ON COMPATIBLE MAPPINGS OF TYPE (I) AND (II) IN INTUITIONISTIC FUZZY METRIC SPACES

Transcription

1 Commun. Korean Mah. Soc. 23 (2008), No. 3, pp ON COMPATIBLE MAPPINGS OF TYPE (I) AND (II) IN INTUITIONISTIC FUZZY METRIC SPACES Cihangir Alaca, Ishak Alun, Duran Turkoglu Reprined from he Communicaions of he Korean Mahemaical Sociey Vol. 23, No. 3, July 2008 c 2008 The Korean Mahemaical Sociey

2 Commun. Korean Mah. Soc. 23 (2008), No. 3, pp ON COMPATIBLE MAPPINGS OF TYPE (I) AND (II) IN INTUITIONISTIC FUZZY METRIC SPACES Cihangir Alaca, Ishak Alun, Duran Turkoglu Absrac. In his paper, we give some new definiions of compaible mappings in inuiionisic fuzzy meric spaces we prove a common fixed poin heorem for four mappings under he condiion of compaible mappings of ype (I) of ype (II) in complee inuiionisic fuzzy meric spaces. 1. Inroducion The noion of fuzzy ses was inroduced by Zadeh [23] in Since hen, o use his concep in opology analysis, many auhors have expansively developed he heory of fuzzy ses applicaions. For example, Deng [3], Erceg [5], Fang [6], George Veeramani [7], Kaleva Seikkala [11], Kramosil Michalek [12] have inroduced he concep of fuzzy meric spaces in differen ways. They also showed ha every meric induces a fuzzy meric. Miheţ [14] obained some new resuls of modifying he noion of convergence in fuzzy meric spaces. Turkoglu e al. [22] gave definiions of compaible mappings of ype (I), (II) some examples for various compaible mappings in fuzzy meric spaces. Afer han, we prove common fixed poin heorem for four mappings saisfying some condiions in fuzzy meric spaces. Park [16] using he idea of inuiionisic fuzzy ses, defined he noion of inuiionisic fuzzy meric spaces wih he help of coninuous -norm coninuous -conorm as a generalizaion of fuzzy meric space due o George Veeramani [7] inroduced he noion of Cauchy sequences in an inuiionisic fuzzy meric space proved he Baire s heorem finding a necessary sufficien condiion for an inuiionisic fuzzy meric space o be complee shown ha every separable inuiionisic fuzzy meric space is second counable ha every subspace of an inuiionisic fuzzy meric space is separable proved he uniform limi heorem for inuiionisic fuzzy meric spaces. Alaca e al. [1] using he idea of inuiionisic fuzzy ses, hey defined he noion Received November 13, Mahemaics Subjec Classificaion. 54H25, 54A40, 47H10. Key words phrases. inuiionisic fuzzy meric space, common fixed poin, compaible mappings compaible mappings of ypes (α), (β), (I) (II). 427 c 2008 The Korean Mahemaical Sociey

3 428 CIHANGIR ALACA, ISHAK ALTUN, AND DURAN TURKOGLU of inuiionisic fuzzy meric space as Park [16] wih he help of coninuous - norms coninuous -conorms as a generalizaion of fuzzy meric space due o Kramosil Michalek [12]. Furher, hey inroduced he noion of Cauchy sequences in an inuiionisic fuzzy meric spaces proved he well-known fixed poin heorems of Banach [2] Edelsein [4] exended o inuiionisic fuzzy meric spaces wih he help of Grabiec [8]. Turkoglu e al. [20] gave generalizaion of Jungck s common fixed poin heorem [10] o inuiionisic fuzzy meric spaces. They firs formulae he definiion of weakly commuing R-weakly commuing mappings in inuiionisic fuzzy meric spaces proved he inuiionisic fuzzy version of Pan s heorem [15]. Turkoglu e al. [21] inroduced he concep of compaible maps compaible maps of ypes (α) (β) in inuiionisic fuzzy meric spaces gave some relaions beween he conceps of compaible maps compaible maps of ypes (α) (β). Many auhors sudied he concep of inuiionisic fuzzy meric space is applicaions [9, 17, 18]. The purpose of his paper, we give conceps of A-compaible B-compaible laer aferwards inroduce definiions of compaible mappings of ype (I), (II) some examples for various compaible mappings in inuiionisic fuzzy meric spaces. Afer han, we prove common fixed poin heorem for four mappings saisfying some condiions in inuiionisic fuzzy meric spaces give an example o validae our main heorem. 2. Inuiionisic fuzzy meric spaces Definiion 1 ([19]). A binary operaion : [0, 1] [0, 1] [0, 1] is coninuous -norm if is saisfying he following condiions: (i) is commuaive associaive; (ii) is coninuous; (iii) a 1 = a for all a [0, 1]; (iv) a b c d whenever a c b d for all a, b, c, d [0, 1]. Definiion 2 ([19]). A binary operaion : [0, 1] [0, 1] [0, 1] is coninuous -conorm if is saisfying he following condiions: (i) is commuaive associaive; (ii) is coninuous; (iii) a 0 = a for all a [0, 1]; (iv) a b c d whenever a c b d for all a, b, c, d [0, 1]. The following definiion was inroduced by Alaca e al. [1]. Definiion 3 ([1]). A 5-uple (X, M, N,, ) is said o be an inuiionisic fuzzy meric space if X is an arbirary se, is a coninuous -norm, is a coninuous -conorm M, N are fuzzy ses on X 2 [0, ) saisfying he following condiions: (i) M(x, y, ) + N(x, y, ) 1 for all x, y X > 0;

