Fixed Point Results in Quasi-Cone Metric Spaces

Abstract

The main aim of this paper is to prove fixed point theorems in quasi-cone metric spaces which extend the Banach contraction mapping and others. This is achieved by introducing different kinds of Cauchy sequences in quasi-cone metric spaces.

1. Introduction and Preliminaries

The Banach contraction principle is a fundamental result in fixed point theory. Due to its importance, several authors have obtained many interesting extensions and generalizations (see, e.g., [1–12]).

A quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric: it can be regarded as an “asymmetric metric.” In fact, quasi-metric space is more comprehensive than metric space. As metric space is important and has numerous applications, Huang and Zhang [13] have announced the concept of the cone metric spaces, replacing the set of real numbers by an ordered Banach space. They have proved some fixed point theorems for contractive-type mappings on cone metric spaces, whereas Rezapour and Hamlbarani [14] omitted the assumption of normality in cone metric spa ces, which is a milestone in developing fixed point theory in cone metric spaces. Since then, numerous authors have started to generalize fixed point theorems in cone metric spaces in many various directions. For some recent results (see, e.g., [15–25]) and for a current survey of the latest results in cone metric spaces, see Janković et al. [26].

Very recently, some authors generalized the contractive conditions in the literature by replacing the constants with functions. Using these generalizations, they have proved the existence and uniqueness of the fixed point in cone metric spaces; for more details see [27, 28]. Because quasi-metric space is more general than metric space and is a subject of intensive research in the context of topology and theoretical computer science, Abdeljawad and Karapinar [29] and Sonmez [30] have given a definition of quasi-cone metric space which extends the quasi-metric space.

In this paper, we also introduce the concept of a quasi-cone metric space which is somewhat different from that of Abdeljawad and Karapinar [29] and Sonmez [30]. Then we establish four kinds of Cauchy sequences in this space according to Reilly et al. [31]. Furthermore, we extend and generalize the Banach contraction principle and some results in the literature to this space. We support our results by examples. In this paper we do not impose the normality condition for the cones, and the only assumption is that the cone 𝑃 is, solid; that is int𝑃≠∅. Now we recall some known notions, definitions, and results which will be used in this work.

Definition 1. Let 𝐸 be a real Banach space and 𝑃 be a subset of 𝐸. 𝑃 is called a cone if and only if (i)𝑃is closed, 𝑃≠∅, 𝑃≠{0}; (ii)for all 𝑥,𝑦∈𝑃⇒𝛼𝑥+𝛽𝑦∈𝑃, where 𝛼,𝛽∈ℝ⁺; (iii)𝑥∈𝑃 and −𝑥∈𝑃⇒𝑥=0.

For a given cone 𝑃⊂𝐸, we define a partial ordering ⪯ with respect to 𝑃 by the following: for 𝑥,𝑦∈𝐸, we say that 𝑥⪯𝑦 if and only if 𝑦−𝑥∈𝑃. Also, we write 𝑥≪𝑦 for 𝑦−𝑥∈int𝑃, where int𝑃 denotes the interior of 𝑃. The cone 𝑃 is called normal if there is a number 𝐾>0 such that for all 𝑥,𝑦∈𝐸0⪯𝑥⪯𝑦⟹‖𝑥‖≤𝐾𝑦. (1) The least positive number 𝐾 satisfying this is called the normal constant of 𝑃. The cone 𝑃 is called regular if every increasing sequence which is bounded above is convergent; that is, if 𝑥_𝑛 is a sequence such that 𝑥₁⪯𝑥₂⪯⋯⪯𝑥_𝑛⪯⋯⪯𝑦 (2) for some 𝑦∈𝐸, then there is 𝑥∈𝐸 such that ‖𝑥_𝑛−𝑥‖→0 as 𝑛→∞. Equivalently, the cone 𝑃 is regular if every decreasing sequence which is bounded below is convergent (for details, see [13]). In this paper, we always suppose that 𝐸 is a real Banach space, 𝑃 is a cone in 𝐸 with int𝑃≠∅, and ⪯ is a partial ordering with respect to 𝑃.

Definition 2 (see [13]). Let 𝑋 be a nonempty set. Suppose the mapping 𝑑∶𝑋×𝑋→𝐸 satisfies (d1)0⪯𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋, and 𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦; (d2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋; (d3)𝑑(𝑥,𝑦)⪯𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋.
Then, 𝑑 is called a cone metric on 𝑋, and (𝑋,𝑑) is called a cone metric space.

