Lacunary Invariant Statistical Convergence of Fuzzy Numbers

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www.i-csrs.org

Available free online at http://www.geman.in

Lacunary Invariant Statistical Convergence of Fuzzy Numbers

Mehmet A¸cıkg¨oz¹ and Ayhan Esi²

1University of Gaziantep, Faculty of Science and Arts Department of Mathematics, 27310 Gaziantep, Turkey

E-mail: [email protected]

2Adiyaman University, Science and Art Faculty Department of Mathematics, 02040 Adiyaman-Turkey

E-mail: [email protected], [email protected] (Received: 10-12-11/ Accepted: 15-1-12)

Abstract

In this paper, we introduce the concepts of invariant convergence, lacunary invariant statistical convergence of sequences of fuzzy numbers and lacunary strongly invariant convergence of sequences of fuzzy numbers. We give some relations related to these concepts.

Keywords: Fuzzy numbers, lacunary sequence, almost convergence, statistical convergence, invariant mean.

1 Introduction

Schaefer [17] defined the σ-convergence as follows:

Let σ be a one- to- one mapping of the set of positive integers into itself such thatσ^k(n) = σσ^k−1(n),k = 1,2,3,· · ·.A continuous linear functional ϕon l∞, the set of all bounded sequences, is said to be an invariant mean or aσ-mean if and only if

(i)ϕ(x)≥0 when the sequence x= (x_n) has x_n≥0 for all n, (ii)ϕ(e) = 1, wheree = (1,1,1,· · ·) and

(iii)ϕ({x_σ(n)}) = ϕ({x_n}) all x= (x_n)∈l_∞.

For certain kinds of mappings σ, every invariant mean ϕ extends the limit functional on the space c, the set of all convergent sequences, in the sense

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that ϕ(x) = limx for all x = (x_n) ∈ c. Consequently, c ⊂ v_σ, where v_σ is the set of bounded sequences all of whose σ-means are equal. In the case σ is the translation mapping n → n + 1, a σ-mean is often called a Banach limit and v_σ is the set of almost convergent sequences. If x = (x_k), write T x= (T x_k) = x_σ(k). It can be shown that

v_σ =

x∈l∞: lim

k→∞t_kn(x) =l, unif ormly in n, l =σ−limx

, where t_kn(x) = (k+ 1)⁻¹ ^P^k

i=0

x_σⁱ_(n), here σ^k(n) denotes the k^th iterate of the mapping σ at n. By a lacunary sequence we mean an increasing integer sequence θ = (k_r) such that k_o = 0 and h_r = k_r −kr−1 → ∞ as r → ∞.

Throughout this paper the intervals determined by θ = (k_r) will be denoted by Ir = (kr−1, kr] and the ratio _k^k^r

r−1 will be abbreviated as qr. Lacunary sequences have been discussed by [3], [7], [9], [11] and many others.

The notion of statistical convergence was introduced by Fast [6] and Schoen- berg [18], independently. Over the years and under different names statistical convergence has been discussed in the different theories such as the theory of Fourier analysis, ergodic theory and number theory. Later on, it was further investigated from the sequence space point of view and linked with summabil- ity theory by Fridy [8], Salat [16], Connor [2] and many others. This concept extended the idea to apply to sequences of fuzzy numbers with Nuray [14], Kwon et al. [11], Altin et al. [1], Nuray and Sava¸s [15] and many others.

A sequence x = (x_k) is said to be statistically convergent to l if for every ε >0,

n→∞lim n⁻¹|{k ≤n:|x_k−l| ≥ε}|= 0

where the vertical bars denote the cardinality of the set which they enclose, in which case we writeS−limx=l. The concept of fuzzy sets was first introduced by Zadeh [19]. Bounded and convergent sequences of fuzzy numbers were introduced by Matloka [12]. Matloka show that every convergent sequences of fuzzy numbers is bounded. Later on sequences of fuzzy numbers have been discussed by Nanda [13], Nuray [14], [15], [10], Esi [5] and many others. Briefly, we recall some of the basic notations in the theory of fuzzy numbers and we refer readers to Matloka [12] and Diamond and Kloeden [4] for more details.

