Common fixed point theorem on Proinov type mappings via simulation function

Abstract

In this paper, we aim to discuss the common fixed point of Proinov type mapping via simulation function. The presented results not only generalize, but also unify the corresponding results in this direction. We also consider an example to indicate the validity of the obtained results.

1 Introduction and preliminaries

Fixed point theory is one of the dynamic research topics of the last decades due to its vast application potential on several distinct disciplines; see e.g. [1–4]. Very recently, Proinov [5] introduced new classes of auxiliary function to propose a new metric fixed point theorem that covers many existing fixed point theorems, mostly having appeared in the last decades. Proinov [5] also showed that recently declared theorems are in fact equivalent to the special cases of Skof’s theorem [6]. Recently, Proinov type contractions have attracted the attention of some authors; see e.g. [7–9] On the other hand, another interesting improvement was reported in 2015: simulation functions were proposed first by Khojasteh et al. [10] to unify some well-known fixed point theorems. This approach has been considered and improved by several authors; see e.g. [11–20].

In this paper, we combine the notions of simulation functions and Proinov type contraction to get a more general framework to guarantee the existence of a fixed point. We investigate the common fixed point of new types mapping under this construction in the context of complete metric space.

We shall first recall the notations we shall use: $\mathbb{R},\mathbb{R}^{+},\mathbb{N}$ for the reals, nonnegative real numbers and natural numbers, $\mathbb{R}^{+}_{0}=\mathbb{R}^{+} \cup \{0\}=[0,\infty )$ and $\mathbb{N}_{0}=\mathbb{N} \cup \{0\}$ and $\Theta = \{ \vartheta:(0,\infty )\rightarrow \mathbb{R} \}$.

Definition 1

(See [10])

A function $\eta:\mathbb{R}^{+}_{0}\times \mathbb{R}^{+}_{0}\rightarrow \mathbb{R}$ is called a simulation function if the following conditions hold:

$(\eta _{1})$:

for all ;

$(\eta _{2})$:

if in $(0,\infty )$ are two sequences such that , then

(1.1)

For the set of all functions simulation functions η, we employ the symbol Z.

Theorem 2

([5])

Let the metric space $(X, \mathsf {d})$ and the mapping $P: X\rightarrow X$ such that

where $\vartheta ,\psi:(0,\infty )\rightarrow \mathbb{R}$ are such that the following conditions hold:

• (a) , for any ;

• (b) , for any ;

• (c) if the sequences $\{ \vartheta (a_{m}) \}$ and $\{ \psi (a_{m}) \}$ are convergent with the same limit and $\{ \vartheta (a_{m}) \}$ is strictly decreasing, then $a_{m}\rightarrow 0$ as $m\rightarrow \infty$;

• (d) or for any ;

• (e) for any .

Then the mapping P possesses exactly one fixed point.

We mention here the following lemmas which will be useful in the sequel.

Lemma 3

([21])

Let be a sequence in a metric space $(X,\mathsf {d})$ such that . If the sequence is not Cauchy then there exist $\mathsf{e}_{0}>0$ and the sequences $\{ m_{l} \}$, $\{ p_{l} \}$ of positive integers such that $m_{l}$ is the smallest index for which $m_{l}>p_{l}>l$, and

(1.2)

Lemma 4

([5])

For $\vartheta:(0,\infty )\rightarrow \mathbb{R}$ the following conditions are equivalent:

• (1) for every .

• (2) for every .

• (3) $\liminf_{m\rightarrow \infty }\vartheta (a_{m})=- \infty$ implies $\lim_{m\rightarrow \infty }a_{m}=0$.

2 Main results

In what follows, we shall consider that $P, Q: X\rightarrow X$ and $\mathsf {m}, \mathsf {r}_{1}, \mathsf {r}_{2}:X\times X\rightarrow \mathbb{R}_{0}^{+}$ are defined as

(2.1)

(2.2)

and

(2.3)

for any such that .

Theorem 5

Let $(X, \mathsf {d})$ be a complete metric space and two mappings $P, Q: X\rightarrow X$. Assume that there exists a function $\eta \in Z$ such that

(2.4)

where $\vartheta ,\psi \in \Theta$. Then the mappings $P,Q$ have a unique fixed point provided that the following conditions are satisfied:

:

, for any ;

:

, for any ;

:

if $\{ a_{m} \}$, $\{ b_{m} \}$ are two convergent sequences with $\lim_{m\rightarrow \infty }a_{m}=\lim_{m \rightarrow \infty }b_{m}>0$ then the sequences $\{ \vartheta (a_{m}) \}$, $\{ \vartheta (b_{m}) \}$ are convergent and $\lim_{m\rightarrow \infty }\vartheta (a_{m})= \lim_{m\rightarrow \infty }\vartheta (b_{m})>0$.

