# 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)
Asymptotically almost periodic solutions of stochastic functional differential equations. (English) Zbl 1230.60058

Summary: We investigate a class of stochastic functional differential equations of the form

$dx\left(t\right)=\left(Ax\left(t\right)+F\left(t,x\left(t\right),{x}_{t}\right)\right)dt+G\left(t,x\left(t\right),{x}_{t}\right)\circ dW\left(t\right)·$

Our main results concern the existence and exponential stability of quadratic-mean asymptotically almost periodic mild solutions. An example is given to illustrate our results.

##### MSC:
 60H10 Stochastic ordinary differential equations
##### References:
 [1] Fink, A.: Almost periodic differential equations, Lecture notes in mathematics 37 (1974) [2] Arendt, W.; Batty, C.: Almost periodic solutions of first and second order Cauchy problems, J. differential equations 137, 363-383 (1997) · Zbl 0879.34046 · doi:10.1006/jdeq.1997.3266 [3] Hino, Y.; Murakami, S.; Yoshizawa, T.: Almost periodic solutions of abstract functional differential equations with infinite delay, Nonlinear anal. 30, 853-864 (1997) · Zbl 0891.34076 · doi:10.1016/S0362-546X(96)00196-4 [4] Levitan, B.; Zhikov, V.: Almost periodic functions and differential equations, (1982) · Zbl 0499.43005 [5] Yoshizawa, T.: Stability theory and the existence of periodic solutions and almost periodic solutions, Appl. math. Sci. 14 (1975) · Zbl 0304.34051 [6] Prüss, J.: Evolutionary integral equations and applications, (1993) [7] Zhang, C.: Almost periodic type functions and ergodicity, (2003) [8] Engel, K.; Nagel, R.: One-parameter semigroups for linear evolution equations, (1999) [9] Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Applied mathematical science 44 (1983) · Zbl 0516.47023 [10] Slutsky, E.: Sur LES fonctions aléatoires presque périodiques et sur la decomposition des functions aléatoires, Actualités sceintiques et industrielles 738 (1938) · Zbl 64.0534.03 [11] Udagawa, M.: Asymptotic properties of distributions of some functionals of random variable, Rep. statist. Appl. res. Union jap. Sci. eng. 2, 1-98 (1952) [12] Kawata, T.: Almost periodic weakly stationary processes, Statistics and probability: essays in honour of CR Rao, 383-396 (1982) · Zbl 0484.60028 [13] Swift, R.: Almost periodic harmonizable processes, Georgian math. J. 3, 275-292 (1996) · Zbl 0854.60036 · doi:10.1007/BF02280009 · doi:emis:journals/GMJ/vol3/contents.htm [14] Bezandry, P.; Diagana, T.: Existence of almost periodic solutions to some stochastic differential equations, Appl. anal. 117, 1-10 (2007) · Zbl 1138.60323 · doi:emis:journals/EJDE/Volumes/2007/117/abstr.html [15] Huang, Z.; Yang, Q.: Existence and exponential stability of almost periodic solution for stochastic cellular neural networks with delay, Chaos solitons fract. 42, 773-780 (2009) · Zbl 1198.60024 · doi:10.1016/j.chaos.2009.02.008 [16] Christopher, T.; Baker, H.; Buckwar, E.: Exponential stability in pth mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. comput. Appl. math. 184, 404-427 (2005) · Zbl 1081.65011 · doi:10.1016/j.cam.2005.01.018 [17] Bao, H.; Cao, J.: Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay, Appl. math. Comput. 215, 1732-1743 (2009) · Zbl 1195.34123 · doi:10.1016/j.amc.2009.07.025 [18] Ren, Y.; Xia, N.: A note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay, Appl. math. Comput. 214, 457-461 (2009) · Zbl 1221.34222 · doi:10.1016/j.amc.2009.04.013 [19] Luo, J.: A note on exponential stability in pth mean of solutions of stochastic delay differential equations, J. comput. Appl. math. 198, 143-148 (2007) · Zbl 1110.65009 · doi:10.1016/j.cam.2005.11.019 [20] Ren, Y.; Xia, N.: Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay, Appl. math. Comput. 210, 72-79 (2009) · Zbl 1167.34389 · doi:10.1016/j.amc.2008.11.009 [21] Ruess, W.; Summers, W.: Asymptotic almost periodicity and motions of semigroups of operators, Linear algebra appl. 84, 335-351 (1986) · Zbl 0616.47047 · doi:10.1016/0024-3795(86)90325-3 [22] Ruess, W.; Vu, Q.: Asymptotically almost periodic solutions of evolution equations in Banach spaces, J. differ. Eqs. 122, 282-301 (1995) · Zbl 0837.34067 · doi:10.1006/jdeq.1995.1149 [23] Arendt, W.; Batty, C.: Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line, Bull. London math. Soc. 31, 291-304 (1999) · Zbl 0952.34048 · doi:10.1112/S0024609398005657 [24] Minh, N.; N’guérékata, G.; Yuan, R.: Lectures on the asymptotically behavior of solutions of differential equations, (2008) [25] Liu, Q.; Yuan, R.: Asymptotic behavior of solutions to abstract functional differential equations, J. math. Anal. appl. 356, 405-417 (2009) · Zbl 1209.34094 · doi:10.1016/j.jmaa.2009.03.029