Advanced Search

The semilinear Cauchy problem (1) u’(t) = Au(t) + G(u(t)), $u\left(0\right)=x\in \overline{D\left(A\right)}$, with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that ||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A) with some linear B: D(A) → X, $B{(\lambda -A)}^{-1}x=\lambda {\int }_0^{\infty }{e}^{-\lambda t}V\left(s\right)xds$, where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on $[0,\infty )\times \overline{D\left(A\right)}$ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)| ≤ Bv whenever...