Tauberian theorems for statistical Cesàro and statistical logarithmic summability of sequences in intuitionistic fuzzy normed spaces
aa r X i v : . [ m a t h . G M ] J a n Tauberian theorems for statistical Ces`aro and statistical logarithmicsummability of sequences in intuitionistic fuzzy normed spaces
Enes Yavuz
Department of Mathematics, Manisa Celal Bayar University, Manisa, TurkeyE-mail: [email protected]
Abstract:
We define statistical Ces`aro and statistical logarithmic summability methods of sequences in intuitionis-tic fuzzy normed spaces(
IF N S ) and give slowly oscillating type and Hardy type Tauberian conditions under whichstatistical Ces`aro summability and statistical logarithmic summability imply convergence in
IF N S . Besides, weobtain analogous results for the higher order summability methods as corollaries. Also, two theorems concerningthe convergence of statically convergent sequences in
IF N S are proved in the paper.
Events and problems that humankind encounter in real world are commonly complex and imprecise due to un-certainty of parameters and indefiniteness of the objects involved. Researchers proposed various theories of un-certainty to handle such events and problems involving incomplete information. Among them, Lotfi A. Zadehproposed fuzzy logic as an extension Boolean logic with the introduction of the mathematical concept of fuzzysets [1]. In [1], Zadeh extended classical sets to fuzzy sets by using gradual memberships taking real values in theinterval [0 , , instead of Boolean memberships which take only integer values { , } . Following its introduction,fuzzy sets are utilized by many researchers in different fields of science to process non-categorical data. Besides,motivated by fuzzy sets, Atanassov [2, 3] introduced the concept of intuitionistic fuzzy sets( IF S ) by consideringgradual non-memberships as well as Zadeh’s gradual memberships. Like fuzzy sets, IFS have also been applied inmany areas of science such as control systems, robotics, computer, medical diagnosis, education, etc. Theoreticalbasis of intuitionistic fuzzy set theory have also developed in time. In particular, Park [4] defined IF − metric andit was pursued by IF − norm [5]. Convergence of sequences with respect to IF − norm is defined and various con-vergence methods in the statistical sense are proposed to achieve a limit where ordinary convergence fails [6–10].Also, there are recent studies applying some weighted mean summation methods in IF N S to achieve limit ofsequences and providing Tauberian conditions which guarantee ordinary convergence [11, 12].In some cases of sequences as in Example 2.2, Example 2.9 and Example 3.2, a limit can not be achieved viastatistical convergence or via weighted mean summation methods in
IF N S . In such cases, we need to use somestronger methods of convergence to achieve a limit in
IF N S , and to investigate whether ordinary convergencecan be guaranteed with some extra conditions. For this aim, in this study we introduce statistical Ces`aro andstatistical logarithmic summability of sequences in
IF N S , and obtain Tauberian conditions of slowly oscillatingtype and of Hardy type under which ordinary convergence in
IF N S follows from statistical Ces`aro and statisticallogarithmic summability. Besides, we show that statistical convergence of slowly oscillating sequences in
IF N S implies ordinary convergence in the space. Examples in the paper provide also new types of sequences for statisticalCes`aro and statistical logarithmic summation methods in classical normed spaces. Before to continue main results,we now give some preliminaries.
Keywords: intuitionistic fuzzy normed space, Tauberian theorem, Ces`aro and logarithmic summability methods, statistical convergence,slow oscillation
Mathematics Subject Classification: efinition 1.1. [13] The triplicate ( N, µ, ν ) is said to be an IF N S if N is a real vector space, and µ, ν are fuzzysets on N × R satisfying the following conditions for every u, w ∈ N and t, s ∈ R :(a) µ ( u, t ) = 0 for t ≤ ,(b) µ ( u, t ) = 1 for all t ∈ R + if and only if u = θ (c) µ ( cu, t ) = µ (cid:16) u, t | c | (cid:17) for all t ∈ R + and c = 0 ,(d) µ ( u + w, t + s ) ≥ min { µ ( u, t ) , µ ( w, s ) } ,(e) lim t →∞ µ ( u, t ) = 1 and lim t → µ ( u, t ) = 0 ,(f) ν ( u, t ) = 1 for t ≤ ,(g) ν ( u, t ) = 0 for all t ∈ R + if and only if u = θ (h) ν ( cu, t ) = ν (cid:16) u, t | c | (cid:17) for all t ∈ R + and c = 0 ,(i) max { ν ( u, t ) , ν ( w, s ) } ≥ ν ( u + w, t + s ) ,(j) lim t →∞ ν ( u, t ) = 0 and lim t → ν ( u, t ) = 1 .We call ( µ, ν ) an IF − norm on N . Example 1.2.
