In this paper, we have established some fixed-point results for the class of multivalued φ-contractions in the setting of extended b-metric space. An example is furnished to show the validity of our results. The results we have obtained generalize/extend many recent results by Asl, Bota, Samreen et al., and those contained therein.

1. Introduction and Preliminaries

One of the important and pioneering results is the celebrated Banach contraction in metric fixed-point theory. Generalizations in the existence of solutions of differential, integral, and integrodifferential equations are mostly based on creating outstanding generalizations in the metric fixed-point theory. These generalizations are obtained by enriching metric structure of underlying space and/or generalizing contraction condition. Bakhtin [1] and Czerwik [2] extended first time the idea of metric space by modifying the triangle inequality and called it a b-metric space. Kamran et al. [3], in 2017, further generalized the idea of b-metric and introduced an extended b-metric (Eb-metric) space. They weakened the triangle inequality of metric and established fixed-point results for a class of contractions. Following the idea of Eb-metric space, a number of authors have published several results in this direction (see, e.g., [4, 5]). To have some insight about miscellaneous generalizations of metric, we refer the readers to a recent article [6] and for some work on b-metric, see [7–17].

In 1976, Nadler [18] extended first time the idea of Banach contraction principle for multivalued mappings. He used the set of all closed and bounded subsets of a metric space P and the Hausdorff metric on it. Some of the important generalization of Nadler’s result can be seen in ([19–21]). Subashi and Gjini [22] further generalized the concept of extended b-metric space to multivalued mappings by using extended Hausdorff b-metric. Unlike Nadler, they used HP, the set of all compact subsets of an Eb-metric space P.

In this paper, we have discussed the multivalued φ-contractions on Eb-metric spaces and proved some fixed-point results. The first section of the paper consists of some essential definitions and preliminaries. The second section is dedicated to some fixed-point results for multivalued mappings where extended b-comparison function φ has been used. In the last section, some well-known theorems are mentioned which are direct consequences of our main result.

The core reason behind adding this section is to recollect some essential concepts and results which are valuable throughout this paper.

Definition 1.

([23], Czerwik) For any nonempty set P, a b-metric on P is a function db:P×P⟶R∪0 such that the following axioms hold:

B1: dbp,v=0 if and only if p=v:∀p,v∈P.

B2: dbp,v=dbv,p: ∀p,v∈P.

B3: ∃b≥1 such thatdbp,u≤bdbp,v+dbv,u:∀p,v,u∈P.

The pair P,db is then termed as b-metric space with coefficient b. Evidently, we can see that the collection of b-metric spaces is a superclass of the collection of metric spaces.

A comparison function is an increasing function φ:R∪0⟶R∪0 such that for all l∈R∪0,limr→∞φrl=0 ([24]).

A nonnegative real-valued function φ on R∪0 is called a c-comparison function if it is increasing, and for every l>0 and r=1,2,3,⋯, the series ∑φrl converges.

It is evident from the definition that a c-comparison function is a comparison function but the converse may not be true in general (see for example [25]).

Let us consider a b-metric space P,db and an increasing nonnegative function φ on R∪0. We call a map φ to be a b-comparison function if for all l∈R∪0, the series ∑r=0∞brφrl converges ([25, 26]).

The function φl=jl is an example of b-comparison function if 0<j<1/b for a b-metric space P,db. Note that for b=1, the defined b-comparison function becomes equivalent to the definition of a comparison function.

In the following, the authors enriched the notion of b-metric space by amending the triangle inequality

Definition 2.

([3], Kamran al.) Consider a map s:P×P⟶1,∞ where P≠ϕ. An extended b-metric (Eb-metric) on P is a function ds:P×P⟶0,∞ which satisfies

EB_{1}: dsp,v=0 if and only if p=v:∀p,v∈P.

EB_{2}: dsp,v=dsv,p∀p,v∈P.

EB_{3}: dsp,u≤sp,udsp,t+dst,v: ∀p,t,v∈P.

The pair P,ds is then termed as an extended b-metric (Eb-metric) space.

If sp1,p2=b for some b≥1, then Definition 2 reduces to the definition of b-metric space with coefficient b.

Definition 3.

([3], Kamran al.) Let us consider an Eb-metric spaceP,ds. A sequence ϖr in P is said to be

convergent which converges to ϖ in P if and only if dsϖr,ϖ⟶0 as r⟶∞; we write limr→∞ϖr=ϖ

a Cauchy sequence if dsϖr,ϖk⟶0 as r,k⟶∞

We say that an Eb-metric space P,ds is complete if every Cauchy sequence in P converges in P. We note that the extended b-metric ds is not a continuous functional in general and every convergent sequence converges to a single point.

