Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1187.34038
Existence of solutions for impulsive integral boundary value problems of fractional order.
(English)
[J] Nonlinear Anal., Hybrid Syst. 4, No. 1, 134-141 (2010). ISSN 1751-570X

Summary: We study a nonlinear impulsive boundary value problem for differential equations of fractional order with boundary conditions given by $$\cases ^cD^qx(t)=f(t,x(t)),\quad 1<q\le 2,\ t\in {\cal J}_1=[0,1]\setminus \{t_1,t_2,\dots,t_p\},\\ \Delta x(t_k)=I_k(x(t^-_k)),\quad \Delta x'(t_k)=J_k(x(t^-_k)),\ t_k\in (0,1),\ k=1,2,\dots,p,\\ \alpha x(0)+\beta x'(0)=\int^1_0 q_1(x(s))\,ds,\quad \alpha x(1) +\beta x'(1)=\int^1_0 q_2(x(s))\,ds,\endcases\tag 1$$ where $^cD$ is the Caputo fractional derivative, $f:J\times \Bbb R\to\Bbb R$ is a continuous function, $J=[0,1]$, $I_j,J_jk:\Bbb R\to\Bbb R$, $\Delta x(t_k)=x(t^+_k)-x(t^-_k)=\lim_{h\to0^+}x(t_k+h)$, $x(t^-_k)=\lim_{h\to 0^-}x(t_k+h)$, $k=1,2,\dots,p$ for $0=t_0<t_1<t_2<\cdots<t_p<t_{p+1}=1$ and $q_1,q_2:\Bbb R\to\Bbb R$ and $\alpha>0$, $\beta\ge 0$ are real numbers. We prove some existence results by applying the contraction mapping principle and Krasnoselskii's fixed point theorem.
MSC 2000:
*34B37 Boundary value problems with impulses
47N20 Appl. of operator theory to differential and integral equations
34B10 Multipoint boundary value problems

Keywords: fractional differential equations; impulse; integral boundary conditions; existence; fixed point theorem

Highlights
Master Server