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摘 要：各类微分方程是基于不同实际问题而建立的数学模型，研究方程的各种解的存在问题引起了国内外数学学者的关注。利用Banach压缩映射原理、概自守型函数的有关理论以及卷积族的指数二分性，针对一类具有延迟的中立型微分方程的渐近概自守温和解的存在唯一性问题进行研究。渐近概自守温和解比概自守温和解更具有一般性，因此本文所研究问题会使这类方程的应用范围更加广泛。
关键词：渐近概自守温和解;中立型微分方程;Banach压缩映射原理
DOI：10.15938/j.jhust.2021.04.021
中图分类号：O177.92
文献标志码：A
文章编号：1007-2683（2021）04-0153-06
Abstract：All kinds of differential equations as mathematical models have been built up due to different practical problems， so the problem of studying the existence of various solutions has attracted the attention of mathematical scholars at home and abroad. The problem on the existence and uniqueness of asymptotically almost automorphic mild solutions for a class of neutral differential equations with delay are researched by using Banach compression mapping principle， related theorems of almost automorphic type functions and the exponential dichotomy of convolution family in this paper. Asymptotically almost automorphic mild solutions are more general than almost automorphic mild solutions， so the research of this paper will make the scope of application on this kind of differential equations more extensive.
Keywords：asymptotically almost automorphic mild solutions; neutral differential equations; Banach compression mapping principle
0 引 言
微分方程是多學科研究领域常用的工具，如数学、物理学、化学、生物学等。其中，要研究的多数问题都可以转化为探讨微分方程的各类解的存在性问题。目前为止已有大量文献对各类微分方程的概周期型解[1-6]，概自守温和解[7-11]、以及伪概自守温和解[12-15]进行了研究。
3 结 论
本文利用Banach压缩映射原理、卷积族的指数二分性及概自守型函数的有关性质证明了一类具有延迟的中立型微分方程在适当的条件下存在唯一的渐近概自守温和解。
参 考 文 献：
[1] MYSLO Y M， TKACHENKO V I. Asymptotically Almost Periodic Solutions of Equations with Delays and Nonfixed Times of Pulse Action[J]. Journal of Mathematical Sciences， 2018， 228（3）：290.
[2] SLYUSARCHUK V Y. Almost Periodic Solutions of Functional Equations[J]. Journal of Mathematical Sciences， 2017，222（3）：359.
[3] YU YUEHUA， GONG SHUHUA. Pseudo Almost Periodic Solutions for First-Order Neutral Differential Equations[J]. Advances in Difference Equations， 2018， 2018（1）：1.
[4] DIAGANA T， HERNANDEZ E， RABELLO M. Pseudo Almost Periodic Solutions to Some Nonautomous Neutral Functional Differential Equations with UnboundedDelay[J]. Mathematical and Computer Modelling. 2007， 45（9/10）：1241.
[5] ZHAO Z H， CHANG Y K， LI W S. Asymptotically Almost Periodic， Almost Periodic and Pseudo-Almost Periodic Mild Solutions for Neutral Differential Equations[J]. Nonlinear Analysis：Real World Applications， 2010， 11（4）：3037. [6] 姚慧丽，孙海彤，卜宪江.一类可变延迟细胞神经网络的渐近概周期解[J]. 哈尔滨理工大学学报， 2017， 22（5）：130.
YAO HUILI， SUN HAITONG， BU XIANJIANG. Asymptotically Almost Periodic Solutions for a Class of Cellular Neural Networks with Varying Delays[J]. Journal of Harbin University of Science and Technology， 2017， 22（5）：130.
[7] REVATHIP， SAKTHI VEL R， REN Y， et al. Existence of Almost Automorphic Mild Solution to Nonautonomous Neutral Stochastic Differential Equation[J]. Applied Mathematics and Computation， 2014，230：639.
[8] DIAGANA T， HERNAN R. HENRIQUEZ， EDUARDO M. HERNANDEZ. Almost Automorphic Mild Solutions t Some Partial Neutral Functional-Differential Equations and Applications[J]. Nonlinear Analysis Theory Methods & Applications， 2008， 69（5-6）：1485.
[9] MISHRA， BAHUGUNA， ABBAS. Existence of Almost Automorphic Solutions of Neutral Functional Differentia Equation[J]. Nonlinear Dynamics & Systems Theory， 2011， 2011（2）：165.
[10]BOULITE S，MANIAR L.， NGURKATA G M.Almost automorphic solutions for hyperbolic semilinear evolution equations[J]. Semigroup Forum，2005，71：231.
