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Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators. (English) Zbl 0790.45011
Bierstedt, Klaus D. (ed.) et al., Functional analysis. Proceedings of the Essen Conference, held in Essen, Germany, November 24 - 30, 1991. New York, NY: Dekker. Lect. Notes Pure Appl. Math. 150, 51-70 (1993).
In this paper, the inhomogeneous Cauchy problem $x'(t)=Ax(t)+f(t),\;t \geq 0,\;x(0)=x, \tag{DE}$ and the Volterra integro-differential equation $x'(t)=Ax (t)+ \int^ t_ 0 a(t-s) Ax(s)ds+f(t),\;t \geq 0,\;x(0)=x, \tag{VE}$ are studied. Here, $$a(\cdot)$$ is a scalar function and $$A$$ is a Hille-Yosida operator on a Banach space $$X$$, i.e. a (not necessarily densely defined) linear operator on $$X$$ which satisfies the usual Hille- Yosida type estimates. The approach adopted by the authors is a combination of extrapolation and operator matrix techniques, and can be described as follows. First, $$A$$ is extended to an operator $$A_{-1}$$ on the so-called extrapolation space $$X_{-1}$$, in which $$X$$ is continuously embedded. This extension is the generator of a $$C_ 0$$- semigroup $$\{T_{-1} (t)\}_{t \geq 0}$$ on $$X_{-1}$$. The problems (DE) and (VE) can then be studied in $$X_{-1}$$ for the operator $${\mathcal A}_{-1}$$. After this reduction to the generator case, (DE) and (VE) are further reduced to the homogeneous case by considering the operator $${\mathcal A}_{-1}$$ on the product space $$X_{-1} \times L^ 1(\mathbb{R}_ +;X_{-1})$$ defined by $D({\mathcal A}_{-1})=X_ 0 \times W^{1,1} (\mathbb{R}_ +;X_{-1}),\quad {\mathcal A}_{-1} {x \choose f}={A_{-1}x+f(0) \choose f'}.$ This operator is shown to generate a $$C_ 0$$-semigroup on $$X_{-1} \times L^ 1(\mathbb{R}_ +;X_{-1})$$, and solutions of the problem $U'(t)={\mathcal A}_{-1} U(t),\;t \geq 0,\;U(0)={x \choose f},$ correspond to solutions of (DE) in $$X_{-1}$$. By restriction to the invariant subspaces $$X_ 0 \times L^ 1(\mathbb{R}_ +;X)$$, a new proof of the Da Prato-Sinestrari theorem for the problem (DE) is obtained. By restriction to $$X_ 0 \times L^ 1(\mathbb{R}_ +;F_{-1})$$, where $$F_{- 1}$$ is the Favard class of $$\{T_{-1} (t)\}_{t \geq 0}$$, an analogue for the $$F_{-1}$$-valued case is obtained. In the final section, similar techniques are applied to prove analogous results for the problem (VE).
For the entire collection see [Zbl 0782.00051].

MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations 45D05 Volterra integral equations