Existence of solutions for some systems of integro-differential equations with transport and superdiffusion

Abstract

We establish the existence in the sense of sequences of solutions for certain systems of integro-differential equations which involve the drift terms and the square root of the one dimensional negative Laplace operator, on the whole real line or on a finite interval with periodic boundary conditions in the corresponding $H^{2}$ spaces. The argument is based on the fixed point technique when the elliptic systems contain first order differential operators with and without Fredholm property. It is proven that, under the reasonable technical conditions, the convergence in $L^{1}$ of the integral kernels yields the existence and convergence in $H^{2}$ of the solutions. We emphasize that the study of the systems is more complicated than of the scalar case and requires to overcome more cumbersome technicalities.

Appendix

Let $G_{k}(x)$ be a function, $G_{k}(x): {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$, for which we denote its standard Fourier transform using the hat symbol as

$$\begin{aligned} \widehat{G_{k}}(p):={1\over \sqrt{2\pi }}\int _{-\infty }^{\infty }G_{k}(x) e^{-ipx}dx, \quad p\in {{\mathbb {R}}}. \end{aligned}$$

(5.1)

Clearly,

$$\begin{aligned} \Vert \widehat{G_{k}}(p)\Vert _{L^{\infty }({{\mathbb {R}}})}\le {1\over \sqrt{2\pi }} \Vert G_{k}(x)\Vert _{L^{1}({{\mathbb {R}}})} \end{aligned}$$

(5.2)

and $\displaystyle {G_{k}(x)={1\over \sqrt{2\pi }}\int _{-\infty }^{\infty } \widehat{G_{k}}(q)e^{iqx}dq, \ x\in {{\mathbb {R}}}.}$ By means of (5.2), we have

$$\begin{aligned} \Vert p\widehat{G_{k}}(p)\Vert _{L^{\infty }({{\mathbb {R}}})}\le {1\over \sqrt{2\pi }} \Bigg \Vert \frac{dG_{k}(x)}{dx}\Bigg \Vert _{L^{1}({{\mathbb {R}}})}. \end{aligned}$$

(5.3)

For the technical purposes we will use the auxiliary quantities

$$\begin{aligned} N_{a, \ b, \ k}:=max\Big \{ \Big \Vert \frac{\widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}\Big \Vert _{L^{\infty }({{\mathbb {R}}})}, \quad \Big \Vert \frac{p^{2}\widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}\Big \Vert _{L^{\infty }({{\mathbb {R}}})} \Big \}, \end{aligned}$$

(5.4)

where $a_{k}\ge 0, \ b_{k}\in {{\mathbb {R}}}, \ b_{k}\ne 0$ are the constants, $1\le k\le N, \ N\ge 2$. Under the assumptions of Lemma A1 below, all the quantities (5.4) will be finite, so that

$$\begin{aligned} N_{a, \ b}:=max_{1\le k\le N}N_{a, \ b, \ k}<\infty . \end{aligned}$$

(5.5)

The auxiliary lemmas below are the adaptations of the ones proved in [16] in order to study the single integro-differential equation with drift and superdiffusion, analogical to system (1.2). Let us provide them for the convenience of the readers.

Lemma A1

Let $N\ge 2, \ 1\le k\le N, \ b_{k}\in {{\mathbb {R}}}, \ b_{k}\ne 0$ and $G_{k}(x): {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}, \ G_{k}(x)\in W^{1, 1}({{\mathbb {R}}})$ and $1\le l\le N-1$.

• a) Let $a_{k}>0$ for $1\le k\le l$. Then $N_{a, \ b, \ k}<\infty$.

• b) Let $a_{k}=0$ for $l+1\le k\le N$ and additionally $xG_{k}(x)\in L^{1}({{\mathbb {R}}})$. Then $N_{0, \ b, \ k}<\infty$ if and only if

  $$\begin{aligned} (G_{k}(x),1)_{L^{2}({{\mathbb {R}}})}=0 \end{aligned}$$

  (5.6)

  is valid.

