Coupled Fixed-Point Theorems in Theta-Cone-Metric Spaces

This paper gives further generalizations of some well-known coupled fixed-point theorems. Specifically, Theorem 3 of the paper is the generalization of the Baskar–Lackshmikantham coupled fixed-point theorem, and Theorem 5 is the generalization of the Sahar Mohamed Ali Abou Bakr fixed-point theorem, where the underlying space is complete 
 
 θ
 
 -cone-metric space.


Introduction and Preliminaries
Since 1922, the pioneering fixed-point principle of Banach [1] showed exclusive interest of researchers because it has many applications, including variational linear inequalities and optimization, and applications in differential equations, in the field of approximation theory, and in minimum norm problems.
Since then, several types of contraction mappings have been introduced and many research papers have been written to generalize this Banach contraction principle.
In 1987, Guo and Lakshmikantham [2] introduced one of the most interesting concepts of coupled fixed point. In 2006, Bhaskar and Lakshmikantham [3] introduced the concept of the mixed monotone property as follows.
Definition 2. Let (E, ≤ ) be a partially ordered set and T be a mapping from E × E to E. en, (1) T is said to be monotone nondecreasing in x if and only if, for any y ∈ E, if x 1 , x 2 ∈ E and x 1 ≤ x 2 , then T x 1 , y ≤ T x 2 , y , (2) T is said to be monotone nonincreasing in y if and only if, for any x ∈ E, if y 1 , y 2 ∈ E and y 1 ≤ y 2 , then T x, y 1 ≥ T x, y 2 , (3) T is said to have a mixed monotone property if and only if T(x, y) is both monotone nondecreasing in x and monotone nonincreasing in y Definition 3. An element (x 0 , y 0 ) ∈ E × E is said to be a lower-anti-upper coupled point of the mapping T: E × E ⟶ E if and only if sequence in E such that x n n∈N converges strongly to x, then x n ≤ x for all n ∈ N (2) E is said to be a sequentially upper-ordered space if it fulfills the condition: If y n n∈N is a nonincreasing sequence in E such that y n n∈N converges strongly to y, then y n ≥ y for all n ∈ N (3) E is said to be a sequentially lower-upper ordered space if it is both a lower-and upper-ordered space In 2006, Bhaskar and Lakshmikantham [3] proved the existence of coupled fixed points for mixed monotone mappings with weak contractivity assumption in a partialordered Banach space (E, ‖.‖, ≤ ) as follows.
Theorem 1 (see [3]). Let E be a sequentially both lower-and upper-ordered Banach space and T: E × E ⟶ E be a mapping with mixed monotone and lower-upper properties. If there is a real number 0 ≤ k < 1 such that then T has coupled fixed points in E.
In 2013, Mohamed Ali [4] introduced novel contraction type of mappings and proved the following fixed-point theorem.
Recently, more advanced approaches for studying coupled fixed points have been presented by the authors in [11][12][13].
In 2007, Huang and Zhang [14] introduced the concept of cone-metric spaces as follows: First, a subset M of the real Banach space E is said to be a cone in E if and only if If intM is the set of all interior points of M, then a cone M in a normed space E induces the following ordered relations: If E is a nonempty set, the distance d(x, y) between any two elements x, y ∈ E is defined to be a vector in the cone M, and the space (E, d) is said to be a cone-metric space if and only if d satisfied the three axioms of metric but using the ordered relation ≺ induced by M for the triangle inequality instead.
ey studied the topological characterizations of such a defined space, and then, they applied their concept to have more generalizations of some previous fixed-point theorems for contractive type of mappings.
A mapping T: E ⟶ E is said to be a contraction if and only if there is a constant α ∈ [0, 1) such that In 2019, Mohamed Ali Abou Bakr [15] proved the existence of a unique common fixed point of generalized joint cyclic Banach algebra contractions and Banach algebra Kannan type of mappings on cone quasimetric spaces.
In 2013, Khojasteh et al. [10] introduced the notion of θ-action function, θ: , the concept of θ-metric, and then, they studied the topological structures of θ-metric spaces in detail. eir work led to a step-forward generalization of metric spaces.
In 2020, Mohamed Ali Abou Bakr [16] replaced [0, ∞) by a cone M in a normed space and used the ordered relation induced by this cone to introduce the following analogous generalization of θ-action function.
Definition 5. Let (E, ≺ ) be an ordered normed space, where ≺ is an ordered relation induced by some cone M ⊂ E and θ: M × M ⟶ M be a continuous mapping with respect to each variable, and we denote en, θ is said to be an ordered action mapping on E if and only if it satisfies the following conditions: (3) For every u ∈ Im(θ) and every

Journal of Mathematics
In addition, Mohamed Ali Abou Bakr [16] gave further replacement, replaced the set of nonnegative real numbers R + by a cone M in a normed space, and used θ-ordered actions to introduce the concept of θ-cone-metric space as follows.
Definition 6 (see [16]). Let (E, ≺ ) be an ordered normed space, where ≺ is the ordered relation induced by some cone M ⊂ E, and θ be an ordered action on E. If E is a nonempty set, then the function d θ : E × E ⟶ M is said to be a θ-cone-metric on E if and only if d θ satisfies the following conditions: e author has further given some topological characterizations of this space and then generalized some previous fixed-point theorems in this setting.
In this paper, we extend and generalize the coupled fixed-point theorem of Baskar-Lackshmikantham (1.5) to a more general one (2.1), where the underlying space (E, d θ ) is a complete θ-cone-metric space. On the other side, if T: E × E ⟶ E is a continuous mapping in the second argument and there are three constants a, b, c ∈ [0, 1) and a + b + c < 1 such that then we proved that T has a unique fixed point in the sense that there is a unique point We also claim that some results of [6][7][8][9][10]17] can be proved in the case of θ-cone-metric spaces.

