非自治发展方程与分数次微分方程的解
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摘要
本文主要分两部分。第一部分通过引进双概自守函数得到了非自治发展方程的伪概自守的存在唯一性。第二部分利用相空间来研究带无穷时滞的抽象分数次微分方程的解,得到了一些新的存在唯一性定理。
In this thesis, we have two objectives. One is obtaining new existence and uniqueness theorems for pseudo-almost automorphic mild solutions to several nonautonomous differential equations by introducing a new concept of bi-almost automorphic functions. The other one is investigating the mild solutions of a Cauchy problem for abstract fractional differential equations with infinite delay. Several theorems of existence and uniqueness of mild solutions are obtained.
In this thesis, we have two objectives. One is obtaining new existence and uniqueness theorems for pseudo-almost automorphic mild solutions to several nonautonomous differential equations by introducing a new concept of bi-almost automorphic functions. The other one is investigating the mild solutions of a Cauchy problem for abstract fractional differential equations with infinite delay. Several theorems of existence and uniqueness of mild solutions are obtained.
引文
[1]P.Acquistapace,Evolution operators and strong solution of abstract parabolic equations,Differential Integral Equations 1(1988) 433-457.
[2]P.Acquistapace,B.Terreni,A unified approach to abstract linear parabolic equations,Rend.Sem.Mat.Univ.Padova 78(1987) 47-107.
[3]E.Ait Dads,O.Arino,Exponential dichotomy and existence of pseudo almost-periodic solutions of some differential equations,Nonlinear Anal.TMA 27(1996),369-386.
[4]E.Ait Dads,K.Ezzinbi,Almost periodic solution for some neutral nonlinear integral equation,Nonlinear Anal.TMA 28(1997),1479-1489.
[5]E.Ait Dads,K.Ezzinbi,Existence of positive pseudo-almost-periodic solution for some nonlinear infinite delay integral equations arising in epidemic problems,Nonlinear Anal.TMA 41(2000),1-13.
[6]E.Ait Dads,K.Ezzinbi,Existence of pseudo almost periodic solution for some abstract semilinear functional differential equation,Dynam.Systems Appl.11(2002),493-498.
[7]A.Atlas,Regularity of the attractor for symmetric regularized wave equation,Communications on Pure and Applied Analysis 4(2005) 695-704.
[8]B.Amir,L.Maniar,Composition of pseudo almost periodic functions and Cauchy problems with operator of nondense domain,Ann.Math.Blaise Pascal 6(1999),1-11.
[9]M.Benchohra,J.Henderson,S.K.Ntouyas,A.Ouahab,Existence results for fractional order differential equations with infinite delay.J.Math.Aanl.Appl.338(2008),pp 1340-1350.
[10]S.Bochner,Uniform convergence of monotone sequences of functions,Proc.Natl.Acad.Sci.USA 47(1961) 582-585.
[11]S.Bochner,A new approach to almost periodicity,Proc.Natl.Acad.Sci.USA 48(1962) 2039-2043.
[12]S.Bochner,Continuous mappings of almost automorphic and almost periodic functions,Proc.Natl.Acad.Sci.USA 52(1964) 907-910.
[13]C.Chicone,Y.Latushkin,Evolution Semigroups in Dynamical Systems and Differential Equations,Amer.Math.Soc.,Providence,RI.1999.
[14]C.Cuevas,C.Lizama,Almost automorphic solutions to a class of semilinear fractional differential equations,Appl.Math.Lett.21(2008),no.12,1315-1319.
[15]T.Diagana,Stepanov-like pseudo almost periodicity and its applications to some nonautonomous differential equations,Nonlinear Anal.TMA 69(2008),no.12,4277-4285.
[16]T.Diagana,Existence of pseudo-almost automorphic solutions to some abstract differential equations with S~p-pseudo-almost automorphic coefficients,Nonlinear Anal.TMA,in press.
[17]T.Dianaga,G.M.N'Guerekata,Almost automorphic solutions to some classes of partial evolution equations,Appl.Math.Lett.20(2007),no.4,462-466.
[18]H.S.Ding,J.Liang,G.M.N'Guerekata,T.J.Xiao,Mild pseudo-almost periodic solutions of nonautonomous semilinear evolution equations,Math.Comput.Model.,45(2007),579-584.
[19]H.S.Ding,J.Liang,Gaston M.N'Guerekata,T.J.Xiao,Pseudo almost periodicity of some nonautonomous evolution equations with delay,Nonlinear Anal.TMA 67(2007) 1412-1418.
[20]C.G.Gal,Almost automorphic mild solutions to some semilinear abstract differential eqautions with deviated argument,J.Integral Equations Appl.17(2005) 391-396.
[21]J.K.Hale,J.Kato,Phase space fbr retarded equations with infinite delay.Funkcial.Evac.21(1978) pp 11-41.
[22]何崇佑,概周期微分方程,高等教育出版社,1992.
[23]V.Lakshmikantham,Theory of fractional differential equations,Nonlinear Anal.TMA 69(2008),no.10,3337-3343.
