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

Authors: Sahar Mohamed Ali Abou Bakr
Journal: Journal of Mathematics
 (2021)

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## Abstract

This paper gives further generalizations of some well-known coupled fixed-point theorems. Specifically, Theorem3 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.


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## Body

## 1. 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.Definition 1.
An elementx,y∈E×E is said to be a coupled fixed point of the mapping T:E×E⟶E if and only if Tx,y=x and Ty,x=y.In 2006, Bhaskar and Lakshmikantham [3] introduced the concept of the mixed monotone property as follows.Definition 2.
LetE,≤ be a partially ordered set and T be a mapping from E×E to E. Then,(1)
T is said to be monotone nondecreasing in x if and only if, for any y∈E,(1)ifx1,x2∈Eandx1≤x2,thenTx1,y≤Tx2,y,(2)
T is said to be monotone nonincreasing in y if and only if, for any x∈E,(2)ify1,y2∈Eandy1≤y2,thenTx,y1≥Tx,y2,(3)
T is said to have a mixed monotone property if and only if Tx,y is both monotone nondecreasing in x and monotone nonincreasing in yDefinition 3.
An elementx0,y0∈E×E is said to be a lower-anti-upper coupled point of the mapping T:E×E⟶E if and only if(3)x0≤Tx0,y0andy0≥Ty0,x0.
A mappingT:E×E⟶E is said to have a lower-upper property if and only if T has at least one lower-anti-upper coupled point.Definition 4.
LetE,≤,. be a partially ordered normed space. Then,(1)
E is said to be a sequentially lower ordered space if it fulfills the condition: If xnn∈N is a nondecreasing sequence in E such that xnn∈N converges strongly to x, then xn≤x for all n∈N(2)
E is said to be a sequentially upper-ordered space if it fulfills the condition: If ynn∈N is a nonincreasing sequence in E such that ynn∈N converges strongly to y, then yn≥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 spaceIn 2006, Bhaskar and Lakshmikantham [3] proved the existence of coupled fixed points for mixed monotone mappings with weak contractivity assumption in a partial-ordered 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(4)Tx,y−Tz,w≤k2x−z+y−w,∀x,y,z,w∈E,z≤x,andy≤w,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.Theorem 2.
(see [4]). Let E,. be a Banach space and T be a mapping from E×E into E, and we suppose there are three constants a,b,c∈0,1 and a+b+c<1 such that(5)Tx,y−Ty,z≤ax−y+bTx,y−x+cTy,z−y,∀x,y,z∈E.Then, there is a unique pointx0∈E such that Tx0,x0=x0.There are many interesting coupled fixed-point theorems concerning some other type of contraction mappings, see [5–10].Recently, more advanced approaches for studying coupled fixed points have been presented by the authors in [11–13].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(1)
M is nonempty closed and M≠Θ, where Θ is the zero (neutral) element of E(2)
λM+μM⊂M for all nonnegative real numbers λ,μ(3)
M∩−M=ΘIfintM is the set of all interior points of M, then a cone M in a normed space E induces the following ordered relations:(6)u≺¯v⇔v−u∈M,u<v⇔v−u∈M,andu≠v,u≺¯≠v⇔v−u∈intM.IfE is a nonempty set, the distance dx,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. They 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 mappingT:E⟶E is said to be a contraction if and only if there is a constant α∈0,1 such that(7)dTx,Ty≺¯αdx,y,∀x,y∈E.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:0,∞×0,∞⟶0,∞, the concept of θ-metric, and then, they studied the topological structures of θ-metric spaces in detail. Their 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.
LetE,≺¯ 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(8)Imθ=t:t∈Msuch that∃u0,v0∈E,θu0,v0=t.
Then,θ is said to be an ordered action mapping on E if and only if it satisfies the following conditions:(1)
θΘ,Θ=Θ and θu,v=θv,u for every u,v∈M(2)
(9)θu,v<θw,tif eitheru<wandv≺¯t;oru≺¯wandv<t(3)
For everyu∈Imθ and every Θ≺¯v≺¯u, there is Θ≺¯w≺¯u such that θv,w=u(4)
θu,Θ≺¯u for every u∈M/ΘBecausex−Θ∈M for every x∈M, one can write instead Θ≺¯x for every x∈M, (Θ<x for every x∈M/Θ).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 dθ:E×E⟶M is said to be a θ-cone-metric on E if and only if dθ satisfies the following conditions:(1)
dθx,y=Θ⇔x=y(2)
dθx,y=dθy,x,∀x,y∈E(3)
dθx,y≺¯θdθx,z,dθz,y,∀x,y,z∈E
The doubleE,dθ is defined to be a θ-cone-metric space.The author has further given some topological characterizations of this space and then generalized some previous fixed-point theorems in this setting.Remark 1.
Ifθu,v=u+v, then we have a cone-metric space.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 spaceE,dθ 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(10)dθTx,y,Ty,z≺¯adθx,y+bdθTx,y,x+cdθTy,z,y,∀x,y,z∈E,then we proved that T has a unique fixed point in the sense that there is a unique point x∈E such that Tx,x=x.We also claim that some results of [6–10, 17] can be proved in the case of θ-cone-metric spaces.


