# Common Fixed Point Theorem for Multivalued Mappings Using Ternary Relation inG-Metric Space with an Application

Authors: Lucas Wangwe
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

This paper proves a common fixed point theorem for single-valued and multivalued mappings by using ternary relation inG-metric spaces. Henceforth, results obtained will be verified with the help of illustrative examples. Also, we demonstrate the results with an application.


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## Body

## 1. Introduction

In 1969, Nadler [1] introduced multivalued contraction mappings using the Hausdorff metric and extended Banach’s contraction principle [2] from single-valued to multivalued mappings. Since then, several researchers have generalized these results for multivalued mappings in various spaces. As a result, the multivalued mapping theory has many applications in diverse areas, such as control theory, approximation theory, differential equations, and economics.Definition 1 (see [1]).
LetX,d be a metric space. A map T:X⟶X is said to be a multivalued contraction if there exists 0≤λ≤1 such that
(1)HTx,Ty≤θdx,y,for all x,y∈X.Kaneko and Sessa [3] extended the concept of compatible mappings due to Jungck [4] to include multivalued mappings as well as single-valued mappings. They followed the works of Kubiak [5] and Nadler [1] and proved coincidence and fixed point theorems for hybrid pair of compatible mappings. Pathak [6] extended the concept of compatible hybrid mappings to f-weak compatible hybrid mappings and proved a coincidence theorem, an extension of the results of Kaneko and Sessa [3].In 2006, Mustafa and Sims [7, 8] gave a generalization of D-metric space to G-metric space soon after identifying some shortcomings concerning the fundamental topological structure on D-metric spaces. They defined several notions, such as continuity, completeness, compactness, convergence, and space product in the G-metric space setting. Abbas and Rhoades [9] established the concept of a common fixed point in G-metric spaces. In addition, Kaewcharoen and Kaewkhao [10] presented some common fixed points for single-valued and multivalued mappings in G-metric spaces. At the same time, Tahat et al. [11] proved common fixed points for single-valued and multivalued maps satisfying a generalized contraction in G-metric spaces. Recently, Mustafa et al. [12] gave the results in common fixed points for multivalued mappings in G-metric spaces with applications using F-contraction concept, and Shoaib and Shahzad [13] proved the common fixed point of multivalued mappings in ordered dislocated quasi G-metric spaces.On the other hand, Alam and Imdad [14] gave a generalization of the Banach contraction principle in a complete metric space equipped with binary relation. Their results show that the contraction condition holds only for those elements linked with the binary relation, not for every pair of elements. For more results on binary relation, one can see [15–20] and the references therein. Perveen et al. [21] gave the prove in relation theoretic common fixed point results for generalized weak nonlinear contractions with an application. Hossain et al. [22] gave the study of relation-theoretic metrical fixed point theorem for rational type contraction mapping with an application. Further, Gaba et al. [23] extended the works of Alam and Imdad [14] by using the Banach contraction mapping principle in generalized metric spaces with a ternary relation. Furthermore, Radha and Singh [24] proved a novel approach to G-metric spaces by using ternary relations. Badshah et al. [25] proved some common fixed point theorems for contractive mappings in cone 2-metric spaces equipped with a ternary relation.This paper is aimed at proving a common fixed point theorem for single-valued and multivalued mappings by using the ternary relation concept in theG-metric space setting. Specifically, we improve and extend the works due to Alam and Imdad [14], Ahmadullah et al. [15, 16], Eke et al. [18], and Radha and Singh [24]. In doing so, we will generalize several other works in the literature having the same directions.


