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Existence of semilinear differential equations with nonlocal initial conditions. (English) Zbl 1129.34041

The author considers the existence of mild solutions for semilinear Cauchy problems

u ' (t)=Au(t)+f(t,u(t)),t[0,b]a.e.,u(0)=g(u)+u 0 ,

where A is an infinitesimal generator of a strongly continuous semigroup T(t) of bounded linear operators in a Banach space X, f:[0,b]×XX, gC([0,b];X) are given X-valued functionals. Certain assumptions are imposed on the nonlinear terms which allow for using a suitable fixed point theorem in proving the existence result. The map g does not need to be compact in order to reach the existence conclusion. Instead some other weaker condition is assumed.

34G20Nonlinear ODE in abstract spaces
47D06One-parameter semigroups and linear evolution equations
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