Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem

Abstract

We study an abstract elliptic Cauchy problem associated with an unbounded self-adjoint positive operator which has a continuous spectrum. It is well-known that such a problem is severely ill-posed; that is, the solution does not depend continuously on the Cauchy data. We propose two spectral regularization methods to construct an approximate stable solution to our original problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.

1. Introduction

Throughout this paper 𝐻 denotes a complex Hilbert space endowed with the inner product (⋅,⋅), and the norm ‖⋅‖, ℒ(𝐻) stands for the Banach algebra of bounded linear operators on 𝐻.

Let 𝐴 be a linear unbounded operator with dense domain 𝐷(𝐴). Assume that 𝐴 is self-adjoint, positive definite in 𝐻, which has a continuous spectrum 𝜎(𝐴)=[𝛾,+∞[, 𝛾=inf(𝜎(𝐴))>0.

We consider the elliptic Cauchy problem (ECP) of finding 𝑢∶[0,𝐿]→𝐻 such that 𝑢_𝑦𝑦𝑦𝑦𝑢(0)𝑢=𝐴𝑢,0<𝑦<𝐿,=𝑓,_𝑦(0)=0, (1) where 𝑓 is some prescribed data in the Hilbert space 𝐻.

This problem is an abstract version of Cauchy problem, which generalizes Cauchy problem for second-order elliptic partial differential equations in a cylindrical domain; for example, we mention the following problem.

Example 1. An example of (1) is the Cauchy problem for the modified Helmholtz equation in the infinite strip ℝ×(0,1) [1]: 𝑢_𝑦𝑦𝑥,𝑦+𝑢_𝑥𝑥(),𝑢()(𝑥)𝑥,𝑦−𝛾𝑢𝑥,𝑦=0,𝑥∈ℝ,𝑦∈0,𝐿𝑥,0=𝑓,𝑢_𝑦(𝑥,0)=0,𝑥∈ℝ, (2) where the operator 𝐴_𝛾 is given by 𝐴_𝛾𝜕=−²𝜕𝑥²𝐷𝐴+𝛾𝐼,_𝛾=𝐻²(ℝ)⊂𝐻=𝐿²(ℝ). (3) It is well known that this operator is self-adjoint with continuous spectrum 𝜎𝐴_𝛾=𝜎𝐴₀+𝛾=[0,+∞[+𝛾=𝛾,+∞. (4) We note here that the discrete eigenfunctions expansion method cannot be used, but we can use the Fourier diagonalization method to deal with this kind of problems.

Such problem arises in many practical situations, nondestructive testing techniques [2], geophysics [3], cardiology [4], and other applications. There are many various monographs about the historical development of this topic, for more details, we refer the reader to [5, 6]. Recently there has been an excellent topic review [7] of this problem.

Because problem (1) is severely ill-posed; that is, a small perturbation in the given Cauchy data may result in a very large error on the solution. In order to overcome this instability character, the regularization methods are required.

Some regularization methods for the Cauchy problem for elliptic equations have been proposed by many authors. For instance: Tikhonov regularization method [8], the quasi-reversibility method [9], the quasi-boundary-value method [10–13], Kozlov-Maz’ya iteration method [14], and the mollification method [15].

This work is mainly devoted to theoretical aspects of the spectral regularization methods to problem (1) in the abstract setting, by considering more general self-adjoint operators when 𝐴 is positive and induces the elliptic case, that is, has the following properties: for any 𝜆∈(−∞,0], the resolvent 𝑅(𝜆;𝐴)=(𝐴−𝜆𝐼)⁻¹ exists and satisfies the estimates ∃𝑀>0∶∀𝜆≥0,(𝐴+𝜆𝐼)⁻¹≤𝑀(1+𝜆)⁻¹. (5)

In the case when 𝐴 is a linear positive self-adjoint operator with compact inverse, problem (1) has been treated by a different method and there is a large literature in this direction. However, in the case where 𝐴 has a continuous spectrum the literatures are quite scarce.

In the present paper we shall use two spectral regularization methods to construct a stable solution to our original ill-posed problem.

2. Preliminaries and Basic Results

In this section we present the notation and the functional setting which will be used in this paper and prepare some material which will be used in our analysis.

2.1. Spectral Theorem and Properties

By the spectral theorem, for each positive self-adjoint operator 𝐴, there is a unique right continuous family {𝐸_𝜆}_{𝜆∈[0,∞[}∶[0,∞[→ℒ(𝐻) of orthogonal projection operators such that 𝐴=∫^∞₀𝜆𝑑𝐸_𝜆 with 𝐷(𝐴)=𝑣∈𝐻∶^∞₀𝜆²𝑑𝐸_𝜆𝑣,𝑣<∞. (6)

In our case, we have 𝐴=∫^∞_𝛾𝜆𝑑𝐸_𝜆 because 𝐴≥𝛾𝐼, 𝛾>0.

Theorem 2 (see [16, Theorem 6, XII.2.5, pages 1196–1198]). Let {𝐸_𝜆,𝜆≥𝛾>0} be the spectral resolution of the identity associate to 𝐴 and let 𝜙 be a complex Borel function defined 𝐸-almost everywhere on the real axis. Then 𝜙(𝐴) is a closed operator with dense domain. Moreover(i)𝐷(𝜙(𝐴))∶={𝑣∈𝐻∶∫^∞_𝛾|𝜙(𝜆)|²𝑑(𝐸_𝜆𝑣,𝑣)<∞},(ii)(𝜙(𝐴)ℎ,𝑦)=∫^∞_𝛾𝜙(𝜆)𝑑(𝐸_𝜆ℎ,𝑦), ℎ∈𝐷(𝜙(𝐴)), 𝑦∈𝐻,(iii)‖𝜙(𝐴)ℎ‖²=∫^∞_𝛾|𝜙(𝜆)|²𝑑(𝐸_𝜆ℎ,ℎ), ℎ∈𝐷(𝜙(𝐴)),(iv)𝜙(𝐴)^∗=𝜙(𝐴). In particular, if 𝜙 is real Borel function, then 𝜙(𝐴) is self-adjoint, (v)the operator 𝜙(𝐴) is bounded if and only if 𝜙(𝜆) is bounded on 𝜎(𝐴)=[𝛾,+∞[. In this case, ‖𝜙(𝐴)‖=sup_{𝜆∈[𝛾,+∞[}|𝜙(𝜆)|.

We denote by 𝑆(𝑦)=𝑒^{√−𝑦𝐴}=∫^∞_𝛾𝑒^{√−𝑦𝜆}𝑑𝐸_𝜆∈ℒ(𝐻), 𝑦≥0, the 𝐶₀-semigroup generated by −√𝐴. Some basic properties of 𝑆(𝑦) are listed in the following theorem.

