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

Authors: Yan Han; Shaoyuan Xu
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we firstly introduce the generalized Reich‐Ćirić‐Rus-type and Kannan-type contractions in coneb-metric spaces over Banach algebras and then obtain some fixed point theorems satisfying these generalized contractive conditions, without appealing to the compactness of X. 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 b-metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in X. 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.


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## Body

## 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.
LetX,d be a complete metric space and let T:X⟶X be a mapping such that there exists K<1/2 satisfying(1)dTx,Ty≤Kdx,Tx+dy,Ty,for all x,y∈X. Then T has a unique fixed point z∈X and for each x∈X, the iterated sequence Tnx converges to z.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.
LetX,d be a complete metric space and let T:X⟶X be a mapping such that there exist a1,a2,a3≥0,a1+a2+a3<1 satisfying(2)dTx,Ty≤a1dx,Tx+a2dy,Ty+a3dx,y,for all x,y∈X. Then T has a unique fixed point z∈X and for each x∈X the iterated sequence Tnx converges to z.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 T is a Reich–Ćirić–Rus-type contraction mapping. Recently, Górnicki [6] proved the following theorems in compact metric spaces.Theorem 3.
LetX,d be a compact metric space and let T:X⟶X be a continuous mapping satisfying(3)dTx,Ty<12dx,Tx+dy,Ty,for all x,y∈X and x≠y. Then T has a unique fixed point z∈X and for each x∈X, the iterated sequence Tnx converges to z.Theorem 4.
LetX,d be a compact metric space and let T:X⟶X be a continuous mapping such that there exist a1,a2,a3≥0,a1+a2+a3=1 satisfying(4)dTx,Ty<a1dx,Tx+a2dy,Ty+a3dx,y,for all x,y∈X and x≠y. Then T has a unique fixed point z∈X and for each x∈X, the iterated sequence Tnx converges to z.Note that the continuity of the mapping and the compactness of the metric space are essential conditions in Theorems3 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 T-orbital compactness. Afterwards, Haokip and Goswami [8] extended some related results in b-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 ofb-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 b-metric is not a continuous function and thus is a generation of metric [11]. Subsequently, Hussian and Shah [12] introduced cone b-metric space which extended cone metric space [13] and b-metric space. In cone b-metric space, the distance between x and y 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 b-metric space over a Banach algebra was defined by Huang and Radenović [16], which is also nonequivalent to b-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 coneb-metric spaces over Banach algebras by introducing the notions of bounded compactness, T-orbital compactness, orbital continuity, orbital completeness, and asymptotic regularity in cone b-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 b-metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in X. 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 coneb-metric spaces.LetA be a real Banach algebra; i.e., A is a real Banach space in which an operation of multiplication is defined, subject to the following properties: for all, x,y,z∈A,a∈ℝ(1)
xyz=xyz(2)
xy+z=xy+xz and x+yz=xz+yz(3)
axy=axy=xay(4)
xy≤xyIn this paper, we shall assume that the Banach algebraA has a unit (i.e., a multiplicative identity) e such that ex=xe=x for all x∈A. An element x∈A is said to be invertible if there is an inverse element y∈A such that xy=yx=e. The inverse of x is denoted by x−1. For more details, we refer to [22].A subsetP of A is called a cone if(i)
P is nonempty and closed and θ,e⊂P, where θ denotes the zero element of A(ii)
αP+βP∈P for all nonnegative real numbers α,β(iii)
P2=PP⊂P(iv)
P∩−P=θFor a given coneP⊂A, we can define a partial ordering ≼ with respect to P by x≼y if and only if y−x∈P. We shall write x≺y if x≼y and x≠y, while x≪y will stand for y−x∈intP, where intP denotes the interior of P.A coneP is called normal if there is a number K>0 such that for all x,y∈A,(5)θ≼x≼yimpliesx≤Ky.The least positive number satisfying the above inequality is called the normal constant ofP. Indeed, the number K cannot be less than 1; see [23]. A cone P is called regular if every increasing sequence which is bounded from above is convergent. In other words, if there is a y∈A such that(6)x1≼x2≼⋯≼xn≼⋯≼y,then there exists x∈A such that limn⟶∞xn−x=0. Equivalently, a cone P 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 coneP is called strongly minihedral if each subset of A which is bounded from above has a supremum. If P is a strongly minihedral cone, then every subset of A bounded below has an infimum (see [24, 25]).Throughout this paper, we always assume thatP is a cone over Banach algebra A with intP≠∅ and ≼ is the partial ordering with respect to P.Definition 1 (see [12, 16, 17, 18]).
