The article discusses inverse problems for the fractional diffusion equation with the Hilfer operator in time. The direct problem is the initial-boundary value problem for this equation with Cauchy-type initial data and Dirichlet boundary conditions. The first inverse problem, which involves determining a time-dependent coefficient, is reduced to an equivalent Volterra-type integral equation. The existence and uniqueness of the solution are proven using the contraction mapping principle. The second inverse problem involves determining a function dependent on the spatial variable on the right-hand side of the equation. This problem is studied using the Fourier method and the properties of the Mittag-Leffler function. The solution is constructed in the form of a series based on eigenfunctions.

In a half-space, for hyperbolic equations with translation operators in lower-order derivatives acting in all coordinate directions, a multiparameter family of solutions is constructed in explicit form using integral transforms. A theorem is proved stating that the obtained solutions are classical provided that the real part of the symbol of the differential–difference operator is positive.

We propose a class of dual gradient Uzawa type methods for general convex constrained optimization problems. In order to provide stable convergence we utilize the partial regularization in primal variables and additional constraints for dual variables. Convergence of the method is established under rather

The present work deals with the existence and periodicity of solutions to non-autonomous partial differential equations of retarded, infinite, and neutral type in the framework of fading memory space, which is defined axiomatically. In our strategy, we rely on Sadovskii's fixed-point theorem. On the other hand, the family of linear operators $A(\cdot)$ is assumed to be non-densely defined and verified by the Acquistapace-Terreni conditions. Finally, we propose an application to illustrate our results.

This paper studies pursuit–evasion differential games and a game with a «life line» for the case where the inertial movements of the players are realized using controls subject to repulsive forces. For solving the pursuit problem and the problem with a «life line», the main tool remains the parallel pursuit strategy (in short, the $\bf{\Pi}$-strategy). With the help of this $\bf{\Pi}$-strategy, necessary and sufficient conditions for the solvability of the pursuit problem are obtained, and the capture set or the players’ reachability set is constructed. For solving the problem with a «life line» in favor of the pursuer, a monotone (with respect to set inclusion) decrease over time of the players’ reachability set is proved. In solving the evasion problem, lower bounds for the distances between the players are obtained, and for a game with a «life line» in this case a set is constructed that the evading player can reach without being captured, under arbitrary control of the pursuer. The obtained results are illustrated by representative examples.

We study commutative local rings over which every upper-triangular matrix is the sum of an idempotent and a $q$-potent that commute. For Galois rings and rings of the form $\mathbb{F}_{p^{k}}[x]/\langle x^{r} \rangle$, necessary and sufficient criterion are provided.

Numerical integration of definite integrals is important in fundamental and applied sciences. The error of approximate integral calculations depends on the initial data and specific requirements, which leads to the imposition of various conditions on the obtained calculations.

In this paper, we consider the problem of constructing an optimal quadrature formula for the approximate calculation of Fourier integrals using the $\varphi$-function method. The error of the quadrature formula is estimated from above using the integral of the square of the function $\varphi$ from the Hilbert space. Next, a function $\varphi$ is chosen for which the integral of the square of the function on this interval takes the smallest value. The coefficients of the optimal quadrature formula are calculated using the obtained $\varphi$ function. The optimal quadrature formula is exact for the functions $e^{\sigma x}$ and $e^{-\sigma x}$, where $\sigma$ is a nonzero real parameter.

In this paper, a family of block operator matrices ${\mathcal A}_{\rm h}(K),$ $K \in (-\pi/{\rm h}; \pi/{\rm h}]^3$, associated with the Hamiltonian of a system with a non-conserved number of particles not exceeding three on a non-integer lattice $({\rm h} {\Bbb Z})^3$ with step ${\rm h}>0$, is considered. It is established that the operator ${\mathcal A}_{\rm h}({\bf 0}),$ ${\bf 0}:=(0,0,0),$ has a finite number of negative eigenvalues if the corresponding generalized Friedrichs model has a zero eigenvalue. It is shown that the operator ${\mathcal A}_{\rm h}({\bf 0})$ possesses an infinite number of negative eigenvalues accumulating at zero (the Efimov effect) if the generalized Friedrichs model has a zero-energy resonance. An asymptotic formula is obtained for the number $N_{\rm h}(z)$ of eigenvalues of the operator ${\mathcal A}_{\rm h}({\bf 0})$ lying below $z,$ $z \leq 0$ as the spectral parameter $z\to -0.$