Mild Solution to Fractional Boundary Value Problem with Nonlinear Boundary Conditions
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Authors
Shirin Eivani
- Department of Mathematics, Urmia University, Urmia, Iran.
Abstract
In this paper, we consider a system of boundary value problems for fractional differential equation given by
\[
\begin{cases}
^cD^\alpha_{0^+}u(t)=A u(t)+f(t,u(t))+\int^t_0 q(t-s)g(s,u(s))ds, t\in I= [0,T], q\in (0,T), t\neq t_k\\
\Delta u(t_k)=I_k(u(t^-_k)), \Delta \acute{u}(t_k)=J_k(u(t^-_k)), k=1,2,...,m,\\
u(0)=u_0\in X,\quad \acute{u}(0)=u_1\in X,
\end{cases}
\]
where \(^cD^\alpha_{0^+}\) is Caputo's fractional derivative of order \(\alpha, A: D(A)\subset X\rightarrow X\) is a sectorial operator of type (\(M,\theta,\alpha,\mu\)) on a Banach space \(X, 0=t_0<t_1<t_2<...<t_m<t_{m+1}=T,I_k,J_k: \mathbb{R}\rightarrow \mathbb{R}, \Delta x(t_k):= x(t^+_k)-x(t^-_k),x(t^-_k)=\lim_{h\rightarrow 0^-}x(t_k+h),x(t^+_k)=\lim_{h\rightarrow 0^+}x(t_k+h), I_k,J_k\in C(X,X)(k=1,2,...,m)\) are bounded function, the functions \(f,g: I\times X\rightarrow X\) are given operators satisfying some assumptions and \(q: I\rightarrow X\) is a integrable function on I. Several existence results of mild solutions are obtained.
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ISRP Style
Shirin Eivani, Mild Solution to Fractional Boundary Value Problem with Nonlinear Boundary Conditions, Journal of Mathematics and Computer Science, 13 (2014), no. 3, 257-280
AMA Style
Eivani Shirin, Mild Solution to Fractional Boundary Value Problem with Nonlinear Boundary Conditions. J Math Comput SCI-JM. (2014); 13(3):257-280
Chicago/Turabian Style
Eivani, Shirin. "Mild Solution to Fractional Boundary Value Problem with Nonlinear Boundary Conditions." Journal of Mathematics and Computer Science, 13, no. 3 (2014): 257-280
Keywords
- Caputo's fractional derivative
- Sectorial operator
- Mild solution
- Analytic solution operators.
MSC
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