Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T06:01:37.525Z Has data issue: false hasContentIssue false
  • English
  • Français

ON PERIODIC SOLUTIONS FOR FIRST-ORDER DIFFERENTIAL EQUATIONS INVOLVING THE DISTRIBUTIONAL HENSTOCK–KURZWEIL INTEGRAL

Part of: Measures, integration, derivative, holomorphy Real functions Difference and functional equations Distributions, generalized functions, distribution spaces Difference equations

Published online by Cambridge University Press:  06 February 2012

WEI LIU ,
GUOJU YE ,
YING WANG  and
XUEYUAN ZHOU
WEI LIU*
Affiliation:
Department of Mathematics, Hohai University, Nanjing 210098, PR China (email: liuw626@hhu.edu.cn)
GUOJU YE
Affiliation:
Department of Mathematics, Hohai University, Nanjing 210098, PR China (email: yegj@hhu.edu.cn)
YING WANG
Affiliation:
Department of Mathematics, Hohai University, Nanjing 210098, PR China
XUEYUAN ZHOU
Affiliation:
Department of Mathematics, Hohai University, Nanjing 210098, PR China
*
✉For correspondence; e-mail: liuw626@hhu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to study the existence of periodic solutions and the topological structure of the solution set of first-order differential equations involving the distributional Henstock–Kurzweil integral. The distributional Henstock–Kurzweil integral is a general integral, which includes the Lebesgue and Henstock–Kurzweil integrals. The main results extend some previously known results in the literature.

Keywords

periodic boundary value problemdistributional Henstock–Kurzweil integraldistributional derivativeextremal solutionsfixed point

MSC classification

Secondary: 26A39: Denjoy and Perron integrals, other special integrals 39A23: Periodic solutions 46F05: Topological linear spaces of test functions, distributions and ultradistributions 46G12: Measures and integration on abstract linear spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 2 , October 2012 , pp. 327 - 338
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

Supported by NNSF of China (10871059) and the Fundamental Research Funds for the Central Universities.

References

[1]Ang, D. D., Schmitt, K. and Vy, L. K., ‘A multidimensional analogue of the Denjoy–Perron–Henstock–Kurzweil integral’, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), 355–371.CrossRefGoogle Scholar
[2]Chew, T. S. and Lee, P. Y., ‘The topology of the space of Denjoy integrable functions’, Bull. Aust. Math. Soc. 42 (1990), 517–524.Google Scholar
[3]Hardy, G. H., ‘Weierstrass’s non-differentiable function’, Trans. Amer. Math. Soc. 17(3) (1916), 301–325.Google Scholar
[4]Lakshmikantham, V. and Leela, S., ‘Existence and monotone method for periodic solutions of first-order differential equations’, J. Math. Anal. Appl. 91 (1983), 237–243.CrossRefGoogle Scholar
[5]Lakshmikantham, V. and Leela, S., ‘Remarks on first and second order periodic boundary value problems’, Nonlinear Anal. 8 (1984), 281–287.CrossRefGoogle Scholar
[6]Lee, P. Y., Lanzhou Lecture on Henstock Integration (World Scientific, Singapore, 1989).Google Scholar
[7]Lee, T. Y., ‘A full descriptive definition of the Henstock–Kurzweil integral in the Euclidean space’, Proc. London Math. Soc. 87 (2003), 677–700.Google Scholar
[8]Liu, Q. L. and Ye, G. J., ‘Some problems on the convergence of the distributional Denjoy integral’, Acta Math. Sin. (Chin. Ser.) 54(4) (2011), 659–664 (in Chinese).Google Scholar
[9]Lu, Y. P., Ye, G. J., Wang, Y. and Liu, W., ‘The Darboux problem involving the distributional Henstock–Kurzweil integral’, Proc. Edinb. Math. Soc. (2), in press.Google Scholar
[10]Nieto, J. J., ‘Existence of periodic solutions for first-order differential equations’, Appl. Math. Comput. 15 (1984), 221–232.Google Scholar
[11]Nieto, J. J., ‘On the structure of the solution set for first-order differential equations’, Appl. Math. Comput. 16 (1985), 177–187.Google Scholar
[12]Rudin, W., Real and Complex Analysis, 3rd edn (McGraw-Hill, New York, 1987).Google Scholar
[13]Schechter, M., An Introduction to Nonlinear Analysis (Cambridge University Press, Cambridge, 2004).Google Scholar
[14]Schwabik, Š. and Ye, G. J., Topics in Banach Space Integration (World Scientific, Singapore, 2005).CrossRefGoogle Scholar
[15]Talvila, E., ‘The distributional Denjoy integral’, Real Anal. Exchange 33 (2008), 51–82.CrossRefGoogle Scholar
[16]Tisdell, C. C., ‘Existence of solutions to first-order periodic boundary value problems’, J. Math. Anal. Appl. 323 (2006), 1325–1332.CrossRefGoogle Scholar
[17]Vidossich, G., ‘A fixed-point theorem for funtion spaces’, J. Math. Anal. Appl. 36 (1971), 581–587.CrossRefGoogle Scholar