Soft Comput DOI 10.1007/s00500-015-1840-0
METHODOLOGIES AND APPLICATION
Power series methods of summability for series of fuzzy numbers and related Tauberian Theorems ˙ Sefa Anıl Sezer1,2 · Ibrahim Çanak3
© Springer-Verlag Berlin Heidelberg 2015
Abstract In this paper, we initiate the study of power series methods of summability for series of fuzzy numbers and establish some Tauberian conditions to obtain convergence of a series of fuzzy numbers from its summability by power series methods. Keywords Sequences and series of fuzzy numbers · Tauberian theorems · power series methods of summability
1 Introduction The theory of fuzzy set has attracted the attention of many researchers from various branches of mathematics since its first introduction by Zadeh (1965). Dubois and Prade (1978) introduced the concept of fuzzy numbers and defined the basic operations such as addition, subtraction, multiplication and division. Matloka (1986) introduced the concepts of con-
Communicated by V. Loia.
B
Sefa Anıl Sezer
[email protected];
[email protected] ˙Ibrahim Çanak
[email protected]
1
Department of Mathematics, ˙Istanbul Medeniyet University, ˙Istanbul, Turkey
2
Present Address: Department of Mathematics, Ege University, ˙Izmir, Turkey
3
Department of Mathematics, Ege University, ˙Izmir, Turkey
vergent and bounded sequences of fuzzy numbers, presented some of their properties and proved that every convergent sequence of fuzzy numbers is bounded. Nanda (1989) proved that every Cauchy sequence of fuzzy numbers is convergent. Later on, difference sequences of fuzzy numbers have been discussed by Altin et al. (2007), Altinok et al. (2012) and many authors. The summability methods of sequences of fuzzy numbers have been introduced and fuzzy analogs of Tauberian theorems for different summability methods have been studied by many researchers. Firstly, Subrahmanyam (1999) have defined the concept of the Cesàro convergence of a sequence of fuzzy numbers and obtained analogs of classical Tauberian theorems related to Cesàro summability method. Talo and Çakan (2012), Talo and Ba¸sar (2013) and Çanak (2014a, b, c) have established some Tauberian theorems for Cesàro summability of sequences of fuzzy numbers. Altin et al. (2010) have defined the statistical Cesàro summability and obtained some Tauberian results for sequences of fuzzy numbers. Tripathy and Baruah (2010) have introduced Nörlund and Riesz summability methods for sequences of fuzzy numbers and determined Tauberian conditions for the Riesz summability method. Later, Çanak (2014d) and Önder et al. (2015) have proved some Tauberian type theorems for the weighted mean method of summability of sequences of fuzzy numbers. Recently, Yavuz and Talo (2014) have presented the concept of Abel summability for sequences and series of fuzzy numbers and obtained fuzzy analogs of some Tauberian results. In this paper, we introduce the concept of the power series method of summability for a series of fuzzy numbers and generalize the Tauberian results in Ishiguro (1964, 1965) to fuzzy analysis. As a special case, we obtain the results in Yavuz and Talo (2014).
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2 Preliminaries
and scalar multiplication on the set of fuzzy numbers are defined as follows:
In this section, we recall the basic definitions and notations dealing with fuzzy numbers. For the sake of completeness of the paper, we give our study in the main results section. A fuzzy number is a fuzzy set u : R → [0, 1], which satisfies the following properties (Kaleva 1987): 1. u is normal; i.e., there is a unique element x0 ∈ R such that u(x0 ) = 1. 2. u is fuzzy convex; i.e., for any x, y ∈ R and for any λ ∈ [0, 1], u(λx + (1 − λ)y) ≥ min{u(x), u(y)}. 3. u is upper semicontinuous. 4. The support of u, [u]0 := {x ∈ R : u(x) > 0} is compact, where {x ∈ R : u(x) > 0} denotes the closure of the set {x ∈ R : u(x) > 0} in the usual topology of R. We denote the set of all fuzzy numbers by E 1 and call it the space of fuzzy numbers. For a fuzzy set u, we define α-level sets of u as follows: [u]α :=
{x ∈ R : u(x) ≥ α},
(0 < α ≤ 1)
{x ∈ R : u(x) > α},
(α = 0).