Proof

First of all, it can be trivially checked that in both cases a) and b) of the lemma, under our assumptions the expressions

$$\begin{aligned} \frac{p^{2}\widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}\in L^{\infty }({{\mathbb {R}}}), \quad 1\le k\le N. \end{aligned}$$

(5.7)

Evidently, the functions $\displaystyle {\frac{p}{|p|-a_{k}-ib_{k}p}}$ are bounded and $p\widehat{G_{k}}(p)\in L^{\infty }({{\mathbb {R}}})$ via inequality (5.3) above, which yields (5.7). We turn our attention to establishing the result of the part a) of our lemma. Let us estimate the expressions

$$\begin{aligned} \frac{|\widehat{G_{k}}(p)|}{\sqrt{(|p|-a_{k})^{2}+b_{k}^{2}p^{2}}}, \quad 1\le k\le l. \end{aligned}$$

(5.8)

Clearly, the numerator of (5.8) can be bounded from above via (5.2) and the denominator in (5.8) can be trivially estimated below by a finite, positive constant, so that

$$\begin{aligned} \Bigg |\frac{\widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}\Bigg |\le C\Vert G_{k}(x)\Vert _ {L^{1}({{\mathbb {R}}})}<\infty \end{aligned}$$

as assumed. Here and below C will stand for a finite, positive constant. This implies that under the given conditions, if $a_{k}>0$ we have $N_{a, \ b, \ k}<\infty$. In the cases of $a_{k}=0$, we will use that

$$\begin{aligned} \widehat{G_{k}}(p)=\widehat{G_{k}}(0)+\int _{0}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\widehat{G_{k}}(p)}{|p|-ib_{k}p}=\frac{\widehat{G_{k}}(0)}{|p|-ib_{k}p}+ \frac{\int _{0}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds}{|p|-ib_{k}p}. \end{aligned}$$

(5.9)

Using definition (5.1) of the standard Fourier transform, we easily obtain

$$\begin{aligned} \Bigg |\frac{d\widehat{G_{k}}(p)}{dp}\Bigg |\le \frac{1}{\sqrt{2\pi }} \Vert xG_{k}(x)\Vert _{L^{1}({{\mathbb {R}}})}. \end{aligned}$$

Hence,

$$\begin{aligned} \Bigg |\frac{\int _{0}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds}{|p|-ib_{k}p}\Bigg |\le \frac{\Vert xG_{k}(x)\Vert _{L^{1}({{\mathbb {R}}})}}{\sqrt{2\pi (1+b_{k}^{2})}}<\infty \end{aligned}$$

due to our assumptions. Therefore, the expression in the left side of (5.9) is bounded if and only if $\widehat{G_{k}}(0)=0$, which is equivalent to orthogonality relation (5.6). $\square$

We introduce the following technical expressions, which will help us to study systems (2.8).

$$\begin{aligned} N_{a, \ b, \ k}^{(m)}:=max\Bigg \{ \Big \Vert \frac{\widehat{G_{k, m}}(p)}{|p|-a_{k}-ib_{k}p}\Big \Vert _{L^{\infty }({{\mathbb {R}}})}, \quad \Big \Vert \frac{p^{2}\widehat{G_{k, m}}(p)}{|p|-a_{k}-ib_{k}p}\Big \Vert _{L^{\infty }({{\mathbb {R}}})}\Bigg \},\nonumber \\ \end{aligned}$$

(5.10)

where $a_{k}\ge 0, \ b_{k}\in {{\mathbb {R}}}, \ b_{k}\ne 0$ are the constants, $1\le k\le N, \ N\ge 2$ and $m\in {{\mathbb {N}}}$. Under the conditions of Lemma A2 below, expressions (5.10) will be finite. This will enable us to define

$$\begin{aligned} N_{a, \ b}^{(m)}:=max_{1\le k\le N} N_{a, \ b, \ k}^{(m)}<\infty \end{aligned}$$

(5.11)

with $m\in {{\mathbb {N}}}$. We have the following technical proposition.

Lemma A2

Let $m\in {{\mathbb {N}}}, \ N\ge 2, \ 1\le k\le N, \ b_{k}\in {{\mathbb {R}}}, \ b_{k}\ne 0$ and $G_{k, m}(x): {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}, \ G_{k, m}(x)\in W^{1, 1}({{\mathbb {R}}})$, so that $G_{k, m}(x)\rightarrow G_{k}(x)$ in $W^{1, 1}({{\mathbb {R}}})$ as $m\rightarrow \infty$ and $1\le l\le N-1$.

