Authors : G.Zh. Berdenova, A.A. Utemissova and A.A. Zhikeyev
Abstract: The asymptotic behavior of the fundamental system of solutions of two fourth-order singular differential equations for large values of the spectral parameter is investigated in this article. The asymptotic formulas for the fundamental system of solutions are determined uniformly with respect to x when ly = λy, λ∈Γ, λ→∞ in the case of slow rotation of the eigenvectors of the real symmetric matrix Q(x) with twice continuously differentiable elements. Replacing the variables in the system of equations of the fourth order allows us to pass to a system of equations of the first order with a new unknown vector function. An orthogonal matrix is introduced which can be reduced to diagonal form by means of transformations. For the system of equations in the space of vector-functions, asymptotic formulas are obtained and proved. Due to the uniformity of the asymptotic formulas, the asymptotics of the spectrum of the corresponding differential operator is calculated in this study. Using the obtained formulas, the defect indices of the corresponding differential operators are calculated.
G.Zh. Berdenova, A.A. Utemissova and A.A. Zhikeyev, 2019. On the Behavior of Solutions of a Fourth-Order Differential System at Infinity. Journal of Engineering and Applied Sciences, 14: 725-733.