On Principal Parts-Extension for a Noether Operator A

Abstract

The main purpose of this work is to realize and establish the extension of a noether operator A defined by a third kind singular integral equation. The extended operator of the initial operator A is noted  and acting from a new specific functional space constructed denoted Vm=C−11−11⊕∑k=1mαkF.p1xk, into the same previous space C0P−11. We investigate the noetherity of the extended initial noether operator noted Â, when we realized the extension taking the unknown function φx from the spaceVm. The index of such noether operator χ is calculated and therefore, the conditions of the noetherity nature of the extended operator are established.

1. Introduction

The construction of noether theory for some integrodifferential operators defined by linear third-kind integral equations in some specific functional spaces is well known and still interests many scientists around the world. Various scientific works dedicated to the noetherity of integrodifferential operators have been published by many researchers investigating such topics.

In our previous works, while constructing noether theory for integrodifferential operators defined by the third kind integral equations, we approached the question of the solvability of linear integral equations of the third kind $a\left(x\right)I+K,$ where $K$ is an integral operator and $a\left(x\right)$ is a given function vanishing on some set of points. That was illustrated in many works (see for example, papers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and scientific collections [13, 14]. We also recall that such equations arise frequently in various applications and by their particular nature are connected with singularities of the function a(x). In connexion with integral equations of the third kind, we can note that they have been investigated at the beginning of the centenary. In addition, there is a particular interest linked to them in relation to the requirement of transport theory without forgetting the theory of elliptical-hyperbolic type equations. In most scientific research works, the solution of the integral equation of the third kind seems to be the space of continuous functions on the closed interval $\left[a,b\right].$

Specific approaches are needed when constructing noether theory for integrodifferential operators and, once we succeed to establish the noetherity of the considered operator, we can always set to the problem of the investigation of the extension of the studied already noether operator. For some details see [15, 16, 17, 18].

The illustration of the great importance of the construction and establishment of noetherity of some integrodifferential operators defined by third-kind integral equations is clearly presented through the works of many scientists. For full details on such investigations and results, we can refer to works such as [3, 4, 5, 15, 16, 19, 20, 21, 22, 23, 24, 25, 26]. Let us recall that among many others, Bart G.R, Warnock R.L., Roghozin V., Raslambekov V.S., and Gobbassov N.S., respectively, in their scientific works, had constructed noether theory for some integrodifferential operators defined similarly but having some specificities on the considered integral equation. They have realized some cases of extension and established noetherity of the considered extended operator.

We recall that the noetherity of the initial operator $\left(\mathit{A\phi }\right)\left(x\right)=x^p{\phi }^{\prime }\left(x\right)+{\int }_{-1}^{1}K\left(x,t\right)\phi \left(t\right)\mathit{dt}=f\left(x\right);x\in \left[-1,1\right]$, with $\phi \in C_{-1}^1\left[-1,1\right]$, $f\left(x\right)\in C_0^{\left\{P\right\}}\left[-1,1\right]$ and $K\left(x,t\right)\in C_0^{\left\{P\right\}}\left[-1,1\right]\mathrm{X}\mathrm{C}\left[-1,1\right]$ is completely established in [4, 5].

E.Tompé and al, in their recent published article titled “On Delta-Extension for a Noether Operator” have realized the extension of the initial operator $\left(\mathit{A\phi }\right)\left(x\right)=x^p{\phi }^{\prime }\left(x\right)+{\int }_{-1}^{1}K\left(x,t\right)\phi \left(t\right)\mathit{dt}=$ $f\left(x\right);x\in \left[-1,1\right],$ when the unknown function rather than $\phi \in C_{-1}^1\left[-1,1\right],$ was took from the space $D_m=C_{-1}^1\left[-1,1\right]\oplus \left\{{\sum }_{k=0}^m{\alpha }_k{\delta }^{\left\{k\right\}}\left(x\right)\right\}$ with the conditions $0\le m\le p-2$. For full details see [22].

Just recently their paper titled “Noetherity of a Dirac Delta-Extension for a Noether Operator” was published in the International Journal of Theoretical and Applied Mathematics. Vol. 8, No. 3, 2022, pp. 51–57. doi: 10.11648/j.ijtam.20220803.11, Abdourahman, Ecclésiaste Tompé Weimbapou, and Emmanuel Kengne completely covered the investigation of the noetherity of the extended operator $\overline{A}$ of the initial operator $\left(\mathit{A\phi }\right)\left(x\right)=x^p{\phi }^{\prime }\left(x\right)+{\int }_{-1}^{1}K\left(x,t\right)\phi \left(t\right)\mathit{dt}=f\left(x\right);x\in \left[-1,1\right],$ previously started, when also, at this time the unknown function rather than $\phi \in C_{-1}^1\left[-1,1\right]$has been taken as following $\phi \in D_m=C_{-1}^1\left[-1,1\right]\oplus \left\{{\sum }_{k=0}^n{\alpha }_k{\delta }^{\left\{k\right\}}\left(x\right)\right\}\text{with supplementary condition}m>p-2$.

The noetherity, in both two cases, investigated, of the extended operator noted $\overline{A}$ was established and its index $\chi \left(\overline{A}\right)$ is calculated.

Following such previous research cited and other works are done by many scientists, related to the realization of various types of extensions of noether operators, we are conducting the work to realize a particular type of extension when we add, at this time, functions from the space of principal-parts values of the following indicated form $\left\{{\sum }_{k=1}^m{\alpha }_{k}F.p\frac{1}{x^k}\right\}$.

Namely here in this paper, we realize the extension of the following noether operator defined by the third kind singular integral equation

$$\left(\mathit{A\phi }\right)\left(x\right)=x^p{\phi }^{\prime }\left(x\right)+{\int }_{-1}^{1}K\left(x,t\right)\phi \left(t\right)\mathit{dt}=f\left(x\right);x\in \left[-1,1\right]$$E1

where $\phi \in C_{-1}^1\left[-1,1\right],f\left(x\right)\in C_0^{\left\{P\right\}}\left[-1,1\right]$ and $K\left(x,t\right)\in C_0^{\left\{P\right\}}\left[-1,1\right]\mathrm{X}\mathrm{C}\left[-1,1\right]$ with principal parts functions, i.e., $\phi \in V_m=C_{-1}^1\left[-1,1\right]\oplus \left\{{\sum }_{k=1}^m{\alpha }_{k}F.p\frac{1}{x^k}\right\}$ and next, we establish the noetherity of the extended operator.

