Some Generalized Fixed-Point Theorems on Complex Valued S-Metric Spaces

(1)

Some Generalized Fixed-Point Theorems on Complex Valued S-Metric Spaces

Nihal Y¬lmaz Özgür

^y

, Nihal Ta¸ s

^z

Received 27 February 2019

Abstract

In this paper, we de…ne new contractive conditions on a complex valued S-metric space. These contractive conditions generalize the classical Rhoades’ contractive condition, Nemytskii-Edelstein contractive condition and ´Ciri´c’s contractive condition. Also we prove some …xed-point theorems using these contractive conditions on a complex valuedS-metric space.

1 Introduction and Mathematical Preliminaries

It is a very famous problem studying the existence and uniqueness …xed-point theorems for a self-mapping on various metric spaces. Recently, new generalized metric spaces such asS-metric,G-metric,b-metric spaces have been presented and some …xed-point theorems have been proved for self-mappings on these generalized metric spaces (for example, see [1, 2, 4, 8, 9, 10,12, 13,14, 15,16, 17,19]). In 2012, Sedghi et al. de…ned the notion of an S-metric space and proved some …xed-point theorems such as the Banach’s contraction principle and the Nemytskii-Edelstein …xed-point theorem on an S-metric space [15]. In 2014, Sedghi and Dung proved new generalized …xed-point theorems such as the ´Ciri´c’s …xed-point result on an S-metric space [16]. The present authors obtained the generalizations of the Banach’s contraction principle and the Rhoades’condition on anS-metric space (see [12,13] for more details).

In 2011, Azam et al. introduced the notion of a complex valued metric space [3]. In 2013, Verma and Pathak de…ned the concept of property (E:A) on a complex valued metric space to obtain some common

…xed-point results for two pairs of weakly compatible mappings, satisfying a contractive condition “max”

type [18]. More recent studies in this context can be found in [5,6]. In 2014, Mlaiki presented the notion of a complex valuedS-metric space as a generalization of a complex valued metric space [7]. Also the present authors proved new …xed-point theorems on a complex valuedS-metric space (see [11] for more details).

At …rst, we recall some known de…nitions and lemmas before stating our aims. Let C be the set of complex numbers andz1,z22C. The partial order-is de…ned onCas follows:

z1-z2 if and only ifRe(z1) Re(z2),Im(z1) Im(z2) and

z1 z2 if and only ifRe(z1)< Re(z2),Im(z1)< Im(z2).

Also we writez₁-z₂ if one of the following conditions holds:

1. Re(z₁) =Re(z₂)andIm(z₁)< Im(z₂), 2. Re(z1)< Re(z2)andIm(z1) =Im(z2), 3. Re(z1) =Re(z2)andIm(z1) =Im(z2).

Mathematics Sub ject Classi…cations: 47H10, 54H25.

yDepartment of Mathematics, Bal¬kesir University, Bal¬kesir 10145, Turkey

zDepartment of Mathematics, Bal¬kesir University, Bal¬kesir 10145, Turkey

70

(2)

Note that

0-z1 z2) jz1j<jz2j and

z1-z2,z2 z3)z1 z3.

De…nition 1 ([7]) LetXbe a nonempty set. A complex valuedS-metric onXis a functionS:X X X ! Cthat satis…es the following conditions for all z; w; q; t2X:

(CS1) 0-S(z; w; q),

(CS2) S(z; w; q) = 0 if and only ifz=w=q, (CS3) S(z; w; q)-S(z; z; t) +S(w; w; t) +S(q; q; t).

The pair(X; S) is called a complex valuedS-metric space.

De…nition 2 ([7]) Let(X; S)be a complex valuedS-metric space.

1. A sequencefzng inX converges toz if and only if for all "such that0 "2Cthere exists a natural number n₀ such that for alln n₀, we haveS(z_n; z_n; z) "and it is denoted by lim

n!1z_n=z.

2. A sequencefznginX is called a Cauchy sequence if for all"such that0 "2Cthere exists a natural number n0 such that for alln; m n0, we haveS(zn; zn; zm) ".

3. A complex valuedS-metric space(X; S) is called complete if every Cauchy sequence is convergent.

Lemma 1 ([7]) Let (X; S) be a complex valued S-metric space and fz_ng a sequence in X. Then fz_ng converges to z if and only ifjS(zn; zn; z)j !0 asn! 1.

Lemma 2 ([7]) Let (X; S) be a complex valuedS-metric space and fz_ng a sequence in X. Then fz_ng is a Cauchy sequence if and only if jS(zn; zn; zn+m)j !0 asn! 1.

