In this part, some consequences are discussed in F-MS.

Theorem 3.1

Let ((Upsilon , mathbb{Q} )) be an FMS, set (Im :Upsilon rightarrow Upsilon ) and (Gamma :Upsilon rightarrow CB ( Upsilon ) ). Presume that there exist functions (beta in mho ), (Lambda in Phi ), and (psi in Psi ) such that (forall gamma ,delta in Upsilon ),

$$ Lambda bigl( H_{mathbb{Q} }(Gamma gamma ,Gamma delta ) bigr) leq psi bigl( Lambda bigl[ beta bigl( aleph _{Im }( gamma ,delta ) bigr) aleph _{Im }(gamma ,delta ) bigr] bigr) , $$

(3.1)

where

$$ aleph _{Im }(gamma ,delta )=max biggl{ mathbb{Q} (Im gamma ,Im delta ),mathbb{Q} (Im gamma ,Gamma gamma ),mathbb{Q} (Im delta ,Gamma delta ),frac{mathbb{Q} (Im gamma ,Gamma delta )+mathbb{Q} (Im delta ,Gamma gamma )}{2} biggr} . $$

If for any (gamma in Upsilon ), (Gamma Upsilon subseteq Im Upsilon ) and ϒ is an Fcomplete subspace of ϒ.

Then, Γ and have a unique point of coincidence. Indeed, if Γ and are weakly compatible, then Γ and have a unique common fixed point (gamma ^{ast }in Upsilon ).

Proof

Let (gamma _{0}in Upsilon ), since (Gamma Upsilon subseteq Im Upsilon ), we can construct a sequence ({ delta _{zeta } } _{zeta in mathbb{N}}) by

$$ delta _{zeta }in Gamma gamma _{zeta -1}=Im gamma _{zeta },quad forall zeta in mathbb{N}. $$

(3.2)

Now, if there exists some (zeta _{0}in mathbb{N} ) such that (mathbb{Q} ( delta _{zeta _{0}},delta _{zeta _{0}+1} ) =0), then (delta _{zeta _{0}}=delta _{zeta _{0}+1}), which implies that (Im gamma _{zeta _{0}}=Gamma gamma _{zeta _{0}}), thus (gamma _{zeta _{0}}) is a coincidence point of Γ and , so (w_{0}in Im gamma _{zeta _{0}}=Gamma gamma _{zeta _{0}}) is the point of coincidence of Γ and . We postulate that (mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) >0) (forall zeta in mathbb{N}). From (3.1) and (3.2), we have

$$begin{aligned} Lambda bigl( mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) bigr) leq &Lambda bigl( H_{mathbb{Q} } ( Gamma gamma _{zeta -1},Gamma gamma _{zeta } ) bigr) \ leq &psi bigl( Lambda bigl( beta bigl( aleph _{Im }(gamma _{zeta -1},gamma _{zeta }) bigr) aleph _{Im }( gamma _{zeta -1},gamma _{zeta }) bigr) bigr) , end{aligned}$$

(3.3)

where

$$begin{aligned} aleph _{Im }(gamma _{zeta -1},gamma _{zeta }) =& max left { textstylebegin{array}{c} mathbb{Q} ( Im gamma _{zeta -1},Im gamma _{zeta } ) ,mathbb{Q} ( Im gamma _{zeta -1},Gamma gamma _{zeta -1} ) , \ mathbb{Q} ( Im gamma _{zeta },Gamma gamma _{zeta } ) ,frac{mathbb{Q} ( Im gamma _{zeta -1},Gamma gamma _{zeta } ) +mathbb{Q} ( Im gamma _{zeta },Gamma gamma _{zeta -1} ) }{2}end{array}displaystyle right } \ =&max left { textstylebegin{array}{c} mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) ,mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) ,mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) , \ frac{mathbb{Q} ( delta _{zeta -1},delta _{zeta +1} ) +mathbb{Q} ( y_{zeta },y_{zeta } ) }{2}end{array}displaystyle right } \ =&max bigl{ mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) ,mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) bigr} . end{aligned}$$

