Abstract
We introduce the concept of subordinate semimetric space. Such notion includes the concept of RS-space introduced by Roldán and Shahzad; therefore the concepts of Branciari’s generalized metric space and Jleli and Samet’s generalized metric space are particular cases. For such spaces we prove a version of Matkowski’s fixed point theorem, and introducing the concept of -contraction we get a fixed point theorem of Kannan-Ćirić type. Moreover, using such result we characterize complete subordinate semimetric spaces.
1. Introduction
Fixed point theory has an enormous number of applications in ordinary (or partial) differential equations, game theory, functional analysis, calculus of several variables (the classical implicit function theorem), etc. This is one of the reasons why we always try to have a fixed point result in a general context, since it could be applied more broadly.
Generalizations of the concept of metric space are currently one of the most active branches of functional analysis (see [1–4] or [5], where the notion of quasimetric space is studied). In fact, in Table 1 we present a nonexhaustive review of some generalizations.
Our work is related to the notion of generalized metric space introduced by Jleli and Samet [1]. Such concept was immediately generalized by Roldán and Shahzad [2] as follows.
Definition 1. An RS-space is a pair where is a nonempty set and is a function such that the following properties are fulfilled:(i)if then ,(ii) for every ,(iii)there exists such that if are two points and is an infinite sequence with and then
The notions of modular space, quasimetric space, dislocated metric space, and generalized metric space (in the Branciari sense) are particular cases of the notion of RS-space (see [1] or [2]). In this paper we introduce the notion of subordinate semimetric space; such concept is a generalization of the notion of RS-space. On the other hand, we give in Example 5 a pair which is a subordinate semimetric space that is not an RS-space. In the context of subordinate semimetric spaces we prove a version of Matkowski’s fixed point theorem and introducing the notion of -contraction we obtain a version of the fixed point theorem of Kannan-Ćirić. Moreover, we prove that if in a subordinate semimetric space every -contraction has a fixed point then this must be complete.
2. Subordinate Metric Spaces
We start with a generalization of the common notion of semimetric space (see [6]).
Definition 2. A semimetric space is a pair where is a nonempty set and is a function that satisfies the following: (i)for every , we have(ii)for every , we have
Some important concepts can be introduced with this general notion.
Definition 3. Let be a semimetric space.(i)A sequence in converges to if(ii)A sequence in is a Cauchy sequence if(iii) is complete if every Cauchy sequence in is convergent.
In order to get a rich mathematical structure we introduce a substitute of triangle inequality (for a metric).
Definition 4. We say that a semimetric space is subordinate if there is a function such that (i) is nondecreasing; ,(ii)for every , with , and being an infinity Cauchy sequence in such that converges to we haveIn this case we say that is subordinate to or that is a subordinate semimetric space.
It is clear that every RS-space is a subordinate semimetric space (just take ), and we give an example of a subordinate semimetric space that is not an RS-space.
Example 5. Let and define asSincethe sequence is an infinite Cauchy sequence that converges to Suppose that there is a constant such thatthen , for all . In this way is not an RS-space. However, the semimetric space is subordinate, for example, to the function .
With the subordination concept we will see that we are able to prove some important fixed point theorems. Then a natural question arises: What conditions implies that a semimetric space is subordinate?
The answer will give us fixed point theorems in a more general scheme.
3. Fixed Point Theorems
We now introduce a type of Kannan-Ćirić contraction condition.
Definition 6. Let be a semimetric space. A mapping is said to be a -contraction, with , ifholds for every .
Let be a function. For each we define recursively as and , for each . With the above notation we have a version of the Kannan-Ćirić’s fixed point theorem.
Theorem 7. Let be a -contraction on a complete semimetric space .
If there is an such thatthen converge to some .
Suppose that is subordinate to andIf then is the unique fixed point of
Proof. Let us set , for each . The -contraction property of impliesThe hypothesis yields .
Now let us see that is a Cauchy sequence. Let , then there is such thatTherefore, if , thenThen there is such that .
Let us see that is a fixed point for . If the set is finite, the Cauchy property of implies that there exists such that , for all , then . On the other hand, if is an infinite set then there is an infinite Cauchy subsequence of such that . If , then there is such thatand in this wayIf , then ; therefore or , but this is not possible, because Thus is a fixed point. If is other fixed point, thenFrom this the uniqueness of follows.
Now let us give an example where condition (12) is necessary to obtain a fixed point.
Example 8. Let be defined as , with . Given there is such that . Let us consider the set with the semimetricSince the sequence is an infinite Cauchy sequence that converge to (this follows from ) we haveTherefore, the semimetric space is subordinate to . The function , defined asdoes not have fixed points. On the other hand, for ,this impliesand, for ,Therefore, is a -contraction on the complete semimetric space subordinated to without fixed point.
Right away we will try a version of Matkowski’s theorem in the context of subordinate semimetric spaces.
Theorem 9. Let be a complete semimetric space subordinated to and . Suppose that there exists a nondecreasing function such that , for all , andIf there is an such thatthen converges to some . Moreover, is the unique fixed point of .
Proof. Let us take . Suppose , then The hypothesis implies that is a Cauchy sequence, then there is such that . Suppose that for some , then , and this yieldsHence all the terms in the Cauchy sequence are different. MoreoverThus is a fixed point of . If is other fixed point, thenFrom this the uniqueness of follows.
It is clear that a semimetric space for which , for each , does not have fixed points, then (26) is a necessary condition in order to have a function and a fixed point.
Proposition 10. Let be a semimetric space subordinated to . Let be a Cauchy sequence in with whenever . If there is a subsequence of such that , then converge to .
Proof. Let , then there is such that , for each . On the other hand, there exists for whichTherefore,Since is nondecreasingHence
With the next result we characterize when a subordinate semimetric space is complete; the corresponding result for metric spaces is due to Subrahmanyam [12].
Theorem 11. Let be a semimetric space subordinated to . Let ; if every -contraction has a fixed point then is complete.
Proof. Let us suppose that is not complete, then there is a nonconvergent Cauchy sequence . The Cauchy property implies that we can take, if necessary, a subsequence with all the elements different, so we suppose that are distinct (see the second part in the proof of Theorem 7). Let us define the sets , for each . If then Proposition 10 implies . The Cauchy property of implies that the sets , for all , are not empty. Let us define , thenFor each there is such that ; thus is well defined. The function , defined as , does not have fixed points. Indeed, by definition and , since . Moreover, for , we assume, without loss of generality, that , then and (34) implies Thus is a -contraction without fixed points; this contradicts the hypothesis.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that he has no conflicts of interest.
Acknowledgments
The author was partially supported by Centro de Ciencias Básicas and the Grant PIM18-2 of Universidad Autónoma de Aguascalientes.