# Fixed-Point Theorems for Multivalued Mappings in Modular Metric Spaces

Authors: Parin Chaipunya; Chirasak Mongkolkeha; Wutiphol Sintunavarat; Poom Kumam
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We give some initial properties of a subset of modular metric spaces and introduce
some fixed-point theorems for multivalued mappings under the setting of contraction type. An appropriate example is as well provided. The stability of fixed points in our main
theorems is also studied.


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## Body

## 1. Introduction and Preliminaries

The field of metric fixed-point theory has been widely investigated since 1922, when Banach [1] had proved his contraction principle. We are going to recall this well-known theorem before we continue over on.A self-mappingf on a metric space (X,d) is called a contraction if there exists 0≤k<1 such that(1.1)d(fx,fy)≤kd(x,y)
for all x,y∈X. The contraction principle simply stated that, if (X,d) is complete, such a mapping has a unique fixed point.One of the most influenced generalizations of Banach’s theorem is traced to Nadler [2]. In 1969, via Hausdorff’s concept of a distance between two arbitrary sets, Nadler proved the contraction principle for multivalued mappings in complete metric spaces. Also, some authors extended Nadler’s principle and established fixed-point theorems for multivalued mappings in metric spaces and other spaces (see [3–9]). One of the most interesting studies are the extensions of such principle in modular spaces and modular function spaces (see [10–12] and references therein).Lately, in 2010, Chistyakov [13] introduced the notion of a modular metric space which is a new generalization of a metric space. We will give a short revisit to modular and modular metric spaces as follows.Definition 1.1.
LetX be a linear space over ℝ with θ∈X as its zero element. A functional ρ:X→[0,+∞] is said to be a modular on X if for any x,y∈X, the following conditions hold:(i)
ρ(x)=0 if and only if x=θ,(ii)
ρ(x)=ρ(-x),(iii)
ρ(αx+βy)≤ρ(x)+ρ(y) whenever α,β≥0 and α+β=1.The linear subspace Xρ:={x∈X:limλ→0ρ(λx)=0} is called a modular space.Definition 1.2 (see [13]).
LetX be a nonempty set. A function ω:(0,+∞)×X×X→[0,+∞] is said to be a metric modular on X if satisfying, for all x,y,z∈X, the following conditions hold:(i)
ωλ(x,y)=0 for all λ>0 if and only if x=y,(ii)
ωλ(x,y)=ωλ(y,x) for all λ>0,(iii)
ωλ+μ(x,y)≤ωλ(x,z)+ωμ(z,y) for all λ,μ>0.Suppose xι∈X, the set Xω(xι)={x∈X:limλ→+∞ωλ(x,xι)=0} is called a modular metric space generated by xι and induced by ω. If its generator xι does not play any role in the situation, we will write Xω instead of Xω(xι).
Observe that a metric modularω on X is nonincreasing with respect to λ>0. We can simply show this assertion by using the condition (iii) itself. For any x,y∈X and 0<μ<λ, we have
(1.2)ωλ(x,y)≤ωλ-μ(x,x)+ωμ(x,y)=ωμ(x,y).
For each x,y∈X and λ>0, we set ωλ+(x,y):=limϵ↓0ωλ+ϵ(x,y) and ωλ-(x,y):=limϵ↓0ωλ-ϵ(x,y). Consequently, from (1.2), we have ωλ+(x,y)≤ωλ(x,y)≤ωλ-(x,y).
If, for anyx,y∈X, a metric modular ω on X possesses a finite value and ωλ(x,y)=ωμ(x,y) for all λ,μ>0, then d(x,y):=ωλ(x,y) is a metric on X.Recently, Mongkolkeha et al. [14] have introduced some notions and established some fixed-point results in modular metric spaces. We now state some notions and results in [14] in the following.Definition 1.3 (see [14]).
LetXω be a modular metric space. (i)
The sequence{xn}n∈ℕ in Xω is said to be convergent if there exists x∈Xω such that ωλ(xn,x)→0, as n→∞ for all λ>0.(ii)
The sequence{xn}n∈ℕ in Xω is said to be a Cauchy sequence if ωλ(xm,xn)→0, as m,n→∞ for all λ>0.(iii)
Xω is said to be complete if every Cauchy sequence in Xω converges.(iv)
A subsetC of Xω is said to be closed if the limit of a convergent sequence of C always belongs to C.(v)
A subsetC of Xω is said to be bounded if, for all λ>0, ϕλ(C) = sup{ωλ(x,y):x,y∈C}<+∞.Along this paper, we will use the following alternative notions of convergence and Cauchyness, which are equivalent to the notions given above.
