Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients

Shaoyuan Xu; Suyu Cheng; Yan Han

doi:10.1515/math-2022-0569

Open Access Published by De Gruyter Open Access April 4, 2023

Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients

Shaoyuan Xu , Suyu Cheng and Yan Han

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0569

Abstract

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 fixed point theorem of contractive conditions with vector-valued coefficients in the non-solid cone b -metric spaces over Banach algebras, where the coefficients s ≥ 1 . As consequences, we obtain a number of fixed point theorems of contractions with vector-valued coefficients, especially the versions of Banach contraction principle, Kannan’s and Chatterjea’s fixed point theorems in non-solid cone b -metric spaces over Banach algebras. Moreover, some valid examples are presented to support our main results.

Keywords: non-solid cone b-metric spaces over Banach algebras; contractions with vector-valued coefficients; coincidence point; fixed point theorems; wrtn-convergence and wrtn-completeness

MSC 2010: 47H10; 54H25

1 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 , 1 2 satisfying

d ( T x , T y ) ≤ γ ( d ( x , T x ) + d ( y , T y ) ) f o r a l l x , y ∈ X ,

d ( T x , T y ) ≤ γ ( d ( x , T y ) + d ( y , T x ) ) f o r a l l x , y ∈ X .

Then T has a unique fixed point in X.

Some interesting fixed point theorems in K -metric and K -normed linear spaces were obtained by Perov [4], Vandergraft [5], Zabrejko [6], and relevant literature therein. In 2007, these spaces was reintroduced by Huang and Zhang [7] under the definition of cone metric spaces and some fixed point theorems in such spaces were discussed. Since then, the cone metric fixed point theory is prompted to investigate by lots of authors; for detail, see [8–27] and the references therein.

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 [7–28]), 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 e : Y → R . 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.

2 Preliminaries

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

Let A be a Banach algebra over K = R or C . That is, A is a Banach space in which an operation of multiplication is defined, subject to the following properties for all x , y , z ∈ A , α ∈ K :

( x y ) z = x ( y z ) ;
x ( y + z ) = x y + x z and ( x + y ) z = x z + y z ;
α ( x y ) = ( α x ) y = x ( α y ) ;
‖ x y ‖ ⩽ ‖ x ‖ ‖ y ‖ .

The vector e ∈ A is called a unit (i.e., a multiplicative identity) in A if e x = x e = x for all x ∈ A . A Banach algebra is called unital if it has a unit. An element x ∈ A is said to be invertible if there is an inverse element y ∈ A such that x y = y x = e . The inverse of x is denoted by x − 1 . Here, x n = x ⋅ x ⋅ ⋯ ⋅ x is the n -fold product of x with itself, and x 0 ≔ e . It is obvious that ∥ x n ∥ ≤ ∥ x ∥ n for all n ∈ N . Let x ∈ A . The spectrum σ ( x ) of x is the set of all λ ∈ K such that λ e − x is not invertible. The spectral radius r A ( x ) of x is defined by

r A ( x ) = sup { ∣ λ ∣ : λ ∈ σ ( x ) } .

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 A is called a cone if:

P is non-empty closed and { θ , e } ⊂ P , where θ denotes the null of the Banach algebra A ;
α P + β P ⊂ P for all non-negative real numbers α , β ;
P 2 = P P ⊂ P ;
P ∩ ( − P ) = { θ } .

For a given cone P ⊂ A , 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 ∈ int P , where int P denotes the interior of P .

The cone P is called normal if there is a number M > 0 such that for all x , y ∈ A ,

θ ≼ x ≼ y implies ‖ x ‖ ⩽ M ‖ y ‖ .

The least positive number satisfying above is called the normal constant of P .

In the following, unless otherwise specified, we always assume that A is a real unital Banach algebra (the term “real” means that the algebra is over R ) with a unit, P is a cone in Banach algebra A and ≼ is the partial ordering with respect to P .

Definition 2.1

(See [9,10,13]) Let X be a nonempty set and s ≥ 1 be a given real number. A mapping d : X × X → A is said to be cone b -metric if and only if for all x , y , z ∈ X the following conditions are satisfied:

θ ≺ d ( x , y ) with x ≠ y and d ( x , y ) = θ if and only if x = y ;
d ( x , y ) = d ( y , x ) ;
d ( x , y ) ≼ s [ d ( x , z ) + d ( z , y ) ] .

The pair ( X , d ) is called a cone b -metric space over a Banach algebra A .

Definition 2.2

(See [9,10,13]) Let ( X , d ) be a cone b -metric space over a Banach algebra A 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

{ 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 ∈ A with θ ≪ ε there is a positive number N ∈ N such that d ( x n , x ) ≪ ε for all n ≥ N . (Note that here “wrts” means “with respect to solidness”.) We denote this by x n → ≪ x ( n → + ∞ ) .
{ x n } is a wrts-Cauchy sequence whenever for every ε ∈ A with θ ≪ ε there is a positive number N ∈ N such that d ( x n , x m ) ≪ ε for all n , m ≥ N .
( X , d ) is wrts-complete if every wrts-Cauchy sequence is wrts-convergent.