LetX be a nonempty set and s≥1 be a constant. Suppose that the mapping d:X×X⟶A satisfies the following:(d1) θ≼dx,y for all x,y∈X and dx,y=θ if and only if x=y(d2) dx,y=dy,x for all x,y∈X(d3) dx,y≼sdx,z+dz,y for all x,y,z∈X
Thend is called a cone b-metric on X and X,d is called a cone b-metric space over Banach algebra A.Definition 2 (see [14, 16]).
LetX,d be a cone b-metric space over a Banach algebra A, x∈X and xn a sequence in X. Then,(i)
xn converges to x if, for every c∈A with c≫θ, there is a natural number N such that for all n>N,dxn,x≪c(ii)
xn is a Cauchy sequence if, for every c∈A with c≫θ, there is a natural number N such that for all n,m>N,dxn,xm≪c(iii)
X,d is a complete cone b-metric space if every Cauchy sequence is convergent in XIt is worth mentioning that unlike the usual metric or cone metric with a normal cone, coneb-metric is not necessarily continuous in general even if the cone is normal, as the following example shows.Example 1.
LetA=ℝ2 with the norm x1,x2=x1+x2. The multiplication is defined by(7)xy=x1,x2y1,y2=x1y1,x1y2+x2y1,where x=x1,x2,y=y1,y2∈A. It follows that A is a Banach algebra with a unit e=1,0. Let P=x1,x2∈ℝ2:x1,x2≥0. Then P is a normal cone with a normal constant K=1. Let X=ℕ∪∞×ℕ∪∞ for all x=x1,x2,y=y1,y2. Define the cone b-metric d:X×X⟶A by(8)dx1,x2,y1,y2=0,0,x1=y1,x2=y2;1x1−1y1,1x2−1y2,one ofx1,x2andy1,y2is odd and the other is odd or∞;6,6,one ofx1,x2andy1,y2is even and the other is even or∞;3,3,otherwise.
Note that for the definition of the coneb-metric d above, x1,x2 is odd if both x1 and x2 are odd; x1,x2 is ∞ if both x1 and x2 are ∞. Then it is sufficient to check that X,d is a cone b-metric space over Banach algebra A with the coefficient s=2.
Letzn=2n+1,2n+3 for each n∈ℕ. Then(9)dzn,∞=d2n+1,2n+3,∞,∞=12n+1−1∞,12n+3−1∞=12n+1,12n+3⟶0,0,n⟶∞.
This giveszn⟶∞. However,(10)dzn,2=d2n+1,2n+3,2,2=3,3↛6,6=d∞,∞,2,2=d∞,2,as n⟶∞. Therefore, the cone b-metric is not continuous even though the cone is normal.Definition 3 (see [26]).
LetP be a solid cone in a Banach algebra A. A sequence un⊂P is a c-sequence if for each c≫θ there exists n0∈ℕ such that un≪c for n≥n0.Lemma 1 (see [16]).
LetP be a solid cone in a Banach algebra A and let xn and yn be sequences in P. If xn and yn are c-sequences and α,β∈P, then αxn+βyn is a c-sequence.Lemma 2 (see [22]).
LetA be a Banach algebra with a unit e and x∈A. If the spectral radius rx of x is less than 1, i.e.,(11)rx=limn⟶∞xn1/n=infn≥1xn1/n<1,then e−x is invertible. Actually, e−x−1=∑i=0∞xi.Inspired by Definition3 in [24], we introduce a new notion of the distance between a set and a singleton in cone b-metric space over a Banach algebra A.Definition 4.