It is well known that u is a fuzzy number if and only if [u]α is a closed, bounded, and nonempty interval for each α ∈ [0, 1] with [u]β ⊂ [u]α if α < β. From this characterization of fuzzy numbers, it follows that a fuzzy number u is completely determined by the end points of the intervals [u]α := [u − (α), u + (α)]. Theorem 1 (Goetschel and Voxman 1986) Let u ∈ E 1 and [u]α = [u − (α), u + (α)]. Then the functions u − , u + : [0, 1] → R, defining the end points of the α−level sets, satisfy the following conditions: 1. u − (α) ∈ R is a bounded, non-decreasing and left continuous function on (0, 1]. 2. u + (α) ∈ R is a bounded, non-increasing and left continuous function on (0, 1]. 3. The functions u − (α) and u + (α) are right continuous at α = 0. 4. u − (1) ≤ u + (1). 5. If the pair of functions u − (α) and u + (α) satisfy the above conditions (1)–(4), then there exists a unique v ∈ E 1 such that [v]α := [u − (α), u + (α)]. By the above theorem we can identify a fuzzy number u with the parameterized representation [u − (α), u + (α)]. Suppose that u, v ∈ E 1 are fuzzy numbers represented by [u − (α), u + (α)] and [v − (α), v + (α)], respectively, and k ∈ R, α ∈ [0, 1]. The operations of addition, subtraction
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[u + v]α := u − (α) + v − (α), u + (α) + v + (α) , [u − v]α := u − (α) − v + (α), u + (α) − v − (α) , [ku]α := k[u]α . The set of all real numbers can be embedded in E 1 . For r ∈ R, r¯ ∈ E 1 is defined by r¯ (x) :=
1, 0,
x = r, x = r.
Lemma 1 (Diamond and Kloeden 1994; Bede 2013) 1. The addition of fuzzy numbers is associative and commutative, i.e., u + v = v + u and u + (v + w) = (u + v) + w, for all u, v, w ∈ E 1 . 2. 0¯ ∈ E 1 is neutral element with respect to +, i.e., u + 0¯ = 0¯ + u = u, for any u ∈ E 1 . 3. With respect to +, none of u ∈ E 1 \R has an opposite in E 1. 4. For any a, b ∈ R with a.b ≥ 0 and any u ∈ E 1 , we have (a + b)u = a.u + b.u. For general a, b ∈ R, this property does not hold. 5. For any a ∈ R and u, v ∈ E 1 , we have a.(u + v) = a.u + a.v. 6. For any a, b ∈ R and any u ∈ E 1 , we have (a.b).u = a.(b.u). As a conclusion from the previous Lemma, we obtain that the space of fuzzy numbers is not a linear space. Additionally, we note that if ak ∈ R and a k ≥ 0 for all k, then for any u ∈ E 1 we have nk=0 uak = u nk=0 ak from (4) of Lemma 1. The most well known and also the most employed metric in the space of fuzzy number is the Hausdorff distance. Let us recall the definition of the Hausdorff distance in the case when A = [A, A], B = [B, B] are two intervals (we denote A, B ∈ W , where W is the set of all closed and bounded intervals), then the Hausdorff distance on W is d(A, B) := max A − B , A − B . It is known that with respect to the Hausdorff distance, W is a complete separable metric space (cf. Nanda 1989). Now, we may define the metric D on the space of fuzzy numbers. Definition 1 (Puri and Ralescu 1986; Diamond and Kloeden 1994) Let D : E 1 × E 1 → R+ ,
Power series methods of summability for series of fuzzy…
D(u, v) := sup max u − (α)−v − (α) , u + (α)−v + (α)
n
α∈[0,1]
:= sup d([u]α , [v]α ).