Theorem 3 (see [17, chapter 2, Theorem 6.13, page 74]). For this family of operators one has:(1)‖𝑆(𝑦)‖≤1, ∀𝑦≥0; (2)the function 𝑦↦𝑆(𝑦), 𝑦>0, is analytic; (3)for every real 𝑟≥0 and 𝑦>0, the operator 𝑆(𝑦)∈ℒ(𝐻,𝒟(𝐴^𝑟/2)); (4)for every integer 𝑘≥0 and 𝑦>0, ‖𝑆^(𝑘)(𝑦)‖=‖𝐴^𝑘/2𝑆(𝑦)‖≤𝑐(𝑘)𝑦^−𝑘; (5)for every 𝑥∈𝒟(𝐴^𝑟/2), 𝑟≥0, one has 𝑆(𝑡)𝐴^𝑟/2𝑥=𝐴^𝑟/2𝑆(𝑦)𝑥.

Theorem 4. For 𝑦>0, 𝑆(𝑦) is self-adjoint and one to one operator with dense range (𝑆(𝑦)=𝑆(𝑦)^∗, ℛ(𝑆(𝑦))=𝐻).

Proof. Let 𝜙_𝑦∶[𝛾,+∞[→ℝ, 𝑠↦𝜙_𝑦(𝑠)=𝑒^−𝑦𝑠. Then by virtue of (iv) of Theorem 2, we can write 𝑆(𝑦)^∗=𝜙_𝑦(𝐴)=𝜙_𝑦(𝐴)=𝑒^{√−𝑦𝐴}=𝑆(𝑦).
Let ℎ∈𝑁(𝑆(𝑦₀)), 𝑦₀>0, then 𝑆(𝑦₀)ℎ=0, which implies that 𝑆(𝑦)𝑆(𝑦₀)ℎ=𝑆(𝑦+𝑦₀)ℎ=0, 𝑦≥0. Using analyticity, we obtain that 𝑆(𝑦)ℎ=0, 𝑦≥0. Strong continuity at 0 now gives ℎ=0. This shows that 𝑁(𝑆(𝑦₀))={0}.
Thanks to 𝑅𝑆𝑦₀=𝑁𝑆𝑦₀^⟂={0}^⟂=𝐻, (7) we conclude that 𝑅(𝑆(𝑦₀)) is dense in 𝐻.

Remark 5. For 𝑦=𝑝𝐿, 𝑝≥1, this Theorem ensures us that 𝑆(𝑝𝐿) is self-adjoint and one to one operator with dense range 𝑅(𝑆(𝑝𝐿)). Then we can define its inverse 𝑆(𝑝𝐿)⁻¹=𝑒^{√𝑝𝐿𝐴}, which is an unbounded self-adjoint strictly positive definite operator in 𝐻 with dense domain 𝐷𝑆𝑝𝐿⁻¹𝑆=𝑒=𝑅𝑝𝐿ℎ∈𝐻∶^{√𝑝𝐿𝐴}ℎ²=_𝛾^+∞𝑒^{√2𝑝𝐿𝜆}𝑑𝐸_𝜆ℎ²<+∞. (8)

Definition 6 (Hilbert scales). Let 𝐵 be an unbounded self-adjoint strictly positive definite operator in 𝐻. Following the definition in [18], one introduces the Hilbert scale {𝒢_𝜃}_𝜃≥0 according to definition 𝒢₀=𝐻,𝒢_𝜃=𝐷𝐵^𝜃. (9) Here 𝒢_𝜃 is a Hilbert space with inner product (𝜉,𝜁)_𝜃=(𝐵^𝜃𝜉,𝐵^𝜃𝜁) and norm ‖𝜉‖²_𝜃=(𝐵^𝜃𝜉,𝐵^𝜃𝜉).
In our setting we take 𝐵=𝑆(𝐿)⁻¹=𝑒^𝐿√𝐴. In this case, one has 𝒢_𝜃=ℎ∈𝐻∶‖ℎ‖²_𝜃=𝑒^{√𝜃𝐿𝐴}ℎ²=_𝛾^+∞𝑒^{√2𝐿𝜃𝜆}𝑑𝐸_𝜆ℎ²<+∞. (10)
Following [15], one defines the following.

Definition 7 (mollification operator). For 𝛼>0 and 𝑝≥1, one introduces the Yosida approximation of identity 𝑀_𝛼=𝐼+𝛼𝑒^{√𝑝𝐿𝐴−1}=_𝛾^+∞1+𝛼𝑒^{√𝑝𝐿𝜆−1}𝑑𝐸_𝜆. (11)

Remark 8. The idea of the mollification method is very simple and natural: if the data are given inexactly, then we try to find a sequence of mollification operators which map the improper data into well-posed classes of the problem (mollify the improper data). Within these mollified data our problem becomes well-posed.

Theorem 9. One has(1)𝑀_𝛼∈ℒ(𝐻) and ‖𝑀_𝛼‖≤1, (2)for all ℎ∈𝐻, 𝑀_𝛼ℎ∈𝒢_𝑝,(3)for all ℎ∈𝐻, lim_𝛼→0‖𝑀_𝛼ℎ−ℎ‖=0.

Proof. (1) ‖𝑀_𝛼‖=sup_𝜆≥𝛾1/(1+𝛼𝑒^{√𝑝𝐿𝜆})=1/(1+𝛼𝑒^{√𝑝𝐿𝛾})<1.
(2) ‖𝑀_𝛼ℎ‖²_𝑝=∫_𝛾^+∞(𝑒^{√𝑝𝐿𝜆}/(1+𝛼𝑒^{√𝑝𝐿𝜆}))²𝑑‖𝐸_𝜆ℎ‖²≤(1/𝛼)²‖ℎ‖²<∞⇔𝑀_𝛼ℎ∈𝒢_𝑝.
(3) Assume that ℎ∈𝒢_𝑝. Then 𝑀_𝛼ℎ−ℎ²=_𝛾^+∞11+𝛼𝑒^{√𝑝𝐿𝜆}−1²𝑑𝐸_𝜆ℎ²=_𝛾^+∞𝛼𝑒^{√𝑝𝐿𝜆}1+𝛼𝑒^{√𝑝𝐿𝜆2}𝑑𝐸_𝜆ℎ²≤𝛼²‖ℎ‖²_𝑝, (12) and thus, lim_𝛼→0‖𝑀_𝛼ℎ−ℎ‖=0.
Now for ℎ∈𝐻, there exists ℎ_𝜀∈𝒢_𝑝 such that ‖ℎ−ℎ_𝜀‖≤𝜀, 𝜀>0 (since 𝒢_𝑝 is dense in 𝐻 (see Remark 5)). We have 𝑀_𝛼=𝑀ℎ−ℎ_𝛼ℎ−𝑀_𝛼ℎ_𝜀+𝑀_𝛼ℎ_𝜀−ℎ_𝜀+ℎ_𝜀≤𝑀−ℎ_𝛼ℎ−ℎ_𝜀+𝑀_𝛼ℎ_𝜀−ℎ_𝜀+ℎ_𝜀𝑀−ℎ≤2𝜀+_𝛼ℎ_𝜀−ℎ_𝜀. (13) Thus, lim_𝛼→0‖𝑀_𝛼ℎ−ℎ‖≤2𝜀, ∀𝜀>0, and so lim_𝛼→0‖𝑀_𝛼ℎ−ℎ‖=0.

