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International Journal of Differential Equations
VolumeΒ 2011Β (2011), Article IDΒ 913125, 11 pages
http://dx.doi.org/10.1155/2011/913125
Research Article

Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator

Laboratoire Equations DiffΓ©rentielles, DΓ©partement de MathΓ©matiques, FacultΓ© des Sciences Exactes, UniversitΓ© Mentouri Constantine, Constantine 25000, Algeria

Received 29 May 2011; Revised 17 August 2011; Accepted 26 September 2011

Academic Editor: AlbertoΒ Cabada

Copyright Β© 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.

Abstract

We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space 𝐻, 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 𝑑=𝑇 and a solution for 0≀𝑑<𝑇 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.

1. Introduction

Let 𝐴 be a positive (we suppose that 𝐴β‰₯πœ‚>0), self-adjoint unbounded linear operator which has a continuous spectrum on a Hilbert space 𝐻 such that βˆ’π΄ generates a contraction 𝐢0-semigroup on 𝐻. Let 𝑇 be a positive real number. We consider the final value problem (FVP) of finding π‘’βˆΆ[0,𝑇]→𝐻 such thatπ‘’ξ…ž(𝑑)+𝐴𝑒(𝑑)=0,0≀𝑑<𝑇,(1.1)𝑒(𝑇)=πœ™,(1.2) for some prescribed final value πœ™ in 𝐻. Such problems are not well posed; that is, even if a unique solution exists on [0,𝑇], it need not depend continuously on the finial value πœ™.

This type of problems, in the case where 𝐴 has a discrete spectrum, has been considered by many authors using different approaches. Such authors as Lattès and Lions [1], Lavrentiev [2], Miller [3], Payne [4], and Showalter [5] have approximated the final value problem (FVP) (1.1), (1.2) by perturbing the operator 𝐴.

A similar problem is treated in a different way; see [6–8]. By perturbing the final value condition, they approximated the problem (1.1), (1.2) withπ‘’ξ…ž(𝑑)+𝐴𝑒(𝑑)=0,0≀𝑑<𝑇.𝛼𝑒(0)+𝑒(𝑇)=πœ™.(1.3)

A similar approach known as the method of auxiliary boundary conditions was given in [9–11]. 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 [16] dealing with inhomogeneous ill-posed problems in Banach space. We also mention the very recent papers by Tuan [17] and Tuan et al. [18] which deal with similar ill-posed problems using different approaches. For some comments on the results presented in paper [17] 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 𝛼 and 𝛽, with boundary condition containing a derivative of the same order than the equation as follows:π‘£ξ…žπœŽ(𝑑)+π΄π›Όπ‘£πœŽπ‘£(𝑑)=0,0≀𝑑<𝑇,πœŽξ€·π‘£(𝑇)+π›½πœŽ(0)βˆ’π‘£ξ…žπœŽξ€Έ(0)=πœ™,(1.4) where the operator 𝐴 is replaced by the operator 𝐴𝛼=𝐴(𝐼+𝛼𝐴)βˆ’1 and 𝑒(𝑇)=πœ™ by π‘’πœŽ(𝑇)+𝛽(π‘’πœŽ(0)βˆ’π‘’ξ…žπœŽ(0))=πœ™, and 𝜎=(𝛼,𝛽), where 𝛼>0, 𝛽>0.

We show that the approximate problems are well posed and that their solutions π‘£πœŽ 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, 19–21]. Finally, we obtain several other results, including some explicit convergence rates.