LetXω be a modular metric space and {xn}n∈ℕ be a sequence in Xω.(i)
A pointx∈Xω is called a limit of {xn}n∈ℕ if for each λ,ϵ>0, there exists n0∈ℕ such that ωλ(xn,x)<ϵ for every n∈ℕ with n≥n0. A sequence that has a limit is said to be convergent (or converges to x) and will be written as  limn→+∞xn=x.(ii)
A sequence{xn}n∈ℕ in Xω is said to be a Cauchy sequence if, for each λ,ϵ>0, there exists n0∈ℕ such that ωλ(xn,xm)<ϵ for every m,n∈ℕ with m,n≥n0.Moreover, we observe that the limit of any sequence in Xω is unique.Definition 1.4 (see [14]).
LetXω be a modular metric space. A self-mapping f on Xω is said to be a contraction if there exists 0≤k<1 such that
(1.3)ωλ(fx,fy)≤kωλ(x,y)
for all x,y∈Xω and λ>0.Theorem 1.5 (see [14]).
LetXω be a complete modular metric space and f a contraction on Xω. Then, the sequence {fnx}n∈ℕ converges to the unique fixed point of f in Xω for any initial x∈Xω.The purpose of this paper is to study some properties of a subset of modular metric spaces, establish and extend some fixed-point theorems of Mongkolkeha et al. [14] to multivalued mappings in modular metric spaces.


## 2. Some Properties of a Subset of Modular Metric Spaces

In this section, we study some properties of a subset of modular metric spaces, some of which will take advantages in the proof of our main theorems. Throughout this paper, let𝒞ℬ(Xω) denotes the set of all nonempty closed bounded subsets of Xω and 𝒞(X) denotes the set of all nonempty closed subsets of X.LetA be a non-empty subset of a modular metric space Xω. For x∈Xω, we denotes ωλ(x,A):=infy∈Aωλ(x,y).ForA,B∈𝒞ℬ(Xω), define δλ(A,B):=supx∈Aωλ(x,B) and the Hausdorff metric modular Ωλ(A,B):=max{δλ(A,B),δλ(B,A)}. Notice that δλ is not symmetric.Proposition 2.1.
LetXω be a modular metric space and A,B,C∈𝒞ℬ(Xω). Then, the following properties hold.(i)
δλ(A,B)=0 for all λ>0⇔A⊆B.(ii)
B⊆C⇒δλ(A,C)≤δλ(A,B) for all λ>0.(iii)
δλ(A∪B,C)=max{δλ(A,C),δλ(B,C)} for all λ>0.(iv)
δλ+μ(A,B)≤δλ(A,C)+δμ(C,B) for all λ,μ>0.Proof.
(i) By the definition ofδλ, we have, for all λ>0, that
(2.1)δλ(A,B)=0⟺sup⁡x∈A⁡ωλ(x,B)=0⟺ωλ(x,B)=0,∀x∈A.
Since B is closed in Xω, we get ωλ(x,B)=0 for all λ>0⇔x∈B. That is, δλ(A,B)=0 for all λ>0⇔A⊆B.
(ii) It is obvious thatωλ(x,C)≤ωλ(x,B) for all x∈Xω and λ>0. Hence, δλ(A,C)≤δλ(A,B).
(iii) Observe that, ifB⊆C, then
(2.2)δλ(A∪B,C)=sup⁡x∈A∪B⁡ωλ(x,C)=max⁡{sup⁡x∈A⁡ωλ(x,C),sup⁡x∈B⁡ωλ(x,C)}.
(iv) Leta∈A, b∈B, and c∈C. Then,
(2.3)ωλ+μ(a,b)≤ωλ(a,c)+ωμ(c,b),
which implies that
(2.4)ωλ+μ(a,B)≤ωλ(a,c)+ωμ(c,B)≤ωλ(a,c)+δμ(C,B).
Since c∈C is arbitrary, we have
(2.5)ωλ+μ(a,B)≤ωλ(a,C)+δμ(C,B).
Similarly, since a∈A is arbitrary, we can deduce that
(2.6)δλ+μ(A,B)≤δλ(A,C)+δμ(C,B).Proposition 2.2.