LetC(Rⁿ) ={A⊂Rⁿ:A is compact and convex set}. The spaceC(Rⁿ) has a linear structure induced by the operations

A+B ={a+b :a ∈A, b∈B} and γA={γa :a∈A}

for A, B ∈ C(Rⁿ) and γ ∈ R. The Hausdorff distance between A and B in C(Rⁿ) is defined by

δ∞(A, B) = max

(

sup

a∈A

b∈Binfka−bk,sup

b∈B

a∈Ainf ka−bk

)

.

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It is well-known that (C(Rⁿ), δ∞) is a complete metric space. A fuzzy number is a functionX fromRⁿto [0,1] which is normal, fuzzy convex, upper semicon- tinuous and the closure of{X ∈Rⁿ:X(x)>0}is compact. These properties imply that for each 0 < α ≤ 1, the α-level set X^α = {X ∈Rⁿ:X(x)≥α}

is a non-empty compact, convex subset of Rⁿ, with support X⁰. Let L(Rⁿ) denote the set of all fuzzy numbers. The linear structure ofL(Rⁿ) induces the addition X+Y and the scalar multiplication λX in terms of α-level sets, by

[X+Y]^α = [X]^α+ [Y]^α and [λX]^α =λ[X]^α

for each 0≤α≤1. Consider the Hausdorff metricsd_q and d∞ defined by

d_q(X, Y) =





1

Z

0

δ∞(X^α, Y^α)dα





1 q

(1≤q≤ ∞)

and

d∞(X, Y) = sup

0≤α≤1

δ∞(X^α, y^α).

Clearly,d∞(X, Y) = limq→∞d_q(X, Y) withd_q(X, Y)≤d_s(X, Y) if q≤s, [4].

For simplicity in notation, we shall write throughout d instead of d_q with 1≤q ≤ ∞. The metric d has the following properties:

d(cX, cY) =|c|d(X, Y) (1)

forc∈R and

d(X+Y, Z+W)≤d(X, Z) +d(Y, W). (2) A metric on L(R) is said to be translation invariant d(X+Z, Y +Z) = d(X, Y) for all X, Y, Z ∈ L(R). A sequence X = (X_k) of fuzzy numbers is a functionX from the set N of natural numbers into L(R). The fuzzy number X_k denotes the value of the function at k ∈ N [12]. We denote by w^F the set of all sequences X = (X_k) of fuzzy numbers. A sequence X = (X_k) of fuzzy numbers is said to be bounded if the set{Xk :k ∈N} of fuzzy numbers is bounded [2]. We denote by l^F_∞ the set of all bounded sequences X = (X_k) of fuzzy numbers. A sequence X = (X_k) of fuzzy numbers is said to be convergent to a fuzzy numberXo if for every ε > 0 there is a positive integer k_o such that d(X_k, X_o) < ε for k > k_o [12]. We denote by c^F the set of all convergent sequencesX = (X_k) of fuzzy numbers. It is straightforward to see thatc^F ⊂l^F_∞⊂w^F. In [13], it was shown thatc^F and l^F_∞ are complete metric spaces.

In the present note, we introduce and examine the concepts of invariant convergence of sequences, lacunary invariant statistical convergence of sequences

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of fuzzy numbers and lacunary strongly invariant convergence of sequences of fuzzy numbers. We give some relations related to these concepts. Let p = (pk) ∈ l∞, then the following well-known inequality will be used in the paper: For sequences (a_k) and (b_k) of complex numbers we have

|a_k+b_k|^p^k ≤K(|a_k|^p^k +|b_k|^p^k), (3) whereK = max1,2^H−1 and H = sup_kp_k<∞.