## 2. Preliminaries

In this section, we introduce some definitions, theorems, and preliminary results, which will be helpful in developing the main result.Motivated by Mustafa and Sims [8], we can formulate the axioms of G-metric spaces as follows:Definition 2 (see [8]).
LetX be a nonempty set and G:X×X×X⟶ℝ+ be a function satisfying the following conditions:
G1Gx,y,z=0 if x=y=z
G20<Gx,x,y, for all x,y∈X with z=y
G3Gx,x,y≤Gx,y,z for all x,y∈X with z=y
G4Gx,y,z=Gx,z,y=Gy,z,x=⋯ (symmetry in all three variables)
G5Gx,y,z≤Gx,a,a+Ga,y,z, for all x,y,z,a∈X (rectangle inequality)The functionG is called a generalized metric or G-metric, and the pair X,G is called a G-metric space.The following example satisfies the above axioms.Example 1 (see [8]).
LetX=ℝ be the set of all real number. Define G:ℝ3⟶ℝ+ by
(2)Gx,y,z=x−y+y−z+x−z,for all x,y,z∈X. Then, it is clear that X,G is a G-metric space with a G-metric on X.Note that ifGx,y,z=0 then x=y=z.Mustafa and Sims [8] proved the following proposition satisfying a G-metric properties.Proposition 3 (see [8]).
LetX,G be a G-metric space; then, the metric associated with G satisfies the following:
(i)
Gx,y,z≤Gx,x,y+Gx,x,z(ii)
Gx,y,y≤2Gy,x,x(iii)
Gx,y,z≤Gx,a,a+Gy,a,a+Gz,a,a
for allx,y,z,a∈X.Inspired by Mustafa and Sims [8], we established some topological properties such as convergence, completeness, and continuity in G-metric spaces as follows:Definition 4 (see [8]).
LetX,G be a G-metric space. A sequence xn∈X is said to be the following:
(i)
G-convergent to x∈X if for any ε>0, there exists k∈ℕ such that Gx,xn,xn<ε for all n≥k(ii)
G-Cauchy if for ε>0, there exists k∈ℕ such that for all xn in X is called Cauchy sequence if for each ε>0, there exists n0∈ℕ such that Gxn,xm,xp<ε for all n,m,p≥n0, i.e., Gxn,xm,xp⟶0 as n,m,p⟶∞Definition 5 (see [8]).
AG-metric space is said to be G-complete if every G-Cauchy sequence in X is G-convergent. Every G-metric on X defines a metric
(3)dGx,y=Gx,y,y+Gy,x,x,for all x,y∈X.Proposition 6 (see [8]).
LetX,G be a G-metric space. Then, the following properties are equivalent:
(i)
xn is G-convergent to x(ii)
Gxn,xm,xn⟶0 as n,m⟶∞(iii)
Gxn,x,x⟶0 as n⟶∞(iv)
xn is a G-Cauchy sequence(v)
For everyε>0, there exists k∈ℕ such that Gxn,xm,xm<ε for n,m>kDefinition 7 (see [8]).
LetX,G and X′,G′ be two G-metric spaces, and let T:X,G⟶X′,G′. Then, the map T is said to be G-continuous at x∈X if for ε>0, there exists δ>0 such that for all x,y∈X and Ga,x,y<δ, we have G′Ta,Tx,Ty<ε. The function T is G-continuous if it is G-continuous for each a∈X.Proposition 8 (see [8]).
LetX,G and X′,G′ be two G-metric spaces, and let T:X,G⟶X′,G′. Then, the map T is said to be G-continuous at x∈T if and only if T is sequentially continuous, i.e., whenever xn is G-convergent to x, the sequence Txn is G-convergent to Tx.Lemma 9 (see [8]).
LetX,G be a G-metric space, then the function Gx,y,zis jointly continuous in all three of its variables.Later, Kaewcharoen and Kaewkhao [10] established the multivalued notion in G-metric space. Let X be a G-metric space. Denote CBX be the class of all nonempty, closed, and bounded subsets of X. Let HG.,.,. be the Haursdorff-G-distance on CBX, that is, for A,B,C∈CBX, define
(4)HGA,B,C=maxsupx∈AGx,B,C,supx∈BGx,C,A,supx∈CGx,A,B,where
(5)Gx,B,C=dGx,B+dGB,C+dGx,C,(6)dGx,B=infdGx,y,y∈B,(7)dGA,B=infdGa,b,a∈A,b∈B,(8)dGx,y,C=infGx,y,z,z∈C.Lemma 10 (see [10]).
LetX,G be a G-metric space and A,B∈CBX. Then, for each a∈A, we have
(9)Ga,B,B≤HGA,B,B.Lemma 11 (see [11]).
LetX,G be a G-metric space and A,B∈CBX; then, for each ε>0, there exists b∈B such that
(10)Ga,b,b≤HGA,B,B+θ.Lemma 12 (see [26]).
Letxn1,xn2,⋯,xnN⊆X be N-sequence on a G-metric space X,G such that
(11)limn⟶∞Gxni,xn+1i,xn+1i=0,for all i∈1,2,⋯,N.Lemma 13 (see [26]).
Ifψ is a c-comparison function, for each t>0(12)limn⟶+∞ψnt=0⟹ψt=t.Definition 14 (see [10]).
LetX be a nonempty set. Assume SX⟶X and T:X⟶2X are two mappings. If w=Sx∈Tx for some x∈X, then x is called a coincidence point of the pair T,S and w is a point of coincidence of S and T. The mapping S and T are called weakly compatible if Sx∈Tx for some x∈X implies STx⊆TSx.Proposition 15 (see [10]).
LetX be a nonempty set. Assume S:X⟶X and T:X⟶2X are weakly compatible mappings. If S and T have a unique point of coincidence u=Sx∈Tx, then u is a unique common fixed point of S and T.Definition 16 (see [27]).
LetT,S be a pair of self mappings on an ordered metric space X,d,⪯ with TX⊆SX. For every x0∈X, consider the sequence xn∈X defined by Txn=Sxn+1, for all n∈ℕ0. Then, Txn is called T-S sequence with initial point x0.


