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

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 𝑡=𝑇 and a solution for 0≤𝑡<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 𝐴≥𝜂>0), self-adjoint unbounded linear operator which has a continuous spectrum on a Hilbert space 𝐻 such that −𝐴 generates a contraction 𝐶₀-semigroup on 𝐻. Let 𝑇 be a positive real number. We consider the final value problem (FVP) of finding 𝑢∶[0,𝑇]→𝐻 such that𝑢^(𝑡)+𝐴𝑢𝑢(𝑡()𝑇=)=0,𝜙,0≤𝑡<𝑇,((11..12)) for some prescribed final value 𝜙 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 𝜙.

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) with𝑢^(𝑡)+𝛼𝑢𝐴(𝑢0()𝑡+)=𝑢(0𝑇,)=0≤𝜙.𝑡<𝑇.(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:𝑣𝑣^𝜎(𝑡)+𝐴_𝛼𝑣_𝜎(𝑡)=0,0≤𝑡<𝑇,_𝜎(𝑇)+𝛽𝑣_𝜎(0)−𝑣^𝜎(0)=𝜙,(1.4) where the operator 𝐴 is replaced by the operator 𝐴_𝛼=𝐴(𝐼+𝛼𝐴)⁻¹ and 𝑢(𝑇)=𝜙 by 𝑢_𝜎(𝑇)+𝛽(𝑢_𝜎(0)−𝑢^𝜎(0))=𝜙, and 𝜎=(𝛼,𝛽), where 𝛼>0, 𝛽>0.

We show that the approximate problems are well posed and that their solutions 𝑣_𝜎 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, {𝐸_𝜆,𝜆≥𝜂>0} the resolution of the identity associated with the positive unbounded self-adjoint operator 𝐴. So the spectral representation of 𝐶₀ semigroup 𝑆(𝑡)=𝑒^−𝑡𝐴 (resp., 𝐴) is given by 𝑆(𝑡)=𝑒^−𝑡𝐴=∫^𝜂+∞𝑒^−𝑡𝜆𝑑𝐸_𝜆 (resp., 𝐴=∫^𝜂+∞𝜆𝑑𝐸_𝜆), and so for all 𝑢∈𝐷(𝐴),𝐴𝑢=∫^𝜂+∞𝜆𝑑𝐸_𝜆𝑢, and this is characterized by 𝑢∈𝐷(𝐴)iff‖𝐴𝑢‖²=^𝜂+∞𝜆²𝑑‖‖𝐸_𝜆𝑢‖‖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 𝑢∈𝐶([0,𝑇];𝐻)∩𝐶¹(]0,𝑇[;𝐻) such that, for all 𝑡∈]0,𝑇[,𝑢(𝑡)∈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 𝑢 if and only if ∫^𝜂+∞𝑒^2𝜆𝑇𝑑‖𝐸_𝜆𝜙‖²<∞, and this unique solution is represented by 𝑢(𝑡)=∫^𝜂+∞𝑒^{(𝑇−𝑡)𝜆}𝑑𝐸_𝜆𝜙<∞ (see [19], Lemma 1]).

2. The Approximate Problem

We approximate the final value problem (1.1), (1.2), by the following perturbed problem: 𝑣𝑣^𝜎(𝑡)+𝐴_𝛼𝑣_𝜎(𝑡)=0,0⩽𝑡<𝑇,_𝜎(𝑇)+𝛽𝑣_𝜎(0)−𝑣^𝜎(0)=𝜙,(2.1) where 𝐴 is as above and 𝐴_𝛼 is the Yosida approximation of operator 𝐴, and 𝜎=(𝛼,𝛽) where 𝛼>0,𝛽>0.

Definition 2.1. Define the function 𝑣_𝜎(𝑡)=𝑆_𝛼(𝑡)𝛽𝐼+𝐴_𝛼+𝑆_𝛼(𝑇)⁻¹𝜙,(2.2) for 𝜙∈𝐻,𝜎=(𝛼,𝛽) and 𝛼>0,𝛽>0,𝑡∈[0,𝑇], where 𝑆_𝛼(𝑡) is the semigroup generated by −𝐴_𝛼.

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

Theorem 2.2. The function 𝑣_𝜎(𝑡) is the unique solution of the perturbed problem (2.1), and it depends continuously on 𝜙.

