October  2020, 13(10): 2719-2734. doi: 10.3934/dcdss.2020220

Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential equations

1. 

Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA

2. 

Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad-500078, Telangana, India

* Corresponding author: Paul Webster Eloe

Received  February 2019 Revised  July 2019 Published  December 2019

The quasilinearization method is applied to a boundary value problem at resonance for a Riemann-Liouville fractional differential equation. Under suitable hypotheses, the method of upper and lower solutions is employed to establish uniqueness of solutions. A shift method, coupled with the method of upper and lower solutions, is applied to establish existence of solutions. The quasilinearization algorithm is then applied to obtain sequences of lower and upper solutions that converge monotonically and quadratically to the unique solution of the boundary value problem at resonance.

Citation: Paul Eloe, Jaganmohan Jonnalagadda. Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2719-2734. doi: 10.3934/dcdss.2020220
References:
[1]

R. P. AgarwalB. Ahmad and A. Alsaedi, Method of quasilinearization for a nonlocal singular boundary value problem in weighted spaces, Bound. Value Probl., 261 (2013), 17 pp.  doi: 10.1186/1687-2770-2013-261.  Google Scholar

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E. Akin-Bohner and F. M. Atici, A quasilinearization approach for two-point nonlinear boundary value problems on time scales, Rocky Mountain J. Math., 35 (2005), 19-45.  doi: 10.1216/rmjm/1181069766.  Google Scholar

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J. Aljedani and P. Eloe, Uniqueness of solutions of boundary value problems at resonance, Advances in the Theory of Nonlinear Analysis and its Application, 2 (2018), 168-183.  doi: 10.31197/atnaa.453919.  Google Scholar

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K. AlanaziM. Alshammari and P. Eloe, Quasilinearization and Boundary Value Problems at Resonance, Georgian Mathematics J., (2019).  doi: 10.1515/gmj-2019-2058.  Google Scholar

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V. Antony Vijesh, A short note on the quasilinearization method for fractional differential equations, Numer. Funct. Anal. Optim., 37 (2016), 1158-1167.  doi: 10.1080/01630563.2016.1188827.  Google Scholar

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R. Bellman, Methods of Nonlinear Analysis. Vol. II, Mathematics in Science and Engineering, Vol. 61-II, Academic Press, New York-London, 1973.  Google Scholar

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R. Bellman and R. Kalba, Quasilinearization and Nonlinear Boundary Value Problems, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3. American Elsevier Publishing Co., Inc., New York, 1965.  Google Scholar

[9]

P. Eloe and J. Jonnalagadda, Quasilinearization and boundary value problems for Riemann-Liouville fractional differential equations, Electron. J. Differential Equations, (2019), 15 pp.   Google Scholar

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P. Eloe and Y. Gao, The method of quasilinearization and a three-point boundary value problem, J. Korean Math. Soc., 39 (2002), 319-330.  doi: 10.4134/JKMS.2002.39.2.319.  Google Scholar

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P. W. Eloe and Y. Z. Zhang, A quadratic monotone iteration scheme for two-point boundary value problems for ordinary differential equations, Nonlinear Anal., 33 (1998), 443-453.  doi: 10.1016/S0362-546X(97)00633-0.  Google Scholar

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P. GuoC. P. Li and G. R. Chen, On the fractional mean-value theorem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 6 pp.  doi: 10.1142/S0218127412501040.  Google Scholar

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H. J. HauberkA. M. Mathai and R. K. Saxena, Mittag - Leffler functions and their applications, Journal of Applied Mathematics, 2011 (2011), 51 pp.  doi: 10.1155/2011/298628.  Google Scholar

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G. InfanteP. Pietramala and F. A.F. Tojo, Nontrivial solutions of local and nonlocal Neumann boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 337-369.  doi: 10.1017/S0308210515000499.  Google Scholar

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R. A. Khan, Existence and approximation of solutions to three-point boundary value problems for fractional differential equations, Electron. J. Qual. Theory Differ. Equ., 2011 (2011), 8 pp.  doi: 10.14232/ejqtde.2011.1.58.  Google Scholar

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V. LakshmikanthamS. Leela and F. A. McRae, Improved generalized quasilinearization (GQL) method, Nonlinear Anal., 24 (1995), 1627-1637.  doi: 10.1016/0362-546X(94)E0090-4.  Google Scholar