The structure of this chapter is the following: Section 2 is devoted to some fundamental well-known notions and concepts of noether theory, Fredholm third kind integral equation, Taylor derivatives, associated spaces, and associated operators. Section 3 presenting the main results of the chapter deals properly with the realization of the extension of the operator A when taking the unknown function from the space $V_m.$ Lastly, after making a small important remark, we conclude our chapter in Section 4, followed by some recommendations for the follow-up or future scientific works to undertake, as stated in Section 5.

2. Preliminaries

Before presenting our main results in full detail, the following definitions and concepts well known from the noether theory of operators are required for the realization of this research. We also recall the notions of Taylor derivatives and linear Fredholm integral equation of the third kind, widely studied in many works done by different authors among many of them Bart GR, Sukavanam N, Shulaia D.A, Gobbassov N.S. See [3, 27, 28, 29, 30, 31] for more details.

First of all, let us move to the following concept.

1. Noether operator

   Definition 1. Let $X$, Y be Banach spaces, $A\in l\left(X,Y\right)$ a linear operator. The quotient space$\mathit{\text{coker}}A=Y/\mathit{imA}$ is called the cokernel of the operator $A$. The dimensions $\alpha \left(A\right)=dim\mathit{\text{kerA}},\beta \left(A\right)=dim\mathit{\text{cokerA}}$ are called the nullity and the deficiency of the operator $A$, respectively. If at least one of the numbers $\alpha \left(A\right)$ or $\beta \left(A\right)$ is finite, then the difference $\mathit{Ind}A=\alpha \left(A\right)-\beta \left(A\right)$ is called the index of the operator $A$.

   Definition 2. Let $X$, $Y$ be Banach spaces, $A\in l\left(X,Y\right)$ is said to be normally solvable if it possesses the following property: The equation $\mathit{Ax}=y\left(y\in Y\right)\left(y\in Y\right)$ has at least one solution $x\in D\left(A\right)$ ($D\left(A\right)$ is the domain of $A$) if and only if $<y,f>=0\forall f\in {\left(\mathit{im}A\right)}^{\perp }$ holds.

   We recall that by the definition of the adjoint operator ${\left(\mathit{im}A\right)}^{\perp }=ker{A}^{\ast }$ and it’s proven in [4] that The operator $A$ is normally solvable if and only if its image space $\mathit{imA}$ is closed.

   Definition 3. A closed normally solvable operator A is called a Noether operator if its index is finite.

   By the way, we briefly review these important notions of Taylor derivatives which are widely used when constructing noether theory of the considered operator A.

   Definition 4. A Continuous function $\phi \left(x\right)ϵC\left[-1,1\right]$ admits at the point $x=0$ Taylor derivative up to the order $p\in \mathbb{N}\text{if there exists recurrently for}k=1,2,\dots ,p,$ the following limits:

   $${\phi }^{\left\{k\right\}}\left(0\right)=k!\underset{x\to 0}{lim}{x}^{-k}\left[\phi \left(x\right)-\sum _{j=0}^{k-1}\frac{{\phi }^{\left\{j\right\}}\left(0\right)}{j!}{x}^j\right]$$E2

   The class of such functions $\phi \left(x\right)$ is denoted $C_0^{\left\{p\right\}}\left[-1,1\right]$.

   Next, let us move to the following part.

   Let $C^m\left[-1,1\right],m\in {\mathbb{Z}}_{+},$ noted the Banach space of continuous functions on $\left[-1,1\right],$ having continuous derivatives up to order $m$, for which the norm is defined as follows:

   $${‖\phi \left(x\right)‖}_{C^m\left[-1,1\right]}=\sum _{\mathrm{j}=0}^{\mathrm{m}}\underset{-1\le \mathrm{x}\le 1}{max}\left|{\mathrm{\phi }}^{\left(\mathrm{j}\right)}\left(\mathrm{x}\right)\right|$$E3

   Therefore, we can consider ${\phi }^{\left\{k\right\}}\left(0\right)$ are defined for all $k=1,2,\dots ,p$.

   We define $C_0^{\left\{p\right\}}\left[-1,1\right]$ as a subspace of continuous functions, having finite Taylor derivatives up to order $p\in {\mathbb{Z}}_{+}$; and when $p=0,\text{we}\mathrm{put}\left(C_0^{\left\{p\right\}}\left[-1,1\right]=C_0^{\left\{0\right\}}\left[-1,1\right]=C\left[-1,1\right]\right)$.

   Let us also define a linear operator $N^k$ on the space $C_0^{\left\{p\right\}}\left[-1,1\right]$ by the formula:

   $$\left(N^k\phi \right)\left(x\right)=\frac{\phi \left(x\right)-{\sum }_{j=0}^{k-1}\frac{{\phi }^{\left\{j\right\}}\left(0\right)}{j!}{x}^j}{x^k},k=1,2,\dots ,p$$E4

   One can easily verify the property $N^k=N^{k_1}{N}^{k-k_1},0\le k_1\le k,k,k_1\in {\mathbb{Z}}_{+},$ where we put $N^0=I.$

   Definition 5. The operator $N^p$ is called the characteristical operator of the space $C_0^{\left\{p\right\}}\left[-1,1\right].$

   Remark: The sense of the previous definition can be seen from the verification of the following lemma and also for more details see [23, 25, 26].

   Lemma 2.1. A function $\phi \left(x\right)$ belongs to $C_0^{\left\{p\right\}}\left[-1,1\right]$ if and only if, the following representation

   $$\phi \left(x\right)=x^p\varphi \left(x\right)+\sum _{k=0}^{p-1}{\alpha }_k{x}^k$$E5

   holds with the function $\varphi \left(x\right)\in C\left[-1,1\right]$, and ${\alpha }_k$ being constants.