• (a) Let $a_{k}>0$ for $1\le k\le l$.

• (b) Let $a_{k}=0$ for $l+1\le k\le N$ and in addition $xG_{k, m}(x)\in L^{1}({{\mathbb {R}}})$, so that $xG_{k, m}(x)\rightarrow xG_{k}(x)$ in $L^{1}({{\mathbb {R}}})$ as $m\rightarrow \infty$ and

  $$\begin{aligned} (G_{k, m}(x), 1)_{L^{2}({{\mathbb {R}}})}=0, \quad m\in {{\mathbb {N}}} \end{aligned}$$

  (5.12)

is valid. Let in addition

$$\begin{aligned} 2\sqrt{\pi }N_{a, \ b}^{(m)}L\le 1-\varepsilon \end{aligned}$$

(5.13)

for all $m\in {{\mathbb {N}}}$ as well with a certain fixed $0<\varepsilon <1$. Then, for all $1\le k\le N$, we have

$$\begin{aligned} \frac{\widehat{G_{k, m}}(p)}{|p|-a_{k}-ib_{k}p}\rightarrow \frac{\widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}, \quad m\rightarrow \infty , \end{aligned}$$

(5.14)

$$\begin{aligned} \frac{p^{2}\widehat{G_{k, m}}(p)}{|p|-a_{k}-ib_{k}p}\rightarrow \frac{p^{2}\widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}, \quad m\rightarrow \infty \end{aligned}$$

(5.15)

in $L^{\infty }({{\mathbb {R}}})$, so that

$$\begin{aligned} \Bigg \Vert \frac{\widehat{G_{k, m}}(p)}{|p|-a_{k}-ib_{k}p}\Bigg \Vert _{L^{\infty }({{\mathbb {R}}})}\rightarrow \Bigg \Vert \frac{\widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}\Bigg \Vert _{L^{\infty }({{\mathbb {R}}})}, \quad m\rightarrow \infty , \end{aligned}$$

(5.16)

$$\begin{aligned} \Bigg \Vert \frac{p^{2}\widehat{G_{k, m}}(p)}{|p|-a_{k}-ib_{k}p}\Bigg \Vert _ {L^{\infty }({{\mathbb {R}}})}\rightarrow \Bigg \Vert \frac{p^{2}\widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}\Bigg \Vert _ {L^{\infty }({{\mathbb {R}}})}, \quad m\rightarrow \infty . \end{aligned}$$

(5.17)

Furthermore,

$$\begin{aligned} 2\sqrt{\pi }N_{a, \ b}L\le 1-\varepsilon . \end{aligned}$$

(5.18)

Proof

By means of inequality (5.2), we easily obtain for $1\le k\le N$ that

$$\begin{aligned} \Vert \widehat{G_{k, m}}(p)-\widehat{G_{k}}(p)\Vert _{L^{\infty }({{\mathbb {R}}})}\le \frac{1}{\sqrt{2\pi }}\Vert G_{k, m}(x)-G_{k}(x)\Vert _{L^{1}({{\mathbb {R}}})}\rightarrow 0, \quad m\rightarrow \infty \nonumber \\ \end{aligned}$$

(5.19)

due to the one of our assumptions. Evidently, (5.16) and (5.17) will trivially follow from the statements of (5.14) and (5.15) respectively by virtue of the standard triangle inequality.

We use the fact that the functions $\displaystyle {\frac{p}{|p|-a_{k}-ib_{k}p}\in L^{\infty }({{\mathbb {R}}})}$ along with the analog of bound (5.3). This yields

$$\begin{aligned} \Bigg |\frac{p^{2}\widehat{G_{k, m}}(p)}{|p|-a_{k}-ib_{k}p}- \frac{p^{2}\widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}\Bigg |\le & {} C\Vert p[\widehat{G_{k, m}}(p)-\widehat{G_{k}}(p)]\Vert _ {L^{\infty }({{\mathbb {R}}})}\\\le & {} \frac{C}{\sqrt{2\pi }}\Bigg \Vert \frac{dG_{k, m}(x)}{dx}-\frac{dG_{k}(x)}{dx}\Bigg \Vert _{L^{1}({{\mathbb {R}}})}. \end{aligned}$$

