# Fixed Point Results for Weakφ-Contractions in Cone Metric Spaces over Banach Algebras and Applications

Authors: Biwen Li; Huaping Huang
Journal: Journal of Function Spaces
 (2017)

---

## Abstract

By using a nontrivial proof method, the purpose of this paper is to obtain some fixed point results for weakφ-contractions in cone metric spaces over Banach algebras. Several examples and applications to the existence and uniqueness of a solution to two classes of equations are also given.


---

## Body

## 1. Introduction

Fixed point theory is without doubt one of the most important tools of modern mathematics as attested by Browder [1], who is considered as one of the pioneers in the development of the nonlinear functional analysis. The flourishing field of fixed point theory started in the early days of topology through the work of Poincaré [2], Lefschetz-Hopf, and Leray-Schauder, for example. Fixed point theory is widely used in different areas such as ordinary and partial differential equations, economics, logic programming, convex optimization, and control theory. In metric fixed point theory, successive approximations are rooted in the work of Cauchy, Fredholm, Liouville, Lipschitz, Peano, and Picard. It is well accepted among experts of this subarea that Banach is responsible for laying the ground for an abstract framework well beyond the scope of elementary differential and integral equations. In 1922, Banach [3] proved the most influential and celebrating fixed point theorem, which was Banach fixed point theorem (i.e., Banach contraction principle). Since then, fixed point theory has had a rapid development. In [4], Huang and Zhang introduced cone metric space and generalized Banach fixed point theorem in such spaces. Subsequently, many people were interested in fixed point results in cone metric spaces (see [5–10] and the references therein). In [11–13], Rus and Berinde introduced the notion of φ-contraction and also generalized Banach fixed point theorem in usual metric spaces. Recently, Liu and Xu [14] introduced the concept of a cone metric space over Banach algebra, which is an interesting generalization of classic metric spaces. From then on, many authors focused on the investigation of fixed point in such spaces (see [15–20]). Stimulated and motivated by the previous work, throughout this paper, we introduce weak φ-contractions in the setting of cone metric spaces over Banach algebras and present some fixed point theorems for weak φ-contractions. Our results improve and weaken the conditions of the vector-valued version of Banach fixed point theorem. To the best of our knowledge, our methods are new. In addition, by using our results, we give the existence and uniqueness of a solution to elementary equations and to integral equations.The following notions and facts will be needed in this paper.Definition 1 (see [12]).
A functionφ:[0,∞)→[0,∞) is called a comparison if it satisfies the following two conditions:(1)
φ is monotone nondecreasing; that is, 0≤t1≤t2⇒φ(t1)≤φ(t2).(2)
φnt(t>0) converges to 0 as n→∞.Remark 2.
By Definition1, it is sufficient to get that φ(t)<t for each t>0, φ(0)=0, and limt→0φ(t)=0.Definition 3 (see [12]).
Let(X,d) be a metric space. A mapping T:X→X is called a φ-contraction if there exists a comparison φ:[0,∞)→[0,∞) such that (1)dTx,Ty≤φdx,y∀x,y∈X.The following theorem generalizes Banach fixed point theorem.Theorem 4 (see [12]).
Let(X,d) be a metric space and T:X→X be a φ-contraction. Then T has a unique fixed point in X. Moreover, for any x∈X, the iterative sequence {Tnx} converges to the fixed point.In the following, we consider our results in the framework of cone metric spaces over Banach algebras. For the reader who is unfamiliar with cone metric space over Banach algebra, we recall some of its notions and results as follows.Definition 5 (see [14]).
LetA be a Banach algebra with a unit e and a zero element θ. A nonempty closed subset P of A is called a cone if the following conditions hold:(1)
θ,e⊂P.(2)
∀α,β∈[0,∞)⇒αP+βP⊆P.(3)
P2=PP⊂P.(4)
P∩(-P)={θ}.
A coneP is called a solid cone if intP≠∅, where intP stands for the interior of P.
On this basis, we define a partial ordering⪯ with respect to P by x⪯y if and only if y-x∈P. We shall write x≪y to indicate that y-x∈ intP. We shall also write · as the norm on A. A cone P is called normal if there is a number K>0 such that, for all x,y∈A, θ⪯x⪯y implies x≤Ky.
In the sequel, unless otherwise specified, we always suppose thatA is a Banach algebra with a unit e, P is a solid cone in A, and ⪯ and ≪ are partial orderings with respect to P. We always write N and R as the set of all natural numbers and the set of all real numbers, respectively.Definition 6 (see [14]).
LetX be a nonempty set and A be a Banach algebra. A mapping d:X×X→A is called a cone metric if it satisfies(i)
θ⪯d(x,y)(∀x,y∈X), d(x,y)=θ⇔x=y;(ii)
d(x,y)=d(y,x)(∀x,y∈X);(iii)
d(x,y)⪯d(x,z)+d(z,y)(∀x,y,z∈X).
In this case, the pair(X,d) is called a cone metric space over Banach algebra.Definition 7 (see [8]).
A sequence{un} in a Banach algebra A is said to be a c-sequence if, for each c≫θ, there exists N∈N such that un≪c for all n>N.
The introduction ofc-sequence is an interesting increase since, by using c-sequence, many intricate concepts may be simplified. For example, Definitions  2–4 of [4] are abbreviated with the following.Definition 8 (see [17]).
Let(X,d) be a cone metric space over Banach algebra A and {xn} be a sequence in X. We say that(i)
{xn} converges to x∈X if dxn,x is a c-sequence;(ii)
{xn} is a Cauchy sequence if {d(xn,xm)} is a c-sequence for n,m;(iii)
(X,d) is complete if every Cauchy sequence in X is convergent.Lemma 9 (see [9]).
LetA be a Banach algebra and u,v,w∈A. Then(1)
u≪w if u⪯v≪w or u≪v⪯w;(2)
u=θ if θ⪯u≪c for each c≫θ.Lemma 10 (see [21]).
LetA be a Banach algebra with its unit e. Then the spectral radius of u∈A equals ρ(u)=limn→∞un1/n.Lemma 11 (see [16]).
LetP be a cone in a Banach algebra A, {un} and {vn} be two c-sequences in A, and α,β∈P be vectors; then {αun+βvn} is a c-sequence in A.Lemma 12 (see [15]).
LetP be a cone and k∈P with ρ(k)<1. Then {kn} is a c-sequence.


