COMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS

PK ISSN 0022-2941; CODEN JNSMAC Vol. 49, No.1 & 2 (April & October 2009) PP 33-47 COMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS *M. A. KHAN, *SUMITRA AND ** R. CHUGH *Departmet of Mathematics, Eritrea Istitute of Techology, Asmara, Eritrea (N. E. Africa) **Maharishi Dayaad Uiversity, Rohtak-Haryaa, Idia E-mail Addresses: alam3333@gmail.com (Received: February 08, 2009) ABSTRACT: The itet of this paper is to establish some commo fixed poit theorems through semi-compatibility for quadruple self maps, two of which are semi-compatible ad remaiig two are weak compatible. I our theorems the completeess of the space X ad the cotiuity of maps is replaced with a set of four alterative coditios for fuctios satisfyig implicit relatios. These theorems exted ad improve the results of B. Sigh ad S. Jai [1-4] ad Mishra et. al. [5]. 2000 AMS Classificatio: 47H10, 54H25. Keywords: Compatible maps, Semi-compatible maps, weak compatible maps, fixed poits ad fuzzy metric space. 1. INTRODUCTION Cho et. al. [6] itroduced the otio of semi-compatible maps i d- topological spaces. Various authors like Saliga [7], Sharma et al. [8] ad Popa [9] proved some iterestig fixed poit results usig implicit real fuctios ad semi-compatibility i d-complete topological spaces. Recetly B. Sigh ad S. Jai [1,3,4] itroduced the cocept of semicompatibility i fuzzy metric spaces, D-metric spaces, 2-metric spaces ad proved fixed poit results usig implicit relatios i these spaces. The mai objective of this paper is to obtai some fixed poit theorems i the settig of fuzzy metric spaces usig weak compatibility, semicompatibility ad a implicit relatio without cosiderig the completeess of the space X ad cotiuity of maps. The relatioship betwee compatible, weak compatible ad semi-compatible maps has also bee established. We first give some prelimiaries ad defiitios usig [10,11].

34 M. A. KHAN, SUMITRA AND R. CHUGH 2. PRELIMINARIES DEFINITION 2.1: A biary operatio : 0,1 0,1 0,1 is cotiuous t orm if satisfies the followig coditios: (i) is commutative ad associative (ii) is cotiuous (iii) a 1a for all a 0,1 (iv) a b c d wheever a c ad b d, a, b, c, d 0,1. DEFINITION 2.2: The triplet (X, M, *) is said to be a fuzzy metric space if X is a arbitrary set, is a cotiuous t [0,) satisfyig the followig; (FM-1) M x, y, t 0 M x, y, t 1 if ad oly if x y. (FM-2) (FM-3) M x, y, t M y, x, t (FM-4) M x, y, tm y, z, s M x, z, t s orm ad M is a fuzzy set o X 2 (FM-5) M x, y, : 0, 0,1 is left cotiuous for all x, y, z X ad st, 0. Note that M (x, y, t) ca be thought of as the degree of earess betwee x ad y with respect to t. Example 1([12]): (Iduced fuzzy metric space) Let (X, d) be a metric space ad a * b = a b for all a, b [0, 1] ad let M d be fuzzy set o X 2 [0,) defied as follows; Md x, y, t t. t d x, y The (X, M d, *) is a fuzzy metric space. We call this fuzzy metric iduced by a metric d.

COMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS 35 Example 2 ([12]): Let X = N. Defie a * b = max {0, a + b 1} for all a, b [0, 1] ad let, M be a fuzzy set o X 2 [0,) as follows; x if x y y M x, y, t for all x, yx. y if y x x The (X, M, *) is a fuzzy metric space. Note that i the above example, there exists o metric d o X, satisfyig,, t M x y t t d x, y, Where (X, M, *) is defied i above example. Also ote that the above fuctio M is ot a fuzzy metric with the t {a, b}. orm defied as a * b = mi DEFINITION 2.3: Let (X, M, *) be a fuzzy metric space. The (a) A sequece {x } i X is said to be coverget to x if for each > 0 ad each t > 0, there exists 0 N such that M x, x, t 1 for all 0 (b) The sequece {x } i X is said to be Cauchy if for each 0 ad for each t > 0, there exists 0 N such that, m, 1, 0 M x x t for all m. (c) A fuzzy metric space i which every Cauchy sequece is coverget is said to be complete. PREPOSITION 1 ([12]): I a fuzzy metric space (X, M, *) if a * a a for all a [0, 1], a * b = mi {a, b} for all a, b [0, 1]. Let (X, M, *) be a fuzzy metric space with the coditio; lim t M x, y, t 1 for all x, y i X. (FM-6) Lemma 1 ([13]): Let {y } be a sequece i a fuzzy metric space (X, M, *) with the coditio (FM-6). If there exists a k (0, 1) such that M (y, y +1, kt) M(y -1, y, t) for all t > 0 ad N, the {y } is a Cauchy sequece i X.

36 M. A. KHAN, SUMITRA AND R. CHUGH DEFINITION 2.4: Two self mappigs A ad S of a fuzzy metric space (X, M, *) are called compatible if lim M ASx, SAx, t 1, Where {x } is a sequece i X such that lim Ax lim Sx x, for some x i X. DEFINITION 2.5: Two self mappigs A ad S of a fuzzy metric space (X, M, *) are called weakly compatible if they commute at their coicidece poits i.e., if A x = S x the S A x = A S x. DEFINITION 2.6: Two self mappigs A, B : X X are said to be semicompatible if lim M ABx, Bx, t 1, for all t 0 wheever {x } is a sequece i X such that lim Ax lim Bx x, for some x X. It follows that if (A, B) is semi-compatible ad A y = B y the A B y = B A y. Thus semi-compatibility implies weak compatibility but coverse is ot true. Remark 1: Compatibility implies weak compatibility but the coverse is ot true as show i the followig example: Example 3: Let (X, M, *) be a fuzzy metric space where X = [0, 2] ad t orm be defied as a * b = mi {a, b} where a, b [0, 1] ad Defie self maps A, S: X M x, y, t t for all x, y X, t 0 t x y X as 2 x, 0 x 1 x, 0 x 1 Ax, S x 2 1 x 2 2 1 x 2.. Cosider the sequece x 1 1 1 1, the Ax 2 1 1.

COMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS 37 1 1 1 1 S x 1, ASx A1 2 1 1. Thus, SAx 1 S 1 2. 1 lim M ASx, SAx, t lim M 1,2, t 1 for all t 0. This implies that (A, S) is ot compatible. But the set of coicidece poits of A ad S is [1, 2]. x 1,2, Ax Sx 2 ad ASx A 2 2 SAx. Thus, A Now, for ay ad S are weak compatible but ot compatible. Also 1 Ax, Sx 1 ad lim M ASx, S1, t lim M 1, 2, t 1 This implies that (A, S) is ot semi-compatible. But (A, S) is weak compatible ad thus weak compatibility ot implies the semi-compatibility. Agai lim M SAx, A1, t lim M 2, 2, t 1 Which implies that (S, A) is semi-compatible but (A, S) is ot semicompatible. Thus, semi-compatibility of (S, A) does ot imply the semicompatibility of (A, S). Now, we show that compatibility does ot imply semi-compatibility. Example 4: Let X= [0, 1] with t-orm Set a, b mi a, b, a, b 0,1. M x, y, t t for all x, y X ad t 0. t x y The, (X, M, *) is fuzzy metric space.

