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Introduction to FK Spaces

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Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations

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Abstract

In this chapter, we study the concepts of linear metric spaces, paranormed spaces, FK spaces and BK spaces which play an important role in our studies on sequence spaces. One of the main advantage of FK space theory is that it provides easy and short proofs of numerous classical results of summability theory and is the most powerful and widely used tool in the characterization of matrix mappings between sequence spaces. Moreover, it enables us to obtain the most important result on the continuity of matrix mappings between FK spaces. The results of the theory of FK and BK spaces are also applied to characterize the matrix classes.

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References

  1. Wilansky, A.: Summability through Functional Analysis. North-Holland Mathematics Studies 85. Elsevier Science Publishers, Amesterdam (1984)

    Google Scholar 

  2. Zeller, K.: Allgemeine eigenschaften von limitierungsverfahren. Math. Z. 53, 463–487 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  3. Lascarides, C.G., Maddox, I.J.: Matrix transformations between some classes of sequences. Proc. Camb. Phil. Soc. 68, 99–104 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  4. Simons, S.: The sequence spaces \(\ell (p_{\nu })\) and \( m(p_{\nu })\). Proc. Lond. Math. Soc. 15(3), 422–436 (1965)

    Google Scholar 

  5. Mursaleen, M.: Elements of Metric Spaces. Anamaya Publishers, New Delhi (2005)

    Google Scholar 

  6. Maddox, I.J.: Elements of Functional Analysis, 2nd edn. The University Press, Cambridge (1988)

    Google Scholar 

  7. Malkowsky, E., Rakočević, V.: An introduction into the theory of sequence spaces and measures of noncompactness. Zbornik Radova Mat. Institut SANU (Beograd) 9(17), 143–234 (2000)

    Google Scholar 

  8. Choudhary, B., Nanda, S.: Functional Analysis with Applications. Wiley, New Delhi (1989)

    MATH  Google Scholar 

  9. Das, G., Kuttner, B., Nanda, S.: On absolute almost convergence. J. Math. Anal. Appl. 164, 381–398 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cooke, R.G.: Infinite Matrices and Sequence Spaces. Dover Publication Inc., New York (1955)

    Google Scholar 

  11. Başar, F.: Summability Theory and its Applications. Bentham Science Publishers, Istanbul (2011). (e-books, Monographs)

    Google Scholar 

  12. Boos, J.: Classical and Modern Methods in Summability. Oxford University Press, New York (2000)

    MATH  Google Scholar 

  13. Knopp, K.: Theory and Application of Infinite Series. Dover Publications, Inc., New York (1990)

    Google Scholar 

  14. Başar, F.: Infinite matrices and almost boundedness. Boll. Unione Mat. Ital. A 6(7), 395–402 (1992)

    Google Scholar 

  15. Mursaleen, M., Noman, A.K.: On the spaces of \(\lambda \) -convergent and bounded sequences. Thai J. Math. 8(2), 311–329 (2010)

    MATH  MathSciNet  Google Scholar 

  16. Mursaleen, M., Noman, A.K.: On generalized means and some related sequence spaces. Comput. Math. Appl. 61, 988–999 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Wang, C.S.: On Nörlund sequence spaces. Tamkang J. Math. 9, 269–274 (1978)

    MathSciNet  Google Scholar 

  18. Ng, P.N., Lee, P.Y.: Cesàro sequence spaces of non-absolute type. Comment. Math. Prace Mat. 20(2), 429–433 (1978)

    MATH  MathSciNet  Google Scholar 

  19. Malkowsky, E.: Recent results in the theory of matrix transformations in sequence spaces. Mat. Vesnik 49, 187–196 (1997)

    MATH  MathSciNet  Google Scholar 

  20. Başar, F., Altay, B.: On the space of sequences of \(p\) -bounded variation and related matrix mappings. Ukrainian Math. J. 55(1), 136–147 (2003)

    Article  MathSciNet  Google Scholar 

  21. Malkowsky, E., Savaş, E.: Matrix transformations between sequence spaces of generalized weighted means. Appl. Math. Comput. 147(2), 333–345 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Aydın, C., Başar, F.: On the new sequence spaces which include the spaces \(c_{0}\) and \(c\). Hokkaido Math. J. 33(2), 383–398 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Aydın, C., Başar, F.: Some new paranormed sequence spaces. Information Sci. 160(1–4), 27–40 (2004)

