#### Abstract

Introducing the concept of -semicompleteness in semimetric spaces, we extend Caristi’s fixed point theorem to -semicomplete semimetric spaces. Via this extension, we characterize -semicompleteness. We also give generalizations of the Banach contraction principle.

#### 1. Introduction

The following, very famous theorem is referred to as Caristi’s fixed point theorem. See also [110] and references therein.

Theorem 1 (Theorem 1 in [11]). Let be a complete metric space and let be a mapping on . Let be a lower semicontinuous function from into . Assume for all . Then has a fixed point.

In 1976, Kirk proved that Caristi’s fixed point theorem characterizes the metric completeness.

Theorem 2 (Theorem 2 in [12]). Let be a metric space. Then the following are equivalent:(i) is complete.(ii)Every mapping on has a fixed point provided there exists a lower semicontinuous function from into satisfying for all .

Very recently, Theorem 1 was extended to semimetric spaces.

Theorem 3 (see [13]). Let be a (, )-complete semimetric space and let be a mapping on . Let be a function from into which is proper, bounded from below, and sequentially lower semicontinuous from above in the sense of Definition 6. Assume for all . Then has a fixed point.

Remark 4. See Definitions 5 and 6 for the definitions of (,  )-completeness and others.

It is a very natural question of whether Theorem 3 characterizes (, )-completeness of the underlying space.

In this paper, we give a negative answer to this question (see Example 17). Motivated by this fact, we introduce the concept of -semicompleteness and extend Theorem 3 to -semicomplete semimetric spaces (see Corollary 12). And we characterize the -semicompleteness via Corollary 12 (see Theorem 13). Also we give generalizations of the Banach contraction principle (see Section 4).

#### 2. Preliminaries

Throughout this paper, we denote by , , and the sets of all positive integers, all rational numbers, and all real numbers, respectively. For an arbitrary set , we also denote by the cardinal number of .

In this section, we give some preliminaries.

Definition 5. Let be a nonempty set and let be a function from into . Then is said to be a semimetric space if the following hold: (D1).(D2).(D3) (symmetry).

Definition 6. Let be a semimetric space, let be a sequence in , and let . Let and let be a function from into .(i) is said to converge to if .(ii) is said to be Cauchy if .(iii) is said to be -Cauchy if .(iv) is said to be (, )-Cauchy if are all different and .(v) is said to be Hausdorff if and imply .(vi) is said to be -Hausdorff if implies , where (vii) is said to be complete if every Cauchy sequence converges.(viii) is said to be -complete if every -Cauchy sequence converges.(ix) is said to be (, )-complete if every (, )-Cauchy sequence converges.(x) is said to be semicomplete if every Cauchy sequence has a convergent subsequence.(xi) is said to be -semicomplete if every -Cauchy sequence has a convergent subsequence.(xii) is said to be (, )-semicomplete if every (, )-Cauchy sequence has a convergent subsequence.(xiii) is said to be sequentially lower semicontinuous if provided converges to and converges to .(xiv) is said to be sequentially lower semicontinuous if provided converges to .(xv) is said to be sequentially lower semicontinuous from above if provided converges to and is strictly decreasing.(xvi) is said to be proper if .

Remark 7. (i)The definitions of -Hausdorffness and -semicompleteness are new.(ii)It is obvious that is Hausdorff is 1-Hausdorff.(iii)It is also obvious that is -Hausdorff is -Hausdorff provided .

Proposition 8. Let be a semimetric space. Then the following implications hold: (I)(i) (ii) (iii) (iv) (vi).(II)(i) (v) (vi).(i) is ∑-complete.(ii) is (, )-complete.(iii) is -semicomplete.(iv) is (, )-semicomplete.(v) is complete.(vi) is semicomplete.

Proof. (i) (ii) (iv), (iii) (iv), and (v) (vi) obviously hold. We have already proved (i) (v) in [13].
In order to prove (iv) (iii), we assume (iv). Let be a -Cauchy sequence in . We consider the following two cases: (a)There exists satisfying .(b)For any , . In the first case, some subsequence of converges to . In the second case, we define a subsequence of the sequence in as follows: . We assume that is defined. Then we define by By induction, we have defined . We note that are all different. We also have Thus, is (, )-Cauchy. From (iv), there exists a subsequence of which converges. It is obvious that the subsequence is also one of subsequences of . Therefore we have shown (iii).
Let us prove (iii) (vi). We assume (iii). Let be a Cauchy sequence in . Choose a subsequence of in satisfying for any with . Then satisfies thus is -Cauchy. From (iii), there exist and a subsequence of in satisfying . Since is also a subsequence of , we obtain (vi).

Proposition 9. Let be a semimetric space. Assume that is sequentially lower semicontinuous. Then is 2-Hausdorff.

Proof. Suppose Then converges to and converges to . So we have Thus, we obtain .

Proposition 10. Let be a -complete semimetric space. Then is -Hausdorff for any .

Proof. Suppose Choose a subsequence of in satisfying for . Define a sequence in as follows: for and . That is, is as follows: We have Thus, is -Cauchy. Since is -complete, converges to some . From the definition of , we have . Thus, is -Hausdorff.

#### 3. Caristi’s Theorem

In this section, we first prove a Kirk-Saliga type fixed point theorem [14] in -semicomplete semimetric spaces. See also [13].

Let be an ordinal number. We denote by and the successor and the predecessor of , respectively. is said to be isolated if exists. On the other hand, is said to be limit if holds and does not exist. For , we define by

Theorem 11. Let be a -semicomplete semimetric space and let be a mapping on . Let be a function from into which is proper and bounded from below. Assume that is sequentially lower semicontinuous from above in the sense of Definition 6. Assume also that there exists satisfying the following: (i) for all .(ii) for all . Then has a fixed point.

