On strongly almost lacunary statistical A -convergence defined by Musielak-Orlicz function
aa r X i v : . [ m a t h . F A ] A ug On strongly almost lacunary statistical A -convergence defined by Musielak-Orlicz function Ekrem Savas¸Istanbul Commerce University, 34840 Istanbul, TurkeyE-mail : [email protected] Borgohain ∗ Department of MathematicsIndian Institute of Technology, BombayPowai:400076, Mumbai, Maharashtra; INDIA.E-mail:[email protected]
Abstract:
We study some new strongly almost lacunary statistical A -convergent sequence spaceof order α defined by a Musielak-Orlicz function. We also give some inclusion relations betweenthe newly introduced class of sequences with the spaces of strongly almost lacunary A -convergentsequence of order α . Moreover we have examined some results on Musielak-Orlicz function withrespect to these spaces. Key Words:
Almost convergence; Statistical convergence; Lacunary seqence; Musielak-Orliczfunction; A -covergence. AMS Classification No: 40A05; 40A25; 40A30; 40C05.
The concept of statistical convergence was initially introduced by Fast [2], whichis closely related to the concept of natural density or asymtotic density of subsetsof the set of natural numbers N . Later on, it was studied as asummability methodby Fridy [4], Fridy and Orhan [6], Freedman and Sember [3], Schoenberg [18],Malafosse and Rako ˇ c evi ´ c [10] and many more mathematicians. Moreover, inrecent years, generalizations of statistical convergence have appeared in the studyof strong integral summability and the structure of ideals of bounded continuous The work of the authors was carried under the Post Doctoral Fellow under National Board ofHigher Mathematics, DAE, project No. NBHM / PDF.50 / / x = ( x k ) ∈ ℓ ∞ if itsBanach limit coincides. The set ˆ c denotes set of all almost convergent sequences .Lorentz [8] proved that,ˆ c = { x ∈ ℓ ∞ : lim m t mn ( x ) exist uniformly in n } , where t mn ( x ) = x n + x n + + .... + x n + m m + . Similarly, the space of strongly almost convergent sequence was definedas, [ ˆ c ] = { x ∈ ℓ ∞ : lim m t m , n ( | x − Le | ) exists uniformly in n for some L } , where, e = (1 , , ... ). (see Maddox [9])A lacunary sequence is defined as an increasing integer sequence θ = ( k r ) suchthat k = h r = k r − k r − → ∞ as r → ∞ . Note: Throughout this paper, the intervals determined by θ will be denoted by J r = ( k r − , k r ] and the ratio k r k r − will be defined by φ r . Let 0 < α ≤ x k ) is said to be statistically convergent oforder α if there is a real number L such that,lim n →∞ n α |{ k ≤ n : | x k − L | ≥ ε }| = , for every ε >
0. In this case, we write S α − lim x k = L . The set of all statisticallyconvergent sequences of order α will be denoted by S α .For any lacunary sequence θ = ( k r ), the space N θ defined as, (Freedman etal.[3]) 2 θ = ( x k ) : lim r →∞ h − r X k ∈ J r | x k − L | = , for some L . The space N θ is a BK space with the norm, k ( x k ) k θ = sup r h − r X k ∈ J r | x k | . Let θ = ( k r ) be a lacunary sequence and 0 < α ≤ x = ( x k ) ∈ w is said to be S αθ -statistically convergent (or lacunary statisticallyconvergent sequence of order α ) if there is a real number L such thatlim r →∞ h α r |{ k ∈ I r : | x k − L | ≥ ε }| = , where I r = ( k r − , k r ] and h α r denotes the α -th power ( h r ) α of h r , that is, h α = ( h α r ) = ( h α , h α , ... h α r , ... ). We write S αθ − lim x k = L . The set of all S αθ -statistically convergentsequences will be denoted by S αθ .By an Orlicz function , we mean a function M : [0 , ∞ ) → [0 , ∞ ), which iscontinuous, non-decreasing and convex with M (0) = , M ( x ) >