Using the change of variables 𝑈𝑦=12𝑢𝑦−𝐴^−1/2𝑢^′𝑦,𝑊𝑦=12𝑢𝑦+𝐴^−1/2𝑢^′𝑦. (14)

Cauchy’s problem (1) reduces to the two Cauchy problems 𝑈^′𝑦=−𝐴^1/2𝑈𝑦𝑈(0)=1,0<𝑦<𝐿,2𝑢(0)−𝐴^−1/2𝑢^′(0)=12𝑊𝑓,_′𝑦=𝐴^1/2𝑊𝑦𝑊(0)=1,0<𝑦<𝐿,2𝑢(0)+𝐴^−1/2𝑢^′(0)=12𝑓. (15)(16) Thus, the solution of the original problem (1) can be written in the form 𝑢𝑦=𝑈𝑦+𝑊𝑦,0≤𝑦≤𝐿. (17) It is well known that the operator −𝐴^1/2 generates a strongly continuous analytic semigroup 𝑆_−1/2(𝑦)=∫_𝛾^+∞𝑒^{√−𝑦𝜆}𝑑𝐸_𝜆. In addition, the spectral radius of the semigroup ‖𝑆_−1/2(𝑦)‖<1 for any 𝑦>0. Hence, it follows that the Cauchy problem (15) is well-posed and its solution may be written in the form 𝑈𝑦=𝑆_−1/2𝑦𝑈(0)=12𝑒^{√−𝑦𝐴}1𝑓=2_𝛾^+∞𝑒^{√−𝑦𝜆}𝑑𝐸_𝜆𝑓. (18) As opposed to problem (15), Cauchy problem (15) (backward parabolic equation) is not correctly posed, and its (unique) formal solution is given by 𝑊𝑦=𝑆_+1/2𝑦𝑊(0)=12𝑒^𝑦√𝐴1𝑓=2_𝛾^+∞𝑒^𝑦√𝜆𝑑𝐸_𝜆𝑓. (19)

Remark 10 (see [19, page 375]). The uniqueness solvability of problem (16) results from the logarithmic convexity of the function 𝑦↦‖𝑊(𝑦)‖: ∀𝑦∈[0,𝐿],𝑊𝑦≤‖𝑊(0)‖^{1−𝑦/𝐿}‖𝑊(𝐿)‖^𝑦/𝐿. (20)
A useful characterization of the admissible set for which problem (16) has a solution is as follows.

Lemma 11. Problem (16) has a solution 𝑊∈𝐶([0,𝐿];𝐻) if and only if 𝑓∈𝒢₁, and its unique solution is represented by 𝑊𝑦=𝑆_+1/2𝑦𝑊(0)=12𝑒^𝑦√𝐴1𝑓=2_𝛾^+∞𝑒^𝑦√𝜆𝑑𝐸_𝜆𝑓. (21)

Proof. Suppose that problem (16) has a solution 𝑊∈𝒞([0,𝐿];𝐻). Then 𝑊𝑦=12_𝛾^+∞𝑒^𝑦√𝜆𝑑𝐸_𝜆𝑓. (22) The function 𝑊∈𝒞([0,𝐿];𝐻) if and only if sup_{𝑦∈[0,𝐿]}𝑊(𝑦)²=14_𝛾^+∞𝑒^√2𝑦𝜆𝑑𝐸_𝜆𝑓²<∞. (23) Observing that the function 𝑦↦‖𝑊(𝐿)‖² is increasing, then sup_{𝑦∈[0,𝐿]}𝑊(𝑦)²=‖𝑊(𝐿)‖²=14_𝛾^+∞𝑒^√2𝐿𝜆𝑑𝐸_𝜆𝑓²<∞. (24) This last inequality is exactly equivalent to 𝑓∈𝒢₁.

As a consequence, we have the following corollary.

Corollary 12. Problem (1) has a solution 𝑢∈𝐶([0,𝐿];𝐻) if and only if 𝑓∈𝒢₁, and its unique solution is represented by 𝑢𝑦𝑦𝑦=1=𝑈+𝑊2_𝛾^+∞𝑒^𝑦√𝜆+𝑒^{√−𝑦𝜆}𝑑𝐸_𝜆𝑓. (25)

3. Regularization and Error Estimates

3.1. The Truncation Method

From (25) we can see that the term 𝑒^𝑦√𝜆 is the cause of unstability. In order to overcome the ill-posedness of problem (1), we modify the solution by filtering the high frequencies using a suitable method and instead consider (25) only for 𝜆≤𝛽(𝛿), where 𝛽(𝛿) is some constant which satisfies lim_𝛿→0𝛽(𝛿)=+∞.

According to spectral theory of self-adjoint operators [20], for any bounded Borel set Δ_𝛽={𝛾≤𝑡≤𝛽}⊆𝜎(𝐴)=[𝛾,+∞[, we can define the orthogonal projection 𝟏_Δ𝛽=_𝛾^+∞𝟏_Δ𝛽(𝜆)𝑑𝐸_𝜆=𝐸_𝛽,∀ℎ∈𝐻,ℎ_𝛽=𝐸_𝛽ℎ⟶ℎ,𝛽⟶+∞. (26)

To solve (1) in a stable way we approximate 𝑓 by its projection 𝑓_𝛽, and instead of considering (1) with 𝑓 we take its projected version 𝑢_𝛽𝑦=cosh𝑦√𝐴𝑓_𝛽=12_𝛾^+∞𝑒^𝑦√𝜆+𝑒^{√−𝑦𝜆}𝟏_[𝛾,𝛽]𝑑𝐸_𝜆𝑓, (27) where 𝟏_[𝑎,𝑏] is the characteristic function of the interval [𝑎,𝑏] for 𝑎<𝑏. The quantity 𝛽 is referred to as a cut-off frequency.

Let 𝑓 (resp., 𝑓_𝛿) be the exact (resp., the measured data) at 𝑦=0, such that ‖𝑓−𝑓_𝛿‖≤𝛿.

The approximated solution 𝑣^𝛿_𝛽 corresponding to the measured data 𝑓_𝛿 is denoted by 𝑣^𝛿_𝛽𝑦=12_𝛾^+∞𝑒^𝑦√𝜆+𝑒^{√−𝑦𝜆}𝟏_[𝛾,𝛽]𝑑𝐸_𝜆𝑓_𝛿. (28) For simplicity, we denote the solution of problem (1) by 𝑢(𝑦), and the regularized solution associated to the data 𝑓_𝛿 by 𝑣^𝛿_𝛽(𝑦).

Our first main theorem is the following theorem.

Theorem 13. The solution defined in (27) depends continuously in 𝒞([0,𝐿];𝐻) on the data 𝑓; that is, if 𝑢¹_𝛽 and 𝑢²_𝛽 are two regularized solutions corresponding to 𝑓₁ and 𝑓₂, respectively, then one has 𝑢¹_𝛽𝑦−𝑢²_𝛽𝑦≤𝑒^𝑦𝛽𝑓₁−𝑓₂. (29)

This inequality implies that the solution of the regularized problem (27) depends continuously on the data 𝑓.

