Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data The work of MVK and AEK was supported by US Army Research Laboratory and US Army Research Office grant W911NF-15-1-0233 and by the Office of Naval Research grant N00014-15-1-2330.

# Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data1

## Abstract

A new numerical method is proposed for a 1-D inverse medium scattering problem with multi-frequency data. This method is based on the construction of a weighted cost functional. The weight is a Carleman Weight Function (CWF). In other words, this is the function, which is present in the Carleman estimate for the undelying differential operator. The presence of the CWF makes this functional strictly convex on any a priori chosen ball with the center at in an appropriate Hilbert space. Convergence of the gradient minimization method to the exact solution starting from any point of that ball is proven. Computational results for both computationally simulated and experimental data show a good accuracy of this method.

Key Words: global convergence, coefficient inverse problem, multi-frequency data, Carleman weight function

2010 Mathematics Subject Classification: 35R30.

## 1 Introduction

The experimental data used in this paper were collected by the Forward Looking Radar of the US Army Research Laboratory [40]. That radar was built for detection and possible identification of shallow explosive-like targets. Since targets are three dimensional objects, one needs to measure a three dimensional information about each target. However, the radar measures only one time dependent curve for each target, see Figure 5. Therefore, one can hope to reconstruct only a very limited information about each target. So, we reconstruct only an estimate of the dielectric constant of each target. For each target, our estimate likely provides a sort of an average of values of its spatially distributed dielectric constant. But even this information can be potentially very useful for engineers. Indeed, currently the radar community is relying only on the energy information of radar images, see, e.g. [47]. Estimates of dielectric constants of targets, if taken alone, cannot improve the current false alarm rate. However, these estimates can be potentially used as an additional piece of information. Being combined with the currently used energy information, this piece of the information might result in the future in new classification algorithms, which might improve the current false alarm rate.

An Inverse Medium Scattering Problem (IMSP) is often also called a Coefficient Inverse Problem (CIP). IMSPs/CIPs are both ill-posed and highly nonlinear. Therefore, an important question to address in a numerical treatment of such a problem is: How to reach a sufficiently small neighborhood of the exact coefficient without any advanced knowledge of this neighborhood? The size of this neighborhood should depend only on the level of noise in the data and on approximation errors. We call a numerical method, which has a rigorous guarantee of achieving this goal, globally convergent method (GCM).

In this paper we develop analytically a new globally convergent method for a 1-D Inverse Medium Scattering Problem (IMSP) with the data generated by multiple frequencies. In addition to the analytical study, we test this method numerically using both computationally simulated and the above mentioned experimental data.

First, we derive a nonlinear integro-differential equation in which the unknown coefficient is not present. The new element of this paper is the method of the solution of this equation. This method is based on the construction of a weighted least squares cost functional. The key point of this functional is the presence of the Carleman Weight Function (CWF) in it. This is the function, which is involved in the Carleman estimate for the underlying differential operator. We prove that, given a closed ball of an arbitrary radius with the center at in an appropriate Hilbert space, one can choose the parameter of the CWF in such a way that this functional becomes strictly convex on that ball.

The existence of the unique minimizer on that closed ball as well as convergence of minimizers to the exact solution when the level of noise in the data tends to zero are proven. In addition,  it is proven that the gradient projection method reaches a sufficiently small neighborhood of the exact coefficient if its starting point is an arbitrary point of that ball. The size of that neighborhood is proportional to the level of noise in the data. Therefore, since restrictions on are not imposed in our method, then this is a globally convergent numerical method. We note that in the conventional case of a non convex cost functional a gradient-like method converges to the exact solution only if its starting point is located in a sufficiently small neighborhood of this solution: this is due to the phenomenon of multiple local minima and ravines of such functionals.

Unlike previously developed globally convergent numerical methods of the first type for CIPs (see this section below), the convergence analysis for the technique of the current paper does not impose a smallness condition on the interval of the variations of the wave numbers .