k=0
α∈[0,1]
Then, D is called Hausdorff distance between fuzzy numbers. Proposition 1 (Diamond and Kloeden 1994) Let u, v, w, z ∈ E 1 and k ∈ R. Then the following statements hold true. 1. (E 1 , D) is a complete metric space. 2. D(u + w, v + w) = D(u, v), i.e., D is translation invariant. 3. D(k.u, k.v) = |k| D(u, v). 4. D(u + v, w + z) ≤ D(u, w) + D(v, z). We now quote the following definitions which will be needed in the sequel. Definition 2 (Matloka 1986) A sequence u = (u n ) of fuzzy numbers is a function u from the set N of all positive integers into E 1 . The fuzzy number u n denotes the value of the function at n ∈ N and is called the n-th term of the sequence. The set of all sequences of fuzzy numbers is denoted by ω(F). Definition 3 (Matloka 1986) A sequence u = (u n ) of fuzzy numbers is said to be convergent to the fuzzy number L, written as limn u n = L, if for all > 0 there exists a positive integer n 0 such that
n
+ u+ k (α) → L (α)
k=0
uniformly in α ∈ [0, 1]. (α), u + Conversely, if u k = {(u − k k (α)) : α ∈ [0, 1]} ∞ − are fuzzy numbers such that k=0 u − k (α) = L (α) and ∞ + + k=0 u k (α) = L (α) converge uniformly in α, then L = {(L − (α), L + (α)) : α ∈ [0, 1]} is a fuzzy number and ∞ k=0 u k = L . ∞ Lemma If the series uk ∞2 (Talo and Ba¸sar 2009) ∞ k=0 ∞ ≤ and v converge, then D k=0 u k , k=0 vk ∞ k=0 k k=0 D(u k , vk ).
3 Main Results Let ∞ k=0 u k be a series of fuzzy numbers with partial sums (sn ). Let p = ( pn ) be a sequence of nonnegative real numbers with p0 > 0 and Pn =
n
pk → ∞, n → ∞.
(1)
k=0
Assume that the power series p(x) =
D(u n , L) < for n > n 0 .
∞
pk x k
(2)
k=0
A sequence u = (u n ) of fuzzy numbers is said to be bounded if there exists a positive number M such that D(u n , 0) < M for all n ∈ N. A sequence (u n ) ∈ ω(F) is said to be a Cauchy sequence if for all > 0 there exists a positive integer n 0 such that D(u n , u m ) < for n, m > n 0 . Remark 1 (Talo and Ba¸sar 2009) If the sequence (u n ) ∈ ω(F) converges to a fuzzy number u, then by the defini+ tion of metric D, {u − n (α)} and {u n (α)} converge uniformly − + to u (α) and u (α) on [0, 1], respectively. Definition 4 (Kim and Ghil 1997) Let (u k ) ∈ ω(F) and of partial L ∈ E 1 . The sequence ∞ (sn ) denotes the sequence sums of the series ∞ k=0 u k . The series k=0 u k is said to be convergent to L if D(sn , L) → 0 as n → ∞. In this case, we call L the sum of the series write ∞ k=0 u k = L which implies
− u− k (α) → L (α) and
∞
k=0 u k
and
has the radius of convergence 1. Throughout this paper, M denotes a positive constant, possibly different at each occurrence and the symbols u n = o(1) and u n = O(1) mean, respectively, that u n → 0 as n → ∞ and (u n ) is bounded for large enough n. 1 ∞ k Definition 5 Suppose that ps (x) = p(x) k=0 pk sk x exists for each x ∈ (0, 1). If lim ps (x) = L, then we x→1− ∞ say that the series k=0 u k is summable to L ∈ E 1 by the power ∞ series method determined by the sequencep;∞in short, k=0 u k is (J, p) summable to L and write k=0 u k = L (J, p). If pk = 1 for all nonnegative integers k, then the corresponding power series method (J, 1) is the Abel summability 1 1 , the method (J, k+1 ) is called the logmethod; if pk = k+1 arithmic summability method. Our first result is the regularity theorem for (J, p) summability of series of fuzzy numbers. Theorem 2 If the series ∞ k=0 u k of fuzzy numbers converges to a fuzzy number L, then it is (J, p) summable to L.