Thus,

$$\begin{aligned} \Bigg \Vert \frac{p^{2}\widehat{G_{k, m}}(p)}{|p|-a_{k}-ib_{k}p}-\frac{p^{2} \widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}\Bigg \Vert _{L^{\infty }({{\mathbb {R}}})}\le \frac{C}{\sqrt{2\pi }}\Bigg \Vert \frac{dG_{k, m}(x)}{dx}-\frac{dG_{k}(x)}{dx}\Bigg \Vert _{L^{1}({{\mathbb {R}}})}\rightarrow 0 \end{aligned}$$

as $m\rightarrow \infty$ via the one of our assumptions, so that (5.15) is valid. Let us establish (5.14) in the situation a) when $a_{k}>0$. For that purpose we need to consider

$$\begin{aligned} \frac{|\widehat{G_{k, m}}(p)-\widehat{G_{k}}(p)|}{\sqrt{(|p|-a_{k})^{2}+ b_{k}^{2}p^{2}}}, \quad 1\le k\le l. \end{aligned}$$

(5.20)

Evidently, the denominator in fraction (5.20) can be bounded from below by a positive constant and the numerator in (5.20) can be estimated from above by means of (5.19). Hence,

$$\begin{aligned} \Bigg \Vert \frac{\widehat{G_{k, m}}(p)}{|p|-a_{k}-ib_{k}p}-\frac{\widehat{G_{k}}(p)}{|p|-a_{k}-ib_{k}p}\Bigg \Vert _{L^{\infty }({{\mathbb {R}}})}\le C \Vert G_{k, m}(x)-G_{k}(x)\Vert _{L^{1}({{\mathbb {R}}})}\rightarrow 0 \end{aligned}$$

as $m\rightarrow \infty$ due to the one of the assumptions, so that (5.14) is valid in the case a) of the lemma. Then we turn our attention to proving (5.14) in the situation b) when $a_{k}=0$. In this case orthogonality conditions (5.12) are valid as assumed. We easily derive that the analogical statements will hold in the limit. Evidently,

$$\begin{aligned} |(G_{k}(x), 1)_{L^{2}({{\mathbb {R}}})}|=|(G_{k}(x)-G_{k, m}(x), 1)_{L^{2}({{\mathbb {R}}})}|\le \Vert G_{k, m}(x)-G_{k}(x)\Vert _{L^{1}({{\mathbb {R}}})}\rightarrow 0 \end{aligned}$$

as $m\rightarrow \infty$ by virtue of the one of our assumptions. Thus,

$$\begin{aligned} (G_{k}(x), 1)_{L^{2}({{\mathbb {R}}})}=0, \quad l+1\le k\le N \end{aligned}$$

(5.21)

is valid. Obviously, we have

$$\begin{aligned} \widehat{G_{k}}(p)=\widehat{G_{k}}(0)+\int _{0}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds, \quad \widehat{G_{k, m}}(p)=\widehat{G_{k, m}}(0)+\int _{0}^{p}\frac{d\widehat{G_{k, m}}(s)}{ds}ds, \end{aligned}$$

with $l+1\le k\le N, \ m\in {{\mathbb {N}}}$. Formulas (5.21) and (5.12) imply that

$$\begin{aligned} \widehat{G_{k}}(0)=0, \quad \widehat{G_{k, m}}(0)=0, \quad l+1\le k\le N, \quad m\in {{\mathbb {N}}}. \end{aligned}$$

Hence,

$$\begin{aligned} \Bigg |\frac{\widehat{G_{k, m}}(p)}{|p|-ib_{k}p}-\frac{\widehat{G_{k}}(p)}{|p|-ib_{k}p}\Bigg |= \Bigg |\frac{\int _{0}^{p}\Big [\frac{d\widehat{G_{k, m}}(s)}{ds}- \frac{d\widehat{G_{k}}(s)}{ds}\Big ]{ds}}{|p|-ib_{k}p}\Bigg |. \end{aligned}$$

(5.22)

Using the definition of the standard Fourier transform (5.1) we easily derive

$$\begin{aligned} \Bigg |\frac{d\widehat{G_{k, m}}(p)}{dp}-\frac{d\widehat{G_{k}}(p)}{dp}\Bigg |\le \frac{1}{\sqrt{2\pi }}\Vert xG_{k, m}(x)-xG_{k}(x)\Vert _{L^{1}({{\mathbb {R}}})}. \end{aligned}$$

