Abstract
In the present research paper, Chatterjea type contraction is defined and discussed in the framework of quasi-partial b-metric space. Further, some common fixed point results are proved using the notion of interpolation. The results are extended to fixed point theorems for modified Chatterjea type Suzuki contraction using w-admissible maps. The results proved are new and unique supported by application which will enrich the existing literature.
1. Introduction and Preliminaries
In 1922, Banach [1] gave one of the important result called Banach contraction principle and discussed its importance in metric fixed point theory, i.e., let be a self map on a nonempty set and be a complete metric space. If there exists a constant such thatthen it possesses a unique fixed point in . Afterwards, many generalizations of Banach contraction principle came in the literature (see [2–4]) In 1968, Kannan [5] removed the continuity property from this principle and defined a new variant.
Theorem 1 (see [5]). Let be a complete metric space and a self map be a Kannan contraction mapping, i.e.,for all , where . Then, admits a unique fixed point in .
In 1972, Chatterjea [6] generalized the contraction mapping on a complete metric space which is stated as follows.
Theorem 2 (see [6]). Let be a complete metric space. A self map be a Chatterjea type contraction. Iffor all , where , then has a unique fixed point.
In 2015, quasi-partial metric space was generalized by Gupta and Gautam [7] to quasi-partial b-metric space. In 2018, Karapinar [8] enriched the concept of interpolation and established a new contraction known as interpolative Kannan-type contractive mapping.
Theorem 3 (see [8]). Let (G, d) be a metric space; a self mapping is said to be an interpolative Kannan-type contraction map if there exist a constant (0, 1) and (0, 1) such thatwhere .
Numerous studies have been done on interpolation inequalities in the literature. Many new research papers based on the concept of interpolation came into existence (for example, see [9–11]). Noorwali [12], in the year 2019, extended the Kannan interpolative contraction by proving common fixed point result for two self maps. Several authors [9, 13–25] have contributed immensely to enrich this field.
Definition 1 (see [26]). A quasi-partial metric on a nonempty set is a function such that for all :(1) implies (2)(3)(4)The pair is called a quasi-partial metric space.
Definition 2 (see [7]). A quasi-partial b-metric on a nonempty set is a function such that for some real number and all :(1) implies (2)(3)(4)The pair is called a quasi-partial b-metric space. The number is called the coefficient of . Let be a quasi-partial b-metric on the set . Then,
Example 1. Let be endowed with metric defined by . Dominance of (G, ) for the special case is visually checked in Figure 1.
Here, as .
Again, as and similarly .
Also, .
Let . If , thenThus, is a quasi-partial b-metric space.
Lemma 1 (see [23]). Let be a quasi-partial b-metric space. Then, the following holds:(i)If (, ) = 0, then (ii)If , then (, ) and (, )
Definition 3 (see [7]). Let be a quasi-partial b-metric. Then,(1)A sequence converges to if and only if .(2)A sequence is called a Cauchy sequence if and only if exists.(3)A quasi-partial b-metric space is said to be complete if every Cauchy sequence converges with respect to to a point such that(4)A mapping is said to be continuous at if, for every , there exists
Lemma 2 (see [18]). Let (G, ) be a quasi-partial b-metric space and (G, ) be the corresponding b-metric space. Then, (G, ) is complete if (G, ) is complete.
Definition 4 (see [23]). Let be a quasi-partial b-metric space and be a given mapping. Then, is said to be sequentially continuous at if for each sequence in converging to , we have , that is, . Similarly, let be a given mapping. is said to be sequentially continuous at if for each sequence in converging to , we have .
2. Main Results
In this paper, the notion of interpolative Chatterjea type contraction is given in the setting of quasi-partial b-metric space.
Definition 5. Let be a complete quasi-partial b-metric space. We say that the self mappings are an interpolative Chatterjea contraction if there exists , such thatfor all such that whenever .
Theorem 4. Let be a complete quasi-partial b-metric space and H and K be an interpolative Chatterjea type contraction. Then, and posses a common fixed point.
Proof. Let . Define the sequence by , for all . If there exist , such that , then is a common fixed point of and . Suppose that there are no such three consecutive identical terms in the sequence and that . Now, using equation (9), we deduce thatBy using Definition 2,Therefore,Suppose thatBy using equation (12),which is a contradiction. Thus, we haveUsing equation (12),Hence,Therefore,Similarly,Hence,This yields thatFrom equations (18) and (21), we deduce thatTo prove sequence is Cauchy, let From (23),Therefore,We conclude that is a Cauchy sequence. Since is complete, there exists such that . Next, we shall prove that is a common fixed point of and .Letting , we get . Similarly,Letting , we get . Hence, and attain a common fixed point.
We can justify this result by illustrating it with an example.
Example 2. Let . Define complete quasi-partial b-metric as , that is:
We define self mappings and on as , . Now, consider the value and .
| 0 | 1 | 2 | 3 | |
| 0 | 0 | 1 | 4 | 9 |
| 1 | 2 | 1 | 2 | 5 |
| 2 | 6 | 3 | 2 | 3 |
| 3 | 12 | 7 | 4 | 3 |
Case 1. Let . Without loss of generality, we haveDominance of right-hand side of equation (9) is easily visually checked in Figure 2. Thus, the inequality required in equation (9) holds for .
