Common Fixed Point Theorems for Occasionally Weakly. Compatible Maps in Fuzzy Metric Spaces

Transcription

1 International athematical Forum, Vol. 6, 2011, no. 37, Common Fixed Point Theorems for Occasionally Weakly Compatible aps in Fuzzy etric Spaces. Alamgir Khan 1 Department of athematics, Eritrea Institute of Technology Asmara, Eritrea ( N. E. Africa ) alam3333@gmail.com Sumitra 2 Department of athematics, Eritrea Institute of Technology Asmara, Eritrea ( N. E. Africa ) mathsqueen_d@yahoo.com Renu Chugh 3 Department of athematics,.d.u, Rohtak-Haryana, India chughrenu@yahoo.com Abstract. The intent of this paper is to introduce the notion of occasionally weakly compatible (owc) maps and prove common fixed point theorems for single and set valued maps by using a contractive condition of integral type in fuzzy metric spaces. Our results are independent of the continuity requirement of the maps and completeness of the space. Several known results are generalized in this note. athematics Subject Classification: 47H10, 54H25 Keywords: Weak Compatible maps, Occasionally weakly compatible maps, fixed points and fuzzy metric space

2 1826. Alamgir Khan, Sumitra, R. Chugh 1. Introduction The concept of fuzzy set coined by Zadeh [15] proved a turning point in the development of athematics and laid the foundation of fuzzy mathematics. Cosequently, the last three decades remained productive for various authors like Deng [9], Erceg [10], Kaleva and Siekkala [12], Kramosil and ichalek [11], Pant [13], S. Sessa [14] etc. In recent years several fixed point theorems for single and set valued maps are proved and have numerous applications and by now, there exists a considerable and rich literature in this domain. Various authors have discussed and studied extensively various results on coincidence, existence and uniqueness of fixed and common fixed points by using the concept of weak commutativity, compatibility, non-compatibility and weak compatibility for single and set valued maps satisfying certain contractive conditions in different spaces and they have been applied to diverse problems. Note that common fixed point theorems for single and set valued maps are interesting and play a major role in many areas. ore recently, Al-Thagafi and N. Shahzad [2] weakened the concept of compatibility by giving a new notion of occasionally weakly compatible (owc) maps which is most general among all the commutativity concepts. No wonder that the notion of occasionally weakly compatible (owc) maps bas become an area of interest for specialists in fixed point theory; see [1], [3], [4], [5], [6], [7] and [8] The main purpose of our paper is to introduce the concept of occasionally weakly compatible (owc) maps in fuzzy metric space and to prove common fixed point theorems for single and set valued maps under strict contractive condition of integral type. Our improvements in this paper are five-fold as; (i) Relaxed the continuity of maps completely (ii) Completeness of the space removed (iii) inimal type contractive condition used (iv) The condition ( x y lim t,, = 1 not used (v) Weakened the concept of compatibility by a more general concept of occasionally weak compatible (owc) maps.

3 Common fixed point theorems 1827 We first give some preliminaries and definitions. 2. Preliminaries Definition 2.1. A binary operation :[ 0,1] [ 0,1] [ 0,1] is continuous t norm if satisfies the following conditions: (i) is commutative and associative (ii) is continuous a 1= a for all a 0,1 (iii) [ ] (iv) a b c d whenever a c and b d, a, b, c, d [ 0,1]. Definition 2.2. A triplet ( X,, ) is said to be a fuzzy metric space if X is an arbitrary set, is a continuous t norm and is a fuzzy set on X 2 [ 0, ) satisfying the following; (F-1) ( x, y, t ) > 0 (F-2) ( x, y, = 1 if and only if x = y. (F-3) ( xyt,, ) = ( yxt,, ) (F-4) ( xyt,, ) ( yzs,, ) ( xzt,, + s) (F-5) ( x, y, ):[ 0, ) [ 0,1] is left continuous for all x, y, z Note that ( xyt,, ) respect to t. X and st>, 0. can be thought of as the degree of nearness between x and y with Example 1. (Induced fuzzy metric space) Let (, ) X d be a metric space and a b = ab for all ab, [ 0,1] andletd befuzzy set on X 2 [ 0, ) defined as follows; t d ( x, y, = t + d( x, y). Then ( X, d, ) is a fuzzy metric space. We call this fuzzy metric induced by a metric d. Through out the paper X will represent the fuzzy metric space ( X,, ) and CB ( X ) set of all non-empty closed and bounded sub-sets of X. For A, B CB ( X ) t > 0, denote H ABt,, = sup abt (,, ); a Ab, B { }, the and for every

