The linear homogeneous differential equation with variable delays $ẏ\left(t\right)={\sum }_{j=1}^n{\alpha }_j\left(t\right)[y\left(t\right)-y(t-{\tau }_j\left(t\right))]$ is considered, where ${\alpha }_j\in C(I,ℝ͞͞⁺)$, I = [t₀,∞), ℝ⁺ = (0,∞), ${\sum }_{j=1}^n{\alpha }_j\left(t\right)>0$ on I, ${\tau }_j\in C(I,ℝ⁺),$ the functions $t-{\tau }_j\left(t\right)$, j=1,...,n, are increasing and the delays ${\tau }_j$ are bounded. A criterion and some sufficient conditions for convergence of all solutions of this equation are proved. The related problem of nonconvergence is also discussed. Some comparisons to known results are given.

We establish some results that concern the Cauchy-Peano problem in Banach spaces. We first prove that a Banach space contains a nontrivial separable quotient iff its dual admits a weak*-transfinite Schauder frame. We then use this to recover some previous results on quotient spaces. In particular, by applying a recent result of Hájek-Johanis, we find a new perspective for proving the failure of the weak form of Peano's theorem in general Banach spaces. Next, we study a kind of algebraic genericity...

The main objective of the present paper is to study the approximate solutions for integrodifferential equations of the neutral type with given initial condition. A variant of a certain fundamental integral inequality with explicit estimate is used to establish the results. The discrete analogues of the main results are also given.

In the present paper we study the approximate solutions of a certain difference-differential equation under the given initial conditions. The well known Gronwall-Bellman integral inequality is used to establish the results. Applications to a Volterra type difference-integral equation are also given.