Soft Comput DOI 10.1007/s00500-017-2606-7

METHODOLOGIES AND APPLICATION

Tauberian theorems for weighted mean summability method of improper Riemann integrals of fuzzy-number-valued functions Cemal Belen1

© Springer-Verlag Berlin Heidelberg 2017

Abstract We consider the concept of weighted mean summability method of improper Riemann integrals of fuzzy-number-valued functions with the help of fuzzy Riemann–Stieltjes integrals introduced by Ren and Wu (Int J Theor Phys 52:2134–2151, 2013). The convergence of the improper Rieman integral implies its summability by weighted mean method, but inverse requirement is not true in general and is realized by means of supplementary conditions known as Tauberian conditions. Keywords Tauberian theorems · Fuzzy-number-valued function · Fuzzy Riemann–Stieltjes integral · Weighted mean method

1 Introduction and preliminaries The study of classical Tauberian theory is well-established area of research in summability theory which deals with the problem of determining conditions to obtain convergence of sequences (or functions) from any summability method. In recent years, this theory has also been used for any summability method of sequences of fuzzy numbers. For instance, Altın et al. (2010), Çanak (2014), Çanak (2016), Önder et al. (2015), Sezer and Çanak (2017), Talo and Çakan (2012), Talo and Ba¸sar (2013), Talo and Bal (2016), Tripathy and Baruah (2010) and Yavuz and Ço¸skun (2016) obtained analogues of Tauberian theorems for certain summability methods of fuzzy number sequences. The main idea of this paper is to

prove Tauberian theorems for weighted mean summability of improper Riemann integrals of fuzzy-number-valued functions as an adaptation of the results proved in Móricz (2004). A fuzzy set of R is a mapping u : R → [0, 1]. Then, the mapping u is a fuzzy number with the following properties (see, for example, Dubois and Prade 1980; Diamond and Kloeden 1990): (i) u is normal, i.e. there exists x0 ∈ R such that u (x0 ) = 1; (ii) u is fuzzy convex, i.e. u (λx + (1 − λ) y) ≥ min {u (x) , u (y)} for all x, y ∈ R and for all λ ∈ [0, 1] . (iii) u is upper semi continuous, i.e. {x : u (x) ≥ α} is closed for every α; (iv) [u]0 = {x ∈ R : u (x) > 0} is a compact set, where A denotes the closure of the set A in the usual topology of R. We denote the set of all fuzzy numbers on R by E 1 and call it fuzzy number space. α-level set [u]α of u ∈ E 1 is defined by  [u]α =

{x ∈ R : u (x) ≥ α} , 0 < α ≤ 1 {x ∈ R : u (x) > α}, α = 0.

Note that [u]0 is called the support of u ∈ E 1 and [u]0 =



[u]α .

α∈(0,1]

Hence for all 0 ≤ α ≤ β ≤ 1, we have Communicated by V. Loia.

B 1

Cemal Belen [email protected] Faculty of Education, Ordu University, 52200 Ordu, Turkey

[u]β ⊂ [u]α ⊂ [u]0

(1)

(see Wu and Ma 1991). The property (i) implies that [u]1 = ∅, so by (1) [u]α = ∅ for all α ∈ [0, 1]. Also the property

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C. Belen

(ii) means that the set [u]α is convex for any α ∈ (0, 1] . Moreover, from the properties (iii)–(iv) and the inclusion (1) andbounded non-empty subset we all also say [u]α is closed  + 1 , of R defined by [u]α = u − α u α . Hence, u ∈ E is a fuzzy number if and only if u is normal and [u]α is compact and convex subset of R for all α ∈ (0, 1] . The addition and product with real scalars in E 1 are defined by + : E 1 × E 1 → E 1 ,

(i) D (λu, λv) = |λ| D (u, v) for any u, v ∈ E 1 and λ ∈ R; (ii) D (u + w, v + w) = D (u, v) for any u, v, w ∈ E 1 ; (iii) D (u + v, w + z) ≤ D (u, w) + D (v, z) for any u, v, w, z ∈ E 1 .