## 2. Main Results

In this section, we introduce weakφ-contractions in the framework of a cone metric space over Banach algebra and obtain some corresponding fixed point theorems. Moreover, we present some examples to illustrate the superiority of the results.Definition 13.
LetA be a Banach algebra and P be a cone in A. A mapping φ:P→P is called a weak comparison if the following conditions hold:(i)
φ is nondecreasing with respect to ⪯; namely, t1,t2∈P, t1⪯t2⇒φ(t1)⪯φ(t2).(ii)
{φn(t)}(t∈P) is a c-sequence in P.(iii)
if{un} is a c-sequence in P, then {φ(un)} is also a c-sequence in P.Remark 14.
By Definition13, it suffices to show that φ(θ)=θ. Indeed, by (i) of Definition 13, we have θ⪯φ(θ)⪯φn(θ). Since {φn(θ)} is a c-sequence, then, by Lemma 9, it may be verified that φ(θ)=θ.Remark 15.
IfA=R1 and P=[0,∞), then Definition 13 is reduced to Definition 1. In other words, Definition 13 is a generalization of Definition 1.The following examples are trivial, whose proofs are straightforward and are therefore omitted.Example 16.
LetA be a Banach algebra, P be a cone in A, and k∈P. Take φ(t)=kt(t∈P), where ρ(k)<1. Then by Lemmas 11 and 12, φ is a weak comparison.Example 17.
LetM be a compact set of Rn and A=C(M), where C(M) denotes the set of all continuous functions on M. Let P=u∈A:u≥0 and define a mapping φ:P→P by φ(u)=u/u+1. Then φ is a weak comparison.Definition 18.
Let(X,d) be a cone metric space over Banach algebra. Let P be a cone and φ:P→P be a weak comparison. Then a mapping T:X→X is called a weak φ-contraction if(2)dTx,Ty⪯φdx,y∀x,y∈X.Remark 19.
By Remark15, Clearly, Definition 18 generalizes Definition 3.The following theorem generalizes Theorem4 and its proof method is nontrivial.Theorem 20.
Let(X,d) be a complete cone metric space over Banach algebra and T:X→X be a weak φ-contraction. Then T has a unique fixed point in X. Moreover, for any x∈X, the iterative sequence {Tnx} converges to the fixed point.Proof.
Choosex0∈X and put xn=Tnx0,n≥1. For any c≫θ, by (ii) of Definition 13, there exists n0∈N such that φn0(c)≪c. Making the most of (2), it is not hard to verify that(3)dxn,xn+n0⪯φndx0,xn0.Since {φn(d(x0,xn0))} is a c-sequence, then, by Lemma 9, {d(xn,xn+n0)} is also a c-sequence. Hence, there exists N1∈N such that (4)dxn,xn+n0≪c-φn0cn≥N1.Let(5)Bxn,c=x∈X:dxn,x≪cn≥N1.Choosing x∈B(xn,c)(n≥N1), by (3) and (5), it is established that (6)dxn,Tn0x⪯dxn,xn+n0+dxn+n0,Tn0x≪c-φn0c+φn0dxn,x⪯c-φn0c+φn0c=c.This means that B(xn,c)(n≥N1) is Tn0-invariant. Accordingly, for any k∈N, we have d(xn,xn+kn0)≪c.
