A novel study for hybrid pair of multivalued dominated mappings in b-multiplicative metric space with applications – Journal of Inequalities and Applications

Let ((Lambda ,sigma )) be a BMMS, (hslash _{0}in Lambda ), and let ({check{S}},mathcal{H} :Lambda rightarrow wp (Lambda )) be multivalued mappings on Λ. Let (hslash _{1}in {check{S}}hslash _{0}) be an element such that (sigma (hslash _{0},{check{S}}hslash _{0})=sigma (hslash _{0}, hslash _{1})). Let (hslash _{2}in mathcal{H} hslash _{1}) be such that (sigma (hslash _{1},mathcal{H} hslash _{1})=sigma (hslash _{1}, hslash _{2})). Let (hslash _{3}in {check{S}}hslash _{2}) be such that (sigma (hslash _{2},{check{S}}hslash _{2})=sigma (hslash _{2}, hslash _{3})). In this way, we get a sequence ({ mathcal{H} {check{S}}(hslash _{n})}) in Λ, where (hslash _{2n+1}in {check{S}}hslash _{2n}), (hslash _{2n+2}in mathcal{H} hslash _{2n+1}),

$n\in \mathbb{N}$

. Also (sigma (hslash _{2n},{check{S}}hslash _{2n})=sigma (hslash _{2n}, hslash _{2n+1})), (sigma (hslash _{2n+1},mathcal{H} hslash _{2n+1})=sigma ( hslash _{2n+1},hslash _{2n+2})). Then, ({ mathcal{H} {check{S}}(hslash _{n})}) is said to be a sequence in Λ generated by (hslash _{0}). If ({check{S}}=mathcal{H} ), then we say ({ Lambda {check{S}}(hslash _{n})}) instead of ({ mathcal{H} {check{S}}(hslash _{n})}). For (e,vin Lambda ), (kappa in (0,frac{1}{2})), we define (nabla (e,y)) as

Theorem 2.1

Let ((Lambda ,sigma )) be a complete BMMS. Suppose there exists a function (alpha :Lambda times Lambda rightarrow {}[ 0,infty )). Let (r>0), (hslash _{0}in overline{beta _{sigma _{m}}(hslash _{0},r)} subseteq Lambda ), and ({check{S}},mathcal{H}:Lambda rightarrow wp (Lambda )) be (alpha _{ast })–dominated mappings on (overline{beta _{sigma _{m}}(hslash _{0},r)}). Assume that (tau >0), and there exists (kappa in (0,frac{1}{ell })) with (eta =frac{kappa }{1-kappa }), and (mathcal{F}) is a strictly increasing function satisfying:

for all (e,yin overline{beta _{sigma _{m}}(hslash _{0},r)}cap {mathcal{H}{check{S}}(hslash _{n})}), (alpha (e,y)geq 1), and (H_{sigma }({check{S}}e,mathcal{H}y)>0) such that,

Then, ({mathcal{H}{check{S}}(hslash _{n})}) is a sequence in (overline{beta _{sigma _{m}}(hslash _{0},r)}), (alpha (hslash _{n},hslash _{n+1})geq 1) for all

$n\in \mathbb{N}$

and ({mathcal{H}{check{S}}(hslash _{n})}rightarrow {mu}in overline{beta _{sigma _{m}}(hslash _{0},r)}). Also, if μ satisfies (2.1), (alpha (hslash _{n}, {mu})geq 1), and (alpha ({mu} ,hslash _{n})geq 1) for all integers (ngeq 0), then Š and (mathcal{H}) have a common fixed point μ in (overline{beta _{sigma _{m}}(hslash _{0},r)}).

Proof

Consider a sequence ({mathcal{H}{check{S}}(hslash _{n})} ). From (2.2), we get

It follows that,

Let (hslash _{2},ldots ,hslash _{j}in overline{beta _{sigma _{m}}(hslash _{0},r)}) for some

$j\in \mathbb{N}$

. If j is odd, then (j=2grave{imath}+1) for some

$\stackrel{}{}$
ı
`
∈N

. Since ({check{S}},mathcal{H}:Lambda rightarrow wp (Lambda )) are (alpha _{ast })-dominated mappings on (overline{beta _{sigma _{m}}(hslash _{0},r)}), so (alpha _{ast }(hslash _{2grave{imath}},{check{S}}hslash _{2grave{imath}})geq 1) and (alpha _{ast }(hslash _{2grave{imath}+1},mathcal{H}hslash _{2grave{imath}+1})geq 1). As (alpha _{ast }(hslash _{2grave{imath}},{check{S}}hslash _{2 grave{imath}})geq 1), this implies (inf {alpha (hslash _{2grave{imath} },b):bin {check{S}}hslash _{2grave{imath}}}geq 1). Also (hslash _{2grave{imath}+1}in {check{S}}hslash _{2grave{imath}}), so (alpha (hslash _{2grave{imath}},hslash _{2grave{imath}+1}) geq 1) and (hslash _{2grave{imath}+2}in mathcal{H}hslash _{2grave{imath}+1}). Now using Lemma 1.12, we have