Proof. Define a function form into by where is the identity mapping on . We have from (ii) Arguing by contradiction, we assume for any . Let be the first uncountable ordinal number. Using transfinite induction, we will define a net satisfying the following: ( : ) and for any with .( : ) for any with .( : )For any and for any with , there exists a finite sequence satisfying Fix with . It follows from (i) that holds. Put . Then obviously hold.
Fix with and assume that hold for . We consider the following two cases: (a) is isolated.(b) is limit.In the first case, we put . For any , since and hold, we have by , (i) and (16) Thus, we have shown . For with , we have by and (ii) Thus, we have shown . Fix and with . In the case where , putting and , we have by (16) In the other case, where , from , there exists a finite sequence satisfying Putting , we have by (16) Thus, we have shown . Therefore we have defined satisfying in the first case.
In the second case, let be a strictly increasing sequence in converging to ; that is, the following hold: (j) for .(jj)For any , there exists satisfying . For any , from , we can choose a finite sequence satisfying Since is bounded from below, is also bounded from below. So we have Since is -semicomplete, the sequence has a subsequence such that converges to some . We note that is strictly increasing and it converges to . Taking a subsequence, without loss of generality, we may assume for . We have by Thus, is strictly decreasing. Fix and with . We can choose satisfying Since is sequentially lower semicontinuous from above, we have from (i) We have shown and . We can choose a finite sequence satisfying Putting , we have by Thus we have defined satisfying in the second case.
Therefore by transfinite induction, we have defined the net satisfying for any . Since the net is strictly decreasing, we obtain which implies a contradiction. Therefore there exists a fixed point of .

Using Theorem 11, we can generalize Theorem 3.

Corollary 12. Let be a -semicomplete semimetric space and let be a mapping on . Let be a function from into which is proper and bounded from below. Assume that is sequentially lower semicontinuous from above in the sense of Definition 6. Assume also for all . Then has a fixed point.

Via Corollary 12, we characterize the -semicompleteness of .

Theorem 13. Let be a semimetric space. Then the following are equivalent: (i) is -semicomplete.(ii)Every mapping on has a fixed point provided there exists a function from into such that is proper and sequentially lower semicontinuous and (32) holds for all .

Proof. By Corollary 12, we obtain (i) (ii).
In order to prove (ii) (i), we will show (i) (ii). We assume that is not -semicomplete. Then by Proposition 8, is not (, )-semicomplete. So there exists a sequence in such that are all different, holds, and there does not exist a subsequence which converges. Define a mapping on and a function from into by We note that and are well defined because are all different and holds. Then is proper, (32) holds for any , and does not have a fixed point. Let be a sequence in converging to some . Arguing by contradiction, we assume Then from the definition of , there exists a subsequence of in such that for any , where we put by Define a function from into by . We consider the following two cases: (a).(b). In the first case, has a subsequence converging to . This is a contradiction. In the second case, we have which implies . This is also a contradiction. Therefore we obtain . Thus, is sequentially lower semicontinuous.

#### 4. Banach’s Theorem

The author has extended the Banach contraction principle [15, 16] to semicomplete semimetric spaces. Such a result will be published somewhere else. See also [17]. In this section, we give other generalizations.

Theorem 14. Let be a 2-Hausdorff -semicomplete semimetric space. Let be a contraction on . Then has a unique fixed point . Moreover, converges to for all .

Proof. Let satisfy Fix . Then we have Since is -semicomplete, there exists a subsequence of in such that converges to some . We have Thus, converges to . So we have From 2-Hausdorffness of , we obtain .
For any , we have The uniqueness of the fixed point follows from (41).

Theorem 15. Let be a (, )-complete Hausdorff semimetric space. Let be a contraction on . Then has a unique fixed point . Moreover, converges to for all .

Proof. Let satisfy (37). Fix . We consider the following two cases: (a)There exists satisfying .(b) for any . In the first case, is a fixed point of . In the second case, we have for any . Hence are all different. We also have Since is (, )-complete, converges to some . We have Thus, converges to . Since is Hausdorff, we obtain . Therefore in both cases, there exists a fixed point of . As in the proof of Theorem 14, we can prove the remainder.

By Propositions 810 and Theorems 14 and 15, we obtain the following corollary.

Corollary 16. Let be a semimetric space. Assume that either of the following holds: (i) is -complete.(ii) is -semicomplete and is sequentially lower and semicontinuous. Let be a contraction on . Then has a unique fixed point . Moreover, converges to for all .

#### 5. Example

We finally give an example which tells that Theorem 3 does not characterize (, )-completeness of the underlying space.

Example 17. Put . Define a function from into as follows: Then the following assertions holds: (i) is a semimetric space.(ii) is not (, )-semicomplete.(iii) is -semicomplete.(iv)Every mapping on has a fixed point provided there exists a function from into such that is proper and sequentially lower semicontinuous and (32) holds for all .

Proof. (i) obviously holds. (iv) follows from (iii) and Corollary 12.
We will show (ii). It is obvious that is a (, )-Cauchy sequence. However, we have for all . Therefore we obtain (ii).
In order to prove (iii), we will show that is (, )-semicomplete. Let be a (, )-Cauchy sequence in . We can choose satisfying From the definition of , we have for . So has a subsequence converging to . We have shown that is (, )-semicomplete. By Proposition 8, is -semicomplete.

#### Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The author is supported in part by JSPS KAKENHI Grant no. 16K05207 from Japan Society for the Promotion of Science.