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(π‘₯,𝑦)≀𝑑(π‘₯,𝑧)+𝑑(𝑧,𝑦) is replaced by quasi-triangular inequality:[]𝑑(π‘₯,𝑦)β‰€π‘˜π‘‘(π‘₯,𝑧)+𝑑(𝑧,𝑦),π‘˜β‰₯1.(1.1)

In 2000 Branciari [7] introduced the concept of generalized metric spaces (gms) (the triangular inequality 𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑧)+𝑑(𝑧,𝑦) is replaced by tetrahedral inequality𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑧)+𝑑(𝑧,𝑀)+𝑑(𝑀,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space (gqms) on the lines of quasimetric space, where the tetrahedral inequality𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑧)+𝑑(𝑧,𝑀)+𝑑(𝑀,𝑦) has been replaced by quasitetrahedral inequality𝑑(π‘₯,𝑦)β‰€π‘˜[𝑑(π‘₯,𝑧)+𝑑(𝑧,𝑀)+𝑑(𝑀,𝑦)].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 π‘˜=1). Also, every qms is a gqms, while the converse is not true.

In gqms, contrary to a metric space, the β€œopen” balls 𝐡(π‘Ž,π‘Ÿ)={π‘₯βˆˆπ‘‹βˆΆπ‘‘(π‘₯,π‘Ž)<π‘Ÿ} 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 (π‘₯𝑛) 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 π‘‘βˆΆπ‘‹2→𝑅+ a mapping such that for all π‘₯,π‘¦βˆˆπ‘‹ and for all distinct points 𝑧,π‘€βˆˆπ‘‹, each of them different from π‘₯ and 𝑦, one has (a)𝑑(π‘₯,𝑦)=0 if and only if π‘₯=𝑦,(b)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯),(c)𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑧)+𝑑(𝑧,𝑀)+𝑑(𝑀,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 π‘˜β‰₯1 such that for all π‘₯,π‘¦βˆˆπ‘‹ and for all distinct points 𝑧,π‘€βˆˆπ‘‹, each of them different from π‘₯ and 𝑦, the following conditions hold:
(i)𝑑(π‘₯,𝑦)=0⇔π‘₯=𝑦; (ii)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯); (iii)𝑑(π‘₯,𝑦)β‰€π‘˜[𝑑(π‘₯,𝑧)+𝑑(𝑧,𝑀)+𝑑(𝑀,𝑦)].
Inequality (2.7) is often called quasitetrahedral inequality and π‘˜ 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 𝐡(π‘Ž,π‘Ÿ)={π‘₯βˆˆπ‘‹βˆΆπ‘‘(π‘₯,π‘Ž)<π‘Ÿ} is called β€œopen” ball with center π‘Žβˆˆπ‘‹ and radius π‘Ÿ>0.
The family 𝜏={π‘„βŠ‚π‘‹βˆΆβˆ€π‘Žβˆˆπ‘„,βˆƒπ‘Ÿ>0,𝐡(π‘Ž,π‘Ÿ)βŠ‚π‘„} 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 constantπ‘˜β‰₯1.

Example 1.3 (see [14]). Let𝑋={1βˆ’(1/𝑛)βˆΆπ‘›=1,2,…}βˆͺ{1,2}. Define π‘‘βˆΆπ‘‹β‹…π‘‹β†’π‘… as follows: ⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩1𝑑(π‘₯,𝑦)=0,forπ‘₯=𝑦,𝑛1,forπ‘₯∈{1,2},𝑦=1βˆ’π‘›1orπ‘¦βˆˆ{1,2},π‘₯=1βˆ’π‘›,π‘₯≠𝑦,3π‘˜,forπ‘₯,π‘¦βˆˆ{1,2},π‘₯≠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 π‘˜>1).

Note that the sequence (π‘₯𝑛)=(1βˆ’(1/𝑛)) converges to both 1 and 2 and it is not a Cauchy sequence:𝑑π‘₯𝑛,π‘₯π‘šξ€Έξ‚€1=𝑑1βˆ’π‘›1,1βˆ’π‘šξ‚=1,βˆ€π‘›,π‘šβˆˆπ‘.(1.3)

Since 𝐡(1,π‘Ÿ)∩𝐡(2,π‘Ÿ)β‰ πœ™ for all π‘Ÿ>0, the (𝑋,𝑑) is non-Hausdorff generalized metric space.