Proof. We consider the following classical Cauchy problem: 𝑣𝑣^𝜎(𝑡)+𝐴_𝛼𝑣_𝜎(𝑡)=0,0<𝑡<𝑇,_𝜎(0)=𝛽𝐼+𝐴_𝛼+𝑆_𝛼(𝑇)⁻¹𝜙.(2.3) It is clear that 𝑣_𝜎(𝑡) is the unique solution and 𝑣_𝜎(𝑇)+𝛽𝑣_𝜎(0)−𝑣^𝜎(0)=+𝑆_𝛼𝛽(𝑇𝐼)+𝛽𝐴_𝛼𝐼+𝐴_𝛼+𝑆_𝛼(𝑇)⁻¹𝜙=𝛽𝐼+𝐴_𝛼𝛽𝐼+𝐴_𝛼+𝑆_𝛼(𝑇)⁻¹𝜙+𝑆_𝛼(𝑇)𝛽𝐼+𝐴_𝛼+𝑆_𝛼(𝑇)⁻¹𝜙=𝜙.(2.4) The continuous dependence of 𝑣_𝜎 on 𝜙 is obtained by showing that ‖‖𝑣_𝜎(𝑡)‖‖=⩽‖‖‖‖𝑆𝛽_𝛼(𝑡𝐼)+𝛽𝐴𝐼_𝛼+𝐴_𝛼+𝑆_𝛼(𝑇)⁻¹𝜙‖‖⩽1+𝑆_𝛼(𝑇)⁻¹𝜙‖‖𝛽+(𝛽𝜂/(1+𝛼𝜂))+𝑒^−𝑇/𝛼‖𝜙‖.(2.5)

Now, we consider the following problem:𝑢𝑢^𝜎(𝑡)+𝐴𝑢_𝜎(𝑡)=0,_𝜎(0)=𝜙_𝜎=𝛽𝐼+𝐴_𝛼+𝑆_𝛼(𝑇)⁻¹𝜙.(2.6)

Theorem 2.3. The problem (2.6) is well posed, and its solution is given by 𝑢_𝜎(𝑡)=𝑆(𝑡)𝛽𝐼+𝐴_𝛼+𝑆_𝛼(𝑇)⁻¹𝜙,(2.7) furthermore, ‖‖𝑢_𝜎(𝑡)‖‖⩽1𝛽(1+(𝜂/(1+𝛼𝜂)))+𝑒^−𝑇/𝛼^{(𝑇−𝑡)/𝑇}‖𝜙‖.(2.8)

Proof. Since ‖‖𝑢_𝜎(𝑡)‖‖²=^𝜂+∞𝑒^−2𝜆𝑇𝛽(1+(𝜆/(1+𝛼𝜆)))+𝑒^{−(𝜆𝑇)/(1+𝛼𝜆)}²𝑑‖‖𝐸_𝜆𝜙‖‖²≤^𝜂+∞𝛽(1+(𝜂/(1+1𝛼𝜂)))+𝑒^−𝑇/𝛼²^{𝑇−𝑡/𝑇}𝑑‖‖𝐸_𝜆𝜙‖‖²,(2.9) then, ‖‖𝑢_𝜎(𝑡)‖‖⩽1𝛽(1+(𝜂/(1+𝛼𝜂)))+𝑒^−𝑇/𝛼^{𝑇−𝑡/𝑇}‖𝜙‖.(2.10)

Now we give some convergence results.

3. The Convergence Results

Theorem 3.1. For all 𝜙∈𝐻,‖𝑢_𝜎(𝑇)−𝜙‖→0, as |𝜎|→0.

