Some Notes on Fixed Point Theorems in ν-generalized Metric Spaces

Some Notes on Fixed Point Theorems in ν

-generalized Metric Spaces

著者

Suzuki Tomonari, Alamri Badriah, Khan Liaqat

Ali

journal or

publication title

Bulletin of the Kyushu Institute of

Technology. Pure and applied mathematics

volume

62

page range

15-23

year

2015-03-31

Bull. Kyushu Inst. Tech.

Pure Appl. Math. No. 62, 2015, pp. 15–23

SOME NOTES ON FIXED POINT THEOREMS IN

n-GENERALIZED METRIC SPACES

Tomonari Suzuki, Badriah Alamri, and Liaqat Ali Khan

Abstract

We studyn-generalized metric spaces. We first study the concept of Cauchy sequence. We next give a proof of the Banach contraction principle in n-generalized metric spaces. The proof is similar to the proof of the original Banach contraction principle in metric spaces. Also, we give proofs of Kannan’s and C´ iric´’s fixed point theorems in n-generalized metric spaces.

1. Introduction

Throughout this paper we denote by N the set of all positive integers.

In 2000, Branciari in [3] introduced the following, very interesting concept. See also [6, 8] and others.

Definition 1 (Branciari [3]). Let X be a set, let d be a function from XX into ½0;y_Þ and let nAN. Then ðX;dÞ is said to be a n-generalized metric space if the following hold:

(N1) dðx;yÞ ¼0 i¤ x¼y for any x;yAX. (N2) dðx;yÞ ¼dðy;xÞ for any x;yA_X.

(N3) dðx;yÞadðx;u1Þ þdðu1;u2Þ þ þdðun;yÞ for any x;u1;u2;. . .;un;yAX

such that x;u1;u2;. . .;un;y are all di¤erent.

It is obvious that ðX;dÞ is a metric space if and only if ðX;dÞ is a 1-generalized metric space. Very recently, in [11], we found that not every generalized metric space has the compatible topology. See also [12]. In [1], we discussed the completeness of

n-generalized metric spaces.

In this paper, we study n-generalized metric spaces. We first study the concept of Cauchy sequence. We next give a proof of the Banach contraction principle in

n-generalized metric spaces. The proof is similar to the proof of the original Banach

2010 Mathematics Subject Classification. Primary 54E99, Secondary 54H25.

Key words and phrases. n-generalized metric space, Cauchy sequence, the Banach contraction principle, Kannan’s fixed point theorem, C´ iric´’s fixed point theorem.

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (35–130–35–HiCi). The authors, therefore, acknowledge technical and financial support of KAU. The first author is supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.

contraction principle in metric spaces. Also, we give proofs of Kannan’s and C´ iric´’s fixed point theorems in n-generalized metric spaces.

2. Preliminaries

In this section, we study the concept of Cauchy sequence. As mentioned above, in general, n-generalized metric spaces do not necessarily have the compatible topology. So we have to define the concept concerning the convergence. See [1, 3, 13] and others.

Definition 2. Let ðX;dÞ be a n-generalized metric space.

_{A sequence fxng in X is said to be Cauchy i¤}

lim

n!ysup_m>n dðxm;xnÞ ¼0

holds.

_{A sequence fxng in X is said to be 2-Cauchy i¤}

lim

n!ysupfdðxn;xnþ1þ2jÞ: j¼0;1;2;. . .g ¼0

holds.

_{A sequence fxng in X is said to converge to x i¤ limndðx;xnÞ ¼0 holds. A sequence fxng in X is said to converge to x in the strong sense i¤ fxng is}

Cauchy and fxng converges to x.

Definition3. Let ðX;dÞ be a n-generalized metric space. Then X is complete i¤

every Cauchy sequence converges.

It is obvious that every Cauchy sequence is 2-Cauchy. If we assume something additional, then the converse holds.

Lemma 4. Let ðX;dÞ be a n-generalized metric space. Let fx_ng be a 2-Cauchy sequence such that xn are all di¤erent and

lim

n!ydðxn;xnþ2Þ ¼0:

Then fxng is Cauchy.

Proof. Fix e>0. Then there exists lAN such that dðxn;xnþ1þ2jÞ<e and dðxn;xnþ2Þ<e

for any jANU_f0g and nAN with nbl. Fix jANU_f0g and nAN with nbl. Then in the case where n¼1, we have

In the other case, where nb2, we have by (N3) dðxn;xnþ2þ2jÞadðxn;xnþ1þ2jÞ þdðxnþ1þ2j;xnþ2jþ2nÞ þX n1 i¼1 dðxnþ2jþ2i;xnþ2jþ2iþ2Þ <ðnþ1Þe: Therefore fxng is Cauchy. r

The following is essentially proved in [1].

