In three-dimensional domain the single-value solvability of a mixed problem for a Hilfer type nonlinear partial differential equation of the even order with small positive parameters in mixed derivatives is considered. The regular solution of this fractional differential equation is studied in the case \(0<\alpha\leq 1\). The Fourier series method is used and a countable system of nonlinear ordinary differential equations is obtained. The ordinary differential equation is integrated and the obtained constants are found by the aid of given initial value conditions. The countable system of nonlinear functional-integral equations is solved by the method of successive approximations. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proved.

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.

In this paper, on the base of Pontryagin maximum principle, the optimal control problem with concentrated parameters for a degenerate differential equation with the Caputo operator and with coefficients from the Lebesgue space is studied. The efficiency indicator of the considered optimal control problem has an integral form of fractional order. A new version of the method of increments is applied and the concept of a conjugate equation with an integral form is essentially used.

In this paper, we establish necessary and sufficient conditions on monotone weight functions for the boundedness for Hausdorff operators of special kind in weighted Lebesgue spaces. In particular, we get similar results for important operators of harmonic analysis which are special cases of the Hausdorff operators. The weights are illustrated by examples at the end of the paper.

The questions of solvability and construction of solutions of a nonlocal direct and inverse problems for a second-order nonhomogeneous Fredholm integro-differential equation with a degenerate kernel, two redefinition data and two real parameters are considered. The degenerate kernel method is used. The features that have arisen in the construction of solutions and are associated with the determination of the integration coefficients and redefinition data are studied. The values of the parameters are calculated for which the solvability of direct nonlocal problem is established and the corresponding solutions are constructed. It is studied the case when inverse problem has a unique solution.