Main Results
Let (E, d θ , ≤ ) be a partially ordered θ-cone-metric space. en, the following relation defines a partial-ordered relation on E × E: We have the following coupled fixed-point theorem.  (G(x, y), en, G has coupled fixed points in E.
Proof. Since G has a lower-upper property, then there exist x 0 , y 0 ∈ E such that We denote x 1 � G(x 0 , y 0 ) and y 1 � G(y 0 , x 0 ) and then give notations for the elements of the following inductively constructed sequences: x n+1 � G x n , y n ≔ G n+1 x 0 , y 0 , y n+1 � G y n , x n ≔ G n+1 y 0 , x 0 , . . .

(14)
Using the mixed monotonicity property of G insures that each step leads to the next step in each of the following: e mixed monotonicity property, the contractiveness of G, and the inductive process prove the following for every n ∈ N:

Journal of Mathematics
Consequently, we have Hence, we claim that both G n (x 0 , y 0 ) n∈N and G n (y 0 , x 0 ) n∈N are Cauchy sequences in E. Indeed, if one of them, say G n (x 0 , y 0 ) n∈N , is not Cauchy, then there exist v ∈ Im(θ), Θ < v and sequences of natural numbers i n n∈N and j n n∈N such that, for any i n > j n > n, v ≺ d θ G i n x 0 , y 0 , G j n x 0 , y 0 , d θ G i n − 1 x 0 , y 0 , G j n x 0 , y 0 < v.

(18)
Since any subsequence of d θ (G n+1 (x 0 , y 0 ), G n (x 0 , y 0 ))} n∈N is convergent to Θ, the properties of θ imply the following contradiction: Similarly, the sequence G n (y 0 , x 0 ) n∈N is also Cauchy. Since E is a complete θ-cone-metric space, there exist Now, we are going to show that (x, y) is a coupled fixed point of G. Since the sequence G n (x 0 , y 0 ) � x n n∈N is nondecreasing with lim n⟶∞ G n (x 0 , y 0 ) � x, then G n (x 0 , y 0 ) ≤ x, and since the sequence G n (y 0 , x 0 ) � y n n∈N is nonincreasing with lim n⟶∞ G n (y 0 , x 0 ) � y, then y ≤ G n (y 0 , x 0 ) for every n ∈ N, and accordingly, we have Taking the limit as n ⟶ ∞ with the help of equation (20), we find that then the following theorem is similarly proved.
then G has coupled fixed points in E.

Corollary 1. Let E be a sequentially both lower-and upperordered Banach space and T: E × E ⟶ E be a mapping with mixed monotone and lower-upper properties. If there is a real
∀x, y, z, w ∈ E, z ≤ x, and y ≤ w, then T has coupled fixed point in E.
Proof. We just notice that any Banach space On the other side, we have the following results: If x 1 and x 2 are arbitrary elements in E, then the sequence x n ∞ n�3 defined iteratively by x n � T x n−1 , x n−2 , ∀n ∈ N, n > 2, which satisfies the following: d θ x n+1 , x n ≺ td θ x n , x n−1 , ∀n > 2, where t � (a + c/1 − b). Moreover, the sequence x n n∈N is a Cauchy sequence.
Proof. Using the contractiveness property of the given mapping gives ≺ ad θ x n , x n−1 + bd θ x n+1 , x n + cd θ x n , x n−1 .

(30)
Hence, and repeating the last step n − 2 times with the term d θ (x n , x n−1 ) proves the inequalities given in (29). To prove that the sequence (27) is Cauchy, we take the limit of both sides of (29) as n ⟶ ∞ gives lim n⟶∞ d θ (x n+1 , x n ) � Θ and suppose that x n n∈N is not Cauchy; then, there exist v ∈ Im(θ), Θ < v and sequences of natural numbers i n n∈N and j n n∈N such that, for any i n > j n > n, v ≺ d θ x i n , x j n , Since any subsequence of d θ (x n+1 , x n ) n∈N is convergent to Θ, the continuity and the properties of θ imply the following contradiction: and then, T has a unique fixed point in the sense that there is a unique point Proof. Since (E, d θ ) is complete, the Cauchy sequence x n ∞ n�3 given in Lemma 1 is converging to some element x 0 in E. We show that x 0 is fixed point of T. Using the properties of θ and the continuity of T, we see that T(x 0 , x 0 ) � x 0 . Now, let x and y be two arbitrarily distinct elements in E with T(x, x) � x and T(y, y) � y, and we have us, d θ (T(x, y), x) � Θ, that is, T(x, y) � x. Similarly, we get T(y, x) � y; therefore, (x, y) is a coupled fixed point of T. On the other hand, we have the following contradiction: ≺ θ Θ, ad θ (x, y) + Θ ≺ ad θ (x, y).

Data Availability
No data were used to support this study.

Conflicts of Interest
e author has no conflicts of interest. 6 Journal of Mathematics