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)
Fixed points and exponential stability for stochastic Volterra-Levin equations. (English) Zbl 1197.60053
Using the contraction fixed point principle Luo studies the exponential stability of the stochastic Volterra-Levin equation. Conditions are given to ensure that the equation is exponentially stable in mean square and is also almost surely exponentially stable.

MSC:
60H10Stochastic ordinary differential equations
34K50Stochastic functional-differential equations
WorldCat.org
Full Text: DOI
References:
[1] Burton, T. A.: Stability by fixed point theory for functional differential equations, (2006) · Zbl 1160.34001
[2] Becker, L. C.; Burton, T. A.: Stability, fixed points and inverses of delays, Proc. roy. Soc. Edinburgh sect. A 136, 245-275 (2006) · Zbl 1112.34054 · doi:10.1017/S0308210500004546
[3] Burton, T. A.: Fixed points, stability, and exact linearization, Nonlinear anal. 61, 857-870 (2005) · Zbl 1067.34077 · doi:10.1016/j.na.2005.01.079
[4] Burton, T. A.: Fixed points, Volterra equations, and Becker’s resolvent, Acta math. Hungar. 108, 261-281 (2005) · Zbl 1091.34040 · doi:10.1007/s10474-005-0224-9
[5] Burton, T. A.: Fixed points and stability of a nonconvolution equation, Proc. amer. Math. soc. 132, 3679-3687 (2004) · Zbl 1050.34110 · doi:10.1090/S0002-9939-04-07497-0
[6] Burton, T. A.: Perron-type stability theorems for neutral equations, Nonlinear anal. 55, 285-297 (2003) · Zbl 1044.34028 · doi:10.1016/S0362-546X(03)00240-2
[7] Burton, T. A.: Integral equations, implicit functions, and fixed points, Proc. amer. Math. soc. 124, 2383-2390 (1996) · Zbl 0873.45003 · doi:10.1090/S0002-9939-96-03533-2
[8] Burton, T. A.; Furumochi, Tetsuo: Krasnoselskii’s fixed point theorem and stability, Nonlinear anal. 49, 445-454 (2002) · Zbl 1015.34046 · doi:10.1016/S0362-546X(01)00111-0
[9] Burton, T. A.; Zhang, Bo: Fixed points and stability of an integral equation: nonuniqueness, Appl. math. Lett. 17, 839-846 (2004) · Zbl 1066.45002 · doi:10.1016/j.aml.2004.06.015
[10] Furumochi, Tetsuo: Stabilities in fdes by Schauder’s theorem, Nonlinear anal. 63, e217-e224 (2005) · Zbl 1159.39301 · doi:10.1016/j.na.2005.02.057
[11] Jin, Chuahua; Luo, Jiaowan: Fixed points and stability in neutral differential equations with variable delays, Proc. amer. Math. soc. 136, 909-918 (2008) · Zbl 1136.34059 · doi:10.1090/S0002-9939-07-09089-2
[12] Raffoul, Y. N.: Stability in neutral nonlinear differential equations with functional delays using fixed-point theory, Math. comput. Modelling 40, 691-700 (2004) · Zbl 1083.34536 · doi:10.1016/j.mcm.2004.10.001
[13] Zhang, Bo: Fixed points and stability in differential equations with variable delays, Nonlinear anal. 63, e233-e242 (2005) · Zbl 1159.34348 · doi:10.1016/j.na.2005.02.081
[14] J.A.D. Appleby, Fixed points, stability and harmless stochastic perturbations, preprint.
[15] Luo, Jiaowan: Fixed points and stability of neutral stochastic delay differential equations, J. math. Anal. appl. 334, 431-440 (2007) · Zbl 1160.60020 · doi:10.1016/j.jmaa.2006.12.058
[16] Luo, Jiaowan: Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. math. Anal. appl. 342, 753-760 (2008) · Zbl 1157.60065 · doi:10.1016/j.jmaa.2007.11.019
[17] Luo, Jiaowan: Stability of stochastic partial differential equations with infinite delays, J. comput. Appl. math. 222, 364-371 (2008) · Zbl 1151.60336 · doi:10.1016/j.cam.2007.11.002
[18] Luo, Jiaowan; Taniguchi, Takeshi: Fixed points and stability of stochastic neutral partial differential equations with infinite delays, Stoch. anal. Appl. 27, 1163-1173 (2009) · Zbl 1177.93094 · doi:10.1080/07362990903259371
[19] Arnold, L.: Stochastic differential equations: theory and applications, (1972) · Zbl 0216.45001
[20] Khasminskii, R. Z.: Stochastic stability of differential equations, (1981) · Zbl 1259.60058
[21] Kolmanovskii, V. B.; Myshkis, A.: Applied theory of functional differential equations, (1992) · Zbl 0917.34001
[22] Kushner, H. J.: Stochastic stability and control, (1967) · Zbl 0244.93065
[23] Ladde, G. S.; Lakshmikantham, V.: Random differential inequalities, (1980) · Zbl 0488.60069
[24] Liu, Kai: Stability of infinite dimensional stochastic differential equations with applications, Pitman monographs series in pure and applied mathematics 135 (2006) · Zbl 1085.60003 · doi:10.1201/9781420034820
[25] Mao, Xuerong: Stability of stochastic differential equations with respect to semimartingales, (1991) · Zbl 0724.60059
[26] Mao, Xuerong: Exponential stability of stochastic differential equations, (1994) · Zbl 0806.60044
[27] Mao, Xuerong: Stochastic differential equations and their applications, (1997) · Zbl 0892.60057
[28] Mao, Xuerong; Yuan, Chenggui: Stochastic differential equations with Markovian switching, (2006) · Zbl 1126.60002
[29] Mohammed, S. -E.A.: Stochastic functional differential equations, (1986)
[30] Mao, Xuerong: Almost sure exponential stability of delay equations with damped stochastic perturbation, Stoch. anal. Appl. 19, 67-84 (2001) · Zbl 0980.60081 · doi:10.1081/SAP-100001183
[31] Volterra, V.: Sur la théorie mathématique des phénom.es héréditaires, J. math. Pures appl. 7, 249-298 (1928) · Zbl 54.0934.06
[32] Levin, J. J.: The asymptotic behavior of Volterra equation, Proc. amer. Math. soc. 14, 434-451 (1963) · Zbl 0115.32403