We start this part with the following definition:
Definition 2.1
We say that a mapping \(\beth :\mho ^{z}\rightarrow \mho \) is a Prešić-type rational η-contraction (PTR η-C, for short) if there is some \(\gamma \in (0,1)\) so that
$$ \eta \bigl( \varpi \bigl( \beth ( \zeta _{1},\dots ,\zeta _{z} ) ,\beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \bigr) \bigr) \leq \biggl\{ \eta \biggl( \max \biggl\{ \frac{\varpi ( \zeta _{j},\zeta _{j+1} ) }{1+\varpi ( \zeta _{j},\zeta _{j+1} ) }:1 \leq j \leq z \biggr\} \biggr) \biggr\} ^{\gamma } $$
(2.1)
for each \(( \zeta _{1},\dots ,\zeta _{z+1} ) \in \mho ^{z+1}\) with \(\beth ( \zeta _{1},\dots ,\zeta _{z} ) \neq \beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \).
It should be noted that if \(\eta (r)=e^{\sqrt{r}}\), then PTR η-C reduces to
$$ \varpi \bigl( \beth ( \zeta _{1},\dots ,\zeta _{z} ) , \beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \bigr) \leq \gamma ^{2} \biggl( \max \biggl\{ \frac{\varpi ( \zeta _{j},\zeta _{j+1} ) }{1+\varpi ( \zeta _{j},\zeta _{j+1} ) }:1\leq j\leq z \biggr\} \biggr) , $$
(2.2)
for each \(( \zeta _{1},\dots ,\zeta _{z+1} ) \in \mho ^{z+1}\), \(\beth ( \zeta _{1},\dots ,\zeta _{z} ) \neq \beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \).
In addition, if \(( \zeta _{1},\dots ,\zeta _{z+1} ) \in \mho ^{z+1}\) is such that \(\beth ( \zeta _{1},\dots ,\zeta _{z} ) =\beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \), then condition (2.2) is more general than (1.3), so the mapping ℶ in (2.2) extends and unifies Cirić–Prešić contraction.
Remark 2.2
Every PTR η-C ℶ is a Prešić mapping by \((\eta _{1})\) and (1.4), that is,
$$\begin{aligned} \varpi \bigl( \beth ( \zeta _{1},\dots ,\zeta _{z} ) , \beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \bigr) &\leq \gamma \max \biggl\{ \frac{\varpi ( \zeta _{j},\zeta _{j+1} ) }{1+\varpi ( \zeta _{j},\zeta _{j+1} ) }:1 \leq j\leq z \biggr\} \\ &< \max \bigl\{ \varpi ( \zeta _{j},\zeta _{j+1} ) :1 \leq j \leq z \bigr\} . \end{aligned}$$
for each \(( \zeta _{1},\dots ,\zeta _{z+1} ) \in \mho ^{z+1}\) with \(\beth ( \zeta _{1},\dots ,\zeta _{z} ) \neq \beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \). Thus, each PTR η-C ℶ is a continuous function.
Now, our first result is as follows:
Theorem 2.3
Suppose that \(\beth :\mho ^{z}\rightarrow \mho \) is a PTR η-C. Then for any chosen points \(\zeta _{1},\dots ,\zeta _{z}\in \mho \), the sequence \(\{\zeta _{l}\}\) described in (1.2) is convergent to \(\zeta ^{\ast }\in \mho \) and \(\zeta ^{\ast }\) is an FP of ℶ. In addition, if \(\beth ( \zeta ^{\ast },\dots ,\zeta ^{\ast } ) \neq \beth ( \zeta ^{\prime },\dots ,\zeta ^{{\prime }} ) \) with
$$ \eta \bigl( \varpi \bigl( \beth \bigl( \zeta ^{\ast },\dots , \zeta ^{\ast } \bigr) ,\beth \bigl( \zeta ^{{\prime }},\dots , \zeta ^{\prime } \bigr) \bigr) \bigr) \leq \bigl[ \eta \bigl( \varpi \bigl( \zeta ^{\ast },\zeta ^{\prime } \bigr) \bigr) \bigr] ^{\gamma } $$
for \(\zeta ^{\ast },\zeta ^{{\prime }}\in \mho \) such that \(\zeta ^{\ast }\neq \zeta ^{{\prime }}\), then the point \(\zeta ^{\ast }\) is unique.