LetX,d be a cone b-metric space over a Banach algebra A and M be a nonempty subset of X. Let P be a normal and strongly minihedral cone. The distance between the set M and the singleton x is defined as follows:(12)dx,M=infdx,y:y∈M.


## 2. Bounded Compactness andT-Orbital Compactness

The concepts of bounded compactness andT-orbital compactness were discussed in usual metric spaces [7] and b-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 T-orbital compactness in the framework of cone b-metric spaces over Banach algebras, which are generalizations of metric spaces and b-metric spaces.Definition 5.
LetX,d be a cone b-metric space over a Banach algebra A with a unit e. The mapping T:X⟶X is said to be a generalized Kannan-type contraction, if it satisfies(13)dTx,Ty≺e2dx,Tx+dy,Ty,for all x,y∈X with x≠y.Definition 6.
LetX,d be a cone b-metric space over a Banach algebra A with a unit e. The mapping T:X⟶X is said to be a generalized Reich–Ćirić–Rus-type contraction, if it satisfies(14)dTx,Ty≺a1dx,Tx+a2dy,Ty+a3dx,y,for all x,y∈X with x≠y, where a1,a2,a3∈P with a1+a2+a3=e.Definition 7.
LetX,d be a cone b-metric space over a Banach algebra A and T be a self-mapping on X. Let x∈X and OTx=x,Tx,T2x,T3x,….
The spaceX,d is said to be boundedly compact, if every bounded sequence in X has a convergent subsequence.
The mappingT is said to be orbitally continuous at a point z∈X if for any sequence xn⊂OTx (for all x∈X), xn⟶z as n⟶∞ implies Txn⟶Tz as n⟶∞. Clearly, every continuous mapping is orbitally continuous, but not the converse.
The setX is said to be T-orbitally compact set, if every sequence in OTx has a convergent subsequence for all x∈X.Example 2.
LetA=Cℝ10,1×Cℝ10,1 with the norm(15)x1,x2=x1∞+x2∞+x1′∞+x2′∞.
Define the multiplication by(16)xy=x1,x2y1,y2=x1y1,x1y2+x2y1,where x=x1,x2,y=y1,y2∈A. Then A is a Banach algebra with a unit e=1,0. Let P=x1t,x2t∈A:x1t≥0,x2t≥0,t∈0,1.(1)
LetX=0,∞×0,∞ and define the cone b-metric d:X×X⟶A by dx1,x2,y1,y2t=x1−y12,x2−y22⋅αt∈P,∀x=x1,x2,y=y1,y2∈X,α>0. Then X,d is a complete cone b-metric space over Banach algebra A with s=2. Define mappings T1,T2:X⟶X as(17)T1x1,x2=x1n+13,x2m+13,n≤x1<n+1,m≤x2<m+1,T2x1,x2=3x1+1,3x2+2,for allx∈X and n∈ℕ. This clearly gives that X is T1-orbitally compact and boundedly compact but not T2-orbitally compact.(2)
LetX=0,1×0,1. The cone b-metric is defined the same as above and T:X⟶X is defined by Tx1,x2=x1/2,x2/3. We deduce that X is T-orbitally compact but not complete.(3)
LetX=−2,2×−2,2. The cone b-metric is defined the same as above and T:X⟶X is defined by(18)Tx1,x2=0,0,x1,x2∈−1,1×−1,1;x14,x26,otherwise.Then, for anyz0=x1,x2∈X,n∈ℕ,zn=Tzn−1,zn⟶θ implies Tzn⟶Tθ=θθ=0,0. So T is orbitally continuous but not continuous in X.In the rest of this section, we always assume thatX,d is a cone b-metric space over Banach algebra A with regular cone P such that dx,y∈intP for all x,y∈X with x≠y and the cone b-metric d is continuous.Theorem 5.