Proof

Let be an arbitrary, but fixed point and the sequence defined as follows:

for each $m\in \mathbb{N}_{0}$. First of all, let us remark that, if there exists $m_{0}\in \mathbb{N}$ such that , then is a fixed point of P (in the case that $m_{0}$ is even) or Q (if $m_{0}$ is odd). Moreover, supposing, for example, that is a fixed point of the mapping P but is not a common fixed point of P and Q (this means ), we get and

since

and taking into account we deduce that

which is a contradiction. Therefore, without loss of generality, we can suppose that for any $m\in \mathbb{N}_{0}$. Thus, supposing that $m=2i$, we have and from (2.4) and $(\eta _{1})$, we have

(2.5)

and using we deduce

(2.6)

In the case that there exists $i_{0}\in \mathbb{N}$ such that , the inequality (2.6) leads to , which is a contradiction. Accordingly,

(2.7)

and then, for any even natural number m, the sequence is non-increasing and positive. Of course, using the same argument, there follows a similar conclusion when m is an odd natural number. Therefore, we can find $\mathsf{D}\geq 0$ such that . Assuming that $\mathsf{D}>0$ by (2.6) we have

(2.8)

which shows us that the sequence is decreasing and moreover, taking into account, it is bounded below. Thus, letting $m\rightarrow \infty$ in (2.8), it follows that the sequences and are convergent to the same limit. Therefore, by $(\eta _{2})$ we get

(2.9)

On the other hand, taking $(\eta _{1})$ into account, (2.4) implies

and

which contradicts (2.9). Thus,

(2.10)

Next, we claim that the sequence is Cauchy. Reasoning by contradiction, if is not Cauchy, by Lemma 3, we can find and the sequences $\{ m_{l} \}$, $\{ p_{l} \}$ of positive integers such that the equalities (1.2) hold, where $m_{l}$ is smallest index for which $m_{l}>p_{l}>l$, for all $l\geq 1$. Replacing in (2.1) by and by we have

and taking into account (2.1) and (2.10) it follows that

(2.11)

So, and by we get

(2.12)

Since by (2.4) we have

(2.13)

or, taking $(\eta _{1})$ and into account,

Using (2.12) we get . Thus, by ($\eta _{2}$) we have

(2.14)

which leads to a contradiction, since by (2.13), we have

Thereupon, is a Cauchy sequence. Moreover, since X is a complete metric space, we can find such that

(2.15)

and we claim that this is a common fixed point of the mappings Q and P. From the point of view of a previous remark, it is enough to prove that is a fixed point of Q (or P). Indeed, supposing , we see that as $m\rightarrow \infty$ and then for infinitely many values of $m\in \mathbb{N}$. Hence, from (2.4) we have

or

(2.16)

where

Thus,

and in view of , . Therefore, letting $m\rightarrow \infty$ in (2.16), we get and using $(\eta _{1})$ and $(\eta _{2})$ we obtain

a contradiction. Thereupon, , which means that is a fixed point of Q and then a common fixed point of P and Q.

Finally, we have to show the uniqueness of this point. If on the contrary, there exists another point , different by , such that , since , we have

which in view of $(\eta _{1})$ becomes

which is obviously a contradiction. □

Corollary 6

Let $(X, \mathsf {d})$ be a complete metric space and a mapping $P: X\rightarrow X$. Assume that there exists a function $\eta \in Z$ such that

(2.17)

for any with , where $\vartheta ,\psi \in \Theta$. Then the mapping P has a unique fixed point provided that the following conditions are satisfied:

:

, for any ;

:

, for any ;

:

if $\{ a_{m} \}$, $\{ b_{m} \}$ are two convergent sequences with $\lim_{m\rightarrow \infty }a_{m}=\lim_{m \rightarrow \infty }b_{m}>0$ then the sequences $\{ \vartheta (a_{m}) \}$, $\{ \vartheta (b_{m}) \}$ are convergent and $\lim_{m\rightarrow \infty }\vartheta (a_{m})= \lim_{m\rightarrow \infty }\vartheta (b_{m})>0$.

