On second order nonlinear boundary value problems and the distributional Henstock-Kurzweil integral

Abstract

In the present paper, we investigate the existence of solutions to second order nonlinear boundary value problems (BVPs) involving the distributional Henstock-Kurzweil integral. The present results in this article are generalizations of previous results in the literature.

Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations

“…Here, we use integrable distributions which are a special class of distributions that will be important for applications of distribution theory to partial differential equations, for details see [13,14,21]. ere are several articles on solving ordinary and partial differential equations using integrable distributions [16,17,19]. Integrable distribution has used to find a distributional solution for Poisson's equation with Dirichlet boundary condition for the upper half plane.…”

“…It seems the concept of integrable distribution is first introduced by Mikusiński and Ostaszewski [13] and further developed by several authors [14,15]. In recent years, integrable distributional solution for differential equations has been used by many authors [16][17][18][19]. We define derivative (in distributional sense) of A C and the integral on it.…”

The Distributional Henstock-Kurzweil Integral and Applications: A Survey

“…Remark 5.3. The Schauder's fixed point theorem and the Vidossich theorem can also be applied to study the existence of solutions and the structure of the set of solutions of the wave equation [27], the periodic boundary value problem [24], the nonlinear multi-point boundary value problem [23].…”

“…Let T be a continuous map of M into a compact subset K of M. Then T has a fixed point. Now, we show the applications of the D HK integral in integral and differential equations[20,[22][23][24][25][26][27].…”

The Application of Distributional Henstock-Kurzweil Integral on Third-order Three-Point Nonlinear Boundary-Value Problems

“…7, No. 4;2015 where r is the distributional derivative of the Weiertrass function R(t) = ∑ ∞ n=0 a n sin b n t,and 0 < a < 1 < b, ab > 1 in [G. H. Hardy. (1916)].…”