若干分数阶积微分方程温性解的存在唯一性
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摘要
本文主要利用算子理论和不动点定理研究了若干分数阶积微分方程温性解的存在唯一性.
在第二章中,研究了带非局部初始条件的分数阶积微分方程的Cauchy问题,其中线性部分是复Banach空间中的一个解算子的无穷小生成元,利用积微分方程的Laplace变换给出了Cauchy问题的温性解定义,在各种新的条件下,证明了该Cauchy问题的温性解的存在唯一性.最后,给出了一个具体的例子来阐述其对应的抽象结果的可行性.
在第三章中,研究了非局部分数阶偏中立微分方程的Cauchy问题的温性解的存在性,在非局部项上减弱了紧性和Lipschitz条件的情况下,证明了Cauchy问题的温性解的存在性.最后,给出了一个具体的例子来阐述其对应的抽象结果的可行性.
在第四章中,利用分数幂算子的性质和Sadovskii不动点定理确立了一类带状态时滞的q(0 在第五章中,讨论了一类时滞偏中立泛函积微分系统的可控性问题.利用预解算子理论、Sadovskii不动点定理、Leray-Schauder二择一不动点定理确立了带状态时滞的偏中立泛函积微分系统可控时的两个充分条件.推广和改进了已有文献的相关结果.
In this paper, we study the existence and uniqueness of mild solutions of some fractional integro-differential equations by using the theory of operator and fixed point theorem mainly.
Chapter 2 focus on a Cauchy problem for fractional impulsive integro-differential equations involving nonlocal initial conditions, where the linear part is a generator of a solution operator on a complex Banach space. A suitable mild solution for the Cauchy problem is introduced by applying the Laplace transform of integro differential equations. The existence and uniqueness of mild solutions for the Cauchy problem, under various criterions, are proved. In the last part of this chapter, we construct an example to illustrate the feasibility of our results.
In Chapter 3, we study the existence of mild solution for a Cauchy problem of fractionalpartial neutral differential equation with nonlocal initial condition. A new existence theorem which allows us to relax the compactness and Lipschitz continuity on nonlocal item, ensuring the existence of mild solution for the Cauchy problem, is presented. In the last part of Chapter 3, we give an example to illustrate the feasibility of abstract results.
In Chapter 4, we establish a sufficient condition for the controllability for Caputo fractional derivative of order q (0< q< 1) neutral differential systems with state-dependent delay by using the fractional power of operators, Sadovskii's fixed point theorem and Leray-Schauder Alternative's fixed point theorem.
In Chapter 5, we study the controllability for a class of partial neutral functional integeo-differential systems with delay. By using the resolvent operator, Sadovskii's fixed point theorem and Leray-Schauder Alternative's fixed point theorem, two sufficient conditions for the controllability of partial neutral functional integeo-differential systems with state-dependent delay in a Banach space are established. The results extend and improve the related reports in the literatures.
在第二章中,研究了带非局部初始条件的分数阶积微分方程的Cauchy问题,其中线性部分是复Banach空间中的一个解算子的无穷小生成元,利用积微分方程的Laplace变换给出了Cauchy问题的温性解定义,在各种新的条件下,证明了该Cauchy问题的温性解的存在唯一性.最后,给出了一个具体的例子来阐述其对应的抽象结果的可行性.
在第三章中,研究了非局部分数阶偏中立微分方程的Cauchy问题的温性解的存在性,在非局部项上减弱了紧性和Lipschitz条件的情况下,证明了Cauchy问题的温性解的存在性.最后,给出了一个具体的例子来阐述其对应的抽象结果的可行性.
在第四章中,利用分数幂算子的性质和Sadovskii不动点定理确立了一类带状态时滞的q(0
In this paper, we study the existence and uniqueness of mild solutions of some fractional integro-differential equations by using the theory of operator and fixed point theorem mainly.
Chapter 2 focus on a Cauchy problem for fractional impulsive integro-differential equations involving nonlocal initial conditions, where the linear part is a generator of a solution operator on a complex Banach space. A suitable mild solution for the Cauchy problem is introduced by applying the Laplace transform of integro differential equations. The existence and uniqueness of mild solutions for the Cauchy problem, under various criterions, are proved. In the last part of this chapter, we construct an example to illustrate the feasibility of our results.
