Generalized Reich–Ćirić–Rus-Type and Kannan-Type Contractions in Cone <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.20973 12.533" width="8.20973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-</span>Metric Spaces over Banach Algebras

Han, Yan; Xu, Shaoyuan

doi:https://doi.org/10.1155/2021/7549981

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Recent Advances in Fixed Point Theory in Abstract Spaces

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Volume 2021 | Article ID 7549981 | https://doi.org/10.1155/2021/7549981

Generalized Reich–Ćirić–Rus-Type and Kannan-Type Contractions in Cone -Metric Spaces over Banach Algebras

Yan Han¹and Shaoyuan Xu²

Academic Editor: Zoran Mitrovic

Received15 Jul 2021

Revised10 Aug 2021

Accepted30 Aug 2021

Published27 Sept 2021

Abstract

In this paper, we firstly introduce the generalized Reich‐Ćirić‐Rus-type and Kannan-type contractions in cone -metric spaces over Banach algebras and then obtain some fixed point theorems satisfying these generalized contractive conditions, without appealing to the compactness of . Secondly, we prove the existence and uniqueness results for fixed points of asymptotically regular mappings with generalized Lipschitz constants. The continuity of the mappings is deleted or relaxed. At last, we prove that the completeness of cone -metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in . Our results greatly extend several important results in the literature. Moreover, we present some nontrivial examples to support the new concepts and our fixed point theorems.

1. Introduction and Preliminaries

It is well known that the fixed point theory is widely applied to almost all fields of quantitative sciences such as computer science, physics, and biology, especially since the famous Banach contraction principle was introduced in 1922 [1]. In 1968, Kannan [2] studied the following meaningful fixed point theorem, which is a generalization of Banach contraction principle.

Theorem 1. Let be a complete metric space and let be a mapping such that there exists satisfyingfor all . Then has a unique fixed point and for each , the iterated sequence converges to .

The mapping satisfying the contractive condition is known as Kannan-type contraction mapping, which is highly interesting since the contraction mapping does not need to be continuous. In 1971, Reich [3] further extended the Banach and Kannan fixed point theorems as follows.

Theorem 2. Let be a complete metric space and let be a mapping such that there exist satisfyingfor all . Then has a unique fixed point and for each the iterated sequence converges to .

The mapping satisfying (2) was originally called Reich-type contraction mapping. Since the importance of the Reich-type contraction is simultaneously proved by Ćirić [4] and Rus [5], we say that the mapping is a Reich–Ćirić–Rus-type contraction mapping. Recently, Górnicki [6] proved the following theorems in compact metric spaces.

Theorem 3. Let be a compact metric space and let be a continuous mapping satisfyingfor all and . Then has a unique fixed point and for each , the iterated sequence converges to .

Theorem 4. Let be a compact metric space and let be a continuous mapping such that there exist satisfyingfor all and . Then has a unique fixed point and for each , the iterated sequence converges to .

Note that the continuity of the mapping and the compactness of the metric space are essential conditions in Theorems 3 and 4. In order to improve these theorems, Garai et al. [7] investigated some meaningful fixed point theorems of Kannan-type contractive mappings in metric spaces by using the notions of bounded compactness, orbital continuity, and -orbital compactness. Afterwards, Haokip and Goswami [8] extended some related results in -metric spaces by using a subadditive altering distance function. In this paper, we further study the fixed point theorems about Kannan-type and Reich–Ćirić–Rus-type contractions in a much broader space.

The concept of -metric space was derived from the work of Bakhtin [9] and Czerwik [10]. They gave a weaker condition than the triangular inequality, with the aim of extending Banach contraction principle. Moreover, in general, a -metric is not a continuous function and thus is a generation of metric [11]. Subsequently, Hussian and Shah [12] introduced cone -metric space which extended cone metric space [13] and -metric space. In cone -metric space, the distance between and is defined by a vector in an ordered Banach space, instead of the usual real line (see [14]). In 2013, Liu and Xu [15] introduced the concept of cone metric space over a Banach algebra by replacing Banach spaces with Banach algebras and considering the contractive constants to be vectors. Moreover, in their paper, it is significant to prove the nonequivalence of fixed point results between metric spaces and cone metric spaces over Banach algebras by some valid examples. In a similar way, the notion of cone -metric space over a Banach algebra was defined by Huang and Radenović [16], which is also nonequivalent to -metric space in terms of the existence of the fixed points of contractions with vector-valued coefficients. Since then, the fixed point theory in these abstract spaces is prompted to be investigated by lots of authors; for detail, see [17–19] and references therein.