Now, we state our definition which is more general than cone metric space.

Definition 3. Let 𝑋 be a nonempty set. Suppose the mapping 𝑞∶𝑋×𝑋→𝐸 satisfies(q1)0⪯𝑞(𝑥,𝑦) for all 𝑥,𝑦∈𝑋; (q2)𝑞(𝑥,𝑦)=0=𝑞(𝑦,𝑥) if and only if 𝑥=𝑦; (q3)𝑞(𝑥,𝑦)⪯𝑞(𝑥,𝑧)+𝑞(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋.
Then, 𝑞 is called a quasi-cone metric on 𝑋, and (𝑋,𝑞) is called a quasi-cone metric space.

Remark 4. Note that in [30] Sonmez defined the quasi-cone metric space as follows.
A quasi-cone metric space on a nonempty 𝑋 is a function 𝑞∶𝑋×𝑋→𝐸 such that for all 𝑥,𝑦,𝑧∈𝑋; (1)𝑞(𝑥,𝑦)=𝑞(𝑦,𝑥)=0⇔𝑥=𝑦, (2)𝑞(𝑥,𝑦)⪯𝑞(𝑥,𝑧)+𝑞(𝑧,𝑦).
A quasi-cone metric space is a pair (𝑋,𝑞) such that 𝑋 is a nonempty set and 𝑞 is a quasi-cone metric on 𝑋.
In fact, it has not mentioned that 𝑞 takes value in 𝑃, but in this paper we require this condition.

Remark 5. Abdeljawad and Karapinar’s definition of quasi-cone metric space [29] is as follows.
Let 𝑋 be a nonempty set. Suppose that the mapping 𝑞∶𝑋×𝑋→𝐸 satisfies the following: (q1)0⪯𝑞(𝑥,𝑦) for all 𝑥,𝑦∈𝑋; (q2)𝑞(𝑥,𝑦)=0⇔𝑥=𝑦; (q3)𝑞(𝑥,𝑦)⪯𝑞(𝑥,𝑧)+𝑞(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋.
Then 𝑑 is said to be a quasi-cone metric on 𝑋, and the pair (𝑋,𝑞) is called a quasi-cone metric space.
The following example indicates that our definition is more general than the one given in [29].

Example 6. Let 𝑋=(0,∞), 𝐸=ℝ², 𝑃={(𝑥,𝑦)∈𝐸∶𝑥,𝑦≥0}, and 𝑞∶𝑋×𝑋→𝐸 defined by 𝑞=1𝑥,𝑦𝑦−1𝑥(),𝑥,if𝑦<𝑥,0,0,if𝑦≥𝑥. (3) Then (𝑋,𝑞) satisfies our definition of a quasi-cone metric space but not the definition in [29] because if 𝑞(𝑥,𝑦)=0 then 𝑥=𝑦 or 𝑦>𝑥.

Remark 7. Note that any cone metric space is a quasi-cone metric space.

2. Necessary Facts and Statements

By considering the established notions in metric spaces [31], we introduce the appropriate generalization in cone metric spaces.

Definition 8. Let (𝑋,𝑞) be a quasi-cone metric space. A sequence {𝑥_𝑛} in 𝑋 is said to be (a)𝑄-Cauchy or bi-Cauchy if for each 𝑐∈int𝑃, there is 𝑛₀∈ℕ such that 𝑞(𝑥_𝑛,𝑥_𝑚)≪𝑐 for all 𝑛,𝑚≥𝑛₀; (b)left (right) Cauchy if for any 𝑐∈int𝑃, there is 𝑛₀∈ℕ such that 𝑞(𝑥_𝑚,𝑥_𝑛)≪𝑐 (𝑞(𝑥_𝑛,𝑥_𝑚)≪𝑐, resp.) for all 𝑛≥𝑚≥𝑛₀; (c)weakly left (right) Cauchy if for each 𝑐∈int𝑃, there is 𝑛₀∈ℕ such that 𝑞(𝑥_𝑛0,𝑥_𝑛)≪𝑐 (𝑞(𝑥_𝑛,𝑥_𝑛0)≪𝑐, resp.) for all 𝑛≥𝑛₀; (d)left (right) 𝑞-Cauchy if for every 𝑐∈int𝑃, there exist 𝑥∈𝑋 and 𝑛₀∈ℕ such that 𝑞(𝑥,𝑥_𝑛)≪𝑐 (𝑞(𝑥_𝑛,𝑥)≪𝑐, resp.) for all 𝑛≥𝑛₀.

Remark 9. These notions in quasi-cone metric space are related in the following way: (i)𝑄-Cauchy ⇒ left (right) Cauchy ⇒ weakly left (right) Cauchy ⇒ left (right) 𝑞-Cauchy; (ii)a sequence is 𝑄-Cauchy if and only if it is both left and right Cauchy.

In this paper, we use the notion of left Cauchy.

Definition 10. Let (𝑋,𝑞) be a quasi-cone metric space. Let {𝑥_𝑛}_𝑛≥1 be a sequence in 𝑋. We say that the sequence {𝑥_𝑛}_𝑛≥1 left converges to 𝑥∈𝑋 if 𝑞(𝑥_𝑛,𝑥)→0. One denotes this by lim_𝑛→∞𝑥_𝑛=𝑥or𝑥_𝑛⟶𝑥. (4)

We will utilize the word converges instead of left converges for simplicity.

Example 11. Let 𝑋=[0,1], 𝐸=ℝ², 𝑃={(𝑥,𝑦)∈𝐸∶𝑥,𝑦≥0}, and 𝑞∶𝑋×𝑋→𝐸 defined by 𝑞=()𝑥,𝑦𝑥−𝑦,𝛼𝑥−𝑦,if𝑥≥𝑦,𝛼,1,if𝑥<𝑦, (5) where 0≤𝛼<1. 𝑞 is a quasi-cone metric on 𝑋. Considering a sequence 𝑥_𝑛=1/𝑛, then {𝑥_𝑛}_𝑛≥1 is left Cauchy and is convergent to {0} due to 𝑞𝑥_𝑛1,𝑥=𝑞𝑛=1,0𝑛,𝛼𝑛⟶(0,0). (6) On the other hand, it is not right Cauchy.

Definition 12. A quasi-cone metric space (𝑋,𝑞) is called left complete if every left Cauchy sequence in 𝑋 converges.

Definition 13. Let (𝑋,𝑞) be a quasi-cone metric space. A function 𝑓∶𝑋→𝑋 is called (1) continuous if for any convergent sequence {𝑥_𝑛}_𝑛≥1 in 𝑋 with lim_𝑛→∞𝑥_𝑛=𝑥, the sequence {𝑓(𝑥_𝑛)}_𝑛≥1 is convergent and lim_𝑛→∞𝑓(𝑥_𝑛)=𝑓(𝑥);(2) contractive if there exists some 0≤𝜅<1 such that 𝑞𝑓(𝑥),𝑓𝑦⪯𝜅𝑞𝑥,𝑦∀𝑥,𝑦∈𝑋, (7) and if 𝜅=1, then 𝑓 is nonexpansive.

The following example shows that there exists a contractive function in quasi-cone metric space which is not continuous.

Example 14. Let 𝑋={0,1,2,…}∪{∞}, 𝐸=ℝ², 𝑃={(𝑥,𝑦)∈𝐸∶𝑥,𝑦≥0}, and 𝑞∶𝑋×𝑋→𝐸 defined by 𝑞=()𝑥,𝑦10,0,if𝑥≥𝑦,2𝑦,𝑦,if𝑥<𝑦, (8) and 𝑓∶X→𝑋 defined by 𝑓(𝑥)=0,if𝑥≥0,1,if𝑥=∞. (9) Then (𝑋,𝑞) is a quasi-cone metric space and 𝑓 is a contractive map but not continuous due to lim_𝑥→∞𝑓(𝑥)≠𝑓(∞).

3. Fixed Point Theorems

In this section, we prove some fixed point results in quasi-cone metric space. Also, we generalize the contractive conditions in the literature by replacing the constants with functions. First, we state the following useful lemma.

Lemma 15. Let (𝑋,𝑞) be a quasi-cone metric space and {𝑥_𝑛}_𝑛≥1 a sequence in 𝑋. Suppose there exist a sequence of nonnegative real numbers {𝜆_𝑛}_𝑛≥1 such that ∑^∞_𝑛=1𝜆_𝑛<∞, in which 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝜆_𝑛𝑀, (10) for some 𝑀∈𝑃, and for all 𝑛∈ℕ. Then the sequence {𝑥_𝑛}_𝑛≥1 is left Cauchy sequence in (𝑋,𝑞).

