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

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

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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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Keywords

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