In Chapter 3, we study the existence of mild solution for a Cauchy problem of fractionalpartial neutral differential equation with nonlocal initial condition. A new existence theorem which allows us to relax the compactness and Lipschitz continuity on nonlocal item, ensuring the existence of mild solution for the Cauchy problem, is presented. In the last part of Chapter 3, we give an example to illustrate the feasibility of abstract results.
In Chapter 4, we establish a sufficient condition for the controllability for Caputo fractional derivative of order q (0< q< 1) neutral differential systems with state-dependent delay by using the fractional power of operators, Sadovskii's fixed point theorem and Leray-Schauder Alternative's fixed point theorem.
In Chapter 5, we study the controllability for a class of partial neutral functional integeo-differential systems with delay. By using the resolvent operator, Sadovskii's fixed point theorem and Leray-Schauder Alternative's fixed point theorem, two sufficient conditions for the controllability of partial neutral functional integeo-differential systems with state-dependent delay in a Banach space are established. The results extend and improve the related reports in the literatures.
引文
[1]V. V. Ahn and R. Mcvinish, Fractional differential equations driven by Levy noise[J]. J. Appl. Stoch. Anal.,16 (2) (2003):97-119.
[2]E. G. Bajlekova, Fractional evolution equations in Banach spaces[M]. Ph. D. Thesis, Eindhoven University of Technology,2001.
[3]I. Podlubny, Fractional Differential Equations[M]. vol.198 of Mathematics in Science and Engineering, Academic Press. San Diego, Calif, USA,1999.
[4]V. V. Anh and N. N. Leonenko, Spectral analysis of fractional kinetic equations with random data[J]. Journal of Statistical Physics,104 (2001):1349-1387.
[5]V. Lakshmikantham, Theory of fractional functional differential equations[J]. Nonlinear Analysis. Theory, Methods & Applications,60 (2008):3337-3343.
[6]V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations[J]. Nonlinear Analysis. Theory, Methods & Applications,69 (2008):2677-2682.
[7]Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations[J]. Computers& Mathematics with Applications,59 (2010):1063-1077.
[8]Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations[J]. Nonlinear Analysis. Real World Applications,11 (2010):4465-4475.
[9]E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations[J]. Discrete and Continuous Dynamical Systems. Series A, (2007):277-285.
[10]H.-S. Ding, J. Liang, and T.-J. Xiao, Positive almost automorphic solutions for a class of non-linear delay integral equations[J]. Applicable Analysis,88 (2009):231-242.
[11]D.-J. Guo and X. Liu, Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces[J]. Journal of Mathematical Analysis and Applications,177 (1993): 538-552.
[12]J. Liang, J. H. Liu, and T.-J. Xiao, Nonlocal problems for integro-differential equations[J]. Dynamics of Continuous. Discrete & Impulsive Systems. Series A,15 (2008):815-824.
[13]J. Liang and T.-J. Xiao, Semilinear integro-differential equations with nonlocal initial conditions[J]. Computers & Mathematics with Applications,47 (2004):863-875.
[14]T.-J. Xiao, J. Liang, and J. van Casteren, Time dependent Desch-Schappacher type perturbations of Volterra integral equations[J]. Integral Equations and Operator Theory,44 (2002):494-506.
[15]L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolutionnonlocal Cauchy problem[J]. Journal of Mathematical Analysis and Applications, 162 (1991):494-505.
[16]L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space[J]. Applicable Analysis, 40(1991):11-19.
[17]J. Liang, J. van Casteren, and T.-J. Xiao, Nonlocal Cauchy problems for semilinear evolution equations[J]. Nonlinear Analysis. Theory, Methods & Applications,50 (2002):173-189.
[18]K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions[J]. J. Math. Anal. Appl.,179 (1993):630-637.
[19]A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations[M]. vol.14, World Scientific Publishing, River Edge, NJ, USA,1995.
[20]M. Benchohra, E. P. Gatsori, J. Henderson, and S. K. Ntouyas, Nondensely defined evolution impulsive differential inclusions with nonlocal conditions[J]. Journal of Mathematical Analysis and Applications,286 (2003):307-325.