LetX,d be a boundedly compact cone b-metric space over Banach algebra A with a unit e and the coefficient s≥1. Let T:X⟶X be a generalized Reich–Ćirić–Rus-type contraction mapping and orbitally continuous. If e−a2−1 and e−s2a3−1 exist, then T has a unique fixed point z∈X and for each x∈X the iterated sequence Tnx converges to z; i.e., T is a Picard operator.Proof.
For an arbitraryx0∈X, let xn=Txn−1=Tnx0,n≥1. We assume that xn≠xn+1 for all n∈ℕ. Indeed, if for some n∈ℕ, xn=xn+1=Txn, then xn is the fixed point of T. Denote tn=dxn,xn+1 for each n∈ℕ. By (14), we have(19)tn=dxn,xn+1=dTxn−1,Txn≺a1dxn−1,xn+a2dxn,xn+1+a3dxn−1,xn=a1tn−1+a2tn+a3tn−1,which gives that tn≤e−a2−1a1+a3tn−1. Let h=e−a2−1a1+a3, then h=e by a1+a2+a3=e. Therefore, we have(20)θ≺⋯≺tn≺tn−1≺⋯≺t0=dx0,x1.
Because the cone is regular, there existsb≽θ in A such that tn⟶bn⟶∞. Thus, for all n,m∈ℕ, consider(21)dxn,xm≼sdxn,xn+1+dxn+1,xm≼sdxn,xn+1+s2dxn+1,xm+1+s2dxm+1,xm≺sdxn,xn+1+s2a1dxn,xn+1+a2dxm,xm+1+a3dxn,xm+s2dxm+1,xm=se+s2a1dxn,xn+1+s2a2+s2edxm,xm+1+s2a3dxn,xm,which implies that(22)dxn,xm≼e−s2a3−1se+s2a1dxn,xn+1+s2e+s2a2dxm,xm+1≺e−s2a3−1se+s2a1+s2e+s2a2t0.
Therefore,xn is bounded. Since X is boundedly compact, there is a convergent subsequence xni of xn and z∈X such that xni⟶zasi⟶∞. So Txni⟶Tz by the orbital continuity of T. If b≻θ, then(23)θ≺b=limi⟶∞dxni,Txni=dz,Tz.
Moreover,(24)θ≺b=limi⟶∞tni=limi⟶∞dTxni,T2xni=dTz,T2z≺a1dz,Tz+a2dTz,T2z+a3dz,Tz,which means(25)b=dTz,T2z≺dz,Tz=b,a contradiction. Thus, b=θ and z=Tz. That is, z is a fixed point of T. Then, the inequality (14) implies(26)dxn+1,z=dTxn,Tz≺a1dxn,Txn+a2dz,Tz+a3dxn,z=a1dxn,xn+1+a3dxn,z≼a1dxn,xn+1+a3s2dxn,xn+1+dxn+1,z.
Hence,e−s2a3dxn+1,z≼a1+s2a3dxn,xn+1⟶θ. This gives Tnx⟶z; i.e., T is a Picard operator.
Finally, the uniqueness of the fixed point can be obtained by (14). If y is another fixed point of T, then(27)dy,z=dTy,Tz≺a1dy,Ty+a2dz,Tz+a3dy,z=a3dy,z,leading to a contradiction. Therefore, z is the unique fixed point of T.Theorem 6.
LetX,d be a T-orbitally compact cone b-metric space over Banach algebra A with a unit e and the coefficient s≥1, where T:X⟶X is a generalized Reich–Ćirić–Rus-type contraction mapping and orbitally continuous. If e−a2−1 and e−s2a3−1 exist, then T has a unique fixed point z∈X, and for each x∈X the iterated sequence Tnx converges to z; i.e., T is a Picard operator.Proof.
