# On Delta-Extension for a Noether Operator

## Abstract

We examine a third kind integral equation in the class of generalized functions. We show that the considered equation has similar solvability properties as the Fredholm equation of the second kind.

## 1. INTRODUCTION

The linear Fredholm integral equation of the third kind is an equation of the form

$$A_{n}\varphi=f ,$$
(1)

where f is a given function of $$x\in \lbrack a,b]$$, $$\varphi$$ is the unknown function of $$x\in \lbrack a,b]$$, and operator $$A_{n}$$ is defined by

$$A_{n}\varphi =g_{n}(x)\varphi (x)-\int_{a}^{b}K(x,t)\varphi (t)dt;$$
(2)

here, $$g_{n}(x)=\Pi _{k=1}^{n}(x-x_{k})$$ is a given function of variable $$x\in \lbrack a,b]$$ with $$x_{k}\in ]a,b[$$, and $$K(x,t)$$ is a given function of variables $$\left( x,t\right) \in \lbrack a,b]\times \lbrack a,b]$$. In transport theory [1] the natural class of solutions of Eq. (1) is a class of generalized functions of the form

$$\varphi (x)=y(x)+\sum_{k=1}^{n}\alpha _{k}P\frac{1}{x-x_{k}} +\sum_{k=1}^{n}\omega _{k}\delta (x-x_{k}),$$

where $$y(x)\in C[a,b]$$, $$\delta$$ is the Dirac delta function, and P indicates that Cauchy's principal value is used for integrations of $${1 }/({x-x_{k}})$$. Since the early works of Hilbert [2] and Picard [3] on integral equation of the third kind, a lot of papers have been devoted on the topic. Bart and Warnock [4], [5] have considered Eq. (1) in the class of generalized functions in the form

$$\varphi (x)=y(x)+\sum_{k=1}^{n}\omega _{k}\delta (x-x_{k}),\quad y(x)\in C[a,b],$$

and have shown that under the condition $$\det(k(x_{i},x_{j}))_{1\leq i,j\leq n}\neq 0$$ three theorems of Fredholm hold. Following Bart and Warnock, Sukavanam [6] has studied Eq. (1) with some additional assumptions on such kernels. In fact, two theorems for Fredholm-type theory for two different types of kernels have been proved.

In Refs. [7], [8], Shulaia has found a necessary and sufficient condition for solvability of different types of Eq. (1) in the class of Hölder functions. Most recently, he has given a necessary and sufficient condition for solvability of equation Eq. (1) in the class of H ölder functions, with the coefficients which are piecewise strictly monotone functions [9]. Noether theory for the operator $$A_{n}$$ defined in Eq. (2) with $$\varphi \in C^{1}[-1,1]$$ has been investigated in [10]. Following Abdourahman [11], [12], we consider in this work the linear Fredholm integral equation of the third kind $$A\varphi =f$$, where A is defined by

$$(A\varphi )(x)=x^{p}\varphi ^{\prime }(x)+\int_{-1}^{1}K(x,t)\varphi (t)dt, \text{ }x\in \lbrack -1,1],$$
(3)

where $$p\in \mathbf{N}$$, f and K are given functions, and $$\varphi$$ is the unknown function. Function f is continuous in $$[-1,1]$$, while K is continuous in the rectangle $$[-1,1]\times \lbrack -1,1]$$. We find similar properties of Noether as the equation of second kind, by taking

$$\varphi \in D_{m}=C_{-1}^{1}[-1,1]\oplus \left\{ \sum_{k=0}^{m}\alpha _{k}\delta ^{\{k\}}(x)\right\} ,$$

that is

$$\varphi (t)=\varphi _{0}(t)+\sum_{k=0}^{m}\alpha _{k}\delta ^{\{k\}}(x),$$

with $$\varphi _{0}\in C_{-1}^{1}[-1,1]$$ and $$\varphi _{0}(-1)=0$$, $$\delta (t)$$ and $$\delta ^{\{i\}}(t)$$ being respectively the Dirac delta function and its Taylor derivatives of order i. We establish the conditions under which, when extending the operator by taking the unknown function $$\varphi$$ from the space of generalized functions $$D_{m}$$, operator A is Noether, as the previous authors, Gabbassov and Raslambekov, have done in their works [10], [13, 14, 15, 16]. This paper is structured as follows: in section 2, we recall some fundamental concepts of Noether theory and theory of third kind integral equation. Section 3 is devoted to some fundamental spaces such as $$D_{m}$$ and $$P^{1}$$ including definitions and some characterizations. In addition, we recall the concept of associated spaces and associated operators. In section 4 we investigate the properties of Noether of the extended operator A defined by Eq. (3) in space $$D_{m}$$. We conclude our paper in section 5.

## 2. PRELIMINARIES

In this section we briefly review the notion of Noether operators and integral equation of third kind (we refer to [17, 18, 19, 20] for a detailed study).

### Definition 2.1.

A bounded linear operator $$A:X\rightarrow Y$$, acting between complex Banach spaces X and Y, is called a Fredholm operator or Noether operator or $$\Phi$$-operator if its range A is closed and the numbers $$\alpha (A)=\operatorname{dim}\operatorname{Ker}A\text{ and }\beta (A)=\operatorname{dim}(Y/Im A)$$ are finite. The set of all $$\Phi$$-operators is denoted by $$\Phi(X,Y)$$. The number $$\chi (A)=\alpha (A)-\beta (A)$$ is called the index of operator A. The numbers $$\alpha (A)$$ and $$\beta (A)$$ are called deficiency numbers.

An equivalent condition of closedness of the range A is the normal solvability.