The majority of currently known numerical methods of solutions of nonlinear ill-posed problems use the nonlinear optimization. In other words, a least squares cost functional is minimized in each problem, see, e.g. [14, 17, 18, 19]. However, the major problem with these functionals is that they are usually non convex. Figure 1 of the paper [44] presents a numerical example of multiple local minima and ravines of non-convex least squares cost functionals for some CIPs. Hence, convergence of the optimization process of such a functional to the exact solution can be guaranteed only if a good approximation for that solution is known in advance. However, such an approximation is rarely available in applications. This prompts the development of globally convergent numerical methods for CIPs, see, e.g. [8, 9, 10, 15, 23, 24, 25, 26, 29, 31, 32, 33, 34, 35, 39, 48].

The first author with coauthors has proposed two types of GCM for CIPs with single measurement data. The GCM of the first type is reasonable to call the “tail functions method”. This development has started from the work [8] and has been continued since then, see, e.g. [9, 15, 31, 32, 34, 35, 39, 48] and references cited therein. In this case, on each step of an iterative process one solves the Dirichlet boundary value problem for a certain linear elliptic PDE, which depends on that iterative step. The solution of this PDE allows one to update the unknown coefficient first and then to update a certain function, which is called “the tail function”. The convergence theorems for this method impose a smallness condition on the interval of the variation of either the parameter of the Laplace transform of the solution of a hyperbolic equation or of the wave number in the Helmholtz equation. Recall that the method of this paper does not impose the latter assumption.

In this paper we present a new version of the GCM of the second type. In any version of the GCM of the second type a weighted cost functional with a CWF in it is constructed. The same properties of the global strict convexity and the global convergence of the gradient projection method hold as the ones indicated above. The GCM of the second type was initiated in [24, 25, 26] with a recently renewed interest in [10, 29, 33]. The idea of any version of the GCM of the second type has direct roots in the method of [11], which is based on Carleman estimates and which was originally designed in [11] only for proofs of uniqueness theorems for CIPs, also see the recent survey in [27].

Another version of the GCM with a CWF in it was recently developed in [6] for a CIP for the hyperbolic equation where is the unknown coefficient. This GCM was tested numerically in [7]. In [6, 7] non-vanishing conditions are imposed: it is assumed that either or or in the entire domain of interest. Similar assumptions are imposed in [10, 29] for the GCM of the second type. On the other hand, we consider in the current paper, so as in [24, 25, 26, 33], the fundamental solution of the corresponding PDE. The differences between the fundamental solutions of those PDEs and solutions satisfying non-vanishing conditions cause quite significant differences between [24, 25, 26, 33] and [6, 7, 10, 29] of corresponding versions of the GCM of the second type.

Recently, the idea of the GCM of the second type was extended to the case of ill-posed Cauchy problems for quasilinear PDEs, see the theory in [28] and some extensions and numerical examples in [4, 30].

CIPs of wave propagation are a part of a bigger subfield, Inverse Scattering Problems (ISPs). ISPs attract a significant attention of the scientific community. In this regard we refer to some direct methods which successfully reconstruct positions, sizes and shapes of scatterers without iterations [12, 13, 20, 22, 36, 37, 38, 45]. We also refer to [3, 37, 41, 42] for some other ISPs in the frequency domain. In addition, we cite some other numerical methods for ISPs considered in [2, 5, 46].

As to the CIPs with multiple measurement, i.e. the Dirichlet-to-Neumann map data, we mention recent works [1, 21, 43] and references cited therein, where reconstruction procedures are developed, which do not require a priori knowledge of a small neighborhood of the exact coefficient.

In section 2 we state our inverse problem. In section 3 we construct that weighted cost functional. In section 4 we prove the main property of this functional: its global strict convexity. In section 5 we prove the global convergence of the gradient projection method of the minimization of this functional. Although this paper is mostly an analytical one (sections 3-5), we complement the theory with computations. In section 6 we test our method on computationally simulated data. In section 7 we test it on experimental data. Concluding remarks are in section 8.

