Recent discovery of cancer stem cells in tumorigenic tissues has raised many questions about their nature, origin, function and their behavior in cell culture. Most of current experiments reporting a dynamics of cancer stem cell populations in culture show the eventual stability of the percentages of these cell populations in the whole population of cancer cells, independently of the starting conditions. In this paper we propose a mathematical model...

The nonimprovable sufficient conditions for the unique solvability of the problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right),u\left(a\right)=c,$$ where $\ell C(I;ℝ)\to L(I;ℝ)$ is a linear bounded operator, $q\in L(I;ℝ)$, $c\in ℝ$, are established which are different from the previous results. More precisely, they are interesting especially in the case where the operator $\ell$ is not of Volterra’s type with respect to the point $a$.

We study existence, uniqueness and form of solutions to the equation $\alpha g-\beta g^{\text{'}}+\gamma g_e=f$ where α, β, γ and f are given, and $g_e$ stands for the even part of a searched-for differentiable function g. This equation emerged naturally as a result of the analysis of the distribution of a certain random process modelling a population genetics phenomenon.

This paper deals with the stability problem for a class of linear neutral delay-differential systems. The time delay is assumed constant and known. Delay-dependent criteria are derived. The criteria are given in the form of linear matrix inequalities which are easy to use when checking the stability of the systems considered. Numerical examples indicate significant improvements over some existing results.

We study conditions of discreteness of spectrum of the functional-differential operator $$ℒu=-u^{\text{'}\text{'}}+p\left(x\right)u\left(x\right)+{\int }_{-\infty }^{\infty }(u\left(x\right)-u\left(s\right)){\mathrm{d}}_{s}r(x,s)$$ on $(-\infty ,\infty )$. In the absence of the integral term this operator is a one-dimensional Schrödinger operator. In this paper we consider a symmetric operator with real spectrum. Conditions of discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain one simple condition for discreteness of spectrum.

Consider boundary value problems for a functional differential equation $$\left\{\begin{array}{cc}{x}^{\left(n\right)}\left(t\right)=\left(T^{+}x\right)\left(t\right)-\left(T^{-}x\right)\left(t\right)+f\left(t\right),\hfill & t\in [a,b],\hfill \\ lx=c,\hfill \end{array}\right.$$ where $T^{+},T^{-}:𝐂[a,b]\to 𝐋[a,b]$ are positive linear operators; $l:{\mathrm{𝐀𝐂}}^{n-1}[a,b]\to {ℝ}^n$ is a linear bounded vector-functional, $f\in 𝐋[a,b]$, $c\in {ℝ}^n$, $n\ge 2$. Let the solvability set be the set of all points $({𝒯}^{+},{𝒯}^{-})\in {ℝ}_2^{+}$ such that for all operators $T^{+}$, $T^{-}$ with $\parallel T^{\pm }{\parallel }_{𝐂\to 𝐋}={𝒯}^{\pm }$ the problems have a unique solution for every $f$ and $c$. A method of finding the solvability sets are proposed. Some new properties of these sets are obtained in various cases. We continue the investigations of the solvability sets started in R. Hakl,...

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $$\dot{x}\left(t\right)-a\left(t\right)\dot{x}\left(g\left(t\right)\right)+b\left(t\right)x\left(h\left(t\right)\right)=0,$$ where $$\left|a\right(t\left)\right|<1,b\left(t\right)\ge 0,h\left(t\right)\le t,g\left(t\right)\le t,$$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.

The paper discusses the asymptotic properties of solutions of the scalar functional differential equation $y^{\text{'}}\left(x\right)=ay\left(\tau \left(x\right)\right)+by\left(x\right),x\in [x_0,\infty ]$. Asymptotic formulas are given in terms of solutions of the appropriate scalar functional nondifferential equation.

Consider the homogeneous equation $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)\text{for}\text{a.e.}t\in [a,b]$$ where $\ell :C\left(\right[a,b];ℝ)\to L\left(\right[a,b];ℝ)$ is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.

The unstable properties of the linear nonautonomous delay system $x^{\text{'}}\left(t\right)=A\left(t\right)x\left(t\right)+B\left(t\right)x(t-r\left(t\right))$, with nonconstant delay $r\left(t\right)$, are studied. It is assumed that the linear system $y^{\text{'}}\left(t\right)=(A\left(t\right)+B\left(t\right))y\left(t\right)$ is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay $r\left(t\right)$ is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function $r\left(t\right)$ and the results depending on the asymptotic properties of the...