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Bounded mild solutions for semilinear integro differential equations in Banach spaces. (English) Zbl 1209.45007

The authors study the structure of several classes of spaces of vector-valued functions (;X); here X denotes a Banach space. The integro-differential equation

u ' (t)=Au(t)+ - t a(t-s)Au(s)ds+f(t,u(t))(1)

is considered, where A is a closed linear operator defined in X and aL loc 1 ( + ) is a scalar-valued kernel. Using a unified approach for various spaces (;X), the authors establish conditions on A and f ensuring that the solution u of (1) exists and has the same asymptotic behaviour as f. In particular, almost automorphic, pseudo-almost automorphic, asymptotically periodic and almost periodic classes of functions are investigated. Moreover, asymptotically compact almost automorphic functions and pseudo compact almost automorphic functions are introduced in the paper.

MSC:
45N05Abstract integral equations, integral equations in abstract spaces
43A60Almost periodic functions on groups, etc.; almost automorphic functions
45J05Integro-ordinary differential equations
45G05Singular nonlinear integral equations
References:
[1]Amir B., Maniar L.: Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain. Ann. Math. Blaise Pascal 6(1), 1–11 (1999)
[2]Andres J., Bersani A.M., Grande R.F.: Hierarchy of almost periodic function spaces. Rend. Mat. Ser. VII Roma 26, 121–188 (2006)
[3]Araya D., Lizama C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69(11), 3692–3705 (2008) · Zbl 1166.34033 · doi:10.1016/j.na.2007.10.004
[4]Blot J., Cieutat P., N’Guérékata G.M., Pennequin D.: Superposition operators between various almost periodic function spaces and applications. Commun. Math. Anal. 6(1), 42–70 (2009)
[5]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
[6]Bohr H.: Zur theorie der fast periodischen funktionen. (German) I. Eine verallgemeinerung der theorie der fourierreihen. Acta Math. 45(1), 29–127 (1925) · Zbl 02595718 · doi:10.1007/BF02395468
[7]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, 1333–1345 (2004)
[8]Clément Ph., Da Prato G.: Existence and regularity results for an integral equation with infinite delay in a Banach space. Integr. Equ. Oper. Theory 11, 480–500 (1988) · Zbl 0668.45010 · doi:10.1007/BF01199303
[9]Coleman B.D., Gurtin M.E.: Equipresence and constitutive equation for rigid heat conductors. Z. Angew. Math. Phys. 18, 199–208 (1967) · doi:10.1007/BF01596912
[10]Cuevas C., Lizama C.: Almost automorphic solutions to a class of semilinear fractional differential equations. Appl. Math. Lett. 21, 1315–1319 (2008) · Zbl 1192.34006 · doi:10.1016/j.aml.2008.02.001
[11]Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297–308 (1970) · Zbl 0214.24503 · doi:10.1007/BF00251609
[12]Dafermos C.M.: An abstract Volterra equation with applications to linear viscoelasticity. J. Differ. Equ. 7, 554–569 (1970) · Zbl 0212.45302 · doi:10.1016/0022-0396(70)90101-4
[13]Da Prato G., Lunardi A.: Periodic solutions for linear integrodifferential equations with infinite delay in Banach spaces. Differential Equations in Banach spaces. Lect. Notes Math. 1223, 49–60 (1985)
[14]Da Prato G., Lunardi A.: Solvability on the real line of a class of linear Volterra integrodifferential equations of parabolic type. Ann. Math. Pura Appl. 4, 67–117 (1988)
[15]De Bruijn N.G.: The asymptotically periodic behavior of the solutions of some linear functional equations. Am. J. Math. 71, 313–330 (1949) · Zbl 0033.27002 · doi:10.2307/2372246
[16]Diagana T.: weighted pseudo almost automorphic functions and applications. C. R. Acad. Sci. Paris Ser. I 343, 643–646 (2006)
[17]Diagana T., Hernández E., Dos Santos J.P.C.: Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations. Nonlinear Anal. 71, 248–257 (2009) · Zbl 1172.45002 · doi:10.1016/j.na.2008.10.046
[18]Diagana T., Henríquez H., Hernández E.: Almost automorphic mild solutions to some partial neutral functional–differential equations and applications. Nonlinear Anal. 69(5-6), 1485–1493 (2008) · Zbl 1162.34062 · doi:10.1016/j.na.2007.06.048