[21]J. Liang, J. H. Liu, and T.-J. Xiao, Nonlocal impulsive problems for nonlinear differential equationsin Banach spaces[J]. Mathematical and Computer Modelling,49 (2009):798-804.
[22]R.-N. Wang, Z.-Q. Li, and X.-H. Ding, Nonlocal Cauchy problems for semilinear evolution equations involving almost sectorial operators[J]. Indian Journal of Pure and Applied Mathematics,39 (2008):333-346.
[23]K. Ezzinbi, S. Ghnimib, M. A. Taoudi, Existence and Regularity of Solutions for Neutral Partial Functional Integro-differential Equations with Infinite Delay[J]. Nonlinear Analysis: Hybrid Systems,4 (2010):54-64.
[24]X.-L. Fu, Controllability of Abstract Neutral Functional Differential Systems with Unbounded Delay[J]. Applied Mathematics and Computation,151 (2004):299-314.
[25]Y.-K. Chang, A. Anguraj, M. Mallika Arjunan, Existence results for non-densely defined neutral impulsive differential inclusions with nonlocal conditions[J]. J. Appl. Math. Comput., 28 (2008):79-91.
[26]B. Liu, Controllability of impulsive neutral functional differential inclusions with infinite delay[J]. Nonlinear Analysis,60(2005):1533-1552.
[27]Z.-Y. Zhang, M. Blanke. A unified apporach to controllability analysis for hybrid systems[J]. Nonlinear Analysis:Hybrid systems 1 (2007):212-222.
[28]Y.-K. Chang, W.-S. Li, Solvability for Impulsive Neutral Integro Differential Equations with State-Dependent Delay via Fractional Operators[J]. J. Optim. Theory Appl.,144 (2010): 445-459.
[29]C. Lizama, Regularized solutions for abstract Volterra equations[J]. Journal of Mathematical Analysisand Applications,243 (2000):278-292.
[30]C. J. K. Batty, J. Liang, and T.-J. Xiao, On the spectral and growth bound of semigroups associatedwith hyperbolic equations[J]. Advances in Mathematics,191 (2005):1-10.
[31]K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations[M]. vol.194 of GraduateTexts in Mathematics, Springer, New York, NY, USA,2000.
[32]H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces[M]. vol.108 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands,1985.
[33]M. E. Gurtin, A. C. Pipkin, A General Theory of Heat Conduction with Finite Wave Speed [J]. Arch Ration Mech Anal..31 (1968):113-126.
[34]J. Liang, R. Nagel, and T.-J. Xiao, Approximation theorems for the propagators of higher orderabstract Cauchy problems[J]. Transactions of the American Mathematical Society,360 (2008):1723-1739.
[35]Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay[M]. in:Lectures Notes in Mathematics, vol.1473, Berlin:Springer,1991.
[36]B. N. Sadovskii, On a fixed point principle[J]. Funct. Anal. Appl.,1 (1967):74-76.
[37]C. Muller, Solving abstract Cauchy problems with closable operators in reflexive spaces via resolvent-free approximation[J]. Forum Mathematicum,19 (2007):1-18.
[38]T.-J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations[M]. vol.1701 of Lecture Notes in Mathematics, Springer, Berlin, Germany,1998.
[39]T.-J. Xiao and J. Liang, A solution to an open problem for wave equations with generalized Wentzell boundary conditions[J]. Mathematische Annalen,327 (2003):351-363.
[40]T.-J. Xiao and J. Liang, Complete second order differential equations in Banach spaces with dynamic boundary conditions[J]. Journal of Differential Equations,200 (2004):105-136.
[41]C. Cuevas and Julio Cesar de Souza, S-asymptotically co-periodic solutions of semilinear fractional integro-differential equations[J]. Appl. Math. Lett.,22 (2009):865-870.
[42]C. Cuevas and Julio Cesar de Souza, Existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay[J]. Nonlinear Analysis,72 (2010):1683-1689.
[43]Z.-B. Fan, Impulsive problems for semilinear differential equations with nonlocal conditions [J]. Nonlinear Analysis,72 (2010):1104-1109.
[44]A. Granas, J. Dugundji, Fixed Point Theory[M]. Springer-Verlag, New York,2003.