This allows us to obtain the estimate from above on the right side of (5.22) as

$$\begin{aligned} \frac{\Vert xG_{k, m}(x)-xG_{k}(x)\Vert _{L^{1}({{\mathbb {R}}})}}{\sqrt{2\pi (1+b_{k}^{2})}}, \end{aligned}$$

such that

$$\begin{aligned} \Bigg \Vert \frac{\widehat{G_{k, m}}(p)}{|p|-ib_{k}p}-\frac{\widehat{G_{k}}(p)}{|p|-ib_{k}p}\Bigg \Vert _{L^{\infty }({{\mathbb {R}}})}\le \frac{\Vert xG_{k, m}(x)-xG_{k}(x)\Vert _{L^{1}({{\mathbb {R}}})}}{\sqrt{2\pi (1+b_{k}^{2})}}\rightarrow 0, \quad m\rightarrow \infty \end{aligned}$$

as assumed. Therefore, (5.14) is valid in the case b) of the lemma when $a_{k}=0$. Evidently, under the stated conditions we have

$$\begin{aligned} N_{a, \ b, \ k}<\infty , \quad N_{a, \ b, \ k}^{(m)}<\infty , \quad m\in {{\mathbb {N}}}, \quad 1\le k\le N, \quad a_{k}\ge 0, \quad b_{k}\in {{\mathbb {R}}}, \quad b_{k}\ne 0 \end{aligned}$$

by means of the result of Lemma A1 above. We have inequalities (5.13). An trivial limiting argument using (5.16) and (5.17) gives us (5.18). $\square$

Consider the function $G_{k}(x): I\rightarrow {{\mathbb {R}}}$, so that $G_{k}(0)=G_{k}(2\pi )$. Its Fourier transform on our finite interval is given by

$$\begin{aligned} G_{k, n}:=\int _{0}^{2\pi }G_{k}(x)\frac{e^{-inx}}{\sqrt{2\pi }}dx, \quad n\in {{\mathbb {Z}}}, \end{aligned}$$

(5.23)

such that $\displaystyle {G_{k}(x)=\sum \nolimits _{n=-\infty }^{\infty }G_{k, n}\frac{e^{inx}}{\sqrt{2\pi }}}$. Obviously, the upper bound

$$\begin{aligned} \Vert G_{k, n}\Vert _{l^{\infty }}\le \frac{1}{\sqrt{2\pi }}\Vert G_{k}(x)\Vert _{L^{1}(I)} \end{aligned}$$

(5.24)

is valid. Evidently, if our function is continuous on the interval I, we have the estimate from above

$$\begin{aligned} \Vert G_{k}(x)\Vert _{L^{1}(I)}\le 2\pi \Vert G_{k}(x)\Vert _{C(I)}. \end{aligned}$$

(5.25)

The upper bound

$$\begin{aligned} \Vert nG_{k, n}\Vert _{l^{\infty }}\le \frac{1}{\sqrt{2\pi }}\Bigg \Vert \frac{dG_{k}(x)}{dx} \Bigg \Vert _{L^{1}(I)} \end{aligned}$$

(5.26)

trivially comes from (5.24). Analogously to the whole real line case, we define

$$\begin{aligned} \mathcal{N}_{a, \ b, \ k}:=max \Bigg \{ \Bigg \Vert \frac{G_{k, n}}{|n|-a_{k}-ib_{k}n} \Bigg \Vert _{l^{\infty }}, \quad \Bigg \Vert \frac{n^{2}G_{k, n}}{|n|-a_{k}-ib_{k}n} \Bigg \Vert _{l^{\infty }}\Bigg \}, \end{aligned}$$

(5.27)

where $a_{k}\ge 0, \ b_{k}\in {{\mathbb {R}}}, \ b_{k}\ne 0$ are the constants, $1\le k\le N, \ N\ge 2$. Let $\mathcal{N}_{0, \ b, \ k}$ denote (5.27) when $a_{k}$ vanishes. Under the conditions of Lemma A3 below, the expressions $\mathcal{N}_{a, \ b, \ k}$ will be finite. This will enable us to introduce

$$\begin{aligned} \mathcal{N}_{a, \ b}:=max_{1\le k\le N}\mathcal{N}_{a, \ b, \ k}<\infty . \end{aligned}$$

(5.28)

We have the following elementary statement.