   To prove Lemma 2.1 it is enough to observe that (5) implies that the Taylor derivatives of $\phi \left(x\right)$ up to the order $p$ exists, and more ${\phi }^{\left\{k\right\}}\left(0\right)=k!{\alpha }_k,k=0,1,2,\dots ,p-1,{\phi }^{\left\{0\right\}}\left(0\right)=p!\varphi \left(0\right)$ with $\varphi \left(x\right)=\left(N^k\phi \right)\left(x\right)$. Conversely, if $\phi \left(x\right)$ belongs to $C_0^{\left\{p\right\}}\left[-1,1\right]$, and we define $\varphi \left(x\right)=\left(N^k\phi \right)\left(x\right)$ with ${\alpha }_k=\frac{{\phi }^{\left\{k\right\}}\left(0\right)}{k!},k=0,1,2,\dots ,p-1,$ then the representation (5) holds. From Lemma 2.1, it follows that for $\phi \left(x\right)\in C_0^{\left\{p\right\}}\left[-1,1\right]$ the inequality

   $$\phi \left(x\right)=x^p\left(N^k\phi \right)\left(x\right)+\sum _{k=0}^{p-1}\frac{{\phi }^{\left\{k\right\}}\left(0\right)}{k!}{x}^k,$$E6

   is valid.

   Consequently, the linear operator $N^p$ establishes a relation between the spaces $C_0^{\left\{p\right\}}\left[-1,1\right]$ and $C\left[-1,1\right]$. The space $C_0^{\left\{p\right\}}\left[-1,1\right]$ with the norm

   $${‖\mathrm{\phi }‖}_{C_0^{\left\{p\right\}}\left[-1,1\right]}={‖N^p\phi ‖}_{C\left[-1,1\right]}+\sum _{k=0}^{p-1}\left|{\phi }^{\left\{k\right\}}\left(0\right)\right|,$$E7

   becomes a Banach space one.

   Let note also that we can define the previous norm in the following way:

   $${‖\mathrm{\phi }‖}_{C_0^{\left\{p\right\}}\left[-1,1\right]}={‖N^p\phi ‖}_{C\left[-1,1\right]}+\sum _{k=0}^{p-1}\left|{\alpha }_k\right|={‖\varphi \left(x\right)‖}_{C\left[-1,1\right]}+\sum _{k=0}^{p-1}\left|{\alpha }_k\right|.$$E8

   Sometimes it is comfortable and suitable to consider as the norm in the space $C_0^{\left\{p\right\}}\left[-1,1\right]$ the equivalent norm is defined as follows:

   $${‖\mathrm{\phi }‖}_{C_0^{\left\{p\right\}}\left[-1,1\right]}=\sum _{j=0}^p{‖N^j\phi ‖}_{C\left[-1,1\right]}$$E9

   We can also note a very useful and clearly helpful next inequality:

   $${‖\mathrm{\phi }‖}_{C\left[-1,1\right]}\le {‖N^p\phi ‖}_{C\left[-1,1\right]}+\sum _{j=0}^{p-1}\left|{\phi }^{\left\{j\right\}}\left(0\right)\right|={‖\mathrm{\phi }‖}_{C_0^{\left\{p\right\}}\left[-1,1\right]}$$E10

   Therefore, it is obvious to see that

   $${‖\mathrm{\phi }‖}_{C\left[-1,1\right]}\le {‖\mathrm{\phi }‖}_{C_0^{\left\{p\right\}}\left[-1,1\right]}.$$E11

   Finally, note that from definition 2.1, we can follow fact that if $\phi \left(x\right)\in C\left[-1,1\right]$, then $x^p\phi \left(x\right)\in C_0^{\left\{p\right\}}\left[-1,1\right]$. This assertion may be generalized as follows:

   Lemma 2.2. Let $p\in \mathbb{N},s\in {\mathbb{Z}}_{+}$. If $\phi \left(x\right)\in C_0^{\left\{s\right\}}\left[-1,1\right]$ then, $x^p\phi \left(x\right)\in C_0^{\left\{p+s\right\}}\left[-1,1\right]$, and the formula holds

   $${\left(x^p\phi \left(x\right)\right)}^{\left\{j\right\}}\left(0\right)=\left\{\begin{array}{c}0,j=0,1,\dots ,p-1,\\ \frac{j!}{\left(j-p\right)!}{\phi }^{\left\{j-p\right\}}\left(0\right),j=p,\dots ,p+s.\end{array}\right.$$E12

   Proof. Note that a stronger assumption on the function $\phi \left(x\right)$, such that $\phi \left(x\right)\in C_0^{\left\{p+s\right\}}\left[-1,1\right]$ would allow us to easily prove the lemma just by applying the Leibniz formula.

   For $s=0$ the statement has already been proved above, so $x^p\phi \left(x\right)\in C_0^{\left\{p\right\}}\left[-1,1\right],$ and ${\left(x^p\phi \left(x\right)\right)}^{\left\{j\right\}}\left(0\right)=0,j=0,\dots ,p-1$ and ${\left(x^p\phi \left(x\right)\right)}^{\left\{p\right\}}\left(0\right)=p!\phi \left(0\right)$. Now let us prove that ${\left(x^p\phi \left(x\right)\right)}^{\left\{j\right\}}\left(0\right)=\frac{j!}{\left(j-p\right)!}{\phi }^{\left\{j-p\right\}}\left(0\right),j=p+1,.\dots ,p+s.$ Since the derivatives are defined recurrently, and (12) is true for $j=p,$ then it is sufficient to verify the passage from $j\text{to}j+1$. We have:

   $${\left(x^p\phi \left(x\right)\right)}^{\left\{j+1\right\}}\left(0\right)=\left(j+1\right)!\underset{x\to 0}{lim}\frac{x^p\phi \left(x\right)-{\sum }_{l=p}^j\frac{x^l}{\left(l-p\right)!}{\phi }^{\left\{l-p\right\}}\left(0\right)}{x^{j+1}}$$E13
   $$=\left(j+1\right)!\underset{x\to 0}{lim}\frac{\phi \left(x\right)-{\sum }_{l=0}^{j-p}\frac{x^l{\phi }^{\left\{l\right\}}\left(0\right)}{l!}}{x^{j+1-p}}=\frac{\left(j+1\right)!}{\left(j+1-p\right)!}{\phi }^{\left\{j+1-p\right\}}\left(0\right).$$E14

   Lemmas 2.1 and 2.2 imply the next important lemma.

