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 [1ā€“6], and so forth and the triangular inequality š‘‘(š‘„,š‘¦)ā‰¤š‘‘(š‘„,š‘§)+š‘‘(š‘§,š‘¦) is replaced by quasi-triangular inequality:[]š‘‘(š‘„,š‘¦)ā‰¤š‘˜š‘‘(š‘„,š‘§)+š‘‘(š‘§,š‘¦),š‘˜ā‰„1.(1.1)

In 2000 Branciari [7] introduced the concept of generalized metric spaces (gms) (the triangular inequality š‘‘(š‘„,š‘¦)ā‰¤š‘‘(š‘„,š‘§)+š‘‘(š‘§,š‘¦) is replaced by tetrahedral inequalityš‘‘(š‘„,š‘¦)ā‰¤š‘‘(š‘„,š‘§)+š‘‘(š‘§,š‘¤)+š‘‘(š‘¤,š‘¦)). Starting with the paper of Branciari, some classical metric fixed point theorems have been transferred to gms (see [8ā€“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space, where the tetrahedral inequalityš‘‘(š‘„,š‘¦)ā‰¤š‘‘(š‘„,š‘§)+š‘‘(š‘§,š‘¤)+š‘‘(š‘¤,š‘¦) has been replaced by quasitetrahedral inequalityš‘‘(š‘„,š‘¦)ā‰¤š‘˜[š‘‘(š‘„,š‘§)+š‘‘(š‘§,š‘¤)+š‘‘(š‘¤,š‘¦)]. 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 š‘˜=1). Also, every qms is a gqms, while the converse is not true.

In gqms, contrary to a metric space, the ā€œopenā€ balls šµ(š‘Ž,š‘Ÿ)={š‘„āˆˆš‘‹āˆ¶š‘‘(š‘„,š‘Ž)<š‘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 (š‘„š‘›) 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 š‘‹ be a set and š‘‘āˆ¶š‘‹2ā†’š‘…+ a mapping such that for all š‘„,š‘¦āˆˆš‘‹ and for all distinct points š‘§,š‘¤āˆˆš‘‹, each of them different from š‘„ and š‘¦, one has (a)š‘‘(š‘„,š‘¦)=0 if and only if š‘„=š‘¦,(b)š‘‘(š‘„,š‘¦)=š‘‘(š‘¦,š‘„),(c)š‘‘(š‘„,š‘¦)ā‰¤š‘‘(š‘„,š‘§)+š‘‘(š‘§,š‘¤)+š‘‘(š‘¤,š‘¦) (tetrahedral inequality).Then š‘‘is called a generalized metric and (š‘‹,š‘‘) is a generalized metric space (or shortly gms).

Definition 1.2 (see [14]). Let š‘‹ be a set. A nonnegative symmetric function š‘‘defined on š‘‹ā‹…š‘‹ is called a generalized quasidistance on š‘‹ if and only if there exists a constant š‘˜ā‰„1 such that for all š‘„,š‘¦āˆˆš‘‹ and for all distinct points š‘§,š‘¤āˆˆš‘‹, each of them different from š‘„ and š‘¦, the following conditions hold:
(i)š‘‘(š‘„,š‘¦)=0ā‡”š‘„=š‘¦; (ii)š‘‘(š‘„,š‘¦)=š‘‘(š‘¦,š‘„); (iii)š‘‘(š‘„,š‘¦)ā‰¤š‘˜[š‘‘(š‘„,š‘§)+š‘‘(š‘§,š‘¤)+š‘‘(š‘¤,š‘¦)].
Inequality (2.7) is often called quasitetrahedral inequality and š‘˜ is often called the coefficient of š‘‘. A pair (š‘‹,š‘‘) is called a generalized quasimetric space if š‘‹ is a set and š‘‘ is a generalized quasidistance on š‘‹.
The set šµ(š‘Ž,š‘Ÿ)={š‘„āˆˆš‘‹āˆ¶š‘‘(š‘„,š‘Ž)<š‘Ÿ} is called ā€œopenā€ ball with center š‘Žāˆˆš‘‹ and radius š‘Ÿ>0.
The family šœ={š‘„āŠ‚š‘‹āˆ¶āˆ€š‘Žāˆˆš‘„,āˆƒš‘Ÿ>0,šµ(š‘Ž,š‘Ÿ)āŠ‚š‘„} is a topology on š‘‹ and it is called induced topology by the generalized quasidistance š‘‘.

