In 1981 Heilpern,S. introduced the concept of fuzzy mappings and prove a fixed point theorem for fuzzy mappings. In 1992 Beg, I and A.Azam [1] made.
Middle-East Journal of Scientific Research 11 (9): 1299-1302, 2012 ISSN 1990-9233 © IDOSI Publications, 2012 DOI: 10.5829/idosi.mejsr.2012.11.09.1460
On the Fixed Point Theorem for Fuzzy Mapping 1
Muhammad Iqbal Chaudhry, 1 Muhammad Naeem, 1 Syed Inayat Ali Shah and 2 Muhammad Arif 1
Department of Mathematics, Islamia College University Peshawar, Pakistan Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan
2
Abstract: In [1] Beg.I and A.Azam has proved some theorem by using complex metric space and fuzzy extension. In this paper we extended this idea to prove Kannan type fixed point theorem for fuzzy mapping satisfying new contractive type conditions. Key words: Fuzzy logic fixed point fuzzy mapping fuzzy set contractive type fuzzy mapping •
•
•
•
m,n ≥ n 0
INTRODUCTION In 1922, Banach a polish mathematician proved a theorem under appropriate conditions and showed the existence and uniqueness of a fixed point this result is called Banach fixed point theorem. This theorem is also applied to prove the existence and uniqueness of the solutions of differential equations. Many authors have made different generalization of Banach fixed theorem. A t the end of 20th century many mathematician have worked on it. In 1965 a well known mathematician Lotfi Zadeh introduced a “fuzzy set theory”. Fuzzy logic has been applied to many fields, from control theory to artificial intelligence. Butrariu [3], Chang [4] and Chitra [5] studied fixed point theorems for fuzzy sets. Weiss [9] initiated the study of fixed point theorems of fuzzy mappings. Heilpern [6] obtained a fixed point theorem for fuzzy contraction mappings, which is a fuzzy analogue of the fixed point theorem for multivalued mappings of Nadler [7]. Bose and Sahani [2] gave an improved version of Heilpern [6]. Park and Jeong [8] studied common fixed point theorems of fuzzy mappings satisfying contractive-type conditions and a rational inequality in comp lete metric spaces. In 1981 Heilpern,S. introduced the concept of fuzzy mappings and prove a fixed point theorem for fuzzy mappings. In 1992 Beg, I and A.Azam [1] made several results on fixed point of contractive type multivalued mappingstostruck the point, we need some basics.
d ( xn , x m ) 0, there exists a number n 0 such that for any
A0 = X / A ( x ) ≥ 0 = B
{
}
where
Corresponding Author: Muhammad Iqbal Chaudhry, Department of Mathematics, Islamia College University Peshawar, Pakistan
1299
Middle-East J. Sci. Res., 11 (9): 1299-1302, 2012 A ⊂ B⇔ A ( X ) ≤ B( X ) for each x∈X. The relation ⊂
B = I{F:F ⊃ B,F is closed in X}
includes a partial order on ∪(X). Definition 6: A fuzzy set A in X is called on approximate quantity if and only if Aα is compact and convex in X for each α∈[0,1] and sup A(X) = 1,α∈X.
Definition 13: Let (X, d) be a metric space and let CB(X) denotes the family of all non-empty bounded closed subsets of X for A,B∈CB(X). Let H(A,B) denote the distance between A and B in Housdorff metric, that is
Definition 7: A subset A α of Rp is convex if whenever x,y∈A α and t ∈ ¡ such that 0≤t≤1, then the point, tx+(1-t)y∈A α. Definition 8: Let (X, d) be a metric space. A mapping L:X→X is said to be contraction in X if there is a positive real number α0, and {x 2m + 2} ≤ L ({ x 2m +1 })
) (
E Lx ,L y ≤ g d ( x,y ) ,P ( x,L x ) ,P y , Ly
and we have d (x 1, x 2 ) ≤ λ d ( x 0 ,x1 ) for
Generally
}
≤ λn + λ n+1 + L + λ l −1 d ( x 0 , x1 )
hen there exist z∈X such that
{x 2m +1} ≤ L ({ x 2m })
}
K, d( x l −1, xl ) ≤ λn , λ n+1,L, λ l −1 d (x 0, x 1 )
Theorem 1: Let X be a complete metric linear space and let L be fuzzy mapping from X into ∪(X) if there is a continuous mapping g∈G such that for all x,y∈X.