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∞ Proof Suppose k=0 u k converges to L, then we have limn→∞ sn = L . Since (sn ) converges to L, there exists n 0 such that D(sn , L) ≤ 2 if n ≥ n 0 . Then choose δ < 1 so that if δ < x < 1, 0 1 D (sk , L) pk x k ≤ p(x) 2
However, it is both logarithmic and Abel summable to the 1 , same fuzzy number. Then, when pk = k+1 [ ps (x)]− (α) =
n
since (1) is satisfied. Therefore, if δ < x < 1, D ( ps (x), L) = D =D ≤
∞
1 sk p k x k , L p(x)
k=0
1 p(x)
1 p(x)
∞
1 = p(x)
∞ k=0
∞
1 sk p k x , L pk x k p(x)
k=0
D(sk , L) pk x k
and [ ps (x)]+ (α) =
D(sk , L) pk x
∞ 1 + pk x k 2 2 p(x)
1 + 2 2 p(x)
k=n 0 +1 ∞
pk x k = .
It is evident by the following example that the converse of Theorem 2 is not necessarily true. Example 1 Let ∞ k=0 u k be a series of fuzzy numbers with partial sums (sn ) such that
1 n+1
if (−1)n ≤ t ≤ (−1)n + 3
n+1
n+1 if (−1)n + 13 ≤ t ≤ 5 − 13
1 n+1 if 5 − 3 ≤t ≤5 otherwise.
We can construct the α-level sets of this sequence as follows: α n+1 α n+1 [sn ]α = (−1)n + . ,5 − 3 3
n+1 Since {sn− (α)} = {(−1)n + α3 } does not converge uniformly in α ∈ [0, 1], (sn ) is divergent by Remark 1. So, the series ∞ k=0 u k is divergent by Definition 4.
123
1 − log(1 − x)
converge uniformly in α. So, by Definition 4, ps (x) exists for each x ∈ (0, 1) and
k=0
1 ∞ k Thus, lim x→1− p(x) k=0 sk pk x = L, which completes the proof of theorem.
⎧ √ ⎪ 3 n+1 t − (−1)n ⎪ ⎪ ⎪ ⎨1 sn (t) = √ ⎪ 3 n+1 5 − t ⎪ ⎪ ⎪ ⎩ 0
∞
5 x k+1 k+1 k=0 ∞ α k+1 1 3 x k+1 − − log(1 − x) k+1 k=0
log 1 − αx 3 =5− log(1 − x) =
k=0
k=n 0 +1
≤
∞
s + (α) 1 k x k+1 − log(1 − x) k+1 k=0
k
∞ 1 + D(sk , L) pk x k p(x)
≤
1 − log(1 − x)
k
k=0 n0
k=0 ∞
(−1)k k+1 x k+1 k=0 ∞ α k+1 1 3 + x k+1 − log(1 − x) k+1 k=0
log 1 − αx log(x + 1) 3 + = − log(1 − x) log(1 − x) =
k=0
∞
s − (α) 1 k x k+1 −log(1−x) k +1
log 1− αx log(x +1) log 1− αx 3 3 + , 5− [ ps (x)]α = . −log(1−x) log(1 − x) log(1−x) We require to prove that lim x→1− ps (x) = L, where [L]α = [0, 5]. Now, since D( ps (x), L) = sup d([ ps (x)]α , [L]α ) α∈[0,1]
= sup max [ ps (x)]− (α) − L − (α) , α∈[0,1]
[ ps (x)]+ (α) − L + (α) log(x +1) log 1− αx 3 = sup max + , −log(1−x) log(1−x) α∈[0,1] log 1 − αx 3 log(1 − x)
log(x + 1) log 1 − αx 3 = sup + log(1 − x) α∈[0,1] − log(1 − x)
log 1 − x3 log(x + 1) = + , − log(1 − x) log(1 − x)
Power series methods of summability for series of fuzzy…
we obtain lim x→1− D( ps (x), L) = 0. That is, the series ∞ 1 k=0 u k is logarithmic summable to L ∈ E . Besides, if pk = 1 for all k, [ ps (x)]− (α) = (1 − x)
∞
∞
sk− (α)x k
k=0
k=0 ∞ ∞ α k+1 k = (1 − x) (−x)k + (1 − x) x 3 k=0
=
