Asymptotic Integration Of Nonoscillatory Differential Equations: A Unified Approach
Journal Of Dynamical And Control Systems
Mathematics and Computer Science
We consider the equation [r(t)x′]′ + f(t)x = 0 as a perturbation of the equation [r(t)y′]′ + g(t)y = 0, where the latter is assumed to be nonoscillatory at infinity. The functions r and g are real-valued, r is positive, and f is complex-valued. The problem of the asymptotic integration of the perturbed equation in comparison with solutions of the unperturbed equation has been studied by many mathematicians, including Hartman and Wintner, Trench, ˇSimˇsa, Chen, and Chernyavskaya and Shuster. Here we apply a unified approach. Working in a matrix setting, we use preliminary and so-called conditioning transformations to bring the system in the formz⃗ =[Λ(t)+R(t)]z⃗ , where Λ is a certain diagonal matrix and R is an absolutely integrable perturbation. This allows us to use Levinson’s fundamental theorem to find the asymptotic behavior of solutions and, in addition, to estimate the error involved. This method allows us to derive these known results in a more unified setting and to weaken the hypotheses in some instances.
Bodine, Sigrun, and D. A. Lutz. 2011. "Asymptotic integration of nonoscillatory differential equations: a unified approach." Journal Of Dynamical And Control Systems 17(3): 329-358.