Vol 59, No 9 (2023)

We study integrally bounded solutions of the differential equation A(x)=z, where  A  is a linear differential operator of order l  defined on functions x:R→H (R=(−∞,∞), H () and H  is a finite-dimensional Euclidean space). The right-hand side z is an integrally bounded function on R ranging in H  and satisfying the inequality (ψ(t),z(t))≥δ|z(t)|, t∈R, δ>0. Conditions are given on the operator A  and the function ψ:R→H  that guarantee an inverse inequality of the following form for the solutions x  under consideration: ∫τ+1τ|x(l)(t)|dt≤c∫τ+2τ−1|x(t)|dt, where the constant  is independent of the choice of a real number t  and function x.

A fourth-order nonlinear differential equation on a network that is a model of a system of Euler–Bernoulli rods is considered. Based on the monotone iteration method, the existence of a solution of a boundary value problem on a graph for this equation is established using the positiveness of the Green’s function and the maximum principle for the corresponding linear differential equation. An example is given to illustrate the results.

We consider the problem of leaky waves in an inhomogeneous waveguide structure covered with a layer of graphene, which is reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. A variational statement of the problem is used to determine the solution. The variational problem is reduced to the study of an operator function. The properties of the operator function necessary for the analysis of its spectral properties are investigated. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of the operator function on the complex plane are proved.

For the telegraph equation with a nonlinear potential, we consider a mixed problem in the first quadrant in which the Cauchy conditions are specified on the spatial semiaxis and the Neumann condition is set on the temporal semiaxis. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of some integral equations. The solvability of these equations is studied, as well as the dependence of the solutions on the smoothness of the initial data. For the problem under consideration, the uniqueness of the solution is proved and conditions are established under which a classical solution exists. If the matching conditions are not met, then a problem with conjugation conditions is constructed, and if the data is not smooth enough, then a mild solution is constructed.

Explicit solutions of model and nonmodel integral equations of Volterra type with two boundary and one interior singular point are obtained, and the properties of the resulting solutions are studied. The well-posed statement of problems with conditions specified on singular manifolds is found in the case where the solution of the model equation contains an arbitrary constant.

We consider shallow elastic shells with a given circular boundary and seek an axisymmetric shell shape minimizing the weight at a given fundamental frequency of shell vibrations. Using the resulting formula for the linear part of the increment of the frequency functional, the multiplicity of the minimum natural frequency of vibrations of the shell is estimated. The Fréchet differentiability of the frequency functional is also established, and optimal conditions for minimizing the weight of the shell at a given fundamental vibration frequency are obtained.