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}
DOI: 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 spaceMSC
35j15 35m12 35p10 35c10 35j561 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.
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Authors’ Affiliations
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