About this Journal Submit a Manuscript Table of Contents
ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 457846, 9 pages
http://dx.doi.org/10.5402/2012/457846
Research Article

A New Fixed Point Theorem on Generalized Quasimetric Spaces

1Department of Mathematics and Computer Sciences, University of Gjirokastra, Gjirokastra, Albania
2Department of Mathematics, University of Tirana, Tirana, Albania

Received 2 November 2011; Accepted 30 November 2011

Academic Editor: R. Avery

Copyright © 2012 Luljeta Kikina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain a new fixed point theorem in generalized quasimetric spaces. This result generalizes, unify, enrich, and extend some theorems of well-known authors from metric spaces to generalized quasimetric spaces.

1. Introduction and Preliminaries

The concept of metric space, as an ambient space in fixed point theory, has been generalized in several directions. Some of such generalizations are quasimetric spaces, generalized metric spaces, and generalized quasimetric spaces.

The concept of quasimetric spaces is treated differently by many authors. In this paper our concept is in line with this treated in [16], and so forth and the triangular inequality 𝑑(𝑥,𝑦)𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦) is replaced by quasi-triangular inequality:[]𝑑(𝑥,𝑦)𝑘𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦),𝑘1.(1.1)

In 2000 Branciari [7] introduced the concept of generalized metric spaces (gms) (the triangular inequality 𝑑(𝑥,𝑦)𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦) is replaced by tetrahedral inequality𝑑(𝑥,𝑦)𝑑(𝑥,𝑧)+𝑑(𝑧,𝑤)+𝑑(𝑤,𝑦)). Starting with the paper of Branciari, some classical metric fixed point theorems have been transferred to gms (see [813]).

Recently L. kikina and k. kikina [14] introduced the concept of generalized quasimetric space (gqms) on the lines of quasimetric space, where the tetrahedral inequality𝑑(𝑥,𝑦)𝑑(𝑥,𝑧)+𝑑(𝑧,𝑤)+𝑑(𝑤,𝑦) has been replaced by quasitetrahedral inequality𝑑(𝑥,𝑦)𝑘[𝑑(𝑥,𝑧)+𝑑(𝑧,𝑤)+𝑑(𝑤,𝑦)]. The well-known fixed point theorems of Banach and of Kannan have been transferred to such a space.

The metric spaces are a special case of generalized metric spaces and generalized metric spaces are a special case of generalized quasimetric spaces (for 𝑘=1). Also, every qms is a gqms, while the converse is not true.

In gqms, contrary to a metric space, the “open” balls 𝐵(𝑎,𝑟)={𝑥𝑋𝑑(𝑥,𝑎)<𝑟} are not always open sets, and consequently, a generalized quasidistance is not always continuous of its variables. The gqms is not always a Hausdorff space and the convergent sequence (𝑥𝑛) in gqms is not always a Cauchy sequence (see Example 1.3).

Under this situation, it is reasonable to consider if some well-known fixed point theorems can be obtained in generalized quasimetric space.

The aim of this paper is to generalize, unify, and extend some theorems of well-known authors such as of Fisher and Popa, from metric spaces to generalized quasimetric spaces.

Let us start with the main definitions.

Definition 1.1 (see [7]). Let 𝑋 be a set and 𝑑𝑋2𝑅+ a mapping such that for all 𝑥,𝑦𝑋 and for all distinct points 𝑧,𝑤𝑋, each of them different from 𝑥 and 𝑦, one has (a)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦,(b)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥),(c)𝑑(𝑥,𝑦)𝑑(𝑥,𝑧)+𝑑(𝑧,𝑤)+𝑑(𝑤,𝑦) (tetrahedral inequality).Then 𝑑is called a generalized metric and (𝑋,𝑑) is a generalized metric space (or shortly gms).

