On some existence results of mild solutions for nonlocal integrodifferential Cauchy problems in Banach spaces. (English) Zbl 1242.45010

Various existence results of a mild solution of the integro-differential equation \[ u'(t)=Au(t)+f\Bigl(t,u(t),\int_0^tk(t,s,u(s))ds\Bigr)\quad(0<t<b) \] with a nonlocal initial value condition \[ u(0)=g(u)+u_0 \] are obtained. Here, \(A\) is the generator of a \(C_0\)-semigroup. Recall that a mild solution is a function \(u\) which formally satisfies the corresponding variation-of-constants formula. The main hypotheses are some growth estimates for \(k\), \(f\), and \(g\) (leading to a-priori bounds), compactness of either \(f\) or of the semigroup, and either compactness of \(g\) or that \(g\) is Lipschitz with a sufficiently small constant. The proofs use Schauder’s or Schaefer’s fixed point theorem.


45J05 Integro-ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
45N05 Abstract integral equations, integral equations in abstract spaces
45G10 Other nonlinear integral equations
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