Let ( N, k · k ) be a normed space and µ , ν be fuzzy sets on N × R defined by µ ( u, t ) = ( , t ≤ , tt + k u k , t > , ν ( u, t ) = ( , t ≤ , k u k t + k u k , t > . Then ( µ , ν ) is IF − norm on N .Throughout the paper ( N, µ, ν ) will denote an IF N S . Definition 1.3. [13] A sequence ( u k ) in ( N, µ, ν ) is said to be convergent to a ∈ N and denoted by u k → a if forevery ε > and t > there exists k ∈ N such that µ ( u k − a, t ) > − ε and ν ( u k − a, t ) < ε for all k ≥ k . Definition 1.4. [6] Let sequence ( u k ) be in ( N, µ, ν ) . We say that ( u k ) is statistically convergent to a ∈ N withrespect to fuzzy norm ( µ, ν ) provided that, for every ε > and t > , lim n →∞ n |{ k ≤ n : µ ( u k − a, t ) ≤ − ε or ν ( u k − u n , t ) ≥ ε }| = 0 . In this case we write st µ,ν − lim u = a . Theorem 1.5. [6] Let sequence ( u k ) be in ( N, µ, ν ) and a ∈ N . Then, st µ,ν − lim u = a if and only if st − lim µ ( u k − a, t ) = 1 and st − lim ν ( u k − a, t ) = 0 for each t . Theorem 1.6. [6] Let sequence ( u k ) be in ( N, µ, ν ) and a ∈ N . If ( u k ) is convergent to a , then ( u k ) is statisticallyconvergent to a . Definition 1.7. [14] A sequence ( u k ) in ( N, µ, ν ) is called q-bounded if lim t →∞ inf k ∈ N µ ( u k , t ) = 1 and lim t →∞ sup k ∈ N ν ( u k , t ) = 0 . Definition 1.8. [11] A sequence ( u k ) is slowly oscillating if and only if for all t > and for all ε ∈ (0 , thereexist λ > and m ∈ N , depending on t and ε , such that µ ( u k − u n , t ) > − ε and ν ( u k − u n , t ) < ε whenever m ≤ n < k ≤ ⌊ λn ⌋ . 2 efinition 1.9. [12] A sequence ( u k ) is slowly oscillating with respect to logarithmic summability if and only iffor all t > and for all ε ∈ (0 , there exist λ > and m ∈ N , depending on t and ε , such that µ ( u k − u n , t ) > − ε and ν ( u k − u n , t ) < ε whenever m ≤ n < k ≤ ⌊ n λ ⌋ . Theorem 1.10. [11] Let sequence ( u k ) be in ( N, µ, ν ) . If { k ( u k − u k − ) } is q-bounded, then ( u k ) is slowlyoscillating. Theorem 1.11. [12] Let sequence ( u k ) be in ( N, µ, ν ) . If { k ln k ( u k − u k − ) } is q-bounded, then ( u k ) is slowlyoscillating with respect to logarithmic summability. Theorem 1.12. [11] Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is Ces`aro summable to some a ∈ N and slowlyoscillating, then ( u k ) converges to a . Theorem 1.13. [12] Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is logarithmic summable to some a ∈ N and slowlyoscillating with respect to logarithmic summability, then ( u k ) converges to a . I F N S
We define statistical Ces`aro summability method in
IF N S as the following.
Definition 2.1.
Let ( u k ) be a sequence in ( N, µ, ν ) . Ces`aro means σ k of ( u k ) is defined by σ k = 1 k k X j =1 u j . We say that ( u k ) is statistically Ces`aro summable to a ∈ N if st µ,ν − lim σ = a .In view of Theorem 1.6 and [11, Theorem 3.2], convergence implies statistical Ces`aro summability in IF N S .But converse statement is not true in general by the next example.
Example 2.2.
