In the paper, for a fractional operator polynomial P_n(A^α) , α∈(0,1) , with a weakly positive operator A acting on a Banach space, the problem of its bounded invertibility is posed and solved, which is equivalent to the Hadamard well-posed solvability of the corresponding operator equation. The problem is solved by the Maslov–Heaviside method, which was previously used by the authors in the case of integer powers of the operator A . This enables us to obtain an integral representation of the inverse operator of P_n(A^α) using the strongly continuous semigroup U(t_1-A) with the generator -A and to indicate a well-posedness estimate for this operator; this estimate connects the type of the semigroup with the roots of the scalar polynomial P_n(x) (which was called the symbol of the operator polynomial P_n(A) by V. P. Maslov). Using examples, we show the naturalness of applying the Maslov–Heaviside method to the study of the well-posed solvability of problems for differential equations.

In this paper, the methods of the general theory of differential equations are used to study the equation describing the flow of a gas jet around an airplane wing. For different parameter values, this equation describes a subsonic flow and has an elliptical type, or has a hyperbolic type and describes a supersonic flow. In the course of the study, the correct solvability of the Cauchy problem is established for the study of the equation in the hyperbolic case using strongly continuous operator semigroups, groups and cosine functions. In this article, the concept of the Chebyshev operator cosine function is introduced for the first time and the representation of the solution of the Cauchy problem for the equation is indicated in terms of orthogonal Chebyshev polynomials of the first kind.

The concept of intermediate asymptotics for the solution of an evolution equation with initial data and a related solution obtained without initial conditions was introduced by G.N. Barenblatt and Ya.B. Zeldovich in the context of extending the concept of strict determinism in statistical physics and quantum mechanics. Here, according to V.P. Maslov, to axiomatize the mathematical theory, we need to know the conditions satisfied by the initial data of the problem. We show that the correct solvability of a problem without initial conditions for fractional differential equations in a Banach space is a necessary, but not sufficient, condition for intermediate asymptotics. Examples of intermediate asymptotics are given.

Using the operator functional relation Sh(t+s)+Sh(t−s) = 2[I +2Sh2($$ \frac{t}{2} $$)] Sh(s), Sh(0) = 0, we introduce and study the strongly continuous sine function Sh(t), t ∈ (−∞, ∞), of linear bounded transformations acting in a complex Banach space E. Also, we study the cosine function Ch(t) given by the equation Ch(t) = I + 2Sh2($$ \frac{t}{2} $$), where I is the identity operator in E. The pair Ch(t) and Sh(t) is called an exponential trigonometric pair (ETP, in brief). For such pairs, we determine the generating operator (generator) by the equation Sh″ (0)φ = Ch″ (0)φ = Aφ and we give a criterion for A to be the generator of ETP. We find a connection between Sh(t) and the uniform well-posedness of the Cauchy problem with Krein’s condition for the equation $$ \frac{d^2u(t)}{dt^2} $$ = Au(t). This problem is uniformly well-posed if and only if A is the exponent generator of the sine function Sh(t). We introduce the concept of a bundle of several ETPs, which also forms an ETP, and we give a representation for bundle’s generator. The facts obtained significantly expand the applicability of operator methods to study the well-posedness of initial-boundary value problems.

Theorems on the existence and form of solutions of variational problems with circular symmetry, such as the deflection problem of an elastic plate and the capillarity problem, were obtained.

For the first time, the theory of strongly continuous cosine operator functions (COF) has been applied to study the correct solvability of boundary value problems for second-order linear differential equations in a Banach space (elliptic case). The correct solvability of the Cauchy problem (hyperbolic case) is usually formulated in COF terms. The conditions on the order of COF growth are specified under which the Dirichlet boundary value problem is correct on a finite interval. An integral representation of the solution and its sharp estimate are given.

This paper presents results of the investigation of bifurcations of stationary solutions of the Swift-Hohenberg equation and dynamic descent to the points of minimal values of the functional of energy for this equation, obtained with the use of the modification of the Lyapunov-Schmidt variation method and some methods from the theory of singularities of smooth functions. Nonstationary case is investigated by the construction of paths of descent along the trajectories of the infinite-dimensional SH dynamical system from arbitrary initial states to points of the minimum energy.

Methods are given for the approximate calculation of a branch of a resonance oscillation when it bifurcates from a stationary point and for optimizing this branch with respect to the nonsymmetry coefficient, which is defined as the ratio between the largest and the smallest values of the amplitude. It is shown that the optimal values of the base amplitudes are the coefficients of the corresponding Fejer series. The largest value of the nonsymmetry coefficient is calculated exactly.

The study of well-solvable operator equations in a Banach space, which was initiated by the authors in [4, 5], is continued. Namely, it is proved by means of Maslov's operator method that a polynomial equation with abstract Newton polynomials is well solvable in the sense of Hadamard. The obtained results are applied to prove that a large class of problems for differential equations with variable coefficient having a singularity (such equations are called generalized Euler equations in the paper) are well solvable.

In the paper, we establish uniformly well-posed solvability of boundary value problems for equations , in the Banach space where the operator A is a generator of semi-group U(x) of the class with the estimate , , M does not depend on x. The results are applied to the investigation of well-posed solvability of boundary value problems for differential equations with fractional derivatives in weight spaces of continuous functions and investigation of problems of heat-and-mass-transfer and of radio physics.