[45]H. Henriquez, M. Pierri, P. Taboas, On S-asymptotically ω-periodic functions on Banach spaces and applications[J]. J. Math. Anal. Appl.,343 (2008):1119-1130.
[46]V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations[M]. Modern Applied Mathematics, vol.6, World Scientific Publishing, New Jersey,1989.
[47]B. M. Levitan, V. V. Zhikov, Almost periodic functions and differential equations[M]. Cambridge Univ. Press,1982.
[48]G. M. Mophou, Existence and uniqueness of mild solutions to impulsive fractional differential equations[J]. Nonlinear Analysis,72 (2010):1604-1615.
[49]J. W. Nunziato, On heat conduction in materials with memory[J]. Quart. Appl. Math.,29 (1971):187-204.
[50]J. Banas and K. Goebe, Measure of Noncompactness in Banach Spaces[M]. Lect. Notes Pure Appl. Math., vol.60, Marcel Dekker, New York,1980.
[51]J.-W. Luo. Exponential Stability for Stochastic Neutral Partial Functional Differential Equations[J]. J Math Anal Appl,355 (2009):414-425
[52]T. Diagana, E. Hernandez, J. P. C. Dos Santos, Existence of Asymptotically Almost Automorphic Solutions to Some Abstract Partial Neutral Integro-differential Equations[J]. Nonlinear Analysis,71 (2009):248-257.
[53]Z.-R. Hu, Z. Jin, Necessary and Sufficient Conditions for the Regularity and Stability of Solutions for Some Partial Neutral Functional Differential Equations with Infinite Delay[J]. Nonlinear Analysis,73 (2010):2752-2765.
[54]K. Balachandran, J. P. Dauer, Controllability of Nonlinear Systems in Banach Spaces:A survey[J]. J Optim Theory Appl,115 (2002):7-28.
[55]K. Balachandran, R. Sakthivel. Controllability of Neutral Functional Integeodifferential Systems in Banach Spaces[J]. Computers and Mathematics with Applications,39 (2000): 117-126.
[56]M. Bartha, Periodic Solutions for Sifferential Squations with State-dependent Selay and Sositive Feedback[J]. Nonlinear Anal. T. M. A.,53 (6) (2003):839-857.
[57]J. P. C. Dos Santos, On State-dependent Selay Partial Neutral Functional Integro-differential Equations[J]. Applied Mathematics and Computeation,216 (2010):1637-1644.
[58]R. Grimmer, A. Pritchard, Analytic Resolvent Operators for Integral Equatins in Banach Spaces[J]. J. Differ. Equ.,50 (2) (1983):233-259.
[59]A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations [M]. in:Applied Mathematical Sciences, vol.44, Springer-Verlag, New York,1983.
[60]叶其孝,李正元,反应扩散方程引论[M].科学出版社,1994.
[2]E. G. Bajlekova, Fractional evolution equations in Banach spaces[M]. Ph. D. Thesis, Eindhoven University of Technology,2001.
[3]I. Podlubny, Fractional Differential Equations[M]. vol.198 of Mathematics in Science and Engineering, Academic Press. San Diego, Calif, USA,1999.
[4]V. V. Anh and N. N. Leonenko, Spectral analysis of fractional kinetic equations with random data[J]. Journal of Statistical Physics,104 (2001):1349-1387.
[5]V. Lakshmikantham, Theory of fractional functional differential equations[J]. Nonlinear Analysis. Theory, Methods & Applications,60 (2008):3337-3343.
[6]V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations[J]. Nonlinear Analysis. Theory, Methods & Applications,69 (2008):2677-2682.
[7]Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations[J]. Computers& Mathematics with Applications,59 (2010):1063-1077.
[8]Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations[J]. Nonlinear Analysis. Real World Applications,11 (2010):4465-4475.
[9]E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations[J]. Discrete and Continuous Dynamical Systems. Series A, (2007):277-285.
[10]H.-S. Ding, J. Liang, and T.-J. Xiao, Positive almost automorphic solutions for a class of non-linear delay integral equations[J]. Applicable Analysis,88 (2009):231-242.
[11]D.-J. Guo and X. Liu, Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces[J]. Journal of Mathematical Analysis and Applications,177 (1993): 538-552.
