Two main results are presented: 1) a new class of applied problems that lead to equations with (p,q)-Laplace is presented; 2) a method for solving nonlinear boundary value problems involving (p,q)-Laplace with measurable unbounded coefficients is introduced. In the main result, the existence, uniqueness, and stability of the nonnegative weak solution to the equations of the form - div(ρ|∇ u|^q-2∇ u)- div(|∇ u|^p-2∇ u)=λ b |u|^q-2u,  p>q are proven. Additionally, an explicit formula that expresses the solution of the equation through the inverse optimal solution of the spectral problem - div(ρ|∇ϕ|^q-2∇ϕ)=λ b|ϕ|^q-2ϕ is presented. The advantage of the method is that the inverse optimal problem has a visible geometry and a simple variational structure, which makes it easy to solve it and, as a consequence, find a solution to the associated nonlinear boundary value problem.

We develop the Perron-Frobenius theory using a variational approach and extend it to a set of arbitrary matrices, including those that are neither irreducible nor essentially positive, and non-preserved cones. We introduce a new concept called a quasi-eigenvalue of a matrix, which is invariant under orthogonal transformations of variables, and has various useful properties, such as determining the largest value of the real parts of the eigenvalues of a matrix. We extend Weyl's inequality for the eigenvalues to the set of arbitrary matrices and prove the new stability result to the Perron root of irreducible nonnegative matrices under arbitrary perturbations. As well as this, we obtain new types of estimates for the ranges of the sets of eigenvalues and their real parts.

The paper discusses the development of a method for constructing asymptotic formulas for x→+∞of a fundamental system of solutions of two-term singular symmetric differential equations of odd order with coefficients from a wide class of functions that allow oscillation (with weakened regularity conditions that do not satisfy the classical Titchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation (i2)[(p(x)y')''+(p(x)y'')']+q(x)y=λythe asymptotics of solutions in the case of different behavior of the coefficients qxis studied, hx−1+1px. New asymptotic formulas are obtained for the case when hx∉L1∞.

This work is devoted to the development of methods for constructing asymptotic formulas as x→∞ of a fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of odd order. The coefficients of the differential expression belong to classes of functions that allow oscillation (for example, those that do not satisfy the classical Titchmarsh–Levitan regularity conditions). As a model equation, the fifth-order equation i2p(x)y‴″+p(x)y″‴+q(x)y=λy, along with various behaviors of coefficients p(x),q(x), is investigated. New asymptotic formulas are obtained for the case when the function h(x)=−1+p−1/2(x)∉L1[1,∞) significantly influences the asymptotics of solutions to the equation. The case when the equation contains a nontrivial bifurcation parameter is studied.

We develop the Perron-Frobenius theory using a variational approach and extend it to a set of arbitrary matrices, including those that are neither irreducible nor essentially positive, and do not preserve a cone. We introduce a new concept called a "quasi-eigenvalue of a matrix," which is invariant under orthogonal transformations of variables, and has various useful properties, such as determining the largest value of the real parts of the eigenvalues of a matrix. We extend Weyl's inequality for the eigenvalues to the set of arbitrary matrices and prove the new stability result to the Perron root of irreducible nonnegative matrices under arbitrary perturbations. As well as this, we obtain new types of estimates for the ranges of the sets of eigenvalues and their real parts.

We investigate the statement of the optimization inverse spectral problem with incomplete spectral data for the one-dimensional Schr¨odinger operator on the entire axis: for a given potential ????0, find the closest function ????^ such that the first ???? eigenvalues of the Schrodinger operator with potential ????^ coincided with the given values ????* ∈ R, ???? = 1, ????.

The article is focused on the development of a method allowing one to construct asymptotics for solutions to ODEs of arbitrary order with oscillating coefficients on the semiaxis. The idea of the method is presented on the example of studying the asymptotics of the Sturm–Liouville equation.

In this paper, we show the development of a method that allows one to construct asymptotics for solutions to ordinary differential equations of arbitrary order with oscillating coefficients on the semiaxis. The idea of the method is presented on the example of studying the asymptotics of the Sturm-Liouville equation solutions.

We consider the following optimization inverse spectral problem: for a given matrix potential Q_0(x) , find a matrix function Q̂(x) closest to it and such that the first eigenvalue of the Sturm–Liouville matrix operator with the potential Q̂(x) coincides with a given value λ _1^*∈ℝ . The main result of the paper is to establish a new type of connection between the specified inverse spectral problem and systems of second-order nonlinear differential equations known as systems of nonlinear Schrödinger equations in mathematical physics.

A new class of inverse problems is considered. In the context of classical theory, inverse problems are concerned with finding a model that has the observed measurements. It is well known that such problems usually are ill-posed. At the same time, it is often the case when there is some a priori information about the system. This naturally leads to the following inverse optimal problem: find data Fˆ of a model which is the nearest to a priori given data F0 and sufficient to ensure the model has the observed measurements S.In this note, an approach to a complete solution to such a problem is developed. Within the framework of this approach, we consider a model problem of recovering the potential field Vˆ from the m observed eigenvalues of the Schrödinger operator, provided that such potential field is at the minimum distance from a priori given potential Va. In the main result, we establish a new type of relationship between the linear spectral problems and systems of nonlinear differential equations which enables us to find a solution to the inverse optimal spectral problem and obtain novel results on the existence of solutions to nonlinear problems as well.