Next, we define the concept of γ-orbital lower semicontinuity (lsc in short) in the case of Eb-metric space which we will use

Definition 4.

[27]. Let γ:D⊂P⟶P, ϖ0∈D, and the orbit of ϖ0∈D, Oϖ0=ϖ0,γϖ0,γ2ϖ0,⋯⊂D. A real-valued function G on D is said to be a γ-orbitally lsc at p∈D if ϖr⟶p and ϖr⊂Oϖ0 implies Gs≤limr→∞infGϖr. In case if γ:D⊂P⟶PP is multivalued, then the orbit of γ at ϖ0 is given as Oϖ0=ϖr:ϖr∈γϖr−1.

2. Main Results

For some technical reasons, Samreen et al., introduced another class of comparison functions for Eb-metric spaces given as follows

Definition 5.

Let P,ds be an Eb-metric space. A nonnegative increasing real-valued function φ on R∪0 is called an extended b-comparison function if there exists a mapping γ:D⊂P⟶P such that for some ϖ0∈D,Oϖ0⊂D and the infinite series ∑r=0∞φrl∏i=1rsϖi,ϖk converges for all l∈R∪0 and for every k∈N. Here, ϖr=γrϖ0 for r=1,2,⋯. We say that φ is an extended b-comparison function for γ at ϖ0.

Remark 6.

It can be easily seen that by taking sp1,p2=b≥1 (a constant), Definition 5 coincides with the definition of a b-comparison function for an arbitrary self-map γ on P. Every extended b-comparison function is also a comparison function for some b; i.e., if sp1,p2≥1 for every p1,p2∈P, then by setting b=infp1,p2∈Psp1,p2, we have
(1)∑r=0∞brφrl≤∑r=0∞φrl∏i=1rsϖi,ϖk.

Example 7.

Let P,ds be an Eb-metric space, γ a self-map on P, and ϖ0∈P,limr,k→∞sϖr,ϖk exists for ϖr=γrϖ0. Define φ:0,∞⟶0,∞ as
(2)φl=jl,suchthatlimr,k→∞sϖr,ϖk<1/j.

Then, by using ratio test, one can easily see that the series ∑r=1∞φrl∏i=1rsϖi,ϖk converges.where dsu,Z=infdsu,z: z∈Z is a distance from a point u∈P to a set Z and sW,Z=supsw,z: w∈W,z∈Z.

Definition 8.

[22] Let P,ds be an Eb-metric space and A,B∈HP. An extended Pompeiu-Hausdorff metric induced by ds is a function Hs:HP×HP⟶R∪0 defined as:
(3)HsW,Z=maxsupw∈Wdsw,Z,supz∈ZdsW,z,

Theorem 9.

[22] Let P,ds be a complete Eb-metric space. Then, HP is a complete Eb-metric space with respect to the metric Hs

The following lemma is trivial.

Lemma 10.

Let P,ds be an Eb-metric space and W,Z∈HP. Then, for any β>0 and for every z∈Z, there exist w∈W such that
(4)dsw,z≤HsW,Z+β.

Now we are able to state our main result.

Theorem 11.

Let ds be a continuous functional on P such that P,ds is an Eb-metric space. Let D be a closed subset of P and γ:D⟶HP be such that Oϖ0⊂D. Assume that for all p∈Oϖ0 and t∈γp;
(5)Hsγp,γt≤φdsp,t.

Moreover, the inequality (1) strictly holds if and only if p≠t and φ is an extended b-comparison function for γ at ϖ0∈D. Then, there exists ϖ in P such that ϖr⟶ϖ, where ϖr∈Tϖr−1. Furthermore, ϖ∈P is a point fixed under the map γ if and only if the map Gl=dsl,γl is γ-orbitally lsc at ϖ.

Proof.

Let ϖ0∈D and ϖ1∈γϖ0. Then, ϖ0≠ϖ1 because if it is equal, then ϖ0 is a fixed point of γ. By using (1) for γϖ0,γϖ1∈HP, we obtain
(6)Hsγϖ0,γϖ1<φdsϖ0,ϖ1.

Choose ε1>0 such that
(7)Hsγϖ0,γϖ1+ε1≤φdsϖ0,ϖ1.

Now, ϖ1∈γϖ0 and ε1>0; then, by Lemma 2, there exists ϖ2∈γϖ1 such that
(8)dsϖ1,ϖ2≤Hsγϖ0,γϖ1+ε1≤φdsϖ0,ϖ1.

Again, ϖ1≠ϖ2; otherwise, ϖ1 is fixed under the map γ. By using (1), we obtain
(9)Hsγϖ1,γϖ2<φdsϖ1,ϖ2.