Proof. For 𝑛>𝑚, we get 𝑞𝑥_𝑚,𝑥_𝑛⪯𝑞𝑥_𝑚,𝑥_𝑚+1+𝑞𝑥_𝑚+1,𝑥_𝑚+2+⋯+𝑞𝑥_𝑛−1,𝑥_𝑛⪯𝑀^∞_𝑖=𝑚𝜆_𝑖. (11) Let 𝑐∈int𝑃 and choose 𝛿>0 such that 𝑐+𝑁_𝛿(0)⊂𝑃 where 𝑁_𝛿(0)={𝑦∈𝐸∶‖𝑦‖<𝛿}. Since ∑^∞_𝑛=1𝜆_𝑛<∞, there exists a natural number 𝑛₀ such that for all 𝑚≥𝑛₀𝑀∑^∞_𝑖=𝑚𝜆_𝑖∈𝑁_𝛿(0), also −𝑀∑^∞_𝑖=𝑚𝜆_𝑖∈𝑁_𝛿(0). Since 𝑐+𝑁_𝛿(0) is open, therefore 𝑐+𝑁_𝛿(0)∈int𝑃; that is 𝑐−𝑀∑^∞_𝑖=𝑚𝜆_𝑖∈int𝑃. Thus, 𝑀∑^∞_𝑖=𝑚𝜆_𝑖≪𝑐 for 𝑚≥𝑛₀ and so 𝑞𝑥_𝑚,𝑥_𝑛≪𝑐for𝑛>𝑚≥𝑛₀. (12) Thus, {𝑥_𝑛}_𝑛≥1 is a left Cauchy sequence.

We are now in a position to state the main fixed point theorem in the context of quasi-cone metric spaces. We will need the notion of Hausdorff in quasi-cone metric space. A quasi-cone metric space (𝑋,𝑑) is Hausdorff if for each pair 𝑥₁,𝑥₂ of distinct points of 𝑋, there exist neighborhoods 𝑈₁ and 𝑈₂ of 𝑥₁ and 𝑥₂, respectively, that are disjoint.

Theorem 16. Let (𝑋,𝑞) be a left complete Hausdorff quasi-cone metric space and let 𝑓∶𝑋→𝑋 be a continuous function. Suppose that there exist functions 𝜂,𝜆,𝜁,𝜇,𝜉∶𝑋→[0,1) which satisfy the following for 𝑥,𝑦∈𝑋: (1)𝜂(𝑓(𝑥))≤𝜂(𝑥),𝜆(𝑓(𝑥))≤𝜆(𝑥),𝜁(𝑓(𝑥))≤𝜁(𝑥),𝜇(𝑓(𝑥))≤𝜇(𝑥) and 𝜉(𝑓(𝑥))≤𝜉(𝑥); (2)𝜂(𝑥)+𝜆(𝑥)+𝜁(𝑥)+𝜇(𝑥)+2𝜉(𝑥)<1; (3)𝑞(𝑓(𝑥),𝑓(𝑦))⪯𝜂(𝑥)𝑞(𝑥,𝑦)+𝜆(𝑥)𝑞(𝑥,𝑓(𝑥))+𝜁(𝑥)𝑞(𝑦,𝑓(𝑦))+𝜇(𝑥)𝑞(𝑓(𝑥),𝑦)+𝜉(𝑥)𝑞(𝑥,𝑓(𝑦)).
Then, 𝑓 has a unique fixed point.

