C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 67, N umb er 2, Pages 165–177 (2018) D O I: 10.1501/C om mua1_ 0000000871 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
C*-ALGEBRA-VALUED S-METRIC SPACES
MELTEM ERDEN EGE AND CIHANGIR ALACA
Abstract. In this study, we present the concept of a C*-algebra-valued S-metric space. We prove Banach contraction principle in this space. Finally, we prove a common …xed point theorem in C*-algebra-valued S-metric spaces de…ning new notions such as L-condition and k-contraction.
1. Introduction
As we have known, Banach contraction principle has very useful structure. For this reason, it has been used in various areas such as modern analysis, applied mathematics and …xed point theory. The main goal of researchers is to obtain new results in di¤erent metric spaces. On the other hand, coupled …xed point theorems have been given in di¤erent metric spaces [12, 23, 31].
The notion of S-metric space was presented by Sedghi et al. [24]. Then, Chouhan [6] proved a common unique …xed point theorem for expansive mappings in S-metric space. Sedghi and Dung [25] proved a general …xed point theorem in S-metric spaces.
Hieu et al. [11] gave a …xed point theorem for a class of maps depending on another map on S-metric spaces. Afra [2] introduced double contractive mappings. For other important papers related to S-metric spaces, see [1, 7, 8, 9, 10, 26].
After studying the operator-valued metric spaces in [17], Ma et al. [18] intro-duced the concept of C*-valued metric spaces and give a …xed point theorem for C*-valued contraction mappings. In [19], C*-algebra-valued b-metric spaces were presented and some applications related to operator and integral equations were given. Coincidence and common …xed point theorems for two mappings in com-plete C*-algebra-valued metric spaces were proved in [22].
Batul and Kamran [5] generalized the notion of C*-valued contraction mappings and established a …xed point theorem for such mappings. In [29], Caristi’s …xed point theorem was given for C*-algebra-valued metric spaces. Kamran et al. [14]
Received by the editors: November 20, 2016, Accepted: July 19, 2017. 2010 Mathematics Subject Classi…cation. Primary 47H10; Secondary 54H25. Key words and phrases. Fixed point, C*-algebra, S-metric.
c 2 0 1 8 A n ka ra U n ive rsity. C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a t h e m a tic s a n d S t a tis tic s . C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra -S é rie s A 1 M a t h e m a tic s a n d S t a tis t ic s .
gave the Banach contraction principle in C*-algebra-valued b-metric spaces with application. Bai [4] presented coupled …xed point theorems in C*-algebra-valued b-metric spaces. For other works, see [3, 13, 15, 21, 28, 30, 32, 33].
In this work, we introduce C*-algebra-valued S-metric spaces and prove Banach contraction principle. We also prove a coupled …xed point theorem in C*-algebra-valued S-metric spaces. For this purpose, we give some de…nitions such as coupled …xed point, L-condition and k-contraction.
2. Preliminaries
In this section, we give some basic de…nitions and theorems from [18] which will be used later. Throughout this paper, A will denote a unital C*-algebra with a unit I. An involution on A is a conjugate linear map a 7! a on A such that
a = a and (ab) = b a
for all a; b 2 A. The pair (A; ) is called a -algebra. A Banach -algebra is a -algebra A together with a complete submultiplicative norm such that
ka k = kak (8a 2 A):
A C*-algebra is a Banach -algebra such that ka ak = kak2.
Set Ah= fx 2 A : x = x g. An element x 2 A is said to be a positive element,
denoted by x , if x 2 Ah and (x) R+ = [0; 1), where (x) is the spectrum
of x. A partial ordering on Ah can be de…ned with these positive elements as
follows:
x y if and only if y x ;
where means the zero element in A. The set fx 2 A : x g will be denoted by A+.
When A is a unital C*-algebra, then for any x 2 A+we have x I , kxk 1
and jxj = (x x)12.
