Fixed Point Theorems of Quasicontractions on Cone Metric Spaces with Banach Algebras

Abstract

We introduce the concept of quasicontractions on cone metric spaces with Banach algebras, and by a new method of proof, we will prove the existence and uniqueness of fixed points of such mappings. The main result generalizes the well-known theorem of Ćirić (Ćirić 1974).

1. Introduction

Let(𝑋,𝑑)be a complete metric space. Recall that a mapping𝑇∶𝑋→𝑋is called a quasicontraction if, for some𝑘∈(0,1)and for all𝑥,𝑦∈𝑋, one has 𝑑𝑑(),𝑑.𝑇𝑥,𝑇𝑦⩽𝑘max𝑥,𝑦,𝑑𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥 (1) Ćirić [1] introduced and studied quasicontractions as one of the most general classes of contractive-type mappings. He proved the well-known theorem that any quasicontraction𝑇has a unique fixed point. Recently, scholars obtained various similar results on cone metric spaces. See, for instance, [2–5].

In this paper, we study the quasicontractions on metric spaces with Banach algebras, which are introduced in [6] and turn out to be an interesting generalization of classic metric spaces. By a new method of proof, we generalize Ćirić theorem.

Let𝐴always be a real Banach algebra with a multiplication unit𝑒; that is,𝑒𝑥=𝑥𝑒=𝑥for all𝑥∈𝐴. An element𝑥∈𝐴is said to be invertible if there is an inverse element𝑦∈𝐴such that𝑥𝑦=𝑦𝑥=𝑒. The inverse of𝑥is denoted by𝑥⁻¹. For more details, we refer to [7].

The following proposition is well known (see [7]).

Proposition 1 (see [7]). Let𝐴be a Banach algebra with a unit𝑒, and let 𝑥∈𝐴. If the spectral radius𝜌(𝑥)of𝑥is less than 1, that is, 𝜌(𝑥)=lim_𝑛→∞𝑥^𝑛1/𝑛=inf_𝑛⩾1𝑥^𝑛1/𝑛<1, (2) then𝑒−𝑥is invertible. Actually, (𝑒−𝑥)⁻¹=^∞_𝑖=0𝑥^𝑖. (3)

A subset𝑃of𝐴is called a cone if (1)𝑃is nonempty closed and{0,𝑒}⊂𝑃; (2)𝛼𝑃+𝛽𝑃⊂𝑃for all nonnegative real numbers𝛼,𝛽; (3)𝑃²=𝑃𝑃⊂𝑃; (4)𝑃∩(−𝑃)={0}.

For a given cone 𝑃⊂𝐴, we can define a partial ordering⩽with respect to𝑃by𝑥⩽𝑦if and only if 𝑦−𝑥∈𝑃. And 𝑥≨𝑦 will stand for𝑥⩽𝑦and𝑥≠𝑦, while𝑥<𝑦will stand for 𝑦−𝑥∈int𝑃, where int𝑃 denotes the interior of𝑃.

Remark 2. In the literature on cone metric spaces, authors use𝑥<𝑦to mean𝑥⩽𝑦and𝑥≠𝑦 and𝑥≪𝑦to mean𝑦−𝑥∈int𝑃. To our knowledge, and from a topological point of view, the order relation𝑦−𝑥∈int𝑃plays a very similar role in cone metric spaces as𝑥<𝑦does inℝ.

The cone𝑃is called normal if there is a number𝑀>0such that for all𝑥,𝑦∈𝐴, 0⩽𝑥⩽𝑦⟹‖𝑥‖⩽𝑀𝑦. (4) The least positive number satisfying above is called the normal constant of𝑃(see [8]).

In the following, we always assume that𝑃is a cone in𝐴with int𝑃≠∅ and⩽is partial ordering with respect to𝑃.

Definition 3 (see [8]). Let𝑋be a nonempty set. Suppose the mapping𝑑∶𝑋×𝑋→𝐴satisfies (1)0⩽𝑑(𝑥,𝑦)for all𝑥,𝑦∈𝑋and𝑑(𝑥,𝑦)=0if and only if𝑥=𝑦; (2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥)for all𝑥,𝑦∈𝑋; (3)𝑑(𝑥,𝑦)⩽𝑑(𝑥,𝑧)+𝑑(𝑧,𝑥)for all𝑥,𝑦,𝑧∈𝑋. Then, 𝑑 is called a cone metric on𝑋, and(𝑋,𝑑)is called a cone metric space (with Banach algebra𝐴).

For more details about cone metric spaces with Banach algebras, we refer the readers to [6].

Definition 4 (see [8]). Let(𝑋,𝑑)be a cone metric space, and let 𝑥∈𝑋and{𝑥_𝑛} be a sequence in𝑋. Then, (1){𝑥_𝑛}converges to𝑥whenever for each 𝑐∈𝐴 with 0<𝑐 there is a natural number𝑁such that 𝑑(𝑥_𝑛,𝑥)<𝑐 for all 𝑛⩾𝑁. We denote this by lim_𝑛→∞𝑥_𝑛=𝑥 or 𝑥_𝑛→𝑥; (2){𝑥_𝑛}is a Cauchy sequence whenever for each 𝑐∈𝐴 with 0<𝑐 there is a natural number 𝑁 such that 𝑑(𝑥_𝑛,𝑥_𝑚)<𝑐 for all 𝑛,𝑚⩾𝑁; (3)(𝑋,𝑑)is a complete cone metric space if every Cauchy sequence is convergent.

The following facts are often used.

Proposition 5 (see [8]). Let(𝑋,𝑑)be a cone metric space, let 𝑃be a normal cone with normal constant𝑀, and let{𝑥_𝑛}be a sequence in𝑋. Then, {𝑥_𝑛}converges to𝑥if and only if𝑑(𝑥_𝑛,𝑥)→0(𝑛→∞).

Proposition 6 (see [8]). Let(𝑋,𝑑)be a cone metric space, let 𝑃be a normal cone with normal constant𝑀, and let{𝑥_𝑛}be a sequence in𝑋. Then, {𝑥_𝑛}is a Cauchy sequence if and only if 𝑑(𝑥_𝑛,𝑥_𝑚)→0(𝑛,𝑚→∞).

2. Main Results

In this section we will define quasicontractions in the setting of cone metric spaces with Banach algebras and prove the fixed point theorem of such mappings.

Definition 7. Let(𝑋,𝑑)be a cone metric space with Banach algebra𝐴. A mapping𝑇∶𝑋→𝑋is called a quasicontraction if for some𝑘∈𝑃with𝜌(𝑘)<1and for all𝑥,𝑦∈𝑋, one has 𝑑𝑇𝑥,𝑇𝑦⩽𝑘𝑢, (5) where 𝑢∈𝑑𝑥,𝑦,𝑑(𝑥,𝑇𝑥),𝑑𝑦,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥. (6)

Remark 8. In Definition 7, we only suppose the spectral radius of𝑘is less than 1, while neither𝑘<𝑒nor‖𝑘‖<1is assumed. In fact, the condition𝜌(𝑘)<1is weaker than that‖𝑘‖<1. See the example in [6].

Theorem 9. Let(𝑋,𝑑)be a complete cone metric space with a Banach algebra𝐴, and let 𝑃be a normal cone with normal constant𝑀. If the mapping𝑇∶𝑋→𝑋is a quasicontraction, then𝑇has a unique fixed point in𝑋. And for any𝑥∈𝑋, iterative sequence{𝑇^𝑛𝑥}converges to the fixed point.

In the rest of the paper, we choose𝑥₀∈𝑋 and denote𝑥_𝑛=𝑇^𝑛𝑥₀. For the sake of clarity, we divide the proof into several steps.

Lemma 10. Assume that the hypotheses in Theorem 9 are satisfied. Then, for each𝑛⩾1, and for all𝑖,𝑗such that1⩽𝑖,𝑗⩽𝑛, one has 𝑑𝑥_𝑖,𝑥_𝑗⩽𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁. (7)

Proof. We present the proof by induction.
When𝑛=1, which implies𝑖=𝑗=1, the conclusion is trivial.
Assume that the statement is true for𝑛=𝑚; that is, 𝑑𝑥_𝑖,𝑥_𝑗⩽𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁,for1⩽𝑖,𝑗⩽𝑚. (8) Now, we will prove that the statement is true for𝑛=𝑚+1. Note that in this case, if1⩽𝑖,𝑗⩽𝑚, then the statement is just (8). Thus, without loss of generality, we suppose that𝑗=𝑚+1and1⩽𝑖⩽𝑚 and denote𝑖=𝑖₀.
By the definition of quasicontraction, we have 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑢, (9) where 𝑢∈𝑑𝑥_𝑖0−1,𝑥_𝑚,𝑑𝑥_𝑖0−1,𝑥_𝑖0,𝑑𝑥_𝑚,𝑥_𝑚+1,𝑑𝑥_𝑖0−1,𝑥_𝑚+1,𝑑𝑥_𝑖0,𝑥_𝑚. (10)
Firstly, we consider the case that𝑖₀=1; that is, 𝑢∈𝑑𝑥₀,𝑥_𝑚,𝑑𝑥₀,𝑥₁,𝑑𝑥_𝑚,𝑥_𝑚+1,𝑑𝑥₀,𝑥_𝑚+1,𝑑𝑥₁,𝑥_𝑚. (11)
If𝑢=𝑑(𝑥₀,𝑥_𝑚), then 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑑𝑥₀,𝑥_𝑚⩽𝑘𝑑𝑥₀,𝑥₁+𝑑𝑥₁,𝑥_𝑚⩽𝑘𝑑𝑥₀,𝑥₁+𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁=𝑘𝑒+𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁=𝑘𝑒+^∞_𝑡=1𝑘^𝑡𝑑𝑥₀,𝑥₁=𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁, (12) and the statement follows.
If𝑢=𝑑(𝑥₀,𝑥₁), then 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑑𝑥₀,𝑥₁⩽^∞_𝑡=1𝑘^𝑡𝑑𝑥₀,𝑥₁=𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁, (13) and the statement also follows.
If𝑢=𝑑(𝑥_𝑚,𝑥_𝑚+1), then we set𝑖₁=𝑚and we have 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑑𝑥_𝑖1,𝑥_𝑚+1. (14)
If𝑢=𝑑(𝑥₀,𝑥_𝑚+1), then 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑑𝑥₀,𝑥_𝑚+1⩽𝑘𝑑𝑥₀,𝑥₁+𝑑𝑥₁,𝑥_𝑚+1=𝑘𝑑𝑥₀,𝑥₁+𝑑𝑥_𝑖0,𝑥_𝑚+1, (15) which implies (𝑒−𝑘)𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑑𝑥₀,𝑥₁. (16) Note that(𝑒−𝑘)⁻¹=∑^∞_𝑡=0𝑘^𝑡⩾0 and that𝑘and(𝑒−𝑘)⁻¹commute. Multiplying both sides by(𝑒−𝑘)⁻¹, we have 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁, (17) and the statement also follows.
If𝑢=𝑑(𝑥_𝑖0,𝑥_𝑚), then 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑑𝑥_𝑖0,𝑥_𝑚⩽𝑘²(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁=^∞_𝑡=2𝑘^𝑡𝑑𝑥₀,𝑥₁⩽^∞_𝑡=1𝑘^𝑡𝑑𝑥₀,𝑥₁=𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁, (18) and the statement also follows.
Secondly, we consider the case that2⩽𝑖₀⩽𝑚.
If𝑢=𝑑(𝑥_𝑖0−1,𝑥_𝑚) or𝑢=𝑑(𝑥_𝑖0−1,𝑥_𝑖0) or𝑢=𝑑(𝑥_𝑖0,𝑥_𝑚), then, by (8), we have 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑢⩽𝑘²(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁=^∞_𝑡=2𝑘^𝑡𝑑𝑥₀,𝑥₁⩽𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁, (19) and the statement follows.
If𝑢=𝑑(𝑥_𝑚,𝑥_𝑚+1) or𝑢=𝑑(𝑥_𝑖0−1,𝑥_𝑚+1), then we set𝑖₁=𝑚or𝑖₁=𝑖₀−1⩾1, respectively. And we have 𝑑𝑥_𝑖0,𝑥_𝑚+1𝑥⩽𝑘𝑢=𝑘𝑑_𝑖1,𝑥_𝑚+1. (20)
In conclusion from discussions of both cases, it results that either the proof is complete, that is, 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁, (21) or there exists an integer𝑖₁such that 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑑𝑥_𝑖1,𝑥_𝑚+1,1⩽𝑖₁⩽𝑚. (22)
As for the latter situation, we continue in a similar way, and come to the result that either 𝑑𝑥_𝑖𝑖,𝑥_𝑚+1⩽𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁, (23) which implies that 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑑𝑥_𝑖1,𝑥_𝑚+1⩽𝑘²(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁⩽𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁, (24) and the proof is complete, or there exists an integer𝑖₂such that 𝑑𝑥_𝑖1,𝑥_𝑚+1⩽𝑘𝑑𝑥_𝑖2,𝑥_𝑚+1,1⩽𝑖₂⩽𝑚, (25) which implies that 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘²𝑑𝑥_𝑖2,𝑥_𝑚+1,1⩽𝑖₂⩽𝑚. (26)
Generally, if the procedure ends by the ℓ-th step with ℓ⩽𝑚−1, that is, there exist ℓ+1integers 𝑖₀,𝑖₁,…,𝑖_ℓ∈{1,…,𝑚}, (27) such that 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑑𝑥_𝑖1,𝑥_𝑚+1⩽⋯⩽𝑘^ℓ𝑑𝑥_𝑖𝑙,𝑥_𝑚+1, (28) and such that 𝑑𝑥_𝑖𝑙,𝑥_𝑚+1⩽𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁, (29) then 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘^ℓ+1(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁=^∞_𝑡=ℓ+1𝑘^𝑡𝑑𝑥₀,𝑥₁⩽𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁. (30) Hence, the proof is complete.
Finally, if the procedure continues more than𝑚steps, then there exist𝑚+1integers 𝑖₀,𝑖₁,…,𝑖_𝑚∈{1,…,𝑚}, (31) such that 𝑑𝑥_𝑖0,𝑥_𝑚+1⩽𝑘𝑑𝑥_𝑖1,𝑥_𝑚+1⩽⋯⩽𝑘^𝑚𝑑𝑥_𝑖𝑚,𝑥_𝑚+1. (32) Thus, there must exist two integers,𝑝and𝑞, say, such that 0⩽𝑝<𝑞⩽𝑚,𝑖_𝑝=𝑖_𝑞. (33) From (32), one sees that 𝑑𝑥_𝑖𝑝,𝑥_𝑖𝑚+1⩽𝑘^𝑞−𝑝𝑑𝑥_𝑖𝑞,𝑥_𝑚+1=𝑘^𝑞−𝑝𝑑𝑥_𝑖𝑝,𝑥_𝑚+1, (34) and therefore 𝑒−𝑘^𝑞−𝑝𝑑𝑥_𝑖𝑝,𝑥_𝑚+1⩽0. (35) Note that 𝜌𝑘^𝑞−𝑝⩽𝜌(𝑘)^𝑞−𝑝<1, (36) which implies𝑒−𝑘^𝑞−𝑝is invertible. And since that 𝑒−𝑘^{𝑞−𝑝−1}=^∞_𝑡=0𝑘^{(𝑞−𝑝)𝑡}⩾0, (37) we have 𝑑𝑥_𝑖𝑝,𝑥_𝑚+1⩽0. (38) So, 𝑑𝑥_𝑖𝑝,𝑥_𝑚+1𝑑𝑥=0,_𝑖0,𝑥_𝑚+1⩽𝑘^𝑝𝑑𝑥_𝑖𝑝,𝑥_𝑚+1()=0⩽𝑘𝑒−𝑘⁻¹𝑑𝑥₀,𝑥₁. (39)(40)
Therefore, by induction, the statement is proved.