### Definition 2.2.

The operator A is called Hausdorff normally solvable or normally solvable if the equation $$A\varphi=f$$ is solvable in X if and only if $$\psi(f)=0$$ for all functionals $$\psi\in \operatorname{Ker} A^{*}$$. Where $$A^{*}$$ is the adjoint operator of A in the dual space $$X^{*}$$.

In sequel, we give some cases when operator A is Noether follow from [18].

### Theorem 2.1.

Let $$A:X\rightarrow Y$$ be a closed operator and $$B\in l(X,Z)$$ a continuous operator with a finite-dimensional kernel. Then $$BA\in\Phi(X,Y)$$ implies $$A\in\Phi(X,Y)$$.

### Theorem 2.2.

For a closed operator $$A:X\rightarrow Y$$ to be a $$\Phi$$-operator it is necessary and sufficient that there exist two continuous operators $$B_{1},B_{2}\in l(X,Y)$$ such that $$\operatorname{Im} B_{2}\subset D(A)$$ and $$B_{1}A=I-T_{1}$$, $$AB_{2}=I-T_{2}$$ where $$T_{1}$$ is a compact operator on X, and $$T_{2}$$ is a compact operator on Y. Operators $$B_{1}$$ and $$B_{2}$$ can be chosen such that $$B_{1}=B_{2}$$ holds and $$T_{1},T_{2}$$ are continuous projections of finite rank.

### Theorem 2.3.

Let $$A\in \Phi(X,Y)$$ be a $$\Phi$$-operator with a dense domain. Then $$A^{*}$$ is normally solvable, i. e.,

$$\operatorname{Im} A^{*}=\{f\in X^{*}:\quad <x,f>=0\quad \forall x\in \operatorname{Ker} A\}.$$

Moreover $$A^{*}$$ has a finite index, and we have

$$\alpha(A^{*})=\beta(A),\quad \beta(A^{*})=\alpha(A).$$

### Theorem 2.4.

Let $$A\in \Phi(X,Y)$$ and $$T\in K(X,Y)$$. Then $$A+T\in \Phi(X,Y)$$ and $$\operatorname{Ind}(A+T)=\operatorname{Ind} A$$.

Some fundamental result in theory of third kind integral equation is known as Alternative of Fredholm. Let us consider the following Fredholm integral equation

$$x(t)=f(t)+\lambda\int_{a}^{b}K(t,s)x(s)ds$$
(4)

where f and K are given functions and x the unknown function. Function K is assumed to be square integrable on the rectangle $$[a,b]\times [a,b]$$. The transpose of (4) is given by

$$y(t)=g(t)+\overline{\lambda}\int_{a}^{b}\overline{K(t,s)}y(s)ds.$$
(5)

### Theorem 2.5.

Either equation (4) and (5) have a unique solution in $$L^{2}([a,b],\mathcal{C})$$ where $$\mathcal{C}$$ stands for the complex number field, and $$f,g \in L^{2}([a,b],\mathcal{C})$$, in which case the homogeneous equations have only a trivial solution, or homogeneous equations have nonzero solutions. In this case, equation (4) has solutions if and only if f is orthogonal to every solution y(t) of the adjoint homogeneous equation;

$$\int_{a}^{b}f(t)\overline{y(t)}dt=0.$$

The proof of Theorem 2.1 given in [20] consists of three steps. In first step the theorem is proved in the special case when the kernel K has finite rank:

$$K(t,s)=\sum_{i=1}^{n}a_{i}(t)\overline{b_{i}(s)}.$$

where functions $$a_{i}(t)$$, $$i=1,2,\ldots,n$$ are linearly independent. In the second step, it is shown that any kernel K can be represented in the form

$$K(t,s)=K_{n}(t,s)+R_{n}(t,s)$$

where $$K_{n}(t,s)$$ is a kernel of finite rank n and $$R_{n}(t,s)$$ is such that

$$\int_{a}^{b}\int_{a}^{b}|R_{n}(t,s)|^{2}dtds=r_{n}^{2}.$$

The third step focuses on the proof of the Fredholm alternative for square integrable kernels on the rectangle $$[a,b]\times [a,b]$$.

Let us recall the Taylor derivative. Which appears in [12] and which we will use in this work.

### Definition 2.3.

We say that a function $$\varphi(x)$$ from $$C[0,1]$$ has at the point $$x=0$$ the Taylor derivatives up to order $$p\in \mathbb{N}$$, if recurrently there exist the limits $$\varphi^{\{k\}}(0)$$ defined by

$$\varphi^{\{k\}}(0)=k!\lim_{x\rightarrow 0}\frac{\varphi(x)-\sum_{j=0}^{k-1}\frac{\varphi^{\{j\}}(0)x^{j}}{j!}}{x^{k}},\quad k=1,2,\ldots,p,$$

and $$\varphi^{\{0\}}(0)=\varphi(0).$$

### Definition 2.4.

The subspace of continuous functions which have finite Taylor derivatives up to order $$p\in\mathbb{Z}_{+}$$ is denoted by $$C_{0}^{\{p\}}[0,1]$$.

Note that when $$p=0$$ we have $$C_{0}^{\{p\}}[0,1]=C[0,1]$$, and linear operator $$N^{k}$$ on the space $$C_{0}^{\{p\}}[0,1]$$ has been defined by the formula

$$(N^{k}\varphi)(x)=\frac{\varphi(x)-\sum_{j=0}^{k-1}\frac{\varphi^{\{j\}}(0)x^{j}}{j!}}{x^{k}},\quad k=1,2,\ldots,p.$$

In sequel we give some results which characterizes the space $$C_{0}^{\{p\}}[0,1]$$.