In this paper, we prove some fixed point theorems about generalized Kannan-type and Reich–Ćirić–Rus-type contractions in cone -metric spaces over Banach algebras by introducing the notions of bounded compactness, -orbital compactness, orbital continuity, orbital completeness, and asymptotic regularity in cone -metric spaces over Banach algebras. The main conclusion improves and extends some important known results in the literature [1–3, 6, 7, 20, 21]. Moreover, we prove that the completeness of cone -metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in . Furthermore, there are some examples to present that our new notions and main conclusions are genuine improvements and extensions of the corresponding notions and works in the literature.

First, let us recall some preliminary concepts of Banach algebras and cone -metric spaces.

Let be a real Banach algebra; i.e., is a real Banach space in which an operation of multiplication is defined, subject to the following properties: for all, (1)(2) and (3)(4)

In this paper, we shall assume that the Banach algebra has a unit (i.e., a multiplicative identity) such that for all . An element is said to be invertible if there is an inverse element such that . The inverse of is denoted by . For more details, we refer to [22].

A subset of is called a cone if(i) is nonempty and closed and , where denotes the zero element of (ii) for all nonnegative real numbers (iii)(iv)

For a given cone , we can define a partial ordering with respect to by if and only if . We shall write if and , while will stand for , where denotes the interior of .

A cone is called normal if there is a number such that for all ,

The least positive number satisfying the above inequality is called the normal constant of . Indeed, the number cannot be less than 1; see [23]. A cone is called regular if every increasing sequence which is bounded from above is convergent. In other words, if there is a such thatthen there exists such that . Equivalently, a cone is regular if and only if every decreasing sequence which is bounded from below is convergent. It is well known that every regular cone is normal.

A cone is called strongly minihedral if each subset of which is bounded from above has a supremum. If is a strongly minihedral cone, then every subset of bounded below has an infimum (see [24, 25]).

Throughout this paper, we always assume that is a cone over Banach algebra with and is the partial ordering with respect to .

Definition 1 (see [12, 16, 17, 18]). Let be a nonempty set and be a constant. Suppose that the mapping satisfies the following: (d1) for all and if and only if (d2) for all (d3) for all Then is called a cone -metric on and is called a cone -metric space over Banach algebra .

Definition 2 (see [14, 16]). Let be a cone -metric space over a Banach algebra , and a sequence in . Then,(i) converges to if, for every with , there is a natural number such that for all (ii) is a Cauchy sequence if, for every with , there is a natural number such that for all (iii) is a complete cone -metric space if every Cauchy sequence is convergent in

It is worth mentioning that unlike the usual metric or cone metric with a normal cone, cone -metric is not necessarily continuous in general even if the cone is normal, as the following example shows.

Example 1. Let with the norm The multiplication is defined bywhere . It follows that is a Banach algebra with a unit . Let . Then is a normal cone with a normal constant . Let for all . Define the cone -metric byNote that for the definition of the cone -metric above, is odd if both and are odd; is if both and are . Then it is sufficient to check that is a cone -metric space over Banach algebra with the coefficient .
Let for each . ThenThis gives . However,as . Therefore, the cone -metric is not continuous even though the cone is normal.

Definition 3 (see [26]). Let be a solid cone in a Banach algebra . A sequence is a -sequence if for each there exists such that for .

Lemma 1 (see [16]). Let be a solid cone in a Banach algebra and let and be sequences in . If and are -sequences and , then is a -sequence.

Lemma 2 (see [22]). Let be a Banach algebra with a unit and . If the spectral radius of is less than 1, i.e.,then is invertible. Actually, .