[19]Ding H.-S., Liang J., Xiao T.-J.: Asymptotically almost automorphic solutions for some integro-differential equations with nonlocal conditions. J. Math. Anal. Appl. 338(1), 141–151 (2008) · Zbl 1142.45005 · doi:10.1016/j.jmaa.2007.05.014
[20]Fašangová E., Prüss J.: Asymptotic behaviour of a semilinear viscoelastic beam model. Arch. Math. (Basel) 77, 488–497 (2001)
[21]Fink A.M.: Almost automorphic and almost periodic solutions which minimize functionals. Tôhoku Math. J. 20(2), 323–332 (1968) · Zbl 0177.12102 · doi:10.2748/tmj/1178243139
[22]Fréchet M.: Les fonctions asymptotiquement presque-periodiques continues (French). C. R. Acad. Sci. Paris 213, 520–522 (1941)
[23]Gao H., Wang K., Wei F., Ding X.: Massera-type theorem and asymptotically periodic logistic equations. Nonlinear Anal. Real World Appl. 7, 1268–1283 (2006) · Zbl 1162.34325 · doi:10.1016/j.nonrwa.2005.11.008
[24]Gripenberg G., Londen S.-O., Staffans O.: Volterra Integral And Functional Equations. In: Encyclopedia of Mathematics and its Applications, vol. 34. Cambridge University Press, Cambridge (1990)
[25]Henríquez H.R., Lizama C.: Compact almost periodic solutions to integral equations with infinite delay. Nonlinear Anal. 71, 6029–6037 (2009) · Zbl 1179.43004 · doi:10.1016/j.na.2009.05.042
[26]Henríquez H.R., Pierri M., Táboas P.: Existence of S-asymptotically ω-periodic solutions for abstract neutral equations. Bull. Aust. Math. Soc. 78, 365–382 (2008) · Zbl 1183.34122 · doi:10.1017/S0004972708000713
[27]Henríquez H.R., Pierri M., Táboas P.: On S-asymptotically ω-periodic functions on Banach spaces and applications. J. Math. Anal. Appl. 343(2), 1119–1130 (2008) · Zbl 1146.43004 · doi:10.1016/j.jmaa.2008.02.023
[28]Levitan, B.M., Zhikov, V.V.: Almost Periodic Functions and Differential Equations. Moscow University House, Moscow (1978) (English Translation by Cambridge University Press, 1982)
[29]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
[30]Liang J., Zhang J., Xiao T.J.: Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J. Math. Anal. Appl. 340(2), 1493–1499 (2008) · Zbl 1134.43001 · doi:10.1016/j.jmaa.2007.09.065
[31]Lizama C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, 278–292 (2000) · Zbl 0952.45005 · doi:10.1006/jmaa.1999.6668
[32]Mophou, G.M., N’Guérékata, G.M.: On some classes of almost automorphic functions and applications to fractional differential equations (2010, to appear)
[33]N’Guérékata G.M.: Topics in Almost Automorphy. Springer, New York (2005)
[34]N’Guérékata G.M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer, New York (2001)
[35]N’Guérékata G.M.: Quelques remarques sur les fonctions asymptotiquement presque automorphes (French). Ann. Sci. Math. Quebec 7(2), 185–191 (1983)
[36]N’Guérékata G.M.: Comments on almost automorphic and almost periodic functions in Banach spaces. Far East J. Math. Sci. (FJMS) 17(3), 337–344 (2005)
[37]N’Guérékata G.M.: Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations. Semigroup Forum 69, 80–86 (2004) · Zbl 1077.47058 · doi:10.1007/s00233-003-0021-0
[38]Nguyen V.M., Naito T., N’Guérékata G.M.: A spectral countability condition for almost automorphy of solutions of differential equations. Proc. Am. Math. Soc. 134, 3257–3266 (2006) · Zbl 1120.34044 · doi:10.1090/S0002-9939-06-08528-5
[39]Nunziato J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–304 (1971)
[40]Prüss, J.: Evolutionary integral equations and applications. In: Monographs in Mathematics, vol. 87. Birkhäuser, Boston (1993)
[41]Renardy, M., Hrusa, W.J., Nohel, J.A.: Mathematical problems in viscoelasticity. In: Pitman Monographs Pure Applied Mathematics, vol. 35. Longman Sci. Tech., Harlow, Essex (1988)
[42]Sforza D.: Existence in the large for a semilinear integrodifferential equation with infinite delay. J. Differ. Equ. 120, 289–303 (1995) · Zbl 0832.45012 · doi:10.1006/jdeq.1995.1113
[43]Xiao T.J., Liang J., Zhang J.: Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 76(3), 518–524 (2008) · Zbl 1154.46023 · doi:10.1007/s00233-007-9011-y
[44]Yuan R.: Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument. Nonlinear Anal. 41(7–8), 871–890 (2000) · Zbl 1024.34068 · doi:10.1016/S0362-546X(98)00316-2
[45]Zhang C.Y.: Integration of vector-valued pseudo-almost periodic functions. Proc. Am. Math. Soc. 121(1), 167–174 (1994) · doi:10.1090/S0002-9939-1994-1186140-8