Now we compute the difference between the original solution 𝑢=𝑢(𝑦;𝑓) and the approximate solution 𝑣^𝛿_𝛽=𝑣^𝛿_𝛽(𝑦;𝑓_𝛿).

Theorem 14. Let 𝑢∈𝐶([0,𝐿];𝐻) be a solution of problem (1) with the exact data 𝑓∈𝐻; then the following estimate holds: 𝑢𝑦−𝑢_𝛽𝑦≤2𝑒^{(𝐿−𝑦)𝛽}‖𝑢(𝐿)‖. (30)

Proof. From relations (25) and (27) we have 𝑢𝑦−𝑢_𝛽(𝑡)=_𝛽^+∞cosh𝑦√𝜆𝑑𝐸_𝜆𝑓=_𝛾^+∞cosh𝑦√𝜆𝟏_[𝛽,+∞]𝑑𝐸_𝜆𝑓, (31) then 𝑢𝑦−𝑢_𝛽𝑦=_𝛾^+∞cosh𝑦√𝜆cosh𝐿√𝜆𝟏_[𝛽,+∞]cosh𝐿√𝜆𝑑𝐸_𝜆𝑓,𝑢(𝑦)−𝑢_𝛽(𝑦)²≤_𝛾^+∞cosh𝑦√𝜆cosh𝐿√𝜆𝟏_[𝛽,+∞]²cosh𝐿√𝜆²𝑑𝐸_𝜆𝑓². (32)
Using the inequality cosh𝑦√𝜆cosh𝐿√𝜆𝟏_[𝛽,+∞]²≤4𝑒^{2(𝐿−𝑦)𝛽}_𝛽+∞,cosh²𝐿√𝜆𝑑𝐸_𝜆𝑓²≤‖𝑢(𝐿)‖², (33) we derive 𝑢𝑦−𝑢_𝛽𝑦²≤4𝑒^{2(𝐿−𝑦)𝛽}‖𝑢(𝐿)‖². (34)
Using (29), (30) and the triangle inequality, we obtain 𝑢𝑦−𝑣^𝛿_𝛽𝑦≤𝑢𝑦−𝑢_𝛽𝑦+𝑢_𝛽𝑦−𝑣^𝛿_𝛽𝑦≤2𝑒^{(𝐿−𝑦)𝛽}‖𝑢(𝐿)‖+𝑒^𝑦𝛽𝛿. (35)
This completes the proof.

Remark 15. If we choose 𝛽=(1/𝐿)log(𝑀/𝛿), where ‖𝑢(𝐿)‖=𝑀, then we have the error bound 𝑢𝑦−𝑣^𝛿_𝛽𝑦≤3𝑀^{(𝐿−𝑦)/𝐿}𝛿^𝑦/𝐿. (36)
From (36) we see that (28) is an approximation of the exact solution 𝑢(𝑦). The approximation error depends continuously on the measurement error for fixed 0<𝑦<𝐿. However, as 𝑦→𝐿, the accuracy of the regularized solution becomes progressively lower. Consequently, we have not any information about the continuous dependence of the solution if 𝑦 is close to 𝐿.
In the theory of ill-posed Cauchy problems, we can often obtain continuous dependence on the data for the closed interval [0,𝐿] by assuming additional smoothness and using a stronger norm.
Now we show two error estimates under the following conditions:(H1)𝑢(𝐿)∈𝐷(𝐴^𝑝),(H2)𝑢(𝐿)∈𝒢_𝑝, 𝑝>0.

Remark 16. In practice, we know that it is very difficult to verify the conditions (H1) and (H2), so we give different assumptions on the given data 𝑓 as follows: 𝑢(𝐿)∈𝐷𝐴^𝑝⟺_𝛾^+∞𝜆^2𝑝cosh²𝐿√𝜆𝑑𝐸_𝜆𝑓²⟺<∞_𝛾+∞𝜆^2𝑝𝑒^√2𝐿𝜆𝑑𝐸_𝜆𝑓²𝑢(𝐿)<∞,∈𝒢_𝑝⟺_𝛾^+∞𝑒^{√2𝑝𝐿𝜆}cosh²𝐿√𝜆𝑑𝐸_𝜆𝑓²⟺<∞_𝛾+∞𝑒^{√2(1+𝑝)𝐿𝜆}𝑑𝐸_𝜆𝑓²<∞⟺𝑓∈𝒢_𝑝+1. (37)

Theorem 17. If ∫_𝛾^+∞𝜆^𝑝𝑒^√2𝐿𝜆𝑑‖𝐸_𝜆𝑓‖²<𝐸²₁ (resp., ∫_𝛾^+∞𝑒^{√2(1+𝑞)𝐿𝜆}𝑑‖𝐸_𝜆𝑓‖²<𝐸²₂), 𝑝>0, 𝑞>0, then one has the following estimates: 𝑢𝑦−𝑣^𝛿_𝛽𝑦≤𝐿𝑎^𝑝𝐸₁1log𝛿^−𝑝+𝛿^{1−𝑦𝑎/𝐿}𝑢𝑦,0<𝑎≤1,−𝑣_𝛽𝑦≤𝑒^−𝑞𝛽𝐸₂+𝑒^𝑦𝛽𝛿. (38)

Proof. From the expansions 𝑢𝑦=^∞_𝛾cosh𝑦√𝜆𝑑𝐸_𝜆𝑢𝑓,_𝛽𝑦=^∞_𝛾cosh𝑦√𝜆𝟏_[𝛾,𝛽]𝑑𝐸_𝜆𝑓, (39) we have 𝑢𝑦−𝑢_𝛽𝑦=^∞_𝛾cosh𝑦√𝜆𝟏_[𝛽,+∞]𝑑𝐸_𝜆𝑓. (40)
Then 𝑢𝑦−𝑢_𝛽𝑦²=^∞_𝛾𝜆^−𝑝/2𝟏_[𝛽,+∞]²cosh²𝑦√𝜆𝜆^𝑝𝑑𝐸_𝜆𝑓²≤^∞_𝛾𝜆^−𝑝/2𝟏_[𝛽,+∞]²𝑒^√2𝐿𝜆𝜆^𝑝𝑑𝐸_𝜆𝑓²≤𝛽^−𝑝∞_𝛾𝑒^√2𝐿𝜆𝜆^𝑝𝑑𝐸_𝜆𝑓²≤𝛽^−2𝑝𝐸²₁. (41) Using Theorem 17 and the triangle inequality, we can write 𝑢𝑦−𝑣^𝛿_𝛽𝑦≤𝑢𝑦−𝑢_𝛽𝑦+𝑢_𝛽𝑦−𝑣^𝛿_𝛽𝑦≤𝛽^−𝑝𝐸₁+𝑒^𝑦𝛽𝛿. (42) By choosing 𝛽=(𝑎/𝐿)log(1/𝛿), we obtain the desired inequality.
Using the same techniques we have 𝑢𝑦−𝑢_𝛽𝑦=^∞_𝛾𝑒^{√−𝑞𝜆}cosh𝑦√𝜆𝑒^𝑞√𝜆𝟏_[𝛽,+∞]𝑑𝐸_𝜆𝑓, (43) hence 𝑢𝑦−𝑢_𝛽𝑦²=^∞_𝛾𝑒^{√−𝑞𝜆}𝟏_[𝛽,+∞]²cosh(𝑦√𝜆)𝑒^{√2𝑞𝜆2}𝑑𝐸_𝜆𝑓²≤𝑒^{−2𝑞𝛽∞}_𝛾𝑒^√2𝑞𝜆𝑑𝐸_𝜆𝑢(𝑦)²≤𝑒^−2𝑞𝛽𝐸²₂. (44) Using (29) and the triangle inequality, we obtain 𝑢𝑦−𝑣^𝛿_𝛽𝑦≤𝑢𝑦−𝑢_𝛽𝑦+𝑢_𝛽𝑦−𝑣^𝛿_𝛽𝑦≤𝑒^−𝑞𝛽𝐸₂+𝑒^𝑦𝛽𝛿. (45) By choosing 𝛽=(𝑎/𝐿)log(1/𝛿), we obtain 𝑢𝑦−𝑣^𝛿_𝛽𝑦≤𝛿^{𝑎𝑞/𝐿}𝐸₂+𝛿^{1−𝑎𝑦/𝐿}. (46)

3.2. The Mollification Method

Now, we approximate the original problem (1) by the sequence of problems 𝑢_𝑦𝑦𝑢(0)=𝐴𝑢,0<𝑦<𝐿,=𝑓_𝛼=𝑀_𝛼𝑢𝑓,_𝑦(0)=0. (47)

Theorem 18. If 𝑓∈𝐻 the approximate Cauchy problem (47) admits a unique solution 𝑢_𝛼, which depends continuously upon the data 𝑓 with respect to uniform topology of 𝐶([0,𝐿];𝐻).

Proof. From the representation 𝑢_𝛼𝑦=cosh𝑦√𝐴𝑓_𝛼=_𝛾^+∞cosh𝑦√𝜆1+𝛼𝑒^{√𝑝𝐿𝜆−1}𝑑𝐸_𝜆𝑓, (48) we have 𝑢_𝛼(𝑦)²=_𝛾^+∞cosh(𝑦√𝜆)1+𝛼𝑒^{√𝑝𝐿𝜆2}𝑑𝐸_𝜆𝑓²≤_𝛾^+∞𝑒^𝐿√𝜆1+𝛼𝑒^{√𝑝𝐿𝜆2}𝑑𝐸_𝜆𝑓². (49)(i)If 𝑝=1, we obtain sup_{𝑦∈[0,𝐿]}𝑢_𝛼(𝑦)²≤1𝛼𝑓. (50)(ii)If 𝑝>1, the function 𝑀(𝑠)=𝑒^𝐿𝑠/(1+𝛼𝑒^𝑝𝐿𝑠) with 𝑠=√𝜆≥√𝛾 achieves its maximum at 𝑠^∗=(1/𝑝𝐿)log(1/𝛼(𝑝−1)), 𝑝>1, from which we deduce 𝑀_∞=𝑀𝑠^∗𝑝1=𝑐𝛼^1/𝑝,𝑐𝑝=𝑝⁻¹𝑝−1^1−1/𝑝≤1. (51)
From this bound, we derive sup_{𝑦∈[0,𝐿]}𝑢_𝛼𝑦≤1𝛼^1/𝑝𝑓. (52)
From the linear property of our problem, stability estimate of problem (47) may be written precisely in the following corollary.

Corollary 19. If 𝑢_𝛼,1(𝑦;𝑓₁) (resp., 𝑢_𝛼,2(𝑦;𝑓₂)) is the approximate solution corresponding to 𝑓₁ (resp., 𝑓₂), then sup_{𝑦∈[0,𝐿]}𝑢_𝛼,1𝑦−𝑢_𝛼,2𝑦≤1𝛼^1/𝑝𝑓₁−𝑓₂. (53)

Remark 20. We have 𝑁(𝑠)=𝑠^𝑟𝑒^𝑇𝑠1+𝛼𝑒^𝑝𝑇𝑠≤1𝛼𝐾(𝑠)=1𝛼𝑠^𝑟𝑒^{−(𝑝−1)𝑇𝑠}√,𝑝>1,𝑠=𝜆≥√𝛾. (54) It is easy to show that 𝐾(𝑠)≤𝐾𝑠^∗=𝑟𝐿=𝑟𝑝−1𝐿𝑝−1^𝑟𝑒^−𝑟=𝜅𝑟,𝑝,𝐿<∞. (55)
This remark shows that 𝑢_𝛼(𝑦)∈𝐷(𝐴^𝑟/2) for all 𝑦∈[0,𝐿].

Proof. The inclusion 𝑢_𝛼(𝑦)∈𝐷(𝐴^𝑟/2) is equivalent to ‖𝐴^𝑟/2𝑢_𝛼(𝑦)‖<∞. We have 𝐴^𝑟/2𝑢_𝛼𝑦²=_𝛾^+∞𝜆^𝑟cosh𝑦√𝜆1+𝛼𝑒^{√𝑝𝐿𝜆2}𝑑𝐸_𝜆𝑓²≤1𝛼²_𝛾^+∞√𝜆^𝑟𝑒^{√−(𝑝−1)𝐿𝜆2}𝑑𝐸_𝜆𝑓²≤1𝛼²𝜅𝑟,𝑝,𝐿²𝑓²<∞, (56) where 𝜅(𝑟,𝑝,𝐿)=sup_𝜆≥𝛾√𝜆^𝑟𝑒^{√−(𝑝−1)𝐿𝜆}=𝑟(𝑝−1)𝐿^𝑟𝑒^−𝑟.

Theorem 21. If 𝑓∈𝒢₁, then sup_{𝑦∈[0,𝐿]}𝑢𝑦−𝑢_𝛼𝑦⟶0,𝛼⟶0. (57)

Proof. We compute 𝑢(𝑦)−𝑢_𝛼𝑦²=_𝛾^+∞1−𝑀_𝛼(𝜆)²cosh²𝑦√𝜆𝑑𝐸_𝜆𝑓²≤_𝛾^+∞1−𝑀_𝛼(𝜆)²cosh²𝐿√𝜆𝑑𝐸_𝜆𝑓²≤_𝛾^+∞1−𝑀_𝛼(𝜆)²𝑒^√2𝐿𝜆𝑑𝐸_𝜆𝑓²=𝐼−𝑀_𝛼𝑓², (58) where 𝑓=𝑒^𝐿√𝐴𝑓 and ‖𝑓‖²=∫_𝛾^+∞𝑒^√2𝐿𝜆𝑑‖𝐸_𝜆𝑓‖²<∞.
This implies that sup_{𝑦∈[0,𝐿]}‖𝑢(𝑦)−𝑢_𝛼(𝑦)‖≤‖(𝐼−𝑀_𝛼)𝑓‖ and by virtue of (3) of Theorem 9, we conclude the desired convergence.