## 2. Main Results

LetE,dθ,≤ be a partially ordered θ-cone-metric space. Then, the following relation defines a partial-ordered relation on E×E:(11)x,y≪z,w⇔x≤z,andw≤y.We have the following coupled fixed-point theorem.Theorem 3.
LetE,dθ,≤ be a partially ordered, sequentially lower-upper ordered complete θ-cone-metric space and G:E×E⟶E be a mapping having mixed monotone and lower-upper properties on E. We assume that there exists r∈0,1 with(12)dθGx,y,Gz,w≺¯r2dθx,z+dθy,w,∀x,y≪z,w.Then,G has coupled fixed points in E.Proof.
SinceG has a lower-upper property, then there exist x0,y0∈E such that(13)x0≤Gx0,y0andGy0,x0≤y0.
We denotex1=Gx0,y0 and y1=Gy0,x0 and then give notations for the elements of the following inductively constructed sequences:(14)x2=Gx1,y1≔G2x0,y0,y2=Gy1,x1≔G2y0,x0,x3=Gx2,y2≔G3x0,y0,y3=Gy2,x2≔G3y0,x0,…xn+1=Gxn,yn≔Gn+1x0,y0,yn+1=Gyn,xn≔Gn+1y0,x0,…
Using the mixed monotonicity property ofG insures that each step leads to the next step in each of the following:(15)x0≤x1=Gx0,y0≤Gx1,y0≤Gx1,y1=x2,y2=Gy1,x1≤Gy1,x0≤Gy0,x0=y1≤y0,x1≤x2=Gx1,y1≤Gx2,y2=x3,y3=Gy2,x2≤Gy1,x1=y2≤y1…xn+1=Gxn,yn≤Gxn−1,yn−1=xn,yn+1=Gyn,xn≤Gyn−1,xn−1=yn,…
The mixed monotonicity property, the contractiveness ofG, and the inductive process prove the following for every n∈N:(16)dθGn+1x0,y0,Gnx0,y0≺¯r2ndθGx0,y0,x0+dθGy0,x0,y0,dθGn+1y0,x0,Gny0,x0≺¯r2ndθGy0,x0,y0+dθGx0,y0,x0.
Consequently, we have(17)limn⟶∞dθGn+1x0,y0,Gnx0,y0=limn⟶∞dθGn+1y0,x0,Gny0,x0=Θ.
Hence, we claim that bothGnx0,y0n∈N and Gny0,x0n∈N are Cauchy sequences in E. Indeed, if one of them, say Gnx0,y0n∈N, is not Cauchy, then there exist v∈Imθ, Θ<v and sequences of natural numbers inn∈N and jnn∈N such that, for any in>jn>n,(18)v≺¯dθGinx0,y0,Gjnx0,y0,dθGin−1x0,y0,Gjnx0,y0<v.
Since any subsequence ofdθGn+1x0,y0,Gnx0,y0n∈N is convergent to Θ, the properties of θ imply the following contradiction:(19)v≺¯dθGinx0,y0,Gjnx0,y0≺¯θdθGin−1x0,y0,Gjnx0,y0,dθGinx0,y0,Gin−1x0,y0<θv,dθGinx0,y0,Gin−1x0,y0<θv,limn⟶∞dθGinx0,y0,Gin−1x0,y0<θv,Θ<v.
Similarly, the sequenceGny0,x0n∈N is also Cauchy. Since E is a complete θ-cone-metric space, there exist x,y∈E such that(20)limn⟶∞dθGnx0,y0,x=Θ,limn⟶∞dθGny0,x0,y=Θ.
Now, we are going to show thatx,y is a coupled fixed point of G. Since the sequence Gnx0,y0=xnn∈N is nondecreasing with limn⟶∞Gnx0,y0=x, then Gnx0,y0≤x, and since the sequence Gny0,x0=ynn∈N is nonincreasing with limn⟶∞Gny0,x0=y, then y≤Gny0,x0 for every n∈N, and accordingly, we have(21)dθGx,y,x≺¯θdθGx,y,Gn+1x0,y0,dθGn+1x0,y0,x,=θdθGx,y,Gxn,yn,dθGn+1x0,y0,x≺¯θr2dθx,xn+dθy,yn,dθGn+1x0,y0,x≺¯θr2dθx,Gnx0,y0+dθy,Gny0,x0,dθGn+1x0,y0,x.