4 1828. Alamgir Khan, Sumitra, R. Chugh and δ ( A, Bt, ) = Inf{ ( abt,, ); a Ab, B}. If A consists of a single point a, we write δ ( A, Bt, ) δ ( abt,, ) a single point b, we write δ ( A, Bt, ) = ( abt,, ). It follows immediately from definition that =. If B also consists of δ ( AB δ ( BA,, =,, 0 δ (,, ) = 1 = = { } for all A, B CB ( X ) A Bt A B a Definition 2.3. A point x X is called a coincidence point (resp. fixed poin of A : X X B : X CB X if Ax Bx (resp. x = Ax Bx ), Definition 2.4. aps A : X X and B : X CB ( X ) are said to be compatible if ABx CB ( X ) for all x X and linn H ( ABx n, BAx n, t ) = 1 whenever { n } sequence in X such that Bx CB ( x ) Ax x. n and n Definition 2.5. aps A : X X : they commute at coincidence points. i.e., if ABx and B X CB ( X ) x is a are said to be weakly compatible if = BAx, whenever Ax Bx. Definition 2.6. aps A : X X and B : X CB ( X ) are said to be occasionally weakly compatible (owc) if there exists some point x X such that Ax Bx and ABx BAx. Clearly weakly compatible maps are occasionally weakly compatible (owc). However, the converse is not true in general as shown in the following example. Example 2. Let X = [ 0, ) with a b = min { a, b} for all, [ 0,1] ( x, y, Define the maps A : X = t t + d ( x, y ) for all t > 0. X B : X CB X by setting and ab and Ax 0,0 x < 1 x + 1,1 x <, Bx { } [ x ] 0,0 x < 1 1, + 2,1 x < Here '1' is a coincidence point of A and B but A and B are not weakly compatible as

5 Common fixed point theorems 1829 BA () 1 [ 1,4] AB () 1 [ 2,4] = =. Hence A and B are not weakly compatible. But A and B are occasionally weakly compatible (owc) as A and B are weakly compatible at 0 A 0 B 0 AB 0 BA 0 A 0 = 0 B 0 = 0 x = as and. i.e., { } { } Now, we prove our main result. 3. ain Result Theorem 1. Let ( X,, ) be a fuzzy metric space with t t = t for all t [ 0,1] A, B : X X and ST, : X CB( X). Let be single and set valued mappings respectively such that the maps (A,S) and (B,T) are (owc) and satisfy the inequality (1.1) φ Sx, Ty, kt m x, y, t tdt> φ tdt for all x, y X 0,1 φ :0,1 0,1 0 0 is a function which is sum able, Lebseque integrable, non-negative and such that ε φ tdt > 0 for each ε > 0. 0 where, k where [ ] [ ] ( Ax, By,, H ( Ax, Sx,, (1.2) m( x, y, min H ( By, Ty,, H ( Ax, Ty, α H ( By, Sx,( 2 α) t ) for every x, y X, t 0, α ( 0, 2) in X. >. Then A, B, S and T have unique common fixed point Proof. Since the pairs (A,S) and (B,T) are occasionally weakly compatible (owc) maps, therefore, there exist two elements uvin, X such that Au Su, ASu SAu and Bv Tv, BTv TBv. First we prove that Au As Au = Bv. Su so AAu ASu SAu, Bv Tv so BBv BTv TBv and hence 2 2 (,, ) δ (,, ) AuBvt SAuTBvt and if Au Bv, then δ ( SAu TBv t ),, < 1. Using

