July  2021, 14(7): 2417-2434. doi: 10.3934/dcdss.2020171

A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators

1. 

Mechatronic Engineering Department, University of Turkish Aeronautical Association, 06790, Ankara, Turkey

2. 

Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia

3. 

Department of Medical Research, China Medical University, Taichung 40402, Taiwan

4. 

Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

5. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588-0130, USA

* Corresponding author: T. Abdeljawad (tabdeljawad@psu.edu.sa)

Received  March 2019 Revised  April 2019 Published  July 2021 Early access  December 2019

In this work, we use integration by parts formulas derived for fractional operators with Mittag-Leffler kernels to formulate and investigate fractional Sturm-Liouville Problems ($ FSLPs $). We analyze the self-adjointness, eigenvalue and eigenfunction properties of the associated Fractional Sturm-Liouville Operators ($ FSLOs $). The discrete analogue of the obtained results is formulated and analyzed by following nabla analysis.

Citation: Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2417-2434. doi: 10.3934/dcdss.2020171
References:
[1]

T. Abdeljawad, On Delta and Nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013), Art. ID 406910, 12 pp. doi: 10.1155/2013/406910.

[2]

T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ., 2013 (2013), 16 pp. doi: 10.1186/1687-1847-2013-36.

[3]

T. Abdeljawad, F. Jarad and D. Baleanu, A semigroup-like property for discrete Mittag-Leffler functions, Adv. Differ. Equ., 2012 (2012), 7 pp. doi: 10.1186/1687-1847-2012-72.

[4]

T. Abdeljawad and F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), Art. ID 406757, 13 pp. doi: 10.1155/2012/406757.

[5]

T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.

[6]

T. Abdeljawad and D. Baleanu, Fractional differences and integration by parts, J. Comput. Anal. Appl., 13 (2011), 574-582. 

[7]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107.  doi: 10.22436/jnsa.010.03.20.

[8]

T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.

[9]

T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), 11 pp. doi: 10.1186/s13660-017-1400-5.

[10]

T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Differ. Equ., 2017 (2017), 11 pp. doi: 10.1186/s13662-017-1285-0.

[11]

T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J. Comput. Appl. Math., 339 (2018), 218-230.  doi: 10.1016/j.cam.2017.10.021.

[12]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ., 2016 (2016), 18 pp. doi: 10.1186/s13662-016-0949-5.

[13]

T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 2017 (2017), 9 pp. doi: 10.1186/s13662-017-1126-1.

[14]

T. Abdeljawad and D. Baleanu, Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel, Chaos Soliton. Fract., 102 (2017), 106-110.  doi: 10.1016/j.chaos.2017.04.006.

[15]

T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017), Art. ID 4149320, 8 pp. doi: 10.1155/2017/4149320.

[16]

T. Abdeljawad and F. Madjidi, A Lyaponuv inequality for fractional difference operators with discrete Mittag-Leffler kernel of order 2 ≤ ν < 5/2, Eur. Phys. J., 226 (2017), 3355-3368. 

[17]

T. Abdeljawad, Different type kernel h-fractional differences and their fractional h-sums, Chaos, Solitons and Fractals, 116 (2018), 146-156.  doi: 10.1016/j.chaos.2018.09.022.

[18]

T. Abdeljawad, R. Mert and A. Peterson, Sturm-Liouville equations in the frame of fractional operators with exponential kernels and their discrete versions, Quaestiones Mathematicae, (2018). doi: 10.2989/16073606.2018.1514540.

[19]

T. Abdeljawad, R. Mert and A. Peterson, Sturm-Liouville equations in the frame of fractional operators with exponential kernels and their discrete versions, Quaestiones Mathematicae, (2018). doi: 10.2989/16073606.2018.1514540.

[20]

B. S. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos Soliton. Fract., 89 (2016), 547-551. 

[21]

Q. M. Al-Mdallal, On the numerical solution of fractional Sturm-Liouville problems, International Journal of Computer Mathematics, 87 (2010), 2837-2845.  doi: 10.1080/00207160802562549.

[22]

Q. M. Al-Mdallal, An efficient method for solving fractional Sturm-Liouville problems, Chaos, Solitons and Fractals, 40 (2009), 183-189.  doi: 10.1016/j.chaos.2007.07.041.

[23]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769. 

[24]

A. Atangana and J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Advances in Mechanical Engineering, 7 (2015). doi: 10.1177/1687814015613758.

[25]

A. Atangana and S. Jain, Models of fluid flowing in non-conventional media: New numerical analysis, Discrete and Continuous Dynamical Systems-S, (2019), 757–763. doi: 10.3934/dcdss.2020026.

[26]

A. Atangana and D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech., 143 (2017). doi: 10.1061/(ASCE)EM.1943-7889.0001091.

[27]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Soliton. Fract., 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.

[28]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.

[29]

A. Atangana and J. F. Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos, Solitons and Fractals, 114 (2018), 516-535.  doi: 10.1016/j.chaos.2018.07.033.