Proof. Let 𝑥₀∈𝑋 be arbitrary and fixed, and we consider the sequence 𝑥_𝑛=𝑓(𝑥_𝑛−1) for all 𝑛∈ℕ. If we take 𝑥=𝑥_𝑛−1 and 𝑦=𝑥_𝑛 in (3) we have 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛⪯𝜂𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑥_𝑛+𝜆𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑓𝑥_𝑛−1+𝜁𝑥_𝑛−1𝑞𝑥_𝑛,𝑓𝑥_𝑛+𝜇𝑥_𝑛−1𝑞𝑓𝑥_𝑛−1,𝑥_𝑛+𝜉𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑓𝑥_𝑛=𝜂𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝜆𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝜁𝑓𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1+𝜇𝑓𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+𝜉𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+1⪯𝜂𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝜆𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝜁𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1+𝜉𝑥_𝑛−2×𝑞𝑥_𝑛−1,𝑥_𝑛+𝑞𝑥_𝑛,𝑥_𝑛+1⋮⪯𝜂𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝜆𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝜁𝑥₀𝑞𝑥_𝑛,𝑥_𝑛+1+𝜉𝑥₀×𝑞𝑥_𝑛−1,𝑥_𝑛+𝑞𝑥_𝑛,𝑥_𝑛+1. (13) So, 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝜂𝑥₀+𝜆𝑥₀+𝜉𝑥₀1−𝜁𝑥₀−𝜉𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛=ℎ𝑞𝑥_𝑛−1,𝑥_𝑛⪯ℎ²𝑞𝑥_𝑛−2,𝑥_𝑛−1⋮⪯ℎ^𝑛𝑞𝑥₀,𝑥₁, (14) where ℎ=(𝜂(𝑥₀)+𝜆(𝑥₀)+𝜉(𝑥₀)/1−𝜁(𝑥₀)−𝜉(𝑥₀)). Thus, by Lemma 15, {𝑥_𝑛}_𝑛≥1 is left Cauchy in 𝑋. Because of completeness of 𝑋 and continuity of 𝑓, there exists 𝑥^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗ and 𝑥_𝑛+1=𝑓(𝑥_𝑛)→𝑓(𝑥^∗). Since 𝑋 is Hausdorff, 𝑓(𝑥^∗)=𝑥^∗.
Uniqueness. Let 𝑦^∗ be another fixed point of 𝑓, then 𝑞𝑥^∗,𝑦^∗=𝑞𝑓𝑥^∗,𝑓𝑦^∗⪯𝜂𝑥^∗𝑞𝑥^∗,𝑦^∗+𝜆𝑥^∗𝑞𝑥^∗,𝑓𝑥^∗+𝜁𝑥^∗𝑞𝑦^∗,𝑓𝑦^∗+𝜇𝑥^∗𝑞𝑓𝑥^∗,𝑦^∗+𝜉𝑥^∗𝑞𝑥^∗,𝑓𝑦^∗=𝜂𝑥^∗𝑞𝑥^∗,𝑦^∗+𝜇𝑥^∗𝑞𝑥^∗,𝑦^∗+𝜉𝑥^∗𝑞𝑥^∗,𝑦^∗=𝜂𝑥^∗+𝜇𝑥^∗+𝜉𝑥^∗𝑞𝑥^∗,𝑦^∗. (15) Therefore, 𝑞(𝑥^∗,𝑦^∗)=0 due to 𝜂(𝑥^∗)+𝜇(𝑥^∗)+𝜉(𝑥^∗)<1. Similarly, 𝑞(𝑦^∗,𝑥^∗)=0. Hence, 𝑥^∗=𝑦^∗.

Corollary 17. Let (𝑋,𝑞) be a left complete Hausdorff quasi-cone metric space and let 𝑓∶𝑋→𝑋 be a continuous function. Suppose that there exist functions 𝜂,𝜆,𝜇∶𝑋→[0,1) which satisfy the following for 𝑥,𝑦∈𝑋: (1)𝜂(𝑓(𝑥))≤𝜂(𝑥),𝜆(𝑓(𝑥))≤𝜆(𝑥) and 𝜇(𝑓(𝑥))≤𝜇(𝑥); (2)𝜂(𝑥)+2𝜆(𝑥)+2𝜇(𝑥)<1; (3)𝑞(𝑓(𝑥),𝑓(𝑦))⪯𝜂(𝑥)𝑞(𝑥,𝑦)+𝜆(𝑥)(𝑞(𝑥,𝑓(𝑥))+𝑞(𝑦,𝑓(𝑦)))+𝜇(𝑥)(𝑞(𝑓(𝑥),𝑦)+𝑞(𝑥,𝑓(𝑦))).
Then, 𝑓 has a unique fixed point.

Proof. We can prove this result by applying Theorem 16 to 𝜆(𝑥)=𝜁(𝑥) and 𝜇(𝑥)=𝜉(𝑥).

