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N / should be such that N rapidly decreases towards K. 0/ D N ; (17) where K and r are positive constants. 1 N=K/. Note that K is the population size at which G is zero and therefore dN=dt D 0 when N D K. Equation (17) is called the (continuous time) logistic growth equation or Verhulst equation, the constant K is called the Applying Functional Analytic Techniques to Evolution Equations 25 carrying capacity of the environment, and r is the unrestricted growth rate. 1 ; ı (18) K= N / exp. t/ ! K as t !

X/ D e nx ; x 0; we have kfn k D 1; while kAf n k D n; hence, there is no constant M such that kAf n k Ä M kfn k: Thus, it is not clear whether the series (2) converges. In fact, its nth term is defined merely for f 2 C n Œ0; 1 (n-times continuously differentiable functions with the nth order derivative in C Œ0; 1). x C t/; with the last equality holding provided f is analytic. , that the semigroup property holds [since we are dealing with an unbounded operator, the semigroup property is not a consequence of (4)].

Banasiak, On an extension of the Kato–Voigt perturbation theorem for substochastic semigroups and its application. Taiwanese J. Math. 5, 169–191 (2001) 3. J. Banasiak, Kinetic models in natural sciences, in Evolutionary Equations with Applications to Natural Sciences, ed. by J. Banasiak, M. Mokhtar-Kharroubi. Lecture Notes in Mathematics (Springer, Berlin, 2014) 4. J. Banasiak, L. Arlotti, Perturbations of Positive Semigroups with Applications (Springer, New York, 2006) 5. J. Banasiak, W. Lamb, The discrete fragmentation equation: semigroups, compactness and asynchronous exponential growth.