### Lemma 2.1.

A function $$\varphi(x)$$ belongs to $$C_{0}^{\{p\}}[0,1]$$ if and only if the following representation

$$\varphi(x)=x^{p}\Phi(x)+\sum_{k=0}^{p-1}\alpha_{k}x^{k}$$
(6)

holds with $$\Phi\in C[0,1]$$ and $$\alpha_{k}$$, $$k=0,1,\ldots,p-1$$, are constants.

Proof. Formula (6) implies that the Taylor derivatives of $$\varphi$$ up to order p exist, and $$\varphi^{\{k\}}(0)=k!\alpha_{k}$$, $$k=0,1,\ldots,p-1$$, $$\varphi^{\{p\}}(0)=p!\Phi(0)$$ with $$\Phi(x)=(N^{p}\varphi)(x)$$, conversely, if $$\varphi\in C_{0}^{\{p\}}[0,1]$$ and we define $$\Phi(x)=(N^{p}\varphi)(x)$$ with $$\alpha_{k}=\frac{\varphi^{\{k\}}(0)}{k!}$$, $$k=0,1,\ldots,p-1$$, then representation (6) holds.

We deduce from the above result that for $$\varphi(x)\in C_{0}^{\{p\}}[0,1]$$ we have $$\varphi(x)=x^{p}(N^{p}\varphi)(x)+\displaystyle\sum\limits_{k=0}^{p-1}\dfrac{\varphi^{\{k\}}(0)x^{k}}{k!}$$. Consequently, linear operator $$N^{p}$$ establishes a relation between the spaces $$C_{0}^{\{p\}}[0,1]$$ and $$C[0,1]$$. The space $$C_{0}^{\{p\}}[0,1]$$ with the norm

$$\|\varphi\|_{C_{0}^{\{p\}}[0,1]}=\|N^{p}\varphi\|_{C[0,1]}+\sum_{k=0}^{p-1}|\varphi^{\{k\}}(0)|$$

becomes a Banach one.

### Lemma 2.2.

Let $$p\in\mathbb{N}$$, $$s\in \mathbb{Z}_{+}$$. If $$\varphi(x)\in C_{0}^{\{s\}}[0,1]$$ then $$x^{p}\varphi(x)\in C_{0}^{\{s+p\}}[0,1]$$, and we have

$$\displaystyle (x^{p}\varphi)^{\{j\}}(0)= \begin{cases} 0,& j=0,1,\ldots,p-1,\\ \dfrac{j!}{(j-p)!}\varphi^{\{j-p\}}(0),& j=p,\ldots,p+s. \end{cases}$$

The proof of Lemma 2.2 is based on the Leibniz formula.

## 3. FUNDAMENTAL SPACES, ASSOCIATED SPACES AND ASSOCIATED OPERATORS

### Definition 3.1.

By

$$D_{m}=C_{-1}^{1}[-1,1]\oplus \left\{\sum_{k=0}^{m}\alpha _{k}\delta ^{\{k\}}(x)\right\}$$

we denote the space of function $$\varphi$$ represented by the following formula

$$\varphi (x)=\varphi _{0}(x)+\sum_{k=0}^{m}\alpha _{k}\delta ^{\{k\}}(x),$$

where $$\varphi _{0}\in C^{1}[-1,1]$$ with $$\varphi _{0}(-1)=0$$ and $$\delta ^{\{k\}}$$ Taylor derivative of $$\delta$$.

### Proposition 3.1.

The space $$D_{m}$$ with the norm

$$\Vert \varphi \Vert _{D_{m}}=\Vert \varphi _{0}\Vert _{C^{1}[-1,1]}+\sum_{k=0}^{m}|\alpha _{k}|$$

is a Banach space.

Proof. To prove this result, we will use the following result.

### Lemma 3.1.

Let $$(f_{j})$$ be a sequence of functions. If $$(f_{j})$$ is continuously differentiable and converges uniformly as well as its derivative sequence, then its limit f is continuously differentiable and its derivative $$f^{\prime }$$ is the limit of $$f_{j}^{\prime }$$.

Let $$(\varphi _{n}+\sum_{k=0}^{m}\alpha _{k}\delta ^{\{k\}})_{n}\subset D_{m}$$ be a Cauchy sequence, then we have

$$\forall \epsilon >0\quad\exists N>0\quad \forall m>n>N\quad\left\Vert \varphi _{m}-\varphi _{n}+\sum_{k=0}^{m}\alpha _{k}\delta ^{\{k\}}-\sum_{k=0}^{m}\beta _{k}\delta ^{\{k\}}\right\Vert _{D_{m}}\leq \epsilon$$

which implies

$$\forall \epsilon >0\quad \exists N>0\quad \forall m>n>N$$
$$\Vert \varphi _{m}-\varphi _{n}\Vert _{C^{1}[-1,1]} \leq \Vert \varphi _{m}-\varphi _{n}\Vert _{C^{1}[-1,1]}+\sum_{k=0}^{m}|\alpha _{k}-\beta _{k}|\leq \epsilon,$$
$$\text{i. e.},\ \forall \epsilon >0\quad \exists N>0\quad \forall m>n>N\ \ \Vert \ \varphi _{m}-\varphi _{n}\Vert _{C^{1}[-1,1]}\leq \epsilon,$$
$$\text{i. e},\ \forall \epsilon >0\quad \exists N>0\quad \forall m>n>N\ \sum_{j=0}^{1}\max_{-1\leq x\leq 1}|\varphi _{m}^{(j)}(x)-\varphi _{n}^{(j)}(x)|\leq \epsilon,$$

this $$\max$$ is well defined because $$\varphi$$ is continuous on the compact$$[-1,1]$$. We deduce that

