In this article, the concept of a Hausdorff fuzzy b-metric space is introduced. The new notion is used to establish some fixed point results for multivalued mappings in G-complete fuzzy b-metric spaces satisfying a suitable requirement of contractiveness. An illustrative example is formulated to support the results. Eventually, an application for the existence of a solution for an integral inclusion is established which involves showing the materiality of the obtained results. These results are more general and some theorems proved by of Shehzad et al. are their special cases.
Citation: Samina Batul, Faisar Mehmood, Azhar Hussain, Dur-e-Shehwar Sagheer, Hassen Aydi, Aiman Mukheimer. Multivalued contraction maps on fuzzy b-metric spaces and an application[J]. AIMS Mathematics, 2022, 7(4): 5925-5942. doi: 10.3934/math.2022330
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In this article, the concept of a Hausdorff fuzzy b-metric space is introduced. The new notion is used to establish some fixed point results for multivalued mappings in G-complete fuzzy b-metric spaces satisfying a suitable requirement of contractiveness. An illustrative example is formulated to support the results. Eventually, an application for the existence of a solution for an integral inclusion is established which involves showing the materiality of the obtained results. These results are more general and some theorems proved by of Shehzad et al. are their special cases.
Fixed point theory has a vital role in mathematics and applied sciences. Also, this theory has lot of applications in differential equations and integral equations to guarantee the existence and uniqueness of the solutions [1,2]. The Banach contraction principle [3] has an imperative role in fixed point theory. Since after the appearance of this principle, it has become very popular and there has been a lot of activity in this area. On the other hand, to establish Banach contraction principle in a more general structure, the notion of a metric space was generalized by Bakhtin [4] in 1989 by introducing the idea of a b-metric space. Afterwards, the same idea was further investigated by Czerwik [5] to establish different results in b-metric spaces. The study of b-metric spaces holds a prominent place in fixed point theory. Banach contraction principle is generalized in many ways by changing the main platform of the metric space [6,7,8,9].
Zadeh [10] introduced the notion of a fuzzy set theory to deal with the uncertain states in daily life. Motivated by the concept, Kramosil and Michálek [11] defined the idea of fuzzy metric spaces. Grabiec [12] gave contractive mappings on a fuzzy metric space and extended fixed point theorems of Banach and Edelstein in such spaces. Successively, George and Veeramani [13] slightly altered the concept of a fuzzy metric space introduced by Kramosil and Michálek [11] and then attained a Hausdorff topology and a first countable topology on it. Numerous fixed point theorems have been constructed in fuzzy metric spaces. For instance, see [14,15,16,17,18,19,20].
Nǎdǎban [21] studied the notion of a fuzzy b-metric space and proved some results. Rakić et al. [22] (see also [23]) proved some new fixed point results in b-fuzzy metric spaces. The notion of a Hausdorff fuzzy metric on compact sets is introduced in [24] and recently studied by Shahzad et al. [25] to establish fixed point theorems for multivalued mappings in complete fuzzy metric spaces. In this paper, we use the idea of a fuzzy b-metric space and establish some fixed point results for multivalued mappings in Hausdorff fuzzy b-metric spaces. Some fixed point theorems are also derived from these results. Finally, we investigate the applicability of the obtained results to integral equations.
Throughout the article, fuzzy metric space and fuzzy b-metric space are denoted by FMS and FBMS, respectively.
Bakhtin [4] defined the notion of a b-metric space as follows:
Definition 2.1. [4] Let Ω be a non-empty set. For any real number b≥1, a function db:Ω×Ω⟶R is called a b-metric if it satisfies the following properties for all ζ1,ζ2,ζ3∈Ω:
BM1 : db(ζ1,ζ2)≥0;
BM2 : db(ζ1,ζ2)=0 if and only if ζ1=ζ2;
BM3 : db(ζ1,ζ2)=db(ζ2,ζ1) for all ζ1,ζ2∈Ω;
BM4 : db(ζ1,ζ3)≤b[db(ζ1,ζ2)+db(ζ2,ζ3)].
The pair (Ω,db) is called a b-metric space.
Definition 2.2. [26] A binary operation ∗:[0,1]×[0,1]→[0,1] is called a continuous t-norm if it satisfies the following conditions:
(1) ∗ is associative and commutative;
(2) ∗ is continuous;
(3) ζ∗1=ζ for all ζ∈[0,1];
(4) ζ1∗ζ2≤ζ3∗ζ4whereverζ1≤ζ3andζ2≤ζ4, for all ζ1,ζ2,ζ3,ζ4∈[0,1].
