三类具有适型分数阶导数的边值问题的正解

发布时间：2018-04-24 16:22

本文选题：适型分数阶导数 + 边值问题　；参考：《山东科技大学》2017年硕士论文

【摘要】：本文用锥上的不动点理论考虑具有适型分数阶导数的几类分数阶边值问题的正解的存在性和多解性,并给出相应问题的Laplace变换.根据内容,全文共六章:第一章,介绍微分方程边值问题的研究背景与发展概况.第二章,介绍分数阶微分方程的相关概念与性质;给出具有适型分数阶微分方程边值问题的Green函数以及具有适型分数阶导数的线性齐次问题Laplace变换;最后给出本文所需要的一些工具.第三章,研究具有适型分数阶导数的非线性边值问题的正解.首先给出此问题的积分算子并证明其全连续性;然后给出研究结果,即对非线性项进行控制,再利用锥压缩拉伸不动点定理证明研究问题的正解的存在性和多解性;最后给出所研究问题的Laplace变换.两个例子说明主要结果.第四章,在上一章的基础上,研究具有适型分数阶导数的非线性特征值问题.解的存在性结果是通过给出特征值准则而得到的,证明过程也利用不动点定理.第五章,研究具有适型分数阶导数的P-Laplacian边值问题的正解.给出边值问题所对应的积分算子,利用Laplace算子的性质证得积分算子的全连续性,进而利用不动点定理证明研究问题的正解的存在性和多解性.第六章,本文的总结和展望.
[Abstract]:In this paper, we use the fixed point theory on cone to consider the existence and multiple solvability of the positive solutions of several class of fractional boundary value problems with a suitable fractional derivative, and give the Laplace transformation of the corresponding problems. In the first chapter, the research background and development of the boundary value problems of differential equations are introduced. The second chapter introduces fractional differential. The related concepts and properties of the equation; give the Green function of the boundary value problem with an adaptive fractional differential equation and the linear homogeneous problem Laplace transformation with a suitable fractional derivative. Finally, some tools needed in this paper are given. The third chapter, the positive solution of the nonlinear boundary value problem with an appropriate fractional order number is studied. The integral operator of the problem is proved to be full continuity, and then the research results are given, that is, to control the nonlinear term, and then to prove the existence and multi solution of the positive solution of the research problem by using the fixed point theorem of the cone compression drawing. Finally, the Laplace transformation of the problem is given. Two examples are given to explain the main results. The fourth chapter is the basis of the previous chapter. The nonlinear eigenvalue problem with a suitable fractional derivative is studied. The existence result of the solution is obtained by giving the eigenvalue criterion. It is proved that the process also uses the fixed point theorem. In the fifth chapter, the positive solution of the P-Laplacian boundary value problem with a suitable fractional derivative is studied. The integral operator corresponding to the boundary value problem is given, and Lapla is used. The nature of the CE operator proves the full continuity of the integral operator, and then uses the fixed point theorem to prove the existence and multi solvability of the positive solution of the research problem. The sixth chapter, the summary and the prospect of this paper.

【学位授予单位】：山东科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8

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