Definition 1.2 (see [14]). Let 𝑋 be a set. A nonnegative symmetric function 𝑑defined on 𝑋𝑋 is called a generalized quasidistance on 𝑋 if and only if there exists a constant 𝑘1 such that for all 𝑥,𝑦𝑋 and for all distinct points 𝑧,𝑤𝑋, each of them different from 𝑥 and 𝑦, the following conditions hold:
(i)𝑑(𝑥,𝑦)=0𝑥=𝑦; (ii)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥); (iii)𝑑(𝑥,𝑦)𝑘[𝑑(𝑥,𝑧)+𝑑(𝑧,𝑤)+𝑑(𝑤,𝑦)].
Inequality (2.7) is often called quasitetrahedral inequality and 𝑘 is often called the coefficient of 𝑑. A pair (𝑋,𝑑) is called a generalized quasimetric space if 𝑋 is a set and 𝑑 is a generalized quasidistance on 𝑋.
The set 𝐵(𝑎,𝑟)={𝑥𝑋𝑑(𝑥,𝑎)<𝑟} is called “open” ball with center 𝑎𝑋 and radius 𝑟>0.
The family 𝜏={𝑄𝑋𝑎𝑄,𝑟>0,𝐵(𝑎,𝑟)𝑄} is a topology on 𝑋 and it is called induced topology by the generalized quasidistance 𝑑.

The following example illustrates the existence of the generalized quasimetric space for an arbitrary constant𝑘1.

Example 1.3 (see [14]). Let𝑋={1(1/𝑛)𝑛=1,2,}{1,2}. Define 𝑑𝑋𝑋𝑅 as follows: 1𝑑(𝑥,𝑦)=0,for𝑥=𝑦,𝑛1,for𝑥{1,2},𝑦=1𝑛1or𝑦{1,2},𝑥=1𝑛,𝑥𝑦,3𝑘,for𝑥,𝑦{1,2},𝑥𝑦,1,otherwise.(1.2) Then it is easy to see that (𝑋,𝑑) is a generalized quasimetric space and is not a generalized metric space (for 𝑘>1).

Note that the sequence (𝑥𝑛)=(1(1/𝑛)) converges to both 1 and 2 and it is not a Cauchy sequence:𝑑𝑥𝑛,𝑥𝑚1=𝑑1𝑛1,1𝑚=1,𝑛,𝑚𝑁.(1.3)

Since 𝐵(1,𝑟)𝐵(2,𝑟)𝜙 for all 𝑟>0, the (𝑋,𝑑) is non-Hausdorff generalized metric space.

The function 𝑑 is not continuous: 1=lim𝑛𝑑(1(1/𝑛),1/2)𝑑(1,1/2)=1/2.

In [14] the following is proved.

Proposition 1.4. If (𝑋,𝑑) is a quasimetric space, then (𝑋,𝑑) is a generalized quasimetric space. The converse proposition does not hold true.

Definition 1.5. A sequence {𝑥𝑛} in a generalized quasimetric space (𝑋,𝑑) is called Cauchy sequence if lim𝑛,𝑚𝑑(𝑥𝑛,𝑥𝑚)=0.

Definition 1.6. Let (𝑋,𝑑) be a generalized quasimetric space. Then one has the following.(1)A sequence {𝑥𝑛} in 𝑋 is said to be convergent to a point 𝑥𝑋 (denoted by lim𝑛𝑥𝑛=𝑥) if lim𝑛𝑑(𝑥𝑛,𝑥)=0.(2)It is called compact if every sequence contains a convergent subsequence.

Definition 1.7. A generalized quasimetric space (𝑋,𝑑) is called complete, if every Cauchy sequence is convergent.

Definition 1.8. Let (𝑋,𝑑) be a gqms and the coefficient of 𝑑 is 𝑘.
A map 𝑇𝑋𝑋 is called contraction if there exists 0<𝑐<1/𝑘 such that𝑑(𝑇𝑥,𝑇𝑦)𝑐𝑑(𝑥,𝑦)𝑥,𝑦𝑋.(1.4)

Definition 1.9. Let 𝑇𝑋𝑋 be a mapping where 𝑋 is a gqms. For each 𝑥𝑋, let 𝑂(𝑥)=𝑥,𝑇𝑥,𝑇2𝑥,,(1.5) which will be called the orbit of 𝑇 at 𝑥. The space 𝑋 is said to be 𝑇-orbitally complete if and only if every Cauchy sequence which is contained in 𝑂(𝑥) converges to a point in 𝑋.