### 2.1. Ternary Relation-Theoretic inG-Metric Spaces

In this part, we recall some definition, proposition, and preliminary results inG-metric space related to ternary relation. We define ℕ0=0∪ℕ, where ℕ is a set of natural numbers in X.Motivated by Nová and Novotný [28], NovotnỲ [29], Šlapal [30], and Alam and Imdad [14], we can introduce the property of ternary relation as follows:Definition 17 (see [30]).
LetX be a set and R⊆X×X×X . Then, R is said to be ternary relation on X. A ternary relation R defined on a nonempty set G is called
(1)
Reflexive ifx,x,x∈R for all x∈X(2)
Symmetric if and only ifx,y,z∈R⟹y,z,x∈R for any x,y,z∈G(3)
Transitive if and only ifx,y,z∈R, y,z,u∈R imply x,z,u∈R, for any x,y,z,u in G(4)
Asymmetric if and only ifx,y,z∈R⟹y,z,x∉R(5)
Irreflexive ifx,y,z∈R⟹x≠y≠z(6)
Irreversible ifx,y,z∈R⟹y,x,p∉R for any p∈G(7)
Feebly regular ifx,y,p∈R⟹y,z,q∉Rx,y,z∈R(8)
Regular ifx,y,p∈R, y,z,q∈R⟹x,y,z∈R(9)
Feebly translative ifx,y,z∈R, y,p,q∈R⟹∃r∈G:x,p,r∉R(10)
Translative ifR is feebly translative and x,y,z∈R,⟹∃r∈G:x,z,r∈R or y,z,r∈R(11)
Cyclic ifx,y,z∈R⟹y,z,x∈R for any x,y,z∈R(12)
Complete ifx,y,z∈R∀x,y,z∈X(13)
Weakly complete ifx,y,z∈R or x=y=z, ∀x,y,z∈XDefinition 18 (see [14]).
A relationR among sets A, B, and C is a subset of the Cartesian product A, B, and C, R⊆A×B×C.Proposition 19 (see [24]).
For any ternary relationR defined on a nonempty set X,
(13)x,y,z∈Rs⟺x,y,z∈R.Definition 20 (see [14]).
LetX be a nonempty set and T a self- mapping on X. A ternary relation R defined on X is called T-closed if for x,y,z∈X,
(14)x,y,z∈R⟹Tx,Ty,Tz∈R.Motivated by Alam and Imdad [14] and Perveen et al. [21], we introduce the following definitions and proposition for multivalued mapping in the context of G-metric space setting.Definition 21.
LetT and S be two multivalued mappings defined on a nonempty set X. Then, a ternary relation R defined on X is called T,S-closed if Sx,Sy,Sz∈R⟹Tx,Ty,Tz∈R,∀x,y,z∈X.Proposition 22.
LetX be nonempty set, R a ternary relation on X, and T,S two multivalued mappings on X. If R is T,S-closed, then Rs is T,S-closed.Definition 23.
LetX,G be a G-metric space endowed with a ternary relation R and T,S:X⟶CBX. T and S are said to be R-compatible if for any sequence xn such that Txn and Sxn are R-preserving and limn⟶∞Txn=limn⟶∞Sxn, we have
(15)limn⟶∞GTSxn,STxn,STxn=0.Inspired by Kolman et al. [19], we introduce the following definition in G-metric space setting.Definition 24.
LetX,G be a G-metric space, R a ternary relation defined on X, and x,y a pair of points in X. Then, a finite sequence z0,z1,z2,⋯,zl∈X is said to be a path of length lwherel∈N joining x to y in R if z0=x,zl=y and zi,zi+1,zi+1∈R for each i∈1,2,3,⋯l−1.Next, we state some preliminary results which will be helpful to develop our main results.Ahmadullah et al. [17] proved the results in metric-like space as well as partial metric spaces equipped with a binary relation as follows:Theorem 25 (see [17]).
LetX,σ, be a metric-like spaces equipped with a binary relation R defined on X and f a self-mapping on X. Suppose that the following conditions are satisfied:
(a)
There exists a subsetY⊆X with fX⊆Y such that Y,σ is R-complete(b)
There existsx0 such that x0,fx0∈R(c)
R is f-closed(d)
Eitherf is R-continuous-like or RY is σ-self-closed(e)
There exists a constantk∈0,1 such that (∀x,y∈X with x,y∈R)(16)σfx,fy≤kσx,y.
Then,f has a fixed point. Moreover, if
(f)
Yfx,fy,Rs is nonempty, for each x,y∈X. Then, f has a unique fixed pointRadha and Singh [24] proved the following theorem in G-metric spaces with ternary relation as follows:Theorem 26 (see [24]).
LetX,G be a complete G-metric space, R a ternary relation on X, and T a self-mapping on X. Suppose that the following conditions hold:
(i)
XT,R is nonempty(ii)
R is T-closed(iii)
EitherT is continuous or R is G-self-closed(iv)
There exists a constantθ∈0,1 such that (∀x,y,z∈X with x,y,z∈R)(17)GTx,Ty,Tz≤θGx,y,z.
Then,T has a fixed point. Moreover, if
(v)
γx,y,z,Rs is nonempty, for each x,y,z∈X. Then, T has a unique fixed point