Proof

Put $Q=P$ in Theorem 5. □

Theorem 7

Let $(X, \mathsf {d})$ be a complete metric space, two mappings $P, Q: X\rightarrow X$ and a function $\eta \in Z$ such that

(2.18)

where $\vartheta ,\psi \in \Theta$. Suppose that

:

, for any ;

:

, for any ;

:

if $\{ a_{m} \}$, $\{ b_{m} \}$ are convergent sequences with with $\lim_{m\rightarrow \infty }a_{m}=\lim_{m \rightarrow \infty }b_{m}>0$ then the sequences $\{ \vartheta (a_{m}) \}$, $\{ \vartheta (b_{m}) \}$, are convergent and $\lim_{m\rightarrow \infty }\vartheta (a_{m})= \lim_{m\rightarrow \infty }\vartheta (b_{m})$.

Then the mappings $P,Q$ have a unique fixed point.

Proof

Let be an arbitrary point and the sequence in X, defined as follows:

(2.19)

for every $m\in \mathbb{N}$. In what follows, we shall suppose that for any $m\in \mathbb{N}$ (using the same arguments as in the previous proof).

Let , $m\in \mathbb{N}$. First of all, we claim that $o_{m+1}< o_{m}$, for all $m\in \mathbb{N}$. For this purpose, we shall distinguish two situations:

(1) If $m=2i$, $i\in \mathbb{N}$ we have

Since $o_{m}>0$ for any $m\in \mathbb{N}$, we see that and by (2.18) we have

Moreover, from $(\eta _{1})$ and it follows

(2.20)

If for some $i_{0}\in \mathbb{N}$, the inequality (2.20) leads to a contradiction. Therefore, for any $i\in \mathbb{N}$.

(2) If $m=2i-1$, $i\in \mathbb{N}$,

and using the same arguments it follows that , for any $i\in \mathbb{N}$. Therefore, we conclude that the sequence $\{ o_{m} \}$ is convergent with the limit $\mathsf {D}\geq 0$ (being decreasing and bounded below by 0). Moreover, from (2.20) together with $(\eta _{1})$ and we get

$$\begin{aligned} \vartheta (o_{2i+1})< \psi (o_{2i})< \vartheta (o_{2i}). \end{aligned}$$

(2.21)

From our considerations, we conclude that the sequence $\{ \vartheta (o_{2i}) \}$ is convergent (being decreasing and taking into account). Thereupon, by (2.21), the sequence $\{ \psi (o_{2i}) \} )$ is convergent and has the same limit as $\{ \vartheta (o_{2i}) \} )$. If we suppose that $\mathsf {D}>0$, on the one hand, by $(\eta _{1})$ we have

$$\begin{aligned} \lim_{i\rightarrow \infty }\eta \bigl(\vartheta (o_{2i+1}), \psi (o_{2i})\bigr)\geq 0. \end{aligned}$$

On the other hand, taking $(\eta _{2})$ into account we get

$$\begin{aligned} \lim_{i\rightarrow \infty }\eta \bigl(\vartheta (o_{2i+1}), \psi (o_{2i})\bigr)< 0. \end{aligned}$$

This is a contradiction. Therefore $\mathsf {D}=0$, so,

(2.22)

We shall prove that is a Cauchy sequence. Arguing by contradiction, if is not Cauchy, by Lemma 3, we can find two sequences $\{ m_{l} \}$, $\{ p_{l} \}$ of positive integers and such that $m_{l}$ is smallest index for which $m_{l}>p_{l}>l$ and (1.2) hold. Letting , respectively, in (2.2) we have

and then . Moreover, by ,

(2.23)

Plugging this into (2.18), we have

(2.24)

or, taking $(\eta _{1})$ and into account

Thus, by (2.23), we get

which implies, by $(\eta _{2})$,

On the other hand, letting $l\rightarrow \infty$ in (2.24), we have

which contradicts the previous inequality. Therefore, the sequence is Cauchy, and by the completeness of the space X it is a convergent sequence. Let such that . We claim that is a common fixed point of P and Q. First of all, we prove that is a fixed point of Q. If for infinitely many values of m, , then

and , so that .

If for any $m\in \mathbb{N}$, by (2.18) we have

(2.25)

or, equivalently

Since and

we see that . Therefore, by it follows that and taking $(\eta _{2})$ into account,

(2.26)

But, letting $m\rightarrow \infty$ in (2.25),

This is a contradiction; consequently, and is a fixed point of Q and we assume, by “reductio ad absurdum”, that is not a fixed point of P. Then and (2.18) gives us

which is equivalent with

(2.27)

which is a contradiction. Therefore, by it follows that and then is a common fixed point of P and Q.