The following example illustrates the existence of the generalized quasimetric space for an arbitrary constantš‘˜ā‰„1.

Example 1.3 (see [14]). Letš‘‹={1āˆ’(1/š‘›)āˆ¶š‘›=1,2,ā€¦}āˆŖ{1,2}. Define š‘‘āˆ¶š‘‹ā‹…š‘‹ā†’š‘… as follows: āŽ§āŽŖāŽŖāŽØāŽŖāŽŖāŽ©1š‘‘(š‘„,š‘¦)=0,forš‘„=š‘¦,š‘›1,forš‘„āˆˆ{1,2},š‘¦=1āˆ’š‘›1orš‘¦āˆˆ{1,2},š‘„=1āˆ’š‘›,š‘„ā‰ š‘¦,3š‘˜,forš‘„,š‘¦āˆˆ{1,2},š‘„ā‰ š‘¦,1,otherwise.(1.2) Then it is easy to see that (š‘‹,š‘‘) is a generalized quasimetric space and is not a generalized metric space (for š‘˜>1).

Note that the sequence (š‘„š‘›)=(1āˆ’(1/š‘›)) converges to both 1 and 2 and it is not a Cauchy sequence:š‘‘ī€·š‘„š‘›,š‘„š‘šī€øī‚€1=š‘‘1āˆ’š‘›1,1āˆ’š‘šī‚=1,āˆ€š‘›,š‘šāˆˆš‘.(1.3)

Since šµ(1,š‘Ÿ)āˆ©šµ(2,š‘Ÿ)ā‰ šœ™ for all š‘Ÿ>0, the (š‘‹,š‘‘) is non-Hausdorff generalized metric space.

The function š‘‘ is not continuous: 1=limš‘›ā†’āˆžš‘‘(1āˆ’(1/š‘›),1/2)ā‰ š‘‘(1,1/2)=1/2.

In [14] the following is proved.

Proposition 1.4. If (š‘‹,š‘‘) is a quasimetric space, then (š‘‹,š‘‘) is a generalized quasimetric space. The converse proposition does not hold true.

Definition 1.5. A sequence {š‘„š‘›} in a generalized quasimetric space (š‘‹,š‘‘) is called Cauchy sequence if limš‘›,š‘šā†’āˆžš‘‘(š‘„š‘›,š‘„š‘š)=0.

Definition 1.6. Let (š‘‹,š‘‘) be a generalized quasimetric space. Then one has the following.(1)A sequence {š‘„š‘›} in š‘‹ is said to be convergent to a point š‘„āˆˆš‘‹ (denoted by limš‘›ā†’āˆžš‘„š‘›=š‘„) if limš‘›ā†’āˆžš‘‘(š‘„š‘›,š‘„)=0.(2)It is called compact if every sequence contains a convergent subsequence.

Definition 1.7. A generalized quasimetric space (š‘‹,š‘‘) is called complete, if every Cauchy sequence is convergent.

Definition 1.8. Let (š‘‹,š‘‘) be a gqms and the coefficient of š‘‘ is š‘˜.
A map š‘‡āˆ¶š‘‹ā†’š‘‹ is called contraction if there exists 0<š‘<1/š‘˜ such thatš‘‘(š‘‡š‘„,š‘‡š‘¦)ā‰¤š‘š‘‘(š‘„,š‘¦)āˆ€š‘„,š‘¦āˆˆš‘‹.(1.4)

Definition 1.9. Let š‘‡āˆ¶š‘‹ā†’š‘‹ be a mapping where š‘‹ is a gqms. For each š‘„āˆˆš‘‹, let ī€½š‘‚(š‘„)=š‘„,š‘‡š‘„,š‘‡2ī€¾š‘„,ā€¦,(1.5) which will be called the orbit of š‘‡ at š‘„. The space š‘‹ is said to be š‘‡-orbitally complete if and only if every Cauchy sequence which is contained in š‘‚(š‘„) converges to a point in š‘‹.

