Compact perturbations of accretive operators in Hilbert spaces. (English) Zbl 0594.47046
Summary: Let H be a real Hilbert space with inner product (.,.), U a nonempty and open subset in H, \(A: D(A)\subset H\to 2^ H\) an m-accretive operator and B: [0,T]\(\times U\to H\) a given function. Let us consider the strongly nonlinear perturbed evolution equation
\[
(1)\quad \frac{du}{dt}(t)+Au(t)\ni B(t,u(t)),\quad 0\leq t\leq T,\quad u(0)=u_ 0,
\]
where \(u_ 0\in U\cap \overline{D(A)}\). The aim of this note is to state a local existence result concerning integral solutions for (1) which generalizes a previous theorem due to E. Schechter [Isr. J. Math. 43, 49-61 (1982; Zbl 0516.34060)].
MSC:
47H06 | Nonlinear accretive operators, dissipative operators, etc. |
34G20 | Nonlinear differential equations in abstract spaces |
47A55 | Perturbation theory of linear operators |
47H20 | Semigroups of nonlinear operators |
47E05 | General theory of ordinary differential operators |