Lemma A3

Let $N\ge 2, \ 1\le k\le N, \ b_{k}\in {{\mathbb {R}}}, \ b_{k}\ne 0, \ 1\le l\le N-1$ and

${G_{k}(x): I\rightarrow {{\mathbb {R}}}, \ G_{k}(x)\in C(I), \ \frac{dG_{k}(x)}{dx}\in L^{1}(I), \ G_{k}(0)=G_{k}(2\pi ).}$

• (a) Let $a_{k}>0$ for $1\le k\le l$. Then $\mathcal{N}_{a, \ b, \ k}<\infty .$

• (b) If $a_{k}=0$ for $l+1\le k\le N$ then $\mathcal{N}_{0, \ b, \ k}<\infty$ if and only if the orthogonality relation

  $$\begin{aligned} (G_{k}(x),1)_{L^{2}(I)}=0 \end{aligned}$$

  (5.29)

  holds.

Proof

It can be easily checked that in both cases (a) and (b) of our lemma under the given conditions we have

$$\begin{aligned} \frac{n^{2}G_{k, n}}{|n|-a_{k}-ib_{k}n}\in l^{\infty }, \quad 1\le k\le N. \end{aligned}$$

(5.30)

Clearly, ${\frac{n}{|n|-a_{k}-ib_{k}n}\in l^{\infty }}$ and $nG_{k, n}\in l^{\infty }$ via inequality (5.26) along with the one of the stated assumptions. Hence (5.30) is valid.

Let us establish the statement of the part a) of the lemma. For that purpose, we need to consider the expression

$$\begin{aligned} \frac{|G_{k, n}|}{\sqrt{(|n|-a_{k})^{2}+b_{k}^{2}n^{2}}}, \quad 1\le k\le l. \end{aligned}$$

(5.31)

Evidently, the denominator in (5.31) can be easily bounded from below by a positive constant. The numerator in (5.31) can be trivially estimated from above by means of (5.24) along with (5.25). Hence, $\mathcal{N}_{a, \ b, \ k}<\infty$ in the case when $a_{k}>0$. Let us demonstrate the validity of the result of the lemma in the situation when $a_{k}=0$. Obviously,

$$\begin{aligned} \Bigg |\frac{G_{k, n}}{|n|-ib_{k}n}\Bigg |, \quad l+1\le k\le N \end{aligned}$$

is bounded if and only if $G_{k, 0}=0$. This is equivalent to orthogonality condition (5.29). In this case we easily arrive at for $l+1\le k\le N$ that

$$\begin{aligned} \Bigg |\frac{G_{k, n}}{|n|-ib_{k}n}\Bigg |\le \frac{1}{\sqrt{2\pi }|n|} \frac{\Vert G_{k}(x)\Vert _{L^{1}(I)}}{\sqrt{1+b_{k}^{2}}}\le \sqrt{2\pi }\frac{\Vert G_{k}(x)\Vert _{C(I)}}{\sqrt{1+b_{k}^{2}}}<\infty \end{aligned}$$

by virtue of (5.24) and (5.25) under our assumptions. $\square$

In order to study the systems of Eq. (2.10), we will use

$$\begin{aligned} \mathcal{N}_{a, \ b, \ k}^{(m)}:=max \Bigg \{ \Bigg \Vert \frac{G_{k, m, n}}{|n|-a_{k}-ib_{k}n} \Bigg \Vert _{l^{\infty }}, \quad \Bigg \Vert \frac{n^{2}G_{k, m, n}}{|n|-a_{k}-ib_{k}n}\Bigg \Vert _{l^{\infty }}\Bigg \}, \end{aligned}$$

(5.32)

where $a_{k}\ge 0, \ b_{k}\in {{\mathbb {R}}}, \ b_{k}\ne 0$ are the constants, $1\le k\le N, \ N\ge 2$ and $m\in {{\mathbb {N}}}$. Under the assumptions of Lemma A4 below, we have that all $\mathcal{N}_{a, \ b, \ k}^{(m)}<\infty$. This will allow us to introduce

$$\begin{aligned} \mathcal{N}_{a, \ b}^{(m)}=max_{1\le k\le N}\mathcal{N}_{a, \ b, \ k}^{(m)}, \quad m\in {{\mathbb {N}}}. \end{aligned}$$

(5.33)

We conclude the work with the following auxiliary proposition.