Definition 1.1 A sequence X = (X_k)of fuzzy numbers is said to be invariant convergent to fuzzy number X_o if

k→∞lim d(t_kn(X), X_o) = 0 (4) uniformly in n, where t_kn(X) = (k+ 1)⁻¹^P^k_i=0X_σⁱ_(n).

This means that for everyε >0, there exists ako ∈N such thatd(tkn(X), Xo)

< ε whenever k ≥ k_o and for all n. If the limit in (4) exists, then we write v_σ^F−limX =X_o. Let v_σ^F be the space of all invariant convergent sequences of fuzzy numbers. It is evident that v^F_σ,0 ⊂ v_σ^F, where v^F_σ,0 denotes the classes of all invariant convergent to zero of fuzzy numbers.

Definition 1.2 Let θ = (k_r) be lacunary sequence. A sequence X = (X_k) of fuzzy numbers is said to be lacunary invariant statistically convergent to fuzzy number X_o if, for every ε >0

r→∞lim h⁻¹_r |{k ∈I_r :d(t_kn(X), X_o)≥ε}|= 0

uniformly in n. In this case we write X_k →X_oS^b_θ^σ or S^b_θ^σ −limX_k=X_o. The set of all lacunary invariant statistically convergent sequences of fuzzy numbers is denoted byS_θ^σ. In special case θ = (2^r), we shall write S^b^σ instead of S^b_θ^σ.

Definition 1.3 Let θ = (k_r) be lacunary sequence and p = (p_k) be any sequence of strictly positive real numbers. A sequence X = (Xk) is said to be lacunary strongly invariant convergent if there is a fuzzy numberX_o such that

r→∞lim h⁻¹_r ^X

k∈Ir

[d(t_kn(X), X_o)]^p^k = 0

uniformly in n. In this case we write X_k →X_oM^c_θ^σ, p.

The set of all lacunary strongly invariant convergent sequences of fuzzy numbers is denoted by M^c_θ^σ, p. In special case θ = (2^r) and p_k = 1 for all k∈N, we shall write C^b^σ, pand C^b_θ^σ instead ofM^c_θ^σ, p, respectively.

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2 Main Results

In this section, we prove the results of this paper.

Theorem 2.1 Let X = (X_k) and Y = (Y_k) be sequences of fuzzy numbers.

(i) If S^b_θ^σ−limX_k =X_o and a∈R, then S^b_θ^σ−limaX_k=aX_o.

(ii) S^b_θ^σ −limX_k = X_o and S^b_θ^σ −limY_k = Y_o, then S^b_θ^σ −lim (X_k+Y_k) = X_o+Y_o.

Proof. (i) Let X_k → X_oS^b_θ^σ and a ∈ R. Then by taking into account the properties (1) and (2) of the metric d, we have

h⁻¹_r |{k∈Ir :d(tkn(aX), aX0)≥ε}| ≤h⁻¹_r

k ∈Ir :d(tkn(X), Xo)≥ ε a

which yields thatS^b_θ^σ−limaX_k=aX_o.

(ii) By combining the Minkowski’s inequality with the property (2) of the metricd, we derive that

d(tkn(X) +tkn(Y), Xo+Yo)≤d(tkn(X), Xo) +d(tkn(Y), Yo). Therefore givenε >0 we have,

h⁻¹_r |{k ∈I_r :d(t_kn(X) +t_kn(Y), X_o+Y_o)≥ε}|

≤h⁻¹_r |{k ∈I_r :d(t_kn(X), X_o) +d(t_kn(Y), Y_o)≥ε}|

≤h⁻¹_r

k∈I_r :d(t_kn(X), X_o)≥ ε 2

+h⁻¹_r

k ∈I_r :d(t_kn(X), Y_o)≥ ε 2

, which yields that thatS^b_θ^σ−lim (X_k+Y_k) = X_o+Y_o.

The following result is a consequence of Theorem 2.1.

Corollary 2.2 Let X = (X_k) andY = (Y_k)be sequences of fuzzy numbers.

