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Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces. (English) Zbl 1227.34060

The authors study the existence of solutions to the following problem

u ' (t)=Au(t)+f(t,u(t)),t(0,T],u(0)=g(u),

where A is the generator of a linear semigroup, f,g satisfy Lipschitz conditions. When g is a constant function this is a Cauchy problem associated with a semilinear evolution equation. If g is an arbitrary Lipschitz function this problem is called “nonlocal” Cauchy problem. The authors use the measure of noncompactness and a fixed point theorem to prove an existence result.

MSC:
34G10Linear ODE in abstract spaces
47D06One-parameter semigroups and linear evolution equations
47N20Applications of operator theory to differential and integral equations
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