Remark 11. Lemma 10 simply says that 𝑑𝑥_𝑖,𝑥_𝑗⩽𝑘(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁,∀𝑖,𝑗⩾1. (41)

Lemma 12. Assume that the hypotheses in Theorem 9 are satisfied. Then, {𝑥_𝑛}is a Cauchy sequence.

Proof. For1<𝑚<𝑛, denote that 𝐶(𝑚,𝑛)=𝑑𝑥_𝑖,𝑥_𝑗∣𝑚⩽𝑖,𝑗⩽𝑛. (42) By the definition of quasicontraction, it follows that, for each𝑢∈𝐶(𝑚,𝑛), there exists𝑣∈𝐶(𝑚−1,𝑛), such that 𝑢⩽𝑘𝑣. (43) Consequently, 𝑑𝑥_𝑚,𝑥_𝑛⩽𝑘𝑢₁⩽𝑘²𝑢₂⩽⋯⩽𝑘^𝑚−1𝑢_𝑚−1⩽𝑘^𝑚(𝑒−𝑘)⁻¹𝑑𝑥₀,𝑥₁, (44) where 𝑢₁(),𝑢∈𝐶𝑚−1,𝑛₂∈𝐶(𝑚−2,𝑛),…,𝑢_𝑚−1∈𝐶(1,𝑛), (45)(46) and the last inequality comes from Lemma 10.
By the normality of𝑃, and noting that‖𝑘^𝑚‖→0(𝑚→∞), we have 𝑑𝑥_𝑚,𝑥_𝑛⩽𝑀𝑘^𝑚(𝑒−𝑘)⁻¹×𝑑𝑥₀,𝑥₁⟶0(𝑛>𝑚⟶∞). (47) The proof is complete.