(iii) If S^b^σ−limX_k =X_o and a∈R, then S^b^σ−limaX_k=aX_o.

(iv) S^b^σ −limX_k = X_o and S^b^σ −limY_k = Y_o, then S^b^σ −lim (X_k+Y_k) = X_o+Y_o.

Theorem 2.3 Let θ = (kr) be lacunary sequence and let X = (Xk) be a sequence of fuzzy numbers. Then,

(i) For lim infq_r >1, we have C^b^σ, p⊂M^c_θ^σ, p. (ii) For lim supq_r <∞, we have C^b^σ, p⊃M^c_θ^σ, p. (iii) C^b^σ, p=M^c_θ^σ, p if 1<lim infq_r ≤lim supq_r <∞.

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Proof. (i) Let lim infq_r >1, then there exists δ >0 such that q_r ≥ 1 +δ for all r≥1. Then for X = (X_k)∈C^b^σ, p, we write

h⁻¹_r ^X

k∈Ir

[d(t_kn(X), X_o)]^p^k = h⁻¹_r

kr

X

k=1

[d(t_kn(X), X_o)]^p^k

−h⁻¹_r

kr−1

X

k=1

= k_rh⁻¹_r



k⁻¹_r

kr

X

k=1





−kr−1h⁻¹_r



k⁻¹_r−1

kr−1

X

k=1



.

Sinceh_r=k_r−kr−1, we havek_rh⁻¹_r ≤(1 +δ)^δ−1andkr−1h⁻¹_r ≤δ⁻¹. The terms k⁻¹_r ^P^k_k=1^r [d(t_kn(X), X_o)]^p^k andk_r−1⁻¹ ^P^k_k=1^r [d(t_kn(X), X_o)]^p^k both converge to 0 as r→ ∞ uniformly inn. Therefore X = (X_k)∈M^c_θ^σ, p.

(ii) Now suppose that lim supq_r < ∞, then there exists A > 0 such that q_r < Afor allr ≥1. LetX = (X_k)∈M^c_θ^σ, pand ε >0 be given. Then there exists a number R >0 such that

Ai =h⁻¹_i ^X

k=Ir

[d(tkn(X), Xo)]^p^k < ε

for every i ≥ R and all n. We can also find K > 0 such that A_i < K for all i = 1,2,3,· · ·. Now let m be any integer with kr−1 < m ≤ kr, where r > R.

Then

m⁻¹

m

X

k=1

[d(t_kn(X), X_o)]^p^k ≤k_r−1⁻¹

kr

X

k=1

= k⁻¹_r−1

" P

k=I1[d(t_kn(X), X_o)]^p^k +^P_k=I₂[d(t_kn(X), X_o)]^p^k +· · · +^P_k=I_r[d(t_kn(X), X_o)]^p^k

#

= (k₁ −k_o)k_r−1⁻¹ k⁻¹₁ ^X

k=I1

[d(t_kn(X), X_o)]^p^k + (k₂−k₁)k_r−1⁻¹ k₂⁻¹(k₂−k₁)⁻¹ ^X

k=I2

[d(t_kn(X), X_o)]^p^k +· · · + (k_R−kR−1)k_r−1⁻¹ (k_R−kR−1)⁻¹ ^X

k=IR

[d(t_kn(X), X_o)]^p^k+· · · + (k_r−kr−1)k_r−1⁻¹ (k_r−kr−1)⁻¹ ^X

k=Ir

= k1k⁻¹_r−1A1+ (k2−k1)k_r−1⁻¹ A2+· · ·+ (kR−kR−1)k_r−1⁻¹ AR+· · · + (k_r−kr−1)k_r−1⁻¹ A_r

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≤ sup

i≥1

A_i

!

k_Rk_r−1⁻¹ + sup

i≥R

A_i

!

(k_r−k_R)k⁻¹_r−1 < Kk_Rk_r−1⁻¹ +εA.