[12]J. Liang, J. H. Liu, and T.-J. Xiao, Nonlocal problems for integro-differential equations[J]. Dynamics of Continuous. Discrete & Impulsive Systems. Series A,15 (2008):815-824.
[13]J. Liang and T.-J. Xiao, Semilinear integro-differential equations with nonlocal initial conditions[J]. Computers & Mathematics with Applications,47 (2004):863-875.
[14]T.-J. Xiao, J. Liang, and J. van Casteren, Time dependent Desch-Schappacher type perturbations of Volterra integral equations[J]. Integral Equations and Operator Theory,44 (2002):494-506.
[15]L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolutionnonlocal Cauchy problem[J]. Journal of Mathematical Analysis and Applications, 162 (1991):494-505.
[16]L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space[J]. Applicable Analysis, 40(1991):11-19.
[17]J. Liang, J. van Casteren, and T.-J. Xiao, Nonlocal Cauchy problems for semilinear evolution equations[J]. Nonlinear Analysis. Theory, Methods & Applications,50 (2002):173-189.
[18]K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions[J]. J. Math. Anal. Appl.,179 (1993):630-637.
[19]A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations[M]. vol.14, World Scientific Publishing, River Edge, NJ, USA,1995.
[20]M. Benchohra, E. P. Gatsori, J. Henderson, and S. K. Ntouyas, Nondensely defined evolution impulsive differential inclusions with nonlocal conditions[J]. Journal of Mathematical Analysis and Applications,286 (2003):307-325.
[21]J. Liang, J. H. Liu, and T.-J. Xiao, Nonlocal impulsive problems for nonlinear differential equationsin Banach spaces[J]. Mathematical and Computer Modelling,49 (2009):798-804.
[22]R.-N. Wang, Z.-Q. Li, and X.-H. Ding, Nonlocal Cauchy problems for semilinear evolution equations involving almost sectorial operators[J]. Indian Journal of Pure and Applied Mathematics,39 (2008):333-346.
[23]K. Ezzinbi, S. Ghnimib, M. A. Taoudi, Existence and Regularity of Solutions for Neutral Partial Functional Integro-differential Equations with Infinite Delay[J]. Nonlinear Analysis: Hybrid Systems,4 (2010):54-64.
[24]X.-L. Fu, Controllability of Abstract Neutral Functional Differential Systems with Unbounded Delay[J]. Applied Mathematics and Computation,151 (2004):299-314.
[25]Y.-K. Chang, A. Anguraj, M. Mallika Arjunan, Existence results for non-densely defined neutral impulsive differential inclusions with nonlocal conditions[J]. J. Appl. Math. Comput., 28 (2008):79-91.
[26]B. Liu, Controllability of impulsive neutral functional differential inclusions with infinite delay[J]. Nonlinear Analysis,60(2005):1533-1552.
[27]Z.-Y. Zhang, M. Blanke. A unified apporach to controllability analysis for hybrid systems[J]. Nonlinear Analysis:Hybrid systems 1 (2007):212-222.
[28]Y.-K. Chang, W.-S. Li, Solvability for Impulsive Neutral Integro Differential Equations with State-Dependent Delay via Fractional Operators[J]. J. Optim. Theory Appl.,144 (2010): 445-459.
[29]C. Lizama, Regularized solutions for abstract Volterra equations[J]. Journal of Mathematical Analysisand Applications,243 (2000):278-292.
[30]C. J. K. Batty, J. Liang, and T.-J. Xiao, On the spectral and growth bound of semigroups associatedwith hyperbolic equations[J]. Advances in Mathematics,191 (2005):1-10.
[31]K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations[M]. vol.194 of GraduateTexts in Mathematics, Springer, New York, NY, USA,2000.
[32]H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces[M]. vol.108 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands,1985.
[33]M. E. Gurtin, A. C. Pipkin, A General Theory of Heat Conduction with Finite Wave Speed [J]. Arch Ration Mech Anal..31 (1968):113-126.
[34]J. Liang, R. Nagel, and T.-J. Xiao, Approximation theorems for the propagators of higher orderabstract Cauchy problems[J]. Transactions of the American Mathematical Society,360 (2008):1723-1739.
[35]Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay[M]. in:Lectures Notes in Mathematics, vol.1473, Berlin:Springer,1991.
