# On the completeness and the basis property of the modified Frankl problem with a nonlocal oddness condition in the Sobolev space $$(W^{1}_{p}(0,\pi))$$

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## Abstract

In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal oddness condition of the first kind in the Sobolev space $$(W^{1}_{p}(0,\pi))$$. We analyze the completeness, the basis property, and the minimality of the eigenfunctions in the space $$(W^{1}_{p}(0,\pi))$$.

## Introduction

The classical Frankl problem was considered in . The problem was further developed in , pp.339-345, , pp.235-252. The modified Frankl problem with a nonlocal boundary condition of the first kind was studied in [4, 5]. The basis property of eigenfunctions of the Frankl problem with nonlocal parity conditions in the Sobolev space was studied in . The coefficients β are found by Theorem 1 in , using the results of , pp.177-179. In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal oddness condition of the first kind. We analyze the completeness, the basis property, and the minimality of the eigenfunctions in the space $$({W}^{1}_{p}(0,\pi))$$, where $$({W}^{1}_{p}(0,\pi))$$ is the space of absolutely continuous functions on $$[0,\pi]$$. So we can obtain new results by the expansion into cosines that are related to new coefficients which we calculated. This analysis and results may be of interest in itself.

## Statement of the modified Frankl problem

### Definition 2.1

In the domain $$D=(D_{+}\cup D_{-1}\cup D_{-2})$$, we seek a solution of the modified generalized Frankl problem

$$u_{xx}+\operatorname{sgn}(y) u_{yy}+ \mu^{2}\operatorname{sgn}(x+y)u=0 \quad\mbox{in } (D_{+}\cup D_{-1}\cup D_{-2}),$$
(1)

with the boundary conditions

\begin{aligned}& u(1,\theta)=0,\quad \theta\in\biggl[0,\frac{\pi}{2}\biggr], \end{aligned}
(2)
\begin{aligned}& \frac{\partial u}{\partial x}(0,y)=0,\quad y\in(-1,0)\cup(0,1), \end{aligned}
(3)
\begin{aligned}& u(0,y)=u(0,-y),\quad y\in[0,1], \end{aligned}
(4)

where $$u(x,y)$$ is a regular solution in the class

$$u\in C^{0} (\overline{D_{+}\cup D_{-1}\cup D_{-2}})\cap C^{2}(\overline{D_{-1}})\cap C^{2}(\overline{D_{-2}}),$$

and where

\begin{aligned} &D_{+} =\biggl\{ (r,\theta): 0< r< 1, 0< \theta< \frac{\pi}{2}\biggr\} , \\ &D_{-1} =\biggl\{ (x,y): -y< x< y+1, \frac{-1}{2}< y< 0\biggr\} , \\ &D_{-2} =\biggl\{ (x,y): x-1< y< -x, 0< x< \frac{1}{2}\biggr\} , \\ &\kappa\frac{\partial u}{\partial y}(x,+0)=\frac{\partial u}{\partial y}(x,-0),\quad -\infty< \kappa< \infty, 0< x< 1. \end{aligned}
(5)

### Theorem 2.2

()

The eigenvalues and eigenfunctions of problem (1)-(5) can be written out in two series. In the first series, the eigenvalues $$\lambda= \mu^{2}_{nk}$$ are found from the equation

$$J_{4n}(\mu_{nk})= 0,$$
(6)

where $$\mu_{nk}$$, $$n =0, 1,2, \ldots$$ , $$k = 1,2,\ldots$$ , are roots of the Bessel equation (6), $$J_{\alpha}(z)$$ is the Bessel function , and the eigenfunctions are given by the formula

$$u_{nk}= \textstyle\begin{cases} A_{nk}J_{4n}(\mu_{nk}r)\cos(4n)(\frac{\pi}{2}-\theta) & \textit{in } D^{+}; \\ A_{nk}J_{4n}(\mu_{nk}\rho)\cosh(4n)\psi & \textit{in } D_{-1}; \\ A_{nk}J_{4n}(\mu_{nk}R)\cosh(4n)\varphi & \textit{in } D_{-2}, \end{cases}$$
(7)