The analysis is similar to that in the proof of Theorem5. We firstly get the sequence xn=Txn−1=Tnx0,n≥1. If there is an integer n∈ℕ such that xn=xn+1=Txn, then xn is the fixed point. Without loss of generality, we assume that xn≠xn+1,∀n∈ℕ. We can prove that tn⟶b≽θ∼n⟶∞, where tn and b are the same as above. As X is T-orbitally compact, there is a convergent subsequence xni of xn and z∈X such that xni⟶zasn⟶∞. By orbital continuity of T, we obtain Txni⟶Tz. The rest proof is similar to Theorem 5.Corollary 1.
LetX,d be a boundedly compact cone b-metric space over Banach algebra A with a unit e and the coefficient s≥1. Let T:X⟶X be a generalized Kannan-type contraction mapping which is orbitally continuous. Then T has a unique fixed point z∈X and for each x∈X the iterated sequence Tnx converges to z; i.e., T is a Picard operator.Proof.
Takinga1=a2=e/2 and a3=θ, we obtain the conclusion by Theorem 5.Corollary 2.
LetX,d be a T-orbitally compact cone b-metric space over Banach algebra A with a unit e and the coefficient s≥1, where T:X⟶X is a generalized Kannan-type contraction mapping and orbitally continuous. Then T has a unique fixed point z∈X and for each x∈X, the iterated sequence Tnx converges to z; i.e., T is a Picard operator.Proof.
The proof is analogous.Remark 1.
Theorems5 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 T-orbital compactness and T-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.
LetA=ℝ2 with the norm x1,x2=x1+x2. The multiplication is defined by(28)xy=x1,x2y1,y2=x1y1,x1y2+x2y1,where x=x1,x2,y=y1,y2∈A. It follows that A is a Banach algebra with a unit e=1,0. Let P=x1,x2∈ℝ2:x1,x2≥0. Then P is a normal cone with a normal constant K=1. Let X=−1,2∪3×−1,2∪3 and define the cone b-metric d:X×X⟶A by(29)dx1,x2,y1,y2=k1x1−y12,k2x2−y22∈P,for all x=x1,x2,y=y1,y2 in X, where k1,k2>0 are constants. Furthermore, define the mapping T:X⟶X by(30)Tx=17x1,15x2,x≠3;0,0,otherwise,where x≠3 is equivalent to x=x1,x2≠3,x2 and x=x1,x2≠x1,3. Obviously, T is not continuous but T-orbitally continuous. Moreover, X,d is an incomplete cone b-metric space over Banach algebra A with s=2 but T-orbitally compact. Let a1=4/9,0,a2=1/3,0,a3=2/9,0, then a1+a2+a3=e and e−a2−1,e−s2a3−1 exist. In order to check the generalized Reich–Ćirić–Rus-type contraction, we have the following three cases:(i)
Ifx≠3 and y=3, then(31)dTx,T3=k117x1−02,k215x2−02=149k1x12,125k2x22≺49,03649k1x12,1625k2x22+13,00,0+29,0k1x1−22,k2x2−22=49,0k1x1−17x12,k2x2−15x22+13,00,0+29,0k1x1−22,k2x2−22=a1dx,Tx+a2d3,T3+a3dx,y.(ii)
Ifx,y∈X with x≠3,y≠3 and x≠y, then(32)dTx,Ty=k117x1−17y1|2,k2|15x2−15y22≺49,03649k1x12,1625k2x22+13,03649k1y12,1625k2y22+29,0k1|x1−y1|2,k2|x2−y2|2=49,0k1|x1−17x1|2,k2|x2−15x2|2+13,0k1|y1−17y1|2,k2|y2−17y2|2+29,0k1|x1−y1|2,k2|x2−y2|2=a1dx,Tx+a2dy,Ty+a3dx,y.(iii)
Ifx=3 and y=3, then(33)dTx,Ty=0,0≺a1dx,Tx+a2dy,Ty+a3dx,y,is clearly true. Therefore, the mappingT has a unique fixed point in X 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 coneb-metric spaces over Banach algebras, under the condition of asymptotic regularity. The regularity or normality of the cone and the continuity of the cone b-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]).
