Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the $\alpha $-norm.

*(English)*Zbl 1185.34112The authors consider the existence of solutions of the neutral partial differential equation with nonlocal conditions:

$$d/dt\left(x\left(t\right)+F(t,x(t\left)\right)\right)=-Ax\left(t\right)+G(t,x\left(t\right)),\phantom{\rule{1.em}{0ex}}t\ge 0,$$

$$x\left(0\right)+g\left(x\right)={x}_{0}\in X,$$

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.

Reviewer: Miklavž Mastinšek (Maribor)

##### MSC:

34K30 | Functional-differential equations in abstract spaces |

34K40 | Neutral functional-differential equations |

35R10 | Partial functional-differential equations |

47D06 | One-parameter semigroups and linear evolution equations |