Lemma A4

Let $m\in {{\mathbb {N}}}, \ N\ge 2, \ 1\le k\le N, \ b_{k}\in {{\mathbb {R}}}, \ b_{k}\ne 0, \ 1\le l\le N-1$ and

$$\begin{aligned} G_{k, m}(x): I\rightarrow {{\mathbb {R}}}, \quad G_{k, m}(x)\in C(I), \quad \frac{dG_{k, m}(x)}{dx}\in L^{1}(I), \quad G_{k, m}(0)=G_{k, m}(2\pi ), \end{aligned}$$

and

$$\begin{aligned} G_{k, m}(x)\rightarrow G_{k}(x) \quad in \quad C(I), \quad \frac{dG_{k, m}(x)}{dx}\rightarrow \frac{dG_{k}(x)}{dx} \quad in \quad L^{1}(I) \end{aligned}$$

as $m\rightarrow \infty$.

• (a) Let $a_{k}>0$ for $1\le k\le l$.

• (b) Let $a_{k}=0$ for $l+1\le k\le N$ and in addition

  $$\begin{aligned} (G_{k, m}(x), 1)_{L^{2}(I)}=0, \quad m\in {{\mathbb {N}}}. \end{aligned}$$

  (5.34)

We also assume that

$$\begin{aligned} 2\sqrt{\pi }\mathcal{N}_{a, \ b}^{(m)}L\le 1-\varepsilon \end{aligned}$$

(5.35)

is valid for all $m\in {{\mathbb {N}}}$ as well with some fixed $0<\varepsilon <1$. Then, for all $1\le k\le N$, we have

$$\begin{aligned} \frac{G_{k, m, n}}{|n|-a_{k}-ib_{k}n}\rightarrow \frac{G_{k, n}}{|n|-a_{k}-ib_{k}n}, \quad m\rightarrow \infty ,\end{aligned}$$

(5.36)

$$\begin{aligned} \frac{n^{2}G_{k, m, n}}{|n|-a_{k}-ib_{k}n}\rightarrow \frac{n^{2}G_{k, n}}{|n|-a_{k}-ib_{k}n}, \quad m\rightarrow \infty \end{aligned}$$

(5.37)

in $l^{\infty }$, so that

$$\begin{aligned} \Bigg \Vert \frac{G_{k, m, n}}{|n|-a_{k}-ib_{k}n}\Bigg \Vert _{l^{\infty }}\rightarrow \Bigg \Vert \frac{G_{k, n}}{|n|-a_{k}-ib_{k}n}\Bigg \Vert _{l^{\infty }}, \quad m\rightarrow \infty , \end{aligned}$$

(5.38)

$$\begin{aligned} \Bigg \Vert \frac{n^{2}G_{k, m, n}}{|n|-a_{k}-ib_{k}n}\Bigg \Vert _{l^{\infty }}\rightarrow \Bigg \Vert \frac{n^{2}G_{k, n}}{|n|-a_{k}-ib_{k}n}\Bigg \Vert _{l^{\infty }}, \quad m\rightarrow \infty . \end{aligned}$$

(5.39)

Furthermore, the estimate

$$\begin{aligned} 2\sqrt{\pi }\mathcal{N}_{a, \ b}L\le 1-\varepsilon \end{aligned}$$

(5.40)

holds.

Proof

Obviously, under the stated assumptions, the limiting kernels $G_{k}(x), \ 1\le k\le N$ are periodic as well. Indeed, we easily obtain

$$\begin{aligned} |G_{k}(0)-G_{k}(2\pi )|\le & {} |G_{k}(0)-G_{k, m}(0)|\nonumber \\&+\,|G_{k,m}(2\pi )-G_{k}(2\pi )|\le 2\Vert G_{k, m}(x)-G_{k}(x)\Vert _{C(I)}\rightarrow 0 \end{aligned}$$

as $m\rightarrow \infty$ as assumed. Thus, $G_{k}(0)=G_{k}(2\pi ), \ 1\le k\le N$. By virtue of (5.24) along with (5.25) we arrive at

$$\begin{aligned} \Vert G_{k, m, n}-G_{k, n}\Vert _{l^{\infty }}\le & {} \frac{1}{\sqrt{2\pi }}\Vert G_{k, m}-G_{k}\Vert _{L^{1}(I)}\nonumber \\\le & {} \sqrt{2\pi }\Vert G_{k, m}-G_{k}\Vert _{C(I)}\rightarrow 0, \quad m\rightarrow \infty \end{aligned}$$