[11]GISELE M M， GASTON M N. Almost Automorphic Solutions of Neutral Functional Differential Equations[J]. Electronic Journal of Differential Equation， 2010， 69：1.
[12]GU Y， REN Y， SAKTHIVEL R. Square-Mean Pseudo Almost Automorphic Mild Solutions for Stochastic Evolution Equations Driven By Brownian Motion[J]. Stochastic Analysis and Applications， 2016， 34（3）：528.
[13]XIAO T J， ZHU X X， LIANG J. Pseudo-Almost Autonomorphic Mild Solutions to Nonautonomous Differential Equations and Applications[J]. Nonlinear Analysis Theory Methods & Applications， 2008， 70（11）：4079.
[14]CHANG Y K， FENG T W. Properties on Measure Pseudo Almost Automorphic Functions and Applications to Fractional Differential Equations in Banach Space[J]. Electronic Journal of Differential Equations， 2018， 2018（47）：1.
[15]CHANG Y K， ZHENG S. Weighted Pseudo Almost Automorphic Solutions to Functional Differential Equations with Infinite Delay[J]. Electronic Journal of Differential Equations， 2016， 2016（286）：1.
[16]BOCHNER S. A New Approach to Almost Periodicity[J]. Proc. Nat. Acad. Sci.U. S. A， 1962， 48：2039.
[17]N′GUEREKATA， GASTON MANDATA. Some Remarks on Asymptotically Almost Automorphic Function[J]. Rivista Di Matematica Della Università Di Parma， 1988， 13（4）：301.
[18]GISELE M M，GASTON M N. Almost Automorphic Solutions of Neutral Functional Differential Equations[J]. Electronic Journal of Differential Equation， 2010，69：1.
[19]YAGII A.Abstract Quasilinear Evolution Equations of Parabolic Type in Banachspaces[J]. Boll.Un. Mat. Ital.B，1991，7（5）：341.
[20]CHICONE C，LATUSHKIN Y.Evolution semigroups in dynamical systems and differential equations[J].Amer.Math.Soc.， 1999，40：65.
[21]LIANG J， ZHANG JUN，XIAO T-J. Composition of Pseudo Almost Automorphic and Asymptotically Almost Autoorphic Functions[J]. Math.Anal.Appl， 2008，340（2）：1493.
[22]DING H-S，NGURKATA G M，LONG W.Almost Automorphic Solutions of Nonautonomous Evolution Equations[J]. Nonlinear Analysis， 2009， 70（12）：4158.
（編辑：温泽宇）
关键词：渐近概自守温和解;中立型微分方程;Banach压缩映射原理
DOI：10.15938/j.jhust.2021.04.021
中图分类号：O177.92
文献标志码：A
文章编号：1007-2683（2021）04-0153-06
Abstract：All kinds of differential equations as mathematical models have been built up due to different practical problems， so the problem of studying the existence of various solutions has attracted the attention of mathematical scholars at home and abroad. The problem on the existence and uniqueness of asymptotically almost automorphic mild solutions for a class of neutral differential equations with delay are researched by using Banach compression mapping principle， related theorems of almost automorphic type functions and the exponential dichotomy of convolution family in this paper. Asymptotically almost automorphic mild solutions are more general than almost automorphic mild solutions， so the research of this paper will make the scope of application on this kind of differential equations more extensive.
Keywords：asymptotically almost automorphic mild solutions; neutral differential equations; Banach compression mapping principle
0 引 言
微分方程是多學科研究领域常用的工具，如数学、物理学、化学、生物学等。其中，要研究的多数问题都可以转化为探讨微分方程的各类解的存在性问题。目前为止已有大量文献对各类微分方程的概周期型解[1-6]，概自守温和解[7-11]、以及伪概自守温和解[12-15]进行了研究。
3 结 论
本文利用Banach压缩映射原理、卷积族的指数二分性及概自守型函数的有关性质证明了一类具有延迟的中立型微分方程在适当的条件下存在唯一的渐近概自守温和解。
参 考 文 献：
[1] MYSLO Y M， TKACHENKO V I. Asymptotically Almost Periodic Solutions of Equations with Delays and Nonfixed Times of Pulse Action[J]. Journal of Mathematical Sciences， 2018， 228（3）：290.
[2] SLYUSARCHUK V Y. Almost Periodic Solutions of Functional Equations[J]. Journal of Mathematical Sciences， 2017，222（3）：359.
[3] YU YUEHUA， GONG SHUHUA. Pseudo Almost Periodic Solutions for First-Order Neutral Differential Equations[J]. Advances in Difference Equations， 2018， 2018（1）：1.
