×

Retarded differential equations with piecewise constant delays. (English) Zbl 0557.34059

Functional differential equations with piecewise constant delays are studied. They are closely related to impulse, loaded and, especially, to difference equations, and have the structure of continuous dynamical systems within intervals of unit length. Continuity of a solution at a point joining any two consecutive intervals implies recursion relations for the solution at such points. The equations are thus similar in structure to those found in certain ”sequential-continuous” models of disease dynamics. The main feature of equations with piecewise constant delays is that it is natural to pose initial-value and boundary-value problems for them not on intervals but at a certain number of individual points. Existence and uniqueness theorems are established for equations with bounded and unbounded operators. A general estimate of the solutions’ growth as \(t\to +\infty\) is found. Special consideration is given to the problem of stability, and oscillatory properties of solutions are studied, too. The instrument of continued fractions plays an important role in the computation of solutions and in the analysis of their asymptotic behavior. Equations with unbounded delay arise in cases of several argument deviations. In such problems, the initial function is prescribed on (-\(\infty,0]\) and the solution is sought for \(t>0\).

MSC:

34K05 General theory of functional-differential equations
34K20 Stability theory of functional-differential equations
34K30 Functional-differential equations in abstract spaces
39A10 Additive difference equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Pelyukh, G.P; Sharkovskii, A.N, Introduction to the theory of functional equations, (1974), Naukova Dumka Kiev, [in Russian]
[2] Halanay, A; Wexler, D, Teoria calitativǎ a sistemelor cu impulsuri, (1968), Acad. RSR Bucuresti · Zbl 0176.05202
[3] Myshkis, A.D, On certain problems in the theory of differential equations with deviating argument, Uspekhi mat. nauk, 32, 173-202, (1977) · Zbl 0356.34082
[4] Busenberg, S; Cooke, K.L, Models of vertically transmitted diseases with sequential-continuous dynamics, (), 179-187
[5] Corduneanu, C; Lakshmikantham, V, Equations with unbounded delay, Nonlinear anal., 4, 831-877, (1980) · Zbl 0449.34048
[6] Hale, J.K, Theory of functional differential equations, (1977), Springer-Verlag New York · Zbl 0425.34048
[7] Khinchine, A.Ja, Continued fractions, (1961), Fizmatgiz Moscow, [in Russian]
[8] Chebotarev, N.G, Theory of continued fractions, (1938), [in Russian]
[9] Bodnarcǔk, P.I; Skorobogat’ko, V.Ja, Branching continued fractions and their applications, (1974), Naukova Dumka Kiev, [in Russian]
[10] Pfluger, P, Matrizenkettenbrüche, (1966), Juris Druck Verlag Zürich
[11] Parodi, M, La localisation des valeurs caractéristiques des matrices et ses applications, (1959), Gauthier-Villars Paris · Zbl 0087.01602
[12] Krein, S.G, Linear differential equations in Banach spaces, () · Zbl 0636.34056
[13] Mizohata, S, The theory of partial differential equations, (1973), University Press Cambridge · Zbl 0263.35001
[14] Cooke, K.L; Wiener, J, Distributional and analytic solutions of functional differential equations, J. math. anal. appl., 98, 111-112, (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.