Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions
Abstract
UDC 517.9
We introduced a class of generalized vector-valued paranormed sequence space $X[E,A,\Delta_v^m,M,p]$ by using a sequence of Orlicz functions $M=(M_k),$ a non-negative infinite matrix $A=[a_{nk}],$ generalized difference operator $\Delta_v^m$ and bounded sequence of positive real numbers $p_k$ with $\inf_k p_k>0.$ Properties related to this space are studied under certain conditions. Some inclusion relations are obtained and a result related to subspace with Orlicz functions satisfying $\Delta_2$-condition has also been proved.
References
Y. Altin, M. Et, B. C. Tripathy, The sequence space $|N_p|(M,r,q,s)$ on seminormed spaces, Appl. Math. and Comput., 154, 423 – 430 (2004), https://doi.org/10.1016/S0096-3003(03)00722-7 DOI: https://doi.org/10.1016/S0096-3003(03)00722-7
Ç. A. Bektaş, Y. Altin, The sequence space $l_M(p,q,s)$ on seminormed spaces, Indian J. Pure and Appl. Math., 34, № 4, 529 – 534 (2003).
Ç. A. Bektaş, On some new generalized difference sequence spaces on seminormed spaces defined by a sequence of Orlicz functions, Math. Slovaca, 61, № 2, 227 – 234 (2011), https://doi.org/10.2478/s12175-011-0007-4 DOI: https://doi.org/10.2478/s12175-011-0007-4
M. Et, R. Çolak, On some generalized difference sequence spaces, Soochow J. Math., 21, № 4, 377 – 386 (1995).
M. Et, A. Esi, On Kothe-Toeplitz duals of generalized difference sequence spaces, Bull. Malays. Math. Sci. Soc., 23, № 1, 25 – 32 (2000).
M. Et, L. P. Yee, B. C. Tripathy, Strongly almost $(V, λ)(Δ^r)$ - summable sequences defined by Orlicz functions, Hokkaido Math. J., 35, 197 – 213 (2006), https://doi.org/10.14492/hokmj/1285766306 DOI: https://doi.org/10.14492/hokmj/1285766306
A. Esi, Some new sequence spaces defined by Orlicz functions, Bull. Inst. Math. Acad. Sin., 27, № 1, 71 – 76 (1999).
P. K. Kamthan, M. Gupta, Sequence spaces and series, Marcel Dekker Inc., New York, Basel (1981).
H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24, № 2, 169 – 176 (1981), https://doi.org/10.4153/CMB-1981-027-5 DOI: https://doi.org/10.4153/CMB-1981-027-5
M. A. Krasnosel’skii, Y. B. Rutickii, Convex functions and Orlicz spaces, Noordhoff, Groningen (1961).
J. Lindenstrauss, L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10, 379 – 390 (1971), https://doi.org/10.1007/BF02771656 DOI: https://doi.org/10.1007/BF02771656
I. J. Maddox, Elements of functional analysis, Cambridge Univ. Press (1970).
I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math., 18, № 2, 345 – 355 (1967), https://doi.org/10.1093/qmath/18.1.345 DOI: https://doi.org/10.1093/qmath/18.1.345
I. J. Maddox, Paranormed sequence spaces generated by infinite matrices, Math. Proc. Cambridge Phil. Soc., 64, 335 – 340 (1968), https://doi.org/10.1017/s0305004100042894 DOI: https://doi.org/10.1017/S0305004100042894
M. Mursaleen, M. A. Khan, Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstr. Math., 32, № 1, 145 – 150 (1999). DOI: https://doi.org/10.1515/dema-1999-0115
S. D. Parashar, B. Choudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure and Appl. Math., 25, № 4, 419 – 428 (1994).
B. C. Tripathy, Y. Altin, M. Et, Generalized difference sequence spaces on seminormed space defined by Orlicz functions, Math. Slovaca, 58, № 3, 315 – 324 (2008), https://doi.org/10.2478/s12175-008-0077-0 DOI: https://doi.org/10.2478/s12175-008-0077-0
B. C. Tripathy, P. Chandra, On some generalized difference paranormed sequence spaces associated with multiplier sequence defined by modulus function, Anal. Theory and Appl., 27, № 1, 21 – 27 (2011), https://doi.org/10.1007/s10496-011-0021-y DOI: https://doi.org/10.1007/s10496-011-0021-y
B. C. Tripathy, H. Dutta, On some new paranormed difference sequence spaces defined by Orlicz functions, Kyungpook Math. J., 50, № 1, 59 – 69 (2010), https://doi.org/10.5666/KMJ.2010.50.1.059 DOI: https://doi.org/10.5666/KMJ.2010.50.1.059
B. C. Tripathy, S. Mahanta, On a class of vector-valued sequences associated with multiplier sequences, Acta Math. Appl. Sin. Engl. Ser., 20, № 3, 487 – 494 (2004), https://doi.org/10.5666/KMJ.2010.50.1.059 DOI: https://doi.org/10.1007/s10255-004-0186-7
B. C. Tripathy, S. Mahanta, On a class of difference sequences related to the $l^p$ space defined by Orlicz functions, Math. Slovaca, 57, № 2, 171 – 178 (2007), https://doi.org/10.2478/s12175-007-0007-6 DOI: https://doi.org/10.2478/s12175-007-0007-6
B. C. Tripathy, B. Sarma, Some classes of difference paranormed sequence spaces defined by Orlicz functions, Thai J. Math., 3, № 2, 209 – 218 (2005)
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