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

2017; Springer Nature; Volume: 2017; Issue: 1 Linguagem: Inglês

10.1186/s13661-017-0755-3

1687-2770

Guoqing Chai, Jinghua Chen,

Fractional Differential Equations Solutions

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.