Lemma 3 ([7]) If(X; S)be a complex valuedS-metric space, thenS(z; z; w) =S(w; w; z)for allz; w2X.

De…nition 3 ([18]) The “max” function is de…ned for the partial order relation-as follow:

1. maxfz₁; z₂g=z₂,z₁-z₂.

2. z1-maxfz2; z3g )z1-z2 orz1-z3. 3. maxfz1; z2g=z2,z1-z2 orjz1j<jz2j.

Lemma 4 ([18]) Letz1; z2; z3; : : :2Cand the partial order relation -be de…ned onC. Then the following statements are satis…ed:

1. Ifz1-maxfz2; z3g thenz1-z2 ifz3-z2,

2. Ifz1-maxfz2; z3; z4gthen z1-z2 if maxfz3; z4g-z2,

3. Ifz₁-maxfz₂; z₃; z₄; z₅g thenz₁-z₂ ifmaxfz₃; z₄; z₅g-z₂, and so on.

Motivated by the above studies, we de…ne some new contractive conditions on a complex valued S- metric space. These contractive conditions generalize the classical Rhoades’contractive condition, Nemytskii- Edelstein contractive condition and Ćirić’s contractive condition on a complex valuedS-metric space. We investigate the relationships among these contractive conditions with counterexamples. Also we prove some …xed-point theorems as generalizations of the classical …xed-point theorems (for example, Nemytskii- Edelstein …xed-point theorem and Ćirić’s …xed-point result) on a complex valuedS-metric space.

(3)

2 Some Fixed-Point Results on Complex Valued S-Metric Spaces

At …rst, we de…ne the Rhoades’condition on a complex valuedS-metric space.

De…nition 4 Let (X; S)be a complex valued S-metric space andT a self-mapping of X. We de…ne S(T z; T z; T w) maxfS(z; z; w); S(T z; T z; z); S(T w; T w; w); S(T w; T w; z); S(T z; T z; w)g, (1) for allz; w2X withz6=w.

Now we introduce the notion of diameter on a complex valuedS-metric space and present a generalization of the condition (1).

De…nition 5 Let(X; S)be a complex valuedS-metric space andAa nonempty subset ofX. Then we de…ne diamfAg= supfjS(z; z; w)j:z; w2Ag,

which is called the diameter ofA. IfA is a bounded set, then we will writediamfAg<1.

De…nition 6 Let(X; S)be a complex valuedS-metric space,T a self-mapping ofXandUz=fTⁿz:n2Ng, diamfUzg<1 anddiamfUwg<1. We de…ne

jS(T z; T z; T w)j< diamfUz[Uwg, (2) for allz; w2X withz6=w.

In the following proposition, we give the relationship between the conditions (1) and (2).

Proposition 1 Let(X; S)be a complex valuedS-metric space andT a self-mapping ofX. IfT satis…es the condition (1); thenT satis…es the condition (2).

Proof. Suppose that the condition (1) is satis…ed byT. Then we get

S(T z; T z; T w) maxfS(z; z; w); S(T z; T z; z); S(T w; T w; w); S(T w; T w; z); S(T z; T z; w)g= and so we obtain

jS(T z; T z; T w)j<j j< diamfU_z[U_wg. Hence the condition (2) is satis…ed.

In the following example, we see that the converse of Proposition 1is not always true.

Example 1 Let X = (0;1) with the complex valuedS-metric de…ned as S(z; w; q) = 5e^ik(jz qj+jz+q 2wj) k2h

0; 2 i

, for allz; w; q2X. Let us de…ne the functionT :X !X as

T z= 8<

:

z ifz2(0;1); z6=¹₂; z6=¹₃;

1

3 ifz= ¹₂;

1

2 ifz= ¹₃;

for allz2X. ThenT is a self-mapping on the complex valuedS-metric space(X; S). Forz=¹₄; w=¹₅ 2X we have

S(T z; T z; T w) = e^ik

2 , S(T w; T w; w) = 0, S(z; z; w) = e^ik 2 ,

(4)

S(T w; T w; z) = e^ik

2 , S(T z; T z; z) = 0, S(T z; T z; w) = e^ik 2 and so we get

S(T z; T z; T w) = e^ik

2 max e^ik

2 ;0;0;e^ik 2 ;e^ik

2 , which implies

jS(T z; T z; T w)j=1 2 < e^ik

2 =1 2.

Therefore T does not satisfy the condition (1). It can be easily seen that T satis…es the condition (2) since sup (0;1) = 1.

We call the complex valued S-metric space X as compact if every sequence in X has a convergent subsequence.