As a last step in our proof, we shall prove the uniqueness of the common fixed point. Indeed, if there exists another point, for example such that and , then, since , from (2.18) we have

which is a contradiction. Thereupon, , so the fixed point of the mappings Q and P is unique. □

Example 8

Let the set $X= \{ \mathsf {a}_{1},\mathsf {a}_{2},\mathsf {a}_{3},\mathsf {a}_{4} \}$ and $\mathsf {d}:X\times X\rightarrow [0,+\infty )$ be defined as follows:

$$\begin{aligned} &\mathsf {d}(\mathsf {a}_{1}, \mathsf {a}_{2})=\mathsf {d}( \mathsf {a}_{2}, \mathsf {a}_{1})=2, \qquad \mathsf {d}(\mathsf {a}_{1}, \mathsf {a}_{3})=\mathsf {d}(\mathsf {a}_{3}, \mathsf {a}_{1})=3, \qquad \mathsf {d}(\mathsf {a}_{1}, \mathsf {a}_{4})=\mathsf {d}( \mathsf {a}_{4}, \mathsf {a}_{1})=5; \\ &\mathsf {d}(\mathsf {a}_{2}, \mathsf {a}_{3})=\mathsf {d}(\mathsf {a}_{3}, \mathsf {a}_{2})=5,\qquad \mathsf {d}(\mathsf {a}_{3}, \mathsf {a}_{4})=\mathsf {d}(\mathsf {a}_{4}, \mathsf {a}_{3})=8, \qquad \mathsf {d}(\mathsf {a}_{2}, \mathsf {a}_{4})= \mathsf {d}( \mathsf {a}_{4}, \mathsf {a}_{2})=3; \\ &\mathsf {d}(\mathsf {a}_{1}, \mathsf {a}_{1})=\mathsf {d}(\mathsf {a}_{2}, \mathsf {a}_{2})= \mathsf {d}(\mathsf {a}_{3}, \mathsf {a}_{3})=\mathsf {d}(\mathsf {a}_{4}, \mathsf {a}_{4})=0. \end{aligned}$$

Let $Q,P: X\rightarrow X$ be two mappings where

$$\begin{aligned} & P\mathsf {a}_{1}=P\mathsf {a}_{2}= P\mathsf {a}_{4}= \mathsf {a}_{1},\qquad P\mathsf {a}_{3}=\mathsf {a}_{2}; \\ &Q\mathsf {a}_{1}=Q\mathsf {a}_{2}=Q\mathsf {a}_{3}= \mathsf {a}_{1}, \qquad Q\mathsf {a}_{4}=\mathsf {a}_{2}, \end{aligned}$$

and we choose the functions $\eta \in Z$ and $\vartheta ,\psi \in \Theta$, with

Of course, we can easily see that, with these choices, the assumptions – of Theorem 7 are obviously satisfied. Thus, we shall check that (2.18) holds for any , such that . We discuss then the following situations:

• , ,

  $$\begin{aligned} &\mathsf {d}(P\mathsf {a}_{1}, Q\mathsf {a}_{4})=\mathsf {d}( \mathsf {a}_{1}, \mathsf {a}_{2})=2, \\ &\begin{aligned}[b] \mathsf {r}_{1}(\mathsf {a}_{1}, \mathsf {a}_{4})&=\max \biggl\{ \mathsf {d}( \mathsf {a}_{1},\mathsf {a}_{4}), \frac{(1+\mathsf {d}(\mathsf {a}_{1},P\mathsf {a}_{1}))\mathsf {d}(\mathsf {a}_{4}, Q\mathsf {a}_{4})}{1+\mathsf {d}(\mathsf {a}_{1}, \mathsf {a}_{4})}, \frac{\mathsf {d}(\mathsf {a}_{1},Q\mathsf {a}_{4})+\mathsf {d}(\mathsf {a}_{4},P\mathsf {a}_{1})}{2} \biggr\} \\ &=\max \biggl\{ \mathsf {d}(\mathsf {a}_{1},\mathsf {a}_{4}), \frac{(1+\mathsf {d}(\mathsf {a}_{1},\mathsf {a}_{1}))\mathsf {d}(\mathsf {a}_{4}, \mathsf {a}_{2})}{1+\mathsf {d}(\mathsf {a}_{1}, \mathsf {a}_{4})}, \frac{\mathsf {d}(\mathsf {a}_{1},\mathsf {a}_{2})+\mathsf {d}(\mathsf {a}_{4},\mathsf {a}_{1})}{2} \biggr\} \\ &=\max \biggl\{ 5, \frac{5}{6},\frac{7}{2} \biggr\} =5 \end{aligned} \end{aligned}$$