Example 2.1. Define a mapping ∗:[0,1]×[0,1]→[0,1] by
ζ1∗ζ2=ζ1ζ2forζ1,ζ2∈[0,1]. |
It is then obvious that ∗ is a continuous t-norm, known as the product norm.
George and Veermani [13] defined a fuzzy metric space as follows:
Definition 2.3. [13] Consider a nonempty set Ω, then (Ω,F,∗) is a fuzzy metric space if ∗ is a continuous t-norm and F is a fuzzy set on Ω×Ω×[0,+∞) satisfying the following for all ζ1,ζ2,ζ3∈Ω and α,β>0:
[F1]: F(ζ1,ζ2,α)>0;
[F2]: F(ζ1,ζ2,α)=1 if and only if ζ1=ζ2;
[F3]: F(ζ1,ζ2,α)=F(ζ2,ζ1,α);
[F4]: F(ζ1,ζ3,α+β)≥F(ζ1,ζ2,α)∗F(ζ2,ζ3,β);
[F5]: F(ζ1,ζ2,.):[0,+∞)→[0,1] is continuous.
In [27], the idea of a fuzzy b-metric space is given as:
Definition 2.4. [27] Let Ω≠ϕ be a set, b≥1 be a real number and ∗ be a continuous t-norm. A fuzzy set Fb on Ω×Ω×[0,+∞) is called a fuzzy b-metric on Ω if for all ζ1,ζ2,ζ3∈Ω, the following conditions hold:
[Fb1]: Fb(ζ1,ζ2,α)>0;
[Fb2]: Fb(ζ1,ζ2,α)=1,forallα>0 if and only if ζ1=ζ2;
[Fb3]: Fb(ζ1,ζ2,α)=Fb(ζ2,ζ1,α);
[Fb4]: Fb(ζ1,ζ3,b(α+β))≥Fb(ζ1,ζ2,α)∗Fb(ζ2,ζ3,β) forallα,β≥0;
[Fb5]: Fb(ζ1,ζ2,.):(0,+∞)→[0,1] is left continuous.
Example 2.2. Let (Ω,db) be a b-metric space. Define a mapping Fb:Ω×Ω×[0,+∞)→[0,1] by
Fb(ζ1,ζ2,α)={αα+db(ζ1,ζ2)ifα>00ifα=0. |
Then (Ω,Fb,∧) is a fuzzy b-metric space, where
ζ1∧ζ2=min{ζ1,ζ2}forallζ1,ζ2∈[0,1]. |
∧ is a t-norm, known as the minimum t-norm.
Following Grabiec [12], we extend the idea of a G-Cauchy sequence and the notion of completeness in the FBMS as follows:
Definition 2.5. Let (Ω,Fb,∗) be a FBMS.
1) A sequence {ζn} in Ω is said to be a G-Cauchy sequence if limn→+∞Fb(ζn,ζn+q,α)=1 for α>0 and q>0.
2) A FBMS in which every G-Cauchy sequence is convergent is called a G-complete FBMS.
Similarly, for a FBMS (Ω,Fb,∗), a sequence {ζn} in Ω is said to be convergent if there exits ζ∈Ω such that for all α>0,
limn→+∞Fb(ζn,ζ,α)=1. |
Definition 2.6. [25] Let ℧≠ϕ be a subset of a FBMS (Ω,F,∗) and α>0, then the fuzzy distance F of an element ϱ1∈Ω and the subset ℧⊂Ω is
F(ϱ1,℧,α)=sup{F(ϱ1,ϱ2,α):ϱ2∈℧}. |
Note that F(ϱ1,℧,α)=F(℧,ϱ1,α).
Lemma 2.1. [28] If Λ∈CB(Ω), then ζ1∈Λ if and only if F(Λ,ζ1,α)=1 for all α>0, where CB(Ω) is the collection of closed bounded subsets of Ω.
Definition 2.7. [25] Let (Ω,F,∗) be a FMS. Define a function ΘF on ^C0(Ω)×^C0(Ω)×(0,+∞) by
ΘF(Λ,℧,α)=min{ infϱ1∈ΛF(ϱ1,℧,α),infϱ2∈℧F(Λ,ϱ2,α)} |
for all Λ,℧∈^C0(Ω) and α>0, where ^C0(Ω) is the collection of all nonemty compact subsets of Ω.
Lemma 2.2. [29] Let (Ω,F,∗) be a complete FMS and F(ζ1,ζ2,kα)≥F(ζ1,ζ2,α) for all ζ1,ζ2∈Ω,k∈(0,1) and α>0 then ζ1=ζ2.