Proof. We have ‖‖𝑢_𝜎(𝑇)−𝜙‖‖²=^𝜂+∞𝛽(1+(𝜆/(1+𝑒𝛼𝜆)^−𝜆𝑇))+𝑒^{−(𝜆𝑇)/(1+𝛼𝜆)}−1²𝑑‖‖𝐸_𝜆𝜙‖‖²=^𝜂+∞𝛽(1+(𝛽𝜆(/1(1++(𝜆𝛼/𝜆()1))++𝛼𝑒𝜆)))^{−(𝜆𝑇/(1+𝛼𝜆))}+𝑒^{−(𝜆𝑇/(1+𝛼𝜆))}−𝑒^−𝜆𝑇𝛽(1+(𝜆/(1+𝛼𝜆)))+𝑒^{−(𝜆𝑇/(1+𝛼𝜆))}²𝑑‖‖𝐸_𝜆𝜙(‖‖3².1.) Putting 𝑀_𝜎(𝜆)=𝛽(1+(𝜆/(1+𝛼𝜆)))𝛽(1+(𝜆/(1+𝛼𝜆)))+𝑒^{−(𝜆𝑇/(1+𝛼𝜆))}𝑁_𝜎,(𝜆)=𝑒^{−(𝜆𝑇/(1+𝛼𝜆))}−𝑒^−𝜆𝑇𝛽(1+(𝜆/(1+𝛼𝜆)))+𝑒^{−(𝜆𝑇/(1+𝛼𝜆))},(3.2) and if we put 𝑃_𝜎(𝜆)=𝑀_𝜎(𝜆)+𝑁_𝜎(𝜆),(3.3) then we get ‖‖𝑢_𝜎(𝑇)−𝜙‖‖²=^𝜂+∞𝑃_𝜎(𝜆)²𝑑‖‖𝐸_𝜆𝜙‖‖²⩽2^𝜂+∞𝑀^2𝜎(𝜆)+𝑁^2𝜎(𝜆)𝑑‖‖𝐸_𝜆𝜙‖‖².(3.4) It is clear that, for all 𝜀>0, there exists 𝑘∈ℕ^∗/∫^𝑘+∞𝑑‖𝐸_𝜆𝜙‖²<𝜀/8 and 𝑀_𝜎(𝜆)≤1,𝑁_𝜎(𝜆)⩽1.
If we put 𝐼𝐽_𝜎𝜎=^𝑘𝜂𝑀^2𝜎(𝜆)𝑑‖‖𝐸_𝜆𝜙‖‖²+^𝑘+∞𝑀^2𝜎(𝜆)𝑑‖‖𝐸_𝜆𝜙‖‖²,=^𝑘𝜂𝑁^2𝜎(𝜆)𝑑‖‖𝐸_𝜆𝜙‖‖²+^𝑘+∞𝑁^2𝜎(𝜆)𝑑‖‖𝐸_𝜆𝜙‖‖².(3.5) then we have 𝐼_𝜎⩽𝜀8+𝛽²1+𝛼1²𝐽_𝜎𝑒^2𝑘𝑇‖𝜙‖²,⩽𝜀8+𝛼²𝑇²𝑘⁴‖𝜙‖².(3.6) Finally, choosing 𝜎 such that |𝜎|²=𝛼²+𝛽²⩽(1/‖𝜙‖²)(1/𝑇²𝑘⁴+1/𝑒^2𝑘𝑇)(𝜀/8), we obtain the following estimate ‖‖𝑢_𝜎(𝑇)−𝜙‖‖²⩽𝜖2+2𝛽²1+𝛼1²𝑒^2𝑘𝑇+𝛼²𝑇²𝑘⁴‖𝜙‖²,(3.7) which gives the desired result.

Let us denote by 𝐶_𝜃(𝐴), 𝜃≥0 the following set:𝐶_𝜃(𝐴)=ℎ∈𝐻∶‖ℎ‖^2𝐶_𝜃=^𝜂+∞𝑒^{2𝑇𝜃𝜆}𝑑‖‖𝐸_𝜆𝜙‖‖²<+∞.(3.8)

It is clear that the following proprieties hold𝐶_𝜃1(𝐶𝐴_𝜃)⊆𝐶_𝜃2(𝐴),𝜃₂⩾𝜃₁,(𝐴)⊂𝐻,𝜃>0.(3.9) Now, we give some convergence results with explicit convergence rates.

