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In this paper a method for studying stability of the equation $x^{\prime \prime }(t)+\sum_{i=1}^{m}a_{i}(t)x(t- \tau_{i}(t))=0$ not including explicitly the first derivative is proposed. We demonstrate that although the corresponding ordinary differential equation $x^{\prime \prime}(t)+\sum_{i=1}^{m}a_{i}(t)x(t)=0$ is not exponentially stable, the delay equation can be exponentially stable. [ABSTRACT FROM AUTHOR]

Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation x''(t) + c(t)x(t) = 0 can be oscillating and asymptotically unstable, the delay equation x''(t) + a(t)x(t - h(t)) - b(t)x(t - g(t)) = 0, where c(t) = a(t) - b(t), can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper refute this delusion. [ABSTRACT FROM AUTHOR]

We obtain the maximum principles for the first-order neutral functional differential equation (Mx)(t) = x'(t) - (Sx')(t) - (Ax)(t) + (Bx)(t) - f(t), t ∈ [0, ω], where A : C[0,ω] → L∞ [0,ω], B : C[0,ω] → L∞[0,ω], and S : L∞[0,ω] → L∞[0,ω] are linear continuous operators, A and B are positive operators, C[0,ω] is the space of continuous functions, and L∞[0,ω] is the space of essentially bounded functions defined on [0,ω]. New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles. [ABSTRACT FROM AUTHOR]

The article presents a study on the mathematical equation of two-dimensional linear differential systems with deviating arguments using Kneser type solutions. It presents an equation with positive coefficient and two linearly independent solutions. The article also evaluates the existence and uniqueness of these types of solutions. It proves that Kneser type solution is equivalent to nonvanishing of the Wronskian fundamental system and positivity of Green's function of the problem.

There are almost no results in mathematical literature on the exponential stability of third-order delay differential equations. One of the main purposes of the paper is to fill this gap. We propose an approach to the study of stability for third-order delay differential equations.On the basis of these results, new possibilities of stabilization by delay feedback input control are proposed. [ABSTRACT FROM AUTHOR]