Abstract
In this paper, we introduce some common fixed point theorems for interpolative contraction operators using Perov operator which satisfy Suzuki type mappings. Further, some results are given. These results generalize several new results present in the literature.
1. Introduction
Banach [1] introduced the Banach contraction principle that generalized in various wide directions by many researchers. One of the generalizations was supposed by Kannan [2] in 1968 and later with other researchers such as iri Reich Rus. In 2018, Karapınar [3] adopted the interpolative approach to define the generalized Kannan type contraction on a complete metric space and proved the following.
A mapping on a complete metric space such that where and , for each . Then, has a unique fixed point in . Afterward, this concept has been extended in different aspects for example [4–14] and also see e.g. [15–19].
On the other hand, Perov [20, 21] gave a characterization of Banach contraction principle in the framework vector-valued metric space.
For a nonempty set , a function is called a vector-valued metric on if the followings are fulfilled: (1) for all ; if , then (2) for all (3) for all (4)where . We mention that, for , that is, and
The notations (respectively, ) denote the collection of all square matrices of real numbers (respectively, nonnegative real numbers) where (respectively, ) is the set of real numbers (respectively, nonnegative real numbers). Furthermore, denotes the complex numbers, as usual.
A matrix converges to zero if and only if its spectral radius is strictly less than 1, that is, , see, e.g., [22]. It is equivalent to saying that all the eigenvalues of are in the open unit disc, that is, , for every with , where denotes the unit matrix of
Theorem 1 (see, e.g., [22, 23]). Let . Then, the following assertions are equivalent: (i)A converges to zero(ii) as (iii)The matrix is nonsingular and(iv)The matrix is nonsingular and has nonnegative elements(v) and are convergent towards zero as , for each
Note also that if with (in the component-wise meaning), then, implies .
Theorem 2 (Perov Cauchy, perov1966certain). Let be a complete vector-valued metric space and the operator with the property that there exists a matrix convergent towards zero such that Then, possesses a unique fixed point.
Ali et al. [24] defined admissible that a generalized of -admissible given by Samet et al. [25].
Definition 3 (see [24]). Let , . A mapping
is said to be -admissible if
where is the identity matrix and the inequality between matrices mans entrywise inequality.
We define some related to -admissible the following concept of admissibility using the above definition and given by some authors [25–30].
Definition 4. Let , , and be a mapping. We say that is an -orbital admissible mapping if Moreover, an -orbital admissible mapping is said to be triangular -orbital admissible if for all we have
Definition 5. For a nonempty set , let and be mappings. We say that is a generalized -admissible pair if for all , we have
Lemma 6. Let and be a triangular -admissible map. Suppose that there exists such that Identify a sequence using Thus, we have for all with .
Lemma 7. Let and be a triangular -admissible mapping. Assume that there exists such that Identify sequence and where So, we have for all with .
Recently, one of the interesting generalizations was given by Suzuki [31, 32] which characterizes the completeness of underlying metric spaces. Suzuki introduced and generalized versions of Banach’s and Edelstein’s basic results. In addition, Popescu [33] has modified the nonexpansiveness situation with the weaker -condition presented by Suzuki. As stated, the existence of fixed points of maps satisfying the -condition has been extensively studied; see [34–38]. Karapnar [39] investigated the definition of a nonexpansive mapping satisfying the -condition.
Definition 8. A mapping on a metric space satisfies the -condition if for each
Theorem 9. Let be a compact metric space and be a mapping satisfying condition for all Then, has a unique fixed point.
2. Main Results
For the rest of the paper, we use the following notation: , where
Definition 10. Let be a complete vector-valued metric space and , be mappings. We say that forms a pair of Perov-interpolative iri-Reich-Rus contractions of Suzuki type, if there exist converges towards zero, (where , ) such that for each , where , are such that .
Theorem 11. Let be a complete vector-valued metric space and be two mappings such that is a pair of Perov-interpolative iri-Reich-Rus contractions of Suzuki type. Assume that (i) is a generalized -admissible pair(ii)There exists such that and (iii) and are continuous mappingsThen, and have a common fixed point.
Proof. Let such that . We define the sequence in as following
for every . From (i) and (ii), and , and then and . Similarly, we get and . Repeating this process, we write
for every .
On the other hand, we have
So, since the mappings forms a pair of Perov-interpolative contractions of Suzuki type, we get
Therefore, it follows that
or equivalent
where Then,
Letting and , since
similarly, we get
Thus, combining (18) and (20), we have that
Take into account (21) and (22), we obtain that for each where By this way, using triangular inequality and (23), for , we get
Because is convergent to zero, we attain that is nonsingular and
Therefore,
So, the sequence is a fundamental (Cauchy), and using the completeness of the space , there exists such that
We claim that the point is a common fixed point of and If (iii.) is provide, that is, the mapping and are continuous, we have
then, . Also, similarly, we get
. Therefore, is a common fixed point of and The proof is complete.☐
In the following theorem, we remove the assumption of the continuity of the mappings and
Theorem 12. Besides the hypothesis (i) and (ii) of Theorem 11, if we assume that the condition: (i)If is a sequence in such that as and, there exists a subsequence of such that holds, then, the mappings and have a common fixed point.
Proof. From Theorem 11, the sequence defined by (12) is a Cauchy sequence and converges to some . Similarly, using (13) and the condition , there exists a subsequence of such that and for all . We claim that for all Supposing on the contrary, we get and letting , we acquire that a contradiction. Therefore, the condition (31) holds, and from (11), we obtain On the taking tend to infinity, it follows that we get . Similarly, we assert that, for all Supposing on the contrary, and, so taking , we obtain that a contradiction. Hence, condition (35) is true and from (11), we obtain Letting tend to infinity, it follows that we acquire . Thus, is a common fixed point of and ☐
Corollary 13. Let be a complete vector-valued metric space and be two continuous mappings such that for each , where , , converges towards zero and , are such that . Then, and have a common fixed point.
Corollary 14. Let be a complete vector-valued metric space and the mappings and such that for each , where , , converges towards zero and the constants , are such that . Assume that (i) is a generalized -admissible pair(ii)There exists such that and (iii)The condition holds or and are continuous mappingsThen, and have a common fixed point.
Letting in Theorem 11, we obtain the next results.
Corollary 15. Let be a generalized metric spaces and . Let be a -orbital admissible mapping such that for each , where , , converges towards zero, and are constants, such that , , and . Assume that (i) is a triangular -orbital admissible(ii)There exists such that and (iii)Either is continuous, or the condition holdsThen, has a fixed point.
Definition 16. Let be a vector-valued metric space and , , be mappings. We say that are Perov-interpolative Kannan contractions of Suzuki type, if there exist a real number and converges towards zero, where , such that for each .
Theorem 17. Let be a complete vector-valued metric space and be Perov-interpolative Kannan contractions of Suzuki type. Assume that (i) is a generalized -admissible pair(ii)There exists such that and (iii)Either, and are continuous mappings or, the condition holdsThen, and have a common fixed point.
Proof. Taking in Theorem 11.☐
Remark 18. If and in the above Theorems, then, we find the concept of the usual metric spaces and interpolative Kannan contraction of Suzuki type and interpolative Ćirić–Reich–Rus contraction of Suzuki type.
Example 19. Let , , where , and two mappings , defined as respectively, We choose , and Let also , where Then, we have to check that (11) holds. We have to examine the following cases: (1). Let and . For and since the inequality (11) holds. (2). Similarly, since the relation (11) holds(3)For and Then, (11) holds. Consequently, the assumptions of Theorem 11 being satisfied, it follows that the mappings and have a fixed point, which is .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.