Abstract
The aim of the paper is to introduce the concept of new hybrid contractions that combine Jaggi hybrid type contractions and Suzuki type contractions with -orbital admissible. We investigate the existence and uniqueness of such new hybrid contractions in theorems and results. Further, an illustrated example is given. With the results of this study, we generalize several well-known results in the recent fixed point literature.
1. Introduction and Preliminaries
In the last three to four decades, metric fixed point theory was one of the crucial and useful topics of functional analysis. It has attracted the attention of researchers from several distinct disciplines. Almost a century ago, Banach [1] initiated the metric fixed point theory with a magnificently simple but enormously useful result, known as the “Banach contraction principle.” It is one of the useful theorems for solving differential integral equations to guarantee both the existence and the uniqueness of the solution. A wide variety of problems arising in different areas of pure and applied mathematics, such as differential equations, discrete and continuous dynamical systems, and nonlinear analysis, can be modeled as fixed point equations of the form . Soon afterward, the Banach contraction principle was extended by Caccioppoli [2] to the setting of a complete metric space, which states that, in a complete metric space , any Banach contraction is a Picard operator; we refer to it as the “Banach-Caccioppoli theorem.” These main theorems have been generalized in different directions by many researchers. Among all, the authors observed a very general result in a general setting by using a simulation function. For more interesting results, see, e.g., [3, 4].
Jaggi [5] demonstrated the following theorem satisfying a contractive condition of a rational type.
Theorem 1 (see [5]). Let be a continuous self-map defined on complete metric space . Further, let satisfy the following condition: for all distinct points and where with . Then, has a unique fixed point in .
Very recently in 2018, Karapinar [6] obtained a new interesting type of contraction generalized from the well-known Kannan contraction by adopting an interpolative approach. In [7], a common fixed point of the interpolative Kannan contraction was considered.
Karapinar’s definition and theorem are listed as follows.
Definition 2 (see [6]). Let be a metric space. A self-mapping is said to be an interpolative Kannan-type contraction if there are two constants and such that for all with .
Theorem 3 (see [6]). Let be a complete metric space and be an interpolative Kannan-type contraction. Then, has a unique fixed point in .
Recall the notion of -admissibility introduced by Samet et al. [8].
Definition 4. A mapping is called -admissible if, for all , we have where is a given function.
Afterward, as a modification in the concept of -admissible maps, Popescu [9] introduced -orbital admissible maps.
Definition 5. Let be a mapping and . A self-mapping is called -orbital admissible if, for all , we have
The following condition has often been considered in order to avoid the continuity of the involved contractive mappings.
space is called -regular, if whenever is a sequence in such that for all and as , then there exists a subsequence of such that for all .
In 2008, Suzuki [10] published one of the most comprehensive generalizations of Banach’s and Edelstein’s basic results. Suzuki contraction is when the contractive condition required to satisfy is not for all points of the domain. The existence and uniqueness of fixed points of maps satisfying a Suzuki type contraction has been extensively studied (see [11–15]). Later, Popescu [9] modified the nonexpansiveness situation with the weaker -condition presented by Suzuki. The -condition is defined as follows.
Definition 6. A mapping on a metric space satisfies the -condition if for each .
Recently, Mitrovic et al. [16] used interpolation contraction and Reich contraction together and combined these two contractions in -metric spaces. The combination of these two types of contractions has been called the new hybrid type contraction. In the last years, inspired by the result in [6], Karapinar and Fulga [17] introduced a new hybrid-type contraction that combines Jaggi type contractions and interpolative-type contractions in the framework of complete metric spaces. In this paper, we investigate the Jaggi-Suzuki type hybrid contraction, inspired by the new contraction in [17], which is a combination of Jaggi hybrid type contractions and Suzuki type contractions with -orbital admissible in the framework of complete metric spaces. We introduce the existence and uniqueness of a fixed point for this contraction.
2. Main Results
We start by recalling the definition of the family as the set of all nondecreasing self-mappings on such that for every . Notice that for , we have and for all (see [18]).
Here are our main definition and theorem.
