On Some New Paranormed Sequence Spaces and Their Topological Properties

Yıl 2014, Cilt: 3 Sayı: 6, 32 - 43, 01.06.2014

Osman Duyar

Öz

In this study, we define new paranormed sequence spaces c0(u, v; p, G) and c(u, v; p, G) by combining a generalized weighted mean and a generalized difference operatorB = B(r, s, t). Furthermore, we compute the α− and β− dualsand obtain bases for these sequence spaces. Finally, we characterize the classes of matrix mappings from the new paranormedsequence spaces to the spaces µ(q) for µ ∈ {c, ,∞}

Anahtar Kelimeler

-

Kaynakça

• C.-S. Wang, On N¨orlund sequence spaces, Tamkang J. Math. 9(1978) 269-274.
• P.-N. Ng, P.-Y. Lee, Ces`aro sequence spaces of non-absolute type, Comment. Math. (Prace Mat.) 20(2)(1978) 429-433.
• E. Malkowsky, Recent results in the theory of matrix transformations in sequence spaces, Mat. Vesnik 49(1997) 187-196.
• A. M. Jarrah and E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations,Filomat 17(2003), 59-78
• B. Altay, F. Ba¸sar, Some Euler sequence spaces of non-absolute type, Ukrainian Math. J. 57(1)(2005) 1-17.
• B. Altay, F. Ba¸sar, Some paranormed sequence spaces of non-absolute type derived by weighted mean, J. Math. Anal. Appl. 319 (2006), 494-508.
• H. Polat, V. Karakaya and N. S¸im¸sek, Difference sequence spaces derived by gen- eralized weighted mean, Appl. Math. Lett. 24(2011), no.5, 608-614.
• B. Altay, F. Ba¸sar, Generalization of the space (p) derived by weighted mean , J. Math. Anal. Appl 330 (2007) 174-185.
• E. Malkowsky, E. Sava¸s, Matrix transformations between sequence spaces of gen- eralized weighted means, Appl. Math. Comput. 147(2004) 333-345.
• M. Ba¸sarır, On some new sequence spaces and related matrix transformations, Indian J. Pure Appl. Math. 26(10) (1995) 1003-1010.
• F. Ba¸sar, B. Altay, M. Mursaleen, Some generalizations of the space bvpof p− bounded variation sequences, Nonlinear Anal. 68(2)(2008) 273-287.
• B. Altay, On the space of p− summable difference sequences of order m, (1 ≤ p < ∞), Studia Sci. Math. Hungar. 43(4)(2006) 387-402.
• E. Malkowsky, M. Mursaleen, S. Suantai, The dual spaces of sets of difference sequences of order m and matrix transformations, Acta Math. Sin. Eng. Ser. 23(3)(2007) 521-532.
• Demiriz, S., C¸ akan, C., On some new paranormed Euler sequence spaces and Euler core , Acta Math. Sin. Eng. Ser. 26(7)(2010), 1207-1222. Demiriz, S., ¨
• Ozdemir O., Duyar O., On some new generalized difference sequence spaces of non-absolute type, arXiv:1309.3903, 2013.
• Duyar O., Demiriz, S., On some new generalized difference sequence spaces and their topological properties, Journal of New Result in Science, 6(2014) 1-14.
• B. Altay, F. Ba¸sar, Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, J. Math. Anal. Appl. 336(1)(2007) 632- 6
• H. Kızmaz, On certain sequence spaces, Canad. Math. Bull. 24(2)(1981) 169-176.
• C. Aydın, F. Ba¸sar, Some new sequence spaces which include the spaces pand ∞, Demonstratio Math. 38(3)(2005) 641-656.
• C. Aydın, F. Ba¸sar, Some generalizations of the sequence space ar, Iran. J. Sci. p Technol. A Sci. 30(No. A2)(2006) 175-190.
• F. Ba¸sar, B. Altay, On the space of sequences of p− bounded variation and related matrix mappings, Ukrainian Math. J. 55(1) (2003) 136-147.
• E. Malkowsky, M. Mursaleen, Some matrix transformations between the difference sequence spaces ∆c0(p), ∆c(p) and ∆∞(p), Filomat, 15(2001) 353-363.
• Z. U. Ahmad, M. Mursaleen, K¨othe-Toeplitz duals of some new sequence spaces and their matrix maps, Publ. Inst. Math. (Belgrad) 42(1987) 57-61.
• A. S¨onmez, F. Ba¸sar, Generalized difference spaces of non-absolute type of con- vergent and null sequences, Abstract and Applied Analysis Volume 2012 , Article ID 435076, (2012) 20 pages.
• M. Ba¸sarır, E. E. Kara, On the mthorder difference sequence space of generalized weighted mean and compact operators, Acta Mathematica Scientia, 33(B3)(2013), 1
• M. Ba¸sarır, E. E. Kara, On the B− difference sequence space derived by generalized weighted mean and compact operators, Journal of Mathematical Analysis and Applications, 391(2012), 67-81.
• E. E. Kara, Some topological and geometrical properties of new Banach sequence space, Journal of Inequalities and Applications, 2013(38) (2013), 15 pages.
• M. Ba¸sarır, E. E. Kara, On some difference sequence space of weighted means and compact operators, Annals of Functional Analysis, 2(2)(2011), 116-131.
• E. E. Kara, M. Ba¸sarır, On compact operators and some Euler B(m)− difference sequence spaces, Journal of Mathematical Analysis and Applications, 379(2011), 499-5
• M. Mursaleen, A.K. Noman, On the spaces of λ−convergent and bounded se- quences, Thai Journal of Mahematics, 8(2) (2010), 311-329.
• M. Mursaleen, Abdullah K. Noman, On some new sequence spaces of non-absolute type related to the spaces pand ∞I, Filomat 25:2 (2011), 33-51.
• A. S¨onmez, Some new sequence spaces derived by the domain of the triple band matrix, Comput. Math. Appl., 62 (2011) 641-650.
• K.-G.Grosse-Erdmann, Matrix transformations between the sequence spaves of Maddox, J. Math. Anal. Appl. 180(1993), 223-238.
• C. G. Lascarides, I.J. Maddox, Matrix transformations between some classes of sequences, Proc. Camb. Phil. Soc. 68(1970), 99-104.
• I.J. Maddox, Elements of Functional Analysis, second ed., The Cambridge Univer- sity Press, Cambridge, 1988.
• I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Camb. Phios. Soc. 64(1968), 335-340.
• I.J.Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford 18(2)(1967), 345-355.
• H. Nakano, Modulared sequence spaces, Proc. Jpn. Acad. 27(2)(1951), 508-512.
• S. Simons, The sequence spaces (pv) and m(pv), Proc. London Math. Soc. 15(3)(1965), 422-436.
• F. Ba¸sar, Summability Theory and Its Appliactions, Bentham Science Publishers, ISBN:978-1-60805-252-3, 2011.
• A. Wilansky, Summability through Functional Analysis, North-Holland Mathe- matics Studies , Amsterdam, 85 1984.