Proof
Let \(\zeta _{1},\dots ,\zeta _{z}\) be arbitrary z elements in ℧ and for \(l\in \mathbb{N} \) the sequence \(\{\zeta _{l}\}\) is defined in (1.2). If for some \(l_{0}=\{1,2,\dots ,z\}\) one has \(\zeta _{l_{0}}=\zeta _{l_{0}+1}\), then
$$ \zeta _{l_{0}+z}=\beth ( \zeta _{l_{0}},\zeta _{l_{0}+1},\dots , \zeta _{l_{0}+z-1} ) =\beth ( \zeta _{l_{0}+z},\zeta _{l_{0}+z}, \dots ,\zeta _{l_{0}+z} ) , $$
which means that \(\zeta _{l_{0}+z}\) is an FP of ℶ and there is no further proof needed. So, we consider \(\zeta _{l+z}\neq \zeta _{l+z+1}\) for all \(l\in \mathbb{N} \). Put \(\gimel _{l+z}=\varpi ( \zeta _{l+z},\zeta _{l+z+1} ) \) and
$$ \phi =\max \biggl\{ \frac{\varpi ( \zeta _{1},\zeta _{2} ) }{1+\varpi ( \zeta _{1},\zeta _{2} ) }, \frac{\varpi ( \zeta _{2},\zeta _{3} ) }{1+\varpi ( \zeta _{2},\zeta _{3} ) }, \dots ,\frac{\varpi ( \zeta _{z},\zeta _{z+1} ) }{1+\varpi ( \zeta _{z},\zeta _{z+1} ) } \biggr\} . $$
Then for all \(l\in \mathbb{N} \) and \(\phi >0\), we have \(\gimel _{l+z}>0\). Thus, for \(l\leq z\), we obtain
$$\begin{aligned} 1 &< \eta ( \gimel _{z+1} ) \\ & =\eta \bigl( \varpi ( \zeta _{z+1}, \zeta _{z+2} ) \bigr) \\ &=\eta \bigl( \varpi \bigl( \beth ( \zeta _{1},\zeta _{2}, \dots ,\zeta _{z} ) ,\beth ( \zeta _{2},\zeta _{3},\dots , \zeta _{z+1} ) \bigr) \bigr) \\ &\leq \biggl[ \eta \biggl( \max \biggl\{ \frac{\varpi ( \zeta _{j},\zeta _{j+1} ) }{1+\varpi ( \zeta _{j},\zeta _{j+1} ) }:1 \leq j\leq z \biggr\} \biggr) \biggr] ^{\gamma } \\ &= \bigl[ \eta ( \phi ) \bigr] ^{\gamma }. \end{aligned}$$
Also,
$$\begin{aligned} 1 &< \eta ( \gimel _{z+2} ) \\ &=\eta \bigl( \varpi ( \zeta _{z+2}, \zeta _{z+3} ) \bigr) \\ &=\eta \bigl( \varpi \bigl( \beth ( \zeta _{2},\zeta _{3}, \dots ,\zeta _{z+1} ) ,\beth ( \zeta _{3},\zeta _{4}, \dots ,\zeta _{z+2} ) \bigr) \bigr) \\ &\leq \biggl[ \eta \biggl( \max \biggl\{ \frac{\varpi ( \zeta _{j},\zeta _{j+1} ) }{1+\varpi ( \zeta _{j},\zeta _{j+1} ) }:2 \leq j\leq z+1 \biggr\} \biggr) \biggr] ^{\gamma } \\ &= \bigl[ \eta ( \phi ) \bigr] ^{\gamma ^{2}}. \end{aligned}$$
Continuing in the same pattern, for \(l\geq 1\), we get
$$ \begin{aligned} 1 &< \eta ( \gimel _{z+l} ) \\ & =\eta \bigl( \varpi ( \zeta _{l+z},\zeta _{l+z+1} ) \bigr) \\ &=\eta \bigl( \varpi \bigl( \beth ( \zeta _{l},\zeta _{l+1}, \dots ,\zeta _{l+z-1} ) ,\beth ( \zeta _{l+1},\zeta _{l+2}, \dots ,\zeta _{l+z} ) \bigr) \bigr) \\ &\leq \bigl[ \eta ( \phi ) \bigr] ^{\gamma ^{l}}. \end{aligned} $$
(2.3)
Taking \(l\rightarrow \infty \) in (2.3) and using \((\eta _{2})\), we have
$$ \lim_{l\rightarrow \infty }\eta ( \gimel _{z+l} ) =1 \quad \Longleftrightarrow\quad \lim_{l\rightarrow \infty }\gimel _{z+l}=0. $$
Based on \((\eta _{3})\), there are \(\ell \in (0,1)\) and \(u\in (0,\infty )\) so that
$$ \lim_{l\rightarrow \infty } \biggl( \frac{\eta ( \gimel _{z+l} ) -1}{\gimel _{z+l}^{\ell }} \biggr) =u. $$
Assume that \(u<\infty \) and \(v=\frac{u}{2}>0\). By the definition of the limit, there is \(l_{1}\in \mathbb{N} \) such that
$$ \biggl\vert \frac{\eta ( \gimel _{z+l} ) -1}{\gimel _{z+l}^{\ell }}-u \biggr\vert \leq v,\quad \forall l>l_{1}. $$
It follows that
$$ \frac{\eta ( \gimel _{z+l} ) -1}{\gimel _{z+l}^{\ell }} \geq u-v=\frac{u}{2}=v,\quad \forall l>l_{1}. $$
Set \(\frac{1}{v}=q\), then
$$ l\gimel _{z+l}^{\ell }\leq lq \bigl( \eta ( \gimel _{z+l} ) -1 \bigr) , \quad \forall l>l_{1}. $$
Suppose that \(u=\infty \) and \(v>0\). By the definition of the limit, there is \(l_{1}\in \mathbb{N} \) such that
$$ v\leq \frac{\eta ( \gimel _{z+l} ) -1}{\gimel _{z+l}^{\ell }},\quad \forall l>l_{1}. $$
This implies after taking \(\frac{1}{v}=q\) that
