### Correction and addition to my paper “The normal form and the stability of solutions of a system of differential equations in the complex domain”

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Boundary value problems for generalized linear differential equations and the corresponding controllability problems are dealt with. The adjoint problems are introduced in such a way that the usual duality theorems are valid. As a special case the interface boundary value problems are included. In contrast to the earlier papers by the author the right-hand side of the generalized differential equations as well as the solutions of this equation can be in general regulated functions (not necessarily...

The paper deals with the linear differential equation (0.1) ${\left(p{u}^{\text{'}}\right)}^{\text{'}}+{q}^{\text{'}}u={f}^{\text{'}\text{'}}$ with distributional coefficients and solutions from the space of regulated functions. Our aim is to get the basic existence and uniqueness results for the equation (0.1) and to generalize the known results due to F. V. Atkinson [At], J. Ligeza [Li1]-[Li3], R. Pfaff ([Pf1], [Pf2]), A. B. Mingarelli [Mi] as well as the results from the paper [Pe-Tv] concerning the equation (0.1).

n this paper we investigate systems of linear integral equations in the space ${\mathbb{G}}_{L}^{n}$ of $n$-vector valued functions which are regulated on the closed interval $[0,1]$ (i.e. such that can have only discontinuities of the first kind in $[0,1]$) and left-continuous in the corresponding open interval $(0,1).$ In particular, we are interested in systems of the form x(t) - A(t)x(0) - 01B(t,s)[d x(s)] = f(t), where $f\in {\mathbb{G}}_{L}^{n}$, the columns of the $n\times n$-matrix valued function $A$ belong to ${\mathbb{G}}_{L}^{n}$, the entries of $B(t,.)$ have a bounded variation on $[0,1]$ for any...

In the paper existence and uniqueness results for the linear differential system on the interval [0,1] ${A}_{1}{\left({A}_{0}x\right)}^{\text{'}}-{A}_{2}^{\text{'}}x={f}^{\text{'}}$ with distributional coefficients and solutions from the space of regulated functions are obtained.

From the fact that the unique solution of a homogeneous linear algebraic system is the trivial one we can obtain the existence of a solution of the nonhomogeneous system. Coefficients of the systems considered are elements of the Colombeau algebra $\overline{\mathbb{R}}$ of generalized real numbers. It is worth mentioning that the algebra $\overline{\mathbb{R}}$ is not a field.

In this paper we present conditions ensuring the existence and localization of lower and upper functions of the periodic boundary value problem ${u}^{\text{'}\text{'}}+k\phantom{\rule{0.166667em}{0ex}}u=f(t,u)$, $u\left(0\right)=u\left(2\phantom{\rule{0.166667em}{0ex}}\pi \right)$, ${u}^{\text{'}}\left(0\right)={u}^{\text{'}}\left(2\pi \right)$, $k\in \mathbb{R}\phantom{\rule{0.56905pt}{0ex}}$, $k\ne 0.$ These functions are constructed as solutions of some related generalized linear problems and can be nonsmooth in general.

We study the singular periodic boundary value problem of the form $${\left(\phi \left({u}^{\text{'}}\right)\right)}^{\text{'}}+h\left(u\right){u}^{\text{'}}=g\left(u\right)+e\left(t\right),\phantom{\rule{1.0em}{0ex}}u\left(0\right)=u\left(T\right),\phantom{\rule{1.0em}{0ex}}{u}^{\text{'}}\left(0\right)={u}^{\text{'}}\left(T\right),$$ where $\phi \phantom{\rule{0.222222em}{0ex}}\mathbb{R}\to \mathbb{R}$ is an increasing and odd homeomorphism such that $\phi \left(\mathbb{R}\right)=\mathbb{R},$ $h\in C[0,\infty ),$ $e\in {L}_{1}J$ and $g\in C(0,\infty )$ can have a space singularity at $x=0,$ i.e. ${lim\; sup}_{x\to 0+}\left|g\left(x\right)\right|=\infty $ may hold. We prove new existence results both for the case of an attractive singularity, when ${lim\; inf}_{x\to 0+}g\left(x\right)=-\infty ,$ and for the case of a strong repulsive singularity, when ${lim}_{x\to 0+}{\int}_{x}^{1}g\left(\xi \right)\phantom{\rule{0.56905pt}{0ex}}\text{d}\xi =\infty .$ In the latter case we assume that $\phi \left(y\right)={\phi}_{p}\left(y\right)={\left|y\right|}^{p-2}y,$ $p>1,$ is the well-known $p$-Laplacian. Our results extend and complete those obtained recently...

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