Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the \(\alpha \)-norm
The authors consider the existence of solutions of the neutral partial differential equation with nonlocal conditions: \[ d/dt\bigl(x(t)+ F(t, x(t))\bigr)= -Ax(t)+ G(t, x(t)),\quad t\geq 0, \] \[ x(0)+ g(x)= x_0\in X, \] where \(-A\) generates an analytic compact semigroup on a Banach space \(X\) and where the functions \(F\), \(G\) and \(g\) satisfy specific continuity and measurability constraints. Fractional powers of \(-A\) are used. By the fixed point theorem of Sadovskii existence of a mild solution is obtained. When \(X\) is a reflexive Banach space and \(G\) is Lipschitz continuous, a strong solution is obtained. The results are applied to a class of partial differential equations with nonlocal conditions.
scientific article; zbMATH DE number 5577443