\rho ),d(f\xi ,f\tau )\})\\&\quad -\psi (\max \{d(f\upsilon ,f\rho ),d(f\xi ,f\tau )\}) \end{aligned}$$

is equivalent to,

$$\begin{aligned} d(S(\upsilon ,\xi ),S(\rho ,\tau ))\le \varphi (\max \{d(f\upsilon ,f\rho ),d(f\xi ,f\tau )\}), \end{aligned}$$

where $\phi \in \Phi$, $\psi \in \Psi$ and $\varphi :[0,+\infty )\rightarrow [0,+\infty )$ is continuous, $\varphi (t)<t$ for all $t>0$ and $\varphi (t)=0$ if and only if $t=0$. So, in view of above our results generalize and extend the results of [2, 30, 33, 34] and several other comparable results.

Corollary 2.14

Suppose $(P,d,s,\preceq )$ be a complete partially ordered b-metric space with parameter $s > 1$. Let $S:P \rightarrow P$ be a continuous, nondecreasing map with regards to $\preceq$ such that there exists $\upsilon _0 \in P$ with $\upsilon _0 \preceq S\upsilon _0$. Suppose that

$$\begin{aligned} \phi (sd(S\upsilon ,S\xi ))\le \phi (M(\upsilon ,\xi ))-\psi (M(\upsilon ,\xi )), \end{aligned}$$

(57)

where $M(\upsilon ,\xi )$ and the conditions upon $\phi , \psi$ are same as in Theorem 2.1. Then S has a fixed point in P.

Proof

Set $L=0$ in a contraction condition (3) and apply Theorem 2.1, we have the required proof. $\square$

Note 1

Besides, if P satisfies the assumptions in Theorem 2.2, then a nondecreasing mapping S has a fixed point in P. Also, if P satisfies the hypothesis (14), then one obtains uniqueness of the fixed point.

Note 2

Setting $L=0$ and following the proofs of Theorems 2.5 and 2.6, we can find the coincidence point for S and f in P. Similarly, from Theorem 2.8, 2.11 and 2.12, one can obtain a coupled coincidence point and its uniqueness, and a unique common fixed point for mappings S and f in $P\times P$ and P satisfying a generalized contraction condition (57), where $M_f(\upsilon ,\xi )$, $M_f(\upsilon ,\xi , \rho , \tau )$ and the conditions upon $\phi , \psi$ are same as above defined.

Corollary 2.15

Suppose that $(P,d,s,\preceq )$ be a complete partially ordered b-metric space with $s > 1$. Let $S:P \rightarrow P$ be a continuous, nondecreasing mapping with regards to $\preceq$. If there exists $k \in [0,1)$ and for any $\upsilon ,\xi \in P$ with $\upsilon \preceq \xi$ such that

$$\begin{aligned} d(S\upsilon ,S\xi )\le & {} \frac{k}{s}\max \Bigg \{\frac{d(\xi ,S\xi ) \left[ 1+d(\upsilon ,S\upsilon )\right] }{1+d(\upsilon ,\xi )},\frac{d(\upsilon ,S\xi )+ d(\xi ,S\upsilon )}{2s},\nonumber \\&d(\upsilon ,S\upsilon ),d(\xi ,S\xi ), d(\upsilon ,\xi )\Bigg \}. \end{aligned}$$

(58)

If there exists $\upsilon _0 \in P$ with $\upsilon _0 \preceq S\upsilon _0$, then S has a fixed point in P.

Proof

Set $\phi (t)=t$ and $\psi (t)=(1-k)t$, for all $t \in (0, +\infty )$ in Corollary 2.14. $\square$

Note 3

Relaxing the continuity of a map S in Corollary 2.15, one can obtains a fixed point for S on taking a nondecreasing sequence $\{\upsilon _n\}$ in P by following the proof of Theorem 2.2.

We illustrate the usefulness of the obtained results in different cases such as continuity and discontinuity of a metric d in a space P.

Example 2.16

Define a metric $d:P\times P \rightarrow P$ as below and $\le$ is an usual order on P, where $P=\{1,2,3,4,5,6\}$

$$\begin{aligned} \begin{aligned}&d(\upsilon , \xi )=d(\xi ,\upsilon )=0, ~\text {if}~ \upsilon , \xi =1,2,3,4,5,6 ~\text {and}~ \upsilon =\xi ,\\&d(\upsilon , \xi )=d(\xi ,\upsilon )=3, ~if~ \upsilon , \xi =1,2,3,4,5 ~\text {and}~ \upsilon \ne \xi ,\\&d(\upsilon , \xi )=d(\xi ,\upsilon )=12, ~if~ \upsilon =1,2,3,4 ~\text {and}~ \xi =6,\\&d(\upsilon , \xi )=d(\xi ,\upsilon )=20, ~if~ \upsilon =5 ~\text {and}~ \xi =6.\\ \end{aligned} \end{aligned}$$

Define a map $S:P \rightarrow P$ by $S1=S2=S3=S4=S5=1, S6=2$ and let $\phi (t)=\frac{t}{2}$, $\psi (t)=\frac{t}{4}$ for $t \in [0,+\infty )$. Then S has a fixed point in P.

