Unified multivalued interpolative Reich–Rus–Ćirić-type contractions

This article examines new multivalued interpolative Reich–Rus–Ćirić-type contraction conditions and fixed point results for multivalued maps that fulfill these conditions. Earlier defined interpolative contraction type conditions cannot be particularized to any contraction type condition. This slackness of the interpolative contraction type condition is addressed through new multivalued interpolative Reich–Rus–Ćirić-type contraction conditions.


Introduction and preliminaries
A fixed point to a self-mapping L defined on a non-void abstract set B is a solution to an equation Lb = b. Banach's fixed point result [1] is the initial result in the metric fixed point theory which deals with the existence of a solution to the aforementioned equation for a self-map L of a metric space (B, d B ). This result requires the following two conditions to ensure the existence and uniqueness of a solution to an equation Lb = b, equivalently, fixed point of L: (1) The metric space should be complete; (2) L should be contraction map, that is, d B (Lb, Lz) ≤ d B (b, z) for each b, z ∈ B, where ∈ [0, 1).
This article examines new multivalued interpolative Reich-Rus-Ćirić-type contraction maps and fixed point results for such maps. The new multivalued interpolative Reich-Rus-Ćirić-type contraction conditions which are being examined in this article cannot only be particularized to Nadler's type contraction condition but also to some other types of interpolative contraction conditions. Debnath and Sen [16] discussed the existence of fixed points for multivalued interpolative Reich-Rus-Ćirić-type contraction map in bmetric space, by assuming that all bounded and closed subsets of the b-metric space are compact. Readers can see that the restriction of compactness is not required in the presented results of this article.
Before moving towards the main results, we discuss the notion of b-metric spaces, presented by Bakhtin [3] and Czerwik [4], with a few essential concepts.
The following theorem has an important role in the results presented by this article.

Main results
This section begins with the following definition.
The existence of fixed points for the defined notion is discussed as follows.

Theorem 2.2 Assume a complete b-metric space (B, d B , λ) and γ -interpolative Reich-
Rus-Ćirić-I-contraction map L. Also, assume that: By (2.3) and the above fact, we conclude Now, we discuss the proof for the following three choices of τ 3 : is a fixed point of L and it is not possible under the assumption. If τ 3 ∈ (0, 1) in (2.4) then we have the following: since 1τ 3 = τ 1 + τ 2 , thus, by the above inequality, we get Hence, we arrive at As b 1 ∈ Lb 0 , b 2 ∈ Lb 1 and γ (b 0 , b 1 ) = 1, then, by (2), we obtain γ (b 1 , b 2 ) = 1. Again, we assume that Thus, by (2.7) and the above inequality, we get Again, we discuss the proof of the following three choices of τ 3 : a fixed point of L and it is not possible under the assumption. If τ 3 ∈ (0, 1) in (2.8) then we have the following: Thus, we arrive at By (2.10) and (2.6) we obtain Also, we get By the triangle inequality, for n > m, we get Here, we claim that b * ∈ Lb * . Let us suppose that if the claim is wrong then for some natural number n 0 . By (2.1) we get By the triangle inequality and (2.11), we get Suppose that τ 3 = 1 and m → ∞ in the above inequality, then we get d B (b * , Lb * ) = 0, that is, b * ∈ Lb * . Suppose that τ 3 = 1 and m → ∞ in the above inequality, then we get Now, one can calculate the following cases.
These calculations verify the validity of (2.1). The remaining axioms of Theorem 2.2 are also valid. Hence, L has a fixed point.
By assuming τ 1 = 1 and τ 2 = τ 3 = 0 in the above result, we arrive at the following results. (B, d B , λ) and maps L :

Then L has a fixed point in B.
By assuming γ (b, z) = 1 for all b, z ∈ B in the above corollary, we obtain the following result which can be considered as an extended form of Nadler's fixed point theorem.

Corollary 2.5 Assume a complete b-metric space (B, d B , λ) and a map L : B → CB(B) satisfying the following inequality:
where ∈ [0, 1 λ 2 ). Then L has a fixed point in B.
The right side of (2.14) is more analogous to interpolative Reich-Rus-Ćirić-contraction.
Now one can easily understand that Theorem 2.9 is a simple consequence of Theorem 2.2. a complete b-metric space (B, d B , λ) and reduced γ -interpolative Reich-Rus-Ćirić-I-contraction map L. Also, assume that:

Then L has a fixed point in B.
By assuming τ 1 = 0 and τ 2 = τ ∈ (0, 1) in the above result we reach the following result.
Remark 2.11 Inequality (2.15) is a generalized form of improved interpolative Kannan contraction.
The following definition provides another way to generalize interpolative Reich-Rus-Ćirić-contraction maps.
The existence of fixed points for the above defined notion are verified through the following result. a complete b-metric space (B, d B , λ) and γ -interpolative Reich-Rus-Ćirić-II-contraction map L. Also, assume that:

Theorem 2.13 Assume
Then L has a fixed point in B.
The following result is a simple consequence of Theorem 2.13.
Thus, it can be seen that the set-valued versions based on the structure of b-metric spaces for the interpolative contraction type conditions given in [8][9][10][11][12]16], with many other existing interpolative contraction type conditions, are not applicable on the above defined function L with respect to the above d B . Meanwhile, all the axioms of Theorem 2.13 are valid on the above defined functions.

Conclusion
This article presents new multivalued interpolative Reich-Rus-Ćirić-type contraction conditions and fixed point results for multivalued maps which fulfil these conditions in a complete b-metric space. Earlier defined interpolative contraction type conditions cannot be particularized to any contraction type condition. This slackness of interpolative contraction type condition is addressed through the introduction of new multivalued interpolative Reich-Rus-Ćirić-type contraction conditions. A few examples are given to support the findings of this article.