Correction and addition to my paper “The normal form and the stability of solutions of a system of differential equations in the complex domain” (original) (raw)

Stability of solutions of a nonstandard ordinary differential system by Lyapunov's second method

Journal of Applied Mathematics and Stochastic Analysis, 1991

Prasanthinilayam-$15 134, Andhra Pradesh, INDIA Differential equations of the form y: =f(t,y,y) where f is not necessarily linear in its arguments represent certain physical phenomena and are known for quite some time. The well known Clairut's and Chrystal's equations fall into this category. Earlier we established the existence of a (unique) solution of the nonstandard initial value problem y = f(t, y, y), y(to) = Yo under certain natural hypotheses on f. In this paper, we studied the stability of solutions of a nonstandard first order ordinary differential system. Differential equations of the form y= f(t,y,y) where f is not necessarily linear in its arguments represent certain physical phenomena and are known for quite some time. The well known Clairut's and Chrystal's equations fall into this category [4]. A few authors, notably E.L. Inee [5], H.T. Davis [4] et. al. have given some methods for finding solutions of equations of the above type. Apart from these, to the authors knowledge, there does not seem to exist any systematic study of these equations.

Note on the Stability of System of Differential Equations ...

In this paper, we examine the relation between practical stability and Hyers-Ulam-stability and Hyers-Ulam-Rassias stability as well. In addition,by practical stability we gave a sufficient condition in order that the first order nonlinear systems of differential equations has local generalized Hyers-Ulam stability and local generalized Hyers-Ulam-Rassias stability.

Note on the stability for linear systems of differential equations

In this paper, by applying the fixed point alternative method, we give a necessary and sufficient condition in order that the first order linear system of differential equationsż(t) + A(t)z(t) + B(t) = 0 has the Hyers-Ulam-Rassias stability and find Hyers-Ulam stability constant under those conditions. In addition to that we apply this result to a second order differential equationÿ(t) + f (t)ẏ(t) + g(t)y(t) + h(t) = 0.

On the stability of solutions of a certain system of second-order differential equation

2005

The paper deals with the stability of solutions along with their derivatives of a certain system of second-order differential equation with respect to certain perturbation. We consider the system dy(x)/dx + A(x)y(x) = 0, (i) where y(x) = 1 11 12 2 12 22 ( ) ( ) ( ) , ( ) ( ( )) ( ) ( ) ( ) ij Y x a x a x A x a x Y x a x a x     = =         and aij(x), i, j = 1, 2 is real-valued continuous function of x, x ∈ [0, ∞). Let B(x) = (bij(x)), i, j = 1, 2 (bij(x)s being real-valued continuous function of x ∈ [0, ∞)) be a set of perturbations which changes (1) to dy(x)/dx + (A(x) + B(x))y(x) = 0. (ii) In this paper, certain results on the stability of solutions of the system (i) along with their derivatives, which are either bounded or tend to zero as the independent variable x tends to infinity with respect to the perturbation B(x) satisfying some conditions, are achieved.

Asymptotic Stability for Second-Order Differential Equations with Complex Coefficients

Electronic Journal of Differential Equations, 2004

We prove asymptotical stability and instability results for a general second-order differential equations with complex-valued functions as coefficients. To prove asymptotic stability of linear second-order differential equations, we use the technique of asymptotic representations of solutions and error estimates. For nonlinear second-order differential equations, we extend the asymptotic stability theorem of Pucci and Serrin to the case of complex-valued coefficients.

Complex Analysis and Differential Equations

Springer Undergraduate Mathematics Series, 2012

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