https://iopscience.iop.org/article/10.1070/RM1977v032n02ABEH001623/pdf

On certain problems in the theory of differential equations with deviating argument

In this paper we use Rab’s lemma [M. Rab, Uber lineare perturbationen eines systems von linearen differentialgleichungen, Czechoslovak Math. J. 83 (1958) 222–229; M. Rab, Note sur les formules asymptotiques pour les solutions d’un systeme d’equations differentielles lineaires, Czechoslovak Math. J. 91 (1966) 127–129] to obtain new sufficient conditions for the asymptotic equivalence of linear and quasilinear systems of ordinary differential equations. Yakubovich’s result [V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1966; V.A. Yakubovich, On the asymptotic behavior of systems of differential equations, Mat. Sb. 28 (1951) 217–240] on the asymptotic equivalence of a linear and a quasilinear system is developed. On the basis of the equivalence, the existence of asymptotically almost periodic solutions of the systems is investigated. The definitions of biasymptotic equivalence for the equations and biasymptotically almost periodic solutions are introduced. Theorems on the sufficient conditions for the systems to be biasymptotically equivalent and for the existence of biasymptotically almost periodic solutions are obtained. Appropriate examples are constructed.

http://users.metu.edu.tr/marat/papers/ATZ73.pdf

Asymptotic equivalence of differential equations and asymptotically almost periodic solutions

(P) * = L(xt) + f(tf xt) Roughly speaking, we will try to answer the following question: if the perturbation f(t, xt) is small, are the solutions of (P) "close" to the solutions of (L), for large values of ti It is clear that we can compare the solutions of the above systems in many ways, depending on what we mean by "close". Many works have been done in the case where it is required that the difference between the solutions of (L) and the solutions of (P) tends to zero, as t goes to infinity. This can be found, for example in Cooke [2], Evans [5], Kato [8]. In this paper we are interested in the case where the relative error, between the solutions of (L) and the solutions of (P), goes to zero, as t goes to infinity. For ordinary differential equations, results in this direction can be found in Szmidt [11], Onuchic [9], Coppel [3], Brauer and Wong [1], Rodrigues [10], etc. Szmidt uses the topological method of Wazewski and Onuchic [9] uses admissibility theory introduced by Massera and Schaffer. More precisely, we will be concerned with the following problems:

/pdf/on-growth-and-decay-of-solutions-of-perturbed-retarded-gdrycmzbxj.pdf

On growth and decay of solutions of perturbed retarded linear equations

Asymptotic equilibria of linear integro-differential equations and asymptotic relations between solutions of linear homogeneous impulsive differential equations and those of linear integro-differential equations are established. A new Gronwall-Bellman type lemma for integro-differential inequalities is proved. An example is given to demonstrate the validity of one of the results.

http://users.metu.edu.tr/marat/papers/aty85.pdf

Richard B Evans

Papers

Asymptotic equivalence of linear functional differential equations