On Generalized Convergence in Sequence Spaces

Abstract

In the present thesis, we have made efforts to develop some classical methods of summability and introduce some summability theories in differentspaces like 2-normed Spaces, probabilistic normed spaces and double sequence spacesfollowed by characterization of certain properties of these generalized convergence methods in the above-mentioned sequence spaces.The whole work in the thesis has been divided into five chapters. Chapter − I is an introductory one in which, the basic terminology of some generalized summability methods and abstract spaces under consideration in present thesis has been presented. We start with the definition of statistical convergence, which was initially introduced by H. Fast [54] and see how it has been extended in various abstract spaces to resolve many problems in the area of mathematical analysis. We also record some facts related to statistical convergence and its generalizations from the history, which will be needed in the sequel and form the base for the present thesis.Chapter − II is devoted to the development of new sequence spaces V fλ [Δmp, I]0,V fλ [Δmp, I]L and V fλ [Δmp, I]∞. These sequence spaces have been defined by the use ofde la vallee poussin mean, modulus function f and the difference operator Δ. After making some early remarks on these spaces, some of their important properties and inclusion relations have been established in this chapter. Some concrete examples in support of our results have also been provided. Subsequently, the concept of SΔmλ (I)-convergence has also been introduced and a condition is obtained under which SΔm λ (I)-convergence coincides with above-mentioned sequence spaces. In Chapter − III, we consider one of the generalized form of abstract spaces i.e.2-norm space. The aim in this chapter is to extend the sequence spaces V fλ [Δmp, I]0,V fλ [Δmp, I]L and V fλ [Δmp, I]∞ introduced in Chapter-II to the sequence spaces V f λ [∥, ∥,Δm, λ, p]0, V fλ [∥, ∥,Δm, λ, p]L and V fλ [∥, ∥,Δm, λ, p]∞ in 2-norm spaces and have obtained some of their properties. A new class of generalized statistical convergentsequences has also been defined in this chapter with the help of an ideal and differencesequences. Further, we also establish a strong connection between this type of convergence and the aforesaid sequence space. The aim of Chapter − IV is to introduce some new sequence spaces using ideals,modulus function and invariant means and examine some of their topologicalproperties. We start with the introduction of the new sequence spaces wfσ[Δmp, I, θ]0,wfσ[Δmp, I,θ]L and wfσ[Δmp, I, θ]∞ and prove that these are linear spaces. We alsofind some inclusion relations among these spaces by imposing some conditions on the modulus function f. Finally, a new kind of summability method called as SΔmθσ (I)- convergence for sequences of numbers has been presented in this chapter. Towards the end of this chapter, lacunary generalized convergence in probabilistic normed spacehas been developed. The goal of Chapter − V is to introduce the concepts of I−lacunary stronglyCes`aro summability (Nθrs(I2)) and I-lacunary statistical convergence (Sθrs(I2)) for double sequence of numbers with help of double lacunary sequence. Some importantproperties of these notions have been obtained in this chapter and it is proved that the set of all lacunary ideal statistically convergent bounded sequences is closed.Subsequently, a stronger form of Sθrs(I2)-convergence has been defined with the help of ideals. Further, in this chapter some relevant onnections between these notionshave also been established.