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整数阶与分数阶非线性微分方程边值问题正解的存在性摘要：本文主要讨论了整数阶和分数阶非线性微分方程边值问题正解的存在性。
首先介绍了整数阶微分方程边值问题的解法，包括格林函数、变分法、等等。
而对于分数阶微分方程边值问题，基于Caputo导数的求解方法被广泛应用于各种实际问题中。
然后，通过在边值问题的严格数学框架下，该文证明了整数阶和分数阶非线性微分方程边值问题的解在一定条件下存在，这些条件包括边值问题的充分性和DFC（differential inequality of finite difference）条件的满足。
最后，多个实例说明了该文所证明的理论结论的实用性和有效性。
关键词：整数阶微分方程；分数阶微分方程；边值问题；正解存在性；格林函数；变分法；Caputo导数1. 引言微分方程在物理、工程、生物、经济等众多领域中都有重要应用。
边值问题是求解微分方程的一种常用方法，它使用一些限制条件来约束解的特性。
而关于整数阶和分数阶微分方程边值问题正解的存在性，一直是微分方程理论中的经典研究问题。
本文旨在探讨这个问题，并通过实例说明所得结论的实用性和有效性。
2. 整数阶微分方程边值问题的解法对于一般的整数阶微分方程边值问题，我们通常采用格林函数、变分法等方法，来求解其正解存在性。
格林函数是一种特殊的解析函数，在微分方程理论中扮演着重要角色。
变分法是另一种常见的求解方法，它可以转化为极值问题，得到问题的最优解。
3. 分数阶微分方程边值问题的求解方法分数阶微分方程边值问题的求解方法虽然和整数阶微分方程有相似之处，但依然有其特殊之处。
此处我们介绍一种基于Caputo导数的求解方法，它广泛应用于各种实际问题中。
该方法将原问题转化为一个无约束问题，并使用Laplace变换和拉普拉斯逆变换求解。
4. 整数阶和分数阶非线性微分方程边值问题正解的存在性在边值问题的严格数学框架下，我们证明了整数阶和分数阶非线性微分方程边值问题的解在一定条件下存在。
一类Caputo分数阶微分方程正解的存在性作者：李云红吕文静来源：《河北科技大学学报》2016年第06期摘要：为了研究一类带p-Laplacian 算子的Caputo分数阶微分方程边值问题正解的存在性，通过计算得到该问题的格林函数，并讨论其性质。
运用单调迭代方法，得到该边值问题至少存在2个正解，最后通过实例验证了此类方程边值问题正解的存在性。
关键词：常微分方程其他学科；Caputo分数阶微分；正解；单调迭代方法；边值问题中图分类号：O175.1 MSC（2010）主题分类：34B15 文献标志码：AAbstract： In order to investigate the existence of positive solutions to a class of Caputo fractional differential equation boundary value problems with p-Laplacian operator，the Green’s function is obtained by calculus， and its properties are discussed. By using monotone iterative technique， at least two positive solutions are obtained for the boundary value problems. An exampleis given to illustrate the existence of positive solutions to this kind of equation boundary value problems.Keywords：ordinary differential equation； Caputo fractional derivative；positive solution；monotone iteratiation；boundary value problems参考文献/References：[1] AHMAD B， MATAR M， AGARWAL R. Existence results for fractional differential equations of arbitrary order with nonlocal integral boundary conditions[J]. Boundary Value Problem，2015，220：1-13.[2] ZHANG Lihong， AHMAD B， WANG Guotao，et al. Nonlinear fractional integro-differential equations on unbounded domains in a Banach space[J]. Journal of Computational and Applied Mathematics，2013，240：51-56.[3] ALBERTO C， HAMDI Z. Nonlinear fractional differential equations with integral boundary value conditions[J]. Applied Mathematics and Computation，2014，228：251-257.[4] VONG S. Positive solutions of singlular fractional differential equations with integral boundary conditions[J]. Mathematical and Computer Modelling，2013，57： 1053-1059.[5] SOTIRIS K N， SINA E. On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions[J]. Applied Mathematics and Computation，2015，266：235-243.[6] KOU Chunhai， ZHOU Huacheng， YAN Ye. Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis[J]. Nonlinear Analysis，2011，74： 5975-5986.[7] CUI Yujun. Uniqueness of solution for boundary value problems for fractional differential equations[J]. Applied Mathematics Letters，2016，51：48-54.[8] ZHANG Lihong，AHMAD B， WANG Guotao. The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative[J]. Applied Mathematics Letters，2014，31：1-6.[9] LI Yunhong， LI Guogang. Positive solutions of p-Laplacian fractional differential equations with integral boundary value conditions[J]. Journal of Nonlinear Science and Applications，2016，9（3）：717-726.[10]ZHAI Chengbo， XU Li. Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter[J]. Communications in Nonlinear Science Numerical Simulation，2014，19：2820-2827.[11] XU Xiaojie，FEI Xiangli. The positive properties of Green’s function for three poi nt boundary value problems of nonlinear fractional differential equations and its applications[J]. Communications in Nonlinear Science Numerical Simulation，2012，17（4）：1555-1565.[12] NTOUYAS S， ETEMAD S. On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions[J]. Applied Mathematics and Computation，2015，266：235-243.[13] 李云红，李艳. 带p-Laplacian算子的分数阶微分方程的正解[J].河北科技大学学报，2015，36（6）：593-597.LI Yunhong，LI Yan. A positive solution for the fractional differential equation with a p-Laplacian operator[J]. Journal of Hebei University of Science and Technology， 2015，36（6）：593-597.[14] ZHANG Xinguang， LIU Lisan， WIWATANAPATAPHEE B， et al. The eigenvalue fora class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition[J]. Applied Mathematics and Computation，2014，235：412-422.[15]AMKO S G，KILBAS A A，MARICHEV O I. Fractional Integrals and Derivatives：Theory and Applications[M]. Switzerland： Gordon and Breach，1993.[16]PODLUNY I. Fractional Differential Equations， Mathematics in Science and Engineering[M]. New York： Academic Press，1999.。
Caputo分数阶微分方程的发展过程、求解和应用作者：陈果来源：《商情》2020年第34期【摘要】文章研究了关于Caputo分数阶微分方程的发展过程到边值问题的求解，并探索分数阶微分方程的脉冲边值问题的解的存在性、可解性。
分数阶微分方程的不断发展为解决现实问题提出了更多切合实际的数学模型，本文给出了HIV-1动力学的应用，主要是验证了所获理论结果的有效性并为整篇文章做出总结。
【关键词】Caputo分数阶微分; 微分方程; 分数阶微分方程应用1、研究背景从1695年开始，Leibnitz给L' Hospital的信中就提到了分数阶微分的概念。
作为一种新的数学理论和方法，分數阶微分方程和分数微积分解决了很多之前驻足不前的实际问题。
实际上，分数阶微分方程改进了很多实际问题的数学模型，对无穷时滞、脉冲、耦合系统或者含不同算子的方程求解和解的存在性结论提供了很多研究空间。
而本文的研究对象，就是Caputo 型的分数阶微分方程、解的存在性及其应用。
2、Caputo型分数阶微分方程2.1 普托型分数微分方程的非局部多点边值问题求解在文献[1]中，作者对新的非局部多点边值问题，即Caputo型连续分数积分-微分方程解的存在性和唯一性进行了推导。