Lemma 2.3. [25] Let (Ω,F,∗) be a complete FMS such that (^C0,ΘF,∗) is a Hausdorff FMS on ^C0. Then for all Λ,℧∈^C0, for each ζ∈Λ and for α>0, there exists ϱζ∈℧ so that F(ζ,℧,α)=F(ζ,ϱζ,α) then
ΘF(Λ,℧,α)≤F(ζ,ϱζ,α). |
The notion of a Hausdorff FMS in Definition 2.6 of [25] can be extended naturally for a Hausdorff FBMS on ^C0 as follows:
Definition 2.8. Let (Ω,Fb,∗) be a FBMS. Define a function ΘFb on ^C0(Ω)×^C0(Ω)×(0,+∞) by
ΘFb(Λ,℧,α)=min{ infζ∈ΛFb(ζ,℧,α),infϱ∈℧Fb(Λ,ϱ,α)} |
for all Λ,℧∈^C0(Ω) and α>0.
This section deals with the idea of Hausdorff FBMS and certain new fixed point results in a fuzzy FBMS. Note that, one can easily extend Lemma 2.1 to 2.3 in the setting of fuzzy b-metric spaces.
Lemma 3.1. If Λ∈C℧(Ω), then ζ∈Λ if and only if Fb(Λ,ζ,α)=1forallα>0.
Proof. Since
Fb(Λ,ζ,α)=sup{Fb(ζ,ϱ,α):ϱ∈Λ}=1, |
there exists a sequence {ϱn}⊂Λ such that Fb(ζ,ϱn,α)>1−1n. Letting n→+∞, we get ϱn→ζ. From Λ∈C℧(Ω), it follows that ζ∈Λ. Conversely, if ζ∈Λ, we have
Fb(Λ,ζ,α)=sup{Fθ(ζ,ϱ,α):ϱ∈Λ}>Fb(ζ,ζ,α)=1. |
Again, due to [17], the following fact follows from [Fb5].
Lemma 3.2. Let (Ω,Fb,∗) be a G-complete FBMS. If for two elements ζ,ϱ∈Ω and for a numberk<1
Fb(ζ,ϱ,kα)≥Fb(ζ,ϱ,α), |
then ζ=ϱ.
Lemma 3.3. Let (Ω,Fb,∗) be a G-complete FBMS, such that (^C0,ΘFb,∗) is a Hausdorff FBMS on ^C0. Then for all Λ,℧∈^C0, foreach ζ∈Λ and for α>0 there exists ϱζ∈℧, satisfying Fb(ζ,℧,α)=Fb(ζ,ϱζ,α) also
ΘFb(Λ,℧,α)≤Fb(ζ,ϱζ,α). |
Proof. If
ΘFb(Λ,℧,α)=infζ∈ΛFb(ζ,℧,α), |
then
ΘFb(Λ,℧,α)≤Fb(ζ,℧,α). |
Since for each ζ∈Λ, there exists ϱζ∈℧ satisfying
Fb(ζ,℧,α)=Fb(α,ϱζ,α). |
Hence,
ΘFb(Λ,℧,α)≤Fb(ζ,ϱζ,α). |
Now, if
ΘFb(Λ,℧,α)=infϱ∈℧Fb(Λ,ϱ,α)≤infζ∈ΛFb(ζ,℧,α)≤Fb(ζ,℧,α)=Fb(ζ,ϱζ,α), |
this implies
ΘFb(Λ,℧,α)≤Fb(ζ,ϱζ,α) |
for some ϱζ∈℧. Hence, in both cases, the result is proved.
Theorem 3.1. Let (Ω,Fb,∗) be a G-complete FBMS with b⩾1 and ΘFb be aHausdorff FBMS. Let S:Ω→^C0(Ω) be a multivalued mapping satisfying
ΘFb(Sζ,Sϱ,kα)≥Fb(ζ,ϱ,α) | (3.1) |
for all ζ,ϱ∈Ω, where bk<1,then S has a fixed point.