Let u k = ( − k + k , k = n ( − k − ( k − , k = n + 1( − k , otherwise be in IF − normed space ( R , µ , ν ) where µ and ν are as in Example 1.2. Sequence ( u k ) is neither convergentnor statistically convergent in ( R , µ , ν ) . Besides, it is not Ces`aro summable.Let us apply statistical Ces`aro summability to achieve a limit. Ces`aro means ( σ k ) of sequence ( u k ) is σ k = k P kj =1 ( − j + k, k = n k P kj =1 ( − j , otherwise. Sequence ( σ k ) is statistically convergent to 0 since for each t > we have st − lim µ ( σ k , t ) = 1 and st − lim ν ( σ k , t ) = 0 where µ ( σ k , t ) = tt + | k P kj =1 ( − j + k | , k = n tt + | k P kj =1 ( − j | , otherwise and ν ( σ k , t ) = | k P kj =1 ( − j + k | t + | k P kj =1 ( − j + k | , k = n | k P kj =1 ( − j | t + | k P kj =1 ( − j | , otherwise. Hence, sequence ( u k ) is statistically Ces`aro summable to 0 in ( R , µ , ν ) .3n this section, we will give some Tauberian conditions for statistical Ces`aro summability to imply convergencein IF N S . To this end, firstly we now show that statistical convergence of slowly oscillating sequences yieldsconvergence in
IF N S . Theorem 2.3.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is statistically convergent to some a ∈ N and slowlyoscillating, then ( u k ) is convergent to a . Proof.
Let st µ,ν − lim u = a and sequence ( u k ) be slowly oscillating. Then by Theorem 1.5, for every t > wehave st − lim µ ( u k − a, t ) = 1 and st − lim ν ( u k − a, t ) = 0 . Our aim to show that lim k →∞ µ ( u k − a, t ) = 1 and lim k →∞ ν ( u k − a, t ) = 0 .Fix t > . Since st − lim µ (cid:0) u k − a, t (cid:1) = 1 , from the proof of [15, Lemma 6] there is a subsequence ofintegers ≤ l < l < · · · such that for any λ > inequality l m < l m +1 < λl m holds for large enough m ; and lim m →∞ µ (cid:18) u l m − a, t (cid:19) = 1 . So, for given ε > there exists m such that µ (cid:18) u l m − a, t (cid:19) > − ε whenever m > m . (2.1)Besides, since ( u k ) is slowly oscillating there exist m and λ > such that µ (cid:18) u k − u l m , t (cid:19) > − ε whenever m ≤ l m < k ≤ λl m . (2.2)It follows from (2.1)–(2.2) that µ ( u k − a, t ) ≥ min (cid:26) µ (cid:18) u k − u l m , t (cid:19) , µ (cid:18) u l m − a, t (cid:19)(cid:27) > − ε holds for l m < k ≤ l m +1 where m > max { m , m } . By considering all m ’s, we get µ ( u k − a, t ) > − ε f or k > l m , where m = max { m , m } . Hence lim k →∞ µ ( u k − a, t ) = 1 is proved. The proof of lim k →∞ ν ( u k − a, t ) = 0 can be done similarly.We need next lemma to prove main theorem of this section. Lemma 2.4.
Let ( u k ) be slowly oscillating sequence in ( N, µ, ν ) . Let t be an arbitrary but fixed positive number.Then, for each ε > followings hold: µ (cid:18) u n − u k , (cid:26) n/k )ln λ (cid:27) t (cid:19) > − ε, ν (cid:18) u n − u k , (cid:26) n/k )ln λ (cid:27) t (cid:19) < ε (2.3) and µ n ⌊ n/λ ⌋ X k = m ( u n − u k ) , (cid:26) λ (cid:27) t > − ε, ν n ⌊ n/λ ⌋ X k = m ( u n − u k ) , (cid:26) λ (cid:27) t < ε (2.4) where m ≤ k ≤ n/λ , and m = m ( t, ε ) and λ = λ ( t, ε ) are from definition of slow oscillation. Proof.