$$\forall \epsilon >0\quad \exists N>0\quad \forall m>n>N$$
$$|\varphi _{m}(x)-\varphi _{n}(x)| \leq \sum_{j=0}^{1}\max_{-1\leq x\leq 1}|\varphi _{m}^{(j)}(x)-\varphi _{n}^{(j)}(x)|\leq \epsilon$$
$$\text{and }\ |\varphi _{m}(x)^{\prime }-\varphi _{n}(x)^{\prime }| \leq \sum_{j=0}^{1}\max_{-1\leq x\leq 1}|\varphi _{m}^{(j)}(x)-\varphi _{n}^{(j)}(x)|\leq \epsilon,$$

so $$(\varphi _{n}(x))_{n}$$ and $$(\varphi _{n}(x)^{\prime })_{n}$$ are Cauchy sequences in $$\mathbb{R}$$ which is complete so $$(\varphi _{n}(x))_{n}$$ and $$(\varphi _{n}(x)^{\prime })_{n}$$ converge in $$\mathbb{R}$$.

Then there exists $$\varphi$$ and, according to Lemma 3.1, $$\varphi$$ is a continuously differentiable function such that $$\varphi _{n}(x)\rightarrow \varphi (x)$$ and $$\varphi _{n}^{\prime }(x)\rightarrow \varphi ^{\prime }(x)$$, so $$\varphi +\sum_{k=0}^{m}\alpha _{k}\delta ^{\{k\}}\in D_{m}$$ then $$D_{m}$$ is a Banach space. We finish this section by recalling definition of $$P^{1}$$ given in [10].

### Definition 3.2.

By $$P^{1}$$ we denote the space of distributions $$\psi$$ on the space of test functions $$C_{0}^{\{m\}}[-1,1]$$ such that

$$\psi (x)=\dfrac{z(x)}{x^{m}}+\sum_{k=0}^{m-1}\beta _{k}\delta ^{\{k\}}(x),$$
(7)

where $$z(x)\in C_{0}^{\{m\}}[-1,1]\cap C_{-1}^{1}[-1,1]$$ and $$z(1)=0$$, $$\beta _{k}$$ are arbitrary constants, $$\delta ^{\{k\}}(x)$$ Taylor derivative up to order k of the Dirac delta function defined on the test space $$C_{0}^{\{m\}}[-1,1]$$ by

$$(\delta ^{\{k\}}(x),\varphi (x))=\int_{-1}^{1}\delta ^{\{k\}}(x)\varphi (x)dx=(-1)^{k}\varphi ^{\{k\}}(0).$$

In the space $$P^{1}$$, introduce the norm

$$\Vert \psi \Vert _{P^{1}}=\Vert z\Vert _{C_{0}^{\{m\}}[-1,1]}+\sum_{k=0}^{m-1}|\beta _{k}|.$$
(8)

With this norm it was proved in [10] that $$P^{1}$$ is a Banach space.

As Noether property depends on the associated operators and associated spaces, we start by recalling these two concepts.

### Definition 3.3.

The Banach space $$E^{\prime }\subset E^{\ast }$$ is called associated space to a Banach space E if

$$|<f,\varphi >|\leq c\Vert f\Vert _{E^{\prime }}\Vert \varphi \Vert _{E}\quad \forall \varphi \in E,f\in E^{\prime }.$$

### Definition 3.4.

Let $$E_{j},j=1,2$$, be a Banach spaces and $$E_{j}^{\prime }$$ their associated spaces, operators $$A\in l(E_{1},E_{2})$$ and $$A^{\prime }\in l(E_{2}^{\prime },E_{1}^{\prime })$$ are associated if

$$(A^{\prime }f,\varphi )=(f,A\varphi )\quad\forall f\in E_{1}^{\prime } \quad\forall \varphi \in E_{1}.$$

The following result gives Noether properties via associated operator.

### Lemma 3.2.

Let $$E_{j},$$ $$j=1,2$$, be a Banach spaces, $$E_{j}^{\prime }$$ their associated spaces, $$A\in l(E_{1},E_{2})$$ and $$A^{\prime }\in l(E_{1}^{^{\prime }},E_{2}^{^{\prime }})$$ are associated Noether operators. We have $$\chi (A)=-\chi (A^{\prime })$$, and $$A\varphi =f$$ is solvable if and only if $$(f,\psi )=0$$ for all solutions of the associated homogeneous equation $$A^{\prime }\psi =0$$.

We finish these reminders with two results that define the associated spaces of spaces that we will use later.

### Lemma 3.3.

Space $$C_{x_{0}}^{1}[-1,1]$$ is associated to space $$C[-1,1]$$, where $$C_{x_{0}}^{1}[-1,1]$$ means the space of functions $$\varphi \in C[-1,1]$$ satisfying $$\varphi (x_{0})=0$$.

Proof. Let $$f\in C_{x_{0}}^{1}[-1,1]$$ and let $$\varphi \in C[-1,1].$$

$$|<f,\varphi >|=\left|\int_{-1}^{1}f(x)\varphi (x)dx\right|\leq 2\max_{-1\leq x\leq 1}|f(x)|\times \max_{-1\leq x\leq 1}|\varphi (x)|.$$

$$\square$$

### Lemma 3.4.

Space $$P^{1}$$ is Banach associated to space $$C_{0}^{\{m\}}[-1,1].$$

The following lemma helps us in the proof of Lemma 3.4.

