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
$$ varpi bigl( beth ( zeta _{1},dots ,zeta _{z} ) , beth ( zeta _{2},dots ,zeta _{z+1} ) bigr) leq gamma ^{2} biggl( max biggl{ frac{varpi ( zeta _{j},zeta _{j+1} ) }{1+varpi ( zeta _{j},zeta _{j+1} ) }:1leq jleq z biggr} biggr) , $$
(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,
$$begin{aligned} varpi bigl( beth ( zeta _{1},dots ,zeta _{z} ) , beth ( zeta _{2},dots ,zeta _{z+1} ) bigr) &leq gamma max biggl{ frac{varpi ( zeta _{j},zeta _{j+1} ) }{1+varpi ( zeta _{j},zeta _{j+1} ) }:1 leq jleq z biggr} \ &< max bigl{ varpi ( zeta _{j},zeta _{j+1} ) :1 leq j leq z bigr} . end{aligned}$$
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 (lin 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 (lin 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 (lin mathbb{N} ) and (phi >0), we have (gimel _{l+z}>0). Thus, for (lleq z), we obtain
$$begin{aligned} 1 &< eta ( gimel _{z+1} ) \ & =eta bigl( varpi ( zeta _{z+1}, zeta _{z+2} ) bigr) \ &=eta bigl( varpi bigl( beth ( zeta _{1},zeta _{2}, dots ,zeta _{z} ) ,beth ( zeta _{2},zeta _{3},dots , zeta _{z+1} ) bigr) bigr) \ &leq biggl[ eta biggl( max biggl{ frac{varpi ( zeta _{j},zeta _{j+1} ) }{1+varpi ( zeta _{j},zeta _{j+1} ) }:1 leq jleq z biggr} biggr) biggr] ^{gamma } \ &= bigl[ eta ( phi ) bigr] ^{gamma }. end{aligned}$$
Also,
$$begin{aligned} 1 &< eta ( gimel _{z+2} ) \ &=eta bigl( varpi ( zeta _{z+2}, zeta _{z+3} ) bigr) \ &=eta bigl( varpi bigl( beth ( zeta _{2},zeta _{3}, dots ,zeta _{z+1} ) ,beth ( zeta _{3},zeta _{4}, dots ,zeta _{z+2} ) bigr) bigr) \ &leq biggl[ eta biggl( max biggl{ frac{varpi ( zeta _{j},zeta _{j+1} ) }{1+varpi ( zeta _{j},zeta _{j+1} ) }:2 leq jleq z+1 biggr} biggr) biggr] ^{gamma } \ &= bigl[ eta ( phi ) bigr] ^{gamma ^{2}}. end{aligned}$$
Continuing in the same pattern, for (lgeq 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 (lrightarrow infty ) in (2.3) and using ((eta _{2})), we have
$$ lim_{lrightarrow infty }eta ( gimel _{z+l} ) =1 quad Longleftrightarrowquad lim_{lrightarrow infty }gimel _{z+l}=0. $$
Based on ((eta _{3})), there are (ell in (0,1)) and (uin (0,infty )) so that
$$ lim_{lrightarrow infty } biggl( frac{eta ( gimel _{z+l} ) -1}{gimel _{z+l}^{ell }} biggr) =u. $$
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
$$ bigglvert frac{eta ( gimel _{z+l} ) -1}{gimel _{z+l}^{ell }}-u biggrvert leq v,quad forall l>l_{1}. $$
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
$$ lgimel _{z+l}^{ell }leq lq bigl( eta ( gimel _{z+l} ) -1 bigr) , quad forall l>l_{1}. $$
Suppose that (u=infty ) and (v>0). By the definition of the limit, there is (l_{1}in mathbb{N} ) such that
$$ vleq frac{eta ( gimel _{z+l} ) -1}{gimel _{z+l}^{ell }},quad forall l>l_{1}. $$
This implies after taking (frac{1}{v}=q) that
$$ lgimel _{z+l}^{ell }leq lq bigl( eta ( gimel _{z+l} ) -1 bigr) , quad forall l>l_{1}. $$
Thus, in both cases, there are (l_{1}in mathbb{N} ) and (q>0) so that
$$ lgimel _{z+l}^{ell }leq lq bigl( eta ( gimel _{z+l} ) -1 bigr) , quad forall l>l_{1}. $$
Applying (2.3), we get
$$ lgimel _{z+l}^{ell }leq lq bigl( bigl[ eta ( phi ) bigr] ^{gamma ^{l}}-1 bigr) , quad forall l>l_{1}, $$
and, when (lrightarrow infty ), have
$$ lim_{lrightarrow infty }lgimel _{z+l}^{ell }=0. $$
Thus, there is (l_{2}in mathbb{N} ) and (q>0) such that
$$ lgimel _{z+l}^{ell }leq 1,quad forall l>l_{2}. $$
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
$$ lim_{l,brightarrow infty }varpi ( zeta _{l},zeta _{b} ) =lim_{lrightarrow infty }varpi bigl( zeta _{l}, zeta ^{ast } bigr) =0. $$
Because ℶ is continuous, we have
$$begin{aligned} hbar &=lim_{lrightarrow infty }zeta _{l+z} \ &=lim _{lrightarrow infty }beth ( zeta _{l},zeta _{l+1},dots ,zeta _{z+l-1} ) \ &=beth Bigl( lim_{lrightarrow infty }zeta _{l},lim _{l rightarrow infty }zeta _{l+1},dots ,lim_{lrightarrow infty } zeta _{z+l-1} Bigr) \ & =beth bigl( zeta ^{ast },zeta ^{ast }, dots ,zeta ^{ast } bigr) . end{aligned}$$
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:
$$ textstylebegin{cases} zeta _{1}=3, \ zeta _{2}=3+7, \ vdots \ zeta _{l}=3+7+11+cdots + ( 4l-1 ) =l(2l+1).end{cases} $$
Assume that (mho = { zeta _{l}:lin 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
$$ beth ( zeta _{l},widetilde{zeta }_{l},widehat{zeta }_{l} ) = textstylebegin{cases} frac{zeta _{l-1}+widetilde{zeta }_{l-1}+widehat{zeta }_{l-1}}{3}, & text{when }l>1, \ frac{zeta _{1}+widetilde{zeta }_{1}+widehat{zeta }_{1}}{3}, & text{otherwise.}end{cases}$$
For (l>5), we have
$$begin{aligned} &varpi bigl( beth ( zeta _{l-4},zeta _{l-3},zeta _{l-2} ) ,beth ( zeta _{l-2},zeta _{l-1},zeta _{l} ) bigr) \ &quad =varpi biggl( frac{zeta _{l-5}+zeta _{l-4}+zeta _{l-3}}{3}, frac{zeta _{l-3}+zeta _{l-2}+zeta _{l-1}}{3} biggr) \ &quad =frac{1}{3} biglvert bigl( (l-5) (2l-9)+(l-4) (2l-7)+(l-3) (2l-5) bigr) \ & qquad {}- bigl( (l-3) (2l-5)+(l-2) (2l-3)+(l-1) (2l-1) bigr) bigrvert \ &quad =frac{1}{3} biglvert bigl(6l^{2}-45l+88bigr)- bigl(6l^{2}-21l+22bigr) bigrvert \ &quad =frac{1}{3} vert 24l-66 vert =8l-22, end{aligned}$$
and
$$begin{aligned} &max bigl{ varpi bigl( ( zeta _{l-4},zeta _{l-3}, zeta _{l-2} ) , ( zeta _{l-2},zeta _{l-1},zeta _{l} ) bigr) bigr} \ &quad =max begin{Bmatrix} biglvert (l-4) (2l-7)-(l-2) (2l-3) bigrvert , \ biglvert (l-3) (2l-5)-(l-1) (2l-1) bigrvert , \ biglvert (l-2) (2l-3)-l(2l+1) bigrvert end{Bmatrix} \ &quad =max bigl{ ( 8l-22 ) , ( 8l-14 ) ,(6l-6) bigr} = ( 8l-14 ) . end{aligned}$$
Now,
$$ lim_{lrightarrow infty } frac{varpi ( beth ( zeta _{l-4},zeta _{l-3},zeta _{l-2} ) ,beth ( zeta _{l-2},zeta _{l-1},zeta _{l} ) ) }{max { varpi ( ( zeta _{l-4},zeta _{l-3},zeta _{l-2} ) , ( zeta _{l-2},zeta _{l-1},zeta _{l} ) ) } }= lim_{lrightarrow infty } frac{8l-22}{8l-14}=1. $$
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