Throughout this paper, we will denote by 𝐻 a Hilbert space, {πΈπœ†,πœ†β‰₯πœ‚>0} the resolution of the identity associated with the positive unbounded self-adjoint operator 𝐴. So the spectral representation of 𝐢0 semigroup 𝑆(𝑑)=π‘’βˆ’π‘‘π΄ (resp., 𝐴) is given by 𝑆(𝑑)=π‘’βˆ’π‘‘π΄=βˆ«πœ‚+βˆžπ‘’βˆ’π‘‘πœ†π‘‘πΈπœ† (resp., ∫𝐴=πœ‚+βˆžπœ†π‘‘πΈπœ†), and so for all βˆ«π‘’βˆˆπ·(𝐴),𝐴𝑒=πœ‚+βˆžπœ†π‘‘πΈπœ†π‘’, and this is characterized by π‘’βˆˆπ·(𝐴)iff‖𝐴𝑒‖2=ξ€œπœ‚+βˆžπœ†2π‘‘β€–β€–πΈπœ†π‘’β€–β€–2<∞.(1.5) Also, throughout this work, we mean by a solution of problem (1.1), (1.2) on the interval [0,𝑇] a function π‘’βˆˆπΆ([0,𝑇];𝐻)∩𝐢1(]0,𝑇[;𝐻) such that, for all π‘‘βˆˆ]0,𝑇[,𝑒(𝑑)∈𝐷(𝐴) 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 𝑒 if and only if βˆ«πœ‚+βˆžπ‘’2πœ†π‘‡π‘‘β€–πΈπœ†πœ™β€–2<∞, and this unique solution is represented by βˆ«π‘’(𝑑)=πœ‚+βˆžπ‘’(π‘‡βˆ’π‘‘)πœ†π‘‘πΈπœ†πœ™<∞ (see [19], Lemma 1]).

2. The Approximate Problem

We approximate the final value problem (1.1), (1.2), by the following perturbed problem: π‘£ξ…žπœŽ(𝑑)+π΄π›Όπ‘£πœŽπ‘£(𝑑)=0,0⩽𝑑<𝑇,πœŽξ€·π‘£(𝑇)+π›½πœŽ(0)βˆ’π‘£ξ…žπœŽξ€Έ(0)=πœ™,(2.1) where 𝐴 is as above and 𝐴𝛼 is the Yosida approximation of operator 𝐴, and 𝜎=(𝛼,𝛽) where 𝛼>0,𝛽>0.

Definition 2.1. Define the function π‘£πœŽ(𝑑)=𝑆𝛼𝛽(𝑑)𝐼+𝐴𝛼+𝑆𝛼(𝑇)βˆ’1πœ™,(2.2) for πœ™βˆˆπ»,𝜎=(𝛼,𝛽) and 𝛼>0,𝛽>0,π‘‘βˆˆ[0,𝑇], where 𝑆𝛼(𝑑) is the semigroup generated by βˆ’π΄π›Ό.

Now, we give the following theorem where the proof is based on the semigroups theory [22].

Theorem 2.2. The function π‘£πœŽ(𝑑) is the unique solution of the perturbed problem (2.1), and it depends continuously on πœ™.

Proof. We consider the following classical Cauchy problem: π‘£ξ…žπœŽ(𝑑)+π΄π›Όπ‘£πœŽπ‘£(𝑑)=0,0<𝑑<𝑇,πœŽξ€·π›½ξ€·(0)=𝐼+𝐴𝛼+𝑆𝛼(𝑇)βˆ’1πœ™.(2.3) It is clear that π‘£πœŽ(𝑑) is the unique solution and π‘£πœŽξ€·π‘£(𝑇)+π›½πœŽ(0)βˆ’π‘£ξ…žπœŽξ€Έ(0)=𝑆𝛼𝛽(𝑇)𝐼+𝐴𝛼+𝑆𝛼(𝑇)βˆ’1πœ™ξ€·+𝛽𝐼+𝐴𝛼𝛽𝐼+𝐴𝛼+𝑆𝛼(𝑇)βˆ’1πœ™=𝛽𝐼+𝐴𝛼+𝑆𝛼𝛽(𝑇)𝐼+𝐴𝛼+𝑆𝛼(𝑇)βˆ’1πœ™=πœ™.(2.4) The continuous dependence of π‘£πœŽ on πœ™ is obtained by showing that β€–β€–π‘£πœŽβ€–β€–=‖‖𝑆(𝑑)𝛼𝛽(𝑑)𝐼+𝐴𝛼+𝑆𝛼(𝑇)βˆ’1πœ™β€–β€–β©½β€–β€–ξ€·π›½ξ€·πΌ+𝐴𝛼+𝑆𝛼(𝑇)βˆ’1πœ™β€–β€–β©½1𝛽+(π›½πœ‚/(1+π›Όπœ‚))+π‘’βˆ’π‘‡/π›Όβ€–πœ™β€–.(2.5)

