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Zbl 0934.34068
Byszewski, Ludwik; Akca, Haydar
Existence of solutions to a semilinear functional-differential evolution nonlocal problem. (English)
[J] Nonlinear Anal., Theory Methods Appl. 34, No.1, 65-72 (1998). ISSN 0362-546X

The following nonlocal Cauchy problem is considered for the semilinear functional-differential equation $$u'(t)+ Au(t)= f(t,u(t), u(b_1(t)),\dots, u(b_m(t))),\quad t\in (t_0, t_0+ a],\quad m\in\bbfN,\tag 1$$ $$u(t_0)+g(u)= u_0,\tag 2$$ with $t_0\ge 0$, $a>0$, $A$ is the infinitesimal generator of a compact $C_0$ semigroup of operators on a Banach space, $f$, $g$, $b_i$, $i= 1,\dots, m,$ are given functions and $u_0$ is an element of the Banach space. Sufficient conditions for the existence of a so-called mild solution and a classical solution to (1), (2) are proved.
[R.G.Koplatadze (Tbilisi)]

MSC 2000:: *34K30 Functional-differential equations in abstract spaces

Keywords: nonlocal Cauchy problem; infinitesimal generator; existence; mild solution

Cited in: Zbl 1083.34045

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