A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups

Abstract

Let ${\alpha}$ be a bounded linear operator in a Banach space ${\mathbb{X}}$, and let A be a closed operator in this space. Suppose that for ${\Phi_1, \Phi_2}$ mapping D(A) to another Banach space ${\mathbb{Y}}$, ${A_{|{\rm ker}\, \Phi_1}}$ and ${A_{|{\rm ker}\, \Phi_2}}$ are generators of strongly continuous semigroups in ${\mathbb{X}}$. Assume finally that ${A_{|{\rm ker}\, \Phi_\text{a}}}$, where ${\Phi_\text{a} = \Phi_1 \alpha + \Phi_2 \beta}$ and ${\beta = I_\mathbb{X} - \alpha}$, is a generator also. In the case where ${\mathbb{X}}$ is an L ¹-type space, and ${\alpha}$ is an operator of multiplication by a function ${0 \le \alpha \le 1}$, it is tempting to think of the later semigroup as describing dynamics which, while at state x, is subject to the rules of ${A_{|{\rm ker}\, \Phi_1}}$ with probability ${\alpha (x)}$ and is subject to the rules of ${A_{|{\rm ker}\, \Phi_2}}$ with probability ${\beta (x)= 1 - \alpha (x)}$. We provide an approximation (a singular perturbation) of the semigroup generated by ${A_{|{\rm ker}\, \Phi_\text{a}}}$ by semigroups built from those generated by ${A_{|{\rm ker}\, \Phi_1}}$ and ${A_{|{\rm ker}\, \Phi_2}}$ that supports this intuition. This result is motivated by a model of dynamics of Solea solea (Arino et al. in SIAM J Appl Math 60(2):408–436, 1999–2000; Banasiak and Goswami in Discrete Continuous Dyn Syst Ser A 35(2):617–635, 2015; Banasiak et al. in J Evol Equ 11:121–154, 2011, Mediterr J Math 11(2):533–559, 2014; Banasiak and Lachowicz in Methods of small parameter in mathematical biology, Birkhäuser, 2014; Sanchez et al. in J Math Anal Appl 323:680–699, 2006) and is, in a sense, dual to those of Bobrowski (J Evol Equ 7(3):555–565, 2007), Bobrowski and Bogucki (Stud Math 189:287–300, 2008), where semigroups generated by convex combinations of Feller’s generators were studied.