Uniform Convergence of Generalized Fourier Series of Hahn-Sturm-Liouville Problem
In this work, we consider the Hahn-Sturm-Liouville boundary value problem defined by $$ (Ly)\left( x\right) :=\frac{1}{r\left( x\right) }\left[ -q^{-1} D_{-\omega q^{-1},q^{-1}}(p\left( x\right) D_{\omega,q}y\left( x\right) )+v\left( x\right) y\left( x\right) \right] =\lambda y\left( x\right) ,\ x\in J_{\omega_{0},a}^{0}=\{x:x=\omega _{0}+(a-\omega_{0})q^{n}, n=1,2,...\} $$ with the boundary conditions $$ y\left( \omega_{0}\right) -h_{1}p\left( \omega_{0}\right) D_{-\omega q^{-1},q^{-1}}y\left( \omega_{0}\right) =0, y\left( a\right) +h_{2}p\left( h^{-1}\left( a\right) \right) D_{-\omega q^{-1},q^{-1}}y\left( a\right) =0,$$ where $q\in\left( 0,1\right) ,\ \omega>0,\ h_{1},h_{2}>0,\ \lambda$ is a complex eigenvalue parameter, $p,v,r$ are real-valued continuous functions at $\omega_{0},$ defined on $J_{\omega_{0},h^{-1}(a)}$ and $p(x)>0,$ $r\left( x\right) >0,\ v\left( x\right) >0,\ x\in J_{\omega_{0},h^{-1}(a)},$ $h^{-1}\left( a\right) =q^{-1}(a-\omega)>a,$ $h^{-1}\left( \omega _{0}\right) =\omega_{0},$ $J_{\omega_{0},a}=\{x:x=\omega_{0}+(a-\omega _{0})q^{n},$ $n=0,1,2...\}\cup\{\omega_{0}\}.$ The existence of a countably infinite set of eigenvalues and eigenfunctions is proved and a uniformly convergent expansion formula in the eigenfunctions is established.
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- [1] Aldwoah K. A., Generalized time scales and associated di¤erence equa-
tions. Ph.D. Thesis, Cairo University (2009).
- [2] Annaby M. H., Hamza A. E. and Aldwoah K. A., Hahn di¤erence oper-
ator and associated Jackson-Nörlund integrals, J. Optim. Theory Appl.
154 (2012), 133153.
- [3] Annaby M. A. and Hassan H. A., Sampling theorems for Jackson-
Nörlund transforms associated with Hahn-di¤erence operators. J. Math.
Anal. Appl. 464 (2018), no. 1, 493506.
- [4] Hahn W., Über orthogonalpolynome, die q?Di¤erenzengleichungen
genügen, Math. Nachr. 2 (1949), 434.
- [5] Hahn W., Ein Beitrag zur Theorie der Orthogonalpolynome, Monatsh.
Math. 95 (1983), 1924.
- [6] Hamza A. E. and Ahmed S. A., Theory of linear Hahn di¤erence equa-
tions, J. Adv. Math. 4(2) (2013), 440460.
- [7] Hamza A. E. and Ahmed S. A., Existence
and uniqueness of solutions
of Hahn di¤erence equations, Adv. Di¤erence Equations 316 (2013),
115.
- [8] Hamza A. E. and Makharesh S. D., Leibnizrule and Fubinis theorem
associated with Hahn di¤erence operator, Journal of Advanced Mathe-
matical, vol. 12, no. 6, (2016), 63356345.
- [9] Annaby M. H., Hamza A. E. and Makharesh S. D., A Sturm-Liouville
theory for Hahn di¤erence operator, in: Xin Li, Zuhair Nashed (Eds.),
Frontiers of Orthogonal Polynomials and q?Series, World Scienti
c,
Singapore, (2018), 3584.
- [10] Weyl H., Über gewöhnlicke Di¤erentialgleichungen mit Singuritaten
und die zugehörigen Entwicklungen willkürlicher Funktionen, Math.
Annal., 68 (1910), 220-269.
- [11] Titchmarsh E. C., Eigenfunction Expansions Associated with Second-
Order Di¤erential Equations. Part I. Second Edition Clarendon Press,
Oxford, 1962.
- [12] Mukhtarov O. Sh. and AydemirK ., "Basis properties of the eigenfunc-
tions of two-interval SturmLiouville problems." Analysis and Mathe-
matical Physics (2018): 1-20.
- [13] Aydemir K., Ol¼gar H., Muk
htarov O. Sh., Muhtarov F., "Di¤erential
operator equations with interface conditions in modi
ed direct sum
spaces", Filomat, 32(3), (2018), 921-931.
- [14] Olgar H., Mukhtarov O. Sh , and Aydemir K., "Some properties of
eigenvalues and generalized eigenvectors of one boundary value prob-
lem." Filomat 32.3 (2018): 911-920.
- [15] Aydemir K. and Mukhtarov O. Sh., "Class of SturmLiouville Problems
with Eigenparameter Dependent Transmission Conditions." Numerical
Functional Analysis and Optimization 38.10 (2017): 1260-1275.
17
- [16] Guseinov G. Sh., An expansion theorem for a Sturm-Liouville operator
on semi-unbounded time scales. Adv. Dyn. Syst. Appl. 3 (2008), no. 1,
147160.
- [17] Guseinov G. Sh., Eigenfunction expansions for a Sturm-Liouville prob-
lem on time scales. Int. J. Di¤erence Equ. 2 (2007), no. 1, 93104.
- [18] Faydao¼glu, ¸S. and Guseinov G. Sh., An expansion result for a Sturm-
Liouville eigenvalue problem with impulse. Turkish J. Math. 34 (2010),
no. 3, 355366.
- [19] Huseynov A., Eigenfunction expansion associated with the one-
dimensional Schrödinger equation on semi-in
nite time scale intervals.
Rep. Math. Phys. 66 (2010), no. 2, 207235.
- [20] Allahverdiev B. P. and Tuna H., Spectral expansion for singular Dirac
system with impulsive conditions, Turkish J. Math. 42, (2018), no. 5,
2527-2545.
- [21] Allahverdiev B. P. and Tuna H., An expansion theorem for q?Sturm-
Liouville operators on the whole line, Turkish J. Math., 42, (2018), no.
3, 1060-1071.
- [22] Huseynov A. and Bairamov E., On expansions in eigenfunctions for
second order dynamic equations on time scales. Nonlinear Dyn. Syst.
Theory 9 (2009), no. 1, 7788.
- [23] Naimark M. A., Linear di¤erential operators, 2nd edn, Nauka, Moscow,
1969; English
transl. of 1st edn, Parts 1, 2, Ungar, New York, 1967,
1968.
- [24] Yosida K., On Titchmarsh-Kodaira formula concerningWeyl-Stone