Since kr−1 → ∞ as m → ∞, it follows that m⁻¹^P^m_k=1[d(t_kn(X), X_o)]^p^k → 0 uniformly inn. Hence X = (X_k)∈C^b^σ, p.

(iii) Follows from (i) and (ii).

Theorem 2.4 Let θ = (k_r) be lacunary sequence and let X = (X_k) be a sequence of fuzzy numbers. Then,

(i) X_k →X_oM^c_θ^σ, p implies X_k →X_oS^b_θ^σ.

(ii) X = (X_k)∈l_∞^F and X_k→X_oS^b_θ^σ imply X_k→X_oM^c_θ^σ, p. (iii) S^b_θ^σ =M^c_θ^σ, p if X = (X_k)∈l_∞^F,

where 0< h= inf_kp_k ≤p_k≤sup_kp_k =H <∞.

Proof. (i) ε >0 andXk →X0

Mc_θ^σ, p. Then we can write h⁻¹_r ^X

k∈Ir

[d(t_kn(X), X_o)]^p^k = h⁻¹_r ^X

k∈Ir, d(t_kn(X),Xo)≥ε

+h⁻¹_r ^X

k∈Ir, d(t_kn(X),Xo)<ε

≥ h⁻¹_r ^X

k∈I_r, d(tkn(X),Xo)≥ε

≥ h⁻¹_r ^X

k∈I_r, d(tkn(X),Xo)≥ε

[ε]^p^k

≥ h⁻¹_r ^X

k∈Ir, d(tkn(X),Xo)≥ε

min^hε^h, ε^Hⁱ^p^k

≥ h⁻¹_r |{k∈I_r :d(t_kn(X), X_o)≥ε}|min^hε^h, ε^Hⁱ. Hence X_k →X_oS^b_θ^σ.

(ii) Suppose that X = (X_k)∈ l_∞^F and X_k →X_oS^b_θ^σ. Since X = (X_k) ∈ l^F_∞, there is a constant B > 0 such thatd(t_kn(X), X_o)≤B. Given ε >0, we have

h⁻¹_r ^X

k∈Ir

[d(t_kn(X), X_o)]^p^k = h⁻¹_r ^X

+h⁻¹_r ^X

k∈I_r, d(tkn(X),Xo)<ε

≤h⁻¹_r ^X

max^hB^h, B^Hⁱ

+h⁻¹_r ^X

k∈Ir, d(tkn(X),Xo)<ε

[ε]^p^k

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≤ max^hB^h, B^Hⁱh⁻¹_r |{k ∈I_r :d(t_kn(X), X_o)≥ε}|

+ max^hε^h, ε^Hⁱ. Hence X_k →X_oM^c_θ^σ, p.

(iii) Follows from (i) and (ii).

Theorem 2.5 Let θ = (k_r) be lacunary sequence and let X = (X_k) be a sequence of fuzzy numbers. Then,

(i) For lim supq_r <∞, we have S^b_θ^σ ⊂S^b^σ. (ii) For lim infq_r >1, we have S^b^σ ⊂S^b_θ^σ.

(iii) if 1<lim infqr≤lim supqr <∞, then S^b_θ^σ =S^b^σ.

Proof. (i) If lim supq_r < ∞, then there is a B > 0 such that q_r < B for all r ≥ 1. Suppose that X_k → X_oS^b_θ^σ and for each n ≥ 1 set N_rn =

|{k ∈I_r :d(t_kn(X), X_o)≥ε}|. Then there exists an r_o ∈N such that

N_rnh⁻¹_r < ε f or all r > r_o and n≥1. (5) Now letM = max{N_rn : 1≤r ≤r_o} and choose m such that kr−1 < m≤k_r, then for each n≥1 we have

m⁻¹|{k ≤m:d(t_kn(X), X_o)≥ε}|

≤ k_r−1⁻¹ |{k ≤kr :d(tkn(X), Xo)≥ε}|

= k_r−1⁻¹ ⁿN_1n+N_2n+· · ·+N_r_o_n+N_(r_o_+1)n+· · ·+N_rn^o

≤ k_r−1⁻¹ M r_o+k⁻¹_r−1ⁿh_r_o₊₁(h_r_o₊₁)⁻¹N_(r_o_+1)n+· · ·+h_rh⁻¹_r N_rn^o

≤ k_r−1⁻¹ M r_o+k⁻¹_r−1 sup

r>ro

h⁻¹_r N_rn

!