The following technical lemmas play the key role in our analysis and calculations.

Lemma 22. Let [[=1𝜈,+∞∋𝑠⟼𝑄𝑎,𝑟,𝑞,𝐿;𝑠𝛼𝑠^𝑟+𝑎𝑒^{−𝑞𝐿𝑠}, (59) where 𝑎>0, 𝛼>0, 𝜈>0, 𝑞>0, 𝐿>0, and 𝑟≥1. Then one has 𝑄≤1𝑎,𝑟,𝑞,𝐿;𝑠𝛼𝑘₁log𝑘₂(1/𝛼)^𝑟, (60) where 𝑘₁(𝑟,𝑞,𝐿)=𝑟𝑞𝐿, 𝑘₂(𝑞,𝑟,𝐿,𝑎)=𝑞^𝑟𝐿^𝑟−1𝑎/𝑟.

Proof. Differentiating the expression and setting the derivative equal to zero, we find 𝑑𝑄=𝑑𝑠𝑎,𝑟,𝑞,𝐿;𝑠−1𝛼𝑠^𝑟+𝑎𝑒^{−𝑞𝐿𝑠2}𝛼𝑟𝑠^𝑟−1−𝑞𝑎𝑒^{−𝑞𝐿𝑠}=0. (61) The function (𝑑/𝑑𝑠)𝑄({𝑎,𝑟,𝑞,𝐿};𝑠)=0 admits a unique solution 𝑠̂=𝑠⟼𝛼𝑟𝑠^𝑟−1∩𝑠⟼𝑞𝑎𝑒^{−𝑞𝐿𝑠}. (62) Therefore 𝑄;𝑠̂≤1𝑎,𝑟,𝑞,𝐿;𝑠≤𝑄𝑎,𝑟,𝑞,𝐿𝛼𝑠^𝑟̂+𝑎𝑒^−𝑞𝐿𝑠̂≤1𝛼𝑠^𝑟̂. (63) We have 𝛼𝑟𝑠̂^𝑟−1−𝑞𝑎𝑒^−𝑞𝐿𝑠̂=0⟺𝑠^𝑟−1𝑒^𝑞𝐿𝑠̂=𝑞𝑎.𝑟𝛼 (64) By using the inequality (𝑒^𝑡≥𝑡, 𝑡≥0), then for 𝑡=𝑞𝐿𝑠̂, we obtain 𝑒^𝑞𝐿𝑠̂≥𝑞𝐿𝑠̂ and we can write 𝑞𝑎=𝑠̂𝑟𝛼^𝑟−1𝑒^𝑞𝐿𝑠̂≤𝑒^𝑞𝐿𝑠̂𝑒^𝑞𝐿𝑠̂𝑞𝐿^𝑟−1=1𝑞𝐿^𝑟−1𝑒^𝑟𝑞𝐿𝑠,̂ (65) which implies that 𝑠̂≥(1/𝑟𝑞𝐿)log((𝑞^{𝑟𝐿𝑟−1𝑎}/𝑟)(1/𝛼)). Hence, we obtain 𝑄≤1𝑎,𝑟,𝑞,𝐿;𝑠𝛼𝑠^𝑟̂≤1𝛼𝑘₁log𝑘₂(1/𝛼)^𝑟, (66) where 𝑘₁(𝑟,𝑞,𝐿)=𝑟𝑞𝐿, 𝑘₂(𝑞,𝑟,𝐿,𝑎)=𝑞^𝑟𝐿^𝑟−1𝑎/𝑟.

Lemma 23. Let [[=𝑒𝜈,+∞∋𝑠⟼𝑅𝑝,𝑞,𝐿;𝑠_𝑝𝐿𝑠1+𝛼𝑒^𝑝𝐿𝑠𝑒^𝑞𝐿𝑠=1𝑒^{(𝑞−𝑝)𝐿𝑠}+𝛼𝑒^𝑞𝐿𝑠, (67) where 𝑝≥1, 𝑞>0, 𝛼>0, 𝜈>0, and 𝐿>0. Then one has the following.
If 1≤𝑝≤𝑞, then 𝑅𝑝,𝑞,𝐿;𝑠≤𝑒^{−(𝑞−𝑝)𝐿𝑠}≤𝑒^{−(𝑞−𝑝)𝐿𝜈}≤1. (68)
If 0<𝑞<𝑝, 𝑝≥1, 0<𝛼≤(𝑝−𝑞)/𝑞, then 𝑅𝑝,𝑞,𝐿;𝑠≤𝑘₃1𝛼^{(𝑝−𝑞)/𝑝},𝑘₃=𝑞𝑝,𝑞𝑝𝑝−𝑞𝑝^{(𝑝−𝑞)/𝑝}≤1. (69)

Proof. By a simple differential calculus, we show that the function 𝑅({𝑝,𝑞,𝐿};𝑠) achieves its maximum at 𝑠̂=(1/𝑝𝐿)log((𝑝−𝑞)/𝛼𝑞). Consequently 𝑅𝑝,𝑞,𝐿;𝑠≤𝑅𝑝,𝑞,𝐿;𝑠̂=𝑘₃1𝛼^{(𝑝−𝑞)/𝑝}. (70)

Now we assume the following a priori bounds hold: 𝑢(𝐿)∈𝐷𝐴^𝑟/2⟺_𝛾^+∞√𝜆^2𝑟𝑒^√2𝐿𝜆𝑑𝐸_𝜆𝑓²≤𝐸²₁𝑢(𝐿)<∞,∈𝒢_𝑞⟺_𝛾^+∞𝑒^{√2𝐿(1+𝑞)𝜆}𝑑𝐸_𝜆𝑓²≤𝐸²₂<∞. (71)(72)

Theorem 24. Let 𝑢 (resp., 𝑢_𝛼) be the solution of problem (1) (resp., (47)) with the exact data 𝑓. If (71) (resp., (72)) is satisfied, then one has the following error estimates: 𝑢𝑦−𝑢_𝛼𝑦1=𝑂log(1/𝛼)^𝑟,𝑢𝑦−𝑢_𝛼𝑦=𝑂(𝛼)𝑂𝛼,𝑖𝑓1≤𝑝≤𝑞,_𝑞/𝑝,𝑖𝑓0<𝑞<𝑝,𝑝≥1. (73)(74)