Now, we consider the following problem:π‘’ξ…žπœŽ(𝑑)+π΄π‘’πœŽπ‘’(𝑑)=0,𝜎(0)=πœ™πœŽ=𝛽𝐼+𝐴𝛼+𝑆𝛼(𝑇)βˆ’1πœ™.(2.6)

Theorem 2.3. The problem (2.6) is well posed, and its solution is given by π‘’πœŽξ€·π›½ξ€·(𝑑)=𝑆(𝑑)𝐼+𝐴𝛼+𝑆𝛼(𝑇)βˆ’1πœ™,(2.7) furthermore, β€–β€–π‘’πœŽβ€–β€–β©½1(𝑑)𝛽(1+(πœ‚/(1+π›Όπœ‚)))+π‘’βˆ’π‘‡/𝛼(π‘‡βˆ’π‘‘)/π‘‡β€–πœ™β€–.(2.8)

Proof. Since β€–β€–π‘’πœŽβ€–β€–(𝑑)2=ξ€œπœ‚+βˆžπ‘’βˆ’2πœ†π‘‡ξ€·π›½(1+(πœ†/(1+π›Όπœ†)))+π‘’βˆ’(πœ†π‘‡)/(1+π›Όπœ†)ξ€Έ2π‘‘β€–β€–πΈπœ†πœ™β€–β€–2β‰€ξ€œπœ‚+βˆžξƒ¬1𝛽(1+(πœ‚/(1+π›Όπœ‚)))+π‘’βˆ’π‘‡/𝛼2ξƒ­π‘‡βˆ’π‘‘/π‘‡π‘‘β€–β€–πΈπœ†πœ™β€–β€–2,(2.9) then, β€–β€–π‘’πœŽβ€–β€–β©½ξƒ¬1(𝑑)𝛽(1+(πœ‚/(1+π›Όπœ‚)))+π‘’βˆ’π‘‡/π›Όξ€Έξƒ­π‘‡βˆ’π‘‘/π‘‡β€–πœ™β€–.(2.10)

Now we give some convergence results.

3. The Convergence Results

Theorem 3.1. For all πœ™βˆˆπ»,β€–π‘’πœŽ(𝑇)βˆ’πœ™β€–β†’0, as |𝜎|β†’0.

