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Boundary value problem of a nonlinear Langevin equation with two different fractional orders and impulses
© Wang et al.; licensee Springer 2012
Received: 15 March 2012
Accepted: 5 October 2012
Published: 6 November 2012
In this paper, we study a new type of a Langevin equation involving two different fractional orders and impulses. Sufficient conditions are formulated for the existence and uniqueness of solutions of the given problems.
MSC:34A08, 34B10, 34B37, 46N10.
Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. These characteristics of the fractional derivatives make the fractional-order models more realistic and practical than the classical integer-order models. In fact, fractional differential equations appear naturally in a number of fields such as physics, geophysics, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, nonlinear oscillation of earthquake, the fluid-dynamic traffic model, etc. For more details and applications, we refer the reader to the books [1–3]. For some recent development on the topic, see [4–15] and the references therein.
It is well known that a Langevin equation is widely used to describe the evolution of physical phenomena in fluctuating environments [16–18]. However, for the systems in complex media, an integer-order Langevin equation does not provide the correct description of the dynamics. One of the possible generalizations of a Langevin equation is to replace the integer-order derivative by a fractional-order derivative in it. This gives rise to a fractional Langevin equation, see [19–22] and the references therein.
In 2008, Lim, Li and Teo  firstly introduced a new type of a Langevin equation with two different fractional orders. The solution to this new version of a fractional Langevin equation gives a fractional Gaussian process parametrized by two indices, which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. In 2009, Lim and Teo  discussed the fractional oscillator process with two indices. In 2010, by using the contraction mapping principle and Krasnoselskii’s fixed point theorem, Ahmad and Nieto  studied a Langevin equation involving two fractional orders with Dirichlet boundary conditions. Recently, the existence of solutions for a three-point boundary value problem of a Langevin equation with two different fractional orders has also been studied in .
where C D is the Caputo fractional derivative, , , , , , , , , where and denote the right and the left limits of at (), respectively. has a similar meaning for . Let . Evidently, is a Banach space endowed with the sup-norm .
where , , are given constants and .
Impulsive differential equations, which provide a natural description of observed evolution processes, are regarded as important mathematical tools for the better understanding of several real world problems in applied sciences. The theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modeling of a wide variety of practical situations and has emerged as an important area of investigation. The impulsive differential equations of fractional order have also attracted a considerable attention and a variety of results can be found in the papers [41–51].
To the best knowledge of the authors, no paper has considered nonlinear Langevin equations involving two different fractional orders and impulses, i.e., problems (2.1), (3.1) and (3.2). This paper fills this gap in the literature.
This paper is organized as follows. In Section 2, we present some preliminary results. Consequently, problem (2.1) is reduced to an equivalent integral equation. Then, by using the fixed point theory, we study the existence and uniqueness of a Dirichlet boundary value problem for nonlinear Langevin equations involving two different fractional orders and impulses. In Section 3, we indicate some generalizations to nonlocal Dirichlet boundary value problems. The last section is devoted to an example illustrating the applicability of the imposed conditions. These results can be considered as a contribution to this emerging field.
2 Dirichlet boundary value problem
Definition 2.1 A function with its Caputo derivative of fractional order existing on is a solution of (2.1) if it satisfies (2.1).
Lemma 2.1 
Lemma 2.2 
for some , , .
2.1 Existence result
for some .
for some .
for some . Combining with , we get that .
for some .
for some .
Substituting the value of in (2.8) and (2.12) and letting , we can get (2.3). Conversely, assume that u is a solution of the impulsive fractional integral equation (2.3). Then by a direct computation, it follows that the solution given by (2.3) satisfies (2.2). This completes the proof. □
2.2 Nonlinear problem
Theorem 2.1 Assume that
Then problem (2.1) has a unique solution provided , where Λ is given by (2.14).
Then the equation (2.1) has a solution if and only if the operator T has a fixed point.
As , therefore, A is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. □
3 Nonlocal Dirichlet boundary value problems
For the forthcoming analysis, we need the following assumptions:
(H2) There exists a constant such that .
(H3) There exists a constant such that .
Theorem 3.1 Assume (H1), (H2) hold if , then problem (3.1) has a unique solution, where Λ is given by (2.14).
