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Note on multiplicative perturbation of local $$C$$-regularized cosine functions with nondensely defined generators. (English) Zbl 1217.47083
Let $$X$$ be a Banach space, $$A$$ a linear operator in $$X$$. Then it is known that the cosine operator function is the main propagator of the Cauchy problem
$u''(t)=Au(t), \quad t\in (-\infty,\infty),\qquad u(0)=0,\quad u'(0)=u_1.$ If $$A$$ is the generator of a $$C$$-regularized cosine function $$\{C(t)\}_{t\in\mathbb R}$$, then $$u(t)=C^{-1}C(t)u_0+C^{-1}\int^t_0C(s)u_1\,ds$$ is the unique solution of the above Cauchy problem for every pair $$(u_0,u_1)$$ of initial values in $$C(D(A))$$, where $$C:=C(0)$$.
The authors study the multiplicative perturbation of local $$C$$-regularized cosine functions with nondensely defined generators, in the case when the range of the regularizing operator $$C$$ is not dense in $$X$$ and the operator $$C$$ may not commute with the perturbation operator.
The authors also present an application of the obtained abstract result to the Cauchy problem in $$C_0(\Omega)$$ for some integrodifferential equation, and prove that the problem has a unique solution in $$C^2([0,\tau];C_0(\Omega))$$, where $$\Omega$$ is a domain in $$\mathbb R^n$$ and $$C_0(\Omega):=\{f\in C(\Omega)$$: for each $$\varepsilon >0$$ there is a compact $$\Omega_{\varepsilon}\subset\Omega$$ such that $$|f(s)|<\varepsilon$$ for all $$s\in\Omega\setminus \Omega_{\varepsilon}\}$$, for every couple of initial values in a large subset of $$C_0(\Omega)$$.

##### MSC:
 47D09 Operator sine and cosine functions and higher-order Cauchy problems 34G10 Linear differential equations in abstract spaces 35G10 Initial value problems for linear higher-order PDEs
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