De…nition 2.1. [18]. Let X be a nonempty set. Suppose the mapping d : X X ! A satis…es the following:
(1) d(x; y) for all x; y 2 X and d(x; y) = , x = y; (2) d(x; y) = d(y; x) for all x; y 2 X;
(3) d(x; y) d(x; z) + d(z; y) for all x; y; z 2 X.
Then d is called a algebra-valued metric on X and (X; A; d) is called a C*-algebra-valued metric space.
It is obvious that C*-algebra-valued metric spaces generalize the concept of met-ric spaces, replacing the set of real numbers by A+.
De…nition 2.2. [18]. Let (X; A) be a C*-algebra-valued metric space. Suppose that fxng X and x 2 X.
(i) If for any " > 0, there is N such that for all n > N , kd(xn; x)k ", then
fxng is said to be convergent with respect to A and fxng converges to x
and x is the limit of fxng. We denote it by lim
n!1xn= x.
(ii) If for any " > 0, there is N such that for all n; m > N , kd(xn; xm)k ",
then fxng is called a Cauchy sequence with respect to A.
(iii) We say that (X; A; d) is a complete C*-algebra-valued metric space if every Cauchy sequence with respect to A is convergent.
Example 2.3. [18]. Let X = R and A = M2(R). De…ne
d(x; y) = diag(jx yj; jx yj);
where x; y 2 R and 0 is a constant. d is a C*-algebra-valued metric and (X; M2(R); d) is a complete C*-algebra-valued metric space by the completeness of
R.
De…nition 2.4. [18]. Suppose that (X; A; d) is a C*-algebra-valued metric space. We call a mapping T : X ! X is a C*-algebra-valued contractive mapping on X, if there exists an A 2 A with kAk < 1 such that
d(T x; T y) A d(x; y)A for all x; y 2 A.
Theorem 2.5. [18]. If (X; A; d) is a complete C*-algebra-valued metric space and T is a contractive mapping, there exists a unique …xed point in X.
De…nition 2.6. [18]. Let X be a nonempty set. We call a mapping T is a C*-algebra-valued expansion mapping on X, if T : X ! X satis…es:
(1) T (X) = X;
(2) d(T x; T y) A d(x; y)A, 8x; y 2 X,
where A 2 A is an invertible element and kA 1k < 1.
Theorem 2.7. [18]. Let (X; A; d) be a complete C*-algebra-valued metric space. Then for the expansion mapping T , there exists a unique …xed point in X.
Lemma 2.8. [18]. Suppose that A is a unital C*-algebra with a unit I. (1) If a 2 A+ with kak < 12, then I a is invertible and ka(I a) 1k < 1;
(2) Suppose that a; b 2 A with a; b and ab = ba, then ab ; (3) by A0 we denote the set
fa 2 A : ab = ba; 8b 2 Ag:
Let a 2 A0, if b; c 2 A with b c and I a 2 A0+ is an invertible operator, then
Theorem 2.9. [18]. Let (X; A; d) be a complete C*-valued metric space. Suppose the mapping T : X ! X satis…es for all x; y 2 X
d(T x; T y) A(d(T x; y) + d(T y; x));
where A 2 A0+ and kAk < 12. Then there exists a unique …xed point in X.
On the other hand, we need to recall the de…nition of S-metric spaces.
De…nition 2.10. [24]. Let X be a non-empty set. An S-metric on X is a function S : X3! [0; 1) that satis…es the following conditions, for each x; y; z; a 2 X,
(i) S(x; y; z) 0;
(ii) S(x; y; z) = 0 if and only if x = y = z; (iii) S(x; y; z) S(x; x; a) + S(y; y; a) + S(z; z; a). The pair (X; S) is called an S-metric space.
3. Main Results
In this section, we introduce C*-algebra-valued S-metric spaces and give some results on this new space.