Let(X; S)and(Y; S )be two complex valuedS-metric spaces andT :X!Y be a function. ThenT is continuous atx2X if and only ifT xn!T xwheneverxn!x. In the following theorem, we obtain a …xed point theorem for a self-mapping satisfying the condition (2) on a compact complex valuedS-metric space.

Theorem 1 Let (X; S)be a compact complex valuedS-metric space andT a continuous self-mapping ofX satisfying the condition (2). Then T has a unique …xed point.

Proof. There exists a compact subsetY ofX such thatT X Y sinceT is a continuous self-mapping and X is compact. Hence we getT Y Y andZ= ¹T

n=1

TⁿY is a nonempty compact subset ofX. We show that Z is a singleton consisting of the unique …xed point z0 of T. Suppose that Z is not a singleton. Then we getdiamfZg>0. SinceZ is compact subset, there existz; w2Z withjS(z; z; w)j=diamfZg. Also there existz0; w02Z withT z0=z andT w0=wsinceT mapsZ onto itself. From the condition (2), we obtain

diamfZg=jS(z; z; w)j=jS(T z₀; T z₀; T w₀)j< diamfZg, which is a contradiction. Therefore,T has a unique …xed point.

By Proposition1, we deduce the following corollary.

Corollary 1 Let (X; S)be a compact complex valuedS-metric space andT a continuous self-mapping ofX satisfying the condition (1). Then T has a unique …xed point.

In the following proposition, we see that a complex valuedS-metric function is continuous.

Proposition 2 Let(X; S)be a complex valuedS-metric space andfzng,fwngbe two sequences. Iffzng !z andfwng !w; thenS(zn; zn; wn)!S(z; z; w).

Proof. Assume thatfz_ng !z andfw_ng !w. Then there existn₁; n₂2Nsuch that S(zn; zn; z) "

4 for eachn n1

and

S(wn; wn; w) "

4 for each n n2.

If we taken0= maxfn1; n2gthen using the condition (CS3)and Lemma 3, we get S(zn; zn; wn)-2S(zn; zn; z) + 2S(wn; wn; w) +S(z; z; w) "+S(z; z; w)

(5)

and so

S(zn; zn; wn) S(z; z; w) ". (3)

Also we have

S(z; z; w) - 2S(z; z; zn) + 2S(w; w; wn) +S(zn; zn; wn)

"+S(z_n; z_n; w_n) and so

S(z; z; w) S(z_n; z_n; w_n) ". (4)

From the inequalities (3) and (4), we obtain

jS(z_n; z_n; w_n) S(z; z; w)j< ",

that is,S(zn; zn; wn)!S(z; z; w). Consequently, the complex valuedS-metric function is continuous.

Now we introduce the Nemytskii-Edelstein condition on a complex valuedS-metric space.

De…nition 7 Let (X; S)be a complex valued S-metric space andT be a self-mapping ofX. We de…ne

S(T z; T z; T w) S(z; z; w), (5)

for allz; w2X withz6=w.

In the following proposition, we give the relationship between the condition (1) and the condition (5).

Proof. The proof can be easily seen from De…nitions4and7.

Using Propositions 1and3, we deduce the following corollary.

Corollary 2 Let (X; S)be a complex valued S-metric space and T a self-mapping of X. IfT satis…es the condition (5); thenT satis…es the condition (2).

In the following example, we see that the converses of Proposition3and Corollary2are not always true.

Example 2 Let X = [0;1] with complex valued S-metric given in Example 1. Let us de…ne the function T :X !X as

T z= z+⁴₅ if z2 0;¹₅ ; 1 if z2 ¹5;1 ;

for allz2X. ThenT is a self-mapping on the complex valuedS-metric space(X; S). Forz=¹₆; w=¹₇ 2X we have

S(T z; T z; T w) = 5

21e^ik, S(z; z; w) = 5 21e^ik and so we get

S(T z; T z; T w) = 5

21e^ik S(z; z; w) = 5 21e^ik, which implies

jS(T z; T z; T w)j= 5

21 <jS(z; z; w)j= 5 21.

Therefore T does not satisfy the condition (5). It can be easily seen that T satis…es the conditions (1) and (2).

(6)

We prove the classical Nemytskii-Edelstein …xed-point theorem on a compact complex valuedS-metric space.

Theorem 2 Let (X; S) be a compact complex valued S-metric space andT a self-mapping of X satisfying the condition (5). ThenT has a unique …xed point.

Proof. Let us de…ne the function :X ![0;1) as

(z) =jS(z; z; T z)j.