其中研究对象为Caputo型分数积分-微分方程，在Riemann-Liouville的积分边界条件下，利用不动点定理得到了几个存在唯一性的结果。
该方程如下：并服从下述边界条件其中cD（·）表示分数阶（·）的Caputo导数，I（·）表示Riemann-Liouville积分分数阶（·）， f：[0，1]×R3→R是一个给定的连续函数。
λ，ai，i=1，……，m是实常数。
在任意段（0，η）？奂[0，1]上考虑的带式条件可以解释为非局部点处未知函数值的线性组合ζ∈（0，1）。
作者定义了Banach空间：有如下范式求解过程如下，引入算子F：X→X：方程（2.1）-（2.2）的问题即可转换成（2.3），即如果F有不动点，那么原方程（2.1）-（2.2）有解。
第61卷 第3期吉林大学学报(理学版)V o l .61 N o .32023年5月J o u r n a l o f J i l i nU n i v e r s i t y (S c i e n c eE d i t i o n )M a y2023d o i :10.13413/j .c n k i .jd x b l x b .2022299带有S t ie l t j e s 积分边界条件和ψ-C a pu t o 导数的分数阶边值问题解的存在性王 宁,周宗福(安徽大学数学科学学院,合肥230601)摘要:考虑一类带有ψ-C a pu t o 分数阶导数的分数阶微分方程边值问题,其边界条件含有S t i e l t j e s 积分和多个ψ-R i e m a n n -L i o u v i l l e 分数阶积分算子.先给出其相应辅助边值问题解的表达式,再利用B a n a c h 压缩映像原理和S c h a e f e r 不动点定理证明该边值问题解的存在性和唯一性结果,最后举例说明所得结果的有效性.关键词:ψ-C a p u t o 导数;S t i e l t j e s 积分;多点;不动点定理中图分类号:O 175.8 文献标志码:A 文章编号:1671-5489(2023)03-0469-08E x i s t e n c e o f S o l u t i o n s f o rF r a c t i o n a l B o u n d a r y V a l u eP r o b l e m sw i t h ψ-C a p u t oD e r i v a t i v e a n dS t i e l t j e s I n t e g r a l B o u n d a r y C o n d i t i o n s WA N G N i n g ,Z HO UZ o n gf u (S c h o o l o f M a t h e m a t i c a lS c i e n c e s ,A n h u i U n i v e r s i t y ,H e fe i 230601,C h i n a )A b s t r a c t :W e c o n s i d e r e da c l a s sof b o u n d a r y v a l u e p r o b l e m s f o r f r a c t i o n a l d i f f e r e n t i a l e qu a t i o n sw i t h ψ-C a p u t of r a c t i o n a ld e r i v a t i v e ,t h eb o u n d a r y c o n d i t i o n si n c l u d e da S t i e l t j e si n t e g r a la n d m u l t i p l e ψ-R i e m a n n -L i o u v i l l e f r a c t i o n a l i n t e g r a l o p e r a t o r s .F i r s t l y ,t h e e x p r e s s i o no f s o l u t i o no f c o r r e s p o n d i n g a u x i l i a r y b o u n d a r y v a l u e p r o b l e m w a s g i v e n .S e c o n d l y ,t h e e x i s t e n c e a n d u n i qu e n e s s o f s o l u t i o n o f t h e b o u n d a r y v a l u e p r o b l e m w e r e p r o v e db y u s i n g B a n a c hc o m p r e s s i o n m a p p i n gp r i n c i p l ea n dS c h a e f e r f i x e d p o i n t t h e o r e m.F i n a l l y ,w e g a v e a ne x a m p l e t o i l l u s t r a t e t h e v a l i d i t y o f t h e o b t a i n e d r e s u l t s .K e y w o r d s :ψ-C a p u t od e r i v a t i v e ;S t i e l t j e s i n t e g r a l ;m u l t i p o i n t ;f i x e d p o i n t t h e o r e m 收稿日期:2022-07-04.第一作者简介:王 宁(1997 ),女,汉族,硕士研究生,从事微分方程的研究,E -m a i l :2925502108@q q.c o m.通信作者简介:周宗福(1964 ),男,汉族,硕士,教授,从事微分方程的研究,E -m a i l :z h o u z f 12@126.c o m.基金项目:安徽省自然科学基金(批准号:1608085MA 12).0 引 言分数阶微积分是整数阶微积分的推广,由于分数阶导数具有良好的记忆性质和遗传特性,使得其在物理和工程等领域应用广泛[1-3],关于分数阶微分方程的研究目前已取得了丰富成果[4-9].分数阶导数的定义形式繁多,为克服该问题,A l m e i d a [10]将C a p u t o 分数阶导数与带有核的R i e m a n n -L i o u v i l l e 分数阶导数相结合,并引入ψ-C a p u t o 分数阶导数,研究了它的一些性质,从而将多种形式的C a p u t o 分数阶导数统一起来,促进了具有一般导数形式的分数阶微积分的发展.文献[11]研究了下列边值问题:Copyright ©博看网. All Rights Reserved.cD α,ψa +y (t )=f (t ,y (t )), t ɪ[a ,b ],y [k ]ψ(a )=y k a (k =0,1, ,n -2), y [n -1]ψ(b )=y b {,(1)其中c D α,ψa +是α阶ψ-C a p u t o 分数阶导数,n -1<αɤn ,n ȡ2,y k a ,y b ɪℝ.文献[12]研究了如下边值问题:c D α,ψ0+(c D β,ψ0+u (t )+h (t ,u (t )))=f (t ,u (t )), 0<α,βɤ1,u (0)=a 1ʏ10u (s )d A 1(s ),u (1)=a 2ʏ10u (s )d A 2(s )+a 3ðm i =1μi I γ,ψ0+u (ηi ìîíïïïïïï),(2)其中c D α,ψ0+,c D β,ψ0+分别是α阶和β阶ψ-C a p u t o 分数阶导数,a i (i =1,2,3)为实数,μi ,ηi (i =1,2, ,m )为正数.受上述研究工作的启发,本文考虑如下分数阶微分方程边值问题:c D α,ψa +u (t )=f (t ,u (t )), t ɪ[a ,b ],u [k ]ψ(a )=u k , k =0,1, ,n -2,u [n -1]ψ(b )=l ʏbau (τ)d A (τ)+ðm i =1μi I βi ,ψa +u (ηi ìîíïïïïïï),(3)其中:cDα,ψa+为α阶ψ-C a p u t o 分数阶导数,n -1<αɤn ,n ȡ2;f :[a ,b ]ˑℝңℝ连续,u k ɪℝ(k =0,1, ,n -2),ʏb au (τ)d A (τ)是R i e m a n n -S t i e l t j e s 积分,A (τ)是[a ,b ]上的单调递增函数,I βi ,ψa+是ψ-R i e m a n n -L i o u v i l l e 分数阶积分,βi >0,μi ɪℝ,ηi ɪ(a ,b )(i =1,2, ,m );l ɪℝ;ψ:[a ,b ]ңℝ是一个单调递增的可微函数且ψᶄ(t )ʂ0,t ɪ[a ,b ];u [k ]ψ(t )=1ψᶄ(t )d d éëêêùûúút k u (t ),k ɪ{0,1, ,n -2},u [n -1]ψ(t )=1ψᶄ(t )d d éëêêùûúút n -1u (t ).1 预备知识定义1[13] 令α>0,u 为[a ,b ]上的可积函数,u (t )的α阶ψ-R i e m a n n -L i o u v i l l e 分数阶积分定义为I α,ψa +u (t )=1Γ(α)ʏt a ψᶄ(s )(ψ(t )-ψ(s ))α-1u (s )d s . 定义2[13]令α>0,u :[a ,b ]ңℝ为可积函数,u (t )的α阶ψ-R i e m a n n -L i o u v i l l e 分数阶导数定义为D α,ψa+u (t )=1ψᶄ(t )d d éëêêùûúút n I n -α,ψa+u (t ),其中n =[α]+1,α∉ℕ,α,αɪℕ{.定义3[13] 令α>0,若u ɪC n -1[a ,b ],则u (t )的α阶ψ-C a pu t o 分数阶导数定义为c D α,ψa +u (t )=D α,ψa +u (t )-ðn -1k =0u [k ]ψ(a )k !(ψ(t )-ψ(a ))éëêêùûúúk ,其中n =[α]+1,α∉ℕ,α,αɪℕ{.定义4[10] 若u ɪC n[a ,b ]且αɪℕ,则cD α,ψa+u (t )=I n -α,ψa +1ψᶄ(t )d d éëêêùûúút n u (t )=1Γ(n -α)ʏt a ψᶄ(s )(ψ(t )-ψ(s ))n -αu [n ]ψ(s )d s ,若α=n ɪℕ,则c D α,ψa +u (t )=u [n ]ψ(t ).074 吉林大学学报(理学版) 第61卷Copyright ©博看网. All Rights Reserved.引理1[13] 令α>0,u :[a ,b ]ңℝ,则下列结论成立:1)若u ɪC [a ,b ],则c D α,ψa +I α,ψa +u (t )=u (t );2)若u ɪC n -1[a ,b ],则I α,ψa+cD α,ψa+u (t )=u (t )-ðn -1k =0C k (ψ(t )-ψ(a ))k ,其中C k =u [k]ψ(a )k !.引理2[10] 令α,β>0,u :[a ,b ]ңℝ为可积函数,则:1)I α,ψa+[ψ(t )-ψ(a )]β-1=Γ(β)Γ(α+β)[ψ(t )-ψ(a )]α+β-1;2)c D α,ψa +[ψ(t )-ψ(a )]β-1=Γ(β)Γ(β-α)[ψ(t )-ψ(a )]β-α-1,其中βɪℝ,β>n ,n =[α]+1;3)c D α,ψa +[ψ(t )-ψ(a )]k=0,∀k ɪ{0,1, ,n -1},n =[α]+1ɪℕ;4)I α,ψa +I β,ψa +u (t )=I α+β,ψa+u (t ).