We conclude that

$$ aleph _{Im } ( gamma _{zeta -1},gamma _{zeta } ) = max bigl{ mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) ,mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) bigr} . $$

Now, if (max { mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) ,mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) } =mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) ) for (zeta geq 1), then from (3.2), we obtain

$$ Lambda bigl( mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) bigr) leq psi bigl[ Lambda bigl( beta bigl( mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) bigr) mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) bigr) bigr] . $$

Since (beta in mho ) and from (( Phi 1 ) ), we have

$$ mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) < mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) , $$

which is a discrepancy as (mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) geq 0). Therefore,

$$ max bigl{ mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) ,mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) bigr} =mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) , $$

(3.4)

by (3.3) and (3.4), we have

$$begin{aligned}& Lambda bigl( mathbb{Q} ( delta _{zeta },delta _{zeta +1} ) bigr) \& quad =Lambda bigl( mathbb{Q} ( Gamma gamma _{zeta -1},Gamma gamma _{zeta } ) bigr) leq Lambda bigl( H_{mathbb{Q} } ( Gamma gamma _{zeta -1},Gamma gamma _{zeta } ) bigr) \& quad leq psi bigl[ Lambda bigl( beta bigl( mathbb{Q} ( Im gamma _{zeta -1},Im gamma _{zeta } ) bigr) .mathbb{Q} ( Im gamma _{zeta -1},Im gamma _{zeta } ) bigr) bigr] \& quad =psi bigl[ Lambda bigl( beta bigl( mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) bigr) .mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) bigr) bigr] \& quad =psi bigl[ Lambda bigl( beta bigl( mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) bigr) .mathbb{Q} ( Gamma gamma _{zeta -2},Gamma gamma _{zeta -1} ) bigr) bigr] \& quad leq psi bigl[ Lambda bigl( beta bigl( mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) bigr) .H_{mathbb{Q} } ( Gamma gamma _{zeta -2},Gamma gamma _{zeta -1} ) bigr) bigr] \& quad leq psi ^{2} bigl[ Lambda bigl( beta bigl( mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) bigr) beta bigl( mathbb{Q} ( delta _{zeta -2},delta _{zeta -1} ) bigr) mathbb{Q} ( Gamma gamma _{zeta -3}, Gamma gamma _{zeta -2} ) bigr) bigr] \& quad =psi ^{2} bigl[ Lambda bigl( beta bigl( mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) bigr) beta bigl( mathbb{Q} ( delta _{zeta -2},delta _{zeta -1} ) bigr) mathbb{Q} ( delta _{zeta -2},delta _{zeta -1} ) bigr) bigr] \& qquad cdots \& quad leq psi ^{zeta } bigl[ Lambda bigl( beta bigl( mathbb{Q} ( Im gamma _{zeta -1}, gamma _{zeta } ) bigr) beta bigl( mathbb{Q} ( gamma _{zeta -2},gamma _{zeta -1} ) bigr) …beta bigl( mathbb{Q} ( gamma _{0}, gamma _{1} ) bigr) mathbb{Q} ( gamma _{0},gamma _{1} ) bigr) bigr] \& quad =psi ^{zeta } bigl[ Lambda bigl( beta bigl( mathbb{Q} ( delta _{zeta -1},delta _{zeta } ) bigr) beta bigl( mathbb{Q} ( delta _{zeta -2},delta _{zeta -1} ) bigr) … beta bigl( mathbb{Q} ( delta _{0},delta _{1} ) bigr) mathbb{Q} ( delta _{0},delta _{1} ) bigr) bigr] \& quad =psi ^{zeta } Biggl[ Lambda Biggl( Biggl[ prod _{i=1}^{zeta }beta bigl( mathbb{Q} ( delta _{i-1},delta _{i} ) bigr) Biggr] mathbb{Q} ( delta _{0},delta _{1} ) Biggr) Biggr] \& quad < psi ^{zeta } bigl[ Lambda bigl( mathbb{Q} ( delta _{0},delta _{1} ) bigr) bigr], end{aligned}$$

for all (zeta in mathbb{N} ). Let (epsilon >0) be fixed and (( mathcal{L} ,a ) in Xi times {}[ 0,infty )) be such that ((mathbb{Q} 3)) is satisfied. By (( Xi 2 ) ), there exists (eth >0) such that

$$ 0< varsigma < eth quad text{ implies } mathcal{L} ( varsigma ) < mathcal{L} ( epsilon ) -a. $$