Choose ε2>0 such that
(10)Hsγϖ1,γϖ2+ε2≤φdsϖ1,ϖ2≤φφdsϖ0,ϖ1=φ2dsϖ0,ϖ1,while the second inequality is due to (4). By Lemma 10, for ϖ2∈γϖ1 and ε2>0, ∃ϖ3∈γϖ2 such that
(11)dsϖ2,ϖ3≤Hsγϖ1,γϖ2+ε2≤φ2dsϖ0,ϖ1.

Continuing in the same way, we get
(12)dsϖr,ϖr+1≤φrdsϖ0,ϖ1.

If k>r, then by using (6) and the triangle inequality in Eb-metric, we obtain,
(13)dsϖr,ϖk≤sϖr,ϖkdsϖr,ϖr+1+sϖr,ϖksϖr+1,ϖkdsϖr+1,ϖr+2+⋯+sϖr,ϖksϖr+1,ϖk⋯sϖk−1,ϖkdsϖk−1,ϖk≤dsϖr,ϖr+1∏i=1rsϖi,ϖk+dsϖr+1,ϖr+2∏i=1r+1sϖi,ϖk+⋯+dsϖk−1,ϖk∏i=1k−1sϖi,ϖk≤φrdsϖ0,ϖ1∏i=1rsϖi,ϖk+φr+1dsϖ0,ϖ1∏i=1r+1sϖi,ϖk+⋯+φk−1dsϖ0,ϖ1∏i=1k−1sϖi,ϖk.

But φ is an extended b-comparison function, so the series ∑j=1∞φjdsϖ0,ϖ1∏i=1jsϖi,ϖk converges. Let S be the sum of the series. By setting Sn=∑j=1nφjdsϖ0,ϖ1∏i=1jsϖi,ϖk, from inequality (7), we obtain
(14)dsϖr,ϖk≤Sk−1−Sr−1,which further implies that limr,k→∞dsϖr,ϖk⟶0. Hence, ϖr is a Cauchy sequence in D. But D is a closed subset of complete space P so there exists ϖ∈D such that ϖr⟶ϖ.

Using the definition of an extended Hausdorff b-metric Hs and (1), we have
(15)dsϖr,ϖr+1≤Hsγϖr−1,γϖr≤φdsϖr−1,ϖr<dsϖr−1,ϖr.

But ϖr⟶ϖ as r⟶∞ which infers that limr→∞dsϖr,γϖr=0.

Assume that Gϖ=dsϖ,γϖ is γ-orbitally lsc at ϖ. Then,
(16)dsϖ,γϖ=Gϖ≤liminfr→∞Gϖr=liminfr→∞dsϖr,γϖr=0.

Hence, ϖ∈γϖ. But γϖ is closed, so ϖ∈γϖ and thus, ϖ is fixed under the map γ. Conversely, if ϖ is a point fixed under the map γ, then Gϖ=0≤liminfr→∞Gϖr.

Remark 12.

Note that Theorem 11 extends/generalizes the main result by Samreen et al. (, Theorem 15.9) to the case of multivalued mappings. Moreover, Theorem 11 includes main results such as by Czerwik (Theorem 9 [2]) and Samreen et al. (Theorem 3.10 (6) [28]) as special cases when the self-mapping is taken on a b-metric space. It also invokes some of the results by Proinov [29] and Hicks and Rhoades [30] in the case of metric space.

Example 13.

Let P=0,1/4 and ds:P×P⟶R be defined as dsl,m=l−m2. Then, P,ds is an Eb-metric space with sp,q=p+q+2. Define γ:P⟶HP by γp=0,l2; then, for each ϖ0∈P and ϖr∈γϖr−1, we have limr,k→∞sϖr,ϖk=limr,k→∞ϖr+ϖk+2=2<4. For every l∈P and m∈Tl, we obtain
(17)Hsγl,γm=Hs0,l2,0,m2=l2−m22=l+m2l−m2≤14l−m2.

If we define φ:0,∞⟶0,∞ by φj=j/4, then γ fulfilled all the conditions present in our main Theorem 11. So ∃ϖ in P such that ϖ∈γ as we can see here that ϖ=0∈γ0.