   Lemma 2.3. Let$\mathrm{f}\left(x\right)\in C_0^{\left\{p\right\}}\left[-1,1\right],p\in \mathbb{N}\mathit{\text{and}}f\left(0\right)=..\dots =f^{\left\{r-1\right\}}\left(0\right)=0,1\le r\le p.$ Then $\frac{f\left(x\right)}{x^r}\in C_0^{\left\{p-s\right\}}\left[-1,1\right]$.

   We say that the kernel $k\left(x,t\right)$ ∈ $C_0^{\left\{P\right\}}\left[-1,1\right]X\mathrm{C}\left[-1,1\right]$, if and only if $k\left(x,t\right)\in \mathrm{C}\left[-1,1\right]\mathrm{X}\mathrm{C}\left[-1,1\right]$ and admits Taylor derivatives according to the variable x at the point (0,t) whatever t ∈$\left[-1,1\right]$.

2. Associated spaces and associated operators

   Instead of talking about adjoint operators when establishing the noetherity of an operator, we can note that also noether property of an operator may depend on the concept of associated operators and associated spaces. Therefore, we start by recalling these two important concepts and we give some associated spaces that we are going to use later within the work.

   Definition 6.

   The Banach space $E^{’}\subset E^{\ast }$ is called associated space with a Banach space $E$, if

   $$\left|<f,\phi >\right|\le c{‖f‖}_{E^{\prime }}{‖\phi ‖}_E\forall \phi \in E,f\in E^{\prime }.$$E15

   Definition 6. Let $E_j$, (j = 1,2) be Banach spaces and $E_j^{\prime }$ their associated spaces, operators $A\in l\left(E_1,E_2\right)$ and $A^{\prime }\in l\left(E_1^{\prime },E_2^{\prime }\right)$ are associated if and only if:

   $$\left(A^{\prime }f,\phi \right)=\left(f,\mathit{A\phi }\right)\forall f\in E_2^{\prime }\text{and}\phi \in E_1.$$E16

   The following important result gives noether property via an associated operator.

   Lemma 1. Let $E_j$, j = 1,2 be Banach spaces, $E_j^{\prime }$ their associated spaces, $A\in l\left(E_1,E_2\right)$ and $A^{\prime }\in l\left(E_1^{\prime },E_2^{\prime }\right)$ are associated with noether operators, we have $\chi \left(A\right)=-\chi \left(A^{\prime }\right)$ (where $\chi$ means the index), and for the solvability of equation $\mathit{A\phi }=f$ it’s necessary and sufficient that $\left(f,\psi \right)=0$ for all solutions of the associated homogenous equation $A^{\prime }\psi =0$.

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

   Lemma 2. Space $C_{x_0}^1\left[-1,1\right]$ is associated with space $C\left[-1,1\right]$ where $C_{x_0}^1\left[-1,1\right]$ means the space of functions $\phi \in C\left[-1,1\right]$ satisfying $\phi \left(x_0\right)=0.$

   Proof. Let $f\in C_{x_0}^1\left[-1,1\right]$ and $\phi \in C\left[-1,1\right]$. Then we have:

   $$\left|<\mathrm{f},\mathrm{\phi }>\right|=\left|{\int }_{-1}^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{\phi }\left(\mathrm{x}\right)\mathrm{dx}\right|\le 2{max}_{-1\le \mathrm{x}\le 1}\left|\mathrm{f}\left(\mathrm{x}\right)\right|.{max}_{-1\le \mathrm{x}\le 1}\left|\mathrm{\phi }\left(\mathrm{x}\right)\right|.∎$$E17

   Let us also recall the definition of the space of generalized functions $P^1$ given in [12, 22, 23].

   Definition 7. Through $P^1$ we denote the space of distributions $\psi$ on the space of test functions $C_{-1}^{\left\{p\right\}}\left[-1,1\right]$ such that:

   $\psi \left(x\right)=\frac{z\left(x\right)}{x^p}+{\sum }_{k=0}^{p-1}{\beta }_k{\delta }^{\left\{k\right\}}\left(x\right)$ where $\left(x\right)\in C_{-1}^{\left\{p\right\}}\left[-1,1\right]\cap C_{-1}^1\left[-1,1\right]$, ${\beta }_k$ are arbitrary constants, ${\delta }^{\left\{k\right\}}\left(x\right)\text{is the}k-\mathit{th}$ Taylor derivative of the Dirac-delta function defined by:

   $$\left({\delta }^{\left\{k\right\}}\left(x\right),\phi \left(x\right)\right)={\int }_{-\infty }^{+\infty }{\delta }^{\left\{k\right\}}\phi \left(x\right)\mathit{dx}={\left(-1\right)}^k{\phi }^{\left\{k\right\}}\left(0\right).$$E18

   In the space $P^1$ let us introduce the norm in the following way:

   $${‖\psi ‖}_{P^1}={‖z‖}_{C_{-1}^{\left\{p\right\}}\left[-1,1\right]}+{‖z‖}_{C^1\left[-1,1\right]}+\sum _{k=0}^{p-1}\left|{\beta }_k\right|$$E19

   with this norm, it was proved in [12, 22, 23] that the space $P^1$ is a Banach space.

   Lemma 3. The space $P^1$ is a Banach space associated with the space $C_{-1}^{\left\{p\right\}}\left[-1,1\right].$

   Proof. Similar to previous proof and for more details see also [5, 22].

