郭彩霞,任玉崗,郭建敏
(山西大同大學(xué)數(shù)學(xué)與計(jì)算機(jī)科學(xué)學(xué)院,山西大同 037009)
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一類Caputo分?jǐn)?shù)階微分方程邊值問(wèn)題多解的存在性*
郭彩霞,任玉崗,郭建敏
(山西大同大學(xué)數(shù)學(xué)與計(jì)算機(jī)科學(xué)學(xué)院,山西大同 037009)
研究一類Caputo分?jǐn)?shù)階微分方程邊值問(wèn)題:
分?jǐn)?shù)階微分方程 邊值問(wèn)題 Leggett-Williams不動(dòng)點(diǎn)定理
分?jǐn)?shù)階微分方程在工程、化學(xué)、物理、生物等領(lǐng)域有著廣泛應(yīng)用,例如熱傳導(dǎo)領(lǐng)域和流體學(xué)領(lǐng)域[1-3],而且分?jǐn)?shù)階導(dǎo)數(shù)模型克服了經(jīng)典整數(shù)階微分模型理論與實(shí)驗(yàn)結(jié)果不吻合的缺點(diǎn)[4],因此研究分?jǐn)?shù)階微分方程邊值問(wèn)題有著重要的意義.近年來(lái),大量文獻(xiàn)報(bào)道微分方程[5-6]和分?jǐn)?shù)階微分方程[4,7-10]邊值問(wèn)題解的存在性.2005年,當(dāng)1<α≤2時(shí),Bai等[7]推導(dǎo)了分?jǐn)?shù)階微分方程邊值問(wèn)題
目前研究分?jǐn)?shù)階微分方程邊值問(wèn)題的主要工具有錐拉伸與錐壓縮不動(dòng)點(diǎn)原理、Krasnoselskii不動(dòng)點(diǎn)原理、Schauder不動(dòng)點(diǎn)原理上下解等.本文利用Leggett-Williams不動(dòng)點(diǎn)定理,參照文獻(xiàn)[9]中的方法研究Caputo分?jǐn)?shù)階微分方程邊值問(wèn)題
(0.1)
正解的存在性,其中1<α≤2,f:[0,+)×→[0,+)是連續(xù)的,是標(biāo)準(zhǔn)的Caputo微分.一方面,邊值問(wèn)題(1)包含了文獻(xiàn)[9]的整數(shù)階微分方程邊值問(wèn)題,推廣了文獻(xiàn)[9]的結(jié)果;另一方面,非線性項(xiàng)f范圍有所擴(kuò)大.
定義1.1[11]一個(gè)連續(xù)函數(shù)u:(0,+)→的α階Caputo導(dǎo)數(shù)定義為
其中α>0,n=[α]+1,[α]代表實(shí)數(shù)α的整數(shù)部分。上式右邊在(0,+)內(nèi)逐點(diǎn)有定義.
引理1.1[11]令α>0,若u∈ACn[0,1]或u∈Cn[0,1],則
引理1.2 令α∈(1,2],給定h∈C[0,1],則
(1.1)
u′(0)=u(1)=0,
(1.2)
因此,(1.1)~(1.2)式的唯一解是
引理1.3 引理1.2中的G(t,s)有下列性質(zhì):
(i)G(t,s)∈C([0,1]×[0,1],)且G(t,s)>0,t,s∈(0,1);
0≤t≤1,
證明 (i)~(iii)顯然可得,只需證明(iv).
又
令γ,β,θ是錐P上的非負(fù)連續(xù)凸函數(shù),α,ψ是錐P上的非負(fù)連續(xù)凹函數(shù),那么對(duì)非負(fù)實(shí)數(shù)h,a,b,d和c,定義下列凸集:
P(γ,c)={u∈P:γ(u) Q(γ,β,d,c)={u∈P:β(u)≤d,γ(u)≤c}, P(γ,θ,α,a,b,c)={u∈P:a≤α(u),θ(u)≤b,γ(u)≤c}, Q(γ,β,ψ,h,d,c)={u∈P:h≤ψ(u),β(u)≤d,γ(u)≤c}. 定理1.1[12]令E是一個(gè)實(shí)Banach空間,且P?E是一個(gè)錐.假設(shè)存在正數(shù)c和M,使錐P上的非負(fù)連續(xù)凹函數(shù)α,ψ及非負(fù)連續(xù)凸函數(shù)γ,β,θ滿足 (B1){u∈P(γ,θ,α,a,b,c):α(u)>a}≠?且α(F(u))>a,u∈P(γ,θ,α,a,b,c); (B2){u∈Q(γ,β,ψ,h,d,c):β(u) (B3)若u∈P(γ,α,a,c)且θ(F(u))>b,則α(F(u))>a; (B4)若u∈Q(γ,β,d,c)且ψ(F(u)) β(u1) α(u)≤β(u), (2.1) (2.2) (H3)f(t,u(t))≤c αΓ(α),t∈[0,t3]∪[1-t3,1],u(t)∈[0,c]. 那么,邊值問(wèn)題(1)至少有3個(gè)正解u1,u2和u3,滿足 證明 在錐P上定義算子A為 因?yàn)?/p> 所以A:P→P連續(xù).由Arzela-Ascoli定理易證A:P→P是全連續(xù)的. ds=b; α(u1)>b,β(u2)a. 本文研究了一類Caputo分?jǐn)?shù)階微分方程邊值問(wèn)題多解的存在性.證明時(shí),將微分方程邊值問(wèn)題轉(zhuǎn)化為積分方程,進(jìn)一步轉(zhuǎn)化為討論積分算子不動(dòng)點(diǎn)的問(wèn)題,然后通過(guò)運(yùn)用Leggett-Williams不動(dòng)點(diǎn)定理該分?jǐn)?shù)階微分方程邊值問(wèn)題至少有3個(gè)正解存在的結(jié)果,其中格林函數(shù)的性質(zhì)和非線性項(xiàng)的條件至關(guān)重要. [1] PODLUBNY I.Fractional Differential Equations,Mathematics in Science and Engineering[M].New York:Academic Press,1999. [2] ADOMIAN G, ELROD M,RACH R.A new approach to boundary value equations and application to a generalization of Airy’s equation[J].J Math Anal Appl,1989,140(2):554-568. [3] AGARWAL R P,MEEHAN M,O’REGAN D.Fixed Point Theory and Applications[M].Cambridge:Cambridge University Press,2001. [4] ABDELJAWAD T, BALEANU D.Fractional differen- ces and integration by parts[J].Journal of Computational Analysis and Applications,2011,13(3): 574-582. [5] 王勇,韋煜明.