A New Fixed Point Theorem on Generalized Quasimetric Spaces
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 is replaced by quasi-triangular inequality:
In 2000 Branciari  introduced the concept of generalized metric spaces (gms) (the triangular inequality is replaced by tetrahedral inequality). 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  introduced the concept of generalized quasimetric space (gqms) on the lines of quasimetric space, where the tetrahedral inequality has been replaced by quasitetrahedral inequality. 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 ). Also, every qms is a gqms, while the converse is not true.
In gqms, contrary to a metric space, the “open” balls 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 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 ). Let be a set and a mapping such that for all and for all distinct points , each of them different from and , one has (a) if and only if ,(b),(c) (tetrahedral inequality).Then is called a generalized metric and is a generalized metric space (or shortly gms).
Definition 1.2 (see ). Let be a set. A nonnegative symmetric function defined on is called a generalized quasidistance on if and only if there exists a constant such that for all and for all distinct points , each of them different from and , the following conditions hold:
(i); (ii); (iii).
Inequality (2.7) is often called quasitetrahedral inequality and is often called the coefficient of . A pair is called a generalized quasimetric space if is a set and is a generalized quasidistance on .
The set is called “open” ball with center and radius .
The family is a topology on and it is called induced topology by the generalized quasidistance .
The following example illustrates the existence of the generalized quasimetric space for an arbitrary constant.
Example 1.3 (see ). Let. Define as follows: Then it is easy to see that is a generalized quasimetric space and is not a generalized metric space (for ).
Note that the sequence converges to both 1 and 2 and it is not a Cauchy sequence:
Since for all , the is non-Hausdorff generalized metric space.
The function is not continuous: .
In  the following is proved.
Proposition 1.4. If is a quasimetric space, then is a generalized quasimetric space. The converse proposition does not hold true.
Definition 1.5. A sequence in a generalized quasimetric space is called Cauchy sequence if .
Definition 1.6. Let be a generalized quasimetric space. Then one has the following.(1)A sequence in is said to be convergent to a point (denoted by ) if .(2)It is called compact if every sequence contains a convergent subsequence.
Definition 1.7. A generalized quasimetric space is called complete, if every Cauchy sequence is convergent.
Definition 1.8. Let be a gqms and the coefficient of is .
A map is called contraction if there exists such that
Definition 1.9. Let be a mapping where is a gqms. For each , let which will be called the orbit of at . The space is said to be -orbitally complete if and only if every Cauchy sequence which is contained in converges to a point in .
Definition 1.10. The set of all upper semicontinuous functions with 3 variables satisfying the following properties: (a) is nondecreasing in respect to each variable, (b), will be noted by and every such function will be called an -function.
Some examples of -function are as follows:
(1),(2), (3),(4), where and .
2. Main Result
We state the following lemma which we will use for the proof of the main theorem.
Lemma 2.1. Let be a generalized quasimetric space and is a sequence of distinct point for all in . If , , for all and , then is a Cauchy sequence.
Proof. If is odd, then writing , , by quasitetrahedral inequality, we can easily show that
If is even, then writing , and using the same arguments as before we can get And so . It implies that is a Cauchy sequence in . This completes the proof of the lemma.
We state the following theorem.
Theorem 2.2. Let and be two generalized quasimetric spaces with coefficients and , respectively. Let be a mapping of into and a mapping of into satisfying the following inequalities: for all and , where , , . If there exists such that is -orbitally complete in and is -orbitally complete in , then has a unique fixed point in and has a unique fixed point in . Further, and .
Proof. Let be an arbitrary point in . Define the sequences and inductively as follows:
Using the inequality (2) we get
By this inequality and properties of , it follows that
Using the inequality (2.3) we have
and so . By this inequality and (2.7) we obtain
Using the mathematical induction, by the inequalities (2.7) and (2.9), we get
Applying the inequality (2), we get
Similarly, using (2.3), we obtain
Using the mathematical induction, we get
We divide the proof into two cases.
Case 1. Suppose for some , . Let . Then ; that is, where and . Now if , by (2.10), we have Since , . So and hence is a fixed point of .
By the equality it follows that . We take and, in similar way, we prove that is a fixed point of .Case 2. Assume that for all . Then, from (2.10), (2.16), and Lemma 2.1 is derived that is a Cauchy sequence in . Since is ST-orbitally complete, there exists such that . In the same way, we show that the sequence is a Cauchy sequence and there exists a such that .
We now prove that the limits and are unique. Suppose, to the contrary, that is also . Since for all , there exists a subsequence of such that and for all . Without loss of generality, assume that is this subsequence. Then by Tetrahedral property of Definition 1.1 we obtain Letting tend to infinity we get and so , in the same way for .
Let us prove now that is a fixed point of . First we prove that . In contrary, if , the sequence does not converge to and there exists a subsequence of such that for all . Then by Tetrahedral property of Definition 1.1 we obtain Then if , we get Using the inequality (2), for and we obtain Letting tend to infinity we get And so, Since , by (2.23) and (2.20), we have and so It follows similarly that By (2.24) and (2.25) we obtain Thus, we proved that the points and are fixed points of and , respectively.
Let us prove now the uniqueness (for Cases 1 and 2 in the same time). Assume that is also a fixed point of . By (2.3) for and we get And so,we have If , in similar way by (2) for and , we have By (2.28) and (2.29) we get . Thus, we have again . The uniqueness of follows similarly. This completes the proof of the theorem.
3. CorollariesIf , then by Theorem 2.2 we obtain [12, Theorem 2.1], that generalize and extend the well-known Fisher fixed point theorem  from metric space to generalized metric spaces.For different expressions of and in Theorem 2.2 we get different theorems. For , where we have an extension of Fisher’s theorem  in generalized quasimetric spaces. For , where , we have an extension of Popa’s theorem  in generalized quasimetric spaces. For and we obtain an extension of Popa’s Corollary  in generalized quasimetric spaces.
Remark 3.1. We can obtain many other similar results for different .
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