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)
Positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations. (English) Zbl 1189.45014

The paper deals with the study of nonlinear infinite delay integral equation of the form

x(t)= - t a(t,t-s)f(s,x(s))ds,(1)

where f:× + is continuous and a:× + + and the function sa(t,s) is integrable over + for any fixed t. Under some additional assumptions a few sufficient conditions are established which guarantee the existence of almost periodic, asymptotically almost period and pseudo almost periodic solutions of (1). Some applications to other types of integral or differential equations of the obtained results are indicated.

45M15Periodic solutions of integral equations
45G10Nonsingular nonlinear integral equations
34K14Almost and pseudo-periodic solutions of functional differential equations
[1]Yoshizawa, T.: Stability theory by Liapunov’s second method, Publications of the mathematical society of Japan 9 (1966) · Zbl 0144.10802
[2]Cooke, K.; Huang, W.: On the problem for linearization for state-dependent delay differential equations, Proc. amer. Math. soc. 124, 1417-1426 (1996) · Zbl 0844.34075 · doi:10.1090/S0002-9939-96-03437-5
[3]Burton, T. A.; Hatvani, L.: On the existence of periodic solutions of some nonlinear functional-differential equations with unbounded delay, Nonlinear anal. 16, 389-396 (1991) · Zbl 0736.34059 · doi:10.1016/0362-546X(91)90038-3
[4]Gao, G.: Periodic solutions of neutral functional-differential equations, Chinese ann. Math. ser. A 9, 263-269 (1988) · Zbl 0707.34055
[5]Dads, E. Ait; Ezzinbi, K.: Existence of positive pseudo-almost periodic solution for a class of functional equations arising in epidemic problems, Cybern. syst. Anal. 30, 900-910 (1994) · Zbl 0834.45006 · doi:10.1007/BF02366449
[6]Dads, E. Ait; Ezzinbi, K.: Existence of positive pseudo-almost-periodic solution for some nonlinear infinite delay integral equations arising in epidemic problems, Nonlinear anal. 41, 1-13 (2000) · Zbl 0964.45003 · doi:10.1016/S0362-546X(98)00219-3
[7]Xu, B.; Yuan, R.: The existence of positive almost periodic type solutions for some neutral nonlinear integral equation, Nonlinear anal. 60, 669-684 (2005) · Zbl 1063.45004 · doi:10.1016/j.na.2004.09.043
[8]Xu, B.; Yuan, R.: On the positive almost periodic type solutions for some nonlinear delay integral equations, J. math. Anal. appl. 304, 249-268 (2005) · Zbl 1074.45007 · doi:10.1016/j.jmaa.2004.09.025
[9]Ding, H. S.; Liang, J.; N’guerekata, G. M.; Xiao, T. J.: Mild pseudo-almost periodic solutions of nonautonomous semilinear evolution equations, Math. comput. Modelling 45, 579-584 (2007) · Zbl 1165.34387 · doi:10.1016/j.mcm.2006.07.006
[10]Ding, H. S.; Liang, J.; N’guerekata, G. M.; Xiao, T. J.: Pseudo-almost periodicity of some nonautonomous evolution equations with delay, Nonlinear anal. 67, 1412-1418 (2007) · Zbl 1122.34345 · doi:10.1016/j.na.2006.07.026
[11]Liang, J.; Maniar, L.; N’guerekata, G. M.; Xiao, T. J.: Existence and uniqueness of cn-almost periodic solutions to some ordinary differential equations, Nonlinear anal. 66, 1899-1910 (2007) · Zbl 1117.34042 · doi:10.1016/j.na.2006.02.030
[12]Bochner, S.: A new approach to almost periodicity, Proc. natl. Acad. sci. USA 48, 2039-2043 (1962) · Zbl 0112.31401 · doi:10.1073/pnas.48.12.2039
[13]Corduneanu, C.: Almost periodic functions, (1968) · Zbl 0175.09101
[14]Fink, A. M.: Almost periodic differential equations, Lecture notes in mathematics 377 (1974) · Zbl 0325.34039
[15]Yoshizawa, T.: Stability theory and the existence of periodic solutions and almost periodic solutions, (1975)
[16]Zhang, C.: Almost periodic type functions and ergodicity, (2003)
[17]Thompson, A. C.: On certain contraction mappings in a partially ordered vector space, Proc. amer. Math. soc. 14, 438-443 (1963) · Zbl 0147.34903 · doi:10.2307/2033816
[18]Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040
[19]Li, H.; Huang, F.; Li, J.: Composition of pseudo almost-periodic functions and semilinear differential equations, J. math. Anal. appl. 255, 436-446 (2001) · Zbl 1047.47030 · doi:10.1006/jmaa.2000.7225