The concepts of controlled frame and its dual in n-Hilbert space have been introduced and then some of their properties are going to be discussed. Also, we study controlled frame in tensor product of n-Hilbert spaces and establish a relationship between controlled frame and bounded linear operator in tensor product of n-Hilbert spaces. At the end, we consider the direct sum of controlled frames in n-Hilbert space.

Let $f$ be a locally integrable function defined on $\mathbb{R}$, and $(n_k)$ be a lacunary sequence. Define
$$A_nf(x)=\frac{1}{n}\int_0^nf(x-t)\, dt,$$
and let
$$\mathcal{V}_{\rho}f(x)=\left(\sum_{k=1}^\infty|A_{n_k}f(x)-A_{n_{k-1}}f(x)|^{\rho}\right)^{1/\rho}.$$
Suppose that  $w\in A_p$, $1\leq p<\infty$, and $\rho\geq 2$. Then, there exists a positive constant $C$ such that
$$\|\mathcal{V}_{\rho}f\|_{L^1_w}\leq C\|f\|_{H^1_w}$$
for all $f\in H^1_w(\mathbb{R})$.

The paper considers the problem of constructing systems of vector fields that are invariant under the action of the local Lie group of transformations. It is shown that there exists a special class of Lie groups for which this problem can be solved elementarily.

We study the Volterra integral equation of the first kind with an integral operator of order n, a singularity and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms on the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term, we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows one to construct a particular and general solutions to the integro-differential equation in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multi-parameter family of solutions is being constructed for the original integral equation.

The inverse scattering method is used to integrate the Korteweg-de Vries equation with time-dependent coefficients. We derive the evolution of the scattering data of the Sturm–Liouville operator whose coefficient is a solution of the Korteweg-de Vries equation with time-dependent coefficients. An algorithm for constructing exact solutions of the Korteweg-de Vries equation with time-dependent coefficients is also proposed; we reduce it to the inverse problem of scattering theory for the Sturm–Liouville operator. Examples illustrating the stated algorithm are given.

An effective technique is proposed for obtaining exact formulas for estimating the area of flow regions in two-dimensional fluid flow problems with free boundaries, that allow an exact solution in terms of elliptic functions. The effectiveness of the technique is demonstrated using a specific example of the problem of capillary waves on the surface of a liquid of finite depth. This example is characterized by mirror symmetry of the flow region, but the technique can be generalized to the case of other symmetry of the flow region.

We obtain a regularized trace formula for 2m-order differential operator perturbed by a quasi-differential perturbation and with periodic boundary conditions.

For the initial-boundary value problem of the dynamics of a thermoviscoelastic medium of Oldroyd type in the planar case, a nonlocal theorem regarding the existence of a weak solution is established.