0, for x > M ( x ) → ∞ , as x → ∞ .The idea of Orlicz function is used to construct the sequence space, (see Lin-denstrauss and Tzafriri [7]), ℓ M = ( x k ) ∈ w : ∞ X k = M | x | ρ ! < ∞ , for some ρ > . This space ℓ M with the norm, k x k = inf ρ > ∞ X k = M | x k | ρ ! ≤ becomes a Banach space which is called an Orlicz sequnce space.Musielak [12] defined the concept of Musielak-Orlicz function as M = ( M k ).A sequence N = ( N k ) defined by 3 k ( v ) = sup {| v | u − M k ( u ) : u ≥ } , k = , , .. is called the complementary function of a Musielak-Orlicz function M . TheMusielak-Orlicz sequence space t M and its subspace h M are defined as follows: t M = { x ∈ w : I M ( cx ) < ∞ for some c > } , h M = { x ∈ w : I M ( cx ) < ∞ , ∀ c > } , where I M is a convex modular defined by, I M ( x ) = ∞ X k = M k ( x k ) , x = ( x k ) ∈ t M . It is considered t M equipped with the Luxemberg norm k x k = inf (cid:26) k > I M (cid:18) xk (cid:19) ≤ (cid:27) or equiped with the Orlicz norm k x k = inf (cid:26) k (1 + I M ( kx )) : k > (cid:27) . A Musielak-Orlicz function ( M k ) is said to satisfy ∆ -condition if there existconstants a , K > c = ( c k ) ∞ k = ∈ ℓ + ( the positive cone of ℓ ) suchthat the inequality M k (2 u ) ≤ KM k ( u ) + c k holds for all k ∈ N and u ∈ R + , whenever M k ( u ) ≤ a .If A = ( a nk ) ∞ n , k = is an infinite matrix, then Ax is the sequence whose n th term isgiven by A n ( x ) = ∞ X k = a nk x k .We consider a sequence x = ( x k ) which is said to be strongly almost lacunarystatistical A -convergent of order α (or S αθ ( A , M , ( s ))-statistically convergent) if ,lim r →∞ h α r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ∈ I r : X k ∈ I r M k | t km ( A k ( x ) − L ) | ρ ( k ) !! ( s k ) ≥ ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = , uniformly in m , I r = ( k r − , k r ] and h α r denotes the α -th power ( h α r ) of h r , that is, h α = ( h α r ) = ( h α , h α , ... h α r , ... ) and M = ( M k ) is a Musielak-Orlicz function.Also we have introduced the space of strongly almost lacunary A -convergentsequences with respect to Musielak-Orlicz function M = ( M k ) as follows:ˆ N αθ ( A , M , ( s )) = ( x k ) : lim r →∞ h α r X k ∈ I r M k | t km ( A k ( x ) − L ) | ρ ( k ) !! ( s k ) = , for some L and ρ ( k ) > . We give some inclusion relations between the sets of S αθ ( A , M , ( s ))-statisticallyconvergent sequences and strongly almost lacunary A -convergent sequence spaceˆ N αθ ( A , M , ( s )). Also some results defined by Musielak-Orlicz function are studiedwith respect to these sequence spaces. Theorem 3.1.
Let α, β ∈ (0 ,
1] be real numbers such that α ≤ β , M be a Musielak-Orlicz function and θ = ( k r ) be a lacunary sequence, then ˆ N αθ ( A , M , ( s )) ⊂ ˆ S βθ .Proof: Let x ∈ ˆ N αθ ( A , M , ( s )).For ε > P as the sum over k ∈ I r , | t km ( A k ( x ) − L ) | ≥ ε. and P denote the sum over k ∈ I r , | t km ( A k ( x ) − L ) | < ε respectively.As h α r ≤ h β r for each r , we may write, h α r X k ∈ I r " M k | t km ( A k ( x ) − L ) | ρ ( k ) ! s k = h α r hP h M k (cid:16) | t km ( A k ( x ) − L ) | ρ ( k ) (cid:17)i s k + P h M k (cid:16) | t km ( A k ( x ) − L ) | ρ ( k ) (cid:17)i s k i ≥ h β r hP h M k (cid:16) | t km ( A k ( x ) − L ) | ρ ( k ) (cid:17)i s k + P h M k (cid:16) | t km ( A k ( x ) − L ) | ρ ( k ) (cid:17)i s k i ≥ h β r P h M k (cid:16) ερ ( k ) (cid:17)i s k h β r P min([ M k ( ε )] h , [ M k ( ε )] H ) , ε = ερ ( k ) ≥ h β r |{ k ∈ I r : | t km ( A ( x )) − L | ≥ ε }| min([ M k ( ε )] h , [ M k ( ε )] H . As x ∈ ˆ N αθ ( A , M , ( s )), the left hand side of the above inequality tends to zeroas r → ∞ .Therefore, the right hand side of the above inequality tends to zero as r → ∞ ,hence x ∈ ˆ S βθ . Corollary 3.2.
