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Convergence for hyperbolic singular perturbation of integrodifferential equations. (English) Zbl 1188.45006
From the text: By virtue of an operator-theoretical approach, we deal with hyperbolic singular perturbation problems for integrodifferential equations. New convergence theorems for such singular perturbation problems are obtained, which generalize some previous results by H. O. Fattorini [J. Differ. Equ. 70, 1–41 (1987; Zbl 0633.35006)] and J. H. Liu [Electron. J. Differ. Equ. 1993, Art. ID 02, 10 pp. (1993; Zbl 0809.45008)].
In more detail, let \(A\) and \(B\) be linear unbounded operators in a Banach space \(X\), let \(K(t)\) be a linear bounded operator for each \(t\geq 0\) in \(X\), and let \(f (t;\varepsilon)\) and \(f (t)\) be \(X\)-valued functions.We study the convergence of derivatives of solutions of
\[ \varepsilon 2u''(t; \varepsilon)+u'(t; \varepsilon) = (\varepsilon^2A+B)u(t; \varepsilon)+\int^t_0K(t-s)(\varepsilon^2A+B)u(s; \varepsilon)\,ds+ f (t; \varepsilon),\quad t \geq 0, \]
\[ u(0; \varepsilon) = u_0(\varepsilon),\quad u'(0; \varepsilon) = u_1(\varepsilon), \]
to derivatives of solutions of \[ w'(t) = Bw(t)+\int^t_0K(t -s)Bw(s)\,ds+ f (t),\quad t \geq 0,\quad w(0) = w_0, \] as \(\varepsilon\to 0\).

MSC:
45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
34K30 Functional-differential equations in abstract spaces
34G20 Nonlinear differential equations in abstract spaces
34K26 Singular perturbations of functional-differential equations
47N20 Applications of operator theory to differential and integral equations
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References:
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