Continuous dependence of solutions to a linear boundary value problem on parameters. I (Q1390431)

Let \(B\) be a Banach space, \(D\simeq B\times \mathbb{R}^N\), \({\mathcal L}:D\to B\), \(l:B\to \mathbb{R}^N\) be linear bounded operators, \(f\in B\) and \(\alpha\in \mathbb{R}^N\). The authors investigate the dependence of solutions to the abstract functional-differential BVP \[ {\mathcal L}x=f,\quad lx=\alpha \tag{*}, \] on operators \({\mathcal L},l\) and elements \(f,\alpha\). In particular, if \(B_k\) is a sequence of Banach spaces, \(D_k\simeq B_k\times \mathbb{R}^N\), \({\mathcal L}_k:D_k\to B_k\), \(l_k:B_k\to \mathbb{R}^N\), \(f_k\in B_k\) \(\alpha_k\in \mathbb{R}^N\), the conditions on the operators \({\mathcal L}_k,l_k\) and the type of convergence \({\mathcal L}_k\to {\mathcal L}\), \(l_k\to l\) and \(f_k\to f\) are given which guarantee that the unique solvability of (*) implies the unique solvability (for large \(k\)) of \[ {\mathcal L}_kx_k=f_k,\quad l_kx_k=\alpha_k, \] and the convergence \(x_k\to x\), where \(x\) is the solution to (*). This abstract setting covers a large variety of special BVPs for functional-differential equations. For related material the reader consults the monograph of \textit{N. V. Azbelev, V. P. Maksimov} and \textit{L. F. Rakhmatullina} [Introduction to the theory of linear functional differential equations, Nauka, Moskow, 1991. Transl. from the Russian (1995; Zbl 0867.34051)].