Continuous dependence results for inhomogeneous ill-posed problems in Banach space |
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Authors: | Beth M Campbell Hetrick Rhonda J Hughes |
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Institution: | a Department of Mathematical Sciences, The Pennsylvania State University at Harrisburg, Middletown, PA 17057, USA b Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, USA |
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Abstract: | We apply semigroup theory and other operator-theoretic methods to prove Hölder-continuous dependence on modeling for the inhomogeneous ill-posed Cauchy problem in Banach space. The inhomogeneous ill-posed Cauchy problem is given by , u(0)=χ, 0?t<T; where −A is the infinitesimal generator of a holomorphic semigroup on a Banach space X, χ∈X, and . For a suitable function f, the approximate problem is given by , v(0)=χ. Under certain stabilizing conditions, we prove that for a related norm, where and M are computable constants independent of β, 0<β<1, and ω(t) is a harmonic function. These results extend earlier work of Ames and Hughes on the homogeneous ill-posed problem. |
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Keywords: | Abstract Cauchy problem Continuous dependence on modeling Ill-posed problems |
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