Second Order Periodic Boundary Value Problems Involving the Distributional Henstock-Kurzweil Integral

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.

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.

Cite this paper

X. Zhou and G. Ye, "Second Order Periodic Boundary Value Problems Involving the Distributional Henstock-Kurzweil Integral,"*Advances in Pure Mathematics*, Vol. 2 No. 5, 2012, pp. 330-336. doi: 10.4236/apm.2012.25046.

X. Zhou and G. Ye, "Second Order Periodic Boundary Value Problems Involving the Distributional Henstock-Kurzweil Integral,"

References

[1] 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

[2] 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

[3] E. Talvila, “The Distributional Denjoy Integral,” Real Analysis, Exchang, Vol. 33, No. 1, 2008, pp. 51-82.

[4] P. Y. Lee, “Lanzhou Lecture on Henstock Integration,” World Scientific, Singapore City, 1989.

[5] S. Schwabik and G. J. Ye, “Topics in Banach Space Integration,” World Scientific, Singapore City, 2005.

[6] 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.

[7] 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.

[8] 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.

[9] 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.

[10] D. J. Guo, Y. J. Cho and J. Zhu, “Partial Ordering Methods in Nonlinear Problems,” Hauppauge, Nova Science Publishers, New York, 2004.

[1] 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

[2] 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

[3] E. Talvila, “The Distributional Denjoy Integral,” Real Analysis, Exchang, Vol. 33, No. 1, 2008, pp. 51-82.

[4] P. Y. Lee, “Lanzhou Lecture on Henstock Integration,” World Scientific, Singapore City, 1989.

[5] S. Schwabik and G. J. Ye, “Topics in Banach Space Integration,” World Scientific, Singapore City, 2005.

[6] 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.

[7] 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.

[8] 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.

[9] 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.

[10] D. J. Guo, Y. J. Cho and J. Zhu, “Partial Ordering Methods in Nonlinear Problems,” Hauppauge, Nova Science Publishers, New York, 2004.