# zbMATH — the first resource for mathematics

Global existence of mild solutions to semilinear differential equations in Banach spaces. (English) Zbl 0618.34056
Let X be a Banach space and $$\Omega$$ be a subset of [a,b)$$\times X$$. Let A be the infinitesimal generator of a linear contraction semigroup $$\{$$ S(t);t$$\geq 0\}$$ on X of class $$(C_ 0)$$ and let B be a continuous function of $$\Omega$$ into X.
In this paper are established the global existence and uniqueness of mild solutions to a semilinear differential equation in X; $(*)\quad u'(t)=Au(t)+B(t,u(t))\quad \tau <t<b,\quad u(\tau)=z$ under the following conditions:
($$\Omega$$ 1) If $$(t_ n,x_ n)\in \Omega$$, $$t_ n\uparrow t$$ in [a,b) and $$x_ n\to x$$ in X as $$n\to \infty$$, then (t,x)$$\in \Omega.$$
($$\Omega$$ 2) $$\liminf_{h\downarrow 0}h^{-1}dist(S(h)x+hB(t,x),\Omega (t+h))=0$$ for all (t,x)$$\in \Omega$$, where $$\Omega (t)=\{x\in X;(t,x)\in \Omega$$ for $$t\in [a,b)\}.$$
($$\Omega$$ 3) $$[x-y,B(t,x)-B(t,y)]_-\leq g(t,| x-y|)$$ for all (t,x), (t,y)$$\in \Omega$$, where $$[x,y]_-=\lim_{h\uparrow 0}h^{- 1}(| x+hy| -| x|)$$ for x,y$$\in X$$ and g is a function from [a,b)$$\times R$$ into R with the following properties:
(g1) g(t,w) satisfies so-called Caratheodory’s condition.
(g2) $$g(t,0)=0$$; and w(t)$$\equiv 0$$ is the maximal solution to the initial-value problem $$w'(t)=g(t,w(t))$$, $$a<t<b$$, $$w(a)=0.$$
This result extends most of the known results concerning the global existence of mild solutions to semilinear equations of the form (*).

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces