Existence of mild solutions for certain delay semilinear evolution inclusion with nonlocal conditions. (English) Zbl 0974.34076
The existence of mild solutions is proven for the first-order delay semilinear evolution inclusion with nonlocal condition
\[
y'-A(t)y \in F \biggl(t,y \bigl(\sigma(t) \bigr)\biggr),\;t\in J=[0,b], \quad y(0)+ f(y)=y_0,
\]
in a real Banach space \(E\). \(A(t)\) is the infinitesimal generator of a linear semigroup; \(F:J\times E\to 2^E\) is strongly measurable in \(t\), upper semicontinuous in \(y\) and bounded-closed-convex-valued; \(\sigma:J\to J\) is continuous and satisfies \(\sigma(t)\leq t\) for all \(t\in J\); \(f:C(J,E)\to E\) is continuous and bounded; and \(y_0\in E\). The result is proven by applying a fixed-point theorem due to M. Martelli [Boll. Unione Mat. Ital. (4), 11, Suppl. Fasc., No. 3, 70-76 (1975; Zbl 0314.47035)].
Reviewer: Daniel C.Biles (Bowling Green)
MSC:
34K30 | Functional-differential equations in abstract spaces |
34G25 | Evolution inclusions |
34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
47H20 | Semigroups of nonlinear operators |
34A60 | Ordinary differential inclusions |
34G20 | Nonlinear differential equations in abstract spaces |
35R10 | Partial functional-differential equations |