JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi 10.1155/2018/9435470 9435470 Research Article Characterization of -Semicompleteness via Caristi’s Fixed Point Theorem in Semimetric Spaces http://orcid.org/0000-0002-2524-6045 Suzuki Tomonari 1 Shukla Satish Department of Basic Sciences Faculty of Engineering Kyushu Institute of Technology Tobata Kitakyushu 804-8550 Japan kyutech.ac.jp 2018 132018 2018 07 12 2017 02 01 2018 132018 2018 Copyright © 2018 Tomonari Suzuki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Japan Society for the Promotion of Science 16K05207
1. Introduction

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

Theorem 1 (Theorem 1 in [<xref ref-type="bibr" rid="B5">11</xref>]).

Let (X,d) be a complete metric space and let T be a mapping on X. Let h be a lower semicontinuous function from X into [0,). Assume h(Tx)+d(x,Tx)h(x) for all xX. Then T has a fixed point.

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

Theorem 2 (Theorem 2 in [<xref ref-type="bibr" rid="B10">12</xref>]).

Let (X,d) be a metric space. Then the following are equivalent:

X is complete.

Every mapping T on X has a fixed point provided there exists a lower semicontinuous function h from X into [0,) satisfying h(Tx)+dx,Txh(x) for all xX.

Very recently, Theorem 1 was extended to semimetric spaces.

Theorem 3 (see [<xref ref-type="bibr" rid="B17">13</xref>]).

Let X,d be a (, )-complete semimetric space and let T be a mapping on X. Let h be a function from X into -,+ which is proper, bounded from below, and sequentially lower semicontinuous from above in the sense of Definition 6. Assume hTx+dx,Txhx for all xX. Then T 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 N, Q, and R the sets of all positive integers, all rational numbers, and all real numbers, respectively. For an arbitrary set A, we also denote by #A the cardinal number of A.

In this section, we give some preliminaries.

Definition 5.

Let X be a nonempty set and let d be a function from X×X into [0,). Then X,d is said to be a semimetric space if the following hold:

d(x,x)=0.

d(x,y)=0x=y.

d(x,y)=d(y,x) (symmetry).

Definition 6.

Let X,d be a semimetric space, let xn be a sequence in X, and let xX. Let κN and let h be a function from X into (-,+].

{xn} is said to converge to x if limnd(xn,x)=0.

{xn} is said to be Cauchy if limnsupdxn,xm:m>n=0.

{xn} is said to be -Cauchy if n=1d(xn,xn+1)<.

{xn} is said to be (, )-Cauchy if xnnN are all different and n=1d(xn,xn+1)<.

X is said to be Hausdorff if limnd(xn,x)=0 and limnd(xn,y)=0 imply x=y.

X is said to be κ-Hausdorff if (1)limnDx,u1n,,uκn,y=0implies x=y, where (2)Dx,u1n,,uκn,y=dx,u1n+du1n,u2n++duκ-1n,uκn+duκn,y.

X is said to be complete if every Cauchy sequence converges.

X is said to be -complete if every -Cauchy sequence converges.

X is said to be (, )-complete if every (, )-Cauchy sequence converges.

X is said to be semicomplete if every Cauchy sequence has a convergent subsequence.

X is said to be -semicomplete if every -Cauchy sequence has a convergent subsequence.

X is said to be (, )-semicomplete if every (, )-Cauchy sequence has a convergent subsequence.

d is said to be sequentially lower semicontinuous if d(x,y)liminfnd(xn,yn) provided {xn} converges to x and {yn} converges to y.

h is said to be sequentially lower semicontinuous if h(x)liminfnh(xn) provided {xn} converges to x.

h is said to be sequentially lower semicontinuous from above if h(x)limnh(xn) provided {xn} converges to x and {hxn} is strictly decreasing.

h is said to be proper if xX:hxR.

Remark 7.

The definitions of κ-Hausdorffness and -semicompleteness are new.

It is obvious that X is Hausdorff X is 1-Hausdorff.

It is also obvious that X is λ-Hausdorff X is κ-Hausdorff provided κ<λ.

Proposition 8.

Let (X,d) be a semimetric space. Then the following implications hold:

(i) (ii) (iii) (iv) (vi).

(i) (v) (vi).

X is ∑-complete.

X is (, )-complete.

X is -semicomplete.

X is (, )-semicomplete.

X is complete.

X is semicomplete.

Proof.