Taking the limit asn⟶∞ with the help of equation (20), we find that(22)dθGx,y,x≺¯θr2Θ+Θ,Θ=θΘ,Θ=Θ.
Hence,dθGx,y,x=Θ; therefore, Gx,y=x. Similarly, Gy,x=y.If the partial-ordered relation onE×E is defined as(23)x,y≪2z,w⇔x≥z,andy≤w,then the following theorem is similarly proved.Theorem 4.
LetE,dθ,≤ be a partially ordered, sequentially lower-upper ordered complete θ-cone-metric space and G:E×E⟶E be a mapping having mixed monotone property, and we suppose that there are x0,y0∈E such that Tx0,y0≤x0andy0≤Ty0,x0. If there exists r∈0,1 with(24)dθGx,y,Gz,w≺¯r2dθx,z+dθy,w,∀x,y≪2z,w,then G has coupled fixed points in E.Corollary 1.
LetE 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(25)Tx,y−Tz,w≤k2x−z+y−w,∀x,y,z,w∈E,z≤x,andy≤w,then T has coupled fixed point in E.Proof.
We just notice that any Banach spaceE,. is a θ-cone-metric space E,dθ, where E,≺¯=ℝ,. is the Banach space of real numbers with the absolute value metric and with the usual ordered relation of real numbers, θu,v=u+v, θ:0,∞×0,∞⟶0,∞, and the metric dθ is the metric induced by the norm . on E, dθx,y=x−y.Remark 2.
Corollary1 is Baskar–Lackshmikantham coupled fixed-point Theorem 1. This proves that Theorem 3 is a quite good generalization of the Baskar–Lackshmikantham coupled fixed-point theorem.On the other side, we have the following results:Lemma 1.
LetE,dθ be a θ-cone-metric space and T be a mapping, T:E×E⟶E. It is supposed that there are constants a,b,c∈0,1 and a+b+c<1 such that(26)dθTx,y,Ty,z≺¯adθx,y+bdθTx,y,x+cdθTy,z,y,∀x,y,z∈E.Ifx1 and x2 are arbitrary elements in E, then the sequence xnn=3∞ defined iteratively by(27)xn=Txn−1,xn−2,∀n∈N,n>2,which satisfies the following:(28)dθxn+1,xn≺¯tdθxn,xn−1,∀n>2,(29)dθxn+1,xn≺¯tndθx2,x1,∀n>2,where t=a+c/1−b. Moreover, the sequence xnn∈N is a Cauchy sequence.Proof.
Using the contractiveness property of the given mapping gives(30)dθxn+1,xn=dθTxn,xn+1,Txn−1,xn−2≺¯adθxn,xn−1+bdθxn+1,xn+cdθxn,xn−1.
Hence,(31)dθxn+1,xn≺¯a+c1−bdθxn,xn−1,∀n>2,and repeating the last step n−2 times with the term dθxn,xn−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 limn⟶∞dθxn+1,xn=Θ and suppose that xnn∈N is not Cauchy; then, there exist v∈Imθ, Θ<v and sequences of natural numbers inn∈N and jnn∈N such that, for any in>jn>n,(32)v≺¯dθxin,xjn,dθxin−1,xjn<v.
Since any subsequence ofdθxn+1,xnn∈N is convergent to Θ, the continuity and the properties of θ imply the following contradiction:(33)v≺¯dθxin,xjn≺¯θdθxin−1,xjn,dθxin,xin−1<θv,dθxin,xin−1⟶n⟶∞θv,Θ<v.Theorem 5.