## 2 Problem statement

### 2.1 Statement of the inverse problem

Let the function be the spatially distributed dielectric constant of the medium. We assume that

 c∈C2(R),c(x)≥1,∀x∈R, (2.1)
 c(x)=1,∀x∉(0,1). (2.2)

Fix the source position For brevity, we do not indicate below dependence of our functions on Consider the 1-D Helmholtz equation for the function ,

 Extra open brace or missing close brace (2.3)
 limx→∞(u′+iku)=0,limx→−∞(u′−iku)=0. (2.4)

Let be the solution of the problem (2.3), (2.4) for the case Then

 u0(x,k)=exp(−ik|x−x0|)2ik. (2.5)

Our interest is in the following inverse problem:

Inverse Medium Scattering Problem (IMSP). Let  be an interval of wavenumbers . Reconstruct the function  assuming that the following function  is known

 g0(k)=u(0,k)u0(0,k),k∈[k––,¯¯¯k]. (2.6)

Denote

 w(x,k)=u(x,k)u0(x,k). (2.7)

It follows from (2.6), (2.7) and [32] that

 w(0,k)=g0(k),k∈[k––,¯¯¯k], (2.8)
 w′(0,k)=g1(k)=2ik(g0(k)−1),k∈[k––,¯¯¯k]. (2.9)

### 2.2 Some properties of the solution of forward and inverse problems

In this subsection we briefly outline some results of [32], which we use below in this paper. Existence and uniqueness of the solution for each was established in [32]. Also, it was proven in [32] that

 u(x,k)≠0,∀x∈[0,1],∀k>0. (2.10)

In particular, In addition, uniqueness of our IMSP was proven in [32]. Also, the following asymptotic behavior of the function takes place:

 u(x,k)=12ikc1/4(x)exp⎡⎢⎣−ikx∫x0√c(ξ)dξ⎤⎥⎦(1+ˆu(x,k)),k→∞,∀x∈[0,1], (2.11)
 ˆu(x,k)=O(1k),∂kˆu(x,k)=O(1k2),k→∞. (2.12)

Given (2.10) and (2.11) we now can uniquely define the function as in [32]. The difficulty here is in defining since this number is usually defined up to the addition of where is an integer. For sufficiently large values of we define the function using (2.5), (2.7), (2.11) and (2.12) as

 Missing or unrecognized delimiter for \right (2.13)

where

 ˆw(x,k)=O(1k),∂kˆw(x,k)=O(1k2),k→∞. (2.14)

Hence, for sufficiently large ,

 |ˆw(x,k)|<2π, (2.15)

which eliminates the above mentioned ambiguity. Suppose that the number is so large that (2.15) is true for Then is defined as in (2.13). As to not large values of , we define the function (2.13) as

 ψ(x,k)=−¯¯¯k∫k∂kw(x,ξ)w(x,ξ)dξ+logw(x,¯¯¯k). (2.16)

By (2.10) Differentiating both sides of (2.16) with respect to , we obtain

 ∂kw(x,k)−w(x,k)∂kψ(x,k)=0. (2.17)

Multiplying both sides of (2.17) by , we obtain Hence, there exists a function independent on such that

 w(x,k)=C(x)eψ(x,k). (2.18)

Setting in (2.18) and using the fact that by (2.16) , we obtain

 C=C(x)=1,x∈[0,1]. (2.19)

Hence, (2.16) and (2.18) imply that is defined as

## 3 The Weighted Cost Functional

In this section we construct the above mentioned weighted cost functional with the CWF in it.

