FIXED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE

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1 Mohia & Samaa, Vol. 1, No. II, December, 016, pp ORIGINAL RESEARCH ARTICLE OPEN ACCESS FIED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE 1 Mohia S. *, Samaa T. K. 1 Deparme of Mahemaics, Sudhir Memorial Isiue, Kolkaa-70013, Wes Begal, Idia Deparme of Mahemaics, Uluberia College, Uluberia, Howrah, Wes Begal, Idia * Correspodig auhor s sumi.mohia@yahoo.com Received 8 Sepember, 015; Revised 17 February, 016 ABSTRACT I his paper, we have esablished some fixed fuzzy poi heorems ad commo fixed fuzzy poi heorems for fuzzy mappigs saisfyig a coracive ype codiio oher ha fuzzy Baach coracive ype codiio i complee fuzzy meric spaces. 010 Mahemaics Subjec Classificaios: 47H10, 54E35, 54A40 Key words: Fuzzy Se, Fuzzy mappig, Fixed fuzzy poi, Fuzzy meric space. INTRODUCTION I may scieific ad egieerig applicaios, cocep of fuzzy se plays a impora role. I mahemaical programmig, problems are expressed as opimizig some goal fucios uder give cerai cosrais. There are some real-life problems havig muliple objecives. Fuzzy ses are oe of he possible mehods o ge feasible soluios ha brig us o opimum of all objecive fucios. The cocep of fuzzy se was iroduced iiially by Zadeh [10] i Sice he, o use his cocep i opology ad aalysis may auhors have expasively developed he heory of fuzzy ses ad is applicaios. Heliper firs iroduced he cocep of fuzzy mappigs ad proved a fixed-poi heorem for fuzzy mappigs [3]. Sice he, may fixed-poi heorems for fuzzy mappigs have bee obaied by may auhors [5, 1]. Kramosil ad Michalek [4] iroduced he cocep of fuzzy meric spaces (briefly, FM-spaces) i 1975, which opeed a aveue for furher developme of aalysis i such spaces. Laer o, i is modified ha a few coceps of mahemaical aalysis have bee developed by George ad Veeramai []. May auhors have iroduced he cocep of fixed poi heorems i fuzzy meric space i differe ways [7, 8]. I his paper, we have esablished some fixed fuzzy poi heorems ad commo fixed fuzzy poi heorems for fuzzy mappigs saisfyig a coracive ype codiio oher ha fuzzy Baach coracive ype codiio i complee fuzzy meric spaces. 34

2 Mohia & Samaa, Vol. 1, No. II, December, 016, pp PRELIMINARIES We quoe some defiiios ad saemes of a few heorems which will be eeded i he sequel. Defiiio.1 [9]: A biary operaio : 0,1 0,1 0,1 is coiuous he followig codiios: - orm if saisfies i is commuaive ad associaive; ii is coiuous; iii a a a 1 0,1 ; iv a b c d wheever a c, b d Defiiio. []: The 3 uple oempy se, is a coiuous i x y,, 0; ii x y,, 1 if ad oly if x y; iii x y y x,,,, ; ad a b c d,,, 0,1 ;,, is called a fuzzy meric space if - orm ad is a fuzzy se i 0, iv x y s y z x z s,,,,,, ; v x, y,. : 0, 0,1 is coiuous; for all x, y, z is a arbirary saisfyig codiios: ad, s 0. I is oed ha x, y, o. ca be hough of as he degree of earess bewee x ad y wih respec Le be a meric liear space ad,, be a fuzzy meric space. A fuzzy se of is a eleme of I where I 0,1. For A, B I we deoe A Bif ad oly if A x B x for each x. For 0,1 he fuzzy poi x of is he fuzzy se of give by x y if y x ad x y 0 else [1]. The - level se of A, deoe by A, is defied by 35