De…nition 3.1. Let X be a nonempty set. Suppose the mapping S : X X X ! A satis…es the following conditions for each x; y; z; a 2 X:
(i) S(x; y; z) ;
(ii) S(x; y; z) = if and only if x = y = z; (iii) S(x; y; z) S(x; x; a) + S(y; y; a) + S(z; z; a).
Then S is called a algebra-valued S-metric and (X; A; S) is called a C*-algebra-valued S-metric space.
Example 3.2. Let A = M2(R) be all 2 2-matrices with the usual operations of
addition, scalar multiplication and matrix multiplication. It is clear that kAk = ( 2 X i;j=1 jai;jj2) 1 2
de…nes a norm on A where A = (aij) 2 A. : A ! A de…nes an involution on A
where A = A. Then A is a C*-algebra [27]. For A = (aij) and B = (bij) in A, a
partial order on A can be given as follows:
A B , (aij bij) 0 for all i; j = 1; 2:
If we de…ne on A
S(x; y; z) = d(x; z) + d(y; z) 0
0 d(x; z) + d(y; z) ;
then it is a C*-algebra-valued S-metric space.
Proof. By the condition (iii) of C*-algebra-valued S-metric, we obtain S(x; x; y) S(x; x; x) + S(x; x; x) + S(y; y; x) = S(y; y; x) and
S(y; y; x) S(y; y; y) + S(y; y; y) + S(x; x; y) = S(x; x; y): Thus we get S(x; x; y) = S(y; y; x).
De…nition 3.4. Let (X; A; S) be a C*-algebra-valued S-metric space.
(i) A sequence fxng in X converges to x 2 X with respect to A if and only if
S(xn; xn; x) ! 0 as n ! 1.
(ii) A sequence fxng in X is called a Cauchy sequence with respect to A if
for each " > 0, there exists N 2 N such that S(xn; xn; xm) " for each
n; m N .
(iii) We say that (X; A; S) is a complete C*-algebra-valued S-metric space if every Cauchy sequence with respect to A is convergent.
Example 3.5. Let X = R, A = R2 and S(x; y; z) = (jx zj + jy zj; 0) be a
C*-algebra valued S-metric space. Consider a sequence (xn) = (n1). Since
S(xn; xn; xm) = (2j 1 n 1 mj; 0) (2 j 1 nj + j 1 mj ; 0) n;m!1 ! (0; 0); (xn) is a Cauchy sequence. On the other hand, (xn) converges to 0 2 X because
S(xn; xn; 0) = (2j
1 nj; 0)
n!1
! (0; 0):
De…nition 3.6. Let (X; A; S) be a C*-algebra-valued S-metric space. A map T : X ! X is said to be C*-algebra-valued contractive mapping on X, if there exists A 2 A with kAk < 1 such that
S(T x; T x; T y) A S(x; x; y)A (3.1)
for all x; y 2 X.
Example 3.7. Let X = [0; 1] and A = M2(R) with kAk = maxfa1; a2; a3; a4g,
where ai’s are the entries of A. Then (X; A; S) is a C*-algebra-valued S-metric
space, where
S(x; y; z) = jx zj + jy zj 0
0 jx zj + jy zj ;
and partial ordering on A is given by a1 a2
a3 a4
b1 b2
De…ne a map T : X ! X by T (x) = x 4. Since S(T x; T x; T y) = S(x 4; x 4; y 4) = 1 2jx yj 0 0 12jx yj = " 1 p 2 0 0 p1 2 # jx yj 0 0 jx yj " 1 p 2 0 0 p1 2 # = A S(x; x; y)A; where A = " 1 p 2 0 0 p1 2 # and kAk =p1 2 < 1, T is a C*-algebra-valued contractive mapping.
We now prove the Banach’s contraction principle for C*-algebra-valued S-metric spaces.
Theorem 3.8. Let (X; A; S) be a complete C*-algebra-valued S-metric space and T : X ! X be a C*-algebra-valued contractive mapping. Then T has a unique …xed point x02 X.
