Continuous dependence results for inhomogeneous ill-posed problems in Banach space

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 illposed Cauchy problem is given by du dt = Au(t)+h(t), u(0) = χ, 0 t < T; where −A is the infinitesimal generator of a holomorphic semigroup on a Banach space X, χ ∈ X, and h: [0,T )→X. For a suitable function f , the approximate problem is given by dv dt = f (A)v(t) + h(t), v(0) = χ. Under certain stabilizing conditions, we prove that u(t) − v(t) Cβ1−ω(t)Mω(t) for a related norm, where C 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. © 2006 Elsevier Inc. All rights reserved