二階非線性時(shí)滯微分方程邊值問(wèn)題正解的存在性[J].廣西科學(xué),2012,19(1):40-43. WANG Y,WEI Y M.Existence of positive solutions for boundary value problems of nonlinear second-order delay differential equations[J].Guangxi Sciences,2012,19(1):40-43. [6] 嚴(yán)建明.中立型微分方程的正解存在性及非振動(dòng)解的漸近性[J].廣西科學(xué),2008,15(1):7-9. YAN J M.Existence of asymptotic behavious of positive solution of neutral differential equation[J].Guangxi Sciences,2008,15(1):7-9. [7] BAI Z B,LU H S.Positive solutions for boundary value problem of nonlinear fractional differential equation[J].Journal of Mathematical Analysis and Applications, 2005,311(2): 495-505. [8] BAI Z B,QIU T T.Existence of positive solution for singular fractional differential equation[J].Applied Mathematics and Computation,2009,215(7):2761-2767. [9] DOGAN A.On the existence of positive solutions for the second-order boundary value problem[J].Applied Mathematics Letters,2015,49:107-112. [10] XIE W Z,XIAO J,LUO Z G.Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions[J].Applied Mathematics Letters,2015,41:46-51. [11] OLDHAM K B,SPANIER J.The Fractional Calculus [M].New York:Academic Press,1974. [12] AVERY R I.A generalization of the Leggett-Williams fixed point theorem[J].Math Sci Res Hot-Line, 1999,3(7):9-14. (責(zé)任編輯:尹 闖) Existence of Multiple Solutions for a Caputo Fractional Difference Equation Boundary Value Problem GUO Caixia,REN Yugang,GUO Jianmin (School of Mathematics and Computer Science,Datong University,Datong,Shanxi,037009,China) We investigate the existence and multiplicity of positive solutions for nonlinear Caputo fractional differential equation boundary value problem fractional difference equation,boundary value problem,Leggett-Williams fixed point theorems 2016-05-15 郭彩霞(1980-),女,講師,主要從事基礎(chǔ)數(shù)學(xué)方面的研究,E-mail:iris-gcx@163.com(C.Guo)。 *國(guó)家自然科學(xué)基金項(xiàng)目(No.11271235),大同大學(xué)青年科研基金項(xiàng)目(2014Q10)和河南省高等學(xué)校重點(diǎn)科研計(jì)劃項(xiàng)目(15A110047)資助。 網(wǎng)絡(luò)優(yōu)先數(shù)字出版時(shí)間:2016-09-13 【DOI】10.13656/j.cnki.gxkx.20160913.002 http://www.cnki.net/kcms/detail/45.1206.G3.20160913.0948.004.html 多解的存在性,其中1<α≤2,f:[0,+∞)×→[0,+∞)是連續(xù)的,是標(biāo)準(zhǔn)的Caputo微分.先將微分方程邊值問(wèn)題轉(zhuǎn)化為積分方程,再轉(zhuǎn)化為積分算子不動(dòng)點(diǎn)問(wèn)題,最后利用Leggett-Williams不動(dòng)點(diǎn)定理得出Caputo分?jǐn)?shù)階微分方程邊值問(wèn)題至少有3個(gè)正解存在,其中格林函數(shù)的性質(zhì)和非線性項(xiàng)的條件至關(guān)重要. O175.8 A 1005-9164(2016)04-0374-04 Where 1<α≤2,f:[0,+∞)×→[0,+∞) is continuous,andis the standard Caputo differentiation.In the process of proof,we first transform it into integral equation,then differential equation boundary value problem is further converted to discuss the problem of integral operator fixed point. Finally,by means of Leggett-Williams fixed point theorems on cone,existence results of at least three positive solutions are obtained.The properties of the Green function and the conditions of the nonlinear term is very important. 廣西科學(xué)Guangxi Sciences 2016,23(4):374~3772 主要結(jié)果
3 結(jié)論