The function 𝑑 is not continuous: 1=limπ‘›β†’βˆžπ‘‘(1βˆ’(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 {π‘₯𝑛} in a generalized quasimetric space (𝑋,𝑑) is called Cauchy sequence if lim𝑛,π‘šβ†’βˆžπ‘‘(π‘₯𝑛,π‘₯π‘š)=0.

Definition 1.6. Let (𝑋,𝑑) be a generalized quasimetric space. Then one has the following.(1)A sequence {π‘₯𝑛} in 𝑋 is said to be convergent to a point π‘₯βˆˆπ‘‹ (denoted by limπ‘›β†’βˆžπ‘₯𝑛=π‘₯) if limπ‘›β†’βˆžπ‘‘(π‘₯𝑛,π‘₯)=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 π‘˜.
A map π‘‡βˆΆπ‘‹β†’π‘‹ is called contraction if there exists 0<𝑐<1/π‘˜ such that𝑑(𝑇π‘₯,𝑇𝑦)≀𝑐𝑑(π‘₯,𝑦)βˆ€π‘₯,π‘¦βˆˆπ‘‹.(1.4)

Definition 1.9. Let π‘‡βˆΆπ‘‹β†’π‘‹ be a mapping where 𝑋 is a gqms. For each π‘₯βˆˆπ‘‹, let 𝑂(π‘₯)=π‘₯,𝑇π‘₯,𝑇2ξ€Ύπ‘₯,…,(1.5) which will be called the orbit of 𝑇 at π‘₯. The space 𝑋 is said to be 𝑇-orbitally complete if and only if every Cauchy sequence which is contained in 𝑂(π‘₯) converges to a point in 𝑋.

Definition 1.10. The set of all upper semicontinuous functions with 3 variables π‘“βˆΆπ‘…3+→𝑅 satisfying the following properties: (a)𝑓 is nondecreasing in respect to each variable, (b)𝑓(𝑑,𝑑,𝑑)≀𝑑, π‘‘βˆˆπ‘…+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(𝑑1,𝑑2,𝑑3)=max{𝑑1,𝑑2,𝑑3},(2)𝑓(𝑑1,𝑑2,𝑑3)=[max{𝑑1𝑑2,𝑑2𝑑3,𝑑3𝑑1}]1/2, (3)𝑓(𝑑1,𝑑2,𝑑3)=[max{𝑑𝑝1,𝑑𝑝2,𝑑𝑝3}]1/𝑝,𝑝>0,(4)𝑓(𝑑1,𝑑2,𝑑3)=(π‘Žπ‘‘1𝑑2+𝑏𝑑2𝑑3+𝑐𝑑3𝑑1)1/2, where π‘Ž,𝑏,𝑐β‰₯0 and π‘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 {π‘₯𝑛} is a sequence of distinct point (π‘₯𝑛≠π‘₯π‘š for all π‘›β‰ π‘š) in 𝑋. If 𝑑(π‘₯𝑛,π‘₯𝑛+1)≀𝑐𝑛𝑙, 0≀𝑐<1/π‘˜<1, for all π‘›βˆˆπ‘ and limπ‘›β†’βˆžπ‘‘(π‘₯𝑛,π‘₯𝑛+2)=0, then {π‘₯𝑛} is a Cauchy sequence.