引理3(S c h a e f e r 不动点定理)[14]设X 为B a n a c h 空间,T:X ңX 是一个连续的紧映射,{u ɪX u =λT (u ),0<λ<1}有界,则T 至少存在一个不动点.引理4 令n -1<α<n ,g :[a ,b ]ңℝ为连续函数,则分数阶边值问题c D α,ψa +u (t )=g (t ), t ɪ[a ,b ],u [k ]ψ(a )=uk , k =0,1, ,n -2,u [n -1]ψ(b )=l ʏb au (τ)d A (τ)+ðm i =1μi I βi ,ψa +u (ηi ìîíïïïïïï)(4)的解为u (t )=ðn -2k =0u k k !(ψ(t )-ψ(a ))k+1M [P +l ʏba I α,ψa +g (τ)d A (τ)+ðmi =1μi I α+βi ,ψa +g (ηi )+ðn -2k =0λk u k -I α-n +1,ψa +g (b )]ˑ(ψ(t )-ψ(a ))n -1+I α,ψa +g (t),(5)其中M =(n -1)!-l ʏba(ψ(τ)-ψ(a ))n -1d A (τ)-ðmi =1μi I βi ,ψa+(ψ(ηi )-ψ(a ))n -1,P =lʏb a ðn-2k =0u kk !(ψ(τ)-ψ(a ))kd A (τ),λk =ðmi =1μi I βi ,ψa+(ψ(ηi )-ψ(a ))kk !.证明:由引理1知,u (t )=ðn -1k =0C k (ψ(t )-ψ(a ))k+I α,ψa +g (t),(6)其中C k =u [k]ψ(a )k !=u k k !,k =0,1, ,n -2,则u [1]ψ(t )=u ᶄ(t )ψᶄ(t )=ðn -1k =1k C k (ψ(t )-ψ(a ))k -1+1Γ(α-1)ʏta ψᶄ(s )(ψ(t )-ψ(s ))α-2g (s )d s ,u [2]ψ(t )=(u [1]ψ(t ))ᶄψᶄ(t )=ðn -1k =2k (k -1)C k (ψ(t )-ψ(a ))k -2+1Γ(α-2)ʏt a ψᶄ(s )(ψ(t )-ψ(s ))α-3g (s )d s , ︙u [n -1]ψ(t )=(n -1)!C n -1+1Γ(α-n +1)ʏtaψᶄ(s )(ψ(t )-ψ(s ))α-n g (s )d s .由边值条件u [n -1]ψ(b )=l ʏbau (τ)d A (τ)+ðmi =1μi I βi ,ψa+u (ηi )可知,174 第3期 王 宁,等:带有S t i e l t j e s 积分边界条件和ψ-C a pu t o 导数的分数阶边值问题解的存在性 Copyright ©博看网. All Rights Reserved.(n -1)!C n -1+1Γ(α-n +1)ʏb a ψᶄ(s )(ψ(t )-ψ(s ))α-n g (s )d s =l ʏb a u (τ)d A (τ)+ðmi =1μi I βi ,ψa +u (ηi ),因此,C n -1=1(n -1)!l ʏb a u (τ)d A (τ)+ðmi =1μi I βi ,ψa +u (ηi )-I α-n +1,ψa +g (b []).(7)将式(6)代入式(7)可得C n -1=1M P +l ʏb a I α,ψa +g (τ)d A (τ)+ðmi =1μi I α+βi ,ψa +g (ηi )+ðn -1k =0λk u k -I α-n +1,ψa +g (b []).将求出的C 0,C 1, ,C n -1代入(6),即知式(5)成立.证毕.2 主要结果定义算子T :C [a ,b ]ңC [a ,b ],(T u )(t )=ðn -2k =0u k k !(ψ(t )-ψ(a ))k+1M [P +l ʏba I α,ψa +f (τ,u (τ))d A (τ)+ðmi =1μi I α+βi ,ψa +f (ηi ,u (ηi ))+ðn -2k =0λk u k -I α-n +1,ψa +f (b ,u (b ))]ˑ(ψ(t )-ψ(a ))n -1+I α,ψa +f (t ,u (t )).由引理4知,T 的不动点即为边值问题(3)的解.为方便,引入以下记号:Λ1=l (A (b )-A (a ))M Γ(α+1)(ψ(b )-ψ(a ))n -1+1Mðmi =1μi (ψ(b )-ψ(a ))βi +n -1Γ(α+βi +1)+1M Γ(α-n +2)+1Γ(α+1). 假设条件:(H 1)存在常数ξ>0,使得f (t ,x )-f (t ,y )ɤξx -y ,t ɪ[a ,b ],x ,y ɪℝ;(H 2)ξΛ1(ψ(b )-ψ(a ))α<1.定理1 假设条件(H 1),(H 2)成立,则边值问题(3)在[a ,b ]上存在唯一解.证明:令B r 0={u ɪC [a ,b ]: u ɤr 0},其中r 0ȡΛ21-ξΛ1(ψ(b )-ψ(a ))α,Λ2=ðn -2k =0u k k !(ψ(b )-ψ(a ))k +1M P +L l (A (b )-A (a ))(ψ(b )-ψ(a ))αΓ(α+1)éëêê+ðmi =1μiL (ψ(b )-ψ(a ))α+βi Γ(α+βi +1)+ðn -2k =0λk ㊃u ùûúúk (ψ(b )-ψ(a ))n -1+L (ψ(b )-ψ(a ))αM Γ(α-n +2)+L (ψ(b )-ψ(a ))αΓ(α+1),L =s u p t ɪ[a ,b ]f (t ,0). 先证T :B r 0ңB r 0.∀u ɪB r 0及∀t ɪ[a ,b ],有(T u )(t )ɤðn -2k =0u k k!(ψ(t )-ψ(a ))k+1M [P+l ʏb aI α,ψa+f (τ,u (τ))d A (τ)+ðmi =1μi I α+βi ,ψa+f (ηi ,u (ηi ))+ðn -2k =0λk ㊃u k +I α-n +1,ψa +f (b ,u (b ))]ˑ(ψ(t )-ψ(a ))n -1+I α,ψa+f (t ,u (t ))ɤðn -2k =0u k k !(ψ(t )-ψ(a ))k+274 吉林大学学报(理学版) 第61卷Copyright ©博看网. All Rights Reserved.1M{P+l ʏb aI α,ψa+[f (τ,u (τ))-f (τ,0)+f (τ,0)]d A (τ)+ðmi =1μiIα+βi ,ψa+[f (ηi ,u (ηi ))-f (ηi ,0)+f (ηi ,0)]+ðn -2k =0λk ㊃u k +I α-n +1,ψa+f (b ,u (b ))-f (b ,0)+f (b ,0[])}(ψ(t )-ψ(a ))n -1+I α,ψa+[f (t ,u (t ))-f (t ,0)+f (t ,0)]ɤðn -2k =0u k k !(ψ(t )-ψ(a ))k +1M{P+l ʏbaI α,ψa +[ξu (τ)+f (τ,0)]d A (τ)+ðmi =1μi I α+βi ,ψa+[ξu (ηi )+f (ηi ,0)]+ðn -2k =0λk ㊃u k +I α-n +1,ψa+[ξu (b )+f (b ,0)]}ˑ(ψ(t )-ψ(a ))n -1+I α,ψa +[ξu (t )+f (t ,0)]ɤðn -2k =0u k k !(ψ(b )-ψ(a ))k +1M P +l (ξ u +L )(A (b )-A (a ))(ψ(b )-ψ(a ))αΓ(α+1éëêê)+ðm i =1μi (ξ u +L )(ψ(b )-ψ(a ))α+βi Γ(α+βi +1)+ðn -2k =0λk ㊃u ùûúúk ˑ(ψ(b )-ψ(a ))n -1+(ξ u +L )(ψ(b )-ψ(a ))αM Γ(α-n +2)+(ξ u +L )(ψ(b )-ψ(a ))αΓ(α+1)=ξΛ1(ψ(b )-ψ(a ))α u +Λ2.因此, T u ɤξΛ1(ψ(b )-ψ(a ))αr 0+Λ2ɤr 0,即T B r 0⊂B r 0,从而T :B r 0ңB r 0.下证边值问题(3)在[a ,b ]上存在唯一解.令u 1,u 2ɪB r 0,t ɪ[a ,b ],则T u 1(t )-T u 2(t )ɤ1M[l ʏb a Iα,ψa+f (τ,u 1(τ))-f (τ,u 2(τ))d A (τ)+ðmi =1μi I α+βi ,ψa+f (ηi ,u 1(ηi ))-f (ηi ,u 2(ηi ))+I α-n +1,ψa+f (b ,u 1(b ))-f (b ,u 2(b ))]ˑ(ψ(t )-ψ(a ))n -1+I α,ψa+f (t ,u 1(t ))-f (t ,u 2(t ))ɤ1M[ξl ʏb aI α,ψa+u 1(τ)-u 2(τ)d A (τ)+ξðmi =1μiI α+βi,ψa+u 1(ηi )-u 2(ηi )+ξI α-n +1,ψa+u 1(b )-u 2(b )]ˑ(ψ(t )-ψ(a ))n -1+ξI α,ψa+u 1(t )-u 2(t )ɤξ(ψ(b )-ψ(a ))αl (A (b )-A (a ))M Γ(α+1)(ψ(b )-ψ(a ))n -éëêê1+1Mðmi =1μi (ψ(b )-ψ(a ))βi +n -1Γ(α+βi +1)+1M Γ(α-n +2)+1Γ(α+1ùûúú) u 1-u 2 ,于是T u 1-T u 2 ɤξΛ1(ψ(b )-ψ(a ))αu 1-u 2 .再由假设条件(H 2)知,T 为压缩映射.因此,根据B a n a c h 压缩映像原理可知,T 在B r 0中存在唯一的不动点,即边值问题(3)存在唯一解.证毕.下面给出边值问题(3)至少存在一个解的结果.引理5 T :C [a ,b ]ңC [a ,b]为全连续算子.证明:首先证明T 连续.任取{u n }⊂C [a ,b ]且u n ңu ɪC [a ,b ](n ң+ɕ),则∀t ɪ[a ,b ],有T u n (t )-T u (t )ɤ1M[l ʏb aI α,ψa+f (τ,u n (τ))-f (τ,u (τ))d A (τ)+374 第3期 王 宁,等:带有S t i e l t j e s 积分边界条件和ψ-C a pu t o 导数的分数阶边值问题解的存在性 Copyright ©博看网. All Rights Reserved.ðmi =1μi I α+βi,ψa+f (ηi ,u n (ηi ))-f (ηi ,u (ηi ))+I α-n +1,ψa+f (b ,u n (b ))-f (b ,u (b ))]ˑ(ψ(t )-ψ(a ))n -1+I α,ψa+f (t ,u n (t ))-f (t ,u (t ))ɤ (ψ(b )-ψ(a ))αl (A (b )-A (a ))M Γ(α+1)(ψ(b )-ψ(a ))n -1+1Mðmi =1μi (ψ(b )-ψ(a ))βi +n -1Γ(α+βi +1éëêê)+ 1M Γ(α-n +2)+1Γ(α+1ùûúú) f (㊃,u n (㊃))-f (㊃,u (㊃)) ,从而T u n -T u ɤΛ1(ψ(b )-ψ(a ))αf (㊃,u n (㊃))-f (㊃,u (㊃)) .由f 的连续性可知 T u n (t )-T u (t ) ң0(n ң+ɕ),进而T 连续.其次证明T 为紧映射.∀r >0,有界集B r ʒ={u ɪG : u ɤr },令K =s u pa ɤt ɤb ,-r ɤu ɤrf (t ,u ).则对∀u ɪB r ,t ɪ[a ,b ],有(T u )(t )ɤðn -2k =0u k k!