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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. MSC: 34K20

The coefficients and delays in models describing various processes are usually obtained as a results of measurements and can be obtained only approximately. We deal with the question of how to estimate the influence of 'mistakes' in coefficients and delays on solutions' behavior of the delay differential neutral systemThis topic is known in the literature as uncertain systems or systems with interval defined coefficients. The goal of this paper is to obtain stability of uncertain systems and to estimate the difference between solutions of a 'real' system with uncertain coefficients and/or delays and corresponding 'model' system. We develop the so-called Azbelev W-transform, which is a sort of the right regularization allowing researchers to reduce analysis of boundary value problems to study of systems of functional equations in the space of measurable essentially bounded functions. In corresponding cases estimates of norms of auxiliary linear operators (obtained as a result of W-transform) lead researchers to conclusions about existence, uniqueness, positivity and stability of solutions of given boundary value problems. This method works efficiently in the case when a 'model' used in W-transform is 'close' to a given 'real' system. In this paper we choose, as the 'models' , systems for which we know estimates of the resolvent Cauchy operators. We demonstrate that systems with positive Cauchy matrices present a class of convenient 'models' . We use the W-transform and other methods of the general theory of functional differential equations. Positivity of the Cauchy operators is studied and then used in the analysis of stability and estimates of solutions.Results: We propose results about exponential stability of the given system and obtain estimates of difference between the solution of this uncertain system and theNew tests of stability and in the future of existence and uniqueness of boundary value problems for neutral delay systems can be obtained on the basis of this technique.

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