Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed

The paper deals with Cauchy problems for first-order systems of linear ordinary differential equations with unknown data. It is assumed that the right-hand sides of equations belong to certain bounded sets in the space of square-integrable vector-functions, and the information about the initial conditions is absent. From indirect noisy observations of solutions to the Cauchy problems on a finite system of points and intervals, the guaranteed mean square estimates of linear functionals on unknown solutions of the problems under consideration are obtained. Under an assumption that the statistical characteristics of noise in observations are not known exactly, it is proved that such estimates can be expressed in terms of solutions to well-defined boundary value problems for linear systems of impulsive ordinary differential equations.

Relatively Compact Sets in Variable Exponent Morrey Spaces on Metric Spaces

We study a characterization of the precompactness of sets in variable exponent Morrey spaces on bounded metric measure spaces. Totally bounded sets are characterized from several points of view for the case of variable exponent Morrey spaces over metric measure spaces. This characterization is new in the case of constant exponents.

Partial integral operators of Fredholm type on Kaplansky-Hilbert module over $L_0$

The article studies some characteristic properties of self-adjoint partially integral operators of Fredholm type in the Kaplansky-Hilbert module $L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over $L_{0}\left(\Omega_{2}\right)$. Some mathematical tools from the theory of Kaplansky-Hilbert module are used. In the Kaplansky-Hilbert module $L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over $ L_{0} \left (\Omega _ {2} \right)$ we consider the partially integral operator of Fredholm type $T_{1}$ ($ \Omega_{1} $ and $\Omega_{2} $ are closed bounded sets in $ {\mathbb R}^{\nu_{1}}$ and $ {\mathbb R}^{\nu_{2}},$ $\nu_{1}, \nu_{2} \in {\mathbb N} $, respectively). The existence of $ L_{0} \left (\Omega _ {2} \right) $ nonzero eigenvalues for any self-adjoint partially integral operator $T_{1}$ is proved; moreover, it is shown that $T_{1}$ has finite and countable number of real $L_{0}(\Omega_{2})$-eigenvalues. In the latter case, the sequence $ L_{0}(\Omega_{2})$-eigenvalues is order convergent to the zero function. It is also established that the operator $T_{1}$ admits an expansion into a series of $\nabla_{1}$-one-dimensional operators.

SOME REMARKS ON ROUGH STATISTICAL \(\Lambda\)-CONVERGENCE OF ORDER \(\alpha\)

The main purpose of this work is to define Rough Statistical \(\Lambda\)-Convergence of order \(\alpha\) \((0<\alpha\leq1)\) in normed linear spaces. We have proved some basic properties and also provided some examples to show that this method of convergence is more generalized than the rough statistical convergence. Further, we have shown the results related to statistically \(\Lambda\)-bounded sets of order \(\alpha\) and sets of rough statistically \(\Lambda\)-convergent sequences of order \(\alpha\).

Stable Convergence Theorems for Products of Uniformly Continuous Mappings in Metric Spaces

We study the behavior of inexact products of uniformly continuous self-mappings of a complete metric space that is uniformly continuous and bounded on bounded sets. It is shown that previously established convergence theorems for products of non-expansive mappings continue to hold even under the presence of computational errors.

Existence and Uniqueness of Solutions to Third-Order Boundary Value Problems

The purpose of this research is to connect fixed point methods with certain third-order boundary value problems in new and interesting ways. Our strategy involves an analysis of the problem under consideration within closed and bounded sets. We develop sufficient conditions under which the associated mappings will be contractive and invariant in these sets, which generates new advances concerning the existence, uniqueness and approximation of solutions.