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V. LakshmikanthamN. Shahzad and J. J. Nieto, Methods of generalized quasilinearization for periodic boundary value problems, Nonlinear Anal., 27 (1996), 143-151.  doi: 10.1016/0362-546X(95)00021-M.  Google Scholar

[20]

V. LakshmikanthamS. Leela and S. Sivasundaram, Extensions of the methods of quasilinearization, J. Optim. Theory Appl., 87 (1995), 379-401.  doi: 10.1007/BF02192570.  Google Scholar

[21]

J. J. Nieto, Generalized quasilinearization method for a second order ordinary differential equation with Dirichlet boundary conditions, Proc. Amer. Math. Soc., 125 (1997), 2599-2604, http://www.jstor.org/stable/2162028. doi: 10.1090/S0002-9939-97-03976-2.  Google Scholar

[22]

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.  Google Scholar

[23]

N. Shahzad and A. S. Vatsala, Improved generalized quasilinearization method for second order boundary value problems, Dynam. Systems Appl., 4 (1995), 79-85.   Google Scholar

[24]

N. Sveikate, Resonant problems by quasilinearization, Int. J. Differ. Equ., 2014 (2014), 8 pp.  doi: 10.1155/2014/564914.  Google Scholar

[25]

J. Vasundhara DeviF. A. McRae and Z. Drici, Generalized quasilinearization for fractional differential equations, Comput. Math. Appl., 59 (2010), 1057-1062.  doi: 10.1016/j.camwa.2009.05.017.  Google Scholar

[26]

W. Z. XieJ. Xiao and Z. G. Luo, Existence of solutions for Riemann-Liouville fractional boundary value problem, Abstr. Appl. Anal., 2014 (2014), 9 pp.  doi: 10.1155/2014/540351.  Google Scholar

[27]

A. Yakar, Initial time difference quasilinearization for Caputo fractional differential equations, Adv. Difference Equ., 2012 (2012), 9 pp.  doi: 10.1186/1687-1847-2012-92.  Google Scholar

[28]

I. Yermachenko and F. Sadyrbaev, Quasilinearization and multiple solutions solutions of the Emden-Fowler type equations, Math. Model. Anal., 10 (2005), 41-50.   Google Scholar

show all references

References:
[1]

R. P. AgarwalB. Ahmad and A. Alsaedi, Method of quasilinearization for a nonlocal singular boundary value problem in weighted spaces, Bound. Value Probl., 261 (2013), 17 pp.  doi: 10.1186/1687-2770-2013-261.  Google Scholar

[2]

E. Akin-Bohner and F. M. Atici, A quasilinearization approach for two-point nonlinear boundary value problems on time scales, Rocky Mountain J. Math., 35 (2005), 19-45.  doi: 10.1216/rmjm/1181069766.  Google Scholar

[3]

J. Aljedani and P. Eloe, Uniqueness of solutions of boundary value problems at resonance, Advances in the Theory of Nonlinear Analysis and its Application, 2 (2018), 168-183.  doi: 10.31197/atnaa.453919.  Google Scholar

[4]

M. Al-Refai, On the fractional derivatives at extreme points, Electron. J. Qual. Theory Differ. Equ., 2012 (2012), 5 pp.  doi: 10.14232/ejqtde.2012.1.55.  Google Scholar

[5]

K. AlanaziM. Alshammari and P. Eloe, Quasilinearization and Boundary Value Problems at Resonance, Georgian Mathematics J., (2019).  doi: 10.1515/gmj-2019-2058.  Google Scholar

[6]

V. Antony Vijesh, A short note on the quasilinearization method for fractional differential equations, Numer. Funct. Anal. Optim., 37 (2016), 1158-1167.  doi: 10.1080/01630563.2016.1188827.  Google Scholar

[7]

R. Bellman, Methods of Nonlinear Analysis. Vol. II, Mathematics in Science and Engineering, Vol. 61-II, Academic Press, New York-London, 1973.  Google Scholar

[8]

R. Bellman and R. Kalba, Quasilinearization and Nonlinear Boundary Value Problems, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3. American Elsevier Publishing Co., Inc., New York, 1965.  Google Scholar

[9]

