ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS OF INHOMOGENEOUS CAUCHY PROBLEMS ON THE HALF-LINE

WOLFGANG ARENDT; CHARLES J. K. BATTY

doi:10.1112/S0024609398005657

Abstract

Let u be a bounded, uniformly continuous, mild solution of an inhomogeneous Cauchy problem on R+: u′(t)=Au(t)+ϕ(t) (t[ges ]0). Suppose that u has uniformly convergent means, σ(A)∩iR is countable, and ϕ is asymptotically almost periodic. Then u is asymptotically almost periodic. Related results have been obtained by Ruess and Vũ, and by Basit, using different methods. A direct proof is given of a Tauberian theorem of Batty, van Neerven and Räbiger, and applications to Volterra equations are discussed.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Batty, Charles J.K. Hutter, Walter and Räbiger, Frank 1999. Almost Periodicity of Mild Solutions of Inhomogeneous Periodic Cauchy Problems. Journal of Differential Equations, Vol. 156, Issue. 2, p. 309.

Chill, Ralph and Pr�ss, Jan 2001. Asymptotic behaviour of linear evolutionary integral equations. Integral Equations and Operator Theory, Vol. 39, Issue. 2, p. 193.

NAITO, Toshiki MINH, Nguyen Van and SHIN, Jong Son 2002. A Massera type theorem for functional differential equations with infinite delay. Japanese journal of mathematics. New series, Vol. 28, Issue. 1, p. 31.

Batty, Charles J.K. Chill, Ralph and Tomilov, Yuri 2002. Strong Stability of Bounded Evolution Families and Semigroups. Journal of Functional Analysis, Vol. 193, Issue. 1, p. 116.

Nishikawa, Takeshi Van Minh, Nguyen and Naito, Toshiki 2004. On the Asymptotic Periodic Solutions of Abstract Functional Differential Equations. Funkcialaj Ekvacioj, Vol. 47, Issue. 2, p. 307.

Basit, Bolis and Pryde, A. J. 2004. Ergodicity and stability of orbits of unbounded semigroup representations. Journal of the Australian Mathematical Society, Vol. 77, Issue. 2, p. 209.

Basit, Bolis and Günzler, Hans 2008. Relations between different types of spectra and spectral characterizations. Semigroup Forum, Vol. 76, Issue. 2, p. 217.

Liu, Qing and Yuan, Rong 2009. Asymptotic behavior of solutions to abstract functional differential equations. Journal of Mathematical Analysis and Applications, Vol. 356, Issue. 2, p. 405.

Nguyen, Van Minh 2009. A spectral theory of continuous functions and the Loomis–Arendt–Batty–Vu theory on the asymptotic behavior of solutions of evolution equations. Journal of Differential Equations, Vol. 247, Issue. 4, p. 1249.

de Andrade, Bruno and Lizama, Carlos 2011. Existence of asymptotically almost periodic solutions for damped wave equations. Journal of Mathematical Analysis and Applications, Vol. 382, Issue. 2, p. 761.

Cao, Junfei Yang, Qigui Huang, Zaitang and Liu, Qing 2011. Asymptotically almost periodic solutions of stochastic functional differential equations. Applied Mathematics and Computation, Vol. 218, Issue. 5, p. 1499.

Van Minh, Nguyen 2013. On the asymptotic behaviour of Volterra difference equations. Journal of Difference Equations and Applications, Vol. 19, Issue. 8, p. 1317.

Ponce, Rodrigo 2013. Bounded mild solutions to fractional integro-differential equations in Banach spaces. Semigroup Forum, Vol. 87, Issue. 2, p. 377.

Liu, Aimin Liu, Yongjian and Liu, Qun 2014. Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations. Abstract and Applied Analysis, Vol. 2014, Issue. , p. 1.

Basit, Bolis and Pryde, Alan J. 2014. Comparison of spectra of absolutely regular distributions and applications. Rocky Mountain Journal of Mathematics, Vol. 44, Issue. 2,

Dieu, Bui Xuan Siegmund, Stefan and Van Minh, Nguyen 2015. A Katznelson–Tzafriri Type Theorem for Almost Periodic Linear Evolution Equations. Vietnam Journal of Mathematics, Vol. 43, Issue. 2, p. 403.

Dieu, Bui Xuan Duc, Luu Hoang Siegmund, Stefan and Van Minh, Nguyen 2016. Mathematical Sciences with Multidisciplinary Applications. Vol. 157, Issue. , p. 113.

Cao, Junfei Debbouche, Amar and Zhou, Yong 2018. Asymptotically Almost Periodicity for a Class of Weyl–Liouville fractional Evolution Equations. Mediterranean Journal of Mathematics, Vol. 15, Issue. 4,

Baskakov, A. G. Strukova, I. I. and Trishina, I. A. 2018. Solutions Almost Periodic at Infinity to Differential Equations With Unbounded Operator Coefficients. Siberian Mathematical Journal, Vol. 59, Issue. 2, p. 231.

Baskakov, Anatoly Obukhovskii, Valeri and Zecca, Pietro 2018. Almost periodic solutions at infinity of differential equations and inclusions. Journal of Mathematical Analysis and Applications, Vol. 462, Issue. 1, p. 747.

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ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS OF INHOMOGENEOUS CAUCHY PROBLEMS ON THE HALF-LINE

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ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS OF INHOMOGENEOUS CAUCHY PROBLEMS ON THE HALF-LINE

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