Lemma 3 Let ( pk ) be a nonnegative sequence satisfying condition (1), then
k=0
α(1 − x) 1−x + 1+x 3 − αx
Pn
k = O(1) for n → ∞. pk 1 − n1
Proof For n ≥ 2, we have n ∞ 1 n 1 k 1 k Pn 1 − < pk 1 − < pk 1 − n n n k=0
and
and +
[ ps (x)] (α) = (1 − x) = (1 − x)
∞ k=0 ∞
= 5−
1 n 1− > n n
sk+ (α)x k
k=0
5x k − (1 − x)
k=0
∞ k=0
α k+1 3
xk
α(1 − x) 3 − αx
converge uniformly in α. Then, ps (x) converges for all x ∈ (0, 1) and
0 < pk ≤ M for all k ≥ 0, k = O(1), Pk
We want to prove lim x→1− ps (x) = L, where [L]α = [0, 5]. Now, since
D uk , 0 = o
D( ps (x), L)
are satisfied, then
α∈[0,1]
1 − x α(1 − x) α(1 − x) , + = sup max 1+x 3 − αx 3 − αx α∈[0,1] 1 − x α(1 − x) = sup + 1+x 3 − αx α∈[0,1]
1−x 1−x + , 1+x 3−x
− we ∞get lim x→1 D( ps (x), L) = 0.1 That is, the series u is Abel summable to L ∈ E . k=0 k
The question naturally appears as to whether one can determine suitable restrictions on the sequence pn and the general u term u n , so that ∞ n=0 n is convergent to L whenever it is (J, p) summable to L . To answer this question, we prove the following Tauberian theorems. For the proof of our main theorems, firstly, we need to prove the following Lemma.
k=0
Theorem 3 If the series ∞ k=0 u k of fuzzy numbers is (J, p) 1 summable to L ∈ E , and the conditions
and
= sup d ([ ps (x)]α , [L]α )
Pn Pn > ∞ .
k k pk 1 − n1 pk 1 − n1
n Since limn→∞ 1 − n1 = 1e , the proof follows.
α(1 − x) α(1 − x) 1−x + ,5 − . [ ps (x)]α = 1+x 3 − αx 3 − αx
=
k=0
pk Pk
(3) (4)
(5)
∞
k=0 u k
= L.
Proof We establish this theorem by showing that D (sn , ps (x)) tends to zero as n→ ∞.
∞ 1 k sk p k x D (sn , ps (x)) = D sn , p(x) k=0
∞ ∞ 1 1 sn p k x k , sk p k x k =D p(x) p(x) k=0
≤ =
1 p(x) 1 p(x)
+
∞ k=0 n−1
k=0
D(sn , sk ) pk x k D(sn , sk ) pk x k
k=0
∞ 1 D(sn , sk ) pk x k p(x) k=n+1
= I1 + I2 .
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S. A. Sezer, ˙I. Çanak
It suffices to show that I1 , I2 → 0 as n → ∞. Since x < 1, we have 1 D(sn , sk ) pk x k p(x) n−1
I1 = ≤
1 p(x)
k=0 n−1
≤
D(sn , sk ) pk
1 {D(sn , s0 ) p0 + D(sn , s1 ) p1 + · · · = p(x) +D(sn , sn−1 ) pn−1 }
1
D u 1 , 0 p0 + D u 2 , 0 ( p0 + p1 ) + ≤ p(x)
· · · + D u n , 0 ( p0 + p1 + · · · + pn−1 ) n Pn 1
D u k , 0 Pk−1 = p(x) Pn k=1
n 1 D u k , 0 Pk−1 Pn =
pk pk p 1 − n1 Pn Pn p(1− n1 )
= O(1) from
D (sn , sk ) ≤ D u n+1 , 0 + D u n+2 , 0 + · · · + D u k , 0 pn+1 pn+2 pk ≤ + + ··· + Pn+1 Pn+2 Pk ≤ ( pn+1 + pn+2 + · · · + pk ) Pn ≤ (Pk − Pn ) Pn Pk ≤ . Pn Hence, using the above inequality and condition (3),
= ≤
Pn p(x) Pn2
Pn M p(x) Pn2
123
(k + 1)x k
k=0
=
Pn p(1 −
1 n)
M 2n2 Pn2
1 when x = 1 − . Therefore, we obtain from Lemma 3 and n condition (4) that I2 → 0 as n → ∞. This completes the proof.
