非線性分數(shù)階脈沖微分方程邊值問題解的存在性

董　雪

(山東科技大學 數(shù)學與系統(tǒng)科學學院，山東 青島 266510)

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非線性分數(shù)階脈沖微分方程邊值問題解的存在性

董雪

(山東科技大學 數(shù)學與系統(tǒng)科學學院，山東 青島 266510)

摘要：關于脈沖的非線性分數(shù)階微分方程初邊值問題已有研究，但細節(jié)過程仍存在問題.證明了一類非線性分數(shù)階脈沖微分方程在混合邊界條件下解的存在性，結果的證明主要依據(jù)了schaefer不動點定理.最后舉例說明了所得結果的正確性.

關鍵詞：非線性分數(shù)階微分方程；脈沖；混和邊值問題；不動點定理

目前，非線性分數(shù)階微分方程邊值問題已經引起許多研究者的注意.分數(shù)階微分方程作為一種系統(tǒng)和流程的數(shù)學模型出現(xiàn)在許多工程和科學學科領域，如物理、化學、控制理論、生物學、經濟學、血液流動現(xiàn)象、信號和圖像處理、空氣動力學、實驗數(shù)據(jù)擬合等等.關于詳細資料可參考文獻[1-7].脈沖微分方程給出了觀察進化過程的一個自然的描述，作為一個重要的數(shù)學工具可更好地理解應用科學中許多現(xiàn)實問題.目前，整數(shù)階脈沖微分方程邊值問題已經被廣泛的研究，如文獻[8-11].但是，很少有文章考慮關于分數(shù)階的非線性脈沖微分方程邊值問題.

本文中我們研究下述非線性分數(shù)階脈沖微分方程混合邊值問題解的存在性.

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1　預備知識

為了證明主要結果，本章節(jié)介紹一些文中所要用到的預備知識.

定義1階為q的函數(shù)f:[0,]→R的分數(shù)階積分公式定義為

(2)

定義2分數(shù)階為q的函數(shù)f:[0,]→R的Caputo導數(shù)公式定義為

(3)

式中[q]記為實數(shù)q的整數(shù)部分.

引理2令q∈(1,2)且h:J→R是連續(xù)的.函數(shù)u給定如下:

(4)

為下面脈沖問題的唯一解.

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2　主要結果

本章節(jié)我們證明問題(1)解的存在性.

定理1假設

(A1) 函數(shù)f:J×R→R為連續(xù)函數(shù)，且存在常數(shù)N1>0使得

(A2) 函數(shù)Ik,Jk:R→R為連續(xù)函數(shù)，且存在常數(shù)N2,N3>0使得

則問題(1)至少有一個解.

證明第一步：定義算子T:PC(J,R)→PC(J,R)如下:

t∈(tk,tk+1],j=1,2,…,p,k=0,1,2,…,p-1.

本評分模型納入預后因素時未加入雙磷酸鹽的使用，原因是本研究中幾乎所有骨轉移患者皆使用過雙膦酸鹽，且雙膦酸鹽的使用對骨轉移患者預后的影響已基本明確，已成為常規(guī)治療方案[14]。本評分模型的缺點在于只基于一家醫(yī)療機構的數(shù)據(jù)，存在選擇偏倚，且納入因素有九個之多，準確性和實用性尚需更大樣本數(shù)據(jù)的檢驗，需要本機構和其他醫(yī)療機構在日后臨床工作中進行實踐，不斷改進，更加準確的指導臨床。

因為f,Ik,Jk為連續(xù)函數(shù)，所以當n→時有‖Tun-Tu‖→0.所以，T:PC(J,R)→PC(J,R)是連續(xù)的.

由(A1)和(A2)可得，對于任意的t∈J有

第三步：令Ωρ為如第二步中所定義的PC(J,R)上的有界集合，且令u∈Ωρ，則對于任意的t∈Jk,0≤k≤p有

因此，令t″,t′∈Jk,t′

所以，在Jk(k=0,1,2,…,p)上T(Ωρ)是等度連續(xù)的.因此，綜上可得T:PC(J,R)→PC(J,R)是全連續(xù)的.

根據(jù)條件(A1)和(A2)可得，對于任意的t∈J有

因此，對于任意的t∈J有

所以集合V是有界的.所以由引理1可得算子T有一個不動點即為問題(1)的一個解.

3　舉例

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參考文獻：

[1] Zhou W X, Chu Y D, Dumitru B. Uniqueness and existence of positive solutions for a multi-point boundary value problem of singular fractional differential equations[J]. Advances in Difference Equations, 2013(1): 114.

[2] Wang G T,Ravi P, Alberto C,etal. Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations[J]. Applied Mathematics Letters, 2012, 25(6): 1019-1024.

[3] Tadeusz J. Initial value problems for neutral fractional differential equations involving a Riemann-Liouville derivative[J]. Applied Mathematics and Computation, 2013, 219(14): 7772-7776.

[4] Wu J W, Zhang X G, Liu L S,etal. Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure[J]. Advances in Difference Equations, 2012(1):1-14.

[5] Seak W V. Positive solutions of singular fractional differential equations with integral boundary conditions[J].Mathematical and Computer Modelling, 2013, 57(5-6):1053-1059.

[6] Liu Z H, Sun J H, Ivan S. Monotone Iterative Technique for Riemann- Liouville fractional integro-differential equations with advanced arguments[J]. Results in Mathmatics, 2013, 63(3-4): 1277-1287.

[7] LiuR J, Kou C H, Jin R. Multiple positive solutions of boundary value problems for fractional order integro-differential equations in a Banach space[J]. Boundary Value Problems, 2013(1):1-18.

[8] Lee E K, Lee Y H. Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equation[J]. Applied Mathematics and Computation, 2004, 158(3):745-759.

[9] Lin X N, Jiang D Q. Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations[J]. Journal of Mathematical Analysis and Applications, 2006, 321(2): 501-514.

[10] Nieto J J, Lope R R. Boundary value problems for a class of impulsive functional equations[J]. Computers and Mathematics with Applications, 2008, 55(12): 2715-2731.

[11] Shen J H, Wang W B. Impulsive boundary value problems with nonlinear boundary conditions[J]. Nonlinear Analysis, 2008, 69(11):4055-4062.

[12] Sun J X. Nonlinear functional analysis and its application[M]. Beijing: Science Press, 2008.

(編輯：姚佳良)

Existenceofsolutionsforimpulsivenonlinear

fractionaldifferentialequationsboundaryvalueproblems

DONGXue

(CollegeofMathematicsandSystemScience,ShandongUniversityofScienceandTechnology,Qingdao266510,China)

Abstract:The initial value problems and boundary value problems of impulsive nonlinear fractional differential equations had been studied,but there were still some details which were not appropriate. We proved the existence of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order. Our result is based on schaefer′s fixed point theorem. Finally, an example was presented to illustrate the result.

Key words:nonlinear fractional differential equations; impulse; mixed boundary value problem; fixed-point theorem

中圖分類號：O175.8

文獻標志碼：A

文章編號：1672-6197(2015)04-0070-05

作者簡介：董雪，女，dx929wdl66@163.com

收稿日期：2014-11-14