Proof. Let’s …rst prove the existence. We choose x 2 X and show that fTn(x)g is
a Cauchy sequence with respect to A. Using induction, we obtain the following:
S(xn; xn; xn+1) = S(Tn(x); Tn(x); Tn+1(x)) A S(Tn 1(x); Tn 1(x); Tn(x))A (A )2S(Tn 2(x); Tn 2(x); Tn 1(x))A2 .. . (A )nS(x; x; T (x))An
for n = 0; 1; : : :. Therefore for m > n, we get S(xn; xn; xm) = S(Tn(x); Tn(x); Tm(x)) 2 m 2X i=n S(Ti(x); Ti(x); Ti+1(x)) + S(Tm 1(x); Tm 1(x); Tm(x)) 2 m 2X i=n (A )iS(x; x; T x)Ai+ (A )m 1S(x; x; T (x))Am 1 = 2 m 2X i=n (A )iB12B12Ai+ (A )m 1B12B12Am 1 = 2 m 2X i=n (B12Ai) (B12Ai) + (B12Am 1) (B12Am 1) = 2 m 2X i=n jB12Aij2+ jB12Am 1j2 k2 m 2X i=n jB12Aij2+ jB12Am 1j2kI 2 m 2X i=n kB12k2kAik2I + kB12k2kAm 1k2I 2kB12k2 m 2X i=n kAk2iI + kB12k2kAk2m 2I 2kB12k2 kAk 2n 1 kAkI + kB 1 2k2kAk2m 2I m;n!1 !
where B = S(x; x; T x). So fTn(x)g is a Cauchy sequence with respect to A. By the
completeness of (X; A; S), there exists an element x0 2 X with lim n!1T n(x) = x 0. Since S(T x0; T x0; x0) = S(T x0; T x0; T xn) + S(T x0; T x0; T xn) + S(x0; x0; xn) A S(x0; x0; xn)A + A S(x0; x0; xn)A + S(x0; x0; xn) n!1 ! ;
we conclude that T x0= x0, i.e., x0 is a …xed point of T .
Finally we show the uniqueness. Assume that there exists u; v 2 X with u = T (u) and v = T (v). Since T is a C*-algebra-valued contractive mapping, we have
On the other hand, since kAk < 1, we obtain 0 kS(u; u; v)k = kS(T u; T u; T v)k kA S(u; u; v)Ak kA kkS(u; u; v)kkAk = kAk2kS(u; u; v)k < kS(u; u; v)k:
But this is impossible. So S(u; u; v) = and u = v which implies that the …xed point is unique.
Example 3.9. Let X, A, S and T be as in Example 3.7. T satis…es the hypothesis of Theorem 3.8. So 0 is the unique …xed point of T .
De…nition 3.10. Let (X; A; S) be a C*-algebra-valued S-metric space. An element (x; y) 2 X X is called a coupled …xed point of a mapping F : X X ! X if F (x; y) = x and F (y; x) = y.
De…nition 3.11. Let (X; A; S) be a C*-algebra-valued S-metric space. An element (x; y) 2 X X is called a coupled coincidence point of the mappings F : X X ! X and g : X ! X if F (x; y) = gx and F (y; x) = gy.
De…nition 3.12. [1]. Let X be a nonempty set. We say the mappings F : X X ! X and g : X ! X satisfy the L-condition if gF (x; y) = F (gx; gy) for all x; y 2 X. De…nition 3.13. Let (X; A; S) be a C*-algebra-valued S-metric space. We say the mappings F : X X ! X and g : X ! X satisfy the k-contraction if
S(F (x; y); F (x; y); F (z; w)) kA [S(gx; gx; gz) + S(gy; gy; gw)]A (3.2) with respect to A for all x; y; z; w; u; v 2 X.
