姚慧麗 李小桐 王晶囡 李雪鑫

摘 要：各類微分方程是基于不同實(shí)際問(wèn)題而建立的數(shù)學(xué)模型，研究方程的各種解的存在問(wèn)題引起了國(guó)內(nèi)外數(shù)學(xué)學(xué)者的關(guān)注。利用Banach壓縮映射原理、概自守型函數(shù)的有關(guān)理論以及卷積族的指數(shù)二分性，針對(duì)一類具有延遲的中立型微分方程的漸近概自守溫和解的存在唯一性問(wèn)題進(jìn)行研究。漸近概自守溫和解比概自守溫和解更具有一般性，因此本文所研究問(wèn)題會(huì)使這類方程的應(yīng)用范圍更加廣泛。

關(guān)鍵詞：漸近概自守溫和解;中立型微分方程;Banach壓縮映射原理

DOI：10.15938/j.jhust.2021.04.021

中圖分類號(hào)：O177.92

文獻(xiàn)標(biāo)志碼：A

文章編號(hào)：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(xué)科研究領(lǐng)域常用的工具，如數(shù)學(xué)、物理學(xué)、化學(xué)、生物學(xué)等。其中，要研究的多數(shù)問(wèn)題都可以轉(zhuǎn)化為探討微分方程的各類解的存在性問(wèn)題。目前為止已有大量文獻(xiàn)對(duì)各類微分方程的概周期型解[1-6]，概自守溫和解[7-11]、以及偽概自守溫和解[12-15]進(jìn)行了研究。

3 結(jié) 論

本文利用Banach壓縮映射原理、卷積族的指數(shù)二分性及概自守型函數(shù)的有關(guān)性質(zhì)證明了一類具有延遲的中立型微分方程在適當(dāng)?shù)臈l件下存在唯一的漸近概自守溫和解。

參 考 文 獻(xiàn)：

[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] 姚慧麗，孫海彤，卜憲江.一類可變延遲細(xì)胞神經(jīng)網(wǎng)絡(luò)的漸近概周期解[J]. 哈爾濱理工大學(xué)學(xué)報(bào)， 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.

（編輯：溫澤宇）