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Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. (English) Zbl 1193.34056

From the introduction: We consider the following impulsive hybrid boundary value problem for nonlinear fractional differential equations

$\left\{\begin{array}{c}{}^{c}{D}^{q}x\left(t\right)=f\left(t,x\left(t\right)\right),\phantom{\rule{1.em}{0ex}}1

where ${}^{c}D$ is the Caputo fractional derivative $f:\left[0,1\right]×ℝ\to ℝ$ is a continuous function, ${I}_{k},{J}_{k}:ℝ\to ℝ$, ${\Delta }x\left({t}_{k}\right)=x\left({t}_{k}^{+}\right)-x\left({t}_{k}^{-}\right)$ with $x\left({t}_{k}^{+}\right)={lim}_{h\to {0}^{+}}x\left({t}_{k}+h\right),x\left({t}_{k}^{-}\right)={lim}_{h\to {0}^{-}}x\left({t}_{k}+h\right)$, $k=1,2,\cdots ,p$ for $0={t}_{0}<{t}_{1}<{t}_{2}<\cdots <{t}_{p}<{t}_{p+1}=1$.

We prove some existence results by means of the contraction mapping principle and Krasnoselskii’s fixed point theorem.

##### MSC:
 34B37 Boundary value problems for ODE with impulses 34A08 Fractional differential equations 47N20 Applications of operator theory to differential and integral equations