On Generalized Fuzzy Normed Spaces

Transcription

1 International Mathematical Forum, 4, 2009, no. 25, On Generalized Fuzzy Normed Spaces Ioan Goleţ Department of Mathematics Politehnica University Timişoara, Romania Abstract In this paper, generalized fuzzy 2-norms are defined on a set of objects endowed with a structure of linear space. The relationships between these fuzzy 2-norms and fuzzy metrics and between these fuzzy 2-norms and the topological structures induced on a linear space are analyzed. Some open problems in this framework are mentioned. Mathematics Subject Classification: 46A19, 54A40 Keywords: fuzzy metric space, fuzzy 2-normed space 1 Introduction The theory of metric spaces and of normed spaces is of fundamental importance in mathematics, computer science, statistics etc. Many problems can be solved by finding an appropriate metric or norms for making some measurements. Both metric spaces and normed spaces were generalized and many papers and books have been published in this area. The positive number expressing the distance between two points or the absolute value of a vector was replaced by a probabilistic distribution function (in the sense of probabilistic theory [9], [13]) or by a fuzzy set (in the sense of fuzzy theory [6-8],[10-12], [14-16]). The distance between two points was extended to three or more points and the norm of a vector to two or more vectors [5]. These subjects were developed in various important directions into a close connection with another areas of mathematics. Different applications in engineering science and economics were also given. In this paper we give an enlargement of the concepts of fuzzy 2-norm. The relationships between these new concepts and fuzzy metrics, topological structures on the base linear space are analyzed. The previous similar concepts of fuzzy norms are obtained as particular cases.

2 1238 I. Goleţ 2 Preliminary Notes A t-norm is a two place function :[0, 1] [0, 1] [0, 1] which is associative, commutative, non decreasing in each place and such that a 1=a, for all a [0, 1]. The most used t-norms in fuzzy metric spaces theory are : a m b = min{a, b}, a p b = a b and a tm b = max{a + b 1, 0}. By an operation on R + we mean a two place function :[0, ) [0, ) [0, ) which is associative, commutative, non decreasing in each place and such that a 0=a, for all a [0, ). The most used operations on R + are a (s, t) =t + s, m (s, t) =max{s, t} and n (s, t) =(s n + t n ) 1 n [10]. Let ϕ be a function defined on the real field R into itself with the following properties : (a 1 ) ϕ( t) =ϕ(t), for every t R; (a 2 ) ϕ(1) = 1; (a 3 ) ϕ is strict increasing and continuous on (0, ); (a 4 ) lim = 0 and lim =. α 0 α As examples of such functions we have : = α ; = α p,p R + ; = 2α2n,n α +1 N+. The functions s and ϕ allow us to generalize metric and normed fuzzy spaces. Definition 2.1 A fuzzy metric space is an ordered triple (X, M, ), where X is a non-empty set, is a t-norm is an operation and M is a mapping defined on X 2 [0, ) with values into R satisfying the following conditions: (g 1 ) M(x, y, 0) = 0; (g 2 ) M(x, y, t) =1for all t>0 if, and only if, x = y; (g 3 ) M(x, y, t) =M(y, x, t); (g 4 ) M(x, z, t s) M(x, y, t) M(y, z, s); (g 5 ) M(x, y, ) :[0, ) [0, 1] is left continuous. 3 Main Results Let ϕ be a function defined on the real field R into itself with the following properties : (f 1 ) ϕ( t) = ϕ(t), for every t R; (f 2 ) ϕ(1) = 1; (f 3 ) ϕ is strict increasing and continuous on (0, ); (f 4 ) lim =0 α 0 and lim =. As examples of such functions we have : = α ; α = α p,p R + ; = 2α2n,n α +1 N+. Definition 3.1 Let L be a linear space over the field R of a dimension greater than one and let N be a mapping defined on L L [0, ) with values into [0, 1] satisfying the following conditions : for all x, y, z L and s, t [0, ) (a 1 ) N(x, y, 0) = 0,

