Journal Menu

- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 863483, 20 pages

http://dx.doi.org/10.1155/2012/863483

Review Article

## Infinite System of Differential Equations in Some Spaces

^{1}Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India^{2}Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 26 July 2012; Accepted 1 October 2012

Academic Editor: Beata Rzepka

Copyright © 2012 M. Mursaleen and Abdullah Alotaibi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- F. Basar,
*Summability Theory and Its Applications*, Bentham Science Publishers, e-books, Monographs, Istanbul, Turkey, 2011. - M. Mursaleen,
*Elements of Metric Spaces*, Anamaya Publishers, New Delhi, India, 2005. - A. Wilansky,
*Summability through Functional Analysis*, vol. 85 of*North-Holland Mathematics Studies*, North-Holland Publishing, Amsterdam, The Netherlands, 1984. - W. L. C. Sargent, “On compact matrix transformations between sectionally bounded $\mathrm{BK}$-spaces,”
*Journal of the London Mathematical Society*, vol. 41, pp. 79–87, 1966. View at Publisher · View at Google Scholar - E. Malkowsky and Mursaleen, “Matrix transformations between FK-spaces and the sequence spaces $m(\phi )$ and $n(\phi )$,”
*Journal of Mathematical Analysis and Applications*, vol. 196, no. 2, pp. 659–665, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. Malkowsky and M. Mursaleen, “Compact matrix operators between the spaces $m(\varphi )$ and $n(\varphi )$,”
*Bulletin of the Korean Mathematical Society*, vol. 48, no. 5, pp. 1093–1103, 2011. View at Publisher · View at Google Scholar - M. Mursaleen, R. Çolak, and M. Et, “Some geometric inequalities in a new Banach sequence space,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 86757, 6 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Mursaleen, “Some geometric properties of a sequence space related to ${l}^{p}$,”
*Bulletin of the Australian Mathematical Society*, vol. 67, no. 2, pp. 343–347, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Kuratowski, “Sur les espaces complets,”
*Fundamenta Mathematicae*, vol. 15, pp. 301–309, 1930. - L. S. Golden stein, I. T. Gohberg, and A. S. Markus, “Investigations of some properties of bounded linear operators with their
*q*-norms,”*Ucenie Zapiski, Kishinevskii*, vol. 29, pp. 29–36, 1957. - L. S. Goldenštein and A. S. Markus, “On the measure of non-compactness of bounded sets and of linear operators,” in
*Studies in Algebra and Mathematical Analysis*, pp. 45–54, Kishinev, 1965. - R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiĭ,
*Measures of Noncompactness and Condensing Operators*, vol. 55 of*Operator Theory: Advances and Applications*, Birkhäuser, Basel, Switzerland, 1992. View at Zentralblatt MATH - J. Banaś and K. Goebl,
*Measures of Noncompactness in Banach Spaces*, vol. 60 of*Lecture Notes in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1980. - E. Malkowsky and V. Rakočević, “An introduction into the theory of sequence spaces and measures of noncompactness,”
*Zbornik Radova*, vol. 9(17), pp. 143–234, 2000. View at Zentralblatt MATH - F. Başar and E. Malkowsky, “The characterization of compact operators on spaces of strongly summable and bounded sequences,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5199–5207, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Başarir and E. E. Kara, “On compact operators on the Riesz ${B}^{m}$—difference sequence spaces,”
*Iranian Journal of Science & Technology A*, vol. 35, no. 4, pp. 279–285, 2011. - M. Başarir and E. E. Kara, “On some difference sequence spaces of weighted means and compact operators,”
*Annals of Functional Analysis*, vol. 2, no. 2, pp. 114–129, 2011. - I. Djolović and E. Malkowsky, “The Hausdorff measure of noncompactness of operators on the matrix domains of triangles in the spaces of strongly ${C}_{1}$ summable and bounded sequences,”
*Applied Mathematics and Computation*, vol. 216, no. 4, pp. 1122–1130, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Djolović and E. Malkowsky, “Characterizations of compact operators on some Euler spaces of difference sequences of order $m$,”
*Acta Mathematica Scientia B*, vol. 31, no. 4, pp. 1465–1474, 2011. View at Publisher · View at Google Scholar - E. E. Kara and M. Başarir, “On compact operators and some Euler ${B}^{(m)}$-difference sequence spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 379, no. 2, pp. 499–511, 2011. View at Publisher · View at Google Scholar - B. de Malafosse, E. Malkowsky, and V. Rakočević, “Measure of noncompactness of operators and matrices on the spaces $c$ and ${c}_{0}$,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2006, Article ID 46930, 5 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. de Malafosse and V. Rakočević, “Applications of measure of noncompactness in operators on the spaces ${s}_{\alpha},{s}_{\alpha}^{0},{s}_{\alpha}^{(c)},{l}_{\alpha}^{p}$,”
*Journal of Mathematical Analysis and Applications*, vol. 323, no. 1, pp. 131–145, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. Malkowsky and I. Djolović, “Compact operators into the spaces of strongly ${C}_{1}$ summable and bounded sequences,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 11, pp. 3736–3750, 2011. View at Publisher · View at Google Scholar - M. Mursaleen, V. Karakaya, H. Polat, and N. Simşek, “Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means,”
