Non - solid cone b - metric spaces over Banach algebras and ﬁ xed point results of contractions with vector - valued coe ﬃ cients

: In this article, without requiring solidness of the underlying cone, a kind of new convergence for sequences in cone b - metric spaces over Banach algebras and a new kind of completeness for such spaces, namely, wrtn - completeness, are introduced. Under the condition that the cone b - metric spaces are wrtn - complete and the underlying cones are normal, we establish a common ﬁ xed point theorem of contractive conditions with vector - valued coe ﬃ cients in the non - solid cone b - metric spaces over Banach algebras, where the coe ﬃ cients s 1 ≥ . As consequences, we obtain a number of ﬁ xed point theorems of contractions with vector - valued coe ﬃ cients, especially the versions of Banach contraction principle, Kannan ’ s and Chatterjea ’ s ﬁ xed point theorems in non - solid cone b - metric spaces over Banach algebras. Moreover, some valid examples are presented to support our main results.


Introduction
As is known to all, the fixed point theory with regard to modern metric was developed from the classical Banach contraction principle (see [1]), which is important and useful in almost all fields of applied mathematical analysis. Afterwards, Kannan [2] and Chatterjea [3] proved the fixed point theorems as follows.
Theorem. Let X d , ( ) be a complete metric space and T X X : → be a mapping such that there exists γ 0, Then T has a unique fixed point in X.
In the cone metric spaces, the distance between x and y is defined by a vector in an ordered Banach space E, quite different from that which is defined by a non-negative real number in usual metric spaces. They indicated the corresponding version of Banach contraction principle and some preliminary properties in cone metric spaces. Later on, by deleting the normality of the cone in [7], Rezapour and Hamlbarani [8] discussed some fixed point results, which are generalizations of the correlative results in [7]. Since then, a lot of authors have been attracted to the field of fixed point theory in cone metric spaces and more general ones-cone b-metric spaces (see ), while the concept of cone b-metric spaces was defined by Hussain and Shah [9] and Shah et al. [10], which is an extension of cone metric spaces and b-metric spaces. The authors also investigated some meaningful topological properties and fixed point theorems in their work. In 2013, Huang and Xu [11] established some fixed point results and gave the relevant application in cone b-metric spaces. Marian and Branga [12] obtained some common fixed point results for a pair of mappings in b-cone metric spaces, but the main methods in their work relied strongly on a nonlinear scalarization function ξ Y : e → . In [13], Shi and Xu also obtained common fixed point theorems in non-normal but solid cone b-metric spaces. However, these results were dependent on the solidness of the underlying cones.
Recently, Liu and Xu [14] and Han and Xu [15] continued to further investigate cone metric spaces by means of Banach algebras, instead of Banach spaces and considering the contractive constants to be vectors. They presented some fixed point theorems of generalized Lipschitz contractions with the assumption that the underlying cones are solid. Since 1997, in general, almost all the results obtained in the present literature except [16] have been showed in the setting of cone metric spaces (or cone b-metric spaces) with the assumption that the underlying cones are solid. In fact, solidness of the cone is an important (sometime, crucial) condition, since without it the interior points could not be used to define the convergence of the sequences in a natural way. Nevertheless, some results were obtained by Kunze et al. in the case when the cone was not solid in [16]. However, as Janković et al. [17] indicated, the approach appearing in [16] is quite restrictive since some strong assumptions (separability and reflexivity of the space) are used. Because there are many examples of cone metric spaces with empty interiors of the cones, a corresponding fixed point theory would be welcome.
In this article, inspired by [16] and [17], we try to establish fixed point theory for contractions with vector-valued coefficients in the setting of cone b-metric spaces (and as their special cases, cone metric spaces) over Banach algebras by deleting the solidness of the underlying cones. Furthermore, we apply some main results to solve the existence and uniqueness of the solution for nonlinear integral equations.

Preliminaries
Firstly, let us recall some preliminary concepts of Banach algebras and cone b-metric spaces.
