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Zbl 1242.45010
Yang, YanLong; Wang, JinRong
On some existence results of mild solutions for nonlocal integrodifferential Cauchy problems in Banach spaces. (English)
[J] Opusc. Math. 31, No. 3, 443-455 (2011). ISSN 1232-9274

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.
[Martin Väth (Berlin)]

MSC 2000:: *45J05 Integro-ordinary differential equations
34G20 Nonlinear ODE in abstract spaces
45N05 Integral equations in abstract spaces
45G10 Nonsingular nonlinear integral equations

Keywords: integro-differential equation; Volterra equation; semigroup; nonlocal initial condition; Cauchy problems; Banach spaces; mild solution; variation-of-constants formular; compactness

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