Where a = χ{a} for any a ∈ R (see e.g. Dubois and Prade 1987; Wu and Gong 2001). Also if u, v ∈ E 1 , 0 ≤ α ≤ 1, λ ∈ R, then we can write

We say that  f (x) is a fuzzy-number-valued function if  f : [a, b] ⊆ R → E 1 . A fuzzy-number-valued function  f (x) is said to be bounded above on [a, b] if there exists a f (x) , such fuzzy number u ∈ E 1 , called an upper bound of   that f (x) u for all x ∈ [a, b]. u is called the supremum f (x), if u is of  f (x) on [a, b] , denoted as u = supx∈[a,b]  an upper bound of  f (x) and u v for any upper bound v of  f (x) . The lower bound and infimum of  f (x) can be defined similarly.  f (x) is said to be bounded on [a, b] , if it is both bounded above and bounded below.  f : [a, b] → E 1 is continuous at x0 ∈ [a, b] if for any ε > 0, there exists δ (ε) > 0 such that if |x − x0 | < δ (ε) for all x ∈ [a, b] , then

  − + + [u + v]α = [u]α + [v]α = u − α + vα , u α + vα

  D  f (x) ,  f (x0 ) = sup

(u + v) (x) = sup min {u (s) , v (t)} , x=s+t

and by · : R × E 1 → E 1 , (λu) (x) =

 x  u λ , if 0, if

λ = 0 λ = 0.

α∈[0,1]

+

 − f α (x) −  max  f α (x) −  f α− (x0 ) ,  f α+ (x0 ) < ε.

and   + [λu]α = λ [u]α = λu − (λ ≥ 0) α , λu α  + − or λu α , u α (λ < 0) . On the other hand, the partial ordering relation on E 1 is defined as follows: u v ⇔ [u]α [v]α ⇔ and

u+ α

≤

vα+ ,

u− α

≤

vα−

(for any α ∈ [0, 1]).

Note that 1u = u1 = u and u +0 = 0+u, that is, 0 is neutral element in E 1 with respect to the +. Also the following result is known. Lemma 1.1 (see e.g. Anastassiou and Gal 2001; Dubois and Prade 1987) If u, v ∈ E 1 , 0 ≤ α ≤ 1, λ ∈ R, then (i) u + v = v + u; and (ii) λ (u + v) = λu + λv. If we define D :

E1

×

E1

→ [0, +∞) by

If  f (x) is continuous at all points of all points of [a, b], then we say that  f (x) is continuous on [a, b]. We recall the definition of Riemann integral for a fuzzynumber-valued function (see for instance Gal 2000). Definition 1.1 A fuzzy-number-valued function  f : [a, b] 1 → E is Riemann integrable on [a, b] if there exists I ∈ E 1 with the property: for any ε > 0, there exists δ > 0 such that for any division of [a, b] , P : a = x0 < x1 < x2 < · · · < xn = b of norm |P| < δ, and for any points ξi ∈ xi , xi+1 , i = 0, 1, ..., n − 1, we have D

n−1

  f (ξi ) (xi+1 − xi ) , I

< ε.

i=0

In this case, we write I =

b a

 f (x) d x.



 − + + , = sup max u − α − vα , u α − vα

f is Note that if  f : [a, b] → E 1 is continuous, then  fuzzy Riemann integrable on [a, b] (see Gal 2000). We also need the following results concerning the fuzzy Riemann integrable functions.

where d (·, ·) is the ordinary Hausdorff metric, then we have the following.

