VOL. 1, NO. 8, November 2011 ISSN ARPN Journal of Systems and Software AJSS Journal. All rights reserved

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1 VOL., NO. 8, Noveber 0 ISSN ARPN Journal of Syses and Sofware AJSS Journal. All righs reserved hp:// Soe Fixed Poin Theores on Expansion Type Maps in Inuiionisic Fuzzy Meric Spaces Saurabh Manro School of Maheaics and Copuer Applicaions, Thapar Universiy,Paiala (Punjab) {sauravanro@yahoo.co, sauravanro@hoail.co} ABSTRACT In his paper, we obain soe resuls on fixed poins of expansion ype apping in inuiionisic fuzzy eric space. Our resuls are he inuiionisic fuzzy version of soe fixed poin heores for expansion ype appings on differen eric spaces. Keywords: Inuiionisic Fuzzy eric space, Expansion ype aps, Coon fixed poin. INTRODUCTION Aanassove [] inroduced and sudied he concep of inuiionisic fuzzy ses as a generalizaion of fuzzy ses. In 004, Park[5] defined he noion of inuiionisic fuzzy eric space wih he help of coninuous -nors and coninuous -conors. Recenly, in 006, Alaca e al.[] using he idea of Inuiionisic fuzzy ses, defined he noion of inuiionisic fuzzy eric space wih he help of coninuous -nor and coninuous - conors as a generalizaion of fuzzy eric space due o Kraosil and Michalek[3]. In his paper, we obain soe resuls on fixed poins of expansion ype apping in inuiionisic fuzzy eric space. Our resuls are he inuiionisic fuzzy version of soe fixed poin heores for expansion ype appings on differen eric spaces.. PRELIMINARIES The conceps of riangular nors (-nors) and riangular conors (-conors) are known as he axioaic skelon ha we use are characerizaion fuzzy inersecions and union respecively. These conceps were originally inroduced by Menger [4] in sudy of saisical eric spaces. Definiion.. [6] A binary operaion * : [0,] [0,] [0,] is coninuous -nor if * saisfies he following condiions: (i) * is couaive and associaive; (ii) * is coninuous; (iii) a * = a for all a [0,] ; (iv) a * b c * d whenever a c and b d for all abcd,,, [0,]. Definiion.. [6] A binary operaion : [0,] [0,] [0,] is coninuous -conor if saisfies he following condiions: (i) is couaive and associaive; (ii) is coninuous; (iii) a 0 = a for all a [0,] ; (iv) a b c d whenever a c and b d for all abcd,,, [0,]. Alaca e al. [] using he idea of Inuiionisic fuzzy ses, defined he noion of inuiionisic fuzzy eric space wih he help of coninuous -nor and coninuous -conors as a generalizaion of fuzzy eric space due o Kraosil and Michalek [3] as : Definiion.3. [] A 5-uple (X, M, N, *, ) is said o be an inuiionisic fuzzy eric space if X is an arbirary se, * is a coninuous -nor, is a coninuous -conor and M, N are fuzzy ses on X [0, ) saisfying he following condiions: (i) M(x, y, ) + N(x, y, ) for all xy, X and > 0; (ii) M(x, y, 0) = 0 for all xy, X ; (iii) M(x, y, ) = for all xy, X and > 0 if and only if x = y; (iv) M(x, y, ) = M(y, x, ) for all xy, X and > 0; (v) M(x, y, ) * M(y, z, s) M(x, z, + s) for all xyz,, X and s, > 0; (vi) for all xy, X, M(x, y,.) : [0, ) [0, ] is lef coninuous; (vii) li