### A boundary value problem for non-linear differential equations with a retarded argument

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In this paper property (A) of the linear delay differential equation $${L}_{n}u\left(t\right)+p\left(t\right)u\left(\tau \left(t\right)\right)=0,$$ is to deduce from the oscillation of a set of the first order delay differential equations.

The linear homogeneous differential equation with variable delays $\u1e8f\left(t\right)={\sum}_{j=1}^{n}{\alpha}_{j}\left(t\right)[y\left(t\right)-y(t-{\tau}_{j}\left(t\right))]$ is considered, where ${\alpha}_{j}\in C(I,\mathbb{R}\u035e\u035e\u207a)$, I = [t₀,∞), ℝ⁺ = (0,∞), ${\sum}_{j=1}^{n}{\alpha}_{j}\left(t\right)>0$ on I, ${\tau}_{j}\in C(I,\mathbb{R}\u207a),$ 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 give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded ${H}^{\infty}$-calculus and is based on elementary analysis.