Main Article Content
In this paper, we discuss the existence of a unique solution of Caputo-Liouville type Langevin equation involving two fractional orders and finitely many nonlinearities, equipped with nonlocal boundary conditions via Banach contraction mapping principle. The location of the unique solution of the given problem is also presented. In addition, we discuss the existence of solutions for the problem at hand by means of Krasnoselskii's fixed point theorem. Examples are constructed for the illustration of the obtained results. The paper concludes with some interesting remarks.
- R.L. Magin, Fractional Calculus in Bioengineering, Begell House, Chicago, 2006.
- A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier Science B.V, Amsterdam, 2006.
- K. Diethelm, The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, 2004, Springer-Verlag, Berlin, 2010.
- F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, Singapore, 2010.
- H.A. Fallahgoul, S.M. Focardi, F.J. Fabozzi, Fractional Calculus and Fractional Processes with Applications to Financial Economics: Theory and Application, Elsevier/Academic Press, London, 2017.
- P. Langevin, Sur la th ´eorie du mouvement brownien (in French), On the theory of Brownian motion, CR Acad. Sci. Paris, 146 (1908) 530-533.
- R. Klages, G. Radons, I.M. Sokolov, Anomalous transport: foundations and applications, Wiley-VCH, Weinheim, 2008.
- R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys., 29 (1966) 255-84.
- R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Second Ed., Springer-Verlag, Berlin, 1991.
- B.J. West, M. Latka, Fractional Langevin model of gait variability, J. NeuroEng. Rehabil. 2 (2005), 24,
- C.H. Eab, S.C. Lim, Fractional generalized langevin equation approach to single-file diffusion, Phys. A, 389 (2010), 2510- 2521.
- S.F. Kwok, Langevin equation with multiplicative white noise: transformation of diffusion processes into the wiener process in different prescriptions, Ann. Phys., 327 (2012), 1989-1997.
- S. Eule, R. Friedrich, F. Jenko, D. Kleinhans, Langevin approach to fractional diffusion equations including inertial effects, J. Phys. Chem. B, 111 (2007), 11474-11477.
- S.C. Lim, M. Li, L.P. Teo, Langevin equation with two fractional orders, Phys. Lett. A, 372 (2008), 6309-6320.
- B. Ahmad, J.J. Nieto, A. Alsaedi, M. El-Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal. Real World Appl. 13 (2012), 599-606.
- O. Baghani, On fractional Langevin equation involving two fractional orders, Commun. Nonlinear Sci. Numer. Simul. 42 (2017), 675-681.
- B. Li, S. Sun, Y. Sun, Existence of solutions for fractional Langevin equation with infinite-point boundary conditions, J. Appl. Math. Comput. 53 (2017), 683-692.
- H. Fazli, J.J. Nieto, Fractional Langevin equation with anti-periodic boundary conditions, Chaos Solitons Fractals, 114 (2018), 332-337.
- Z. Zhou, Y. Qiao, Solutions for a class of fractional Langevin equations with integral and anti-periodic boundary conditions, Bound. Value Probl. 2018 (2018), 152.
- B. Ahmad, A. Alsaedi, S. Salem, On a nonlocal integral boundary value problem of nonlinear Langevin equation with different fractional orders, Adv. Difference Equ. 2019 (2019), 57.
- Y. Liu, R. Agarwal, Existence of solutions of BVPs for impulsive fractional Langevin equations involving Caputo fractional derivatives, Turk. J. Math. 43 (2019), 2451-2472.
- Z. Laadjal, B. Ahmad, N. Adjeroud, Existence and uniqueness of solutions for multi-term fractional Langevin equation with boundary conditions, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 27 (2020), 339-350.
- A. Wongcharoen, B. Ahmad, S.K. Ntouyas, J. Tariboon, Three-point boundary value problems for the Langevin equation with the Hilfer fractional derivative, Adv. Math. Phys. 2020 (2020), 9606428.
- H. Fazli, H. Sun, J.J. Nieto, New existence and stability results for fractional Langevin equation with three-point boundary conditions, Comput. Appl. Math. 40 (2021), 48.
- F. Jiao, Y. Zhou, Existence of solution for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl. 62 (2011), 1181-1199.
- S. Sun, Y. Zhao, Z. Han, Y. Li, The existence of solutions for boundary-value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul. 14 (2012), 4961-4967.
- J. Henderson, N. Kosmatov, Eigenvalue comparison for fractional boundary value problems with the Caputo derivative, Fract. Calc. Appl. Anal. 17 (2014), 72-880.
- J. Henderson, R. Luca, Nonexistence of positive solutions for a system of coupled fractional boundary value problems, Bound. Value Probl. 2015 (2015), 138.
- B. Ahmad, R. Luca, Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions, Fract. Calc. Appl. Anal. 21 (2018), 423-441.
- R.P. Agarwal, R. Luca, Positive solutions for a semipositone singular Riemann-Liouville fractional differential problem, Int. J. Nonlinear Sci. Numer. Simul. 20 (2019), 823-831.
- M.A. Krasnosel'skii, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk, 10 (1955), 123-127.