CUBO A Mathematical Journal Vol.13, N^o03, (117-139). October 2011

 

Pseudo-Almost Automorphic Solutions to Some Second-Order Differential Equations

 

Toka Diagana and Ahmed Mohamed

Department of Mathematics, Howard University, 2441 6th Street N. W., Washington, D.C. 20059 - USA. email: tdiagana@howard.edu

Department ofMathematics, Howard University, 2441 6th Street N. W., Washington, D.C. 20059 - USA. email: ahmohamed@howard.edu

---

ABSTRACT

In this paper we study and obtain the existence of pseudo-almost automorphic solutions to some classes of second-order abstract differential equations on a Hilbert space. To illustrate our abstract results, we discuss the existence of pseudo almost automorphic solutions to the N-dimensional Sine-Gordon boundary value problem.

Keywords. exponential stability, sectorial operator, hyperbolic semigroup, almost automorphic; pseudo-almost automorphic; autonomous second-order differential equation; Sine-Gordon equation. amenability, Banach modules, module amenability, weak module amenability, semigroup algebra, inverse semigroup.

Mathematics Subject Classification: 43A60; 34B05; 34C27; 42A75; 47D06; 35L90.

---

RESUMEN

En este trabajo se estudia y obtiene la existencia de soluciones casi-seudo automorfas a algunas clases de ecuaciones diferenciales abstractas de segundo orden en un espacio de Hilbert. Para ilustrar nuestros resultados abstractos, se discute la existencia de soluciones casi-seudo automorfas en el problema de contorno N-dimensional de Sine-Gordon

---

References

[1] P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations. Differential Integral Equations 1 (1988), pp. 433-457.         [ Links ]

[2] P. Acquistapace, F. Flandoli, B. Terreni, Initial boundary value problems and optimal control for nonautonomous parabolic systems. SIAM J. Control Optim. 29 (1991), pp. 89-118.         [ Links ]

[3] P. Acquistapace, B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations. Rend. Sem. Mat. Univ. Padova 78 (1987), pp. 47-107.         [ Links ]

[4] J. M. Alonso and R. Ortega, Global Aymptotic Stability of a Forced Newtonian System with Dissipation. J. Math. Anal. Appl. (1995), pp. 965-986.         [ Links ]

[5] J. M. Alonso, J. Mawhin, and R. Ortega, Bounded Solutions of Second-Order Semilinear Evolution Equations and Applications to the Telegraph Equation. J. Math. Pures Appl. (1999), pp. 43-63.         [ Links ]

[6] H. Amann, Linear and Quasilinear Parabolic Problems, Birkhauser, Berlin 1995.         [ Links ]

[7] B. Amir and L. Maniar, Existence and some asymptotic behaviors of solutions to semilinear Cauchy problems with non dense domain via extrapolation spaces, Rend. Circ. Mat. Palermo (2000), pp. 481-496.         [ Links ]

[8] B. Amir and L. Maniar, Composition of Pseudo-Almost Periodic Functions and Cauchy Problems with Perator of Nondense Domain. Ann. Math. Blaise Pascal 6 (1999), no. 1, pp. 1-11.         [ Links ]

[9] W. Arendt, R. Chill, S. Fornaro, and C. Poupaud, L^p-Maximal Regularity for Non-Autonomous Evolution Equations. J. Differential Equations 237 (2007), no. 1, pp. 1-26.         [ Links ]

[10] W. Arendt and C. J. K. Batty, Almost Periodic Solutions of First- and Second-Order Cauchy Problems. J. Differential Equations 137 (1997), no. 2, pp. 363-383.         [ Links ]

[11] J. Blot, P. Cieutat, and J. Mawhin, Almost Periodic Oscillation of Monotone Second-Order Systems. Advances Diff. Equ. 2 (1997), no. 5, pp. 693-714.         [ Links ]

[12] M. Baroun, S. Boulite, T. Diagana, and L. Maniar, Almost periodic solutions to some semi-linear non-autonomous thermoelastic plate equations. J. Math. Anal. Appl. 349(2009), no. 1, pp. 74-84.         [ Links ]

[13] M. Baroun, S. Boulite, G. M. N'Guerékata, and L. Maniar, Almost automorphy of Semilinear Parabolic Equations. Electron. J. Differential Equations 2008(2008), no. 60, pp. 1-9.         [ Links ]

[14] S. Boulite, L. Maniar, and G. M. N'Guerekata, Almost Automorphic Solutions for Hyperbolic Semilniear Evolution Equations, Semigroup Forum. Vol. 71 (2005), pp. 231-240.         [ Links ]

