Existence of solutions for impulsive integral boundary value problems of fractional order. (English) Zbl 1187.34038

Summary: We study a nonlinear impulsive boundary value problem for differential equations of fractional order with boundary conditions given by
\[ \begin{cases} ^cD^qx(t)=f(t,x(t)),\quad 1<q\leq 2,\;t\in {\mathcal J}_1=[0,1]\setminus \{t_1,t_2,\dots,t_p\},\\ \Delta x(t_k)=I_k(x(t^-_k)),\quad \Delta x'(t_k)=J_k(x(t^-_k)),\;t_k\in (0,1),\;k=1,2,\dots,p,\\ \alpha x(0)+\beta x'(0)=\int^1_0 q_1(x(s))\,ds,\quad \alpha x(1) +\beta x'(1)=\int^1_0 q_2(x(s))\,ds,\end{cases}\tag{1} \]
where \(^cD\) is the Caputo fractional derivative, \(f:J\times \mathbb R\to\mathbb R\) is a continuous function, \(J=[0,1]\), \(I_j,J_jk:\mathbb R\to\mathbb R\), \(\Delta x(t_k)=x(t^+_k)-x(t^-_k)=\lim_{h\to0^+}x(t_k+h)\), \(x(t^-_k)=\lim_{h\to 0^-}x(t_k+h)\), \(k=1,2,\dots,p\) for \(0=t_0<t_1<t_2<\cdots<t_p<t_{p+1}=1\) and \(q_1,q_2:\mathbb R\to\mathbb R\) and \(\alpha>0\), \(\beta\geq 0\) are real numbers.
We prove some existence results by applying the contraction mapping principle and Krasnoselskii’s fixed point theorem.


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


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