LetX,d be a metric space. The mapping T:X⟶X is said to be asymptotically regular, if limn⟶∞dTn+1x,Tnx=0 for all x∈X.Definition 9.
LetX,d be a cone b-metric space over a Banach algebra A. The mapping T:X⟶X is said to be asymptotically regular, if for every c∈A with c≫θ, there is a natural number N such that for all n≥N,x∈X,dTn+1x,Tnx≪c. That is, dTn+1x,Tnx is a c-sequence for all x∈X.Compared with Definition8, Definition 9 shows a great generalization. The condition that dTn+1x,Tnx is a c-sequence is a sharp improvement of that limn⟶∞dTn+1x,Tnx=0. 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 b-metric space over Banach algebra A. Inspired by the concept of T-orbitally complete in metric space [4], we give the similar concept in cone b-metric space over a Banach algebra A as follows.Definition 10.
LetX,d be a cone b-metric space over a Banach algebra A. The space X,d is said to be T-orbitally complete, if every Cauchy sequence which is contained in OTx for some x∈X converges in X. Every complete cone b-metric space over Banach algebra A is T-orbitally complete for any T, but a T-orbitally complete cone b-metric space over Banach algebra A needs not be complete.The continuity of the mapping and the coneb-metric is not necessary in the following theorems.Theorem 7.
LetT:X⟶X be an asymptotically regular mapping in the T-orbitally complete cone b-metric space over Banach algebra A with a unit e and the coefficient s≥1. If there exist a1,a2,a3∈P with ra2<1/s and ra3<1/s2 such that(34)dTx,Ty≼a1dx,Tx+a2dy,Ty+a3dx,y,for all x,y∈X, then T has a unique fixed point z∈X and, for each x∈X, the iterated sequence Tnx converges to z; i.e., T is a Picard operator.Proof.
For an arbitraryx0∈X, let xn=Txn−1=Tnx0,n≥1. Without loss of generality, we assume that xn≠xn+1,∀n∈ℕ. Indeed, if for some n∈ℕ, xn=xn+1=Txn, then xn is the fixed point of T. By asymptotic regularity of T, dxn,xn+1 is a c-sequence. For all m>n, we have(35)dxn,xm≼sdxn,xn+1+dxn+1,xm≼sdxn,xn+1+s2dxn+1,xm+1+s2dxm+1,xm≼sdxn,xn+1+s2a1dxn,xn+1+a2dxm,xm+1+a3dxn,xm+s2dxm+1,xm≼sdxn,xn+1+s2a1dxn,xn+1+a2dxm,xm+1+a3dxn,xm+s2dxm+1,xm.
Now,e−s2a3 is invertible, which is due to the fact that rs2a3=s2ra3<1. It follows that(36)dxn,xm≼e−s2a3−1se+s2a1dxn,xn+1+s2e+s2a2dxm,xm+1.
By Lemma1, xn is a Cauchy sequence in X. Since X,d is T-orbitally complete, there exists z∈X such that xn⟶zasn⟶∞. We shall prove Tz=z; i.e., z is the fixed point of T. By the inequality (34), we have(37)dz,Tz≼sdz,Txn+dTxn,Tz≼sdz,Txn+a1dxn,Txn+a2dz,Tz+a3dxn,z,which implies that(38)dz,Tz≼e−sa2−1sdz,xn+1+a1dxn,xn+1+a3dxn,z.
The right side of the above equality is ac-sequence by Lemma 1, so z=Tz. Using a similar analysis to Theorem 5, we can prove that z is unique.Corollary 3.
LetT:X⟶X be an asymptotically regular mapping on the T-orbitally complete cone b-metric space over Banach algebra A with a unit e and the coefficient s≥1. If there exists a∈P with ra<1/s2 such that(39)dTx,Ty≼adx,Tx+dy,Ty+dx,y,for all x,y∈X, then T has a unique fixed point z∈X and for each x∈X the iterated sequence Tnx converges to z; i.e., T is a Picard operator.Now, ifT is orbitally continuous, then the condition ra2<1/s can be deleted.Theorem 8.