Proof

It is apparent that, $(P,d,s,\preceq )$ is a complete partially ordered b-metric space for $s=2$. Consider the possible cases for $\upsilon$, $\xi$ in P:

Case 1. Suppose $\upsilon , \xi \in \{1,2,3,4,5\}$, $\upsilon <\xi$ then $d(S\upsilon ,S\xi )=d(1,1)=0$. Hence,

$$\begin{aligned} \phi (2d(S\upsilon ,S\xi ))=0 \le \phi (M(\upsilon ,\xi ))-\psi (M(\upsilon ,\xi )). \end{aligned}$$

Case 2. Suppose that $\upsilon \in \{1,2,3,4,5\}$ and $\xi =6$, then $d(S\upsilon ,S\xi )=d(1,2)=3$, $M(6,5)=20$ and $M(\upsilon ,6)=12$, for $\upsilon \in \{1,2,3,4\}$. Therefore, we have the following inequality,

$$\begin{aligned} \phi (2d(S\upsilon ,S\xi )) \le \frac{M(\upsilon ,\xi )}{4} =\phi (M(\upsilon ,\xi ))-\psi (M(\upsilon ,\xi )). \end{aligned}$$

Thus, condition (57) of Corollary 2.14 holds. Furthermore, the remaining assumptions in Corollary 2.14 are fulfilled. Hence, S has a fixed point in P as Corollary 2.14 is appropriate to $S, \phi , \psi$ and $(P,d,s,\preceq )$. $\square$

Example 2.17

A metric $d:P\times P \rightarrow P$, where $P=\{0, 1, \frac{1}{2},\frac{1}{3},\frac{1}{4},\ldots \frac{1}{n},\ldots \}$ with usual order $\le$ is as follows

$$\begin{aligned} \begin{aligned} d(\upsilon ,\xi )=\left\{ \begin{array}{ll} 0 , &{} if~ \upsilon =\xi \\ 1 , &{} if~ \upsilon \ne \xi \in \{0,1\} \\ |\upsilon -\xi |, &{} if~ \upsilon ,\xi \in \{0, \frac{1}{2n},\frac{1}{2m}: n \ne m \ge 1\} \\ 2 ,&{} otherwise. \end{array}\right. \end{aligned} \end{aligned}$$

A map $S: P \rightarrow P$ be such that $S0=0, S\frac{1}{n}=\frac{1}{12n}$ for all $n\ge 1$ and let $\phi (t)=t$, $\psi (t)=\frac{4t}{5}$ for $t \in [0,+\infty )$. Then, S has a fixed point in P.

Proof

It is obvious that for $s=\frac{12}{5}$, $(P,d,s,\preceq )$ is a complete partially ordered b-metric space and also by definition, d is discontinuous b-metric space. Now for $\upsilon ,\xi \in P$ with $\upsilon <\xi$, we have the following cases:

Case 1. If $\upsilon =0$ and $\xi =\frac{1}{n}$, $n \ge 1$, then $d(S\upsilon ,S\xi )=d(0,\frac{1}{12n})=\frac{1}{12n}$ and $M(\upsilon ,\xi )=\frac{1}{n}$ or $M(\upsilon ,\xi )= \{1,2\}$. Therefore, we have

$$\begin{aligned} \phi \left( \frac{12}{5}d(S\upsilon ,S\xi )\right) \le \frac{M(\upsilon ,\xi )}{5} =\phi (M(\upsilon ,\xi ))-\psi (M(\upsilon ,\xi )). \end{aligned}$$

Case 2. If $\upsilon =\frac{1}{m}$ and $\xi =\frac{1}{n}$ with $m>n\ge 1$, then

$$\begin{aligned} d(S\upsilon ,S\xi )=d(\frac{1}{12m},\frac{1}{12n}) ~\text {and}~ M(\upsilon ,\xi )\ge \frac{1}{n}-\frac{1}{m}~ \text {or}~ M(\upsilon ,\xi )=2. \end{aligned}$$

Therefore,

$$\begin{aligned} \phi \left( \frac{12}{5}d(S\upsilon ,S\xi )\right) \le \frac{M(\upsilon ,\xi )}{5} =\phi (M(\upsilon ,\xi ))-\psi (M(\upsilon ,\xi )). \end{aligned}$$

Hence, condition (57) of Corollary 2.14 and remaining assumptions are satisfied. Thus, S has a fixed point in P. $\square$

Example 2.18

Let $P=C[a,b]$ be the set of all continuous functions. Let us define a b-metric d on P by

$$\begin{aligned} d(\theta _1,\theta _2)=\sup _{t \in C[a,b]}\{|\theta _1(t)-\theta _2(t)|^2\} \end{aligned}$$

for all $\theta _1,\theta _2 \in P$ with partial order $\preceq$ defined by $\theta _1 \preceq \theta _2$ if $a\le \theta _1(t) \le \theta _2 (t)\le b$, for all $t \in [a,b]$, $0 \le a<b$. Let $S:P \rightarrow P$ be a mapping defined by $S \theta = \frac{\theta }{5}, \theta \in P$ and the two altering distance functions by $\phi (t)=t$, $\psi (t)=\frac{t}{3}$, for any $t \in [0, +\infty ]$. Then S has a unique fixed point in P.

Proof

From the hypotheses, it is clear that $(P,d,s,\preceq )$ is a complete partially ordered b-metric space with parameter $s=2$ and fulfill all conditions of Corollary 2.14 and Note 1. Furthermore, for any $\theta _1,\theta _2 \in P$, the function $\min (\theta _1,\theta _2) (t)=\min \{\theta _1(t),\theta _2(t)\}$ is also continuous and the conditions of Corollary 2.14 and Note 1 are satisfied. Hence, S has a unique fixed point $\theta =0$ in P. $\square$