Lemma 5 ([1]). LetðX;dÞbe a n-generalized metric space. Let fx_ng be a sequence in X such that xn are all di¤erent and

Xy

n¼1

dðxn;xnþ1Þ<y:

Then the following hold:

_{fxng is Cauchy provided n is odd. fxng is 2-Cauchy provided n is even.}

By using these lemmas, we can prove the following, which is useful in this paper.

Proposition6. LetðX;dÞbe an-generalized metric space. Let fx_ng be a sequence in X such that xn are all di¤erent,

Xy

n¼1

dðxn;xnþ1Þ<y and lim

n!ydðxn;xnþ2Þ ¼0:

Then fxng is Cauchy.

Proof. In the case where n is odd, the conclusion obviously holds by Lemma 5.

In the other case, where nis even,fxng is 2-Cauchy by Lemma 5 again. So by Lemma

4, fxng is Cauchy. r

The following is connected with the continuity of d. See also [8].

Proposition 7 ([13]). Let ðX;dÞ be a n-generalized metric space. Let fx_ng and fyng be sequences in X converging to u and v in the strong sense, respectively. Then

dðu;vÞ ¼ lim

n!ydðxn;ynÞ

holds.

3. Fixed point theorems

We give proofs of some fixed point theorems in n-generalized metric spaces. We note that the proofs in this section are similar to the proofs of the original theorems in metric spaces.

Lemma 8. Let ðX;dÞ be a n-generalized metric space and let T be a mapping on X . Assume that

Xy

n¼1

dðTn_u;Tnþ1_uÞ<y

ð1Þ

for some uA_{X . Assume also either of the following:}

_{n is odd. n is even and} lim n!y dðT n_u;Tnþ2_{uÞ ¼0} ð2Þ holds. Then fTn_{ug is Cauchy.}

Proof. We consider the following two cases:

_{There exists k;l}AN such that k<l and Tk_u¼Tl_u.

_Tn_{u are all di¤erent.}

In the first case, we note that Tku is a fixed point of Tlk_{. So we have}

Xy n¼1 dðTn_u;Tnþ1_uÞbX y j¼1 dðTkþjðlkÞ u;Tkþ1þjðlkÞuÞ ¼X y j¼1 dðTk_u;Tkþ1_uÞ:

Hence dðTk_u;Tkþ1_{uÞ>0 contradicts (1). Thus T}k_{u is a fixed point of T. Therefore}

fTn_{ug is Cauchy. In the second case, by Lemma 5 and Proposition 6, fT}n_{ug is}

Cauchy. r

We give a proof of the following fixed point theorem which is a generalization of the Banach contraction principle [2, 4].

Theorem9 (Branciari [3]). Let ðX;dÞ be a completen-generalized metric space and let T be a contraction on X , that is, there exists rA_½0;1Þ such that

dðTx;TyÞardðx;yÞ

for any x;yA_{X . Then T has a unique fixed point z of T. Moreover, for any x}A_{X ,}

fTn_{xg converges to z in the strong sense.}

Proof. Fix uAX. Then we have Xy n¼1 dðTn_u;Tnþ1_uÞaX y n¼1 rn_dðu;TuÞ<y: We also have lim n!ydðT n_u;Tnþ2_{uÞa lim} n!yr n2_dðu;T2_{uÞ ¼0:}

Therefore by Lemma 8, fTn_{ug is Cauchy. Since X is complete, fT}n_{ug converges to}

some zA_{X. By Proposition 7, we have}

dðz;TzÞ ¼ lim n!y dðT n_u;TzÞ a lim n!y rdðT n1_{u;zÞ ¼rdðz;zÞ ¼0;}

which implies z is a fixed point of T. Let y be a fixed point of T. Then we have

dðz;yÞ ¼dðTz;TyÞardðz;yÞ

and hence z¼y holds. So the fixed point z is unique. r

We also give a proof of the following fixed point theorem which is a generalization of Kannan’s fixed point theorem [7].

Theorem 10. Let ðX;dÞ be a complete n-generalized metric space and let T be a Kannan mapping on X , that is, there exists aA_½0;1=2Þ such that

dðTx;TyÞaadðx;TxÞ þadðy;TyÞ

for any x;yA_{X . Then T has a unique fixed point z of T. Moreover, for any x}A_{X ,}

fTn_{xg converges to z in the strong sense.}

Proof. Since

dðTx;T2xÞaadðx;TxÞ þadðTx;T2xÞ;

we have

dðTx;T2xÞardðx;TxÞ

for any xAX, where r:¼a=ð1aÞA_½0;1Þ. Fix uAX. Then we have

Xy n¼1 dðTnu;Tnþ1uÞaX y n¼1 rndðu;TuÞ<y;

which implies limndðTnu;Tnþ1uÞ ¼0. We also have

lim

n!y dðT

n_u;Tnþ2_{uÞa lim} n!yðadðT

n1_u;Tn_{uÞ þadðT}nþ1_u;Tnþ2_{uÞÞ ¼0:}

Therefore by Lemma 8, fTn_{ug is Cauchy. Since X is complete, fT}n_{ug converges to}