Inspired by Definition 3 in [24], we introduce a new notion of the distance between a set and a singleton in cone -metric space over a Banach algebra .

Definition 4. Let be a cone -metric space over a Banach algebra and be a nonempty subset of . Let be a normal and strongly minihedral cone. The distance between the set and the singleton is defined as follows:

2. Bounded Compactness and -Orbital Compactness

The concepts of bounded compactness and -orbital compactness were discussed in usual metric spaces [7] and -metric spaces [8], which were important to weaken the condition of compactness. In the following, we give the notions of generalized Kannan-type and Reich–Ćirić–Rus-type contractions, bounded compactness, and -orbital compactness in the framework of cone -metric spaces over Banach algebras, which are generalizations of metric spaces and -metric spaces.

Definition 5. Let be a cone -metric space over a Banach algebra with a unit . The mapping is said to be a generalized Kannan-type contraction, if it satisfiesfor all with .

Definition 6. Let be a cone -metric space over a Banach algebra with a unit . The mapping is said to be a generalized Reich–Ćirić–Rus-type contraction, if it satisfiesfor all with , where with .

Definition 7. Let be a cone -metric space over a Banach algebra and be a self-mapping on . Let and .
The space is said to be boundedly compact, if every bounded sequence in has a convergent subsequence.
The mapping is said to be orbitally continuous at a point if for any sequence (for all ), as implies as . Clearly, every continuous mapping is orbitally continuous, but not the converse.
The set is said to be -orbitally compact set, if every sequence in has a convergent subsequence for all .

Example 2. Let with the normDefine the multiplication bywhere . Then is a Banach algebra with a unit . Let .(1)Let and define the cone -metric by . Then is a complete cone -metric space over Banach algebra with . Define mappings as for all and . This clearly gives that is -orbitally compact and boundedly compact but not -orbitally compact.(2)Let . The cone -metric is defined the same as above and is defined by . We deduce that is -orbitally compact but not complete.(3)Let . The cone -metric is defined the same as above and is defined by Then, for any implies . So is orbitally continuous but not continuous in .

In the rest of this section, we always assume that is a cone -metric space over Banach algebra with regular cone such that for all with and the cone -metric is continuous.

Theorem 5. Let be a boundedly compact cone -metric space over Banach algebra with a unit and the coefficient . Let be a generalized Reich–Ćirić–Rus-type contraction mapping and orbitally continuous. If and exist, then has a unique fixed point and for each the iterated sequence converges to ; i.e., is a Picard operator.

Proof. For an arbitrary , let . We assume that for all . Indeed, if for some , , then is the fixed point of . Denote for each . By (14), we havewhich gives that . Let , then by . Therefore, we haveBecause the cone is regular, there exists in such that . Thus, for all , considerwhich implies thatTherefore, is bounded. Since is boundedly compact, there is a convergent subsequence of and such that . So by the orbital continuity of . If , thenMoreover,which meansa contradiction. Thus, and . That is, is a fixed point of . Then, the inequality (14) impliesHence, . This gives ; i.e., is a Picard operator.
Finally, the uniqueness of the fixed point can be obtained by (14). If is another fixed point of , thenleading to a contradiction. Therefore, is the unique fixed point of .

Theorem 6. Let be a -orbitally compact cone -metric space over Banach algebra with a unit and the coefficient , where is a generalized Reich–Ćirić–Rus-type contraction mapping and orbitally continuous. If and exist, then has a unique fixed point , and for each the iterated sequence converges to ; i.e., is a Picard operator.

Proof. The analysis is similar to that in the proof of Theorem 5. We firstly get the sequence . If there is an integer such that , then is the fixed point. Without loss of generality, we assume that . We can prove that , where and are the same as above. As is -orbitally compact, there is a convergent subsequence of and such that . By orbital continuity of , we obtain . The rest proof is similar to Theorem 5.

Corollary 1. Let be a boundedly compact cone -metric space over Banach algebra with a unit and the coefficient . Let be a generalized Kannan-type contraction mapping which is orbitally continuous. Then has a unique fixed point and for each the iterated sequence converges to ; i.e., is a Picard operator.