Proof. We have β€–β€–π‘’πœŽβ€–β€–(𝑇)βˆ’πœ™2=ξ€œπœ‚+βˆžξ‚΅π‘’βˆ’πœ†π‘‡π›½(1+(πœ†/(1+π›Όπœ†)))+π‘’βˆ’(πœ†π‘‡)/(1+π›Όπœ†)ξ‚Άβˆ’12π‘‘β€–β€–πΈπœ†πœ™β€–β€–2=ξ€œπœ‚+βˆžξ‚΅π›½(1+(πœ†/(1+π›Όπœ†)))𝛽(1+(πœ†/(1+π›Όπœ†)))+π‘’βˆ’(πœ†π‘‡/(1+π›Όπœ†))+π‘’βˆ’(πœ†π‘‡/(1+π›Όπœ†))βˆ’π‘’βˆ’πœ†π‘‡π›½(1+(πœ†/(1+π›Όπœ†)))+π‘’βˆ’(πœ†π‘‡/(1+π›Όπœ†))ξ‚Ά2π‘‘β€–β€–πΈπœ†πœ™β€–β€–2.(3.1) Putting π‘€πœŽ(πœ†)=𝛽(1+(πœ†/(1+π›Όπœ†)))𝛽(1+(πœ†/(1+π›Όπœ†)))+π‘’βˆ’(πœ†π‘‡/(1+π›Όπœ†)),π‘πœŽπ‘’(πœ†)=βˆ’(πœ†π‘‡/(1+π›Όπœ†))βˆ’π‘’βˆ’πœ†π‘‡π›½(1+(πœ†/(1+π›Όπœ†)))+π‘’βˆ’(πœ†π‘‡/(1+π›Όπœ†)),(3.2) and if we put π‘ƒπœŽ(πœ†)=π‘€πœŽ(πœ†)+π‘πœŽ(πœ†),(3.3) then we get β€–β€–π‘’πœŽβ€–β€–(𝑇)βˆ’πœ™2=ξ€œπœ‚+βˆžξ€·π‘ƒπœŽξ€Έ(πœ†)2π‘‘β€–β€–πΈπœ†πœ™β€–β€–2ξ€œβ©½2πœ‚+βˆžξ€·π‘€2𝜎(πœ†)+𝑁2πœŽξ€Έπ‘‘β€–β€–πΈ(πœ†)πœ†πœ™β€–β€–2.(3.4) It is clear that, for all πœ€>0, there exists π‘˜βˆˆβ„•βˆ—/βˆ«π‘˜+βˆžπ‘‘β€–πΈπœ†πœ™β€–2<πœ€/8 and π‘€πœŽ(πœ†)≀1,π‘πœŽ(πœ†)β©½1.
If we put 𝐼𝜎=ξ€œπ‘˜πœ‚π‘€2πœŽβ€–β€–πΈ(πœ†)π‘‘πœ†πœ™β€–β€–2+ξ€œπ‘˜+βˆžπ‘€2πœŽβ€–β€–πΈ(πœ†)π‘‘πœ†πœ™β€–β€–2,𝐽𝜎=ξ€œπ‘˜πœ‚π‘2πœŽβ€–β€–πΈ(πœ†)π‘‘πœ†πœ™β€–β€–2+ξ€œπ‘˜+βˆžπ‘2πœŽβ€–β€–πΈ(πœ†)π‘‘πœ†πœ™β€–β€–2.(3.5) then we have πΌπœŽβ©½πœ€8+𝛽2ξ‚€11+𝛼2𝑒2π‘˜π‘‡β€–πœ™β€–2,π½πœŽβ©½πœ€8+𝛼2𝑇2π‘˜4β€–πœ™β€–2.(3.6) Finally, choosing 𝜎 such that |𝜎|2=𝛼2+𝛽2β©½(1/β€–πœ™β€–2)(1/𝑇2π‘˜4+1/𝑒2π‘˜π‘‡)(πœ€/8), we obtain the following estimate β€–β€–π‘’πœŽβ€–β€–(𝑇)βˆ’πœ™2β©½ξ‚΅πœ–2𝛽+22ξ‚€11+𝛼2𝑒2π‘˜π‘‡+𝛼2𝑇2π‘˜4ξ‚Άξ‚Άβ€–πœ™β€–2,(3.7) which gives the desired result.

Let us denote by πΆπœƒ(𝐴), πœƒβ‰₯0 the following set:πΆπœƒξ‚»(𝐴)=β„Žβˆˆπ»βˆΆβ€–β„Žβ€–2πΆπœƒ=ξ€œπœ‚+βˆžπ‘’2π‘‡πœƒπœ†π‘‘β€–β€–πΈπœ†πœ™β€–β€–2ξ‚Ό.<+∞(3.8)

It is clear that the following proprieties holdπΆπœƒ1(𝐴)βŠ†πΆπœƒ2(𝐴),πœƒ2β©Ύπœƒ1,πΆπœƒ(𝐴)βŠ‚π»,πœƒ>0.(3.9) Now, we give some convergence results with explicit convergence rates.

