KYUNGPOOK Math. J. 51(2011), 385-393 http://dx.doi.org/10.5666/KMJ.2011.51.4.385 On Some Lacunary Generalized Difference Sequence Spaces of Invariant Means Defined by a Sequence of Modulus Functions G¨ ulcan Atıci Department of Mathematics, Mu¸ s Alparslan University, Mu¸ s, 49100, Turkey e-mail : gatici23@hotmail.com C ¸i˘ gdem Asma Bektas ¸ * Department of Mathematics, Firat University, Elazig, 23119, Turkey e-mail : cigdemas78@hotmail.com Abstract. The aim of this paper is to introduce and study the sequence spaces [w, θ, F, p, q] ∞ (Δ m v ), [w, θ, F, p, q] 1 (Δ m v ) and [w, θ, F, p, q] 0 (Δ m v ), which arise from the no- tions of generalized difference sequence space, lacunary convergence, invariant mean and a sequence of Moduli F =(f k ). We establish some inclusion relations between these spaces under some conditions. 1. Introduction Let ℓ ∞ , c and c 0 denote the Banach spaces of bounded, convergent and null sequences x =(x k ), with x k ∈ R or C, normed by ∥x∥ = sup k |x k |, respectively. Let σ be a mapping of the set of positive integers into itself. A continuous linear functional ϕ on ℓ ∞ , is said to be an invariant mean or σ−mean if and only if (i) ϕ(x) ≥ 0 when the sequence x =(x n ) has x n ≥ 0 for all n, (ii) ϕ(e) = 1, (iii) ϕ(x σ(n) )= ϕ(x) for all x ∈ ℓ ∞ . In case σ is the translation mapping n → n + 1, a σ−mean is often called a Banach limit and V σ , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences. If x =(x k ), write Tx =(Tx k )= (x σ(k) ). It can be shown that V σ = { x ∈ ℓ ∞ : lim k t kn (x)= l, uniformly in n } , * Corresponding Author. Received August 31, 2010; revised October 8, 2010; accepted December 9, 2010. 2010 Mathematics Subject Classification: 46A45, 40A05. Key words and phrases: Invariant mean, Difference sequence spaces, lacunary sequence, modulus function. 385