$$ l\gimel _{z+l}^{\ell }\leq lq \bigl( \eta ( \gimel _{z+l} ) -1 \bigr) , \quad \forall l>l_{1}. $$
Thus, in both cases, there are \(l_{1}\in \mathbb{N} \) and \(q>0\) so that
$$ l\gimel _{z+l}^{\ell }\leq lq \bigl( \eta ( \gimel _{z+l} ) -1 \bigr) , \quad \forall l>l_{1}. $$
Applying (2.3), we get
$$ l\gimel _{z+l}^{\ell }\leq lq \bigl( \bigl[ \eta ( \phi ) \bigr] ^{\gamma ^{l}}-1 \bigr) , \quad \forall l>l_{1}, $$
and, when \(l\rightarrow \infty \), have
$$ \lim_{l\rightarrow \infty }l\gimel _{z+l}^{\ell }=0. $$
Thus, there is \(l_{2}\in \mathbb{N} \) and \(q>0\) such that
$$ l\gimel _{z+l}^{\ell }\leq 1,\quad \forall l>l_{2}. $$
Hence we can write
$$ \gimel _{z+l}\leq \frac{1}{l^{\frac{1}{\ell }}}, \quad \forall l>l_{2}. $$
Now, we clarify that \(\{\zeta _{l}\}\) is a Cauchy sequence. For \(b>l>l_{2}\), one can write
$$\begin{aligned} \varpi ( \zeta _{z+l},\zeta _{z+b} ) ={}&\varpi \bigl( \beth ( \zeta _{l},\dots ,\zeta _{z+l-1} ) ,\beth ( \zeta _{b},\dots ,\zeta _{z+b-1} ) \bigr) \\ \leq{}& \varpi \bigl( \beth ( \zeta _{l},\dots ,\zeta _{z+l-1} ) ,\beth ( \zeta _{l+1},\dots ,\zeta _{z+l} ) \bigr) \\ &{} +\varpi \bigl( \beth ( \zeta _{l+1},\dots ,\zeta _{z+l} ) , \beth ( \zeta _{l+2},\dots ,\zeta _{z+l+1} ) \bigr) \\ & {}+\cdots +\varpi \bigl( \beth ( \zeta _{b-1},\dots , \zeta _{z+b-2} ) ,\beth ( \zeta _{b},\dots ,\zeta _{z+b-1} ) \bigr) \\ ={}&\varpi ( \zeta _{z+l},\zeta _{z+l+1} ) +\varpi ( \zeta _{z+l+1},\zeta _{z+l+2} ) +\cdots +\varpi ( \zeta _{z+b-1}, \zeta _{z+b} ) \\ ={}&\gimel _{l+z}+\gimel _{l+z+1}+\cdots +\gimel _{z+b-1} \\ ={}&\sum_{s=l}^{b-1}\gimel _{s+z}< \sum_{s=l}^{\infty } \gimel _{s+z}\leq \sum_{s=l}^{\infty } \frac{1}{s^{\frac{1}{\ell }}}< \infty , \end{aligned}$$
hence it follows that \(\{\zeta _{l}\}\) is a Cauchy sequence in \((\mho ,\varpi )\). The completeness of ℧ yields that there is \(\zeta ^{\ast }\in \mho \) such that
$$ \lim_{l,b\rightarrow \infty }\varpi ( \zeta _{l},\zeta _{b} ) =\lim_{l\rightarrow \infty }\varpi \bigl( \zeta _{l}, \zeta ^{\ast } \bigr) =0. $$
Because ℶ is continuous, we have
$$\begin{aligned} \hbar &=\lim_{l\rightarrow \infty }\zeta _{l+z} \\ &=\lim _{l\rightarrow \infty }\beth ( \zeta _{l},\zeta _{l+1},\dots ,\zeta _{z+l-1} ) \\ &=\beth \Bigl( \lim_{l\rightarrow \infty }\zeta _{l},\lim _{l \rightarrow \infty }\zeta _{l+1},\dots ,\lim_{l\rightarrow \infty } \zeta _{z+l-1} \Bigr) \\ & =\beth \bigl( \zeta ^{\ast },\zeta ^{\ast }, \dots ,\zeta ^{\ast } \bigr) . \end{aligned}$$
For uniqueness, assume that \(\zeta ^{\ast }\) and \(\zeta ^{{\prime }}\) are two distinct FP of the mapping ℶ, i.e., \(\zeta ^{\ast }=\beth ( \zeta ^{\ast },\zeta ^{\ast },\dots , \zeta ^{\ast } ) \) and \(\zeta ^{{\prime }}=\beth ( \zeta ^{{\prime }},\zeta ^{{\prime }}, \dots ,\zeta ^{{\prime }} ) \) with \(\zeta ^{\ast }\neq \zeta ^{{\prime }}\). Hence, by hypothesis (2.1), we can write
$$\begin{aligned} \eta \bigl( \varpi \bigl( \zeta ^{\ast },\zeta ^{{\prime }} \bigr) \bigr) &=\eta \bigl( \varpi \bigl( \beth \bigl( \zeta ^{\ast }, \zeta ^{\ast },\dots ,\zeta ^{\ast } \bigr) ,\beth \bigl( \zeta ^{{ \prime }},\zeta ^{{\prime }},\dots ,\zeta ^{{\prime }} \bigr) \bigr) \bigr) \\ &\leq \biggl[ \eta \biggl( \frac{\varpi ( \zeta ^{\ast },\zeta ^{{\prime }} ) }{1+\varpi ( \zeta ^{\ast },\zeta ^{{\prime }} ) } \biggr) \biggr] ^{\gamma } \\ &\leq \bigl[ \eta \bigl( \varpi \bigl( \zeta ^{\ast },\zeta ^{{ \prime }} \bigr) \bigr) \bigr] ^{\gamma }, \end{aligned}$$
a contradiction, as \(\gamma \in (0,1)\). Therefore, \(\zeta ^{\ast }=\zeta ^{{\prime }}\). This ends the proof. □
The following examples support Theorem 2.3.