Definition 2.3

Let ( X , d ) be a cone b -metric space over a Banach algebra. Let x ∈ X and { x n } be a sequence in X . Then we say

{ x n } converges to x with respect to the norm of E (for convenience, we may say { x n } wrtn-converges to x ) if and only if ‖ d ( x n , x ) ‖ → 0 as n → + ∞ (Note that here “wrtn” means “with respect to normality.”) We denote this by x n → ‖ ⋅ ‖ x ( n → + ∞ ) .
{ x n } is a wrtn-Cauchy sequence if and only if ‖ d ( x n , x m ) ‖ → 0 as n , m → + ∞ .
( X , d ) is wrtn-complete if every wrtn-Cauchy sequence is wrtn-convergent.

Lemma 2.1

Let ( X , d ) 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 A is unique. That is, if for any given sequence { x n } ( n ∈ N ) ⊂ X , there exist x , y ∈ X such that x n → ‖ ⋅ ‖ x and x n → ‖ ⋅ ‖ y as n → + ∞ , then x = y .

Proof

Since

d ( x , y ) ≼ s [ d ( x n , x ) + d ( x n , y ) ] ,

by the normality of P , we obtain

‖ d ( x , y ) ‖ ≤ s M ‖ d ( x n , x ) + d ( x n , y ) ‖ ≤ s M ( ‖ d ( x n , x ) ‖ + ‖ d ( x n , y ) ‖ ) .

So it follows from the wrtn-convergence of { x n } that d ( x , y ) = θ , i.e., x = y . □

Remark 2.1

Taking s = 1 , 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 x n → ≪ x ( n → + ∞ ) if and only if x n → ‖ ⋅ ‖ x ( n → + ∞ ) ; ( X , d ) is a wrts-complete cone metric space if and and only if ( X , d ) is a wrtn-complete cone metric space.

Lemma 2.2

(See [18]) Let A be a unital Banach algebra and x ∈ A . Then the spectral radius r A ( x ) of x satisfies

r A ( x ) = lim n → + ∞ ‖ x n ‖ 1 n = inf n ∈ N ‖ x n ‖ 1 n .

In particular, r A ( x ) = r A ( − x ) and r A ( x ) ≤ ‖ x ‖ .

Lemma 2.3

(See [18,19]) Let A be a unital Banach algebra with a unit e and x , y ∈ A . If x commutes with y , then the following hold:

r A ( x + y ) ≤ r A ( x ) + r A ( y ) ;
r A ( x y ) ≤ r A ( x ) r A ( y ) ;
∣ r A ( x ) − r A ( y ) ∣ ≤ ∣ r A ( x − y ) ∣ .

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 A be a unital Banach algebra with a unit and x ∈ A . If the spectral radius r A ( x ) of x is less than 1, then e − x is invertible. Moreover,

( e − x ) − 1 = ∑ i = 0 + ∞ x i .

Proof

Let γ ≔ r A ( x ) < 1 . By Lemma 2.2, lim n → + ∞ ‖ x n ‖ 1 n = γ < 1 . Then there is a real number ε > 0 such that γ + ε < 1 . For such ε > 0 , there exists N ε > 0 such that for any n ∈ N with n ≥ N ε , we have

‖ x n ‖ 1 n < γ + ε

(2.1) ‖ x n ‖ < ( γ + ε ) n .

For every n ∈ N , write C n = ∑ i = 0 n x i . For any m , n ∈ N with m > n ≥ N ε , by (2.1), we obtain

∥ C m − C n ∥ = ∑ i = n + 1 m x i ≤ ∑ i = n + 1 + ∞ ∥ x i ∥ ≤ ∑ i = n + 1 + ∞ ( γ + ε ) i = ( γ + ε ) n + 1 ( 1 − γ − ε ) − 1 .

This means { C n } is a Cauchy sequence in A . By the completeness of A , there exists a point C ∈ A such that C = ∑ i = 0 + ∞ x i . Now we begin to prove

C ( e − x ) = ( e − x ) C = e .

It is easy to see that

C n ( e − x ) = ( e − x ) C n = e − x n + 1 for all n ∈ N .

When n is large enough, (2.1) shows

‖ x n + 1 ‖ < ( γ + ε ) n + 1 ,

and hence,

‖ x n + 1 ‖ → 0 as n → + ∞ .

Therefore, we prove C ( e − x ) = ( e − x ) C = e . So e − x is invertible and ( e − x ) − 1 = C = ∑ i = 0 + ∞ x i .□

It is worth mentioning that Lemma 2.4 can be rewritten as follows.

Lemma 2.5

Let A be a unital Banach algebra with a unit e and x ∈ A . If r A ( x ) < ∣ λ ∣ for some non-zero complex constant λ , then λ e − x is invertible. Moreover,

( λ e − x ) − 1 = ∑ i = 0 + ∞ x i λ i + 1 .

By (2.1) and the proof of Lemma 2.4, we are led to the following lemma.

Lemma 2.6

(See [19]) Let A be a Banach algebra and x ∈ A . If r A ( x ) < 1 , then lim n → + ∞ ‖ x n ‖ = 0 .

Lemma 2.7

Let A be a Banach algebra with a normal cone P . If x , y ∈ A satisfy θ ≼ x ≼ y and x commutes with y , then the following hold:

θ ≼ x n ≼ y n for all n ∈ N ;
r A ( x ) ≤ r A ( y ) .

In particular, if A is commutative, then the spectral radius r A 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 P P ⊂ P and x commutes with y , we see that x 2 ∈ P , x y − x 2 = x ( y − x ) ∈ P and y 2 − x y = y ( y − x ) ∈ P , which imply

y 2 − x 2 = ( y 2 − x y ) + ( x y − x 2 ) ∈ P .