38 M. A. KHAN, SUMITRA AND R. CHUGH Defie self mappigs A, B : X X as Takig The 1 x 0 x A 2 x x, Bx 1 1 x 2 x 1 1. 2 1 1 1 1 1 1 2 2 2 2, Bx A x. 1 1 1 1 1 1 1 1 ABx A ad BAx B. 2 2 2 2 Thus lim M ABx, BAx, t 1. Which implies that A ad B are compatible. But 1 1 1 lim M ABx, B, t lim, 1, 2 M 2 t 1. Which implies that (A, B) is ot semi-compatible. Class of Implicit Relatio: Let be the set of all cotiuous fuctios 5 : R R, o-decreasig i first argumet ad satisfyig the followig coditios; (i) For u v u u v u v or u v u v u (ii) u u u, 0,,,,, 0,,,, 0 impliesu v,,1,,1 0 implies u 1. I [1], B. Sigh ad S. Jai proved the followig theorem..

COMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS 39 Theorem 1: Let A, B, S ad T be self maps o a complete fuzzy metric space (X, M, *) satisfyig: (i) A X T X, B X S X (ii) Oe of A or B is cotiuous. (iii) (A, S) is semi-compatible ad (B, T) is weak compatible. (iv) for all x, y X ad 0 t, M Ax, By, t M Sx, Ty, t : 0,1 0,1 is a cotiuous fuctio such that t where 0t 1. The A, B, S ad T have a uique commo fixed poit. t for each U. Mishra et. al. [5] improved the above theorem i the followig form: Theorem 2: Let A, B, S ad T be self maps o a complete fuzzy metric space (X, M, *) where is a cotiuous t orm b] satisfyig: (i) A X T X, B X S X defied by a * b = mi [a, (ii) (B, T) is weak compatible. (iii) for all x, y X ad 0 t, M Ax, By, t M Sx, Ty, t where : 0,1 0,1 is a cotiuous fuctio such that 1 1, 0 0 ad a a for each 0 < a < 1. If (A, S) is semicompatible pair of reciprocal cotiuous maps the, A, B, S ad T have a uique commo fixed poit. Now, we give geeralizatio of the results of [1] ad [17] i the followig way.

40 M. A. KHAN, SUMITRA AND R. CHUGH 3. MAIN RESULTS Theorem 3: Let (X, M, *) be a fuzzy metric space with a * a a for all a [0, 1] ad the coditio (FM-6). Let A, B, S, T : X X be self maps satisfyig (1.1) A X T X, B X S X (1.2) There exists k (0, 1) such that,,,,,,,,,, M Sx, By, 2 t for all x, y X, 0,2 ad t 0. M Ax By kt M Sx Ax t M Ty By t M Sx Ty t M Ty Ax t (1.3) The pair (A, S) is semi-compatible ad the pair (B, T) is weak compatible. (1.4) Oe of A(X), T(X), B(X) or S(X) is complete (1.5) There exists F F 5 such that F M Ax, By, t, M Ax, Ty, t, M Sy, By, t, M Ax, Sy, t, M Sy, Ty, t 0 The A, B, S ad T have a uique commo fixed poit i X. Proof: Let x 0 be ay poit i X, the by coditio (1.1) there exists x 1, x 2 X such that A x 0 = T x 1 = y 0 ad B x 1 = S x 2 = y 1. Iductively, we ca costruct sequeces {x } ad {y } i X such that y 2 Ax 2 Tx 21, y 21 Bx 21 Sx 22, 0,1,2,3... Puttig x x 2, y x 21 ad (1 q) with q 0,1, we get 2, 21, 2, 2, 21, 21, 2, 21, M Tx, Ax, 1 qt M Sx, Bx, 1 qt. M Ax Bx kt M Sx Ax t M Tx Bx t M Sx Tx t ad so, 21 2 2 21

COMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS 41 2, 21, 21, 2, 2, 21, 21, 2, M y2, y2, 1 qt M y21, y21, 1 qt. M y21, y2, t M y2, y21, t M y21, y2, t M y, y, t M y, y, qt. M y y kt M y y t M y y t M y y t Sice t 21 2 2 21 orm is cotiuous, lettig q 1, we have,,,,,, M y2 y21 kt M y21 y2 t M y2 y21 t. It follows that,,,,,, M y2 y21 kt M y21 y2 t M y2 y21 t. Similarly,,,,,,, M y21 y22 kt M y2 y21 t M y21 y22 t. Therefore, for all eve or odd, we have Cosequetly,,,,,,, M y y1 kt M y1 y t M y y1 t., 1, 1,, 1, 1, 1 M y y t M y y k t M y y k t By a simple iductio, we have M y, 1, 1,, 1, 1, m y t M y y k t M y y k t Sice M y, 1, m y k t 1 as m. It follows that. M y, y 1, kt M y 1, y, t for all N ad t 0. Therefore, by lemma (1),yis a Cauchy sequece i X.

42 M. A. KHAN, SUMITRA AND R. CHUGH Case (1): Suppose that S(X) is a complete subspace of X, the the sequece y 2 Sx 2 1 is a Cauchy sequece i S(X) ad hece has a limit z (say). Now, z S(X), so there exists w X such that z = S w. Sice (A, S) are semi-compatible, so lim ASx Sz. Step I. Puttig x Sx, y x i 1.5 1, we get 1 1 1 1,,,,, M ASx, Bx, t, M ASx, Tx, t, M Sx, Bx, t, F 0. M ASx Sx 1 t M Sx 1 Tx 1 t Lettig, we get i.e., F M Sz, z, t, M Sz, z, t, M z, z, t, M Sz, z, t, M z, z, t 0 F M Sz, z, t, M Sz, z, t,1, M Sz, z, t,1 0. Which implies that Sz z. Step II. Puttig x z, y x with 1 i 1.2 1, we get, 1,,, 1, 1,, 1, M Tx, Az, t M Sz, Bx, t. M Az Bx kt M Sz Az t M Tx Bx t M Sz Tx t Takig as, 1 1 M Az, z, kt M z, Az, t M z, z, t M z, z, t M z, Az, t M z, z, t. Which implies that Az z Sz. Step III. As A X T X, there exists some u X, such that z Az Tu. Puttig x x2, y u with 1 i (1.2), we have 2,, 2, 2,,, 2,, M Tu, Ax, t M Sx, Bu, t. M Ax Bu kt M Sx Ax t M Tu Bu t M Sx Tu t As, we get 2 2

COMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS 43 M z, Bu, kt M z, z, t M z, Bu, t M z, z, t M z, z, t M z, Bu, t. Which gives Bu z Tu. But BT, is weak compatible so BTu TBu i. e., Bz Tz. Step IV. By puttig x z, y z with 1 i (1.2) ad assumig that Az Bz, We have M z, z, t M Tz, Tz, t M Az, Bz, t M Bz, Az, t M Az, Bz, t. M Az, Bz, kt M Sz, Az, t M Tz, Bz, t M Sz, Tz, t M Tz, Az, t M Sz, Bz, t. Which is a cotradictio ad gives Az Bz z. Combiig all the results we get z Az Sz Bz Tz. Thus z is a commo fixed poit of A, B, S ad T i this case. Case (2): Let T X be complete. I this case z T X. Hece there exists wx, such that z Tw. Step I. Puttig x x y w with i 2, 1 1.2, we get 2,, 2, 2,,, 2,, M Tw, Ax, tm Sx, Bw, t M Ax Bw kt M Sx Ax t M Tw Bw t M Sx Tw t Lettig, we get 2 2,,,,,,,,,,,, M z Bw kt M z z t M z Bw t M z z t M z z t M z Bw t Which implies that Bwz Tw. But (B, T) is weak compatible so BTw TBw. Which further implies that Bz Tz Step II. As (A, S) is semi compatible so lim ASx Sz. By puttig, x Sx y z (1.5), we get