3 On generalized fuzzy normed spaces 1239 (a 2 ) N(x, y, t) =1, for all t>0 if and only if x, y are linear dependent, (a 3 ) N(x, y, t) =N(y, x, t) for all x, y L, and t>0, (a 4 ) N(x + y, z, t s) N(x, z, t) N(y, z, s), (a 5 ) N(x, y, ) :[0, ) [0, 1] is left continuous. t (a 6 ) N(αx, y, t) =N(x, y, ),α R. The triple (L, N, ) is called fuzzy ϕ-2-normed space under operation. If = Sum then (L, N, ) will be called a fuzzy ϕ-2-normed space. If = Max then (L, N, ) will be called a non-archimedean fuzzy ϕ-2-normed space. N(x, y, t) is called a fuzzy ϕ-2-norm. Note that N(x, y, t) can be thought of as the degree of linear dependence of x and y. N(x, y, t) =1, for all t>0 if, and only if, x and y are linear dependent. So, we identify y = αx with N(x, y, t) =1, for all t>0. From the above conditions it follows that for all x, y L, N(x, y, ) is a nondecreasing mapping. In the following example we will show that every ϕ-2-normed space can be made a fuzzy ϕ-2-normed space. The following definition gives a generalization of ordinary 2-normed spaces in sense of the function ϕ. Definition 3.2 Let L be a linear space over the field R of a dimension greater than one and let, be a mapping defined on L L with values into the field R satisfying the following conditions : for all x, y, z L and s, t [0, ) (b 1 ) x, y =0, if and only if, x, y are linear dependent; (b 2 ) x, y = y, x for all x, y L; (b 3 ) αx, y = x, y, for all α R. (b 4 ) x + y, z x, z + y, z ; The triple (L, N, ) is called ϕ-2-normed space. The mapping, is called a ϕ-2-norm. If = α then one obtains the 2-normed spaces [5]. If = α p, p (0, 1], then one obtains the p-2-normed spaces as a generalization of 2- normed spaces. Example 3.3 Let (L,, ) be a ϕ-2-normed space. Consider t-norms a b = min{a, b} or a b= a b and for all x, y L and t>0, t N(x, y, t) = t + x, y, then (L, N, ) is a fuzzy ϕ-2-normed space. We call this fuzzy ϕ-2-normed space as the standard fuzzy ϕ-2-normed space. Definition 3.4 Let (L, N, ) be a fuzzy ϕ-2-normed space : (a) A sequence {x n } in L is said to be convergent to a point x L (denoted by lim n x n = x), if lim n N(x n x, a, t) =1,

4 1240 I. Goleţ for all a L and t>0 (b) A sequence {x n } in L is called a Cauchy sequence if lim n N(x n+p x n,a,t)=1, for all a Lt>0 and p>0. (c) A fuzzy ϕ-normed space in which every Cauchy sequence is convergent is said to be complete. Theorem 3.5 Let (L, N, ) be a fuzzy ϕ-2-normed space and let M : L L L [0, ) [0, 1], M(x, y, z, t) =N(y x, z x, t) then (L, M, ) is a fuzzy 2-metric space. The following theorem will show that a fuzzy 2-norm N generates a linear topology on L compatible with the above convergence. Let A be the family of all finite and non-empty subsets of the linear space L and A A. Theorem 3.6 Let (L, N, ) be a fuzzy ϕ-2-normed space under a continuous t-norm such that, for all r, t [0, 1], r t Max{r + t 1, 0} then : (a) The following system of neighborhoods V = {V (, λ, A) :>0,λ (0, 1),A A}, where V (, λ, A) ={x L : N(x, a, ) > 1 λ, a A} is a base of the null vector θ, for a linear topology on L, named N-topology generated by the fuzzy ϕ-2-norm N. (b) If = Min, then V (, λ, A) is a convex set. Hence L endowed with the N- topology generated by the fuzzy ϕ-2-norm N is a local convex linear topological space. Proof. (a) Let V ( k,λ k,a k ),k =1, 2 be in V. We consider A = A 1 A 2,= min{ 1, 2 },λ= min{λ 1,λ 2 }, then V (, λ, A) V ( 1,λ 1,A 1 ) V ( 2,λ 2,A 2 ). Let α R such that 0 α 1 and x αv (, λ, A), then x = αy, where y V (, λ, A). For every a A we have N(x, a, ) =N(αy, a, ) =N(y, a, ) N(y, a, ) > 1 λ. This shows us that x V (, λ, A), hence αv (, λ, A) V (, λ, A). Now, let s show that, for every V Vand x L there exists β R, ϕ(β) 0such that βx V. If V Vthen, there exists >0, λ (0, 1) and A Asuch that V = V (, λ, A). Let x be arbitrarily fixed in L and α R, α 0, then N(αx, a, ) =N(x, a, ). Since lim N(x, a, 0 )=1