[36]B. N. Sadovskii, On a fixed point principle[J]. Funct. Anal. Appl.,1 (1967):74-76.
[37]C. Muller, Solving abstract Cauchy problems with closable operators in reflexive spaces via resolvent-free approximation[J]. Forum Mathematicum,19 (2007):1-18.
[38]T.-J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations[M]. vol.1701 of Lecture Notes in Mathematics, Springer, Berlin, Germany,1998.
[39]T.-J. Xiao and J. Liang, A solution to an open problem for wave equations with generalized Wentzell boundary conditions[J]. Mathematische Annalen,327 (2003):351-363.
[40]T.-J. Xiao and J. Liang, Complete second order differential equations in Banach spaces with dynamic boundary conditions[J]. Journal of Differential Equations,200 (2004):105-136.
[41]C. Cuevas and Julio Cesar de Souza, S-asymptotically co-periodic solutions of semilinear fractional integro-differential equations[J]. Appl. Math. Lett.,22 (2009):865-870.
[42]C. Cuevas and Julio Cesar de Souza, Existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay[J]. Nonlinear Analysis,72 (2010):1683-1689.
[43]Z.-B. Fan, Impulsive problems for semilinear differential equations with nonlocal conditions [J]. Nonlinear Analysis,72 (2010):1104-1109.
[44]A. Granas, J. Dugundji, Fixed Point Theory[M]. Springer-Verlag, New York,2003.
[45]H. Henriquez, M. Pierri, P. Taboas, On S-asymptotically ω-periodic functions on Banach spaces and applications[J]. J. Math. Anal. Appl.,343 (2008):1119-1130.
[46]V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations[M]. Modern Applied Mathematics, vol.6, World Scientific Publishing, New Jersey,1989.
[47]B. M. Levitan, V. V. Zhikov, Almost periodic functions and differential equations[M]. Cambridge Univ. Press,1982.
[48]G. M. Mophou, Existence and uniqueness of mild solutions to impulsive fractional differential equations[J]. Nonlinear Analysis,72 (2010):1604-1615.
[49]J. W. Nunziato, On heat conduction in materials with memory[J]. Quart. Appl. Math.,29 (1971):187-204.
[50]J. Banas and K. Goebe, Measure of Noncompactness in Banach Spaces[M]. Lect. Notes Pure Appl. Math., vol.60, Marcel Dekker, New York,1980.
[51]J.-W. Luo. Exponential Stability for Stochastic Neutral Partial Functional Differential Equations[J]. J Math Anal Appl,355 (2009):414-425
[52]T. Diagana, E. Hernandez, J. P. C. Dos Santos, Existence of Asymptotically Almost Automorphic Solutions to Some Abstract Partial Neutral Integro-differential Equations[J]. Nonlinear Analysis,71 (2009):248-257.
[53]Z.-R. Hu, Z. Jin, Necessary and Sufficient Conditions for the Regularity and Stability of Solutions for Some Partial Neutral Functional Differential Equations with Infinite Delay[J]. Nonlinear Analysis,73 (2010):2752-2765.
[54]K. Balachandran, J. P. Dauer, Controllability of Nonlinear Systems in Banach Spaces:A survey[J]. J Optim Theory Appl,115 (2002):7-28.
[55]K. Balachandran, R. Sakthivel. Controllability of Neutral Functional Integeodifferential Systems in Banach Spaces[J]. Computers and Mathematics with Applications,39 (2000): 117-126.
[56]M. Bartha, Periodic Solutions for Sifferential Squations with State-dependent Selay and Sositive Feedback[J]. Nonlinear Anal. T. M. A.,53 (6) (2003):839-857.
[57]J. P. C. Dos Santos, On State-dependent Selay Partial Neutral Functional Integro-differential Equations[J]. Applied Mathematics and Computeation,216 (2010):1637-1644.
[58]R. Grimmer, A. Pritchard, Analytic Resolvent Operators for Integral Equatins in Banach Spaces[J]. J. Differ. Equ.,50 (2) (1983):233-259.
[59]A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations [M]. in:Applied Mathematical Sciences, vol.44, Springer-Verlag, New York,1983.
[60]叶其孝,李正元,反应扩散方程引论[M].科学出版社,1994.
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