(ψ(t )-ψ(a ))k+1M [P +l ʏb aIα,ψa+f (τ,u (τ))d A (τ)+ðmi =1μi I α+βi ,ψa+f (ηi ,u (ηi ))+ðn -2k =0λk ㊃u k +I α-n +1,ψa +f (b ,u (b ))]ˑ(ψ(t )-ψ(a ))n -1+I α,ψa+f (t ,u (t ))ɤðn -2k =0u k k !(ψ(b )-ψ(a ))k+1MP +ðn -2k =0λk ㊃u ()kˑ(ψ(b )-ψ(a ))n -1+K Λ1(ψ(b )-ψ(a ))α.从而T (B r )有界,即T (B r )中的函数一致有界.下证T (B r )中的函数等度连续.定义Q (α)=1,αȡ1,-1,0<α<1{.∀t 1,t 2ɪ[a ,b ],不妨设t 1<t 2,则有(T u )(t 2)-(T u )(t 1)ɤðn -2k =0u k k ![(ψ(t 2)-ψ(a ))k -(ψ(t 1)-ψ(a ))k]+1M[P +l ʏb aI α,ψa+f (τ,u (τ))d A (τ)+ðmi =1μi I α+βi ,ψa+f (ηi ,u (ηi ))+ ðn -2k =0λk ㊃u k +I α-n +1,ψa+f (b ,u (b ))]ˑ[(ψ(t 2)-ψ(a ))n -1-(ψ(t 1)-ψ(a ))n -1]+ I α,ψa +f (t 2,u (t 2))-I α,ψa +f (t1,u (t 1))ɤ ðn -2k =0u k k ![(ψ(t 2)-ψ(a ))k -(ψ(t 1)-ψ(a ))k]+1M P +l K (A (b )-A (a ))Γ(α+1)(ψ(b )-ψ(a ))éëêêα+ K ðmi =1μi Γ(α+βi +1)(ψ(b )-ψ(a ))α+βi + ðn -2k =0λk ㊃u k +K Γ(α-n +2)(ψ(b )-ψ(a ))α-n +ùûú1ˑ [(ψ(t 2)-ψ(a ))n -1-(ψ(t 1)-ψ(a ))n -1]+ 1Γ(α)ʏt 2a ψᶄ(s )(ψ(t 2)-ψ(s ))α-1f (s ,u (s ))d s -ʏt 1aψᶄ(s )(ψ(t 2)-ψ(s ))α-1f (s ,u (s ))d s ɤ474吉林大学学报(理学版)第61卷Copyright ©博看网. All Rights Reserved.ðn -2k =0u k k![(ψ(t 2)-ψ(a ))k -(ψ(t 1)-ψ(a ))k]+1MP +l K (A (b )-A (a ))Γ(α+1)(ψ(b )-ψ(a ))éëêα+ K ðmi =1μi Γ(α+βi +1)(ψ(b )-ψ(a ))α+βi +ðn -2k =0λk ㊃u k +K Γ(α-n +2)(ψ(b )-ψ(a ))α-n +ùûúú1ˑ [(ψ(t 2)-ψ(a ))n -1-(ψ(t 1)-ψ(a ))n -1]+ K Γ(α)ʏt 1a Q (α)ψᶄ(s )[(ψ(t 2)-ψ(s ))α-1-(ψ(t 1)-ψ(s ))α-1]d s +K Γ(α)ʏt 2t 1ψᶄ(s )(ψ(t 2)-ψ(t 1))α-1d s ɤ ðn -2k =0u k k ![(ψ(t 2)-ψ(a ))k -(ψ(t 1)-ψ(a ))k]+ 1M P +l K (A (b )-A (a ))Γ(α+1)(ψ(b )-ψ(a ))éëêα+K ðmi =1μi Γ(α+βi +1)(ψ(b )-ψ(a ))α+βi + ðn -2k =0λk ㊃u k +K Γ(α-n +2)(ψ(b )-ψ(a ))α-n +ùûúú1[(ψ(t 2)-ψ(a ))n -1-(ψ(t 1)-ψ(a ))n -1]+ K Γ(α+1)Q (α)[(ψ(t 2)-ψ(a ))α-(ψ(t 2)-ψ(t 1))α-(ψ(t 1)-ψ(a ))α]+K Γ(α+1)(ψ(t 2)-ψ(t 1))α,当t 2ңt 1时,(T u )(t 2)-(T u )(t 1)ң0,因此T (B r )等度连续.由A r z e l a -A s c o l i 定理可知T (B r )列紧,故T 为紧算子,所以T :C [a ,b ]ңC [a ,b ]为全连续算子.证毕.假设条件:(H 3)存在常数r 1>0及连续单调递增函数ϕ:[0,+ɕ)ң[0,+ɕ),使得f (t ,v )ɤϕ(v )+r 1, ∀t ɪ[a ,b ], ∀v ɪℝ; (H 4)存在常数r 2>0,使得当r >r 2时,r >Ω1+Ω2(ϕ(r )+r 1),其中Ω1=ðn -2k =0u k k !(ψ(b )-ψ(a ))k +1MP +ðn -2k =0λk ㊃u ()k(ψ(b )-ψ(a ))n -1,Ω2=Λ1(ψ(b )-ψ(a ))α. 定理2 假设条件(H 3),(H 4)成立,则边值问题(3)在[a ,b ]上至少有一个解.证明:由引理5知,T :C [a ,b ]ңC [a ,b ]为全连续算子.令Δ={u ɪC [a ,b ]:u =λT u ,λɪ(0,1)},下证Δ有界.由Δ的定义知,∀u ɪΔ,∃λɪ(0,1),使得u =λT u ,故∀t ɪ[a ,b ],u (t )=λT u (t ),有 u ɤ T u .由引理5的证明知, T u ɤΩ1+Ω2 f (㊃,u (㊃)) .利用假设条件(H 3)可得, u ɤΩ1+Ω2(ϕ(u )+r 1),由假设条件(H 4)知, u ɤr 2.所以Δ有界.由S c h a e f e r 不动点定理可知T 在C [a ,b ]中至少存在一个不动点,即边值问题(3)在[a ,b ]上至少有一个解.证毕.3 应用实例考虑下列边值问题:c D α,ψa +u (t )=f (t ,u (t )), t ɪ[0,1],u [k ]ψ(0)=0, k =0,1,u [n -1]ψ(1)=34ʏ10u (τ)d τ+17I 1/2,ψ0+u æèçöø÷15+37I 1/3,ψ0+u æèçöø÷45ìîíïïïïï,(8)其中α=52,n =3,m =2,f (t ,u )=t 2000s i n u 1+s i n 2æèçöø÷u +t 2,ψ(t )=2t +2.∀u ,v ɪℝ,t ɪ[0,1],有574 第3期 王 宁,等:带有S t i e l t j e s 积分边界条件和ψ-C a pu t o 导数的分数阶边值问题解的存在性 Copyright ©博看网. All Rights Reserved.f (t ,u )-f (t ,v )=t 2000s i n u 1+s i n 2æèçöø÷u -t 2000s i n v 1+si n 2æèçöø÷v =t 2000(s i n u -s i n v )(1-s i n u s i n v )(1+s i n 2u )(1+s i n 2v )ɤ11000s i n u -s i n v ɤ11000u -v .因此,可取ξ=11000.计算可知,ξ(ψ(b )-ψ(a ))αˑl (A (b )-A (a ))M Γ(α+1)(ψ(b )-ψ(a ))n -1+1Mðmi =1μi (ψ(b )-ψ(a ))βi +n -1Γ(α+βi +1)éëêê+1M Γ(α-n +2)+1Γ(α+1ùûúú)ʈ0.2202<1.由定理1可知边值问题(8)在[0,1]上存在唯一解.参考文献[1] Z H O UP ,M AJ ,T A N G J .C l a r i f y t h eP h y s i c a lP r o c e s s f o rF r a c t i o n a lD y n a m i c a l S y s t e m s [J ].N o n l i n e a rD y n a m i c s ,2020,100:2353-2364.[2] B A L E A N U D ,V A C A R U SI .F r a c t i o n a l A n a l o g o u s M o d e l si n M e c h a n i c sa n d G r a v i t y T h e o r i e s [C ]//F r a c t i o n a l D y n a m i c s a n dC o n t r o l .N e wY o r k :S p r i n ge r ,2012:199-207.[3] S A B A T I E RJ .B e y o n d t h eP a r t i c u l a rC a s e of C i r c u i t sw i t hG e o m e t r i c a l l y D i s t r i b u t e dC o m p o n e n t s f o rA p pr o x i m a t i o n o fF r a c t i o n a lO r d e rM o d e l s :A p p l i c a t i o n t oaN e w C l a s so fM o d e l f o rP o w e rL a w T y p eL o n g M e m o r y Be h a v i o u r M o d e l l i n g [J ].J o u r n a l o fA d v a n c e dR e s e a r c h ,2020,25:243-255.[4] WA N G Y Q ,L I U L S .P o s i t i v eS o l u t i o n sf o raC l a s so fF r a c t i o n a l i nF i n i t e -P o i n tB o u n d a r y V a l u eP r o b l e m s [J /O L ].B o u n d a r y V a l u eP r o b l e m s ,(2018-07-21)[2021-09-30].h t t p s ://d o i .o r g/10.1186/s 13661-018-1035-6.[5] S I N G H B K ,A G R AWA L S .S t u d y o f T i m e F r a c t i o n a lP r o p o r t i o n a lD e l a y e d M u l t i -p a n t o g r a p h S y s t e m a n d I n t e g r o -D i f f e r e n t i a l E q u a t i o n s [J ].M a t h e m a t i c a lM e t h o d s i n t h eA p p l i e dS c i e n c e s ,2022,45(13):8305-8328.[6] S AMA D IA ,N T O U Y A SSK ,T A R I B O O NJ .N o n l o c a l C o u p l e dS y s t e mf o r (k ,ϕ)-H i l f e rF r a c t i o n a lD i f f e r e n t i a l E qu a t i o n s [J ].F r a c t a l a n dF r a c t i o n a l ,2022,6(5):234-1-234-17.[7] J I A N GJQ ,WA N G H C .E x i s t e n c e a n dU n i q u e n e s s o f S o l u t i o n s f o r aF r a c t i o n a l D i e r e n t i a l E q u a t i o nw i t h M u l t i -p o i n tB o u n d a r y V a l u eP r o b l e m s [J ].J o u r n a l o fA p p l i e dA n a l y s i s&C o m p u t a t i o n ,2019,9(6):2156-2168.