Theorem 3.2. If πœ™βˆˆπΆπœƒ(𝐴), then one has β€–β€–π‘’πœŽβ€–β€–(𝑇)βˆ’πœ™2𝐢⩽221𝛽(πœƒ)2πœƒπ›Ό2ξ‚΅πœ‚ξ‚Ά1+π›Όπœ‚2(πœƒβˆ’1)+𝐢22(πœƒ,𝑇)𝛼2ξƒͺβ€–πœ™β€–2πΆπœƒ,(3.10) for 0<πœƒ<1 and β€–β€–π‘’πœŽβ€–β€–(𝑇)βˆ’πœ™2𝛽⩽22𝛼2+𝐢22(πœƒ,𝑇)𝛼2ξ‚Άβ€–πœ™β€–2πΆπœƒ,forπœƒβ©Ύ1,(3.11) such that 𝐢1(πœƒ)=(1βˆ’πœƒ)1βˆ’πœƒπœƒπœƒβ©½1,𝐢21(πœƒ,𝑇)=π‘‡πœƒ2.(3.12)

Proof. Using the proof of the previous theorem, we have β€–β€–π‘’πœŽβ€–β€–(𝑇)βˆ’πœ™2=ξ€œπœ‚+βˆžπ‘ƒ2πœŽβ€–β€–πΈ(πœ†)π‘‘πœ†πœ™β€–β€–2=ξ€œπœ‚+βˆžπ‘ƒ2𝜎(πœ†)π‘’βˆ’2π‘‡πœƒπœ†π‘’2π‘‡πœƒπœ†π‘‘β€–β€–πΈπœ†πœ™β€–β€–2𝑀⩽22𝜎,πœƒ,∞+𝑁2𝜎,πœƒ,βˆžξ€Έβ€–πœ‘β€–2πΆπœƒ,(3.13) where π‘€πœŽ,πœƒ,∞=supπœ†β©Ύπœ‚π‘€πœŽπ‘’βˆ’π‘‡πœƒπœ†,π‘πœŽ,πœƒ,∞=supπœ†β©Ύπœ‚π‘πœŽπ‘’βˆ’π‘‡πœƒπœ†.(3.14) If 0<πœƒ<1, then we have π‘€πœŽ,πœƒ,βˆžβ‰€2π›Όπ›½πœƒπΆ1(πœƒ).(3.15) And, for πœƒβ©Ύ1, we have π‘€πœŽ,πœƒ,βˆžπ›½β‰€2𝛼.(3.16) We also have π‘πœŽ,πœƒ,∞⩽4π›Όπ‘‡πœƒ2𝑒2β©½π›Όπ‘‡πœƒ2,βˆ€πœƒ>0.(3.17) Then, using the above estimates, we get the desired results.

Now, let 𝐹 be the function defined byπΉβˆΆβ„+×ℝ+ξ‚»π‘’βŸΆπ»,𝜎=(𝛼,𝛽)⟢𝐹(𝜎)=𝜎(0)=πœ™πœŽ,πœŽβ‰ (0,0),𝑒(0)=πœ™0,𝜎=(0,0).(3.18)

Theorem 3.3. For all πœ™βˆˆπ», the problem (1.1), (1.2) has a solution 𝑒(𝑑) if and only if the function 𝐹 is continious at (0,0). Furthermore, π‘’πœŽ(𝑑)→𝑒(𝑑),π‘Žπ‘ |𝜎|β†’0, uniformly in 𝑑.