{h_r_o₊₁+· · ·+h_r}

≤ k_r−1⁻¹ M r_o+k⁻¹_r−1ε(k_r−k_r_o) by(5)

≤ k_r−1⁻¹ M r_o+εq_r

≤ k_r−1⁻¹ M r_o+Bε.

This completes the proof.

(ii) Suppose that lim inf_q_r > 1. Then there exists a δ > 0 such that q_r ≥ 1 +δ for sufficiently large r ≥1. Since h_r =k_r−kr−1, we have h_rk_r⁻¹ ≤ (1 +δ)⁻¹δ. Let X_k →X_oS^b^σ. Then for every ε >0 and for all n, we have

k⁻¹_r |{k ≤k_r :d(t_kn(X), X_o)≥ε}| ≥k_r⁻¹|{k ∈I_r :d(t_kn(X), X_o)≥ε}|

≥(1 +δ)⁻¹δh⁻¹_r |{k∈I_r :d(t_kn(X), X_o)≥ε}|

Hence S^b^σ ⊂S^b_θ^σ.

(iii) Follows from (i) and (iii).

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Theorem 2.6 Let 0< p_k ≤ q_k and p_kq⁻¹_k be bounded. Then M^c_θ^σ, q ⊂

Mc_θ^σ, p.

Proof. Let X = (x_k) ∈ M^c_θ^σ, q. Let w_k,n = (d(t_kn(X), X_o))^q^k and λ_k = p_kq_k⁻¹ for all k ∈ N. We define the sequences (u_k,n) as follows: For w_k,n ≥ 1, let u_k,n = w_k,n and v_k,n = 0 and for w_k,n < 1, let u_k,n = 0 and v_k,n = w_k,n. Then it is clear that for all k ∈ N, we have w_k,n = u_k,n +v_k,n and w_k,n^λ^k =u^λ_k,n^k +v_k,n^λ^k. Now it follows that u^λ_k,n^k ≤u_k,n ≤w_k,n and v_k,n^λ^k ≤ v_k,n^λ . Therefore

h⁻¹_r ^X

k∈Ir

w_n,m^λⁿ =h⁻¹_r ^X

k∈Ir

(u_n,m +v_n,m)^λⁿ ≤h⁻¹_r ^X

k∈Ir

w_n,m+h⁻¹_r v^λ_n,m.

Now for each r, h⁻¹_r ^X

k∈Ir

v^λ_n,m = ^X

k∈Ir

h⁻¹_r v^λ_n,m^λh⁻¹_r ^1−λ

≤



 X

k∈Ir

h⁻¹_r v^λ_n,m^λ

¹_λ





λ

 X

k∈Ir

h⁻¹_r v^λ_n,m^λ

¹_λ





λ

=



h⁻¹_r ^X

k∈Ir

vn,m





λ

and so

h⁻¹_r ^X

k∈I_r

w_n,m^λⁿ ≤h⁻¹_r ^X

k∈I_r

wn,m+



h⁻¹_r ^X

k∈I_r

vn,m





λ

.

Hence X = (X_k) ∈ M^c_θ^σ, p, i.e. M^c_θ^σ, q ⊂ M^c_θ^σ, p. The proof of the following result is easy and thus is omitted.

Theorem 2.7 (i) Let 0<inf_kp_k ≤1, then M^c_θ^σ, p⊂M^c_θ^σ. (ii) Let 0< p_k ≤sup_kp_k ≤ ∞, then M^c_θ^σ⊂M^c_θ^σ, p.

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