4 INTUITIONISTIC FUZZY METRIC SPACES 429 (ii) M(x, y, 0) = 0 for all x, y X; (iii) M(x, y, ) = 1 for all x, y X > 0 if only if x = y; (iv) M(x, y, ) = M(y, x, ) for all x, y X > 0; (v) M(x, y, ) M(y, z, s) M(x, z, + s) for all x, y, z X s, > 0; (vi) for all x, y X, M(x, y,.) : [0, ) [0, 1] is lef coninuous; (vii) lim M(x, y, ) = 1 for all x, y X > 0; (viii) N(x, y, 0) = 1 for all x, y X; (ix) N(x, y, ) = 0 for all x, y X > 0 if only if x = y; (x) N(x, y, ) = N(y, x, ) for all x, y X > 0; (xi) N(x, y, ) N(y, z, s) N(x, z, + s) for all x, y, z X s, > 0; (xii) for all x, y X, N(x, y,.) : [0, ) [0, 1] is righ coninuous; (xiii) lim N(x, y, ) = 0 for all x, y in X. Then (M, N) is called an inuiionisic fuzzy meric on X. The funcions M(x, y, ) N(x, y, ) denoe he degree of nearness he degree of nonnearness beween x y wih respec o, respecively. Remark 1. Every fuzzy meric space (X, M, ) is an inuiionisic fuzzy meric space of he form (X, M, 1 M,, ) such ha -norm -conorm are associaed ([13]), i.e., x y = 1 ((1 x) (1 y)) for all x, y X. Remark 2. In inuiionisic fuzzy meric space X, M(x, y,.) is non-decreasing N(x, y,.) is non-increasing for all x, y X. Definiion 4 ([1]). Le (X, M, N,, ) be an inuiionisic fuzzy meric space. Then (a) a sequence {x n } in X is said o be Cauchy sequence if, for all > 0 p > 0, lim M(x n+p, x n, ) = 1, lim N(x n+p, x n, ) = 0. (b) a sequence {x n } in X is said o be convergen o a poin x X if, for all > 0, lim M(x n, x, ) = 1, lim N(x n, x, ) = 0. Since are coninuous, he limi is uniquely deermined from (v) (xi), respecively. Definiion 5 ([1]). An inuiionisic fuzzy meric space (X, M, N,, ) is said o be complee if only if every Cauchy sequence in X is convergen. Definiion 6 ([1]). An inuiionisic fuzzy meric space (X, M, N,, ) is said o be compac if every sequence in X conains a convergen subsequence. Lemma 1. Le (X, M, N,, ) be an inuiionisic fuzzy meric space {y n } be a sequence in X. If here exiss a number k (0, 1) such ha (2.1) M(y n+2, y n+1, k) M(y n+1, y n, ), N(y n+2, y n+1, k) N(y n+1, y n, ) for all > 0 n = 1, 2,..., hen {y n } is a Cauchy sequence in X.

5 430 CIHANGIR ALACA, ISHAK ALTUN, AND DURAN TURKOGLU Proof. By he simple inducion wih he condiion (2.1) wih he help of Alaca e al. [1], we have, for all > 0 n = 1, 2,..., (2.2) M(y n+1, y n+2, ) M ( y 1, y 2, ) ( k n, N(y n+1, y n+2, ) N y 1, y 2, ) k n. Thus, by (2.2) Definiion 3 ((v) (xi)), for any posiive ineger p real number > 0, we have M(y n, y n+p, ) M(y n, y n+1, p ) p imes M(y n+p 1, y n+p, p ) M(y 1, y 2, p imes ) M(y pkn 1 1, y 2, pk n+p 2 ) N(y n, y n+p, ) N(y n, y n+1, p )p imes N(y n+p 1, y n+p, p ) N(y 1, y 2, )p imes N(y pkn 1 1, y 2, Therefore, by Definiion 3 ((vii) (xiii)), we have lim M(y n, y n+p, ) 1 p imes 1 1 lim N(y n, y n+p, ) 0 p imes 0 0, pk n+p 2 ). which implies ha {y n } is a Cauchy sequence in X. This complees he proof. Lemma 2. Le (X, M, N,, ) be an inuiionisic fuzzy meric space for all x, y X, > 0 if for a number k (0, 1), (2.3) M(x, y, k) M(x, y, ) N(x, y, k) N(x, y, ) hen x = y. Proof. Since > 0 k (0, 1), we ge > k. Using Remark 2 (In inuiionisic fuzzy meric space X, M(x, y, ) is non-decreasing N(x, y, ) is non-increasing for all x, y X.), we have, M(x, y, ) M(x, y, k) N(x, y, ) N(x, y, k). Using (2.3) he definiion of inuiionisic fuzzy meric, we have, x = y.