   Definition 8. An equation of the form

   $${\mathrm{A}}_{\mathrm{n}}\mathrm{\phi }=\mathrm{f},$$E20

   where $\mathrm{f}$ is a given function of the variable $\mathrm{x}\in \left[\mathrm{a},\mathrm{b}\right]$ and $\mathrm{\phi }$ the unknown function of $\mathrm{x}\in \left[\mathrm{a},\mathrm{b}\right]$ when the operator ${\mathrm{A}}_{\mathrm{n}}$ is defined by

   $${\mathrm{A}}_{\mathrm{n}}\mathrm{\phi }={\mathrm{g}}_{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{\phi }\left(\mathrm{x}\right)-{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{k}\left(\mathrm{x},\mathrm{t}\right)\mathrm{\phi }\left(\mathrm{t}\right)\mathrm{dt}$$E21

   is called a linear Fredholm integral equation of the third kind.

   In this case, ${\mathrm{g}}_{\mathrm{n}}\left(\mathrm{x}\right)=\prod _{\mathrm{k}=1}^{\mathrm{n}}\left(\mathrm{x}-{\mathrm{x}}_{\mathrm{k}}\right)$ is a given function of the variable $\mathrm{x}\in \left[\mathrm{a},\mathrm{b}\right]$ with ${\mathrm{x}}_{\mathrm{k}}\in \left]\mathrm{a},\mathrm{b}\right[$ and $\mathrm{k}\left(\mathrm{x},\mathrm{t}\right)$ is a given function of variables $\left(\mathrm{x},\mathrm{t}\right)\in \left[\mathrm{a},\mathrm{b}\right]\mathrm{X}\left[\mathrm{a},\mathrm{b}\right]$.

   Related to this notion of linear Fredholm integral equation of the third kind with full details also can be found in [5, 22].

   Now, we can move to the presentation of the general results of our investigation stated in the following section.

3. Main results

Before we state the content of our results within the whole investigation, let’s also give some needed definitions related to the main space and to singular integral for functions we used in the work.

Definition 9. We denote through $V_m=C_{-1}^1\left[-1,1\right]\oplus \left\{{\sum }_{k=1}^m{\alpha }_{k}F.p.\left(\frac{1}{x^k}\right)\right\}$ the space of all functions presented as follows:

$$\phi \left(x\right)={\phi }_0\left(x\right)+\sum _{k=1}^m{\alpha }_{k}F.p.\left(\frac{1}{x^k}\right)$$E22

where ${\phi }_0\left(x\right)\in C\left[-1,1\right]$ and ${\phi }_0\left(-1\right)=0$ with the natural norm

$$‖\phi ‖={‖{\phi }_0‖}_{C\left[-1,1\right]}+\sum _{k=1}^m\left|{\alpha }_k\right|..$$E23

Next, we move to the following important concept.

1. Singular integral for functions from the space $C_0^{\left\{p\right\}}\left[-1,1\right].$

   If the function $g\left(x\right)$ has feature (singularity) in $x=0$, then we say that ${\int }_{-1}^1\mathrm{g}\left(x\right)\mathit{dx}$ exists in the sense of Hadamard if it is true the following representation:

   $${\int }_{-1}^{-\mathrm{\epsilon }}\mathrm{g}\left(x\right)\mathit{dx}+{\int }_{\mathrm{\epsilon }}^1\mathrm{g}\left(x\right)\mathit{dx}=a+\sum _{k=1}^l{a}_k{\epsilon }^{-k}+a_{l+1}\mathit{ln}\frac{1}{\epsilon }+0\left(\epsilon \right),\epsilon \to 0.$$E24

   In this case, we put $\mathrm{F}.\mathrm{p}.{\int }_{-1}^1\mathrm{g}\left(x\right)\mathit{dx}=a$, i.e., it remains the finite parts. Note that under the definition of convergence by Hadamard, we often take $a_{l+1}=0$, but we do not exclude that possibility as this can allow us to consider the convergence $\left(V.p.\right)$ in the sense of Cauchy principal part as a particular case of convergence in the sense of Hadamard.

   Now let $\phi \left(x\right)\in C_0^{\left\{p\right\}}\left[-1,1\right],p\in \mathbb{N}$ and consider ${\int }_{-1}^1\frac{\phi \left(x\right)}{x^p}\mathit{dx},p\in \mathbb{N}$.

   Lemma 4. Let $\phi \left(x\right)\in C_0^{\left\{p\right\}}\left[-1,1\right],p\in \mathbb{N}$. Then it takes place the following relationships:

   $$\left\{\begin{array}{c}\mathrm{F}.\mathrm{p}.{\int }_{-1}^1\frac{\phi \left(x\right)}{x^p}\mathit{dx}={\int }_{-1}^1\left({\mathrm{N}}^{\mathrm{p}}\phi \right)\left(x\right)\mathit{dx}+\sum _{k=0}^{p-2}\frac{{\phi }^{\left\{k\right\}}\left(0\right)\left(1-{\left(-1\right)}^{k-p+1}\right)}{k-p+1}\\ \text{under}p\ge 2,\\ \text{and}\\ \mathrm{F}.\mathrm{p}.{\int }_{-1}^1\frac{\phi \left(x\right)}{x}\mathit{dx}=V.p.{\int }_{-1}^1\frac{\phi \left(x\right)}{x}\mathit{dx}={\int }_{-1}^1\left(\mathrm{N}\phi \right)\left(x\right)\mathit{dx}.\end{array}\right.$$E25

   Proof. For the proof, we note that by virtue of lemma 2.1 we have $\phi \left(x\right)=x^p\left({\mathrm{N}}^{\mathrm{p}}\phi \right)\left(x\right)+{\sum }_{k=0}^{p-1}\frac{{\phi }^{\left\{k\right\}}\left(0\right)}{k!}{x}^k$ and next, it remains to note that

   $$\left\{\begin{array}{c}{\int }_{-1}^{-\mathrm{\epsilon }}\frac{\mathit{dx}}{x^{p-k}}+{\int }_{\mathrm{\epsilon }}^1\frac{\mathit{dx}}{x^{p-k}}=\frac{1-{\left(-1\right)}^{k-p+1}}{k-p+1}+\frac{-{\mathrm{\epsilon }}^{k-p+1}+{\left(-\mathrm{\epsilon }\right)}^{k-p+1}}{k-p+1}\\ \text{when}k\ne p-1\\ \text{and}\\ \text{and}{\int }_{-1}^{-\mathrm{\epsilon }}\frac{\mathit{dx}}{x}+{\int }_{\mathrm{\epsilon }}^1\frac{\mathit{dx}}{x}=0.\end{array}\right.$$E26

   Consequently,

   $$\mathrm{F}.\mathrm{p}.{\int }_{-1}^1\frac{\mathit{dx}}{x^{p-k}}=\frac{1-{\left(-1\right)}^{k-p+1}}{k-p+1},k=0,1,\dots \dots ,p-2.$$E27

   That is leading us to the first assertion.