Thus, θ ≼ x 2 ≼ y 2 and conclusion (1) is true for n = 2 . Assume that conclusion (1) is true for n = k ∈ N . Then x k ∈ P and y k − x k ∈ P . Since

x k y − x k + 1 = x k ( y − x ) ∈ P

and

y k + 1 − x k y = y ( y k − x k ) ∈ P ,

we obtain

y k + 1 − x k + 1 = ( y k + 1 − x k y ) + ( x k y − x k + 1 ) ∈ P ,

which means x k + 1 ≼ y k + 1 holds. Hence, conclusion (1) is also true for n = k + 1 . This means conclusion (1) is true for all n ∈ N .

To see conclusion (2), let M be the normal constant of P . By conclusion (1), we have ‖ x n ‖ ≤ M ‖ y n ‖ for all n ∈ N . So it follows that

r A ( x ) = lim n → + ∞ ‖ x n ‖ 1 n ≤ lim n → + ∞ ( M ‖ y n ‖ ) 1 n ≤ lim n → + ∞ M 1 n ⋅ lim n → + ∞ ( ‖ y n ‖ ) 1 n = r A ( y ) ,

which completes the proof.□

Definition 2.4

(See [20]) Let X be a set. The mappings f , g : X → X are said to be weakly compatible, if for every x ∈ X holds f g x = g f x whenever f x = g x .

Definition 2.5

(See [21]) Let f and g be self-maps of a set X . If w = f x = g x 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 .

Lemma 2.8

(See [21]) Let f and g be weakly compatible self-maps of a set X. If f and g have a unique point of coincidence w = f x = g x , then w is the unique common fixed point 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 self-mappings in non-solid cone b -metric spaces over Banach algebras are presented.

Theorem 3.1

Let ( X , d ) be a wrtn-complete cone b-metric space over a Banach algebra A with the coefficient s ≥ 1 and the mappings f , g : X → X . Let a 1 , a 2 , a 3 , a 4 , a 5 ∈ P and let k = a 2 + a 3 + s ( a 4 + a 5 ) satisfying

(3.1) 2 s r A ( a 1 ) + ( s + 1 ) r A ( k ) < 2 .

Assume that

the range of g contains the range of f , that is, f ( X ) ⊆ g ( X ) ,
g ( X ) or f ( X ) is a complete subspace of X,
a 1 commutes with k ,
for all x , y ∈ X , one has
(3.2) d ( f x , f y ) ≼ a 1 d ( g x , g y ) + a 2 d ( g x , f x ) + a 3 d ( g y , f y ) + a 4 d ( g x , f y ) + a 5 d ( g y , f x ) .

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 0 ∈ X be given. By (H1), there exists an x 1 ∈ X such that f x 0 = g x 1 , then we can choose a sequence { x n } n ∈ N ∪ { 0 } such that f x n = g x n + 1 for n ∈ N ∪ { 0 } by induction. If for some natural number n ^ , g x n ^ = g x n ^ + 1 = f x n ^ , then x n ^ is a coincidence point of f and g in X . We assume that g x n + 1 ≠ g x n for all n ∈ N . For any n ∈ N , by (H4), we have

d ( g x n + 1 , g x n ) = d ( f x n , f x n − 1 ) ≼ a 1 d ( g x n , g x n − 1 ) + a 2 d ( g x n , f x n ) + a 3 d ( g x n − 1 , f x n − 1 ) + a 4 d ( g x n , f x n − 1 ) + a 5 d ( g x n − 1 , f x n ) = a 1 d ( g x n , g x n − 1 ) + a 2 d ( g x n , g x n + 1 ) + a 3 d ( g x n − 1 , g x n ) + a 5 d ( g x n − 1 , g x n + 1 ) ≼ ( a 1 + a 3 + s a 5 ) d ( g x n − 1 , g x n ) + ( a 2 + s a 5 ) d ( g x n , g x n + 1 )

and

d ( g x n , g x n + 1 ) = ( f x n − 1 , f x n ) ≼ a 1 d ( g x n − 1 , g x n ) + a 2 d ( g x n − 1 , f x n − 1 ) + a 3 d ( g x n , f x n ) + a 4 d ( g x n − 1 , f x n ) + a 5 d ( g x n , f x n − 1 ) = a 1 d ( g x n − 1 , g x n ) + a 2 d ( g x n − 1 , g x n ) + a 3 d ( g x n , g x n + 1 ) + a 4 d ( g x n − 1 , g x n + 1 ) ≼ ( a 1 + a 2 + s a 4 ) d ( g x n − 1 , g x n ) + ( a 3 + s a 4 ) d ( g x n , g x n + 1 ) .

Hence, for any n ∈ N , we obtain

2 d ( g x n , g x n + 1 ) ≼ ( 2 a 1 + a 2 + a 3 + s a 4 + s a 5 ) d ( g x n − 1 , g x n ) + ( a 2 + a 3 + s a 4 + s a 5 ) d ( g x n , g x n + 1 ) .

Since k = a 2 + a 3 + s a 4 + s a 5 , it follows from the last inequality that

(3.3) ( 2 e − k ) d ( g x n , g x n + 1 ) ≼ ( 2 a 1 + k ) d ( g x n − 1 , g x n ) for any n ∈ N .