5 On generalized fuzzy normed spaces 1241 it follows that, for every a A there exists α(a) K such that N(x, a, ) > α(a) 1 λ. If we choose β = min{ϕ(α(a)) : a A}, then we have N(βx,a,)=N(x, a, ) N(x, a, β ϕ(α(a)) ) > 1 λ, for all a A, hence βx V. Let us prove that, for each V V there exists V 0 V such that V 0 + V 0 V. Let V = V (, λ, A) and x V (, λ, A), then there exists η > 0 such that N(x, a, ) > 1 η>1 λ, for all a A. We now consider η 0 = sup{η : N(x, a, ) > 1 η>1 λ, x V 0,a A}. Now let V 0 = V (, η 0 ). If x, y V 2 3 0, by the triangle inequality (f 3 ) N(x + y, a, ) N(x, a, 2 ) N(y, a, 2 )) Max{1 2η 0 3, 0} > 1 η 0 1 λ. The above inequalities show us that V 0 + V 0 V. Let us show that V Vand α R,α 0implies αv V. Remark that αv = αv (, λ, A) ={αx : N(x, a, ) > 1 λ}. Since N(x, a, ) =N(x, a, )=N(αx, a, ), we have αv = {αx : N(αx, A, ) > 1 λ. So, we have αv V (, λ, A). Now let y V (, λ, A), then N(y, a, ) > 1 λ. But, N(y, a, ) =N( αy α,a,) =N( y α,a,). So, we have the inequality N( y,a,) > 1 λ. This shows us that y V (, λ, A), α α that is, y αv (, λ, A). Finally, we have that αv (, λ, A) =V (, λ, A) V. The above statements show us that V is a base for a system of neighborhoods of the origin in the linear space L. The topology generated by this system on the linear space L is named N-topology on L. References [1] T. Bag and S. K. Samanta, Fuzyy bounded linear operator, Fuzzy Sets and Systems, 151 (2005), [2] Li Chen, H. Kou, M. KongLuo, W. Zhang, Discrete dynamical systens in topological spaces, Fuzzy Sets and Systems, 156 (2005),

6 1242 I. Goleţ [3] J. Fang and G. Song, PHI-contractor and the solutions for nonlinear operator equations in fuzzy normed spaces, Fuzzy Sets and Systems, 121 (2001), [4] J. Fang, A note on the completions of fuzzy metric spaces and fuzzy normed spaces, Fuzzy Sets and Systems, 131 (2002), [5] S. Gähler, Lineare 2-nomierte Raume, Math. Nachr., 28 (1964), [6] I. Goleţ, On fuzzy normed spaces, Southeast Asian Bull. of Math., 131 (2007), [7] I. Goleţ and Iţ. Goleţ, Approximation of Fuzzy Normed Space-Valued Functions, American Institute of Physics Conference Proceedings, 936 (2007), [8] V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 125 (2002), [9] O. Hadžić, E. Pap, Fixed point theory in probabilistic metric spaces, Kluver Academic Publishers, Dordrecht, (2001). [10] N. Huang and H. Lan, A couple of nonlinear equations with fuzzy mappings in fuzzy normed spaces, Fuzzy Sets and Systems, 152 (2005), [11] H. T. Nguyen, Fuzzy and random sets, Fuzzy Sets and Systems, 156 (2005), [12] O. Sarma, On fuzzy metric space, Southeast Assian Bull. of Math., 26 (2002), [13] B. Schweizer, A. Sklar, Probabilistic metric spaces North Holland, New York, Amsterdam, Oxford, (1983). [14] J-Z. Xiao and X-H. Zhu, Topological degree theory and fixed point theorems in fuzzy normed spaces, Fuzzy Sets and Systems, 147 (2004), [15] J-Z. Xiao and X-H. Zhu, On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets and Systems, 125, (2002), [16] J-Z. Xiao and X-H. Zhu, Fuzzy normed space of operators and its completeness, Fuzzy Sets and Systems, 133 (2004), Received: November, 2008