International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 457846 | 9 pages | https://doi.org/10.5402/2012/457846

A New Fixed Point Theorem on Generalized Quasimetric Spaces

Academic Editor: R. Avery
Received02 Nov 2011
Accepted30 Nov 2011
Published26 Jan 2012

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 [1–6], 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),𝑘≥1.(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 [8–13]).

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space (gqms) on the lines of quasimetric space, where the tetrahedral inequality𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑤)+𝑑(𝑤,𝑦) has been replaced by quasitetrahedral inequality𝑑(𝑥,𝑦)≤𝑘[𝑑(𝑥,𝑧)+𝑑(𝑧,𝑤)+𝑑(𝑤,𝑦)].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(ğ‘Ž,𝑟)={𝑥∈𝑋∶𝑑(𝑥,ğ‘Ž)<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 𝑑∶𝑋2→𝑅+ a mapping such that for all 𝑥,𝑦∈𝑋 and for all distinct points 𝑧,𝑤∈𝑋, 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 𝑋⋅𝑋 is called a generalized quasidistance on 𝑋 if and only if there exists a constant 𝑘≥1 such that for all 𝑥,𝑦∈𝑋 and for all distinct points 𝑧,𝑤∈𝑋, each of them different from 𝑥 and 𝑦, the following conditions hold:
(i)𝑑(𝑥,𝑦)=0⇔𝑥=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(ğ‘Ž,𝑟)={𝑥∈𝑋∶𝑑(𝑥,ğ‘Ž)<𝑟} is called “open” ball with center ğ‘Žâˆˆğ‘‹ and radius 𝑟>0.
The family 𝜏={ğ‘„âŠ‚ğ‘‹âˆ¶âˆ€ğ‘Žâˆˆğ‘„,∃𝑟>0,𝐵(ğ‘Ž,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 constant𝑘≥1.

Example 1.3 (see [14]). Let𝑋={1−(1/𝑛)∶𝑛=1,2,…}∪{1,2}. Define 𝑑∶𝑋⋅𝑋→𝑅 as follows: ⎧⎪⎪⎨⎪⎪⎩1𝑑(𝑥,𝑦)=0,for𝑥=𝑦,𝑛1,for𝑥∈{1,2},𝑦=1−𝑛1or𝑦∈{1,2},𝑥=1−𝑛,𝑥≠𝑦,3𝑘,for𝑥,𝑦∈{1,2},𝑥≠𝑦,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:𝑑𝑥𝑛,𝑥𝑚1=𝑑1−𝑛1,1−𝑚=1,∀𝑛,𝑚∈𝑁.(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=limğ‘›â†’âˆžğ‘‘(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,ğ‘šâ†’âˆžğ‘‘(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 𝑥∈𝑋 (denoted by limğ‘›â†’âˆžğ‘¥ğ‘›=𝑥) if limğ‘›â†’âˆžğ‘‘(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 𝑇∶𝑋→𝑋 is called contraction if there exists 0<𝑐<1/𝑘 such that𝑑(𝑇𝑥,𝑇𝑦)≤𝑐𝑑(𝑥,𝑦)∀𝑥,𝑦∈𝑋.(1.4)

Definition 1.9. Let 𝑇∶𝑋→𝑋 be a mapping where 𝑋 is a gqms. For each 𝑥∈𝑋, let 𝑂(𝑥)=𝑥,𝑇𝑥,𝑇2𝑥,…,(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 𝑓∶𝑅3+→𝑅 satisfying the following properties: (a)𝑓 is nondecreasing in respect to each variable, (b)𝑓(𝑡,𝑡,𝑡)≤𝑡, 𝑡∈𝑅+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)=(ğ‘Žğ‘¡1𝑡2+𝑏𝑡2𝑡3+𝑐𝑡3𝑡1)1/2, where ğ‘Ž,𝑏,𝑐≥0 and ğ‘Ž+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 (𝑥𝑛≠𝑥𝑚 for all 𝑛≠𝑚) in 𝑋. If 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝑐𝑛𝑙, 0≤𝑐<1/𝑘<1, for all 𝑛∈𝑁 and limğ‘›â†’âˆžğ‘‘(𝑥𝑛,𝑥𝑛+2)=0, then {𝑥𝑛} is a Cauchy sequence.

