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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Identification of unknown terms in convolution integro-differential equations in a Banach space

Alfredo Lorenzi1 / Gianluca Mola2

1Department of Mathematics “F. Enriques”, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy. E-mail:

2Department of Mathematics “F. Enriques”, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy. E-mail:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 18, Issue 3, Pages 321–355, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/jiip.2010.013, July 2010

Publication History

Received:
2007-04-12
Revised:
2007-07-13
Published Online:
2010-07-14

Abstract

We consider the following identification problems in a general Banach space X: find a function u : [0, T] → X and a vector zX such that the initial-value problems

and

are fulfilled, along with the nonlocal additional condition [0,T] u(t)(t) = φ ∈ X, for some probability Borel probability measure μ on the interval [0, T]. Here A : D(A) ⊂ XX is a (possibly unbounded) closed linear operator, h, k and ƒ are scalar functions and g is a X-valued source term. We recall that the same problem with h = k = 0 has been previously studied by Anikonov and Lorenzi in [J. Inverse Ill-posed Probl. 7: 669–681, 2007], Prilepko, Piskarev and Shaw in [J. Inverse Ill-Posed Probl. 15: 831–851, 2007], and subsequently generalized by Lorenzi and Vrabie in [Discr. Continuous Dynam. Syst., 2011]. Under suitable assumptions on the structural data of the problem, we prove local-in-time existence and uniqueness for the function u, and an explicit representation formula for z depending on u. Also, a continuous dependence of Lipschitz type of the solution (u, z) on the data is provided. Finally, two applications to parabolic integro-differential boundary value problems are considered.

Keywords.: Linear first-order integro-differential equations in Banach spaces; recovering an unknown vector in the source; analytic semigroup theory; existence; uniqueness and continuous dependence results; applications to linear integro-differential parabolic equations

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