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 (*).

34G20 Nonlinear differential equations in abstract spaces