$$begin{aligned} &eta bigl( varpi bigl( beth ( zeta _{1},dots ,zeta _{l} ) ,beth ( zeta _{2},dots ,zeta _{l+1} ) bigr) bigr) \ &quad =e^{sqrt{ frac{varpi ( beth ( zeta _{1},dots ,zeta _{l} ) ,beth ( zeta _{2},dots ,zeta _{l+1} ) ) }{1+varpi ( beth ( zeta _{1},dots ,zeta _{l} ) ,beth ( zeta _{2},dots ,zeta _{l+1} ) ) }}} \ &quad =e^{sqrt{ frac{ ( frac{1}{8l} ) vert ( zeta _{1}-zeta _{2} ) + ( zeta _{l}-zeta _{l+1} ) vert }{1+ vert ( zeta _{1}-zeta _{2} ) + ( zeta _{l}-zeta _{l+1} ) vert }}} \ &quad =e^{ ( frac{1}{2sqrt{2l}} ) sqrt{ frac{ vert ( zeta _{1}-zeta _{2} ) + ( zeta _{l}-zeta _{l+1} ) vert }{1+ vert ( zeta _{1}-zeta _{2} ) + ( zeta _{l}-zeta _{l+1} ) vert }}} \ &quad leq e^{ ( frac{1}{sqrt{2}} ) sqrt{ frac{max { varpi ( zeta _{1},zeta _{2} ) ,varpi (zeta _{l},zeta _{l+1}) } }{1+max { varpi ( zeta _{1},zeta _{2} ) ,varpi (zeta _{l},zeta _{l+1}) } }}} \ &quad leq e^{ ( frac{1}{sqrt{2}} ) sqrt{max { frac{varpi ( zeta _{j},zeta _{j+1} ) }{1+varpi ( zeta _{j},zeta _{j+1} ) }:1 leq jleq z } }} \ &quad = biggl[ eta biggl( max biggl{ frac{varpi ( zeta _{j},zeta _{j+1} ) }{1+varpi ( zeta _{j},zeta _{j+1} ) }:1 leq jleq z biggr} biggr) biggr] ^{gamma }, end{aligned}$$
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
$$begin{aligned} eta bigl( varpi bigl( beth bigl( zeta ^{ast },zeta ^{ ast },dots ,zeta ^{ast } bigr) ,beth bigl( zeta ^{prime }, zeta ^{prime },dots ,zeta ^{prime } bigr) bigr) bigr) &= eta biggl( frac{ vert zeta ^{ast }-zeta ^{prime } vert }{8l} biggr) \ &=e^{ sqrt{ ( frac{frac{ vert zeta ^{ast }-zeta ^{prime } vert }{8l}}{1+frac{ vert zeta ^{ast }-zeta ^{prime } vert }{8l}} ) }} \ &leq e^{ ( frac{1}{2sqrt{2l}} ) sqrt{ ( frac{ vert zeta ^{ast }-zeta ^{prime } vert }{1+ vert zeta ^{ast }-zeta ^{prime } vert } ) }} \ &leq e^{frac{1}{sqrt{2}}sqrt{ ( frac{ vert zeta ^{ast }-zeta ^{prime } vert }{1+ vert zeta ^{ast }-zeta ^{prime } vert } ) }} \ &= bigl[ eta bigl( varpi bigl( zeta ^{ast },zeta ^{prime } bigr) bigr) bigr] ^{gamma }, end{aligned}$$
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
$$ varpi bigl( beth ( zeta _{1},dots ,zeta _{z} ) , beth ( zeta _{2},dots ,zeta _{z+1} ) bigr) leq gamma ^{2} biggl( max biggl{ frac{varpi ( zeta _{j},zeta _{j+1} ) }{1+varpi ( zeta _{j},zeta _{j+1} ) }:1leq jleq z biggr} biggr) . $$
(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
$$begin{aligned} &2-frac{2}{pi }arctan biggl( frac{1}{ [ varpi ( beth ( zeta _{1},dots ,zeta _{z} ) ,beth ( zeta _{2},dots ,zeta _{z+1} ) ) ] ^{theta }} biggr) \ &quad leq biggl[ 2-frac{2}{pi }arctan biggl( frac{1}{ [ max { frac{varpi ( zeta _{j},zeta _{j+1} ) }{1+varpi ( zeta _{j},zeta _{j+1} ) }:1leq jleq z } ] ^{theta }} biggr) biggr] ^{gamma }, end{aligned}$$
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:
Our Theorem 2.3 unifies and extends Theorem 1.3 in [10] and Theorem 1.2 in [9].
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).
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].
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