# C ⁎-Algebra-Valued G-Metric Spaces and Related Fixed-Point Theorems

Authors: Congcong Shen; Lining Jiang; Zhenhua Ma
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We introduce the notion of theC⁎-algebra-valued G-metric space. The existence and uniqueness of some fixed-point theorems for self-mappings with contractive or expansive conditions on complete C⁎-algebra-valued G-metric spaces are proved. As an application, we prove the existence and uniqueness of the solution of a type of differential equations.


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## Body

## 1. Introduction

As is known to all, the proverbial fixed-point theorem of Banach has been widely used in many branches of mathematics and physics. There are a large number of generalizations for such a theorem. In general, the theorem has been extended in two directions. On the one hand, the usual contractive condition is replaced with weakly contractive conditions [1–7]. On the other hand, the action spaces are replaced with different types of metric spaces. Particularly, in 2006, Mustafa and Sims [8] introduced the concept of generalized metric spaces (G-metric spaces). Since then, many scholars studied fixed-point theory in G-metric spaces and many meaningful results are obtained.Let us recall the basic definitions and conclusions onG-metric spaces. Details can be seen in [8–20].Definition 1 (see [8]).
LetX be a nonempty set. Suppose that G:X×X×X→R+ is a mapping satisfying(1)
G(x,y,z)=0 if x=y=z;(2)
0<G(x,x,y) for all x,y∈X with x≠y;(3)
G(x,x,y)≤G(x,y,z) for all x,y,z∈X with z≠y;(4)
G(x,y,z)=G(x,z,y)=G(y,z,x)=··· (symmetry in all three variables);(5)
G(x,y,z)≤G(x,a,a)+G(a,y,z) for all x,y,z,a∈X(rectangle inequality).
ThenG is called a generalized metric, or a G-metric on X. The pair (X,G) is called a G-metric space.Definition 2 (see [8]).
Let(X,G) be a G-metric space, {xn}⊆X. If there exists x∈X, such that limn,m→∞⁡G(x,xn,xm)=0, we say that the sequence {xn} is called G-convergent to x. If for any ε>0, there is an N∈N, such that, for all m,n,l>N, (1)Gxn,xm,xl<ε,{xn} is called a G-Cauchy sequence.Notice that, in aG-metric space (X,G), the following statements are equivalent:(1)
{xn} is G-convergent to x;(2)
G(xn,xn,x)→0 as n→∞;(3)
G(xn,x,x)→0 as n→∞.As we have known thatC∗-algebra, which was first proposed for its use in quantum mechanics to model algebras of physical observable, is an important research field of modern mathematics [21–29]. In 1947, I. Segal [28] introduced the term “C∗-algebra” to describe a “uniformly closed, self-adjoint algebra of bounded operators on a Hilbert space”.Throughout this paper,A will denote a unital C∗-algebra with a unit I; namely, A is a unital Banach algebra with an involution ∗ such that A∗A=A2 (A∈A). Let H be a Hilbert space and B(H) the set of all bounded linear operators on H, then B(H) is a C∗-algebra with the operator norm. Let Asa be the set of all self-adjoint elements in A, and define the spectrum of A∈A to be the set σ(A)=λ∈C∣λI-Ais not invertible. An element A∈A is positive (denoted by A≥θ) if A∈Asa and σ(A)⊆R+, set A+=A∈A∣A≥θ, then A+=A∗A∣A∈A [27]. Using the positive element, one can define a partial ordering “≤” on Asa as follows: A≤B if and only if B-A≥θ. It is clear that if A,B∈Asa and C∈A, then A≤B⇒C∗AC≤C∗BC, and that if A,B∈A+ are invertible, then A≤B⇒θ≤B-1≤A-1.Using the partial ordering “≤” on Asa, Ma introduced the notion of C∗-algebra-valued metric spaces and gave the fixed-point theory for contractive or expansion mapping on such a space [30, 31]. Let us recall the definition first.Definition 3 (see [30]).