  and

  $$\begin{aligned} \eta \bigl(\vartheta \bigl(\mathsf {d}(P\mathsf {a}_{1},Q\mathsf {a}_{4})\bigr), \psi \bigl(\mathsf {r}_{1}(\mathsf {a}_{1},\mathsf {a}_{4})\bigr) \bigr)=0.88\cdot 0.91 \cdot 5-2=2.004>0. \end{aligned}$$

• , ,

  $$\begin{aligned} &\mathsf {d}(P\mathsf {a}_{2}, Q\mathsf {a}_{4})=\mathsf {d}( \mathsf {a}_{1}, \mathsf {a}_{2})=2, \\ &\begin{aligned}[b] \mathsf {r}_{1}(\mathsf {a}_{2}, \mathsf {a}_{4})&=\max \biggl\{ \mathsf {d}( \mathsf {a}_{2},\mathsf {a}_{4}), \frac{(1+\mathsf {d}(\mathsf {a}_{2},P\mathsf {a}_{2}))\mathsf {d}(\mathsf {a}_{4}, Q\mathsf {a}_{4})}{1+\mathsf {d}(\mathsf {a}_{2}, \mathsf {a}_{4})}, \frac{\mathsf {d}(\mathsf {a}_{2},Q\mathsf {a}_{4})+\mathsf {d}(\mathsf {a}_{4},P\mathsf {a}_{2})}{2} \biggr\} \\ &=\max \biggl\{ \mathsf {d}(\mathsf {a}_{2},\mathsf {a}_{4}), \frac{(1+\mathsf {d}(\mathsf {a}_{2},\mathsf {a}_{1}))\mathsf {d}(\mathsf {a}_{4}, \mathsf {a}_{2})}{1+\mathsf {d}(\mathsf {a}_{2}, \mathsf {a}_{4})}, \frac{\mathsf {d}(\mathsf {a}_{1},\mathsf {a}_{2})+\mathsf {d}(\mathsf {a}_{4},\mathsf {a}_{1})}{2} \biggr\} \\ &=\max \biggl\{ 3, \frac{9}{4},\frac{7}{2} \biggr\} = \frac{7}{2} \end{aligned} \end{aligned}$$

  and

  $$\begin{aligned} \eta \bigl(\vartheta \bigl(\mathsf {d}(P\mathsf {a}_{2},Q\mathsf {a}_{4})\bigr), \psi \bigl(\mathsf {r}_{1}(\mathsf {a}_{2},\mathsf {a}_{4})\bigr) \bigr)=0.88\cdot 0,91 \cdot 3.5-2=0.8>0. \end{aligned}$$

• , ,

  $$\begin{aligned} &\mathsf {d}(P\mathsf {a}_{3}, Q\mathsf {a}_{1})=\mathsf {d}( \mathsf {a}_{2}, \mathsf {a}_{1})=2, \\ &\begin{aligned}[b] \mathsf {r}_{1}(\mathsf {a}_{3}, \mathsf {a}_{1})&=\max \biggl\{ \mathsf {d}( \mathsf {a}_{3},\mathsf {a}_{1}), \frac{(1+\mathsf {d}(\mathsf {a}_{3},P\mathsf {a}_{3}))\mathsf {d}(\mathsf {a}_{1}, Q\mathsf {a}_{1})}{1+\mathsf {d}(\mathsf {a}_{3}, \mathsf {a}_{1})}, \frac{\mathsf {d}(\mathsf {a}_{3},Q\mathsf {a}_{1})+\mathsf {d}(\mathsf {a}_{1},P\mathsf {a}_{3})}{2} \biggr\} \\ &=\max \biggl\{ \mathsf {d}(\mathsf {a}_{3},\mathsf {a}_{1}), \frac{(1+\mathsf {d}(\mathsf {a}_{3},\mathsf {a}_{2}))\mathsf {d}(\mathsf {a}_{1}, \mathsf {a}_{1})}{1+\mathsf {d}(\mathsf {a}_{2}, \mathsf {a}_{4})}, \frac{\mathsf {d}(\mathsf {a}_{3},\mathsf {a}_{1})+\mathsf {d}(\mathsf {a}_{1},\mathsf {a}_{1})}{2} \biggr\} \\ &=\max \biggl\{ 3, 0,\frac{3}{2} \biggr\} =3 \end{aligned} \end{aligned}$$

  and

  $$\begin{aligned} \eta \bigl(\vartheta \bigl(\mathsf {d}(P\mathsf {a}_{3},Q\mathsf {a}_{1})\bigr), \psi \bigl(\mathsf {r}_{1}(\mathsf {a}_{3},\mathsf {a}_{1})\bigr) \bigr)=0.88\cdot 0,91 \cdot 3-2=0.4024>0. \end{aligned}$$