[24]V.Lakshmikantham,A.S.Vatsala,Basic theory of fractional differential equations,Nonlinear Anal.TMA 69(2008),no.8,2677-2682.
[25]V.Lakshmikantham,A.S.Vatsala,Theory of fractional differential inequalities and applications,Commun.Appl.Anal.11(2007),no.3-4,395-402.
[26]J.Liang and T.J.Xiao,Functional differential equations with infinite delay in Banach spaces,Internat.J.Math.Math.Sci.14(1991),no.3,pp 497-508.
[27]J.Liang and T.J.Xiao,The Cauchy problem for nonlinear abstract functional differential equations with infinite delay,Comput.Math.Appl.40(2000),no.6-7,pp 693-703.
[28]J.Liang,T.J.Xiao and J.Van Casteren,A note on semilinear abstract functional differential and integrodifferential equations with infinite delay,Appl.Math.Lett.17(2004),no.4,pp 473-477.
[29]J.Liang,J.Zhang,T.J.Xiao,Composition of pseudo almost automorphic functions,J.Math.Anal.Appl.340(2008),1493-1499.
[30]L.Maniar,R.Schnaubelt,Almost periodicity of inhomogeneous parabolic evolution equations,in:Lecture Notes in Pure and Appl.Math.,vol.234,Dekker,New York,2003,pp.299-318.
[31]K.S.Miller,B.Ross,An introduction to the fractional calculus and differential equations,John Wiley,New York,1993.
[32]G.M.N'Guerekata,A Cauchy problem fbr some fractional abstract differential equation with non local onditions,Nonlinear Anal.TMA(2008),doi:10.1016/j.na.2008.02.087.
[33]G.M.N'Guerekata,Almost automorphic and almost periodic functions in Abstract Space,Kluwer Academic/Plenum Publishers,New York,(2001).
[34]G.M.N'Guerekata,Topics in Almost Automorphy,Springer-Verlag,New York,2005.
[35]A.Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations,Springer-Verlag,New York,1983.
[36]I.Podlubny,Fractional differential equations,Academic Press,San Diego,1999.
[37]S.G.Samko,A.A.Kilbas,O.I.Marichev,Fractional integrals and derivatives:Theory and applications,Gordon and Breach Yverden,1993.
[38]K.Schumacher,Existence and continuous dependence for functional differential equations with unbounded delay,Arch.Rat.Mech.Ana.67(1978) pp 315-335.
[39]Shih-tong Tu,Shy-der Lin,Yu-tan Huang,H.M.Srivastava,Solutions of a Certain Class of Fractional Differintegral Equations,Appl.Math.Lett.14(2001), pp 223-229.
[40]W.A.Veech,Almost automorphic functions on groups,Amer.J.Math.87(1965),719-751.
[41]T.J.Xiao,J.Liang,J.Zhang,Pseudo almost automorphic solutions to semilinear differential equations in Banach space,Semigroup Forum 76(2008),no.3,518-524.
[42]T.J.Xiao,J.Liang,The Cauchy Problem for Higher-Order Abstract Differential Equations,Lecture Notes in Mathematics,Vol.1701,Springer,Berlin,1998.
[2]P.Acquistapace,B.Terreni,A unified approach to abstract linear parabolic equations,Rend.Sem.Mat.Univ.Padova 78(1987) 47-107.
[3]E.Ait Dads,O.Arino,Exponential dichotomy and existence of pseudo almost-periodic solutions of some differential equations,Nonlinear Anal.TMA 27(1996),369-386.
[4]E.Ait Dads,K.Ezzinbi,Almost periodic solution for some neutral nonlinear integral equation,Nonlinear Anal.TMA 28(1997),1479-1489.
[5]E.Ait Dads,K.Ezzinbi,Existence of positive pseudo-almost-periodic solution for some nonlinear infinite delay integral equations arising in epidemic problems,Nonlinear Anal.TMA 41(2000),1-13.
[6]E.Ait Dads,K.Ezzinbi,Existence of pseudo almost periodic solution for some abstract semilinear functional differential equation,Dynam.Systems Appl.11(2002),493-498.
[7]A.Atlas,Regularity of the attractor for symmetric regularized wave equation,Communications on Pure and Applied Analysis 4(2005) 695-704.
[8]B.Amir,L.Maniar,Composition of pseudo almost periodic functions and Cauchy problems with operator of nondense domain,Ann.Math.Blaise Pascal 6(1999),1-11.
[9]M.Benchohra,J.Henderson,S.K.Ntouyas,A.Ouahab,Existence results for fractional order differential equations with infinite delay.J.Math.Aanl.Appl.338(2008),pp 1340-1350.
[10]S.Bochner,Uniform convergence of monotone sequences of functions,Proc.Natl.Acad.Sci.USA 47(1961) 582-585.
[11]S.Bochner,A new approach to almost periodicity,Proc.Natl.Acad.Sci.USA 48(1962) 2039-2043.
[12]S.Bochner,Continuous mappings of almost automorphic and almost periodic functions,Proc.Natl.Acad.Sci.USA 52(1964) 907-910.