Proof. For a0∈Ω, we choose a sequence {ζn} in Ω as follows: Let a1∈Ω such that a1∈Sa0. By using Lemma 6, we can choose a2∈Sa1 such that
Fb(a1,a2,α)⩾ΘFb(Sa0,Sa1,α)forallα>0. |
By induction, we have an+1∈San satisfying
Fb(an,an+1,α)⩾ΘFb(San−1,San,α)foralln∈N. |
Now, by (3.1) together with Lemma 3.3, we have
Fb(an,an+1,α)≥ΘFb(San−1,San,α)≥Fb(an−1,an,αk)≥ΘFb(San−2,San−1,αk)≥Fb(an−2,an−1,αk2)⋮≥ΘFb(Sa0,Sa1,αkn−1)≥Fb(a0,a1,αkn). | (3.2) |
For any q∈N, writing α=αqb+α(q−1)qb and using [Fb4] to get
Fb(an,an+q,α)≥Fb(an,an+1,αqb)∗Fb(an+1,an+q,(q−1)αqb). |
Again, writing (q−1)αqb=αqb+α(q−2)qb together with [Fb4], we have
Fb(an,an+q,α)≥Fb(an,an+1,αqb)∗Fb(an+1,an+2,αqb2)∗Fb(an+2,an+q,(q−2)αqb2). |
Continuing in the same way and using [Fb4] repeatedly for (q−2) more steps, we obtain
Fb(an,an+q,α)≥Fb(an,an+1,αqb)∗Fb(an+1,an+2,αqb2)∗…∗Fb(an+q−1,an+q,αqbq). |
Using (3.2) and [Fb5], we get
Fb(an,an+q,α)≥Fb(a0,a1,αqbkn)∗Fb(a0,a1,αq(b)2kn+1)∗Fb(a0,a1,αq(b)3kn+2)∗…∗Fb(a0,a1,αq(b)qkn+q−1). |
Consequently,
Fb(an,an+q,α)≥Fb(a0,a1,αq(bk)kn−1)∗Fb(a0,a1,αq(bk)2kn−1)∗Fb(a0,a1,αq(bk)3kn−1)∗…∗Fb(a0,a1,αq(bk)qkn−1). |
Since for all n,q∈N, we have bk<1, taking limit as n→+∞, we get
limn→+∞Fb(an,an+q,α)=1∗1∗…∗1=1. |
Hence, {an} is a G-Cauchy sequence. Then, the G-completeness of Ω implies that there exists z∈Ω such that
Fb(z,Sz,α)≥Fb(z,an+1,α2b)∗Fb(an+1,Sz,α2b)≥Fb(z,an+1,α2b)∗ΘFb(San,Sz,α2b)≥Fb(z,an+1,α2b)∗Fb(an,z,α2bk)⟶1asn→+∞. |
By Lemma 3.1, we have z∈Sz. Hence, z is a fixed point for S.
Example 3.1. Let Ω=[0,1] and Fb(ζ,ϱ,α)=αα+(ζ−ϱ)2.
It is easy to verify that (Ω,Fb,∗) is a G-complete FBMS with b≥1.
For k∈(0,1), define a mapping S:Ω→^C0(Ω) by
S(ζ)={{0}ifζ=0{0,√kζ2}otherwise. |
In the case ζ=ϱ, we have
ΘFb(Sζ,Sϱ,kα)=1=Fb(ζ,ϱ,α). |
For ζ≠ϱ, we have the following cases:
If ζ=0 and ϱ∈(0,1], we have
ΘFb(S(0),S(ϱ),kα)=min{ infa∈S(0)Fb(a,S(ϱ),kα),infb∈S(ϱ)Fb(S(0),b,kα)}=min{infa∈S(0)Fb(a,{0,√kϱ2},kα),infb∈S(ϱ)Fb({0},b,kα)}=min{inf{Fb(0,{0,√kϱ2},kα)},inf{Fb({0},0,kα),Fb({0},√kϱ2,kα)}}=min{inf{sup{Fb(0,0,kα),Fb(0,√kϱ2,kα)}},inf{Fb(0,0,kα),Fb(0,√kϱ2,kα)}}=min{inf{sup{1,αα+ϱ24}},inf{1,αα+ϱ24}}=min{inf{1},αα+ϱ24}=min{1,αα+ϱ24}=αα+ϱ24. |
It follows that ΘFb(S(0),S(ϱ),kα)>Fb(0,ϱ,α)=αα+ϱ2.
If ζ and ϱ∈(0,1], an easy calculation with either possibility of supremum and infimum, yield that
ΘFb(S(ζ),S(ϱ),kα)=min{sup{αα+ζ24,αα+(ζ−ϱ)24},sup{αα+ϱ24,αα+(ζ−ϱ)24}}≥αα+(ζ−ϱ)24>αα+(ζ−ϱ)2=Fb(ζ,ϱ,α). |
Thus, for all cases, we have
ΘFb(Sζ,Sϱ,kα)≥Fb(ζ,ϱ,α). |
Hence, all the conditions of Theorem 3.1 are satisfied and 0 is a fixed point of S.
Theorem 3.2. Let (Ω,Fb,∗) be a G-complete FBMS with b⩾1 and ΘFb be a Hausdorff FBMS. Let S:Ω→^C0(Ω) be a multivalued mapping which satisfies
ΘFb(Sζ,Sϱ,kα)≥min{Fb(ϱ,Sϱ,α)[1+Fb(ζ,Sϱ,α)]1+Fb(ζ,ϱ,α),Fb(ζ,ϱ,α)} | (3.3) |
for all ζ,ϱ∈Ω, where bk<1,then S has a fixed point.