   Analogously we can prove the second assertion.

   As previously indicated, we note through $\widehat{A}$ the extension of the operator A onto the space $C_{-1}^1\left[-1,1\right]$.

   We will also note it in the following way:

   $$F.p\left(\frac{1}{x^k}\right)=P\left(\frac{1}{x^k}\right)$$E28

   and use the properties

   $$\frac{d^l}{{\mathit{dx}}^l}P\left(\frac{1}{x^k}\right)=P\left(\frac{d^l}{{\mathit{dx}}^l}\frac{1}{x^k}\right).$$E29
   $$x^{l}P\left(\frac{1}{x^k}\right)=\left\{\begin{array}{c}{x}^{l-k},l>k\\ 1,l=k\\ P\left(\frac{1}{x^{k-l}}\right),l<k.\end{array}\right..$$E30

   In the following part, we consider our operator defined by the integral equation in the case to be investigated.

2. Integral equation in the case $m=p-1$

   Let $\widehat{A}-$ be the extension of the operator $A$ defined by the Eq. (1) and $A^{\prime }$ $$ is the associated operator to $A$. Let us explain under which conditions the operators $\widehat{A}$ and ${\widehat{A}}^{\prime }$ are at least formally associated operators.

   Let $m=p-1$. So that we have immediately considered computations:

   $$\phi \left(x\right)={\phi }_0\left(x\right)+\sum _{k=1}^m{\alpha }_{k}P\left(\frac{1}{x^k}\right)={\phi }_0\left(x\right)+\sum _{k=1}^{p-1}{\alpha }_{k}P\left(\frac{1}{x^k}\right).$$E31

   First of all, we calculate $\left(\widehat{A}\phi ,\Psi \right)$. Then we have

   $$\begin{array}{rcl}\left(\widehat{A}\phi ,\Psi \right)& =& (x^p\phi {\prime }_0\left(x\right)-\sum _{k=1}^{p-1}k{\alpha }_k{x}^{p-k-1}+{\int }_{-1}^{1}K\left(x,t\right){\phi }_0\left(t\right)\mathit{dt}+\sum _{k=1}^m{\alpha }_k{\int }_{-1}^1\frac{K\left(x,t\right)}{t^k}\mathit{dt},\\ & & \frac{z\left(x\right)}{x^p}+\sum _{n=0}^{p-1}{\beta }_n{\delta }^{\left\{n\right\}}\left(x\right))\\ & =& \left(\phi {\prime }_0\left(x\right),z\left(x\right)\right)-\sum _{k=1}^{p-1}k{\alpha }_k{\int }_{-1}^1\frac{z\left(x\right)}{x^{k+1}}\mathit{dx}+\left({\int }_{-1}^{1}K\left(x,t\right){\phi }_0\left(t\right)\mathit{dt},\frac{z\left(x\right)}{x^p}\right)\\ & & +\sum _{k=1}^{p-1}{\alpha }_k\left({\int }_{-1}^1\frac{K\left(x,t\right)}{t^k}\mathit{dt},\frac{z\left(x\right)}{x^p}\right)+\sum _{n=0}^{p-1}{\left(-1\right)}^n{\beta }_n{\int }_{-1}^1{K}_1^{\left\{n\right\}}\left(0,t\right){\phi }_0\left(t\right)\mathit{dt}\\ & & +\sum _{k=1}^{p-1}{\alpha }_k\sum _{n=0}^{p-1}{\left(-1\right)}^n{\beta }_n{\int }_{-1}^1\frac{K_1^{\left\{n\right\}}\left(0,t\right)}{t^k}\mathit{dt}+\left(-\sum _{k=1}^{p-1}k{\alpha }_k{x}^{p-k-1},\sum _{n=0}^{p-1}{\beta }_n{\delta }^{\left\{n\right\}}\left(x\right)\right).\end{array}$$E32

   Next, let us also separately calculate on the other side the following expression:

   $$\begin{array}{rcl}\left(\widehat{A}\phi ,\Psi \right)& =& (x^p\phi {\prime }_0\left(x\right)-\sum _{k=1}^{p-1}k{\alpha }_k{x}^{p-k-1}+{\int }_{-1}^{1}K\left(x,t\right){\phi }_0\left(t\right)\mathit{dt}+\sum _{k=1}^m{\alpha }_k{\int }_{-1}^1\frac{K\left(x,t\right)}{t^k}\mathit{dt},\\ & & \frac{z\left(x\right)}{x^p}+\sum _{n=0}^{p-1}{\beta }_n{\delta }^{\left\{n\right\}}\left(x\right))\\ & =& \left(\phi {\prime }_0\left(x\right),z\left(x\right)\right)-\sum _{k=1}^{p-1}k{\alpha }_k{\int }_{-1}^1\frac{z\left(x\right)}{x^{k+1}}\mathit{dx}+\left({\int }_{-1}^{1}K\left(x,t\right){\phi }_0\left(t\right)\mathit{dt},\frac{z\left(x\right)}{x^p}\right)\\ & & +\sum _{k=1}^{p-1}{\alpha }_k\left({\int }_{-1}^1\frac{K\left(x,t\right)}{t^k}\mathit{dt},\frac{z\left(x\right)}{x^p}\right)+\sum _{n=0}^{p-1}{\left(-1\right)}^n{\beta }_n{\int }_{-1}^1{K}_1^{\left\{n\right\}}\left(0,t\right){\phi }_0\left(t\right)\mathit{dt}\\ & & +\sum _{k=1}^{p-1}{\alpha }_k\sum _{n=0}^{p-1}{\left(-1\right)}^n{\beta }_n{\int }_{-1}^1\frac{K_1^{\left\{n\right\}}\left(0,t\right)}{t^k}\mathit{dt}+\left(-\sum _{k=1}^{p-1}k{\alpha }_k{x}^{p-k-1},\sum _{n=0}^{p-1}{\beta }_n{\delta }^{\left\{n\right\}}\left(x\right)\right).\end{array}$$E33

