Multivalued φ-Contractions on Extended b-Metric Spaces

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 Ebmetric 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 HðPÞ, 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.


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 Ebmetric 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][8][9][10][11][12][13][14][15][16][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][20][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 HðPÞ, 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. The pair ðP, d b Þ 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 nonnegative real-valued function φ on R ∪ f0g is called a c-comparison function if it is increasing, and for every l > 0 and r = 1, 2, 3, ⋯, the series ∑φ r ðlÞ 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 ðP, d b Þ and an increasing nonnegative function φ on R ∪ f0g. We call a map φ to be a b-comparison function if for all l ∈ R ∪ f0g, the series ∑ ∞ r=0 b r φ r ðlÞ converges ( [25,26]). The function φðlÞ = jl is an example of b-comparison 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 bmetric space by amending the triangle inequality The pair ðP, d s Þ is then termed as an extended b-metric (Eb-metric) space.
If sðp 1 , p 2 Þ = b for some b ≥ 1, then Definition 2 reduces to the definition of b-metric space with coefficient b.
We say that an Eb-metric space ðP, d s Þ is complete if every Cauchy sequence in P converges in P. We note that the extended b-metric d s is not a continuous functional in general and every convergent sequence converges to a single point.

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 ðP, d s Þ be an Eb-metric space. A nonnegative increasing real-valued function φ on R ∪ f0g is called an extended b-comparison function if there exists a mapping γ : D ⊂ P ⟶ P such that for some ϖ 0 ∈ D, Oðϖ 0 Þ ⊂ D and the infinite series ∑ ∞ r=0 φ r ðlÞ Q r i=1 sðϖ i , ϖ k Þ converges for all l ∈ R ∪ f0g and for every k ∈ N. Here, ϖ r = γ r ϖ 0 for r = 1, 2, ⋯: We say that φ is an extended b-comparison function for γ at ϖ 0 . Remark 6. It can be easily seen that by taking sðp 1 , p 2 Þ = 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 sðp 1 , p 2 Þ ≥ 1 for every p 1 , p 2 ∈ P, then by setting b = inf p 1 ,p 2 ∈P sðp 1 , p 2 Þ, we have Example 7. Let ðP, d s Þ be an Eb-metric space, γ a self-map on P, and ϖ 0 ∈ P, lim r,k→∞ sðϖ r , ϖ k Þ exists for ϖ r = γ r ϖ 0 . Define φ : ½0,∞Þ ⟶ ½0,∞Þ as Then, by using ratio test, one can easily see that the series ∑ ∞ r=1 φ r ðlÞ where d s ðu, ZÞ = inf fd s ðu, zÞ: z ∈ Zg is a distance from a point u ∈ P to a set Z and sðW, ZÞ = sup fsðw, zÞ: w ∈ W, z ∈ Zg.
Moreover, the inequality (1) strictly holds if and only if p ≠ t and φ is an extended b-comparison function for γ at ϖ 0 ∈ D. Then, there exists ϖ in P such that ϖ r ⟶ ϖ, where ϖ r ∈ Tðϖ r−1 Þ. Furthermore, ϖ ∈ P is a point fixed under the map γ if and only if the map GðlÞ = d s ðl, γðlÞÞ is γ-orbitally lsc at ϖ.
Assume that GðϖÞ = d s ðϖ, γϖÞ is γ-orbitally lsc at ϖ. Then, Hence, ϖ ∈ γðϖÞ. But γðϖÞ is closed, so ϖ ∈ γðϖÞ and thus, ϖ is fixed under the map γ. Conversely, if ϖ is a point fixed under the map γ, then GðϖÞ = 0 ≤ liminf r→∞ Gðϖ r Þ. 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.

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 ϖ 0 . Then, ∃ϖ in P such that γ r ϖ 0 ⟶ ϖð as r ⟶ ∞Þ. Additionally, ϖ is a point in P fixed under the map γ if and only if the map GðlÞ = d s ðl, γlÞ is γ-orbitally lsc at ϖ .for every p ∈ Oðϖ 0 Þ. Then, γ r ϖ 0 ⟶ ϖ ∈ P as r ⟶ ∞. Additionally, ϖ is a point fixed under the map γ if and only if the map GðlÞ = d s ðl, γðlÞÞ 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 d s be a continuous functional on P such that ðP, d s Þ is a complete Eb-metric space. Suppose γ : P ⟶ HðPÞ is a β * − φ contractive multivalued operator of type (Eb) satisfies the following: (i) γ is β * -admissible (ii) There exist ϖ 0 ∈ P and ϖ 1 ∈ γðϖ 0 Þ such that βðϖ 0 , ϖ 1 Þ ≥ 1 Corollary 14. (Theorem 3.9) Let d s be a continuous functional on P such that ðP, d s Þ is a complete Eb-metric space. Let γ : D ⊂ P ⟶ P be a map such that Oðϖ 0 Þ ⊆ D. Assume that for every q ∈ Oðϖ 0 Þ d s γq, Proof. The assertion simply follows by taking γ a self-map and then using Theorem 11.