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Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the α-norm. (English) Zbl 1185.34112

The authors consider the existence of solutions of the neutral partial differential equation with nonlocal conditions:

d/dtx ( t ) + F ( t , x ( t ) )=-Ax(t)+G(t,x(t)),t0,
x(0)+g(x)=x 0 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.

MSC:
34K30Functional-differential equations in abstract spaces
34K40Neutral functional-differential equations
35R10Partial functional-differential equations
47D06One-parameter semigroups and linear evolution equations