Proof. In the same way as in Theorem 3.1 for a0∈Ω, we choose a sequence {an} in Ω as follows: Let a1∈Ω such that a1∈Sa0. By Lemma 2.3, we can choose a2∈Sa1 such that
Fb(a1,a2,α)⩾ΘFb(Sa0,Sa1,α)forallα>0. |
By induction, we have an+1∈San satisfying
Fb(an,an+1,α)⩾ΘFb(San−1,San,α)foralln∈N. |
Now, by (3.3) together with Lemma 3.3, we have
Fb(an,an+1,α)≥ΘFb(San−1,San,α)≥min{Fb(an,San,αk)[1+Fb(an−1,San−1,αk)]1+Fb(an−1,an,αk),Fb(an−1,an,αk)}≥min{Fb(an,an+1,αk)[1+Fb(an−1,an,αk)]1+Fb(an−1,an,αk),Fb(an−1,an,αk)}≥min{Fb(an,an+1,αk),Fb(an−1,an,αk)}. | (3.4) |
If
min{Fb(an,an+1,αk),Fb(an−1,an,αk)}=Fb(an,an+1,αk), |
then (3.4) implies
Fb(an,an+1,α)≥Fb(an,an+1,αk). |
Then nothing to prove by Lemma 3.2. If
min{Fb(an,an+1,αk),Fb(an−1,an,αk)}=Fb(an−1,an,αk), |
then from (3.4) we have
Fb(an,an+1,α)≥Fb(an−1,an,αk)⩾…⩾Fb(a0,a1,αkn). |
By adopting the same procedure as in Theorem 3.1 after inequality (3.2), we can complete the proof.
Remark 3.1. By taking b=1 in Theorem 3.2, Theorem 2.1 of [25] can be obtained.
Theorem 3.3. Let (Ω,Fb,∗) be a G-complete FBMS (with b⩾1) and ΘFb be a Hausdorff FBMS. Let S:Ω→ˆC0(Ω) be a multivalued map which satisfies
ΘFb(Sζ,Sϱ,kα)≥min{Fb(ϱ,Sϱ,α)[1+Fb(ζ,Sζ,α)+Fb(ϱ,Sζ,α)]2+Fb(ζ,ϱ,α),Fb(ζ,ϱ,α)} | (3.5) |
for all ζ,ϱ∈Ω, where bk<1,then S has a fixed point.
Proof. Starting same way as in Theorem 3.1, we have
Fb(a1,a2,α)⩾ΘFb(Sa0,Sa1,α)forallα>0. |
By induction, we have an+1∈San satisfying
Fb(an,an+1,α)⩾ΘFb(San−1,San,α)foralln∈N. |
Now, by (3.5) together with Lemma 3.3, we have
Fb(an,an+1,α)≥ΘFb(San−1,San,α)≥min{Fb(an,San,αk)[1+Fb(an−1,San−1,αk)+Fb(an,San−1,αk)]2+Fb(an−1,an,αk),Fb(an−1,an,αk)}≥min{Fb(an,an+1,αk)[1+Fb(an−1,an,αk)+Fb(an,an,αk)]2+Fb(an−1,an,αk),Fb(an−1,an,αk)}≥min{Fb(an,an+1,αk)[1+Fb(an−1,an,αk)+1]2+Fb(an−1,an,αk),Fb(an−1,an,αk)}≥min{Fb(an,an+1,αk)[2+Fb(an−1,an,αk)]2+Fb(an−1,an,αk),Fb(an−1,an,αk)}≥min{Fb(an,an+1,αk),Fb(an−1,an,αk)}. | (3.6) |
If
min{Fb(an,an+1,αk),Fb(an−1,an,αk)}=Fb(an,an+1,αk), |
then (3.6) implies
Fb(an,an+1,α)≥Fb(an,an+1,αk). |
Then nothing to prove by Lemma 3.2.
If
min{Fb(an,an+1,αk),Fb(an−1,an,αk)}=Fb(an−1,an,αk), |
then from (3.6), we have
Fb(an,an+1,α)≥Fb(an−1,an,αk)⩾…⩾Fb(a0,a1,αkn). |
By adopting the same procedure as in Theorem 3.1 after inequality (3.2), we can complete the proof.
Next, a corollary of Theorem 3.3 is given.
Corollary 3.1. Let (Ω,F,∗) be a G-complete FMS and ΘF be a Hausdorff FMS. Let S:Ω→ˆC0(Ω) be a multivalued mapping satisfying
ΘF(Sζ,Sϱ,kα)≥min{F(ϱ,Sϱ,α)[1+F(ζ,Sζ,α)+F(ϱ,Sζ,α)]2+F(ζ,ϱ,α),F(ζ,ϱ,α)} |
for all ζ,ϱ∈Ω, where 0<k<1, then S has a fixed point.