(3.5)

Let (ell ( epsilon ) in mathbb{N} ) such that (0<sum_{zeta geq ell ( epsilon ) }psi ^{zeta } [ Lambda ( mathbb{Q} ( delta _{0},delta _{1} ) ) ] <Lambda ( eth )). Hence, by using the properties of ψ, (3.5), and (( Xi 1 ) ), we have

$$begin{aligned} mathcal{L} Biggl( sum_{j=zeta }^{eta -1} psi ^{j} bigl[ Lambda bigl( mathbb{Q} ( delta _{0},delta _{1} ) bigr) bigr] Biggr) leq & mathcal{L} biggl( sum_{zeta geq ell }psi ^{zeta } bigl[ Lambda bigl( mathbb{Q} ( delta _{0},delta _{1} ) bigr) bigr] biggr) \ < & mathcal{L} bigl( Lambda ( epsilon ) bigr) -a, end{aligned}$$

(3.6)

where (eta >zeta >ell ) with (mathbb{Q} ( delta _{zeta },delta _{eta } ) >0), using (( mathbb{Q} 3 ) ) and (3.6), we have

$$begin{aligned} mathcal{L} bigl( Lambda bigl( mathbb{Q} ( delta _{zeta },delta _{eta } ) bigr) bigr) leq & mathcal{L} Biggl( sum_{j=zeta }^{eta -1} psi ^{j} bigl[ Lambda bigl( mathbb{Q} ( delta _{0},delta _{1} ) bigr) bigr] Biggr) +a \ leq & mathcal{L} biggl( sum_{zeta geq ell }psi ^{zeta } bigl[ Lambda bigl( mathbb{Q} ( delta _{0},delta _{1} ) bigr) bigr] biggr) +a \ < & mathcal{L} bigl( Lambda ( epsilon ) bigr) -a+a \ =& mathcal{L} bigl( Lambda ( epsilon ) bigr) , end{aligned}$$

which yields by (( Xi 1 ) ) and (( Phi 1 ) ) that

$$ mathbb{Q} ( delta _{zeta }, delta _{eta } ) < epsilon ,quad forall eta >zeta >ell. $$

Therefore, ({ delta _{zeta } } = { Im gamma _{zeta } } ) is an F-Cauchy sequence in ϒ. Since ϒ is F-complete, there exists (v^{ast },u^{ast }in Upsilon ) such that (v^{ast }=Im u^{ast }), which implies

$$ lim_{zeta rightarrow infty } mathbb{Q} bigl( v^{ast },delta _{zeta } bigr) =0=lim_{eta ,zeta rightarrow infty } mathbb{Q} ( delta _{eta },delta _{zeta } ) = lim_{zeta rightarrow infty } mathbb{Q} bigl( Im u^{ast },delta _{zeta } bigr) =0. $$

(3.7)

Now, we show that (v^{ast }in Gamma u^{ast }). Postulating that (mathbb{Q} ( v^{ast },Gamma u^{ast } ) >0), by (3.1), we have

$$begin{aligned} Lambda bigl( mathbb{Q} bigl( delta _{zeta },Gamma u^{ast } bigr) bigr) =&Lambda bigl( mathbb{Q} bigl(Gamma gamma _{zeta -1},Gamma u^{ast }bigr) bigr) leq Lambda bigl( H_{mathbb{Q} }bigl(Gamma gamma _{zeta -1},Gamma u^{ast } bigr) bigr) \ leq &psi bigl( Lambda bigl[ beta bigl( aleph _{Im }bigl( gamma _{zeta -1},u^{ast }bigr) bigr) .aleph _{Im } bigl(gamma _{zeta -1},u^{ast }bigr) bigr] bigr) , end{aligned}$$

