## 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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### References:

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