[30]

F. M. Atici and P. W. Eloe, A Transform method in discrete fractional calculus, International Journal of Difference Equations, 2 (2007), 165-176. 

[31]

F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, P. Amer. Math. Soc., 137 (2009), 981-989.  doi: 10.1090/S0002-9939-08-09626-3.

[32]

F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., Spec. Ed. I, 2009 (2009), 12 pp. doi: 10.14232/ejqtde.2009.4.3.

[33]

E. Bas and R. Ozarslana, Sturm-Liouville problem via Coulomb type in difference equations, Filomat, 31 (2017), 989-998.  doi: 10.2298/FIL1704989B.

[34]

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, Inc., New York-London-Sydney, 1965.

[35]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85. 

[36]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.  doi: 10.18576/pfda/020101.

[37]

C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, Cham, 2015. doi: 10.1007/978-3-319-25562-0.

[38]

H. L. Gray and N. F. Zhang, On a new definition of the fractional difference, Math. Comp., 50 (1988), 513-529.  doi: 10.1090/S0025-5718-1988-0929549-2.

[39]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos, Solitons and Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.

[40]

B. Karaagac, Analysis of the cable equation with non-local and non-singular kernel fractional derivative, Eur. Phys. J. Plus, 133 (2018). doi: 10.1140/epjp/i2018-11916-1.

[41]

A. A. Kilbas, M. H. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[42]

A. A. KilbasM. Saigo and R. K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms Spec. Funct., 15 (2004), 31-49.  doi: 10.1080/10652460310001600717.

[43]

M. Klimek and O. P. Agrawal, Fractional Sturm-Liouville problem, Comput. Math. Appl., 66 (2013), 795-812.  doi: 10.1016/j.camwa.2012.12.011.

[44]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87-92. 

[45]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006.

[46]

K. S. Miller and B. Ross, Fractional difference calculus, Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Ser. Math. Appl., Horwood, Chichester, (1989), 139–152.

[47]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.

[48]

M. RiveroJ. J. Trujillo and M. P. Velasco, A fractional approach to the Sturm-Liouville problem, Cent. Eur. J. Phys., 11 (2013), 1246-1254.  doi: 10.2478/s11534-013-0216-2.

[49]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[50]

I. SuwanT. Abdeljawad and F. Jarad, Monotonicity analysis for nabla h-discrete fractional Atangana-Baleanu differences, Chaos, Solitons and Fractals, 117 (2018), 50-59.  doi: 10.1016/j.chaos.2018.10.010.

[51]

M. I. Syam, Q. M. Al-Mdallal and M. Al-Refai, A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems, Communications in Numerical Analysis, 2017 (2017), Art. ID cna-00334, 217–232. doi: 10.5899/2017/cna-00334.

[52]

M. Zayernouri and G. E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, Journal of Computational Physics, 252 (2013), 495-517.  doi: 10.1016/j.jcp.2013.06.031.

[53]

A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005.

show all references

References:
[1]

T. Abdeljawad, On Delta and Nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013), Art. ID 406910, 12 pp. doi: 10.1155/2013/406910.

[2]

T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ., 2013 (2013), 16 pp. doi: 10.1186/1687-1847-2013-36.

[3]

T. Abdeljawad, F. Jarad and D. Baleanu, A semigroup-like property for discrete Mittag-Leffler functions, Adv. Differ. Equ., 2012 (2012), 7 pp. doi: 10.1186/1687-1847-2012-72.

[4]

T. Abdeljawad and F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), Art. ID 406757, 13 pp. doi: 10.1155/2012/406757.

[5]

T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.

[6]

T. Abdeljawad and D. Baleanu, Fractional differences and integration by parts, J. Comput. Anal. Appl., 13 (2011), 574-582. 

[7]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107.  doi: 10.22436/jnsa.010.03.20.

[8]

T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.

[9]

T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), 11 pp. doi: 10.1186/s13660-017-1400-5.

[10]

T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Differ. Equ., 2017 (2017), 11 pp. doi: 10.1186/s13662-017-1285-0.

[11]

T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J. Comput. Appl. Math., 339 (2018), 218-230.  doi: 10.1016/j.cam.2017.10.021.

[12]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ., 2016 (2016), 18 pp. doi: 10.1186/s13662-016-0949-5.

[13]

T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 2017 (2017), 9 pp. doi: 10.1186/s13662-017-1126-1.

[14]

T. Abdeljawad and D. Baleanu, Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel, Chaos Soliton. Fract., 102 (2017), 106-110.  doi: 10.1016/j.chaos.2017.04.006.

[15]

T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017), Art. ID 4149320, 8 pp. doi: 10.1155/2017/4149320.

[16]

T. Abdeljawad and F. Madjidi, A Lyaponuv inequality for fractional difference operators with discrete Mittag-Leffler kernel of order 2 ≤ ν < 5/2, Eur. Phys. J., 226 (2017), 3355-3368. 

