Nonlocal conformable-fractional differential equations with a measure of noncompactness in Banach spaces (Q2176060)

Summary: This paper deals with the existence of mild solutions for the following Cauchy problem: \[ d^\alpha(t)/dt^\alpha = Ax(t) + f( t,x(t)),\quad x(0) = x_0+g(x),\ t\in [0,\tau], \] where \(d^\alpha(\cdot)/dt^\alpha\) is the so-called conformable fractional derivative. The linear part \(A\) is the infinitesimal generator of a uniformly continuous semigroup \((T(t))_{t \geq 0}\) on a Banach space \(X, f\) and \(g\) are given functions. The main result is proved by using the Darbo-Sadovskii fixed point theorem without assuming the compactness of the family \((T(t))_{t > 0}\) and the Lipschitz condition on the nonlocal part \(g\).