Lemma 1.3 Anastassiou (2002) If  f , g : [a, b] → E 1 are continuous, then  the function  F : [a, b] → [0, ∞) defined by F (x) = D  f (x) ,  g (x) is continuous on [a, b] and

D (u, v) = sup d ([u]α , [v]α ) α∈[0,1] α∈[0,1]

Lemma  1  1.2 (see e.g. Ming 1993; Wu and Gong 2001) E , D is a complete metric space and in addition D has the following properties:

123

 D a

b

 f (x) d x,

 a

b

  g (x) d x

 ≤ a

b

  D  f (x) ,  g (x) d x.

Tauberian theorems for weighted mean summability method of improper Riemann integrals…

Lemma 1.4 Anastassiou (2002) If  f : [a, b] → E 1 is continuous, then 

x

 s (x) =

 f (t) dt

a

is a continuous fuzzy-number-valued function in x ∈ [a, b]. 1  Definition 1.2  t (see e.g. Anastassiou 2004). Let f : R → E be such that a  f (x) d x exists for every real number such that t ≥ a. Then,



∞

 f (x) d x = lim



t→+∞ a

a

t

 f (x) d x

    (iv) If  f , g ∈ FRS [a, b] ,  h, g ∈ FRS [a, b] and b b  f dg a  hdg. f (t)  h (t) for all t ∈ [a, b], then a  (v) Let  f : [a, b] → E 1 is continuous and g is an increasing real function on [a, b] . Then  f , g ∈ f α+ , g are FRS [a, b] if and only if  f α− , g and  uniformly RS-integrable   for α ∈ [0, 1] on [a, b] ,  b  b − b + and a f dg = a f α dg, a f α dg . That is, α    f , g ∈ FRS [a, b] if and only if for any ε > 0, there exists δ (ε) > 0 such that for every partition T with |T | := max1≤i≤n (xi − xi−1 ) < δ (ε) , and for every choice of points ξi ∈ xi , xi+1 , we have  n−1 b

  − −   f dg − f α (ξi ) g (xi ) − g (xi−1 ) < ε, a α i=0  n−1 b

  + +   f dg − f α (ξi ) g (xi ) − g (xi−1 ) < ε, a α

E1

with provided the limit exists in  ∞respect to the D-metric. f (x) d x is convergent The improper Riemann integral a  if the corresponding limit exists in E 1 and divergent if the limit does not exist in E 1 .

i=0

The definition of Riemann–Stieltjes integral of a fuzzynumber- valued function was first given by Nanda (1989) and Wu (1998) in different ways, but both of these definitions were too complex. So, Ren and Wu (2013) presented a new definition for the fuzzy Riemann–Stieltjes integral through the integral sum which provides easiness to investigate and characterize the properties of this integral. Let  f : [a, b] → E 1 be a bounded function, g be an increasing real function on [a, b] and w ∈ E 1 . Also let T be any partition of [a, b] such that T : a =  x0 < x1 < · · · < xn = b. Choose any point ξi ∈ xi , xi+1 , i = 0, 1, . . . , n − 1, and form the fuzzy summation  sT =

n−1

   f (ξi ) g (xi ) − g (xi−1 ) .

for all α ∈ [0, 1] . (vi) If  f ,g ∈ FRS [a, b], then  for any c ∈ (a, b) , we have  f , g ∈ FRS [a, c] ,  f , g ∈ FRS [c, b] and b c b    a f dg = a f dg + c f dg. Let  f , h : [a, b] → E 1 are continuous and g is an increasing real function on [a, b] . Then, we have 

 f dg,

a



b

  hdg = sup

α∈[0,1]

a

   b +  − − +     f α − h α dg , f α − h α dg a a   b  b − + +  dg,   ≤ sup max dg h− − h fα −  f α α α   max

