On one approach to the study of the asymptotic behaviour of the Riccati equation with complex-valued coefficients
type: Item
instance of: scholarly article
title: On one approach to the study of the asymptotic behaviour of the Riccati equation with complex-valued coefficients
zbMATH Open document ID: 0814.34029
DOI: 10.1007/BF01765633
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1995-06-12T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34M99
zbMATH DE Number: 711477
zbMATH Keywords: Riccati equation with complex valued coefficients
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: bounded solutions
author: J. Kalas
reviewed by: Vasile Dragan
MaRDI profile type: Publication
cites work: Sur un mouvement plan
cites work: On continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals
cites work: On the monotonicity property for a certain class of second order differential equations
cites work: Q5330830
cites work: Q3946463
cites work: Q3931728
cites work: Q3660095
cites work: Q3963384
cites work: On certain asymptotic properties of the solutions of the equation $\dot z=f(t,z)$ with a complex-valued function $f$
cites work: Q3677999
cites work: Contributions to the asymptotic behaviour of the equation $\dot z=f(t,z)$ with a complex-valued function $f$
cites work: Bounded Solutions of Dynamical Systems in the Plane under the Conditions of Instability
cites work: ASYMPTOTIC PROPERTIES OF DYNAMICAL SYSTEMS IN THE PLANE
cites work: Q5685429
cites work: The Riccati differential equation with complex-valued coefficients
cites work: Equation $Z'=A(t)-Z^2$ coefficient of which has a small modulus
cites work: Geometrical approach to the study of the Riccati differential equation with complex-valued coefficients
cites work: Stability of dynamical systems in the plane
cites work: Q3660132
cites work: Q5790518
description: scientific article; zbMATH DE number 711477