Proof. Taking and , we obtain the conclusion by Theorem 5.

Corollary 2. Let be a -orbitally compact cone -metric space over Banach algebra with a unit and the coefficient , where is a generalized Kannan-type contraction mapping and orbitally continuous. Then has a unique fixed point and for each , the iterated sequence converges to ; i.e., is a Picard operator.

Proof. The proof is analogous.

Remark 1. Theorems 5 and 6 greatly improve Theorem 2.3 in [6]. The assumptions of compactness and continuity considered in Theorem 2.3 of [6] are relaxed by bounded compactness or -orbital compactness and -orbital continuity, respectively. Corollaries 1 and 2 mainly improve and generalize Theorem 2.2 in [6] and Theorem 2.1 and Theorem 2.2 in [7].

Example 3. Let with the norm . The multiplication is defined bywhere . It follows that is a Banach algebra with a unit . Let . Then is a normal cone with a normal constant . Let and define the cone -metric byfor all in , where are constants. Furthermore, define the mapping bywhere is equivalent to and . Obviously, is not continuous but -orbitally continuous. Moreover, is an incomplete cone -metric space over Banach algebra with but -orbitally compact. Let , then and exist. In order to check the generalized Reich–Ćirić–Rus-type contraction, we have the following three cases:(i)If and , then(ii)If with and , then(iii)If and , then is clearly true. Therefore, the mapping has a unique fixed point in by Theorem 6.

3. Asymptotic Regularity and Orbital Completeness

In the following, we obtain some fixed point theorems of generalized contractive mappings in orbitally complete cone -metric spaces over Banach algebras, under the condition of asymptotic regularity. The regularity or normality of the cone and the continuity of the cone -metric are not necessary. Now, we give the definition of asymptotic regularity, which is a generalization of the counterpart in metric spaces.

Definition 8 (see [27]). Let be a metric space. The mapping is said to be asymptotically regular, if for all .

Definition 9. Let be a cone -metric space over a Banach algebra . The mapping is said to be asymptotically regular, if for every with , there is a natural number such that for all . That is, is a -sequence for all .

Compared with Definition 8, Definition 9 shows a great generalization. The condition that is a -sequence is a sharp improvement of that . The latter is established only under normal cones (see Proposition 2.5 in [28]) or usual metric spaces (see [6, 20, 21, 29]), while the following theorems are established in nonnormal cone -metric space over Banach algebra . Inspired by the concept of -orbitally complete in metric space [4], we give the similar concept in cone -metric space over a Banach algebra as follows.

Definition 10. Let be a cone -metric space over a Banach algebra . The space is said to be -orbitally complete, if every Cauchy sequence which is contained in for some converges in . Every complete cone -metric space over Banach algebra is -orbitally complete for any , but a -orbitally complete cone -metric space over Banach algebra needs not be complete.

The continuity of the mapping and the cone -metric is not necessary in the following theorems.

Theorem 7. Let be an asymptotically regular mapping in the -orbitally complete cone -metric space over Banach algebra with a unit and the coefficient . If there exist with and such thatfor all , then has a unique fixed point and, for each , the iterated sequence converges to ; i.e., is a Picard operator.

Proof. For an arbitrary , let . Without loss of generality, we assume that . Indeed, if for some , , then is the fixed point of . By asymptotic regularity of , is a -sequence. For all , we haveNow, is invertible, which is due to the fact that . It follows thatBy Lemma 1, is a Cauchy sequence in . Since is -orbitally complete, there exists such that . We shall prove ; i.e., is the fixed point of . By the inequality (34), we havewhich implies thatThe right side of the above equality is a -sequence by Lemma 1, so . Using a similar analysis to Theorem 5, we can prove that is unique.

Corollary 3. Let be an asymptotically regular mapping on the -orbitally complete cone -metric space over Banach algebra with a unit and the coefficient . If there exists with such thatfor all , then has a unique fixed point and for each the iterated sequence converges to ; i.e., is a Picard operator.

Now, if is orbitally continuous, then the condition can be deleted.