By (H3), it is easy to see that 2 s a 1 commutes with ( s + 1 ) k and k commutes with 2 s a 1 + ( s + 1 ) k . Since

k ≼ 2 s a 1 + ( s + 1 ) k ,

and by Lemma 2.3 and (3.1), we obtain

r A ( k ) ≤ r A ( 2 s a 1 + ( s + 1 ) k ) ≤ 2 s r A ( a 1 ) + ( s + 1 ) r A ( k ) < 2 .

By Lemma 2.5, 2 e − k is invertible and

( 2 e − k ) − 1 = ∑ i = 0 + ∞ k i 2 i + 1 .

We claim

(3.4) r A ( ( 2 e − k ) − 1 ) ≤ ∑ i = 0 + ∞ [ r A ( k ) ] i 2 i + 1 .

In fact, for any n ∈ N , by Lemma 2.3, we have

r A ∑ i = 0 n k i 2 i + 1 ≤ ∑ i = 0 n r A k i 2 i + 1 ≤ ∑ i = 0 n [ r A ( k ) ] i 2 i + 1 .

Since ∑ i = 0 + ∞ k i 2 i + 1 commutes with k , we have ∑ i = 0 + ∞ k i 2 i + 1 commutes with ∑ i = 0 n k i 2 i + 1 for any n ∈ N . So Lemmas 2.3 and 2.7 show

r A ( ( 2 e − k ) − 1 ) = r A ∑ i = 0 + ∞ k i 2 i + 1 ≤ ∑ i = 0 + ∞ [ r A ( k ) ] i 2 i + 1 .

Let λ ≔ ( 2 e − k ) − 1 ( 2 a 1 + k ) . By (3.3), we obtain

(3.5) d ( g x n , g x n + 1 ) ≼ ( 2 e − k ) − 1 ( 2 a 1 + k ) d ( g x n − 1 , g x n ) = λ d ( g x n − 1 , g x n ) for any n ∈ N .

For any n ∈ N , from (3.5), we have

(3.6) d ( g x n , g x n + 1 ) ≼ λ d ( g x n − 1 , g x n ) ≼ λ 2 d ( g x n − 2 , g x n − 1 ) ≼ ⋯ ≼ λ n − 1 d ( g x 1 , g x 2 ) .

Since a 1 commutes with k , it suffices to prove that ( 2 e − k ) − 1 commutes with 2 a 1 + k . Then (3.4) and Lemma 2.3 lead to

r A ( λ ) = r A ( ( 2 e − k ) − 1 ( 2 a 1 + k ) ) ≤ r A ( ( 2 e − k ) − 1 ) r A ( 2 a 1 + k ) ≤ ∑ i = 0 + ∞ [ r A ( k ) ] i 2 i + 1 [ 2 r A ( a 1 ) + r A ( k ) ] ≤ 1 2 − r A ( k ) [ 2 r A ( a 1 ) + r A ( k ) ] .

By (3.1), we have

s [ 2 r A ( a 1 ) + r A ( k ) ] < 2 − r A ( k ) .

So, it follows that

r A ( λ ) ≤ 1 2 − r A ( k ) [ 2 r A ( a 1 ) + r A ( k ) ] < 1 s ≤ 1 .

Hence, by Lemmas 2.5 and 2.6, ( e − s λ ) − 1 exists and lim n → + ∞ ‖ λ n ‖ = 0 . For any positive integers m and n with m > n , by (3.6), we have

d ( g x n , g x m ) ≼ s d ( g x n , g x n + 1 ) + s d ( g x n + 1 , g x m ) ≼ s d ( g x n , g x n + 1 ) + s 2 d ( g x n + 1 , g x n + 2 ) + s 2 d ( g x n + 2 , g x m ) ≼ s d ( g x n , g x n + 1 ) + s 2 d ( g x n + 1 , g x n + 2 ) + s 3 d ( g x n + 2 , g x n + 3 ) + ⋯ + s m − n − 1 d ( g x m − 2 , g x m − 1 ) + s m − n − 1 d ( g x m − 1 , g x m ) ≼ s d ( g x n , g x n + 1 ) + s 2 d ( g x n + 1 , g x n + 2 ) + s 3 d ( g x n + 2 , g x n + 3 ) + ⋯ + s m − n − 1 d ( g x m − 2 , g x m − 1 ) + s m − n d ( g x m − 1 , g x m ) ≼ ( s λ n − 1 + s 2 λ n + ⋯ + s m − n λ m − 2 ) d ( g x 1 , g x 2 ) ≼ s λ n − 1 ( e + s λ + ( s λ ) 2 + ⋯ + ( s λ ) m − n − 1 + ⋯ ) d ( g x 1 , g x 2 ) = s λ n − 1 ( e − s λ ) − 1 d ( g x 1 , g x 2 ) .

Hence, by the normality of P , we obtain

‖ d ( g x n , g x m ) ‖ ≤ M ‖ s λ n − 1 ( e − s λ ) − 1 d ( g x 1 , g x 2 ) ‖ ≤ M s ‖ ‖ λ n − 1 ‖ ‖ ( e − s λ ) − 1 d ( g x 1 , g x 2 ) ‖ .

Therefore, it follows that ‖ d ( g x n , g x m ) ‖ → 0 ( n , m → + ∞ ) . That is, { g x n } is a wrtn-Cauchy sequence in g ( X ) .