Let 0 < α ≤ M be a Musielak-Orlicz function and θ = ( k r ) bea lacunary sequence, then ˆ N αθ ( A , M , ( s )) ⊂ ˆ S αθ . Theorem 3.3.
Let M be a Musielak-Orlicz function, x = ( x k ) be a boundedsequence and θ = ( k r ) be a lacunary sequence. If lim r →∞ h r h α r = x ∈ ˆ S αθ ⇒ x ∈ ˆ N αθ ( A , M , ( s )).Proof: Suppose that x = ( x k ) be a bounded sequence that is x ∈ ℓ ∞ andˆ S αθ − lim x k = L .As x ∈ ℓ ∞ , then there is a constant T > | x k | ≤ T . Given ε >
0, we have, h α r X k ∈ I r " M k | t km ( A k ( x ) − L ) | ρ ( k ) ! s k h α r P h M k (cid:16) | t km ( A k ( x ) − L ) | ρ ( k ) (cid:17)i s k + h α r P h M k (cid:16) | t km ( A k ( x ) − L ) | ρ ( k ) (cid:17)i s k ≤ h α r P max (cid:26)h M k (cid:16) T ρ ( k ) (cid:17)i h , h M k (cid:16) T ρ ( k ) (cid:17)i H (cid:27) + h α r P h M k (cid:16) ερ ( k ) (cid:17)i s k ≤ max { [ M k ( K )] h , [ M k ( K )] H } h α r |{ k ∈ I r : | t km ( A k ( x ) − L ) | ≥ ε }| + h r h α r max { [ M k ( ε )] h , [ M k ( ε )] H } , T ρ ( k ) = K , ερ ( k ) = ε . Hence, x ∈ ˆ N αθ ( A , M , ( s )). 6 heorem 3.4. If lim s k > x = ( x k ) is strongly ˆ N αθ ( A , M , ( s ))-summableto L with respect to the Musielak-Orlicz function M , then ˆ N αθ ( A , M , ( s )) − lim x k isunique.Proof: Let lim s k = s >
0. Suppose that ˆ N αθ ( A , M , ( s )) − lim x k = L , andˆ N αθ ( A , M , ( s )) − lim x k = L . Then,lim r →∞ h α r X k ∈ I r M k | t km ( A k ( x ) − L ) | ρ ( k )1 s k = , for some ρ ( k )1 > r →∞ h α r X k ∈ I r M k | t km ( A k ( x ) − L ) | ρ ( k )2 s k = , for some ρ ( k )2 > . Define ρ ( k ) = max(2 ρ ( k )1 , ρ ( k )2 ). As M is nondecreasing and convex, we have, h α r X k ∈ I r " M k | L − L | ρ ( k ) ! s k ≤ Dh α r X k ∈ I r s k M k | t km ( A k ( x ) − L ) | ρ ( k )1 s k + M k | t km ( A k ( x ) − L ) | ρ ( k )2 s k ≤ Dh α r X k ∈ I r M k | t km ( A k ( x ) − L ) | ρ ( k )1 s k + Dh α r X k ∈ I r M k | t km ( A k ( x ) − L ) | ρ ( k )2 s k → , ( r → ∞ ) , where sup k s k = H and D = max(1 , H − ). Hence,lim r →∞ h α r X k ∈ I r " M k | L − L | ρ ( k ) ! s k = . As lim k →∞ s k = s , we have,lim k →∞ " M k | L − L | ρ ( k ) ! s k = " M k | L − L | ρ ( k ) ! s and so L = L .Thus the lmit is unique. Theorem 3.5.