i=0

Then, we say that w is the Riemann–Stieltjes integral of  f with respect to the function g if for any ε > 0, there exists δ (ε) > 0 such that for every partition T with |T | := max1≤i≤n (xi − xi−1 ) < δ (ε) , and for every choice of sT ) < ε. In this case we write w = points ξi , we have D (w, b   f dg. If the Riemann–Stieltjes integral a  of f with respect to the function g exists, then we write  f , g ∈ FRS [a, b]. The important properties of the fuzzy Riemann–Stieltjes integral are summarized in the following theorem (see Ren and Wu 2013). b f dg exists and c is a positive constant, Lemma 1.5 (i) If a  b  b b   f dg = c a  then a c f dg exists and a c  f dg.   1 f,g ∈ (ii) If  f (x) = u ∈ E for all x ∈ [a, b] , then  b FRS [a, b] and a  f dg = u (g (b) − g (a)) . (iii) If  f : [a, b] → E 1 is continuous  and  g is an increasing real function on [a, b] , then  f , g ∈ FRS [a, b] .

b

D

b



α∈[0,1]



=

b

a

a

+

 −  + dg, sup max  fα −  h− α , fα − h α

a α∈[0,1]

that is  D a

b

 f dg,

 a

b

   hdg ≤

b

  D  f , h dg

(2)

a

where the right-hand side integral  exists in usual  Riemann– Stieltjes sense since F(x) = D  f (x) ,  h (x) is continuous real function on [a, b] (by Lemma 1.3) and g is monotone on [a, b]

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C. Belen

2 Main results Let 0 = p : R+ → R+ be an increasing function such that p (0) = 0, p (t) → ∞ as t → ∞ and lim inf t→∞

p (λt) > 1 for every λ > 1. p (t)

(3)

Assume that  f : [0, ∞) → E 1 is a continuous fuzzynumber-valued function and let 

t

 s (t) =

 f (u) du and  σ (t) =

0

1 p (t)



t

 s (x) dp (x) ,

0

where the second integral exists in the fuzzy Riemann– Stieltjes sense. The fuzzy-number-valued function  s (t) is said to be summable to a fuzzy number L ∈ E 1 with  the weighted mean method determined by p, in short N , p summable to L ∈ E 1 , if lim D ( σ (t) , L) = 0.

t→∞

(4)

Note that in special cases p (t) = t and p (t) = log(1 + t), weighted mean summability method reduces to Cesàro and logarithmic summability methods, respectively. Cesàro summability of integrals of fuzzy-number-valued functions has been studied by Yavuz et al. (2016) recently. By Definition 1.2, if the limit

D ( σ (t) , L)    t 1  s (x) dp (x) , L =D p (t) 0    t  t 1 1  s (x) dp (x) , Ldp (x) =D p (t) 0 p (t) 0   t  t 1 D =  s (x) dp (x) , Ldp (x) p (t) 0 0  t 1 ≤ D ( s (x) , L) dp (x) p (t) 0  t1 1 = D ( s (x) , L) dp (x) p (t) 0  t 1 + D ( s (x) , L) dp (x) p (t) t1   p (t1 ) p (t1 ) ε + 1− <ε 1

s (t) = L lim 



t→∞

lim D

exists in D-metric, i.e. limt→∞ D ( s (t) , L) = 0, then the improper fuzzy Riemann integral 

∞

 f (u) du

0

also exists and equals to L . Furthermore, we can prove the following: Theorem s (t) = L ∈ E 1 , then  s (t) is  2.1 If limt→∞   N , p summable to L . Proof Let limt→∞  s (t) = L ∈ E 1 . Then for each ε > s (t) , L) < ε/2 for all t > 0 there exists t1 such that D ( s (t) , L) implies its t1 . Furthermore, the continuity of D ( s (t) , L) . boundedness on [0, t1 ], say M := max0≤t≤t1 D ( Hence, by using Lemma 1.5 (ii)–(vi), Lemma 1.2 (i)–(ii) and the inequality (2), we have

123

t→∞



1 p (λt) − p (t)

λt

  s (x) dp (x) , L

=0

(5)

= 0.