Definition 1.10. The set of all upper semicontinuous functions with 3 variables š‘“āˆ¶š‘…3+ā†’š‘… satisfying the following properties: (a)š‘“ is nondecreasing in respect to each variable, (b)š‘“(š‘”,š‘”,š‘”)ā‰¤š‘”, š‘”āˆˆš‘…+will be noted by š”½3 and every such function will be called an š”½3-function.
Some examples of š”½3-function are as follows:
(1)š‘“(š‘”1,š‘”2,š‘”3)=max{š‘”1,š‘”2,š‘”3},(2)š‘“(š‘”1,š‘”2,š‘”3)=[max{š‘”1š‘”2,š‘”2š‘”3,š‘”3š‘”1}]1/2, (3)š‘“(š‘”1,š‘”2,š‘”3)=[max{š‘”š‘1,š‘”š‘2,š‘”š‘3}]1/š‘,š‘>0,(4)š‘“(š‘”1,š‘”2,š‘”3)=(š‘Žš‘”1š‘”2+š‘š‘”2š‘”3+š‘š‘”3š‘”1)1/2, where š‘Ž,š‘,š‘ā‰„0 and š‘Ž+š‘+š‘<1.

2. Main Result

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

Lemma 2.1. Let (š‘‹,š‘‘) be a generalized quasimetric space and {š‘„š‘›} is a sequence of distinct point (š‘„š‘›ā‰ š‘„š‘š for all š‘›ā‰ š‘š) in š‘‹. If š‘‘(š‘„š‘›,š‘„š‘›+1)ā‰¤š‘š‘›š‘™, 0ā‰¤š‘<1/š‘˜<1, for all š‘›āˆˆš‘ and limš‘›ā†’āˆžš‘‘(š‘„š‘›,š‘„š‘›+2)=0, then {š‘„š‘›} is a Cauchy sequence.

