# Article

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Keywords:
Functional-differential equation; singular case; transformation; canonical form
Summary:
The paper contains applications of Schrőder’s equation to differential equations with a deviating argument. There are derived conditions under which a considered equation with a deviating argument intersecting the identity $y=x$ can be transformed into an equation with a deviation of the form $\tau (x)=\lambda x$. Moreover, if the investigated equation is linear and homogeneous, we introduce a special form for such an equation. This special form may serve as a canonical form suitable for the investigation of oscillatory and asymptotic properties of the considered equation.
References:
[1] Eĺsgolc, L. E.: Vvedenije v teoriju differencialnych uravnenij s otklonjajuščimsa argumentom. Nauka, Moscow, 1964. (Russian) MR 0170049
[2] Heard, M. L.: Asymptotic behavior of solutions of the functional differential equation $x^{\prime }(t)=ax(t)+bx(t^{\alpha }), \alpha >1$. J.Math.Anal.Appl. 44 (1973), 745–757. MR 0333405 | Zbl 0289.34115
[3] Kato, T., McLeod, J. B.: The functional-differential equation $y^{\prime }(x)=ay(\lambda x)+by(x)$. Bull. Amer. Math. Soc. 77 (1971), 891–937. MR 0283338
[4] Kuczma, M.: Functional Equations in a Single Variable. Polish Scient.Publ., Warszawa, 1968. MR 0228862 | Zbl 0196.16403
[5] Lade, G. S., Lakshmikantham, V., Zhang, B. G.: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker, New York, 1983.
[6] Lim, E.-B.: Asymptotic behavior of solutions of the functional differential equation $x^{\prime }(t)=Ax(\lambda t)+Bx(t), \lambda >0$. J.Math.Anal.Appl. 55 (1976), 794–806. MR 0447749 | Zbl 0336.34060
[7] Neuman, F.: On transformations of differential equations and systems with deviating argument. Czechoslovak Math.J. 31(106) (1981), 87-90. MR 0604115 | Zbl 0463.34051
[8] Neuman, F.: Transformation and canonical forms of functional-differential equation. Proc. Roy.Soc.Edinburgh 115A (1990), 349-357. MR 1069527
[9] Pandofi, L.: Some observations on the asymptotic behaviors of the solutions of the equation $x^{\prime }(t)=A(t)x(\lambda t)+B(t)x(t), \lambda >0$. J.Math.Anal.Appl. 67 (1979), 483–489. MR 0528702
[10] Szekeres, G.: Regular iteration of real and complex functions. Acta Math. 100 (1958), 203–258. MR 0107016 | Zbl 0145.07903
[11] Tryhuk, V.: The most general transformation of homogeneous retarded linear differential equations of the $n$-th order. Math.Slovaka 33 (1983), 15–21. MR 0689272 | Zbl 0514.34058

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