Now, we finish the remaining part of the proof of Theorem 9.

Proof. By Lemma 12 and the completeness of(𝑋,𝑑), there is𝑥^∗∈𝑋such that𝑥_𝑛→𝑥^∗(𝑛→∞). Then, 𝑑𝑥^∗,𝑇𝑥^∗⩽𝑑𝑥^∗,𝑥_𝑛+𝑑𝑥_𝑛,𝑇𝑥^∗⩽𝑑𝑥^∗,𝑥_𝑛+𝑘𝑢, (48) where 𝑢∈𝑑𝑥_𝑛−1,𝑥^∗,𝑑𝑥_𝑛−1,𝑥_𝑛,𝑑𝑥^∗,𝑇𝑥^∗,𝑑𝑥_𝑛−1,𝑇𝑥^∗,𝑑𝑥^∗,𝑥_𝑛. (49)
If𝑢=𝑑(𝑥_𝑛−1,𝑥^∗) or𝑢=𝑑(𝑥_𝑛−1,𝑥_𝑛) or𝑢=𝑑(𝑥^∗,𝑥_𝑛), then‖𝑢‖→0(𝑛→∞). Hence, 𝑑𝑥^∗,𝑇𝑥^∗⩽𝑀𝑑𝑥^∗,𝑥_𝑛+‖𝑘𝑢‖().‖‖⟶0𝑛⟶∞ (50)
If𝑢=𝑑(𝑥^∗,𝑇𝑥^∗), then (𝑒−𝑘)𝑑𝑥^∗,𝑇𝑥^∗⩽𝑑𝑥^∗,𝑥_𝑛. (51) Hence, 𝑑𝑥^∗,𝑇𝑥^∗⩽𝑀(𝑒−𝑘)⁻¹𝑑𝑥^∗,𝑥_𝑛().⟶0𝑛⟶∞ (52)
If𝑢=𝑑(𝑥_𝑛−1,𝑇𝑥^∗), then 𝑑𝑥^∗,𝑇𝑥^∗⩽𝑑𝑥^∗,𝑥_𝑛+𝑘𝑑𝑥_𝑛−1,𝑇𝑥^∗⩽𝑑𝑥^∗,𝑥_𝑛+𝑘𝑑𝑥_𝑛−1,𝑥^∗+𝑘𝑑𝑥^∗,𝑇𝑥^∗. (53) Hence, 𝑑𝑥^∗,𝑇𝑥^∗⩽𝑀(𝑒−𝑘)⁻¹×𝑑𝑥^∗,𝑥_𝑛+‖𝑘‖𝑑𝑥_𝑛−1,𝑥^∗⟶0, (54) as𝑛→∞.
In each case, we have‖𝑑(𝑥^∗,𝑇𝑥^∗)‖=0. Thus, 𝑇𝑥^∗=𝑥^∗.
Now, if𝑦^∗is another fixed point, then 𝑑𝑥^∗,𝑦^∗=𝑑𝑇𝑥^∗,𝑇𝑦^∗⩽𝑘𝑢, (55) where 𝑢∈𝑑𝑥^∗,𝑦^∗,𝑑𝑥^∗,𝑇𝑥^∗,𝑑𝑦^∗,𝑇𝑦^∗,𝑑𝑥^∗,𝑇𝑦^∗,𝑑𝑦^∗,𝑇𝑥^∗. (56)
If𝑢=𝑑(𝑥^∗,𝑇𝑥^∗)=𝑑(𝑦^∗,𝑇𝑦^∗)=0, then𝑑(𝑥^∗,𝑦^∗)=0.
If𝑢=𝑑(𝑥^∗,𝑦^∗)=𝑑(𝑥^∗,𝑇𝑦^∗)=𝑑(𝑦^∗,𝑇𝑥^∗), then (𝑒−𝑘)𝑑𝑥^∗,𝑦^∗⩽0, (57) which implies 𝑑𝑥^∗,𝑦^∗=0. (58)
Thus, the fixed point is unique. And we obtain Theorem 9.

Acknowledgments

The authors are extremely grateful to the referees for their useful comments and suggestions. The research is partially supported by Doctoral Initial Foundation of Hanshan Normal University, China (no. QD20110920).