The existence of real solutions of a nonlinear equation in a neighborhood of an abnormal (degenerate) point is studied. We prove that if the mapping describing this equation admits a lambda-truncation that is regular in some direction, then there exist solutions for all right-hand sides in some neighborhood of the image of the abnormal point. Under the same assumption, we prove that there exist nontrivial smooth curves passing through the abnormal point and lying on the level set of the map in question.

For nonlinear mappings acting in Banach spaces, we examine inverse and implicit function theorems under various smoothness assumptions. For various regularity (normality) conditions imposed on such mappings, we prove that the corresponding equations have solutions under any sufficiently small (in the norm) completely continuous perturbations. A priori estimates for these solutions are obtained.

We present Battle–Lemari´e wavelet systems of natural orders. Our main result is a decomposition theorem in Besov and Triebel–Lizorkin spaces with local Muckenhoupt weights, which is formulated in terms of bases generated by systems of such a type. The Battle–Lemari´e wavelets are splines and suit very well the study of integration operators.

Рассмотрены слоистые изотропные структуры, включающие слои метаматериала. Двухслойные и многослойные симметричные структуры, а также структура с непрерывным изменением параметров материала были исследованы. Впервые условия прозрачности таких конструкций и сооружений выведены из них представлены. Показано, что границы областей непрохождения волны соответствуют условиям прозрачности таких конструкций. Показано, что для симметричных структур прозрачности и четности-времени (ПП) в электродинамике и оптике требуются дополнительные условия, в отличие от представленных структур.

In the paper, a delayed control for an even-order chain of integrators is proposed. It is known that for many differential models, in particular, for models of mechanical systems, the control problem is reduced to the stabilization of chain of integrators. The proposed method use a simple inductive design and cascade systems’ asymptotic stability property. For any order of the integrator the control is a linear combination of delayed state coordinates with odd indices, and it depends on three numerical parameters. The parameters are selected from the stability region of a second-order linear delayed system and they can vary within a wide range. The result obtained can be applied to the stabilization problem for systems with incomplete measurements. In particular, for some class of mechanical systems, the proposed control can be a derivative-free alternative of stabilizing control.