Definition 1.10. The set of all upper semicontinuous functions with 3 variables 𝑓𝑅3+𝑅 satisfying the following properties: (a)𝑓 is nondecreasing in respect to each variable, (b)𝑓(𝑡,𝑡,𝑡)𝑡, 𝑡𝑅+will be noted by 𝔽3 and every such function will be called an 𝔽3-function.
Some examples of 𝔽3-function are as follows:
(1)𝑓(𝑡1,𝑡2,𝑡3)=max{𝑡1,𝑡2,𝑡3},(2)𝑓(𝑡1,𝑡2,𝑡3)=[max{𝑡1𝑡2,𝑡2𝑡3,𝑡3𝑡1}]1/2, (3)𝑓(𝑡1,𝑡2,𝑡3)=[max{𝑡𝑝1,𝑡𝑝2,𝑡𝑝3}]1/𝑝,𝑝>0,(4)𝑓(𝑡1,𝑡2,𝑡3)=(𝑎𝑡1𝑡2+𝑏𝑡2𝑡3+𝑐𝑡3𝑡1)1/2, where 𝑎,𝑏,𝑐0 and 𝑎+𝑏+𝑐<1.

2. Main Result

We state the following lemma which we will use for the proof of the main theorem.

Lemma 2.1. Let (𝑋,𝑑) be a generalized quasimetric space and {𝑥𝑛} is a sequence of distinct point (𝑥𝑛𝑥𝑚 for all 𝑛𝑚) in 𝑋. If 𝑑(𝑥𝑛,𝑥𝑛+1)𝑐𝑛𝑙, 0𝑐<1/𝑘<1, for all 𝑛𝑁 and lim𝑛𝑑(𝑥𝑛,𝑥𝑛+2)=0, then {𝑥𝑛} is a Cauchy sequence.

Proof. If 𝑚>2 is odd, then writing 𝑚=2𝑝+1, 𝑝1, by quasitetrahedral inequality, we can easily show that 𝑑𝑥𝑛,𝑥𝑛+𝑚𝑑𝑥𝑘𝑛,𝑥𝑛+1𝑥+𝑑𝑛+1,𝑥𝑛+2𝑥+𝑑𝑛+2,𝑥𝑛+𝑚𝑥𝑘𝑑𝑛,𝑥𝑛+1+𝑘2𝑑𝑥𝑛+1,𝑥𝑛+2+𝑘2𝑑𝑥𝑛+2,𝑥𝑛+𝑚𝑘𝑐𝑛𝑙+𝑘2𝑐𝑛+1𝑙+𝑘2𝑑𝑥𝑛+2,𝑥𝑛+𝑚𝑘𝑐𝑛𝑙+𝑘2𝑐𝑛+1𝑙+𝑘3𝑐𝑛+2𝑙++𝑘𝑚1𝑐𝑛+𝑚2𝑙+𝑘𝑚1𝑐𝑛+𝑚1𝑙𝑘𝑐𝑛𝑙+𝑘2𝑐𝑛+1𝑙+𝑘3𝑐𝑛+2𝑙++𝑘𝑚1𝑐𝑛+𝑚2𝑙+𝑘𝑚𝑐𝑛+𝑚1𝑙𝑘𝑐𝑛𝑙1+𝑘𝑐++(𝑘𝑐)𝑚1=𝑘𝑐𝑛𝑙1(𝑘𝑐)𝑚1𝑘𝑐<𝑘𝑐𝑛𝑙1.1𝑘𝑐(2.1) Therefore, lim𝑛𝑑(𝑥𝑛,𝑥𝑛+𝑚)=0.
If 𝑚>2 is even, then writing 𝑚=2𝑝, 𝑝2 and using the same arguments as before we can get 𝑑𝑥𝑛,𝑥𝑛+𝑚𝑑𝑥𝑘𝑛,𝑥𝑛+2𝑥+𝑑𝑛+2,𝑥𝑛+3𝑥+𝑑𝑛+3,𝑥𝑛+𝑚𝑥𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑥𝑙+𝑘𝑑𝑛+3,𝑥𝑛+𝑚𝑥𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑙+𝑘2𝑑𝑥𝑛+3,𝑥𝑛+4𝑥+𝑑𝑛+4,𝑥𝑛+5𝑥+𝑑𝑛+5,𝑥𝑛+𝑚𝑥𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑙+𝑘2𝑐𝑛+3𝑙++𝑘𝑚2𝑐𝑛+𝑚1𝑙𝑥=𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑙1+𝑘𝑐++(𝑘𝑐)𝑚3𝑥=𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑙1(𝑘𝑐)𝑚2𝑥1𝑘𝑐<𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑙1.1𝑘𝑐(2.2) And so lim𝑛𝑑(𝑥𝑛,𝑥𝑛+𝑚)=0. It implies that {𝑥𝑛} is a Cauchy sequence in 𝑋. This completes the proof of the lemma.

