Existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions (Q1881340)

The authors study the existence and regularity of solutions for neutral partial differential equations with nonlocal conditions of the form \[ \frac{d}{dt}[x(t)-F(t,x(h_1(t)))]= A[x(t)-F(t,x(h_1(t)))]+G(t,x(h_2(t))), \quad 0\leq t\leq a, \] \[ x(0)+g(x)=x_0\in X, \] where \(A\) generates a strongly continuous semigroup \((T(t))_{t\geq 0}\) on a Banach space \(X,\) \(F, G: [0,a]\times X\to X\) are continuous and Lipschitzian with respect to the second argument, \(g: C([0,a],X)\to X\) is Lipschitz continuous and \(h_i\in C([0,a],[0,a]), i=1,2.\) Optimal regularity of the mild solutions is given when the semigroup \((T(t))_{t\geq 0}\) is analytic. They also study the special case of a nonlocal condition when \(g(x)=\sum_{i=0}^{p}g_i(x(t_i)),\) where \(g_i\) are Lipschitz continuous from \(X\) to \(X\) and \(t_0<\ldots<t_p\) are given numbers on \([0,a],\) and give sufficient conditions for the existence of mild solutions, which are better than in general case. Some examples to illustrate the abstract results are included, two.