3 Mohia & Samaa, Vol. 1, No. II, December, 016, pp : if 0,1 A x A x A x : A x 0 where B deoes he closure of he (o-fuzzy) se B. Heilper [3] called a fuzzy mappig, a mappig from he se of as follows: A W( ) if ad oly if A is a compac ad covex i i o a family W( ) sup A x : x 1. I his coex, we have give he followig defiiios: Defiiio.3: Le A, B W( ) 0,1 ad. Defie P ( A, B, ) sup ( a, b, ) : a A, b B ad 0 D ( A, B, ) H ( A, B, ) I defied 0,1 for each H ( A, B, ) if if sup x, y,, if sup x, y, x B y A x A y B The fucio P is called a - space. I is easy o see ha P is o-decreasig fucio of. H is he Hausdorff fuzzy meric. Noaio.4: Le be a meric space ad 0,1 W ( ) A I : A is oempy, compac ad covex. Cosider he followig family W ( ) : Defiiio.5: Le x be a fuzzy poi of. We will say ha x is a fixed fuzzy poi of he fuzzy mappig F over if x F x (i.e., he fixed degree of x is he leas ). ad Defiiio.6: Le,, be a fuzzy meric space, x, 0,1 r, 0,,, /,, 1. The,, B x r y x y r B x r is called o ope ball ceered a x of radius r wih respec o. 36

4 Mohia & Samaa, Vol. 1, No. II, December, 016, pp Defiiio.7: Le,, be a fuzzy meric space ad P. is said o be a closed se i,, if ad oly if sequece i P coverges o x P i.e., iff lim x, x, 1 x. xp x Defiiio.8: Le,, be a fuzzy meric space, S x, r, y / x, y, 1 r. Hece,, x of radius some y S x, r,. r wih respec o P x r 0,1,, 0, S x r is said o be a closed ball ceered a iff. Ay sequece x i S x, r, coverges o Defiiio.9: A sequece lim x, y, 1. x x i fuzzy meric space is said o coverge o x if ad oly if A sequece x lim x p, x, 1. x i fuzzy meric space is said o be a Cauchy sequece if ad oly if A fuzzy meric space coverge i.,, is said o be complee if ad oly if every Cauchy sequece i is Lemma.10 [6]: Le x, y, x, y,,, be fuzzy meric space. If x x ad y y i,, he as for all 0 i, he se of all real umbers. Lemma.11: Le x, A W characerisic fucio of se x. If x A Proof: If x A he x A ad x be a fuzzy se wih membership fucio equal a if ad oly if P x, A, 1 for each 0, 1 for each 0,1, P x A x y Coversely, if P x, A, 1, he x y each 0, 1. Thus x A..,, sup,, 1. y A sup,, 1. I follows ha x A A for y A 37

5 Mohia & Samaa, Vol. 1, No. II, December, 016, pp P x, A, x, y, P y, A, Lemma.1: x, y P x, A, sup x, z, sup x, y, y, z, z A z A Proof: x, y, P y, A, Lemma.13: If x 0 A, he P x 0, B, D A, B, Proof: P x 0, B, sup x 0, y, y B x y D A B if sup,,,, x A z B MAIN RESULTS for each B W Theorem 3.1: Le from o W,, be a complee fuzzy meric space ad F saisfyig he followig codiio: be a coiuous fuzzy mappig D F x, F y, k mi x, y,, P x, F x,, P y, F y,, P x, F y,, P y, F x, (1) for all x, y, 0, 1 ad 1 k 0, 4. The here exiss x Proof: Le x0 such ha x is a fixed fuzzy poi of F. suppose ha here exiss x 1 F x 0 compac subse of, he here exiss x F x 1. Sice Fx 1 ad by lemma.13 is a oempy 38

6 Mohia & Samaa, Vol. 1, No. II, December, 016, pp we ge, x 1, x, k P x 1, F x 1, k D F x 0, F x 1, k By iducio we cosruc a sequece x i such ha x F x 1 x, x 1, k P x, F x, k D F x 1, F x, k mi x, x,, P x, F x,, P x, F x,, 1 1 P x, F x,, P x, F x, x 1 x x 1 x P x F x 1 mi,,,,,,,, ad x, x 1, P x 1, F x,, x 1, x, 1 P x, F x,, P x, F x, x 1 x x 1 x x x 1 mi,,,,, 1,,, 1, x 1, x, x, x 1, P x 1, F x,,1 4 4 x 1 x x 1 x x x 1 mi,,,,,,,,, x 1, x, x, x 1, 4 which implies ha x 1, x, x, x 1, 1,

7 Mohia & Samaa, Vol. 1, No. II, December, 016, pp x, x 1, k x 1, x, x, x 1, 4 x, x 1, x 1, x, x, x 1, k 4k x 1, x, x, x 1, k 4k x, x 1, 1 We ow verify ha x is a Cauchy sequece i,,. Le 1. p x, x p, x, x 1, 1 x 1, x, 1 x, x p, 1 x. is Cauchy sequece i,, x p 1, x p, 1 Sice is a complee, here exiss x such ha x x i,,. Now by Lemmas.1 ad.13 we have k k P x, F x, k x, x, P x, F x, 1 k k x, x, D F x, F x, k x, x, mi x 1, x,, P x 1, F x 1,, 40