Proof. If š‘š>2 is odd, then writing š‘š=2š‘+1, š‘ā‰„1, by quasitetrahedral inequality, we can easily show that š‘‘ī€·š‘„š‘›,š‘„š‘›+š‘šī€øī€ŗš‘‘ī€·š‘„ā‰¤š‘˜š‘›,š‘„š‘›+1ī€øī€·š‘„+š‘‘š‘›+1,š‘„š‘›+2ī€øī€·š‘„+š‘‘š‘›+2,š‘„š‘›+š‘šī€·š‘„ī€øī€»ā‰¤š‘˜š‘‘š‘›,š‘„š‘›+1ī€ø+š‘˜2š‘‘ī€·š‘„š‘›+1,š‘„š‘›+2ī€ø+š‘˜2š‘‘ī€·š‘„š‘›+2,š‘„š‘›+š‘šī€øā‰¤š‘˜š‘š‘›š‘™+š‘˜2š‘š‘›+1š‘™+š‘˜2š‘‘ī€·š‘„š‘›+2,š‘„š‘›+š‘šī€øā‰¤ā‹Æā‰¤š‘˜š‘š‘›š‘™+š‘˜2š‘š‘›+1š‘™+š‘˜3š‘š‘›+2š‘™+ā‹Æ+š‘˜š‘šāˆ’1š‘š‘›+š‘šāˆ’2š‘™+š‘˜š‘šāˆ’1š‘š‘›+š‘šāˆ’1š‘™ā‰¤š‘˜š‘š‘›š‘™+š‘˜2š‘š‘›+1š‘™+š‘˜3š‘š‘›+2š‘™+ā‹Æ+š‘˜š‘šāˆ’1š‘š‘›+š‘šāˆ’2š‘™+š‘˜š‘šš‘š‘›+š‘šāˆ’1š‘™ā‰¤š‘˜š‘š‘›š‘™ī€ŗ1+š‘˜š‘+ā‹Æ+(š‘˜š‘)š‘šāˆ’1ī€»=š‘˜š‘š‘›š‘™1āˆ’(š‘˜š‘)š‘š1āˆ’š‘˜š‘<š‘˜š‘š‘›š‘™1.1āˆ’š‘˜š‘(2.1) Therefore, limš‘›ā†’āˆžš‘‘(š‘„š‘›,š‘„š‘›+š‘š)=0.
If š‘š>2 is even, then writing š‘š=2š‘, š‘ā‰„2 and using the same arguments as before we can get š‘‘ī€·š‘„š‘›,š‘„š‘›+š‘šī€øī€ŗš‘‘ī€·š‘„ā‰¤š‘˜š‘›,š‘„š‘›+2ī€øī€·š‘„+š‘‘š‘›+2,š‘„š‘›+3ī€øī€·š‘„+š‘‘š‘›+3,š‘„š‘›+š‘šī€·š‘„ī€øī€»ā‰¤š‘˜š‘‘š‘›,š‘„š‘›+2ī€ø+š‘˜š‘š‘›+2ī€·š‘„š‘™+š‘˜š‘‘š‘›+3,š‘„š‘›+š‘šī€øī€·š‘„ā‰¤š‘˜š‘‘š‘›,š‘„š‘›+2ī€ø+š‘˜š‘š‘›+2š‘™+š‘˜2ī€ŗš‘‘ī€·š‘„š‘›+3,š‘„š‘›+4ī€øī€·š‘„+š‘‘š‘›+4,š‘„š‘›+5ī€øī€·š‘„+š‘‘š‘›+5,š‘„š‘›+š‘šī€·š‘„ī€øī€»ā‰¤ā‹Æā‰¤š‘˜š‘‘š‘›,š‘„š‘›+2ī€ø+š‘˜š‘š‘›+2š‘™+š‘˜2š‘š‘›+3š‘™+ā‹Æ+š‘˜š‘šāˆ’2š‘š‘›+š‘šāˆ’1š‘™ī€·š‘„=š‘˜š‘‘š‘›,š‘„š‘›+2ī€ø+š‘˜š‘š‘›+2š‘™ī€ŗ1+š‘˜š‘+ā‹Æ+(š‘˜š‘)š‘šāˆ’3ī€»ī€·š‘„=š‘˜š‘‘š‘›,š‘„š‘›+2ī€ø+š‘˜š‘š‘›+2š‘™1āˆ’(š‘˜š‘)š‘šāˆ’2ī€·š‘„1āˆ’š‘˜š‘<š‘˜š‘‘š‘›,š‘„š‘›+2ī€ø+š‘˜š‘š‘›+2š‘™1.1āˆ’š‘˜š‘(2.2) And so limš‘›ā†’āˆžš‘‘(š‘„š‘›,š‘„š‘›+š‘š)=0. It implies that {š‘„š‘›} is a Cauchy sequence in š‘‹. This completes the proof of the lemma.

We state the following theorem.

Theorem 2.2. Let (š‘‹,š‘‘) and (š‘Œ,šœŒ) be two generalized quasimetric spaces with coefficients š‘˜1 and š‘˜2, respectively. Let š‘‡ be a mapping of š‘‹ into š‘Œ and š‘† a mapping of š‘Œ into š‘‹ satisfying the following inequalities: š‘‘(š‘†š‘¦,š‘†š‘‡š‘„)ā‰¤š‘š‘“1šœŒ{š‘‘(š‘„,š‘†š‘¦),š‘‘(š‘„,š‘†š‘‡š‘„),šœŒ(š‘¦,š‘‡š‘„)},(š‘‡š‘„,š‘‡š‘†š‘¦)ā‰¤š‘š‘“2{šœŒ(š‘¦,š‘‡š‘„),šœŒ(š‘¦,š‘‡š‘†š‘¦),š‘‘(š‘„,š‘†š‘¦)},(2.3) for all š‘„āˆˆš‘‹ and š‘¦āˆˆš‘Œ, where 0<š‘<1/š‘˜ā‰¤1, š‘˜=max{š‘˜1,š‘˜2}, š‘“1,š‘“2āˆˆš”½3. If there exists š‘„0āˆˆš‘‹ such that š‘‚(š‘„0) is š‘†š‘‡-orbitally complete in š‘‹ and š‘‚(š‘‡š‘„0) is š‘‡š‘†-orbitally complete in š‘Œ, then š‘†š‘‡ has a unique fixed point š›¼ in š‘‹ and š‘‡š‘† has a unique fixed point š›½ in š‘Œ. Further, š‘‡š›¼=š›½ and š‘†š›½=š›¼.

