Langevin equation involving one fractional order with threepoint boundary conditions

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Authors
Ahmed Salem
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia.
Faris Alzahrani
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia.
Lamya Almaghamsi
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia.
 Department of Mathematics, University of Jeddah, 41510, Saudi Arabia.
Abstract
In this paper, we investigate a class of nonlinear Langevin equation involving one fractional order \(\alpha\in(0, 1]\) with threepoint boundary conditions. By the Banach contraction principle and Krasnoselskii's fixed point theorem, the existence and uniqueness results of solutions are obtained. Two examples are given to show the applicability of our main results.
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ISRP Style
Ahmed Salem, Faris Alzahrani, Lamya Almaghamsi, Langevin equation involving one fractional order with threepoint boundary conditions, Journal of Nonlinear Sciences and Applications, 12 (2019), no. 12, 791798
AMA Style
Salem Ahmed, Alzahrani Faris, Almaghamsi Lamya, Langevin equation involving one fractional order with threepoint boundary conditions. J. Nonlinear Sci. Appl. (2019); 12(12):791798
Chicago/Turabian Style
Salem, Ahmed, Alzahrani, Faris, Almaghamsi, Lamya. "Langevin equation involving one fractional order with threepoint boundary conditions." Journal of Nonlinear Sciences and Applications, 12, no. 12 (2019): 791798
Keywords
 Fractional Langevin equations
 fixed point theorem
 existence and uniqueness
MSC
References

[1]
B. Ahmad, J. J. Nieto, Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions, Int. J. Differ. Equ., 2010 (2010), 10 pages

[2]
B. Ahmad, J.J. Nieto, A. Alsaedi, M. ElShahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal. Real World Appl., 13 (2012), 599606

[3]
O. Baghani, On fractional Langevin equation involving two fractional orders, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 675681

[4]
H. Baghani, Existence and uniqueness of solutions to fractional Langevin equations involving two fractional orders, J. Fixed Point Theory Appl., 20 (2018), 7 pages

[5]
A. Chen, Y. Chen, Existence of Solutions to Nonlinear Langevin Equation Involving Two Fractional Orders with Boundary Value Conditions, Bound. Value Probl., 2011 (2011), 17 pages

[6]
W. T. Coffey, Y. P. Kalmykov, J. T. Waldron, The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, World Scientific Publishing Co., River Edge (2004)

[7]
H. Fazli, J. J. Nieto, Fractional Langevin equation with antiperiodic boundary conditions, Chaos Solitons Fractals, 114 (2018), 332337

[8]
Z. Y. Gao, X. L. Yu, J. R. Wang, Nonlocal problems for Langevintype differential equations with two fractionalorder derivatives, Bound. Value Probl., 2016 (2016), 21 pages

[9]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam (2006)

[10]
M. A. Krasnoselskii, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk (N.S.), 10 (1955), 123127

[11]
X. Z. Li, M. Medved', J. R. Wang, Generalized Boundary Value Problems for Nonlinear Fractional Langevin Equations, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 53 (2014), 85100

[12]
B. X. Li, S. R. Sun, Y. Sun, Existence of solutions for fractional Langevin equation with infinitepoint boundary conditions, J. Appl. Math. Comput., 53 (2017), 683692

[13]
S. J. Linz, J. C. Sprott, Elementary chaotic flow, Phys. Lett. A, 259 (1999), 240245

[14]
F. Mainradi, P. Pironi, The fractional Langevin equation: Brownian motion revisited, Extracta Math., 11 (1996), 140154

[15]
T. Muensawat, S. K. Ntouyas, J. Tariboon, Systems of generalized SturmLiouville and Langevin fractional differential equations, Adv. Difference Equ., 2017 (2017), 15 pages

[16]
I. Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego (1999)

[17]
A. Salem, F. Alzahrani, L. Almaghamsi, Fractional Langevin equation with nonlocal integral boundary condition, Mathematics, 7 (2019), 110

[18]
J. C. Sprott, Some simple chaotic jerk functions, Amer. J. Phys., 65 (2017), 537543

[19]
J. C. Sprott, A new class of chaotic circuit, Phys. Lett. A, 266 (2000), 1923

[20]
J. C. Sprott, A new chaotic Jerk circuit, IEEE Tran. Citcuit Syst. A, 58 (2011), 240243

[21]
W. Sudsutad, J. Tariboon, Nonlinear fractional integrodifferential Langevin equation involving two fractional orders with threepoint multiterm fractional integral boundary conditions, J. Appl. Math. Comput., 43 (2013), 507522

[22]
T. Yu, K. Deng, M. Luo, Existence and uniqueness of solutions of initial value problems for nonlinear Langevin equation involving two fractional orders, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 16611668

[23]
W. Yukunthorn, S. K. Ntouyas, J. Tariboon, Nonlinear fractional CaputoLangevin equation with nonlocal RiemannLiouville fractional integral conditions, Adv. Difference Equ., 2014 (2014), 18 pages

[24]
C. B. Zhai, P. P. Li, Nonnegative Solutions of Initial Value Problems for Langevin Equations Involving Two Fractional Orders, Mediterr. J. Math., 15 (2018), 11 pages

[25]
C. B. Zhai, P. P. Li, H. Y. Li, Single uppersolution or lowersolution method for Langevin equations with two fractional orders, Adv. Difference Equ., 2018 (2018), 10 pages

[26]
K. H. Zhao, P. Gong, Existence of positive solutions for a class of higherorder Caputo fractional differential equation, Qual. Theory Dyn. Syst., 14 (2015), 157171

[27]
Z. F. Zhou, Y. Qiao, Solutions for a class of fractional Langevin equations with integral and antiperiodic boundary conditions, Bound. Value Probl., 2018 (2018), 10 pages