### Lemma 3.5.

Let f have the Taylor derivative up to order n then, we have the following formula

$$f(x)\delta ^{\{n\}}(x)=(-1)^{n}f^{\{n\}}(0)\delta (x)+(-1)^{n-1}nf^{\{n-1\}}(0)\delta ^{\{1\}}(x)+\ldots$$
$$+\ (-1)^{n-2}\frac{n(n-1)}{ 2!}f^{\{n-2\}}(0)\delta ^{\{2\}}(x)+\ldots+f(0)\delta ^{\{n\}}(x)$$

In particular,

$$\displaystyle t^{m}\delta ^{\{n\}}(x)= \begin{cases}0 & \text{ for }\ m>n, \\ \frac{(-1)^{m}n!}{(n-m)!}\delta ^{\{n-m\}}(t) & \text{ for }\ m\leq n, \end{cases}$$
(9)

Proof. Let $$\Phi$$ be a test function, we have

$$\int_{-\infty }^{+\infty }[f(x)\delta ^{\{n\}}(x)]\Phi (x)dx =[(f(x)\Phi (x))\delta ^{\{n-1\}}(x)]_{-\infty }^{+\infty }-\int_{-\infty }^{+\infty }(f(x)\Phi (x))^{\prime }\delta ^{\{n-1\}}(x)dx$$
$$=-\int_{-\infty }^{+\infty }(f(x)\Phi (x))^{\prime }\delta ^{\{n-1\}}(x)dx.$$

After a similar successive integrating by parts we have

$$\int_{-\infty }^{+\infty }[f(x)\Phi (x)]\delta ^{\{n\}}(x)dx =(-1)^{n}\int_{-\infty }^{+\infty }[f(x)\Phi (x)]^{\{n\}}\delta (x)dx,$$

according to Leibnitz's formula we finally have

$$\int_{-\infty }^{+\infty }[f(x)\Phi (x)]\delta ^{\{n\}}(x)dx =(-1)^{n}\Bigg[f^{\{0\}}\Phi (0)+nf^{\{n-1\}}(0)\Phi ^{\prime }(0)+\frac{n(n-1) }{2!}f^{\{n-2\}}(0)\Phi ^{\{2\}}(0)\\ +\ldots+f(0)\Phi ^{\{n\}}(0)\Bigg].$$

On the other hand we have

$$<f(x)\delta ^{\{k\}}(x),\Phi (x)>=<(-1)^{n}f^{\{n\}}(0)\delta (x)$$
$$+\ (-1)^{n-1}nf^{\{n-1\}}(0)\delta ^{\{1\}}(x)+\frac{n(n-1)}{2!} (-1)^{n-2}f^{\{n-2\}}(0)+\ldots+f(0)\delta ^{\{n\}}(x),\Phi (x)>.$$

Finally we deduce the following formula

$$f(x)\delta ^{\{n\}}(x)=(-1)^{n}f^{\{n\}}(0)\delta (x)+(-1)^{n-1}nf^{\{n-1\}}(0)\delta ^{\{1\}}(x)+\ldots$$
$$+\ (-1)^{n-2}\frac{n(n-1)}{ 2!}f^{\{n-2\}}(0)\delta ^{\{2\}}(x)+\ldots+f(0)\delta ^{\{n\}}(x).$$

In addition, the case where $$f(t)=t^{m}$$, leads us to

$$t^{m}\delta ^{\{n\}}(x) =(-1)^{n}(t^{m})^{\{n\}}(0)\delta (x)+(-1)^{n-1}n(t^{m})^{\{n-1\}}(0)\delta ^{\{1\}}(x)$$
$$+(-1)^{n-2}\frac{n(n-1)}{2!}(t^{m})^{\{n-2\}}(0)\delta ^{\{2\}}(x)+\ldots+(t^{m})(0)\delta ^{\{n\}}(x)$$
$$=\sum_{i=0}^{n}(-1)^{n-i}C_{n}^{i}(t^{m})^{\{n-i\}}(\delta (x))^{\{i\}}.$$

We note that

in case $$n-i>m$$, i. e., $$i\in \{0,\ldots,n-m-1\}$$, we have $$(t^{m})^{\{n-i\}}=0$$,

and in case $$n-i<m$$, i. e., $$\ i\in \{n-m+1,\ldots,n\}$$, we have $$(t^{m})^{\{n-i\}}=0.$$

In addition, by taking in account the following formula $$(t^{m})^{\{n-i\}} =\dfrac{m!}{(m-n+i)!}t^{m-n+i}$$, we have in case $$i=n-m$$ the equality below

$$(t^{m})^{\{n-(n-m)\}}(\delta (x))^{\{n-m\}}=\frac{(-1)^{m}n!}{(n-m)!}(\delta (x))^{\{n-m\}}.$$

Finally we deduce the relationship below

$$\displaystyle t^{m}\delta ^{\{n\}}(x)= \begin{cases}0 & \text{ for }\ m>n, \\ \dfrac{(-1)^{m}n!}{(n-m)!}\delta ^{\{n-m\}}(t) & \text{ for }\ m\leq n. \end{cases}$$

$$\square$$

We continue with the proof of Lemma 3.4.

Proof of Lemma 3.4. Obviously $$P^{1}$$ is a Banach space with the norm (8).

