Some New Difference Sequence Spaces of Invariant Means Defined by Ideal and Modulus Function

The main objective of this paper is to introduce a new kind of sequence spaces by combining the concepts of modulus function, invariant means, difference sequences, and ideal convergence. We also examine some topological properties of the resulting sequence spaces. Further, we introduce a new concept of -convergence and obtain a condition under which this convergence coincides with above-mentioned sequence spaces. 1. Introduction and Background Let and be the Banach spaces of real bounded and convergent sequences with the usual supremum norm. Let be the mapping of the set of all positive integers into itself. A continuous linear functional on is said to be an invariant mean or -mean if and only if(i) , when the sequence has , for all ;(ii) , where ;(iii) , for all . If , we write . It can be shown by Schaefer [1] that where . In case is the translation mapping , -mean is often called a Banach limit and , the set of bounded sequences all whose invariant means are equal, is the set of almost convergent sequences (see Lorentz [2]). Using the concept of invariant means Mursaleen et al. [3] introduced the following sequence spaces as a generalization of Das and Sahoo [4]: and investigated some of its properties. The notion of statistical convergence for number sequences was studied at the initial stage by Fast [5] and later investigated by Connor [6], Fridy [7], Maddox [8], ？alát [9], and many others. Definition 1 (see [5]). A number sequence is said to be statistically convergent to a number (denoted by ) provided that, for every , where the vertical bars denote the cardinality of the enclosed set. By a lacunary sequence, we mean an increasing sequence of positive integers such that and as . The intervals determined by will be denoted by , where the ratio is denoted by . The space of lacunary strongly convergent sequence was defined by Freedman et al. [10] as follows: Fridy and Orhan [11] generalized the concept of statistical convergence by using lacunary sequence which is called lacunary statistical convergence. Further, lacunary sequences have been studied by Fridy and Orhan [12], Pehlivan and Fisher [13], Et and G？khan [14], and Tripathy and Dutta [15]. Quite recently, Karakaya [16] combined the approach of lacunary sequence with invariant means and introduced the notion of strong -lacunary statistically convergence as follows. Definition 2 (see [16]). Let be a lacunary sequence. A sequence is said to be lacunary strong -lacunary statistically convergent if, for every , where denotes the set of all lacunary strong -lacunary statistically convergent