If g ( X ) ⊂ X is wrtn-complete, there exist q ∈ g ( X ) and p ∈ X such that g x n → ‖ ⋅ ‖ q as n → + ∞ and g p = q (if f ( X ) is wrtn-complete, there exists q ∈ f ( X ) such that f x n → ‖ ⋅ ‖ q as n → + ∞ . Since f ( X ) ⊂ g ( X ) , we can find p ∈ X such that g p = q ).

Now, we need to show that f p = q . By (3.2), we see

d ( g x n + 2 , f p ) = d ( f x n + 1 , f p ) ≼ a 1 d ( g x n + 1 , q ) + a 2 d ( g x n + 1 , g x n + 2 ) + a 3 d ( q , f p ) + a 4 d ( g x n + 1 , f p ) + a 5 d ( q , g x n + 2 ) .

Similarly,

d ( f p , g x n + 2 ) = d ( f p , f x n + 1 ) ≼ a 1 d ( q , g x n + 1 ) + a 2 d ( q , f p ) + a 3 d ( g x n + 1 , g x n + 2 ) + a 4 d ( q , g x n + 2 ) + a 5 d ( g x n + 1 , f p ) .

Thus, we have

2 d ( g x n + 2 , f p ) ≼ 2 a 1 d ( g x n + 1 , q ) + ( a 2 + a 3 ) d ( g x n + 1 , g x n + 2 ) + ( a 2 + a 3 ) d ( q , f p ) + ( a 4 + a 5 ) d ( g x n + 1 , f p ) + ( a 4 + a 5 ) d ( q , g x n + 2 ) ≼ ( 2 s a 1 + s a 2 + s a 3 + a 4 + a 5 ) d ( g x n + 2 , q ) + ( s a 2 + s a 3 + s a 4 + s a 5 ) d ( g x n + 2 , f p ) + ( 2 s a 1 + a 2 + a 3 + s a 4 + s a 5 ) d ( g x n + 1 , g x n + 2 ) .

Note that by Lemma 2.7 and (3.1), we obtain

r A ( a 2 + a 3 + a 4 + a 5 ) ≤ r A ( a 2 + a 3 + s ( a 4 + a 5 ) ) < 2 s + 1 < 2 s ,

so ( 2 e − s a 2 − s a 3 − s a 4 − s a 5 ) − 1 exists. Hence, we obtain

d ( g x n + 2 , f p ) ≼ ( 2 e − s a 2 − s a 3 − s a 4 − s a 5 ) − 1 ( 2 s a 1 + s a 2 + s a 3 + a 4 + a 5 ) d ( g x n + 2 , q ) + ( 2 e − s a 2 − s a 3 − s a 4 − s a 5 ) − 1 ( 2 s a 1 + a 2 + a 3 + s a 4 + s a 5 ) d ( g x n + 1 , g x n + 2 ) .

Similarly, by the normality of P and the fact that { g x n } is a wrtn-Cauchy sequence and g x n → ‖ ⋅ ‖ q ( n → + ∞ ), we easily deduce that ‖ d ( g x n + 2 , f p ) ‖ → 0 as n → + ∞ . This is, g x n → ‖ ⋅ ‖ f p ( n → + ∞ ) . Therefore, by Lemma 2.1, we have f p = q = g p .

Next, let us check the uniqueness of the point of coincidence for the mappings f and g . If there exist u , w in X such that f u = g u = w , by (3.2), we see that

d ( g u , g p ) = d ( f u , f p ) ≼ a 1 d ( g u , g p ) + a 2 d ( f u , g u ) + a 3 d ( f p , g p ) + a 4 d ( f p , g u ) + a 5 d ( f u , g p ) = ( a 1 + a 4 + a 5 ) d ( g u , g p ) .

As r A ( a 1 + a 4 + a 5 ) ≤ r A ( a 1 ) + r A ( a 4 + a 5 ) < 1 , we can deduce that d ( g u , g p ) = θ , i.e., w = g u = g p = q . 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.

Corollary 3.1

Let ( X , d ) be a wrtn-complete cone b-metric space over a Banach algebra A with coefficient s ≥ 1 . Suppose the mapping T : X → X satisfies the generalized Banach contractive condition

d ( T x , T y ) ≼ k d ( x , y ) , f o r a l l x , y ∈ X ,

where k ∈ P is a constant vector satisfying r A ( k ) ∈ ( 0 , 1 s ) . Then T has a unique fixed point in X. And for any x ∈ X , iteration sequence { T n x } wrtn-converges to the fixed point.

Corollary 3.2

Let ( X , d ) be a wrtn-complete cone metric space over a Banach algebra A . Suppose the mapping T : X → X satisfies the generalized Banach contractive condition

d ( T x , T y ) ≼ k d ( x , y ) , f o r a l l x , y ∈ X ,

where k ∈ P is a constant vector satisfying r A ( k ) ∈ ( 0 , 1 ) . Then T has a unique fixed point in X. And for any x ∈ X , iteration sequence { T n x } wrtn-converges to the fixed point.