Proof. Taking b=1 in Theorem 3.3, one can complete the proof.
Theorem 3.4. Let (Ω,Fb,∗) be a G-complete FBMS with b⩾1 and ΘFb be a Hausdorff FBMS. Let S:Ω→ˆC0(Ω) be a multivalued map which satisfies
ΘFb(Sζ,Sϱ,kα)≥min{Fb(ζ,Sζ,α)[1+Fb(ϱ,Sϱ,α)]1+Fb(Sζ,Sϱ,α),Fb(ζ,Sϱ,α)[1+Fb(ζ,Sζ,α)]1+Fb(ζ,ϱ,α),Fb(ζ,Sζ,α)[2+Fb(ζ,Sϱ,α)]1+Fb(ζ,Sϱ,α)+Fb(ϱ,Sζ,α),Fb(ζ,ϱ,α)} | (3.7) |
for all ζ,ϱ∈Ω, where bk<1,then S has a fixed point.
Proof. For a0∈Ω, we choose a sequence {xn} in Ω as follows: Let a1∈Ω such that a1∈Sa0. By using Lemma 2.3, we can choose a2∈Sa1 such that
Fb(a1,a2,α)⩾ΘFb(Sa0,Sa1,α)forallα>0. |
By induction, we have an+1∈San satisfying
Fb(an,an+1,α)⩾ΘFb(San−1,San,α)foralln∈N. |
Now, by (3.7) together with 3.3, we have
Fb(an,an+1,α)≥ΘFb(San−1,San,α)≥min{Fb(an−1,San−1,αk)[1+Fb(an,San,αk)]1+Fb(San−1,San,αk),Fb(an,San,αk)[1+Fb(an−1,San−1,αk)]1+Fb(an−1,an,αk),Fb(an−1,San−1,αk)[2+Fb(an−1,San,αk)]1+Fb(an−1,San,αk)+Fb(an,San−1,αk),Fb(an−1,an,αk)} |
≥min{Fb(an−1,an,αk)[1+Fb(an,an+1,αk)]1+Fb(an,an+1,αk),Fb(an,an+1,αk)[1+Fb(an−1,an,αk)]1+Fb(an−1,an,αk),Fb(an−1,an,αk)[2+Fb(an−1,an+1,αk)]1+Fb(an−1,an+1,αk)+Fb(an,an,αk),Fb(an−1,an,αk)} |
Fb(an,an+1,α)≥min{Fb(an,an+1,αk),Fb(an−1,an,αk)}. | (3.8) |
If
min{Fb(an,an+1,αk),Fb(an−1,an,αk)}=Fb(an,an+1,αk), |
so (3.8) implies
Fb(an,an+1,α)≥Fb(an,an+1,αk). |
Then nothing to prove by Lemma 3.2. While, if
min{Fb(an,an+1,αk),Fb(an−1,an,αk)}=Fb(an−1,an,αk), |
then from (3.6) we have
Fb(an,an+1,α)≥Fb(an−1,an,αk)⩾…⩾Fb(a0,a1,αkn). |
By adopting the same procedure as in Theorem 3.1 after inequality (3.2) we can complete the proof.
The same result in fuzzy metric spaces is stated as follows.
Corollary 3.2. Let (Ω,F,∗) be a G-complete FMS and ΘF be a Hausdorff FMS. Let S:Ω→^C0(Ω) be a multivalued mapping satisfying
ΘF(Sζ,Sϱ,kα)≥min{F(ζ,Sζ,α)[1+F(ϱ,Sϱ,α)]1+F(Sζ,Sϱ,α),F(ϱ,Sϱ,α)[1+F(ζ,Sζ,α)]1+F(ζ,ϱ,α),F(ζ,Sζ,α)[2+F(ζ,Sϱ,α)]1+F(ζ,Sϱ,α)+F(ϱ,Sζ,α),F(ζ,ϱ,α)} |
for all ζ,ϱ∈Ω,then S has a fixed point.
Proof. Taking b=1 in Theorem 3.4, one can complete the proof.
This section is about the construction of some fixed point results involving integral inequalities as consequences of our results. Define a function τ:[0,+∞)→[0,+∞) by
τ(α)=∫α0ψ(α)dαforallα>0, | (4.1) |
where τ(α) is a non-decreasing and continuous function. Moreover, ψ(α)>0 for α>0 and ψ(α)=0 if and only if α=0.
