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On one approach to the study of the asymptotic behaviour of the Riccati equation with complex-valued coefficients. (English) Zbl 0814.34029
The Riccati equation with complex valued coefficients \[ z'= p(t) (a(t)- z)(b(t)- z)\tag{1} \] is considered. The asymptotic behaviour of the solutions of this equation, especially the existence of bounded solutions and of solutions satisfying conditions of the form \[ \lim_{t\to\infty} z(t)= a_ 0\quad\text{or}\quad \int^ \infty_{t_ 0} | z(t)- a_ 0| dt< \infty, \] is studied. The investigation is based on the reduction of the Riccati equation to the equation (2) \(z'= zg(t, z)+ h(t, z)\). Sufficient conditions for the equation (2) to have bounded solutions (as \(t\to\infty\)) and solutions satisfying \[ \lim_{t\to\infty} z(t)= 0\quad\text{or}\quad \int^ \infty_{t_ 0} | z(t)| dt <\infty \] are obtained.

MSC:
34D05 Asymptotic properties of solutions to ordinary differential equations
34M99 Ordinary differential equations in the complex domain
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