本文關(guān)鍵詞： 分?jǐn)?shù)階微分方程 Caputo分?jǐn)?shù)階導(dǎo)數(shù) 多點(diǎn)邊值問題 共振 重合度理論 Mawhin定理　出處：《華中科技大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：分?jǐn)?shù)階微分方程邊值問題的理論與應(yīng)用研究已受到了人們廣泛的關(guān)注,得到了長足的進(jìn)步與發(fā)展.分?jǐn)?shù)階微分方程作為一個(gè)有實(shí)用價(jià)值的的工具,廣泛應(yīng)用于許多科學(xué)領(lǐng)域.共振條件下的分?jǐn)?shù)階微分方程邊值問題引起了廣大學(xué)者的興趣,本文利用重合度理論的Mawhin定理,討論在共振條件下的分?jǐn)?shù)階微分方程邊值問題的解的存在性.首先,介紹分?jǐn)?shù)階微分方程的發(fā)展現(xiàn)狀和Caputo分?jǐn)?shù)階導(dǎo)數(shù)的理論和應(yīng)用.列出前人在分?jǐn)?shù)階微分方程上研究的成果,并說明本文要討論的分?jǐn)?shù)階微分方程邊值問題.其次,在參考文獻(xiàn)的基礎(chǔ)上,改變分?jǐn)?shù)階微分方程,使其更具有普遍性.給出關(guān)于分?jǐn)?shù)階微分方程邊值問題的基本假設(shè)和相關(guān)引理,在Caputo分?jǐn)?shù)階導(dǎo)數(shù)1α≤2,0β≤1的前提下,建立合適的算子,運(yùn)用重合度理論Mawhin定理,討論其共振條件下分?jǐn)?shù)階微分方程邊值問題在核維數(shù)是1的解存在性,得到了解的存在性條件.再次,在此前基礎(chǔ)上研究一類新的邊值條件,使其更具有研究意義.給出關(guān)于分?jǐn)?shù)階微分方程邊值問題的基本假設(shè)和相關(guān)引理,在Caputo分?jǐn)?shù)階導(dǎo)數(shù)1α≤2,0β≤1的前提下,改變邊值條件,建立合適的算子,運(yùn)用重合度理論Mawhin定理,考慮了共振條件下分?jǐn)?shù)階微分方程多點(diǎn)邊值問題在核維數(shù)是2的解存在性,建立了解的存在性條件.最后,總結(jié)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 偽 鈮，

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