Definition 7. Let be a metric space. We say that the mapping is a Jaggi-Suzuki type hybrid contraction if there exist and such that for each where and , , such that with ,
Theorem 8. Let be a complete metric space and be a Jaggi-Suzuki-type hybrid contraction. Assume also that is -orbital admissible mapping and for some . Then, has a fixed point in provided that at least one of the following conditions holds:
(j1) is -regular
(j2) is continuous
(j3) is continuous and where
Proof. Let the sequence in be constructed by for all positive integer , where such that .
If for some integer , then is a fixed point of , so we shall assume that for all positive integer Since is -orbital admissible, so implies that . Continuing this argument, we get
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Case 1. , by taking choosing and in (6), we get where
Hereafter, we find that
Suppose that , so, using (11), we write which is a contradiction. Then, we get
Eventually, from (11), we have and by repeating this process, we find that
for any . We claim that is a Cauchy sequence in . Then, using the triangle inequality with (15), we can write
where But, , so the series is convergent, then there exists a positive real number such that . As a result, letting in the above inequality, we obtain
Thus, is a Cauccess of the space ; it follows that there exists such that
We assert that is a fixed point of .
If the assumption holds, we get , and we assert that
for all . Since, if we suppose that
then, taking into account the triangle inequality and the fact that the sequence is decreasing, we obtain that
a contradiction. Thus, for all , either or holds. In case that (22) holds, we get
If the second condition, (23), holds, we can write
Hence, taking in (24) and (25), which is a contraction. Therefore, we get that , that is,
If the assumption is true, that is, the mapping is continuous, we obtain
In case that last assumption holds, from above, we get and we aim to prove that also . Assuming on the contrary that , from using (6), we obtain that
a contradiction. As a result, , that is, is a fixed point of the mapping .
Case 2. , by taking choosing and in (6), we obtain
Due to the above inequality, we obtain and from , we attain that for every . Using (31), we get and using previous reasoning
By using the same methods as in the case of , we obviously obtain that forms a Cauchy sequence in complete metric space. Subsequently, there exists such that , and we claim that this is a fixed point of . In case the space is -regular and verifies (8), that is, for every we get . On the other hand, we know (see the proof of case 1) that either or
holds, for all . If (35) is taken, we get
If the second condition, (36), is taken, we get
Hence, taking in the last two inequalities, we find that that is, If the mapping is continuous, we obtain that we know (see the proof of case 1).
If the last assumption (j3) holds, as above, we get , and we aim to prove that . Assuming, on the contrary, that , from using (6), we find that
a contradiction. As a result, , that is, is a fixed point of the mapping .
Theorem 9. Besides the hypothesis of Theorem 8, if we suppose that for any Thereupon, has a unique fixed point in .
Proof. We shall show that is a unique fixed point of . Assume that another is fixed point of , that is, with In the case that , hence, from (6), we have Then, which is a contradiction. In the case that , thus, from (6), we get that a contradiction. Consequently, , that is, has a unique fixed point in .☐
Example 10. Let and be the usual metric on and the mapping be determined as
Make allowances for the function with and , such that
The mapping is not continuous, but is a continuous mapping. We choose , , and , then we have
We obtain that the following cases.
Case 1. For , we get , then (6) holds.
Case 2. For and ,
Other cases are provided, since Eventually, the presumptions of Theorem 8, being satisfied, and is the fixed point of .
Corollary 11. Let be a complete metric space and a continuous mapping. If there exist such that
for each where ; ; and with and therefore, has a fixed point in .
Proof. Case 1: for , it follows from Theorem 8, letting for .
Case 2: for , it follows from Theorem 8, letting for and , .☐
Corollary 12. Let be a complete metric space and a continuous mapping. If there exist such that
for each where ; ; and with and therefore, has a fixed point in .
Proof. Case 1: for , it follows from Corollary 11, letting for any .
Case 2: for , it follows from Corollary 11, letting for any and , .☐
Corollary 13. Let be a complete metric space and a continuous mapping. If there exist such that
for each ; therefore, has a fixed point in .
Proof. For , it follows from Corollary 12, letting .☐
Corollary 14. Let be a complete metric space. A continuous mapping such that for each , where , . Thus, has a fixed point in .
Proof. For , it follows from Corollary 12, letting and with for .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Authors’ Contributions
Both authors have contributed equally to this work. Both authors have read and approved the final manuscript.