Existence of solutions of BVPs for fractional Langevin equations involving Caputo fractional derivatives

• Autorzy: Kiyamehr Zohre , Baghani Hamid

Identyfikatory

DOI

10.1515/jaa-2020-2029

• Języki publikacji: EN

Abstrakty

EN

This article investigates a nonlinear fractional Caputo-Langevin equation D^β(D^α + λ)x(t) = f(t, x(t)), 0 < t < 1, 0 < α ≤ 1, 1 < β ≤ 2, subject to the multi-point boundary conditions x(0) = 0, D^2αx(1) + λD^αx(1) = 0, x(1) =^η∫₀ x(τ) dτ for some 0 < η < 1, where D^α is the Caputo fractional derivative of order α, f : [0, 1] × ℝ → ℝ is a given continuous function, and λ is a real number. Some new existence and uniqueness results are obtained by applying an interesting fixed point theorem.

Słowa kluczowe

EN

Caputo fractional derivative; three-point boundary conditions; existence and uniqueness; fractional Langevin equation

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2021
• Tom: Vol. 27, nr 1
• Strony: 47--55
• Opis fizyczny: Bibliogr. 26 poz.

Twórcy

autor

Kiyamehr Zohre

• Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, University of Sistan and Baluchestan, Zahedan, Iran

autor

Baghani Hamid

• Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, University of Sistan and Baluchestan, Zahedan, Iran

Bibliografia

• [1] R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109 (2010), no. 3, 973-1033.
• [2] A. Amini-Harandi and H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. 72 (2010), no. 5, 2238-2242.
• [3] H. Baghani, Existence and uniqueness of solutions to fractional Langevin equations involving two fractional orders, J. Fixed Point Theory Appl. 20 (2018), no. 2, Paper No. 63.
• [4] H. Baghani and J. J. Nieto, On fractional Langevin equation involving two fractional orders in different intervals, Nonlinear Anal. Model. Control 24 (2019), no. 6, 884-897.
• [5] D. Băleanu, O. G. Mustafa and R. P. Agarwal, An existence result for a superlinear fractional differential equation, Appl. Math. Lett. 23 (2010), no. 9, 1129-1132.
• [6] J. Deng and Z. Deng, Existence of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett. 32 (2014), 6-12.
• [7] K. Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), no. 2, 229-248.
• [8] H. Fazli and J. J. Nieto, Fractional Langevin equation with anti-periodic boundary conditions, Chaos Solitons Fractals 114 (2018), 332-337.
• [9] P. A. Górka and P. Rybka, Existence and uniqueness of solutions to singular ODE’s, Arch. Math. (Basel) 94 (2010), no. 3, 227-233.
• [10] P. Guo, C. Zeng, C. Li and Y. Chen, Numerics for the fractional Langevin equation driven by the fractional Brownian motion, Fract. Calc. Appl. Anal. 16 (2013), no. 1, 123-141.
• [11] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, 2000.
• [12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006.
• [13] N. Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Anal. 70 (2009), no. 7, 2521-2529.
• [14] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal. 69 (2008), no. 10, 3337-3343.
• [15] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, 2009.
• [16] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. 69 (2008), no. 8, 2677-2682.
• [17] V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett. 21 (2008), no. 8, 828-834.
• [18] B. Li, S. Sun and Y. Sun, Existence of solutions for fractional Langevin equation with infinite-point boundary conditions, J. Appl. Math. Comput. 53 (2017), no. 1-2, 683-692.
• [19] W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl. 332 (2007), no. 1, 709-726.
• [20] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993.
• [21] S. P. Nasholm and S. Holm, Linking multiple relaxation, power-law, attenuation, and fractional wave equations, J. Acoust. Soc. Amer. 130 (2011), 3038-3045.
• [22] D. R. Owen and K. Wang, Weakly Lipschitzian mappings and restricted uniqueness of solutions of ordinary differential equations, J. Differential Equations 95 (1992), no. 2, 385-398.
• [23] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999.
• [24] A. Salem, F. Alzahrani and L. Almaghamsi, Fractional Langevin equations with nonlocal integral boundary conditions, Mathematics 7 (2019), Paper No. 402.
• [25] T. Yu, K. Deng and 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), no. 6, 1661-1668.
• [26] W.-X. Zhou and Y.-D. Chu, Existence of solutions for fractional differential equations with multi-point boundary conditions, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 3, 1142-1148.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021)