## Rocky Mountain Journal of Mathematics

### Existence of pseudo almost automorphic mild solutions to some nonautonomous second order differential equations

Toka Diagana

#### Article information

Source
Rocky Mountain J. Math. Volume 43, Number 3 (2013), 793-824.

Dates
First available in Project Euclid: 1 August 2013

https://projecteuclid.org/euclid.rmjm/1375361975

Digital Object Identifier
doi:10.1216/RMJ-2013-43-3-793

Mathematical Reviews number (MathSciNet)
MR3093266

Zentralblatt MATH identifier
1291.34102

#### Citation

Diagana, Toka. Existence of pseudo almost automorphic mild solutions to some nonautonomous second order differential equations. Rocky Mountain J. Math. 43 (2013), no. 3, 793--824. doi:10.1216/RMJ-2013-43-3-793. https://projecteuclid.org/euclid.rmjm/1375361975

#### References

• P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integ. Equat. 1 (1988), 433-457.
• P. Acquistapace, F. Flandoli and B. Terreni, Initial boundary value problems and optimal control for nonautonomous parabolic systems, SIAM J. Contr. Optim. 29 (1991), 89-118.
• P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova 78 (1987), 47-107.
• H. Amann, Linear and quasilinear parabolic problems, Birkhäuser, Berlin, 1995.
• M. Baroun, S. Boulite, T. Diagana and L. Maniar, Almost periodic solutions to some semilinear non-autonomous thermoelastic plate equations, J. Math. Anal. Appl. 349 (2009), 74-84.
• M. Baroun, S. Boulite, G.M. N'Guérékata and L. Maniar, Almost automorphy of semilinear parabolic equations, Electron. J. Differ. Equat. 2008 (2008), 1-9.
• C.J.K. Batty, J. Liang and T.J. Xiao, On the spectral and growth bound of semigroups associated with hyperbolic equations, Adv. Math. 191 (2005), 1-10.
• S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Nat. Acad. Sci. 52 (1964), 907-910.
• D. Bugajewski and T. Diagana, Almost automorphy of the convolution operator and applications to differential and functional-differential equations, Nonlinear Stud. 13 (2006), 129-140.
• D. Bugajewski, T. Diagana and C.M. Mahop, Asymptotic and pseudo almost periodicity of the convolution operator and applications to differential and integral equations, Z. Anal. Anwend. 25 (2006), 327-340.
• P. Cieutat and K. Ezzinbi, Existence, uniqueness and attractiveness of a pseudo almost automorphic solutions for some dissipative differential equations in Banach spaces, J. Math. Anal. Appl. 354 (2009), 494-506.
• T. Diagana, Existence of pseudo-almost automorphic solutions to some abstract differential equations with $S^p$-pseudo-almost automorphic coefficients, Nonlinear Anal. 70 (2009), 3781-3790.
• –––, Almost automorphic solutions to some damped second-order differential equations, Comm. Nonlin. Sci. Num. Sim. 17 (2012), 4074-4084.
• –––, Pseudo almost periodic functions in Banach spaces, Nova Sci. Publ., Inc., New York, 2007.
• T. Diagana, Existence of almost automorphic solutions to some classes of nonautonomous higher-order differential equations, Electron. J. Qual. Theor. Diff. Equat. 22 (2010), 1-26.
• –––, Almost automorphic mild solutions to some classes of nonautonomous higher-order differential equations, Semigroup Forum 82 (2011), 455-477.
• –––, Erratum to: Almost automorphic mild solutions to some classes of nonautonomous higher-order differential equations, Semigroup Forum 87 (2013), 275-276.
• K.J. Engel and R. Nagel, One parameter semigroups for linear evolution equations, Grad. Texts Math., Springer Verlag, New York, 1999.
• K. Ezzinbi, S. Fatajou and G.M. N’Guérékata, Pseudo almost automorphic solutions to some neutral partial functional differential equations in Banach space, Nonlinear Anal. 70 (2009), 1641-1647.
• K. Ezzinbi, S. Fatajou and G.M. N’Guérékata, Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces, J. Math. Anal. Appl. 351 (2009), 765-772.
• J.A. Goldstein and G.M. N'Guérékata, Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc. 133 (2005), 2401-2408.
• –––, Corrigendum on Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc. 140 (2012), 1111-1112.
• H. Leiva, Existence of bounded solutions solutions of a second-order system with dissipation, J. Math. Anal. Appl. 237 (1999), 288-302.
• J. Liang, R. Nagel and T.J. Xiao, Nonautonomous heat equations with generalized Wentzell boundary conditions, J. Evol. Equat. 3 (2003), 321-331.
• –––, Approximation theorems for the propagators of higher order abstract Cauchy problems, Trans. Amer. Math. Soc. 360 (2008), 1723-1739.
• J. Liang, G.M. N'Guérékata, T-J. Xiao and J. Zhang, Some properties of pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Anal. 70 (2009), 2731-2735.
• J. Liang, J. Zhang and T-J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2008), 1493-1499.
• A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, PNLDE 16, Birkhäauser Verlag, Basel, 1995.
• G.M. N'Guérékata, Almost automorphic functions and almost periodic functions in abstract spaces, Kluwer Academic/Plenum Publishers, New York, 2001.
• –––, Topics in almost automorphy, Springer, New York, 2005.
• R. Schnaubelt, Sufficient conditions for exponential stability and dichotomy of evolution equations, Forum Math. 11 (1999), 543-566.
• T.J. Xiao and J. Liang, The Cauchy problem for higher-order abstract differential equations, Lect. Notes Math. 1701, Springer-Verlag, Berlin, 1998.
• –––, A solution to an open problem for wave equations with generalized Wentzell boundary conditions, Math. Ann. 327 (2003), 351-363.
• –––, Complete second order differential equations in Banach spaces with dynamic boundary conditions, J. Differential Equat. 200 (2004), 105-136.
• –––, Complete second order differential equations in Banach spaces with dynamic boundary conditions, J. Differential Equat. 200 (2004), 105-136.
• –––, Second order differential operators with Feller-Wentzell type boundary conditions, J. Funct. Anal. 254 (2008), 1467-1486.
• T-J. Xiao, J. Liang and J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum 76 (2008), 518-524.
• Ti-J. Xiao, X-X. Zhu and J. Liang, Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear Anal. 70 (2009), 4079-4085.
• A. Yagi, Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II, Funk. Ekvac. 33 (1990), 139-150.
• –––, Abstract quasilinear evolution equations of parabolic type in Banach spaces, Boll. Un. Mat. Ital. 5 (1991), 341-368. \noindentstyle