Let be a Banach algebra over = or . That is, is a Banach space in which an operation of multiplication is defined, subject to the following properties for all x y z α , , , ∈ ∈ : The vector e ∈ is called a unit (i.e., a multiplicative identity) in if ex xe x = = for all x ∈ . A Banach algebra is called unital if it has a unit. An element x ∈ is said to be invertible if there is an inverse element y ∈ such that xy yx e = = . The inverse of x is denoted by x 1 − . Here, x x x x n = ⋅ ⋅⋯⋅ is the n-fold product of x with itself, and x e 0 ≔ . It is obvious that x x n n ∥ ∥ ∥ ∥ ≤ for all n ∈ . Let x ∈ . The spectrum σ x ( ) of x is the set of all λ ∈ such that λe x − is not invertible. The spectral radius r x ( ) of x is defined by For more details, we refer to [18]. A well-known fact is that the spectrum is a non-empty compact subset of the complex plane. The following proposition for the spectral radius formula is well known (see [21]). A subset P of is called a cone if: (1) P is non-empty closed and θ e P , { } ⊂ , where θ denotes the null of the Banach algebra ; (2) αP βP P + ⊂ for all non-negative real numbers α β , ; For a given cone P ⊂ , we can define a partial ordering ≼ with respect to P by x y ≼ if and only if y x P − ∈ . x y ≺ will stand for x y ≼ and x y ≠ , while x y ≪ will stand for y x P int − ∈ , where P int denotes the interior of P.
The cone P is called normal if there is a number M 0 > such that for all x y , ∈ , The least positive number satisfying above is called the normal constant of P.
In the following, unless otherwise specified, we always assume that is a real unital Banach algebra (the term "real" means that the algebra is over ) with a unit, P is a cone in Banach algebra and ≼ is the partial ordering with respect to P. [9,10,13]) Let X be a nonempty set and s 1 ≥ be a given real number. A mapping d X X : × → is said to be cone b-metric if and only if for all x y z X , , ∈ the following conditions are satisfied: ) is called a cone b-metric space over a Banach algebra .
Definition 2.2. (See [9,10,13]) Let X d , ( ) be a cone b-metric space over a Banach algebra with the coefficient s 1 ≥ , x X ∈ where the underlying cone P is solid and let x n { } be a sequence in X. Then we say (i) x n { } converges to x with respect to the solidness of P (for convenience, we say x n { } wrts-converges to x) whenever for every ε ∈ with θ ε ≪ there is a positive number N ∈ such that d x x ε , n ( ) ≪ for all n N ≥ . (Note that here "wrts" means "with respect to solidness".) We denote this by ) is wrts-complete if every wrts-Cauchy sequence is wrts-convergent.
) be a cone b-metric space over a Banach algebra. Let x X ∈ and x n { } be a sequence in X. Then ) is wrtn-complete if every wrtn-Cauchy sequence is wrtn-convergent.
) be a cone b-metric space over a Banach algebra with coefficient s 1 ≥ . Let P be a normal cone with the normal constant M. The limit of a wrtn-convergent sequence in a cone b-metric space over Banach algebra is unique. That is, if for any given sequence x n X by the normality of P, we obtain we can obtain the corresponding definitions and properties in non-solid cone metric spaces over Banach algebras. Similar to the ones presented earlier, they can be obtained in non-solid cone metric spaces and non-solid cone b-metric spaces. Moreover, by [7,Lemmas 1,4], if the cone P is solid and normal, we can check that ) is a wrts-complete cone metric space if and and only if X d , ( ) is a wrtn-complete cone metric space.  [18,19]) Let be a unital Banach algebra with a unit e and x y , ∈ . If x commutes with y, then the following hold: In the following, we establish some useful and auxiliary well-known results on Banach algebra. Their proofs can be found in some literature, but we show them for the sake of completeness and the readers' convenience.
Lemma 2.4. Let be a unital Banach algebra with a unit and x ∈ . If the spectral radius r x ( ) of x is less than 1, then e x − is invertible. Moreover, For every n ∈ , write C x n i n i 0 = ∑ = . For any m n , ∈ with m n N ε > ≥ , by (2.1), we obtain and hence, x n 0 as .
It is worth mentioning that Lemma 2.4 can be rewritten as follows.
Lemma 2.5. Let be a unital Banach algebra with a unit e and x ∈ . Lemma 2.7. Let be a Banach algebra with a normal cone P. If x y , ∈ satisfy θ x y ≼ ≼ and x commutes with y, then the following hold: In particular, if A is commutative, then the spectral radius r is monotone with respect to P.
Proof. The proof of conclusion (1) is obtained by mathematic induction on n. Clearly, conclusion (1) is true for n 1 = .
Since θ x y ≼ ≼ , we have x P ∈ , y P ∈ and y x P − ∈ . By PP P ⊂ and x commutes with y, we see that and y x y y y x P, which means x y k k 1 1 ≼ + + holds. Hence, conclusion (1) is also true for n k 1 = + . This means conclusion (1) is true for all n ∈ .