some zA_{X. By Proposition 7, we have}

dðz;TzÞ ¼ lim

n!ydðT

n_{u;TzÞa lim} n!yðadðT

n1_u;Tn_{uÞ þadðz;TzÞÞ ¼adðz;TzÞ;}

which implies dðz;TzÞ ¼0. Hence z is a fixed point of T. Let y be a fixed point of T. Then we have

dðz;yÞ ¼dðTz;TyÞaadðz;TzÞ þadðy;TyÞ ¼0

and hence z¼y holds. So the fixed point z is unique. r

We finally give a proof of the following fixed point theorem which is a gener-alization of C´ iric´’s fixed point theorem [5] and Theorems 9 and 10.

Theorem 11. Let ðX;dÞ be a complete n-generalized metric space and let T be a mapping on X such that there exists rA_½0;1Þ satisfying

dðTx;TyÞarmaxfdðx;yÞ;dðx;TxÞ;dðy;TyÞ;dðx;TyÞ;dðy;TxÞg ð3Þ

for any x;yA_{X . Then T has a unique fixed point z of T. Moreover, for any x}A_{X ,}

fTn_{xg converges to z in the strong sense.}

Proof. Fix uAX and put

Aðm;nÞ ¼ fTj_{u: jANUf0g;majang;}

Aðm;y_{Þ ¼ fT}ju: jANU_f0g;majg;

Dðm;nÞ ¼supfdðx;yÞ:x;yA_Aðm;nÞg

and

Dðm;y_{Þ ¼supfdðx};yÞ:x;yA_Aðm;y_Þg

for m;nANU_{f0g with man, where T}0 _{is the identity mapping on X. Thus Dðm;nÞ}

and Dðm;y_Þ are the diameter of Aðm;nÞ and Aðm;y_Þ, respectively. By (3), we note

Dðm;nÞarDðm1;nÞ ð4Þ

for m;nA_{N with man. We also note by (3)}

maxfdðu;Tj_{uÞ:1ajang ¼Dð0;nÞ}

ð5Þ

for nAN. We consider the following two cases:

_{There exists k};lA_{N such that k}<l and Tk_u¼Tl_u.

In the first case, we note

Dðk;l1Þ ¼Dðkþ1;lÞarDðk;lÞ ¼rDðk;l1Þ

by (4). Hence

Dðk;y_{Þ ¼Dðk};l1Þ ¼0;

which implies Tk _{is a fixed point of T. We next consider the second case. Fix nAN}

with n>n. By (5), there exists lAN with lan such that dðu;Tl_{uÞ ¼Dð0;nÞ. If}

l>n, then we have Dð0;nÞ ¼dðu;Tl_uÞ aX n1 j¼0 dðTj_u;Tjþ1_{uÞ þdðT}n_u;Tl_uÞ anDð0;nÞ þDðn;lÞ anDð0;nÞ þrnDð0;lÞ anDð0;nÞ þrnDð0;nÞ and hence Dð0;nÞa n 1rnDð0;nÞ: ð6Þ

If lan, then (6) obviously holds. SincenANis arbitrary, fDð0;nÞgis bounded, which is equivalent to Dð0;y_Þ<y_{. By (3), we note}

Dðm;y_ÞarDðm1;y_Þa armDð0;y_Þ

formAN, which implies that fTn_{ug is Cauchy. Since X is complete, fT}n_{ug converges}

to some zA_{X. By Proposition 7, we have}

dðz;TzÞ ¼ lim n!ydðT n_u;TzÞ a lim n!yrmaxfdðT n1_u;zÞ;dðTn1_u;Tn_{uÞ;dðz;TzÞ;dðT}n1_u;TzÞ;dðTn_u;zÞg ¼rdðz;TzÞ

and hencedðz;TzÞ ¼0, thus,zis a fixed point ofT. We have shown that there exists a fixed point in both cases. As in the proof of Theorem 9, we can prove that the fixed

point is unique. r

References

[ 1 ] B. Alamri, T. Suzuki and L. A. Khan, Caristi’s fixed point theorem and Subrahmanyam’s fixed point theorem in n-generalized metric spaces, submitted.

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[13] ———, Edelstein’s fixed point theorem in generalized metric spaces, submitted.

(T. Suzuki)

Department of Basic Sciences Faculty of Engineering Kyushu Institute of Technology Tobata, Kitakyushu 804-8550, Japan

Department of Mathematics Faculty of Science King Abdulaziz University

Jeddah, Saudi Arabia E-mail: [email protected]

(B. Alamri)

Department of Mathematics Faculty of Science King Abdulaziz University

Jeddah, Saudi Arabia E-mail: [email protected]

(L. A. Khan) Department of Mathematics

Faculty of Science King Abdulaziz University

Jeddah, Saudi Arabia E-mail: [email protected]