[8] H R I S T O V A S ,T E R S I A N S ,T E R Z I E V A R.L i p s c h i t z S t a b i l i t y i n T i m ef o r R i e m a n n -L i o u v i l l e F r a c t i o n a l D i f f e r e n t i a l E qu a t i o n s [J ].F r a c t a l a n dF r a c t i o n a l ,2021,5(2):37-1-37-13.[9] S U T A RST ,K U C C H E K D.O n N o n l i n e a r H y b r i dF r a c t i o n a lD i f f e r e n t i a lE q u a t i o n s w i t h A t a n g a n a -B a l e a n u -C a p u t oD e r i v a t i v e [J ].C h a o s ,S o l i t o n sF r a c t a l s ,2021,143:110557-1-110557-11.[10] A L M E I D A R.A C a p u t o F r a c t i o n a l D e r i v a t i v e o f a F u n c t i o n w i t h R e s p e c t t o A n o t h e r F u n c t i o n [J ].C o mm u n i c a t i o n s i nN o n l i n e a r S c i e n c e a n dN u m e r i c a l S i m u l a t i o n ,2017,44:460-481.[11] A B D O M S ,P A N C HA L S K ,S A E E D A M.F r a c t i o n a lB o u n d a r y V a l u eP r o b l e m w i t h ψ-C a p u t o F r a c t i o n a l D e r i v a t i v e [J ].P r o c e e d i n g s o f t h e I n d i a nA c a d e m y o f S c i e n c e s :M a t h e m a t i c a l S c i e n c e s ,2019,129(5):1-14.[12] S HAMMA K H W ,A L Z UM IHZ ,A l Q A H T A N I BA.A N e wC l a s s o f ψ-C a p u t oF r a c t i o n a l D i f f e r e n t i a l E q u a t i o n s a n dI n c l u s i o n [J /O L ].J o u r n a lo f M a t h e m a t i c s ,(2021-01-18)[2021-09-30].h t t p s ://d o i .o r g /10.1155/2021/6677959.[13] A L M E I D A R ,MA L I N OW S K A A B ,MO N T E I R O M T T.F r a c t i o n a lD i f f e r e n t i a lE q u a t i o n s w i t ha C a pu t o D e r i v a t i v ew i t hR e s p e c t t o aK e r n e lF u n c t i o na n dT h e i rA p p l i c a t i o n s [J ].M a t h e m a t i c a lM e t h o d s i nt h eA p pl i e d S c i e n c e ,2018,41(1):336-352.[14] S C HA E F E R H.Üb e r d i eM e t h o d e d e r aP r i o r i -S c h r a n k e n [J ].M a t h e m a t i s c h eA n n a l e n ,1955,129:415-416.(责任编辑:赵立芹)674 吉林大学学报(理学版) 第61卷Copyright ©博看网. 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Caputo-Hadamard型分数阶微分方程边值问题解的存在唯一性师琳斐;李成福【摘要】本文研究了一类Caputo-Hadamard型分数阶微分方程边值问题的解.利用Banach不动点定理和上下解方法,获得了解的存在性和唯一性,推广了常微分方程边值问题的一些结果.作为应用,给出了两个例子来说明我们的主要结果.%In this paper,we study a class of Caputo-Hadamard fractional differential equations with boundary value problems.By using Banach fixed point theorem and the method of upper and lower solutions method,the existence and uniqueness results of the solutions are obtained,which generalizes some results about ordinary differential equations with boundary value problems.As an application,two examples are given to illustrate our main results.【期刊名称】《数学杂志》【年(卷),期】2019(039)004【总页数】11页(P493-503)【关键词】分数阶微分方程;Caputo-Hadamard导数;Banach不动点定理;上下解【作者】师琳斐;李成福【作者单位】湘潭大学数学与计算科学学院,湖南湘潭411105;湘潭大学数学与计算科学学院,湖南湘潭411105【正文语种】中文【中图分类】O175.81 IntroductionOver the past few decades,the fractional calculus made great progress,and it was widely used in various fields of science and engineering. There were numbers applications in electromagnetics,control theory,viscoelasticity and so on.There was a high-speed development in fractional differential equations in recent years,and we referred the reader to the monographs Podlubny[1],Kilbas et al.[2]and Zhou[3].In the current theory of fractional differential equations,much of the work is based on Riemann-Liouville and Caputo fractional derivatives,but the research of Caputo-Hadamard fractional derivatives of differential equations is very few,which includes logarithmic function and arbitrary exponents.Motivated by this fact,we consider a class of Caputo-Hadamard fractional differential equations with boundary value problems(BVPs).Nowadays,some authors studied the existence and uniqueness of solutions for nonlinear fractional differential equation with boundary value problems.For the recent development of the topic,we referred the reader to a series papers by Ahmad et al.[4–6],Mahmudov et al.[7]and the references therein.Details and properties of the Hadamard fractional derivative and integral can be found in[8–12].Wafa Shammakh[13]studied the existence and uniqueness results for thefollowing three-point BVPswhere is the Caputo-Hadamard fractional derivative of order 1 ≤α ≤2,0≤γ<1,and f:[1,e]→[0,∞).Yacine Arioua and Nouredine Benhamidouche[14]studied the existence of solutions for the following BVPs of nonlinear fractional differential equationswhere is the Caputo-Hadamard fractional derivative of order α,andf:[1,e]×R →R is a given continuous function.Yunru Bai and Hua Kong[15]used the method of upper and lower solutions,proved the existence of solutions to nonlinear Caputo-Hadamard fractional differential equationswhere and stand for the Caputo-Hadamard fractional derivative and Hadamard integral operators,f:[a,b]×R×R →R and 1<a<b<∞.The purpose of this paper is to discuss the existence and uniqueness of solutions for nonlinear Caputo-Hadamard fractional differential equationswhere is the Caputo-Hadamard fractional derivative of order 2<α ≤3,and f is a continuous function.2 PreliminariesIn this