Proof. Let š‘„0 be an arbitrary point in š‘‹. Define the sequences (š‘„š‘›) and (š‘¦š‘›) inductively as follows: š‘„š‘›=š‘†š‘¦š‘›=(š‘†š‘‡)š‘›š‘„0,š‘¦1=š‘‡š‘„0,š‘¦š‘›+1=š‘‡š‘„š‘›=(š‘‡š‘†)š‘›š‘¦1,š‘›ā‰„1.(2.4) Denote š‘‘š‘›ī€·š‘„=š‘‘š‘›,š‘„š‘›+1ī€ø,šœŒš‘›ī€·š‘¦=šœŒš‘›,š‘¦š‘›+1ī€ø,š‘›=1,2,ā€¦.(2.5) Using the inequality (2) we get šœŒš‘›ī€·š‘¦=šœŒš‘›,š‘¦š‘›+1ī€øī€·=šœŒš‘‡š‘„š‘›āˆ’1,š‘‡š‘†š‘¦š‘›ī€øā‰¤š‘š‘“2ī€·šœŒī€·š‘¦š‘›,š‘¦š‘›ī€øī€·š‘¦,šœŒš‘›,š‘¦š‘›+1ī€øī€·š‘„,š‘‘š‘›āˆ’1,š‘„š‘›ī€øī€ø=š‘š‘“2ī€·0,šœŒš‘›,š‘‘š‘›āˆ’1ī€ø.(2.6) By this inequality and properties of š‘“2, it follows that šœŒš‘›ā‰¤š‘š‘‘š‘›āˆ’1.(2.7) Using the inequality (2.3) we have š‘‘š‘›ī€·š‘„=š‘‘š‘›,š‘„š‘›+1ī€øī€·=š‘‘š‘†š‘¦š‘›,š‘†š‘‡š‘„š‘›ī€øā‰¤š‘š‘“1ī€·š‘‘ī€·š‘„š‘›,š‘„š‘›ī€øī€·š‘„,š‘‘š‘›,š‘„š‘›+1ī€øī€·š‘¦,šœŒš‘›,š‘¦š‘›+1ī€øī€ø=š‘š‘“1ī€·0,š‘‘š‘›,šœŒš‘›ī€ø,(2.8) and so š‘‘š‘›ā‰¤š‘šœŒš‘›. By this inequality and (2.7) we obtain š‘‘š‘›ā‰¤š‘2š‘‘š‘›āˆ’1ā‰¤š‘š‘‘š‘›āˆ’1.(2.9) Using the mathematical induction, by the inequalities (2.7) and (2.9), we get š‘‘š‘›ā‰¤š‘š‘›š‘‘ī€·š‘„0,š‘„1ī€ø,šœŒš‘›ā‰¤š‘š‘›š‘‘ī€·š‘„0,š‘„1ī€ø.(2.10) So limš‘›ā†’āˆžš‘‘ī€·š‘„š‘›,š‘„š‘›+1ī€ø=limš‘›ā†’āˆžšœŒī€·š‘¦š‘›,š‘¦š‘›+1ī€ø=0.(2.11) Applying the inequality (2), we get šœŒī€·š‘¦š‘›,š‘¦š‘›+2ī€øī€·=šœŒš‘‡š‘„š‘›āˆ’1,š‘‡š‘†š‘¦š‘›+1ī€øā‰¤š‘š‘“2ī€·šœŒī€·š‘¦š‘›+1,š‘¦š‘›ī€øī€·š‘¦,šœŒš‘›+1,š‘¦š‘›+2ī€øī€·š‘„,š‘‘š‘›āˆ’1,š‘„š‘›+1ī€øī€ø=š‘š‘“2ī€·šœŒš‘›,šœŒš‘›+1ī€·š‘„,š‘‘š‘›āˆ’1,š‘„š‘›+1ī€½š‘ī€øī€øā‰¤š‘maxš‘›š‘‘ī€·š‘„0,š‘„1ī€øī€·š‘„,š‘‘š‘›āˆ’1,š‘„š‘›+1,ī€øī€¾(2.12) and so šœŒī€·š‘¦š‘›,š‘¦š‘›+2ī€øī€½š‘ā‰¤maxš‘›š‘‘ī€·š‘„0,š‘„1ī€øī€·š‘„,š‘š‘‘š‘›āˆ’1,š‘„š‘›+1ī€øī€¾.(2.13) Similarly, using (2.3), we obtain š‘‘ī€·š‘„š‘›,š‘„š‘›+2ī€øī€½š‘ā‰¤maxš‘›š‘‘ī€·š‘„0,š‘„1ī€øī€·š‘„,š‘š‘‘š‘›āˆ’1,š‘„š‘›+1ī€øī€¾.(2.14) Using the mathematical induction, we get š‘‘ī€·š‘„š‘›,š‘„š‘›+2ī€øī€½š‘ā‰¤maxš‘›š‘‘ī€·š‘„0,š‘„1ī€øī€·š‘„,š‘š‘‘š‘›āˆ’1,š‘„š‘›+1ī€½š‘ī€øī€¾ā‰¤maxš‘›š‘‘ī€·š‘„0,š‘„1ī€ø,š‘2š‘‘ī€·š‘„š‘›āˆ’2,š‘„š‘›ī€½š‘ī€øī€¾ā‰¤ā‹Æā‰¤maxš‘›š‘‘ī€·š‘„0,š‘„1ī€ø,š‘š‘›š‘‘ī€·š‘„0,š‘„2ī€øī€¾=š‘š‘›ī€½š‘‘ī€·š‘„max0,š‘„1ī€øī€·š‘„,š‘‘0,š‘„2ī€øī€¾=š‘š‘›š‘™,(2.15) and so š‘‘ī€·š‘„š‘›,š‘„š‘›+2ī€øā‰¤š‘š‘›ī€·š‘¦š‘™,similarlyšœŒš‘›,š‘¦š‘›+2ī€øā‰¤š‘š‘›š‘™,(2.16) where š‘™=max{š‘‘(š‘„0,š‘„1),š‘‘(š‘„0,š‘„2)}.