Theorem 4.1. Let (Ω,Fb,∗) be a complete fuzzy b-metric space and ΘFb be a Hausdorff fuzzy b-metric space. Let S:Ω→ˆC0(Ω) be a multivalued mapping satisfying
∫ΘFb(Sζ,Sϱ,kα)0ψ(α)dα≥∫Fb(ζ,ϱ,α)0ψ(α)dα | (4.2) |
for all ζ,ϱ∈Ω, where bk<1, then S has a fixed point.
Proof. Taking (4.1) in account, (4.2) implies that
τ(ΘFb(Sζ,Sϱ,kα))⩾τ(Fb(ζ,ϱ,α)). |
Since τ is continuous and non-decreasing, we have
ΘFb(Sζ,Sϱ,kα)⩾Fb(ζ,ϱ,α). |
The rest of the proof follows immediately from Theorem 3.1.
A more general form of Theorem 4.1 can be stated as an immediate consequence of Theorem 3.2.
Theorem 4.2. Let (Ω,Fb,∗) be a complete FBMS and ΘFb be a Hausdorff FBMS. Let S:Ω→ˆC0(Ω) be a multivalued mapping satisfying
∫ΘFb(Sζ,Sϱ,kα)0ψ(α)dα≥∫β(ζ,ϱ,α)0ψ(α)dα, | (4.3) |
where
β(ζ,ϱ,α)=min{Fb(ϱ,Sϱ,α)[1+Fb(ζ,Sϱ,α)]1+Fb(ζ,ϱ,α),Fb(ζ,ϱ,α)} |
for all ζ,ϱ∈Ω, where bk<1, then S has a fixed point.
Proof. Taking (4.1) in account, (4.3) implies that
τ(ΘFb(Sζ,Sϱ,kα))⩾τ(β(ζ,ϱ,α). |
Since τ is continuous and non-decreasing, we have
ΘFb(Sζ,Sϱ,kα)⩾β(ζ,ϱ,α). |
The rest of the proof follows immediately from Theorem 3.2.
Remark 4.1. By taking b=1 in Theorem 4.2, Theorem 3.1 of [25] can be obtained.
Theorem 4.3. Let (Ω,Fb,∗) be a complete FBMS and ΘFb be a Hausdorff FBMS. Let S:Ω→ˆC0(Ω) be a multivalued mapping satisfying
∫ΘFb(Sζ,Sϱ,kα)0ψ(α)dα≥∫β(ζ,ϱ,α)0ψ(α)dα, |
where
β(ζ,ϱ,α)=min{Fb(ϱ,Sϱ,α)[1+Fb(ζ,Sζ,α)+Fb(ϱ,Sζ,α)]2+Fb(ζ,ϱ,α),Fb(ζ,ϱ,α)} |
for all ζ,ϱ∈Ω, where bk<1, then S has a fixed point.
Note that if β(ζ,ϱ,α)=Fb(ζ,ϱ,α), then the above result follows from Theorem 4.1. Similar results on integral inequalities can be obtained as a consequence of Theorem 3.4.
Theorem 4.4. Let (Ω,Fb,∗) be a complete FBMS and ΘFb be a Hausdorff FBMS. Let S:Ω→ˆC0(Ω) be a multivalued mapping satisfying
∫ΘFb(Sζ,Sϱ,kα)0ψ(α)dα≥∫γ(ζ,ϱ,α)0ψ(α)dα, |
where
γ(ζ,ϱ,α)=min{Fb(ζ,Sζ,α)[1+Fb(ϱ,Sϱ,α)]1+Fb(Sζ,Sϱ,α),Fb(ϱ,Sϱ,α)[1+Fb(ζ,Sζ,α)]1+Fb(ζ,ϱ,α),Fb(ζ,Sζ,α)[2+Fb(ζ,Sϱ,α)]1+Fb(ζ,Sϱ,α)+Fb(ϱ,Sζ,α),Fb(ζ,ϱ,α)} |
for all ζ,ϱ∈Ω, where bk<1, then S has a fixed point.
Nonlinear integral equations arise in a variety of fields of physical science, engineering, biology, and applied mathematics [31,32]. This theory in abstract spaces is a rapidly growing field with lot of applications in analysis as well as other branches of mathematics [33].
Fixed point theory is a valuable tool for the existence of a solution of different kinds of integral as well as differential inclusions, such as [33,34,35]. Many authors provided a solution of different integral inclusions in this context, for instance see [30,36,37,38,39,40]. In this section, a Volterra-type integral inclusion as an application of Theorem 3.1 is studied.
Consider Ω=C([0,1],R) as the space of all continuous functions defined on [0,1] and define the G-complete fuzzy b-metric on Ω by
Fb(ξ,ϱ,α)=e−supε∈[0,1]|ξ(ε)−ϱ(ε)|2α |
for all α>0 and ξ,ϱ∈Ω.