(i) (ii) (iv), (iii) (iv), and (v) (vi) obviously hold. We have already proved (i) (v) in .

In order to prove (iv) (iii), we assume (iv). Let {xn} be a -Cauchy sequence in X. We consider the following two cases:

There exists zX satisfying #nN:xn=z=.

For any xX, #nN:xn=x<.

In the first case, some subsequence of xn converges to z. In the second case, we define a subsequence fn of the sequence n in N as follows: f1=1. We assume that fn is defined. Then we define fn+1 by (3)fn+1=maxkN:xk=xfn+1.By induction, we have defined fn. We note that xf(n)(nN) are all different. We also have (4)n=1dxfn,xfn+1n=1dxn,xn+1<.Thus, xfn is (, )-Cauchy. From (iv), there exists a subsequence of xfn which converges. It is obvious that the subsequence is also one of subsequences of xn. Therefore we have shown (iii).

Let us prove (iii) (vi). We assume (iii). Let xn be a Cauchy sequence in X. Choose a subsequence fn of n in N satisfying (5)supdxl,xm:m>l<2-nfor any l,nN with lf(n). Then xfn satisfies (6)n=1dxfn,xfn+1n=12-n=1<,thus xfn is -Cauchy. From (iii), there exist zX and a subsequence gn of n in N satisfying limnd(xfgn,z)=0. Since {xfg(n)} is also a subsequence of {xn}, we obtain (vi).

Proposition 9.

Let X,d be a semimetric space. Assume that d is sequentially lower semicontinuous. Then X is 2-Hausdorff.

Proof.

Suppose (7)limndx,un+dun,vn+dvn,y=0.Then un converges to x and vn converges to y. So we have (8)dx,yliminfndun,vn=0.Thus, we obtain x=y.

Proposition 10.

Let X,d be a -complete semimetric space. Then X is κ-Hausdorff for any κN.

Proof.

Suppose (9)limnDx,u1n,,uκn,y=0.Choose a subsequence fn of n in N satisfying (10)Dx,u1fn,,uκfn,y<2-nfor nN. Define a sequence {vn} in X as follows: (11)v2κ+2i+1=x,v2κ+2i+j+1=ujf2i+1,v2κ+2i+κ+2=y,v2κ+2i+2κ-j+3=ujf2i+2for iN0 and j1,,κ. That is, vn is as follows: (12)x,u1f1,,uκf1,y,uκf2,,u1f2,x,u1f3,,uκf3,y,uκf4,,u1f4,x,.We have (13)i=1dvi,vi+1=i=0Dvκ+1i+1,,vκ+1i+1+1<i=12-i=1<.Thus, {vn} is -Cauchy. Since X is -complete, vn converges to some z. From the definition of vn, we have x=z=y. Thus, X is κ-Hausdorff.

3. Caristi’s Theorem

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

Let α be an ordinal number. We denote by α+ and α- the successor and the predecessor of α, respectively. α is said to be isolated if α- exists. On the other hand, α is said to be limit if α0 holds and α- does not exist. For κN, we define α+κ by (14)α+κ=α++κ.

Theorem 11.

Let X,d be a -semicomplete semimetric space and let T be a mapping on X. Let h be a function from X into -,+ which is proper and bounded from below. Assume that h is sequentially lower semicontinuous from above in the sense of Definition 6. Assume also that there exists κN satisfying the following:

hTxh(x) for all xX.

hTκx+dx,Txhx for all xX.

Then T has a fixed point.

Proof.

Define a function H form X into (-,+] by (15)Hx=j=0κ-1hTjx,where T0 is the identity mapping on X. We have from (ii) (16)HTx+dx,TxHxforxX.

Arguing by contradiction, we assume Txx for any xX. Let Ω be the first uncountable ordinal number. Using transfinite induction, we will define a net uα:αΩ satisfying the following:

huαhuβ and Huα<Huβ for any βΩ with β<α.

huα<huβ for any βΩ with β+κα.

For any ε>0 and for any βΩ with β<α, there exists a finite sequence (γ0,,γn)Ωn+1 satisfying (17)β=γ0<γ1<<γn=α,Huα+j=0n-1duγj,uγj+1<Huβ+ε.

Fix uX with hu<. It follows from (i) that Hu< holds. Put u0=u. Then (P1:0)(P3:0) obviously hold.

Fix αΩ with 0<α and assume that (P1:β)(P3:β) hold for β<α. We consider the following two cases:

α is isolated.

α is limit.