We state the following theorem.

Theorem 2.2. Let (𝑋,𝑑) and (𝑌,𝜌) be two generalized quasimetric spaces with coefficients 𝑘1 and 𝑘2, respectively. Let 𝑇 be a mapping of 𝑋 into 𝑌 and 𝑆 a mapping of 𝑌 into 𝑋 satisfying the following inequalities: 𝑑(𝑆𝑦,𝑆𝑇𝑥)𝑐𝑓1𝜌{𝑑(𝑥,𝑆𝑦),𝑑(𝑥,𝑆𝑇𝑥),𝜌(𝑦,𝑇𝑥)},(𝑇𝑥,𝑇𝑆𝑦)𝑐𝑓2{𝜌(𝑦,𝑇𝑥),𝜌(𝑦,𝑇𝑆𝑦),𝑑(𝑥,𝑆𝑦)},(2.3) for all 𝑥𝑋 and 𝑦𝑌, where 0<𝑐<1/𝑘1, 𝑘=max{𝑘1,𝑘2}, 𝑓1,𝑓2𝔽3. If there exists 𝑥0𝑋 such that 𝑂(𝑥0) is 𝑆𝑇-orbitally complete in 𝑋 and 𝑂(𝑇𝑥0) is 𝑇𝑆-orbitally complete in 𝑌, then 𝑆𝑇 has a unique fixed point 𝛼 in 𝑋 and 𝑇𝑆 has a unique fixed point 𝛽 in 𝑌. Further, 𝑇𝛼=𝛽 and 𝑆𝛽=𝛼.

