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A modified quasi-boundary value method for ill-posed problems. (English) Zbl 1084.34536

Consider the final value problem (FVP) \[ u'(t)+Au(t) =0, \text{ } 0\leq t\leq T, \text{ } u(T)=f, \] in a Hilbert space \(H\), where \(A\) is a selfadjoint operator on \(H\) with eigenvalues \(\lambda_i \to \infty\). It is well known that (FVP) is an ill-posed problem. In this interesting paper, the authors perturb the final condition and reduce (FVP) to the nonlocal problem \[ u'(t)+Au(t) =0, \text{ } 0\leq t\leq T, \text{ } u(T)-\alpha u'(0)=f.\eqno(1) \]
It is shown that (1) is well-posed for each \(\alpha >0\) and that their solutions \(u_{\alpha}\) converge in \(C^1([0,T], H)\) as \(\alpha \to 0\) if and only if (FVP) has a classical solution. The rates of convergence are also given.

MSC:

34G10 Linear differential equations in abstract spaces
47A52 Linear operators and ill-posed problems, regularization
47N20 Applications of operator theory to differential and integral equations
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