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The modulus semigroup for linear delay equations. (English) Zbl 1065.34054

The authors explicitly construct the generator of the modulus semigroup of the \(C_0\)-semigroup associated to the delay equation \[ u'(t)= Au(t)+ Lu_1, \quad t\geq 0, \qquad u(0)= x\in \mathbb{R}^n, \quad u_0= f\in L_p((-h,0), \mathbb{R}^n), \] in the Banach lattice \(\mathbb{R}^n\times L_p((-h,0), \mathbb{R}^n)\), where \(1\leq p< \infty\), \(A\in \mathbb{R}^{n\times n}\), and \(L: C([-h,0]; \mathbb{R}^n)\to \mathbb{R}^n\) is the bounded linear operator given by \[ Lf:= \int_{[-h,0]} d\eta(\theta) f(\theta), \quad f\in C([-h,0]; \mathbb{R}^n) \] where \(\eta: [-h,0]\to \mathbb{R}^{n\times n}\) is a function of bounded variation.

MSC:

34K06 Linear functional-differential equations
47D06 One-parameter semigroups and linear evolution equations
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