Consider the integral inclusion:
ξ(ε)∈∫u0G(ε,σ,ξ(σ))dσ+h(ε)forallε,σ∈[0,1]andh,ξ∈C([0,1], | (5.1) |
where G:[0,1]×[0,1]×R→Pcv(R) is a multivalued continuous function.
For the above integral inclusion, we define a multivalued operator S:Ω→^C0(Ω) by
S(ξ(ε))={w∈Ω:w∈∫u0G(ε,σ,ξ(σ))dσ+h(ε),ε∈[0,1]}. |
The next result proves the existence of a solution of the integral inclusion (5.1).
Theorem 5.1. Let S:Ω→^C0(Ω) be the multivalued integral operator given by
S(ξ(ε))={w∈Ω:w∈∫ε0G(ε,σ,ξ(σ))dσ+h(ε),ε∈[0,1]}. |
Suppose the following conditions are satisfied:
1) G:[0,1]×[0,1]×R→Pcv(R) is such that G(ε,σ,ξ(σ)) is lower semi-continuous in [0,1]×[0,1];
2) For all ε,σ∈[0,1], f(ε,σ)∈Ω and for all ξ,ϱ∈Ω, we have
|G(ε,σ,ξ(σ))−G(ε,σ,ϱ(σ))|2≤f2(ε,σ)|ξ(σ)−ϱ(σ)|2, |
where f:[0,1]→[0,+∞) is continuous;
3) There exists 0<k<1 such that
supε∈[0,1]∫ε0f2(ε,σ)dσ≤k. |
Then the integral inclusion (5.1) has the solution in Ω.
Proof. For G:[0,1]×[0,1]×R→Pcv(R), it follows from Michael's selection theorem that there exists a continuous operator
Gi:[0,1]×[0,1]×R→R |
such that Gi(ε,σ,ξ(σ))∈G(ε,σ,ξ(σ)) for all ε,σ∈[0,1]. It follows that
ξ(ε)∈∫ε0Gi(ε,σ,ξ(σ))dσ+h(ε))∈S(ξ(ε)) |
hence S(ξ(ε))≠∅ and closed. Moreover, since h(ε) is continuous on [0,1], and G is continuous, their ranges are bounded. This means that S(ξ(ε)) is bounded and S(ξ(ε))∈^C0(Ω). For q,r∈Ω, there exist q(ε)∈S(ξ(ε)) and r(ε)∈S(ϱ(ε)) such that
q(ξ(ε))={w∈Ω:w∈∫ε0Gi(ε,σ,ξ(σ))dσ+h(σ),ε∈[0,1]} |
and
r(ϱ(u))={w∈Ω:w∈∫ε0Gi(ε,σ,ϱ(σ))dσ+h(ε),ε∈[0,1]}. |
It follows from item 5.1 that
|Gi(ε,σ,ξ(σ))−Gi(ε,σ,ϱ(σ))|2≤f2(ε,σ)|ξ(σ)−ϱ(σ)|2. |
Now,
e−supt∈[0,1]|q(ε)−r(ε))|2kα≥e−supε∈[0,1]∫ε0|Gi(ε,σ,ξ(σ))−Gi(ε,σ,ϱ(σ))|2dσkα≥e−supε∈[0,1]∫ε0f2(ε,σ)|ξ(σ)−ϱ(σ)|2dσkα≥e−|ξ(σ)−ϱ(σ)|2supε∈[0,1]∫ε0f2(ε,σ)dσkα≥e−k|ξ(σ)−ϱ(σ)|2kα=e−|ξ(σ)−ϱ(σ)|2α≥e−supσ∈[0,1]|ξ(σ)−ϱ(σ)|2α=Fb(ξ,ϱ,α). |
So, we have
Fb(q,r,kα)≥Fb(ξ,ϱ,α). |
By interchanging the roll of ξ and ϱ, we reach to
ΘFb(Sξ,Sϱ,kα)≥Fb(ξ,ϱ,α). |
Hence, S has a fixed point in Ω, which is a solution of the integral inclusion (5.1).
In this article we proved certain fixed point results for Hausdorff fuzzy b-metric spaces. The main results are validated by an example. Theorem 3.2 generalizes the result of [25]. These results extend the theory of fixed points for multivalued mappings in a more general class of fuzzy b-metric spaces. For instance, some fixed point results can be obtained by taking b=1 (corresponding to G-complete FMSs). An application for the existence of a solution for a Volterra type integral inclusion is also provided.
The author Aiman Mukheimer would like to thank Prince Sultan University for paying APC and for the support through the TAS research LAB.
The authors declare no conflict of interest.
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