∞ Corollary 1 Let the series k=0 u k of fuzzy numbers is (J, p) summable to L ∈ E 1 . If pk Pk
and there exists positive constants, γ1 and γ2 , such that
there exists an n 0 such that for k > n 0 , we obtain D(u k , 0) ≤ Ppkk . Assume that k > n > n 0 . Then, we have by condition (5) that
k=n+1 ∞
Pn p(x) Pn2
D(u k , 0) = o
Lemma 3. Moreover, using condition (5) and the fact that the weighted mean matrix W , with wn,k = Ppnk , is regular, we get I1 → 0 as n → ∞. Fix > 0 and consider I2 . Since D(u k , 0) = o Ppkk ,
∞ 1 Pk pk x k p(x) Pn
k=n+1 ∞ 2 M
k=1
I2 ≤
∞ Pn M 2 (k + 1)x k p(x) Pn2
1 Pn M 2 = 2 p(x) Pn (1 − x)2
k=0
if x is chosen to be equal to 1− n1 . Now,
≤
γ1 ≤ pk ≤ γ2 for all k ≥ 0, then
∞
k=0 u k
(6)
= L.
Proof It is sufficient to prove that (6) implies (4). From condition (6), we get 1 k 1 ≤ ≤ . 2γ2 Pk γ1
Therefore, the proof follows. Corollary 2 (Yavuz and Talo 2014) If the series fuzzy numbers is Abel summable to L ∈ E 1 and
∞
k=0 u k
of
k D(u k , 0) = o(1), then
∞
k=0 u k
= L.
Proof Take pk = 1 for all nonnegative integers k in Theorem 3.
∞ Theorem 4 If the series k=0 u k of fuzzy numbers is (J, p) summable to L ∈ E 1 , and the conditions kpk = O(1)
(7)
and Pk pk x k
k=n+1 ∞ k=n+1
D(u k , 0) = o
Pk x k
pk Pk
are satisfied, then
(8)
∞
k=0 u k
= L.
Power series methods of summability for series of fuzzy…
Proof As in the proof of Theorem 3, let D (sn , ps (x)) = I1 + I2 . By Lemma 3 and condition (8), we obtain limn I1 = 0. Now, we shall estimate I2 . Using conditions (7) and (8), we have
D (sn , sk ) ≤ D u n+1 , 0 + D u n+2 , 0 + · · · + D u k , 0 pn+1 pn+2 pk ≤ + + ··· + Pn+1 Pn+2 Pk ≤ ( pn+1 + pn+2 + · · · + pk ) Pn 1 M 1 1 ≤ + + ··· + Pn n + 1 n + 2 k M (k + 1) . ≤ Pn n Hence by using (7), I2 ≤ ≤ ≤
∞ 1 M (k + 1) pk x k p(x) n Pn k=n+1 ∞
M2
1 p(x) n 2 Pn
M2
Pn p(x) n 2 Pn2
(k + 1)x k
k=n+1 ∞
(k + 1)x k
k=0
1 Pn M 2 = 2 2 p(x) n Pn (1 − x)2 =
M2 , p(1 − n1 ) Pn2 Pn
1 when x = 1− . Thus, we obtain from Lemma 3 that I2 → 0 n as n → ∞. This completes the proof.
Corollary 3 If the series ∞ k=0 u k of fuzzy numbers is logarithmic summable to L ∈ E 1 and k log k D(u k , 0) = o(1), then
∞
k=0 u k
= L.
Proof Take pk = orem 4.
1 k+1
for all nonnegative integers k in The
4 Conclusion In this study, we have introduced the notion of (J, p) summability of series in fuzzy analysis. We have obtained that if a series of fuzzy numbers converges to L ∈ E 1 , then it is (J, p)
summable to L. Then, we have constructed examples to show that the converse of this statement is not true in general and later we have proved fuzzy analogs of several Tauberian theorems for (J, p) summability to establish that the converse is only true under some additional conditions called Tauberian conditions. Moreover, we have obtained Tauberian results for some special cases of power series methods such as Abel summability method and logarithmic summability method. The notion of summability of series has applications in almost all areas of science. The newly introduced concept of (J, p) summability of series of fuzzy numbers will help the researchers of other areas of science who are working on the fuzzy analogs of existing results.
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