Proof. If 𝑚>2 is odd, then writing 𝑚=2𝑝+1, 𝑝≥1, by quasitetrahedral inequality, we can easily show that 𝑑𝑥𝑛,𝑥𝑛+𝑚𝑑𝑥≤𝑘𝑛,𝑥𝑛+1𝑥+𝑑𝑛+1,𝑥𝑛+2𝑥+𝑑𝑛+2,𝑥𝑛+𝑚𝑥≤𝑘𝑑𝑛,𝑥𝑛+1+𝑘2𝑑𝑥𝑛+1,𝑥𝑛+2+𝑘2𝑑𝑥𝑛+2,𝑥𝑛+𝑚≤𝑘𝑐𝑛𝑙+𝑘2𝑐𝑛+1𝑙+𝑘2𝑑𝑥𝑛+2,𝑥𝑛+𝑚≤⋯≤𝑘𝑐𝑛𝑙+𝑘2𝑐𝑛+1𝑙+𝑘3𝑐𝑛+2𝑙+⋯+𝑘𝑚−1𝑐𝑛+𝑚−2𝑙+𝑘𝑚−1𝑐𝑛+𝑚−1𝑙≤𝑘𝑐𝑛𝑙+𝑘2𝑐𝑛+1𝑙+𝑘3𝑐𝑛+2𝑙+⋯+𝑘𝑚−1𝑐𝑛+𝑚−2𝑙+𝑘𝑚𝑐𝑛+𝑚−1𝑙≤𝑘𝑐𝑛𝑙1+𝑘𝑐+⋯+(𝑘𝑐)𝑚−1=𝑘𝑐𝑛𝑙1−(𝑘𝑐)𝑚1−𝑘𝑐<𝑘𝑐𝑛𝑙1.1−𝑘𝑐(2.1) Therefore, limğ‘›â†’âˆžğ‘‘(𝑥𝑛,𝑥𝑛+𝑚)=0.
If 𝑚>2 is even, then writing 𝑚=2𝑝, 𝑝≥2 and using the same arguments as before we can get 𝑑𝑥𝑛,𝑥𝑛+𝑚𝑑𝑥≤𝑘𝑛,𝑥𝑛+2𝑥+𝑑𝑛+2,𝑥𝑛+3𝑥+𝑑𝑛+3,𝑥𝑛+𝑚𝑥≤𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑥𝑙+𝑘𝑑𝑛+3,𝑥𝑛+𝑚𝑥≤𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑙+𝑘2𝑑𝑥𝑛+3,𝑥𝑛+4𝑥+𝑑𝑛+4,𝑥𝑛+5𝑥+𝑑𝑛+5,𝑥𝑛+𝑚𝑥≤⋯≤𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑙+𝑘2𝑐𝑛+3𝑙+⋯+𝑘𝑚−2𝑐𝑛+𝑚−1𝑙𝑥=𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑙1+𝑘𝑐+⋯+(𝑘𝑐)𝑚−3𝑥=𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑙1−(𝑘𝑐)𝑚−2𝑥1−𝑘𝑐<𝑘𝑑𝑛,𝑥𝑛+2+𝑘𝑐𝑛+2𝑙1.1−𝑘𝑐(2.2) And so limğ‘›â†’âˆžğ‘‘(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 𝑥∈𝑋 and 𝑦∈𝑌, where 0<𝑐<1/𝑘≤1, 𝑘=max{𝑘1,𝑘2}, 𝑓1,𝑓2∈𝔽3. If there exists 𝑥0∈𝑋 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,𝑛≥1.(2.4) Denote 𝑑𝑛𝑥=𝑑𝑛,𝑥𝑛+1,𝜌𝑛𝑦=𝜌𝑛,𝑦𝑛+1,𝑛=1,2,….(2.5) Using the inequality (2) we get 𝜌𝑛𝑦=𝜌𝑛,𝑦𝑛+1=𝜌𝑇𝑥𝑛−1,𝑇𝑆𝑦𝑛≤𝑐𝑓2𝜌𝑦𝑛,𝑦𝑛𝑦,𝜌𝑛,𝑦𝑛+1𝑥,𝑑𝑛−1,𝑥𝑛=𝑐𝑓20,𝜌𝑛,𝑑𝑛−1.(2.6) By this inequality and properties of 𝑓2, it follows that 𝜌𝑛≤𝑐𝑑𝑛−1.(2.7) Using the inequality (2.3) we have 𝑑𝑛𝑥=𝑑𝑛,𝑥𝑛+1=𝑑𝑆𝑦𝑛,𝑆𝑇𝑥𝑛≤𝑐𝑓1𝑑𝑥𝑛,𝑥𝑛𝑥,𝑑𝑛,𝑥𝑛+1𝑦,𝜌𝑛,𝑦𝑛+1=𝑐𝑓10,𝑑𝑛,𝜌𝑛,(2.8) and so 𝑑𝑛≤𝑐𝜌𝑛. By this inequality and (2.7) we obtain 𝑑𝑛≤𝑐2𝑑𝑛−1≤𝑐𝑑𝑛−1.(2.9) Using the mathematical induction, by the inequalities (2.7) and (2.9), we get 𝑑𝑛≤𝑐𝑛𝑑𝑥0,𝑥1,𝜌𝑛≤𝑐𝑛𝑑𝑥0,𝑥1.