section,we introduce some necessary definitions,lemmas and notations that will be used later.Definition 2.1[2]The Hadamard fractional integral of order α ∈R+for a continuous function g:[1,∞)→R is given bywhere Γ(·)stands for the Gamma function.Definition 2.2[2]The Hada mard fractional derivative of order α ∈R+for a continuous function g:[1,∞)→R is given bywhere n−1<α<n,n=[α]+1,δ=and[α]denotes the integer part of the real number α.Definition 2.3 [16,17]The Caputo-Hadamard fractional derivative of order α ∈R+for at leas t n-times differentiable function g:[1,∞)→R is defined asLemma 2.4 [16,17]Let ([1,e],R),thenhere ([1,e],R)={u:[1,e]→R:δn−1u ∈C([1,e],R)}.Lemma 2.5 Let h ∈C([1,e],R),u ∈([1,e],R). Then the unique solution of the linear Caputo-Hadamard fractional differential equationis equivalent to the following integral equationwhereProof In view of Lemma 2.4,applying to both sides of(2.2),where c0,c1,c2 ∈R.The boundary condition u(1)=u'(1)=0 implies that c0=c1=0.ThusIn view of the boundary conditionwe conclude thatSubstituting(2.5)in(2.4),we obtain(2.3).This completes the proof.Based on Lemma 2.5,the solution of problems(1.1)–(1.2)can be expressed as3 Main ResultsLet E:=C([1,e],R)be the Banach space of all continuous functionsfrom[1,e]to R with the norm Due to Lemma 2.5,we define an operator A:E →E asIt should be noticed that BVPs(1.1)has solutions if and only if the operator A has fixed points.First,we obtain the existence and uniqueness results via Banach fixed point theorem.Theorem 3.1 Assume th at f:[1,e]×R →R is a continuous function,and there exists a constant L>0 such that(H1)|f(t,u)−f(t,v)|≤L|u −v|,∀t ∈[1,e],u,v ∈R.Ifthen problem(1.1)has a unique solution on[1,e].Proof Denote we set Br:={u ∈C([1,e],R):u≤r}and choose where Obviously it is concluded thatNow,we show that For any u ∈Br,t ∈[1,e],we havewhich implies that ABr ⊆Br.Let u,v ∈Br,and for each t ∈[1,e],we haveTherefore,From assumption(3.2),it follows that A is a contraction mapping.Hence problem(1.1)has a unique solution by using Banach fixed point theorem.This completes the proof.Next,we will use the method of upper and lower solutions to obtain the existence result of BVPs(1.1).Definition 3.2 Functionsare called upper and lower solutions of fractional integral equation(2.6),respactively,if it satisfies for any t ∈[1,e],Define={u ∈C([1,e],R):(t)≤u(t)≤(t),t ∈[1,e],u is the solution of(2.6)}.Theorem 3.3 Let f ∈C([1,e]×R,R). Assume that∈C([1,e],R)are upper and lower solutions of fractional integral equation(2.6)with for t ∈[1,e]. If f is nondecreasing with respect to u that is f(t,u1)≤f(t,u2),u1 ≤u2,then there exist maximal and minimal solutions in moreover,for each ,one hasProof Constructing two sequences{pn},{qn}as followsThis proof divides into three steps.Step 1 Finding the monotonicity of the two sequences,that is,the sequences{pn},{qn}satisfy the following relationfor t ∈[1,e].First,we verify that the sequence{pn}is nondecreasing and satisfiesSinceare upper and lower solutions respectively,we knowthat(t)=p0(t)≤(t)=q0(t)for t ∈[1,e],Since f is nondecreasing respect to the second variable,this implies thatThis deducesTherefore,we assume inductivelyIn view of definition of{pn},{qn},we haveBy means of the monotonicity of f,it is obvious thatWe show thatFor n=0,it is obvious that for all t ∈[1,e]. Now,we also suppose inductivelyAnalogously,we easily conclude from the monotonicity of f with respect to the second variables thatIn a similar way,we know that the sequence{qn}is nonincreasing.Step 2 The sequences constructed by(3.3),(3.4)are both relatively compact in C([1,e],R).According to that f is continuous and∈C([1,e],R),from Step 1,wehave{pn}and{qn}also belong to C([1,e],R).Moreover,it followsfrom(3.5)that{pn}and{qn}are uniformly bounded. For any t1,t2∈[1,e],without loss of generality,let t1 ≤t2,we know thatapproaches zero as t2 −t1 →0,where W>0 is a constant independent of n,t1 and t2,|f(t,pn(t))|≤W. It implies that{pn}is equicontinuous in C([1,e],R). By Arzelà-Ascoli theorem,we imply that{pn}is relatively compact inC([1,e],R). In the same way,we conclude that{qn}is also relatively compact in C([1,e],R).Step 3 There exist maximal and minimal solutions inThe sequences{pn}and{qn}are both monotone and relatively compact in C([1,e],R)by Step 1 and Step 2. There exist continuous functions p and q such that pn(t)≤p(t)≤q(t)≤qn(t)for all t ∈[1,e]and n ∈N.