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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 Applications20152015:263

https://doi.org/10.1186/s13660-015-0782-5

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 problemLebesgue integralHölder inequalityBessel equationSobolev space

MSC

35j1535m1235p1035c1035j56

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

([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
$$ J_{4n}(\mu_{nk})= 0, $$
(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 [8], 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 [5])

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 \(\Delta \in(0,\frac{3}{4})\).

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
$$ f(\theta)=\sum^{\infty}_{n=1}B_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta+B_{0}, $$
(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 [9]). 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 [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
$$ f(\theta)-f(0)=\sum^{\infty}_{n=1}B_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta -\sum^{\infty}_{n=1}B_{n}, $$
(14)
which has a meaning if the following series converges
$$ \sum^{\infty}_{n=1}B_{n}. $$
(15)
By using the results of [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
$$B_{0}=f(0)-\sum^{\infty}_{n=1}B_{n}. $$
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 [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
$$f(\theta)=\sum^{\infty}_{n=0}D_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta, $$
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 [10].

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 [9] 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 [9], 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 [9] 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 [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. □

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

(1)
Department of Mathematics, Lorestan University

References

  1. Frankl, F: On the problems of Chaplygin for mixed sub- and supersonic flows. Tech. Memos. Nat. Adv. Comm. Aeronaut. 1947, 1155 (1947), 32 pp. (4 plates) Google Scholar
  2. Bitsadze, AV: Nekotorye Klassy Uravnenii v Chastnykh Proizvodnykh. Nauka, Moscow (1981) Google Scholar
  3. Smirnov, MM: Uravneniya Smeshannogo Tipa. Nauka, Moscow (1970) Google Scholar
  4. Abbasi, N: Basis property and completeness of the eigenfunctions of the Frankl’ problem. Dokl. Akad. Nauk SSSR 425(3), 295-298 (2009) MathSciNetGoogle Scholar
  5. Moiseev, EI, Abbasi, N: The basis property of the eigenfunctions of a generalized gas-dynamic problem of Frankl’ with a nonlocal parity condition and a discontinuity in the solution gradient. Differ. Uravn. 45(10), 1452-1456 (2009) MathSciNetGoogle Scholar
  6. Moiseev, EI: On the basis property of a system of sines. Differ. Uravn. 23(1), 177-179 (1987) MathSciNetMATHGoogle Scholar
  7. Moiseev, EI: On the basis property of sine and cosine systems. Dokl. Akad. Nauk SSSR 275(4), 794-798 (1984) MathSciNetGoogle Scholar
  8. Erdélyi, A, Magnus, W, Oberhettinger, F, Tricomi, FG, Bateman, H: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953) Google Scholar
  9. Moiseev, EI, Abbasi, N: The basis property of an eigenfunction of the Frankl problem with a nonlocal parity condition in the space Sobolev \((W^{1}_{p}(0,\pi))\). Integral Transforms Spec. Funct. 22(6), 415-421 (2011) MathSciNetView ArticleMATHGoogle Scholar
  10. Abbasi, N, Moiseev, EI: Basis property of eigenfunctions of the generalized gasedynamic problem of Frankl with a nonlocal oddness condition. Integral Transforms Spec. Funct. 21(3-4), 289-294 (2010) MathSciNetView ArticleMATHGoogle Scholar

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© Abbasi and Shakori 2015