LetE,dθ be a complete θ-cone-metric space and T:E×E⟶E be a continuous mapping in the second argument, and we suppose there are three constants a,b,c∈0,1 and a+b+c<1 such that(34)dθTx,y,Ty,z≺¯adθx,y+bdθTx,y,x+cdθTy,z,y,∀x,y,z∈E,and then, T has a unique fixed point in the sense that there is a unique point x0∈E such that Tx0,x0=x0.Proof.
SinceE,dθ is complete, the Cauchy sequence xnn=3∞ given in Lemma 1 is converging to some element x0 in E. We show that x0 is fixed point of T. Using the properties of θ and the continuity of T, we see that(35)dθTx0,x0,x0≺¯θdθTx0,x0,xn,dθxn,x0≺¯θθdθTx0,x0,Tx0,xn−1,dθTx0,xn−1,xn,dθxn,x0,=θθdθTx0,xn−1,Txn−1,xn−2,dθTx0,x0,Tx0,xn−1,=θθdθTx0,xn−1,Txn−1,xn−2,dθTx0,x0,Tx0,xn−1dθxn,x0≺¯θθadθx0,xn−1+bdθTx0,xn−1,x0+cdθTxn−1,xn−2,xn−1,dθTx0,x0,Tx0,xn−1,dθxn,x0≺¯θθadθx0,xn−1+bdθTx0,xn−1,x0+cdθxn,xn−1,dθTx0,x0,Tx0,xn−1,dθxn,x0⟶n⟶∞≺¯θθaΘ+bdθTx0,x0,x0+cΘ,dθTx0,x0,Tx0,x0,Θ≺¯θθaΘ+bdθTx0,x0,x0+cΘ,Θ,Θ,=θθbdθTx0,x0,x0,Θ,Θ≺¯θbdθTx0,x0,x0,Θ≺¯bdθTx0,x0,x0.
Sinceb<1, we get dθTx0,x0,x0=Θ; consequently, Tx0,x0=x0. Now, let x and y be two arbitrarily distinct elements in E with Tx,x=x and Ty,y=y, and we have(36)dθTx,y,x≺¯θdθTx,x,Tx,y,dθTx,x,x≺¯θbdθTx,x,x+cdθTx,y,x,dθTx,x,x≺¯θbΘ+cdθTx,y,x,Θ≺¯cdθTx,y,x.
Thus,dθTx,y,x=Θ, that is, Tx,y=x. Similarly, we get Ty,x=y; therefore, x,y is a coupled fixed point of T. On the other hand, we have the following contradiction:(37)dθx,y=dθTx,x,Ty,y≺¯θdθTx,x,Tx,y,dθTx,y,Ty,y≺¯θcdθTx,y,x,adθx,y+bdθTx,y,x≺¯θcdθx,x,adθx,y+bdθx,x≺¯θΘ,adθx,y+Θ≺¯adθx,y.
Sincea<1, we have dθx,y; consequently, x=y.We conclude the following.Corollary 2.
LetE,. be a Banach space and T be a mapping from E×E into E, and we suppose that there are three constants a,b,c∈0,1 and a+b+c<1 such that(38)Tx,y−Ty,z≤ax−y+bTx,y−x+cTy,z−y,∀x,y,z∈E.Then, there is a unique pointx0∈E such that Tx0,x0=x0.Proof.
It can be proved in a similar way of Corollary1 with the same notice.Remark 3.
Corollary2 is the fixed-point theorem of Mohamed Ali Abou Bakr; accordingly, Theorem 5 is a generalization of fixed-point Theorem 2 in the setting of a complete θ-cone-metric space.


## 3. Conclusions

This paper gives further generalizations of some well-known coupled fixed-point theorems. Specifically, Theorem3 generalizes the Baskar–Lackshmikantham coupled fixed-point theorem [3], and Theorem 5 generalizes the Sahar Mohamed Ali Abou Bakr fixed-point theorem [4]; the underlying space E,dθ is a complete θ-cone-metric space, and we claim that some results of [6–10] can be proved in the case of θ-cone-metric spaces.



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