• , ,

  $$\begin{aligned} &\mathsf {d}(P\mathsf {a}_{3}, Q\mathsf {a}_{2})=\mathsf {d}( \mathsf {a}_{2}, \mathsf {a}_{1})=2, \\ &\begin{aligned}[b] \mathsf {r}_{1}(\mathsf {a}_{3}, \mathsf {a}_{2})&=\max \biggl\{ \mathsf {d}( \mathsf {a}_{3},\mathsf {a}_{2}), \frac{(1+\mathsf {d}(\mathsf {a}_{3},P\mathsf {a}_{3}))\mathsf {d}(\mathsf {a}_{2}, Q\mathsf {a}_{2})}{1+\mathsf {d}(\mathsf {a}_{3}, \mathsf {a}_{2})}, \frac{\mathsf {d}(\mathsf {a}_{3},Q\mathsf {a}_{2})+\mathsf {d}(\mathsf {a}_{2},P\mathsf {a}_{3})}{2} \biggr\} \\ &=\max \biggl\{ \mathsf {d}(\mathsf {a}_{3},\mathsf {a}_{2}), \frac{(1+\mathsf {d}(\mathsf {a}_{3},\mathsf {a}_{2}))\mathsf {d}(\mathsf {a}_{1}, \mathsf {a}_{1})}{1+\mathsf {d}(\mathsf {a}_{2}, \mathsf {a}_{4})}, \frac{\mathsf {d}(\mathsf {a}_{3},\mathsf {a}_{1})+\mathsf {d}(\mathsf {a}_{1},\mathsf {a}_{1})}{2} \biggr\} \\ &=\max \biggl\{ 5, 0,\frac{3}{2} \biggr\} =5 \end{aligned} \end{aligned}$$

  and

  $$\begin{aligned} \eta \bigl(\vartheta \bigl(\mathsf {d}(P\mathsf {a}_{3},Q\mathsf {a}_{2})\bigr), \psi \bigl(\mathsf {r}_{1}(\mathsf {a}_{3},\mathsf {a}_{2})\bigr) \bigr)=0.88\cdot 0,91 \cdot 5-2=2.004>0. \end{aligned}$$

• , ,

  $$\begin{aligned} &\mathsf {d}(P\mathsf {a}_{3}, Q\mathsf {a}_{3})=\mathsf {d}( \mathsf {a}_{2}, \mathsf {a}_{1})=2, \\ &\begin{aligned}[b] \mathsf {r}_{1}(\mathsf {a}_{3}, \mathsf {a}_{3})&=\max \biggl\{ \mathsf {d}( \mathsf {a}_{3},\mathsf {a}_{3}), \frac{(1+\mathsf {d}(\mathsf {a}_{3},P\mathsf {a}_{3}))\mathsf {d}(\mathsf {a}_{3}, Q\mathsf {a}_{3})}{1+\mathsf {d}(\mathsf {a}_{3}, \mathsf {a}_{3})}, \frac{\mathsf {d}(\mathsf {a}_{3},Q\mathsf {a}_{3})+\mathsf {d}(\mathsf {a}_{3},P\mathsf {a}_{3})}{2} \biggr\} \\ &=\max \biggl\{ \mathsf {d}(\mathsf {a}_{3},\mathsf {a}_{3}), \frac{(1+\mathsf {d}(\mathsf {a}_{3},\mathsf {a}_{2}))\mathsf {d}(\mathsf {a}_{3}, \mathsf {a}_{1})}{1+\mathsf {d}(\mathsf {a}_{3}, \mathsf {a}_{3})}, \frac{\mathsf {d}(\mathsf {a}_{3},\mathsf {a}_{1})+\mathsf {d}(\mathsf {a}_{3},\mathsf {a}_{1})}{2} \biggr\} \\ &=\max \biggl\{ 0, 18,\frac{6}{2} \biggr\} =18 \end{aligned} \end{aligned}$$

  and

  $$\begin{aligned} \eta \bigl(\vartheta \bigl(\mathsf {d}(P\mathsf {a}_{3},Q\mathsf {a}_{3})\bigr), \psi \bigl(\mathsf {r}_{1}(\mathsf {a}_{3},\mathsf {a}_{3})\bigr) \bigr)=0.88\cdot 0,91 \cdot 18-2=12.41>0. \end{aligned}$$

