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ON FUZZY UNIFORM CONVERGENCE Jae Eung Kong

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ON FUZZY UNIFORM CONVERGENCE Jae Eung Kong Kangweon-Kyungki Math. Jour. 5 (1997), No. 1, pp. 1–8 ON FUZZY UNIFORM CONVERGENCE Jae Eung Kong and Sung Ki Cho Abstract. In this note, we study on fuzzy uniform convergences of sequences of fuzzy numbers, and sequences of fuzzy functions. 1. Introduction Zhang [1] provided “The Cauchy criterion for sequences of fuzzy numbers” under a restricted condition. In this note, we prove the criterion of fuzzy uniform convergences for fuzzy numbers and fuzzy functions. 2. Preliminaries All fuzzy sets, considered in this paper, are functions defined in the set R of real numbers to [0, 1]. Definition 2.1. [1] A fuzzy set A is called a fuzzy number if the following conditions are satisfied: (1) there exists x ∈ R such that A(x) = 1; (2) for any λ ∈ (0, 1], the set {x|A(x) ≥ λ} is a closed interval, − + denoted by [Aλ ,Aλ ]. Note that every fuzzy point a1 (a ∈ R) defined by 1 for x = a a1(x) = 0 for x 6= a Received July 15, 1996. 1991 Mathematics Subject Classification: 40A30. Key words and phrases: fuzzy Cauchy sequence, uniform fuzzy Cauchy se- quence, pointwise fuzzy convergent sequence, uniform fuzzy convergent sequence, uniform fuzzy convergent series. 2 Jae Eung Kong and Sung Ki Cho is a fuzzy number. Let F(R) be the set of all fuzzy numbers. Remark that for any A ∈ F(R), A = sup λχ[A−,A+], λ∈[0,1] λ λ where each χ − + denotes the characteristic function. For notational [Aλ ,Aλ ] − + convenience, we shall denote χ − + by [A ,A ]. [Aλ ,Aλ ] λ λ Definition 2.2. [1] For any a ∈ R, any {Ak|k = 1, ··· , n} ⊂ F(R) and any A, B, C ∈ F(R), − − − + (1) C = A ± B if for every λ ∈ (0, 1], Cλ = Aλ ± Bλ and Cλ = + + Aλ ± Bλ . We denote n X A1 + ··· + An = Ak; k=1 − − − + (2) C = A ∗ B if for every λ ∈ (0, 1], Cλ = Aλ Bλ and Cλ = + + Aλ Bλ . We denote A ∗ · · · ∗ A = An; | {z } n-copies − − + + (3) A ≤ B if for every λ ∈ (0, 1], Aλ ≤ Bλ and Aλ ≤ Bλ ; (4) A < B if A ≤ B and there exists λ ∈ (0, 1] such that A− < 0 λ0 B− or A+ < B+ ; λ0 λ0 λ0 (5) A = B if A ≤ B and B ≤ A. (6) ( − + supλ∈[0,1] λ[aAλ , aAλ ] for a ≥ 0 aA = + − supλ∈[0,1] λ[aAλ , aAλ ] for a < 0. Lemma 2.3. For any A ∈ F(R) and any a, b ∈ R, (1) A + 01 = A; (2) A ∗ 11 = A; (3) a1 + b1 = (a + b)1; (4) a1 ≤ b1, if a ≤ b in R. On fuzzy uniform convergence 3 Definition 2.4. [1] A fuzzy number A in F(R) is said to belong to fuzzy infinity, denoted by A ∈ ∞, if for any positive real number M, there exists λ ∈ (0, 1] such that A− ≤ −M or A+ ≥ M. 0 λ0 λ0 Definition 2.5. [1] A function d : F(R) × F(R) → F(R) is called a fuzzy distance on F(R) if (1) d(A, B) ≥ 01, d(A, B) = 01 if and only if A = B; (2) d(A, B) = d(B, A); (3) d(A, B) ≤ d(A, C) + d(C, B). Lemma 2.6. [1] The function ρ : F(R) × F(R) → F(R), defined by − − − − + + ρ(A, B) = sup λ[|A1 − B1 |, sup max{|Aµ − Bµ |, |Aµ − Bµ |}] λ∈[0,1] λ≤µ≤1 is a fuzzy distance on F(R). Throughout this paper, F(R) is the set of all fuzzy numbers with a fuzzy distance ρ defined in the above lemma. The equality ρ(A ± C, B ± C) = ρ(A, B) for A, B, C ∈ F(R) ([1]) is needed in the proof of Theorem 3.7. Definition 2.7. [1] Let {An} ⊂ F(R) and A ∈ F(R). {An} is said to converge to A, denoted by limn→∞ An = A, if for any > 0, there exists a positive integer N such that ρ(An,A) < 1 for every n ≥ N. Lemma 2.8. [1] Let {An} ⊂ F(R). Then {An} converges to a fuzzy − + number A if and only if {(An)λ } and {(An)λ } converges uniformly to − + Aλ and Aλ , respectively, for every λ ∈ (0, 1] in the usual distance of real numbers. Definition 2.9. [1] Let A ⊂ F(R). A mapping f : A → F(R) is called a fuzzy function on A. For any fuzzy functions f, g : A → F(R), we define for any A ∈ A,(f ± g)(A) = f(A) ± g(A) and (f ∗ g)(A) = f(A) ∗ g(A). 