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)
Attractors for differential equations with unbounded delays. (English) Zbl 1135.34040
For some differential equations with infinite delay (e.g., of logistic or Volterra-Lotka-type), boundedness properties of the solutions are proved; such as compact attractors, absorbing sets, and pullback attractors in the non-autonomous case. The paper starts with a section that includes a set-valued context. The main techniques are assumptions on the nonlinearity like \quad $ <F(x), x> \; \leq\; - \text{const}\cdot \vert \vert x\vert \vert ^2$, Gronwall type estimates, and the Arzelà-Ascoli theorem.

34K25Asymptotic theory of functional-differential equations
37L30Attractors and their dimensions, Lyapunov exponents
Full Text: DOI
[1] Atkinson, F. V.; Haddock, J. R.: On determining phase spaces for functional differential equations. Funkcial. ekvac. 31, 331-347 (1988) · Zbl 0665.45004
[2] Babin, A. V.; Vishik, M. I.: Attractors of evolution equations. Stud. math. Appl. 25 (1992) · Zbl 0778.58002
[3] Ball, J. M.: Global attractors for damped semilinear wave equations. Discrete contin. Dyn. syst. 10, No. 1 -- 2, 31-52 (2004) · Zbl 1056.37084
[4] Burton, T.; Hutson, V.: Repellers in systems with infinite delay. J. math. Anal. appl. 137, 240-263 (1989) · Zbl 0677.92016
[5] Caraballo, T.; Langa, J. A.; Valero, J.: Global attractors for multi-valued random dynamical systems. Nonlinear anal. 48, 805-829 (2002) · Zbl 1004.37035
[6] Caraballo, T.; Langa, J. A.; Melnik, V. S.; Valero, J.: Pullback attractors of non-autonomous and stochastic multi-valued dynamical systems. Set-valued anal. 11, No. 2, 153-201 (2003) · Zbl 1018.37048
[7] Caraballo, T.; Langa, J. A.; Robinson, J.: Attractors for differential equations with variable delays. J. math. Anal. appl. 260, No. 2, 421-438 (2001) · Zbl 0997.34073
[8] Caraballo, T.; łukaszewicz, G.; Real, J.: Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear anal. 64, No. 3, 484-498 (2006) · Zbl 1128.37019
[9] Caraballo, T.; Marín-Rubio, P.; Valero, J.: Autonomous and non-autonomous attractors for differential equations with delays. J. differential equations 208, No. 1, 9-41 (2005) · Zbl 1074.34070
[10] Caraballo, T.; Real, J.: Attractors for 2D-Navier -- Stokes models with delays. J. differential equations 205, 271-297 (2004) · Zbl 1068.35088
[11] Cheban, D. N.; Kloeden, P. E.; Schmalfuß, B.: The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems. Nonlinear dyn. Syst. theory 2, No. 2, 9-28 (2002) · Zbl 1054.34087
[12] Chepyzhov, V. V.; Vishik, M. I.: Evolution equations and their trajectory attractors. J. math. Pures appl. 76, 913-964 (1997) · Zbl 0896.35032
[13] Cushing, J. M.: Integrodifferential equations and delay models in population dynamics. Lecture notes in biomath. 20 (1979)
[14] Fox, L.; Mayers, D. F.; Ockendon, J. R.; Tayler, A. B.: On a functional differential equation. J. inst. Math. appl. 8, 271-307 (1971) · Zbl 0251.34045
[15] Hale, J. K.: Asymptotic behavior of dissipative systems. Math. surveys monogr. 25 (1988) · Zbl 0642.58013
[16] Hale, J. K.; Kato, J.: Phase space for retarded equations with infinite delay. Funkcial. ekvac. 21, 11-41 (1978) · Zbl 0383.34055
[17] Hale, J. K.; Lunel, S. M. V.: Introduction to functional -- differential equations. Appl. math. Sci. (1993) · Zbl 0787.34002
[18] Hino, Y.; Murakami, S.; Naito, T.: Functional differential equations with infinite delay. Lecture notes in math. 1473 (1991) · Zbl 0732.34051
[19] Kato, J.: Stability problem in functional differential equations with infinite delay. Funkcial. ekvac. 21, 63-80 (1978) · Zbl 0413.34076
[20] Kato, S.: Existence, uniqueness, and continuous dependence of solutions of delay-differential equations with infinite delays in a Banach space. J. math. Anal. appl. 195, 82-91 (1995) · Zbl 0846.34086
[21] Kato, T.; Mcleod, J. B.: The functional -- differential equation $y^{\prime}(x)=ay(\lambda x)+by(x)$. Bull. amer. Math. soc. 77, 891-937 (1971) · Zbl 0236.34064
[22] Kloeden, P. E.; Schmalfuß, B.: Non-autonomous systems, cocycle attractors and variable time-step discretization. Numer. algorithms 14, No. 1 -- 3, 141-152 (1997) · Zbl 0886.65077
[23] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002
[24] Ladyzhenskaya, O.: Attractors for semigroups and evolution equations. Lincei lectures 25 (1991) · Zbl 0755.47049
[25] Marín-Rubio, P.; Robinson, J.: Attractors for the stochastic 3D Navier -- Stokes equations. Stoch. dyn. 3, 279-297 (2003) · Zbl 1059.35100
[26] Melnik, V. S.; Valero, J.: On attractors of multi-valued semi-flows and differential inclusions. Set-valued anal. 6, 83-111 (1998)
[27] Murray, J. D.: Mathematical biology. (1993) · Zbl 0779.92001
[28] Péics, H.: On the asymptotic behaviour of a pantograph-type difference equation. J. difference equations appl. 6, No. 3, 257-273 (2000) · Zbl 0963.39019
[29] Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Appl. math. Sci. 68 (1997) · Zbl 0871.35001
[30] Wang, L.; Xu, D.: Asymptotic behavior of a class of reaction -- diffusion equations with delays. J. math. Anal. appl. 281, No. 2, 439-453 (2003) · Zbl 1031.35065