• , ,

  $$\begin{aligned} &\mathsf {d}(P\mathsf {a}_{4}, Q\mathsf {a}_{4})=\mathsf {d}( \mathsf {a}_{1}, \mathsf {a}_{2})=2, \\ &\begin{aligned}[b] \mathsf {r}_{1}(\mathsf {a}_{4}, \mathsf {a}_{4})&=\max \biggl\{ \mathsf {d}( \mathsf {a}_{4},\mathsf {a}_{4}), \frac{(1+\mathsf {d}(\mathsf {a}_{4},P\mathsf {a}_{4}))\mathsf {d}(\mathsf {a}_{4}, Q\mathsf {a}_{4})}{1+\mathsf {d}(\mathsf {a}_{4}, \mathsf {a}_{4})}, \frac{\mathsf {d}(\mathsf {a}_{4},Q\mathsf {a}_{4})+\mathsf {d}(\mathsf {a}_{4},P\mathsf {a}_{4})}{2} \biggr\} \\ &=\max \biggl\{ \mathsf {d}(\mathsf {a}_{4},\mathsf {a}_{4}), \frac{(1+\mathsf {d}(\mathsf {a}_{4},\mathsf {a}_{1}))\mathsf {d}(\mathsf {a}_{4}, \mathsf {a}_{2})}{1+\mathsf {d}(\mathsf {a}_{4}, \mathsf {a}_{4})}, \frac{\mathsf {d}(\mathsf {a}_{4},\mathsf {a}_{2})+\mathsf {d}(\mathsf {a}_{4},\mathsf {a}_{1})}{2} \biggr\} \\ &=\max \{ 0, 18,4 \} =18 \end{aligned} \end{aligned}$$

  and

  $$\begin{aligned} \eta \bigl(\vartheta \bigl(\mathsf {d}(P\mathsf {a}_{4},Q\mathsf {a}_{4})\bigr), \psi \bigl(\mathsf {r}_{1}(\mathsf {a}_{4},\mathsf {a}_{4})\bigr) \bigr)=0.88\cdot 0.91 \cdot 18-2=12.41>0. \end{aligned}$$

Therefore, all the assumptions of Theorem 7 are satisfied; $\mathsf {a}_{1}$ is the unique common fixed point of the mappings P and Q.

Corollary 9

Let $(X, \mathsf {d})$ be a complete metric space, a mapping $P: X\rightarrow X$ and a function $\eta \in Z$ such that

(2.28)

for any with , where $\vartheta ,\psi \in \Theta$. Suppose that

:

, for any ;

:

, for any ;

:

if $\{ a_{m} \}$, $\{ b_{m} \}$ are convergent sequences with with $\lim_{m\rightarrow \infty }a_{m}=\lim_{m \rightarrow \infty }b_{m}>0$ then the sequences $\{ \vartheta (a_{m}) \}$, $\{ \vartheta (b_{m}) \}$, are convergent and $\lim_{m\rightarrow \infty }\vartheta (a_{m})= \lim_{m\rightarrow \infty }\vartheta (b_{m})$.

Then the mapping P possesses a unique fixed point.

Proof

Put $Q=P$ in Theorem 7. □

Theorem 10

Let $(X, \mathsf {d})$ be a complete metric space, two mappings $P, Q: X\rightarrow X$, the functions $\vartheta ,\psi \in \Theta$ and a function such that

(2.29)

when and when . Suppose that

:

, for any ;

:

, for any ;

:

if $\{ a_{m} \}$, $\{ b_{m} \}$ are convergent sequences with with $\lim_{m\rightarrow \infty }a_{m}=\lim_{m \rightarrow \infty }b_{m}>0$ then the sequences $\{ \vartheta (a_{m}) \}$, $\{ \vartheta (b_{m}) \}$, are convergent and $\lim_{m\rightarrow \infty }\vartheta (a_{m})= \lim_{m\rightarrow \infty }\vartheta (b_{m})$;

:

if $\{ \vartheta (a_{m}) \}$ is a strictly decreasing sequence and $\{ \vartheta (a_{m}) \}$, $\{ \psi (a_{m}) \}$ are convergent with the same limit then $\lim_{m\rightarrow \infty }a_{m}=0$;

:

, for any .