Let $$f\in C_{0}^{\{m\}}[-1,1]$$ and $$\psi \in P^{1}$$, where f and $$\psi$$ are defined respectively by (6) and (7). According to Lemma 3.5, (6) and (7) we have

$$|<f,\psi >|=\Bigg|\int_{-1}^{1}z(x)g(x)dx+\int_{-1}^{1}\dfrac{z(x)}{x^{p}} \sum_{k=0}^{m-1}c_{k}x^{k}dx+\int_{-1}^{1}\sum_{n=0}^{m-1}c_{n}x^{n} \sum_{k=0}^{m-1}\beta _{k}\delta ^{\{k\}}(x)dx\Bigg|$$
$$\leq\ \int_{-1}^{1}|z(x)g(x)|dx+\sum_{k=0}^{m-1}|c_{k}|\Bigg|\int_{-1}^{1}\dfrac{ z(x)}{x^{p-k}}dx\Bigg|$$
$$+\ \left|\sum_{n=0}^{m-1}c_{n}\sum_{k=n}^{m-1}\beta _{k}\dfrac{(-1)^{n}k!}{(k-n)!} \int_{-1}^{1}\delta ^{\{k-n\}}(x)dx\right|=I_{1}+I_{2}+I_{3}.$$

Let us make three estimations with $$I_{1},I_{2}$$ and $$I_{3}$$.

$$|I_{1}| = \Bigg|\int_{-1}^{1}z(x)g(x)dx\Bigg| \leq \int_{-1}^{1}|z(x)||g(x)|dx \leq \Vert z\Vert _{C[-1,1]}\Vert g\Vert _{C[-1,1]} \leq \Vert \psi \Vert _{p^{1}}\Vert f\Vert _{C_{0}^{\{m\}}[-1,1]}.$$
$$|I_{2}| =\Bigg|\sum_{k=0}^{m-1}c_{k}\Bigg|\Bigg|\int_{-1}^{1}\dfrac{z(x)}{x^{p-k}}dx\Bigg|,$$
$$|I_{2}| \leq \sum_{k=0}^{m-1}|c_{k}|M_{k}\Vert z\Vert _{C_{0}^{\{m\}}[-1,1]},\quad \text{ where }M_{k}=\int_{-1}^{1}\dfrac{1}{x^{p-k}}dx,$$
$$\leq M\Vert f\Vert _{C_{0}^{\{m\}}[-1,1]}\Vert \psi \Vert _{P^{1}},\quad \text{ where }M=\max_{0\leq k\leq p-1}M_{k}.$$
$$|I_{3}| =|\int_{-1}^{1}\sum_{n=0}^{m-1}c_{n}\sum_{k=n}^{m-1}\beta _{k} \dfrac{(-1)^{n}k!}{(k-n)!}\delta ^{\{k-n\}}(x)dx|$$
$$\leq \sum_{n=0}^{m-1}|c_{n}|\sum_{k=n}^{m-1}|\beta _{k}|\dfrac{(-1)^{n}k!}{ (k-n)!}\int_{-1}^{1}\delta ^{\{k-n\}}(x)dx$$
$$\leq N_{k}\sum_{n=0}^{m-1}|c_{n}|\sum_{k=n}^{m-1}|\beta _{k}|,\quad \text{ where }N_{k}=\dfrac{(-1)^{n}k!}{(k-n)!}\int_{-1}^{1}\delta ^{\{k-n\}}(x)dx,\quad \text{ hence, }$$
$$|I_{3}| \leq N_{k}\Vert f\Vert _{C_{0}^{\{p\}}[-1,1]}\Vert \psi \Vert _{P^{1}},\quad \text{ where }N=\max_{0\leq k\leq p-1}N_{k}.$$

Finally we deduce that

$$|<f,\varphi >|\leq C\Vert f\Vert _{C_{0}^{\{m\}}[-1,1]}\Vert \psi \Vert _{P^{1}},\quad \text{ where }C=\max\{N,M,1\}.$$

$$\square$$

## 4. EXTENSION OF THE OPERATOR A

In this section, we investigate Noetherity of the operator A defined by equation (3), through the associated operators and associated spaces.

### Theorem 4.1.

Let $$\overline{A}$$ be the extension of operator A defined by (3) on $$D_{m}$$ with $$m=p-2$$. Let $$\overline{P^{1}}$$ be the restriction of $$P^{1}$$ by the conditions

$$z^{\{1\}}(0)=z^{\{2\}}(0)=\cdots =z^{\{p-1\}}(0)=0.$$
(10)

Let $$\overline{A^{\prime }}$$ be the restriction of the operator $$A^{\prime }$$ defined on $$\overline{P^{1}}$$ by

$$A^{\prime }\psi =-(x^{p}\psi )^{\prime }+\int_{-1}^{1}K(t,x)\psi (t)dt.$$

Then $$\overline{A}$$ and $$\overline{A^\prime}$$ are associated operators and we have

$$(\overline{A}\varphi ,\psi )=(\varphi ,\overline{A^\prime }\psi ).$$

Proof. Let us compute by taking in account formula (9)

$$(A\varphi ,\psi ) =\left(x^{p}(\varphi _{0}^{\prime }(x)+\sum_{k=0}^{p-2}\alpha _{k}\delta ^{\{k+1\}}(x))+\int_{-1}^{1}K(x,t)\left[\varphi _{0}(t)+\sum_{k=0}^{p-2}\alpha _{k}\delta ^{\{k\}}(t)\right]dt,\right.$$
$$\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\dfrac{z(x)}{x^{p}} +\sum_{i=0}^{p-1}\omega _{i}\delta ^{\{i\}}(x)\right)$$
$$=\left(x^{p}\varphi _{0}^{\prime }(x)+\int_{-1}^{1}K(x,t)\varphi _{0}(t)dt+\sum_{k=0}^{p-2}\alpha _{k}(-1)^{k}k_{2}^{\{k\}}(x,0),\dfrac{z(x)}{ x^{p}}+\sum_{i=0}^{p-1}\omega _{i}\delta ^{\{i\}}(x)\right)$$
$$=(\varphi _{0}^{\prime }(x),z(x))+\int_{-1}^{1}\dfrac{z(x)}{x^{p}} dx\int_{-1}^{1}K(x,t)\varphi _{0}(t)dt+\int_{-1}^{1}\sum_{i=0}^{p-1}(-1)^{i}\omega _{i}\int_{-1}^{1}K_{i}^{\{i\}}(0,t)\varphi _{0}(t)dt$$
$$+\sum_{k=0}^{p-2}(-1)^{k}\alpha _{k}\int_{-1}^{1}K_{2}^{\{k\}}(x,0)\dfrac{ z(x)}{x^{p}}dx+\sum_{k=0}^{p-2}(-1)^{k}\alpha _{k}\sum_{i=0}^{p-1}(-1)^{i}\omega _{i}K_{21}^{\{k\}\{i\}}(0,0).$$

