Growth bounds of solutions of abstract nonlinear differential equations (Q1328952)

The equation (1) \(z'(t)= Az(t)+ Fz(t)\), \(t\geq 0\), is investigated. It is assumed that \(A\) is the infinitesimal generator of a strongly continuous semigroup \(T(t)\), \(t\geq 0\), of bounded linear operators in a Banach space \(X\) and \(F\) is a nonlinear operator in \(X\). Moreover, the following two hypotheses are employed: (a) there exist a nonnegative integer \(n\), a nonnegative constant \(w\) and a positive constant \(M\) such that \(\limsup_{t\to\infty} e^{-wt}| T(t)|/ t^ n\leq M\), (b) \(F\) is bounded on bounded sets, continuous and \(\| F(x)\|\leq c(\| x\|)\| x\|\) for some nonincreasing function \(c: [r_ 0,\infty)\to [0,\infty)\), where \(r_ 0\) is a number, \(r_ 0> 1\). Assuming additionally that \(c(r)\) tends to 0 in a special way as \(r\to\infty\), some growth estimates are provided for mild solutions of the equation (1). Five examples illustrate the results of this interesting paper.