Case 2. Let .Dominance of right-hand side of equation (9) is easily visually checked in Figure 3. Thus, the inequality required in equation (9) holds for .
Case 3. Let .Dominance of right-hand side of equation (9) is easily visually checked in Figure 4. Thus, the inequality required in equation (9) holds for .
Therefore, 0 is the common fixed point of and in the setting of interpolative Chatterjea type contraction. Many more common fixed points can be obtained in similar manner. Hence, a fixed point exists but is not unique.
Open problem 1. Let be a complete quasi-partial b-metric space. Let us consider a family of self mappings : , , and such thatWhat are the conditions on for to have a common fixed point?
3. Modified Chatterjea Type Suzuki Contraction
In our next result, we present the existence of interpolative Chatterjea contraction via Suzuki map. First, let us recall w-admissible map.
Definition 6. Let us consider a self map H defined on G and be a map where G. H is said to be w-orbital admissible if for all G, we haveR Condition. If is a sequence in G such that for each n and as , then there exists from such that , for each k.
Definition 7. Let (G, ) be a quasi-partial b-metric space. A self map H defined on G satisfies the C-condition iffor each , G.
Definition 8 (see [15,22]). Let be the set of all nondecreasing self mappings on such thatNotice that for , we have and for all .
Definition 9. Let us consider a quasi-partial b-metric space (G, ). A map GG is said to be a w--interpolative Chatterjea contraction of Suzuki type if there exist and a self-mapping, GG [0, ) and a real number [0, 1), such thatfor each , G.
Theorem 5. Let (G, ) be a complete quasi-partial b-metric space and HGG be a --interpolative Chatterjea contraction of Suzuki type. Assume that H is -orbital admissible mapping and (, H) 1 for some G. Then, H has a fixed point in G provided that at least one of the following conditions holds:(1)(G, ) is w-regular(2)H is continuous(3) is continuous and (, H) 1 where Fix
Proof. Let us consider an initial point G such that (, H) 1 and now construct a sequence by using = for each positive integer n. If = for some , we get = H that means we get is a fixed point of H. Then, we can assume that for each positive integer n. As H is -orbital admissible, we have the condition (, H) = (, ) 1 implies that (, H) = (, ) 1. In continuation, we haveHence, choosing = and = H in (54), we getwhich impliesHence, it follows thator equivalentlyThus, it follows that the sequence is a nonincreasing sequence of positive terms and = 1. From the above equations and the nondecreasing nature of function , we obtainNow, applying triangular inequality, for all j 1, we getwhere . However, the series is convergent, so there exists a positive real number P such that = P. Letting n and j in the above inequality, we getTherefore, is a Cauchy sequence, and using the completeness property of space, it follows that there exists G such thatand we claim that is the fixed point of H.
In case that the assumption (1) holds, we have 1 and we claim thatfor every n. SupposeBy triangular inequality in space, we havewhich is a contradiction. Therefore, for every nN, eitherholds. If the first condition holds, we obtainIf the second condition holds, we getTherefore, letting n, we get that , that is, .
In case that assumption (2) holds, that is, the mapping H is continuous, we getIf the last assumption (3) holds, we have = = and we show that, . Suppose on the contrary, that sinceBy (54), we getwhich is a contradiction. So, , i.e., is the fixed point of the mapping H.
4. Application
In this section, we have discussed the existence of the solution for the nonlinear matrix equation by using interpolative Chatterjea mapping in the notion of quasi-partial b-metric space. Consider a nonlinear matrix equation aswhere denotes the set of Hermitian matrices and denotes the set of positive definite matrices. Here, , , and is matrices. Let us suppose a continuous order preserving map with .
Define a trace norm on the set that is a complete quasi-partial b-metric space and partially ordered with partial ordering , where .
Lemma 3. If are matrices, then
Lemma 4. If , then
Theorem 6. Let a nonlinear matrix equation (54) and suppose the following conditions hold:(1)There exists with(2)For all ,(3)There exists for which and such that(4)There exists and such thatThen, equation (54) owns a solution in .
Proof. Consider a quasi-partial b-metric asDefine a continuous map asfor all . Then, the solution of the matrix equation (54) is the fixed point of the map H.
On taking, with , then . Thus, for . Now, (1) (4) of Theorem 6 yields the following:From equation (64) (66) and (61), we haveWe deduce thatThus,Since , we conclude that .Thus, by using Theorem 4, H owns a fixed point in .
5. Conclusion
The significant contribution of this manuscript is to provide the existence of common fixed points that are nonunique for interpolative Chatterjea contraction mappings in the realm of quasi-partial b-metric space. Further, we have given examples to show that the new results are applicable. In metric fixed point theory, interpolative contraction is a generalization of Kannan-type contraction. Additionally, these proposed contractions can be generalized in other well-known spaces and can give new fixed point results. The uniqueness property of obtained fixed points for these mappings and their application in the study of nonlinear integral equations will be an interesting concept for subsequent work.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.