On nonlocal integral boundary value problems for impulsive nonlinear differential equations of fractional order (Q2925780)

The authors consider the nonlinear impulsive fractional order boundary value problem NEWLINE\[NEWLINE^CD^qx(t)=f(t,x(t)),~~1<q\leq2,~t\in J',NEWLINE\]NEWLINE NEWLINE\[NEWLINE\Delta x(t_k)=I_k(x(t_k)),~~\Delta x'(t_k)=I^{*}_k(x(t_k)),~k=1,2,\dots,p, NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(0)=0,~ x(1)=\beta\int_{0}^{\eta}x(s)ds, ~0<\eta<1,NEWLINE\]NEWLINE where \(^CD^q\) is the Caputo fractional derivative. They establish the existence of solution for the boundary value problem by using some standard fixed point theorems and also establish the uniqueness of the solution of the boundary value problem by applying the contraction mapping theorem.