Proof. Let 𝑥0 be an arbitrary point in 𝑋. Define the sequences (𝑥𝑛) and (𝑦𝑛) inductively as follows: 𝑥𝑛=𝑆𝑦𝑛=(𝑆𝑇)𝑛𝑥0,𝑦1=𝑇𝑥0,𝑦𝑛+1=𝑇𝑥𝑛=(𝑇𝑆)𝑛𝑦1,𝑛1.(2.4) Denote 𝑑𝑛𝑥=𝑑𝑛,𝑥𝑛+1,𝜌𝑛𝑦=𝜌𝑛,𝑦𝑛+1,𝑛=1,2,.(2.5) Using the inequality (2) we get 𝜌𝑛𝑦=𝜌𝑛,𝑦𝑛+1=𝜌𝑇𝑥𝑛1,𝑇𝑆𝑦𝑛𝑐𝑓2𝜌𝑦𝑛,𝑦𝑛𝑦,𝜌𝑛,𝑦𝑛+1𝑥,𝑑𝑛1,𝑥𝑛=𝑐𝑓20,𝜌𝑛,𝑑𝑛1.(2.6) By this inequality and properties of 𝑓2, it follows that 𝜌𝑛𝑐𝑑𝑛1.(2.7) Using the inequality (2.3) we have 𝑑𝑛𝑥=𝑑𝑛,𝑥𝑛+1=𝑑𝑆𝑦𝑛,𝑆𝑇𝑥𝑛𝑐𝑓1𝑑𝑥𝑛,𝑥𝑛𝑥,𝑑𝑛,𝑥𝑛+1𝑦,𝜌𝑛,𝑦𝑛+1=𝑐𝑓10,𝑑𝑛,𝜌𝑛,(2.8) and so 𝑑𝑛𝑐𝜌𝑛. By this inequality and (2.7) we obtain 𝑑𝑛𝑐2𝑑𝑛1𝑐𝑑𝑛1.(2.9) Using the mathematical induction, by the inequalities (2.7) and (2.9), we get 𝑑𝑛𝑐𝑛𝑑𝑥0,𝑥1,𝜌𝑛𝑐𝑛𝑑𝑥0,𝑥1.(2.10) So lim𝑛𝑑𝑥𝑛,𝑥𝑛+1=lim𝑛𝜌𝑦𝑛,𝑦𝑛+1=0.(2.11) Applying the inequality (2), we get 𝜌𝑦𝑛,𝑦𝑛+2=𝜌𝑇𝑥𝑛1,𝑇𝑆𝑦𝑛+1𝑐𝑓2𝜌𝑦𝑛+1,𝑦𝑛𝑦,𝜌𝑛+1,𝑦𝑛+2𝑥,𝑑𝑛1,𝑥𝑛+1=𝑐𝑓2𝜌𝑛,𝜌𝑛+1𝑥,𝑑𝑛1,𝑥𝑛+1𝑐𝑐max𝑛𝑑𝑥0,𝑥1𝑥,𝑑𝑛1,𝑥𝑛+1,(2.12) and so 𝜌𝑦𝑛,𝑦𝑛+2𝑐max𝑛𝑑𝑥0,𝑥1𝑥,𝑐𝑑𝑛1,𝑥𝑛+1.(2.13) Similarly, using (2.3), we obtain 𝑑𝑥𝑛,𝑥𝑛+2𝑐max𝑛𝑑𝑥0,𝑥1𝑥,𝑐𝑑𝑛1,𝑥𝑛+1.(2.14) Using the mathematical induction, we get 𝑑𝑥𝑛,𝑥𝑛+2𝑐max𝑛𝑑𝑥0,𝑥1𝑥,𝑐𝑑𝑛1,𝑥𝑛+1𝑐max𝑛𝑑𝑥0,𝑥1,𝑐2𝑑𝑥𝑛2,𝑥𝑛𝑐max𝑛𝑑𝑥0,𝑥1,𝑐𝑛𝑑𝑥0,𝑥2=𝑐𝑛𝑑𝑥max0,𝑥1𝑥,𝑑0,𝑥2=𝑐𝑛𝑙,(2.15) and so 𝑑𝑥𝑛,𝑥𝑛+2𝑐𝑛𝑦𝑙,similarly𝜌𝑛,𝑦𝑛+2𝑐𝑛𝑙,(2.16) where 𝑙=max{𝑑(𝑥0,𝑥1),𝑑(𝑥0,𝑥2)}.
We divide the proof into two cases.
Case 1. Suppose 𝑥𝑝=𝑥𝑞 for some 𝑝,𝑞𝑁, 𝑝𝑞. Let 𝑝>𝑞. Then (𝑆𝑇)𝑝𝑥0=(𝑆𝑇)𝑝𝑞(𝑆𝑇)𝑞𝑥0=(𝑆𝑇)𝑞𝑥0; that is, (𝑆𝑇)𝑛𝛼=𝛼 where 𝑛=𝑝𝑞 and (𝑆𝑇)𝑞𝑥0=𝛼. Now if 𝑛>1, by (2.10), we have 𝑑(𝛼,𝑆𝑇𝛼)=𝑑(𝑆𝑇)𝑛𝛼,(𝑆𝑇)𝑛+1𝛼𝑐𝑛𝑑(𝛼,𝑆𝑇𝛼).(2.17) Since 0<𝑐<1, 𝑑(𝛼,𝑆𝑇𝛼)=0. So 𝑆𝑇𝛼=𝛼 and hence 𝛼 is a fixed point of 𝑆𝑇.
By the equality 𝑥𝑝=𝑥𝑞 it follows that 𝑦𝑝+1=𝑦𝑞+1. We take 𝛽=(𝑇𝑆)𝑞𝑇𝑥0 and, in similar way, we prove that 𝛽 is a fixed point of 𝑇𝑆.
Case 2. Assume that 𝑥𝑛𝑥𝑚 for all 𝑛𝑚. Then, from (2.10), (2.16), and Lemma 2.1 is derived that {𝑥𝑛} is a Cauchy sequence in 𝑋. Since 𝑂(𝑥0) is ST-orbitally complete, there exists 𝛼𝑋 such that lim𝑛𝑥𝑛=𝛼. In the same way, we show that the sequence (𝑦𝑛) is a Cauchy sequence and there exists a 𝛽𝑌 such that lim𝑛𝑦𝑛=𝛽.
