【摘要】：近半個世紀以來由于分數(shù)階微積分理論的發(fā)展和廣泛應(yīng)用,分數(shù)階微分方程的理論研究得到迅猛發(fā)展,分數(shù)階微分方程的一些模型也被廣泛的應(yīng)用在具體的科學(xué)領(lǐng)域中,例如于經(jīng)濟、化學(xué)、生物、物理、醫(yī)學(xué)等學(xué)科.因此,在具體的學(xué)術(shù)研究中一個可解的分數(shù)階微分方程模型能在現(xiàn)代社會中產(chǎn)生巨大的影響.本文對三類非線性分數(shù)階微分方程邊值問題正解的存在性進行了研究.第一類研究的是分數(shù)階微分方程多點邊值問題迭代正解的存在性.本文運用迭代技巧和0u正算子研究了下列多點邊值問題正解的存在唯一性:得到的結(jié)論是如果函數(shù)f(t,u(t))滿足Lipschitz條件,并且在Lipschitz常數(shù)滿足一定條件下,就可以得到正解的存在性和唯一性.第二類是對以下帶有p-Laplacian算子的分數(shù)階微分方程多點邊值問題正解的存在性和不存在性進行了研究,本文討論了參數(shù)l的取值范圍,通過運用Guo-Krasnoselskii不動點定理得到正解存在性和不存在性的充分條件.第三類研究的是以下帶有p-Laplacian算子的分數(shù)階奇異微分方程積分邊值問題正解的存在性:文中通過運用上下解的方法和Schauder不動點定理,通過證明修正后微分方程邊值問題存在正解,規(guī)避了方程的奇異性,得到了當(dāng)參數(shù)l在特定范圍時正解的存在性.
[Abstract]:In the last half century, with the development and wide application of fractional calculus theory, the theoretical research of fractional differential equations has been developed rapidly, and some models of fractional differential equations have also been widely used in specific scientific fields. Such as economics, chemistry, biology, physics, medicine and so on. Therefore, a solvable fractional differential equation model can have a great influence in modern society. In this paper, the existence of positive solutions for boundary value problems of three nonlinear fractional differential equations is studied. The first is the existence of iterative positive solutions for multipoint boundary value problems of fractional differential equations. In this paper, the existence and uniqueness of the positive solution of the following multipoint boundary value problems are studied by means of iterative technique and 0u positive operator: the conclusion is obtained that if the function f (t u (t) satisfies the Lipschitz condition and the Lipschitz constant satisfies some conditions), The existence and uniqueness of positive solution can be obtained. The second is to study the existence and nonexistence of positive solutions for multipoint boundary value problems of fractional differential equations with p-Laplacian operator. In this paper, the range of parameter l is discussed. By using Guo-Krasnoselskii fixed point theorem, sufficient conditions for the existence and non-existence of positive solutions are obtained. In the third category, the existence of positive solutions for integral boundary value problems of fractional singular differential equations with p-Laplacian operator is studied. By using the method of upper and lower solutions and Schauder fixed point theorem, the existence of positive solutions to the boundary value problems of modified differential equations is proved. The singularity of the equation is avoided and the existence of positive solution is obtained when the parameter l is in a specific range.
【學(xué)位授予單位】：華北電力大學(xué)(北京)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.8

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本文編號：2212717