[17]

T. Abdeljawad, Different type kernel h-fractional differences and their fractional h-sums, Chaos, Solitons and Fractals, 116 (2018), 146-156.  doi: 10.1016/j.chaos.2018.09.022.

[18]

T. Abdeljawad, R. Mert and A. Peterson, Sturm-Liouville equations in the frame of fractional operators with exponential kernels and their discrete versions, Quaestiones Mathematicae, (2018). doi: 10.2989/16073606.2018.1514540.

[19]

T. Abdeljawad, R. Mert and A. Peterson, Sturm-Liouville equations in the frame of fractional operators with exponential kernels and their discrete versions, Quaestiones Mathematicae, (2018). doi: 10.2989/16073606.2018.1514540.

[20]

B. S. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos Soliton. Fract., 89 (2016), 547-551. 

[21]

Q. M. Al-Mdallal, On the numerical solution of fractional Sturm-Liouville problems, International Journal of Computer Mathematics, 87 (2010), 2837-2845.  doi: 10.1080/00207160802562549.

[22]

Q. M. Al-Mdallal, An efficient method for solving fractional Sturm-Liouville problems, Chaos, Solitons and Fractals, 40 (2009), 183-189.  doi: 10.1016/j.chaos.2007.07.041.

[23]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769. 

[24]

A. Atangana and J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Advances in Mechanical Engineering, 7 (2015). doi: 10.1177/1687814015613758.

[25]

A. Atangana and S. Jain, Models of fluid flowing in non-conventional media: New numerical analysis, Discrete and Continuous Dynamical Systems-S, (2019), 757–763. doi: 10.3934/dcdss.2020026.

[26]

A. Atangana and D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech., 143 (2017). doi: 10.1061/(ASCE)EM.1943-7889.0001091.

[27]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Soliton. Fract., 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.

[28]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.

[29]

A. Atangana and J. F. Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos, Solitons and Fractals, 114 (2018), 516-535.  doi: 10.1016/j.chaos.2018.07.033.

[30]

F. M. Atici and P. W. Eloe, A Transform method in discrete fractional calculus, International Journal of Difference Equations, 2 (2007), 165-176. 

[31]

F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, P. Amer. Math. Soc., 137 (2009), 981-989.  doi: 10.1090/S0002-9939-08-09626-3.

[32]

F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., Spec. Ed. I, 2009 (2009), 12 pp. doi: 10.14232/ejqtde.2009.4.3.

[33]

E. Bas and R. Ozarslana, Sturm-Liouville problem via Coulomb type in difference equations, Filomat, 31 (2017), 989-998.  doi: 10.2298/FIL1704989B.

[34]

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, Inc., New York-London-Sydney, 1965.

[35]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85. 

[36]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.  doi: 10.18576/pfda/020101.

[37]

C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, Cham, 2015. doi: 10.1007/978-3-319-25562-0.

[38]

H. L. Gray and N. F. Zhang, On a new definition of the fractional difference, Math. Comp., 50 (1988), 513-529.  doi: 10.1090/S0025-5718-1988-0929549-2.

[39]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos, Solitons and Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.

[40]

B. Karaagac, Analysis of the cable equation with non-local and non-singular kernel fractional derivative, Eur. Phys. J. Plus, 133 (2018). doi: 10.1140/epjp/i2018-11916-1.

[41]

A. A. Kilbas, M. H. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[42]

A. A. KilbasM. Saigo and R. K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms Spec. Funct., 15 (2004), 31-49.  doi: 10.1080/10652460310001600717.

[43]

M. Klimek and O. P. Agrawal, Fractional Sturm-Liouville problem, Comput. Math. Appl., 66 (2013), 795-812.  doi: 10.1016/j.camwa.2012.12.011.

[44]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87-92. 

[45]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006.

[46]

K. S. Miller and B. Ross, Fractional difference calculus, Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Ser. Math. Appl., Horwood, Chichester, (1989), 139–152.

[47]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.

[48]

M. RiveroJ. J. Trujillo and M. P. Velasco, A fractional approach to the Sturm-Liouville problem, Cent. Eur. J. Phys., 11 (2013), 1246-1254.  doi: 10.2478/s11534-013-0216-2.

[49]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[50]

I. SuwanT. Abdeljawad and F. Jarad, Monotonicity analysis for nabla h-discrete fractional Atangana-Baleanu differences, Chaos, Solitons and Fractals, 117 (2018), 50-59.  doi: 10.1016/j.chaos.2018.10.010.

[51]

M. I. Syam, Q. M. Al-Mdallal and M. Al-Refai, A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems, Communications in Numerical Analysis, 2017 (2017), Art. ID cna-00334, 217–232. doi: 10.5899/2017/cna-00334.

[52]

M. Zayernouri and G. E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, Journal of Computational Physics, 252 (2013), 495-517.  doi: 10.1016/j.jcp.2013.06.031.

[53]

A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005.

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