Some new sequence spaces defined by special functions

In this study, the definitions of the sequence spaces w A, p, f, q, s ,ˆ ˆ ˆw A, p, M, q, s , w ∆r , A, p, M, q, s and w ∆r , A, p, f, q, s are given by using fmodulus and M Orlicz functions on a semi-normed space (X, q) . This work consistsof five chapters.First chapter is devoted to the fundamental concepts and some FunctionalAnalysis tools which will be used in the following chapters.In the second chapter, some properties of modulus function are established.Furthermore in the case of (X, q) is complete, we showed that the sequence spaceˆw0 A, p, f, q, s is complete and in addition to this some inclusion relations are alsoˆgiven. Finally, by constructing the sequence spaces w A, p, f v , q, s , the relationsˆ ˆbetween the sequence spaces w A, p, f v , q, s and w A, p, f m , q, s are examined,where v, m ∈ N.In the third chapter, certain attributes of Orlicz functions which are taken part inthe special classes of convex functions and studied in detail by M.A. Krasnosel?skii& Y.B. Rutickii and later M.M. Rao & Z.D. Ren, are introduced. Most of theˆchapter devoted to investigation of the space w A, p, M, q, s which is a subspaceof s defined by an Orlicz function.In the fourth chapter, the generalized difference sequence spacesˆ ˆ ˆw0 ∆r , A, p, M, q, s , w ∆r , A, p, M, q, s and w∞ ∆r , A, p, M, q, s are definediiiand some of their properties are investigated, where ∆x = xk − xk+1 , ∆r xk =∆r−1 xk − ∆r−1 xk+1 , r ∈ N and M is an Orlicz function.ˆIn the last chapter, the generalized difference sequence spaces w0 ∆r , A, p, f, q, s ,ˆ ˆw ∆r , A, p, f, q, s and w∞ ∆r , A, p, f, q, s are defined and some of their propertiesare investigated, where ∆x = xk − xk+1 , ∆r xk = ∆r−1 xk − ∆r−1 xk+1 , r ∈ N and fis a modulus function.KEY WORDS: Sequence space, Modulus function, Orlicz function, paranormedspace, difference sequence space.