Making the best of (2), it ensures us that (7)dxn,xn+1+dxn+1,xn+2+⋯+dxn+n0-1,xn+n0⪯φndx0,x1+φndx1,x2+⋯+φndxn0-1,xn0.Now that {φn(d(xi,xi+1))}(i=0,1,…,n0-1) are c-sequences, then, by Lemma 11, {φn(d(x0,x1))+φn(d(x1,x2))+⋯+φn(d(xn0-1,xn0))} is a c-sequence. Next, by Lemma 9, {d(xn,xn+1)+d(xn+1,xn+2)+⋯+d(xn+n0-1,xn+n0)} is also a c-sequence. So, for the above c≫θ, there exists N2∈N such that (8)dxn,xn+1+dxn+1,xn+2+⋯+dxn+n0-1,xn+n0≪cn>N2.
DenoteN=maxN1,N2, for m,n>N; let (9)km=m-Nn0,kn=n-Nn0,where · stands for the integer part. Because (10)N≤m-kmn0<N+n0,N≤n-knn0<N+n0,we have (11)dxm,xn⪯dxm,xm-kmn0+dxm-kmn0,xn-knn0+dxn-knn0,xn≪3c.Then, by Lemma 9, we claim that {xn} is a Cauchy sequence. Since (X,d) is complete, then there exists x∗∈X such that xn→x∗(n→∞).
We prove thatx∗ is the fixed point of T. Actually, by (2), we arrive at (12)dTx∗,x∗⪯dTx∗,xn+dxn,x∗=dTx∗,Txn-1+dxn,x∗⪯φdx∗,xn-1+dx∗,xn.Now that dx∗,xn-1 is a c-sequence, it follows immediately from (iii) of Definition 13 that φdx∗,xn-1 is a c-sequence. Thus, by Lemma 11, we speculate that {φ(d(x∗,xn-1))+d(x∗,xn)} is also a c-sequence. Next by Lemma 9, it may be verified that d(Tx∗,x∗)=θ; that is, Tx∗=x∗.
Finally, we prove that the fixed point is unique. Assume thaty∗ is another fixed point of T; then, by (2) and the monotonicity of φ, we deduce that (13)dx∗,y∗=dTnx∗,Tny∗⪯φndx∗,y∗.Since φndx∗,y∗ is a c-sequence, then, by Lemma 9, it is obvious that d(x∗,y∗)=θ. As a result, x∗=y∗.Corollary 21.
Let(X,d) be a complete cone metric space over Banach algebra A and P be a cone in A. Suppose that k∈P and T:X→X is a mapping satisfying (14)dTx,Ty⪯kdx,y∀x,y∈X.If ρ(k)<1, then T has a unique fixed point in X. Moreover, for any x∈X, the iterative sequence {Tnx} converges to the fixed point.Proof.
Letφ(t)=kt(t∈P); then, by Theorem 20, we get the desired result.Remark 22.
Corollary21 is called the vector-valued version of Banach fixed point theorem. It generalizes Theorem  2.1 of [14] because it deletes the assumption of normality of cones of Theorem  2.1 of [14].The following example shows the superiority of Theorem20.Example 23.
Under the conditions of Example17, let X=A and define a mapping d:X×X→A by d(u,v)=|u-v|. Then (X,d) is a complete cone metric space over Banach algebra. Define a mapping T:X→X by (15)Tu=uu+1u∈X.Clearly, T is not a contraction; that is, there is no k∈A with ρ(k)<1 such that the contractive condition of Theorem  2.1 of [14] holds.
However, for allu,v∈X, we have (16)dTu,Tv=uu+1-vv+1≤u-vu-v+1=du,vdu,v+1.By Example 17, we know that T is a weak φ-contraction and then, by Theorem 20, T has a unique fixed point.