Lemma 3.1 (Carleman estimate). For any complex valued function  with  and for any parameter  the following Carleman estimate holds

 1∫0∣∣u′′∣∣2e−2λxdx≥C⎡⎢⎣1∫0|u′′|2e−2λxdx+λ1∫0|u′|2e−2λxdx+λ31∫0|u|2e−2λxdx⎤⎥⎦, (3.1)

where the constant is independent of  and

Proof. In the case when the integral with is absent in the right hand side of (3.1) this lemma was proved in [32]. To incorporate this integral, we note that

 21∫0∣∣u′′∣∣2e−2λxdx≥1∫0∣∣u′′∣∣2e−2λxdx+C⎡⎢⎣λ1∫0|u′|2e−2λxdx+λ31∫0|u|2e−2λxdx⎤⎥⎦. (3.2)

Let Then (3.2) implies (3.1) where is replaced with

### 3.1 Nonlinear integro-differential equation

For consider the function and its derivative , where

 v(x,k)=logw(x,k)k2, q(x,k)=∂kv(x,k). (3.3)

Hence,

 v(x,k)=−¯¯¯k∫kq(x,τ)dτ+v(x,¯¯¯k). (3.4)

Consider the function , which we call the “tail function”, and this function is unknown,

 V(x)=v(x,¯¯¯k). (3.5)

Let Note that since for then equation (2.3) and the first condition (2.4) imply that for Hence, (2.5) and (2.7) imply that for It follows from (2.3), (2.5), (2.7)–(2.9), (2.18) and (2.19) that

 w′′−2ikw′+k2β(x)w=0,x∈(0,1), (3.6)
 w(0,k)=g0(k),w′(0,k)=g1(k),w′(1,k)=0. (3.7)

Using (2.18), (2.19), (3.3) and (3.6), we obtain

 v′′+k2(v′)2−2ikv′=−β(x). (3.8)

Differentiate (3.8) with respect to and use (3.3)-(3.7). We obtain

 Missing or unrecognized delimiter for \right
 +2k⎛⎜ ⎜⎝−¯¯¯k∫kq′(x,τ)dτ+V′(x)⎞⎟ ⎟⎠2=0, x∈(0,1),k∈(k––,¯¯¯k), (3.9)
 q(0,k)=p0(k),q′(0,k)=p1(k),q′(1,k)=0,k∈(k––,¯¯¯k), (3.10)

where

 p0(k)=∂∂k(logg0(k)k2),p1(k)=∂∂k[2ik(1−1g0(k))],k∈[k––,¯¯¯k]. (3.11)

We have obtained an integro-differential equation (3.9) for the function with the overdetermined boundary conditions (3.10). The tail function is also unknown. First, we will approximate the tail function . Next, we will solve the problem (3.9), (3.10) for the function . To solve this problem, we will construct the above mentioned weighted cost functional with the CWF in it, see (3.1). This construction, combined with corresponding analytical results, is the central part of our paper. Thus, even though the problem (3.9)-(3.11) is the same as the problem (65), (66) in [32], the numerical method of the solution of the problem (3.9)-(3.11) is radically different from the one in [32].

Now, suppose that we have obtained approximations for both functions and . Then we obtain the unknown coefficient via backwards calculations. First, we calculate the approximation for the function via (3.4) and (3.5). Next, we calculate the function via (3.8). We have learned from our numerical experience that the best value of to use in (3.8) for the latter calculation is

### 3.2 Approximation for the tail function V(x)

The approximation for the tail function is done here the same way as the approximation for the so-called “first tail function” in section 4.2 of [32]. However, while tail functions are updated in [32], we are not doing such updates here.

It follows from (2.7)-(2.14) and (3.3)-(3.5) that there exists a function such that

 V(x,k)=r(x)k+O(1k2), q(x,k)=−r(x)k2+O(1k3),k→∞,x∈(0,1). (3.12)

Hence, assuming that the number is sufficiently large, we drop terms and in (3.12). Next, we set

 V(x,k)=r(x)k, q(x,k)=−r(x)k2,k≥¯¯¯k,x∈(0,1). (3.13)

Set in (3.9) and (3.10). Next, substitute (3.13) in (3.9) and (3.10) at . We obtain Recall that functions and are linked via (2.9). Thus,