The function takes on its minimum value since(X; S)is a compact complex valuedS-metric space. That is, there existsz02X such that

jS(z0; z0; T z0)j<jS(z; z; T z)j,

for allz 2X. Now we prove thatz0 is a …xed point ofT. Suppose that z0 is not …xed point ofT, that is, T z06=z0. Using the condition (5), we get

S(T z0; T z0; T T z0) S(z0; z0; T z0) and so

jS(T z0; T z0; T T z0)j<jS(z0; z0; T z0)j,

which contradicts the minimality of jS(z0; z0; T z0)j among alljS(z; z; T z)j. Therefore,z0 is a …xed point of T. We now show that the …xed point z0 is unique. Assume that w0 is another …xed point of T, that is, T w0=w0 andz06=w0. Using the condition(5), we obtain

S(z0; z0; w0) =S(T z0; T z0; T w0) S(z0; z0; w0) and so

jS(z0; z0; w0)j<jS(z0; z0; w0)j, which impliesz0=w0. Consequently, z0is a unique …xed point of T.

Remark 1 We can deduce the following results for a continuous self-mapping on a compact complex valued S-metric space.

1. Corollary1is a generalization of Theorem2.

2. Theorem1is another generalization of Theorem 2by Proposition1.

3. If we consider Example 2then T has a unique …xed pointz = 1 since the conditions (1) and (2) are satis…ed.

4. If we take the metric function asS:X X X ![0;1)in Theorem 2then we get Theorem 3.3 given in [15].

Finally we introduce the ´Ciri´c’s condition on a complex valuedS-metric space.

De…nition 8 Let (X; S)be a complex valued S-metric space andT a self-mapping of X. We de…ne S(T z; T z; T w)-hmaxfS(z; z; w); S(T z; T z; z); S(T w; T w; w); S(T w; T w; z); S(T z; T z; w)g, (6) for allz; w2X and someh2 0;¹₃ .

In the following proposition, we give the relationship between the condition (1)and(6).

(7)

Proof. The proof can be easily seen from De…nitions4and8.

Using Propositions 1and4, we deduce the following corollary.

Corollary 3 Let (X; S)be a complex valued S-metric space and T a self-mapping of X. IfT satis…es the condition (6); thenT satis…es the condition (2).

We note that the self-mappingT de…ned in Example 2 satis…es the conditions (1) and (2) but does not satisfy the condition (6).

We prove the ´Ciri´c’s …xed-point result on a complete complex valuedS-metric space.

Theorem 3 Let (X; S)be a complete complex valued S-metric space and T a self-mapping ofX satisfying the condition (6). ThenT has a unique …xed point.

Proof. Letz₀2X and the sequencefz_ngbe de…ned as follows:

T zn=zn+1,n= 0;1;2; : : :.

Assume thatz_n6=z_n+1 for alln. By the condition (6) and Lemma 3, we get S(zn; zn; zn+1)

= S(T z_n ₁; T z_n ₁; T z_n)

- hmaxfS(zn 1; zn 1; zn); S(zn; zn; zn 1); S(zn+1; zn+1; zn); S(zn+1; zn+1; zn 1); S(zn; zn; zn)g

= hmaxfS(z_n ₁; z_n ₁; z_n); S(z_n+1; z_n+1; z_n); S(z_n+1; z_n+1; z_n ₁)g

= h

and so

jS(z_n; z_n; z_n+1)j hj j 2hjS(z_n+1; z_n+1; z_n)j+hjS(z_n ₁; z_n ₁; z_n)j, which implies

jS(zn; zn; zn+1)j h

1 2hjS(zn 1; zn 1; zn)j. (7) Let a = _{1 2h}^h . Then we have a < 1 since 3h < 1. We note that 1 2h 6= 0 since 0 h < ¹₃. Using mathematical induction and the inequality (7), we obtain

jS(z_n; z_n; z_n+1)j aⁿjS(z₀; z₀; z₁)j. (8) We now prove that the sequencefzng is Cauchy. For alln; m2N, n < m, using the inequality (8) and the condition(CS3), we get

jS(zn; zn; zm)j aⁿ

1 ajS(z0; z0; z1)j.

Hence jS(zn; zn; zm)j ! 0 as n; m ! 1. Therefore fzng is a Cauchy sequence. Using the completeness hypothesis, there existsz2X such thatfzng !z.

