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Zbl 1095.34040
Xue, Xingmei
Nonlinear differential equations with nonlocal conditions in Banach spaces.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 63, No. 4, A, 575-586 (2005). ISSN 0362-546X

The author studies the following nonlocal initial value problem $$u'(t)\in Au(t)+f(t,u(t)) ,\quad t \in (0,b) , u(0)=g(u) ,$$ where $A$ is a nonlinear, $m$-dissipative multi-valued operator which generates a contraction semigroup $T(t)$ in a Banach space $X$, $f:[0,b] \times D \to X, g: C([0,b],D) \to \overline{D(A)}$ are functions with $D(A) \subset D \subset X$. Using fixed-point theorems, he proves some sufficient conditions for the existence of integral solutions, by assuming different hypotheses on $f,g$ and $A$.
[Francesca Papalini (Ancona)]
MSC 2000:
*34G25 Evolution inclusions
47H20 Semigroups of nonlinear operators

Keywords: nonlocal initial condition; nonlinear compact semigroup; $m$-dissipative operator; fixed-point

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