We now prove that the limits 𝛼 and 𝛽 are unique. Suppose, to the contrary, that 𝛼𝛼 is also lim𝑛𝑥𝑛. Since 𝑥𝑛𝑥𝑚 for all 𝑛𝑚, there exists a subsequence (𝑥𝑛𝑘) of (𝑥𝑛) such that 𝑥𝑛𝑘𝛼 and 𝑥𝑛𝑘𝛼 for all 𝑘𝑁. Without loss of generality, assume that (𝑥𝑛) is this subsequence. Then by Tetrahedral property of Definition 1.1 we obtain𝑑𝛼,𝛼𝑑𝑘𝛼,𝑥𝑛𝑥+𝑑𝑛,𝑥𝑛+1𝑥+𝑑𝑛+1,𝛼.(2.18) Letting 𝑛 tend to infinity we get 𝑑(𝛼,𝛼)=0 and so 𝛼=𝛼, in the same way for 𝛽.
Let us prove now that 𝛼 is a fixed point of 𝑆𝑇. First we prove that 𝛽=𝑇𝛼. In contrary, if 𝛽𝑇𝛼, the sequence (𝑦𝑛) does not converge to 𝑇𝛼 and there exists a subsequence (𝑦𝑛𝑞) of (𝑦𝑛) such that 𝑦𝑛𝑞𝑇𝛼 for all 𝑞𝑁. Then by Tetrahedral property of Definition 1.1 we obtain𝜌𝜌(𝛽,𝑇𝛼)𝑘𝛽,𝑦𝑛𝑞1𝑦+𝜌𝑛𝑞1,𝑦𝑛𝑞𝑦+𝜌𝑛𝑞,𝑇𝛼.(2.19) Then if 𝑞, we get 𝜌(𝛽,𝑇𝛼)𝑘___lim𝑞𝜌𝑦𝑛𝑞,𝑇𝛼.(2.20) Using the inequality (2), for 𝑥=𝛼 and 𝑦=𝑦𝑛1 we obtain 𝜌𝑇𝛼,𝑦𝑛=𝜌𝑇𝛼,𝑇𝑆𝑦𝑛1𝑐𝑓2𝜌𝑦𝑛1𝑦,𝑇𝛼,𝜌𝑛1,𝑇𝑆𝑦𝑛1,𝑑𝛼,𝑆𝑦𝑛1=𝑐𝑓2𝜌𝑦𝑛1𝑦,𝑇𝛼,𝜌𝑛1,𝑦𝑛,𝑑𝛼,𝑥𝑛1.(2.21) Letting 𝑛 tend to infinity we get ___lim𝑛𝜌𝑇𝛼,𝑦𝑛𝑐𝑓2___lim𝑛𝜌𝑦𝑛1,𝑇𝛼,0,0.(2.22) And so, ___lim𝑛𝜌𝑇𝛼,𝑦𝑛=0.(2.23) Since ___lim𝑞𝜌(𝑦𝑛𝑞,𝑇𝛼)___lim𝑛𝜌(𝑇𝛼,𝑦𝑛), by (2.23) and (2.20), we have 𝜌(𝛽,𝑇𝛼)=0 and so 𝑇𝛼=𝛽.(2.24) It follows similarly that 𝑆𝛽=𝛼.(2.25) By (2.24) and (2.25) we obtain 𝑆𝑇𝛼=𝑆𝛽=𝛼,𝑇𝑆𝛽=𝑇𝛼=𝛽.(2.26) Thus, we proved that the points 𝛼 and 𝛽 are fixed points of 𝑆𝑇 and 𝑇𝑆, respectively.
Let us prove now the uniqueness (for Cases 1 and 2 in the same time). Assume that 𝛼𝛼 is also a fixed point of 𝑆𝑇. By (2.3) for 𝑥=𝛼 and 𝑦=𝛽 we get 𝑑𝛼,𝛼=𝑑𝑆𝛽,𝑆𝑇𝛼𝑐𝑓1𝑑𝛼,𝛼,0,𝜌𝑇𝛼,𝑇𝛼.(2.27) And so,we have 𝑑𝛼,𝛼𝑐𝜌𝑇𝛼,𝑇𝛼.(2.28) If 𝑇𝛼𝑇𝛼, in similar way by (2) for 𝑥=𝑆𝑇𝛼 and 𝑦=𝑇𝛼, we have 𝜌𝑇𝛼,𝑇𝛼𝑐𝑑𝛼,𝛼.(2.29) By (2.28) and (2.29) we get 𝑑(𝛼,𝛼)=0. Thus, we have again 𝛼=𝛼. The uniqueness of 𝛽 follows similarly. This completes the proof of the theorem.