LetT:X⟶X be an asymptotically regular mapping on the T-orbitally complete cone b-metric space over Banach algebra A with a unit e and the coefficient s≥1. There exist a1,a2,a3∈P with ra3<1/s2 such that(40)dTx,Ty≼a1dx,Tx+a2dy,Ty+a3dx,y,for all x,y∈X. If T is orbitally continuous, then T has a unique fixed point z∈X and for each x∈X the iterated sequence Tnx converges to z; i.e., T is a Picard operator.Proof.
According to Theorem7, we see that there exists z∈X such that xn⟶zasn⟶∞. Because T is orbitally continuous, we have xn+1=Txn⟶Tzasn⟶∞. Then, z=Tz. Similar to Theorem 7, the conclusion is true.Corollary 4.
LetT:X⟶X be an asymptotically regular mapping on the T-orbitally complete cone b-metric space over Banach algebra A with a unit e and the coefficient s≥1. There exist a,b∈P with rb<1/s2 such that(41)dTx,Ty≼adx,Tx+dy,Ty+bdx,y,for all x,y∈X. If T is orbitally continuous, then T has a unique fixed point z∈X and for each x∈X the iterated sequence Tnx converges to z; i.e., T is a Picard operator.Remark 2.
Corollary3 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 b-metric spaces or b-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.
LetA=Cℝ10,1 and X=0,1. For each x∈A, x=x∞+x′∞. The multiplication is defined by its usual pointwise multiplication. Then A is a Banach algebra with a unit e=1. Define dx,yt=x−y2ϕ for all x,y∈X and ϕ∈P=ft∈A:ft≥0,t∈0,1. Then P is a nonnormal cone and X,d is a complete cone b-metric space over Banach algebra A with coefficient s=2. Choose a1t=4t+3,a2t=5t+2 and a3t=1/9t+1/9. We deduce that(42)a3nt=19t+19n,a3nt′=n919t+19n−1,hence that (t = 1)(43)a3n=a3n∞+a3n′∞=29n+n929n−1=n929n−19n⋅29+1=n929n−11+2n,and finally that(44)ra3=limn⟶∞a3n1/n=limn⟶∞n91/n29n−1/n1+2n1/n=29.
This means(45)ra3=29<14=1s2.
Define the mappingT:X⟶X by(46)Tx=x10sinx9,x∈ℚ∩X;log1+x4,x∈ℝ\ℚ∩X.
ThenT 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 allx,y∈ℚ∩X,(47)dTx,Tyt=x10sinx9−y10siny92ϕ≼190x+1102x−y2ϕ≼4t+3x−x10sinx92ϕ+5t+2y−log1+y42ϕ+19t+19x−y2ϕ=a1dx,Txt+a2dy,Tyt+a3dx,yt.(2)
For allx,y∈ℝ\ℚ∩X,(48)dTx,Tyt=log1+x4−log1+y42ϕ≼142x4−y42ϕ≼4t+3x−log1+x42ϕ+5t+2y−log1+y42ϕ+19t+19x−y2ϕ=a1dx,Txt+a2dy,Tyt+a3dx,yt.(3)
For allx∈ℚ∩X, y∈ℝ\ℚ∩X,(49)dTx,Tyt=x10sinx9−log1+y42ϕ≼x10sinx92ϕ+log1+y42ϕ≼x102ϕ+y42ϕ≼4t+3x−x10sinx92ϕ+5t+2y−log1+y42ϕ+19t+19x−y2ϕ=a1dx,Txt+a2dy,Tyt+a3dx,yt.
Similarly, we can also prove thatdTx,Tyt≼a1dx,Txt+a2dy,Tyt+a3dx,yt for all x∈ℝ\ℚ∩X, y∈ℚ∩X. Therefore, T has a unique fixed point in X by Theorem 8.