To see conclusion (2), let M be the normal constant of P. By conclusion (1) [20]) Let X be a set. The mappings f g X X , : → are said to be weakly compatible, if for every x X ∈ holds fgx gfx = whenever fx gx = .
Definition 2.5. (See [21]) Let f and g be self-maps of a set X. If w fx gx = = for some x in X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g . 3 Fixed point theorems in non-solid cone b-metric spaces over Banach algebras In this section, we always suppose that P is a normal cone with normal constant M 1 ≥ . Under the aforementioned definitions and lemmas, some common fixed point results for two weakly compatible selfmappings in non-solid cone b-metric spaces over Banach algebras are presented.
) be a wrtn-complete cone b-metric space over a Banach algebra with the coefficient s 1 ≥ and the mappings f g X X , : → . Let a a a a a P , , , Then f and g have a unique coincidence point in X. Furthermore, if f and g are weakly compatible, then they have a unique common fixed point in X.
Proof. Let x X 0 ∈ be given. By (H1), there exists an x X 1 ∈ such that fx gx If g X X ( ) ⊂ is wrtn-complete, there exist q g X ( ) ∈ and p X ∈ such that gx q n → ‖⋅‖ as n → +∞ and gp q = Similarly, by the normality of P and the fact that gx n { } is a wrtn-Cauchy sequence and gx q n → ‖⋅‖ ( n → +∞),  Furthermore, if f and g are weakly compatible, by Lemma 2.8, we conclude that q is the unique common fixed point of f and g . □ It is sufficient to obtain the following fixed point theorems of contractions with vector-valued coefficients in the setting of non-solid cone b-metric or cone metric spaces over Banach algebras from Theorem 3.1, so we omit their proofs.
) be a wrtn-complete cone b-metric space over a Banach algebra with coefficient s 1 ≥ . Suppose the mapping T X X : → satisfies the generalized Banach contractive condition d Tx Ty kd x y for all x y X , , , , , . Then T has a unique fixed point in X. And for any where k P ∈ is a constant vector satisfying r k 0, . Then T has a unique fixed point in X. And for any x X, ∈ iteration sequence T x n { } wrtn-converges to the fixed point.
Remark 3.1. We emphasize that one cannot use the existed fixed point results in b-metric spaces or metric spaces to deduce the main results presented in the setting of wrtn-complete non-solid cone b-metric spaces or cone metric spaces over Banach algebras by virtue of the method from [17].
) is a wrtn-complete cone metric space over a Banach algebra and the cone P is normal with normal constant M. Suppose the mapping T X X : → satisfies the generalized Banach contractive condition d Tx Ty kd x y x y X , , , f o r a l l , , where k P ∈ is a constant vector satisfying r k 0, 1 ( ) ( ) ∈ . By [17], we can suppose that M 1. = Therefore, according to (3.7), we have is a complete metric space, but one cannot conclude that T has a unique fixed point in X by using the famous Banach contraction principle, since in general one cannot assert k 1 ∥ ∥ < though r k 0, 1 ( ) ( ) ∈ . These results improve and extend many relevant results in metric spaces [30,31].
In this section, we will present some examples to show our main results are effective tools to verify the uniqueness of solutions to fixed point equalities whether the underlying cones are solid or non-solid.
) is a normal and solid cone b -metric space over a Banach algebra with the coefficient s 2 1 Considering a simple inequality Then P is a normal but non-solid cone of the real Banach algebra with the operations as follows: It is easy shown that X d , ( ) is a wrtn-complete cone metric space over Banach algebra with the norm   Remark 4.1. Recently, many authors investigated the problem of whether all the fixed point results in cone b-metric spaces (cone metric spaces) are equivalent to that in b-metric spaces (metric spaces). The equivalent relationship was established between the fixed point results in metric and in cone metric spaces, see [22][23][24][25][26][27][28][29]. However, it is significant to point out that one is unable to show that the non-solid cone b-metric spaces (cone metric spaces) over Banach algebras introduced in this article are equivalent to any b-metric spaces (metric spaces), even though the cone is normal. This is due to the fact that all the methods to prove such equivalence appearing in the literature strongly rely on the solidness of the cones, which shows that these methods together with the corresponding techniques are invalid.
Remark 4.2. The main results in this article are a valuable addition to the existing literature concerning fixed point theory in the context of metric spaces or abstract metric spaces such as references [30][31][32][33][34].
Funding information: The research was partially supported by the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities' Association (No. 202101BA070001-045).