3. Corollaries

(1) If 𝑘1=𝑘2=1, then by Theorem 2.2 we obtain [12, Theorem  2.1], that generalize and extend the well-known Fisher fixed point theorem [15] from metric space to generalized metric spaces.For different expressions of 𝑓1 and 𝑓2 in Theorem 2.2 we get different theorems.(2) For 𝑓1=𝑓2=𝑓, where 𝑓(𝑡1,𝑡2,𝑡3)=max{𝑡1,𝑡2,𝑡3} we have an extension of Fisher’s theorem [15] in generalized quasimetric spaces.(3) For 𝑓1=𝑓2=𝑓, where 𝑓(𝑡1,𝑡2,𝑡3)=[max{𝑡1𝑡2,𝑡2𝑡3,𝑡3𝑡1}]1/2, we have an extension of Popa’s theorem [13] in generalized quasimetric spaces.(4) For 𝑓1(𝑡1,𝑡2,𝑡3)=(𝑎1𝑡1𝑡2+𝑏1𝑡2𝑡3+𝑐1𝑡3𝑡1)1/2 and 𝑓2(𝑡1,𝑡2,𝑡3)=(𝑎2𝑡1𝑡2+𝑏2𝑡2𝑡3+𝑐2𝑡3𝑡1)1/2 we obtain an extension of Popa’s Corollary [13] in generalized quasimetric spaces.

Remark 3.1. We can obtain many other similar results for different 𝑓.

References

  1. M. Bramanti and L. Brandolini, “Schauder estimates for parabolic nondivergence operators of Hörmander type,” Journal of Differential Equations, vol. 234, no. 1, pp. 177–245, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. L. Kikina and K. Kikina, “A related fixed point theorem for m mappings on m complete quasi-metric spaces,” Mathematica Cluj. In press.
  3. L. Kikina and K. Kikina, “Generalized fixed point theorem for three mappings on three quasi-metric spaces,” Journal of Computational Analysis and Applications. In press.
  4. B. Pepo, “Fixed points for contractive mappings of third order in pseudo-quasimetric spaces,” Indagationes Mathematicae, vol. 1, no. 4, pp. 473–481, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. C. Peppo, “Fixed point theorems for (ϕ,k,i,j)-mappings,” Nonlinear Analysis, vol. 72, no. 2, pp. 562–570, 2010. View at Publisher · View at Google Scholar
  6. Q. Xia, “The geodesic problem in quasimetric spaces,” Journal of Geometric Analysis, vol. 19, no. 2, pp. 452–479, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. A. Branciari, “A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces,” Publicationes Mathematicae Debrecen, vol. 57, no. 1-2, pp. 31–37, 2000. View at Zentralblatt MATH
  8. A. Azam and M. Arshad, “Kannan fixed point theorem on generalized metric spaces,” Journal of Nonlinear Sciences and Its Applications, vol. 1, no. 1, pp. 45–48, 2008. View at Zentralblatt MATH
  9. P. Das, “A fixed point theorem on a class of generalized metric spaces,” Korean Journal of Mathematical Sciences, vol. 1, pp. 29–33, 2002.
  10. P. Das and L. K. Dey, “A fixed point theorem in a generalized metric space,” Soochow Journal of Mathematics, vol. 33, no. 1, pp. 33–39, 2007. View at Zentralblatt MATH
  11. L. Kikina and K. Kikina, “A fixed point theorem in generalized metric spaces,” Demonstratio Mathematica. In press.
  12. L. Kikina and K. Kikina, “Fixed points on two generalized metric spaces,” International Journal of Mathematical Analysis, vol. 5, no. 29-32, pp. 1459–1467, 2011.
  13. I. R. Sarma, J. M. Rao, and S. S. Rao, “Contractions over generalized metric spaces,” Journal of Nonlinear Science and its Applications, vol. 2, no. 3, pp. 180–182, 2009. View at Zentralblatt MATH
  14. L. Kikina and K. Kikina, “Two fixed point theorems on a class of generalized quasi-metric spaces,” Journal of Computational Analysis and Applications. In press.
  15. B. Fisher, “Fixed points on two metric spaces,” Glasnik Matematički, vol. 16, no. 36, pp. 333–337, 1981. View at Zentralblatt MATH