Corollary 18. Let (𝑋,𝑞) be a left complete Hausdorff quasi-cone metric space, and let 𝑓∶𝑋→𝑋 be a continuous function and 𝑞𝑓(𝑥)𝑦(𝑥)𝑦𝑓(𝑥)𝑦,𝑓⪯𝛼𝑞𝑥,𝑦+𝛽𝑞𝑥,𝑓+𝛾𝑞𝑦,𝑓+𝑘𝑞,𝑦+ℓ𝑞𝑥,𝑓 (16) for all 𝑥,𝑦∈𝑋 and 𝛼,𝛽,𝛾,𝑘,ℓ≥0 with 𝛼+𝛽+𝛾+𝑘+2ℓ<1. Then, 𝑓 has a unique fixed point.

Proof. We can prove this result by applying Theorem 16 to 𝜂(𝑥)=𝛼,𝜆(𝑥)=𝛽,𝜁(𝑥)=𝛾,𝜇(𝑥)=𝑘 and 𝜉(𝑥)=ℓ.

The following corollaries generalize some results of [14] in cone metric spaces to quasi-cone metric spaces.

Corollary 19. Let (𝑋,𝑞) be a left complete Hausdorff quasi-cone metric space, and let 𝑓∶𝑋→𝑋 be a continuous function and 𝑞𝑓(𝑥),𝑓𝑦⪯𝛼𝑞𝑥,𝑓(𝑥)+𝑞𝑦,𝑓𝑦 (17) for all 𝑥,𝑦∈𝑋 and 𝛼∈[0,1/2). Then, 𝑓 has a unique fixed point.

Corollary 20. Let (𝑋,𝑞) be a left complete Hausdorff quasi-cone metric space, and let 𝑓∶𝑋→𝑋 be a continuous function and 𝑞𝑓(𝑥),𝑓𝑦⪯𝛾𝑞𝑓(𝑥),𝑦+𝑞𝑥,𝑓𝑦 (18) for all 𝑥,𝑦∈𝑋 and 𝛾∈[0,1/2). Then, 𝑓 has a unique fixed point.

Corollary 21. Let (𝑋,𝑞) be a left complete Hausdorff quasi-cone metric space, and let 𝑓∶𝑋→𝑋 be a continuous function and 𝑞𝑓(𝑥),𝑓𝑦⪯𝛼𝑞𝑥,𝑦+𝛽𝑞𝑓(𝑥),𝑦, (19) for all 𝑥,𝑦∈𝑋 and 𝛼,𝛽∈[0,1) with 𝛼+𝛽<1. Then, 𝑓 has a unique fixed point.

The next corollary is a generalization of Banach contraction principle.

Corollary 22. Let (𝑋,𝑞) be a left complete Hausdorff quasi-cone metric space, and let 𝑓∶𝑋→𝑋 be a continuous function and 𝑞𝑓(𝑥),𝑓𝑦⪯𝛼𝑞𝑥,𝑦, (20) for all 𝑥,𝑦∈𝑋 and 𝛼∈[0,1). Then, 𝑓 has a unique fixed point.

The example in [31] shows that the Hausdorff condition is necessary for quasi-metric spaces and is so for quasi-cone metric spaces. Now, we present two examples. The first one fulfills Theorem 16in which 𝑃 is normal. The second example satisfies Corollary 18 without normality of 𝑃.