In the first case, we put uα=Tuα-. For any β<α, since βα- and uα-uα hold, we have by P1:α-, (i) and (16) (18)huαhuα-huβ,Huα<Huα+duα-,uαHuα-Huβ.Thus, we have shown P1:α. For βΩ with β+κα, we have by P1:α and (ii) (19)huαhuβ+κ=hTκuβ<hTκuβ+duβ,Tuβhuβ.Thus, we have shown P2:α. Fix ε>0 and βΩ with β<α. In the case where β=α-, putting γ0=β and γ1=α, we have by (16) (20)Huα+j=0n-1duγj,uγj+1=HTuβ+duβ,TuβHuβ<Huβ+ε.In the other case, where β<α-, from P3:α-, there exists a finite sequence γ0,,γnΩn+1 satisfying (21)β=γ0<γ1<<γn=α-,Huα-+j=0n-1duγj,uγj+1<Huβ+ε.Putting γn+1=α, we have by (16) (22)Huα+j=0nduγj,uγj+1Huα-+j=0n-1duγj,uγj+1<Huβ+ε.Thus, we have shown (P3:α). Therefore we have defined uα satisfying P1:αP3:α in the first case.

In the second case, let βn be a strictly increasing sequence in Ω converging to α; that is, the following hold:

βn<α for nN.

For any β<α, there exists nN satisfying β<βn.

For any nN, from P3:βn+1, we can choose a finite sequence γn0,,γnνnΩνn+1 satisfying (23)βn=γn0<γn1<<γnνn=βn+1,j=0νn-1duγnj,uγnj+1<Huβn-Huβn+1+2-n.Since h is bounded from below, H is also bounded from below. So we have (24)n=1j=0νn-1duγnj,uγnj+1<Huβ1-limnHuβn+1<.Since X is -semicomplete, the sequence (25)β1=γ10,,γ1ν1=β2=γ20,γ21,,γ2ν2=β3=γ30,γ31,has a subsequence δn such that uδn converges to some uαX. We note that δn is strictly increasing and it converges to α. Taking a subsequence, without loss of generality, we may assume δn+κδn+1 for nN. We have by P2:δn+1(26)huδn+1<huδnfornN.Thus, huδn is strictly decreasing. Fix ε>0 and βΩ with β<α. We can choose νN satisfying (27)β<δν,duδν,uα<ε.Since h is sequentially lower semicontinuous from above, we have from (i) (28)huαlimnhuδn<huδνhuβ,Huακhuακlimnhuδn=κinfhuγ:γ<α=limnHuδn<Huδν<Huβ.We have shown P1:α and P2:α. We can choose a finite sequence (γ0,,γn)Ωn+1 satisfying (29)β=γ0<γ1<<γn=δν,Huδν+j=0n-1duγj,uγj+1<Huβ+ε.Putting γn+1=α, we have by (P1:α)(30)Huα+j=0nduγj,uγj+1<Huδν+j=0n-1duγj,uγj+1+ε<Huβ+2ε.Thus we have defined uα satisfying (P1:α)(P3:α) in the second case.

Therefore by transfinite induction, we have defined the net uα:αΩ satisfying P1:αP3:α for any αΩ. Since the net Huα:αΩ is strictly decreasing, we obtain (31)#Q=#N<#Ω#Q,which implies a contradiction. Therefore there exists a fixed point of T.

Using Theorem 11, we can generalize Theorem 3.

Corollary 12.

Let (X,d) be a -semicomplete semimetric space and let T be a mapping on X. Let h be a function from X into -,+ which is proper and bounded from below. Assume that h is sequentially lower semicontinuous from above in the sense of Definition 6. Assume also (32)hTx+dx,Txhxfor all xX. Then T has a fixed point.

Via Corollary 12, we characterize the -semicompleteness of X.

Theorem 13.

Let X,d be a semimetric space. Then the following are equivalent:

X is -semicomplete.

Every mapping T on X has a fixed point provided there exists a function h from X into 0, such that h is proper and sequentially lower semicontinuous and (32) holds for all xX.

Proof.

By Corollary 12, we obtain (i) (ii).