P. Eloe and J. Jonnalagadda, Quasilinearization and boundary value problems for Riemann-Liouville fractional differential equations, Electron. J. Differential Equations, (2019), 15 pp.   Google Scholar

[10]

P. Eloe and Y. Gao, The method of quasilinearization and a three-point boundary value problem, J. Korean Math. Soc., 39 (2002), 319-330.  doi: 10.4134/JKMS.2002.39.2.319.  Google Scholar

[11]

P. W. Eloe and Y. Z. Zhang, A quadratic monotone iteration scheme for two-point boundary value problems for ordinary differential equations, Nonlinear Anal., 33 (1998), 443-453.  doi: 10.1016/S0362-546X(97)00633-0.  Google Scholar

[12]

P. GuoC. P. Li and G. R. Chen, On the fractional mean-value theorem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 6 pp.  doi: 10.1142/S0218127412501040.  Google Scholar

[13]

H. J. HauberkA. M. Mathai and R. K. Saxena, Mittag - Leffler functions and their applications, Journal of Applied Mathematics, 2011 (2011), 51 pp.  doi: 10.1155/2011/298628.  Google Scholar

[14]

G. InfanteP. Pietramala and F. A.F. Tojo, Nontrivial solutions of local and nonlocal Neumann boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 337-369.  doi: 10.1017/S0308210515000499.  Google Scholar

[15]

R. A. Khan, Existence and approximation of solutions to three-point boundary value problems for fractional differential equations, Electron. J. Qual. Theory Differ. Equ., 2011 (2011), 8 pp.  doi: 10.14232/ejqtde.2011.1.58.  Google Scholar

[16]

R. A. Khan and R. Rodríguez-López, Existence and approximation of solutions of second-order nonlinear four point boundary value problems, Nonlinear Anal., 63 (2005), 1094-1115.  doi: 10.1016/j.na.2005.05.030.  Google Scholar

[17]

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

[18]

V. LakshmikanthamS. Leela and F. A. McRae, Improved generalized quasilinearization (GQL) method, Nonlinear Anal., 24 (1995), 1627-1637.  doi: 10.1016/0362-546X(94)E0090-4.  Google Scholar

[19]

V. LakshmikanthamN. Shahzad and J. J. Nieto, Methods of generalized quasilinearization for periodic boundary value problems, Nonlinear Anal., 27 (1996), 143-151.  doi: 10.1016/0362-546X(95)00021-M.  Google Scholar

[20]

V. LakshmikanthamS. Leela and S. Sivasundaram, Extensions of the methods of quasilinearization, J. Optim. Theory Appl., 87 (1995), 379-401.  doi: 10.1007/BF02192570.  Google Scholar

[21]

J. J. Nieto, Generalized quasilinearization method for a second order ordinary differential equation with Dirichlet boundary conditions, Proc. Amer. Math. Soc., 125 (1997), 2599-2604, http://www.jstor.org/stable/2162028. doi: 10.1090/S0002-9939-97-03976-2.  Google Scholar

[22]

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.  Google Scholar

[23]

N. Shahzad and A. S. Vatsala, Improved generalized quasilinearization method for second order boundary value problems, Dynam. Systems Appl., 4 (1995), 79-85.   Google Scholar

[24]

N. Sveikate, Resonant problems by quasilinearization, Int. J. Differ. Equ., 2014 (2014), 8 pp.  doi: 10.1155/2014/564914.  Google Scholar

[25]

J. Vasundhara DeviF. A. McRae and Z. Drici, Generalized quasilinearization for fractional differential equations, Comput. Math. Appl., 59 (2010), 1057-1062.  doi: 10.1016/j.camwa.2009.05.017.  Google Scholar

[26]

W. Z. XieJ. Xiao and Z. G. Luo, Existence of solutions for Riemann-Liouville fractional boundary value problem, Abstr. Appl. Anal., 2014 (2014), 9 pp.  doi: 10.1155/2014/540351.  Google Scholar

[27]

A. Yakar, Initial time difference quasilinearization for Caputo fractional differential equations, Adv. Difference Equ., 2012 (2012), 9 pp.  doi: 10.1186/1687-1847-2012-92.  Google Scholar

[28]

I. Yermachenko and F. Sadyrbaev, Quasilinearization and multiple solutions solutions of the Emden-Fowler type equations, Math. Model. Anal., 10 (2005), 41-50.   Google Scholar

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