Lemma 3.14. Let (X; A; S) be a C*-algebra-valued S-metric space. Suppose that F : X X ! X and g : X ! X satis…es the k-contraction for k 2 (0;12). If (x; y) is a coupled coincidence point of the mappings F and g, then
F (x; y) = gx = gy = F (y; x):
Proof. We have gx = F (x; y) and gy = F (y; x) because (x; y) is the coupled coin-cidence point of the mappings F and g. If we assume gx 6= gy, then we obtain
S(gx; gx; gy) = S(F (x; y); F (x; y); F (y; x))
kA [S(gx; gx; gy) + S(gy; gy; gx)]A = 2kA S(gx; gx; gy)A
and
kS(gx; gx; gy)k 2kkAk2kS(gx; gx; gy)k < kS(gx; gx; gy)k
by (3.2) and Lemma 3.3. But it is a contradiction. Therefore gx = gy and F (x; y) = gx = gy = F (y; x):
Theorem 3.15. Let (X; A; S) be a C*-algebra-valued S-metric space. Suppose that F : X X ! X and g : X ! X are mappings satisfying k-contraction for k 2 (0;12) and L-condition. If g(X) is continuous with closed range such that
F (X X) g(X), then there is a unique x in X such that gx = F (x; x) = x. Proof. Let x0; y0 2 X. By the fact that F (X X) g(X), two elements x1; y1
could be chosen as follows:
gx1= F (x0; y0) and gy1= F (y0; x0):
Starting from the pair (x1; y1), two sequences fxng and fyng in X can be obtained
such that
gxn+1= F (xn; yn) and gyn+1= F (yn; xn):
The inequality (3.2) gives the following for n 2 N:
S(gxn 1; gxn 1; gxn) kA [S(gxn 2; gxn 2; gxn 1) + S(gyn 2; gyn 2; gyn 1)]A:
(3.3) On the other hand, we get
F (yn 2; xn 2) = S(gyn 1; gyn 1; gyn) kA [S(gyn 2; gyn 2; gyn 1)
+ S(gxn 2; gxn 2; gxn 1)]A:
(3.4) If we sum (3.3) and (3.4), we get
S(gxn 1; gxn 1; gxn) + S(gyn 1; gyn 1; gyn) 2kA [S(gxn 2; gxn 2; gxn 1)
+ S(gyn 2; gyn 2; gyn 1)]A
for all n 2 N. If (3.2) is applied adequately,
S(gxn; gxn; gxn+1) 2k2(A )2[S(gxn 2; gxn 2; gxn 1) + S(gyn 2; gyn 2; gyn 1)]A2 : : : 1 2k n(p2A )n[S(gx 0; gx0; gx1) + S(gy0; gy0; gy1)]( p 2A)n: Using the de…nition of C*-algebra-valued S-metric space and Lemma 3.3,
S(gxn; gxn; gxm) 2 m 2X i=n S(gxi; gxi; gxi+1) + S(gxm 1; gxm 1; gxm) 2 m 2X i=n 1 2k i(p2A )i[S(gx 0; gx0; gx1) + S(gy0; gy0; gy1)]( p 2A)i +1 2k m 1(p2A )m 1[S(gx 0; gx0; gx1) + S(gy0; gy0; gy1)]( p 2A)m 1;
where m; n 2 N, m > n + 2, then we conclude that kS(gxn; gxn; gxm)k m 2X i=n kikp2Ak2i[S(gx0; gx0; gx1) + S(gy0; gy0; gy1)] +1 2k m 1 kp2Ak2m 2[S(gx0; gx0; gx1) + S(gy0; gy0; gy1)]: Since kAk < p1 2, when n; m ! 1, we get kS(gxn; gxn; gxm)k ! 0. So fgxng
is a Cauchy sequence. In a similar way, fgyng is a Cauchy sequence. From the
closedness of g(X), fgxng and fgyng are convergent to x 2 X and y 2 X. Since g
is continuous, fg(gxn)g is convergent to gx and fg(gyn)g is convergent to gy. Since
F and g satisfy the L-condition, we get
g(gxn+1) = g(F (xn; yn)) = F (gxn; gyn)
g(gyn+1) = g(F (yn; xn)) = F (gyn; gxn):
This shows that the following inequalities:
S(g(gxn+1); g(gxn+1); F (x; y)) kA [S(g(gxn); g(gxn); gx)+S(g(gyn); g(gyn); gy)]A