## 2.1. Ternary Relation-Theoretic inG-Metric Spaces

In this part, we recall some definition, proposition, and preliminary results inG-metric space related to ternary relation. We define ℕ0=0∪ℕ, where ℕ is a set of natural numbers in X.Motivated by Nová and Novotný [28], NovotnỲ [29], Šlapal [30], and Alam and Imdad [14], we can introduce the property of ternary relation as follows:Definition 17 (see [30]).
LetX be a set and R⊆X×X×X . Then, R is said to be ternary relation on X. A ternary relation R defined on a nonempty set G is called
(1)
Reflexive ifx,x,x∈R for all x∈X(2)
Symmetric if and only ifx,y,z∈R⟹y,z,x∈R for any x,y,z∈G(3)
Transitive if and only ifx,y,z∈R, y,z,u∈R imply x,z,u∈R, for any x,y,z,u in G(4)
Asymmetric if and only ifx,y,z∈R⟹y,z,x∉R(5)
Irreflexive ifx,y,z∈R⟹x≠y≠z(6)
Irreversible ifx,y,z∈R⟹y,x,p∉R for any p∈G(7)
Feebly regular ifx,y,p∈R⟹y,z,q∉Rx,y,z∈R(8)
Regular ifx,y,p∈R, y,z,q∈R⟹x,y,z∈R(9)
Feebly translative ifx,y,z∈R, y,p,q∈R⟹∃r∈G:x,p,r∉R(10)
Translative ifR is feebly translative and x,y,z∈R,⟹∃r∈G:x,z,r∈R or y,z,r∈R(11)
Cyclic ifx,y,z∈R⟹y,z,x∈R for any x,y,z∈R(12)
Complete ifx,y,z∈R∀x,y,z∈X(13)
Weakly complete ifx,y,z∈R or x=y=z, ∀x,y,z∈XDefinition 18 (see [14]).
A relationR among sets A, B, and C is a subset of the Cartesian product A, B, and C, R⊆A×B×C.Proposition 19 (see [24]).
For any ternary relationR defined on a nonempty set X,
(13)x,y,z∈Rs⟺x,y,z∈R.Definition 20 (see [14]).
LetX be a nonempty set and T a self- mapping on X. A ternary relation R defined on X is called T-closed if for x,y,z∈X,
(14)x,y,z∈R⟹Tx,Ty,Tz∈R.Motivated by Alam and Imdad [14] and Perveen et al. [21], we introduce the following definitions and proposition for multivalued mapping in the context of G-metric space setting.Definition 21.
LetT and S be two multivalued mappings defined on a nonempty set X. Then, a ternary relation R defined on X is called T,S-closed if Sx,Sy,Sz∈R⟹Tx,Ty,Tz∈R,∀x,y,z∈X.Proposition 22.
LetX be nonempty set, R a ternary relation on X, and T,S two multivalued mappings on X. If R is T,S-closed, then Rs is T,S-closed.Definition 23.
LetX,G be a G-metric space endowed with a ternary relation R and T,S:X⟶CBX. T and S are said to be R-compatible if for any sequence xn such that Txn and Sxn are R-preserving and limn⟶∞Txn=limn⟶∞Sxn, we have
(15)limn⟶∞GTSxn,STxn,STxn=0.Inspired by Kolman et al. [19], we introduce the following definition in G-metric space setting.Definition 24.
LetX,G be a G-metric space, R a ternary relation defined on X, and x,y a pair of points in X. Then, a finite sequence z0,z1,z2,⋯,zl∈X is said to be a path of length lwherel∈N joining x to y in R if z0=x,zl=y and zi,zi+1,zi+1∈R for each i∈1,2,3,⋯l−1.Next, we state some preliminary results which will be helpful to develop our main results.Ahmadullah et al. [17] proved the results in metric-like space as well as partial metric spaces equipped with a binary relation as follows:Theorem 25 (see [17]).
LetX,σ, be a metric-like spaces equipped with a binary relation R defined on X and f a self-mapping on X. Suppose that the following conditions are satisfied:
(a)
There exists a subsetY⊆X with fX⊆Y such that Y,σ is R-complete(b)
There existsx0 such that x0,fx0∈R(c)
R is f-closed(d)
Eitherf is R-continuous-like or RY is σ-self-closed(e)
There exists a constantk∈0,1 such that (∀x,y∈X with x,y∈R)(16)σfx,fy≤kσx,y.
Then,f has a fixed point. Moreover, if
(f)
Yfx,fy,Rs is nonempty, for each x,y∈X. Then, f has a unique fixed pointRadha and Singh [24] proved the following theorem in G-metric spaces with ternary relation as follows:Theorem 26 (see [24]).
LetX,G be a complete G-metric space, R a ternary relation on X, and T a self-mapping on X. Suppose that the following conditions hold:
(i)
XT,R is nonempty(ii)
R is T-closed(iii)
EitherT is continuous or R is G-self-closed(iv)
There exists a constantθ∈0,1 such that (∀x,y,z∈X with x,y,z∈R)(17)GTx,Ty,Tz≤θGx,y,z.
Then,T has a fixed point. Moreover, if
(v)
γx,y,z,Rs is nonempty, for each x,y,z∈X. Then, T has a unique fixed point