(2.10) So limğ‘›â†’âˆžğ‘‘î€·ğ‘¥ğ‘›,𝑥𝑛+1=limğ‘›â†’âˆžğœŒî€·ğ‘¦ğ‘›,𝑦𝑛+1=0.(2.11) Applying the inequality (2), we get 𝜌𝑦𝑛,𝑦𝑛+2=𝜌𝑇𝑥𝑛−1,𝑇𝑆𝑦𝑛+1≤𝑐𝑓2𝜌𝑦𝑛+1,𝑦𝑛𝑦,𝜌𝑛+1,𝑦𝑛+2𝑥,𝑑𝑛−1,𝑥𝑛+1=𝑐𝑓2𝜌𝑛,𝜌𝑛+1𝑥,𝑑𝑛−1,𝑥𝑛+1𝑐≤𝑐max𝑛𝑑𝑥0,𝑥1𝑥,𝑑𝑛−1,𝑥𝑛+1,(2.12) and so 𝜌𝑦𝑛,𝑦𝑛+2𝑐≤max𝑛𝑑𝑥0,𝑥1𝑥,𝑐𝑑𝑛−1,𝑥𝑛+1.(2.13) Similarly, using (2.3), we obtain 𝑑𝑥𝑛,𝑥𝑛+2𝑐≤max𝑛𝑑𝑥0,𝑥1𝑥,𝑐𝑑𝑛−1,𝑥𝑛+1.(2.14) Using the mathematical induction, we get 𝑑𝑥𝑛,𝑥𝑛+2𝑐≤max𝑛𝑑𝑥0,𝑥1𝑥,𝑐𝑑𝑛−1,𝑥𝑛+1𝑐≤max𝑛𝑑𝑥0,𝑥1,𝑐2𝑑𝑥𝑛−2,𝑥𝑛𝑐≤⋯≤max𝑛𝑑𝑥0,𝑥1,𝑐𝑛𝑑𝑥0,𝑥2=𝑐𝑛𝑑𝑥max0,𝑥1𝑥,𝑑0,𝑥2=𝑐𝑛𝑙,(2.15) and so 𝑑𝑥𝑛,𝑥𝑛+2≤𝑐𝑛𝑦𝑙,similarly𝜌𝑛,𝑦𝑛+2≤𝑐𝑛𝑙,(2.16) where 𝑙=max{𝑑(𝑥0,𝑥1),𝑑(𝑥0,𝑥2)}.
We divide the proof into two cases.
Case 1. Suppose 𝑥𝑝=ğ‘¥ğ‘ž for some 𝑝,ğ‘žâˆˆğ‘, ğ‘â‰ ğ‘ž. Let 𝑝>ğ‘ž. Then (𝑆𝑇)𝑝𝑥0=(𝑆𝑇)ğ‘âˆ’ğ‘ž(𝑆𝑇)ğ‘žğ‘¥0=(𝑆𝑇)ğ‘žğ‘¥0; that is, (𝑆𝑇)𝑛𝛼=𝛼 where 𝑛=ğ‘âˆ’ğ‘ž and (𝑆𝑇)ğ‘žğ‘¥0=𝛼. Now if 𝑛>1, by (2.10), we have 𝑑(𝛼,𝑆𝑇𝛼)=𝑑(𝑆𝑇)𝑛𝛼,(𝑆𝑇)𝑛+1𝛼≤𝑐𝑛𝑑(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=ğ‘¥ğ‘ž it follows that 𝑦𝑝+1=ğ‘¦ğ‘ž+1. We take 𝛽=(𝑇𝑆)ğ‘žğ‘‡ğ‘¥0 and, in similar way, we prove that 𝛽 is a fixed point of 𝑇𝑆.
Case 2. Assume that 𝑥𝑛≠𝑥𝑚 for all 𝑛≠𝑚. 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 𝛼∈𝑋 such that limğ‘›â†’âˆžğ‘¥ğ‘›=𝛼. In the same way, we show that the sequence (𝑦𝑛) is a Cauchy sequence and there exists a 𝛽∈𝑌 such that limğ‘›â†’âˆžğ‘¦ğ‘›=𝛽.
We now prove that the limits 𝛼 and 𝛽 are unique. Suppose, to the contrary, that ğ›¼î…žâ‰ ğ›¼ is also limğ‘›â†’âˆžğ‘¥ğ‘›. Since 𝑥𝑛≠𝑥𝑚 for all 𝑛≠𝑚, there exists a subsequence (𝑥𝑛𝑘) of (𝑥𝑛) such that 𝑥𝑛𝑘≠𝛼 and ğ‘¥ğ‘›ğ‘˜â‰ ğ›¼î…ž for all 𝑘∈𝑁. Without loss of generality, assume that (𝑥𝑛) is this subsequence. Then by Tetrahedral property of Definition 1.1 we obtain𝑑𝛼,ğ›¼î…žî€¸î€ºğ‘‘î€·â‰¤ğ‘˜ğ›¼,𝑥𝑛𝑥+𝑑𝑛,𝑥𝑛+1𝑥+𝑑𝑛+1,ğ›¼î…žî€¸î€».