Proof. If π‘š>2 is odd, then writing π‘š=2𝑝+1, 𝑝β‰₯1, by quasitetrahedral inequality, we can easily show that 𝑑π‘₯𝑛,π‘₯𝑛+π‘šξ€Έξ€Ίπ‘‘ξ€·π‘₯β‰€π‘˜π‘›,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€·π‘₯+𝑑𝑛+2,π‘₯𝑛+π‘šξ€·π‘₯ξ€Έξ€»β‰€π‘˜π‘‘π‘›,π‘₯𝑛+1ξ€Έ+π‘˜2𝑑π‘₯𝑛+1,π‘₯𝑛+2ξ€Έ+π‘˜2𝑑π‘₯𝑛+2,π‘₯𝑛+π‘šξ€Έβ‰€π‘˜π‘π‘›π‘™+π‘˜2𝑐𝑛+1𝑙+π‘˜2𝑑π‘₯𝑛+2,π‘₯𝑛+π‘šξ€Έβ‰€β‹―β‰€π‘˜π‘π‘›π‘™+π‘˜2𝑐𝑛+1𝑙+π‘˜3𝑐𝑛+2𝑙+β‹―+π‘˜π‘šβˆ’1𝑐𝑛+π‘šβˆ’2𝑙+π‘˜π‘šβˆ’1𝑐𝑛+π‘šβˆ’1π‘™β‰€π‘˜π‘π‘›π‘™+π‘˜2𝑐𝑛+1𝑙+π‘˜3𝑐𝑛+2𝑙+β‹―+π‘˜π‘šβˆ’1𝑐𝑛+π‘šβˆ’2𝑙+π‘˜π‘šπ‘π‘›+π‘šβˆ’1π‘™β‰€π‘˜π‘π‘›π‘™ξ€Ί1+π‘˜π‘+β‹―+(π‘˜π‘)π‘šβˆ’1ξ€»=π‘˜π‘π‘›π‘™1βˆ’(π‘˜π‘)π‘š1βˆ’π‘˜π‘<π‘˜π‘π‘›π‘™1.1βˆ’π‘˜π‘(2.1) Therefore, limπ‘›β†’βˆžπ‘‘(π‘₯𝑛,π‘₯𝑛+π‘š)=0.
If π‘š>2 is even, then writing π‘š=2𝑝, 𝑝β‰₯2 and using the same arguments as before we can get 𝑑π‘₯𝑛,π‘₯𝑛+π‘šξ€Έξ€Ίπ‘‘ξ€·π‘₯β‰€π‘˜π‘›,π‘₯𝑛+2ξ€Έξ€·π‘₯+𝑑𝑛+2,π‘₯𝑛+3ξ€Έξ€·π‘₯+𝑑𝑛+3,π‘₯𝑛+π‘šξ€·π‘₯ξ€Έξ€»β‰€π‘˜π‘‘π‘›,π‘₯𝑛+2ξ€Έ+π‘˜π‘π‘›+2ξ€·π‘₯𝑙+π‘˜π‘‘π‘›+3,π‘₯𝑛+π‘šξ€Έξ€·π‘₯β‰€π‘˜π‘‘π‘›,π‘₯𝑛+2ξ€Έ+π‘˜π‘π‘›+2𝑙+π‘˜2𝑑π‘₯𝑛+3,π‘₯𝑛+4ξ€Έξ€·π‘₯+𝑑𝑛+4,π‘₯𝑛+5ξ€Έξ€·π‘₯+𝑑𝑛+5,π‘₯𝑛+π‘šξ€·π‘₯ξ€Έξ€»β‰€β‹―β‰€π‘˜π‘‘π‘›,π‘₯𝑛+2ξ€Έ+π‘˜π‘π‘›+2𝑙+π‘˜2𝑐𝑛+3𝑙+β‹―+π‘˜π‘šβˆ’2𝑐𝑛+π‘šβˆ’1𝑙π‘₯=π‘˜π‘‘π‘›,π‘₯𝑛+2ξ€Έ+π‘˜π‘π‘›+2𝑙1+π‘˜π‘+β‹―+(π‘˜π‘)π‘šβˆ’3ξ€»ξ€·π‘₯=π‘˜π‘‘π‘›,π‘₯𝑛+2ξ€Έ+π‘˜π‘π‘›+2𝑙1βˆ’(π‘˜π‘)π‘šβˆ’2ξ€·π‘₯1βˆ’π‘˜π‘<π‘˜π‘‘π‘›,π‘₯𝑛+2ξ€Έ+π‘˜π‘π‘›+2𝑙1.1βˆ’π‘˜π‘(2.2) And so limπ‘›β†’βˆžπ‘‘(π‘₯𝑛,π‘₯𝑛+π‘š)=0. It implies that {π‘₯𝑛} 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 (π‘Œ,𝜌) be two generalized quasimetric spaces with coefficients π‘˜1 and π‘˜2, respectively. Let 𝑇 be a mapping of 𝑋 into π‘Œ and 𝑆 a mapping of π‘Œ into 𝑋 satisfying the following inequalities: 𝑑(𝑆𝑦,𝑆𝑇π‘₯)≀𝑐𝑓1𝜌{𝑑(π‘₯,𝑆𝑦),𝑑(π‘₯,𝑆𝑇π‘₯),𝜌(𝑦,𝑇π‘₯)},(𝑇π‘₯,𝑇𝑆𝑦)≀𝑐𝑓2{𝜌(𝑦,𝑇π‘₯),𝜌(𝑦,𝑇𝑆𝑦),𝑑(π‘₯,𝑆𝑦)},(2.3) for all π‘₯βˆˆπ‘‹ and π‘¦βˆˆπ‘Œ, where 0<𝑐<1/π‘˜β‰€1, π‘˜=max{π‘˜1,π‘˜2}, 𝑓1,𝑓2βˆˆπ”½3. If there exists π‘₯0βˆˆπ‘‹ such that 𝑂(π‘₯0) is 𝑆𝑇-orbitally complete in 𝑋 and 𝑂(𝑇π‘₯0) is 𝑇𝑆-orbitally complete in π‘Œ, then 𝑆𝑇 has a unique fixed point 𝛼 in 𝑋 and 𝑇𝑆 has a unique fixed point 𝛽 in π‘Œ. Further, 𝑇𝛼=𝛽 and 𝑆𝛽=𝛼.

