# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Exponential solutions of equation $\dot y(t)=\beta(t)[y(t-\delta)-y(t-\tau)]$. (English) Zbl 1058.34099
The authors study the asymptotic behaviour as $t\to\infty$ of the first-order differential equation with two deviating arguments $$\dot y= \beta(t)[y(t -\delta)- y(t- \tau)].\tag{*}$$ The relation between solutions of $(*)$ and of the inequality $$\dot y\le\beta(t)[y(t- \delta)- y(t- \tau)]\tag{**}$$ plays a basic role. A criterion for representing solutions of $(*)$ in exponential form is obtained. A sufficient condition for the existence of unbounded solutions is derived, and an illustrative example is given.

##### MSC:
 34K25 Asymptotic theory of functional-differential equations
Full Text:
##### References:
 [1] Arino, O.; Györi, I.; Pituk, M.: Asymptotically diagonal delay differential systems. J. math. Anal. appl. 204, 701-728 (1996) · Zbl 0876.34078 [2] Arino, O.; Pituk, M.: Convergence in asymptotically autonomous functional differential equations. J. math. Anal. appl. 237, 376-392 (1999) · Zbl 0936.34064 [3] Arino, O.; Pituk, M.: More on linear differential systems with small delays. J. differential equations 170, 381-407 (2001) · Zbl 0989.34053 [4] Atkinson, F. V.; Haddock, J. R.: Criteria for asymptotic constancy of solutions of functional differential equations. J. math. Anal. appl. 91, 410-423 (1983) · Zbl 0529.34065 [5] Bellman, R.; Cooke, K. L.: Differential--difference equations. Mathematics in science and engineering (1963) [6] H. Bereketoglu, M. Pituk, Asymptotic constancy or nonhomogeneous linear differential equations with unbounded delays, Preprint No. 096, Department of Mathematics and Computing, University of Veszprém, 2002, pp. 1--8 [7] Castillo, S.; Pinto, M.: Lp perturbations in delay differential equations. Electron. J. Differential equations 2001, 1-11 (2001) · Zbl 0997.34072 [8] Čermák, J.: On the asymptotic behaviour of solutions of certain functional differential equations. Math. slovaca 48, 187-212 (1998) · Zbl 0942.34060 [9] Čermák, J.: The asymptotic bounds of solutions of linear delay systems. J. math. Anal. appl. 225, 373-388 (1998) · Zbl 0913.34063 [10] Čermák, J.: A change of variables in the asymptotic theory of differential equations with unbounded delays. J. comput. Appl. math. 143, 81-93 (2002) · Zbl 1016.34077 [11] Cooke, K.; Yorke, J.: Some equations modelling growth processes and gonorrhea epidemics. Math. biosci. 16, 75-101 (1973) · Zbl 0251.92011 [12] Domshlak, Y.; Stavroulakis, I. P.: Oscillations of differential equations with deviating arguments in a critical state. Dynam. systems appl. 7, 405-414 (1998) · Zbl 0916.34062 [13] Džurina, J.: Comparison theorems for functional differential equations. (2002) [14] Erbe, L. H.; Kong, Q.; Zhang, B. G.: Oscillation theory for functional differential equations. (1995) · Zbl 0821.34067 [15] Györi, I.; Ladas, G.: Oscillation theory of delay differential equations. (1991) · Zbl 0780.34048 [16] Györi, I.; Pituk, M.: Comparison theorems and asymptotic equilibrium for delay differential and difference equations. Dynam. systems appl. 5, 277-302 (1996) · Zbl 0859.34053 [17] Györi, I.; Pituk, M.: L2-perturbation of a linear delay differential equation. J. math. Anal. appl. 195, 415-427 (1995) · Zbl 0853.34070 [18] Györi, I.; Pituk, M.: Special solutions for neutral functional differential equations. J. inequal. Appl. 6, 99-117 (2001) · Zbl 1004.34068 [19] Krisztin, T.: Asymptotic estimation for functional differential equations via Lyapunov functions. Colloq. math. Soc. jános bolyai 53, 365-376 (1988) [20] Mahler, K.: On a special functional equation. J. London math. Soc. 15, 115-123 (1940) · Zbl 0027.15704 [21] Murakami, K.: Asymptotic constancy for systems of delay differential equations. Nonlinear anal. 30, 4595-4606 (1997) · Zbl 0959.34058 [22] Diblı\acute{}k, J.: Asymptotic convergence criteria of solutions of delayed functional differential equations. J. math. Anal. appl. 274, 349-373 (2002) · Zbl 1025.34062 [23] Diblı\acute{}k, J.: Asymptotic representation of solutions of equation y \dot{}$(t)={\beta}(t)[y(t)$-y(t-${\tau}(t))$]. J. math. Anal. appl. 217, 200-215 (1998) · Zbl 0892.34067