Abstract

Debnath and De La Sen introduced the notion of set valued interpolative Hardy-Rogers type contraction mappings on b-metric spaces and proved that on a complete b-metric space, whose all closed and bounded subsets are compact, the set valued interpolative Hardy-Rogers type contraction mapping has a fixed point. This article presents generalizations of above results by omitting the assumption that all closed and bounded subsets are compact.

1. Introduction

There are numerous studies on interpolation inequalities in literature. In 1999, Chua [1] gave some weighted Sobolev interpolation inequalities on product spaces. Badr and Russ [2] proved some Littlewood-Paley inequalities and interpolation results for Sobolev spaces. Interpolation is considered as one of the central concepts in pure logic. Various interpolation properties find their applications in computer science and have many deep purely logical consequences (see [3, 4]). Gogatishvili and Koskela [5] presented variant interpolation properties of Besov spaces defined on metric spaces. Going in the same direction in the setting of metric spaces via contraction mappings, Karapinar [6] presented the concept of an interpolative Kannan contraction mapping and proved that this mapping admits a fixed point on complete metric spaces. Later on, this notation has been extended into several directions (see [7–18]).

In [6], Karapinar presented the interpolative Kannan contraction as follows: a mapping is an interpolative Kannan contraction if for all with , where and . This inequality was further refined by Karapinar et al. [7] by for all , where , , and .

Gaba and Karapinar [9] further modified the interpolative Kannan contraction concept in the following way: a mapping is a -interpolative Kannan contraction, if for all , where , with . Karapinar et al. [10] gave the interpolative Hardy-Rogers type contraction as follows: a mapping is called an interpolative Hardy-Rogers type contraction if for each , where and with .

Later on, Debnath and De La Sen [12] extended the above definition to set valued interpolative Hardy-Rogers type contraction mappings on b-metric spaces and proved that on complete b-metric spaces, whose all closed and bounded subsets are compact, the set valued interpolative Hardy-Rogers type contraction mapping has a fixed point.

On the other hand, Bakhtin [19] and Czerwik [20] introduced the notion of b-metric spaces.

Definition 1 (see [19, 20]). Let be a nonempty set and be a function so that for all and some , Then, is a b-metric on , and is called a b-metric space with a coefficient .
For related works in this setting, see [21–23]. From now on, is a b-metric space with a coefficient . In the whole paper, is the coefficient of the b-metric space.

Definition 2 (see [20]). We have the following: (a)A sequence in is said to be Cauchy if (b)A sequence in is said to be convergent to if (c) is said to be complete if every Cauchy sequence in is convergentDenote by the set of nonempty closed bounded subsets of . For , consider where . The functional defined by is known as the Pompieu-Hausdorff b-metric on . We state the following known lemma.

Lemma 3 (see [24]). Let be a b-metric space . Let and . We have the two following statements:
(i) For each , there is so that (ii) For each , there is so that This article presents two new generalizations of set valued interpolative Hardy-Rogers type contraction mappings. Namely, we ensure the existence of fixed points of such maps on a complete b-metric space without considering the assumption that all closed and bounded subsets must be compact. Two examples are also presented.

2. Main Results

First, we define the notion of -interpolative Hardy-Rogers type contractions.

Definition 4. Consider a b-metric space . Also, consider maps and . Such a map is called an -interpolative Hardy-Rogers type contraction if for each with where and with .

The following result ensures the existence of a fixed point of -interpolative Hardy-Rogers type contractions.

Theorem 5. Consider a complete b-metric space and consider an -interpolative Hardy-Rogers type contraction map . Also, consider the given assertions. (I)There must exist and such that (II)For each with , we have (III)For each in with and , we have Then, must have a fixed point in .

Proof. By assertion (I) there are and with . If then has a fixed point. Suppose that By (10), we obtain This leads to Since , there is such that Thus, by (15), Note that . Hence, by (17), we get Now, we consider . Then, by (18), we get This implies Note that when we take in (18), then we get , that is, ; hence, this choice is not possible. As and and , then by assertion (II), we get . Again, we consider then by (10), we get Since , there is such that Thus, by (22), we conclude Note that . Hence, by (24), we get Now, we consider . Then, by (18), we get This yields that Note that if we take in (25), then , that is, , which is not possible. From (27) and (20), we get Proceeding in this way, we can obtain a sequence in with , for all and Also, by the construction of , we get By a triangular inequality, we have for , Since the above series is convergent, is a Cauchy sequence in . Completeness of gives in such that . By considering assertion (III), we get . Here, we claim . If the claim is wrong, then for all , for some . From (10), we get From the above, we get . By the triangular inequality, we have By taking the limit , we get , that is, . Therefore, our claim is valid.