Proof. Let π‘₯0 be an arbitrary point in 𝑋. Define the sequences (π‘₯𝑛) and (𝑦𝑛) inductively as follows: π‘₯𝑛=𝑆𝑦𝑛=(𝑆𝑇)𝑛π‘₯0,𝑦1=𝑇π‘₯0,𝑦𝑛+1=𝑇π‘₯𝑛=(𝑇𝑆)𝑛𝑦1,𝑛β‰₯1.(2.4) Denote 𝑑𝑛π‘₯=𝑑𝑛,π‘₯𝑛+1ξ€Έ,πœŒπ‘›ξ€·π‘¦=πœŒπ‘›,𝑦𝑛+1ξ€Έ,𝑛=1,2,….(2.5) Using the inequality (2) we get πœŒπ‘›ξ€·π‘¦=πœŒπ‘›,𝑦𝑛+1ξ€Έξ€·=πœŒπ‘‡π‘₯π‘›βˆ’1,𝑇𝑆𝑦𝑛≀𝑐𝑓2ξ€·πœŒξ€·π‘¦π‘›,𝑦𝑛𝑦,πœŒπ‘›,𝑦𝑛+1ξ€Έξ€·π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛=𝑐𝑓2ξ€·0,πœŒπ‘›,π‘‘π‘›βˆ’1ξ€Έ.(2.6) By this inequality and properties of 𝑓2, it follows that πœŒπ‘›β‰€π‘π‘‘π‘›βˆ’1.(2.7) Using the inequality (2.3) we have 𝑑𝑛π‘₯=𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·=𝑑𝑆𝑦𝑛,𝑆𝑇π‘₯𝑛≀𝑐𝑓1𝑑π‘₯𝑛,π‘₯𝑛π‘₯,𝑑𝑛,π‘₯𝑛+1𝑦,πœŒπ‘›,𝑦𝑛+1ξ€Έξ€Έ=𝑐𝑓1ξ€·0,𝑑𝑛,πœŒπ‘›ξ€Έ,(2.8) and so π‘‘π‘›β‰€π‘πœŒπ‘›. By this inequality and (2.7) we obtain 𝑑𝑛≀𝑐2π‘‘π‘›βˆ’1β‰€π‘π‘‘π‘›βˆ’1.(2.9) Using the mathematical induction, by the inequalities (2.7) and (2.9), we get 𝑑𝑛≀𝑐𝑛𝑑π‘₯0,π‘₯1ξ€Έ,πœŒπ‘›β‰€π‘π‘›π‘‘ξ€·π‘₯0,π‘₯1ξ€Έ.(2.10) So limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑛,π‘₯𝑛+1ξ€Έ=limπ‘›β†’βˆžπœŒξ€·π‘¦π‘›,𝑦𝑛+1ξ€Έ=0.(2.11) Applying the inequality (2), we get πœŒξ€·π‘¦π‘›,𝑦𝑛+2ξ€Έξ€·=πœŒπ‘‡π‘₯π‘›βˆ’1,𝑇𝑆𝑦𝑛+1≀𝑐𝑓2ξ€·πœŒξ€·π‘¦π‘›+1,𝑦𝑛𝑦,πœŒπ‘›+1,𝑦𝑛+2ξ€Έξ€·π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛+1ξ€Έξ€Έ=𝑐𝑓2ξ€·πœŒπ‘›,πœŒπ‘›+1ξ€·π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛+1𝑐≀𝑐max𝑛𝑑π‘₯0,π‘₯1ξ€Έξ€·π‘₯,π‘‘π‘›βˆ’1,π‘₯𝑛+1,ξ€Έξ€Ύ(2.12) and so πœŒξ€·π‘¦π‘›,𝑦𝑛+2𝑐≀max𝑛𝑑π‘₯0,π‘₯1ξ€Έξ€·π‘₯,π‘π‘‘π‘›βˆ’1,π‘₯𝑛+1ξ€Έξ€Ύ.(2.13) Similarly, using (2.3), we obtain 𝑑π‘₯𝑛,π‘₯𝑛+2𝑐≀max𝑛𝑑π‘₯0,π‘₯1ξ€Έξ€·π‘₯,π‘π‘‘π‘›βˆ’1,π‘₯𝑛+1ξ€Έξ€Ύ.(2.14) Using the mathematical induction, we get 𝑑π‘₯𝑛,π‘₯𝑛+2𝑐≀max𝑛𝑑π‘₯0,π‘₯1ξ€Έξ€·π‘₯,π‘π‘‘π‘›βˆ’1,π‘₯𝑛+1𝑐≀max𝑛𝑑π‘₯0,π‘₯1ξ€Έ,𝑐2𝑑π‘₯π‘›βˆ’2,π‘₯𝑛𝑐≀⋯≀max𝑛𝑑π‘₯0,π‘₯1ξ€Έ,𝑐𝑛𝑑π‘₯0,π‘₯2ξ€Έξ€Ύ=𝑐𝑛𝑑π‘₯max0,π‘₯1ξ€Έξ€·π‘₯,𝑑0,π‘₯2ξ€Έξ€Ύ=𝑐𝑛𝑙,(2.15) and so 𝑑π‘₯𝑛,π‘₯𝑛+2≀𝑐𝑛𝑦𝑙,similarlyπœŒπ‘›,𝑦𝑛+2≀𝑐𝑛𝑙,(2.16) where 𝑙=max{𝑑(π‘₯0,π‘₯1),𝑑(π‘₯0,π‘₯2)}.