{pn}and{qn}converge uniformly to p and q inC([1,e],R),severally.Therefore,p and q are two solutions of(2.6),i.e.,for t ∈[1,e].However,fact(3.5)determines thatFinally,we shall show that p and q are the minimal and maximal solutions in respectively.For any then we haveBecause f is nondecreasing with respect to the second parameter,we concludeTaking limits as n →∞into the above inequality,we havewhich means that uL=p and uM=q are the minimal and maximal solutions in This completes the proof.Theorem 3.4 Assume that assumptions of Theorem 3.3 are satisfied.Then fractional nonlinear differential equation(1.1)has at least one solution inC([1,e],R).Proof By the hypotheses and Theorem 3.3,we induct then the solution set of fractional integral equation(2.6)is nonempty in C([1,e],R). It follows from the solution set of(2.6)together with Lemma 2.5 that problem(1.1)has at least one solution in C([1,e],R).This completes the proof.4 ExamplesIn this section,we present two examples to explain our main results. Example 1 Consider the following nonlinear Caputo-Hadamard fractional differential equationhere α=,λ=1,f(t,u)= 1 ≤t ≤e.One can easily calcula te Q=Clearly f is acontinuous function and we haveTherefore LQ<1.Thus all conditions of Theorem 3.1 satisfy which implies the existence of uniqueness solution of the the boundary value problem(4.1).Example 2 Consider the problemProof Where t ∈[1,e],f is continuous and nondecreasing with respect to u.ThusIt is easy to check that=(0,(ln t)3)is a pair of upper and lower solutions of(4.3)and that all assumptions of Theorem 3.2 are satisfied.So uL=p and uM=q are the minimal and maximal solutions of the boundary problem(4.3),and the iteration sequences is as followsand Applying Theorem 3.4,this boundary value problem(4.2)has at least one solution.References【相关文献】[1] Podlubny I.Fractional differential equations[M].San Diego:Academic Press,1999.[2] Kilbas A A,Srivastava H M,Trujillo J J.Theory and application of fractional differential equations[M].New York:North-Holland,2006.[3] Zhou Yong.Basic theory of fractional differential equations[M].Singapore:World Scientific,2014.[4] Ahmad B,Ntouyas S K.Boundary value problems of Hadamard-type fractional differential equations and inclusions with nonlocal conditions[J].VietnamJ.Math.,2017,45(3):409–423.[5] Ahmad B,Ntouyas S K.Existence and uniqueness of solutions for Caputo-Hadamard sequential fractional order neutral functional differentialequations[J].Electron.J.Diff.Equ.,2017,36:1–11.[6] Ahmad B,Ntouyas S K,Tariboon J.Existence results for mixed Hadamard and Riemann-Liouville fractional integro-differential equations[J].Adv.Diff.Equ.,2015,293:1–10.[7] Mahmudov N I,Awadalla M.Hadamard and Caputo-Hadamard FDE’s with three point integral boundary conditions[J].Nonl.Anal.Differ.Equ.,2017,5(6):271–282.[8] Butzer P L,Kilbas A A,Trujillo J positions of Hadamard-type fractional integration operators and the semigroup property[J].J.Math.Anal.Appl.,2002,269(2):387–400.[9] Butzer P L,Kilbas A A,Trujillo J J.Fractional calculus in the Mellin setting and Hadamard-type fractional integrals[J].J.Math.Anal.Appl.,2002,269(1):1–27.[10] Butzer P L,Kilbas A A,Trujillo J J.Mellin transform analysis and integration by parts for Hadamard-type fractional integrals[J].J.Math.Anal.Appl.,2002,270(1):1–15.[11] Anatoly A K.Hadamard-type fractional calculus[J].J.Korean Math.Soc.,2001,38(6):1191–1204.[12] Kilbas A,Trujillo J.Hadamard-type integrals as G-transforms[J].IntegralTrans.Spec.F.,2003,14(5):413–427.[13] Shammakh W.Existence and uniqueness results for three-point boundary value problems for Caputo-Hadamard-type fractional differentialequations[J].Nonl.Anal.Diff.Equ.,2016,4(5):207–217.[14] Arioua Y,Benhamidouche N.Boundry value problem for Caputo-Hadamard fractional differential equations[J].Surv.Math.Appl.,2017,12:103–115.[15] Bai Yunru,Kong Hua.Existence of solutions for nonlinear Caputo-Hadamard fractional differential equations via the method of upper and lowersolutions[J].J.Nonl.Sci.Appl.,2017,10:5744–5752.[16] Gambo Y Y,Jarad F,Baleanu D.On Caputo modification of the Hadamard fractional derivatives[J].Adv.DifferenceEqu.,2014,10(1):1–12.[17] Jarad F,Abdeljawad T,Baleanu D.Caputo-type modification of the Hadamard fractional derivatives[J].Adv.Diff.Equ.,2012,142:1–8.。
一类分数阶时滞微分系统的精确解及稳定性
邬忆萱;寇春海
【期刊名称】《东华大学学报(自然科学版)》
【年(卷),期】2024(50)1
【摘要】为拓展整数阶微分系统的已有结果,研究了一类具有多个时滞的Caputo 分数阶线性微分系统。
运用不可交换矩阵的多项式定理,在不要求系数矩阵可交换的前提下,得到了系统的精确解表示。
研究结果表明,该系统在有限时间内Hyers-Ulam稳定。
【总页数】11页(P152-162)
【作者】邬忆萱;寇春海
【作者单位】东华大学理学院
【正文语种】中文
【中图分类】O175.13
【相关文献】
1.一类分数阶时滞微分系统的两度量稳定性
2.一类分数阶时滞微分系统的解
3.一类非局部非自治分数阶时滞微分系统的稳定性
4.一类分数阶时滞微分方程解的存在性及稳定性
5.一类推广的非线性Hilfer分数阶时滞微分方程解的存在性
因版权原因，仅展示原文概要，查看原文内容请购买。
《分数阶微分方程边值问题解的存在性》篇一一、引言分数阶微分方程在许多领域中有着广泛的应用，包括物理、工程、经济和社会科学等。
这些方程能更好地描述具有记忆效应和历史依赖性的过程。
因此，分数阶微分方程边值问题的解的存在性成为了近年来研究的热点。
本文将针对一类特定的分数阶微分方程边值问题，探讨其解的存在性。
二、问题描述考虑如下形式的分数阶微分方程边值问题：Dαu(x) = f(x, u(x), Du(x), ..., Dn-1u(x)), 0 < x < 1, 其中Dα表示分数阶导数，f是已知的函数，u(x)是未知的函数。
在区间[0, 1]的端点处，给定边值条件u(0) = α, u(1) = β。
我们的目标是证明在满足一定条件下，该方程存在解。
三、解的存在性证明（一）定义与符号的介绍首先需要了解分数阶微分方程的基本概念和性质，如Caputo 导数、分数阶Sobolev空间等。
同时，需要引入一些重要的符号和定义，如Banach空间、压缩映射原理等。
（二）构造算子为了证明解的存在性，我们需要将原问题转化为一个算子方程。
我们定义一个算子L，使得L(u) = u - Kf(x, u, Du, ..., Dn-1u)，其中K是一个依赖于问题的常数。
这样，原问题就转化为寻找L 的不动点问题。
（三）不动点定理的应用我们可以使用Banach空间中的压缩映射原理或Schauder不动点定理来证明算子L在某个闭球上存在不动点。
首先需要证明L是一个压缩映射，然后根据不动点定理得出L存在不动点。
这等价于原问题存在解。
（四）证明解的唯一性除了证明解的存在性，我们还需要证明解的唯一性。
这通常需要利用更强的条件或额外的假设。
例如，我们可以假设f满足某种单调性或Lipschitz条件，从而保证解的唯一性。
四、结论通过上述证明过程，我们得出了该类分数阶微分方程边值问题解的存在性。
这为解决具有记忆效应和历史依赖性的实际问题提供了理论依据。
一类 Caputo 分数阶微分方程初值问题解的存在性杨帅;蔡宁宁【摘要】将一类Caputo分数阶微分方程初值问题转化为等价的Volterra积分方程，通过构造一个特殊的Banach空间，在此Banach空间上定义算子，将求解Volterra积分方程转化为求算子的不动点问题，应用Schauder 不动点定理证明了其解的存在性。