LetXω be a modular metric space. Then,
(2.7)Ωλ(A∪B,C∪D)≤max⁡⁡{Ωλ(A,C),Ωλ(B,D)}
for all A,B,C,D∈𝒞ℬ(Xω).Proof.
Supposeλ>0 is arbitrary. For a∈A and b∈B, we have ωλ(a,C∪D)≤ωλ(a,C) and ωλ(b,C∪D)≤ωλ(b,D). Hence, we get
(2.8)δλ(A∪B,C∪D)=max⁡{δλ(A,C∪D),δλ(B,C∪D)}≤max⁡{δλ(A,C),δλ(B,D)}≤max⁡{Ωλ(A,C),Ωλ(B,D)}.
Similarly, we have
(2.9)δλ(C∪D,A∪B)≤max⁡{Ωλ(A,C),Ωλ(B,D)}.
Hence, we have
(2.10)Ωλ(A∪B,C∪D)=max⁡{δλ(A∪B,C∪D),δλ(C∪D,A∪B)}≤max⁡{Ωλ(A,C),Ωλ(B,D)}.Proposition 2.3.
LetXω be a modular metric space generated by xι. Then, 𝒞ℬ(Xω) is a modular metric space generated by {xι} and is induced by Ω.Proof.
For{xι},A∈𝒞ℬ(Xω), we have
(2.11)Ωλ(A,{xι})=max⁡{sup⁡x∈A⁡ωλ(x,{xι}),sup⁡x∈{xι}⁡ωλ(x,A)}=max⁡{sup⁡x∈A⁡ωλ(x,xι),inf⁡x∈A⁡ωλ(xι,x)}=sup⁡x∈A⁡ωλ(x,xι).
Since x∈A⊆Xω and limλ→+∞wλ(x,xι)=0, we have limλ→+∞Ωλ(A,{xι})=0.
By the definition ofΩ and Proposition 2.1, it is clear that Ωλ(A,B)=Ωλ(B,A)≥0 for all λ>0 and Ωλ(A,B)=0 for all λ>0 if and only if A=B.
Again, by Proposition2.1, we have
(2.12)Ωλ+μ(A,B)=max⁡{δλ+μ(A,B),δλ+μ(B,A)}≤max⁡{δλ(A,C)+δμ(C,B),δμ(B,C)+δλ(C,A)}≤max⁡{δλ(A,C),δλ(C,A)}+max⁡{δμ(B,C),δμ(C,B)}=Ωλ(A,C)+Ωμ(C,B)
for all λ,μ>0. Therefore, 𝒞ℬ(Xω) is a modular metric space generated by {xι} and is induced by Ω.Remark 2.4.
Note that the metric modularΩ depends on ω, so the completeness of Xω implies the completeness of 𝒞ℬ(Xω).Now, we are arriving at the most important lemma used in our proof of main theorems.Lemma 2.5.
LetA,B∈𝒞ℬ(Xω) and a∈A. Then, for ϵ>0, there exists a point bϵ∈B such that ωλ(a,bϵ)≤Ωλ(A,B)+ϵ.Proof.
Leta∈A, ϵ,λ>0 be arbitrary. Since ωλ(a,B)=infb∈Bωλ(a,b), we claim that ωλ(a,B)+ϵ is not a lower bound of the set {ωλ(a,b):b∈B}. Therefore, there exists bϵ∈B for which ωλ(a,bϵ)≤ωλ(a,B)+ϵ and hence ωλ(a,bϵ)≤Ωλ(A,B)+ϵ.


## 3. Fixed-Point Theorems for Multivalued Mappings

In this section, we extend the result by Mongkolkeha et al. [14] under the multivalued setting and hereby obtain some corollaries. Beforehand, we will give the notion of a multivalued ω-contraction in modular metric spaces.Definition 3.1.
LetXω be a modular metric space. A multivalued mapping F:Xω→𝒞ℬ(Xω) is said to be a multivaluedω-contraction if there exists 0≤k<1 such that
(3.1)Ωλ(Fx,Fy)≤kωλ(x,y)
for all x,y∈Xω and λ>0. In this case, the least number k which satisfies the inequality (3.1) is said to be the contraction constant.Remark 3.2.
For a sequence{xn}n∈ℕ in Xω, it is obvious that, if limn→+∞xn=x and F is a multivalued ω-contraction on Xω, then limn→+∞Fxn=Fx.Theorem 3.3.