Theorem 3.2. If 𝜙∈𝐶_𝜃(𝐴), then one has ‖‖𝑢_𝜎(𝑇)−𝜙‖‖²⩽2𝐶²¹(𝜃)𝛽^2𝜃𝛼²1+𝜂𝛼𝜂^2(𝜃−1)+𝐶²²(𝜃,𝑇)𝛼²‖𝜙‖^2𝐶𝜃,(3.10) for 0<𝜃<1 and ‖‖𝑢_𝜎(𝑇)−𝜙‖‖²⩽2𝛽²𝛼²+𝐶²²(𝜃,𝑇)𝛼²‖𝜙‖^2𝐶𝜃,for𝜃⩾1,(3.11) such that 𝐶₁(𝜃)=(1−𝜃)^1−𝜃𝜃^𝜃⩽1,𝐶₂(𝜃,𝑇)=1𝑇𝜃².(3.12)

Proof. Using the proof of the previous theorem, we have ‖‖𝑢_𝜎(𝑇)−𝜙‖‖²=^𝜂+∞𝑃^2𝜎(𝜆)𝑑‖‖𝐸_𝜆𝜙‖‖²=^𝜂+∞𝑃^2𝜎(𝜆)𝑒^{−2𝑇𝜃𝜆}𝑒^{2𝑇𝜃𝜆}𝑑‖‖𝐸_𝜆𝜙‖‖²⩽2𝑀^{2𝜎,𝜃,∞}+𝑁^{2𝜎,𝜃,∞}‖𝜑‖^2𝐶_𝜃,(3.13) where 𝑀_{𝜎,𝜃,∞}=sup_𝜆⩾𝜂𝑀_𝜎𝑒^{−𝑇𝜃𝜆},𝑁_{𝜎,𝜃,∞}=sup_𝜆⩾𝜂𝑁_𝜎𝑒^{−𝑇𝜃𝜆}.(3.14) If 0<𝜃<1, then we have 𝑀_{𝜎,𝜃,∞}≤2𝛼𝛽^𝜃𝐶₁(𝜃).(3.15) And, for 𝜃⩾1, we have 𝑀_{𝜎,𝜃,∞}≤2𝛽𝛼.(3.16) We also have 𝑁_{𝜎,𝜃,∞}⩽4𝛼𝑇𝜃²𝑒²⩽𝛼𝑇𝜃²,∀𝜃>0.(3.17) Then, using the above estimates, we get the desired results.

Now, let 𝐹 be the function defined by𝐹𝜎=∶ℝ(𝛼⁺,𝛽×)ℝ⟶⁺⟶𝐹(𝐻𝜎),=𝑢_𝜎𝑢(0(0))==𝜙𝜙_𝜎0,𝜎≠(0,0),,𝜎=(0,0).(3.18)

Theorem 3.3. For all 𝜙∈𝐻, the problem (1.1), (1.2) has a solution 𝑢(𝑡) if and only if the function 𝐹 is continious at (0,0). Furthermore, 𝑢_𝜎(𝑡)→𝑢(𝑡),𝑎𝑠|𝜎|→0, uniformly in 𝑡.

