郝晓红,周宗福.一类分数阶微分方程多点边值问题解的存在性[J].数学研究及应用,2013,33(2):175~188
一类分数阶微分方程多点边值问题解的存在性
The Existence of Solutions to a Class of Multi-point Boundary Value Problem of Fractional Differential Equation
投稿时间:2011-11-26  最后修改时间:2012-10-09
DOI:10.3770/j.issn:2095-2651.2013.02.005
中文关键词:  分数阶微分方程  多点边值问题  重合度.
英文关键词:fractional differential equation  multi-point boundary value problem  coincidence degree.
基金项目:国家自然科学基金(Grant No.11071001), 安徽省自然科学基金(Grant No.1208085MA13), 安徽大学211工程项目(Grant No.KJTD002B).
作者单位
郝晓红 安徽大学数学科学学院, 安徽 合肥 230601 
周宗福 安徽大学数学科学学院, 安徽 合肥 230601 
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中文摘要:
      本文讨论下面一类分数阶微分方程多点边值问题 $$\align &D^{\alpha}_{0+}u(t) = f(t, u(t),~D^{\alpha-1}_{0+}u(t), D^{\alpha-2}_{0+}u(t), D^{\alpha-3}_{0+}u(t)),~~t\in(0,1), \\&I^{4-\alpha}_{0+}u(0) = 0, ~D^{\alpha-1}_{0+}u(0)=\displaystyle{\sum_{i=1}^{m}}\alpha_{i}D^{\alpha-1}_{0+}u(\xi_{i}),\\&D^{\alpha-2}_{0+}u(1)=\sum\limits_ {j=1}^{n}\beta_{j} D^{\alpha-2}_{0+}u(\eta_{j}),~D^{\alpha-3}_{0+}u(1)-D^{\alpha-3}_{0+}u(0)=D^{\alpha-2}_{0+}u(\frac{1}{2}),\endalign$$其中$3<\alpha \leq 4$是一个实数.通过应用Mawhin重合度理论和构建适当的算子,得到了该边值问题解的存在性结果.
英文摘要:
      In this paper, we consider the following multi-point boundary value problem of fractional differential equation $$\align &D^{\alpha}_{0+}u(t) = f(t, u(t),~D^{\alpha-1}_{0+}u(t), D^{\alpha-2}_{0+}u(t), D^{\alpha-3}_{0+}u(t)),~~t\in(0,1), \\&I^{4-\alpha}_{0+}u(0) = 0, ~D^{\alpha-1}_{0+}u(0)=\displaystyle{\sum_{i=1}^{m}}\alpha_{i}D^{\alpha-1}_{0+}u(\xi_{i}),\\&D^{\alpha-2}_{0+}u(1)=\sum\limits_ {j=1}^{n}\beta_{j} D^{\alpha-2}_{0+}u(\eta_{j}),~D^{\alpha-3}_{0+}u(1)-D^{\alpha-3}_{0+}u(0)=D^{\alpha-2}_{0+}u(\frac{1}{2}),\endalign$$ where $3<\alpha \leq 4$ is a real number. By applying Mawhin coincidence degree theory and constructing suitable operators, some existence results of solutions can be established.
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