Let A = ( a mk ) be an infinite matrix of complex numbersand let M = ( M k ) be a Musielak-Orlicz function satisfying ∆ -condition . If x
7s strongly almost lacunary A -convergent sequences with respect to M , thenˆ N αθ ( A ) ⊂ ˆ N αθ ( A , M ).Proof: Let x ∈ ˆ N αθ ( A ).Then, lim r →∞ h α r X k ∈ I r | t km ( A ( x ) − L ) | =
0, uniformly in m .Let us define two sequences y and z such that,( | t km ( A k ( y ) − L ) | ) = ( | t km ( A k ( x ) − L ) | ) if ( | t km ( A k ( x ) − L ) | ) > θ if ( | t km ( A k ( x ) − L ) | ) ≤ . ( | t km ( A k ( z ) − L ) | ) = θ if ( | t km ( A k ( x ) − L ) | ) > | t km ( A k ( x ) − L ) | ) if ( | t km ( A k ( x ) − L ) | ) ≤ . Hence, ( | t km ( A k ( x ) − L ) | ) = ( | t km ( A k ( y ) − L ) | ) + ( | t km ( A k ( z ) − L ) | ) . Also, ( | t km ( A k ( y ) − L ) | ) ≤ ( | t km ( A k ( x ) − L ) | ) and ( | t km ( A k ( z ) − L ) | ) ≤ ( | t km ( A k ( x ) − L ) | ).Since, ˆ N αθ ( A ) is normal, so we have y , z ∈ ˆ N αθ ( A ).Let sup k M k (2) = T Then, h α r X k ∈ I r " M k | t km ( A k ( x ) − L ) | ρ ( k ) ! = h α r X k ∈ I r " M k | t km ( A k ( y ) − L ) | + | t km ( A k ( z ) − L ) | ρ ( k ) ! ≤ h α r X k ∈ I r " M k | t km ( A k ( y ) − L ) | ρ ( k ) ! + M k | t km ( A k ( z ) − L ) | ρ ( k ) ! <
12 1 h α r X k ∈ I r K | t km ( A k ( y ) − L ) | ρ ( k ) ! M k (2) +
12 1 h α r X k ∈ I r K | t km ( A k ( z ) − L ) | ρ ( k ) ! M k (2)8
12 1 h α r X k ∈ I r K | t km ( A k ( y ) − L ) | ρ ( k ) ! sup M k (2) +
12 1 h α r X k ∈ I r K | t km ( A k ( z ) − L ) | ρ ( k ) ! sup M k (2) → r → ∞ . Hence x ∈ ˆ N αθ ( A , M ). This completes the proof. Theorem 3.6.
Let A = ( a mk ) be an infinite matrix of complex numbers and let M = ( M k ) be a Musielak-Orlicz function satisfying ∆ -condition. Iflim ν →∞ inf k M k (cid:16) νρ ( k ) (cid:17) νρ ( k ) > , for some ρ ( k ) > , then, ˆ N αθ ( A ) = ˆ N αθ ( A , M ).Proof: If ˆ N αθ ( A ) = ˆ N αθ ( A , M ) for some ρ ( k ) >
0, then there exists a number γ > M k νρ ( k ) ! ≥ γ νρ ( k ) ! , ∀ ν > , and some ρ ( k ) > . Let x ∈ ˆ N αθ ( A , M ) . Then,1 h α r X k ∈ I r " M k | t km ( A k ( x ) − L ) | ρ ( k ) ! ≥ h α r X k ∈ I r ν " γ t km ( A k ( x ) − L ) | ρ ( k ) ! = γ h α r X k ∈ I r | t km ( A k ( x ) − L ) | ρ ( k ) ! Hence, x ∈ ˆ N αθ ( A ). This completes the proof. Theorem 3.7.