(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 𝛽≠𝑇𝛼, the sequence (𝑦𝑛) does not converge to 𝑇𝛼 and there exists a subsequence (ğ‘¦ğ‘›ğ‘ž) of (𝑦𝑛) such that ğ‘¦ğ‘›ğ‘žâ‰ ğ‘‡ğ›¼ for all ğ‘žâˆˆğ‘. Then by Tetrahedral property of Definition 1.1 we obtain𝜌𝜌(𝛽,𝑇𝛼)≤𝑘𝛽,ğ‘¦ğ‘›ğ‘žâˆ’1𝑦+ğœŒğ‘›ğ‘žâˆ’1,ğ‘¦ğ‘›ğ‘žî‚î‚€ğ‘¦+ğœŒğ‘›ğ‘ž,𝑇𝛼.(2.19) Then if ğ‘žâ†’âˆž, we get 𝜌(𝛽,𝑇𝛼)≤𝑘___limğ‘žâ†’âˆžğœŒî‚€ğ‘¦ğ‘›ğ‘žî‚,𝑇𝛼.(2.20) Using the inequality (2), for 𝑥=𝛼 and 𝑦=𝑦𝑛−1 we obtain 𝜌𝑇𝛼,𝑦𝑛=𝜌𝑇𝛼,𝑇𝑆𝑦𝑛−1≤𝑐𝑓2𝜌𝑦𝑛−1𝑦,𝑇𝛼,𝜌𝑛−1,𝑇𝑆𝑦𝑛−1,𝑑𝛼,𝑆𝑦𝑛−1=𝑐𝑓2𝜌𝑦𝑛−1𝑦,𝑇𝛼,𝜌𝑛−1,𝑦𝑛,𝑑𝛼,𝑥𝑛−1.(2.21) Letting 𝑛 tend to infinity we get ___limğ‘›â†’âˆžğœŒî€·ğ‘‡ğ›¼,𝑦𝑛≤𝑐𝑓2___limğ‘›â†’âˆžğœŒî€·ğ‘¦ğ‘›âˆ’1,𝑇𝛼,0,0.(2.22) And so, ___limğ‘›â†’âˆžğœŒîƒ©ğ‘‡ğ›¼,𝑦𝑛=0.(2.23) Since ___limğ‘žâ†’âˆžğœŒ(ğ‘¦ğ‘›ğ‘ž,𝑇𝛼)≤___limğ‘›â†’âˆžğœŒ(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 ğ›¼î…žâ‰ ğ›¼ is also a fixed point of 𝑆𝑇. By (2.3) for 𝑥=ğ›¼î…ž and 𝑦=𝛽 we get 𝑑𝛼,ğ›¼î…žî€¸î€·=𝑑𝑆𝛽,ğ‘†ğ‘‡ğ›¼î…žî€¸â‰¤ğ‘ğ‘“1î€·ğ‘‘î€·ğ›¼î…žî€¸î€·,𝛼,0,𝜌𝑇𝛼,ğ‘‡ğ›¼î…žî€¸î€¸.(2.27) And so,we have 𝑑𝛼,ğ›¼î…žî€¸î€·â‰¤ğ‘ğœŒğ‘‡ğ›¼,ğ‘‡ğ›¼î…žî€¸.(2.28) If 𝑇𝛼≠𝑇𝛼′, in similar way by (2) for 𝑥=𝑆𝑇𝛼 and 𝑦=ğ‘‡ğ›¼î…ž, we have 𝜌𝑇𝛼,ğ‘‡ğ›¼î…žî€¸î€·â‰¤ğ‘ğ‘‘ğ›¼,ğ›¼î…žî€¸.(2.29) By (2.28) and (2.29) we get 𝑑(𝛼,ğ›¼î…ž)=0. Thus, we have again 𝛼=ğ›¼î…ž. 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)=(ğ‘Ž1𝑡1𝑡2+𝑏1𝑡2𝑡3+𝑐1𝑡3𝑡1)1/2 and 𝑓2(𝑡1,𝑡2,𝑡3)=(ğ‘Ž2𝑡1𝑡2+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.

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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.


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