6 INTUITIONISTIC FUZZY METRIC SPACES Various definiions of compaibiliy In his secion, we give some definiions of compaible mappings, some properies some examples in inuiionisic fuzzy meric spaces. Definiion 7 ([21]). Le A B be maps from an inuiionisic fuzzy meric space (IFM-space) (X, M, N,, ) ino iself. The maps A B are said o be compaible if, for all > 0, lim M(ABx n, BAx n, ) = 1 lim N(ABx n, BAx n, ) = 0 whenever {x n } is a sequence in X such ha lim Ax n = lim Bx n = z for some z X. Definiion 8 ([21]). Le A B be maps from an IFM-space (X, M, N,, ) ino iself. The maps A B are said o be compaible of ype (α) if, for all > 0, lim n, BBx n, ) = 1 lim n, BBx n, ) = 0, lim n, AAx n, ) = 1 lim n, AAx n, ) = 0 whenever {x n } is a sequence in X such ha lim Ax n = lim Bx n = z for some z X. Definiion 9 ([21]). Le A B be maps from an IFM-space (X, M, N,, ) ino iself. The maps A B are said o be compaible of ype (β) if, for all > 0, lim M(AAx n, BBx n, ) = 1 lim N(AAx n, BBx n, ) = 0 whenever {x n } is a sequence in X such ha lim Ax n = lim Bx n = z for some z X. Proposiion 1. Le (X, M, N,, ) be an IFM-space wih (1 ) (1 ) (1 ) for all [0, 1] A B be coninuous mappings from X ino iself. Then A B are compaible if only if hey are compaible mappings of ype (α). Proof. Suppose ha A B are compaible le {x n } be a sequence in X such ha lim Ax n = lim Bx n = z for some z X. Since A B are coninuous, we have lim n = lim n = Az, lim n = lim n = Bz. Furher, since A B are compaible, lim M(ABx n, BAx n, ) = 1 lim N(ABx n, BAx n, ) = 0

7 432 CIHANGIR ALACA, ISHAK ALTUN, AND DURAN TURKOGLU for all > 0. Now, since we have M(ABx n, BBx n, ) M(ABx n, BAx n, 2 ) M(BAx n, BBx n, 2 ) N(ABx n, BBx n, ) N(ABx n, BAx n, 2 ) N(BAx n, BBx n, 2 ) for all > 0, i follows ha which implies ha lim M(ABx n, BBx n, ) lim N(ABx n, BBx n, ) lim M(ABx n, BBx n, ) = 1 lim N(ABx n, BBx n, ) = 0. Similarly, we have also, for all > 0, lim M(BAx n, AAx n, ) = 1 lim N(BAx n, AAx n, ) = 0. Therefore, A B are compaible of ype (α). Conversely, suppose ha A B are compaible of ype (α) le {x n } be a sequence in X such ha lim Ax n = lim Bx n = z for some z X. Since A B are coninuous, we have lim n = lim n = Az, lim n = lim n = Bz. Furher, since A B are compaible of ype (α), we have, for all > 0, lim n, BBx n, ) = 1 lim n, BBx n, ) = 0, lim n, AAx n, ) = 1 lim n, AAx n, ) = 0. Thus, from he inequaliy M(ABx n, BAx n, ) M(ABx n, BBx n, 2 ) M(BBx n, BAx n, 2 ) N(ABx n, BAx n, ) N(ABx n, BBx n, 2 ) N(BBx n, BAx n, 2 ) for all > 0, i follows ha lim M(ABx n, BAx n, ) lim N(ABx n, BAx n, ) 0 0 0

8 INTUITIONISTIC FUZZY METRIC SPACES 433 for all > 0, which implies ha lim M(ABx n, BAx n, ) = 1 lim N(ABx n, BAx n, ) = 0. Therefore, A B are compaible. This complees he proof. Proposiion 2 ([21]). Le (X, M, N,, ) be an IFM-space wih (1 ) (1 ) (1 ) for all [0, 1] le A B be coninuous maps from X ino iself. Then A B are compaible if only if hey are compaible maps of ype (β). Proposiion 3 ([21]). Le (X, M, N,, ) be an IFM-space wih (1 ) (1 ) (1 ) for all [0, 1] le A B be coninuous maps from X ino iself. Then A B are compaible maps of ype (β) if only if hey are compaible maps of ype (α). We now inroduce he following definiions. Definiion 10. Le A B be mappings from an IF M-space (X, M, N,, ) ino iself. Then he pair (A, B) is called A-compaible if, for all > 0, lim M(ABx n, BBx n, ) = 1 lim N(ABx n, BBx n, ) = 0 whenever {x n } is a sequence in X such ha lim Ax n = lim Bx n = z for some z X. Definiion 11. Le A B be mappings from an IF M-space (X, M, N,, ) ino iself. Then he pair (A, B) is called B-compaible if only if (B, A) is B-compaible. Definiion 12. Le A B be mappings from an IF M-space (X, M, N,, ) ino iself. Then he pair (A, B) is said o be compaible of ype (I) if, for all > 0, lim M(ABx n, z, λ) M(Bz, z, ) lim N(ABx n, z, λ) N(Bz, z, ) whenever λ (0, 1] {x n } is a sequence in X such ha lim Ax n = lim = z for some z X. Bx n Definiion 13. Le A B be mappings from an IF M-space (X, M, N,, ) ino iself. Then he pair (A, B) is said o be compaible of ype (II) if only if (B, A) is compaible of ype (I). Proposiion 4. Le (X, M, N,, ) be an IF M-space A, B, be mappings from X ino iself such ha B (resp., A) is coninuous. If he pair (A, B) is A-compaible (resp., B-compaible), hen i is compaible of ype (I) (resp., of ype (II)).