LetXω be a complete modular metric space and F a multivalued ω-contraction on Xω with contraction constant k. Then,  F has a fixed point in Xω.Proof.
Letx0∈Xω be arbitrary and x1∈Fx0. By Lemma 2.5, there exists x2∈Fx1 such that
(3.2)ωλ(x1,x2)≤Ωλ(Fx0,Fx1)+k.
Similarly, by this procedure, we define a sequence {xn}n∈ℕ in Xω such that xn∈Fxn-1 and
(3.3)ωλ(xn,xn+1)≤Ωλ(Fxn-1,Fxn)+kn
for all n∈ℕ. Hence, by the multivalued ω-contractivity, we have
(3.4)ωλ(xn,xn+1)≤Ωλ(Fxn-1,Fxn)+kn≤kωλ(xn-1,xn)+kn≤k[kωλ(xn-2,xn-1)+kn-1]+kn≤k2ωλ(xn-2,xn-1)+2kn.
Thus, by induction, we deduce that
(3.5)ωλ(xn,xn+1)≤knωλ(x0,x1)+nkn.
Notice that ∑n∈ℕkn<+∞ and ∑n∈ℕnkn<+∞. Now, since
(3.6)∑n∈Nωλ(xn,xn+1)≤ωλ(x0,x1)∑n∈Nkn+∑n∈Nnkn<+∞,
for all λ>0. Without loss of generality, suppose m,n∈ℕ and m>n. Observe that, for arbitrary ϵ>0,
(3.7)ωλ(xn,xm)≤ωλ/(m-n)(xn,xn+1)+ωλ/(m-n)(xn+1n,xn+2)+⋯+ωλ/(m-n)(xm-1,xm)≤ωλ/m(xn,xn+1)+ωλ/m(xn+1n,xn+2)+⋯+ωλ/m(xm-1,xm)≤∑n=n*+∞ωλ/m(xn,xn+1)<ϵ
for all m>n≥n* for some n*∈ℕ, and hence {xn}n∈ℕ is a Cauchy sequence. Then, the completeness of Xω implies that limn→+∞xn=x for some x∈Xω. Consequently, the sequence {Fxn}n∈ℕ converges to Fx, that is, limn→+∞Ωλ(Fxn,Fx)=0 for all λ>0. Since xn∈Fxn-1, we have
(3.8)0≤ωλ(xn+1,Fx)≤δλ(Fxn,Fx)≤Ωλ(Fxn,Fx)
which implies that ωλ(x,Fx)=0. Since Fx is closed, it follows that x∈Fx.Example 3.4.
LetX=[0,1], ω:(0,+∞)×X×X→[0,+∞] defined by ωλ(x,y)∶=(1/(1+λ))|x-y|. Clearly, Xω=[0,1] for any generator xι∈X. Now, we define a multivalued mapping F:Xω→𝒞ℬ(Xω) given by
(3.9)Fx={0,x+12}.
We have Ωλ(Fx,Fy)=(1/2(1+λ))|x-y|≤(1/2)ωλ(x,y). Therefore, F is a multivalued ω-contraction with contraction constant k=1/2, and we have that 0 and 1 are fixed points of F.Remark 3.5.
Note that our result does not assure the uniqueness of a fixed point, as illustrated in the above example.We next present the local version of Theorem3.3.Theorem 3.6.
LetXω be a complete modular metric space,
(3.10)Bω(x0,γ):={x∈Xω:ωλ(x,x0)≤γ,∀λ>0},
and F:ℬω(x0,γ)→𝒞ℬ(Xω). Suppose there exists 0≤k<1 for which
(3.11)Ωλ(Fx,Fy)≤kωλ(x,y)
for all x,y∈ℬω(x0,γ), λ>0 and
(3.12)Ωλ(Fx0,{x0})≤(1-k)γ
for all λ>0. Then, F has a fixed point in ℬω(x0,γ).Proof.
To prove this theorem, we only need to show thatℬω(x0,γ) is complete and Fx⊆ℬω(x0,γ). To show that ℬω(x0,γ) is complete, suppose that {xn}n∈ℕ is a Cauchy sequence in ℬω(x0,γ). Since Xω is complete, limn→+∞ωλ(xn,x)=0 for some x∈Xω for all λ>0. Since, for each n∈ℕ, xn∈ℬω(x0,γ), we get
(3.13)ωλ(x0,x)≤ωλ/2(x0,xn)+ωλ/2(xn,x)≤γ+ωλ/2(xn,x).