In order to prove (ii) (i), we will show ¬ (i) ¬ (ii). We assume that X is not -semicomplete. Then by Proposition 8, X is not (, )-semicomplete. So there exists a sequence xn in X such that xn(nN) are all different, n=1d(xn,xn+1)< holds, and there does not exist a subsequence which converges. Define a mapping T on X and a function h from X into 0, by (33)Tx=xn+1if x=xn for some nNx1ifxxn:nN,hx=j=ndxj,xj+1ifx=xnforsomenNifxxn:nN.We note that T and h are well defined because xn(nN) are all different and n=1dxn,xn+1< holds. Then h is proper, (32) holds for any xX, and T does not have a fixed point. Let yn be a sequence in X converging to some yX. Arguing by contradiction, we assume (34)hy>liminfnhyn.Then from the definition of h, there exists a subsequence fn of n in N such that yf(n)xm:m>ν for any nN, where we put νN0 by (35)ν=nify=xnforsomenN0ifyxn:nN.Define a function g from N into mN:m>ν by yfn=xgn. We consider the following two cases:

limsupng(n)=.

μlimsupng(n)<.

In the first case, xn has a subsequence converging to y. This is a contradiction. In the second case, we have (36)=#nN:gn=μ=#nN:yfn=xμ#nN:yn=xμ,which implies y=xμ. This is also a contradiction. Therefore we obtain h(y)liminfnh(yn). Thus, h 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 . In this section, we give other generalizations.

Theorem 14.

Let X,d be a 2-Hausdorff -semicomplete semimetric space. Let T be a contraction on X. Then T has a unique fixed point z. Moreover, Tnx converges to z for all xX.

Proof.

Let r0,1 satisfy (37)dTx,Tyrdx,yforx,yX.Fix uX. Then we have (38)n=1dTnu,Tn+1un=1rndu,Tu<.Since X is -semicomplete, there exists a subsequence fn of n in N such that Tfnu converges to some zX. We have (39)limndTfn+1u,TzlimnrdTfnu,z=0.Thus, Tfn+1u converges to Tz. So we have (40)limnDz,Tfnu,Tfn+1u,Tz=0.From 2-Hausdorffness of X, we obtain Tz=z.

For any xX, we have (41)limndTnx,z=limndTnx,Tnzlimnrndx,z=0.The uniqueness of the fixed point follows from (41).

Theorem 15.

Let X,d be a (, )-complete Hausdorff semimetric space. Let T be a contraction on X. Then T has a unique fixed point z. Moreover, Tnx converges to z for all xX.

Proof.

Let r0,1 satisfy (37). Fix uX. We consider the following two cases:

There exists νN satisfying Tν+1u=Tνu.

TnuTn+1u for any nN.

In the first case, Tνu is a fixed point of T. In the second case, we have (42)dTn+1u,Tn+2urdTnu,Tn+1u<dTnu,Tn+1ufor any nN. Hence TnunN are all different. We also have (43)n=1dTnu,Tn+1un=1rndu,Tu<.Since X is (, )-complete, Tnu converges to some zX. We have (44)limndTn+1u,TzlimnrdTnu,z=0.

Thus, Tn+1u converges to Tz. Since X is Hausdorff, we obtain Tz=z. Therefore in both cases, there exists a fixed point of T. 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 (X,d) be a semimetric space. Assume that either of the following holds:

X is -complete.

X is -semicomplete and d is sequentially lower and semicontinuous.

Let T be a contraction on X. Then T has a unique fixed point z. Moreover, {Tnx} converges to z for all xX.

5. Example

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

Example 17.

Put X=02-n:nN. Define a function d from X×X into 0, as follows: (45)dx,x=0,d2-n,2-n+1=d2-n+1,2-n=2-n+1fornN,d2-2n,0=d0,2-2n=2-2nfornN,dx,y=1otherwise.Then the following assertions holds:

(X,d) is a semimetric space.

X is not (, )-semicomplete.

X is -semicomplete.

Every mapping T on X has a fixed point provided there exists a function h from X into 0, such that h is proper and sequentially lower semicontinuous and (32) holds for all xX.

Proof.

(i) obviously holds. (iv) follows from (iii) and Corollary 12.

We will show (ii). It is obvious that {2-n} is a (, )-Cauchy sequence. However, we have limsupnd2-n,x=1 for all xX. Therefore we obtain (ii).

In order to prove (iii), we will show that X is (, )-semicomplete. Let xn be a (, )-Cauchy sequence in X. We can choose μ,νN satisfying (46)0xn:nν,xν=2-μ.From the definition of d, we have xν+j=2-μ+j for jN. So {xn} has a subsequence converging to 0. We have shown that X is (, )-semicomplete. By Proposition 8, X 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.

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