    Article  MATH  Google Scholar 

  24. Aydın, C., Başar, F.: Some new difference sequence spaces. Appl. Math. Comput. 157(3), 677–693 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Aydın, C., Başar, F.: Some new sequence spaces which include the spaces \(\ell _{p}\) and \(\ell _{\infty }\). Demonstratio Mathematica 38(3), 641–656 (2005)

    MATH  MathSciNet  Google Scholar 

  26. Altay, B., Başar, F.: Some Euler sequence spaces of non-absolute type. Ukrainian Math. J. 57(1), 1–17 (2005)

    Article  MathSciNet  Google Scholar 

  27. Altay, B., Başar, F., Mursaleen, M.: On the Euler sequence spaces which include the spaces \(\ell _{p}\) and \(\ell _{\infty }\). Inf. Sci. 176(10), 1450–1462 (2006)

    Article  MATH  Google Scholar 

  28. Mursaleen, M., Başar, F., Altay, B.: On the Euler sequence spaces which include the spaces \(\ell _{p}\) and \(\ell _{\infty }\) II. Nonlinear Anal. 65(3), 707–717 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. Mursaleen, M., Noman, A.K.: On some new difference sequence spaces of non-absolute type. Math. Comput. Modell. 52, 603–617 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Mursaleen, M., Noman, A.K.: On some new sequence spaces of non-absolute type related to the spaces \(\ell _{p}\) and \(\ell _{\infty }\) \(I\). Filomat 25(2), 33–51 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mursaleen, M., Noman, A.K.: On some new sequence spaces of non-absolute type related to the spaces \(\ell _{p}\) and \(\ell _{\infty }\) \(II\). Math. Commun. 16(2), 383–398 (2011)

    MATH  MathSciNet  Google Scholar 

  32. Başarır, M.: On some new sequence spaces and related matrix transformations. Indian J. Pure Appl. Math. 26(10), 1003–1010 (1995)

    MATH  MathSciNet  Google Scholar 

  33. Et, M.: On some generalized Cesàro difference sequence spaces. İstanbul Üniv. Fen Fak. Mat. Derg. 55–56, 221–229 (1996–1997)

    Google Scholar 

  34. Kızmaz, H.: On certain sequence spaces. Canad. Math. Bull. 24(2), 169–176 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  35. Et, M.: On some difference sequence spaces. Turkish J. Math. 17, 18–24 (1993)

    MATH  MathSciNet  Google Scholar 

  36. Mursaleen, M.: Generalized spaces of difference sequences. J. Math. Anal. Appl. 203(3), 738–745 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  37. Kirişçi, M., Başar, F.: Some new sequence spaces derived by the domain of generalized difference matrix. Comput. Math. Appl. 60(5), 1299–1309 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Şengönül, M., Başar, F.: Some new Cesàro sequence spaces of non-absolute type which include the spaces \(c_{0}\) and \(c\). Soochow J. Math. 31(1), 107–119 (2005)

    MATH  MathSciNet  Google Scholar 

  39. Kara, E.E., Başarır, M.: On compact operators and some Euler \(B^{(m)}\)-difference sequence spaces. J. Math. Anal. Appl. 379, 499–511 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  40. Rhoades, B.E.: Some sequence spaces which include \(c_{0}\) and \( c\). Hokkaido Math. J. 35, 587–599 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  41. Khan, F.M., Rahman, M.F.: Matrix transformations on Ces àro sequence spaces of nonabsolute type. J. Anal. 4, 97–101 (1996)

    MATH  MathSciNet  Google Scholar 

  42. Çolak, R., Et, M., Malkowsky, E.: Some Topics of Sequence Spaces. Lecture Notes in Mathematics, Fırat University Elâzıg, Turkey, 2004, pp. 1–63. Fırat University Press (2004), ISBN: 975-394-038-6

    Google Scholar 

  43. Et, M., Çolak, R.: On some generalized difference sequence spaces. Soochow J. Math. 21(4), 377–386 (1995)

    MATH  MathSciNet  Google Scholar 

  44. Çolak, R., Et, M.: On some generalized difference sequence spaces and related matrix transformations. Hokkaido Math. J. 26(3), 483–492 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  45. Altay, B.: On the space of p-summable difference sequences of order \(m\), \((1\le p<1)\). Stud. Sci. Math. Hungar. 43(4), 387–402 (2006)