Thus,

for all (iin N), where (eta =frac{kappa }{1-kappa }). As

$\mathcal{F}:{}_{}$
R
+
→R

is a strictly increasing function then

Similarly, if j is even, we have

Now, we have

Therefore,

Now,

Then, we have

which implies (hslash _{j+1}in overline{beta _{sigma _{m}}(hslash _{0},r)}). Hence, by induction (hslash _{n}in overline{beta _{sigma _{m}}(hslash _{0},r)}) for all

$n\in \mathbb{N}$

. Also, (alpha (hslash _{n},hslash _{n+1})geq 1) for all

$n\in \mathbb{N}\cup \{0\}$

. Now,

Now, for any positive integers m, n ((n>m)), we have

Clearly, (sigma (hslash _{m},hslash _{n})rightarrow 1) as (m,nrightarrow infty ). Hence, ({mathcal{H}{check{S}}(hslash _{n})}) is a Cauchy sequence in a complete multiplicative metric space ((overline{beta _{sigma _{m}}(hslash _{0},r)},sigma )). So, there is a ({mu} in overline{beta _{sigma _{m}}(hslash _{0},r)}) and ({mathcal{H}{check{S}}(hslash _{n})}rightarrow {mu}) such that (nrightarrow infty ). Then

Now

So, there exists (hslash _{2n+1}in {check{S}}hslash _{2n}) and (sigma (hslash _{2n},{check{S}}hslash _{2n})=sigma (hslash _{2n}, hslash _{2n+1})). Using Lemma 1.12 and (2.1), we obtain

By assumption, (alpha (hslash _{n},{mu})geq 1). Suppose that (sigma ({mu},mathcal{H}{mu})>0), there exists a positive integer k such that (sigma (hslash _{n},mathcal{H}{mu})>0) for all (ngeq k). For (ngeq k), we have

Taking limit (nrightarrow infty ) and inequality (2.8) from both sides of (2.9), we get (sigma ({mu},mathcal{H}{mu})<sigma ({mu},mathcal{H}{mu})^{ell kappa }) that is not true in general. Our supposition is wrong because (ell kappa <1). Hence, (sigma ({mu},mathcal{H}{mu})=1) or ({mu}in mathcal{H}{mu}). Similarly, adopting the similar way and using Lemma 1.12 and inequality (2.8), we can get (sigma ({mu},{check{S}}{mu})=1) or ({mu}in {check{S}}{mu}). So, Š and (mathcal{H}) have a common fixed point μ in (overline{beta _{sigma _{m}}(hslash _{0},r)}). Now,

This implies that (sigma ({mu},{mu})=1). □

Example 2.2

Let (Lambda =R^{+}cup {0}) and the function (sigma :Lambda times Lambda rightarrow Lambda ) defined by

Define the mappings ({check{S}},mathcal{H}:Lambda times Lambda rightarrow wp ( Lambda )) by

and,

Suppose that, (hslash _{0}=1), (ell =2), (r=81), (overline{B_{sigma }(hslash _{0},r)}=[0,15]cap Lambda ). Now, (sigma (hslash _{0},{check{S}}hslash _{0})=sigma (1,{check{S}}1)=sigma (1,frac{1}{5})). So (hslash _{1}=frac{1}{5}). Now, (sigma (hslash _{1},mathcal{H}hslash _{1})=sigma (frac{1}{5}, mathcal{H}frac{1}{5})=sigma (frac{1}{5},frac{1}{35})). So (hslash _{2}=frac{1}{35}). Now, (sigma (hslash _{2},{check{S}}hslash _{2})=sigma (frac{1}{35},{check{S}}frac{1}{35})=sigma ( frac{1}{35},frac{1}{175})). So (hslash _{3}=frac{1}{175}). Continuing in this way, we have ({mathcal{H}{check{S}}(hslash _{n})}={1,frac{1}{5}, frac{1}{35}, frac{1}{175},ldots}). Moreover, taking (kappa =frac{7}{23}in (0,frac{1}{2})) and (eta =frac{7}{17}in (0,1)). From (2.2), we also have