As n→+∞, we have ωλ(x0,x)≤γ. Therefore, ℬω(x0,γ) is complete.
Now, we prove the latter. For anyx∈ℬω(x0,γ), let y∈Fx. Observe that, for all λ>0,
(3.14)ωλ(y,x0)=δλ({y},{x0})≤δλ/3({y},Fx)+δλ/3(Fx,Fx0)+δλ/3(Fx0,{x0})≤Ωλ/3(Fx,Fx0)+Ωλ/3(Fx0,{x0})≤kωλ(x,x0)+(1-k)γ≤γ.
This implies that Fx⊆ℬω(x0,γ) for all x∈ℬω(x0,γ). Applying Theorem 3.3 to complete the proof.In the following theorem, we prove the existence of fixed points for a mapping introduced in 1969 by Kannan [15] in view of multivalued mappings in modular metric spaces.Theorem 3.7.
LetXω be a complete modular metric space and F:Xω→𝒞ℬ(Xω) a multivalued mapping such that there exists 0≤k<1/2 such that
(3.15)Ωλ(Fx,Fy)≤k[ω2λ(x,Fx)+ω2λ(y,Fy)]
for all x,y∈Xω and λ>0. Then, F has a fixed point in Xω.Proof.
Let the sequence{xn}n∈ℕ be constructed as in the proof of Theorem 3.3, so we get, for all λ>0,
(3.16)ωλ(xn,xn+1)≤Ωλ(Fxn-1,Fxn)+kn
for all n∈ℕ. Observe that
(3.17)ωλ(xn,xn+1)≤Ωλ(Fxn-1,Fxn)+kn≤k[ω2λ(xn-1,Fxn-1)+ω2λ(xn,Fxn)]+kn≤k[ωλ(xn-1,Fxn-1)+ωλ(xn,Fxn)]+kn≤k[ωλ(xn-1,Fxn-1)+ωλ(xn,xn+1)]+kn.
Further, set ξ:=k/(1-k)<1, we obtain
(3.18)ωλ(xn,xn+1)≤ξωλ(xn-1,xn)+kn1-k≤ξ2ωλ(xn-2,xn-1)+kn(1-k)2+kn(1-k)≤ξ2ωλ(xn-2,xn-1)+2kn(1-k)2⋮≤ξnωλ(x0,x1)+nξn.
As in the proof of Theorem 3.3, we conclude that {xn}n∈ℕ is a Cauchy sequence. The completeness of Xω implies that limn→+∞xn=x for some x∈Xω.
Now, we show thatx is a fixed point of F. Observe that
(3.19)ωλ(x,Fx)=δλ({x},Fx)≤δλ/2({x},Fxn)+δλ/2(Fxn,Fx)=ωλ/2(x,Fxn)+δλ/2(Fxn,Fx)≤ωλ/2(x,xn+1)+Ωλ/2(Fxn,Fx)≤ωλ/2(x,xn+1)+k[ωλ(xn,Fxn)+ωλ(x,Fx)].
Again, we have that
(3.20)ωλ(x,Fx)≤11-kωλ/2(x,xn+1)+k1-kωλ(xn,Fxn).
As n→+∞, we have ωλ(x,Fx)=0. Since Fx is closed, we have x∈Fx. Therefore, x is a fixed point of F in Xω.


## 4. Stability of Fixed Points

In this section, we discuss some stability of fixed points in Theorems3.3 and 3.7. In this context, Fix(F) will denote the set of all fixed points of a self-mapping F on Xω.Theorem 4.1.
LetXω be a complete modular metric space, and let F,G:Xω→𝒞ℬ(Xω) be two multivalued ω-contractions having the same contraction constant k. If, for any A,B∈𝒞ℬ(Xω), limλ↓0Ωλ(A,B)=ζ(A,B)<+∞, then Ωλ(Fix(F),Fix(G))≤(1-k)-1supx∈Xωζ(Fx,Gx).Proof.
Supposeλ>0, by Theorem 3.3, we can conclude that Fix(F)≠∅≠Fix(G). Let ϵ>0 be arbitrary, and let  γ>0 be such that γ∑n∈ℕnkn<1. For x0∈Fix(F), choose x1∈Gx0 such that
(4.1)ωλ(x0,x1)≤Ωλ(Fx0,Gx0)+ϵ.