(3.8)

where

$$begin{aligned} aleph _{Im }(gamma _{zeta -1},u) =&max left { textstylebegin{array}{c} mathbb{Q} (Im gamma _{zeta -1},Im u^{ast }),mathbb{Q} (Im gamma _{zeta -1},Gamma gamma _{zeta -1}), \ mathbb{Q} (Im u^{ast },Gamma u^{ast }),frac{mathbb{Q} (Im gamma _{zeta -1},Gamma u^{ast })+mathbb{Q} (Im u^{ast },Gamma gamma _{zeta -1})}{2}end{array}displaystyle right } \ =&max left { textstylebegin{array}{c} mathbb{Q} (delta _{zeta -1},v^{ast }),mathbb{Q} (delta _{zeta -1},delta _{zeta }),mathbb{Q} (v^{ast },Gamma u^{ast }), \ frac{mathbb{Q} (delta _{zeta -1},Gamma u^{ast })+mathbb{Q} (v^{ast },delta _{zeta })}{2}end{array}displaystyle right } . end{aligned}$$

(3.9)

Since (beta in mho ), from (( Phi 1 ) ), letting (zeta rightarrow infty ) in (3.8) and applying (3.9), we obtain

$$ mathbb{Q} bigl( v^{ast },Gamma u^{ast } bigr) < mathbb{Q} bigl(v^{ast },Gamma u^{ast }bigr), $$

which is a discrepancy. Therefore, (mathbb{Q} ( v^{ast },Gamma u^{ast } ) =0), which implies that (v^{ast }in Gamma u^{ast }). Thus, (v^{ast }=Im u^{ast }in Gamma u^{ast }), and hence Γ and have a coincidence point (u^{ast }), and (v^{ast }) is a point of coincidence of Γ and . By (( mathbb{Q} 1 ) ), we have (mathbb{Q} ( v^{ast },v^{ast } ) =0). Postulating that (v_{1}^{ast }) is another point of coincidence of Γ and such that we can find (u_{1}^{ast }in Upsilon ), such that (v_{1}^{ast }=Im u_{1}^{ast }in Gamma u_{1}^{ast }) and by (( mathbb{Q} 1 ) ), (mathbb{Q} ( v_{1}^{ast },v_{1}^{ast } ) =0). Now, we prove that (mathbb{Q} ( v^{ast },v_{1}^{ast } ) =0) by contrast. Assume that (mathbb{Q} ( v^{ast },v_{1}^{ast } ) >0), from (3.1)

$$begin{aligned}& begin{aligned}[b] Lambda bigl( mathbb{Q} bigl( v^{ast },v_{1}^{ast } bigr) bigr) &leq Lambda bigl( mathbb{Q} bigl( Gamma u^{ast },Gamma u_{1}^{ast } bigr) bigr) leq Lambda bigl( H_{mathbb{Q} } bigl( Gamma u^{ast },Gamma u_{1}^{ast } bigr) bigr) \ &leq psi bigl( Lambda bigl[ beta bigl( aleph _{Im } bigl( u^{ast },u_{1}^{ast } bigr) bigr) aleph _{Im } bigl( u^{ast },u_{1}^{ast } bigr) bigr] bigr) , end{aligned} end{aligned}$$

(3.10)

$$begin{aligned}& begin{aligned}[b] aleph _{Im } bigl( u^{ast },u_{1}^{ast } bigr) &=max left { textstylebegin{array}{c} mathbb{Q} ( Im u^{ast },Im u_{1}^{ast } ) ,mathbb{Q} ( Im u^{ast },Gamma u^{ast } ) , \ mathbb{Q} ( Im u_{1}^{ast },Gamma u_{1}^{ast } ) ,frac{mathbb{Q} ( Im u^{ast },Gamma u_{1}^{ast } ) +mathbb{Q} ( Gamma u^{ast },Im u_{1}^{ast } ) }{2}end{array}displaystyle right } \ &=max left { textstylebegin{array}{c} mathbb{Q} ( v^{ast },v_{1}^{ast } ) ,mathbb{Q} ( v^{ast },v^{ast } ) ,mathbb{Q} ( v_{1}^{ast },v_{1}^{ast } ) , \ frac{mathbb{Q} ( v^{ast },v_{1}^{ast } ) +mathbb{Q} ( v^{ast },v_{1}^{ast } ) }{2}end{array}displaystyle right } \ &=mathbb{Q} bigl( v^{ast },v_{1}^{ast } bigr) . end{aligned} end{aligned}$$