3. Consequences

In this section, we will discuss an important consequence of Theorem 11 which involves β∗−φ multivalued contractions on Eb-metric spaces. The obtained result generalizes some results by Asl et al. (Theorem 2.1 [31]) and Bota et al. (Theorem 9 [32]).where φ is an extended b-comparison function for γ at ϖ0. Then, ∃ϖ in P such that γrϖ0⟶ϖasr⟶∞. Additionally, ϖ is a point in P fixed under the map γ if and only if the map Gl=dsl,γl is γ-orbitally lsc at ϖ.for every p∈Oϖ0. Then, γrϖ0⟶ϖ∈P as r⟶∞. Additionally, ϖ is a point fixed under the map γ if and only if the map Gl=dsl,γl is γ-orbitally lsc at ϖ.for all m,u∈P. Here, Φ˘Eb denotes the class of all extended b-comparison functions.Theorem 4. Let ds be a continuous functional on P such that P,ds is a complete Eb-metric space. Suppose γ:P⟶HP is a β∗−φ contractive multivalued operator of type (Eb) satisfies the following:

γ is β∗-admissible

There exist ϖ0∈P and ϖ1∈γϖ0 such that βϖ0,ϖ1≥1

Corollary 14.

(Theorem 3.9) Let ds be a continuous functional on P such that P,ds is a complete Eb-metric space. Let γ:D⊂P⟶P be a map such that Oϖ0⊆D. Assume that for every q∈Oϖ0(18)dsγq,γ2q≤φdsq,γq,

Proof.

The assertion simply follows by taking γ a self-map and then using Theorem 11.

Theorem 15.

Let ds be a continuous functional on P such that P,ds is a complete Eb-metric space. Let γ:D⊂P⟶P be such that the orbit of ϖ0, Oϖ0 is a subset of D. Suppose that limr,k→∞sϖr,ϖk exists and j is a constant so that limr,k→∞sϖr,ϖk<1/j for all ϖr,ϖk∈Oϖ0. Assume that
(19)dsγp,γ2p≤jdsp,γp,

Proof.

Define φ:R∪0⟶R∪0 by φl=jl. By taking γ a self-map, Example 7 invokes that φ is an extended b-comparison function for γ at ϖ0. Hence, the result follows from Theorem 11.

Remark 16.

Note that Theorem 15 generalizes Theorem 9 [30] for multivalued mappings in the case of Eb-metric spaces.

Definition 17.

Let s:P×P⟶1,∞ be a map such that P is an Eb-metric space. A multivalued mapping γ:P⟶PP is said to be a β∗-admissible map if there exists a real-valued mapping β on P×P which is nonnegative and βp,q≥1 implies that β∗γp,γq≥1 for all p,q∈P. Note that β∗:PP×PP⟶R∪0 is defined by
(20)β∗W,Z=infβp,q: p∈W,q∈Z.

Definition 18.

[32] Let P,ds be an Eb-metric space. A multivalued mapping γ:P⟶PP is said to be a β∗−φ-contractive multivalued operator of type (Eb) if there exist two functions β:P×P⟶R∪0 and φ∈Φ˘Eb such that [33]
(21)φdsm,u≥β∗γm,γuHsγm,γu,

Then, ∃ϖ∈P such that ϖr⟶ϖ as rt∞ where ϖr∈γϖr−1. Furthermore, the point ϖ is fixed under the map γ if and only if the function Gl=dsl,γl is γ-orbitally lsc at ϖ.

Proof.

Since γ is β∗-admissible and βϖ0,ϖ1≥1 for ϖ1∈γϖ0, so β∗γϖ0,γϖ1≥1. By using infimum property, for ϖ1∈γϖ0 and ϖ2∈γϖ1,
(22)βϖ1,ϖ2≥β∗γϖ0,γϖ1.

Thus, βϖ1,ϖ2≥1 which further implies that β∗γϖ1,γϖ2≥1. Again, by using the same property, for ϖ2∈γϖ1 and ϖ3∈Tϖ2, βϖ2,ϖ3≥β∗γϖ1,γϖ2≥1. Continue the similar process to obtain
(23)β∗γϖr,γϖr+1≥1,r=1,2,3,⋯.

The contractive condition (8) thus implies
(24)Hsγϖr,γϖr+1≤β∗γϖr,γϖr+1Hsγϖr,γϖr+1≤φdsγr−1ϖ0,γrϖ0,which becomes equivalent to the following condition:
(25)Hsγp1,γp2≤φdsp1,p2,for every p1∈Oϖ0 and p2∈γp1. Thus, all the conditions of Theorem 11 are satisfied and so the assertions follow.

Remark 19.

1. Note that Theorem 4.2 in becomes a special case of Theorem 4 for a self-map. Also, for a selfmap γ and sp1,p2=1, Theorem 4 reduces to Theorem 2, 1 [33].

Data Availability

No data is used.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

M. S., W. U., and E. K. contributed in writing, reviewing, and editing the manuscript. All authors contributed equally and significantly in writing this article. All authors have read and agreed to the published version of the manuscript.

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