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Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations. (English) Zbl 1172.45002

The authors consider the equation
\[ \frac{d}{dt}D(t,u_t)= AD(t,u_t)+\int^t_0B(t-s)D(s,u_s)\,ds+g(t,u_t), \quad t\in[\sigma,\sigma +a), \quad u_\sigma=\varphi\in{\mathcal B}, \]
in a Banach space \(X\), where \(D(t,x_t)=x(0)+f(t,x_t)\), \(A,B(t)\) being bounded operators on \(X\), with common domain \(D(A)\), independent on \(t\). The history space \({\mathcal B}\) is the fading memory space (Hale and Kato). The functions \(f\) and \(g\) are subject to several conditions. By denoting \(D(t,u_t)= v(t)\), one obtains an integrodifferential equation for which a resolvent formula is applied. That’s how the mild solutions are defined. The almost automorphic functions are those representable as \(u=v+h\), where \(v\) is automorphic while \(h\in C_0\). The uniqueness is also discussed and an example is provided.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
47G20 Integro-differential operators
45K05 Integro-partial differential equations
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[1] Bochner, S., A new approach to almost periodicity, Proc. Natl. Acad. Sci. USA, 48, 2039-2043 (1962) · Zbl 0112.31401
[2] Bugajewski, D.; N’Guérékata, G. M., On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces, Nonlinear Anal., 59, 8, 1333-1345 (2004) · Zbl 1071.34055
[3] Bugajewski, D.; Diagana, T., Almost automorphy of the convolution operator and applications to differential and functional-differential equations, Nonlinear Stud., 13, 2, 129-140 (2006) · Zbl 1102.44007
[4] Chen, G., Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim., 17, 1, 66-81 (1979) · Zbl 0402.93016
[5] Desch, W.; Grimmer, R.; Schappacher, W., Well-posedness and wave propagation for a class of integrodifferential equations in Banach space, J. Differential Equations, 74, 20, 391-411 (1988) · Zbl 0663.45008
[6] Diagana, T.; Henríquez, H. R.; Hernández, E., Almost automorphic mild solutions to some partial neutral functional-differential Equations and Applications., Nonlinear Anal., 69, 5, 1485-1493 (2008) · Zbl 1162.34062
[7] Diagana, T.; N’Guérékata, G. M., Almost automorphic solutions to semilinear evolution equations, Funct. Differ. Equ., 13, 2, 195-206 (2006) · Zbl 1102.34044
[8] Diagana, T.; N’Guérékata, G. M., Almost automorphic solutions to some classes of partial evolution equations, Appl. Math. Lett., 20, 4, 462-466 (2007) · Zbl 1169.35300
[9] 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
[10] Ding, H. S.; Xiao, T.; Liang, J., Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, J. Math. Anal. Appl., 338, 1, 141-151 (2008) · Zbl 1142.45005
[11] Ezzinbi, K.; N’Guérékata, G. M., A Massera type theorem for almost automorphic solutions of functional differential equations of neutral type, J. Math. Anal. Appl., 316, 707-721 (2006) · Zbl 1122.34052
[12] Grimmer, R., Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273, 1, 333-349 (1982) · Zbl 0493.45015
[13] Grimmer, R.; Prüss, J., On linear Volterra equations in Banach spaces. Hyperbolic partial differential equations II, Comput. Math. Appl., 11, 189-205 (1985) · Zbl 0569.45020
[14] 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
[15] Hale, J. K., Partial neutral functional-differential equations, Rev. Roumaine Math. Pures Appl., 39, 4, 339-344 (1994) · Zbl 0817.35119
[16] Henríquez, H. R.; Hernández, E.; dos Santos, J. P.C., Asymptotically almost periodic and almost periodic solutions for partial neutral integrodifferential equations, Z. Anal. Anwend., 26, 3, 261-375 (2007) · Zbl 1139.34051
[17] 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
[18] 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
[19] Hernández, E.; dos Santos, J. P.C., Existence results for partial neutral integro differential equations with unbounded delay, Appl. Anal., 86, 2, 223-237 (2007) · Zbl 1168.47306
[20] Hernández, E., Existence results for partial neutral integrodifferential equations with unbounded delay, J. Math. Anal. Appl., 292, 1, 194-210 (2004) · Zbl 1056.45012
[21] Hino, Y.; Murakami, S., Almost automorphic solutions for abstract functional differential equations, J. Math. Anal. Appl., 286, 741-752 (2003) · Zbl 1046.34088
[22] 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
[23] N’Guérékata, G. M., Sur les solutions presque automorphes d’équations différentielles abstraites, Ann. Sci. Math. Québec., 1, 69-79 (1981) · Zbl 0494.34045
[24] 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
[25] N’Guérékata, G. M., Topics in Almost Automorphy (2005), Springer: Springer New York, Boston, Dordrecht, London, Moscow · Zbl 1073.43004
[26] Veech, W. A., Almost automorphic functions, Proc. Natl. Acad. Sci. USA, 49, 462-464 (1963) · Zbl 0173.33402
[27] Wu, J.; Xia, H., Rotating waves in neutral partial functional-differential equations, J. Dynam. Differential Equations, 11, 2, 209-238 (1999) · Zbl 0939.35188
[28] 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
[29] 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
[30] 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
[31] Zaki, M., Almost automorphic solutions of certain abstract differential equations, Ann. Mat. Pura Appl., 101, 4, 91-114 (1974) · Zbl 0304.42028
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