We divide the proof into two cases.
Case 1. Suppose š‘„š‘=š‘„š‘ž for some š‘,š‘žāˆˆš‘, š‘ā‰ š‘ž. Let š‘>š‘ž. Then (š‘†š‘‡)š‘š‘„0=(š‘†š‘‡)š‘āˆ’š‘ž(š‘†š‘‡)š‘žš‘„0=(š‘†š‘‡)š‘žš‘„0; that is, (š‘†š‘‡)š‘›š›¼=š›¼ where š‘›=š‘āˆ’š‘ž and (š‘†š‘‡)š‘žš‘„0=š›¼. Now if š‘›>1, by (2.10), we have ī€ŗš‘‘(š›¼,š‘†š‘‡š›¼)=š‘‘(š‘†š‘‡)š‘›š›¼,(š‘†š‘‡)š‘›+1š›¼ī€»ā‰¤š‘š‘›š‘‘(š›¼,š‘†š‘‡š›¼).(2.17) Since 0<š‘<1, š‘‘(š›¼,š‘†š‘‡š›¼)=0. So š‘†š‘‡š›¼=š›¼ and hence š›¼ is a fixed point of š‘†š‘‡.
By the equality š‘„š‘=š‘„š‘ž it follows that š‘¦š‘+1=š‘¦š‘ž+1. We take š›½=(š‘‡š‘†)š‘žš‘‡š‘„0 and, in similar way, we prove that š›½ is a fixed point of š‘‡š‘†.
Case 2. Assume that š‘„š‘›ā‰ š‘„š‘š for all š‘›ā‰ š‘š. Then, from (2.10), (2.16), and Lemma 2.1 is derived that {š‘„š‘›} is a Cauchy sequence in š‘‹. Since š‘‚(š‘„0) is ST-orbitally complete, there exists š›¼āˆˆš‘‹ such that limš‘›ā†’āˆžš‘„š‘›=š›¼. In the same way, we show that the sequence (š‘¦š‘›) is a Cauchy sequence and there exists a š›½āˆˆš‘Œ such that limš‘›ā†’āˆžš‘¦š‘›=š›½.
We now prove that the limits š›¼ and š›½ are unique. Suppose, to the contrary, that š›¼ī…žā‰ š›¼ is also limš‘›ā†’āˆžš‘„š‘›. Since š‘„š‘›ā‰ š‘„š‘š for all š‘›ā‰ š‘š, there exists a subsequence (š‘„š‘›š‘˜) of (š‘„š‘›) such that š‘„š‘›š‘˜ā‰ š›¼ and š‘„š‘›š‘˜ā‰ š›¼ī…ž for all š‘˜āˆˆš‘. Without loss of generality, assume that (š‘„š‘›) is this subsequence. Then by Tetrahedral property of Definition 1.1 we obtainš‘‘ī€·š›¼,š›¼ī…žī€øī€ŗš‘‘ī€·ā‰¤š‘˜š›¼,š‘„š‘›ī€øī€·š‘„+š‘‘š‘›,š‘„š‘›+1ī€øī€·š‘„+š‘‘š‘›+1,š›¼ī…žī€øī€».(2.18) Letting š‘› tend to infinity we get š‘‘(š›¼,š›¼ā€²)=0 and so š›¼=š›¼ā€², in the same way for š›½.