Example 2.4
Let \(\{\zeta _{l}\}\) be a sequence defined as follows:
$$ \textstyle\begin{cases} \zeta _{1}=3, \\ \zeta _{2}=3+7, \\ \vdots \\ \zeta _{l}=3+7+11+\cdots + ( 4l-1 ) =l(2l+1).\end{cases} $$
Assume that \(\mho = \{ \zeta _{l}:l\in \mathbb{N} \} \) and \(\varpi ( \widetilde{\zeta },\widehat{\zeta } ) = \vert \widetilde{\zeta }-\widehat{\zeta } \vert \). Clearly, \(( \mho ,\varpi ) \) is a complete metric space. Define a mapping \(\beth :\mho ^{3}\rightarrow \mho \) by
$$ \beth ( \zeta _{l},\widetilde{\zeta }_{l},\widehat{\zeta }_{l} ) = \textstyle\begin{cases} \frac{\zeta _{l-1}+\widetilde{\zeta }_{l-1}+\widehat{\zeta }_{l-1}}{3}, & \text{when }l>1, \\ \frac{\zeta _{1}+\widetilde{\zeta }_{1}+\widehat{\zeta }_{1}}{3}, & \text{otherwise.}\end{cases}$$
For \(l>5\), we have
$$\begin{aligned} &\varpi \bigl( \beth ( \zeta _{l-4},\zeta _{l-3},\zeta _{l-2} ) ,\beth ( \zeta _{l-2},\zeta _{l-1},\zeta _{l} ) \bigr) \\ &\quad =\varpi \biggl( \frac{\zeta _{l-5}+\zeta _{l-4}+\zeta _{l-3}}{3}, \frac{\zeta _{l-3}+\zeta _{l-2}+\zeta _{l-1}}{3} \biggr) \\ &\quad =\frac{1}{3} \bigl\vert \bigl( (l-5) (2l-9)+(l-4) (2l-7)+(l-3) (2l-5) \bigr) \\ & \qquad {}- \bigl( (l-3) (2l-5)+(l-2) (2l-3)+(l-1) (2l-1) \bigr) \bigr\vert \\ &\quad =\frac{1}{3} \bigl\vert \bigl(6l^{2}-45l+88\bigr)- \bigl(6l^{2}-21l+22\bigr) \bigr\vert \\ &\quad =\frac{1}{3} \vert 24l-66 \vert =8l-22, \end{aligned}$$
and
$$\begin{aligned} &\max \bigl\{ \varpi \bigl( ( \zeta _{l-4},\zeta _{l-3}, \zeta _{l-2} ) , ( \zeta _{l-2},\zeta _{l-1},\zeta _{l} ) \bigr) \bigr\} \\ &\quad =\max \begin{Bmatrix} \bigl\vert (l-4) (2l-7)-(l-2) (2l-3) \bigr\vert , \\ \bigl\vert (l-3) (2l-5)-(l-1) (2l-1) \bigr\vert , \\ \bigl\vert (l-2) (2l-3)-l(2l+1) \bigr\vert \end{Bmatrix} \\ &\quad =\max \bigl\{ ( 8l-22 ) , ( 8l-14 ) ,(6l-6) \bigr\} = ( 8l-14 ) . \end{aligned}$$
Now,
$$ \lim_{l\rightarrow \infty } \frac{\varpi ( \beth ( \zeta _{l-4},\zeta _{l-3},\zeta _{l-2} ) ,\beth ( \zeta _{l-2},\zeta _{l-1},\zeta _{l} ) ) }{\max \{ \varpi ( ( \zeta _{l-4},\zeta _{l-3},\zeta _{l-2} ) , ( \zeta _{l-2},\zeta _{l-1},\zeta _{l} ) ) \} }= \lim_{l\rightarrow \infty } \frac{8l-22}{8l-14}=1. $$
Thus,
$$ \varpi \bigl( \beth ( \zeta _{l-4},\zeta _{l-3},\zeta _{l-2} ) ,\beth ( \zeta _{l-2},\zeta _{l-1},\zeta _{l} ) \bigr) \leq \gamma \max \bigl\{ \varpi \bigl( ( \zeta _{l-4}, \zeta _{l-3},\zeta _{l-2} ) , ( \zeta _{l-2},\zeta _{l-1}, \zeta _{l} ) \bigr) \bigr\} $$
does not hold for \(\gamma \in (0,1)\), which implies that assumption (1.1) of Theorem 1.1 is not fulfilled. Now, define the mapping \(\eta :(0,\infty )\rightarrow (1,\infty )\) by \(\eta (s)=e^{\frac{se^{s}}{1+s}}\). We can easily verify that \(\eta \in \nabla \) and ℶ is PTR η-C. Indeed, the inequality
$$ \begin{aligned} &e^{\sqrt{\varpi ( \beth ( \zeta _{i},\zeta _{i+1}, \zeta _{i+2} ) ,\beth ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) \frac{e^{\varpi ( \beth ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) ,\beth ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) }}{1+\varpi ( \beth ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) ,\beth ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) }}} \\ &\quad \leq e^{\gamma \sqrt{\varpi ( ( \zeta _{i},\zeta _{i+1}, \zeta _{i+2} ) , ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) \frac{e^{\varpi ( ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) , ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) }}{1+\varpi ( ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) , ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) }}}, \end{aligned} $$
(2.4)
holds for \(\beth ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) \neq \beth ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) \), \(i=1,2,\dots \), and for some \(\gamma \in (0,1)\). Inequality (1.1) is equivalent to
