Vấn đề giá trị biên của phương trình Langevin phi tuyến với hai bậc phân số khác nhau và các dao động

Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006. Podlubny I Mathematics in Science and Engineering. In Fractional Differential Equations. Academic Press, New York; 1999. Sabatier J, Agrawal OP, Machado JAT (Eds): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht; 2007. Ahmad B, Nieto JJ: Sequential fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 2012, 64: 3046–3052. 10.1016/j.camwa.2012.02.036 Kaslik E, Sivasundaram S: Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions. Nonlinear Anal., Real World Appl. 2012, 13: 1489–1497. 10.1016/j.nonrwa.2011.11.013 Guezane-Lakoud A, Khaldi R: Solvability of a fractional boundary value problem with fractional integral condition. Nonlinear Anal. 2012, 75: 2692–2700. 10.1016/j.na.2011.11.014 Zhou P, Zhu W: Function projective synchronization for fractional-order chaotic systems. Nonlinear Anal., Real World Appl. 2011, 12: 811–816. 10.1016/j.nonrwa.2010.08.008 Goodrich CS: Existence of a positive solution to systems of differential equations of fractional order. Comput. Math. Appl. 2011, 62: 1251–1268. 10.1016/j.camwa.2011.02.039 Agarwal RP, Benchohra M, Hamani S, Pinelas S: Boundary value problems for differential equations involving Riemann-Liouville fractional derivative on the half-line. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 2011, 18: 235–244. Baleanu D, Agarwal RP, Mustafa OG, Cosulschi M: Asymptotic integration of some nonlinear differential equations with fractional time derivative. J. Phys. A 2011., 44: Article ID 055203 Agarwal RP, Zhou Y, Wang J, Luo X: Fractional functional differential equations with causal operators in Banach spaces. Math. Comput. Model. 2011, 54: 1440–1452. 10.1016/j.mcm.2011.04.016 Wang G, Agarwal RP, Cabada A: Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Lett. 2012, 25: 1019–1024. 10.1016/j.aml.2011.09.078 Chen F, Nieto JJ, Zhou Y: Global attractivity for nonlinear fractional differential equations. Nonlinear Anal., Real World Appl. 2012, 13: 287–298. 10.1016/j.nonrwa.2011.07.034 Wang J, Zhou Y: A class of fractional evolution equations and optimal controls. Nonlinear Anal. 2011, 12: 262–272. 10.1016/j.nonrwa.2010.06.013 Wang G, Ntouyas SK, Zhang L: Positive solutions of the three-point boundary value problem for fractional-order differential equations with an advanced argument. Adv. Differ. Equ. 2011., 2011: Article ID 2 Wax N (Ed): Selected Papers on Noise and Stochastic Processes. Dover, New York; 1954. Mazo R: Brownian Motion: Fluctuations, Dynamics and Applications. Oxford Univ. Press, Oxford; 2002. Coffey WT, Kalmykov YP, Waldron JT: The Langevin Equation. 2nd edition. World Scientific, Singapore; 2004. Kobolev V, Romanov E: Fractional Langevin equation to describe anomalous diffusion. Prog. Theor. Phys. Suppl. 2000, 139: 470–476. Lim SC, Muniandy SV: Self-similar Gaussian processes for modeling anomalous diffusion. Phys. Rev. E 2002., 66: Article ID 021114 Picozzi S, West B: Fractional Langevin model of memory in financial markets. Phys. Rev. E 2002., 66: Article ID 046118 Lim SC, Li M, Teo LP: Locally self-similar fractional oscillator processes. Fluct. Noise Lett. 2007, 7: 169–179. 10.1142/S0219477507003817 Lim SC, Li M, Teo LP: Langevin equation with two fractional orders. Phys. Lett. A 2008, 372(42):6309–6320. 10.1016/j.physleta.2008.08.045 Lim SC, Teo LP: The fractional oscillator process with two indices. J. Phys. A 2009., 42(6): Article ID 065208 Ahmad B, Nieto JJ: Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions. Int. J. Differ. Equ. 2010., 2010: Article ID 649486 Ahmad B, Nieto JJ, Alsaedi A, El-Shahed M: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl. 2012, 13: 599–606. 10.1016/j.nonrwa.2011.07.052 Byszewski L, Lakshmikantham V: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 1991, 40: 11–19. 10.1080/00036819008839989 Byszewski L: Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 1991, 162: 494–505. 10.1016/0022-247X(91)90164-U Balachandran K, Trujillo JJ: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Anal. 2010, 72: 4587–4593. 10.1016/j.na.2010.02.035 N’Guerekata GM: A Cauchy problem for some fractional abstract differential equation with nonlocal condition. Nonlinear Anal. 2009, 70: 1873–1876. 10.1016/j.na.2008.02.087 Feng M, Zhang X, Ge W: New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 720702 Salem HAH: Fractional order boundary value problem with integral boundary conditions involving Pettis integral. Acta Math. Sci. 2011, 31: 661–672. Ahmad B, Nieto JJ, Alsaedi A: Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions. Acta Math. Sci. 2011, 31: 2122–2130. Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. 2009., 2009: Article ID 708576 Ahmad B, Ntouyas SK, Alsaedi A: New existence results for nonlinear fractional differential equations with three-point integral boundary conditions. Adv. Differ. Equ. 2011., 2011: Article ID 107384 Benchohra M, Graef JR, Hamani S: Existence results for boundary value problems with nonlinear fractional differential equations. Appl. Anal. 2008, 87: 851–863. 10.1080/00036810802307579 Hamani S, Benchohra M, Graef JR: Existence results for boundary value problems with nonlinear fractional inclusions and integral conditions. Electron. J. Differ. Equ. 2010, 20: 1–16. Liu X, Jia M, Wu B: Existence and uniqueness of solution for fractional differential equations with integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2009, 69: 1–10. Cabada A, Wang G: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 2012, 389: 403–411. 10.1016/j.jmaa.2011.11.065 Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 2009, 71: 2391–2396. 10.1016/j.na.2009.01.073 Wang, G, Ahmad, B, Zhang, L: New existence results for nonlinear impulsive integro-differential equations of fractional order with nonlocal boundary conditions. Nonlinear Stud. (to appear) Wang G, Ahmad B, Zhang L: Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions. Comput. Math. Appl. 2011, 62: 1389–1397. 10.1016/j.camwa.2011.04.004 Wang G, Ahmad B, Zhang L: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 2011, 74: 792–804. 10.1016/j.na.2010.09.030 Zhang L, Wang G: Existence of solutions for nonlinear fractional differential equations with impulses and anti-periodic boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2011., 2011: Article ID 7 Ahmad B, Wang G: A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations. Comput. Math. Appl. 2011, 62: 1341–1349. 10.1016/j.camwa.2011.04.033 Ahmad B, Nieto JJ: Existence of solutions for impulsive anti-periodic boundary value problems of fractional order. Taiwan. J. Math. 2011, 15: 981–993. Ahmad B, Sivasundaram S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. Hybrid Syst. 2010, 4: 134–141. 10.1016/j.nahs.2009.09.002 Agarwal RP, Ahmad B: Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 2011, 18: 457–470. Mophou GM: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 2010, 72: 1604–1615. 10.1016/j.na.2009.08.046 Tian Y, Bai Z: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comput. Math. Appl. 2010, 59: 2601–2609. 10.1016/j.camwa.2010.01.028 Zhang X, Huang X, Liu Z: The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay. Nonlinear Anal. Hybrid Syst. 2010, 4: 775–781. 10.1016/j.nahs.2010.05.007