   We rewrite this term in the form of a sum and we obtain definitively the equation as follows:

   $$\left(\sum _{n=0}^{p-1}{\beta }_n{\delta }^{\left\{n\right\}}\left(x\right),-\sum _{k=1}^{p-1}k{\alpha }_k{x}^{p-k-1}\right)=-\sum _{k=1}^{p-1}{\left(-1\right)}^{k-1}{\alpha }_{p-k}{\beta }_{k-1}\left(p-k\right)\left(k-1\right)!$$E34

   On the other side, we compute also the following needed expression:

   $$\begin{array}{rcl}\left(\phi ,{\widehat{A}}^{\prime }\Psi \right)& =& \left({\phi }_0\left(x\right)+\sum _{k=1}^{p-1}{\alpha }_k\frac{1}{x^k},-{\left(x^p\Psi \right)}^{\prime }+{\int }_{-1}^{1}K\left(t,x\right)\Psi \left(t\right)\mathit{dt}\right)\\ & =& \left({\phi }_0\left(x\right)+\sum _{k=1}^{p-1}{\alpha }_k\frac{1}{x^k},-z^{\prime }\left(x\right)+{\int }_{-1}^1\frac{K\left(x,t\right)}{t^p}z\left(t\right)\mathit{dt}+\sum _{n=0}^{p-1}{\left(-1\right)}^n{\beta }_n{K}_1^{\left\{n\right\}}\left(0,x\right)\right)\\ & =& \left({\phi }_0\left(x\right),-z^{\prime }\left(x\right)\right)-\sum _{k=1}^{p-1}{\alpha }_k\left(z^{\prime }\left(x\right),\frac{1}{x^k}\right)+\left({\phi }_0\left(x\right),{\int }_{-1}^1\frac{K\left(t,x\right)}{t^p}z\left(t\right)\mathit{dt}\right)\\ & & +\sum _{k=1}^{p-1}{\alpha }_k\left(\frac{1}{x^k},{\int }_{-1}^1\frac{K\left(t,x\right)}{t^p}z\left(t\right)\mathit{dt}\right)+\sum _{n=0}^{p-1}{\beta }_n{\left(-1\right)}^n\left(K_1^{\left\{n\right\}}\left(0,x\right),{\phi }_0\left(x\right)\right)\\ & & +\sum _{k=1}^{p-1}{\alpha }_n\sum _{n=0}^{p-1}{\left(-1\right)}^n{\beta }_n\left(K_1^{\left\{n\right\}}\left(0,x\right),\frac{1}{x^k}\right).\end{array}$$E35

   Now, we are able to compare $\left(\widehat{A}\phi ,\Psi \right)$ and $\left(\phi ,{\widehat{A}}^{\prime }\Psi \right).$ Therefore, we obtain the equality between the terms considered for every $\phi \left(x\right)\in V_m$ and for every $\Psi \in P^1,$ only if and only if it is taking place in the following relationship:

   $$\sum _{k=1}^{p-1}{\alpha }_k\left(z^{\prime }\left(x\right),\frac{1}{x^k}\right)=\sum _{k=1}^{p-1}{\left(-1\right)}^{k-1}{\alpha }_{p-k}{\beta }_{k-1}\left(p-k\right)\left(k-1\right)!+\sum _{k=1}^{p-1}{\mathit{k\alpha }}_k{\int }_{-1}^1\frac{z\left(x\right)}{x^{k+1}}\mathit{dx},$$E36

   where ${\beta }_{p-1}$ is an arbitrary constant.

   In other words, the operators $\widehat{A}$ and ${\widehat{A}}^{\prime }$ are associated operators only if and only, when it is accomplished under the following conditions:

   $${\alpha }_k{\int }_{-1}^1\frac{z^{\prime }\left(x\right)}{x^k}\mathit{dx}={\left(-1\right)}^{k-1}{\alpha }_{p-k}{\beta }_{k-1}\left(p-k\right)\left(k-1\right)!+{\mathit{k\alpha }}_k{\int }_{-1}^1\frac{z\left(x\right)}{x^{k+1}}\mathit{dx}$$E37

   for every $k=1,.\dots ,p-1$.

   From condition (37) we can express the parameters ${\beta }_{k-1}$ through the function $z\left(x\right)$, that is going to give us:

   $${\mathrm{\beta }}_{\mathrm{k}-1}=\frac{{\left(-1\right)}^{\mathrm{k}-1}}{\left(\mathrm{k}-1\right)!\left(\mathrm{p}-\mathrm{k}\right){\mathrm{\alpha }}_{\mathrm{p}-\mathrm{k}}}{\mathrm{\alpha }}_{\mathrm{k}}\left({\int }_{-1}^1\frac{{\mathrm{z}}^{\prime }\left(\mathrm{x}\right)}{{\mathrm{x}}^{\mathrm{k}}}\mathrm{dx}-\mathrm{k}{\int }_{-1}^1\frac{\mathrm{z}\left(\mathrm{x}\right)}{{\mathrm{x}}^{\mathrm{k}+1}}\mathrm{dx}\right),$$
   $$k=1,.\dots ,p-1.$$E38

   Therefore, if we note by $\widehat{P^1}$ the restriction of the space $P^1$ with the condition (37), then it has the following form:

   $$\widehat{P^1}=\left\{\Psi \left(x\right)\in P^1/\Psi \left(x\right)=\frac{z\left(x\right)}{x^p}+{\sum }_{k=0}^{p-2}{\beta }_k{\delta }^{\left\{k\right\}}\left(x\right)+{\beta }_{p-1}{\delta }^{\left\{p-1\right\}}\left(x\right)\right\},$$E39
   where ${\beta }_k,k=0,.\dots ,p-$ 2 are defined by the formula (38) and ${\beta }_{p-1}$ is an arbitrary constant.