Let M = ( M k ) be a Musielak-Orlicz function where ( M k )is pointwise convergent. Then, ˆ N αθ ( A , M , ( s )) ⊂ ˆ S αθ ( A , M , ( s )) if and only iflim k M k νρ ( k ) ! > ν > , ρ ( k ) > ε > x ∈ ˆ N αθ ( A , M , ( s )).Also, if lim k M k νρ ( k ) ! >
0, then there exists a number c > k νρ ( k ) ! ≥ c , for ν > ε. Let us consider, I r = n i ∈ I r : h M k (cid:16) | t km ( A ( x ) − L ) | ρ ( k ) (cid:17)i ≥ ε o .Then, 1 h α r X k ∈ I r " M k | t km ( A ( x ) − L ) | ρ ( k ) ! s k ≥ h α r X k ∈ I r " M k | t km ( A ( x ) − L ) | ρ ( k ) ! s k ≥ c h α r | t km ( A ( ε ) | Hence, it follows that x ∈ ˆ S αθ ( A , M , ( s )).Conversely, let us assume that the condition does not hold good. For a number ν > k M k νρ ( k ) ! = ρ >
0. Now, we select a lacunary sequence θ = ( n r ) such that M k (cid:16) νρ ( k ) (cid:17) < − r for any k > n r .Let A = I and define a sequence x by putting, A k ( x ) = ν if n r − < k ≤ n r + n r − ; θ if n r + n r − < k ≤ n r . Therefore, 1 h α r X k ∈ I r " M k | A k ( x ) | ρ ( k ) ! s k = h α r X n r − < k ≤ ( nr + nr − M k νρ ( k ) ! < h α r r − (cid:20) n r + n r − − n r − (cid:21) = r → r → ∞ . Thus we have x ∈ ˆ N α θ ( A , M , ( s )). 10ut,lim r →∞ h α r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ∈ I r : X k ∈ I r " M k | t km ( A ( x )) | ρ ( k ) ! s k ≥ ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = lim r →∞ h α r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ∈ (cid:18) n r − , n r + n r − (cid:19) : X k ∈ I r " M k νρ ( k ) ! s k ≥ ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = lim r →∞ h α r n r − n r − = . So, x < ˆ S αθ ( A , M , ( s )). Theorem 3.8.
Let M = ( M k ) be a Musielak-Orlicz function. Then ˆ S αθ ( A , M , ( s )) ⊂ ˆ N αθ ( A , M , ( s )) if and only if sup ν sup k M k νρ ( k ) ! < ∞ .Proof: Let x ∈ ˆ S αθ ( A , M , ( s )). Suppose h ( ν ) = sup k M k νρ ( k ) ! and h = sup ν h ( ν ). Let I r = ( k ∈ I r : M k | t km ( A ( x ) − L ) | ρ ( k ) ! < ε ) . Now, M k ( ν ) ≤ h for all k , ν >
0. So,1 h α r X k ∈ I r " M k | t km ( A ( x ) − L ) | ρ ( k ) ! s k = h α r X k ∈ I r " M k | t km ( A ( x ) − L ) | ρ ( k ) ! s k + h α r X k ∈ I r " M k | t km ( A ( x ) − L ) | ρ ( k ) ! s k ≤ h h α r | t km ( A ( ε ) | + h ( ε ) . Hence, as ε →
0, it follows that x ∈ ˆ N αθ ( A , M , ( s )).Conversely, suppose that sup ν sup k M k νρ ( k ) ! = ∞ . Then, we have 0 < ν < ν < ... < ν r − < ν r < ...
11o that M n r (cid:16) ν r ρ ( k ) (cid:17) ≥ h α r for r ≥
1. Let A = I . We set a sequence x = ( x k ) by, A k ( x ) = ν r if k = n r for some r = , , .. ; θ otherwise . Then, lim r →∞ h α r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ∈ I r : X k ∈ I r M k | t km ( A k ( x )) | ρ ( k ) ! s k ≥ ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = lim r →∞ h α r = x ∈ ˆ S αθ ( A , M , ( s )).But, lim r →∞ h α r X k ∈ I r " M k | t km ( A k ( x ) − L ) | ρ ( k ) ! = lim r →∞ h α r " M n r | ν r − L | ρ ( k ) ! ≥ lim r →∞ h α r h α r = x ∈ ˆ N αθ ( A , M , ( s )). References [1] Esi, A. and Gokhan, A., Lacunary strong almost A -Convergence with re-spect to a sequence of Orlicz function, Journal of Computational Analysis andApplications , 12(4)(2010), 853-865.[2] Fast, H., Sur la convergence statistique,
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