[15] D. Bugajewski and T. Diagana, Almost Automorphy of the Convolution Operator and Applications to Differential and Functional-Differential Equations, Nonlinear Stud. 13 (2006), no. 2, pp. 129-140.         [ Links ]

[16] D. Bugajewski, T. Diagana, C. M. Mahop, Asymptotic and Pseudo Almost Periodicity of the Convolution Operator and Applications to Differential and Integral Equations. Z. Anal. Anwend. 25 (2006), no. 3, pp. 327-340.         [ Links ]

[17] C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations. Amer. Math. Soc., 1999.         [ Links ]

[18] P. Cieutat and K. Ezzinbi, Existence, uniqueness and attractiveness of a pseudo almost auto-morphic solutions for some dissipative differential equations in Banach spaces. J. Math. Anal. Appl. 354 (2009), no. 2, pp. 494-506.         [ Links ]

[19] G. Da Prato and P. Grisvard, Equations devolution abstraites non lineaires de type parabolique. Ann. Mat. Pura Appl. (4) 120 (1979), pp. 329-396.

[20] T. Diagana, Existence of pseudo-almost automorphic solutions to some abstract differential equations with S^p-pseudo-almost automorphic coefficients. Nonlinear Anal. 70 (2009), no. 11, pp. 3781-3790.         [ Links ]

[21] T. Diagana, Pseudo almost periodic functions in Banach spaces. Nova Science Publishers, Inc., New York, 2007.         [ Links ]

[22] T. Diagana and E. Hernandez M., Existence and Uniqueness of Pseudo Almost Periodic Solutions to Some Abstract Partial Neutral Functional-Differential Equations and Applications, J. Math. Anal. Appl. 327(2007), no. 2, pp. 776-791.         [ Links ]

[23] T. Diagana, Existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. Electron. J. Qual. Theory Differ. Equ. 2007, No. 3, 12 pp.         [ Links ]

[24] T. Diagana, Existence and Uniqueness of Pseudo Almost Periodic Solutions to Some Classes of Partial Evolution Equations. Nonlinear Anal. 66 (2007), no. 2, pp. 384-395.         [ Links ]

[25] T. Diagana and G. M. N'Guerékata, Pseudo Almost Periodic Mild Solutions To Hyperbolic Evolution Equationa in Abstract Intermediate Banach Spaces. Applicable Anal. 85 (2006), Nos. 6-7, pp. 769-780.         [ Links ]

[26] T. Diagana, N. Henriquez, and E. Hernandez, Almost automorphic mild solutions to some partial neutral functional-differential Equations and Applications. Nonlinear Anal. 69 (2008), no. 5, pp. 1485-1493.         [ Links ]

[27] T. Diagana and G. M. N'Guerékata, Almost automorphic solutions to some classes of partial evolution equations. Appl. Math. Lett. 20 (2007), no. 4, pp. 462-466.         [ Links ]

[28] K. J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate texts in Mathematics, Springer Verlag 1999.         [ Links ]

[29] K. Ezzinbi, S. Fatajou and G. M. NGuerékata, Pseudo almost automorphic solutions to some neutral partial functional differential equations in Banach space. Nonlinear Anal. 70 (2009), no. 4, pp. 1641-1647.         [ Links ]

[30] K. Ezzinbi, S. Fatajou and G. M. NGuerékata, Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces. J. Math. Anal. Appl. 351 (2009), no. 2, pp. 765-772.         [ Links ]

[31] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics 377, Springer-Verlag, New York-Berlin, 1974.         [ Links ]

[32] E. Hernandez and H. R. Henriquez, Existence of Periodic Solutions of Partial neutral Functional Differential Equations with Unbounded Delay. J. Math. Anal. Appl 221 (1998), no. 2, pp. 499-522.         [ Links ]

[33] E. Hernandez, Existence Results for Partial Neutral Integro-differential Equations with Unbounded Delay. J. Math. Anal. Appl 292 (2004), no. 1, pp. 194-210.         [ Links ]

[34] E. Herniandez M., M. L. Pelicer, and J. P. C. dos Santos , Asymptotically Almost Periodic and Almost Periodic Solutions for a Class of Evolution Equations, Electron. J. Diff. Eqns 2004(2004), no. 61, pp. 1-15.         [ Links ]

[35] Y. Hino, T. Naito, N. V. Minh, and J. S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces. Stability and Control: Theory, Methods and Applications, 15. Taylor and Francis, London, 2002.         [ Links ]

