The Ideal Convergence of Strongly of in Metric Spaces Defined by Modulus

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 790950, 8 pages http://dx.doi.org/10.1155/2014/790950 Research Article The Ideal Convergence of Strongly of Γ2 in 𝑝-Metric Spaces Defined by Modulus N. Subramanian,1 K. Balasubramanian,1 and K. Chandrasekhara Rao2 1 2 Department of Mathematics, SASTRA University, Thanjavur 613 401, India Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA University, Kumbakonam 612 001, India Correspondence should be addressed to N. Subramanian; Received 9 January 2014; Accepted 25 April 2014; Published 20 May 2014 Academic Editor: Feyzi Başar Copyright © 2014 N. Subramanian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The aim of this paper is to introduce and study a new concept of the Γ2 space via ideal convergence defined by modulus and also some topological properties of the resulting sequence spaces were examined. The space Γ2 is a metric space with the metric 1. Introduction Let (𝑥𝑚𝑛 ) be a double sequence of real or complex numbers. Then the series ∑∞ 𝑚,𝑛=1 𝑥𝑚𝑛 is called a double series. The double series ∑∞ 𝑚,𝑛=1 𝑥𝑚𝑛 is said to be convergent if and only if the double sequence (𝑆𝑚𝑛 ) is convergent, where 𝑚,𝑛 𝑆𝑚𝑛 = ∑ 𝑥𝑖𝑗 , (𝑚, 𝑛 = 1, 2, 3, . . .) . 𝑖,𝑗=1 (1) We denote 𝑤2 as the class of all complex double sequences (𝑥𝑚𝑛 ). A sequence 𝑥 = (𝑥𝑚𝑛 ) is said to be double analytic if 󵄨 󵄨1/𝑚+𝑛 < ∞. sup󵄨󵄨󵄨𝑥𝑚𝑛 󵄨󵄨󵄨 𝑚𝑛 (2) The vector space of all prime sense double analytic sequences is usually denoted by Λ2 . A sequence 𝑥 = (𝑥𝑚𝑛 ) is called double entire sequence if 󵄨 1/𝑚+𝑛 󵄨 (󵄨󵄨󵄨𝑥𝑚𝑛 󵄨󵄨󵄨) 󳨀→ 0 as 𝑚, 𝑛 󳨀→ ∞. (3) The vector space of all prime sense double entire sequences is usually denoted by Γ2 . The space Λ2 is a metric space with the metric 󵄨󵄨1/𝑚+𝑛 󵄨 𝑑 (𝑥, 𝑦) = sup {󵄨󵄨󵄨𝑥𝑚𝑛 − 𝑦𝑚𝑛 󵄨󵄨 𝑚𝑛 : 𝑚, 𝑛 : 1, 2, 3, . . .} . (4) 󵄨 1/𝑚+𝑛 󵄨 𝑑 (𝑥, 𝑦) = sup {(󵄨󵄨󵄨𝑥𝑚𝑛 − 𝑦𝑚𝑛 󵄨󵄨󵄨) : 𝑚, 𝑛 : 1, 2, 3, . . .} , (5) 𝑚𝑛 for all 𝑥 = {𝑥𝑚𝑛 } and 𝑦 = {𝑦𝑚𝑛 } in Γ2 . Consider a double sequence 𝑥 = (𝑥𝑖𝑗 ). The (𝑚, 𝑛)th section 𝑥[𝑚,𝑛] of the sequence is defined by 𝑥[𝑚,𝑛] = ∑𝑚,𝑛 𝑖,𝑗=0 𝑥𝑖𝑗 𝛿𝑖𝑗 for all 𝑚, 𝑛 ∈ N, 0 0 ⋅⋅⋅ 0 0 ⋅⋅⋅ 0 0 ⋅⋅⋅ 0 0 ⋅⋅⋅ 𝛿𝑚𝑛 = ( ... ), 0 0 ⋅⋅⋅ 1 0 ⋅⋅⋅ 0 0 ⋅⋅⋅ 0 0 ⋅⋅⋅ (6) with 1 in the (𝑚, 𝑛)th position and zero otherwise. An FKspace (or a