×

Almost automorphic mild solutions to some partial neutral functional-differential equations and applications. (English) Zbl 1162.34062

This paper deals with the existence and uniqueness of almost automorhic mild solutions to the first order partial neutral functional-differential equations of the form
\[ \frac{d}{dt}[u(t)+f(t,u_t)]=Au(t)+g(t,u_t),\quad t\in [\sigma , \sigma +a],\quad u_{\phi}=\phi, \]
where \(A:D(A)\subset X\to X\) is the infinitesimal generator of a uniformly exponentially stable semigroup of linear operators on a Banach space \(X\), \(u_t(\theta )= u(t+\theta )\) for \(\theta \in (-\infty, 0]\) and \(f\), \(g\) are continuous functions. Sufficient conditions for the existence of uniqueness of almost automorphic mild solutions to the above equations are obtained. As an application, a first-order boundary value problem arising in control systems is considered.

MSC:

34K30 Functional-differential equations in abstract spaces
35R10 Partial functional-differential equations
34K40 Neutral functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Adimy, M.; Ezzinbi, K., Strict solutions of nonlinear hyperbolic neutral differential equations, Appl. Math. Lett., 12, 1, 107-112 (1999) · Zbl 0941.34075
[2] Bochner, S., A new approach to almost periodicity, Proc. Natl. Acad. Sci. USA, 48, 2039-2043 (1962) · Zbl 0112.31401
[3] Diagana, T.; N’Guérékata, G. M.; Minh, N. V., Almost automorphic solutions of evolution equations, Proc. Amer. Math. Soc., 132, 11, 3289-3298 (2004) · Zbl 1053.34050
[4] Liu, J.; N’Guérékata, G.; van Minh, N., A Massera type theorem for almost automorphic solutions of differential equations, J. Math. Anal. Appl., 299, 2, 587-599 (2004) · Zbl 1081.34054
[5] Hale, J. K.; Verduyn Lunel, S. M., (Introduction to Functional-Differential Equations. Introduction to Functional-Differential Equations, Applied Mathematical Sciences, vol. 99 (1993), Springer-Verlag: Springer-Verlag New York) · Zbl 0787.34002
[6] Hale, J. K., Partial neutral functional-differential equations, Rev. Roumaine Math. Pures Appl., 39, 4, 339-344 (1994) · Zbl 0817.35119
[7] Hernández, E.; Henríquez, H. R., Existence results for partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl., 221, 2, 452-475 (1998) · Zbl 0915.35110
[8] Hernández, E.; Henríquez, H. R., Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl., 221, 2, 499-522 (1998) · Zbl 0926.35151
[9] Hernández, E., Existence results for partial neutral integrodifferential equations with unbounded delay, J. Math. Anal. Appl., 292, 1, 194-210 (2004) · Zbl 1056.45012
[10] Hernández, E. M.; Pelicer, M. L., Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations, Appl. Math. Lett., 18, 11, 1265-1272 (2005) · Zbl 1102.34064
[11] Hino, Y.; Murakami, S., Almost automorphic solutions for abstract functional differential equations, J. Math. Anal. Appl., 286, 741-752 (2003) · Zbl 1046.34088
[12] Hino, Y.; Murakami, S.; Naito, T., (Functional-Differential Equations with Infinite Delay. Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473 (1991), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0732.34051
[13] Minh Man, N.; Van Minh, N., On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations, Commun. Pure Appl. Anal., 3, 2, 291-300 (2004) · Zbl 1083.34053
[14] N’Guérékata, G. M., Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces (2001), Kluwer Academic/Plenum Publishers: Kluwer Academic/Plenum Publishers New York, London, Moscow · Zbl 1001.43001
[15] N’Guérékata, G. M., Topics in Almost Automorphy (2005), Springer: Springer New York, Boston, Dordrecht, London, Moscow · Zbl 1073.43004
[16] Yuan, R., Existence of almost periodic solutions of neutral functional-differential equations via Liapunov-Razumikhin function, Z. Angew. Math. Phys., 49, 1, 113-136 (1998) · Zbl 0895.34056
[17] Wu, J.; Xia, H., Rotating waves in neutral partial functional-differential equations, J. Dynam. Differential Equations, 11, 2, 209-238 (1999) · Zbl 0939.35188
[18] Wu, J.; Xia, H., Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Differential Equations, 124, 1, 247-278 (1996) · Zbl 0840.34080
[19] Wu, J., (Theory and Applications of Partial Functional-Differential Equations. Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Sciences, vol. 119 (1996), Springer-Verlag: Springer-Verlag New York) · Zbl 0870.35116
[20] Veech, W. A., Almost automorphic functions, Proc. Natl. Acad. Sci. USA, 49, 462-464 (1963) · Zbl 0173.33402
[21] Zaidman, S., Almost automorphic solutions of some abstract evolution equations. II, Istit. Lombardo Accad. Sci. Lett. Rend. A, 111, 2, 260-272 (1977) · Zbl 0432.34038
[22] Zaki, M., Almost automorphic solutions of certain abstract differential equations, Ann. Mat. Pura Appl., 101, 4, 91-114 (1974) · Zbl 0304.42028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.