Corollary 3.3

Let ( X , d ) be a wrtn-complete cone metric space over a Banach algebra A . Suppose the mapping T : X → X satisfies the generalized Kannan contractive condition

d ( T x , T y ) ≼ k ( d ( T x , x ) + d ( T y , y ) ) , f o r a l l x , y ∈ X ,

where k ∈ P is a constant vector satisfying r A ( k ) ∈ ( 0 , 1 2 ) . Then T has a unique fixed point in X. And for any x ∈ X , iteration sequence { T n x } wrtn-converges to the fixed point.

Corollary 3.4

Let ( X , d ) be a wrtn-complete cone metric space over a Banach algebra A . Suppose the mapping T : X → X satisfies the generalized Chatterjea contractive condition

d ( T x , T y ) ≼ k ( d ( T x , y ) + d ( T y , x ) ) , f o r a l l x , y ∈ X ,

where k ∈ P is a constant vector satisfying r A ( k ) ∈ ( 0 , 1 2 ) . Then T has a unique fixed point in X. And for any x ∈ X , iteration sequence { T n x } 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].

In fact, if ( X , d ) is a wrtn-complete cone metric space over a Banach algebra A and the cone P is normal with normal constant M . Suppose the mapping T : X → X satisfies the generalized Banach contractive condition

(3.7) d ( T x , T y ) ≼ k d ( x , y ) , for all x , y ∈ X ,

where k ∈ P is a constant vector satisfying r A ( k ) ∈ ( 0 , 1 ) . By [17], we can suppose that M = 1 . Therefore, according to (3.7), we have

∥ d ( T x , T y ) ∥ ≤ ∥ k ∥ ∥ d ( x , y ) ∥ .

Then the space ( X , D ) with D ( x , y ) = ∥ d ( x , y ) ∥ 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 A ( k ) ∈ ( 0 , 1 ) . These results improve and extend many relevant results in metric spaces [30,31].

4 Applications

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.

Example 4.1

Consider the space L p ( 0 < p < 1 ) of all real function x ( t ) ( t ∈ [ 0 , 1 ] ) such that ∫ 0 1 ∣ x ( t ) ∣ p d t < + ∞ . Let X = L p , A = R 2 , P = { ( x , y ) ∈ A ∣ x , y ≥ 0 } ⊂ R 2 , and d : X × X → A such that

d ( x , y ) = α ∫ 0 1 ∣ x ( t ) − y ( t ) ∣ p d t 1 p , β ∫ 0 1 ∣ x ( t ) − y ( t ) ∣ p d t 1 p ,

where α , β ≥ 0 are constants. Then ( X , d ) is a normal and solid cone b -metric space over a Banach algebra A with the coefficient s = 2 1 p − 1 . Define the mapping T : X → X by

T x ( t ) = 1 4 ln ( 1 + ∣ x ( t ) ∣ ) .

Considering a simple inequality 0 < ln ( 1 + x ) < x , where x > 0 , we have

d ( T x , T y ) = α ∫ 0 1 1 4 ln ( 1 + ∣ x ( t ) ∣ ) − 1 4 ln ( 1 + ∣ y ( t ) ∣ ) p d t 1 p , β × ∫ 0 1 1 4 ln ( 1 + ∣ x ( t ) ∣ ) − 1 4 ln ( 1 + ∣ y ( t ) ∣ ) p d t 1 p = 1 4 α ∫ 0 1 ln 1 + ∣ x ( t ) ∣ 1 + ∣ y ( t ) ∣ p d t 1 p , β ∫ 0 1 ln 1 + ∣ x ( t ) ∣ 1 + ∣ y ( t ) ∣ p d t 1 p = 1 4 α ∫ 0 1 ln 1 + ∣ x ( t ) ∣ − ∣ y ( t ) ∣ 1 + ∣ y ( t ) ∣ p d t 1 p , β ∫ 0 1 ln 1 + ∣ x ( t ) ∣ − ∣ y ( t ) ∣ 1 + ∣ y ( t ) ∣ p d t 1 p ≤ 1 4 α ∫ 0 1 ln 1 + ∣ x ( t ) − y ( t ) ∣ 1 + ∣ y ( t ) ∣ p d t 1 p , β ∫ 0 1 ln 1 + ∣ x ( t ) − y ( t ) ∣ 1 + ∣ y ( t ) ∣ p d t 1 p ≤ 1 4 α ∫ 0 1 ∣ x ( t ) − y ( t ) ∣ p d t 1 p , β ∫ 0 1 ∣ x ( t ) − y ( t ) ∣ p d t 1 p = 1 4 d ( x , y ) ,

where p = 1 2 and k = 1 4 ∈ ( 0 , 1 s ) for 1 s = 1 2 1 p − 1 = 1 2 . Consequently, all conditions of Corollary 3.1 are satisfied and then T has a unique fixed point in X .

Example 4.2

Let X = Ł [ 0 , 1 ] denote the set of all generalized real-valued Lebesgue integral functions on [ 0 , 1 ] . Let A = Ł [ 0 , 1 ] . Consider the following nonlinear integral equation:

(4.1) ∫ 0 1 F ( t , f ( s ) ) d s = f ( t ) ,

where F satisfies:

F : [ 0 , 1 ] × R ¯ → R ¯ is a generalized real-valued Lebesgue integral function where R ¯ = [ − ∞ , + ∞ ] denoting the set of all generalized real numbers;
there exists a function G ( x ) with 0 < ∫ 0 1 G ( x ) d x < 1 such that for a.e. x ∈ [ 0 , 1 ] , and a.e. y 1 , y 2 ∈ R ¯ , one has
∣ F ( x , y 1 ) − F ( x , y 2 ) ∣ ≤ G ( x ) ∣ y 1 − y 2 ∣ .