By the multivalued ω-contractivity, it is possible to choose x2∈Gx1 such that
(4.2)ωλ(x1,x2)≤kωλ(x0,x1)+γϵk1-k.
Now, define a sequence {xn}n∈ℕ inductively by xn∈Gxn-1 and
(4.3)ωλ(xn,xn+1)≤kωλ(xn-1,xn)+γϵkn1-k.
Set η:=γϵ/(1-k), it follows that
(4.4)ωλ(xn,xn+1)≤kωλ(xn-1,xn)+ηkn≤k2ωλ(xn-1,xn-1)+2ηkn.
Inductively, we have that
(4.5)ωλ(xn,xn+1)≤knωλ(x0,x1)+nηkn.
Notice that ∑n∈ℕkn<+∞ and ∑n∈ℕnkn<+∞. Now, since
(4.6)∑n∈Nωλ(xn,xn+1)≤ωλ(x0,x1)∑n∈Nkn+η∑n∈Nnkn<+∞,
we can say that {xn}n∈ℕ is a Cauchy sequence. The completeness of Xω implies that limn→+∞xn=x for some x∈Xω. Since limn→+∞Ωλ(Gxn,Gx)=0 and xn∈Gxn-1, we get x∈Fix(G). Now, observe that
(4.7)ωλ(x0,x)≤ωλ/2n(x0,x1)+ωλ/2n(x1,x2)+⋯+ωλ/2n(xn-1,xn)+ωλ/2(xn,x)≤∑m∈Nωλ/2n(xm-1,xm)+ωλ/2(xn,x)≤ωλ/2n(x0,x1)∑m∈Nkm-1+η∑m∈N(m-1)km-1+ωλ/2(xn,x)≤(1-k)-1[ωλ/2n(x0,x1)+ϵ]+ωλ/2(xn,x)≤(1-k)-1[Ωλ/2n(Fx0,Gx0)+2ϵ]+ωλ/2(xn,x).
Since ωλ(x0,x)≥ωλ(x0,Fix(G)) and together with (4.18), we have, as n→+∞, that
(4.8)δλ(Fix⁡(F),Fix⁡(G))≤(1-k)-1[sup⁡x∈Fix⁡(F)⁡ζ(Fx,Gx)+2ϵ]≤(1-k)-1[sup⁡⁡x∈Xω⁡ζ(Fx,Gx)+2ϵ].
Similarly, we have
(4.9)δλ(Fix⁡(G),Fix⁡(F))≤(1-k)-1[sup⁡⁡x∈Xω⁡ζ(Fx,Gx)+2ϵ].
Since ϵ>0 is arbitrary, this completes the proof.Corollary 4.2.
LetXω be a complete modular metric space and Fn:Xω→𝒞ℬ(Xω), for n∈ℕ, multivalued ω-contractions having the same contraction constant k, and for any A,B∈𝒞ℬ(Xω), limλ↓0Ωλ(A,B)=ζ(A,B)<+∞. If limn→+∞ζ(Fnx,Fx)=0 uniformly for x∈Xω, then limn→+∞Ωλ(Fix(Fn),Fix(F))=0.Proof.
Letϵ>0 be arbitrary. Since limn→+∞ζ(Fnx,Fx)=0 uniformly for x∈Xω and λ>0, there exists n0∈ℕ such that
(4.10)sup⁡⁡x∈Xω⁡ζ(Fnx,Fx)<(1-k)ϵ
for all n∈ℕ with n≥n0. By Theorem 4.1, we have
(4.11)Ωλ(Fix⁡(Fn),Fix⁡(F))<ϵ
for all n∈ℕ with n≥n0 and λ>0.Likewise, we can deduce a stability theorem for fixed points in Theorem3.7.Theorem 4.3.
LetXω be a complete modular metric space, and let F,G:Xω→𝒞ℬ(Xω) be two multivalued mappings such that there exists 0≤k<1 such that
(4.12)Ωλ(Fx,Fy)≤k[ω2λ(x,Fx)+ω2λ(y,Fy)],Ωλ(Gx,Gy)≤k[ω2λ(x,Gx)+ω2λ(y,Gy)],
for all x,y∈Xω and λ>0. If, for any A,B∈𝒞ℬ(Xω), limλ↓0Ωλ(A,B)=ζ(A,B)<+∞, then Ωλ(Fix(F),Fix(G))≤(1-k)-1supx∈Xωζ(Fx,Gx).Proof.