[36] H. Leiva, Existence of Bounded Solutions Solutions of a Second-Order System with Dissipation. J. Math. Anal. Appl. 237 (1999), pp. 288-302.         [ Links ]

[37] H. Leiva and Z. Sivoli, Existence, Stability and Smoothness of a Bounded Solution for Nonlinear Time-Varying Theormoelastic Plate Equations. J. Math. Anal. Appl. 285 (1999), pp. 191-211.         [ Links ]

[38] J.-L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation. Inst. Hautes tudes Sci. Publ. Math. no. 19 (1964), pp. 5-68.         [ Links ]

[39] J. Liang, J. Zhang, and T-J. Xiao, Composition of Pseudo Almost Automorphic and Asymptotically almost automorphic functions. J. Math. Anal. Appl. 340 (2008), pp. 1493-1499.         [ Links ]

[40] J. Liang, G. M. N'Guieriekata, T-J. Xiao, and J. Zhang, Some properties of pseudo almost automorphic functions and applications to abstract differential equations. Nonlinear Anal. 70 (2009), no. 7, pp. 2731-2735.         [ Links ]

[41] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, PNLDE Vol. 16, Birkhaauser Verlag, Basel, 1995.         [ Links ]

[42] L. Maniar, R. Schnaubelt, Almost periodicity of inhomogeneous parabolic evolution equations, Lecture Notes in Pure and Appl. Math. vol. 234, Dekker, New York (2003), 299-318.         [ Links ]

[43] J. Mawhin, Bounded Solutions of Second-Order Semicoercive Evolution Equations in a Hilbert Space and Nonlinear Telegraph Equations. Rend. Sem. Mat. Univ. Pol. Torino. 58 (2000), no. 3, pp. 361-374.         [ Links ]

[44] M. G. Naso, A. Benabdallah, Thermoelastic plate with thermal interior control, Mathematical models and methods for smart materials (Cortona, 2001), 247-250, Ser. Adv. Math. Appl. Sci., 62, World Sci. Publ., River Edge, NJ, 2002.

[45] A. W. Naylor and G. R. Sell, Linear Operator Theory in Engineering and Science. Applied Mathematical Sciences 40, Springer-Verlag, 1971.         [ Links ]

[46] G. M. N'Guieriekata, Almost automorphic functions and almost periodic functions in abstract spaces, Kluwer Academic / Plenum Publishers, New York-London-Moscow, 2001.         [ Links ]

[47] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.         [ Links ]

[48] J. Prüss, Evolutionary Integral Equations and Applications, Birklmuser, 1993.         [ Links ]

[49] R. Schnaubelt, Sufficient conditions for exponential stability and dichotomy of evolution equations. Forum Math. 11(1999), pp. 543-566.         [ Links ]

[50] R. Schnaubelt, Asymptotically autonomous parabolic evolution equations, J. Evol. Equ. 1 (2001), pp. 19-37.         [ Links ]

[51] R. Schnaubelt, Asymptotic behavior of parabolic nonautonomous evolution equations, in: M. Iannelli, R. Nagel, S. Piazzera (Eds.), Functional Analytic Methods for Evolution Equations, in: Lecture Notes in Math., 1855, Springer-Verlag, Berlin, 2004, pp. 401-472.

[52] T. J. Xiao, J. Liang, J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 76 (2008), no. 3, pp. 518-524.         [ Links ]

[53] T. J. Xiao, X-X. Zhu, J. Liang, Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications. Nonlinear Anal. 70 (2009), no. 11, pp. 4079-4085.         [ Links ]

[54] T. J. Xiao, J. Liang, Second-order linear differential equations with almost periodic solutions, Acta Math. Svmca (N.S.) 7 (1991), 354-359.         [ Links ]

[55] T. J. Xiao, J. Liang, Complete second-order linear differential equations with almost periodic solutions, J. Math. Anal. Appl. 163 (1992), 136-146.         [ Links ]

[56] T. J. Xiao, J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, Lecture Notes in Mathematics, Vol. 1701, Springer, Berlin, 1998.         [ Links ]

[57] A. Yagi, Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II, Funkcial. Ekvac. 33 (1990), pp. 139-150.         [ Links ]

[58] A. Yagi, Abstract quasilinear evolution equations of parabolic type in Banach spaces, Boll. Un. Mat. Ital. B (7) 5 (1991), pp. 341-368.         [ Links ]

[59] S. Zaidman, Topics in Abstract Differential Equations, Pitman Research Notes in Mathematics Ser. II John Wiley and Sons, New York, 1994-1995.         [ Links ]

---

Received: July 2010. Revised: August 2010.