In this paper it is constructed a new grand grand Sobolev-Morrey $S_{p),\varkappa ),a,\alpha }^{l}W(G)$ spaces with dominant mixed derivatives. With help integral representation of generalized mixed derivatives of functions, defined on $n$-dimensional domains satisfying flexible horn condition, an embedding theorem is proved. In other works, the embedding theorem is proved in these spaces and belonging of the generalized mixed derivatives of functions from these spaces to the Holder class, was studied.

In this paper the controllable spread of some infectious disease is considered. The evolution model of the disease is described by the 3-dimensional nonlinear ordinary differential equations system. Vaccination and treatment are accepted as control parameters of the system. It is assumed that the stocks of vaccination and treatment is limited. Attainable sets of the system are approximately calculated for different control stocks. Graphical results are presented and possible biological applications are discussed.

In this paper we study the fractional maximal operator $\mathcal{M}^{\alpha}$ in the local generalized Morrey space $LM_{p,\varphi}^{\{t_0\}}(\Gamma)$ and the generalized Morrey space $M_{p,\varphi}(\Gamma)$ defined on Carleson curves $\Gamma$, respectively. For the operator $\mathcal{M}^{\alpha}$ we shall give a characterization the strong and weak Spanne-Guliyev type boundedness on $LM_{p,\varphi}^{\{t_0\}}(\Gamma)$ and the strong and weak Adams-Guliyev type boundedness on $M_{p,\varphi}(\Gamma)$.

In this paper the boundedness of multidimensional Hausdorff operator in weighted Lebesgue spaces is proved. In particular, necessary and sufficient condition for the boundedness of multidimensional Hausdorff operator are established in weighted Lebesgue spaces.

We obtain the Lipschitz boundedness for a class of fractional multilinear operators $I_{\Omega,\alpha}^{A,m}$ with rough kernels $\Omega\in L_{s}(\mathbb S^{n-1})$, $s>n/(n-\alpha)$ on the local generalized Morrey spaces $LM_{p,\varphi}^{\{x_0\}}$, generalized Morrey spaces $M_{p,\varphi}$ and vanishing generalized Morrey spaces $VM_{p,\varphi}$, where the functions $A$ belong to homogeneous Lipschitz space $\dot{\Lambda}_{\beta}$, $0<\beta<1$. We find the sufficient conditions on the pair $(\varphi_1,\varphi_2)$ which ensures the boundedness of the operators $I_{\Omega,\alpha}^{A,m}$ from $LM_{p,\varphi_1}^{\{x_0\}}$ to $LM_{q,\varphi_2}^{\{x_0\}}$, from $M_{p,\varphi_1}$ to $M_{q,\varphi_2}$ and from $VM_{p,\varphi_1}$ to $VM_{q,\varphi_2}$ for $1<p<q <\infty$ and $1/p-1/q=(\alpha+\beta)/n$. In all cases the conditions for the boundedness of the operator $I_{\Omega,\alpha}^{A,m}$ is given in terms of Zygmund-type integral inequalities on $(\varphi_1,\varphi_2)$, which do not assume any assumption on monotonicity of $\varphi_1(x,r), \varphi_2(x,r)$ in $r$.

In this paper, we introduce and examine some properties of three sequence spaces defined using lacunary sequence and invariant mean which generalize several known sequence spaces.

In this paper, the authors investigate the boundedness of the oscillatory singular integrals with variable Calderón-Zygmund kernel on generalized Morrey spaces $M^{p,\varphi}(\mathbb R^n)$ and the vanishing generalized Morrey spaces $VM^{p,\varphi}(\mathbb R^n)$. When $1<p<\infty$ and $(\varphi_1,\varphi_2)$ satisfies some conditions, we show that the oscillatory singular integral operators $T_{\lambda}$ and $T_{\lambda}^{*}$ are bounded from $M^{p,\varphi_1}(\mathbb R^n)$ to $M^{p,\varphi_2}(\mathbb R^n)$ and from $VM^{p,\varphi_1}(\mathbb R^n)$ to $VM^{p,\varphi_2}(\mathbb R^n)$. Meanwhile, the corresponding result for the oscillatory singular integrals with standard Calderón-Zygmund kernel are established.

In this study, the generalization of a frictionless contact problem in case of shearing deformation for an elastic inhomogeneous half space is presented. The basic equations of the elasticity theory and Fourier transform technique are applied to the problem to derive the system of singular integral equations. The obtained system of singular integral equations is solved by a quadrature approach. The numerical results are presented for the case of $N=1$, $N=2$, $N=3$, where $N$ denotes the number of the punches whose base are flat.

We show continuity in parabolic generalized Orlicz-Morrey spaces $M^{\Phi,\varphi}$ of commutator of parabolic nonsingular integral operators. We shall give necessary and sufficient conditions for the boundedness of the commutator of parabolic nonsingular integral operator on $M^{\Phi,\varphi}$ spaces with $BMO$ functions.

In this work, by defining Peiffer pairings in the Moore complex of a simplicial algebgroid, we give the close relationship between the category of simplicial algebroids with Moore complex of length 1 and that of internal categories in the category of R-algebroids.

In this paper we investigate the best approximation by trigonometric polynomials in weighted Morrey spaces $\mathcal{M}_{p,\lambda}(I_{0},w)$, where the weight function $w$ is in the Muckenhoupt class $A_{p}(I_{0})$ with $1 < p < \infty$ and $I_{0}=[0, 2\pi]$. We prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces $\mathcal{\widetilde{M}}_{p,\lambda}(I_{0},w)$ the closure of $C^{\infty}(I_{0})$ in $\mathcal{M}_{p,\lambda}(I_{0},w)$. We give the characterization of $K-$functionals in terms of the modulus of smoothness and obtain the Bernstein type inequality for trigonometric polynomials in the spaces $\mathcal{M}_{p,\lambda}(I_{0},w)$.

In this paper, we studied the abstract versions of Korovkin type approximation theorems via statistical relative $\mathcal{A-}$Summation process in modular spaces for double sequences. Then, we discuss the results which are obtained by special choice of the scale function and the matrix sequences and we give an application that shows our results are stronger than studied before. Finally, we study an extension to non-positive linear operators.