9 434 CIHANGIR ALACA, ISHAK ALTUN, AND DURAN TURKOGLU Proof. Suppose ha he pair (A, B) is A-compaible le {x n } be a sequence in X such ha lim Ax n = lim Bx n = z for some z X. Since B is coninuous, we have Furher, lim M(ABx n, BBx n, ) = 1 = lim M(ABx n, Bz, ) lim N(ABx n, BBx n, ) = 0 = lim N(ABx n, Bz, ). M(Bz, z, ) M(ABx n, z, 2 ) M(ABx n, Bz, 2 ), N(Bz, z, ) N(ABx n, z, 2 ) N(ABx n, Bz, 2 ) so M(Bz, z, ) lim (M(ABx n, z, 2 ) M(ABx n, Bz, 2 ) ) = lim M(ABx n, z, 2 ) N(Bz, z, ) lim (N(ABx n, z, 2 ) N(ABx n, Bz, 2 ) ) Hence i follows ha = lim N(ABx n, z, 2 ). lim M(ABx n, z, ) M(Bz, z, ) lim 2 N(ABx n, z, ) N(Bz, z, ). 2 This inequaliy holds for every choice of he sequence {x n } in X wih he corresponding z X so he pair (A, B) is compaible of ype (I). This complees he proof. Proposiion 5. Le (X, M, N,, ) be an IF M-space A, B be mappings from X ino iself wih B (resp., A) is coninuous. If he pair (A, B) is compaible of ype (I) (resp., of ype (II)), for every sequence {x n } in X such ha lim Ax n = lim Bx n = z for some z X, i follows ha lim ABx n = z (resp., lim BAx n = z), hen i is A-compaible (resp., B-compaible). Proof. Suppose ha he pair (A, B) is compaible of ype (I) le {x n } be a sequence in X such ha lim Ax n = lim Bx n = z for some z X. Since B is coninuous, we have M(Bz, z, ) lim M(ABx n, z, λ) N(Bz, z, ) lim N(ABx n, z, λ)

10 INTUITIONISTIC FUZZY METRIC SPACES 435 lim M(BBx n, z, ) lim (M(Bz, z, 2 ) M(BBx n, Bz, 2 ) ) = M(Bz, z, 2 ), so lim N(BBx n, z, ) lim (N(Bz, z, 2 ) N(BBx n, Bz, 2 ) ) = N(Bz, z, 2 ) lim M(BBx n, z, ) lim Now, we have M(ABx n, z, λ 2 lim N(ABx n, z, λ 2 ). ), lim N(BBx n, z, ) M(ABx n, BBx n, ) M(ABx n, z, 2 ) M(BBx n, z, 2 ), Hence, leing n, N(ABx n, BBx n, ) N(ABx n, z, 2 ) N(BBx n, z, 2 ). lim M(ABx n, BBx n, ) lim = 1, lim lim N(ABx n, BBx n, ) lim = 0. lim (M(ABx n, z, 2 ) M(BBx n, z, 2 ) ) ( M(ABx n, z, 2 ) M(ABx n, z, λ ) 4 ) (N(ABx n, z, 2 ) N(BBx n, z, 2 ) ) ( N(ABx n, z, 2 ) N(ABx n, z, λ ) 4 ) Thus lim M(ABx n, BBx n, ) = 1 lim N(ABx n, BBx n, ) = 0. This limi always exiss hese are 1 0 for any sequence {x n } in X wih he corresponding z X. Hence he pair (A, B) is A-compaible. This complees he proof. By unifying Proposiions 4 5, we have he following:

11 436 CIHANGIR ALACA, ISHAK ALTUN, AND DURAN TURKOGLU Proposiion 6. Le (X, M, N,, ) be an IF M-space A, B be mappings from X ino iself wih B (resp., A) is coninuous. Suppose ha, for every sequence {x n } in X such ha lim Ax n = lim Bx n = z for some z X, we have lim ABx n = z (resp., lim BAx n = z). Then he pair (A, B) is compaible of ype (I) (resp., of ype (II)) if only if i is A-compaible (resp., B-compaible). Now, we give some examples. Example 1. Le X = [0, ) wih he meric d defined by d(x, y) = x y. For each (0, ) x, y X, define (M, N) by { [ ( )] 1 M(x, y, ) = exp x y, > 0, 0, = 0, { [ ( ) ] [ ( )] 1 N(x, y, ) = exp x y 1 exp x y, > 0, 1, = 0. Clearly, (X, M, N,, ) is an IF M-space, where are define by a b = min{a, b} a b = max{a, b} respecively. Le A B be defined by Ax = 1 for all x [0, 1], Ax = 1 + x for all x (1, ) Bx = 1 + x for all x [0, 1), Bx = 1 for all x [1, ). Le {x n } be a sequence in X such ha lim Ax n = lim Bx n = z. By definiion of A B, z {1} lim x n = 0. A B boh are disconinuous a z = 1. Therefore, we have lim n, BAx n, ) = 1 lim n, BAx n, ) = 0, lim n, BBx n, ) = 1 lim n, BBx n, ) = 0. Also, we consider he sequence {x n } in X defined by x n = 1 2n, n = 1, 2,.... Then we have lim Ax n = lim Bx n = 1. Furher, for > 0, we have [ ( )] 1 1 lim M(ABx n, BAx n, ) = exp < 1, [ ( ) ] [ ( )] lim N(ABx n, BAx n, ) = exp 1 exp > 0, lim M(ABx n, BBx n, ) = lim N(ABx n, BBx n, ) = [ ( )] 1 1 exp < 1, [ ( ) ] [ 1 exp 1 exp ( )] 1 1 > 0. Therefore, (A, B) is compaible of ype (β) B-compaible, bu hey are neiher compaible nor A-compaible. Moreover, z = 1 is a common fixed

