On the nonexistence of blowing-up solutions to a fractional functional-differential equation (Q2882543)

This paper is concerned with the nonlocal functional-differential problem NEWLINE\[NEWLINE\dot{x}(t)=Ax(t)+f(t,x(t),x_t,(I^\beta g(\cdot,x(\cdot),x_\cdot))(t)),\;t>0,NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(t)=\Phi(t),\;t\in [-r,0],NEWLINE\]NEWLINE where \((I^\beta v)(t)=\int_0^t (t-s)^{\beta-1}v(s)ds,~\beta>0\), \(A\) is the generator of a strongly continuous semigroup. They firstly establish a new integral inequality, which is the generalization of the Henry-Gronwall inequality. By applying this inequality, a sufficient condition for the nonexistence of blowing-up mild solution is given.