6 1830. Alamgir Khan, Sumitra, R. Chugh (1.2) for x = Au, y = Bv (,, α ) (,,( 2 α) ) AAu, BBv, t, H AAu, SAu, t, H BBv, TBv, t, m( Au, Bv, = min H AAu TBv t H BBv SAu t (,, α ) (,,( 2 α) ) AAu, BBv, t, AAu, SAu, t, BBv, TBv, t, min AAu TBv t BBv SAu t (1.3) Since is continuous, letting α 1 in (1.3), we get m( Au, Bv, ( A u, B v,, ( A u, SAu,, ( BvTBvt,, ), ( AuTBvt,, ) ( Bv, SAut, ) { δ ( SAu TBv t ) δ ( SAu TBv t ) δ ( TBv SAu t )} min min,,,1,1,,,,, = ( SAu, TBv, t ) δ (1.4) From (1.1) and (1.4), we have SAu, TBv, kt m Au, Bv, t SAu, TBv, t φ tdt> φ tdt φ tdt, a contradiction Hence Au = Bv. 2 2 Also, ( AuBut,, ) δ ( SAuTut,, ), ( AuTut,, ) δ ( SAuTut,, ). Now, we claim that Au = u. It not, then δ ( SAu, Tu, t ) < 1. Considering (1.2) for Au = x, u = y, α = 1 (,, ) (,, ) AAu, Bu, t, H AAu, SAu, t, H Bu, Tu, t, m( Au, u, = min H AAu Tu t H Bu SAu t = (,, ) (,, ) AuBut,,, H AuSAut,,, H BuTut,,, min H A u Tu t H Bu SAu t

7 Common fixed point theorems (,, ) (,, ) AuBut,,, AuSAut,,, BuTut,,, min A u Tu t Bu SAu t min { δ ( SAuTut,, ),1,1, δ ( SAuTut,, ) δ ( TuSAut,, )} = δ ( SAu, Tu, t ) i.e., m( Au, u, ( SAu, Tu, t ) δ (1.5) From (1.1) and (1.5) we have SAuTu,, kt m Auut,, SAuTut,, φ tdt> φ tdt φ tdt, which is again a contradiction and hence Au = u. Similarly, we can get Bv = v. Thus A, B, S and T have a common fixed point in X. For uniqueness let u u be another fixed point of A, B, S and T, then (1.2) gives (,, α ) (,,( 2 α) ) Au, Bu, t, H Au, Su, t, H Bu, Tu, t, m( u, u, = min H Au Tu t H Bu Su t Letting α 1, δ δ ( Au, Tu, t ) δ ( Bu, Su, t ) Au, Bu, t, Au, Su, t, Bu, Tu, t, m( u, u, = min δ { } (,, ) = min (,, ),1,1, δ (,, ) = ( Su, Tu, t ) m u u t Su Tu t Su Tu t δ (1.6) Again from (1.1) and (1.6), we obtain

8 1832. Alamgir Khan, Sumitra, R. Chugh δ ( Su, Tu, kt ) m( u, u, t ) δ ( Su, Tu, t ) φ tdt> φ tdt φ tdt Which yields Su = Tu. i.e., u = u. Thus, A, B, S and T have unique common fixed point. Now, we furnish our theorem with an example. Example 3. Let ( X,, ) be a fuzzy metric space in which X = R +, a b = min { a, b} t for all ab, [ 0,1] such that such that ( x, y, = for all t > 0. + Define the maps A, B, S and T on X as follows; t x y A x S x 2x 1, x 5 2 x, x > 5 { 1, } x 2 [ ] < 2 x,2x + 5, x 2, B ( x ), T ( x ) 3 2 x, x 1, x + 1, x > 1 { 1, } x 1 [ x x + ] =, 2, otherwise Here the pairs (A,S) and (B,T) are owc. Define φ :0,1 [ ] [ 0,1] as φ ( 0) = 0, φ ( 1) = 1 and ( s ) contractive condition (1.1) is satisfied for all t > 0. φ = s for 0< s < 1, then the Thus all the conditions of our theorem are satisfied and '1' is the common fixed point of A, B, S and T. Corollary 1. Let ( X,, ) be a fuzzy metric space with t t = t for all t [ 0,1] A, B : X X and ST, : X CB( X). Let be single and set valued mappings respectively such that the maps (A,S) and (B,T) are (owc) and satisfy the inequality (1.1) φ Sx, Ty, kt m x, y, t tdt> φ tdt for all x, y X 0,1 φ :0,1 0,1 0 0 is a Lebseque integrable mapping which is sum able, non-negative and such that, k where [ ] [ ]