(5.41)

due to the one of our assumptions. It can be trivially checked that the statements of (5.36) and (5.37) will imply (5.38) and (5.39) respectively via the triangle inequality. Using (5.26), we obtain the estimate from above

$$\begin{aligned} \Bigg \Vert \frac{n^{2}G_{k, m, n}}{|n|-a_{k}-ib_{k}n}-\frac{n^{2}G_{k, n}}{|n|-a_{k}-ib_{k} n}\Bigg \Vert _{l^{\infty }}\le & {} \frac{1}{\sqrt{2\pi }}\Bigg \Vert \frac{n}{|n|-a_{k}-ib_{k}n} \Bigg \Vert _{l^{\infty }}\\&\Bigg \Vert \frac{dG_{k, m}(x)}{dx}-\frac{dG_{k}(x)}{dx}\Bigg \Vert _ {L^{1}(I)}, \end{aligned}$$

which tends to zero as $m\rightarrow \infty$ as assumed, so that (5.37) is valid. Let us establish (5.36) in the situation a) when $a_{k}>0$. For that purpose, we need to treat

$$\begin{aligned} \frac{|G_{k, m, n}-G_{k, n}|}{\sqrt{(|n|-a_{k})^{2}+b_{k}^{2}n^{2}}}, \quad 1\le k \le l. \end{aligned}$$

(5.42)

Obviously, the denominator of (5.42) can be bounded from below by a positive constant and the numerator estimated from above via (5.41). This gives us (5.36) for $a_{k}>0$.

Let us demonstrate the validity of (5.36) in the case case b) when $a_{k}=0$. By means of the one of the given assumptions, we have orthogonality conditions (5.34). It can be trivially checked that the analogical relations holds in the limit. Indeed,

$$\begin{aligned} |(G_{k}(x), 1)_{L^{2}(I)}|= & {} |(G_{k}(x)\nonumber \\&-\,G_{k, m}(x),1)_{L^{2}(I)}|\le 2\pi \Vert G_{k, m}(x)-G_{k}(x)\Vert _{C(I)}\rightarrow 0, \quad m\rightarrow \infty \end{aligned}$$

via the one of our assumptions. Thus,

$$\begin{aligned} (G_{k}(x), 1)_{L^{2}(I)}=0, \quad l+1\le k\le N. \end{aligned}$$

This is equivalent to $G_{k, 0}=0, \ l+1\le k\le N$. Evidently, $G_{k, m, 0}=0, \ l+1\le k\le N, \ m\in {{\mathbb {N}}}$ by virtue of orthogonality condition (5.34). Using (5.41), we easily obtain that

$$\begin{aligned} \Bigg |\frac{G_{k, m, n}-G_{k, n}}{|n|-ib_{k}n}\Bigg |\le \frac{\sqrt{2\pi } \Vert G_{k, m}(x)-G_{k}(x)\Vert _{C(I)}}{\sqrt{1+b_{k}^{2}}}. \end{aligned}$$

Since the norm in the right side of this estimate from above tends to zero as $m\rightarrow \infty$, (5.36) holds in the case when $a_{k}=0$ as well. Clearly, under the stated assumptions we have

$$\begin{aligned} \mathcal{N}_{a, \ b, \ k}<\infty , \quad \mathcal{N}_{a, \ b, \ k}^{(m)}<\infty , \quad m\in {{\mathbb {N}}}, \quad 1\le k\le N, \quad a_{k}\ge 0, \quad b_{k}\in {{\mathbb {R}}}, \quad b_{k}\ne 0 \end{aligned}$$

by virtue of the result of our Lemma A3 above. We assume the validity of upper bound (5.35). A simple limiting argument using (5.38) and (5.39) gives us (5.40). $\square$