Erratum to Inverse Nodal Problem for p-Laplacian energy-dependent Sturm-LiouvilleequationHikmet KOYUNBAKAN, Boundary Value Problems 2013, 2013;272
Hikmet Kemaloglu, Firat University
19 August 2014
Theorem 2.1. The eigenvalues _{n} of the Dirichlet problem (1.3),(1.4) ( in [1]) are _{n}^{2/p}=n_{p}+(1/(p(n_{p})^{p-1}))∫₀¹q(t)dt+(2/(p(n_{p})^{((p-2)/p)}))∫₀¹r(t)dt+O((1/(n^{(p/2)})))
Theorem 2.2. For the problem (1.3), (1.4) ( in [1]), The nodal points expansion satisfies x_{j}ⁿ = (j/n)+(j/(pn^{p+1}(_{p})^{p}))∫₀¹q(t)dt+((2j)/(pn^{((p/2)+1)}(_{p})^{(p/2)}))∫₀¹r(t)dt+(2/((n_{p})^{^{(p/2)}}))∫₀^{x_{j}ⁿ}r(x)S_{p}^{p}dx +(1/((n_{p})^{^{p}}))∫₀^{x_{j}ⁿ}q(x)S_{p}^{p}dx+O((1/(n^{((p/2)+2)}))).
Proof: Let =_{n} and integrating (2.3) from 0 to x_{j}ⁿ, we have ((j._{p})/(_{n}^{2/p}))=x_{j}ⁿ-∫₀^{x_{j}ⁿ}((2r(x))/(_{n}))S_{p}^{p}dx-∫₀^{x_{j}ⁿ}((q(x))/(_{n}²))S_{p}^{p}dx. By using the estimates of eigenvalues as (1/(_{n}^{2/p}))=(1/(n_{p}))+(1/(p(n_{p})^{p+1}))∫₀¹q(t)dt+(2/(p(n_{p})^{((p/2)+1)}))∫₀¹r(t)dt+O((1/(n^{(p/2)+2}))), then we obtain the result easily.
1- H. Koyunbakan, Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation, Boundary Value Problems, 2013, 2013;272
2- C. K. Law, W. C. Lian, W. C. Wang, Inverse Nodal Problem and Ambarzumyan Theorem for the p-Laplacian, Proc. Royal Soc. Edinburgh 139A (2009), 1261-1273
Competing interests
I am declare that they have not competing interests.
Erratum to Inverse Nodal Problem for p-Laplacian energy-dependent Sturm-LiouvilleequationHikmet KOYUNBAKAN, Boundary Value Problems 2013, 2013;272
19 August 2014
Theorem 2.1. The eigenvalues _{n} of the Dirichlet problem (1.3),(1.4) ( in [1]) are
_{n}^{2/p}=n_{p}+(1/(p(n_{p})^{p-1}))∫₀¹q(t)dt+(2/(p(n_{p})^{((p-2)/p)}))∫₀¹r(t)dt+O((1/(n^{(p/2)})))
Theorem 2.2. For the problem (1.3), (1.4) ( in [1]), The nodal points expansion satisfies
x_{j}ⁿ = (j/n)+(j/(pn^{p+1}(_{p})^{p}))∫₀¹q(t)dt+((2j)/(pn^{((p/2)+1)}(_{p})^{(p/2)}))∫₀¹r(t)dt+(2/((n_{p})^{^{(p/2)}}))∫₀^{x_{j}ⁿ}r(x)S_{p}^{p}dx +(1/((n_{p})^{^{p}}))∫₀^{x_{j}ⁿ}q(x)S_{p}^{p}dx+O((1/(n^{((p/2)+2)}))).
Proof: Let =_{n} and integrating (2.3) from 0 to x_{j}ⁿ, we have
((j._{p})/(_{n}^{2/p}))=x_{j}ⁿ-∫₀^{x_{j}ⁿ}((2r(x))/(_{n}))S_{p}^{p}dx-∫₀^{x_{j}ⁿ}((q(x))/(_{n}²))S_{p}^{p}dx.
By using the estimates of eigenvalues as
(1/(_{n}^{2/p}))=(1/(n_{p}))+(1/(p(n_{p})^{p+1}))∫₀¹q(t)dt+(2/(p(n_{p})^{((p/2)+1)}))∫₀¹r(t)dt+O((1/(n^{(p/2)+2}))),
then we obtain the result easily.
1- H. Koyunbakan, Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation, Boundary Value Problems, 2013, 2013;272
2- C. K. Law, W. C. Lian, W. C. Wang, Inverse Nodal Problem and Ambarzumyan Theorem for the p-Laplacian, Proc. Royal Soc. Edinburgh 139A (2009), 1261-1273
Competing interests
I am declare that they have not competing interests.