Let us prove now that š›¼ is a fixed point of š‘†š‘‡. First we prove that š›½=š‘‡š›¼. In contrary, if š›½ā‰ š‘‡š›¼, the sequence (š‘¦š‘›) does not converge to š‘‡š›¼ and there exists a subsequence (š‘¦š‘›š‘ž) of (š‘¦š‘›) such that š‘¦š‘›š‘žā‰ š‘‡š›¼ for all š‘žāˆˆš‘. Then by Tetrahedral property of Definition 1.1 we obtainī‚ƒšœŒī‚€šœŒ(š›½,š‘‡š›¼)ā‰¤š‘˜š›½,š‘¦š‘›š‘žāˆ’1ī‚ī‚€š‘¦+šœŒš‘›š‘žāˆ’1,š‘¦š‘›š‘žī‚ī‚€š‘¦+šœŒš‘›š‘ž,š‘‡š›¼ī‚ī‚„.(2.19) Then if š‘žā†’āˆž, we get šœŒ(š›½,š‘‡š›¼)ā‰¤š‘˜___limš‘žā†’āˆžšœŒī‚€š‘¦š‘›š‘žī‚,š‘‡š›¼.(2.20) Using the inequality (2), for š‘„=š›¼ and š‘¦=š‘¦š‘›āˆ’1 we obtain šœŒī€·š‘‡š›¼,š‘¦š‘›ī€øī€·=šœŒš‘‡š›¼,š‘‡š‘†š‘¦š‘›āˆ’1ī€øā‰¤š‘š‘“2ī€·šœŒī€·š‘¦š‘›āˆ’1ī€øī€·š‘¦,š‘‡š›¼,šœŒš‘›āˆ’1,š‘‡š‘†š‘¦š‘›āˆ’1ī€øī€·,š‘‘š›¼,š‘†š‘¦š‘›āˆ’1ī€øī€ø=š‘š‘“2ī€·šœŒī€·š‘¦š‘›āˆ’1ī€øī€·š‘¦,š‘‡š›¼,šœŒš‘›āˆ’1,š‘¦š‘›ī€øī€·,š‘‘š›¼,š‘„š‘›āˆ’1.ī€øī€ø(2.21) Letting š‘› tend to infinity we get ___limš‘›ā†’āˆžšœŒī€·š‘‡š›¼,š‘¦š‘›ī€øā‰¤š‘š‘“2ī‚µ___limš‘›ā†’āˆžšœŒī€·š‘¦š‘›āˆ’1ī€øī‚¶,š‘‡š›¼,0,0.(2.22) And so, ___limš‘›ā†’āˆžšœŒīƒ©š‘‡š›¼,š‘¦š‘›īƒŖ=0.(2.23) Since ___limš‘žā†’āˆžšœŒ(š‘¦š‘›š‘ž,š‘‡š›¼)ā‰¤___limš‘›ā†’āˆžšœŒ(š‘‡š›¼,š‘¦š‘›), by (2.23) and (2.20), we have šœŒ(š›½,š‘‡š›¼)=0 and so š‘‡š›¼=š›½.(2.24) It follows similarly that š‘†š›½=š›¼.(2.25) By (2.24) and (2.25) we obtain š‘†š‘‡š›¼=š‘†š›½=š›¼,š‘‡š‘†š›½=š‘‡š›¼=š›½.(2.26) Thus, we proved that the points š›¼ and š›½ are fixed points of š‘†š‘‡ and š‘‡š‘†, respectively.
Let us prove now the uniqueness (for Cases 1 and 2 in the same time). Assume that š›¼ī…žā‰ š›¼ is also a fixed point of š‘†š‘‡. By (2.3) for š‘„=š›¼ī…ž and š‘¦=š›½ we get š‘‘ī€·š›¼,š›¼ī…žī€øī€·=š‘‘š‘†š›½,š‘†š‘‡š›¼ī…žī€øā‰¤š‘š‘“1ī€·š‘‘ī€·š›¼ī…žī€øī€·,š›¼,0,šœŒš‘‡š›¼,š‘‡š›¼ī…žī€øī€ø.(2.27) And so,we have š‘‘ī€·š›¼,š›¼ī…žī€øī€·ā‰¤š‘šœŒš‘‡š›¼,š‘‡š›¼ī…žī€ø.(2.28) If š‘‡š›¼ā‰ š‘‡š›¼ā€², in similar way by (2) for š‘„=š‘†š‘‡š›¼ and š‘¦=š‘‡š›¼ī…ž, we have šœŒī€·š‘‡š›¼,š‘‡š›¼ī…žī€øī€·ā‰¤š‘š‘‘š›¼,š›¼ī…žī€ø.(2.29) By (2.28) and (2.29) we get š‘‘(š›¼,š›¼ī…ž)=0. Thus, we have again š›¼=š›¼ī…ž. The uniqueness of š›½ follows similarly. This completes the proof of the theorem.