Then equation (4.1) has a unique non-negative solution in Ł [ 0 , 1 ] .

Now we check all the conditions of Corollary 3.2 are satisfied. Define a norm on A by ∥ f ∥ 1 = ∫ 0 1 ∣ f ( t ) ∣ d t for f ∈ A . Let P = { f ∈ Ł [ 0 , 1 ] ∣ f = f ( t ) ≥ 0 , for a.e. t ∈ [ 0 , 1 ] } . Then P is a normal but non-solid cone of the real Banach algebra A with the operations as follows:

( f + g ) ( t ) = f ( t ) + g ( t ) , ( c f ) ( t ) = c f ( t ) , ( f g ) ( t ) = f ( t ) g ( t ) ,

for all f = f ( t ) , g = g ( t ) ∈ A and c ∈ R . Moreover, A owns the unit element e = e ( t ) = 1 for a.e. t ∈ [ 0 , 1 ] . Let T be a self-mapping of X defined by T f ( t ) = ∫ 0 1 F ( t , f ( s ) ) d s .

It is easy shown that ( X , d ) is a wrtn-complete cone metric space over Banach algebra A with the norm ‖ ⋅ ‖ 1 where the cone metric is defined by d ( f , g ) = e t ∫ 0 1 ∣ f ( t ) − g ( t ) ∣ d t . Now let us check that T : X → X is a generalized Banach contraction with the generalized Lipschitz coefficient L = ∫ 0 1 G ( x ) d x satisfying the spectral radius r ( L ) < 1 . Indeed, by (b) together with the fact that ‖ f g ‖ 1 ≤ ‖ f ‖ 1 ‖ g ‖ 1 , we see

d ( T f ( x ) , T g ( x ) ) = e x ∫ 0 1 ∣ T f ( x ) − T g ( x ) ∣ d x = e x ∫ 0 1 ∫ 0 1 ( F ( x , f ( t ) ) − F ( x , g ( t ) ) ) d t d x ≤ e x ∫ 0 1 ∫ 0 1 ∣ F ( x , f ( t ) ) − F ( x , g ( t ) ) ∣ d t d x ≤ e x ∫ 0 1 L ∣ f ( t ) − g ( t ) ∣ d t ≤ L e x ∫ 0 1 ∣ f ( t ) − g ( t ) ∣ d t = L d ( f ( x ) , g ( x ) ) ,

where the spectral radius r ( L ) of L satisfies r ( L ) ≤ ‖ L ‖ = L ∈ ( 0 , 1 ) . Therefore, it follows from Corollary 3.2 that equation (4.1) has a unique non-negative solution in Ł [ 0 , 1 ] .

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 [22–29]. 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 [30–34].

Funding information: The research was partially supported by the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities’ Association (No. 202101BA070001-045).
Conflict of interest: The authors declare that they have no conflict of interest.
Data availability statement: No data were used to support this study.

References

[1] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1992), 133–181. 10.4064/fm-3-1-133-181Search in Google Scholar

[2] R. Kannan, Some results on fixed points II, Amer. Math. Monthly 76 (1969), 405–408. 10.1080/00029890.1969.12000228Search in Google Scholar

[3] S. K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci. 25 (1972), 727–730. Search in Google Scholar

[4] A. I. Perov, The Cauchy problem for systems of ordinary differential equations, Priblizhen. Metody Reshen. Difer. Uravn. 2 (1964), 115–134 (in Russian). Search in Google Scholar

[5] J. S. Vandergraft, Newton’s method for convex operators in partially ordered spaces, SIAM J. Numer. Anal. 4 (1967), 406–432. 10.1137/0704037Search in Google Scholar

[6] P. P. Zabrejko, K-metric and K-normed linear spaces: survey, Collect. Math. 48 (1997), no. 4–6, 825–859. Search in Google Scholar

[7] L. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1468–1476. 10.1016/j.jmaa.2005.03.087Search in Google Scholar

[8] S. Rezapour and R. Hamlbarani, Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”, J. Math. Anal. Appl. 345 (2008), 719–724. 10.1016/j.jmaa.2008.04.049Search in Google Scholar

[9] N. Hussian and M. H. Shah, KKM mappings in cone b-metric spaces, Comput. Math. Appl. 62 (2011), 1677–1684. 10.1016/j.camwa.2011.06.004Search in Google Scholar

[10] M. H. Shah, S. Simić, N. Hussain, A. Sretenović and S. Radenović, Common fixed point theorems for occasionally weakly compatible pairs on cone metric type spaces, J. Comput. Anal. Appl. 14 (2012), no. 2, 290–297. Search in Google Scholar

[11] H. Huang and S. Xu, Fixed point theorems of contractive mappings in cone b-metric spaces and applications, Fixed Point Theory Appl. 2013 (2013), no. 112, 1–10. 10.1186/1687-1812-2013-112Search in Google Scholar

[12] I. M. Oralu, A. Branga, and A. Oprea, Common fixed point results in b-cone metric or TVS-topological vector spaces, Gen. Math. 20 (2012), no. 1, 57–67. Search in Google Scholar