12 INTUITIONISTIC FUZZY METRIC SPACES 437 poin of A B. Hence he pair (A, B) is compaible of ype (I) as well as of ype (II). Example 2. Le X = [0, 1] le be he coninuous -norm be he coninuous -conorm defined by a b = min{a, b} a b = max{a, b} respecively, for all a, b [0, 1]. For each (0, ) x, y X, define (M, N) by M(x, y, ) = { + x y, > 0, 0, = 0 N(x, y, ) = { x y + x y, > 0, 1, = 0. Clearly, (X, M, N,, ) is an inuiionisic fuzzy meric space, where are defined by a b = min{a, b} a b = max{a, b} respecively. Le A B be defined by Ax = 0 for 1 3 < x < 1 2, Ax = 1 for 0 x x 1 Bx = x for all x X. Le {x n} be a sequence in X such ha lim Ax n = lim Bx n = z. By definiion of A B, z {1} lim x n = 1. Therefore, we have lim n, BAx n, ) = 1 lim n, BAx n, ) = 0, lim n, BAx n, ) = 1 lim n, BAx n, ) = 0, lim n, BBx n, ) = 1 lim n, BBx n, ) = 0, lim n, BBx n, ) = 1 lim n, BBx n, ) = 0. Thus (A, B) is compaible, A-compaible, B-compaible, compaible of ype (α), compaible of ype (β). Moreover, z = 0 is a fixed poin of B z = 1 is a fixed poin of A. Hence he pair (A, B) is compaible of ype (I) as well as of ype (II). In he previous wo examples, he pair (A, B) was compaible of ype (I) as well as of ype (II). Bu he following example shows ha his need no o be case always also shows ha he conclusion of Proposiion 4 need no o be rue B is no coninuous. Example 3. Le X = [0, 2] wih he usual meric. For each > 0 x, y X, define (M, N) by M(x, y, ) = { + x y, > 0, 0, = 0 N(x, y, ) = { x y + x y, > 0, 1, = 0. Clearly, (X, M, N,, ) is an inuiionisic fuzzy meric space, when are defined by a b = ab a b = min{1, a + b} respecively. Le A B be defined as Ax = 1 for all x X Bx = 1 for x 1, Bx = 2 for x = 1. Then B is no coninuous a z = 1. We asser ha he pair (A, B) is compaible of ype (II), bu no of ype (I), of ype (α), of ype (β), A-compaible, B- compaible or compaible.

13 438 CIHANGIR ALACA, ISHAK ALTUN, AND DURAN TURKOGLU To see his, we suppose ha {x n } is a sequence in X such ha lim Ax n = lim Bx n = z. By definiion of A B, z {1}. Since A B agree on X\{1}, we need only consider x n 1. Now, ABx n = 1, BAx n = 2, AAx n = 1, BBx n = 2, A1 = 1 B1 = 2. Thus, for > 0, lim M(ABx n, BAx n, ) = lim M(AAx n, BAx n, ) = lim M(ABx n, BBx n, ) = < 1 lim + 1 N(ABx n, BAx n, ) = > 0, < 1 lim + 1 N(AAx n, BAx n, ) = > 0, < 1 lim + 1 N(ABx n, BBx n, ) = > 0, lim M(AAxn, BBxn, ) = 1 < 1 lim N(AAxn, BBxn, ) = > 0. Therefore, he pair (A, B) is none of compaible of ype (α), of ype (β), A- compaible, B-compaible or compaible. Also, for > 0, M(B1, 1, ) = + 1 < 1 = lim M(ABx n, 1, ) N(B1, 1, ) = > 0 = lim N(ABx n, 1, ), M(A1, 1, ) = 1 > + 1 = lim M(BAx n, 1, ) N(A1, 1, ) = 0 < = lim N(BAx n, 1, ). Therefore, he pair (A, B) compaible of ype (II), bu no of ype (I). Proposiion 7. Le (X, M, N,, ) be an IF M-space A, B be mappings from X ino iself. Suppose ha he pair (A, B) is compaible of ype (I) (resp., of ype (II)) Az = Bz for some z X. Then, for > 0 λ (0, 1], M(Az, BBz, ) M(Az, ABz, λ) N(Az, BBz, ) N(Az, ABz, λ) (resp., M(Bz, AAz, ) M(Bz, BAz, λ) N(Bz, AAz, ) N(Bz, BAz, λ)). Proof. Le {x n } be a sequence in X defined by x n = z for n = 1, 2,... Az = Bz for some z X. Then we have lim Ax n = lim Bx n = z. Suppose he pair (A, B) is compaible of ype (I). Then, for > 0 λ (0, 1], M(Az, BBz, ) lim M(Az, ABx n, λ) = M(Az, ABz, λ) N(Az, BBz, ) lim N(Az, ABx n, λ) = N(Az, ABz, λ).