metric space) 𝑋 is said to have AK property if (𝛿𝑚𝑛 ) is a Schauder basis for 𝑋. Or equivalently 𝑥[𝑚,𝑛] → 𝑥. We need the following inequality in the sequel of the paper. Lemma 1. For 𝑎, 𝑏 ≥ 0 and 0 < 𝑝 < 1, one has (𝑎 + 𝑏)𝑝 ≤ 𝑎𝑝 + 𝑏𝑝 . (7) Some initial work on double sequence spaces is found in Bromwich. Later on it was investigated by Moricz [1], Moricz and Rhoades [2], Basarir and Solancan [3], Tripathy [4], 2 Abstract and Applied Analysis Turkmenoglu [5], Subramanian and Misra [6, 7], and many others. Tripathy and Dutta [8] introduced and investigated different types of fuzzy real valued double sequence spaces. Generalizing the concept of ordinary convergence for real sequences Kostyrko et al. introduced the concept of ideal convergence which is a generalization of statistical convergence, by using the ideal 𝐼 of the subsets of the set of natural numbers. The notion of different sequence spaces (for single sequences) was introduced by Kizmaz [9] as follows: 𝑍 (Δ) = {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : (Δ𝑥𝑘 ) ∈ 𝑍} , (8) for 𝑍 = 𝑐, 𝑐0 and ℓ∞ , where Δ𝑥𝑘 = 𝑥𝑘 − 𝑥𝑘+1 for all 𝑘 ∈ N. Here 𝑤, 𝑐, 𝑐0 , and ℓ∞ denote the classes of all, convergent, null, and bounded scalar valued single sequences, respectively. The above spaces are Banach spaces normed by 󵄨 󵄨 󵄨 󵄨 ‖𝑥‖ = 󵄨󵄨󵄨𝑥1 󵄨󵄨󵄨 + sup 󵄨󵄨󵄨Δ𝑥𝑘 󵄨󵄨󵄨 . (9) 𝑘≥1 Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by 𝑍 (Δ) = {𝑥 = (𝑥𝑚𝑛 ) ∈ 𝑤2 : (Δ𝑥𝑚𝑛 ) ∈ 𝑍} , (10) where 𝑍 = Λ2 and Γ2 , respectively. Δ𝑥𝑚𝑛 = (𝑥𝑚𝑛 − 𝑥𝑚𝑛+1 ) − (𝑥𝑚+1𝑛 − 𝑥𝑚+1𝑛+1 ) = 𝑥𝑚𝑛 − 𝑥𝑚𝑛+1 − 𝑥𝑚+1𝑛 + 𝑥𝑚+1𝑛+1 for all 𝑚, 𝑛 ∈ N. We further generalized this notion and introduced the following notion. For 𝑚, 𝑛 ≥ 1, 𝑍 (Δ𝜇𝛾 ) = {𝑥 = 𝑥𝑚𝑛 : (Δ𝜇𝛾 𝑥𝑚𝑛 ) ∈ 𝑍} , for 𝑍 = Λ2 , Γ2 . (11) An Orlicz function is a function 𝑓 : [0, ∞) → [0, ∞) which is continuous, nondecreasing, and convex with 𝑓(0) = 0, 𝑓(𝑥) > 0, for 𝑥 > 0 and 𝑓(𝑥) → ∞ as 𝑥 → ∞. If convexity of Orlicz function 𝑓 is replaced by 𝑓(𝑥 + 𝑦) ≤ 𝑓(𝑥) + 𝑓(𝑦), then this function is called modulus function. A modulus function 𝑓 is said to satisfy Δ2 -condition for all values 𝑢, if there exists 𝐾 > 0 such that 𝑓(2𝑢) ≤ 𝐾𝑓(𝑢), 𝑢 ≥ 0. Remark 2. A modulus function satisfies the inequality 𝑓(𝜆𝑥) ≤ 𝜆𝑓(𝑥) for all 𝜆 with 0 < 𝜆 < 1. Lemma 3. Let 𝑓 be a modulus function which satisfies Δ2 