Now we show thatz is a …xed point ofT. On the contrary, assume thatzis not a …xed point ofT, that is,T z6=z. Then using the condition (6), we obtain

S(z_n; z_n; z) = S(T z_n ₁; T z_n ₁; T z)

- hmaxfS(zn 1; zn 1; z); S(zn; zn; zn 1); S(T z; T z; z); S(T z; T z; zn 1); S(zn; zn; z)g and so taking the limit forn! 1we have

S(z; z; T z)-hS(T z; T z; z)

(8)

and by Lemma 3, we obtain

jS(z; z; T z)j=jS(T z; T z; z)j hjS(T z; T z; z)j,

which impliesT z=z, that is,zis a …xed point ofT. We prove thatzis the unique …xed point ofT. Assume thatwis another …xed point ofT such thatz6=w. Using the condition (6), we get

S(z; z; w) = S(T z; T z; T w)

- hmaxfS(z; z; w); S(z; z; z); S(w; w; w); S(w; w; z); S(z; z; w)g and so by Lemma3, we …nd

jS(z; z; w)j jS(z; z; w)j,

which impliesz=wsinceh2 0;¹₃ . Consequently, zis the unique …xed point of T.

Remark 2 We can deduce the following results for a continuous self-mapping on a compact complete complex valuedS-metric space.

1. Corollary1 is a generalization of Theorem3.

2. Theorem1 is another generalization of Theorem3 by Proposition1.

3. If we take the metric function asS :X X X ![0;1)in Theorem3, then we get Corollary 2.21 given in [16].

Acknowledgment. The authors would like to thank the anonymous referee for his/her comments that helped us improve this article.

References

[1] G. M. Abd-Elhamed, Fixed point theorems for contractions and generalized contractions in compact G-metric spaces, J. Interpolat. Approx. Sci. Comput., 1(2015), 20–27.

[2] T. Abdeljawad, N. Mlaiki, H. Aydi and N. Souayah, Double controlled metric type spaces and some

…xed point results, Mathematics, 6(2018), 320.

[3] A. Azam, B. Fisher and M. Khan, Common …xed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim., 32(2011), 243–253.

[4] N. T. Hieu, N. T. Ly and N. V. Dung, A generalization of Ciric quasi-contractions for maps onS-metric spaces, Thai J. Math., 13(2015), 369–380.

[5] N. Hussain, A. Azam, J. Ahmad and M. Arshad, Common …xed point results in complex valued metric spaces with application to integral equations, Filomat, 28(2014), 1363–1380.

[6] M. Kumar, P. Kumar, S. Kumar and S. Araci, Weakly compatible maps in complex valued metric spaces and an application to solve Urysohn integral equation, Filomat, 30(2016), 2695–2709.

[7] N. M. Mlaiki, Common …xed points in complex S-metric space, Adv. Fixed Point Theory, 4(2014), 509–524.

[8] N. Mlaiki, H. Aydi, N. Souayah and T. Abdeljawad, Controlled metric type spaces and the related contraction principle, Mathematics, 6(2018), 194.

[9] B. Moeini, P. Kumar and H. Aydi, Zamfrescu type contractions onC -algebra-valued metric spaces, J.

Math. Anal., 9(2018), 150–161.

(9)

[10] B. Moeini, M. Asasi, H. Aydi, H. Alsamir and M. S. Noorani,C -algebra-valuedM-metric spaces and some related …xed point results, Ital. J. Pure Appl. Math., 41(2019), 708–723.

[11] N. Y. Özgür and N. Ta¸s, Some generalizations of the Banach’s contraction principle on a complex valued S -metric space, J. New Theory, 2(2016), 26–36.

[12] N. Y. Özgür and N. Ta¸s, Some …xed point theorems onS-metric spaces, Mat. Vesnik, 69(2017), 39–52.

[13] N. Y. Özgür and N. Ta¸s, Some new contractive mappings on S-metric spaces and their relationships with the mapping (S25), Math. Sci., 11(2017), 7–16.

[14] M. Sarwar and M. U. Rahman, Fixed point theorems for Ciric’s and generalized contractions inb-metric spaces, Int. J. Anal. Appl., 7(2015), 70–78.

[15] S. Sedghi, N. Shobe and A. Aliouche, A generalization of …xed point theorems inS-metric spaces, Mat.

Vesnik, 64(2012), 258–266.

[16] S. Sedghi and N. V. Dung, Fixed point theorems onS-metric spaces, Mat. Vesnik, 66(2014), 113–124.

[17] S. Sedghi, ·I. Altun, N. Shobe and M. Salahshour, Some properties of S-metric space and …xed point results, Kyungpook Math. J., 54(2014), 113–122.

[18] R. K. Verma and H. K. Pathak, Common …xed point theorems using property(E:A)in complex-valued metric spaces, Thai J. Math., 11(2013), 347–355.

[19] N. Ta¸s, Fixed Point Theorems and Their Various Applications, Ph. D. Thesis, 2017.