Mild Solution To Fractional Boundary Value Problem With Nonlinear Boundary Conditions

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.