## 3. Applications

In this section, we give some applications to prove the existence and uniqueness of a solution to some equations. First of all, we consider the following elementary equations:(17)x=sinh-1λx,y=λarctan⁡y+sin⁡x,where 0<λ<1 is a constant.Theorem 24.
The elementary equation (17) has a unique solution in R2.Proof.
LetA=X=R2 and P={(u1,u2)∈A:u1,u2≥0}. Take u=(u1,u2),v=(v1,v2)∈A. Define a norm of A by (u1,u2)=|u1|+|u2|. Define a multiplication of A by (18)uv=u1,u2v1,v2=u1v1,u1v2+u2v1.For any x=(x1,x2),y=(y1,y2)∈X, define a mapping d:X×X→A by (19)dx,y=x1-y1,x2-y2;then (X,d) is a complete cone metric space over Banach algebra. Define a mapping T:X→X by (20)Tx=Tx1,x2=sinh-1λx1,λarctan⁡x2+sin⁡x1.For all x,y∈X, by mean value theorem of differentials, there exists ξ belonging to numbers between x2 and y2, such that (21)dTx,Ty=dTx1,x2,Ty1,y2=sinh-1λx1-sinh-1λy1,λarctan⁡x2-λarctan⁡y2+sin⁡x1-sin⁡y1⪯λx1-y1,λ1+ξ2x2-y2+x1-y1⪯λ,1x1-y1,x2-y2=λ,1dx,y.Choose φ(u)=ku with k=(λ,1). Clearly, φ is a weak comparison. Thus, T is a weak φ-contraction. By Theorem 20, T has a unique fixed point in X. That is to say, the elementary equation (17) has a unique solution in R2.
Secondly, we prove an existence theorem for a solution of the following nonlinear integral equation by using our results in the previous section.(22)xc=ϕc+∫abKc,t,xtdt,where a,b∈R, x∈C[a,b]  (the set of all continuous functions from [a,b] into R), ϕ:[a,b]→R, and K:[a,b]×[a,b]×R→R are given mappings.Theorem 25.
LetLp[a,b]={x=x(t):∫ab‍|x(t)|pdt<∞}(p≥1). If there exists M>0 with M(b-a)1/p<1 such that, for all x,y∈Lp[a,b] and c∈[a,b],(23)Kc,t,xt-Kc,t,yt≤Mxt-ytp,then the integral equation (22) has a unique solution in Lp[a,b].Proof.
LetA=R2 with the same norm, the same multiplication, and the same cone P as stated in the proof of Theorem 24. Then P is a normal cone and A is a Banach algebra with a unit e=(1,0). Let X=Lp[a,b]. We endow X with the cone metric(24)dx,y=∫abxc-ycpdc1/p,∫abxc-ycpdc1/pfor all x,y∈X. It is clear that (X,d) is a complete cone metric space over Banach algebra A. Define the mapping T:X→X by(25)Txc=ϕc+∫abKc,t,xtdtfor all c∈[a,b]. Then the existence of a solution to (22) is equivalent to the existence of fixed point of T. Indeed, by utilizing (23)–(25), we have (26)dTx,Ty=∫ab∫abKc,t,xtdt-∫abKc,t,ytdtpdc1/p,∫ab∫abKc,t,xtdt-∫abKc,t,ytdtpdc1/p=∫ab∫abKc,t,xt-Kc,t,ytdtpdc1/p,∫ab∫abKc,t,xt-Kc,t,ytdtpdc1/p⪯M∫ab∫abxt-ytpdtpdc1/p,M∫ab∫abxt-ytpdtpdc1/p=Mb-a1/p,0∫abxt-ytpdt,∫abxt-ytpdt=φdx,y,where φ(w)=(M(b-a)1/p,0)(w1p,w2p) and w=(w1,w2)∈A.
It is easy to get thatφ is a weak comparison and T is a weak φ-contraction. Accordingly, all the conditions of Theorem 20 are satisfied and then T has a unique fixed point in Lp[a,b]. In other words, the integral equation (22) has a unique solution in Lp[a,b].



---

*Source: 5054603-2017-12-13.xml*