Otherwise we have

$$(\varphi ,A^{\prime }\psi ) =\left(\varphi _{0}(x)+\sum_{k=0}^{p-2}\alpha _{k}\delta ^{\{k\}}(x),-(x^{p}\psi )^{\prime }+\int_{-1}^{1}K(t,x)\psi (t)dt\right)$$
$$=\left(\varphi _{0}(x)+\sum_{k=0}^{p-2}\alpha _{k}\delta ^{\{k\}}(x),-z^{\prime }(x)+\int_{-1}^{1}K(t,x)\dfrac{z(t)}{t^{p}}dt+\sum_{i=0}^{p-1}(-1)^{i}\omega _{i}k_{1}^{\{i\}}(0,x)\right)$$
$$=(\varphi _{0}(x),-z^{\prime }(x))+\int_{-1}^{1}\dfrac{z(x)}{x^{p}} dx\int_{-1}^{1}K(x,t)\varphi _{0}(t)dt+\sum_{i=0}^{p-1}(-1)^{i}\omega _{i}\int_{-1}^{1}\varphi _{0}(x)K_{1}^{\{i\}}(0,x)dx$$
$$-\sum_{k=0}^{p-2}\alpha _{k}(-1)^{k}(z^{\prime })^{\{k\}}(0)+\sum_{k=0}^{p-2}\alpha _{k}(-1)^{k}\int_{-1}^{1}K_{2}^{\{k\}}(t,0)\dfrac{z(t)}{t^{p}}dt$$
$$+\sum_{k=0}^{p-2}\alpha _{k}\sum_{i=0}^{p-1}(-1)^{i}(-1)^{k}\omega _{i}K_{1,2}^{\{i\}\{k\}}(0,0).$$

Comparing the values of $$(A\varphi ,\psi )$$ and $$(\varphi ,A^{\prime }\psi )$$, and according to the mixed kernel derivatives we note that $$(A\varphi ,\psi )=(\varphi ,A^{\prime }\psi )$$ is possible if and only, if $$\sum_{k=0}^{p-2}\alpha _{k}(-1)^{k}(z^{\prime })^{\{k\}}(0)=0$$. It means that $$z^{\{1\}}(0)=z^{\{2\}}(0)=\cdots =z^{\{p-1\}}(0)=0.$$ $$\square$$

We note that condition (10) on the Taylor derivative of function z permits us to have the general form of z in the space $$\overline{P^{1}}$$. In fact according to (10) the general form of z in $$\overline{P^{1}}$$ is

$$z(x)=c_{0}+c_{1}x+\ldots+c_{p-1}x^{p-1}+x^{p}g(x)=c_{0}+x^{p}g(x),$$

where $$c_{0}=-g(1)$$ because $$z(1)=0$$.

Thus we have

$$\overline{P^{1}}=\left\{\psi :\psi (x)=\dfrac{z(x)}{x^{p}}+\sum_{i=0}^{p-1}\omega _{i}\delta ^{\{i\}}(x),\text{ }z(1)=0\right\}=\left\{\psi :\psi (x)=-\dfrac{g(1)}{x^{p}} +g(x)+\sum_{i=0}^{p-1}\omega _{i}\delta ^{\{i\}}(x)\right\},$$

where

$$g(x)=\dfrac{\displaystyle z(x)-\sum\limits_{j=0}^{p-1}\dfrac{z^{\{j\}}(0)x^{j}}{j!}}{ x^{p}}.$$

In the next result, we give associated spaces of $$D_{m}$$ and $$\overline{P^{1}}$$.

### Lemma 4.1.

Spaces $$D_{m}$$ and $$C[-1,1]$$ form a pair of associated Banach spaces. Analogously the pair $$\overline{P^{1}}$$ and $$C_{0}^{\{m\}}[-1,1]$$ also defines a pair of associated Banach spaces.

We state the following theorem.

### Theorem 4.2.

Operator $$\overline{A}:D_{m}\rightarrow C_{0}^{\{m\}}[-1,1]$$ when $$m=p-2$$ is Noether with index $$\chi (\overline{A})=-1.$$

Proof. Let us put $$\overline{L}\varphi =x^{p}\varphi ^{\prime }$$ and $$K=\displaystyle\int_{-1}^{1}K(x,t)\varphi (t)dt$$. Then $$\overline{A}=\overline{L}+K.$$ Now we calculate the deficient numbers of operator $$\overline{L}$$. Let $$\varphi \in D_{m}$$. Function $$\varphi \in \operatorname{Ker}\overline{L }$$ if and only if $$\overline{L}\varphi =0$$. That means $$x^{p}\varphi ^{\prime }=0$$ then $$\varphi(x)=\displaystyle\sum\limits_{k=0}^{p-2}c_{k}\delta ^{\{k\}}(x)$$, where $$c_{k}$$ are constants.