where $$x=r\cos\theta$$, $$y=r\sin\theta$$ for $$0\leq\theta\leq\frac{\pi }{2}$$, $$0< r<1$$, and $$r^{2}=x^{2}+y^{2}$$ in $$D_{+}$$, $$x=\rho\cosh\psi$$, $$y=\rho\sinh\psi$$ for $$0<\rho<1$$, $$-\infty<\psi<0$$, $$\rho^{2}= x^{2}-y^{2}$$ in $$D_{-1}$$ and $$x=R\sinh\varphi$$, $$y=-R\cosh\varphi$$ for $$0<\varphi<+\infty$$, $$R^{2}=y^{2}-x^{2}$$ in $$D_{-2}$$.

In the second series, the eigenvalues $$\tilde{\lambda}=\tilde{\mu }^{2}_{nk}$$ are found from the equation

$$J_{4(n-\Delta )}(\tilde{\mu}_{nk})= 0,$$
(8)

where $$n =1,2, \ldots$$ , and $$k = 1,2,\ldots$$ , and $$(\tilde{\mu}_{nk})$$ are the roots of the Bessel equation (8).

$$\tilde{u}_{nk}= \textstyle\begin{cases} \tilde{A}_{nk} J_{4(n-\Delta )}(\tilde{\mu}_{nk}r)\cos 4(n-\Delta )(\frac{\pi}{2}-\theta) & \textit{in } D^{+}; \\ \tilde{A}_{nk}J_{4(n-\Delta )}(\tilde{\mu}_{nk}\rho)[\cosh 4(n-\Delta )\varphi \cos4(n-\Delta )\frac{\pi}{2} \\ \quad{}+\kappa\sinh4(n-\Delta )\psi\cos4(n-\Delta )] & \textit {in } D_{-1}; \\ \tilde{A}_{nk}J_{4(n-\Delta )}(\tilde{\mu}_{nk}R)\cosh 4(n-\Delta )\varphi[\cos4(n-\Delta )\frac{\pi}{2} \\ \quad{}-\kappa\sin4(n-\Delta )\frac{\pi}{2}] & \textit{in } D_{-2}, \end{cases}$$
(9)

where $$\Delta=\frac{1}{\pi}\arcsin\frac{\kappa}{\sqrt{1+\kappa^{2}}}$$, $$\Delta \in(0,\frac{1}{2})$$, and

\begin{aligned}& A^{2}_{nk}\int^{1}_{0}J^{2}_{4n}( \mu_{nk}r)r\,dr=1, \\& \tilde{A}^{2}_{nk}\int^{1}_{0}J^{2}_{4n-1}( \tilde{\mu}_{nk}r)r\,dr=1, \end{aligned}

$$A_{nk}>0$$ and $$\tilde{A}_{nk}>0$$.

### Theorem 2.3

(see )

The function system

$$\biggl\{ \cos(4n) \biggl(\frac{\pi}{2}-\theta\biggr)\biggr\} ^{\infty}_{n=0}, \qquad \biggl\{ \cos4(n-\Delta ) \biggl( \frac{\pi}{2}-\theta\biggr)\biggr\} ^{\infty}_{n=1}$$
(10)

is a Riesz basis in $$L_{2}(0,\frac{\pi}{2})$$ provided that $$\Delta \in(0,\frac{3}{4})$$.

## The completeness, the basis property and minimality of the eigenfunctions

### Theorem 3.1

The system of functions $$\{\cos(n-\frac{\beta}{2})\theta\}^{\infty}_{n=0}$$ is a Riesz basis in $$(W^{1}_{p}(0,\pi))$$ if and only if $$\beta\in(-\frac{1}{p},2-\frac {1}{p})$$, $$\beta\neq1$$.