Let ( u k ) be slowly oscillating sequence in ( N, µ, ν ) , and m and λ > be from Definition1.8. Let t be anarbitrary but fixed positive number. Fix m ≤ k ≤ n/λ . Consider the sequence(see [15, Proof of Lemma 8]) n := n, n p := 1 + j n p − λ k , p = 1 , , . . . , q + 1 , where q is determined by the condition n q +1 ≤ k < n q . − ε < min (cid:8) µ ( u n − u n , t ) , µ ( u n − u n , t ) , · · · , µ ( u n q − u k , t ) (cid:9) ≤ µ ( u n − u k , ( q + 1) t ) ≤ µ (cid:18) u n − u k , (cid:26) n/k )ln λ (cid:27) t (cid:19) and ε > max (cid:8) ν ( u n − u n , t ) , ν ( u n − u n , t ) , · · · , ν ( u n q − u k , t ) (cid:9) ≥ ν (cid:18) u n − u k , (cid:26) n/k )ln λ (cid:27) t (cid:19) by virtue of the inequality q ≤ λ ln (cid:0) nk (cid:1) which was calculated in [15, Proof of Lemma 8]. This proves (2.3).On the other hand by using (2.3) we have: − ε < min m ≤ k ≤⌊ n/λ ⌋ µ (cid:18) u n − u k , (cid:26) n/k )ln λ (cid:27) t (cid:19) ≤ µ ⌊ n/λ ⌋ X k = m ( u n − u k ) , ⌊ n/λ ⌋ X k = m (cid:18) n/k )ln λ (cid:19) t ≤ µ ⌊ n/λ ⌋ X k = m ( u n − u k ) , ( n X k =1 (cid:18) n/k )ln λ (cid:19)) t ≤ µ ⌊ n/λ ⌋ X k = m ( u n − u k ) , (cid:26) λ (cid:27) nt = µ n ⌊ n/λ ⌋ X k = m ( u n − u k ) , (cid:26) λ (cid:27) t and ε > max m ≤ k ≤⌊ n/λ ⌋ ν (cid:18) u n − u k , (cid:26) n/k )ln λ (cid:27) t (cid:19) ≥ ν n ⌊ n/λ ⌋ X k = m ( u n − u k ) , (cid:26) λ (cid:27) t by virtue of P nk =2 ln k > R n ln( x ) dx , which proves (2.4) Theorem 2.5.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is slowly oscillating, then sequence ( σ k ) of Ces`aro meansis also slowly oscillating. Proof.
Let sequence ( u k ) be slowly oscillating and σ k = k P kj =1 u j . Fix t > . For given ε > there exists m , m ∈ N and < λ < such that• µ ( u k − u n , t/ > − ε and ν ( u k − u n , t/ < ε whenever m ≤ n < k ≤ ⌊ λn ⌋ .• µ k − nkn m − X j =1 ( u n − u j ) , t > − ε and ν k − nkn m − X j =1 ( u n − u j ) , t < ε whenever m ≤ n < k ≤ ⌊ λn ⌋ , by virtue of inequalities in (2.3).5hen, for max { m , m } ≤ n < k ≤ ⌊ λn ⌋ , we get µ ( σ k − σ n , t ) = µ k − nkn n X j =1 ( u n − u j ) + 1 k k X j = n +1 ( u j − u n ) , t ≥ min µ k − nkn m − X j =1 ( u n − u j ) , t , µ k − nkn ⌊ n/λ ⌋ X j = m ( u n − u j ) , t ,µ k − nkn n X j = ⌊ n/λ ⌋ +1 ( u n − u j ) , t , µ k k X j = n +1 ( u j − u n ) , t ≥ min µ k − nkn m − X j =1 ( u n − u j ) , t , µ k − nkn ⌊ n/λ ⌋ X j = m ( u n − u j ) , t , min ⌊ n/λ ⌋≤ j ≤ n µ (cid:18) u n − u j , t (cid:19) , min n +1 ≤ j ≤ k µ (cid:18) u j − u n , t (cid:19)(cid:27) > min µ k − nkn ⌊ n/λ ⌋ X j = m ( u n − u j ) , t , − ε ≥ min µ n ⌊ n/λ ⌋ X j = m ( u n − u j ) , t λ − , − ε ≥ min µ n ⌊ n/λ ⌋ X j = m ( u n − u j ) , (cid:26) λ (cid:27) t , − ε ≥ − ε by virtue of the facts that k − nk < λ − and λ − λ (ln λ + ln 2 + 1) < , and of Lemma 2.4. On the other hand, ν ( σ k − σ n , t ) < ε can be shown similarly. Hence, the proof is completed.Now we give the main theorem of this section. Theorem 2.6.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is statistically Ces`aro summable to some a ∈ N andslowly oscillating, then ( u k ) is convergent to a . Proof.