14 INTUITIONISTIC FUZZY METRIC SPACES Fixed poin heorem In his secion, we prove a fixed poin heorem for four mappings under he condiion of compaible mappings of ype (I) (II) in inuiionisic fuzzy meric spaces. Theorem 1. Le (X, M, N,, ) be an complee IF M-space wih (1 ) (1 ) (1 ) for all [0, 1]. Le A, B, S T be mappings from X ino iself such ha (4.1) A(X) S(X) B(X) T (X) here exiss a consan k (0, 1) such ha (4.2) M(Ax, By, k) M(Sx, T y, ) M(Ax, Sx, ) M(By, T y, ) M(Ax, T y, α) M(By, Sx, (2 α)) N(Ax, By, k) N(Sx, T y, ) N(Ax, Sx, ) N(By, T y, ) N(Ax, T y, α) N(By, Sx, (2 α)) for all x, y X, α (0, 2) > 0. Le A, B, S T are saisfying condiions any one of he following: (C 1 ) A is coninuous he pairs (A, S) (B, T ) are compaible of ype (II). (C 2 ) B is coninuous he pairs (A, S) (B, T ) are compaible of ype (II). (C 3 ) S is coninuous he pairs (A, S) (B, T ) are compaible of ype (I). (C 4 ) T is coninuous he pairs (A, S) (B, T ) are compaible of ype (I). Then A, B, S T have a unique common fixed poin in X. Proof. Le x 0 be an arbirary poin of X. By (4.1), we can consruc a sequence {y n } in X such ha y 2n = T x 2n+1 = Ax 2n, y 2n+1 = Sx 2n+2 = Bx 2n+1 for n = 0, 1,.... Then, by (4.2), for α = 1 q, q (0, 1), we have M(Ax 2n, Bx 2n+1, k) M(Sx 2n, T x 2n+1, ) M(Ax 2n, Sx 2n, ) M(Bx 2n+1, T x 2n+1, ) M(Ax 2n, T x 2n+1, (1 q)) M(Bx 2n+1, Sx 2n, (1 + q))

15 440 CIHANGIR ALACA, ISHAK ALTUN, AND DURAN TURKOGLU so Thus i follows ha N(Ax 2n, Bx 2n+1, k) N(Sx 2n, T x 2n+1, ) N(Ax 2n, Sx 2n, ) N(Bx 2n+1, T x 2n+1, ) N(Ax 2n, T x 2n+1, (1 q)) N(Bx 2n+1, Sx 2n, (1 + q)) M(y 2n, y 2n+1, k) M(y 2n 1, y 2n, ) M(y 2n, y 2n 1, ) M(y 2n+1, y 2n, ) M(y 2n, y 2n, (1 q)) M(y 2n+1, y 2n 1, (1 + q)) M(y 2n 1, y 2n, ) M(y 2n, y 2n+1, ) M(y 2n+1, y 2n, q) M(y 2n, y 2n+1, k) N(y 2n 1, y 2n, ) N(y 2n, y 2n 1, ) N(y 2n+1, y 2n, ) N(y 2n, y 2n, (1 q)) N(y 2n+1, y 2n 1, (1 + q)) N(y 2n 1, y 2n, ) N(y 2n, y 2n+1, ) N(y 2n+1, y 2n, q). M(y 2n, y 2n+1, k) M(y 2n 1, y 2n, ) M(y 2n+1, y 2n, ) M(y 2n+1, y 2n, q) N(y 2n, y 2n+1, k) N(y 2n 1, y 2n, ) N(y 2n+1, y 2n, ) N(y 2n+1, y 2n, q). Since -norm -conorm are coninuous M(x, y, ) N(x, y, ) are coninuous, leing q 1, we have Similarly, we also have M(y 2n, y 2n+1, k) M(y 2n 1, y 2n, ) M(y 2n+1, y 2n, ) N(y 2n, y 2n+1, k) N(y 2n 1, y 2n, ) N(y 2n+1, y 2n, ). M(y 2n+1, y 2n+2, k) M(y 2n, y 2n+1, ) M(y 2n+2, y 2n+1, ) N(y 2n+1, y 2n+2, k) N(y 2n, y 2n+1, ) N(y 2n+2, y 2n+1, ). In general, we have, for m = 1, 2,..., M(y m+1, y m+2, k) M(y m, y m+1, ) M(y m+1, y m+2, )

16 INTUITIONISTIC FUZZY METRIC SPACES 441 N(y m+1, y m+2, k) N(y m, y m+1, ) N(y m+1, y m+2, ). Consequenly, i follows ha, for m, p = 1, 2,..., M(y m+1, y m+2, k) M(y m, y m+1, ) M(y m+1, y m+2, N(y m+1, y m+2, k) N(y m, y m+1, ) N(y m+1, y m+2, k p ) k p ). By noing ha M(y m+1, y m+2, k ) 1 N(y p m+1, y m+2, k ) 0 as p, p we have, for m = 1, 2,..., M(y m+1, y m+2, k) M(y m, y m+1, ) N(y m+1, y m+2, k) N(y m, y m+1, ). Hence, by Lemma 1, {y n } is a Cauchy sequence in X. Since (X, M, N,, ) is complee, i converges o a poin z in X. Since {Ax 2n }, {Bx 2n+1 }, {Sx 2n+2 } {T x 2n+1 } are subsequence of {y n }. Therefore, Ax 2n, Bx 2n+1, Sx 2n+2, T x 2n+1 z as n. Now, suppose ha he condiion (C 4 ) holds. Then, since he pair (B, T ) is compaible of ype (I) T is coninuous, we have M(T z, z, ) lim M(BT x 2n+1, z, λ), N(T z, z, ) lim N(BT x 2n+1, z, λ), T T x 2n+1 T z. Now, for α = 1, seing x = x 2n y = T x 2n+1 in (4.2), we obain (4.3) M(Ax 2n, BT x 2n+1, k) M(Sx 2n, T T x 2n+1, ) M(Ax 2n, Sx 2n, ) M(BT x 2n+1, T T x 2n+1, ) M(Ax 2n, T T x 2n+1, ) M(BT x 2n+1, Sx 2n, ) N(Ax 2n, BT x 2n+1, k) N(Sx 2n, T T x 2n+1, ) N(Ax 2n, Sx 2n, ) N(BT x 2n+1, T T x 2n+1, ) N(Ax 2n, T T x 2n+1, ) NM(BT x 2n+1, Sx 2n, ).

