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Published in:
2018 | OriginalPaper | Chapter
Regularization Results for Inhomogeneous Ill-Posed Problems in Banach Space
Author : Beth M. Campbell Hetrick
Published in: Advances in the Mathematical Sciences
Publisher: Springer International Publishing
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Abstract
We prove continuous dependence on modeling for the inhomogeneous ill-posed Cauchy problem in Banach space X, then use these results to obtain a regularization result. The particular problem we consider is given by \(\frac {\mathrm{d} u(t)}{\mathrm{d} t} = A u(t) + h(t), 0 \leq t <T, u(0) = \chi \), where − A generates a uniformly bounded holomorphic semigroup {e−zA|Re(z) ≥ 0} and h : [0, T) → X. In the approximate problem, the operator A is replaced by the operator fβ(A), β > 0, which approximates A as β goes to 0. We use a logarithmic approximation introduced by Boussetila and Rebbani. Our results extend earlier work of the author together with Fury and Huddell on the homogeneous ill-posed problem.