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.
Highlights
This paper is devoted to the study of the existence of solutions of the second order periodic boundary value problem (PBVP for brevity)D2 x = f t g t, x, Dx, (1.1)(1) converts into x = g t, x, x, (1.2)The existence of solutions of (1.2) have been extensively studied by many authors [1,2]
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
By using the method of upper and lower solutions and a fixed point theorem, we achieve some interesting results which are the generalizations of some corresponding results in the references
Summary
This paper is devoted to the study of the existence of solutions of the second order periodic boundary value problem (PBVP for brevity). The existence of solutions of (1.2) have been extensively studied by many authors [1,2]. It is well-known, the notion of a distributional derivative is a general concept, including ordinary derivatives and approximate derivatives. As far as we know, few papers have applied distributional derivatives to study PBVP. PBVP and obtain some results of the existence of solutions. By using the method of upper and lower solutions and a fixed point theorem, we achieve some interesting results which are the generalizations of some corresponding results in the references
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