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Zbl 1185.34112
Chang, Jung-Chan; Liu, Hsiang
Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the $\alpha $-norm. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 9, A, 3759-3768 (2009). ISSN 0362-546X

The authors consider the existence of solutions of the neutral partial differential equation with nonlocal conditions: $$d/dt\bigl(x(t)+ F(t, x(t))\bigr)= -Ax(t)+ G(t, x(t)),\quad t\ge 0,$$ $$x(0)+ g(x)= x_0\in X,$$ 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.
[Miklavž Mastinšek (Maribor)]

MSC 2000:: *34K30 Functional-differential equations in abstract spaces
34K40 Neutral equations
35R10 Difference-partial differential equations
47D06 One-parameter semigroups and linear evolution equations

Keywords: analytic compact semigroup; nonlocal condition; mild solution; strong solution; $\alpha $-norm

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