Proof. We assume that lim|𝜎|β†’0πœ™πœŽ=πœ™0 and β€–πœ™0β€–<∞. Let 𝑀(𝑑)=𝑆(𝑑)πœ™0. So, we have ‖‖𝑀(𝑑)βˆ’π‘’πœŽβ€–β€–β©½β€–β€–πœ™(𝑑)0βˆ’πœ™πœŽβ€–β€–.(3.19) Hence, sup0⩽𝑑⩽𝑇‖‖𝑀(𝑑)βˆ’π‘’πœŽβ€–β€–β©½β€–β€–πœ™(𝑑)0βˆ’πœ™πœŽβ€–β€–βŸΆ|𝜎|β†’00.(3.20) Since lim|𝜎|β†’0π‘’πœŽ(𝑇)=πœ™ and lim|𝜎|β†’0π‘’πœŽ(𝑇)=𝑀(𝑇) and so by the unicity of the limit, we obtain that 𝑀(𝑑) is a solution to problem (1.1), (1.2).
Now, we suppose that βˆ«π‘’(𝑑)=πœ‚+βˆžπ‘’(π‘‡βˆ’π‘‘)πœ†π‘‘πΈπœ†πœ™ is a solution to problem (1.1), (1.2). Since 𝑒(0)=𝑆(βˆ’π‘‡)πœ™βˆˆπ» (see [19], Lemma 1), then we have ‖‖𝑒(0)2=β€–πœ™β€–2𝐢1=ξ€œπœ‚+βˆžπ‘’2π‘‡πœ†π‘‘β€–β€–πΈπœ†πœ™β€–β€–2<∞.(3.21) Let π‘˜>0 and πœ–>0, such that βˆ«π‘˜+βˆžπ‘’2π‘‡πœ†π‘‘β€–πΈπœ†πœ™β€–2<πœ–/8. If 𝜎1=(𝛼1,𝛽1) and 𝜎2=(𝛼2,𝛽2). Then, β€–β€–π‘’πœŽ1(0)βˆ’π‘’πœŽ2β€–β€–(0)2=ξ€œπœ‚+βˆžξƒ©1𝛽1ξ€·ξ€·ξ€·1+πœ†/1+𝛼1πœ†ξ€Έξ€Έξ€Έ+π‘’βˆ’(πœ†π‘‡/(1+𝛼1πœ†))βˆ’1𝛽2ξ€·ξ€·ξ€·1+πœ†/1+𝛼2πœ†ξ€Έξ€Έξ€Έ+π‘’βˆ’(πœ†π‘‡/1+𝛼2πœ†))ξƒͺ2π‘‘β€–β€–πΈπœ†πœ™β€–β€–2ξ€œβ‰€2πœ‚+βˆžξƒ¬π›½1ξ€·ξ€·ξ€·1+πœ†/1+𝛼2πœ†ξ€Έξ€Έξ€Έβˆ’π›½2ξ€·ξ€·ξ€·1+πœ†/1+𝛼1πœ†ξ€Έξ€Έξ€Έξ€·π›½1ξ€·ξ€·ξ€·1+πœ†/1+𝛼1πœ†ξ€Έξ€Έξ€Έ+π‘’βˆ’(πœ†π‘‡/(1+𝛼1πœ†))𝛽2ξ€·1+πœ†/1+𝛼2πœ†ξ€Έ+π‘’βˆ’(πœ†π‘‡/(1+𝛼2πœ†))ξ€Έξƒ­2π‘‘β€–β€–πΈπœ†πœ™β€–β€–2ξ€œ+2πœ‚+βˆžξƒ¬π‘’βˆ’πœ†π‘‡/1+𝛼1πœ†βˆ’π‘’βˆ’πœ†π‘‡/1+𝛼2πœ†ξ€·π›½1ξ€·ξ€·1+πœ†/1+𝛼1πœ†ξ€Έξ€Έ+π‘’βˆ’(πœ†π‘‡/(1+𝛼1πœ†))𝛽2ξ€·ξ€·1+πœ†/1+𝛼2πœ†ξ€Έξ€Έ+π‘’βˆ’(πœ†π‘‡/(1+𝛼2πœ†))ξ€Έξƒ­2π‘‘β€–β€–πΈπœ†πœ™β€–β€–2.(3.22) If we put π‘€πœŽ1,𝜎2=||𝛽1ξ€·ξ€·ξ€·1+πœ†/1+𝛼2πœ†ξ€Έξ€Έξ€Έβˆ’π›½2ξ€·ξ€·ξ€·1+πœ†/1+𝛼1πœ†||𝛽1ξ€·ξ€·1+πœ†/1+𝛼1πœ†ξ€Έξ€Έ+π‘’βˆ’πœ†π‘‡/1+𝛼1πœ†π›½ξ€Έξ€·2ξ€·ξ€·ξ€·1+πœ†/1+𝛼2πœ†ξ€Έξ€Έξ€Έ+π‘’βˆ’πœ†π‘‡/1+𝛼2πœ†ξ€Έ,π‘πœŽ1,𝜎2=||π‘’βˆ’(πœ†π‘‡/(1+𝛼1πœ†))βˆ’π‘’βˆ’(πœ†π‘‡/(1+𝛼2πœ†))||𝛽1ξ€·ξ€·1+πœ†/1+𝛼1πœ†ξ€Έξ€Έ+π‘’βˆ’(πœ†π‘‡/(1+𝛼1πœ†))𝛽2ξ€·ξ€·ξ€·1+πœ†/1+𝛼2πœ†ξ€Έξ€Έξ€Έ+π‘’βˆ’(πœ†π‘‡/(1+𝛼2πœ†))ξ€Έ,(3.23) then we have β€–β€–π‘’πœŽ1(0)βˆ’π‘’πœŽ2β€–β€–(0)2ξ€œβ©½2πœ‚+βˆžπ‘€2𝜎1,𝜎2‖‖𝐸(πœ†)π‘‘πœ†πœ™β€–β€–2ξ€œ+2πœ‚+βˆžπ‘2𝜎1,𝜎2‖‖𝐸(πœ†)π‘‘πœ†πœ™β€–β€–2.(3.24) Using analogous calculations as in the proof of Theorem 3.1, we obtain β€–β€–π‘’πœŽ1(0)βˆ’π‘’πœŽ2β€–β€–(0)2β©½πœ–2+2𝑒2π‘˜π‘‡β€–πœ™β€–2𝐢1×𝛽2ξ‚΅π‘˜1+1+𝛼2π‘˜ξ‚Άβˆ’π›½1ξ‚΅π‘˜1+1+𝛼1π‘˜ξ‚Άξ‚Ή2+𝑇2π‘˜4𝛼1βˆ’π›Ό2ξ€Έ2ξ‚Ή.(3.25) Now, if 𝜎0=(0,0), then by choosing 𝜎=(𝛼,𝛽) such that |𝜎|2=𝛼2+𝛽2β©½1β€–πœ™β€–2𝐢11𝑒2π‘˜π‘‡ξ‚΅11+𝑇2π‘˜4ξ‚Άπœ–8,(3.26) we find that β€–π‘’πœŽ(0)βˆ’π‘’πœŽ0(0)β€–2=β€–πœ™πœŽβˆ’πœ™0β€–2β‰€πœ–. This means that the function 𝐹 is continuous at (0,0).

