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Existence of solutions for higher-order impulsive boundary value problems of a quasilinear fractional differential equation. (Chinese. English summary) Zbl 1313.34028

Summary: The existence of solutions for higher-order impulsive boundary value problems of the form \[ \begin{aligned} & D^q_{0+}y(t)=A(t,y)y(t)+f(t,y(t),\varPhi y(t),\varPsi y(t)) \forall t\in [0,1], q\in (n-1,n],\\ & y^{(i)}(0)=0,\Delta y^{(i)}|_{t=t_k}=0, 1\leqslant i\leqslant n-2, k=1,2,\cdots, p,\\ & \Delta y|_{t=t_k}=I_k(y(t_k)),\Delta y^{(n-1)}|_{t=t_k}=J_k(y(t_k)), k=1,2,\cdots, p,\\ & y(0)=y_0+g(y),\;y^{(n-1)}(1)=y_1+\sum\limits^{m-2}_{j=1}b_jy^{(n-1)}(\xi_j)\end{aligned} \] is studied. By defining a contraction mapping and using a fixed point theorem, some sufficient conditions for the existence of a unique solution and at least one solution are established. Furthermore, two examples are presented to illustrate the main results.

MSC:

34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B37 Boundary value problems with impulses for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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