Supposeλ>0, by Theorem 3.7, we can conclude that Fix(F)≠∅≠Fix(G). Let ϵ>0 be arbitrary, and let  γ>0 be such that γ∑n∈ℕnkn<1. For x0∈Fix(F), choose x1∈Gx0 such that
(4.13)ωλ(x0,x1)≤Ωλ(Fx0,Gx0)+γϵ1-k.
It is possible to choose x2∈Gx1 such that
(4.14)ωλ(x1,x2)≤Ωλ(Gx0,Gx1)+γϵ1-kk.
By induction, we can construct a sequence {xn}n∈ℕ such that
(4.15)ωλ(xn,xn+1)≤Ωλ(Gxn-1,Gxn)+γϵ1-kkn.
Observe that
(4.16)ωλ(xn,xn+1)≤Ωλ(Fxn-1,Fxn)+γϵ1-kkn≤k[ω2λ(xn-1,Fxn-1)+ω2λ(xn,Fxn)]+γϵ1-kkn≤k[ωλ(xn-1,Fxn-1)+ωλ(xn,Fxn)]+γϵ1-kkn≤k[ωλ(xn-1,Fxn-1)+ωλ(xn,xn+1)]+γϵ1-kkn.
Further, set ξ:=k/(1-k)<1 and η:=γϵ/(1-k), we obtain
(4.17)ωλ(xn,xn+1)≤ξωλ(xn-1,xn)+γϵ(1-k)2kn≤ξ2ωλ(xn-2,xn-1)+2γϵ(1-k)3kn⋮≤ξnωλ(x0,x1)+nηξn.
Similar to the proof of Theorem 4.1, we conclude that {xn}n∈ℕ is a Cauchy sequence. The completeness of Xω implies that {xn}n∈ℕ converges to some limit x∈Xω. We can further see that x∈Fix(G). Now, observe that
(4.18)ωλ(x0,x)≤ωλ/2n(x0,x1)+ωλ/2n(x1,x2)+⋯+ωλ/2n(xn-1,xn)+ωλ/2(xn,x)≤∑m∈Nωλ/2n(xm-1,xm)+ωλ/2(xn,x)≤ωλ/2n(x0,x1)∑m∈Nkm-1+η∑m∈N(m-1)km-1+ωλ/2(xn,x)≤(1-k)-1[ωλ/2n(x0,x1)+ϵ]+ωλ/2(xn,x)≤(1-k)-1[Ωλ/2n(Fx0,Gx0)+2ϵ]+ωλ/2(xn,x).
Since ωλ(x0,x)≥ωλ(x0,Fix(G)) and together with (4.18), we have, as n→+∞, that
(4.19)δλ(Fix⁡(F),Fix⁡(G))≤(1-k)-1[sup⁡x∈Fix⁡(F)⁡ζ(Fx,Gx)+2ϵ]≤(1-k)-1[sup⁡⁡x∈Xω⁡ζ(Fx,Gx)+2ϵ].
Similarly, we have
(4.20)δλ(Fix⁡(G),Fix⁡(F))≤(1-k)-1[sup⁡⁡x∈Xω⁡ζ(Fx,Gx)+2ϵ].
Since ϵ>0 is arbitrary, this completes the proof.Corollary 4.4.
LetXω be a complete modular metric space, and let  Fn:Xω→𝒞ℬ(Xω), for n∈ℕ, be multivalued mappings such that there exists 0≤k<1 such that
(4.21)Ωλ(Fnx,Fny)≤k[ω2λ(x,Fnx)+ω2λ(y,Fny)]
for all x,y∈Xω and λ>0. Suppose for any A,B∈𝒞ℬ(Xω), limλ↓0Ωλ(A,B)=ζ(A,B)<+∞. If limn→+∞ζ(Fnx,Fx)=0 uniformly for x∈Xω, then limn→+∞Ωλ(Fix(Fn),Fix(F))=0.Proof.
Letϵ>0 be arbitrary. Since limn→+∞ζ(Fnx,Fx)=0 uniformly for x∈Xω and λ>0, there exists n0∈ℕ such that
(4.22)sup⁡⁡x∈Xω⁡ζ(Fnx,Fx)<(1-k)ϵ
for all n∈ℕ with n≥n0. By Theorem 4.3, we have
(4.23)Ωλ(Fix⁡(Fn),Fix⁡(F))<ϵ
for all n∈ℕ with n≥n0 and λ>0.



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*Source: 503504-2012-04-10.xml*