Let ( u k ) be statistically Ces`aro summable to some a ∈ N and slowly oscillating. Then, by Theorem 2.5sequence ( σ k ) of Ces`aro means is also slowly oscillating. From Theorem 2.3, ( σ k ) → a . This means that ( u k ) isCes`aro summable to a . Since ( u k ) is slowly oscillating, from Theorem 1.12 ( u k ) is convergent to a .In view of Theorem 1.10 and Theorem 2.6 we get following theorem. Theorem 2.7.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is statistically Ces`aro summable to a ∈ N and { k ( u k − u k − ) } is q-bounded, then ( u k ) converges to a .In some cases of sequences as in Example 2.9, first order Ces`aro means fails to converge both ordinarily andstatistically in IF N S . To handle such sequences, we consider higher order Ces`aro means and define statisticalH ¨older summability method in
IF N S . In the sequel, we give corresponding Tauberian theorems as corollaries.
Definition 2.8.
Let ( u k ) be in ( N, µ, ν ) . m -th order H ¨older means H mk of ( u k ) is defined by H mk = 1 k k X j =1 H m − j , H k = u k . Sequence ( u k ) is said to be ( H, m ) summable to a ∈ N if lim k →∞ H mk = a and it is said to bestatistically ( H, m ) summable to a if st µ,ν − lim H m = a Example 2.9.
Let u k = ( − k k + k , k = n ( − k k − ( k − − ( k − k − ( k − k , k = n + 1( − k k + 3( k − ( k − , k = n + 2( − k k − ( k − ( k − k − , k = n + 3( − k k , otherwise be in IF − normed space ( R , µ , ν ) where µ and ν are as in Example 1.2 and n ≥ . Sequence ( u k ) is neitherconvergent nor statistically convergent in ( R , µ , ν ) . Furthermore, it is neither Ces`aro summable nor statisticallyCes`aro summable.Let us apply statistical ( H, summability to achieve a limit. H ¨older means ( H k ) , ( H k ) and ( H k ) of sequence ( u k ) are H k = σ (1) k + k , k = n σ (1) k − ( k − − ( k − k, k = n + 1 σ (1) k + ( k − ( k − , k = n + 2 σ (1) k , otherwiseH k = σ (2) k + k , k = n σ (2) k − ( k − , k = n + 1 σ (2) k , otherwiseH k = ( σ (3) k + k, k = n σ (3) k , otherwise where sequence (cid:8) σ ( m ) k (cid:9) denotes m − fold Ces`aro means of sequence (cid:8) ( − k k (cid:9) and n ≥ . Sequence ( H k ) isstatistically convergent to 0 since for each t > we have st − lim µ (cid:0) H k , t (cid:1) = 1 and st − lim ν (cid:0) H k , t (cid:1) = 0 where µ (cid:0) H k , t (cid:1) = tt + (cid:12)(cid:12)(cid:12) σ (3) k + k (cid:12)(cid:12)(cid:12) , k = n tt + (cid:12)(cid:12)(cid:12) σ (3) k (cid:12)(cid:12)(cid:12) , otherwise and ν (cid:0) H k , t (cid:1) = (cid:12)(cid:12)(cid:12) σ (3) k + k (cid:12)(cid:12)(cid:12) t + (cid:12)(cid:12)(cid:12) σ (3) k + k (cid:12)(cid:12)(cid:12) , k = n (cid:12)(cid:12)(cid:12) σ (3) k (cid:12)(cid:12)(cid:12) t + (cid:12)(cid:12)(cid:12) σ (3) k (cid:12)(cid:12)(cid:12) , otherwise. in view of the fact that lim k →∞ σ (3) k = 0 . Hence, sequence ( u k ) is statistically ( H, summable to 0 in ( R , µ , ν ) .In view of Theorem 1.12, Theorem 2.5 and Theorem 2.6 we get following Tauberian theorem. Theorem 2.10.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is statistically ( H, m ) summable to some a ∈ N andslowly oscillating, then ( u k ) is convergent to a .Also, in view of theorem above and Theorem 1.10 we get following theorem. Theorem 2.11.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is statistically ( H, m ) summable to some a ∈ N and { k ( u k − u k − ) } is q-bounded, then ( u k ) is convergent to a .7 Tauberian theorems for statistical logarithmic summability in
I F N S
We define statistical logarithmic summability method in
IF N S as the following.