We study the properties of singular Sturm–Liouville operators in Hilbert spaces. Although the literature on the topic is immense, there are a number of questions that have yet to be solved, for example, those pertaining to the behavior of solutions of the Sturm–Liouville equation with an irregular potential at infinity. This problem is topical not only for being of interest in itself but also because it naturally arises when dealing with questions related to the spectral properties of the Sturm–Liouville operator.

This work is aimed at studying optimization inverse spectral problems with a so-called incomplete spectral data. As incomplete spectral data, the partial traces of the Sturm-Liouville operator serve. We study the following formulation of the inverse spectral problem with incomplete data (optimization problem): find a potential (V) over cap closest to a given function V-0 such that a partial trace of the Sturm-Liouville operator with the potential (V) over cap. has a prescribed value. As a main result, we prove the existence and uniqueness theorem for solutions of this optimization inverse spectral problem. A new type of relationship between linear spectral problems and systems of nonlinear differential equations is established. This allows us to find a solution to the inverse optimal spectral problem by solving a boundary value problem for a system of nonlinear differential equations and to obtain a solvability of the system of nonlinear differential equations. To prove the uniqueness of solutions, we use the convexity property of the partial trace of the Sturm-Liouville operator with the potential (V) over cap; the trace is treated as a functional of the potential (V) over cap. We obtain a new generalization of the Lidskii-Wielandt inequality to arbitrary self-adjoint semi-bounded operators with a discrete spectrum.

This paper is devoted to a new statement and the study of direct and inverse spectral problems for small linear oscillations of orthotropic plates that carry concentrated masses at a finite set of points, which, in turn, are connected to a stationary base by elastic springs with known stiffness coefficients.

We study the inverse spectral problem of reconstructing nonsplitting boundary conditions for the Sturm–Liouville operator from a minimal amount of spectral data. Necessary and sufficient conditions are derived for the existence of solutions of the problem under consideration and for these solutions to be isolated or nonisolated. We propose a simple analytical algorithm for reconstructing boundary conditions. (Explicit formulas are provided for the coefficients of the boundary conditions.) It is shown that the condition determining the properties of the problem is the dimension of the linear span of matrices of fundamental solution systems corresponding to the given spectral data.

In this paper, the problems of constructing a mathematical model for small linear oscillations of an orthotropic plate carrying concentrated masses in a finite set of points connected with a fixed base by elastic struts (springs) with certain stiffness coefficients, as well as the direct and inverse spectral problems for such a model in a new formulation are considered.

The purpose of this paper is twofold: firstly, we present a new type of relationship between inverse problems and nonlinear differential equations. Secondly, we introduce a new type of inverse spectral problem, posed as follows: for a priori given potential $V_0$ find the closest function $\hat{V}$ such that $m$ eigenvalues of one-dimensional space Schrodinger operator with potential $\hat{V}$ would coincide with the given values $ E_1 $, $ \ldots $, $ E_m \in \mathbb {R} $. In our main result, we prove the existence of a solution to this problem, and more importantly, we show that such a solution can be directly found by solving a system of nonlinear differential equations.

We establish a relationship between an inverse optimization spectral problem for the N-dimensional Schrödinger equation −Δϕ+q(x)ϕ=λϕ and a solution of the nonlinear boundary value problem −Δu+q(x)u=λu−uγ−1,u>0,u|∂Ω=0. Using this relationship, we find an exact solution for the inverse optimization spectral problem, investigate its stability and obtain new results on the existence and uniqueness of the solution for the nonlinear boundary value problem.

In the present paper, we are concerned with the Sturm-Liouville operator L[q]u := -u '' + q(x)u subject to the separated boundary conditions. We suppose that q is an element of L-2(0,pi) and study a so-called inverse optimization spectral problem: given a potential q(0) and a value lambda(k), where k = 1, 2, . . . , find a potential (q) over cap closest to q(0) in the norm of L-2(0, pi) such that the value lambda(k) coincides with k-th eigenvalue lambda(k) ((q) over cap) of the operator L[(q) over cap]. In the main result, we prove that this problem is related to the existence of a solution to a boundary value problem for the nonlinear equation -u '' + q(0)(x)u = lambda(k)u + sigma u(3) with sigma =1 or sigma = -1. This implies that the minimizing solution of the inverse optimization spectral problem can be obtained by solving the corresponding nonlinear boundary value problem. On the other hand, this relationship allows us to establish an explicit formula for the solution to the nonlinear equation by finding the minimizer of the corresponding inverse optimization spectral problem. As a consequence of this result, a new method of proving the generalized Sturm nodal theorem for the nonlinear boundary value problems is obtained.