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

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→∞

Oscillation of Solutions of a Linear Second-Order Discrete-Delayed Equation

A linear second-order discrete-delayed equation Δx n −p n x n − 1 with a positive coefficient p is considered for n → ∞. This equation is known to have a positive solution if p fulfils an inequality. The goal of the paper is to show that, in the case of the opposite inequality for p, all solutions of the equation considered are oscillating for n → ∞