zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Inequalities and stability for a linear scalar functional differential equation. (English) Zbl 1064.34062
The paper is concerned with the stability of the zero solution for the nonautonomous linear scalar functional-differential equation with one fixed delay $$x'(t) = a(t)x(t) + b(t)x(t-h) \quad\text{for }t > t_0,$$ addressing the cases where $a(t)$ may have variable sign or $b(t)$ may be unbounded. A typical result, assuming $h = 1$ without loss of generality, implies that $$\vert x(t)\vert = O\Biggl(\exp\biggl(\frac{1}{2}\int_{t_0}^{t-1/2} (a(s) + b(s+1)) \,ds\biggr)\Biggr),$$ if $-1/2 \le a(t) + b(t+1) \le -b^2(t+1)$ for all $t$. Lower bounds are also derived. The proofs use Lyapunov functionals.

34K20Stability theory of functional-differential equations
34K06Linear functional-differential equations
34K12Growth, boundedness, comparison of solutions of functional-differential equations
Full Text: DOI
[1] Burton, T. A.: Stability and periodic solutions of ordinary and functional differential equations. (1985) · Zbl 0635.34001
[2] Burton, T. A.; Casal, A.; Somolinos, A.: Upper and lower bounds for Liapunov functionals. Funkcial. ekvac. 32, 23-55 (1989) · Zbl 0687.34067
[3] Busenberg, S. V.; Cooke, K. L.: Stability conditions for linear non-autonomous delay differential equations. Quart. appl. Math. 42, 295-306 (1984) · Zbl 0558.34059
[4] Hale, J.: Theory of functional differential equations. (1977) · Zbl 0352.34001
[5] Hatvani, L.; Krizstin, T.: Asymptotic stability for a differential--difference equation containing terms with and without a delay. Acta sci. Math. (Szeged) 60, 371-384 (1995) · Zbl 0834.34092
[6] Hatvani, L.: Annulus arguments in the stability theory for functional differential equations. Differential integral equations 10, 975-1002 (1997) · Zbl 0897.34060
[7] Sansone, G.; Conti, R.: Nonlinear differential equations. (1964) · Zbl 0128.08403
[8] Wang, T.: The modular estimations of solutions of a type of non-autonomous nonlinear systems. J. east China inst. Tech. 42, 23-33 (1987)
[9] Wang, T.: Weakening the condition W1(|${\phi}(0)|)\leqslantV(t,{\phi})$\leqslantW2(||${\phi}$||) for uniform asymptotic stability. Nonlinear anal. 23, 251-264 (1994)
[10] Wang, T.: Stability in abstract functional differential equations. Part II. Applications. J. math. Anal. appl. 186, 835-861 (1994) · Zbl 0822.34065
[11] Wang, T.: Wazewski’s inequality in a linear Volterra integro-differential equations, Volterra equations and applications. Stability and control: theory, methods and applications 10, 483-492 (2000) · Zbl 0958.45006
[12] Wazewski, T.: Sur la limitation des intégrales des systèmes d’équations différentielles linèaires ordinaires. Studia math. 10, 48-59 (1948) · Zbl 0036.05703