 V′′ =0,in (0,1), (3.14) V(0) =logg0(¯¯¯k)¯¯¯k2,V′(0)=i¯¯¯k(1−1g0(¯¯¯k)),V′(1)=0, (3.15)

where functions and are defined in (2.8) and (2.9) respectively. It seems to be at the first glance that one can find the function   as, for example Cauchy problem for ODE (3.14) with data and However, it was noticed in Remark 5.1 of [34] that this approach, being applied to a similar problem, does not lead to good results. We have the same observation in our numerical studies. This is likely to the approximate nature of (3.13). Thus, just like in [32], we solve the problem (3.14), (3.15) by the Quasi-Reversibility Method (QRM). The boundary condition provides a better stability property.

So, we minimize the following functional on the set , where

 Jα(V)=12(∥V′′∥2L2(0,1)+α∥V∥2H3(0,1)), (3.16)
 V∈W:={V∈H3(0,1):V(0)=logg0(¯¯¯k)¯¯¯k2,V′(0)=i¯¯¯k(1−1g0(¯¯¯k)),V′(1)=0}, (3.17)

where is the regularization parameter. The existence and uniqueness of the solution of this minimization problem as well as convergence of minimizers in the norm to the exact solution of the problem (3.15), (3.16) with the exact data as were proved in [32]. We note that in the regularization theory one always assumes existence of an ideal exact solution with noiseless data [9, 17].

Recall that by the embedding theorem and

 ∥f∥C1[0,1]≤C∥f∥H2(0,1),∀f∈H2(0,1), (3.18)

where is a generic constant Theorem 3.1 is a reformulation of Theorem 4.2 of [32].

Theorem  3.1. Let the function  satisfying conditions (2.1)-(2.2) be the exact solution of our IMSP with the noiseless data , where  and  is the solution of the forward problem (2.3), (2.4). Let the exact tail function  and the function have the form (3.13) with  Assume that for

 ∣∣logg0(k)−logg∗0(k)∣∣≤δ,∣∣g0(k)−g∗0(k)∣∣≤δ,∣∣g′0(k)−(g∗0)′(k)∣∣≤δ, (3.19)

where  is a sufficiently small number, which characterizes the level of the error in the boundary data. Let in (3.16)  Let the function  be the minimizer of the functional (3.16) on the set of functions  defined in (3.17). Then there exists a constant  depending only on  and  such that

 Missing or unrecognized delimiter for \left (3.20)

Remark 3.1. We have also tried to consider two terms in the asymptotic expansion for in (3.12): the second one with This resulted in a nonlinear system of two equations. We have solved it by via minimizing an analog of the functional of section 3.3. However, the quality of resulting images deteriorated as compared with the above function In addition, we have tried to iterate with respect to the tail function . However, the quality of resulting images has also deteriorated.

### 3.3 The weighted cost functional

Consider the function satisfying (3.9)-(3.11). In sections 5.2 and 5.3 we use Lemma 2.1 and Theorem 2.1 of [4]. To apply theorems, we need to have zero boundary conditions at Hence, we introduce the function

 p(x,k)=q(x,k)−(x2−1)2p0−x(x2−1)2p1, where p0=p0(k),p1=p1(k). (3.21)

Denote

 m(x,k)=(x2−1)2p0(k)+x(x2−1)2p1(k). (3.22)

Also, replace in (3.9) with Then (3.9), (3.10) and (3.21) and (3.22) imply that

 L(p)=p′′+m′′−2ik(p′+m′)+2k2(p′+m′)⎛⎜ ⎜⎝−¯¯¯k∫k(p′+m′)(x,τ)dτ+V′α(δ)(x)⎞⎟ ⎟⎠
 −2i⎛⎜ ⎜⎝−¯¯¯k∫k(p′+m′)(x,τ)dτ+V′(x)⎞⎟ ⎟⎠+2k⎛⎜ ⎜⎝−¯¯¯k∫k(p′+m′)(x,τ)dτ+V′α(δ)(x)⎞⎟ ⎟⎠2=0 (3.23)
 p(0,k)=0,p′(0,k)=0,p′(1,k)=0. (3.24)