## 3. Main Results

In this section, we prove the following theorem which is a generalization and improvement of Theorem26.Theorem 27.
LetX,G be a complete G-metric space equipped with a ternary relation R defined on X. Let S:X⟶X and T:X⟶CBX be a pair of hybrid multivalued mapping on X. Suppose that the following conditions hold:
(i)
There existsTX⊆SX such that X,G is R-complete(ii)
There existsx0 such that Sx0,Tx0,Tx0∈R(iii)
XT,S,R is nonempty(iv)
Forn even Sx2n and Tx2n converge to a common fixed point(v)
EitherT,S is R-continuous- or R is G-self-closed(vi)
There exists a constantθ∈0,1 such that (∀x,y,z∈X with x,y,z∈R)(18)HGTx,Ty,Tz≤θMx,y,z,where
(19)Mx,y,z≤maxGSx,Ty,Ty,GSx,Tz,TzGSy,Tx,Tx,GSy,Tz,TzGSz,Tx,Tx,GSz,Ty,Ty.
Then,T and S have a common fixed point. Moreover, if
(vii)
ϒTx,Ty,TzTx,Ty,Tz,Rs is nonempty, for each x,y,z∈X. Then, T and S have a unique common fixed point if z and w are coincidence points of T and S such that dGz,w≤0Proof.
Suppose thatTX⊆SX, TX is a G-complete subspace of X and X,G is R-complete, for x0 with Sx0,Tx0,Tx0∈R. We can construct a T-S-sequence such that Txn∈R with initial point x0 satisfying
(20)Sx2n,Tx2n−1,Txn−1,Sx2n+1,Tx2n,Txn,Sx2n+2,Tx2n+1,Txn+1⋯,
∀n∈ℕ0, such that Tx2n, Sx2n∈TX.
By (iii), letx0 be an arbitrary element of XT,S,R, then Sx0,Tx0,Tx0∈R. If Sx0=Tx0, then x0 is a common fixed point of T and S and proof is completed. On contrary, if Tx0≠Sx0, then SX⊂TX. Now, we choose x1∈X such that Sx1=Tx0. Again, we can choose x2∈X such that Sx2=Tx1. Proceeding the same way and applying Definition 16, we can construct a sequence xn⊂X, such that
(21)Sx2n=Tx2n−1,Sx2n+1=Tx2n,which is equivalently to
(22)Sx2n,Sx2n+1=Tx2n−1,Tx2n.
By substitutingx=x2n−1 and y=x2n in (18), we have
(23)HGTx2n−1,Tx2n,Tz≤θMx2n−1,x2n,z,where
(24)Mx2n−1,x2n,z≤maxGSx2n−1,Tx2n,Tx2nGSx2n−1,Tz,TzGSx2n,Tx2n−1,Tx2n−1GSx2n,Tz,TzGSz,Tx2n−1,Tx2n−1GSz,Tx2n,Tx2n.
Applyingy=z=x2n in (24), we obtain
(25)Mx2n−1,x2n,x2n≤maxGSx2n−1,Tx2n,Tx2nGSx2n−1,Tx2n,Tx2nGSx2n,Tx2n−1,Tx2n−1GSx2n,Tx2n,Tx2nGSx2n,Tx2n−1,Tx2n−1GSx2n,Tx2n,Tx2n,≤maxGSx2n−1,Sx2n+1,Sx2n+1GSx2n−1,Sx2n+1,Sx2n+1GSx2n,Sx2n,Sx2nGSx2n,Sx2n+1,Sx2n+1GSx2n,Sx2n,Sx2nGSx2n,Sx2n+1,Sx2n+1.
ByG5, we have
(26)GSx2n−1,Sx2n+1,Sx2n+1≤GSx2n−1,Sx2n,Sx2n+GSx2n,Sx2n+1,Sx2n+1.
Using (26) in (25), we get
(27)Mx2n−1,x2n,x2n≤maxGSx2n−1,Sx2n,Sx2n+GSx2n,Sx2n+1,Sx2n+1GSx2n−1,Sx2n,Sx2n+GSx2n,Sx2n+1,Sx2n+1GSx2n,Sx2n,Sx2nGSx2n,Sx2n+1,Sx2n+1GSx2n,Sx2n,Sx2nGSx2n,Sx2n+1,Sx2n+1.(28)Mx2n−1,x2n,x2n=GSx2n−1,Sx2n,Sx2n+GSx2n,Sx2n+1,Sx2n+1.
From (23), (27), and Lemma 11, we get
(29)GSx2n,Sx2n+1,Sx2n+1≤θGSx2n−1,Sx2n,Sx2n+θGSx2n,Sx2n+1,Sx2n+1+θ,1−θGSx2n,Sx2n+1,Sx2n+1≤θGSx2n−1,Sx2n,Sx2n+θ,GSx2n,Sx2n+1,Sx2n+1≤θ1−θGSx2n−1,Sx2n,Sx2n+θ1−θ.
Similarly, forSx2n+1∈Tx2n, there exists Sx2n+2∈Tx2n+1 such that
(30)GSx2n,Sx2n+1,Sx2n+1≤HGTx2n,Tx2n+1,Tx2n+1+θ2,≤θ21−θ2GSx2n−1,Sx2n,Sx2n+2θ21−θ2.
Lettingψ=θ/1−θ and (30), we have
(31)GSx2n,Sx2n+1,Sx2n+1≤ψ2GSx2n−1,Sx2n,Sx2n+2ψ2.
Continuing this process, we obtain by mathematical induction a sequencexn with Sx2n+1∈Tx2n such that
(32)GSx2n,Sx2n+1,Sx2n+1≤ψnGSx2n−1,Sx2n,Sx2n+nψn.
It follows that(33)∑n=0∞GSx2n,Sx2n+1,Sx2n+1≤∑n=0∞ψnGSx2n−1,Sx2n,Sx2n+∑n=0∞nψn<∞.
By taking limits on both sides in (33), we obtain
(34)limn⟶∞GSx2n,Sx2n+1,Sx2n+1=GSx0,Sx1,Sx1=0.
Using (32) and G4-symmetric properties, for all n,m∈ℕ0 with m>n, we obtain
(35)GSx2n,Sx2n+m,Sx2n+m≤GSx2n,Sx2n+1,Sx2n+1+GSx2n+1,Sx2n+2,Sx2n+2+⋯+GSx2n+m−2,Sx2n+m−1,Sx2n+m−1+GSx2n+m−1,Sx2n+m,Sx2n+m,≤ψn+ψn+1+ψn+2+⋯+ψn+m−1GSx2n−1,Sx2n,Sx2n+n,≤ψn1+ψn+ψn+1+ψn+2+⋯+ψn+m−1σ,≤ψn1−ψσ,⟶0asn⟶∞.where σ=GSx2n−1,Sx2n,Sx2n+n and 0<ψ<1. This shows that the sequence Sx2n is a G-Cauchy sequence in the complete subspace SX.
The results obey aG-Cauchy sequence properties of completeness. Hence, Sx2n is R-preserving Cauchy sequence. If the pair T,S is closed and R-compatible, using Definition 23, we have