(6)

t

and for each 0 < λ < 1  lim D

t→∞



1 p (t) − p (λt)

t λt

  s (x) dp (x) , L

Proof It is enough to prove the case λ > 1, the case 0 < λ < 1 is similar. By Lemma 1.2 (ii)–(iii), we have  D

1 p (λt) − p (t) 

=D 



λt

  s (x) dp (x) , L

t

1 p (λt) − p (t)

1 p (λt) − p (t) + D ( σ (t) , L)



λt

  s (x) dp (x) +  σ (t) ,  σ (t) + L

t



≤D

t

λt

  s (x) dp (x) ,  σ (t) ,

Tauberian theorems for weighted mean summability method of improper Riemann integrals…

also we have 

  λt 1  s (x) dp (x) ,  σ (t) p (λt) − p (t) t   λt 1 =D  s (x) dp (x) , p (λt) − p (t) t   t 1  s (x) dp (x) p (t) 0   λt 1 D  s (x) dp (x) p (λt) − p (t) t  t 1 +  s (x) dp (x) , p (λt) − p (t) 0   t  t 1 1  s (x) dp (x) +  s (x) dp (x) p (t) 0 p (λt) − p (t) 0   λt 1 =D  s (x) dp (x) , p (λt) − p (t) 0   t p (λt)  s (x) dp (x) ( p (λt) − p (t)) p (t) 0   λt 1 p (λt) D =  s (x) dp (x) , p (λt) − p (t) p (λt) 0   t 1  s (x) dp (x) p (t) 0 p (λt) D ( σ (λt) ,  σ (t)) = p (λt) − p (t) p (λt) ≤ σ (λt) , L) + D ( σ (t) , L)) (D ( p (λt) − p (t)

Sufficiency. Suppose that condition (8) holds. Then for any given ε > 0, there exists λ > 1 such that 

D

Thus, Eq. (5) is obtained by applying lim sup to last inequality and using (4) and (3). 

  Theorem 2.2 Let  s (t) be N , p summable to a fuzzy number L. Then, lim  s (t) = L

(7)

t→∞

holds if and only if  inf lim sup D

λ>1 t→∞

1 p (λt) − p (t)



λt

  s (x) dp (x) , s (t) = 0

t

(8)

 inf lim sup D

t→∞



λt

  s (x) dp (x) , s (t) < ε.

t

Since D ( s (t) , L)  =D  s (t) +

 λt 1  s (x) dp (x) , p (λt) − p (t) t   λt  s (x) dp (x) + L

1 p (λt) − p (t) t    λt 1  s (x) dp (x) ≤D  s (t) , p (λt) − p (t) t    λt 1 +D  s (x) dp (x) , L , p (λt) − p (t) t we get s (t) , L) ≤ ε. lim sup D ( t→∞

Similarly if the condition (9) holds, then for any given ε > 0, there exists 0 < λ < 1 such that    t 1  s (x) dp (x) , s (t) < ε. lim sup D p (t) − p (λt) λt t→∞ Since D ( s (t) , L)  =D  s (t) +

 t 1  s (x) dp (x) , p (t) − p (λt) λt   t 1  s (x) dp (x) + L p (t) − p (λt) λt    t 1  s (x) dp (x) ≤D  s (t) , p (t) − p (λt) λt    t 1 +D  s (x) dp (x) , L p (t) − p (λt) λt

we conclude that s (t) , L) ≤ ε. lim sup D ( t→∞

or

0<λ<1 t→∞

lim sup D

1 p (λt) − p (t)

1 p (t) − p (λt)



t λt

  s (x) d p (x) , s (t) = 0.

(9)   Proof Necessity. Assume that (7) holds. Since s (t) is N , p summable to L , the necessity of conditions (8) or (9) follows from (5 ) or (6), respectively.

From this, we obtain that limt→∞ D ( s (t) , L) = 0 and this completes the proof. 