Proof. We assume that lim_|𝜎|→0𝜙_𝜎=𝜙₀ and ‖𝜙₀‖<∞. Let 𝑤(𝑡)=𝑆(𝑡)𝜙₀. So, we have ‖‖𝑤(𝑡)−𝑢_𝜎(𝑡)‖‖⩽‖‖𝜙₀−𝜙_𝜎‖‖.(3.19) Hence, sup_{0⩽𝑡⩽𝑇}‖‖𝑤(𝑡)−𝑢_𝜎(𝑡)‖‖⩽‖‖𝜙₀−𝜙_𝜎‖‖⟶_|𝜎|→00.(3.20) Since lim_|𝜎|→0𝑢_𝜎(𝑇)=𝜙 and lim_|𝜎|→0𝑢_𝜎(𝑇)=𝑤(𝑇) and so by the unicity of the limit, we obtain that 𝑤(𝑡) is a solution to problem (1.1), (1.2).
Now, we suppose that 𝑢(𝑡)=∫^𝜂+∞𝑒^{(𝑇−𝑡)𝜆}𝑑𝐸_𝜆𝜙 is a solution to problem (1.1), (1.2). Since 𝑢(0)=𝑆(−𝑇)𝜙∈𝐻 (see [19], Lemma 1), then we have ‖𝑢(0)‖²=‖𝜙‖^2𝐶₁=^𝜂+∞𝑒^2𝑇𝜆𝑑‖‖𝐸_𝜆𝜙‖‖²<∞.(3.21) Let 𝑘>0 and 𝜖>0, such that ∫^𝑘+∞𝑒^2𝑇𝜆𝑑‖𝐸_𝜆𝜙‖²<𝜖/8. If 𝜎₁=(𝛼₁,𝛽₁) and 𝜎₂=(𝛼₂,𝛽₂). Then, ‖‖𝑢_𝜎1(0)−𝑢_𝜎2(0)‖‖²=^𝜂+∞𝛽₁1+𝜆/1+𝛼₁1𝜆+𝑒^{−(𝜆𝑇/(1+𝛼1𝜆))}−1𝛽₂1+𝜆/1+𝛼₂𝜆+𝑒^{−(𝜆𝑇/1+𝛼2𝜆))}²𝑑‖‖𝐸_𝜆𝜙‖‖²≤2^𝜂+∞𝛽¹1+𝜆/1+𝛼₂𝜆−𝛽₂1+𝜆/1+𝛼₁𝜆𝛽₁1+𝜆/1+𝛼₁𝜆+𝑒^{−(𝜆𝑇/(1+𝛼1𝜆))}𝛽₂1+𝜆/1+𝛼₂𝜆+𝑒^{−(𝜆𝑇/(1+𝛼2𝜆))}²𝑑‖‖𝐸_𝜆𝜙‖‖²+2^𝜂+∞𝑒^{−𝜆𝑇/1+𝛼1𝜆}−𝑒^{−𝜆𝑇/1+𝛼2𝜆}𝛽₁1+𝜆/1+𝛼₁𝜆+𝑒^{−(𝜆𝑇/(1+𝛼1𝜆))}𝛽₂1+𝜆/1+𝛼₂𝜆+𝑒^{−(𝜆𝑇/(1+𝛼2𝜆))}²𝑑‖‖𝐸_𝜆𝜙‖‖(²3..22) If we put 𝑀_𝜎1,𝜎2=||𝛽¹1+𝜆/1+𝛼₂𝜆−𝛽₂1+𝜆/1+𝛼₁𝜆||𝑁_𝜎1,𝜎2𝛽₁1+𝜆/1+𝛼₁𝜆+𝑒^{−𝜆𝑇/1+𝛼1𝜆}𝛽₂1+𝜆/1+𝛼₂𝜆+𝑒^{−𝜆𝑇/1+𝛼2𝜆},=||𝑒^{−(𝜆𝑇/(1+𝛼1𝜆))}−𝑒^{−(𝜆𝑇/(1+𝛼2𝜆))}||𝛽₁1+𝜆/1+𝛼₁𝜆+𝑒^{−(𝜆𝑇/(1+𝛼1𝜆))}𝛽₂1+𝜆/1+𝛼₂𝜆+𝑒^{−(𝜆𝑇/(1+𝛼2𝜆))},(3.23) then we have ‖‖𝑢_𝜎1(0)−𝑢_𝜎2(0)‖‖²⩽2^𝜂+∞𝑀^2𝜎_1,𝜎2(𝜆)𝑑‖‖𝐸_𝜆𝜙‖‖²+2^𝜂+∞𝑁^2𝜎_1,𝜎2(𝜆)𝑑‖‖𝐸_𝜆𝜙‖‖².(3.24) Using analogous calculations as in the proof of Theorem 3.1, we obtain ‖‖𝑢_𝜎1(0)−𝑢_𝜎2(0)‖‖²⩽𝜖2+2𝑒^2𝑘𝑇‖𝜙‖^2𝐶₁×𝛽₂1+1+𝑘𝛼₂𝑘−𝛽¹1+1+𝑘𝛼₁𝑘²+𝑇²𝑘⁴𝛼₁−𝛼₂².(3.25) Now, if 𝜎₀=(0,0), then by choosing 𝜎=(𝛼,𝛽) such that |𝜎|²=𝛼²+𝛽²⩽1‖𝜙‖^2𝐶₁1𝑒^2𝑘𝑇1+𝑇²1𝑘⁴8𝜖,(3.26) we find that ‖𝑢_𝜎(0)−𝑢_𝜎0(0)‖²=‖𝜙_𝜎−𝜙₀‖²≤𝜖. This means that the function 𝐹 is continuous at (0,0).