$$ \begin{aligned} &\varpi \bigl( \beth ( \zeta _{i},\zeta _{i+1}, \zeta _{i+2} ) ,\beth ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) \bigr) e^{ \frac{\varpi ( \beth ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) ,\beth ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) }{1+\varpi ( \beth ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) ,\beth ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) }} \\ &\quad \leq \gamma ^{2}\max \bigl\{ \varpi \bigl( ( \zeta _{i}, \zeta _{i+1},\zeta _{i+2} ) , ( \zeta _{i+2},\zeta _{i+3}, \zeta _{i+4} ) \bigr) \bigr\} e^{{ \frac{\max \{ \varpi ( ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) , ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) \} }{1+\max \{ \varpi ( ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) , ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) \} }}}. \end{aligned} $$
So, for some \(\gamma \in (0,1)\), we can write
$$ \frac{\varpi ( \beth ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) ,\beth ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) e^{\frac{\varpi ( \beth ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) ,\beth ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) }{1+\varpi ( \beth ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) ,\beth ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) }}}{\max \{ \varpi ( ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) , ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) \} e^{{\frac{\max \{ \varpi ( ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) , ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) \} }{1+\max \{ \varpi ( ( \zeta _{i},\zeta _{i+1},\zeta _{i+2} ) , ( \zeta _{i+2},\zeta _{i+3},\zeta _{i+4} ) ) \} }}}}\leq \gamma ^{2}. $$
Now, we will discuss the following cases:
(i) If \(i=l=1\), we get
$$\begin{aligned} & \frac{\varpi ( \beth ( \zeta _{1},\zeta _{2},\zeta _{3} ) ,\beth ( \zeta _{3},\zeta _{4},\zeta _{5} ) ) e^{\frac{\varpi ( \beth ( \zeta _{1},\zeta _{2},\zeta _{3} ) ,\beth ( \zeta _{3},\zeta _{4},\zeta _{5} ) ) }{1+\varpi ( \beth ( \zeta _{1},\zeta _{2},\zeta _{3} ) ,\beth ( \zeta _{3},\zeta _{4},\zeta _{5} ) ) }}}{\max \{ \varpi ( ( \zeta _{1},\zeta _{2},\zeta _{3} ) , ( \zeta _{3},\zeta _{4},\zeta _{5} ) ) \} e^{{\frac{\max \{ \varpi ( ( \zeta _{1},\zeta _{2},\zeta _{3} ) , ( \zeta _{3},\zeta _{4},\zeta _{5} ) ) \} }{1+\max \{ \varpi ( ( \zeta _{1},\zeta _{2},\zeta _{3} ) , ( \zeta _{3},\zeta _{4},\zeta _{5} ) ) \} }}}} \\ &\quad = \frac{\varpi ( \frac{\zeta _{1}+\zeta _{2}+\zeta _{3}}{3},\frac{\zeta _{3}+\zeta _{4}+\zeta _{5}}{3} ) e^{\frac{\varpi ( \frac{\zeta _{1}+\zeta _{2}+\zeta _{3}}{3},\frac{\zeta _{3}+\zeta _{4}+\zeta _{5}}{3} ) }{1+\varpi ( \frac{\zeta _{1}+\zeta _{2}+\zeta _{3}}{3},\frac{\zeta _{3}+\zeta _{4}+\zeta _{5}}{3} ) }}}{\max \{ \varpi ( ( \zeta _{1},\zeta _{2},\zeta _{3} ) , ( \zeta _{3},\zeta _{4},\zeta _{5} ) ) \} e^{{\frac{\max \{ \varpi ( ( \zeta _{1},\zeta _{2},\zeta _{3} ) , ( \zeta _{3},\zeta _{4},\zeta _{5} ) ) \} }{1+\max \{ \varpi ( ( \zeta _{1},\zeta _{2},\zeta _{3} ) , ( \zeta _{3},\zeta _{4},\zeta _{5} ) ) \} }}}} \\ &\quad = \frac{\varpi ( \frac{34}{3},\frac{112}{3} ) e^{\frac{\varpi ( \frac{34}{3},\frac{112}{3} ) }{1+\varpi ( \frac{34}{3},\frac{112}{3} ) }}}{\max \{ \varpi ( ( 3,10,21 ) , ( 21,36,55 ) ) \} e^{{\frac{\max \{ \varpi ( ( 3,10,21 ) , ( 21,36,55 ) ) \} }{1+\max \{ \varpi ( ( 3,10,21 ) , ( 21,36,55 ) ) \} }}}} \\ &\quad \leq \frac{26e^{26}}{34e^{34}}=\frac{13}{17}e^{-8}< e^{-2}. \end{aligned}$$
(ii) If \(i=l>1\), we obtain
