伪概自守函数及在发展方程中的应用
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摘要
本文主要讨论了伪概自守函数和相关函数的基本性质及其在发展方程中的应用。全文共分五章。
第一章介绍了本文的研究背景和主要工作。
第二章是预备知识,主要介绍了概周期函数、概自守函数、渐近概周期函数、渐近概自守函数、伪概周期函数、伪概自守函数等的概念和基本性质。此外,我们还介绍了C_0半群和cosine算子函数的一些定义和相关性质。
第三章主要研究了概自守函数和伪概自守函数的基本性质。这些性质为研究自守函数在发展方程中的进一步应用奠定了基础。
§3.1主要研究了概自守函数和具有零平均值的函数的一些基本性质。
§3.2中我们主要讨论了在Lipschitz连续性假设下,函数f(t,x)与x(t)复合后保持伪概自守性质不变的条件,并获得了关于伪概自守函数的复合定理。
§3.3讨论了伪概自守函数的分解的唯一性,同时证明了其在范数下的完备性。
§3.4讨论了广义的伪概自守函数,即此时其扰动项并不是有界连续函数的情形。
第四章主要研究了伪概自守函数在线性发展方程u~'(t)=Au(t)+f(t), t∈R和半线性发展方程u~'(t)=Au(t)+f(t,u(t)), t∈R中的应用。我们分别针对线性算子A生成指数稳定的C_0半群和生成紧的C_0半群情形,进行了研究,给出了这两类发展方程的伪概自守的温和解的存在性定理,并在某些情形下得到了解的唯一性。
我们在第五章研究了具有指数增长的渐近概周期函数的性质及在一阶微分方程和非完全二阶算子微分方程中的应用。
Properties of pseudo almost automorphic functions and related ones are discussedin this thesis. Applications of these properties to evolution equations are also presented. This thesis consists of five chapters.
Chapter 1 presents the research background and main results of this thesis.
Chapter 2 is about preliminaries of this thesis. Definitions and basic properties of almost periodic, almost automorphic, asymptotically almost periodic, asymptoticallyalmost automorphic, pseudo almost periodic and pseudo almost automorphic functions are stated. Moreover, definitions of C_0 semigroups and cosine operator functions are revoiewed.
Chapter 3 is concerned with basic properties of almost automorphic functions and pseudo almost automorphic functions, which is a basis for exploring further applications of these functions to evolution equations.
In Section 3.1, some basic properties of almost automorphic functions and functions with 0 mean value are investigated.
In Section 3.2, we study the composition of pseudo almost automorphic functions f(t,x) and x(t) under Lipschitz conditions.
In Section 3.3, we show the uniqueness of decomposition of pseudo almost automorphicfunctions, and then prove that pseudo almost automorphic functions form a Banach space under the norm
Section 3.4 is mainly about generalized pseudo almost automorphic functions, whose corrective terms are not bounded and continuous.
Chapter 4 studies the applications of pseudo almost automorphic functions to linear evolution equations u~'(t) = Au(t)+f(t), t∈R, and semilinear evolution equations u~'(t) = Au(t) + f(t,u(t)), t∈R.
For the case of A generating an exponentially stable C_0 semigroup and the case of A generating a compact C_0 semigroup, respectively, we establish new existence and uniqueness theorems for mild pseudo almost automorphic solutions to these evolution equations.
In Chapter 5, we investigate exponentially asymptotically almost periodic functions.Applications to first order and incomplete second order evolution equations are discussed.
第一章介绍了本文的研究背景和主要工作。
第二章是预备知识,主要介绍了概周期函数、概自守函数、渐近概周期函数、渐近概自守函数、伪概周期函数、伪概自守函数等的概念和基本性质。此外,我们还介绍了C_0半群和cosine算子函数的一些定义和相关性质。
第三章主要研究了概自守函数和伪概自守函数的基本性质。这些性质为研究自守函数在发展方程中的进一步应用奠定了基础。
§3.1主要研究了概自守函数和具有零平均值的函数的一些基本性质。
§3.2中我们主要讨论了在Lipschitz连续性假设下,函数f(t,x)与x(t)复合后保持伪概自守性质不变的条件,并获得了关于伪概自守函数的复合定理。
§3.3讨论了伪概自守函数的分解的唯一性,同时证明了其在范数下的完备性。
§3.4讨论了广义的伪概自守函数,即此时其扰动项并不是有界连续函数的情形。
第四章主要研究了伪概自守函数在线性发展方程u~'(t)=Au(t)+f(t), t∈R和半线性发展方程u~'(t)=Au(t)+f(t,u(t)), t∈R中的应用。我们分别针对线性算子A生成指数稳定的C_0半群和生成紧的C_0半群情形,进行了研究,给出了这两类发展方程的伪概自守的温和解的存在性定理,并在某些情形下得到了解的唯一性。
我们在第五章研究了具有指数增长的渐近概周期函数的性质及在一阶微分方程和非完全二阶算子微分方程中的应用。
Properties of pseudo almost automorphic functions and related ones are discussedin this thesis. Applications of these properties to evolution equations are also presented. This thesis consists of five chapters.
Chapter 1 presents the research background and main results of this thesis.
Chapter 2 is about preliminaries of this thesis. Definitions and basic properties of almost periodic, almost automorphic, asymptotically almost periodic, asymptoticallyalmost automorphic, pseudo almost periodic and pseudo almost automorphic functions are stated. Moreover, definitions of C_0 semigroups and cosine operator functions are revoiewed.
Chapter 3 is concerned with basic properties of almost automorphic functions and pseudo almost automorphic functions, which is a basis for exploring further applications of these functions to evolution equations.