Example 23. Let 𝑋=[0,1], 𝐸=ℝ², 𝑃={(𝑥,𝑦)∈𝐸∶𝑥,𝑦≥0},and𝑞∶𝑋×𝑋→𝐸 such that 𝑞=()𝑥,𝑦𝑥−𝑦,𝛾𝑥−𝑦,if𝑥≥𝑦,0,0,if𝑥<𝑦, (21) where 𝛾∈[0,1]. Suppose 𝑓(𝑥)=𝑥/4,𝜂(𝑥)=𝑥/8,𝜆(𝑥)=1/3+𝑥/24,𝜁(𝑥)=(𝑥+𝑥²)/6,𝜇(𝑥)=1/24 and 𝜉(𝑥)=0. Then for all 𝑥,𝑦∈𝑋, we have the following.(1)𝜂(𝑓(𝑥))=𝑥/32≤𝑥/8=𝜂(𝑥),  𝜆(𝑓(𝑥))=1/3+𝑥/96≤1/3+𝑥/24=𝜆(𝑥), 𝜁(𝑓(𝑥))=(𝑥/4+𝑥²/16)/6≤(𝑥+𝑥²)/6=𝜁(𝑥), 𝜇(𝑓(𝑥))=𝜇(𝑥)=1/24, and 𝜉(𝑓(𝑥))=𝜉(𝑥)=0. (2)𝑥/8+(1/3+𝑥/24)+(𝑥+𝑥²)/6+1/24<1. (3)Condition (3) of Theorem 16 is satisfied. For 𝑥≥𝑦, we have 𝑞𝑓(𝑥)𝑦(𝑥)𝑞(𝑥)𝑞(𝑥)(𝑥)𝑞𝑦(𝑥)𝑞𝑓(𝑥)(𝑥)𝑞𝑦,𝑓≤𝜂𝑥,𝑦+𝜆𝑥,𝑓+𝜁𝑦,𝑓+𝜇,𝑦+𝜉𝑥,𝑓 (22)due to 𝑥4−𝑦4𝑥,𝑟4−𝑦4≤25𝑥−𝑦96+𝑥24²𝑦8+5𝑥²32,𝑟25𝑥−𝑦96+𝑥24²𝑦8+5𝑥²32 (23) and for 𝑥<𝑦 it is trivial.
Therefore, {0} is a fixed point.

Example 24. Let 𝑋=[0,∞), 𝐸=𝐶¹_ℝ[0,1]×𝐶¹_ℝ[0,1], 𝑃={(𝜙,𝜑)∈𝐸∶𝜙(𝑡)and𝜑(𝑡)≥0,𝑡∈[0,1]}, and 𝑞∶𝑋×𝑋→𝐸 defined by 𝑞=𝑥𝑥,𝑦²−𝑦²𝜙,𝑥²−𝑦²𝜙(),if𝑥≥𝑦,0,0,if𝑥≤𝑦, (24) where 𝜙(𝑡)=𝑒^𝑡. Suppose 𝑓(𝑥)=(1/8)𝑥. If we take 𝛼=1/16,𝛽=1/3,𝛾=1/4,𝑘=1/6, and ℓ=1/16, then all the assumptions of Corollary 18 are satisfied. Thus, {0} is a fixed point.

Theorem 25. Let (𝑋,𝑞) be a left complete Hausdorff quasi-cone metric space, and let 𝑓∶𝑋→𝑋 be a continuous function and for all 𝑥,𝑦∈𝑋𝑓(𝑥)𝑦(𝑥)𝑦𝑦(𝑥)𝛼𝑞,𝑓+𝛽𝑞𝑥,𝑓+𝛾𝑞𝑦,𝑓+𝑘𝑞𝑥,𝑓+ℓ𝑞𝑦,𝑓⪯𝑠𝑞𝑥,𝑦+𝑡𝑞𝑥,𝑓²(𝑥), (∗) where 𝑠≥𝛽≥ℓ,𝛾≥𝑘≥𝑡,𝛼+𝑘>0, and 0≤(𝑠−ℓ)/(𝛼+𝑘)<1. Then, 𝑓 has a unique fixed point.

