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Regularization of an ill-posed Cauchy problem for a second order equation in Banach space. (Russian) Zbl 0607.34002

In a Banach space E, the Cauchy (ill-posed) problem \[ d^ 2u/dt^ 2=Au,\quad 0\leq t\leq T,\quad u(0)=f_ 1,\quad u'(0)=f_ 2, \] where A satisfies \(\exists c>0\), \(\forall \lambda <0\), \(\exists R(A,\lambda)\), \(\| R(A,\lambda)\| \leq c/(1+| \lambda |),\overline{D(A)}=E\) is considered. By using the solution \(u_{\alpha}\) of the system \(d^ 2u_{\alpha}/dt^ 2=Au_{\alpha}\), \(0\leq t\leq T,\quad u_{\alpha}(0)+cu_{\alpha}(T)=f_ 1,\) \(u'(0)=f_ 2\), one can construct a family of operators \(R_{\alpha,t}(f_ 1,f_ 2)\) \((\alpha >0\), \(0<t<T)\), which is the regulator of the given problem.
Reviewer: A.Haimovici

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47E05 General theory of ordinary differential operators
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