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**Existence of some semilinear nonlocal functional differential equations of neutral type.**
*(English)*
Zbl 1298.34148

Summary: This paper is concerned with the existence of mild and strong solutions on the interval \([0,T]\) for some neutral partial differential equations with nonlocal conditions. The linear part of the equations is assumed to generate a compact analytic semigroup of bounded linear operators, whereas the nonlinear part satisfies the Carathéodory condition and is bounded by some suitable functions. We first employ the Schauder fixed-point theorem to prove the existence of solution on the interval \([\delta, T]\) for \(\delta > 0\) that is small enough, and, then, by letting \(\delta\to 0\) and using a diagonal argument, we have the existence results on the interval \([0,T]\). This approach allows one to drop the compactness assumption on a nonlocal condition, which generalizes recent conclusions on this topic. The obtained results will be applied to a class of functional partial differential equations with nonlocal conditions.

### MSC:

34K30 | Functional-differential equations in abstract spaces |

34K40 | Neutral functional-differential equations |

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

47D03 | Groups and semigroups of linear operators |

47N20 | Applications of operator theory to differential and integral equations |

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\textit{H. Liu} et al., Abstr. Appl. Anal. 2013, Article ID 503656, 12 p. (2013; Zbl 1298.34148)

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