M( xy,, ) = for all xy, X and > 0; (viii) N(x, y, 0) = for all xy, X ; (ix) N(x, y, ) = 0 for all xy, X and > 0 if and only if x = y; (x) N(x, y, ) = N(y, x, ) for all xy, X and > 0; (xi)n(x, y, ) N(y, z, s) N(x, z, + s) for all xyz,, X and s, > 0; (xii) for all xy, X, N(x, y,.) : [0, ) [0, ] is righ coninuous; (xiii) li N( xy,, ) = 0 for all xy, X. Then (M, N) is called an inuiionisic fuzzy eric space on X. The funcions M(x, y,) and N(x, y, )denoe he degree of nearness and he degree of non-nearness beween x and y w.r.. respecively. Reark.: [] Every fuzzy eric space (X, M, *) is an inuiionisic fuzzy eric space of he for (X, M, - M, *, ) such ha -nor * and -conor are associaed as x y = -((-x) * (-y)) for all xy, X. 58

2 VOL., NO. 8, Noveber 0 ISSN ARPN Journal of Syses and Sofware AJSS Journal. All righs reserved Reark.: [] In inuiionisic fuzzy eric space (X, M, N, *, ), M(x, y, *) is non-decreasing and N(x, y, ) is non-increasing for all xy, X. Alaca, Turkoglu and Yildiz [] inroduced he following noions: Definiion.4:[] Le (X, M, N, *, ) be an inuiionisic fuzzy eric space. Then (a) a sequence {x n } in X is said o be Cauchy sequence if, for all > 0 and p > 0, li n M(x n+p, x n, ) = and li n 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, li n M(x n, x, ) = and li n N(x n, x, ) = 0. Definiion.5: [] An inuiionisic fuzzy eric space (X, M, N, *, ) is said o be coplee if and only if every Cauchy sequence in X is convergen. Exaple.: [] Le X = {/n : n N } {0} and le * be he coninuous -nor and be he coninuous -conor defined by a * b = ab and a b = in{, a+b} respecively, for all ab, [0,]. For each (0, ) and xy, X, define (M, N) by, > 0, M(x, y, ) = = x-y, > 0, + x-y =0 + x-y 0 =0 hp:// and N(x, y, ) Clearly, (X, M, N, *, ) is coplee inuiionisic fuzzy eric space. 3. LEMMA The proof of our resul is based upon he following leas of which he firs wo are due o Alaca e al.[]: Lea 3.. Le (X, M, N, *, ) be inuiionisic fuzzy eric space and for all xy, X, > 0 and if for a nuber k > such ha M(x, y, k) M(x, y, ) and N(x, y, k) N(x, y, ) Then, x = y. Lea 3.. Le (X, M, N, *, ) be inuiionisic fuzzy eric space and for all xy, X, > 0 and if for a nuber k > such ha M( yn+, yn+, ) M( yn+, yn, k), N( yn+, yn+, ) N( yn+, yn, k). Then { y n } is a Cauchy sequence Theore 4.: Le ( X, M, N,*, ) be a coplee inuiionisic fuzzy eric space and f be a self ap of X ono iself. There exis k > such ha M ( fx, fy, k) M ( x, y, ), () N( fx, fy, k) N( x, y, ). > 0. Then f has a unique fixed poin Proof: Le x0 X as f is ono, here is an eleen x f ( x0). In he sae way, xn f ( xn ) for all n =, 3, 4,.... Thus, we ge a sequence{ x n }. If x for soe, hen x is a fixed poin of f. Now, suppose xn xn for all n =,, hen i follows fro () ha for all n =, 3, 4,...., > 0 M ( x, x, k) = M fx, fx, k M ( x, x, ), n n+ n+ n+ n+ n+ N( x, x, k) = N fx, fx, k N( x, x, ). n n+ n+ n+ n+ n+ By Lea 3., { x n } is a Cauchy sequence in X. As X is coplee. Therefore, { x n } has a lii u X. As f is ono, here is an eleen v X such ha v f ( u) M ( x, u, k) = M fx, u, k = M fx, fv, k M ( x, v, ), n n+ n+ n+ N( x, u, k) = N fx, u, k = N fx, fv, k N( x, v, ) n n+ n+ n+ which gives, as n, we ge = Muuk (,, ) Muv (,, ), 0 = Nuuk (,, ) Nuv (,, ). This gives, u = v = f(v). Therefore, u = v is fixed poin of f. Uniqueness: Le u and w be wo fixed poins of f. Then () gives, M ( fu, fw, k) = M ( u, w, k) M ( u, w, ), N( fu, fw, k) = N( u, w, k) N( u, w, ). Therefore, by Lea 3., u = w, which shows ha u is unique fixed poin of f. Theore 4.: Le ( X, M, N,*, ) be a coplee inuiionisic fuzzy eric space wih a* b= in{ ab, } and a b= ax{ ab, } for all ab, [0,] and f be a self ap of X ono iself. There exis k > such ha 4. MAIN RESULTS M ( fx, fy, k) M (, x y,)* M (, x fx,)* M ( y, fy,), N( fx, fy, k) N(, x y,) N(, x fx,) N( y, fy,). () 59

3 VOL., NO. 8, Noveber 0 ISSN ARPN Journal of Syses and Sofware AJSS Journal. All righs reserved hp:// > 0. Then f has a unique fixed poin Proof: Le x0 X as f is ono, here is an eleen x f ( x ). In he sae way, x f ( x ) for 0 n n all n =, 3, 4... Thus, we ge a sequence{ x n }. If x for soe, hen xn xn x is a fixed poin of f. Now, suppose for all n =,, hen i follows fro () ha for all n =, 3, 4,...., > 0 M ( x, x, k) = M fx, fx, k M ( x, x,)* M ( x, fx,)* M ( x, fx,), n n+ n+ n+ n+ n+ n+ n+ n+ n+ N( x, x, k) = N fx, fx, k N( x, x,) N( x, fx,) N( x, fx,). This gives n n+ n+ n+ n+ n+ n+ n+ n+ n+ M( x, x, k) M( x, x,)* M( x, x,), n n+ n+ n n+ n+ N( x, x, k) N( x, x,) N( x, x,). n n+ n+ n n+ n+ (3) Now, Suppose M( x, x,) < M( x, x,), n+ n+ n+ n N( x, x,) > N( x, x,). n+ n+ n+ n Therefore, i follows fro (3) ha M( x, x, k) M( x, x, ), n n+ n+ n+ N( x, x, k) N( x, x, ). n n+ n+ n+ Therefore, by lea 3., we have, xn n +, a conradicion. Hence, M( x, x,) M( x, x,), n+ n+ n+ n N( x, x,) N( x, x,). n+ n+ n+ n Hence, equaion (3) gives, M ( x, x, k) M ( x, x, ), N( x, x, k) N( x, x, ). n n+ n+ n n n+ n+ n Therefore, by lea 3., { x n } is a Cauchy sequence in As f is ono, here is an eleen v X such ha X. As X is coplee. Therefore, { x n } has a lii u v f ( u) M ( x, u, k) = M fx, u, k = M fx, fv, k M ( x,,)* v M ( x, fx,)* M (, v fv,), n n+ n+ n+ n+ n+ N( x, u, k) = N fx, u, k = N fx, fv, k N( x,,) v N( x, fx,) N(, v fv,). n n+ n+ n+ n+ n+ which gives, as n, we ge = Muuk (,, ) Muv (,,)* Muu (,,)* M(, vu,) Muv (,,), 0 = Nuuk (,, ) Nuv (,,) Nuu (,,) Nvu (,,) Nuv (,,). This gives, u = v = fv. Therefore, u = v is fixed poin of f. Uniqueness: Le u and w be wo fixed poins of f. Then () gives, M ( fu, fw, k) = M (, u w, k) M (, u w,)* M (, u fu,)* M ( w, fw,), N( fu, fw, k) = N(, u w, k) N(, u w,) N(, u fu,) N( w, fw,). 60

4 VOL., NO. 8, Noveber 0 ISSN ARPN Journal of Syses and Sofware AJSS Journal. All righs reserved This gives, Muwk (,, ) Muw (,, ), Nuwk (,, ) Nuw (,, ). Therefore, by Lea 3., u = w, which shows ha u is unique fixed poin of f. Theore 4.3: Le ( X, M, N,*, ) be a coplee inuiionisic fuzzy eric space wih a* b= in{ ab, } and a b= ax{ ab, } for all ab, [0,] and f and g be wo self ap of X ono iself. There exis k > such ha hp:// M ( fx, gy, k) M (, x y,)* M (, x