We note that we can easily show thatβ€–β€–π‘’πœŽβ€–β€–(0)βˆ’π‘’(0)2=β€–β€–π‘’πœŽβ€–β€–(𝑇)βˆ’πœ™2𝐢1,(3.27) for πœ™βˆˆπΆ1(𝐴). And by using Theorem 3.1 again, we see thatβ€–β€–π‘’πœŽβ€–β€–(0)βˆ’π‘’(0)2⟢0as|𝜎|⟢0.(3.28)

Theorem 3.4. If πœ™βˆˆπΆπœƒ+1(𝐴), then one has β€–β€–π‘’πœŽβ€–β€–(0)βˆ’π‘’(0)2ξ‚€4β©½2𝛼2𝐢21(πœƒ)𝛽2πœƒ+𝐢22(πœƒ,𝑇)𝛼2ξ‚β€–πœ™β€–2πΆπœƒ+1,(3.29) for 0<πœƒ<1. And β€–β€–π‘’πœŽβ€–β€–(0)βˆ’π‘’(0)2ξ‚΅β©½24𝛽2𝛼2+𝐢22(πœƒ,𝑇)𝛼2ξ‚Άβ€–πœ™β€–2πΆπœƒ+1,forπœƒβ©Ύ1(3.30) such that 𝐢1(πœƒ)=(1βˆ’πœƒ)1βˆ’πœƒπœƒπœƒβ‰€1,𝐢21(πœƒ,𝑇)=π‘‡πœƒ2.(3.31)

Proof. Since β€–β€–π‘’πœŽβ€–β€–(0)βˆ’π‘’(0)2=β€–β€–π‘’πœŽβ€–β€–(𝑇)βˆ’πœ™2𝐢1=ξ€œπœ‚+βˆžπ‘ƒ2𝜎(πœ†)𝑒2(1+πœƒ)π‘‡πœ†π‘’βˆ’2π‘‡πœ†πœƒπ‘‘β€–β€–πΈπœ†πœ™β€–β€–2,𝑀⩽22𝜎,πœƒ,∞+𝑁2𝜎,πœƒ,βˆžξ€Έβ€–πœ™β€–2πΆπœƒ+1,(3.32) then using Theorem 3.2, we get the required result.

Corollary 3.5. If πœ™βˆˆπΆπœƒ+1(𝐴), then one has sup0β©½π‘‘β©½π‘‡β€–β€–π‘’πœŽβ€–β€–(𝑑)βˆ’π‘’(𝑑)2β©½β€–β€–π‘’πœŽβ€–β€–(0)βˆ’π‘’(0)2ξ‚€4β©½2𝛼2𝐢21(πœƒ)𝛽2πœƒ+𝐢22(πœƒ,𝑇)𝛼2ξ‚β€–πœ™β€–2πΆπœƒ+1,(3.33) for 0<πœƒ<1. And sup0β©½π‘‘β©½π‘‡β€–β€–π‘’πœŽβ€–β€–(𝑑)βˆ’π‘’(𝑑)2β©½β€–β€–π‘’πœŽβ€–β€–(0)βˆ’π‘’(0)2ξ‚΅β©½24𝛽2𝛼2+𝐢22(πœƒ,𝑇)𝛼2ξ‚Άβ€–πœ™β€–2πΆπœƒ+1,forπœƒβ©Ύ1.(3.34)

Proof. Using Theorem 3.3, we have sup0β©½π‘‘β©½π‘‡β€–β€–π‘’πœŽβ€–β€–(𝑑)βˆ’π‘’(𝑑)2β©½β€–β€–π‘’πœŽβ€–β€–(0)βˆ’π‘’(0)2.(3.35) And, by Theorem 3.4, we obtain the desired result.

Remark 3.6. We note that in a very recent paper by Tuan [17] 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 𝑑=0 and so he gets the convergence of the approximate solution at 𝑑=0. 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 𝛽(πœ–). 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, 19–21].

Conclusion 1. (1.1) Note that, in this work, the error factor πœ€(𝜎) introduced by small changes in the final value is of order 1/(𝛽(1+(πœ‚/(1+π›Όπœ‚)))+π‘’βˆ’π‘‡/𝛼) and in a recent work [21], by the same authors, the error factor given was of order 1/(π›½πœ‚/(1+π›Όπœ‚)+π‘’βˆ’π‘‡/𝛼)β‰₯1/(𝛽(1+(πœ‚/(1+π›Όπœ‚)))+π‘’βˆ’π‘‡/𝛼).
(1.2) We also note that the error factor 𝑒(𝛽) given in [7] (resp. 𝑒(𝛼) in [1]) is of order 1/𝛽 (resp., 𝑒𝑇/𝛼) and so 1/(π›½πœ‚/(1+π›Όπœ‚)+π‘’βˆ’π‘‡/𝛼)≀1/𝛽 and 1/(π›½πœ‚/(1+π›Όπœ‚)+π‘’βˆ’π‘‡/𝛼)≀𝑒𝑇/𝛼.
(1.3) Also the error factor 𝑒(𝜎) given in Boussetila and Rebbani [19] is of order 1/(𝛽+π‘’βˆ’π‘‡/𝛼) and since 1/(𝛽(1+(πœ‚/(1+π›Όπœ‚)))+π‘’βˆ’π‘‡/𝛼)≀1/(𝛽+π‘’βˆ’π‘‡/𝛼), for 0<𝛼≀1βˆ’1/πœ‚,πœ‚>1 and so for all operators 𝐴(𝐴β‰₯πœ‚>1), considered above, our method of approximation gives a better approximation than the methods given by Boussetila and Rebbani in [19] and other authors, for example, [1, 7].

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