17 442 CIHANGIR ALACA, ISHAK ALTUN, AND DURAN TURKOGLU Thus, by leing he limi inferior on boh sides of (4.3), we have lim M(z, BT x 2n+1, k) M(z, T z, ) M(z, z, ) lim M(T z, BT x 2n+1, ) M(z, T z, ) lim M(z, BT x 2n+1, ) Therefore, i follows ha lim N(z, BT x 2n+1, k) N(z, T z, ) M(z, z, ) lim N(T z, BT x 2n+1, ) N(z, T z, ) lim N(z, BT x 2n+1, ). lim M(z, BT x 2n+1, k) M(z, T z, ) lim M(T z, BT x 2n+1, ) lim M(z, BT x 2n+1, ) M(z, T z, ) M(z, T z, 2 ) lim M(z, BT x 2n+1, 2 ) lim M(z, BT x 2n+1, ) lim M(z, BT x 2n+1, λ) lim M(T z, BT x 2n+1, λ 2 ) lim M(z, BT x 2n+1, 2 ) lim M(z, BT x 2n+1, ) lim N(z, BT x 2n+1, k) N(z, T z, ) lim N(T z, BT x 2n+1, ) lim N(z, BT x 2n+1, ) N(z, T z, ) N(z, T z, ) lim 2 N(z, BT x 2n+1, 2 ) lim N(z, BT x 2n+1, ) lim N(z, BT x 2n+1, λ) lim N(T z, BT x 2n+1, λ 2 ) lim N(z, BT x 2n+1, ) lim 2 N(z, BT x 2n+1, )

18 INTUITIONISTIC FUZZY METRIC SPACES 443, for λ = 1, lim M(z, BT x 2n+1, k) lim M(z, BT x 2n+1, 2 ) lim N(z, BT x 2n+1, k) lim N(z, BT x 2n+1, 2 ). Thus, by Remark 3, i follows ha lim BT x 2n+1 = z. Now using he compaibiliy of ype (I), we have M(T z, z, ) lim M(z, BT x 2n+1, λ) = 1, N(T z, z, ) lim N(z, BT x 2n+1, λ) = 0 so i follows ha T z = z. Again, replacing x by x 2n y by z in (4.2), for α = 1, we have M(Ax 2n, Bz, k) M(Sx 2n, z, ) M(Ax 2n, Sx 2n, ) M(Bz, z, ) M(Ax 2n, z, ) M(Bz, Sx 2n, ) N(Ax 2n, Bz, k) N(Sx 2n, z, ) N(Ax 2n, Sx 2n, ) so, leing n, we have N(Bz, z, ) N(Ax 2n, z, ) N(Bz, Sx 2n, ) M(Bz, z, k) M(Bz, z, ) N(Bz, z, k) N(Bz, z, ), which implies ha, by Lemma 2, Bz = z. Since B(X) S(X), here exiss a poin u X such ha Bz = Su = z. By (4.2), for α = 1, we have so M(Au, z, k) M(Su, z, ) M(Au, Su, ) M(z, z, ) M(Au, z, ) M(z, Su, ) N(Au, z, k) N(Su, z, ) N(Au, Su, ) N(z, z, ) N(Au, z, ) N(z, Su, ) M(Au, z, k) M(Au, z, ) N(Au, z, k) N(Au, z, ) which implies ha, by Lemma 2, Au = z. Since he pair (A, S) is compaible of ype (I) Au = Su = z, by Proposiion 7, we have so M(Au, SSz, ) M(Au, ASz, ) N(Au, SSz, ) N(Au, ASz, ) M(z, Sz, ) M(z, Az, ) N(z, Sz, ) N(z, Az, ).