and
kS(g(gxn+1); g(gxn+1); F (x; y))k kkAk2kS(g(gxn); g(gxn); gx) + S(g(gyn); g(gyn); gy)k:
If we take the limit as n ! 1,
kS(gx; gx; F (x; y))k kkAk2kS(gx; gx; gx)k + kS(gy; gy; gy)k = 0:
So gx = F (x; y). Similarly, gy = F (y; x). From Lemma 3.14, (x; y) is a coupled coincidence point of the mappings F and g. So gx = F (x; y) = F (y; x) = gy. Since
S(gxn+1; gxn+1; gx) = S(F (xn; yn); F (xn; yn); F (x; y))
kA (S(gxn; gxn; gx) + S(gyn; gyn; gy))A
and
S(gyn+1; gyn+1; gy) kA (S(gyn; gyn; gy) + S(gxn; gxn; gx))A;
we have
S(gxn+1; gxn+1; gx)+S(gyn+1; gyn+1; gy) 2kA (S(gxn; gxn; gx)+S(gyn; gyn; gy))A
and
kS(gxn+1; gxn+1; gx)+S(gyn+1; gyn+1; gy)k 2kkA kkS(gxn; gxn; gx)+S(gyn; gyn; gy)kkAk:
Taking the limit as n ! 1, we obtain the following:
kS(x; x; gx) + S(y; y; gy)k 2kkA kkS(x; x; gx) + S(y; y; gy)kkAk = 2kkAk2kS(x; x; gx) + S(y; y; gy)k: Since 2k < 1 and kAk < p1
2, we have S(x; x; gx) = 0 and S(y; y; gy) = 0. So gx = x
To show the uniqueness, assume that there is an element z 6= x in X such that z = gz = F (z; z). We have S(x; x; z) = S(F (x; x); F (x; x); F (z; z)) 2kA S(gx; gx; gz)A = 2kA S(x; x; z)A: Since 2k < 1, kAk < p1 2 and kS(x; x; z)k 2kkAk2kS(x; x; z)k; we conclude that S(x; x; z) = 0, that is, x = z.
The following corollary can be easily deduced from the Theorem 3.15.
Corollary 3.16. Let (X; A; S) be a C*-algebra-valued S-metric space. If a mapping F : X X ! X satis…es the following condition
S(F (x; y); F (u; v); F (z; w)) kA [S(x; u; z) + S(y; v; w)]A
with respect to A for all x; y; z; u; v; w 2 X and k 2 (0;12), then there exists a unique
element x 2 X such that F (x; x) = x.
4. Conclusion
In this work, we investigate whether there are correspondences of some metric and …xed point properties in S-metric spaces taking the domain set of S-metric function as A which is a C*-algebra-valued set, and …rst present C*-algebra-valued S-metric space on the set having this structure using properties of this algebraic notion. This given structure is important in terms of integrating some metric constructions of algebraic topology and …xed point theory.
5. Acknowledgements
The authors express their sincere gratitude to the anonymous referees for their careful reading and suggestions that improved the presentation of this paper.
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Current address : Meltem ERDEN EGE (Corresponding author): Manisa Celal Bayar Univer-sity, Institute of Natural and Applied Sciences, Department of Mathematics, Yunusemre 45140 Manisa, Turkey
E-mail address : mltmrdn@gmail.com
ORCID Address: http://orcid.org/0000-0002-4519-9506
Current address : Cihangir ALACA: Manisa Celal Bayar University, Faculty of Science and Arts, Department of Mathematics, Yunusemre 45140 Manisa, Turkey
E-mail address : cihangiralaca@yahoo.com.tr