(36)TSx2n=STx2n,Tx2n=Sx2n,STx2n=SSx2n,TSx2n=TTx2n.
We can show that forn, even Tx2n and Sx2n converge to a common fixed point of T and S. To see this, let x=Tx2n,y=Sx2n, using (18) which gives
(37)HGTSx2n,TTx2n,TTx2n≤θMSx2n,Tx2n,Tx2n,where
(38)MSx2n,Tx2n,Tx2n≤maxGSSx2n,TTx2n,TTx2nGSSx2n,TTx2n,TTx2nGSTx2n,TSx2n,TSx2nGSTx2n,TTx2n,TTx2nGSTx2n,TSx2n,TSx2nGSTx2n,TTx2n,TTx2n,MSx2n,Tx2n,Tx2n=GSTx2n,TSx2n,TSx2n,
which implies that(39)HGTSx2n,TTx2n,TTx2n≤θGSTx2n,TSx2n,TSx2n.
By using Lemma10 and (39), we obtain
(40)GSx2n,Tx2n,Tx2n≤θGSTx2n,TSx2n,TSx2n.
SinceX is a complete G-metric space and T,S is closed and R-compatible, from Definition 23, we get
(41)limn⟶∞GSTx2n,TSx2n,TSx2n=0.
Assume thatz is a common fixed point of T and S such that limn⟶∞x2n=z. Thus, SSz=SSz=Sz. So Sz is a common fixed point of T and S.
Now we show thatz is the unique common fixed point of T and S. For the uniqueness, take z=Sz as a common fixed point of T and S. Assume that w=Tw and z≠w, and using x=z,y=w in (3.1), we get
(42)HGTz,Tw,Tw≤θMz,w,w,where
(43)Mz,w,w≤maxGSz,Tw,Tw,GSz,Tw,TwGSw,Tz,Tz,GSw,Tw,TwGSw,Tz,Tz,GSw,Tw,Tw=GSz,Tw,Tw.
Consequently,(44)HGTz,Tw,Tw≤θGSz,Tw,Tw.
Using Lemma10 and (44), we get
(45)GSz,Tw,Tw≤θGSz,Tw,Tw.
Hence, we get(46)GSz,Tw,Tw≤0,⟹GSz,Tw,Tw=0,and we conclude that Sz=Tw, which is a contradiction. Therefore, z is a unique common fixed point of T and S.
Using the condition considered in Theorem27, we prove assertion (vii) as follows: we observe that CT,S is nonempty, so let us take a pair of elements say a,b in CT,S such that
(47)Ta=Sa=a¯,Tb=Sb=b¯.
Next, we are required to show thata¯=b¯. By observing the above assertion, there exists an S-path (say, z0,z1,z2,⋯,zl) of length l in Rs from Ta to Tb, with
(48)Sz0=Ta,(49)Szl=Tb,such that
(50)Sz2i,Sz2i+1,Sz2i+1∈Rs⊆R,for all i∈0,1,2,3,⋯l−1.
And(51)Sz2i,Tz2i,Tz2i∈Rs⊆R,for every i∈0,1,2,3,⋯l−1.
Define two constant sequences such that(52)z2n0=a,z2nl=b.
By using (49), for all n∈ℕ, we have
(53)Tz2n0=Ta=a¯,Tz2nl=Tb=b¯.
By usual substitution forz0i=zi for each i∈0,1,2,⋯l, that is,
(54)z01=z1,z02=z2,z03=z3,z04=z4,z0l−1=zl−1.
Recall thatTX⊆SX and Lemma 12. Thus, we construct a sequence
(55)z2n1,z2n2,z2n3,⋯z2ni∈X.
In general,zn1∈X(56)Sz2n+11=Tz2n1,Sz2n+12=Tz2n2,Sz2n+13=Tz2n3,Sz2n+14=Tz2n4,Sz2n+1l−1=Tz2nl−1,∀n∈ℕ.
We obtain(57)Sz2n+1i=Tz2ni,for all i∈0,l−1. Corresponding to each zi, we have Sz0i,Sz1i∈R from (49), (50), and T,S-compactness of R, and we get
(58)limn⟶∞GSz2ni,Sz2n+1i,Sz2n+1i=0,for each i∈1,2,3,⋯l−1.
Thus,R is T,S-closed, and we conclude that Tz2ni,Tz2ni+1,Tz2ni+1∈R, for each i∈0,1,2,3,⋯l−1 and for all n∈ℕ.
Otherwise,Sz2ni,Sz2ni+1,Sz2ni+1∈R, for each i∈0,1,2,3,⋯l−1 and for all n∈ℕ.
DefineG2ni=GSz2ni,Sz2ni+1,Sz2ni+1, for each i∈0,1,2,3,⋯l−1 and for all n∈ℕ. We assert that limn⟶∞G2ni>0. Assume that limn⟶∞G2ni=G>0.
SinceSz2ni,Sz2ni+1,Sz2ni+1∈R, either Sz2ni,Sz2ni+1,Sz2ni+1∈R or Sz2ni+1,Sz2ni,Sz2ni∈R.
IfSz2ni,Sz2ni+1,Sz2ni+1∈R, for x=z2ni and y=z2ni+1, then applying the condition (vi), we have
(59)HGTz2ni,Tz2ni+1,Tz2ni+1≤θMz2ni,z2ni+1,z2ni+1,where
(60)Mz2ni,z2ni+1,z2ni+1≤maxGSz2ni,Tz2ni+1,Tz2ni+1GSz2ni,Tz2ni+1,Tz2ni+1GSz2ni+1,Tz2ni,Tz2niGSz2ni+1,Tz2ni+1,Tz2ni+1GSz2ni+1,Tz2ni,Tz2niGSz2ni+1,Tz2ni+1,Tz2ni+1≤maxGSz2ni,Sz2n+1i+1,Sz2n+1i+1GSz2ni,Sz2n+1i+1,Sz2n+1i+1GSz2ni+1,Sz2n+1i,Sz2n+1iGSz2ni+1,Sz2n+1i+1,Sz2n+1i+1GSz2ni+1,Sz2n+1i,Sz2n+1iGSz2ni+1,Sz2n+1i+1,Sz2n+1i+1=GSz2ni,Sz2n+1i+1,Sz2n+1i+1.
By Lemma10 and (60), we get
(61)GSz2ni,Tz2ni+1,Tz2ni+1≤θGSz2ni,Sz2n+1i+1,Sz2n+1i+1,GSz2ni,Sz2n+1i+1,Sz2n+1i+1≤θGSz2ni,Sz2n+1i+1,Sz2n+1i+1.
Taking lim asn⟶∞ and using limn⟶∞G2ni=G, we get
(62)GSz2ni,Sz2n+1i+1,Sz2n+1i+1≤0,which is contradiction, and hence,
(63)limn⟶∞G2ni=G=0.
The same for symmetric propertyG4, if Sz2ni+1,Sz2n+1i,Sz2n+1i∈R, we have