3. Corollaries

(1) If š‘˜1=š‘˜2=1, then by Theorem 2.2 we obtain [12, Theoremā€‰ā€‰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 š‘“1 and š‘“2 in Theorem 2.2 we get different theorems.(2) For š‘“1=š‘“2=š‘“, where š‘“(š‘”1,š‘”2,š‘”3)=max{š‘”1,š‘”2,š‘”3} we have an extension of Fisherā€™s theorem [15] in generalized quasimetric spaces.(3) For š‘“1=š‘“2=š‘“, where š‘“(š‘”1,š‘”2,š‘”3)=[max{š‘”1š‘”2,š‘”2š‘”3,š‘”3š‘”1}]1/2, we have an extension of Popaā€™s theorem [13] in generalized quasimetric spaces.(4) For š‘“1(š‘”1,š‘”2,š‘”3)=(š‘Ž1š‘”1š‘”2+š‘1š‘”2š‘”3+š‘1š‘”3š‘”1)1/2 and š‘“2(š‘”1,š‘”2,š‘”3)=(š‘Ž2š‘”1š‘”2+š‘2š‘”2š‘”3+š‘2š‘”3š‘”1)1/2 we obtain an extension of Popaā€™s Corollary [13] in generalized quasimetric spaces.

Remark 3.1. We can obtain many other similar results for different š‘“.