    MATH  MathSciNet  Google Scholar 

  46. Malkowsky, E., Parashar, S.D.: Matrix transformations in space of bounded and convergent difference sequences of order \(m\). Analysis 17, 87–97 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  47. Polat, H., Başar, F.: Some Euler spaces of difference sequences of order \(m\). Acta Math. Sci. 27B(2), 254–266 (2007)

    Google Scholar 

  48. Başar, F., Altay, B.: Matrix mappings on the space \( bs(p)\) and its \(\alpha \), \(\beta \) and \(\gamma -\)duals. Aligarh Bull. Math. 21(1), 79–91 (2001)

    Google Scholar 

  49. Ahmad, Z.U., Mursaleen, M.: Köthe-Toeplitz duals of some new sequence spaces and their matrix maps. Publ. Inst. Math. (Beograd) 42, 57–61 (1987)

    MathSciNet  Google Scholar 

  50. Asma, Ç., Çolak, R.: On the Köthe-Toeplitz duals of some generalized sets of difference sequences. Demonstratio Math. 33, 797–803 (2000)

    MATH  MathSciNet  Google Scholar 

  51. Bektas, Ç.A.: On some new generalized sequence spaces. J. Math. Anal. Appl. 277, 681–688 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  52. Altay, B., Başar, F.: Some paranormed sequence spaces of non-absolute type derived by weighted mean. J. Math. Anal. Appl. 319(2), 494–508 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  53. Altay, B., Başar, F.: Generalization of the sequence space \(\ell (p)\) derived by weighted mean. J. Math. Anal. Appl. 330(1), 174–185 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  54. Başar, F., Altay, B., Mursaleen, M.: Some generalizations of the space \(bv_{p}\) of \(p-\)bounded variation sequences. Nonlinear Anal. 68(2), 273–287 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  55. Altay, B., Başar, F.: On the paranormed Riesz sequence spaces of non-absolute type. Southeast Asian Bull. Math. 26(5), 701–715 (2002)

    Google Scholar 

  56. Altay, B., Başar, F.: Some paranormed Riesz sequence spaces of non-absolute type. Southeast Asian Bull. Math. 30(5), 591–608 (2006)

    MATH  MathSciNet  Google Scholar 

  57. Başarır, M.: On the generalized Riesz \(B\)-difference sequence spaces. Filomat 24(4), 35–52 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  58. Başarır, M., Öztürk, M.: On the Riesz difference sequence space. Rend. Circ. Mat. Palermo 57(2), 377–389 (2008)

    Google Scholar 

  59. Başarır, M., Kayıkçı, M.: On the generalized \( B^{m}\)-Riesz sequence space and property. J. Inequal. Appl. Article ID 385029, 18 pp (2009)

    Google Scholar 

  60. Kara, E.E., Öztürk, M., Başarır, M.: Some topological and geometric properties of generalized Euler sequence spaces. Math. Slovaca 60(3), 385–398 (2010)

    Google Scholar 

  61. Et, M., Başarır, M.: On some new generalized difference sequence spaces. Period. Math. Hung. 35(3), 169–175 (1997)

    Google Scholar 

  62. Malkowsky, E., Mursaleen, M., Suantai, S.: The dual spaces of sets of difference sequences of order \(m\) and matrix transformations. Acta Math. Sin. Engl. Ser. 23(3), 521–532 (2007)

    Google Scholar 

  63. Mursaleen, M., Noman, A.K.: Applications of Hausdorff measure of noncompactness in the spaces of generalized means. Math. Ineq. Appl. 16, 207-220 (2013)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Rzeszów University of Technology, al. Powstańców Warszawy 8, 35-959, Rzeszow, Poland

    Józef Banaś

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, Uttar Pradesh, India

    Mohammad Mursaleen

Authors
  1. Józef Banaś
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  2. Mohammad Mursaleen
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Correspondence to Józef Banaś .

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Banaś, J., Mursaleen, M. (2014). Introduction to FK Spaces. In: Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. Springer, New Delhi. https://doi.org/10.1007/978-81-322-1886-9_1

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