8 Mohia & Samaa, Vol. 1, No. II, December, 016, pp P x, F x,, P x, F x,, P x, F x, 1 1 k x, x, mi x 1, x,, x 1, x, 4, 1,,,, P x F x P x F x, x 1, x, 4 4,,,,, P x F x x x P x, F x, k x, x, mi x 1, x,, x 1, x, 1, 4 P x, F x,, x 1, x, x, x, P x, F x,, x, x, 1 4 akig limi as, we have P x, F x, P x, F x, P x, F x, 4 k 4 k P x, F x, 1 ad by lemma.11, x F x. This complees he proof. Example. Le 0,1 ad : where x, y, x y for x, y. 1 Le 0, ad suppose F : I defied by 41

9 Mohia & Samaa, Vol. 1, No. II, December, 016, pp x 0 1 F 0 x x 0, 1 x,1 1 x 0 1 F 1 x x 0, 1 x,1 ad for z 0, 1. 1 x 0 1 F z x x 0, 1 0 x, 1 The, F 0 F z F 1 0 F F z F , ad F 0 F 1 0, 1, F z 1 0, Cosequely, k P1 F x, F y, k sup for x F x, y F 1 y k x y 1 D1 F x, F y, k H F x, F y, k 1 1 ad k k if if sup, if sup x F y 1 y F x k x y x F x 1 1 y F y k x y 1 The LHS of (1), D1 F x, F y, k 1 x, y. 4

10 Mohia & Samaa, Vol. 1, No. II, December, 016, pp ad for all x, y he RHS of (1), x, y, 1, 1,, 1, P x F x P x F y 1,, 1, ad herefore P1 y, F x, 1, P1 y, F y, k 1, mi x, y,, P x, F x,, P y, F y,, P x, F y,, P1 y, F x, 1. Thus, (1) holds. k P F x, F y, k sup for x F x, y F y k x y D F x, F y, k H F x, F y, k k k if if sup, if sup x F y y F x k x y x F x y F y k x y Now, LHS of (1), D F x, F y, k 1 x, y ad he RHS of (1), mi x, y,, P x, F x,, P y, F y,, P x, F y,, P y, F x, 1. Hece, (1) holds. Agai, we see ha k P F x, F y, k sup for x F x, y F y k x y 43

11 Mohia & Samaa, Vol. 1, No. II, December, 016, pp D F x, F y, k H F x, F y, k k k if if sup, if sup x F y y F x k x y x F x y F y k x y Now, he LHS of (1), D F x, F y, k 1 x, y ad he RHS of (1), mi x, y,, P x, F x,, P y, F y,, P x, F y,, P y, F x, 1. Hece, (1) holds. Thus (1) holds ad hece all he codiios of he Theorem (3.1) are saisfied. Applyig he Theorem (3.1), we ca coclude ha F has a fixed fuzzy poi i. Corollary 3.: Le o W,, be a complee fuzzy meric space. Le F saisfyig he followig codiio: There exiss 0,1 ad k 0,1 such ha,,,, k D F x F y x y (1) be a fuzzy mappig from i for all x, y, 0, 1 ad 1 k 0, 4. The here exiss x such ha x is a fixed fuzzy poi of F. 44

12 Mohia & Samaa, Vol. 1, No. II, December, 016, pp Theorem 3.3: Le, ad,, fuzzy mappig from i o saisfyig he codiio (1) whe x, y S x, r, assume ha 0,1 k W 0,1 be a complee fuzzy meric space. Le F be a. Moreover,,, 1 k P x F x r x. The F has fixed fuzzy poi i S x, r,. Proof: x F x 1, x F x,, 1 k P x F x r 1,, x F x 1 P x, F x, for all 1 r 1 r k N. Now,, 1 P x F x r 1 x, x, P x, F x, 1 r x, x 1, 1 r x1 S x, r, Assumig ha x, x, x,, x S x, r,. We show ha x S x, r, , 1, k D F x, F x 1, x, x 1, 1 r k P x F x P x, F x, r 1 r k x, x, 1 r Agai, 1 k P x, F x, k D F x 1, F x, x 1, x, 1 r 45

