共振条件下一类分数阶微分方程边值问题解的存在性

发布时间：2018-01-19 11:21

  本文关键词： 分数阶微分方程 Caputo分数阶导数 多点边值问题 共振 重合度理论 Mawhin定理　出处：《华中科技大学》2015年硕士论文　论文类型：学位论文

【摘要】：分数阶微分方程边值问题的理论与应用研究已受到了人们广泛的关注,得到了长足的进步与发展.分数阶微分方程作为一个有实用价值的的工具,广泛应用于许多科学领域.共振条件下的分数阶微分方程边值问题引起了广大学者的兴趣,本文利用重合度理论的Mawhin定理,讨论在共振条件下的分数阶微分方程边值问题的解的存在性.首先,介绍分数阶微分方程的发展现状和Caputo分数阶导数的理论和应用.列出前人在分数阶微分方程上研究的成果,并说明本文要讨论的分数阶微分方程边值问题.其次,在参考文献的基础上,改变分数阶微分方程,使其更具有普遍性.给出关于分数阶微分方程边值问题的基本假设和相关引理,在Caputo分数阶导数1α≤2,0β≤1的前提下,建立合适的算子,运用重合度理论Mawhin定理,讨论其共振条件下分数阶微分方程边值问题在核维数是1的解存在性,得到了解的存在性条件.再次,在此前基础上研究一类新的边值条件,使其更具有研究意义.给出关于分数阶微分方程边值问题的基本假设和相关引理,在Caputo分数阶导数1α≤2,0β≤1的前提下,改变边值条件,建立合适的算子,运用重合度理论Mawhin定理,考虑了共振条件下分数阶微分方程多点边值问题在核维数是2的解存在性,建立了解的存在性条件.最后,总结全文的主要成果,并提出下一步的研究方向.
[Abstract]:The theory and application of the boundary value problems of fractional differential equations have been paid more and more attention and have made great progress and development. Fractional differential equations as a practical tool. The boundary value problem of fractional differential equation under resonance condition has aroused the interest of many scholars. In this paper, the Mawhin theorem of coincidence degree theory is used. The existence of solutions for boundary value problems of fractional differential equations under resonance conditions is discussed. This paper introduces the development of fractional differential equations and the theory and application of Caputo fractional derivative, and lists the achievements of previous researches on fractional differential equations. The boundary value problem of fractional differential equation is discussed in this paper. Secondly, on the basis of reference, the fractional differential equation is changed. The basic assumptions and relevant Lemma for boundary value problems of fractional differential equations are given. On the premise of Caputo fractional derivative 1 伪 鈮，

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