Theorem 8. Let be an asymptotically regular mapping on the -orbitally complete cone -metric space over Banach algebra with a unit and the coefficient . There exist with such thatfor all . If is orbitally continuous, then has a unique fixed point and for each the iterated sequence converges to ; i.e., is a Picard operator.

Proof. According to Theorem 7, we see that there exists such that . Because is orbitally continuous, we have . Then, . Similar to Theorem 7, the conclusion is true.

Corollary 4. Let be an asymptotically regular mapping on the -orbitally complete cone -metric space over Banach algebra with a unit and the coefficient . There exist with such thatfor all . If is orbitally continuous, then has a unique fixed point and for each the iterated sequence converges to ; i.e., is a Picard operator.

Remark 2. Corollary 3 is a generalization of Theorems 3.1 and 3.3 in [6]. Similarly, Corollary 4 is an extension of Theorem 2.6 in [20] and Theorem 2.1 in [21], since our cone is a nonnormal cone. In fact, we establish the contractive mappings with several generalized Lipschitz constants, where the constants are all vectors but not usual real constants. These results are not equivalent to the theorems in cone -metric spaces or -metric spaces, which may offer us more applications since there are lots of nonnormal cones (see [23]). Moreover, we weaken the continuity of the mapping which is necessary in Theorem 2.6 of [20] by orbital continuity.

Example 4. Let and . For each , . The multiplication is defined by its usual pointwise multiplication. Then is a Banach algebra with a unit . Define for all and . Then is a nonnormal cone and is a complete cone -metric space over Banach algebra with coefficient . Choose and . We deduce thathence that (t = 1)and finally thatThis meansDefine the mapping byThen is asymptotically regular and orbitally continuous but not continuous. Now, we will show that inequality (40) is satisfied in the following three cases:(1)For all ,(2)For all ,(3)For all , ,Similarly, we can also prove that for all , . Therefore, has a unique fixed point in by Theorem 8.

4. Completeness and Fixed Point

By Corollaries 1 and 2, we know that if the generalized Kannan-type contraction mapping is orbitally continuous in boundedly compact or -orbitally compact cone -metric spaces over Banach algebras, then has a unique fixed point. Conversely, if has a unique fixed point in cone -metric spaces over Banach algebras, then what conditions do have to satisfy? Now, we prove an important theorem showing that the completeness of cone -metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in .

Theorem 9. Let be a cone -metric space over Banach algebra with a unit and the coefficient . Let be a normal and strongly minihedral cone. If every self-mapping satisfyingfor all with has a unique fixed point, then must be a complete cone -metric space over Banach algebra .

Proof. On the contrary, suppose that is not complete; then, there exists a Cauchy sequence in , which is not convergent. If it has a convergent subsequence of such that as , thenfor all . This is a -sequence since and are -sequences. Thus, we can assume that all terms of the sequence are distinct. Let ; then, for all by the fact that the sequence does not converge in . Let be an arbitrary point. If , then there is an integer such thatfor all and arbitrary . That isSuppose ; then for some . Since is a Cauchy sequence, we can find some such thatNow, define byFor any with , we divide the following proof into three cases.

Case 1. If , then and . Without loss of generality, we assume that . By (53), we getThis gives .

Case 2. If , then and for some . Then and . Without loss of generality, we assume that . By (54), we deduce thatwhich established the formula .

Case 3. If , then for some . Therefore and . If , by (53), we havewhich yields . If , by (54), we see thatwhich also gives . Therefore, for all with , we always have . That is, is a generalized Kannan-type contraction mapping which has no fixed point in , a contradiction. Hence, the assumption does not hold and the space must be a complete cone -metric space over Banach algebra . The conclusion is true.

Remark 3. According to the proof of Theorem 9, we see at once that inequality (50) can be replaced byfor all and a fixed point .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was partially supported by the Undergraduate Teaching Quality Project of Guangdong Province (no. 520006) and the Basic Education Research Project of Hanshan Normal University in 2018 (no. ZD 201807).

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Copyright © 2021 Yan Han and Shaoyuan Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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