%The initial value problem of a class of Caputo fractional differential equations is trans‐formed into an equivalent Volterra integral equation .By defining a operator on a special Banach space ,the solvability of the Volterra integral equation is transformed to a fixed point problem . The existence of its solution is proved by employing S chauder′s fixed point theorem .【期刊名称】《山东理工大学学报（自然科学版）》【年(卷),期】2016(000)003【总页数】4页(P29-32)【关键词】Caputo分数阶微分方程;初值问题;Volterra积分方程【作者】杨帅;蔡宁宁【作者单位】中国矿业大学北京理学院，北京100083;中国矿业大学北京理学院，北京100083【正文语种】中文近年来，随着相关理论的不断拓展和完善，分数阶微分方程已广泛应用于分数物理学、粘弹性力学、自动控制、混沌与湍流、生物化学、非牛顿流体力学、随机过程等诸多科学领域[1]. 关于分数阶微分方程解的存在性及其求解也取得了丰硕的成果[2-5]. 分数阶微分方程初值问题是非线性微分方程的一个重要研究课题，许多学者都独立地探讨了各类分数阶微分方程初值问题[6-9].本文主要讨论如下一类Caputo分数阶微分方程边值问题式是Caputo分数阶导数，核心思想是将其转化为等价的Volterra积分方程，利用Schauder不动点定理证明其解的存在性.首先，介绍几个基本概念和一些Caputo分数阶导数的性质以及相关引理.定义1[1] 设是R中的有限区间，∀α∈R+，则连续函数f(x)的α阶Riemann-Liouville分数阶积分定义为式中0≤x≤h，右端项在上有定义. α=0时，令=I为恒等算子.定义2[2] 设是R中的有限区间，n-1<α≤n，n≥2，则连续函数f(x)的α阶Caputo分数阶导数定义为式中0≤x≤h，右端项在上有定义.性质1 常数的Caputo分数阶导数为0，即性质2 设f1,f2是两个连续函数，∀几乎处处存在. 设几乎处处存在，且引理1 设α>0，f∈C[0,h]，则有几乎处处成立[2].引理2 设f∈L1[0,h]，∀α≥0，则几乎处处存在，且在L1[0,h]中有界[1].定理1 设f(x,y(x))在[0,1]×R上连续，则Caputo分数阶微分方程边值问题(1)等价于下面的第二类非线性Volterra积分方程证明设y(x)是Caputo分数阶微分方程边值问题(1)的解，则由Caputo分数阶导数的定义有第2个等号两边作用1-α阶的Caputo分数阶导数，由引理1可得即等式两边积分可得y(x)=y(0)+最后一个等号用到了则y(x)是Volterra积分方程(2)的解.另一方面，设y(x)是Volterra积分方程(2)的解，即等号两边作用α(0<α<1)阶的Caputo分数阶导数，则由性质1、2以及引理1可得且在(2)中令x=0可得y(0)=y0，则y(x)是Caputo分数阶微分方程边值问题(1)的解.综上两个方面，得到(1)与(2)等价.定理2 设在[0,1]×R上连续，f(x,0)不恒为零，并且存在非负函数p(x)∈L1[0,1]及q(y)∈C(R)使得，另有(α+1)，则Caputo分数阶微分方程边值问题(1)在C[0,1]中至少有一个非零解.证明由定理1知道，Caputo分数阶微分方程边值问题(1)与Volterra积分方程(2)等价，定义算子A:Ay(x)=y0+则方程的解转化为算子A的不动点问题.由(α+1)知，存在L>0以及0<K<Γ(α+1)，使得当时，有.另外，由于p(x)∈L1[0,1]，则由引理2知，存在N>0，使得N. 则对于(x,y)∈[0,1]×R有定义,其中‖·‖C是Bannach空间C[0,1]上的最大值范数，即，且容易得到U是Bannach空间C[0,1]的有界完备闭凸子集.接下来，分以下几步来证明：第1步，任取y∈U，可以得到即Ay(x)∈U，于是算子A:U→U.第2步，讨论算子A的连续性.∀y1,y2∈U，x∈[0,1]，对于∀ε>0，由f在‖y‖上的一致连续性知，∃δ0>0使得当时，有,∀x∈[0,1]那么则A:U→U连续.第3步，，对于∀z∈A(U)，有即A(U)中诸函数一致有界.第4步，讨论A(U)中诸函数的等度连续性.由U是Bannach空间C[0,1]的有界完备闭凸子集，且在[0,1]×U上连续，则可令∀y∈U,∀x1,x2∈(0,1]，不妨设0<x1≤x2≤1. 对∀ε>0，取，则当时，有由于0<α<1，则dt=于是ε.可知A(U)中诸函数等度连续.由Ascoli-Arzela定理知A(U)是B相对紧集. 因此A:U→U全连续. 根据Schauder 不动点定理知A在U中必有不动点.综上，证明了Caputo分数阶微分初值问题(1)解的存在性，即(1)必有连续解y∈C[0,1].【相关文献】[1] Kilbas A A, Srivastava H M, Trujillo J J. Theory and applications of fractional differential equations [M]. Amsterdam: Elsevier, 2006.[2]Diethelm K. The analysis of fractional differential equations[M]. Heidelberg: Springer, 2010.[3]Sabatier J, Agrawal O P, Tenreiro Machado J A. Advances in fractionalcalculus[M].Nether-Lands: Springer, 2007.[4]Miller K S, Ross B. An introduction to the fractional calculus and fractional differential equations [M]. New York: Wiley, 1993.[5] Podlubny I. Fractional differential equations [M]. London: Academic Press,1999.[6] Zhang S Q. Positive solutions to singular boundary value problem for nonlinear fractional differential equation[J]. Computers and Mathematics with Applications,2009,59 (3):1 300-1 309.[7] Zhang S Q. Positive solution of singular boundary value problem for nonlinear fractional differential equation with nonlinearity that changes sign[J]. Positivity,2012,16(1): 177-193.[8] Su X W. Boundary value problem for a coupled system of nonlinear fractional differential equations.[J]. Appl. Math. Lett.,2009,22: 64-69.[9] Su X W . Positive solutions to singular boundary value problems for fractional functional differential equations with changing sign nonlinearity[J]. Computers and Mathematics with Applications,2012, 64 (10):3 425-3 435.。
一类Caputo分数阶差分方程边值问题解的存在性胡卫敏;苏有慧;贠永震【期刊名称】《兰州理工大学学报》【年(卷),期】2017(43)6【摘要】研究了一类Caputo分数阶差分方程边值问题解的存在性.利用Caputo 分数阶差分方程及边值条件的特性给出了它的Green's函数,借助于Banach压缩映像原理、Krasnosel'skiis不动点定理和Leray-Schauder非线性抉择定理得到边值问题解的存在性,作为应用,给出一个例子验证所得的主要结果.%The existence of solution to boundary value problem of a class of Caputo fractional-order difference equations is studied.The features of these equations and boundary conditions are used to give out their Green's function.The existence of solutions to boundary value problem is obtained by means of Banach's contraction mapping principle,Krasnosel'skiis fixed point theorem,and Leray-Schauder nonlinear alternative theorem,and as an application,an example is given to verify the main result obtained.【总页数】5页(P161-165)【作者】胡卫敏;苏有慧;贠永震【作者单位】伊犁师范学院,数学与统计学院,新疆伊宁835000;徐州工程学院,数学与物理科学学院,江苏徐州221111;伊犁师范学院,数学与统计学院,新疆伊宁835000;徐州工程学院,数学与物理科学学院,江苏徐州221111【正文语种】中文【中图分类】O175.7【相关文献】1.一类带有边值条件的Caputo分数阶差分方程解的存在性 [J], 贠永震;胡卫敏2.一类Caputo分数阶微分方程边值问题解的存在性 [J], 巴哈尔古力;刘洋3.一类Caputo分数阶微分方程边值问题解的存在性 [J], 巴哈尔古力;刘洋4.一类具ψ-Caputo导数的分数阶微分方程边值问题解的存在性 [J], 董伟萍;周宗福5.一类带有p-Laplacian算子与积分边值条件的Caputo分数阶q-差分方程解的存在性 [J], 姜聪颖;候成敏因版权原因，仅展示原文概要，查看原文内容请购买。
分数阶微分方程边值问题解的存在性
高阶微分方程边值问题是指一类用于研究微分方程的解的边值问题，将这类问题的解的存在性问题称为高阶边值问题。
从研究的角度看，边值问题的求解方法大多都依赖于高阶边值问题的解的存在性。
首先要阐明的是，在高阶边值问题中，微分方程的参数多采用多项式，以确保函数的变换性。
其次，高阶边值问题中还要考虑与形式和变换有关的基本假设，因为这些变换会对解的存在性产生影响。
再次，我们还要考虑条件，也就是解存在性和有界性，以确定解的唯一性。
接下来，要进行综合分析，进一步研究高阶边值问题的解的存在性。
要做到这一点，首先要理解高阶微分方程参数的变换特性，以便把握其解的存在性并据此对解以及它们的性质进行研究。
此外，还要引入条件和假设，以检验通过这些变换会出现的一系列后果，并借此确定其解的大致特性和解结构，最终确定其存在性。
最后，要强调的是，高阶边值问题的解的存在性是研究微分方程的方法的基础。
因此，对于任意的微分方程，研究它们的解的存在性，以及若存在的话，怎样求得这些解的存在性，是非常重要的一个问题。
从另一个角度来说，只有了解和深入理解这些解的存在性，才能够在更高的层次上正确解决这些高阶边值问题。