### Proof

Using the formula (20) of , we have the relation

$$f(\theta)=\sum^{\infty}_{n=1}B_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta+B_{0},$$
(11)

where

$$B_{n}=-\int^{\pi}_{0}f'( \theta)h^{\beta}_{n}\,d\theta\biggl(n-\frac{\beta }{2} \biggr)^{-1}\quad (n=1,2,\ldots).$$
(12)

The coefficient $$B_{0}$$ depends on $$B_{n}$$ (see ). Consider the formally differentiated series (11)

$$\sum^{\infty}_{n=1}{B}_{n} \biggl(n-\frac{\beta}{2}\biggr)\sin\biggl(n-\frac{\beta }{2}\biggr)\theta.$$
(13)

Since the coefficient $$B_{n}$$ is found by formula (12), using the results of , we obtain that series (11) converges to $$f'(\theta)$$ in the space $$L_{p}(0,\pi)$$. Integrating series (11) from 0 to θ, we obtain the relation

$$f(\theta)-f(0)=\sum^{\infty}_{n=1}B_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta -\sum^{\infty}_{n=1}B_{n},$$
(14)

which has a meaning if the following series converges

$$\sum^{\infty}_{n=1}B_{n}.$$
(15)

By using the results of , we obtain that the numerical series (15) converges and relation (11) uniformly converges on $$[0,\pi]$$, and therefore it converges in the space $$L_{p}(0,\pi)$$. Now we assume that

$$B_{0}=f(0)-\sum^{\infty}_{n=1}B_{n}.$$

Then expression (14) coincides with expression (11), and therefore series (11) converges to a function in the space $$(W^{1}_{p}(0,\pi))$$.

Now let us show that the coefficients $$B_{n}$$ are uniquely found by using relation (11). Indeed, if series (11) converges in the space $$(W^{1}_{p}(0,\pi))$$, then series (15) converges in the space $$L_{p}(0,\pi)$$ (see ), this implies that $$\lim_{n\to \infty}B_{n}=0$$. For $$\beta\in(-\frac{1}{p},2-\frac{1}{p})$$. Now let us show that the system $$\{\cos(n-\frac{\beta}{2})\theta,1\}^{\infty}_{n=1}$$ does not compose a basis for $$\beta\notin(-\frac{1}{p},2-\frac{1}{p})$$. If $$\beta\in(2-\frac {1}{p},4-\frac{1}{p})$$, then, using the substitution $$\beta-2=\beta'$$ and removing the first cosine, we obtain the cosine system $$\{\cos(n-\frac{\beta'}{2})\theta^{\infty }_{n=1},1\}$$, which, as was proved above, composes a basis in $$(W^{1}_{p}(0,\pi))$$, and therefore the initial cosine system is not minimal in $$(W^{1}_{p}(0,\pi ))$$. Analogously, for $$\beta\in(-2-\frac{1}{p},-\frac{1}{p})$$, the substitution $$\beta+2=\beta'$$ reduces the initial cosine system to the system with $$\beta'\in(-\frac{1}{p},2-\frac{1}{p})$$, in which there is no function $$(\cos(1-\frac{\beta'}{2})\theta)$$, and therefore the initial cosine system is not complete. Other ranges of the parameter $$\beta\in(-\frac{1}{p}+2k,2-\frac{1}{p}+2k)$$, $$k=\pm1,\pm2,\ldots$$ , can be considered analogously. Furthermore, for $$\beta=2-\frac{1}{p}$$ in the space $$(W^{1}_{p}(0,\pi))$$, where $$\hat{p}>p$$, we have $$-\frac {1}{\hat{p}}<\beta<2-\frac{1}{\hat{p}}$$, and therefore the cosine system composes a basis in $$W^{1}_{\hat{p}}(0,\pi)$$, and hence it is complete in the space $$(W^{1}_{p}(0,\pi))$$.

