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International Journal of Differential Equations
Volume 2011 (2011), Article ID 913125, 11 pages
Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator
Laboratoire Equations Diff####^~^~^~^~^~^####xe9;rentielles, D####^~^~^~^~^~^####xe9;partement de Math####^~^~^~^~^~^####xe9;matiques, Facult####^~^~^~^~^~^####xe9; des Sciences Exactes, Universit####^~^~^~^~^~^####xe9; Mentouri Constantine, Constantine 25000, Algeria
Received 29 May 2011; Revised 17 August 2011; Accepted 26 September 2011
Academic Editor: Alberto Cabada
Copyright © 2011 Salah Djezzar and Nihed Teniou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space , where the coefficient operator in the equation is an unbounded self-adjoint positive operator which has a continuous spectrum and the data is given at the final time and a solution for is sought. It is well known that this problem is illposed in the sense that the solution (if it exists) does not depend continuously on the given data. The method of regularization used here consists of perturbing both the equation and the final condition to obtain an approximate nonlocal problem depending on two small parameters. We give some estimates for the solution of the regularized problem, and we also show that the modified problem is stable and its solution is an approximation of the exact solution of the original problem. Finally, some other convergence results including some explicit convergence rates are also provided.
Let be a positive (we suppose that ), self-adjoint unbounded linear operator which has a continuous spectrum on a Hilbert space such that generates a contraction -semigroup on . Let be a positive real number. We consider the final value problem (FVP) of finding such that for some prescribed final value in . Such problems are not well posed; that is, even if a unique solution exists on , it need not depend continuously on the finial value .
This type of problems, in the case where has a discrete spectrum, has been considered by many authors using different approaches. Such authors as Latt####^~^~^~^~^~^####xe8;s and Lions , Lavrentiev , Miller , Payne , and Showalter  have approximated the final value problem (FVP) (1.1), (1.2) by perturbing the operator .
A similar approach known as the method of auxiliary boundary conditions was given in [9####^~^~^~^~^~^####x2013;11]. Also, the nonstandard conditions of the form (1.3) for parabolic equations have been considered in some recent papers [12, 13]. For further results related to these type of problems, we can also see [14, 15]. It is also worth reading the recent paper by Campbell Hetrick and Hunhes  dealing with inhomogeneous ill-posed problems in Banach space. We also mention the very recent papers by Tuan  and Tuan et al.  which deal with similar ill-posed problems using different approaches. For some comments on the results presented in paper  using a different regularization approach (the truncation regularization method), see Remark 3.6 at the end of this paper.
In this paper, we perturb both (1.1) and the final condition (1.2) to form an approximate nonlocal problem depending on two small parameters and , with boundary condition containing a derivative of the same order than the equation as follows: where the operator is replaced by the operator and by , and , where , .
We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also show that this method gives a better approximation than many other quasireversibility and quasiboundary type methods, for example, [1, 6, 7, 19####^~^~^~^~^~^####x2013;21]. Finally, we obtain several other results, including some explicit convergence rates.
Throughout this paper, we will denote by a Hilbert space, the resolution of the identity associated with the positive unbounded self-adjoint operator . So the spectral representation of semigroup (resp., ) is given by (resp., , and so for all , and this is characterized by Also, throughout this work, we mean by a solution of problem (1.1), (1.2) on the interval a function such that, for all and (1.1), (1.2) hold. A useful characterization of the admissible set for which problem (1.1), (1.2) has a solution is as follows: Problem (1.1), (1.2) has a solution if and only if , and this unique solution is represented by (see , Lemma 1]).
2. The Approximate Problem
Definition 2.1. Define the function for and , where is the semigroup generated by .
Now, we give the following theorem where the proof is based on the semigroups theory .
Theorem 2.2. The function is the unique solution of the perturbed problem (2.1), and it depends continuously on .
Proof. We consider the following classical Cauchy problem: It is clear that is the unique solution and The continuous dependence of on is obtained by showing that
Now, we consider the following problem:
Theorem 2.3. The problem (2.6) is well posed, and its solution is given by furthermore,
Proof. Since then,
Now we give some convergence results.
3. The Convergence Results
Theorem 3.1. For all , as .
Proof. We have
and if we put
then we get
It is clear that, for all , there exists and .
If we put then we have Finally, choosing such that , we obtain the following estimate which gives the desired result.
Let us denote by , the following set:
It is clear that the following proprieties hold Now, we give some convergence results with explicit convergence rates.
Theorem 3.2. If , then one has for and such that
Proof. Using the proof of the previous theorem, we have where If , then we have And, for , we have We also have Then, using the above estimates, we get the desired results.
Now, let be the function defined by
Proof. We assume that and . Let . So, we have
Since and and so by the unicity of the limit, we obtain that is a solution to problem (1.1), (1.2).
Now, we suppose that is a solution to problem (1.1), (1.2). Since (see , Lemma 1), then we have Let and , such that . If and . Then, If we put then we have Using analogous calculations as in the proof of Theorem 3.1, we obtain Now, if , then by choosing such that we find that . This means that the function is continuous at .
We note that we can easily show that for . And by using Theorem 3.1 again, we see that
Theorem 3.4. If , then one has for . And such that
Proof. Since then using Theorem 3.2, we get the required result.
Corollary 3.5. If , then one has for . And
Remark 3.6. We note that in a very recent paper by Tuan  a new use of a different regularization method (the truncation method) is introduced for dealing with a similar class of problems. This truncation method consists in eliminating all high frequencies from the solution of the considered ill-posed problem to get an approximate regularized solution together with some stability and error estimates that he indicates to be of Holder type. In particular, the author gives some estimates which hold at and so he gets the convergence of the approximate solution at . For a significant comparison with these results obtained by this truncation regularization method, one needs the determination and selection, for each case, of a possible appropriate regularization parameter . However, the method of regularization presented in our work still gives a better approximation than many other quasireversibility and quasi-boundary type methods, for example, [1, 6, 7, 19####^~^~^~^~^~^####x2013;21].
Conclusion 1. (1.1) Note that, in this work, the error factor introduced by small changes in the final value is of order and in a recent work , by the same authors, the error factor given was of order .
(1.2) We also note that the error factor given in  (resp. in ) is of order (resp., ) and so and .
(1.3) Also the error factor given in Boussetila and Rebbani  is of order and since , for and so for all operators , considered above, our method of approximation gives a better approximation than the methods given by Boussetila and Rebbani in  and other authors, for example, [1, 7].
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