Existence of solutions for impulsive fractional boundary value problems via variational method

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• 作者：Guoqing Chai ; Jinghua Chen
• 关键词：fractional differential equations ; critical point theory ; variational method ; impulsive equation ; iterative technique
• 刊名：Boundary Value Problems
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1525KB
• 刊物主题：Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1687-2770
• 卷排序：2017

文摘

In this paper, the authors consider the following fractional boundary value problem for impulsive fractional differential equations: $$\textstyle\begin{cases} { }_{t}D_{T}^{\alpha}({ }_{0}^{c} D_{t}^{\alpha}u(t))+a(t)u(t)=f(t,u(t),{ }_{0}^{c} D_{t}^{\alpha}u(t)),\quad t\ne t_{j} ,\mbox{a.e. }t\in[0,T], \\ \Delta({ }_{t}D_{T}^{\alpha-1} ({ }_{0}^{c} D_{t}^{\alpha}u))(t_{j} )=I_{j} (u(t_{j} )),\quad j=1,2,\ldots,n, \\ u(0)=u(T)=0, \end{cases} $$ where \(\alpha\in(1/2,1]\), \(0=t_{0} < t_{1} < t_{2} <\cdots<t_{n} <t_{n+1} =T\), \(f:[0,T]\times{\mathbb{R}}\times{\mathbb{R}}\to{\mathbb{R}}\) and \(I_{j} :{\mathbb{R}}\to {\mathbb{R} }\), \(j=1,2,\ldots,n\), are continuous functions, \(a\in C([0,T])\) and $$\begin{aligned}& \Delta \bigl({ }_{t}D_{T}^{\alpha-1} \bigl({ }_{0}^{c} D_{t}^{\alpha}u \bigr) \bigr) (t_{j} )={ }_{t}D_{T}^{\alpha -1} \bigl({ }_{0}^{c} D_{t}^{\alpha}u \bigr) \bigl(t_{j}^{+} \bigr)-{ }_{t}D_{T}^{\alpha-1} \bigl({ }_{0}^{c} D_{t}^{\alpha}u \bigr) \bigl(t_{j}^{-} \bigr), \\& { }_{t}D_{T}^{\alpha-1} \bigl({ }_{0}^{c} D_{t}^{\alpha}u \bigr) \bigl(t_{j} ^{+} \bigr)= \lim _{t\to t_{j}^{+} } { }_{t}D_{T}^{\alpha-1} \bigl({ }_{0}^{c} D_{t}^{\alpha}u \bigr) (t),\quad\quad { }_{t}D_{T}^{\alpha-1} \bigl({ }_{0}^{c} D_{t}^{\alpha}u \bigr) \bigl(t_{j} ^{-} \bigr)= \lim _{t\to t_{j}^{-} } { }_{t}D_{T}^{\alpha-1} \bigl({ }_{0}^{c} D_{t}^{\alpha}u \bigr) (t). \end{aligned}$$ By using the variational method and iterative technique, the authors show the existence of at least one nontrivial solution to the above boundary value problem.Keywordsfractional differential equationscritical point theoryvariational methodimpulsive equationiterative technique1 IntroductionFractional calculus has applications in many areas including fluid flow, electrical networks, probability and statistics, chemical physics and signal processing, etc. For details, see [1–6] and the references therein. In recent years, there are many papers dealing with the existence of solutions of nonlinear initial (or boundary) value problems of fractional equations by applying nonlinear analysis such as fixed point theorems, lower and upper solutions method, monotone iterative method, coincidence degree theory. However, up to now, there are few results on the solutions to fractional boundary value problems that are established by the variational methods; see, for example, [7–18]. It is often very difficult to establish a suitable space and variational functional for fractional boundary value problem, especially for the fractional equations including both left and right fractional derivatives.