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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 \((W^{1}_{p}(0,\pi))\)
- Naser Abbasi^{1}Email author and
- Mahmood Shakori^{1}
https://doi.org/10.1186/s13660-015-0782-5
© Abbasi and Shakori 2015
- Received: 9 May 2015
- Accepted: 12 August 2015
- Published: 28 August 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 \((W^{1}_{p}(0,\pi))\).
Keywords
- Frankl problem
- Lebesgue integral
- Hölder inequality
- Bessel equation
- Sobolev space
MSC
- 35j15
- 35m12
- 35p10
- 35c10
- 35j56
1 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.
2 Statement of the modified Frankl problem
Definition 2.1
Theorem 2.2
([7])
Theorem 2.3
(see [5])
3 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
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
Proof
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 \(({W}^{1}_{p}(0,\pi))\).
Declarations
Acknowledgements
The authors are grateful to EI Moiseev for his interest in this work.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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