9 Common fixed point theorems 1833 ε 0 φ tdt> 0 for each ε > 0. where ( Ax, By, t ), H ( Ax, Sx, αt ) H ( By, Ty, ( 2 α) t ) (1.7) m( x, y, min, H ( Ax, Ty, β H ( By, Sx, ( 2 β) for every x, y X, t 0, αβ, ( 0,2) in X. >. Then A, B, S and T have unique common fixed point If we take A=B and S=T in our theorem, then we get the following results. Corollary 2. Let ( X,, ) be a fuzzy metric space with t t = t for all t [ 0,1] A : X X and S : X CB ( X ). Let be single and set valued mappings respectively such that the pair (A,S) is owc and satisfy the inequality (1.1) φ 0 0 Sx, Ty, kt m x, y, t tdt> φ tdt for all x, y X, ( 0,1) k where φ :0,1 [ ] [ 0,1] is a Lebseque integral mapping which is summable, non-negative and such that ε φ ( tdt ) > 0 for each ε > 0 0 where, ( Ax, Ay, t ), H ( Ax, Sx, βt ) H ( Ay, Sy, ( 2 β) t ), (1.8) m( x, y, = min H ( Ax, Sy, α H ( Ay, Sx, ( 2 α) x, y X, t > 0, αβ, 0,2. Then A and S have unique common fixed point in X. for every Corollary 3. Let ( X,, ) be a fuzzy metric space with t t = t for all t [ 0,1] A : X X and S : X CB ( X ). Let be single and set valued mappings respectively such that the pair (A,S), (A,T) are owc and satisfy the inequality

10 1834. Alamgir Khan, Sumitra, R. Chugh (1.1) φ 0 0 Sx, Ty, kt m x, y, t tdt> φ tdt for all x, y X, ( 0,1) k where φ :0,1 [ ] [ 0,1] tdt > 0 is a Lebseque integral mapping which is sumable, non-negative and such that φ for each ε > 0 ( Ax, Ay,, H ( Ax, Sx,, (1.9) m( x, y, min H ( Ay, Ty, t ), H ( Ax, Ty, αt ) H ( Ay, Sx, ( 2 α) t ) for every x, y X, t 0, α ( 0, 2) X. >. Then A, S and T have unique common fixed point in Now, we give the projection of above results in metric spaces. Let ( X, d ) denotes a metric space and CB ( X ) the family of all non empty closed and bounded subsets of X. Let H be the Hausdorff metric on CB ( X ) induced by the metric d; i.e., { } (, ) max sup x A (, ),sup y B (, ) (, ) = inf { (, ): } H AB = d xb d Ay for, d x A d x y y A A B in ε 0 CB X, where Theorem 2. Let ( X, d ) be a metric space. Let A, B : X X and ST, : X CB( X) be single and set valued mappings respectively such that the maps (A,S) and (B,T) are owc and satisfy the inequality (1) φ H Sx, Ty d x, y 0 0 tdt< k φ tdt d Ax, By, d Ax, Sx, d By, Ty, where d ( x, y ) = max d ( Ax, Ty ) + d ( By, Sx ) for every x, y X, k ( 0,1), and φ : R R is a Lebseque interable mapping which is sum able, non-negative and such that ε 0 φ tdt> 0 for each ε > 0

11 Common fixed point theorems 1835 Then A, B, S and T have unique common fixed point in X. Proof. The proof follows from theorem 1. Considering the induced fuzzy metric space t ( X,, ) where a b = min { a, b} and ( x, y, = t + d x, y Theorem 3. Let ( X, d ) be a metric space. Let A, B : X X and ST, : X CB( X) be single and set valued mappings respectively such that the maps (A,S) and (B,T) are owc and satisfy the inequality (1) φ H Sx, Ty d x, y 0 0 tdt< k φ tdt,,,, where d ( x, y ) (, ), d Ax Sx + d By Ty, d Ax Ty + d Sx By d Ax By = max for every x, y X, k ( 0,1) and φ : R R is a Lebseque integrable mapping which is sum able, non-negative and such that ε φ tdt> 0 for each ε > 0 0 Then A, B, S and T have unique common fixed point in X. Proof. The proof follows from theorem 1. Considering the induced fuzzy metric space t ( X,, ) where a b = min { a, b} and ( x, y, = t + d x, y Remark 1. In view of theorem 2, it is clear that the some results of [1,2,3,4,5,6,7,8] are special cases of our main results in fuzzy metric space References [1]. Abbas and B. E. Rhoades, Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized condition of integral type, Fixed point theory appl., Art. ID 54101, 9pp [2]. A. Al-Thagafi and N. Shahzad, A note on occasionally weakly compatible maps, Int. J. ath. Anal. 3(2)(2009), [3]. A. Al-Thagafi and N. Shahzad, Generalized I-non expansive self maps and invariant approximation, Acta athematica Sinica, English Series 24(2008),

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