condition and let 0 < 𝛿 < 1. Then for each 𝑡 ≥ 𝛿, one has 𝑓(𝑡) < 𝐾𝛿−1 𝑓(2) for some constant 𝐾 > 0. Spaces of strongly summable sequences were discussed by Kuttner, Maddox, and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox as an extension of the definition of strongly Cesàro summable sequences. Connor further extended this definition to a definition of strong 𝐴summability with respect to a modulus where 𝐴 = (𝑎𝑛,𝑘 ) is a nonnegative regular matrix and established some connections between strong 𝐴-summability, strong 𝐴-summability with respect to a modulus, and 𝐴-statistical convergence. The notion of convergence of double sequences was presented by A. Pringsheim. Also, the four-dimensional matrix ∞ 𝑚𝑛 transformation (𝐴𝑥)𝑘,ℓ = ∑∞ 𝑚=1 ∑𝑛=1 𝑎𝑘ℓ 𝑥𝑚𝑛 was studied extensively by Robison and Hamilton. 2. Definitions and Preliminaries Let 𝑋 be a nonempty set. A nonvoid class 𝐼 ⊆ 2𝑋 (power set, of 𝑋) is called an ideal if 𝐼 is additive (i.e., 𝐴, 𝐵 ∈ 𝐼 ⇒ 𝐴 ⋃ 𝐵 ∈ 𝐼) and hereditary (i.e., 𝐴 ∈ 𝐼 and 𝐵 ⊆ 𝐴 ⇒ 𝐵 ∈ 𝐼). A nonempty family of sets 𝐹 ⊆ 2𝑋 is said to be a filter on 𝑋 if 𝜙 ∉ 𝐹; 𝐴, 𝐵 ∈ 𝐹 ⇒ 𝐴 ⋂ 𝐵 ∈ 𝐹 and 𝐴 ∈ 𝐹, 𝐴 ⊆ 𝐵 ⇒ 𝐵 ∈ 𝐹. For each ideal 𝐼 there is a filter 𝐹(𝐼) given by 𝐹(𝐼) = {𝐾 ⊆ 𝑁 : 𝑁 \ 𝐾 ∈ 𝐼}. A nontrivial ideal 𝐼 ⊂ 2𝑋 is called admissible if and only if {{𝑥} : 𝑥 ∈ 𝑋} ⊂ 𝐼. A double sequence space 𝐸 is said to be solid or normal if (𝛼𝑚𝑛 𝑥𝑚𝑛 ) ∈ 𝐸, whenever (𝑥𝑚𝑛 ) ∈ 𝐸 and for all double sequences 𝛼 = (𝛼𝑚𝑛 ) of scalars with |𝛼𝑚𝑛 | ≤ 1, for all 𝑚, 𝑛 ∈ N. Let 𝑛 ∈ N and let 𝑋 be a real vector space of dimension 𝑤, where 𝑛 ≤ 𝑤. A real valued function 𝑑𝑝 (𝑥1 , . . . , 𝑥𝑛 ) = ‖(𝑑1 (𝑥1 ), . . . , 𝑑𝑛 (𝑥𝑛 ))‖𝑝 on 𝑋 satisfies the following four conditions: (i) ‖(𝑑1 (𝑥1 ), . . . , 𝑑𝑛 (𝑥𝑛 ))‖𝑝 = 0 if and and only if 𝑑1 (𝑥1 ), . . . , 𝑑𝑛 (𝑥𝑛 ) are linearly dependent, (ii) ‖(𝑑1 (𝑥1 ), . . . , 𝑑𝑛 (𝑥𝑛 ))‖𝑝 is invariant under permutation, (iii) ‖(𝛼𝑑1 (𝑥1 ), . . . , 𝑑𝑛 (𝑥𝑛 ))‖𝑝 = |𝛼|‖(𝑑1 (𝑥1 ), . . . , 𝑑𝑛 (𝑥𝑛 ))‖𝑝 , 𝛼 ∈ R, (iv) 𝑑𝑝 ((𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 ) ⋅ ⋅ ⋅ (𝑥𝑛 , 𝑦𝑛 )) = (𝑑𝑋 (𝑥1 , 𝑥2 , . . ., 1/𝑝 𝑥𝑛 )𝑝 + 𝑑𝑌 (𝑦1 , 𝑦2 , . . . , 𝑦𝑛 )𝑝 ) for 1 ≤ 𝑝 < ∞, or (v) 𝑑((𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 ), . . . , (𝑥𝑛 , 𝑦𝑛 )) := sup{𝑑𝑋 (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ), 𝑑𝑌 (...truncated)