Proof. Putting 𝐵₁(𝜆)=𝑒^{√𝑝𝐿𝜆}1+𝛼𝑒^{√𝑝𝐿𝜆}1√𝜆^𝑟=1√𝜆^𝑟𝑒^{√−𝑝𝐿𝜆}+𝛼√𝜆^𝑟≤𝐵₂(𝜆)=1√𝛾^𝑟𝑒^{√−𝑝𝐿𝜆}+𝛼√𝜆^𝑟,𝐵₃(𝜆)=𝑒^{√𝑝𝐿𝜆}1+𝛼𝑒^{√𝑝𝐿𝜆}1𝑒^{√𝑞𝐿𝜆}=1𝑒^{√(𝑞−𝑝)𝐿𝜆}+𝛼𝑒^{√𝑞𝐿𝜆}. (75) Using the change of variables 𝑠=√𝜆, we obtain the new expressions 𝐵₂(𝑠)=1√𝛾^𝑟𝑒^{−𝑝𝐿𝑠}+𝛼𝑠^𝑟,𝐵₃(𝑠)=1𝑒^{(𝑞−𝑝)𝐿𝑠}+𝛼𝑒^𝑞𝐿𝑠. (76)(77)
By virtue of Lemma 22 (inequality (60) and Lemma 23 (inequalities (68) and (69)), we can write 𝐵₂(𝑠)≤1𝛼𝑘₁log𝑘₂(1/𝛼)^𝑟, (78) where 𝑘₁(𝑟,𝑝,𝐿)=𝑟𝑞𝐿, 𝑘₂(𝑝,𝑟,𝐿,√𝛾^𝑟)=𝑞^𝑟𝐿^𝑟−1√𝛾^𝑟/𝑟. Consider 𝐵₃(𝑠)≤11,if1≤𝑝≤𝑞,𝛼^{(𝑝−𝑞)/𝑝},if0<𝑞<𝑝,𝑝≥1. (79)
We have 𝑢(𝑦)−𝑢_𝛼(𝑦)²=_𝛾^+∞𝛼𝑒^{√𝑝𝐿𝜆}1+𝛼𝑒^{√𝑝𝐿𝜆2}cosh²𝑦√𝜆𝑑𝐸_𝜆𝑓²≤𝛼²_𝛾^+∞𝐵₂(𝜆)²√𝜆^2𝑟𝑒^√2𝐿𝜆𝑑𝐸_𝜆𝑓²≤𝛼²sup_{√𝑠≥𝛾}𝐵₂(𝑠)²𝐸²₁,𝑢(𝑦)−𝑢_𝛼(𝑦)²=_𝛾^+∞𝛼𝑒^{√𝑝𝐿𝜆}1+𝛼𝑒^{√𝑝𝐿𝜆2}cosh²𝑦√𝜆𝑑𝐸_𝜆𝑓²≤𝛼²_𝛾^+∞𝐵₃(𝜆)²𝑒^{√𝐿(1+𝑞)𝜆}𝑑𝐸_𝜆𝑓²≤𝛼²sup_{√𝑠≥𝛾}𝐵₃(𝑠)²𝐸²₂. (80) Using (78) and (79), we derive 𝑢𝑦−𝑢_𝛼𝑦1≤𝛼𝛼𝑘₁log𝑘₂(1/𝛼)^𝑟1=𝑂log(1/𝛼)^𝑟,𝑢𝑦−𝑢_𝛼𝑦≤𝛼𝛼,if1≤𝑝≤𝑞,_𝑞/𝑝,if0<𝑞<𝑝,𝑝≥1. (81)

Combining (53), (73), and (74) with the help of triangle inequality 𝑢𝑦−𝑢^𝛿_𝛼𝑦≤𝑢𝑦−𝑢_𝛼𝑦+𝑢_𝛼𝑦−𝑢^𝛿_𝛼𝑦=Δ₁+Δ₂, (82) we deduce the following corollary.

Corollary 25. Let 𝑢(𝑦;𝑓) (resp., 𝑢^𝛿_𝛼(𝑦;𝑓_𝛿)) be the solution of problem (1) (resp., (47)) with the exact data 𝑓 (resp., the inexact data 𝑓_𝛿) such that ‖𝑓−𝑓_𝛿‖≤𝛿. If (71) (resp., (72)) is satisfied, then one has the following error estimates: ()𝑢𝑦𝑐𝑎𝑠𝑒𝑟≥1−𝑢^𝛿_𝛼𝑦=𝑂𝜃₁(𝛼+1)𝛼^1/𝑝𝑢𝑦𝛿,𝑐𝑎𝑠𝑒1≤𝑝≤𝑞−𝑢^𝛿_𝛼𝑦=𝑂𝜃₂(𝛼)+1𝛼^1/𝑝𝑢𝑦𝛿,𝑐𝑎𝑠𝑒0<𝑞<𝑝,𝑝≥1−𝑢^𝛿_𝛼𝑦=𝑂𝜃₃(𝛼)+1𝛼^1/𝑝𝛿, (83)(84)(85) where 𝜃₁(𝛼)1=𝑂log(1/𝛼)^𝑟,𝜃₂(𝛼)=𝑂(𝛼),𝜃₂(𝛼)=𝑂𝛼^𝑞/𝑝. (86)

If we choose 𝛼=𝛼(𝛿)=𝛿^𝑝/𝜔 with 𝜔>1, then we have 𝛿1𝛿^{𝑝/𝜔1/𝑝}=𝛿^{(𝜔−1)/𝑤},𝜃₁(𝛼)1=𝑂log1/𝛿^{𝑝/𝜔𝑟},𝜃₂(𝛼)=𝛿^𝑝/𝜔,𝜃₃(𝛼)=𝑂𝛿^𝑞/𝜔. (87)(88)(89)

3.3. Example: Cauchy Problem for the Modified Helmholtz Equation

In this paragraph, we give a concrete example to see how to apply the theoretical results developed in this study.

Let us consider the Cauchy problem (modified Helmholtz equation) in the infinite strip ℝ×(0,1): 𝑢_𝑦𝑦𝑥,𝑦+𝑢_𝑥𝑥(),𝑢()(𝑥)𝑥,𝑦−𝛾𝑢𝑥,𝑦=0,𝑥∈ℝ,𝑦∈0,1𝑥,0=𝑓,𝑢_𝑦(𝑥,0)=0,𝑥∈ℝ, (90) where 𝛾 is a real positive constant.

Let 𝑢(𝜉,𝑦)=(ℱ𝑢)(𝜉,𝑦) be the Fourier transform of 𝑢(𝑥,𝑦): 𝑢=1𝜉,𝑦√2𝜋^ℝ𝑒^{−𝑖𝜉𝑥}𝑢𝑥,𝑦𝑑𝑥. (91) With the help of the Fourier transformation, problem (1) can be transformed to an equivalent problem in the frequency domain: 𝑢_𝑦𝑦𝜉,𝑦−𝜉²𝑢𝑢(),𝑢()=𝑓(𝜉),𝑢𝜉,𝑦−𝛾𝜉,𝑦=0,𝜉∈ℝ,𝑦∈0,1𝜉,0_𝑦(𝜉,0)=0,𝜉∈ℝ. (92) It is easy to check that the formal solution of problem (92) has the form 𝑢𝜉,𝑦=cosh𝑦𝜉²+𝛾𝑓(𝜉), (93) or equivalently, the formal solution of problem (90) is given by 𝑢𝑥,𝑦=ℱ⁻¹𝑢=1𝑥,𝑦√2𝜋^ℝ𝑒^𝑖𝑥𝜉𝑢=1𝜉,𝑦𝑑𝜉√2𝜋^ℝ𝑒^𝑖𝑥𝜉cosh𝑦𝜉²+𝛾𝑓(𝜉)𝑑𝜉. (94)

Putting Θ(𝜉)=(𝜉²+𝛾). Then Θ(𝜉)→+∞ as |𝜉|→+∞. From this remark, it is easy to see that a small perturbation in the data 𝑓(𝜉) may cause a dramatically large error in the solution 𝑢(𝜉,𝜉). In addition, the magnifying factor is Θ(𝜉)∼𝑒^|𝜉|, hence, the problem is severely ill-posed.

