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.
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.
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.