Implicit nonautonomous first-order difference equations in spaces endowed with distance are considered. The distance is only required to be non-negative (the distance need not possess any of the standard properties of a metric). Analogs of classical definitions of stability with respect to initial data (Lyapunov, asymptotic, and exponential stability) are introduced, taking into account the possible non-uniqueness of the solution to the Cauchy problem. Under the assumption that the mapping of two arguments generating the equation under consideration is covering with respect to one argument and Lipschitz with respect to the other, statements are obtained on the stability with respect to the initial data of a particular solution to the Cauchy problem and of all solutions to the equation. As a corollary, the stability of a stationary solution to an implicit autonomous equation is investigated. Conditions for the existence of a stationary solution for such an autonomous equation are also obtained. It is demonstrated how the presented results imply well-known conditions for the exponential stability of equilibria of explicit difference equations in linear normed spaces. A scheme for applying the obtained results to the study of the stability of some functional equations with delay is proposed. An example of a study using this scheme of a specific functional equation in the space of Lebesgue measurable essentially bounded functions is given.

We propose an approach to the analysis of saddle points of locally twice continuously differentiable functions on a subset of a finite-dimensional space. The approach is based on second-order necessary optimality conditions in problems with constraints. We assume that the sets, on the product of which the function has a saddle point, are given as zero sets of certain mappings. The advantage of the proposed approach lies in the absence of regularity of these mappings. We provide various examples of functions having a saddle point and of the corresponding constraints in which the obtained conditions take place. 

We study the simplest problem of the calculus of variations in the space of absolutely continuous functions. In the considered formulation, it is assumed that the Lagrangian is a lower-bounded continuous function taking values in the extended real line, which means that it may take the value +∞. The well-posedness of the problem is shown. In the case where the Lagrangian has the special form L(y, y0 ) = α(y)β(y 0 ), necessary conditions for a weak local minimum and for a strong local minimum are obtained in the form of the Euler equation. An important distinction of the obtained results from those known earlier is as follows. The obtained necessary conditions are applicable in the case where the function y, b at which the minimum is attained has a singularity, i.e., a point x such that L yb(x), yb 0 (x) = +∞. The application of the obtained results is demonstrated on the example of the brachistochrone problem in the class of absolutely continuous functions. Using the necessary conditions for a minimum obtained in the paper, it is shown that, as in the classical formulation of the brachistochrone problem, the cycloid is the only function that can be the only candidate for an extremum. Since this curve has a singularity at zero, the example illustrates the applicability of the obtained results in the presence of singularities.

This paper is devoted to the study of the interrelation of arbitrary motions and the recurrent motions of dynamical system defined in a Hausdorff semi-metric space Γ and endowed with specified topologies of compact and uniform convergence.

Based on the definitions of minimal set and recurrent motion introduced by G.Birkhoff at the beginning of the last century, a new property of motions was obtained. This property rigidly links arbitrary motions and the recurrent motions and establishes that if in the space Γ the positive (negative) semi-trajectory of some motion is relatively compact, then the ω -limit (α -limit) set of this motion is a compact minimal set. Also this property allows us to dentify some important properties of recurrent motions.

This paper investigates the problem of moment stability of solutions to linear Ito functional-integral equations by employing the theory of positively invertible matrices. This class of equations encompasses a broad range of mathematical models, including ordinary and fractional differential equations, both with and without stochastic perturbations, as well as systems with and without time delays. It also includes several relatively new types of equations that have attracted increasing attention in the recent scientific literature.

Traditionally, the stability analysis of specific subclasses of these equations has relied primarily on methods based on Lyapunov–Krasovskii–Razumikhin functionals. In contrast, the theory of deterministic functional differential equations has demonstrated the effectiveness of the auxiliary (comparison) equation approach. In the present work, this methodology is adapted to the investigation of moment stability. Specifically, the analysis is carried out using a modified auxiliary equation method, which is based on representing the solutions of an appropriate auxiliary problem followed by the application of the theory of positively invertible matrices.

Sufficient conditions for the moment stability of solutions to the considered equations are established in terms of the positive invertibility of matrices constructed from the parameters of the corresponding systems. In addition, several illustrative examples are provided to verify the applicability of the proposed conditions in particular cases.

An infinite system of nonlinear integral equations on the whole line with a noncompact and monotone Hammerstein-type operator is investigated. This system, for various particular representations of the Hammerstein operator, arises in many areas of physics and biology. In particular, systems of this nature are encountered in the dynamical theory of p -adic open and open-closed strings for a scalar tachyon field, in the nonlinear theory of energy transport in inhomogeneous media, in the kinetic theory of gases, and in the mathematical theory of the spread of various epidemic diseases within the framework of the standard and modified Diekmann–Kaper model. A constructive theorem for the existence of a nonnegative, nontrivial, and bounded solution is proved. Furthermore, uniform convergence (with a rate of decreasing geometric progression) of a specially chosen iterative process to the solution of this system of nonlinear integral equations is established. In a certain subclass of infinite bounded vector functions, the uniqueness of the constructed solution is also proved. At the end, specific examples of matrix kernels and nonlinearities satisfying all the conditions of the proven theorems are given.

The issues of the structure of maximal linked systems and ultrafilters on widely understood measurable spaces (MS) and on their nonempty subspaces are investigated. The role of Stone-type sets in the representation of ultrafilters on the mentioned subspaces is investigated, including the possibility of topological identification in the case of MS with an algebra or semialgebra of sets. The issues related to the properties and transformations of trivial (fixed) and free ultrafilters on widely understood MS are considered.