For $$\beta=-\frac{1}{p}$$, the cosine system is minimal since, as was proved above, the coefficients $$B_{n}$$ are found by concrete formulas in the form of an integral. Let us show that for $$\beta=2-\frac{1}{p}$$, the cosine system is not minimal. By using the results of , we obtain that for $$\beta=2-\frac{1}{p}$$, the cosine system is complete but not minimal, and hence, for $$\beta=-\frac{1}{p}$$, the cosine system is complete (since it is minimal in this case). Now let us prove that for $$\beta=-\frac{1}{p}$$, the cosine system does not compose a basis. Let $$f(\theta)=\theta$$, then $$f(\theta)\in(W^{1}_{p}(0,\pi))$$, $$f'(\theta)=1$$, and the coefficients $$B_{n}$$ can be calculated by using formula (12) exactly in the same way as in , where it was shown that a series converges to a function not belonging to $$L_{p}(0,\pi)$$, thus Theorem 3.1 is proved. □

### Theorem 3.2

The cosine system $$\{\cos(n-\frac{\beta}{2})\theta\}^{\infty}_{n=0}$$ forms a basis in the space $$(W^{1}_{p}(0,\pi))$$ if and only if $$\beta \in(-\frac{1}{p},2-\frac{1}{p})$$, $$\beta\neq1$$. The expansion into cosines has the form

$$f(\theta)=\sum^{\infty}_{n=0}D_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta,$$

where the coefficients $$D_{n}$$ are calculated according to the formulas

$$D_{0}= \int^{\pi}_{0}f( \theta)H^{\beta}_{0}(\theta) \,d(\theta)$$
(16)

for $$\beta<1$$ and

\begin{aligned} D_{0}&=\frac{8(1-\beta)}{\pi\beta(2-\beta)} \int^{\pi }_{0} \frac{\sin(\theta)\sin(\frac{\beta\theta}{2})}{(2\cos\frac{\theta }{2})^{\beta}} \,d(\theta) \\ & = \int^{\pi}_{0}f(\theta)H^{\beta}_{0}( \theta) \,d(\theta) + \int^{\pi }_{0} \frac{f'(\theta)h^{\beta}_{1}}{1-\frac{\beta}{2}}\,d(\theta) \end{aligned}
(17)

for $$\beta>1$$ and for all $$n\in N$$, $$D_{n}$$ is given by

$$D_{n}=-\int^{\pi}_{0} \biggl(f'+D_{0}\biggl(\frac{\beta}{2}\biggr)\sin\biggl( \frac{\beta\theta }{2}\biggr)\biggr)h^{\beta}_{n}\,d(\theta) \biggl(n-\frac{\beta}{2}\biggr)^{-1},$$
(18)

where $$H^{\beta}_{n}$$ and $$h^{\beta}_{n}(\theta)$$ were studied in .

### Proof

Analogously to the proof of relation (14), we obtain the relation

$$f(\theta)-f(0)=\sum^{\infty}_{n=0}D_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta -\sum^{\infty}_{n=1}D_{n}.$$
(19)

The convergence of numerical series $$\sum^{\infty}_{n=0}D_{n}$$ is proved analogously to the proof of the convergence of series $$\sum^{\infty}_{n=1}B_{n}$$. This implies the uniform convergence of series (19).

First let $$\beta<1$$, then multiply series (19) by $$H^{\beta }_{0}$$. Integrating over the closed interval $$[0,\pi]$$ and taking into account relations (6) of  and (16) or (17), we have the relation

$$f(0)=\sum^{\infty}_{n=0}D_{n}.$$

Therefore, instead of the relation, we can write

$$f(\theta)=\sum^{\infty}_{n=0}D_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta.$$
(20)

For $$\beta>0$$, we multiply series (19) by $$H^{\beta-2}_{0}(\theta )$$ and integrate the obtained relation over the closed interval $$[0,\pi ]$$. Using relation (9) of , we obtain

\begin{aligned} \int^{\pi}_{0}f(\theta)H^{\beta-2}_{0}( \theta) \,d(\theta )={}&D_{0}\int^{\pi}_{0} \cos\frac{\beta\theta}{2}H^{\beta-2}_{0}(\theta) \,d( \theta)+D_{1} \\ &{} + \Biggl(f(0)-\sum^{\infty}_{n=0}D_{n} \Biggr) \int^{\pi}_{0}H^{\beta -2}_{0}( \theta)\, d(\theta). \end{aligned}