(3.11)

Since (beta in mho), from (( Phi 1 )), (3.10), and (3.11), we obtain (mathbb{Q} ( v^{ast },v_{1}^{ast } ) <mathbb{Q} ( v^{ast },v_{1}^{ast } )), which is a discrepancy. Therefore, (mathbb{Q} ( v^{ast },v_{1}^{ast } ) =0) implies that (v^{ast }=v_{1}^{ast }). Thus, Γ and have a unique point of coincidence. Moreover, since Γ and are weakly compatible, we have (Im v^{ast }=Gamma v^{ast }). Now, let (w=Im v^{ast }in Gamma v^{ast }). From the uniqueness of the point of coincidence, we have (w=v=Im v^{ast }in Gamma v^{ast }). Therefore, Γ and have a unique common fixed point. □

Corollary 3.2

Let ((Upsilon ,mathbb{Q} )) be an Fcomplete FMS, (alpha :Upsilon times Upsilon rightarrow {}[ 0,infty )) be a function. A mapping (Gamma :Upsilon rightarrow Upsilon ) is called an improved αGeraghty contraction mapping if there exist (beta in mho ) such that for all (gamma ,delta in Upsilon ),

$$ alpha (gamma ,delta )mathbb{Q} (Gamma gamma ,Gamma delta )leq beta bigl( aleph (gamma , delta ) bigr) .aleph (gamma ,delta ), $$

where

$$ aleph (gamma ,delta )=max biggl{ mathbb{Q} (gamma ,delta ),mathbb{Q} (gamma ,Gamma gamma ),mathbb{Q} (delta ,Gamma delta ),frac{mathbb{Q} (gamma ,Gamma delta )+mathbb{Q} (delta ,Gamma gamma )}{2}biggr} , $$

for all (gamma ,delta in Upsilon ), satisfying the following stipulations:

  1. (1)

    Γ is an improved αGeraghty contraction;

  2. (2)

    Γ is triangular αorbital admissible;

  3. (3)

    there exists an (gamma _{0}in Upsilon ) such that (alpha (gamma _{0},Gamma gamma _{0})geq 1);

  4. (4)

    Γ is continuous.

Then, Γ has a unique fixed point (gamma ^{ast }in Upsilon ).

Corollary 3.3

Let ((Upsilon ,mathbb{Q} )) be an FMS, and (Gamma ,Im :Upsilon rightarrow Upsilon ) be two mappings with (Gamma Upsilon subseteq Im Upsilon ) and ϒ is Fcomplete. The pair (( Gamma ,Im ) ) is an improved Geraghty contraction if there exists (beta in mho ) such that for all (gamma ,delta in Upsilon ),

$$ mathbb{Q} (Gamma gamma ,Gamma delta ) leq beta bigl( aleph _{Im }(gamma ,delta ) bigr) .aleph _{Im }(gamma ,delta ), $$

where

$$ aleph _{Im }(gamma ,delta )=max biggl{ mathbb{Q} (Im gamma ,Im delta ),mathbb{Q} (Im gamma ,Gamma gamma ),mathbb{Q} (Im delta ,Gamma delta ),frac{mathbb{Q} (Im gamma ,Gamma delta )+mathbb{Q} (Im delta ,Gamma gamma )}{2} biggr} . $$

Then, Γ and g have a unique point of coincidence. Indeed, if Γ and are weakly compatible, then Γ and have a unique common fixed point (gamma ^{ast }in Upsilon ).

