Abstract:
The article considers a three-term fourth-order differential equation with unbounded coefficients.
The coefficient of the intermediate term of the equation is assumed to be a smooth and rapidly
increasing function at infinity. This intermediate term, as an operator, does not obey the
differential operator formed by the extreme terms of the equation. This is precisely what makes
the work unique. Using functional methods, sufficient conditions are obtained for a generalized
solution of the equation to exist, be unique and maximally regular. These conditions characterize
the relationship between intermediate and small coefficients. The differential equation under
consideration is caused by problems of practical processes of stochastic analysis, shaft oscillations,
etc. The article uses such methods as obtaining an a priori estimate of the solution, reducing
the problem above to the problem of invertibility of a third-order differential operator with a
potential of constant sign, and constructing a pseudo-resolvent using correct local operators. In
general, the article substantiates an effective method for solving the main problems posed for
differential equations on an infinite interval in the case of a new equation with an unbounded
intermediate coefficient. Although the coefficients are assumed to be smooth, the work does not
impose restrictions on the variation of their derivatives. This, in turn, allows us to cover a wide
class of fourth-order equations. The methods developed in the work and the results obtained can
be used in the study of the qualitative properties of higher-order differential equations.