Then the mappings $P,Q$ have a unique fixed point.

Proof

First of al, by (2.29) and taking $(\eta _{1})$ into account, we have

which can be rewritten as

(2.30)

Let be the sequence defined by (2.19). Since for any $m\in \mathbb{N}_{0}$ (we have already shown this in the proof of Theorem 5), letting $m=2i$, we have

(2.31)

and letting $m=2i-1$,

(2.32)

Thus, from (2.30) and keeping in mind we get

and similarly

Denoting by $o_{m}$, we get

$$\begin{aligned} \begin{aligned}[b] &\vartheta (o_{2i})\leq \psi (o_{2i-1})< \vartheta (o_{2i-1}), \\ &\vartheta (o_{2i+1})\leq \psi (o_{2i})< \vartheta (o_{2i}) \end{aligned} \end{aligned}$$

and we conclude that

$$\begin{aligned} \vartheta (o_{m+1})< \psi (o_{m})< \vartheta (o_{m}) \end{aligned}$$

(2.33)

for any $m\in \mathbb{N}$. Consequently, the sequence $\{ \vartheta (o_{m}) \}$ is convergent, being strictly decreasing and bounded below(from and Lemma 4). Thus, letting $m\rightarrow \infty$ in (2.33) we see that $\{ \psi (o_{m}) \}$ is convergent with the same limit as $\{ \vartheta (o_{m}) \}$. Thereupon, by ,

We claim that the sequence is Cauchy, reasoning by contradiction. Indeed, if we suppose that is not Cauchy, by Lemma 3, we can find and the sequences $\{ m_{l} \}$, $\{ p_{l} \}$ of positive integers such that the equalities (1.2) hold, where $m_{l}$ is the smallest index for which $m_{l}>p_{l}>l$, for all $l\geq 1$. We have

and setting and it follows that . Moreover, by , $\lim_{l\rightarrow \infty }\vartheta (u_{l})= \lim_{l\rightarrow \infty }\vartheta (v_{l})$. Consequently, by (2.33), $\lim_{l\rightarrow \infty }\vartheta (u_{l})= \lim_{l\rightarrow \infty }\psi (v_{l})$ and using $(\eta _{2})$,

$$\begin{aligned} \limsup_{l\rightarrow \infty }\eta \bigl(\vartheta (u_{l}), \psi (v_{l}) \bigr)< 0, \end{aligned}$$

(2.34)

which is a contradiction, since by (2.29)

In this way we proved that is a Cauchy sequence on a complete metric space, so there exists such that .

We shall show that is a common fixed point of P and Q. First of all, we remark that, if for infinitely many values of m, then

That means that .

Therefore, we can suppose that for infinitely many values of m, by (2.29) we have

or equivalently, by

(2.35)

where

as $m\rightarrow \infty$. Let . If we suppose that , from (2.35) and we have

which is a contradiction. Therefore, .

Similarly, choosing and in (2.29), we can show that and we conclude that .

To prove the uniqueness of the common fixed point, we will assume that, on the contrary, there exists another point such that and . Since , we have

which is a contradiction. Hence . □

Corollary 11

Let $(X, \mathsf {d})$ be a complete metric space, a mapping $P: X\rightarrow X$, the functions $\vartheta ,\psi \in \Theta$ and a function $\eta \in Z$ such that

(2.36)

for any with , when and when . Suppose that

:

, for any ;

:

, for any ;

:

if $\{ a_{m} \}$, $\{ b_{m} \}$ are convergent sequences with with $\lim_{m\rightarrow \infty }a_{m}=\lim_{m \rightarrow \infty }b_{m}>0$ then the sequences $\{ \vartheta (a_{m}) \}$, $\{ \vartheta (b_{m}) \}$, are convergent and $\lim_{m\rightarrow \infty }\vartheta (a_{m})= \lim_{m\rightarrow \infty }\vartheta (b_{m})$;

:

if $\{ \vartheta (a_{m}) \}$ is a strictly decreasing sequence and $\{ \vartheta (a_{m}) \}$, $\{ \psi (a_{m}) \}$ are convergent with the same limit then $\lim_{m\rightarrow \infty }a_{m}=0$;

:

, for any .

Then the mapping P has exactly one fixed point.

Proof

Put $Q=P$ in Theorem 10. □