## 4. Completeness and Fixed Point

ByCorollaries 1 and 2, we know that if the generalized Kannan-type contraction mapping T is orbitally continuous in boundedly compact or T-orbitally compact cone b-metric spaces over Banach algebras, then T has a unique fixed point. Conversely, if T has a unique fixed point in cone b-metric spaces over Banach algebras, then what conditions do X,d have to satisfy? Now, we prove an important theorem showing that the completeness of cone b-metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in X.Theorem 9.
LetX,d be a cone b-metric space over Banach algebra A with a unit e and the coefficient s≥1. Let P be a normal and strongly minihedral cone. If every self-mapping T satisfying(50)dTx,Ty≺e2dx,Tx+dy,Ty,for all x,y∈X with x≠y has a unique fixed point, then X,d must be a complete cone b-metric space over Banach algebra A.Proof.
On the contrary, suppose thatX,d is not complete; then, there exists a Cauchy sequence xn in X, which is not convergent. If it has a convergent subsequence xnk of xn such that xnk⟶z∈X as k⟶∞, then(51)dxm,z≼sdxm,xnk+dxnk,z,for all m≥nk. This is a c-sequence since dxm,xnk and dxnk,z are c-sequences. Thus, we can assume that all terms of the sequence xn are distinct. Let M=xn:n∈ℕ; then, dx,M≻θ for all x∈X−M by the fact that the sequence xn does not converge in X. Let x∈X be an arbitrary point. If x∈X−M, then there is an integer nx∈ℕ such that(52)dxm,xnx≺e2dx,M≼e2dx,xn,for all m≥nx and arbitrary n∈ℕ. That is(53)dxm,xnx≺e2dx,xn,∀m≥nxand∀n∈ℕ.
Supposex′∈M; then x′=xn0 for some n0∈ℕ. Since xn is a Cauchy sequence, we can find some n0′∈ℕ such that(54)dxm,xn0′≺e2dxn0′,xn0,∀m≥n0′>n0.
Now, defineT:X⟶X by(55)Tx=xnx,ifx∈X−M;xn0′,ifx∈Mandx=xn0.
For anyx,y∈X with x≠y, we divide the following proof into three cases.Case 1.
Ifx,y∈X−M, then Tx=xnx and Ty=xny. Without loss of generality, we assume that ny≥nx. By (53), we get(56)dTx,Ty=dxnx,xny≺e2dx,xnx=e2dx,Tx.
This givesdTx,Ty≺e/2dx,Tx+dy,Ty.Case 2.
Ifx,y∈M, then x=xn0 and y=xm0 for some n0,m0∈ℕ. Then Tx=xn0′ and Ty=xm0′. Without loss of generality, we assume that m0′≥n0′. By (54), we deduce that(57)dTx,Ty=dxm0′,xn0′≺e2dxn0′,xn0=e2dTx,x,which established the formula dTx,Ty≺e/2dx,Tx+dy,Ty.Case 3.
Ifx∈X−M,y∈M, then y=xn0 for some n0∈ℕ. Therefore Tx=xnx and Ty=xn0′. If n0′≥nx, by (53), we have(58)dTx,Ty=dxnx,xn0′=dxn0′,xnx≺e2dx,xnx=e2dx,Tx,which yields dTx,Ty≤e/2dx,Tx+dy,Ty. If nx≥n0′, by (54), we see that(59)dTx,Ty=dxnx,xn0′≺e2dxn0′,xn0=e2dy,Ty,which also gives dTx,Ty≺e/2dx,Tx+dy,Ty. Therefore, for all x,y∈X with x≠y, we always have dTx,Ty≺e/2dx,Tx+dy,Ty. That is, T is a generalized Kannan-type contraction mapping which has no fixed point in X, a contradiction. Hence, the assumption does not hold and the space X,d must be a complete cone b-metric space over Banach algebra A. The conclusion is true.Remark 3.
According to the proof of Theorem9, we see at once that inequality (50) can be replaced by(60)dTx,Ty≼kdx,Tx+dy,Ty,for all x,y∈X and a fixed point k∈P.



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*Source: 7549981-2021-09-27.xml*