Example 3.4

Let (Upsilon =[0,infty )) and F-M (mathbb{Q} ) given by

$$ mathbb{Q} ( gamma ,delta ) = textstylebegin{cases} e^{ vert gamma -delta vert } & text{if }gamma neq delta ,\ 0 & text{if }gamma =delta, end{cases} $$

with (mathcal{L} ( varsigma ) =frac{-1}{varsigma }) and (a=1). Then, (( Upsilon ,mathbb{Q} ) ) is F-complete F-MS. Define (Im :Upsilon rightarrow Upsilon ) and (Gamma :Upsilon rightarrow CB ( Upsilon ) ) by

$$ Gamma gamma = textstylebegin{cases} { frac{gamma }{8} } , & text{if }gamma in mathbb{N} cup { 0 } ,\ { 0 } & text{otherwise}, end{cases}displaystyle quad text{and}quad Im gamma = textstylebegin{cases} frac{3gamma }{2} & text{if }gamma in mathbb{N} cup { 0 } ,\ 0 & text{otherwise}. end{cases} $$

Clearly, for all (gamma in mathbb{N} cup { 0 } ), (Gamma ( Upsilon ) subseteq Im ( Upsilon ) ) and (Im ( Upsilon ) ) is an F-complete subset of ϒ; let (beta :Upsilon times Upsilon rightarrow {}[ 0,1)) be as (beta ( gamma ,delta ) =frac{1}{2},~Lambda ( t ) =t) and (psi ( t ) =frac{2}{3}t). Now, for all (( gamma ,delta ) in mathbb{N} cup { 0 } ) with (gamma neq delta ), then

$$begin{aligned} Lambda bigl( H_{mathbb{Q} } ( Gamma gamma ,Gamma delta ) bigr) =& Lambda bigl( max bigl( sup_{ain Gamma gamma } mathbb{Q} ( a,Gamma delta ) ,sup_{bin Gamma delta } mathbb{Q} ( Gamma gamma ,b ) bigr) bigr) \ =&Lambda biggl( max biggl( sup_{ain Gamma gamma } mathbb{Q} biggl( a, biggl{ frac{delta }{8} biggr} biggr) ,sup_{bin Gamma delta } mathbb{Q} biggl( biggl{ frac{gamma }{8} biggr} ,b biggr) biggr) biggr) \ =&Lambda biggl( max biggl( mathbb{Q} biggl( frac{gamma }{8}, biggl{ frac{delta }{8} biggr} biggr) ,mathbb{Q} biggl( biggl{ frac{gamma }{8} biggr} ,frac{delta }{8} biggr) biggr) biggr) \ =&Lambda biggl( max biggl( mathbb{Q} biggl( frac{gamma }{8},frac{delta }{8} biggr) ,mathbb{Q} biggl( frac{gamma }{8},frac{delta }{8} biggr) biggr) biggr) \ =&Lambda biggl( mathbb{Q} biggl( frac{gamma }{8},frac{delta }{8} biggr) biggr) \ =&Lambda bigl( e^{ vert frac{gamma }{8}-frac{delta }{8} vert } bigr) =Lambda bigl( e^{frac{1}{4} vert frac{gamma }{2}-frac{delta }{2} vert } bigr) \ leq &frac{2}{3}Lambda biggl( frac{1}{2}e^{ vert frac{3gamma }{2}-frac{3delta }{2} vert } biggr) \ leq &psi bigl( Lambda bigl[ beta bigl( aleph _{Im }(gamma , delta ) bigr) .aleph _{Im }(gamma ,delta ) bigr] bigr) . end{aligned}$$

If (gamma =delta ), then we have

$$ Lambda bigl( H_{mathbb{Q} } ( Gamma gamma ,Gamma delta ) bigr) =0leq psi bigl( Lambda bigl[ beta bigl( aleph _{Im }(gamma ,delta ) bigr) .aleph _{Im }(gamma ,delta ) bigr] bigr) . $$

Otherwise, we have that (3.1) trivially holds. Therefore, all stipulations of Theorem 3.1 are satisfied. Since (Upsilon 0=Im 0=0), thus (gamma =0) is a common fixed point of Γ and .

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