In Section 3.1, some basic properties of almost automorphic functions and functions with 0 mean value are investigated.
In Section 3.2, we study the composition of pseudo almost automorphic functions f(t,x) and x(t) under Lipschitz conditions.
In Section 3.3, we show the uniqueness of decomposition of pseudo almost automorphicfunctions, and then prove that pseudo almost automorphic functions form a Banach space under the norm
Section 3.4 is mainly about generalized pseudo almost automorphic functions, whose corrective terms are not bounded and continuous.
Chapter 4 studies the applications of pseudo almost automorphic functions to linear evolution equations u~'(t) = Au(t)+f(t), t∈R, and semilinear evolution equations u~'(t) = Au(t) + f(t,u(t)), t∈R.
For the case of A generating an exponentially stable C_0 semigroup and the case of A generating a compact C_0 semigroup, respectively, we establish new existence and uniqueness theorems for mild pseudo almost automorphic solutions to these evolution equations.
In Chapter 5, we investigate exponentially asymptotically almost periodic functions.Applications to first order and incomplete second order evolution equations are discussed.
引文
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[3]E.Ait Dads,K.Ezzinbi,Almost periodic solution for some neutral nonlinear integral equation,Nonlinear Anal.Theory Methods Appl.28(1997),1479-1489.
[4]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.41(2000),1-13.
[5]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.
[6]E.Ait Dads,L.Lhachimi,New approach for the existence of pseudo almost periodic solutions for some second order differential equation with piecewise constant argument.Nonlinear Anal.64(2006),no.6,1307-1324.
[7]E.Ait Dads,P.Cieutat,K.Ezzinbi,The existence of pseudo-almost periodic solutions for some nonlinear differential equations in Banach spaces.Nonlinear Anal.69(2008),no.4,1325-1342.
[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.
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[11]A.S.Besicovitch,On Parseval's Theorem for Dirichlet Series.Proc.London Math.Soc,(1927),26,25-34
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[13]S.Bochner,J.V.Neumann,Almost Periodic Functions in Groups Ⅱ.Trans.Amer.Math.Soc.(1935),37(l):21-50
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[15]H.Bohr,On the definition of almost periodicity.J.Analyse Math.1,(1951).11-27.
[16]S.Boulite,L.Maniar,G.M.N'Guerekata,Almost automorphic solutions for hyperbolic semilinear evolution equations.Semigroup Forum 71(2005),no.2,231-240.
[17]D.Bugajewski,G.M.N'Guerekata,On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces,Nonlinear Anal.59(2004),1333-1345.
[18]D.Bugajewski,T.Diagana,Almost automorphy of the convolution operator and applications to differential and functional-differential equations.Nonlinear Studies 13(2006),129-140.
[19]T.A.Burton,Tetsuo Furumochi,Existence theorems and periodic solutions of neutral integral equations,Nonlinear Anal.43(2001),527-546.
[20]T.Caraballo,D.Cheban,Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation conditon.I.J.Differential Equations 246(2009),no.1,108-128.
[21]C.Chicone,Y.Latushkin,Evolution Semigroups in Dynamical Systems and Differential Equations,Amer.Math.Soc,1999.
[22]K.L.Cooke,J.L.Kaplan,A periodicity threshold theorem for epidemics and population growth,Math.Biosci.31(1976),87-104.
[23]C.Corduneanu,Almost Periodic Functions,2nd edition,Chelsea,New york,1989.
[24]C.Cuevas,M.Pinto,Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain,Nonlinear Anal.45(2001),73-83.
[25]C.Cuevas,C.Lizama,Almost automorphic solutions to a class of semilinear fractional differential equations.Appl.Math.Lett.21(2008),no.12,1315-1319.
[26]K.Deimling,Nonlinear Functionl Analysis,Springer-verlay,New York,1985.
[27]M.Frechet,Les Fonctions Asymptotiquement Presque-periodiques Continues.C.R.Acad.Sci.Paris.(1941),213,520-522
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[29]W.Desch,R.Grimmer and W.Schappacher,Some considerations for linear integro-differential equations,J.Math.Anal.Appl.104(1984),219-234.
[30]T.Diagana,Pseudo almost periodic solutions to some differential equations,Nonlinear Anal.60(2005),1277-1286.
[31]T.Diagana,C.M.Mahop,G.M.N'Guerekata,Pseudo almost periodic solution to some semilinear differential equations,Mathematical and Computer Modelling 43(2006),89-96.
[32]T.Diagana,Pseudo almost periodic functions in Banach spaces.Nova Science Publishers,Inc.,New York,2007
[33]T.Diagana,G.M.N'Guerekata,Almost automorphic solutions to some classes of partial evolution equations.Appl.Math.Lett.20(2007),no.4,462-466.
[34]T.Diagana,Existence results for pseudo almost periodic differential,functional,and neutral integral equations.Int.J.Evol.Equ.2(2007),no.2,205-233.
[35]T.Dianaga,Existence of weighted pseudo almost periodic solutions to some non-autonomous differential equations.Int.J.Evol.Equ.2(2008),no.4,397-410.
[36]T.Diagana,Weighted pseudo-almost periodic solutions to a neutral delay integral equation of advanced type.Nonlinear Anal.70(2009),no.1,298-304.
[37] H. S. Ding, J. Liang, G. M. N'Gu(?)r(?)kata, T. J. Xiao, Mild pseudo-almost periodic solutions of nonautonomous semilinear evolution equations, Mathematical and Computer Modelling, 45 (2007), 579-584.
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