In fact,

$$x^{p}\left(\sum_{k=0}^{p-2}c_{k}\delta ^{\{k\}}(x)\right)^{\prime }=\sum_{k=0}^{p-2}c_{k}x^{p}\delta ^{\{k+1\}}(x)=0,\ \text{ because }\ k+1<p,$$

therefore $$\alpha (\overline{L})=\operatorname {dim}\operatorname{Ker}\overline{L}=p-1.$$

Otherwise, the non-homogeneous equation $$\overline{L}\varphi =f(x)$$ with $$f(x)\in C_{0}^{\{m\}}[-1,1]$$ is solvable in the space $$D_{m}$$ if and only if we have $$(f(x),\delta ^{\{k\}}(x))=0$$, $$k=1,2,\ldots,p-1$$.

In fact, $$\overline{L} \varphi =f(x)$$ is solvable if and only if

$$(f(x),\psi(x) )=0\quad \forall \psi \in (\operatorname{Im}L)^{\perp }=\operatorname{Ker}\overline{L^\prime } \text{, i.e., }\ \overline{L^\prime }\psi =0.$$

We have

$$(f(x),\psi (x)) =\left(f(x),\dfrac{z(x)}{x^{p}}+\sum_{k=0}^{p-1}c_{k}\delta ^{\{k\}}(x)\right)=(\varphi ^{\prime },z(x))+\left(f,\sum_{k=0}^{p-1}c_{k}\delta ^{\{k\}}(x)\right)$$
$$=-(\varphi ,z^{\prime }(x))+\sum_{k=0}^{p-1}c_{k}(f,\delta ^{\{k\}}(x)).$$

Otherwise,

$$\overline{L^\prime }\psi =0\Leftrightarrow -z^{\prime }(x)+\sum_{k=0}^{p-1}c_{k}x^{p}\delta ^{\{k\}}(x)=0\Leftrightarrow -z^{\prime }(x)=0.$$

Then, we deduce

$$(f(x),\psi(x) )=\sum_{k=0}^{p-1}c_{k}(f(x),\delta ^{\{k\}}(x))$$

and

$$(f(x),\psi(x) )=0\Leftrightarrow (f(x),\delta ^{\{k\}}(x))=0\quad\forall k=0,1,\ldots,p-1.$$

Consequently, $$\beta (\overline{L})=p,$$ thus,

$$\chi (\overline{L})=\alpha (\overline{L})-\beta (\overline{L})=p-1-p=-1.$$

$$\square$$

The following result deals with the index of associated operator.

### Theorem 4.3.

Operator $$\overline{A^{\prime }}:\overline{P^{1}} \rightarrow C[-1,1]$$ is a Noether operator with index $$\chi (\overline{ A^{\prime }})=1.$$

Proof. According to Theorem 4.2, $$\chi (\overline{A^{\prime }})=-\chi ( \overline{A})=1$$. Note that it is also possible to study the Noetherity of operator $$\overline{A^{\prime }}$$ through the principle part $$\overline{ L^{\prime }}$$ of operator $$\overline{A^{\prime }}$$. In fact $$\overline{ L^{\prime }}\psi =0$$ has in the space $$\overline{P^{1}}$$ exactly p-linear independent solutions, so $$\alpha (\overline{L^{\prime }})=p$$, solution $$\psi$$ has the form

$$\psi (x)=\dfrac{1}{x^{p}}\int_{x}^{1}h(t)dt,$$

and $$\psi (x)\in \overline{P^{1}}$$ if $$z(x)=\displaystyle\int_{x}^{1}h(x)dt$$ under $$p-1$$ conditions

$$z^{\prime }(0)=\cdots =z^{\{p-1\}}(0)=0,$$

i.e.,

$$h(0)=h^{\{1\}}(0)=\cdots =h^{\{p-2\}}(0)=0.$$
(11)

Therefore $$\beta (\overline{L^{\prime }})=p-1$$ and finally

$$\chi (\overline{L^{\prime }})=\chi (\overline{A^{\prime }})=\alpha ( \overline{L^{\prime }})-\beta (\overline{L^{\prime }})=p-(p-1)=1.$$

Note that conditions (11) can be understood like the orthogonality conditions

$$(h(t),\delta ^{\{j\}}(t))=0,\quad j=0,\ldots,p-2,$$

of all solutions of equation $$\overline{L^{\prime }}\psi =0.$$

Note that our investigation is based on the case of $$m=p-2$$. Now, consider $$0\leq m<p-2$$, then by investigating as above, we will have instead of conditions (10) $$m+1$$ conditions

$$z^{\{1\}}(0)=z^{\{2\}}(0)=\cdots =z^{\{m+1\}}(0)=0.$$

$$\square$$

## 5. CONCLUSION

In this paper, we have investigated the Noetherity of extension $$\overline{A}$$ of the operator A defined by (3) in the space $$D_{m}$$, and have studied the associated operator $$\overline{A^{\prime }}$$ of the operator $$\overline{A}$$ by using the principle part under the perturbation with compact operator. The first result we have obtained concerning associated operators $$\overline{A^{\prime }}$$ of $$\overline{A}$$ is proposed in Theorem 4.1. The definition of the associated operator $$\overline{A^{\prime }}$$ has helped us to investigate the Noetherity of operators $$\overline{A}$$ and $$\overline{A^{\prime }}$$ in Theorem 4.2 and Theorem 4.3.

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## Author information

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Correspondence to E. Tompé Weimbapou, Abdourahman or E. Kengne.

Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 11, pp. 40–53.

Tompé Weimbapou, E., Abdourahman & Kengne, E. On Delta-Extension for a Noether Operator. Russ Math. 65, 34–45 (2021). https://doi.org/10.3103/S1066369X21110050

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• DOI: https://doi.org/10.3103/S1066369X21110050

### Keywords

• Integral equation of the third kind
• Characteristic numbers
• fundamental functions
• Singular operator