Introduce the Hilbert space of pairs of real valued functions as

 Missing or unrecognized delimiter for \left (3.25)

Here and below

Based on (3.23) and (3.24), we define our weighted cost functional as

 Jλ(p)=e2λ¯¯¯k∫k––1∫0|L(p)|2e−2λxdxdk,∀p∈H. (3.26)

Let be an arbitrary number. Let be the closure in the norm of the space of the open set of functions defined as

 B(R)={p∈H:∥p∥H

Minimization Problem. Minimize the functional  on the set

Remark 3.1. The analytical part of this paper below is dedicated to this minimization problem. Since we deal with complex valued functions, we consider below as the functional with respect to the 2-D vector of real valued functions Thus, even though we the consider complex conjugations below, this is done only for the convenience of writing. Below is the scalar product in . Even though we use in (3.21) and (3.23) the functions it is always clear from the context below what do we actually mean in each particular case: the first component of of the vector function or the above functions

## 4 The Global Strict Convexity of Jλ(p)

Theorem 4.1 is the main analytical result of this paper.

Theorem 4.1.  Assume that conditions of Theorem 3.1 are satisfied. Then the functional  has the Frechét derivative  for all Also, there exists a sufficiently large number depending only on listed parameters and a generic constant , such that for all  the functional  is strictly convex on  i.e. for all

 Jλ(p+h)−Jλ(p)−J′λ(p)(h)≥C∥h∥2H. (4.1)

Proof. Everywhere below in this paper  denotes different constants depending only on listed parameters. Since conditions of Theorem 3.1 are satisfied, then by (3.20)

 Missing or unrecognized delimiter for \left (4.2)

Let where Then (3.18), (3.25) and (3.27) imply that

 Missing or unrecognized delimiter for \right (4.3)

Using (4.3), we obtain

 ∣∣ ∣ ∣∣¯¯¯k∫kh′(x,k)dk∣∣ ∣ ∣∣2≤(¯¯¯k−k––)¯¯¯k∫k––∣∣h′(x,k)∣∣2dk (4.4)
 ≤(¯¯¯k−k––)¯¯¯k∫k––∥h(x,k)∥2C1[0,1]dk≤C2.

We use the formula

 |a|2−|b|2=(a−b)¯¯¯a+(¯¯¯a−¯¯b)b,∀a,b∈C, (4.5)

where is the complex conjugate of . Denote

 a=L(p+h),b=L(p), (4.6)

Consider functions defined as

 A=|L(p+h)|2−|L(p)|2,A1(x,k)=(a−b)¯¯¯a,A2(x,k)=(¯¯¯a−¯¯b)b. (4.7)

First, using (3.23) and (4.7), we single out in the part, which is linear with respect to the vector function . Then

 a=(p+h)′′+m′−2ik(p′+h′+m′)
 +2k2(p′+h′+m′)⎛⎜ ⎜⎝−¯¯¯k∫k(p′+h′+m′)(x,τ)dτ+V′α(δ)(x)⎞⎟ ⎟⎠ (4.8)
 −2i⎛⎜ ⎜⎝−¯¯¯k∫k(p′+h′+m′)(x,τ)dτ+V′α(δ)(x)⎞⎟ ⎟⎠+2k⎛⎜ ⎜⎝−¯¯¯k∫k(p′+h′+m′)(x,τ)dτ+V′α(δ)(x)⎞⎟ ⎟⎠2.