$$\begin{aligned} & \frac{\varpi ( \beth ( \zeta _{l},\zeta _{l+1},\zeta _{l+2} ) ,\beth ( \zeta _{l+2},\zeta _{l+3},\zeta _{l+4} ) ) e^{\frac{\varpi ( \beth ( \zeta _{l},\zeta _{l+1},\zeta _{l+2} ) ,\beth ( \zeta _{l+2},\zeta _{l+3},\zeta _{l+4} ) ) }{1+\varpi ( \beth ( \zeta _{l},\zeta _{l+1},\zeta _{l+2} ) ,\beth ( \zeta _{l+2},\zeta _{l+3},\zeta _{l+4} ) ) }}}{\max \{ \varpi ( ( \zeta _{l},\zeta _{l+1},\zeta _{l+2} ) , ( \zeta _{l+2},\zeta _{l+3},\zeta _{l+4} ) ) \} e^{{\frac{\max \{ ( \zeta _{l},\zeta _{l+1},\zeta _{l+2} ) , ( \zeta _{l+2},\zeta _{l+3},\zeta _{l+4} ) \} }{1+\max \{ ( \zeta _{l},\zeta _{l+1},\zeta _{l+2} ) , ( \zeta _{l+2},\zeta _{l+3},\zeta _{l+4} ) \} }}}} \\ &\quad = \frac{\varpi ( \frac{\zeta _{l-1}+\zeta _{l}+\zeta _{l+1}}{3},\frac{\zeta _{l+1}+\zeta _{l+2}+\zeta _{l+3}}{3} ) e^{\frac{\varpi ( \frac{\zeta _{l-1}+\zeta _{l}+\zeta _{l+1}}{3},\frac{\zeta _{l+1}+\zeta _{l+2}+\zeta _{l+3}}{3} ) }{1+\varpi ( \frac{\zeta _{l-1}+\zeta _{l}+\zeta _{l+1}}{3},\frac{\zeta _{l+1}+\zeta _{l+2}+\zeta _{l+3}}{3} ) }}}{\max \{ \varpi ( ( \zeta _{l},\zeta _{l+1},\zeta _{l+2} ) , ( \zeta _{l+2},\zeta _{l+3},\zeta _{l+4} ) ) \} e^{{\frac{\max \{ \varpi ( ( \zeta _{l},\zeta _{l+1},\zeta _{l+2} ) , ( \zeta _{l+2},\zeta _{l+3},\zeta _{l+4} ) ) \} }{1+\max \{ \varpi ( ( \zeta _{l},\zeta _{l+1},\zeta _{l+2} ) , ( \zeta _{l+2},\zeta _{l+3},\zeta _{l+4} ) ) \} }}}} \\ &\quad = \frac{ \vert \frac{6l^{2}+3l+4}{3}-\frac{6l^{2}+27l+34}{3} \vert e^{\frac{ \vert \frac{6l^{2}+3l+4}{3}-\frac{6l^{2}+27l+34}{3} \vert }{1+ \vert \frac{6l^{2}+3l+4}{3}-\frac{6l^{2}+27l+34}{3} \vert }}}{\max \{ \vert 8l+10 \vert , \vert 8l+18 \vert , \vert 8l+26 \vert \} e^{{\frac{\max \{ \vert 8l+10 \vert , \vert 8l+18 \vert , \vert 8l+26 \vert \} }{1+\max \{ \vert 8l+10 \vert , \vert 8l+ 18 \vert , \vert 8l+26 \vert \} }}}} \\ &\quad = \frac{ ( 8l+10 ) e^{\frac{ ( 8l+10 ) }{1+ ( 8l+10 ) }}}{ ( 8l+26 ) e^{\frac{ ( 8l+26 ) }{1+ ( 8l+26 ) }}}\leq \frac{ ( 8l+10 ) e^{ ( 8l+10 ) }}{ ( 8l+26 ) e^{ ( 8l+26 ) }}e^{-16}< e^{-2}, \end{aligned}$$
with \(\gamma =\frac{1}{e}\). Hence all requirements of Theorem 2.3 are fulfilled and the point \((1,1,1)\) is the unique FP of ℶ.
Example 2.5
Assume that \(\mho =[0,1]\), \(\varpi ( \widetilde{\zeta },\widehat{\zeta } ) = \vert \widetilde{\zeta }-\widehat{\zeta } \vert \), and \(\beth :\mho ^{z}\rightarrow \mho \) is described by
$$ \beth ( \zeta _{1},\dots ,\zeta _{l} ) = \frac{\zeta _{1}+\zeta _{l}}{8l},\quad \forall \zeta _{1},\dots ,\zeta _{l}\in \mho . $$
Let \(\eta :(0,\infty )\rightarrow (1,\infty )\) be a mapping defined by \(\eta (s)=e^{\sqrt{\frac{s}{1+s}}}\). Since \(e^{\sqrt{\frac{s}{1+s}}}\leq e^{\sqrt{s}}\), we can see from [15] that \(\eta \in \nabla \). Now, for \(\zeta _{1},\zeta _{2},\dots ,\zeta _{l+1}\in \mho \), one can write
$$ \varpi \bigl( \beth ( \zeta _{1},\dots ,\zeta _{l} ) , \beth ( \zeta _{2},\dots ,\zeta _{l+1} ) \bigr) >0, $$
and
$$\begin{aligned} &\eta \bigl( \varpi \bigl( \beth ( \zeta _{1},\dots ,\zeta _{l} ) ,\beth ( \zeta _{2},\dots ,\zeta _{l+1} ) \bigr) \bigr) \\ &\quad =e^{\sqrt{ \frac{\varpi ( \beth ( \zeta _{1},\dots ,\zeta _{l} ) ,\beth ( \zeta _{2},\dots ,\zeta _{l+1} ) ) }{1+\varpi ( \beth ( \zeta _{1},\dots ,\zeta _{l} ) ,\beth ( \zeta _{2},\dots ,\zeta _{l+1} ) ) }}} \\ &\quad =e^{\sqrt{ \frac{ ( \frac{1}{8l} ) \vert ( \zeta _{1}-\zeta _{2} ) + ( \zeta _{l}-\zeta _{l+1} ) \vert }{1+ \vert ( \zeta _{1}-\zeta _{2} ) + ( \zeta _{l}-\zeta _{l+1} ) \vert }}} \\ &\quad =e^{ ( \frac{1}{2\sqrt{2l}} ) \sqrt{ \frac{ \vert ( \zeta _{1}-\zeta _{2} ) + ( \zeta _{l}-\zeta _{l+1} ) \vert }{1+ \vert ( \zeta _{1}-\zeta _{2} ) + ( \zeta _{l}-\zeta _{l+1} ) \vert }}} \\ &\quad \leq e^{ ( \frac{1}{\sqrt{2}} ) \sqrt{ \frac{\max \{ \varpi ( \zeta _{1},\zeta _{2} ) ,\varpi (\zeta _{l},\zeta _{l+1}) \} }{1+\max \{ \varpi ( \zeta _{1},\zeta _{2} ) ,\varpi (\zeta _{l},\zeta _{l+1}) \} }}} \\ &\quad \leq e^{ ( \frac{1}{\sqrt{2}} ) \sqrt{\max \{ \frac{\varpi ( \zeta _{j},\zeta _{j+1} ) }{1+\varpi ( \zeta _{j},\zeta _{j+1} ) }:1 \leq j\leq z \} }} \\ &\quad = \biggl[ \eta \biggl( \max \biggl\{ \frac{\varpi ( \zeta _{j},\zeta _{j+1} ) }{1+\varpi ( \zeta _{j},\zeta _{j+1} ) }:1 \leq j\leq z \biggr\} \biggr) \biggr] ^{\gamma }, \end{aligned}$$