(64)limn⟶∞G2ni=limn⟶∞GSz2ni+1,Sz2n+1i,Sz2n+1i=0,for i∈0,1,2,⋯l−1.
Using (50), limn⟶∞G2ni=0, and (G4), we have
(65)Ga¯,b¯,b¯=GSz2n0,Sz2ni,Sz2ni≤∑i=0l−1GSz2ni,Sz2ni+1,Sz2ni+1≤∑i=0l−1G2ni,⟶0asn⟶∞,
so that(66)Ga¯,b¯,b¯=0,a¯=b¯.
Therefore,(67)Sz=Sw.
Now, suppose thatz=a¯ and w=b¯, we claim that for the existence of unique fixed point dGz,w>0. To see this, we have
(68)dGz,w=Gz,w,w+Gw,z,z.
From(69)Gz,w,w=z−w+w−w+z−w=2z−w,Gw,z,z=w−z+z−z+w−z=2z−w,we have
(70)dGz,w=2z−w+2z−w=4z−w,which is a contradiction. Since a¯=b¯z=w, thus z is a unique common fixed point of T and S, and we have z=b. Thus, the proof is completed.From Theorem27, we can deduce the corollary as follows:
(i)
SX is closed in X,G(ii)
There exists a continuous comparison functionψ∈Ψ such that(71)HGTx,Ty,Tz≤ψMx,y,z,where
(72)Mx,y,z≤GSx,Sy,Sz,GSx,Tx,SzGSz,Tz,Sz,GSy,Sy,Sz.Corollary 28.
LetX,G be a G-metric space and let S:X⟶X and T:X⟶CBX be pair of hybrid mapping such that the following conditions are hold:Then,T and S have the unique common fixed point.Proof.
The proof of the above corollary follows similar steps of Theorem27. Therefore, the proof is completed.The following example is motivated from [31] for ternary relation.Example 2.
ConsiderX=0,1,2,3,⋯14,15, R a ternary relation on X defined by the equation
(73)x2+5y=z,endowed with complete G-metric space, defined by G-metric Gx,y,z=∣x−y∣+∣y−z∣+∣x−z∣ in ℝ3 with ternary relation. Since x2>15 for x>3, we need only to find solutions for y and z when x=1,2,3. This yields the following:
(74)R=1,1,6,1,2,11,2,1,9,2,2,14,3,1,14onX.We claim thatX is either G-complete or R-complete. To observe this, define S:X⟶X by
(75)Sx=x2,∀x∈X,and T:X⟶CBX by
(76)Tx=0,x,∀x∈X.Then,(i)
TX is a G-complete subspace in X(ii)
There exists a constantθ∈0,1 such that (∀x,y,z∈X with x,y,z∈R)(77)HGTx,Ty,Tz≤θMx,y,z,where
(78)Mx,y,z≤maxGSx,Ty,Ty,GSx,Tz,TzGSy,Tx,Tx,GSy,Tz,TzGSz,Tx,Tx,GSz,Ty,Ty.By using (i), we show thatTX is a G-complete subspace in X. By Definition 5, we have
(79)Gx,y,y≤x−y+y−y+x−y=2x−y,(80)Gy,x,x≤y−x+x−x+y−x,=2y−x.Using (79) and (79) in (3), we get
(81)dGx,y≤2x−y+2y−x=4x−y.To prove (ii), letx,y,z∈X. If x=y=z=0, then Tx=Ty=Tz=0 and HGTx,Ty,Tz=0. The proof is completed. On contrary, suppose that the values of x,y,z are not all zero.Forx≤y≤z, we have
(82)HGTx,Ty,Tz=HG0,x,0,y,0,z.By (4), we have
(83)⟹maxsup0≤a≤xGa,0,y,0,zsup0≤b≤yGb,0,x,0,zsup0≤c≤zGc,0,x,0,y.Sincex≤y≤z, then 0,x⊆0,y⊆0,z, using (5), and as a result,
(84)dG0,x,0,y=0,dG0,y,0,z=0,dG0,x,0,z=0.Now for each0≤a≤x and dGx,y=4∣x−y∣ in (5) and (6), we have
(85)Ga,0,y,0,z=dGa,0,y+dG0,y,0,y+dGa,0,z,≤4a−4y+0+4a−4z,=8a−4y−4z,≤8x−4y−4z,=8x−4y−4z.Next, for each0≤b≤y and dGx,y=4∣x−y∣ in (5) and (6), we get
(86)Gb,0,x,0,z=dGb,0,x+dG0,x,0,z+dGb,0,z,≤4b−4x+0+4b−4z,=8b−4x−4z,=8y−4x−4z,=8y−4x+z.Likewise, for each0≤z and (5) and (6), it gives
(87)Gc,0,x,0,y=dGc,0,x+dGx,0,y+dGc,0,y,≤4c−4x+0+4c−4y,=8c−4x−4y,=8z−4x+y.Consequently,(88)HGTx,Ty,Tz≤maxsup0≤a≤x8x−4y+zsup0≤b≤y8y−4z+xsup0≤c≤z8z−4x+y=8z−4x+y.On the other hand, we formulate the followingG-metrics. By (5) and (6), this yields to
(89)GSx,Ty,Ty=Gx2,0,y,0,y,=dGx2,0,y+dG0,y,0,y+dGx2,0,y,≤infdGx2,0,dGx2,y+0+infdGx2,0,dGx2,y,=4x2−y∣+4∣x2−y,=4x−2y.In similar manner, we can calculate the followingG-metrics.
(90)GSy,Tz,Tz=Gy2,0,z,0,z=4y−2z,GSz,Tx,Tx=Gz2,0,x,0,x=4z−2x,GSx,Tz,Tz=Gx2,0,z,0,z=4x−2z,GSy,Tx,Tx=Gy2,0,x,0,x=4y−2x,GSz,Ty,Ty=Gz2,0,y,0,y=4z−2y.Applying the above equality, we obtain(91)Mx,y,z≤max4x−2y,4x−2z4y−2x,4y−2z4z−2x,4z−2y,=4z−2x.From (88) and (91), it follows that
(92)∣8z−4x+y∣≤θ4∣z−2x∣.This shows that allx,y,z∈R and conditions imposed in Theorem 27 are satisfied. Hence, a pair of hybrid mapping S and T in G-metric space has a unique common fixed point. Thus, x=0 is a unique common fixed point of S and T. Clearly, R is T,S-closed, and x0=0,S0,T0,T0∈R.


## 4. An Application to SODE inG-Metric Space

Next, to validate the result of Theorem27, we apply the second-order differential equation (SODE) by transforming it to the system of integral equations. The fixed point theory is involved in physics applications, specifically in the solution of electric equations, and this type of work is motivated from [22, 32, 33]. It is well known that an electric circuit can be represented by ternary relation, R a resistor, L an inductor, and C a capacitor on E an electromotive force in series. On the other hand, they can be presented as ternary relations with XL an inductive reactance, XC a capacitive reactance, and Z an impedance. If the rate of charge q in condenser with respect to time t is denoted by current I, that is amount to say I=dq/dt. We get the following ternary relations:
(93)V=IR,V=qC,V=LdIdt.By Kirchhoff’s law, the sum of voltage drops across the circuit is equal to the supplied voltage. We consider theR−L−C-circuit, and the differential equation for the charge q on the condenser is
(94)IR+qC+LdIdt=Vt,(95)Ld2qdt2+Rdqdt+qC=Vt,where Vt is the applied voltage at time t. If the initial conditions are assumed to be
(96)q0=0,dq0dt=0,R2>4LC,τ=R2L.The differential equation (95) can be represented using the Green function given by
(97)Gt,s=−seτs−t,0≤s≤t≤1,−teτs−t,0≤t≤s≤1.Consider the following systems of equations motivated by [21].
(98)zt=∫0TGt,sKt,s,zsds+gt,wt=∫0TGt,sKt,s,wsds+gt,for all t,s∈X=0,T,T>0, where K:X×X×ℝn⟶ℝn and g:X⟶ℝn. Let R be a ternary relation on ℝn and X=CX,ℝn on X, set of all continuous mappings from X⟶ℝn. Consider the ternary relation R on X as
(99)z,w,w∈R⟺zt,wt,wt∈R,for all t∈X. Now we define
(100)Tzt=∫0TGt,sKt,s,zsds+gt,Swt=∫0TGt,sKt,s,wsds+gt,for all t,s∈X.LetX=C0,T be the set of all continuous function defined on 0,T. Define a G-metric on X, by G:X×X×X⟶ℝ+ and
(101)Gz,w,w≤supt∈0,Tz−w+supt∈0,Tw−w+supt∈0,Tz−w,=2supt∈0,Tz−w.Then,X,G is a G-complete metric space.Theorem 29.
Suppose the following conditions holds:(i)
K:0,T×0,T×ℝ⟶ℝ and g:ℝ⟶ℝ are continuous(ii)
There exists somet,s∈0,T and z∈X such that(102)∫0TGt,sKt,s,zsds+gt,∫0TGt,sKt,s,wsds+gt,∫0TGt,sKt,s,wsds+gt∈R,for all t,s∈X.
(iii)
There exists some a functionG:0,T×0,T×0,T⟶0,ℝ, for each z,w∈R or z,w∈Rs and t,s∈0,T with θ∈0,1 such that(103)∣Kt,s,zs−Kt,s,ws∣≤Gt,sMz,w,w,where Mz,w,w=2∣Szt−Swt and
(104)Mz,w,w=maxGSz,Tw,Tw,GSz,Tw,TwGSw,Tz,Tz,GSw,Tw,TwGSw,Tz,Tz,GSw,Tw,Tw,(iv)
There existsθ∈0,1 such that(105)Gt,s=1−e−τt−τteτ1−tτ2≤θ,where τ=R/2L.Then, the system of equations (95) has a common solution zt which is a solution of the integral equation (98).Proof.
Forz,w∈C0,T with z<w, we claim that HGTz,Tw,Tw≰θMz,w,w. Then, we have
(106)HGTz,Tw,Tw≤2sup0∈0,T∣Szt−Swt∣,≤2sup0∈0,T∣∫0TGt,sKt,s,zsds+gt−∫0TGt,sKt,s,wsds+gt∣,≤2sup0∈0,T∣∫0TGt,sKt,s,zs−Kt,s,wsds∣,≤2sup0∈0,T∣Szt−Swt∣∫0TGt,sds≤2sup0∈0,T∣Szt−Swt∣∫0t−seτs−tds−∫01teτs−tds,≤2sup0∈0,T∣Szt−Swt∣1−e−τt−τteτ1−tτ2,HGTz,Tw,Tw≤θMz,w,w,which is a contradiction. Hence, z is a common fixed of T and S and also a solution to integral equation (98) and a second-order differential equation (95). Hence, we can conclude that all the conditions imposed in Theorems 27 and 29 are satisfied. Therefore, the proof is completed.



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*Source: 3593697-2022-03-02.xml*