Consider the mapping (alpha :Lambda times Lambda rightarrow {}[ 0,infty )) by

Now, if (hslash ,vin overline{beta _{sigma _{m}}(hslash _{0},r)}cap {mathcal{H}{check{S}}(hslash _{n})}) with (alpha (hslash ,v)geq 1), we have

Thus,

this implies that if there is (tau in (0,frac{11}{91}]), and (mathcal{F}) is a strictly increasing function defined as (mathcal{F}(ell )=ln ell +ell ), we have

Note that, taking 17, (18in Lambda ), then (alpha (17,18)geq 1). Now, we have

So, the condition (2.1) is not satisfied on Λ. Hence, Š and (mathcal{H}) are satisfied all conditions of Theorem 2.1 for all (hslash ,vin overline{beta _{sigma _{m}}(hslash _{0},r)}cap { mathcal{H}{check{S}}(hslash _{n})}) with (alpha (hslash ,v)geq 1). Hence, Š and (mathcal{H}) admit a common fixed point.

Corollary 2.3

Let ((Lambda ,sigma )) be a complete BMMS. Suppose there exists a function (alpha :Lambda times Lambda rightarrow {}[ 0,infty )). Let (r>0), (hslash _{0}in overline{beta _{sigma _{m}}(hslash _{0},r)} subseteq Lambda ), and ({check{S}},mathcal{H}:Lambda rightarrow wp (Lambda )) are (alpha _{ast })–dominated multifunctions on (overline{beta _{sigma _{m}}(hslash _{0},r)}). Assume that (tau >0), and there exists (kappa in (0,frac{1}{ell })) with (eta =frac{kappa }{1-kappa }), and (mathcal{F}) be a strictly increasing function satisfying:

for all (e,yin overline{beta _{sigma _{m}}(hslash _{0},r)}cap { hslash _{n}}), (alpha (e,y)geq 1), and (sigma ({check{S}}e,mathcal{H}y)>0) such that,

Then, ({mathcal{H}{check{S}}(hslash _{n})}) is a sequence in (overline{beta _{sigma _{m}}(hslash _{0},r)}), (alpha (hslash _{n},hslash _{n+1})geq 1) for all

$n\in \mathbb{N}$

and ({mathcal{H}{check{S}}(hslash _{n})}rightarrow {mu}in overline{beta _{sigma _{m}}(hslash _{0},r)}). Also, if μ satisfies (2.11) (alpha (hslash _{n},{mu})geq 1) and (alpha ({mu},hslash _{n})geq 1) for all naturals (ngeq 0), then Š and (mathcal{H}) admit a common fixed point μ in (overline{beta _{sigma _{m}}(hslash _{0},r)}).

Corollary 2.4

Let ((Lambda ,sigma )) be a complete BMMS. Suppose there exists a function (alpha :Lambda times Lambda rightarrow {}[ 0,infty )). Let (r>0), (hslash _{0}in overline{beta _{sigma _{m}}(hslash _{0},r)} subseteq Lambda ) and ({check{S}}:Lambda rightarrow wp (Lambda )) be a (alpha _{ast })–dominated multifunction on (overline{beta _{sigma _{m}}(hslash _{0},r)}). Assume that (tau >0), and there exists (kappa in (0,frac{1}{ell })) with (eta =frac{kappa }{1-kappa }), and (mathcal{F}) be a strictly increasing function satisfying:

for all (e,yin overline{beta _{sigma _{m}}(hslash _{0},r)}cap { Lambda {check{S}}(hslash _{n})}), (alpha (e,y)geq 1), and (H_{sigma }({check{S}}e,mathcal{H}y)>0), such that,

Then, ({Lambda {check{S}}(hslash _{n})}) is a sequence in (overline{beta _{sigma _{m}}(hslash _{0},r)}), (alpha (hslash _{n},hslash _{n+1})geq 1) for all

$n\in \mathbb{N}$

and ({Lambda {check{S}}(hslash _{n})}rightarrow {mu}in overline{beta _{sigma _{m}}(hslash _{0},r)}). Also, if μ satisfies (2.12), (alpha (hslash _{n},{mu})geq 1) and (alpha ({mu},hslash _{n})geq 1) for all natural (ngeq 0), then Š and (mathcal{H}) admit a fixed point μ in (overline{beta _{sigma _{m}}(hslash _{0},r)}).