19 444 CIHANGIR ALACA, ISHAK ALTUN, AND DURAN TURKOGLU Again, by (4.2), for α = 1, we have M(Az, z, k) M(Sz, z, ) M(Az, Sz, ) M(z, z, ) M(Az, z, ) M(z, Sz, ) N(Az, z, k) N(Sz, z, ) N(Az, Sz, ) N(z, z, ) N(Az, z, ) N(z, Sz, ). Thus i follows ha M(Az, z, k) M(Sz, z, ) M(Az, Sz, ) M(Az, z, ) M(Az, z, 2 ) N(Az, z, k) N(Sz, z, ) N(Az, Sz, ) N(Az, z, ) N(Az, z, 2 ) so, by Lemma 2, Az = z. Therefore, Az = Bz = Sz = T z = z z is a common fixed poin of A, B, S T. The uniqueness of a common fixed poin can be easily verified by using (4.2). The oher cases (C 1 ), (C 2 ) (C 3 ) can be disposed from a similar argumen as above. Now we give an example o suppor our main heorem. Example 4. Le X = { 1 n : n = 1, 2,...} {0} wih he usual meric, for all > 0 x, y X, define (M, N) by { M(x, y, ) = + x y, > 0, 0, = 0 N(x, y, ) = { x y + x y, > 0, 1, = 0. Clearly, (X, M, N,, ) is a complee inuiionisic fuzzy meric space, where are defined by a b = min{a, b} a b = max{a, b} respecively. Le A, B, S T be define by Ax = x 4, Sx = x 2, Bx = x 6, T x = x 3 for all x X. Then we have A(X) = { 1 1 : n = 1, 2,...} {0} { : n = 1, 2,...} {0} = S(X) 4n 2n B(X) = { 1 1 : n = 1, 2,...} {0} { : n = 1, 2,...} {0} = T (X). 6n 3n Also, he condiion (4.2) of Theorem 1 is saisfied A, B, S T are coninuous. Furher, he pairs (A, S) (B, T ) are compaible of ype (I) of ype (II) if lim n = 0, where {x n } is a sequence in X such ha

20 INTUITIONISTIC FUZZY METRIC SPACES 445 lim Ax n = lim Sx n = lim Bx n = lim T x n = 0 for some 0 X. Thus all he condiions of Theorem 1 are saisfied also 0 is he unique common fixed poin of A, B, S T. Acknowledgemens. The auhors would like o hank Professor Eungil Ko he referees for heir help in he improvemen of his paper. References [1] C. Alaca, D. Turkoglu, C. Yildiz, Fixed poins in inuiionisic fuzzy meric spaces, Chaos, Solions & Fracals 29 (2006), [2] S. Banach, Theorie les operaions lineaires, Manograie Mahemayezne Warsaw Pol, [3] Z. K. Deng, Fuzzy pseudo-meric spaces, J. Mah. Anal. Appl. 86 (1982), [4] M. Edelsein, On fixed periodic poins under conracive mappings, J. London Mah. Soc. 37 (1962), [5] M. A. Erceg, Meric spaces in fuzzy se heory, J. Mah. Anal. Appl. 69 (1979), [6] J. X. Fang, On fixed poin heorems in fuzzy meric spaces, Fuzzy Ses Sysems 46 (1992), [7] A. George P. Veeramani, On some resuls in fuzzy meric spaces, Fuzzy Ses Sysems 64 (1994), [8] M. Grabiec, Fixed poins in fuzzy meric spaces, Fuzzy Ses Sysems 27 (1988), [9] V. Gregori, S. Romaguera, P. Veeramani, A noe on inuiionisic fuzzy meric spaces, Chaos, Solions & Fracals 28 (2006), [10] G. Jungck, Commuing mappings fixed poins, Amer. Mah. Monhly. 83 (1976), [11] O. Kaleva S. Seikkala, On fuzzy meric spaces, Fuzzy Ses Sysems 12 (1984), [12] I. Kramosil J. Michalek, Fuzzy meric Saisical meric spaces, Kyberneica 11 (1975), [13] R. Lowen, Fuzzy Se Theory, Kluwer Academic Pub., Dordrech [14] D. Miheţ, On fuzzy conracive mappings in fuzzy meric spaces, Fuzzy Ses Sysems 158 (2007), [15] R. P. Pan, Common fixed poins of noncommuing mappings, J. Mah. Anal. Appl. 188 (1994), [16] J. H. Park, Inuiionisic fuzzy meric spaces, Chaos, Solions & Fracals 22 (2004), [17] J. S. Park, Y. C. Kwun, J. H. Park, A fixed poin heorem in he inuiionisic fuzzy meric spaces, Far Eas J. Mah. Sci. 16 (2005), [18] R. Saadai J. H. Park, On he inuiionisic fuzzy opological spaces, Chaos, Solions & Fracals 27 (2006), [19] B. Schweizer A. Sklar, Saisical meric spaces, Pacific J. Mah. 10 (1960), [20] D. Turkoglu, C. Alaca, Y. J. Cho, C. Yildiz, Common fixed poin heorems in inuiionisic fuzzy meric spaces, J. Appl. Mah. & Compuing 22 (2006), [21] D. Turkoglu, C. Alaca, C. Yildiz, Compaible maps compaible maps of ypes (α) (β) in inuiionisic fuzzy meric spaces, Demonsraio Mah. 39 (2006), [22] D. Turkoglu, I. Alun, Y. J. Cho, Common fixed poins of compaible mappings of ype (I) (II) in fuzzy meric spaces, J. Fuzzy Mah. 15 (2007), [23] L. A. Zadeh, Fuzzy ses, Inform. Conrol 8 (1965),

21 446 CIHANGIR ALACA, ISHAK ALTUN, AND DURAN TURKOGLU Cihangir Alaca Deparmen of Mahemaics Faculy of Science Ars Sinop Universiy Sinop, Turkey address: Ishak Alun Deparmen of Mahemaics Faculy of Science Ars Kirikkale Universiy Yahsihan, Kirikkale, Turkey address: Duran Turkoglu Deparmen of Mahemaics Faculy of Science Ars Gazi Universiy Teknikokullar, Ankara, Turkey address: