鄭春華,寧艷艷
(陜西工業(yè)職業(yè)技術學院基礎部,陜西咸陽712000)
一類分數(shù)階Laplacian方程邊值問題解的存在性與唯一性
鄭春華,寧艷艷
(陜西工業(yè)職業(yè)技術學院基礎部,陜西咸陽712000)
研究了一類分數(shù)階p-Laplacian方程2點邊值問題解的存在性,利用Leray-Schauder非線性抉擇和Banach壓縮映射原理獲得了該邊值問題解存在性和唯一性的充分條件,得到了一些新的結果.
分數(shù)階微分方程;p-Laplace算子;兩點邊值問題;Leray-Schauder非線性抉擇;Banach壓縮映射原理
根據(jù)理論和應用中的不同需要,整數(shù)階微積分被人們推廣成多種類型的分數(shù)階微積分.由于分數(shù)階微分方程在流變學、材料和力學系統(tǒng)等應用領域的廣泛出現(xiàn),國內外學者開始重視分數(shù)階微分方程的研究[1-3],并在分數(shù)階微分方程的邊值問題方面取得了一些進展[4-9].
文獻[4]利用Krasnoselskii不動點定理和Leggett-Williams不動點定理研究了問題
正解的存在性.
文獻[6]在文獻[5]的基礎上利用Krasnoselskii不動點定理研究了含參數(shù)邊值問題
正解的存在性.
對于含有p-Laplace算子的分數(shù)階微分方程的研究結果還不是很多,在文獻[8]中,作者利用Mawhin連續(xù)性定理研究了含有p-Laplace算子的分數(shù)階微分方程2點共振邊值問題
解的存在性.
受以上研究工作的啟發(fā),本文利用Leray-Schauder非線性抉擇和Banach壓縮映射原理研究分數(shù)階微分方程的2點邊值問題
解的存在性,其中f∈C([0,1]×R,R),0<α,β≤1且1<α+β≤2,ri∈R且ri≠-1(i=1,2).
引理3[10]設X為Banach空間,U是X的有界開子集且0∈U,T: U→X為全連續(xù)算子,則下列結論二擇一.
1)T在U中有不動點;2)存在u∈?U和λ∈(1,+∞)使得λu=Tu.
為了方便討論BVP(1)解的存在性,將它轉化為下面的方程組形式
其中q滿足1/p+1/q=1.顯然,證明BVP(1)有解只需證明邊值問題(2)有解即可.
引理4 設0<α,β≤1且1<α+β≤2,h1,h2∈C[0,1],則BVP
從而可知引理4中G1(t,s)的表達式正確,類似可證G2(t,s)的正確性.
定義算子T:X→X,
引理5 T是X→X的全連續(xù)算子.
定理1 若下列條件成立,
1)存在連續(xù)函數(shù)a(t)和b(t)滿足
則BVP(1)至少存在一個解x(t).
由1<p≤2知2≤q.如果存在x∈?U和λ∈(1,+∞)使得λx=Tx,則利用引理4和條件1)~3)可得
當2<p時,也可以得到同樣的結論.進而可知
定理2 若1<p≤2且下列條件成立,
1)存在連續(xù)函數(shù)c(t)滿足
則BVP(1)存在唯一解x(t).
對?x1=(x11(t),x12(t))T,x2=(x21(t),x22(t))T∈,利用條件1),2)有
[1]MILLER K S,ROSS B.An introduction to the fractional calculus and fractional differential equations[M].New York:Wiley,1993.
[2]PODLUBNY I.Fractional differential equation[M].San Diego:Academic Press,1999.
[3]KILBAS A A,SRIVASTAVA H M,TRUJILLO J J.Theory and applications of fractional differential equations[M].Netherlands:Elsevier,2006.
[4]BAIZhan-bing,LYU Hai-shen.Positive solutions of boundary value problems of nonlinear fractional differential equation[J].Journal of Mathematical Analysis and Applications,2005,311(2):495-505.
[5]ZHANG Shu-qin.Positive solutions for boundary value problems of nonlinear fractional differential equations[J].Electronic Journal of Differential Equations,2006(36):1-12.
[6]ZHAO Yi-ge,SUN Shu-rong,HAN Zhen-lai.Positive solutions for boundary value problems of nonlinear fractional differential equations[J].Applied Mathematics and Computation,2011,217(16):6950-6958.
[7]ZHANG Yin-ghan,BAIZhan-bing.Existence of solutions for nonlinear fractional three-point boundary value problems at resonance[J].Journal of Applied Mathematic and Computing,2011,36(1/2):417-440.
[8]CHEN Tai-yong,LIU Wen-bin,HU Zhi-gang.A boundary value problem for fractional differential equation with p-Laplacian operator at resonance[J].Nonlinear Anal,2012,75(6):3210-3217.
[9]JIANG Wei-hua.Eigenvalue interval for multi-point boundary value problems of fractional differential equations[J].Journal of Applied Mathematic and Computing,2013,219(9):4570-4575.
[10]GUO Da-jun,LAKSHMIKANTHAM V.Nonlinear Problems in Abstract Cones[M].Orlando:Academic Press,1988.
(責任編輯 梁志茂)
Existence and uniqueness of the solutions to a boundaryvalue problem for a class of fractional differentialequations with p-Laplace operator
ZHENG Chun-hua,NING Yan-yan
(Basic Courses Department,Shaanxi Polytechnic Institute,Xianyang 712000,China)
The class of fractional differential equation with a two-point boundary value problem and p-Laplace operator is studied.By using the Leray-Schauder nonlinear alternative theorem and Banach contraction mapping principle,some sufficient conditions concerning the existence and uniqueness of solutions for this boundary value problem are obtained.Some known results are extended and improved.
fractioal order differential equation;p-Laplace operator;two-point boundary value problem;Leray-Schauder nonlinear alternative theorem;Banach contraction mapping principle
O175.8
A
1672-8513(2014)06-0429-05
2014-04-11.
國家自然科學基金(11271364);陜西工業(yè)職業(yè)技術學院科研項目(ZK13-40)
鄭春華(1982-),男,碩士,講師.主要研究方向:微分方程邊值問題.