Proof. Let 𝑥₀∈𝑋 be arbitrary and fixed and 𝑥_𝑛=𝑓(𝑥_𝑛−1) for all 𝑛∈ℕ. If we take 𝑥=𝑥_𝑛−1 and 𝑦=𝑥_𝑛 in (∗), we have 𝛼𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛+𝛽𝑞𝑥_𝑛−1,𝑓𝑥_𝑛−1+𝛾𝑞𝑥_𝑛,𝑓𝑥_𝑛+𝑘𝑞𝑥_𝑛−1,𝑓𝑥_𝑛+ℓ𝑞𝑥_𝑛,𝑓𝑥_𝑛−1⪯𝑠𝑞𝑥_𝑛−1,𝑥_𝑛+𝑡𝑞𝑥_𝑛−1,𝑓²𝑥_𝑛−1. (25) Rewriting this inequality as 𝛼𝑞𝑥_𝑛,𝑥_𝑛+1+𝛽𝑞𝑥_𝑛−1,𝑥_𝑛+𝛾𝑞𝑥_𝑛,𝑥_𝑛+1+𝑘𝑞𝑥_𝑛−1,𝑥_𝑛+1+ℓ𝑞𝑥_𝑛,𝑥_𝑛⪯𝑠𝑞𝑥_𝑛−1,𝑥_𝑛+𝑡𝑞𝑥_𝑛−1,𝑥_𝑛+1 (26) implies that 𝛼+𝛾𝑞𝑥_𝑛,𝑥₊₁+(𝑘−𝑡)𝑞𝑥_𝑛−1,𝑥_𝑛+1⪯𝑠−𝛽𝑞𝑥_𝑛−1,𝑥_𝑛. (27) Since 𝑘≥𝑡, we have 𝛼+𝛾𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑠−𝛽𝑞𝑥_𝑛−1,𝑥_𝑛. (28) Therefore, we obtain 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑠−𝛽𝑞𝑥𝛼+𝛾^𝑛−1,𝑥_𝑛, (29) due to 𝛽≥ℓ,𝛾≥𝑘, and 𝛼+𝛾>0, and we get (𝑠−𝛽)/(𝛼+𝛾)≤(𝑠−ℓ)/(𝛼+𝑘). Therefore, 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑠−ℓ𝑞𝑥𝛼+𝑘^𝑛−1,𝑥_𝑛⪯𝑠−ℓ𝛼+𝑘²𝑞𝑥_𝑛−2,𝑥_𝑛−1⋮⪯𝑠−ℓ𝛼+𝑘^𝑛𝑞𝑥₀,𝑥₁. (30) Thus, by Lemma 15, {𝑥_𝑛}_𝑛≥1 is left Cauchy in 𝑋. Because of completeness of 𝑋 and continuity of 𝑓, there exists 𝑥^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗ and 𝑥_𝑛+1=𝑓(𝑥_𝑛)→𝑓(𝑥^∗). Since 𝑋 is Hausdorff, 𝑓(𝑥^∗)=𝑥^∗.
Uniqueness. Let 𝑦^∗ be another fixed point. Putting 𝑥=𝑥^∗ and 𝑦=𝑦^∗ in (∗), we obtain 𝛼𝑞𝑓𝑥^∗,𝑓𝑦^∗+𝛽𝑞𝑥^∗,𝑓𝑥^∗+𝛾𝑞𝑦^∗,𝑓𝑦^∗+𝑘𝑞𝑥^∗,𝑓𝑦^∗+ℓ𝑞𝑦^∗,𝑓𝑥^∗⪯𝑠𝑞𝑥^∗,𝑦^∗+𝑡𝑞𝑥^∗,𝑓²𝑥^∗. (31) Hence, (𝛼+𝑘)𝑞𝑥^∗,𝑦^∗+ℓ𝑞𝑦^∗,𝑥^∗⪯𝑠𝑞𝑥^∗,𝑦^∗. (32) Similarly, applying (∗) with 𝑥=𝑦^∗ and 𝑦=𝑥^∗, we have (𝛼+𝑘)𝑞𝑦^∗,𝑥^∗+ℓ𝑞𝑥^∗,𝑦^∗⪯𝑠𝑞𝑦^∗,𝑥^∗. (33) Adding up the above two inequalities, we get (𝛼+𝑘)𝑞𝑥^∗,𝑦^∗+𝑞𝑦^∗,𝑥^∗+ℓ𝑞𝑦^∗,𝑥^∗+𝑞𝑥^∗,𝑦^∗⪯𝑠𝑞𝑥^∗,𝑦^∗+𝑞𝑦^∗,𝑥^∗. (34)Subsequently, we obtain (𝛼+𝑘)𝑞𝑥^∗,𝑦^∗+𝑞𝑦^∗,𝑥^∗⪯(𝑠−ℓ)𝑞𝑥^∗,𝑦^∗+𝑞𝑦^∗,𝑥^∗. (35) Thus, 𝑞𝑥^∗,𝑦^∗+𝑞𝑦^∗,𝑥^∗⪯𝑠−ℓ𝑞𝑥𝛼+𝑘^∗,𝑦^∗+𝑞𝑦^∗,𝑥^∗. (36) Hence, 𝑞(𝑥^∗,𝑦^∗)+𝑞(𝑦^∗,𝑥^∗)=0 due to 0≤(𝑠−ℓ)/(𝛼+𝑘)<1. Therefore, 𝑞(𝑥^∗,𝑦^∗)=𝑞(𝑦^∗,𝑥^∗)=0 and 𝑥^∗=𝑦^∗.

Acknowledgment

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant ERGS/1/2011/STG/UKM/01/13.