Novel results of α ∗ $alpha _{ast }$ -ψ-Λ-contraction multivalued mappings in F-metric spaces with an application – Journal of Inequalities and Applications

ByMustafa Mudhesh, Nabil Mlaiki, Muhammad Arshad, Aftab Hussain, Eskandar Ameer, Reny George and Wasfi Shatanawi

Aug 26, 2022

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)

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 .