[4] DIAGANA T， HERNANDEZ E， RABELLO M. Pseudo Almost Periodic Solutions to Some Nonautomous Neutral Functional Differential Equations with UnboundedDelay[J]. Mathematical and Computer Modelling. 2007， 45（9/10）：1241.
[5] ZHAO Z H， CHANG Y K， LI W S. Asymptotically Almost Periodic， Almost Periodic and Pseudo-Almost Periodic Mild Solutions for Neutral Differential Equations[J]. Nonlinear Analysis：Real World Applications， 2010， 11（4）：3037. [6] 姚慧丽，孙海彤，卜宪江.一类可变延迟细胞神经网络的渐近概周期解[J]. 哈尔滨理工大学学报， 2017， 22（5）：130.
YAO HUILI， SUN HAITONG， BU XIANJIANG. Asymptotically Almost Periodic Solutions for a Class of Cellular Neural Networks with Varying Delays[J]. Journal of Harbin University of Science and Technology， 2017， 22（5）：130.
[7] REVATHIP， SAKTHI VEL R， REN Y， et al. Existence of Almost Automorphic Mild Solution to Nonautonomous Neutral Stochastic Differential Equation[J]. Applied Mathematics and Computation， 2014，230：639.
[8] DIAGANA T， HERNAN R. HENRIQUEZ， EDUARDO M. HERNANDEZ. Almost Automorphic Mild Solutions t Some Partial Neutral Functional-Differential Equations and Applications[J]. Nonlinear Analysis Theory Methods & Applications， 2008， 69（5-6）：1485.
[9] MISHRA， BAHUGUNA， ABBAS. Existence of Almost Automorphic Solutions of Neutral Functional Differentia Equation[J]. Nonlinear Dynamics & Systems Theory， 2011， 2011（2）：165.
[10]BOULITE S，MANIAR L.， NGURKATA G M.Almost automorphic solutions for hyperbolic semilinear evolution equations[J]. Semigroup Forum，2005，71：231.
[11]GISELE M M， GASTON M N. Almost Automorphic Solutions of Neutral Functional Differential Equations[J]. Electronic Journal of Differential Equation， 2010， 69：1.
[12]GU Y， REN Y， SAKTHIVEL R. Square-Mean Pseudo Almost Automorphic Mild Solutions for Stochastic Evolution Equations Driven By Brownian Motion[J]. Stochastic Analysis and Applications， 2016， 34（3）：528.
[13]XIAO T J， ZHU X X， LIANG J. Pseudo-Almost Autonomorphic Mild Solutions to Nonautonomous Differential Equations and Applications[J]. Nonlinear Analysis Theory Methods & Applications， 2008， 70（11）：4079.
[14]CHANG Y K， FENG T W. Properties on Measure Pseudo Almost Automorphic Functions and Applications to Fractional Differential Equations in Banach Space[J]. Electronic Journal of Differential Equations， 2018， 2018（47）：1.
[15]CHANG Y K， ZHENG S. Weighted Pseudo Almost Automorphic Solutions to Functional Differential Equations with Infinite Delay[J]. Electronic Journal of Differential Equations， 2016， 2016（286）：1.
[16]BOCHNER S. A New Approach to Almost Periodicity[J]. Proc. Nat. Acad. Sci.U. S. A， 1962， 48：2039.
[17]N′GUEREKATA， GASTON MANDATA. Some Remarks on Asymptotically Almost Automorphic Function[J]. Rivista Di Matematica Della Università Di Parma， 1988， 13（4）：301.
[18]GISELE M M，GASTON M N. Almost Automorphic Solutions of Neutral Functional Differential Equations[J]. Electronic Journal of Differential Equation， 2010，69：1.
[19]YAGII A.Abstract Quasilinear Evolution Equations of Parabolic Type in Banachspaces[J]. Boll.Un. Mat. Ital.B，1991，7（5）：341.
[20]CHICONE C，LATUSHKIN Y.Evolution semigroups in dynamical systems and differential equations[J].Amer.Math.Soc.， 1999，40：65.
[21]LIANG J， ZHANG JUN，XIAO T-J. Composition of Pseudo Almost Automorphic and Asymptotically Almost Autoorphic Functions[J]. Math.Anal.Appl， 2008，340（2）：1493.
[22]DING H-S，NGURKATA G M，LONG W.Almost Automorphic Solutions of Nonautonomous Evolution Equations[J]. Nonlinear Analysis， 2009， 70（12）：4158.
（編辑：温泽宇）