Almost automorphic solutions of semilinear evolution equations
HTML articles powered by AMS MathViewer
- by Jerome A. Goldstein and Gaston M. N’Guérékata PDF
- Proc. Amer. Math. Soc. 133 (2005), 2401-2408 Request permission
Abstract:
We are concerned with the semilinear differential equation in a Banach space $\mathbb {X}$, \[ x’(t)=Ax(t)+F(t,x(t)),\;\ t\in \mathbb {R} ,\] where $A$ generates an exponentially stable $C_0$-semigroup and $F(t,x): \mathbb {R} \times \mathbb {X} \to \mathbb {X}$ is a function of the form $F(t,x)=P(t)Q(x)$. Under appropriate conditions on $P$ and $Q$, and using the Schauder fixed point theorem, we prove the existence of an almost automorphic mild solution to the above equation.References
- S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Nat. Acad. Sci. U.S.A. 52 (1964), 907–910. MR 168997, DOI 10.1073/pnas.52.4.907
- Toka Diagana, Gaston Nguerekata, and Nguyen Van Minh, Almost automorphic solutions of evolution equations, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3289–3298. MR 2073304, DOI 10.1090/S0002-9939-04-07571-9
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
- Gaston M. N’Guérékata, Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup Forum 69 (2004), no. 1, 80–86. MR 2063980, DOI 10.1007/s00233-003-0021-0
- Gaston M. N’Guerekata, Almost automorphic and almost periodic functions in abstract spaces, Kluwer Academic/Plenum Publishers, New York, 2001. MR 1880351, DOI 10.1007/978-1-4757-4482-8
Additional Information
- Jerome A. Goldstein
- Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152-3240
- MR Author ID: 74805
- Email: jgoldste@memphis.edu
- Gaston M. N’Guérékata
- Affiliation: Department of Mathematics, Morgan State University, Baltimore, Maryland 21251
- ORCID: 0000-0001-5765-7175
- Email: gnguerek@jewel.morgan.edu
- Received by editor(s): February 11, 2004
- Received by editor(s) in revised form: April 12, 2004
- Published electronically: March 4, 2005
- Communicated by: Carmen C. Chicone
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 2401-2408
- MSC (2000): Primary 34A05, 34K05, 47D60, 34G20
- DOI: https://doi.org/10.1090/S0002-9939-05-07790-7
- MathSciNet review: 2138883