   The restriction of the operator $A^{\prime }$ on the space $\widehat{P^1}$ we denote by the following way ${\widehat{A}}^{’}$.

   Next, we note $\widehat{A}$ as previously the extension of the operator $A$ on $V_m$. Then the following operators:

   $\widehat{A}$: $V_m\to C_0^{\left\{P\right\}}\left[-1,1\right]$ and ${\widehat{A}}^{’}$:$\widehat{P^1}\to C_0^{\left\{P\right\}}\left[-1,1\right]$ on the basis of previously done computations are verifying the established relationship:

   $$\left(\widehat{A}\phi ,\Psi \right)=\left(\phi ,{\widehat{A}}^{\prime }\Psi \right)$$E40

   for every $\phi \left(x\right)\in V_m$ and for every $\Psi \in \widehat{P^1}$, so that they are associated operators.

   As operator $\widehat{A}$ is the extension of the operator $\mathrm{A}$ on $\left(p-1\right)-$ dimensional space, then the operator $\widehat{A}$: $V_m\to C_0^{\left\{P\right\}}\left[-1,1\right]$ is a noether operator with the index $\chi \left(\widehat{A}\right)=-p+\left(p-1\right)=-1$.

   Next, as the operator ${\widehat{A}}^{\prime }$ is the restriction of the operator ${\mathrm{A}}^{\prime }$ on $\left(p-1\right)-$ conditions (38), then the operator ${\widehat{A}}^{’}$:$\widehat{P^1}\to C_0^{\left\{P\right\}}\left[-1,1\right]$ is also a noether operator and its index $\chi \left({\widehat{A}}^{\prime }\right)=p-\left(p-1\right)=1$.

   All that has been said allow us to formulate the result on noetherity of the extended operator $\widehat{A}$, that is what is given by virtue of Duduchava’s Lemma this following important global theorem:

   Theorem 3.1. The equation $\widehat{A}\phi =f,$ where $\widehat{A}$ is the extended operator of the operator $A$ $$ of the form (1) and $f\left(x\right)\in C_0^{\left\{P\right\}}\left[-1,1\right]$ is solvable in the space $V_m$ only if and only when ${\int }_{-1}^{1}f\left(t\right){\Psi }_k\left(t\right)\mathit{dt}=0,k=1,2,3,.\dots ,\alpha \left({\widehat{A}}^{\prime }\right),$ where $\left\{{\Psi }_k\right\}-$ is the basis of solutions of the associated homogeneous equation ${\widehat{A}}^{\prime }\Psi =0$ in the associated space $\widehat{P^1}$.

   Before concluding, let us make an important remark.

   Remark

   The requirements (38) allow us to write in a more clear way the form of the functions from the associated space $\widehat{P^1}$, i.e.,

   $$\Psi \left(x\right)\in \widehat{P^1}\iff \Psi \left(x\right)=\begin{array}{l}\frac{z\left(x\right)}{x^p}+\sum _{k=1}^{p-1}\frac{{\left(-1\right)}^{k-1}}{\left(k-1\right)!\left(p-k\right){\alpha }_{p-k}}{\alpha }_k\left({\int }_{-1}^1\frac{z^{\prime }\left(x\right)}{x^k}\mathit{dx}-k{\int }_{-1}^1\frac{z\left(x\right)}{x^{k+1}}\mathit{dx}\right){\delta }^{\left\{k-1\right\}}\left(x\right)\\ +{\beta }_{k-1}{\delta }^{\left\{p-1\right\}}\left(x\right),\end{array}$$E41

   where ${\beta }_{p-1}$ is an arbitrary constant.

4. Conclusion

Summarizing our work, we state that we have completely realized the extension of a noether operator $A$ defined by the extended operator $\widehat{A}$ in the space $V_m$. We applied the well-known noether theory for integrodifferential operators defined by a third kind integral equation and, we computed very attentively both the two expressions $\left(\widehat{A}\phi ,\Psi \right)\text{and}\left(\phi ,{\widehat{A}}^{\prime }\Psi \right),$ taking the functions $\phi$ and $\Psi$ respectively from the generalized functional spaces $V_m$ and $\widehat{P^1}$. From the previous, we released the conditions under which the two operators $\widehat{A}$ and ${\widehat{A}}^{\prime }$ are associated operators. Consequently, we formalized within theorem 3.1 the global results of the investigation related to the question of noetherity nature of the extended operator, which as proved is noether operator. The principle of the conservation of noetherity nature of a noether operator after extension by some finite dimensional space of added functions to the initial space is established firmly as proved in theory. We can also note that the index of the initial operator after extension remains the same, i.e., $\chi \left(A\right)=\chi \left(\widehat{A}\right)$ no matter the deficient numbers may be other than the previous before extension, i.e., $\left(\alpha \left(A\right),\beta \left(A\right)\right)\ne \left(\alpha \left(\widehat{A}\right),\beta \left(\widehat{A}\right)\right)$.

5. Recommendations

The achieved researches in this work completed by those already obtained by many scientific researchers related to the question of the noetherity nature of an extended operator of an initial noether operator in some various functional generalized spaces may lead us to project, and to set very interesting and challenging future investigations for noetherity when at this time, we take the unknown generalized function from the space $T_m=C_{-1}^1\left[-1,1\right]\oplus \left\{{\sum }_{k=0}^m{\alpha }_k{\delta }^{\left\{k\right\}}\left(x\right)\right\}\oplus \left\{{\sum }_{k=1}^m{\alpha }_{k}F.p\frac{1}{x^k}\right\},$ as previously done by many scientists in their researchers, namely cited Gobbassov N. S, Raslambekov S. N, Bart G. R, and Warnock R. L.

This will be the next work to be done in a brief future. We underline once more again that the main difficulty appearing when realizing such extension is still and always connected with the derivative of the unknown function within the third kind singular integral equation through which is defined the initial integrodifferential operator to be extended onto the new generalized functional space.

Acknowledgments

The author is very grateful to the late Professor KARAPETYANTS N.K former Head of the Department of Differential and Integral Equations of the Rostov State University in the Russian Federation, for his invaluable support and helpful discussions during that time when conducting such research.