Proof:
Let
xn ∈X
and
y n ∈Y
such
that
x(x n ,y n ) = d(x n ,L x n ). Let z and y& be cluster points of
{xn } and {y n } respectively i.e. {Xn } and {y n } has subsequence which converges to z and y& respectively. It is obvious that z = &y by the uniqueness of cluster point. 1301
Middle-East J. Sci. Res., 11 (9): 1299-1302, 2012
Also
d ( y n , Lz ) ≤ E ( Lx n ,L z ) + ∈
(
(
) (
≤ g d ( xn ,y n ), P xn ,L ( x n ) , P y n ,L ( yn )
y& ∈ Lz ,
since
{Xn }
is
8.
)) ( ii )
asymptotically
9.
L-regular
sequence in X therefore d ( x n ,Lx n ) ≤ 0 and thus (ii)
10.
implies that d ( y n ,L z ) ≤ 0 or d ( z,L z ) → 0. This implies that z∈Lz ⇒ {z} ⊂ L (z ). This completes the proof.
11.
REFERENCES 1.
2.
3. 4. 5.
6. 7.
12.
Beg, I and A-Azam, 1992. Fixed points of asymptotically regular multivalued mappings. J- Austral. Maths soc 53: 313-326. Bose, R.K. and D. Sahani, 1987. Fuzzy mappings and fixed point theorems. Fuzzy Sets and Systems, 21: 53-58. Butnariu, D., 1982. Fixed points for fuzzy mappings. Fuzzy Sets and Systems, 7: 191-207. Chang, S.S., 1985. Fixed point theorems for fuzzy mappings. Fuzzy Sets and Systems, 17: 181-187. Chitra, A., 1986. A note on the fixed point theorems for fuzzy maps on partially ordered topological spaces. Fuzzy Sets and Systems, 19: 305-308. Heilpern, S., 1981. Fuzzy mappings and fixed point theorem. J. Math. Anal. Appl., 83: 566-569. Nadler, S.B., 1969. Multivalued contraction mappings. Pacific J. Math., 30: 475-488.
13.
14.
15.
16.
17.
1302
Park, J.Y. and J.U. Jeong, 1997. Fixed point theorems for fuzzy mappings. Fuzzy Sets and Systems, 87: 111-116. Weiss, M.D., 1975. Fixed points and induced fuzzy topologies for fuzzy sets. J. Math. Anal. Appl., 50: 142-150. Zadeh, L.A. et al., 1996. Fuzzy Sets, Fuzzy Logic, Fuzzy Systems. World Scientific Press, ISBN 9810224214. Zadeh, L.A., 1968. Fuzzy algorithms, Information and Control. doi:10.1016/S0019-9958(68)90211-8. SSN 0019-9958, 12 (2): 94–102 Jungck, G., 1988. Common fixed points for commuting and compatible maps on compacta. Proc. Amer. Math. Soc., 103: 977-983. Caristi, J., 1976. Fixed points theorems for mappings satisfying in wardness condition. Trans. Amer. Math. Soc, 215: 241-251. Pant, R.P., 1994. Common fixed points of non commuting mappings. J. Math, Anal. Appl., 188: 436-440. Rhoades, B.E., S. Park and K.B. Moon, 1990. On genera lization of the mair-keeler type contraction maps. J. Math. Appl., 146: 482-494. Sessa, S., 1982. On a week commutatively condition of mappings in fixed point considerations. Publ. Inst. Math., 32: 149-153. Lee, B.S. and S.J. Cho, 1994. A fixed point theorem for contractive type fuzzy mappings. Fuzzy Sets and Systems, 61: 309-312.