Substituting the expression for $$D_{1}$$ from (18) in the latter relation, we obtain

\begin{aligned} &\int^{\pi}_{0}f(\theta)H^{\beta-2}_{0}( \theta) \,d(\theta )-D_{0}\int^{\pi}_{0}\cos \frac{\beta\theta}{2}H^{\beta-2}_{0}(\theta) \,d(\theta) \\ &\qquad{} + \int^{\pi}_{0}f'( \theta)h^{\beta}_{1}(\theta) \,d(\theta )\frac{1}{1-\frac{\beta}{2}} \\ &\qquad{} +D_{0}\int^{\pi}_{0}\sin \frac{\beta\theta}{2}h^{\beta }_{1}(\theta) \,d(\theta) \frac{\beta}{2(1-\frac{\beta}{2})} \\ &\quad=\Biggl(f(0)-\sum^{\infty}_{n=0}D_{n} \Biggr) \int^{\pi}_{0}H^{\beta-2}_{0}( \theta )\, d(\theta). \end{aligned}
(21)

Now let us show that the left-hand side of relation (21) vanishes, this will imply

$$f(0)=\sum^{\infty}_{n=0}D_{n}.$$

Indeed, integrating relation (9) of  by parts, we obtain the relation

$$\frac{\beta}{2(1-\frac{\beta}{2})}\int^{\pi}_{0}H^{\beta-2}_{0}( \theta )\cos\frac{\beta\theta}{2}\, d(\theta)= \biggl(1-\frac{\beta}{2}\biggr) \frac{2}{\beta }\int^{\pi}_{0}\sin \frac{\beta\theta}{2}h^{\beta}_{1}(\theta) \,d(\theta).$$

Furthermore, substituting this formula in (21), we immediately see that

\begin{aligned} &\int^{\pi}_{0}\biggl(f(\theta)H^{\beta-2}_{0}( \theta)+ \frac{f'(\theta )h^{\beta}_{1}(\theta)}{1-\frac{\beta}{2}}\biggr) \,d(\theta) \\ &\qquad{}+D_{0}\int^{\pi }_{0}\sin \frac{\beta\theta}{2}h^{\beta}_{1}(\theta)\, d(\theta) \biggl( \frac {2}{2-\beta}-\frac{2-\beta}{\beta}\biggr) \\ &\quad = \int^{\pi}_{0}\biggl(f( \theta)H^{\beta-2}_{0}(\theta)+ \frac{f'(\theta )h^{\beta}_{1}(\theta)}{1-\frac{\beta}{2}}\biggr) \,d( \theta) \\ &\qquad{}+ \biggl(\frac{4D_{0}(\beta-1)}{\beta(2-\beta)}\biggr)\int^{\pi}_{0} \sin\frac {\beta\theta}{2}h^{\beta}_{1}(\theta) \,d(\theta). \end{aligned}

By using relations (16) (or (17)) and (9) of , we annihilate the latter relation, i.e., we obtain relation (20) for $$\beta>1$$. The remaining part of Theorem 3.2 is proved analogously to Theorem 3.1. □

### Remark 3.3

In case $$\kappa>0$$. The system of functions (10) is a Riesz basis in $$({W}^{1}_{p}(0,\pi))$$ if $$\Delta \in(\frac {-1}{4},0)\cup(0,\frac{3}{4})$$.

If $$\Delta \geq\frac{3}{4}$$, $$\Delta \neq1,2,3,\ldots$$ , then system (10) is complete but is not minimal in $$({W}^{1}_{p}(0,\pi))$$.

If $$\Delta =\frac{-1}{4}$$, then system (10) is complete and minimal but is not basis in $$({W}^{1}_{p}(0,\pi))$$.

If $$\Delta <\frac{-1}{4}$$, $$\Delta \neq1,2,3,\ldots$$ , then system (10) is not complete but is minimal in $$({W}^{1}_{p}(0,\pi))$$.

### Proof

The proof of Remark 3.3 reproduces that of Theorem 3.1 and Theorem 3.2. □

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## Acknowledgements

The authors are grateful to EI Moiseev for his interest in this work.

## Author information

Correspondence to Naser Abbasi.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The research and writing of this manuscript was a collaborative effort of all the authors. All authors read and approved the final manuscript.

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