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Nonlocal boundary value problem for impulsive differential equations of fractional order. (English) Zbl 1219.34011

Summary: We consider the following problem:
\[ ^cD^qu(t)=f(t,u(t),u'(t)),\quad 1<q\leq 2, \quad t\in{\mathcal J}_1=[0,1]\setminus \{t_1,t_2,\dots,t_p\}, \]
\[ \Delta u(t_k)=I_k(u(t^-_k)),\quad \Delta u'(t_k)={\mathcal J}_k(u(t^-_k)),\quad t_k\in (0,1),\;k=1,2,\dots,p, \]
\[ \alpha u(0)+\beta u'(0)=g_1(u),\quad \alpha u(1)+\beta u'(1)=g_2(u), \]
where \(J=[0,1]\), \(f:J\times \mathbb R\times \mathbb R\to \mathbb R\) is a continuous function, and \(I_k,J_k:\mathbb R\to\mathbb R\) are continuous functions, \(\Delta u(t_k)=u(t^+_k)-u(t^-_k)\), with \(u(t^+_k)=\lim_{h\to 0^+}u(t_k+h)\), \(u(t^-_k)=\lim_{h\to 0^-}u(t_k+h)\), \(k=1,2,\dots,p\), \(0=t_0<t_1<t_2<\cdots<t_p<t_{p+1}=1\), \(\alpha>0\), \(\beta\geq 0\), and \(g_1,g_2:\text{PC}(J,\mathbb R)\to\mathbb R\) are two continuous functions.
By means of a fixed point theorem due to O’Regan, we establish sufficient conditions for the existence of at least one solution of the problem. For the illustration of the main result, an example is given.

MSC:

34A08 Fractional ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B37 Boundary value problems with impulses for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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