By (4.7)

 A1(x,k)=(a−b)¯¯¯a=⎧⎪ ⎪⎨⎪ ⎪⎩h′′−⎡⎢ ⎢⎣2ik+2k2⎛⎜ ⎜⎝¯¯¯k∫k(p′+m′)(x,τ)dτ+V′α(δ)(x)⎞⎟ ⎟⎠⎤⎥ ⎥⎦h′⎫⎪ ⎪⎬⎪ ⎪⎭¯¯¯a
 +⎡⎢ ⎢⎣2k2(p′+m′)+2i−4k⎛⎜ ⎜⎝¯¯¯k∫k(p′+m′)(x,τ)dτ−V′α(δ)(x)⎞⎟ ⎟⎠⎤⎥ ⎥⎦¯¯¯k∫kh′(x,τ)dτ⋅¯¯¯a (4.9)
 +⎡⎢ ⎢ ⎢⎣−2k2h′¯¯¯k∫kh′(x,τ)dτ+2k⎛⎜ ⎜⎝¯¯¯k∫kh′(x,τ)dτ⎞⎟ ⎟⎠2⎤⎥ ⎥ ⎥⎦¯¯¯a.

Hence,

 Missing or unrecognized delimiter for \left
 +⎡⎢ ⎢⎣(p′+m′)+2i−4k⎛⎜ ⎜⎝¯¯¯k∫k(p′+m′)(x,τ)dτ−V′α(δ)(x)⎞⎟ ⎟⎠⎤⎥ ⎥⎦¯¯¯¯¯¯¯¯¯¯¯¯L(p)¯¯¯k∫kh′(x,τ)dτ (4.10)
 +∣∣h′′(x,k)∣∣2+˜A1,p(h)(x,k),

where depends nonlinearly on the vector function . Also, by (4.2)-(4.4) and the Cauchy-Schwarz inequality

 ∣∣˜A1,p(h)∣∣(x,k)≤12∣∣h′′(x,k)∣∣2+C2⎛⎜ ⎜⎝∣∣h′(x,k)∣∣2+¯¯¯k∫k––∣∣h′(x,τ)∣∣2dτ⎞⎟ ⎟⎠. (4.11)

To explain the presence of the multiplier “1/2” at in (4.11), we note that it follows from (4.8) that the term in (4.9) contains the term which is included in (4.10) already, as well as terms

 h′′¯¯¯¯h′, h′′¯¯¯k∫k––¯¯¯¯h′(x,τ)dτ, h′′⎛⎜ ⎜⎝¯¯¯k∫k––¯¯¯¯h′(x,τ)dτ⎞⎟ ⎟⎠2. (4.12)

We now show how do we estimate the third term in (4.12), since estimates of two other terms are simpler. We use the so-called “Cauchy-Schwarz inequality with

 −|(c,d)|≥−ε2|c|2−12ε|d|2,∀c,d∈Rn,∀ε>0,

where is the scalar product in Hence,

 −∣∣ ∣ ∣ ∣∣h′′⎛⎜ ⎜⎝¯¯¯k∫k¯¯¯¯h′(x,τ)dτ⎞⎟ ⎟⎠2∣∣ ∣ ∣ ∣∣≥−ε2∣∣h′′∣∣2−(¯¯¯k−k––)2ε¯¯¯k∫k––∣∣h′(x,τ)∣∣2dτ.

Thus, choosing appropriate numbers we obtain the term in (4.11). The second term in the right hand side of (4.11) is obtained similarly.

Analogously, using (4.5)-(4.7), we obtain

 Missing or unrecognized delimiter for \left
 +¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⎡⎢ ⎢⎣(p′+m′)+2i−4k⎛⎜ ⎜⎝¯¯¯k∫k(p′+m′)(x,τ)dτ−V′α(δ)(x)⎞⎟ ⎟⎠⎤⎥ ⎥⎦¯¯¯k∫kh′(x,τ)dτ⋅L(p) (4.13)
 +˜A2,p(h)(x,k),

where depends nonlinearly on the vector function and similarly with (4.11)

 ∣∣˜A2,p(h)∣∣(x,k)≤C2⎛⎜ ⎜⎝∣∣h′(x,k)∣∣2+¯¯¯k∫k––∣∣h′(x,k)∣∣2dk⎞⎟ ⎟⎠. (4.14)

It is clear from (4.7), (4.10)-(4.14) that the linear with respect to the vector function