with \(\gamma =\frac{1}{\sqrt{2}}\). In addition, for all \(\zeta ^{\ast },\zeta ^{\prime }\in \mho \) with \(\zeta ^{\ast }\neq \zeta ^{\prime }\), we obtain
$$ \varpi \bigl( \beth \bigl( \zeta ^{\ast },\zeta ^{\ast },\dots , \zeta ^{\ast } \bigr) ,\beth \bigl( \zeta ^{\prime },\zeta ^{ \prime },\dots ,\zeta ^{\prime } \bigr) \bigr) = \frac{ \vert \zeta ^{\ast }-\zeta ^{\prime } \vert }{8l}>0, $$
and
$$\begin{aligned} \eta \bigl( \varpi \bigl( \beth \bigl( \zeta ^{\ast },\zeta ^{ \ast },\dots ,\zeta ^{\ast } \bigr) ,\beth \bigl( \zeta ^{\prime }, \zeta ^{\prime },\dots ,\zeta ^{\prime } \bigr) \bigr) \bigr) &= \eta \biggl( \frac{ \vert \zeta ^{\ast }-\zeta ^{\prime } \vert }{8l} \biggr) \\ &=e^{ \sqrt{ ( \frac{\frac{ \vert \zeta ^{\ast }-\zeta ^{\prime } \vert }{8l}}{1+\frac{ \vert \zeta ^{\ast }-\zeta ^{\prime } \vert }{8l}} ) }} \\ &\leq e^{ ( \frac{1}{2\sqrt{2l}} ) \sqrt{ ( \frac{ \vert \zeta ^{\ast }-\zeta ^{\prime } \vert }{1+ \vert \zeta ^{\ast }-\zeta ^{\prime } \vert } ) }} \\ &\leq e^{\frac{1}{\sqrt{2}}\sqrt{ ( \frac{ \vert \zeta ^{\ast }-\zeta ^{\prime } \vert }{1+ \vert \zeta ^{\ast }-\zeta ^{\prime } \vert } ) }} \\ &= \bigl[ \eta \bigl( \varpi \bigl( \zeta ^{\ast },\zeta ^{\prime } \bigr) \bigr) \bigr] ^{\gamma }, \end{aligned}$$
with \(\gamma =\frac{1}{\sqrt{2}}\). Hence, all assumptions of Theorem 2.3 are fulfilled. In addition, for some chosen \(\zeta _{1},\dots ,\zeta _{l}\in \mho \), the sequence \(\{\zeta _{l}\}\) defined in (2.3) converges to \(\zeta ^{\ast }=0\), which is the unique FP of ℶ.
If we put \(\eta (s)=e^{\sqrt{s}}\) in Theorem 2.3, we get the result below.
Corollary 2.6
Consider \(\beth :\mho ^{z}\rightarrow \mho \) is a given mapping and suppose there is \(\gamma \in (0,1)\) such that
$$ \varpi \bigl( \beth ( \zeta _{1},\dots ,\zeta _{z} ) , \beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \bigr) \leq \gamma ^{2} \biggl( \max \biggl\{ \frac{\varpi ( \zeta _{j},\zeta _{j+1} ) }{1+\varpi ( \zeta _{j},\zeta _{j+1} ) }:1\leq j\leq z \biggr\} \biggr) . $$
(2.5)
Then for any chosen points \(\zeta _{1},\dots ,\zeta _{z}\in \mho \), the sequence \(\{\zeta _{l}\}\) described in (1.2) converges to \(\zeta ^{\ast }\in \mho \) and \(\zeta ^{\ast }=\beth (\zeta ^{\ast },\dots ,\zeta ^{\ast })\). Moreover, if
$$ \varpi \bigl( \beth \bigl( \zeta ^{\ast },\dots ,\zeta ^{\ast } \bigr) ,\beth \bigl( \zeta ^{{\prime }},\dots ,\zeta ^{\prime } \bigr) \bigr) \leq \gamma ^{2}\varpi \bigl( \zeta ^{\ast }, \zeta ^{\prime } \bigr) $$
holds for all \(\zeta ^{\ast },\zeta ^{{\prime }}\in \mho \) with \(\zeta ^{\ast }\neq \zeta ^{{\prime }}\), Then the point \(\zeta ^{\ast }\) is a unique FP of the mapping ℶ.
Corollary 2.7
Assume that \(\beth :\mho ^{z}\rightarrow \mho \) is a given mapping and there are nonnegative constants \(\gamma _{1},\gamma _{2},\dots ,\gamma _{z}\) with \(\gamma _{1}+\gamma _{2}+\cdots +\gamma _{z}<1\) such that
$$ \begin{aligned} \varpi \bigl( \beth ( \zeta _{1},\dots , \zeta _{z} ) ,\beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \bigr)\leq {}&\gamma _{1} \frac{\varpi ( \zeta _{1},\zeta _{2} ) }{1+\varpi ( \zeta _{1},\zeta _{2} ) }+ \gamma _{2} \frac{\varpi ( \zeta _{2},\zeta _{3} ) }{1+\varpi ( \zeta _{2},\zeta _{3} ) } \\ &{} +\cdots +\gamma _{z} \frac{\varpi ( \zeta _{z},\zeta _{z+1} ) }{1+\varpi ( \zeta _{z},\zeta _{z+1} ) }, \end{aligned} $$
(2.6)
for each \(( \zeta _{1},\dots ,\zeta _{z+1} ) \in \mho ^{z+1}\) with \(\beth ( \zeta _{1},\dots ,\zeta _{z} ) \neq \beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \). Then for any chosen points \(\zeta _{1},\dots ,\zeta _{z}\in \mho \), the sequence \(\{\zeta _{l}\}\), given by (1.2) converges to \(\zeta ^{\ast }\in \mho \), where \(\zeta ^{\ast }\) is a unique FP of ℶ.
