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Second Order Periodic Boundary Value Problems Involving the Distributional Henstock-Kurzweil Integral
Department of Mathematics, Hohai University, Nanjing, P. R. China
Department of Mathematics, Hohai University, Nanjing, P. R. China
- 1 Department of Mathematics, Hohai University, Nanjing, P. R. China
- 2 Department of Mathematics, Hohai University, Nanjing, P. R. China
Advances in Pure Mathematics·Volume 02 (2012)·Pages 330–336·Published 25 September 2012·DOI10.4236/apm.2012.25046
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Abstract
We apply the distributional derivative to study the existence of solutions of the second order periodic boundary value problems involving the distributional Henstock-Kurzweil integral. The distributional Henstock-Kurzweil integral is a general intergral, which contains the Lebesgue and Henstock-Kurzweil integrals. And the distributional derivative includes ordinary derivatives and approximate derivatives. By using the method of upper and lower solutions and a fixed point theorem, we achieve some results which are the generalizations of some previous results in the literatures.
KeywordsPeriodic Boundary Value ProblemDistributional Henstock-Kurzweil IntegralDistributional DerivativeExistenceUpper and Lower SolutionsFixed Point
- S. Leela, “Monotone Method for Second Order Periodic Boundry Value Problems,” Nonlinear Analysis, Vol. 7, No. 4, 1983, pp. 349-355. doi:10.1016/0362-546X(83)90088-3
- V. Lakshmikantham, V. Sree Hari Rao and A. S. Vatsala, “Monotone Method for a System of Second Order Periodic Boundary Value Problems,” Applied Mathematics Computation, Vol. 15, 1984, pp. 71-83. doi:10.1016/0096-3003(84)90054-7
- E. Talvila, “The Distributional Denjoy Integral,” Real Analysis, Exchang, Vol. 33, No. 1, 2008, pp. 51-82.
- P. Y. Lee, “Lanzhou Lecture on Henstock Integration,” World Scientific, Singapore City, 1989.
- S. Schwabik and G. J. Ye, “Topics in Banach Space Integration,” World Scientific, Singapore City, 2005.
- Y. P. Lu, G. J. Ye and Y. Wang, “The Darboux Problem Involving the Distributional Henstock-Kurzweil Integral,” Proceedings of the Edinburgh Mathematical Society, Vol. 55, No. 1, 2012, pp. 197-205.
- Y. P. Lu, G. J. Ye and W. Liu, “Existence of Solutions of the Wave Equation Involving the Distributional Henstock-Kurzweil Integral,” Differential Integral Equation, Vol. 24, No. 11-12, 2011, pp. 1063-1071.
- Q. L. Liu and G. J. Ye, “Some Problems on the Convergence of the Distributional Denjoy Integral,” Acta Mathematics Sinica, Vol. 54, No. 4, 2011, pp. 659-664.
- D. D. Ang, K. Schmitt and L. K. Vy, “A Multidimensional Analogue of the Denjoy-Perron-Henstock-Kurz- weil Integral,” Bulletin of the Belgian Mathematical Society, Vol. 4, No. 3, 1997, pp. 355-371.
- D. J. Guo, Y. J. Cho and J. Zhu, “Partial Ordering Methods in Nonlinear Problems,” Hauppauge, Nova Science Publishers, New York, 2004.