Growth bounds of solutions of abstract nonlinear differential equations

Growth estimates are provided for solutions of the equation z' = Az + F(z), where A is the infinitesimal generator of a Co-semigroup of operators in a Banach space X and F is a nonlinear operator in X.The rate at which solutions grow to infinity is connected to the rate of growth of the semigroup generated by A and the rate at which IJF(x)ll/llxll converges to 0 as llxiJ goes to infinity.