We divide the proof into two cases.
Case 1. Suppose π‘₯𝑝=π‘₯π‘ž for some 𝑝,π‘žβˆˆπ‘, π‘β‰ π‘ž. Let 𝑝>π‘ž. Then (𝑆𝑇)𝑝π‘₯0=(𝑆𝑇)π‘βˆ’π‘ž(𝑆𝑇)π‘žπ‘₯0=(𝑆𝑇)π‘žπ‘₯0; that is, (𝑆𝑇)𝑛𝛼=𝛼 where 𝑛=π‘βˆ’π‘ž and (𝑆𝑇)π‘žπ‘₯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 π‘₯𝑝=π‘₯π‘ž it follows that 𝑦𝑝+1=π‘¦π‘ž+1. We take 𝛽=(𝑇𝑆)π‘žπ‘‡π‘₯0 and, in similar way, we prove that 𝛽 is a fixed point of 𝑇𝑆.
Case 2. Assume that π‘₯𝑛≠π‘₯π‘š for all π‘›β‰ π‘š. Then, from (2.10), (2.16), and Lemma 2.1 is derived that {π‘₯𝑛} is a Cauchy sequence in 𝑋. Since 𝑂(π‘₯0) is ST-orbitally complete, there exists π›Όβˆˆπ‘‹ such that limπ‘›β†’βˆžπ‘₯𝑛=𝛼. In the same way, we show that the sequence (𝑦𝑛) is a Cauchy sequence and there exists a π›½βˆˆπ‘Œ such that limπ‘›β†’βˆžπ‘¦π‘›=𝛽.
We now prove that the limits 𝛼 and 𝛽 are unique. Suppose, to the contrary, that π›Όξ…žβ‰ π›Ό is also limπ‘›β†’βˆžπ‘₯𝑛. Since π‘₯𝑛≠π‘₯π‘š for all π‘›β‰ π‘š, there exists a subsequence (π‘₯π‘›π‘˜) of (π‘₯𝑛) such that π‘₯π‘›π‘˜β‰ π›Ό and π‘₯π‘›π‘˜β‰ π›Όξ…ž for all π‘˜βˆˆπ‘. Without loss of generality, assume that (π‘₯𝑛) is this subsequence. Then by Tetrahedral property of Definition 1.1 we obtain𝑑𝛼,π›Όξ…žξ€Έξ€Ίπ‘‘ξ€·β‰€π‘˜π›Ό,π‘₯𝑛π‘₯+𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝑑𝑛+1,π›Όξ…žξ€Έξ€».(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 (π‘¦π‘›π‘ž) of (𝑦𝑛) such that π‘¦π‘›π‘žβ‰ π‘‡π›Ό for all π‘žβˆˆπ‘. Then by Tetrahedral property of Definition 1.1 we obtainξ‚ƒπœŒξ‚€πœŒ(𝛽,𝑇𝛼)β‰€π‘˜π›½,π‘¦π‘›π‘žβˆ’1𝑦+πœŒπ‘›π‘žβˆ’1,π‘¦π‘›π‘žξ‚ξ‚€π‘¦+πœŒπ‘›π‘ž,𝑇𝛼.(2.19) Then if π‘žβ†’βˆž, we get 𝜌(𝛽,𝑇𝛼)β‰€π‘˜___limπ‘žβ†’βˆžπœŒξ‚€π‘¦π‘›π‘žξ‚,𝑇𝛼.(2.20) Using the inequality (2), for π‘₯=𝛼 and 𝑦=π‘¦π‘›βˆ’1 we obtain πœŒξ€·π‘‡π›Ό,𝑦𝑛=πœŒπ‘‡π›Ό,π‘‡π‘†π‘¦π‘›βˆ’1≀𝑐𝑓2ξ€·πœŒξ€·π‘¦π‘›βˆ’1𝑦,𝑇𝛼,πœŒπ‘›βˆ’1,π‘‡π‘†π‘¦π‘›βˆ’1ξ€Έξ€·,𝑑𝛼,π‘†π‘¦π‘›βˆ’1ξ€Έξ€Έ=𝑐𝑓2ξ€·πœŒξ€·π‘¦π‘›βˆ’1𝑦,𝑇𝛼,πœŒπ‘›βˆ’1,𝑦𝑛,𝑑𝛼,π‘₯π‘›βˆ’1.ξ€Έξ€Έ(2.21) Letting 𝑛 tend to infinity we get ___limπ‘›β†’βˆžπœŒξ€·π‘‡π›Ό,𝑦𝑛≀𝑐𝑓2ξ‚΅___limπ‘›β†’βˆžπœŒξ€·π‘¦π‘›βˆ’1ξ€Έξ‚Ά,𝑇𝛼,0,0.(2.22) And so, ___limπ‘›β†’βˆžπœŒξƒ©π‘‡π›Ό,𝑦𝑛ξƒͺ=0.(2.23) Since ___limπ‘žβ†’βˆžπœŒ(π‘¦π‘›π‘ž,𝑇𝛼)≀___limπ‘›β†’βˆžπ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 π›Όξ…žβ‰ π›Ό is also a fixed point of 𝑆𝑇. By (2.3) for π‘₯=π›Όξ…ž and 𝑦=𝛽 we get 𝑑𝛼,π›Όξ…žξ€Έξ€·=𝑑𝑆𝛽,π‘†π‘‡π›Όξ…žξ€Έβ‰€π‘π‘“1ξ€·π‘‘ξ€·π›Όξ…žξ€Έξ€·,𝛼,0,πœŒπ‘‡π›Ό,π‘‡π›Όξ…žξ€Έξ€Έ.(2.27) And so,we have 𝑑𝛼,π›Όξ…žξ€Έξ€·β‰€π‘πœŒπ‘‡π›Ό,π‘‡π›Όξ…žξ€Έ.(2.28) If 𝑇𝛼≠𝑇𝛼′, in similar way by (2) for π‘₯=𝑆𝑇𝛼 and 𝑦=π‘‡π›Όξ…ž, we have πœŒξ€·π‘‡π›Ό,π‘‡π›Όξ…žξ€Έξ€·β‰€π‘π‘‘π›Ό,π›Όξ…žξ€Έ.(2.29) By (2.28) and (2.29) we get 𝑑(𝛼,π›Όξ…ž)=0. Thus, we have again 𝛼=π›Όξ…ž. The uniqueness of 𝛽 follows similarly. This completes the proof of the theorem.

3. Corollaries

(1) If π‘˜1=π‘˜2=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(𝑑1,𝑑2,𝑑3)=max{𝑑1,𝑑2,𝑑3} we have an extension of Fisher’s theorem [15] in generalized quasimetric spaces.(3) For 𝑓1=𝑓2=𝑓, where 𝑓(𝑑1,𝑑2,𝑑3)=[max{𝑑1𝑑2,𝑑2𝑑3,𝑑3𝑑1}]1/2, we have an extension of Popa’s theorem [13] in generalized quasimetric spaces.(4) For 𝑓1(𝑑1,𝑑2,𝑑3)=(π‘Ž1𝑑1𝑑2+𝑏1𝑑2𝑑3+𝑐1𝑑3𝑑1)1/2 and 𝑓2(𝑑1,𝑑2,𝑑3)=(π‘Ž2𝑑1𝑑2+𝑏2𝑑2𝑑3+𝑐2𝑑3𝑑1)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.