Definition 3.1.
Let sequence ( u k ) be in ( N, µ, ν ) . Logarithmic mean τ k of ( u k ) is defined by τ k = 1 ℓ k k X j =1 u j j where ℓ k = k X j =1 j · ( u k ) is said to be statistically logarithmic summable to a ∈ N if st µ,ν − lim τ = a .In view of Theorem 1.6 and [12, Theorem 2.2], convergence implies statistical logarithmic summability in IF N S . But converse statement is not true in general by the next example.
Example 3.2.
Consider the vector space C [0 , equipped with the norm k f k = max x ∈ [0 , | f ( x ) | . Let f k ( x ) = ( − x ) k k + k ( ℓ k ) , k = n ( − x ) k k − k ( ℓ k − ) , k = n + 1( − x ) k k, otherwise be in IF − normed space ( C [0 , , µ , ν ) where µ and ν are as in Example 1.2. Sequence ( f k ) is neither con-vergent nor statistically convergent in ( C [0 , , µ , ν ) . Besides, ( f k ) is neither Ces`aro summable nor logarithmicsummable. Furthermore, ( f k ) is not statistically Ces`aro summable which we have defined in previous section.Let us apply statistical logarithmic summability to achieve a limit. Logarithmic means ( τ k ) of sequence ( f k ) is τ k ( x ) = ℓ k P kj =1 ( − x ) j + ℓ k , k = n ℓ k P kj =1 ( − x ) j , otherwise. Hence, ( f k ) is statistically logarithmic summable to 0 since for each t > we have st − lim µ ( τ k , t ) = 1 and st − lim ν ( τ k , t ) = 0 where µ ( τ k , t ) = tt + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓk k P j =1 ( − x ) j + ℓ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , k = n tt + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓk k P j =1 ( − x ) j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , otherwise and ν ( τ k , t ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓk k P j =1 ( − x ) j + ℓ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓk k P j =1 ( − x ) j + ℓ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , k = n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓk k P j =1 ( − x ) j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓk k P j =1 ( − x ) j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , otherwise. In this section, we will give some Tauberian conditions for statistical logarithmic summability to imply conver-gence in
IF N S . To this end, firstly we now show that statistical convergence of slowly oscillating sequences withrespect to logarithmic summability implies convergence in
IF N S . Theorem 3.3.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is statistically convergent to some a ∈ N and slowlyoscillating with respect to logarithmic summability, then ( u k ) is convergent to a . Proof.
Let st µ,ν − lim u = a and sequence ( u k ) be slowly oscillating with respect to logarithmic summability.Then by Theorem 1.5, for every t > we have st − lim µ ( u k − a, t ) = 1 and st − lim ν ( u k − a, t ) = 0 . Our aimto show that lim k →∞ µ ( u k − a, t ) = 1 and lim k →∞ ν ( u k − a, t ) = 0 .Fix t > . Since st − lim µ (cid:0) u k − a, t (cid:1) = 1 , from the proof of [16, Lemma 10] there is a subsequence ofintegers ≤ j < j < · · · such that for any λ > inequality j m < j m +1 < j λm holds for large enough m ; and lim m →∞ µ (cid:18) u j m − a, t (cid:19) = 1 . ε > there exists m such that µ (cid:18) u j m − a, t (cid:19) > − ε whenever m > m . (3.1)Besides, since ( u k ) is slowly oscillating with respect to logarithmic summability there exist m and λ > suchthat µ (cid:18) u k − u j m , t (cid:19) > − ε whenever m ≤ j m < k ≤ j λm . (3.2)It follows from (3.1)–(3.2) that µ ( u k − a, t ) ≥ min (cid:26) µ (cid:18) u k − u j m , t (cid:19) , µ (cid:18) u j m − a, t (cid:19)(cid:27) > − ε holds for j m < k ≤ j m +1 where m > max { m , m } . By considering all m ’s, we get µ ( u k − a, t ) > − ε f or k > j m , where m = max { m , m } . Hence lim k →∞ µ ( u k − a, t ) = 1 is proved. The proof of lim k →∞ ν ( u k − a, t ) = 0 can be done similarly.We need the next lemma to prove main theorem of this section. Lemma 3.4.
Let sequence ( u k ) be slowly oscillating with respect to logarithmic summability in ( N, µ, ν ) . Let t be an arbitrary but fixed positive number. Then, for each ε > followings hold: µ (cid:18) u n − u k , (cid:26) λ ln (cid:18) n ln k (cid:19)(cid:27) t (cid:19) > − ε, ν (cid:18) u n − u k , (cid:26) λ ln (cid:18) n ln k (cid:19)(cid:27) t (cid:19) < ε (3.3) and µ ℓ n ⌊ n /λ ⌋ X k = m (cid:18) u n − u k k (cid:19) , (cid:26) λ (cid:27) t > − ε, ν ℓ n ⌊ n /λ ⌋ X k = m (cid:18) u n − u k k (cid:19) , (cid:26) λ (cid:27) t < ε (3.4) where < m ≤ k ≤ n /λ , and m = m ( t, ε ) and λ = λ ( t, ε ) are from definition of slow oscillation with respectto logarithmic summability. Proof.
Let sequence ( u k ) be slowly oscillating with respect to logarithmic summability in ( N, µ, ν ) , and < m and λ > be from Definition 1.9. Let t be an arbitrary but fixed positive number. Fix m ≤ k ≤ n /λ . Considerthe sequence(see [16, Proof of Lemma 5]) n := n, n p := 1 + j n /λp − k , p = 1 , , . . . , q + 1 , where q is determined by the condition n q +1 ≤ k < n q . Then, we get − ε < min (cid:8) µ ( u n − u n , t ) , µ ( u n − u n , t ) , · · · , µ ( u n q − u k , t ) (cid:9) ≤ µ ( u n − u k , ( q + 1) t ) ≤ µ (cid:18) u n − u k , (cid:26) λ ln (cid:18) n ln k (cid:19)(cid:27) t (cid:19) and ε > max (cid:8) ν ( u n − u n , t ) , ν ( u n − u n , t ) , · · · , ν ( u n q − u k , t ) (cid:9) ≥ ν (cid:18) u n − u k , (cid:26) λ ln (cid:18) n ln k (cid:19)(cid:27) t (cid:19) by virtue of the inequality q ≤ λ ln (cid:0) n ln k (cid:1) which was calculated in [16, Proof of Lemma 5]. This proves (3.3).9n the other hand by using (3.3) we have: − ε < min m ≤ k ≤⌊ n /λ ⌋ µ (cid:18) u n − u k , (cid:26) λ ln (cid:18) n ln k (cid:19)(cid:27) t (cid:19) ≤ µ ⌊ n /λ ⌋ X k = m (cid:18) u n − u k k (cid:19) , ⌊ n /λ ⌋ X k = m k (cid:18) λ ln (cid:18) n ln k (cid:19)(cid:19) t ≤ µ ⌊ n /λ ⌋ X k = m (cid:18) u n − u k k (cid:19) , ( n X k =2 k (cid:18) λ ln (cid:18) n ln k (cid:19)(cid:19)) t ≤ µ ⌊ n /λ ⌋ X k = m (cid:18) u n − u k k (cid:19) , (cid:26) λ (cid:27) ℓ n t = µ ℓ n ⌊ n /λ ⌋ X k = m (cid:18) u n − u k k (cid:19) , (cid:26) λ (cid:27) t and ε > max m ≤ k ≤⌊ n /λ ⌋ ν (cid:18) u n − u k , (cid:26) λ ln (cid:18) n ln k (cid:19)(cid:27) t (cid:19) ≥ ν ℓ n ⌊ n /λ ⌋ X k = m (cid:18) u n − u k k (cid:19) , (cid:26) λ (cid:27) t which proves (3.4). Theorem 3.5.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is slowly oscillating with respect to logarithmic summa-bility, then sequence ( τ k ) of logarithmic means is also slowly oscillating with respect to logarithmic summability. Proof.
Let sequence ( u k ) be slowly oscillating with respect to logarithmic summability and τ k = ℓ k P kj =1 u j j . Fix t > . For given ε > there exists < m , m ∈ N and < λ < such that• µ ( u k − u n , t/ > − ε and ν ( u k − u n , t/ < ε whenever m ≤ n < k ≤ ⌊ n λ ⌋ .• µ (cid:18) ℓ n − ℓ k (cid:19) m − X j =1 (cid:18) u n − u j j (cid:19) , t > − ε and ν (cid:18) ℓ n − ℓ k (cid:19) m − X j =1 (cid:18) u n − u j j (cid:19) , t < ε whenever m ≤ n < k ≤ ⌊ n λ ⌋ , by virtue of inequalities in (3.3).Then, for max { m , m } ≤ n < k ≤ ⌊ n λ ⌋ , we get µ ( τ k − τ n , t ) = µ (cid:18) ℓ n − ℓ k (cid:19) n X j =1 (cid:18) u n − u j j (cid:19) + 1 ℓ k k X j = n +1 (cid:18) u j − u n j (cid:19) , t ≥ min µ (cid:18) ℓ n − ℓ k (cid:19) m − X j =1 (cid:18) u n − u j j (cid:19) , t , µ (cid:18) ℓ n − ℓ k (cid:19) ⌊ n /λ ⌋ X j = m (cid:18) u n − u j j (cid:19) , t ,µ (cid:18) ℓ n − ℓ k (cid:19) n X j = ⌊ n /λ ⌋ +1 (cid:18) u n − u j j (cid:19) , t , µ ℓ k k X j = n +1 (cid:18) u n − u j j (cid:19) , t ≥ min µ (cid:18) ℓ n − ℓ k (cid:19) m − X j =1 (cid:18) u n − u j j (cid:19) , t , µ (cid:18) ℓ n − ℓ k (cid:19) ⌊ n /λ ⌋ X j = m (cid:18) u n − u j j (cid:19) , t , in ⌊ n /λ ⌋≤ j ≤ n µ (cid:18) u n − u j , t (cid:19) , min n +1 ≤ j ≤ k µ (cid:18) u j − u n , t (cid:19)(cid:27) > min µ (cid:18) ℓ n − ℓ k (cid:19) ⌊ n /λ ⌋ X j = m (cid:18) u n − u j j (cid:19) , t , − ε ≥ min µ ℓ n ⌊ n /λ ⌋ X j = m (cid:18) u n − u j j (cid:19) , t λ − , − ε ≥ min µ ℓ n ⌊ n /λ ⌋ X j = m (cid:18) u n − u j j (cid:19) , (cid:26) λ (cid:27) t , − ε ≥ − ε by virtue of the facts that ℓ k − ℓ n ℓ k ≤ ln k − ln n ln k < λ − and λ − λ (ln λ + ln 2 + 2) < , and of Lemma 3.4. On theother hand, ν ( τ k − τ n , t ) < ε can be shown similarly. Hence, the proof is completed.Now we give the main theorem of this section. Theorem 3.6.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is statistically logarithmic summable to some a ∈ N andslowly oscillating with respect to logarithmic summability, then ( u k ) is convergent to a . Proof.
Let ( u k ) be statistically logarithmic summable to some a ∈ N and slowly oscillating with respect tologarithmic summability. Then, by Theorem 3.5 sequence ( τ k ) of logarithmic means is also slowly oscillatingwith respect to logarithmic summability. From Theorem 3.3, ( τ k ) → a . This means that ( u k ) is logarithmicsummable to a . Since ( u k ) is slowly oscillating with respect to logarithmic summability, from Theorem 1.13 ( u k ) is convergent to a .In view of Theorem 1.11 and Theorem 3.6 we get following theorem. Theorem 3.7.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is logarithmic summable to a ∈ N and { k ln k ( u k − u k − ) } is q-bounded, then ( u k ) converges to a .For the case of higher order logarithmic summability methods we can give following Tauberian theorems ascorollaries. Definition 3.8.
Let ( u k ) be in ( N, µ, ν ) . m -th order logarithmic means L mk of ( u k ) is defined by L mk = 1 ℓ k k X j =1 L m − j j , where L k = u k . We say that sequence ( u k ) is statistically ( L, m ) summable to a ∈ N if sequence ( L mk ) isstatistically convergent to a .In view of Theorem 1.13, Theorem 3.5 and Theorem 3.6 we get following Tauberian theorem. Theorem 3.9.
Let sequence ( u k ) be in ( N, µ, ν ) . If ( u k ) is statistically ( L, m ) summable to some a ∈ N andslowly oscillating with respect to logarithmic summability, then ( u k ) is convergent to a .Also, in view of theorem above and Theorem 1.11 we get following theorem. Theorem 3.10.
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