Integral inequalities and mild solutions of semilinear neutral evolution equations (Q1765794)

The paper is mainly concerned with the existence of mild solutions for a nonlocal Cauchy problem governed by a semilinear neutral differential equation in a Banach space \(X\) \[ \frac{d}{dt}[x(t)+g(t,x(t))]= Ax(t)+f(t,x(t)), \quad x(0)=x_0-h(x(t)), \quad t\in [0,T] \] where \(A\) is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators; \(g,f:[0,T]\times X\to X\) and \(h:C([0,T],X)\to X\) are given functions. In order to prove the existence theorem, the authors provide new results on a singular nonlinear integral inequality of Bihari type \[ u(t)\leq l(t)+\int_0^t \frac{1}{(t-s)^{1-\beta}}\alpha_1(s)\omega_1(u(s))\, ds + \int_0^t \alpha_2(s)\omega_2(u(s))\, ds\;. \] Finally, they give an application to partial differential equations.