Existence, uniqueness and regularity of solutions to semilinear nonlocal functional differential problems (Q1881339)

The author studies the following semi-linear abstract differential equation with nonlocal conditions \[ x'(t) = Ax(t)+f(t,x(t), x(b_{1}(t)), x(b_{2}(t)),\dots, x(b_{n}(t))), \quad t\in I=[0,a], \] \[ h(x)= \varphi_{0} \quad\text{on } [-r,0]. \] Here, \(A\) is the infinitesimal generator of a \(C_{0}\)-semigroup of bounded linear operators on a Banach space \(X\), \(\varphi_{0} \in C( [-r,0]; X)\) and \(f:I\times X^{n+1}\to X\), \(b_{i}:[0,a]\to [-r,a] \), \(h:C( [-r,a]; X)\to C( [-r,a]; X) \) are appropriated functions. By using the contraction mapping principle, the author establishes the existence of a mild solution. The existence of a strong solution is considered, too.