zbMATH — the first resource for mathematics

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].

45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45D05 Volterra integral equations
PDF BibTeX Cite