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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

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Kairi Kasemets and Jaan Janno
Abstract and Applied Analysis, 2013, Volume 2013, Page 1
[2]
Kairi Kasemets and Jaan Janno
Mathematical Modelling and Analysis, 2011, Volume 16, Number 2, Page 199
[3]
Yurii E. Anikonov and Shumin Li
Journal of Inverse and Ill-posed Problems, 2011, Volume 19, Number 1, Page 1

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