44 M. A. KHAN, SUMITRA AND R. CHUGH [ M ASx, Bz, t, M ASx, Tz, t, M Sz, Bz, t, M ASx, Sz, t, M Sz, Bz, t ] 0. F That is, F[ M Sz, Bz, t, M Sz, Tz, t, M Sz, Bz, t, M Sz, Sz, t, M Sz, Bz, t] 0. Which implies that Sz Bz Tz. Step III. Now, puttig x y z with 1 i (1.2), we get Thus,,,,,,,, M Tz, Az, tm Sz, Bz, t M Az Bz kt M Sz Az t M Tz Bz t M Sz Tz t M Az Bz t,,. M Az, Bz, t 11,, 1 M Az Bz t Az Bz Sz Tz. Step IV. Agai takig x x y z with i 2, 1 1.2, we get,,,,,,,, M Tz, Ax, tm Sx, Bz, t M Ax Bz kt M Sx Ax t M Tz Bz t M Sx Tz t Lettig, we get,,,,,,,,,,,, M z Bz kt M z z t M Bz Bz t M z Bz t M Bz z t M z Bz t Which gives z Bz. Thus z Bz Az Sz Tz. That is z is a commo fixed poit of A, B, S ad T i this case also. Case (3): Whe A X or B X is complete. As AX T X ad B X S X, therefore, the result follows from Case (1) ad Case (2). Moreover the uiqueess of the fixed poit follows from coditio (1.2).

COMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS 45 Example 5: Let X = [0, 1] equipped with,,, t d x y x y ad defie M x y t for all x,, y X ad t t d x y 0. Clearly (X, M, *) is a fuzzy metric space with a * b = mi {a, b}. Defie self maps A, B, S, T : X X as 1 1 0, 0 x x 0 x 2 A 2 x, B x 1 1 1, x 0 x 2 2 2 S x 1 4 0, x x x 2 5, T x 1 4 1 x x 1 x 2 5 1 1,, 0 0 0. 2 Cosider x the Ax Sx ad ASx A S lim M ASx, S0, t lim M 0, 0, t 1 for all t 0. Thus, This implies that A ad S are semi-compatible. Now, 1 1 Bx Tx for all x 0, ad BTx Bx x Tx TBx for all x 0,. 2 2 Which implies that B ad T are weak compatible. Also, A, B, S ad T are all discotiuous. Moreover, AX T X ad BX S X. Here, A, B, S ad T satisfy all the coditios of our theorem with k (0, 1) ad A, B, S ad T have a uique commo fixed poit x = 1/2. If we take A = B = f ad S = T = g i theorem 1, we get the followig corollary:

46 M. A. KHAN, SUMITRA AND R. CHUGH Corollary 1: Let (X, M, *) be a fuzzy metric space with a * a a for all a [0, 1] ad with coditio (FM-6) ad let A, B, S, T : X X be self maps satisfyig (1.1), (1.2), (1.4), (1.5) ad the pairs (A, S) ad (B, T) are semicompatible. The A, B, S ad T have a uique commo fixed poit i X. Proof: As semi-compatibility implies weak compatibility, the proof follows from theorem 1. O takig A = B i theorem 1, we have the followig corollary. Corollary 2: Let (X, M, *) be a fuzzy metric space with a * a a, a [0, 1] ad A, S ad T : X X satisfyig (1.4) (i) AX TX SX (ii)the pair (A, S) is semi compatible ad (A, T) is weak compatible (iii) There exists k (0, 1) such that M for Ax, Ay, kt M Sx, Ax, t M Ty, Ay, t M Sx, Ty, y M Ty, Ax, t M Sx, Ay, 2 t all x, y X, 0,2 ad t 0. (iv) There exists F F 5 such that,,,,,,,,,,, M Sy, Ty, t M Ax Ay t M Ax Ty t M SY Ay t F 0 M Ax Sy t. The A, S ad T have a uique commo fixed poit i X. Now, takig S = I ad T= I i theorem 1, we get the followig corollary. Corollary 3: Let (X, M, *) be a fuzzy metric space with a * a a for all a [0, 1] ad with coditio (FM-6) ad let A, B: X X be self maps satisfyig (1.1), (1.4) ad (iii) There exists k (0, 1) such that

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