LetX be a nonempty set. If d:X×X→A+ is a mapping satisfying(1)
d(x,y)=θ⇔x=y;(2)
d(x,y)=d(y,x) for all x,y∈X;(3)
d(x,y)≤d(x,z)+d(y,z) for all x,y,z∈X,
thend is called a C∗-algebra-valued metric on X. (X,A,d) is called a C∗-algebra-valued metric space.Definition 4 (see [30]).
Let(X,A,d) be a C∗-algebra-valued metric space, {xn}⊂X,x∈X. If for any ε>0, there is an N∈N, such that for all n>N, d(xn,x)<ε, then we say this {xn} converges to x. We denote it by limn→∞⁡xn=x.
If for anyε>0, there is an N∈N, such that, for all m,n>N,d(xm,xn)<ε, then we say {xn} is a Cauchy sequence with respect to A. We say that (X,A,d) is a complete C∗-algebra-valued metric space if every Cauchy sequence with respect to A is convergent.Definition 5 (see [30]).
Let(X,A,d) be a C∗-algebra-valued metric space. We call a mapping T:X→X is a contractive mapping on (X,A,d) if there exists A∈A with A<1 such that d(Tx,Ty)≤A∗d(x,y)A, ∀x,y∈X.Theorem 6 (see [30]).
If(X,A,d) is a complete C∗-algebra-valued metric space and T is a contractive mapping, there exists a unique fixed point in X.In this paper, we will defineC∗-algebra-valued G-metric spaces and prove some fixed point theorems on such spaces. We also provide an application of the theory for a type of differential equations.


## 2. Main Results

In this section, we first give the definition of aC∗-algebra-valued G-metric spaces.Definition 7.
LetX be a nonempty set and S3 be the permutation group on {1,2,3}. If G:X×X×X→A+ is a mapping satisfying(1)
G(x1,x2,x3)=θ⇔x1=x2=x3;(2)
G(xσ(1),xσ(2),xσ(3))=G(x1,x2,x3) for all x1,x2,x3∈X,σ∈S3;(3)
G(x1,x1,x2)≤G(x1,x2,x3) for all x1,x2,x3∈X with x2≠x3;(4)
G(x1,x2,x3)≤G(x1,a,a)+G(a,x2,x3) for all x1,x2,x3,a∈X,
G is called a C∗-algebra-valued G-metric on X, and (X,A,G) is called a C∗-algebra-valued G-metric space.Example 8.
LetX={x,y}. Notice that M2(C) of 2×2-matrices with entries in C is identified with B(C2). This is a C∗-algebra. Let (2)Gx,x,x=Gy,y,y=θ,Gx,x,y=I,Gx,y,y=2Iand extend G to all of X×X×X by symmetry in the variables. Then G is a C∗-algebra-valued G-metric on X, and (X,M2(C),G) is a C∗-algebra-valued G-metric space.Definition 9.
Let(X,A,G) be a C∗-algebra-valued G-metric space, {xn}⊆X,x∈X. If for any ε>0, there is an N∈N, such that for all m,n>N,G(xm,xn,x)<ε, then we say {xn} is G-convergent to x, and denote by xn→Gx, (n→∞).Proposition 10.
Let(X,A,G) be a C∗-algebra-valued G-metric space, {xn}⊆X,x∈X. The following statements are equivalent:(1)
xn→Gx(n→∞).(2)
G(xn,xn,x)→θ, as n→∞.(3)
G(xn,x,x)→θ, as n→∞.Proof.
( 1 ) ⇒ ( 2 ) If xn→Gx, that is, for all ε>0, ∃N∈N, such that, for all, m,n>N,G(xm,xn,x)<ε, especially, G(xn,xn,x)<ε. Hence G(xn,xn,x)→θ, as n→∞.
( 2 ) ⇒ ( 3 ) If ∀ε>0, ∃N∈N, such that, for all n>N,G(xn,xn,x)<ε/2, then when n>N, (3)Gxn,x,x=Gx,xn,x≤Gx,xn,xn+Gxn,xn,x<ε,that is, G(xn,x,x)→θ, as n→∞.
( 3 ) ⇒ ( 1 ) If G(xn,x,x)→θ, as n→∞, then, for any ε>0, ∃N1∈N, such that for all n>N1,G(xn,x,x)<ε/2; ∃N2∈N, such that, for all m>N2,G(xm,x,x)<ε/2. Let N=N1+N2, for m,n>N, (4)Gxm,xn,x≤Gxm,x,x+Gx,xn,x<ε,that is, xn→Gx(n→∞).Definition 11.
Let(X,A,G) be a C∗-algebra-valued G-metric space, {xn}⊆X. If for any ε>0, there is an N∈N such that for all m,n,l>N,G(xm,xn,xl)<ε, then the sequence {xn} is called a G-Cauchy sequence. If any G-Cauchy sequence in (X,A,G) is G-convergent, then (X,A,G) is called a complete C∗-algebra-valued G-metric space.Proposition 12.
Let(X,A,G) be a C∗-algebra-valued G-metric space, {xn}⊆X. Then {xn} is a G-Cauchy sequence if and only if for any ε>0, there is an N∈N, such that, for all m,n>N,G(xm,xn,xn)<ε.Proof.
It suffices to show the necessity. For anyε>0, ∃N1∈N, such that for all m,n>N1,G(xm,xn,xn)<ε/2; ∃N2∈N, such that for all n,l>N2,G(xl,xn,xn)<ε/2. So for the above ε, let N=N1+N2, when m,n,l>N, (5)Gxm,xn,xl≤Gxm,xn,xn+Gxn,xn,xl<ε,that is, {xn} is a G-Cauchy sequence.Example 13.
SetX=C,A=M3(C). Let α,β>0; set (6)Gx,y,z=gx,y,z000αgx,y,z000βgx,y,z,where g(x,y,z)=x-y+y-z+z-x, then G is a C∗-algebra-valued G-metric on X, and (X,A,G) is a complete C∗-algebra-valued G-metric space.
It is easy to see that(X,A,G) is a C∗-algebra-valued G-metric space. We only need to prove the completeness. Let {xn}⊆(X,A,G) be a G-Cauchy sequence. Then for any ε>0, there is an N∈N, such that for all m,n>N, (7)Gxm,xn,xn=max⁡gxm,xn,xn,αgxm,xn,xn,βgxm,xn,xn<ε.So g(xm,xn,xn)=2xm-xn<ε. Since X is complete, there exists x∈X, such that xn→x. Hence there is N0∈N such that, for any n>N0, xn-x<ε/2. It follows that(8)Gxn,x,x=max⁡2x-xn,2αx-xn,2βx-xn=2max⁡1,α,βxn-x<max⁡1,α,βε.Therefore, xn→Gx, and (X,A,G) is complete.Example 14.
SupposeΩ is a compact Hausdorff space and μ is a positive regular Borel measure on Ω. Let X=L∞(Ω,μ), the set of all essentially bounded complex-valued measurable functions on Ω, then X is a Banach space with the essential supremum norm f∞. Set H=L2(Ω,μ)=f:Ω→C∣∫Ω‍fω2dμω<∞; H is a Hilbert space with the inner-product (9)f,g=∫Ωfωgω¯dμω.
Forf∈L∞(Ω,μ), define(10)Mf:H⟶Hφ⟼fφ.Then Mf is bounded and moreover Mf=f∞. Let G:X×X×X→B(H) by (11)Gf,g,h=Mf-g+g-h+h-f.Then G is a C∗-algebra-valued G-metric and (X,B(H),G) is a complete C∗-algebra-valued G-metric space. We omit its proof and leave it to readers.Next, we define the contractive mapping onC∗-algebra-valued G-metric space and prove the fixed point theorem for contractive mappings.Definition 15.
Let(X,A,G) be a C∗-algebra-valued G-metric space and T:X→X is a mapping. If there exists A∈A with A<1 such that (12)GTx,Ty,Tz≤A∗Gx,y,zA,∀x,y,z∈X,then T is called a contractive mapping on (X,A,G).Theorem 16.
Let(X,A,G) be a complete C∗-algebra-valued G-metric space. If T:X→X is a contractive mapping on (X,A,G), then there is a unique fixed point of T on X.Proof.
Letd(x,y)=G(x,x,y)+G(x,y,y).
We first show thatd is a C∗-algebra-valued metric on X. It suffices to show that (13)dx,y≤dx,z+dy,z,∀x,y,z∈X.That is, (14)Gx,x,y+Gx,y,y≤Gx,x,z+Gx,z,z+Gy,y,z+Gy,z,z.Since(15)Gx,x,y+Gx,y,y=Gy,x,x+Gx,y,y≤Gy,z,z+Gz,x,x+Gx,z,z+Gz,y,y=Gx,x,z+Gx,z,z+Gy,z,z+Gy,y,z,we have d(x,y)≤d(x,z)+d(y,z),∀x,y,z∈X.
Next, we show(X,A,d) is a complete C∗-algebra-valued metric space. Let {xn}⊆X be a Cauchy sequence with respect to A. Then for any ε>0, there is an N∈N such that for all m,n>N, d(xn,xm)<ε, that is, (16)Gxn,xm,xm+Gxn,xn,xm<ε.Since θ≤G(xn,xm,xm)≤G(xn,xm,xm)+G(xn,xn,xm), (17)Gxn,xm,xm≤Gxn,xm,xm+Gxn,xn,xm<ε.So {xn}⊆X is a G-Cauchy sequence. By the completeness of (X,A,G), there exists an x∈X, such that xn→Gx(n→∞). That is, for any ε>0, there is an N1∈N such that, for all n>N1,G(xn,x,x)<ε/2; there is an N2∈N such that for all m>N2,G(xm,xm,x)<ε/2. Let N=N1+N2, then for all n>N, we have (18)Gxn,x,x<ε2,Gxn,xn,x<ε2.Therefore,(19)dxn,x=Gxn,x,x+Gxn,xn,x≤Gxn,x,x+Gxn,xn,x<ε.Hence limn→∞⁡xn=x, and (X,A,d) is complete.
Moreover,T is a contractive mapping on (X,A,d). In fact,(20)dTx,Ty=GTx,Tx,Ty+GTx,Ty,Ty≤A∗Gx,x,yA+A∗Gx,y,yA=A∗Gx,x,y+Gx,y,yA=A∗dx,yA.It follows from Theorem 6 that there is an x∈X such that Tx=x.
Finally, we show the uniqueness of this fixed point. Lety is another fixed point of T. If x≠y, then (21)θ≨Gx,y,y=GTx,Ty,Ty≤A∗Gx,y,yA.Since G(x,y,y)≠0, (22)Gx,y,y≤A∗Gx,y,yA≤A2Gx,y,y<Gx,y,y.This is a contradiction. So x=y.Remark 17.
(1) In the theorem above, the completeness ofX is essential. For example, let X=(0,1)⊆R and G:X×X×X→B(H) satisfy G(x,y,z)=x-y+y-z+z-xI, then (X,B(H),G) is a C∗-algebra-valued G-metric space, but (X,B(H),G) is not complete. Considering the mapping(23)T:X⟶Xx⟶12xT is a contractive mapping, but T has no fixed point.
(2) In the definition of contractive mapping, the elementA∈A does not depend on the choice of x,y,z. If A depends on x,y,z, then T may not have a fixed point.
Indeed, letX=R and G:X×X×X→B(H) be defined by G(x,y,z)=x-y+y-z+z-xI. Then (X,B(H),G) is a complete C∗-algebra-valued G-metric space. Let f(x)=π/2+x-arctan⁡x. If x<y<z, then(24)Gfx,fy,fz=x-y-arctan⁡x+arctan⁡y+y-z-arctan⁡y+arctan⁡z+z-x-arctan⁡z+arctan⁡xI=x-y+y-x1+ξ2+y-z+z-y1+η2+z-x+x-z1+ζ2I=ξ21+ξ2x-y+η21+η2y-z+ζ21+ζ2z-xI≤αx-y+y-z+z-xI=αIGx,y,zαI,where ξ∈(x,y), η∈(y,z), ζ∈(x,z), and α=max⁡ξ2/1+ξ2,η2/1+η2,ζ2/1+ζ2<1 depends on x,y,z, but f has no fixed point.
(3) WhenA=1, T may not have a unique fixed point.
ConsiderX=l∞(N)={x1,x2,⋯∣xn∈C,n∈N and supn⁡xn<+∞}, for x=(x1,x2,⋯)∈l∞(N), and let x∞=supn⁡xn. Define G:X×X×X→M2(C) by (25)Gx,y,z=x-y∞+y-z∞+z-x∞I.Then (X,M2(C),G) is a complete C∗-algebra-valued G-metric space. Let (26)Tx1,x2,x3,⋯=1+x2,12+x3,122+x4,⋯.Then (27)GTx,Ty,Tz≤Gx,y,z.But for each α∈C, x=(α,α-1,α-3/2,⋯,α-∑i=0n-11/2i,⋯) is a fixed point, which means the fixed point is not unique.What follows is the definition of the expansion mapping onC∗-algebra-valued G-metric space and the fixed-point theorem for expansion mappings.Definition 18.
Let(X,A,G) be a C∗-algebra-valued G-metric space. If T:X→X satisfies the condition (28)GTx,Ty,Tz≥A∗Gx,y,zA,where A∈A is invertible and A-1<1, we call T an expansion mapping on (X,A,G).Theorem 19.
Let(X,A,G) be a complete C∗-algebra-valued G-metric space and T:X→X a expansion mapping on (X,A,G). If T is surjective, then there is a unique fixed point for T.Proof.
First we showT is injective. Indeed, if Tx=Ty, then (29)θ=GTx,Ty,Ty≥A∗Gx,y,yA.Since A∗G(x,y,y)A∈A+, A∗G(x,y,y)A=θ, and since A is invertible, G(x,y,y)=θ. Therefore x=y.
Next we show thatT has a unique fixed point in X. In fact, T is bijective and so T is invertible. Since ∀x,y,z∈X, (30)GTx,Ty,Tz≥A∗Gx,y,zA.Replace x,y,z by T-1x, T-1y, T-1z, respectively, we get (31)Gx,y,z≥A∗GT-1x,T-1y,T-1zA.Hence (32)GT-1x,T-1y,T-1z≤A∗-1Gx,y,zA-1.Thus (33)GT-1x,T-1y,T-1z≤A-1∗Gx,y,zA-1.By Theorem 16, there is a unique x∈X, such that T-1x=x, and therefore there is a unique x∈X, such that Tx=x.The following lemma is necessary for another fixed-point theorem, for detail, see [27].Lemma 20.
SetA′={A∈A:AB=BA,∀B∈A}.
(1) IfA,B∈A+ and AB=BA, then AB∈A+.
(2) IfA∈A′, B,C∈A with B≥C≥θ and I-A∈A′ is invertible, then (I-A)-1B≥(I-A)-1C.Theorem 21.
Let(X,A,G) be a complete C∗-algebra-valued G-metric space, T:X→X. If there exists an A∈(A′)+, A<1/6 such that for any x,y∈X, (34)GTx,Ty,Tz≤AGTx,y,z+Gx,Ty,z+Gx,y,Tz,or(35)GTx,Ty,Tz≤AGTx,Ty,z+Gx,Ty,Tz+GTx,y,Tz,then T has a unique fixed point in X.Proof.
Without loss of generality, we can assumeA≠θ.
(1) Suppose thatT satisfies (36)GTx,Ty,Tz≤AGTx,y,z+Gx,Ty,z+Gx,y,Tz,∀x,y,z∈X.Since A∈(A′)+, (37)AGTx,y,z+Gx,Ty,z+Gx,y,Tz≥θ.
Forx0∈X, set xn+1=Txn=Tn+1x0,n=1,2,⋯, and B=G(x1,x0,x0). Then(38)Gxn+1,xn,xn=GTxn,Txn-1,Txn-1≤AGTxn,xn-1,xn-1+Gxn,Txn-1,xn-1+Gxn,xn-1,Txn-1=AGxn+1,xn-1,xn-1+2AGxn,xn,xn-1≤AGxn+1,xn,xn+Gxn,xn-1,xn-1+2AGxn,xn-1,xn-1+Gxn-1,xn,xn-1=AGxn+1,xn,xn+5AGxn,xn-1,xn-1,That is, (39)I-AGxn+1,xn,xn≤5AGxn,xn-1,xn-1.Since A∈(A′)+ with A<1/6, we have (I-A)-1∈(A′)+ and furthermore A(I-A)-1∈(A′)+ with 5A(I-A)-1<1. Therefore, (40)Gxn+1,xn,xn≤5AI-A-1Gxn,xn-1,xn-1.Let t=5A(I-A)-1, for n+1>m,(41)Gxn+1,xm,xm≤Gxn+1,xn,xn+Gxn,xn-1,xn-1⋯+Gxm+1,xm,xm≤tn+tn-1+⋯+tmGx1,x0,x0=∑k=mntkB=∑k=mntk/2B1/22≤∑k=mntk/2B1/22I≤∑k=mntk/22B1/22I≤B1/22∑k=mntkI≤B1/22tm1-tI⟶θm⟶∞.This implies that {xn} is a G-Cauchy sequence in X. Since (X,A,G) is complete, there exists an x∈X such that xn→Gx(n→∞), i.e., Txn-1→Gx(n→∞). Since(42)GTx,x,x≤GTx,Txn,Txn+GTxn,x,x≤AGTx,xn,xn+Gx,Txn,xn+Gx,xn,Txn+Gxn+1,x,x≤AGTx,x,x+AGx,xn,xn+2AGx,xn,xn+1+Gxn+1,x,x.That is(43)I-AGTx,x,x≤AGx,xn,xn+2AGx,xn,xn+1+Gxn+1,x,x.Then(44)GTx,x,x≤AI-A-1Gx,xn,xn+2Gx,xn,xn+1+I-A-1Gxn+1,x,x⟶0n⟶∞,and hence Tx=x and x is the fixed point of T on X.
Next, we show the uniqueness ofx. If there exists another fixed point y∈X, then(45)θ≤Gx,x,y=GTx,Tx,Ty≤AGTx,x,y+Gx,Tx,y+Gx,x,Ty=AGx,x,y+2AGx,x,y,i.e.,(46)Gx,x,y≤2AI-A-1Gx,x,y.Since 5A(I-A)-1<1, 2A(I-A)-1<1. Hence G(x,x,y)=θ, and x=y.
(2) The case when(47)GTx,Ty,Tz≤AGTx,Ty,z+Gx,Ty,Tz+GTx,y,Tz,∀x,y,z∈Xcan be proved similarly and we omitted it.Definition 22.
Let(X,A,G) be a C∗-algebra-valued G-metric space. We say G is symmetric if G(x,x,y)=G(x,y,y) for all x,y∈X.It is easy to show that, in Example8, G is not symmetric and, in Example 13, G is symmetric.Theorem 23.
Let(X,A,G) be a complete C∗-algebra-valued G-metric space and G symmetric. If T:X→X is a mapping satisfying that for x,y∈X(48)GTx,Ty,Tz≤AGTx,y,z+Gx,Ty,z+Gx,y,Tz,or (49)GTx,Ty,Tz≤AGTx,Ty,z+Gx,Ty,Tz+GTx,y,Tz,where A∈(A′)+ and A<1/4, then T has a unique fixed point in X.Proof.
Without loss of generality, one can assumeA≠θ.
We only consider the case whenT satisfies (50)GTx,Ty,Tz≤AGTx,y,z+Gx,Ty,z+Gx,y,Tz.Since A∈(A′)+,A(G(Tx,y,z)+G(x,Ty,z)+G(x,y,Tz))≥θ.
Forx0∈X, set xn+1=Txn=Tn+1x0,n=1,2,⋯, and B=G(x1,x0,x0). If G is symmetric,(51)Gxn+1,xn,xn=GTxn,Txn-1,Txn-1≤AGTxn,xn-1,xn-1+Gxn,Txn-1,xn-1+Gxn,xn-1,Txn-1=AGxn+1,xn-1,xn-1+2AGxn,xn,xn-1≤AGxn+1,xn,xn+Gxn,xn-1,xn-1+2AGxn,xn-1,xn-1=AGxn+1,xn,xn+3AGxn,xn-1,xn-1.That is, (52)I-AGxn+1,xn,xn≤3AGxn,xn-1,xn-1.Since A∈(A′)+ with A<1/4, then I-A is invertible and (I-A)-1∈(A′)+ and 3A(I-A)-1<1. Therefore (53)Gxn+1,xn,xn≤3AI-A-1Gxn,xn-1,xn-1.Just like the proof of Theorem 21, we can prove that, for n+1>m, (54)Gxn+1,xm,xm⟶θm⟶∞.This implies that {xn} is a G-Cauchy sequence in X; thus there exists an x∈X such that xn→Gx(n→∞) i.e., Txn-1→Gx(n→∞). Similarly, we can show that Tx=x, x is the fixed point of T on X.
The uniqueness of the fixed point can be proved similarly.


## 3. Applications

For fixed-point theorems, there are a number of applications in differential equations and integral equations.Consider the second-order differential equation:∗d2udt2=Kt,ut,u0=u1=0,where u:[0,1]→[0,∞) and K:[0,1]×R+→R+ are continuous. Then the equation has a unique solution if and only if u(t)=∫01g(t,s)K(s,u(s))ds,t∈[0,1] has a unique solution, where (55)gt,s=t1-s,0≤s<t≤1;s1-t,0≤t<s≤1.LetX=C([0,1],R+)⊆L∞([0,1],R+),A=B(L2[0,1]),G(u,v,w)=M|u-v|+|v-w|+|w-u|, then (X,A,G) is a complete C∗-algebra-valued G-metric space. Let T:X→X be a mapping defined by(56)Tut=∫01gt,sKs,usds,then ∗ has a unique solution if and only if T has a unique fixed point.Theorem 24.
IfKs,x-Ks,y≤x-y,∀s∈[0,1],x,y∈R+, equation ∗ has a unique solution.Proof.
(57) G T u , T v , T w = M T u - T v + T v - T w + T w - T u = T u - T v ∞ + T v - T w ∞ + T w - T u ∞ = max t ∈ 0,1 ⁡ ∫ 0 1 g t , s K s , u s - K s , v s d s + max t ∈ 0,1 ⁡ ∫ 0 1 g t , s K s , v s - K s , w s d s + max t ∈ 0,1 ⁡ ∫ 0 1 g t , s K s , w s - K s , u s d s ≤ max t ∈ 0,1 ⁡ ∫ 0 1 g t , s K s , u s - K s , v s d s + max t ∈ 0,1 ⁡ ∫ 0 1 g t , s K s , v s - K s , w s d s + max t ∈ 0,1 ⁡ ∫ 0 1 g t , s K s , w s - K s , u s d s ≤ max t ∈ 0,1 ⁡ ∫ 0 1 g t , s u s - v s d s + max t ∈ 0,1 ⁡ ∫ 0 1 g t , s v s - w s d s + max t ∈ 0,1 ⁡ ∫ 0 1 g t , s w s - u s d s = max t ∈ 0,1 ⁡ ∫ 0 1 g t , s u s - v s + v s - w s + w s - u s d s ≤ G u , v , w max t ∈ 0,1 ⁡ ∫ 0 1 g t , s d sFor a fixed t∈[0,1], (58)∫01gt,sds=∫0tt1-sds+∫t1s1-tds=12+t22-t2≤12.Therefore, (59)GTu,Tv,Tw≤12Gu,v,wand T has a unique fixed point.



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*Source: 3257189-2018-07-12.xml*