Using the concept of slowly oscillating function (see, for example, Hardy 1949), we say that a fuzzy-number-valued function  s : [0, ∞) → E 1 is said to be slowly oscillating if s (x) , s (t)) = 0. lim lim sup max D (

λ→1+ t→∞ t
(10)

123

C. Belen

Note that in (10) we can replace limλ→1+ by inf λ>1 , since s (x) , s (t)) g (λ) := lim sup max D ( t→∞ t
Example 2.1 Let  f : [0, ∞) → E 1 be a fuzzy-numbervalued function defined by ⎧ u (1 + t) 2 + sin t ⎪ ⎪ , 0≤u≤ ⎪ ⎪ 2 + sin t 1+t ⎪ ⎪ ⎪ ⎨    u (1+t) 2+ sin t 2 (2+ sin t) f (t) (u) = − 1, ≤u≤ ⎪ ⎪ 2 + sin t 1+t 1+t ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, otherwise t and let  s (t) = 0  f (v) dv. It is clear that  f is continuous and for any α ∈ [0, 1] we have 2 + sin t 2 + sin t  α and  f α− (t) = f α− (t) = (1 + α) . 1+t 1+t Then,  s (t) is slowly oscillating. Indeed, for t < x s + (t)  sα+ (x) −  x  xα dv (1 + α) (2 + sin v) dv ≤ 3 (1 + α) = 1+v t t 1+v   1+x = 3 (1 + α) ln 1+t and 

α (2 + sin v) dv ≤ 3α 1+v t   1+x . ≤ 3 (1 + α) ln 1+t x



x t

dv 1+v

D ( s (x) , s (t))

+

 − sα (x) − sα (x) − = sup max  sα− (t) ,  sα+ (t) 

1 + λt ≤ 6 ln 1+t





λ (1 + t) < 6 ln 1+t

 = 6 ln λ

for any λ > 1 and t < x ≤ λt. This implies that lim lim sup max D ( s (x) , s (t)) = 0.

λ→1+ t→∞ t
123

N, p



  λt 1  s (x) dp (x) , s (t) D p (λt) − p (t) t   λt 1 =D  s (x) dp (x) , p (λt) − p (t) t   λt 1  s (t) dp (x) p (λt) − p (t) t  λt 1 ≤ D ( s (x) , s (t)) dp (x) p (λt) − p (t) t ≤ max D ( s (x) , s (t)) 

t
we obtain that (8) holds. Hence, (7) is satisfied by Theorem 2.2. 

   Note that if the condition  u D f (u) , 0 = O(1) holds for all u > 0,that is u D  f (u) , 0 ≤ C for some C ≥ 0, x then  s (x) = 0  f (u) du is slowly oscillating. Indeed, for any t < x ≤ λt, we have D ( s (x) , s (t))  x   t   =D f (u) du, f (u) du 0 0   t  x  t    f (u) du + f (u) du, f (u) du + 0 =D 0 t  x 0 x    D  f (u) , 0 du f (u) du, 0 ≤ =D t t  x x du = C log ≤C u t t s (x) , s (t)) ≤ C log λ. Hence and then maxt
λ→1+ t→∞ t
which proves our claim.

Then, we get

α∈[0,1]



Proof Assume that  s (t) is slowly oscillating. Since

is an increasing function of λ. It is easy to see that if (10 ) holds if and only if for every ε > 0, there exist some t0 = t0 (ε) ≥ 0 and λ = λ (ε) > 1 such that D ( s (x) , s (t)) ≤ ε whenever t0 ≤ t < x ≤ λt.

 sα− (x) − sα− (t) =

Corollary 2.1 If  s (t) is slowly oscillating and summable to a fuzzy number L, then (7) holds.

   Corollary  x 2.2 If u D f (u) ,0 = O(1) for all u > 0 and f (u) du is N , p summable to a fuzzy number  s (t) = 0  L , then (7) holds. Compliance with ethical standards Conflict of interest The author declares that he has no conflict of interest. Ethical approval This article does not contain any studies with human participants performed by any of the authors.

Tauberian theorems for weighted mean summability method of improper Riemann integrals…

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