[13]C.Chicone,Y.Latushkin,Evolution Semigroups in Dynamical Systems and Differential Equations,Amer.Math.Soc.,Providence,RI.1999.
[14]C.Cuevas,C.Lizama,Almost automorphic solutions to a class of semilinear fractional differential equations,Appl.Math.Lett.21(2008),no.12,1315-1319.
[15]T.Diagana,Stepanov-like pseudo almost periodicity and its applications to some nonautonomous differential equations,Nonlinear Anal.TMA 69(2008),no.12,4277-4285.
[16]T.Diagana,Existence of pseudo-almost automorphic solutions to some abstract differential equations with S~p-pseudo-almost automorphic coefficients,Nonlinear Anal.TMA,in press.
[17]T.Dianaga,G.M.N'Guerekata,Almost automorphic solutions to some classes of partial evolution equations,Appl.Math.Lett.20(2007),no.4,462-466.
[18]H.S.Ding,J.Liang,G.M.N'Guerekata,T.J.Xiao,Mild pseudo-almost periodic solutions of nonautonomous semilinear evolution equations,Math.Comput.Model.,45(2007),579-584.
[19]H.S.Ding,J.Liang,Gaston M.N'Guerekata,T.J.Xiao,Pseudo almost periodicity of some nonautonomous evolution equations with delay,Nonlinear Anal.TMA 67(2007) 1412-1418.
[20]C.G.Gal,Almost automorphic mild solutions to some semilinear abstract differential eqautions with deviated argument,J.Integral Equations Appl.17(2005) 391-396.
[21]J.K.Hale,J.Kato,Phase space fbr retarded equations with infinite delay.Funkcial.Evac.21(1978) pp 11-41.
[22]何崇佑,概周期微分方程,高等教育出版社,1992.
[23]V.Lakshmikantham,Theory of fractional differential equations,Nonlinear Anal.TMA 69(2008),no.10,3337-3343.
[24]V.Lakshmikantham,A.S.Vatsala,Basic theory of fractional differential equations,Nonlinear Anal.TMA 69(2008),no.8,2677-2682.
[25]V.Lakshmikantham,A.S.Vatsala,Theory of fractional differential inequalities and applications,Commun.Appl.Anal.11(2007),no.3-4,395-402.
[26]J.Liang and T.J.Xiao,Functional differential equations with infinite delay in Banach spaces,Internat.J.Math.Math.Sci.14(1991),no.3,pp 497-508.
[27]J.Liang and T.J.Xiao,The Cauchy problem for nonlinear abstract functional differential equations with infinite delay,Comput.Math.Appl.40(2000),no.6-7,pp 693-703.
[28]J.Liang,T.J.Xiao and J.Van Casteren,A note on semilinear abstract functional differential and integrodifferential equations with infinite delay,Appl.Math.Lett.17(2004),no.4,pp 473-477.
[29]J.Liang,J.Zhang,T.J.Xiao,Composition of pseudo almost automorphic functions,J.Math.Anal.Appl.340(2008),1493-1499.
[30]L.Maniar,R.Schnaubelt,Almost periodicity of inhomogeneous parabolic evolution equations,in:Lecture Notes in Pure and Appl.Math.,vol.234,Dekker,New York,2003,pp.299-318.
[31]K.S.Miller,B.Ross,An introduction to the fractional calculus and differential equations,John Wiley,New York,1993.
[32]G.M.N'Guerekata,A Cauchy problem fbr some fractional abstract differential equation with non local onditions,Nonlinear Anal.TMA(2008),doi:10.1016/j.na.2008.02.087.
[33]G.M.N'Guerekata,Almost automorphic and almost periodic functions in Abstract Space,Kluwer Academic/Plenum Publishers,New York,(2001).
[34]G.M.N'Guerekata,Topics in Almost Automorphy,Springer-Verlag,New York,2005.
[35]A.Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations,Springer-Verlag,New York,1983.
[36]I.Podlubny,Fractional differential equations,Academic Press,San Diego,1999.
[37]S.G.Samko,A.A.Kilbas,O.I.Marichev,Fractional integrals and derivatives:Theory and applications,Gordon and Breach Yverden,1993.
[38]K.Schumacher,Existence and continuous dependence for functional differential equations with unbounded delay,Arch.Rat.Mech.Ana.67(1978) pp 315-335.
[39]Shih-tong Tu,Shy-der Lin,Yu-tan Huang,H.M.Srivastava,Solutions of a Certain Class of Fractional Differintegral Equations,Appl.Math.Lett.14(2001), pp 223-229.
[40]W.A.Veech,Almost automorphic functions on groups,Amer.J.Math.87(1965),719-751.
[41]T.J.Xiao,J.Liang,J.Zhang,Pseudo almost automorphic solutions to semilinear differential equations in Banach space,Semigroup Forum 76(2008),no.3,518-524.
[42]T.J.Xiao,J.Liang,The Cauchy Problem for Higher-Order Abstract Differential Equations,Lecture Notes in Mathematics,Vol.1701,Springer,Berlin,1998.