[13] L. Shi and S. Xu, Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces, Fixed Point Theory Appl. 2013 (2013), no. 120, 1–11. 10.1186/1687-1812-2013-120Search in Google Scholar

[14] H. Liu and S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl. 2013 (2013), 320. 10.1186/1687-1812-2013-320Search in Google Scholar

[15] Y. Han and S. Xu, Some new theorems on c-distance without continuity in cone metric spaces over Banach algebras, J. Funct. Spaces 2018 (2018), 7463435. 10.1155/2018/9395057Search in Google Scholar

[16] H. Kunze, D. La Torre, F. Mendivil, and E. R. Vrscay, Generalized fractal transforms and self-similar objects in cone metric spaces, Comput. Math. Appl. 64 (2012), 1761–1769. 10.1016/j.camwa.2012.02.011Search in Google Scholar

[17] S. Janković, Z. Kadelburg, and S. Radenović, On the cone metric space: A survey, Nonlinear Anal. 74 (2011), 2591–2601. 10.1016/j.na.2010.12.014Search in Google Scholar

[18] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York, 1991. Search in Google Scholar

[19] M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008), 416–420. 10.1016/j.jmaa.2007.09.070Search in Google Scholar

[20] G. Song, X. Sun, Y. Zhao, and G. Wang, New common fixed point theorems for maps on cone metric spaces, Appl. Math. Lett. 23 (2010), 1033–1037. 10.1016/j.aml.2010.04.032Search in Google Scholar

[21] M. Abbas, V. Ć. Rajić, T. Nazir, and S. Radenović, Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces, Afr. Mat. 26 (2015), 17–30, DOI: https://doi.org/10.1007/s13370-013-0185-z. 10.1007/s13370-013-0185-zSearch in Google Scholar

[22] W.-S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259–2261. 10.1016/j.na.2009.10.026Search in Google Scholar

[23] W.-S. Du and E. Karapınar, A note on cone b-metric and its related results: generalizations or equivalence? Fixed Point Theory Appl. 2013 (2013), 210. 10.1186/1687-1812-2013-154Search in Google Scholar

[24] Z. Kadelburg, S. Radenović, and V. Rakočević, A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett. 24 (2011), 370–374. 10.1016/j.aml.2010.10.030Search in Google Scholar

[25] Y. Feng and W. Mao, The equivalence of cone metric spaces and metric spaces, Fixed Point Theory 11 (2010), no. 2, 259–264. Search in Google Scholar

[26] H. Çakallı, A. Sönmez, and Ç. Genç, On an equivalence of topological vector space valued cone metric spaces and metric spaces, Appl. Math. Lett. 25 (2012), 429–433. 10.1016/j.aml.2011.09.029Search in Google Scholar

[27] P. Kumam, N. V. Dung, and V. T. L. Hang, Some equivalence between cone b-metric spaces and b-metric spaces, Abstr. Appl. Anal. 2013 (2013), 1–8. 10.1186/1687-1812-2013-5Search in Google Scholar

[28] O. Olmez and S. Aytar, On equivalence cone metric spaces, Ukrainian Math. J. 65 (2014), no. 12, 1898–1903. 10.1007/s11253-014-0905-zSearch in Google Scholar

[29] Z. Ercan, On the end of cone metric spaces, Topology Appl. 166 (2014), 10–14. 10.1016/j.topol.2014.02.004Search in Google Scholar

[30] P. Agarwal, M. Jleli, and B. Samet, Banach Contraction Principle and Applications: Fixed Point Theory in Metric Spaces, Springer, Singapore, 2018. 10.1007/978-981-13-2913-5Search in Google Scholar

[31] D. Gopal, P. Agarwal, and P. Kumam, Metric Structures and Fixed Point Theory, Chapman and Hall/CRC, 2021. 10.1201/9781003139607Search in Google Scholar

[32] P. Debnath, N. Konwar, and S. Radenović, Metric Fixed Point Theory: Applications in Science, Engineering and Behavioural Sciences, Springer, Singapore, 2021. 10.1007/978-981-16-4896-0Search in Google Scholar

[33] F. F. Bonsall, Lectures On Some Fixed Point Theorems of Functional Analysis, Tata Institute of Fundamental Research, Bombay, 1962. Search in Google Scholar

[34] S. Aleksič, Z. Kadelburg, Z. D. Mitrovič, and S. Radenovič, A new survey: Cone metric spaces, J. Int. Math. Virtual Inst. 9 (2019), 93–121. Search in Google Scholar

Received: 2022-03-21

Revised: 2022-08-10

Accepted: 2023-03-02

Published Online: 2023-04-04

This work is licensed under the Creative Commons Attribution 4.0 International License.

Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients

Abstract

1 Introduction

Theorem

2 Preliminaries

Definition 2.1

Definition 2.2

Definition 2.3

Lemma 2.1

Proof

Remark 2.1

Lemma 2.2

Lemma 2.3

Lemma 2.4

Proof

Lemma 2.5

Lemma 2.6

Lemma 2.7

Proof

Definition 2.4

Definition 2.5

Lemma 2.8

3 Fixed point theorems in non-solid cone b -metric spaces over Banach algebras

Theorem 3.1

Proof

Corollary 3.1

Corollary 3.2

Corollary 3.3

Corollary 3.4

Remark 3.1

4 Applications

Example 4.1

Example 4.2

Remark 4.1

Remark 4.2

References

Journal and Issue

Articles in the same Issue