4 Jae Eung Kong and Sung Ki Cho 3. Fuzzy Uniform Convergence Definition 3.1. [1] Let {An} ⊂ F(R). Then {An} is called a fuzzy Cauchy sequence if for any > 0, there exists a positive integer N such that ρ(An,Am) < 1 for all m, n ≥ N. Lemma 3.2. Let {An} ⊂ F(R). Then {An} is fuzzy Cauchy if and − + only if {(An)λ } and {(An)λ } are uniformly Cauchy sequence of real numbers for every λ ∈ (0, 1] in the usual distance of real numbers. Proof. (⇒) Assume that {An} is a fuzzy Cauchy sequence. Then for any > 0, there exists a positive integer N such that for all n, m ≥ N, ρ(An,Am) < (/2)1. This means that for all n, m ≥ N and all λ ∈ (0, 1], − − + + max{|(An)λ − (Am)λ |, |(An)λ − (Am)λ |} ≤ /2 < , and hence the desired result follows. (⇐) Assume the given condition is satisfied. Let > 0 be given. Then there exists a positive real number N such that − − + + |(An)λ − (Am)λ | < /2 and |(An)λ − (Am)λ | < /2 for all n, m ≥ N and all λ ∈ (0, 1]. Thus − − + + supλ≤µ≤1 max{|(An)λ − (Am)λ |, |(An)λ − (Am)λ |} < for all n, m ≥ N and all λ ∈ (0, 1]. Consequently, ρ(An,Am) < 1 for all n, m ≤ N. Theorem 3.3. Every fuzzy Cauchy sequence converges. Proof. Let {An} be a fuzzy Cauchy sequence. By Lemma 3.2, − + {(An)λ } and {(An)λ } are uniformly Cauchy for every λ ∈ (0, 1], in the usual distance of real numbers. By the well known theorem for uniform Cauchy sequence of real valued functions, there exist real valued func- − + tions f(λ) and g(λ) defined on (0, 1] such that {(An)λ } and {(An)λ } − converges uniformly to f(λ) and g(λ), respectively. Let f(λ) = Aλ , + − + g(λ) = Aλ and A = supλ∈[0,1] λ[Aλ ,Aλ ]. Then A is a fuzzy number. By Lemma 2.8, limn→∞ An = A. On fuzzy uniform convergence 5 Definition 3.4. Let A ⊂ F(R) and let {fn} be a sequence of fuzzy functions from A to F(R). (1) We say that {fn} converges pointwise to a fuzzy function f : A → F(R) if for any A ∈ A, {fn(A)} converges to f(A). (2) We say that {fn} converges uniformly to a fuzzy function f : A → F(R) if for any > 0 and any A ∈ A, there exists a positive integer N() such that ρ(fn(A), f(A)) < 1 for all n ≥ N() and all A ∈ A. (3) The sequence {fn} is said to be uniformly fuzzy Cauchy if for any > 0, there exists a positive integer N() such that ρ(fm(A), fn(A)) < 1 for all m, n ≥ N() and all A ∈ A. P∞ (4) An infinite series k=1 fk of fuzzy functions is said to converge uniformly to a fuzzy function f : A → F(R) if the sequence Pn of partial sums {Sn} = { k=1 fk} of the series converges uni- formly to f If we use Theorem 3.3, the proof of the following theorem follows that of corresponding classical theorem in real analysis. Theorem 3.5. Let A ⊂ F(R) and let {fn} be a sequence of fuzzy functions from A to F(R). Then the sequence {fn} is uniformly Cauchy if and only if there exists a fuzzy function f : A → F(R) such that {fn} converges uniformly to f. Lemma 3.6. For any A ∈ F(R), the following are equivalent: (1) A/∈ ∞; (2) there exists a positive real number M such that − + max{|Aλ |, |Aλ |} < M for all λ ∈ (0, 1]; (3) there exists a positive real number M such that ρ(A, 01) < M1. Proof. (1) ⇒ (2). Assume A/∈ ∞. Then there exists a positive real − + number M such that for every λ ∈ (0, 1], −M < Aλ ≤ Aλ < M. Thus, − + max{|Aλ |, |Aλ |} < M for all λ ∈ (0, 1]. 6 Jae Eung Kong and Sung Ki Cho (2) ⇒ (3). If such an M exists, then − − + ρ(A, 01) = sup λ[|A1 |, sup max{|Aµ |, |Aµ |}] λ∈[0,1] λ≤µ≤1 < sup λ[M, M] λ∈[0,1] = M1. (3) ⇒ (1). Assume to the contrary that A ∈ ∞. Then for every positive integer M, there exists λ ∈ (0, 1] such that M ≤ A+ or 0 λ0 A− ≤ −M. Therefore, sup max{|A−|, |A+|} ≥ M, contrary to λ0 λ0≤µ≤1 µ µ the hypothesis. Theorem 3.7. [Weierstrass M-test] Let A ⊂ F(R) and let {fn} be a sequence of fuzzy functions from A to F(R). If for each n, there exists a positive real number Mn such that − + max{|(fn(A))λ |, |(fn(A))λ |} ≤ Mn P∞ for all A ∈ A and all λ ∈ (0, 1], and if the series n=1 Mn converges, P∞ then there exists a fuzzy function f : A → F(R) such that n=1 fn converges uniformly to f. P∞ Proof.
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