THE ASYMPTOTICS OF SOLUTIONS OF THE DIFFERENTIAL EQUATION OF THE FOURTH ORDER IN DEGENERATE CASE
In the theory of singular differential operations, the central place is occupied by the issue of studying the spectral properties of operators, depending on the coefficients of the appropriate differential expression. Many works were devoted to the study of the asymptotic behavior of solutions of differential equations of the form: ly= (-1)ny(2n)+k=1n-1-1k(pn-k(x)y(k))(k)=λy, 0≤x<∞, where x→∞, λ is the complex parameter, pkx are twice continuously differentiable functions. The behavior of the fundamental system of solutions of the equation ly=λy at x→∞ was considered in the case then the contribution of all coefficients in the asymptotic formulas was the same and pnx=0, pn-1(x)≥cx2+ε, ε>0 at x≥ x0. The author of the article studied the differential equation of the fourth order and succeeded in weakening these conditions on the coefficients. The aim of this work is the study of the asymptotics of the fundamental system of solutions Ly = y<sup>IV</sup>-2(p(x)y ׳ ) ׳ = λу, 0 ≤ х ≤ ∞ (1), when х→∞. The author introduced the following notations: F(x, λ, μ) = μ<sup>4</sup>-2р(х)μ<sup>2 </sup>+ λ, (2), where μ<sub>1,</sub> μ<sub>2, </sub>μ<sub>3,</sub>μ<sub>4</sub> are the roots of characteristic equation (2). Let us assume that the following conditions are satisfied: 1. р(х) = a xα, where a, α is a constant: a ≠ 0, α > 0; 2. Re (μ<sub>3</sub>(x, λ) - μ<sub>4</sub>(x, λ)) does not change the sign at sufficiently large х; 3. At sufficiently large х<sub>0</sub> > 0 and then х ≥ х<sub>0</sub>, the following estimate holds b ≤ μ<sub>3</sub>(x, λ) μ<sub>4</sub>(x, λ) ≤ с, where b and с are positive constants. The main result of the article is the theorem, which gives the asymptotic formulas of solutions of the equation (1), when x→∞.