Three-point singular boundary-value problem for a system of three differential equations

A singular Cauchy-Nicoletti problem for a system of three ordinary differential equations is considered. An approach which combines topological method of T. Ważewski and Schauder\u27s principle is used. Theorem concerning the existence of a solution of this problem (a graph of which lies in a given domain) is proved. Moreover, an estimation of its coordinates is obtained

Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices

We represent a solution of an inhomogeneous second-order differential equation with two delays by using matrix functions under the assumption that the linear parts are given by permutable matrices.Зображення розв’язку задачi кошi для коливної системи з двома запiзнюваннями та переставними матрицям

Asymptotic Behavior of Solutions of Delayed Difference Equations

This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques—the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations

On Stability of Linear Delay Differential Equations under Perron's Condition

The stability of the zero solution of a system of first-order linear functional differential equations with nonconstant delay is considered. Sufficient conditions for stability, uniform stability, asymptotic stability, and uniform asymptotic stability are established

An Explicit Criterion for the Existence of Positive Solutions of the Linear Delayed Equation x

The paper investigates an equation with single delay ẋ(t)=-c(t)x(t-τ(t)), where τ:[t0-r,∞)→(0,r], r>0, t0∈R, and c:[t0-r,∞)→(0,∞) are continuous functions, and the difference t-τ(t) is an increasing function. Its purpose is to derive a new explicit integral criterion for the existence of a positive solution in terms of c and τ. An overview of known relevant criteria is provided, and relevant comparisons are also given

A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed Equation Δ

A linear (k+1)th-order discrete delayed equation Δx(n)=−p(n)x(n−k) where p(n) a positive sequence is considered for n→∞. This equation is known to have a positive solution if the sequence p(n) satisfies an inequality. Our aim is to show that, in the case of the opposite inequality for p(n), all solutions of the equation considered are oscillating for n→∞

Construction of the General Solution of Planar Linear Discrete Systems with Constant Coefficients and Weak Delay

Planar linear discrete systems with constant coefficients and weak delay are considered. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, the space of solutions with a given starting dimension is pasted after several steps into a space with dimension less than the starting one. In a sense this situation copies an analogous one known from the theory of linear differential systems with constant coefficients and weak delay when the initially infinite dimensional space of solutions on the initial interval on a reduced interval, turns (after several steps) into a finite dimensional set of solutions. For every possible case, general solutions are constructed and, finally, results on the dimensionality of the space of solutions are deduced