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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 \((W^{1}_{p}(0,\pi))\).
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
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 \(\Delta \in(0,\frac{3}{4})\).
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
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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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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DOI: https://doi.org/10.1186/s13660-015-0782-5