On the Darboux problem involving the distributional Henstock\u2013Kurzweil integral

Guoju Ye,Hui Mei,Rong Cheng,Wei Liu

doi:10.1186/s13661-018-1097-5

Guoju Ye, Hui Mei + Show 2 more

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https://doi.org/10.1186/s13661-018-1097-5

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Abstract

In this paper, we apply the method associated with the technique of measure of noncompactness and the Darbo fixed point theorem to study the existence of solutions of the Darboux problem involving the distributional Henstock–Kurzweil integral. Meanwhile, an example is provided to illustrate our results.

Highlights

IntroductionIn [2], the authors used a fixed point theorem and some properties of measure of weak noncompactness to prove the existence of pseudo-solutions for the Darboux problem in a Banach space E:
1 Introduction The Darboux problems have been studied by many authors
In [2], the authors used a fixed point theorem and some properties of measure of weak noncompactness to prove the existence of pseudo-solutions for the Darboux problem in a Banach space E:

Summary

Introduction

In [2], the authors used a fixed point theorem and some properties of measure of weak noncompactness to prove the existence of pseudo-solutions for the Darboux problem in a Banach space E:. If there exists a continuous function F ∈ C0(Q ) on Q such that f is the distributional derivative of F (the definition of C0(Q ) will be introduced in Sect. In order to prove the existence of solutions of the Darboux problem involving the distributional Henstock–Kurzweil integral, the method associated with the technique of measure of noncompactness and the Darbo fixed point theorem will be used. Lemma 2.10 ([10, Theorem 2, Darbo]) Let Ω be a nonempty, bounded, closed and convex subset of a Banach space E and let T : Ω → Ω be a continuous mapping. According to Theorem 3.2 and (10) the definition of the operator F, we have: Theorem 3.3 Under the assumptions (D1)–(D5), Eq (2) has at least one solution in the space C( )

Application

Conclusions