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...

We consider the numerical solvability of the general linear boundary value problem for the systems of linear ordinary differential equations. Along with the continuous boundary value problem we consider the sequence of the general discrete boundary value problems, i.e. the corresponding general difference schemes. We establish the effective necessary and sufficient (and effective sufficient) conditions for the convergence of the schemes. Moreover, we consider the stability of the solutions of general...

The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $\mathrm{d}x\left(t\right)=\mathrm{d}{A}_0\left(t\right)·x\left(t\right)+\mathrm{d}{f}_0\left(t\right)$, $x\left(t_0\right)=c_0$$(t\in I)$ with a unique solution $x_0$ is considered....

Nonimprovable, in a sense sufficient conditions guaranteeing 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, $q\in L(I,ℝ)$, and $c\in ℝ$, are established.

This paper concerns with the existence of the solutions of a second order impulsive delay differential equation with a piecewise constant argument. Moreover, oscillation, nonoscillation and periodicity of the solutions are investigated.

Our purpose is to analyze a first order nonlinear differential equation with advanced arguments. Then, some sufficient conditions for the oscillatory solutions of this equation are presented. Our results essentially improve two conditions in the paper “Oscillation tests for nonlinear differential equations with several nonmonotone advanced arguments” by N. Kilıç, Ö. Öcalan and U. M. Özkan. Also we give an example to illustrate our results.

2000 Mathematics Subject Classification: 34K15.This paper is concerned with the oscillatory behavior of first-order delay differential equation of the form x'(t) + p(t)x (τ(t)) = 0.

Sufficient oscillation conditions involving $lim\; sup$ and $lim\; inf$ for first-order differential equations with non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the obtained conditions over known ones.

Consider the first-order linear delay (advanced) differential equation$$x^{\text{'}}\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)=0(x^{\text{'}}\left(t\right)-q\left(t\right)x\left(\sigma \left(t\right)\right)=0),t\ge t_0,$$ where $p$$\left(q\right)$ is a continuous function of nonnegative real numbers and the argument $\tau \left(t\right)$$...$

The aim of this paper is to derive sufficient conditions for the linear delay differential equation (r(t)y′(t))′ + p(t)y(τ(t)) = 0 to be oscillatory by using a generalization of the Lagrange mean-value theorem, the Riccati differential inequality and the Sturm comparison theorem.

This paper is concerned with a class of even order nonlinear differential equations of the form $$\frac{d}{dt}\left(|{\left(x\left(t\right)+p\left(t\right)x\left(\tau \right(t\left)\right)\right)}^{(n-1)}{|}^{\alpha -1}{(x\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right))}^{(n-1)}\right)+F(t,x\left(g\left(t\right)\right))=0,$$ where $n$ is even and $t\ge t_0$. By using the generalized Riccati transformation and the averaging technique, new oscillation criteria are obtained which are either extensions of or complementary to a number of existing results. Our results are more general and sharper than some previous results even for second order equations.