fx,)* M ( y, gy,), N( fx, gy, k) N(, x y,) N(, x fx,) N( y, gy,). (4) > 0. Then f has a unique fixed poin Proof: Le x 0 X as f is ono, here is an eleen x f ( x0). Since, g is ono, here is an eleen x saisfying x g ( x). In he sae way, in general, xn+ f ( xn), xn+ g ( xn+ ) for all n = 0,,, 3, 4,.... Thus, we ge a sequence{ x n } we have wo cases as follows: Case- I If x x + for all = 0,,, hen in his case, by (3), we have for all > 0 M( x, x, k) = M fx, g x, k M( x, x,)* M( x, fx,)* M( x, fx,), n n+ n+ n+ n+ n+ n+ n+ n+ n+ N( x, x, k) = N fx, g x, k N( x, x,) N( x, fx,) N( x, fx,). n n+ n+ n+ n+ n+ n+ n+ n+ n+ This gives M( x, x, k) M( x, x,)* M( x, x,), n n+ n+ n+ n+ n N( x, x, k) N( x, x,) N( x, x,). n n+ n+ n+ n+ n (5) Now, Suppose M( x, x,) > M( x, x,), N( x, x,) < N( x, x,). n+ n+ n+ n n+ n+ n+ n Therefore, i follows fro (5) ha M ( x, x, k) M ( x, x, ), N( x, x, k) N( x, x, ). n n+ n+ n n n+ n+ n Therefore, by lea 3., we have, xn n +, a conradicion. Hence, M( x, x,) M( x, x,), N( x, x,) N( x, x,). n+ n+ n+ n n+ n+ n+ n Hence, equaion (5) gives, M ( xn, xn+, k) M ( xn+, xn+, ), N( xn, xn+, k) N( xn+, xn+, ). Therefore, by lea 3., { x n } is a Cauchy sequence As, X is coplee. Therefore, { n } { x n } and { x n + } are sub sequences of{ x n }. Therefore, xn+ ux, n u as n. As f and g are ono funcions, here exiss u and w in X such ha v f ( u) and w g ( u) n n+ n+ n+ n+ n n+ n+ n+ n+ x has a lii poin As M ( x, u, k) = M fx, gw, k M ( x, w,)* M ( x, fx,)* M ( w, fw,), N( x, u, k) = N fx, gw, k N( x, w,) N( x, fx,) N( w, fw,). which gives, as n, we ge = Muuk (,, ) Muw (,,)* Muu (,,)* M( wu,,) Muw (,,), 0 = Nuuk (,, ) Nuw (,,) Nuu (,,) Nwu (,,) Nuw (,,). This gives, u = w. In he siilar paern, aking x = v and y n + in (5) and proceeds as above, we obain u = v. Therefore, u = v = w is coon fixed poin of f and g. Uniqueness, direcly follows fro (5) by aking x = u and y = z. which shows ha u is unique coon fixed poin of f and g. 6

5 VOL., NO. 8, Noveber 0 ISSN ARPN Journal of Syses and Sofware AJSS Journal. All righs reserved Case II- If x + or x, for coe where ay be even or odd posiive ineger. Wihou any loss of generaliy, Suppose is an even ineger say = p, hen x = p xp i.e. gx. p = fxp xp p Therefore, we have xp p p+ =... which shows ha { x n } is a convergen sequence and so { x n } is a Cauchy sequence The res of he proof is siilar o case-i. This coplees he proof. 5. ACKNOWLEDGEMENTS We would like o hank he referee for he criical coens and suggesions for he iproveen of y paper. REFERENCES [] C. Alaca, D. Turkoglu, and C. Yildiz, Fixed poins in Inuiionisic fuzzy eric spaces, Chaos, Solions & Fracals 9, (006). [] K. Aanassov, Inuiionisic Fuzzy ses, Fuzzy ses and syse, 0, 87-96(986). [3] I. Kraosil and J. Michalek, Fuzzy eric and Saisical eric spaces, Kyberneica, (975). [4] K. Menger, Saisical erics, Proc. Na. Acad. Sci. (USA), 8 (94), [5] J. H. Park, Inuiionisic fuzzy eric spaces, Chaos, Solions & Fracals, (004). [6] B. Schweizer and A. Sklar, Probabilisic Meric Spaces, Norh Holland Aserda, 983. hp:// 6