Proof
It is clear that (2.6) implies (2.5) with \(\gamma ^{2}=\gamma _{1}+\gamma _{2}+\cdots +\gamma _{z}\).
Now, suppose that \(\zeta ^{\ast },\zeta ^{{\prime }}\in \mho \) with \(\zeta ^{\ast }\neq \zeta \), Based on (2.6), one can obtain
$$\begin{aligned} &\varpi \bigl( \beth \bigl( \zeta ^{\ast },\zeta ^{\ast },\dots , \zeta ^{\ast } \bigr) ,\beth \bigl( \zeta ^{{\prime }},\zeta ^{{ \prime }},\dots ,\zeta ^{\prime } \bigr) \bigr) \\ &\quad =\varpi \bigl( \beth \bigl( \zeta ^{\ast },\dots ,\zeta ^{\ast } \bigr) ,\beth \bigl( \zeta ^{\ast },\dots ,\zeta ^{\ast },\zeta ^{ \prime } \bigr) \bigr) \\ &\qquad {} +\varpi \bigl( \beth \bigl( \zeta ^{\ast },\dots ,\zeta ^{ \ast }, \zeta ^{\prime } \bigr) ,\beth \bigl( \zeta ^{\ast },\dots , \zeta ^{\ast },\zeta ^{\prime },\zeta ^{\prime } \bigr) \bigr) \\ &\qquad {} +\cdots +\varpi \bigl( \beth \bigl( \zeta ^{\ast },\dots , \zeta ^{\prime },\zeta ^{\prime } \bigr) ,\beth \bigl( \zeta ^{ \prime }, \dots ,\zeta ^{\prime },\zeta ^{\prime } \bigr) \bigr) \\ &\quad \leq ( \gamma _{z}+\gamma _{z-1}+\cdots +\gamma _{z} ) \frac{\varpi ( \zeta ^{\ast },\zeta ^{\prime } ) }{1+\varpi ( \zeta ^{\ast },\zeta ^{\prime } ) } \\ &\quad \leq \gamma ^{2}\varpi \bigl( \zeta ^{\ast },\zeta ^{\prime } \bigr) . \end{aligned}$$
Thus, the conditions of Corollary 2.6 hold. □
If we take a large class of functions ∇, for example,
$$ \eta (s)=2-\frac{2}{\pi }\arctan \biggl( \frac{1}{s^{\theta }} \biggr) , $$
where \(\theta \in (0,1)\) and \(s>0\), we obtain the following theorem from Theorem 2.3.
Theorem 2.8
Suppose that \(\beth :\mho ^{z}\rightarrow \mho \) is a given mapping. If there are a mapping \(\eta \in \nabla \) and constants \(\gamma ,\theta \in (0,1)\) such that
$$\begin{aligned} &2-\frac{2}{\pi }\arctan \biggl( \frac{1}{ [ \varpi ( \beth ( \zeta _{1},\dots ,\zeta _{z} ) ,\beth ( \zeta _{2},\dots ,\zeta _{z+1} ) ) ] ^{\theta }} \biggr) \\ &\quad \leq \biggl[ 2-\frac{2}{\pi }\arctan \biggl( \frac{1}{ [ \max \{ \frac{\varpi ( \zeta _{j},\zeta _{j+1} ) }{1+\varpi ( \zeta _{j},\zeta _{j+1} ) }:1\leq j\leq z \} ] ^{\theta }} \biggr) \biggr] ^{\gamma }, \end{aligned}$$
for each \(( \zeta _{1},\dots ,\zeta _{z+1} ) \in \mho ^{z+1}\) with \(\beth ( \zeta _{1},\dots ,\zeta _{z} ) \neq \beth ( \zeta _{2},\dots ,\zeta _{z+1} ) \), then for any chosen points \(\zeta _{1},\dots ,\zeta _{z}\in \mho \), the sequence \(\{\zeta _{l}\}\), given by (1.2) converges to \(\zeta ^{\ast }\in \mho \). Then \(\zeta ^{\ast }\) is a unique FP of ℶ. Moreover, if
$$\begin{aligned} &2-\frac{2}{\pi }\arctan \biggl( \frac{1}{ [ \varpi ( \beth ( \zeta ^{\ast },\dots ,\zeta ^{\ast } ) ,\beth ( \zeta ^{{\prime }},\dots ,\zeta ^{\prime } ) ) ] ^{\theta }} \biggr) \\ &\quad \leq \biggl[ 2-\frac{2}{\pi }\arctan \biggl( \frac{1}{ ( \varpi ( \zeta ^{\ast },\zeta ^{\prime } ) ) ^{\theta }} \biggr) \biggr] ^{\gamma }, \end{aligned}$$
holds for \(\zeta ^{\ast },\zeta ^{{\prime }}\in \mho \) with \(\zeta ^{\ast }\neq \zeta ^{{\prime }}\), then the point \(\zeta ^{\ast }\) is a unique FP of the mapping ℶ.
Remark 2.9
It should be noted that:
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Our Theorem 2.3 unifies and extends Theorem 1.3 in [10] and Theorem 1.2 in [9].
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Corollary 1 in [15] can be obtained directly from Theorem 2.3 putting \(\gamma =1\) and neglecting the denominator of the contractive condition (2.1).
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If we take \(\gamma =1\) and neglect the denominators of the contractivity conditions of Corollaries 2.6 and 2.7, we obtain the BCP [1].