*Computers & Mathematics with Applications*, vol. 62, no. 2, pp. 814–820, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Mursaleen and A. Latif, “Applications of measure of noncompactness in matrix operators on some sequence spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 378250, 10 pages, 2012. View at Zentralblatt MATH - M. Mursaleen and S. A. Mohiuddine, “Applications of measures of noncompactness to the infinite system of differential equations in ${\ell}_{p}$ spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 4, pp. 2111–2115, 2012. View at Publisher · View at Google Scholar - M. Mursaleen and A. K. Noman, “Compactness by the Hausdorff measure of noncompactness,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 73, no. 8, pp. 2541–2557, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Mursaleen and A. K. Noman, “Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means,”
*Computers & Mathematics with Applications*, vol. 60, no. 5, pp. 1245–1258, 2010. - M. Mursaleen and A. K. Noman, “The Hausdorff measure of noncompactness of matrix operators on some $BK$ spaces,”
*Operators and Matrices*, vol. 5, no. 3, pp. 473–486, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Mursaleen and A. K. Noman, “On $\sigma $-conservative matrices and compact operators on the space ${V}_{\sigma}$,”
*Applied Mathematics Letters*, vol. 24, no. 9, pp. 1554–1560, 2011. View at Publisher · View at Google Scholar - M. Mursaleen and A. K. Noman, “Compactness of matrix operators on some new difference sequence spaces,”
*Linear Algebra and Its Applications*, vol. 436, no. 1, pp. 41–52, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Mursaleen, “Application of measure of noncompactness to infinite systems of differential equations,”
*Canadian Mathematical Bulletin*. In press. View at Publisher · View at Google Scholar - J. Banaś and M. Lecko, “Solvability of infinite systems of differential equations in Banach sequence spaces,”
*Journal of Computational and Applied Mathematics*, vol. 137, no. 2, pp. 363–375, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Bellman,
*Methods of Nonliner Analysis II*, vol. 12, Academic Press, New York, NY, USA, 1973. - K. Deimling,
*Ordinary Differential Equations in Banach Spaces*, vol. 596 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1977. - E. Hille, “Pathology of infinite systems of linear first order differential equations with constant coefficients,”
*Annali di Matematica Pura ed Applicata*, vol. 55, pp. 133–148, 1961. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. N. Oğuztöreli, “On the neural equations of Cowan and Stein. I,”
*Utilitas Mathematica*, vol. 2, pp. 305–317, 1972. View at Zentralblatt MATH - O. A. Žautykov, “Denumerable systems of differential equations and their applications,”
*Differentsialcprime nye Uravneniya*, vol. 1, pp. 162–170, 1965. - K. P. Persidskii, “Countable system of differential equations and stability of their solutions,”
*Izvestiya Akademii Nauk Kazakhskoj SSR*, vol. 7, pp. 52–71, 1959. - K. P. Persidski, “Countable systems of differential equations and stability of their solutions III: Fundamental theorems on stability of solutions of countable many differential equations,”
*Izvestiya Akademii Nauk Kazakhskoj SSR*, vol. 9, pp. 11–34, 1961. - K. G. Zautykov and K. G. Valeev,
*Beskonechnye Sistemy Differentsialnykh Uravnenii*, Izdat, “Nauka” Kazah. SSR, Alma-Ata, 1974. - A. Voigt, “Line method approximations to the Cauchy problem for nonlinear parabolic differential equations,”
*Numerische Mathematik*, vol. 23, pp. 23–36, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Walter,
*Differential and Integral Inequalities*, Springer, New York, NY, USA, 1970. - M. Frigon, “Fixed point results for generalized contractions in gauge spaces and applications,”
*Proceedings of the American Mathematical Society*, vol. 128, no. 10, pp. 2957–2965, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Herzog, “On ordinary linear differential equations in ${C}^{J}$,”
*Demonstratio Mathematica*, vol. 28, no. 2, pp. 383–398, 1995. View at Zentralblatt MATH - G. Herzog, “On Lipschitz conditions for ordinary differential equations in Fréchet spaces,”
*Czechoslovak Mathematical Journal*, vol. 48(123), no. 1, pp. 95–103, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Lemmert and A. Weckbach, “Charakterisierungen zeilenendlicher Matrizen mit abzählbarem Spektrum,”
*Mathematische Zeitschrift*, vol. 188, no. 1, pp. 119–124, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Lemmert, “On ordinary differential equations in locally convex spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 10, no. 12, pp. 1385–1390, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Mlak and C. Olech, “Integration of infinite systems of differential inequalities,”
*Annales Polonici Mathematici*, vol. 13, pp. 105–112, 1963. View at Zentralblatt MATH - K. Moszyński and A. Pokrzywa, “Sur les systèmes infinis d'équations différentielles ordinaires dans certains espaces de Fréchet,”
*Dissertationes Mathematicae*, vol. 115, 29 pages, 1974. View at Zentralblatt MATH - H. Mönch and G.-F. von Harten, “On the Cauchy problem for ordinary differential equations in Banach spaces,”
*Archives Mathématiques*, vol. 39, no. 2, pp. 153–160, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. O'Regan and M. Meehan,
*Existence Theory for Nonlinear Integral and Integrodifferential Equations*, vol. 445 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998. View at Publisher · View at Google Scholar - S. Szufla, “On the existence of solutions of differential equations in Banach spaces,”
*L'Académie Polonaise des Sciences*, vol. 30, no. 11-12, pp. 507–515, 1982. View at Zentralblatt MATH