13 Mohia & Samaa, Vol. 1, No. II, December, 016, pp r P x, F x, 1 r k x, x 3, 1 r Similarly, i ca be show ha, Le 1 p x, x, 3, 3, 1, P x F x r. Thus, we see ha,, P x, F x, 1 r x, x 1, x 1, x, 1,, x x 1 r 1 r 1 r 1 r 1 r x, x, x S x, r, From he lemmas.11,.1 ad.13, we ca say ha he res of he proof is obvious. Therefore, x F x. This complees he proof. Theorem 3.4: Le mappigs from o W,, be a complee fuzzy meric space. Le F ad G saisfyig he followig codiio: be coiuous fuzzy D F x, G y, k mi x, y,, P x, F x,, P y, G y,, P x, G y,, P y, F x, for all x, y, 0, 1 ad 1 k 0, 4. 46

14 Mohia & Samaa, Vol. 1, No. II, December, 016, pp The here exiss x Proof: Le x0 The here exiss such ha x is a fixed fuzzy poi of F, G., Sice Fx 0 x, he here exiss x G x 1 is oempy subse of, he here exiss x 1 F x 0 such ha x F x 1 also sice G x 1 ad by lemma.13, we ge x, x, k P x, G x, k D F x, G x, k mi x 0, x 1,, P x 0, F x 0,, P x 1, G x 1,, P x, G x,, P x, F x, mi x 0, x 1,, x 0, x 1, P x 1, F x 0,, x 1, x, P x, G x 1,, x 0, x 1, P x 1, G x 1,,1 x 0 x 1 x 0 x 1 x 1 x mi,,,,, 1,,, 1, x 0, x 1, x 1, x, 4 x 0, x 1, x 1, x, 1, 1 4. is oempy subse of By iducio we cosruc a sequece x i such ha x F x 1 x 1 G x ad x, x 1, k P x, G x, k D F x 1, G x, k, 47

15 Mohia & Samaa, Vol. 1, No. II, December, 016, pp mi x, x,, P x, F x,, P x, G x,, 1 1 P x, G x,, P x, F x, x, x 1, k x 1, x, x, x 1, 4 As i he above heorem 3.1, he proof is similar. x is Cauchy sequece i,,. Sice is a complee, here exiss x such ha x x i,,. Now by Lemmas.1 ad.13 we have k k P x, G x, k x, x, P x, G x, 1 k k x, x, D F x, G x, k x, x, mi x 1, x,, x 1, x, 1, P x, G x,, 4 1,,,,, 4 8, 4,,, x x x x P x G x x x 4 1 akig limi as, we have P x, G x, P x, G x, P x, G x, 4 k 4 k P x, G x, 1 ad by lemma.11, x G x. Similarly, x F x This complees he proof.. 48

16 Mohia & Samaa, Vol. 1, No. II, December, 016, pp ACKNOWLEDGMENTS The auhors wish o hak he chief edior ad he reviewers for heir valuable suggesios o recify he paper. REFERENCE [1] Fisher B, Fixed poi heorems for fuzzy mappigs, J. Appl. Mah. ad Compuig, 19 (005) (1- ), [] George A & Veeramai P, O Some resul i fuzzy meric space, Fuzzy Se ad sysems, 64 (1994), [3] Heilper S, Fuzzy mappigs ad fixed poi heorem, J. Mah. Aal. Appl., 83 (1981), [4] Kramosil O & Michalek J, Fuzzy meric ad saisical meric spaces, Kybereica, 11 (1975), [5] Rhoades B E, Fixed poi of some fuzzy mappigs, Soocho J. Mah., (1996) 1, [6] Samaa T K & Mohia S, O Fixed poi heorems i Iuiioisic Fuzzy Meric Space I, Ge. Mah. Noes, 3 (011), 1 1. [7] Samaa T K & Mohia S, Fixed poi heorems O Fuzzy Meric Space, Lap Lamber Academic Publishig ( )-ISBN-13: [8] Samaa T K & Mohia S, Commo fixed poi heorems for sigle ad se-valued maps i o Archimedea fuzzy meric spaces, Aca Uiv. Sapieiae, Mahemaica, 4 (01), [9] Schweizer B & Sklar A, Saisical meric spaces, Pacific Joural of Mahemaics, 10 (1960), [10] Zadeh L A, Fuzzy ses, Iformaio ad corol, 8 (1965),