Since the data 𝑓(⋅) are based on (physical) observations and are not known with complete accuracy, we assume that 𝑓 and 𝑓_𝛿 satisfy 𝑓−𝑓_𝛿≤𝛿, (95) where 𝑓 and 𝑓_𝛿 belong to 𝐿²(ℝ), 𝑓_𝛿 denotes the measured data, and 𝛿 denotes the noise level.

For this problem, we define the regularized solutions with noisy data 𝑓_𝛿: 𝑢^𝛿_𝑁=1𝑥,𝑦√2𝜋^ℝ𝑒^𝑖𝑥𝜉cosh𝑦𝜉²+𝛾𝑓_𝛿(𝜉)𝟏_{[−𝑁,𝑁]}(𝜉)=1𝑑𝜉√2𝜋^𝑁−𝑁𝑒^𝑖𝑥𝜉cosh𝑦𝜉²+𝛾𝑓_𝛿(𝜉)𝑑𝜉, (96) where 𝟏_{[−𝑁,𝑁]} is the characteristic function of the interval [−𝑁,𝑁]. Consider 𝑢^𝛿_𝛼=1𝑥,𝑦√2𝜋^ℝ𝑒^𝑖𝑥𝜉cosh𝑦𝜉²+𝛾1+𝛼𝑒^{𝑝(𝜉2+𝛾)}𝑓_𝛿(𝜉)𝑑𝜉, (97) where 𝑝≥1. The quantities 𝛼=𝛼(𝛿) and 𝑁=𝑁(𝛿) are the parameters which were defined in Sections 3.1 and 3.2.

4. The Nonlocal Boundary Value Problem Method and Some Extensions

In this section we give the connection between the mollification method and the nonlocal boundary value problem method; also we give some extensions to our investigation.

4.1. The Nonlocal Boundary Value Problem Method

Consider the following problem: 𝑃𝑎()𝑢𝑎≥1;𝛼,𝛽𝑢(0^𝑦𝑦)()=𝐴𝑢,0<𝑦<𝑎𝐿,+𝛼𝑢𝑎𝐿+𝛽𝐴^−1/2𝑢_𝑦()𝑢𝑎𝐿=𝑓,_𝑦(0)=0. (98)(i)The case 𝛽=0 coincides with the method treated by Hào et al. in [10]. (ii)In the case 𝛼=𝛽, the solution of 𝑃(𝑎;𝛼,𝛼) is 𝑢_𝛼(𝑦)=cosh(𝑦√𝐴)(𝐼+𝛼𝑒^𝑎𝐿)⁻¹𝑓 coincides with the solution resulting from the mollification method. (iii)The error estimates obtained in our analysis by using the mollification method are similar to those obtained in [8, 10].

This shows that our study framework is more general and includes many results obtained in this direction.

4.2. Generalization

Let us consider 𝑃𝑢_𝑦𝑦𝑢(0)𝑢=𝐴𝑢,0<𝑦<𝐿,=𝑓,_𝑦(0)=0, (99) where 𝐴 is a self-adjoint, linear unbounded operator in 𝐻 and changes the sign with 0∈𝜌(𝐴) (𝐴⁻¹ exists and 𝐴⁻¹∈ℒ(𝐻)).

We assume 𝜎(𝐴)=]−∞,−𝛾]∪[𝛾,+∞[, 𝛾>0. The spectral theory of self-adjoint operators enables us to write ℎ=_ℝ𝑑𝐸_𝜆ℎ=^−𝛾_−∞𝑑𝐸_𝜆ℎ+_𝛾^+∞𝑑𝐸_𝜆ℎ=ℎ₋+ℎ₊,ℎ∈𝐻; (100)

that is, the Hilbert space 𝐻 decomposes into the direct sum 𝐻=𝐻₋⊕𝐻₊, and 𝐴=_ℝ𝜆𝑑𝐸_𝜆=^−𝛾_−∞𝜆𝑑𝐸_𝜆+_𝛾^+∞𝜆𝑑𝐸_𝜆=𝐴₋+𝐴₊. (101)

This decomposition gives us two problems: one is elliptic (ill-posed) and the other is hyperbolic (well-posed). Consider EllipticProblem𝑈𝑦𝑦=𝐴₊𝑈(0)𝑈,0<𝑦<𝐿,=𝑓₊,𝑈_𝑦(0)=0,HyperbolicProblem𝑉𝑦𝑦=𝐴₋𝑉(0)𝑉,0<𝑦<𝐿,=𝑓₋,𝑉_𝑦(0)=0. (102)

The formal solution of problem (99) is 𝑢𝑦𝑦𝑦,𝑈𝑦𝑦=𝑈+𝑉=cosh𝐴₊𝑓₊,𝑉𝑦=cos𝑦−𝐴₋𝑓₋. (103)

We define 𝑢_𝛼=cos𝑦−𝐴₋𝑓₋+cosh𝑦𝐴₊𝐼+𝛼𝑒^{𝑝𝐿𝐴+−1}𝑓₊=^−𝛾_−∞cos𝑦√−𝜆𝑑𝐸_𝜆𝑓+_𝛾^+∞cosh𝑦√𝜆1+𝛼𝑒^{√𝑝𝐿𝜆−1}𝑑𝐸_𝜆𝑣𝑓,_𝛼=cos𝑦−𝐴₋𝑓₋+cosh𝑦𝐴₊𝐸_𝛽𝑓₊=^−𝛾_−∞cos𝑦√−𝜆𝑑𝐸_𝜆𝑓+_𝛾^+∞cosh𝑦√𝜆𝟏_[𝛾,𝛽]𝑑𝐸_𝜆𝑓. (104) We follow the same methodology developed in the previous Sections 3.1 and 3.2, we show that 𝑢_𝛼 and 𝑣_𝛼 are two stable approximations to problem (99), and we establish the same results of error estimates.

Remark 26. We define the mollification operator 𝑀_𝛼=(𝐼+𝛼Θ_𝑝(𝐴))⁻¹, where Θ_𝑝∶[𝛾,+∞[→∞ satisfies ∀𝜆∈𝛾,+∞,Θ_𝑝(𝜆)≥𝑒^{√𝑝𝐿𝜆},𝑝≥1. (105) Under certain conditions on Θ_𝑝(⋅) and with the help of a technical calculation, we can extend the results obtained in Section 3.2.

Acknowledgments

The authors are grateful to the editor and the anonymous referees for their valuable comments and helpful suggestions which have much improved the presentation of the paper. This work is supported by the MESRS of Algeria (CNEPRU Project B01120090003).