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

*Journal of Inequalities and Applications*
**volume 2015**, Article number: 263 (2015)

## 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 [1]. The problem was further developed in [2], pp.339-345, [3], 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 [4]. The coefficients *β* are found by Theorem 1 in [6], using the results of [6], 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

with the boundary conditions

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

and where

### Theorem 2.2

([7])

*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*

*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* [8], *and the eigenfunctions are given by the formula*

*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*

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

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

\(A_{nk}>0\)
*and*
\(\tilde{A}_{nk}>0\).

### Theorem 2.3

(see [5])

*The function system*

*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 [9], we have the relation

where

The coefficient \(B_{0}\) depends on \(B_{n}\) (see [9]). Consider the formally differentiated series (11)

Since the coefficient \(B_{n}\) is found by formula (12), using the results of [7], 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

which has a meaning if the following series converges

By using the results of [9], 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

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 [9]), 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 [7], 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 [7], 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*

*where the coefficients*
\(D_{n}\)
*are calculated according to the formulas*

*for*
\(\beta<1\)
*and*

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

*where*
\(H^{\beta}_{n}\)
*and*
\(h^{\beta}_{n}(\theta)\)
*were studied in* [10].

### Proof

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

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 [9] and (16) or (17), we have the relation

Therefore, instead of the relation, we can write

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 [9], we obtain

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

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

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

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

By using relations (16) (or (17)) and (9) of [9], 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.

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### 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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### Cite this article

Abbasi, N., Shakori, M. On the completeness and the basis property of the modified Frankl problem with a nonlocal oddness condition in the Sobolev space \((W^{1}_{p}(0,\pi))\)
.
*J Inequal Appl* **2015, **263 (2015). https://doi.org/10.1186/s13660-015-0782-5

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### MSC

- 35j15
- 35m12
- 35p10
- 35c10
- 35j56

### Keywords

- Frankl problem
- Lebesgue integral
- Hölder inequality
- Bessel equation
- Sobolev space