We note that we can easily show that‖‖𝑢_𝜎(0)−𝑢(0)‖‖²=‖‖𝑢_𝜎(𝑇)−𝜙‖‖^2𝐶₁,(3.27) for 𝜙∈𝐶₁(𝐴). And by using Theorem 3.1 again, we see that‖‖𝑢_𝜎(0)−𝑢(0)‖‖²⟶0as|𝜎|⟶0.(3.28)

Theorem 3.4. If 𝜙∈𝐶_𝜃+1(𝐴), then one has ‖‖𝑢_𝜎(0)−𝑢(0)‖‖²⩽24𝛼²𝐶²¹(𝜃)𝛽^2𝜃+𝐶²²(𝜃,𝑇)𝛼²‖𝜙‖^2𝐶_𝜃+1,(3.29) for 0<𝜃<1. And ‖‖𝑢_𝜎(0)−𝑢(0)‖‖²⩽24𝛽²𝛼²+𝐶²²(𝜃,𝑇)𝛼²‖𝜙‖^2𝐶𝜃+1,for𝜃⩾1(3.30) such that 𝐶₁(𝜃)=(1−𝜃)^1−𝜃𝜃^𝜃≤1,𝐶₂(𝜃,𝑇)=1𝑇𝜃².(3.31)

Proof. Since ‖‖𝑢_𝜎(0)−𝑢(0)‖‖²=‖‖𝑢_𝜎(𝑇)−𝜙‖‖^2𝐶₁=⩽2^𝜂+∞𝑀𝑃^2𝜎(_{2𝜎,𝜃,∞}𝜆)𝑒^{2(1+𝜃)𝑇𝜆}𝑒^{−2𝑇𝜆𝜃}𝑑‖‖𝐸_𝜆𝜙‖‖²,+𝑁^{2𝜎,𝜃,∞}‖𝜙‖^2𝐶_𝜃+1,(3.32) then using Theorem 3.2, we get the required result.

Corollary 3.5. If 𝜙∈𝐶_𝜃+1(𝐴), then one has sup_{0⩽𝑡⩽𝑇}‖‖𝑢_𝜎(𝑡)−𝑢(𝑡)‖‖²⩽‖‖𝑢_𝜎(0)−𝑢(0)‖‖²⩽24𝛼²𝐶²¹(𝜃)𝛽^2𝜃+𝐶²²(𝜃,𝑇)𝛼²‖𝜙‖^2𝐶_𝜃+1,(3.33) for 0<𝜃<1. And sup_{0⩽𝑡⩽𝑇}‖‖𝑢_𝜎(𝑡)−𝑢(𝑡)‖‖²⩽‖‖𝑢_𝜎(0)−𝑢(0)‖‖²⩽24𝛽²𝛼²+𝐶²²(𝜃,𝑇)𝛼²‖𝜙‖^2𝐶𝜃+1,for𝜃⩾1.(3.34)

Proof. Using Theorem 3.3, we have sup_{0⩽𝑡⩽𝑇}‖‖𝑢_𝜎(𝑡)−𝑢(𝑡)‖‖²⩽‖‖𝑢_𝜎(0)−𝑢(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 𝑡=0 and so he gets the convergence of the approximate solution at 𝑡=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(𝜖). 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 𝜀(𝜎) introduced by small changes in the final value is of order 1/(𝛽(1+(𝜂/(1+𝛼𝜂)))+𝑒^−𝑇/𝛼) and in a recent work [21], by the same authors, the error factor given was of order 1/(𝛽𝜂/(1+𝛼𝜂)+𝑒^−𝑇/𝛼)≥1/(𝛽(1+(𝜂/(1+𝛼𝜂)))+𝑒^−𝑇/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., 𝑒^𝑇/𝛼) and so 1/(𝛽𝜂/(1+𝛼𝜂)+𝑒^−𝑇/𝛼)≤1/𝛽 and 1/(𝛽𝜂/(1+𝛼𝜂)+𝑒^−𝑇/𝛼)≤𝑒^𝑇/𝛼.
(1.3) Also the error factor 𝑒(𝜎) given in Boussetila and Rebbani [19] is of order 1/(𝛽+𝑒^−𝑇/𝛼) and since 1/(𝛽(1+(𝜂/(1+𝛼𝜂)))+𝑒^−𝑇/𝛼)≤1/(𝛽+𝑒^−𝑇/𝛼), for 0<𝛼≤1−1/𝜂,𝜂>1 and so for all operators 𝐴(𝐴≥𝜂>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].