Theorem 3.2 Assume (H1), (H3) hold if , then problem (3.2) has a unique solution, where Λ is given by (2.14).
The proofs of Theorem 3.2 and Theorem 3.1 are similar. Here we only prove Theorem 3.1.
The rest of the proof is almost the same as that of Theorem 2.1, so we omit it. □
The following example is a direct application of our main result.
where , , , , , and .
Therefore, by Theorem 2.1, we can get that the above equation (4.1) has a unique solution on .
We would like to express our gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which have improved the quality of the present paper. The research was supported by the Natural Science Foundation for Young Scientists of Shanxi Province (2012021002-3), China.
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Podlubny I Mathematics in Science and Engineering. In Fractional Differential Equations. Academic Press, New York; 1999.Google Scholar
- Sabatier J, Agrawal OP, Machado JAT (Eds): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht; 2007.Google Scholar
- Ahmad B, Nieto JJ: Sequential fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 2012, 64: 3046–3052. 10.1016/j.camwa.2012.02.036MathSciNetView ArticleGoogle Scholar
- 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.2011.11.013MathSciNetView ArticleGoogle Scholar
- 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.2011.11.014MathSciNetView ArticleGoogle Scholar
- 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.2010.08.008MathSciNetView ArticleGoogle Scholar
- 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.2011.02.039MathSciNetView ArticleGoogle Scholar
- 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.MathSciNetGoogle Scholar
- 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 055203Google Scholar
- 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.2011.04.016MathSciNetView ArticleGoogle Scholar
- 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.2011.09.078MathSciNetView ArticleGoogle Scholar
- 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.2011.07.034MathSciNetView ArticleGoogle Scholar
- Wang J, Zhou Y: A class of fractional evolution equations and optimal controls. Nonlinear Anal. 2011, 12: 262–272. 10.1016/j.nonrwa.2010.06.013View ArticleGoogle Scholar
- 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 2Google Scholar
- Wax N (Ed): Selected Papers on Noise and Stochastic Processes. Dover, New York; 1954.Google Scholar
- Mazo R: Brownian Motion: Fluctuations, Dynamics and Applications. Oxford Univ. Press, Oxford; 2002.Google Scholar
- Coffey WT, Kalmykov YP, Waldron JT: The Langevin Equation. 2nd edition. World Scientific, Singapore; 2004.Google Scholar
- Kobolev V, Romanov E: Fractional Langevin equation to describe anomalous diffusion. Prog. Theor. Phys. Suppl. 2000, 139: 470–476.View ArticleGoogle Scholar
- Lim SC, Muniandy SV: Self-similar Gaussian processes for modeling anomalous diffusion. Phys. Rev. E 2002., 66: Article ID 021114Google Scholar
- Picozzi S, West B: Fractional Langevin model of memory in financial markets. Phys. Rev. E 2002., 66: Article ID 046118Google Scholar
- Lim SC, Li M, Teo LP: Locally self-similar fractional oscillator processes. Fluct. Noise Lett. 2007, 7: 169–179. 10.1142/S0219477507003817View ArticleGoogle Scholar
- Lim SC, Li M, Teo LP: Langevin equation with two fractional orders. Phys. Lett. A 2008, 372(42):6309–6320. 10.1016/j.physleta.2008.08.045MathSciNetView ArticleGoogle Scholar
- Lim SC, Teo LP: The fractional oscillator process with two indices. J. Phys. A 2009., 42(6): Article ID 065208Google Scholar
- 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 649486Google Scholar
- 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.2011.07.052MathSciNetView ArticleGoogle Scholar
- 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. 10.1080/00036819008839989MathSciNetView ArticleGoogle Scholar
- 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-UMathSciNetView ArticleGoogle Scholar
- 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.2010.02.035MathSciNetView ArticleGoogle Scholar
- 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.2008.02.087MathSciNetView ArticleGoogle Scholar
- 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 720702Google Scholar
- Salem HAH: Fractional order boundary value problem with integral boundary conditions involving Pettis integral. Acta Math. Sci. 2011, 31: 661–672.MathSciNetView ArticleGoogle Scholar
- 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.MathSciNetView ArticleGoogle Scholar
- 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 708576Google Scholar
- 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 107384Google Scholar
- Benchohra M, Graef JR, Hamani S: Existence results for boundary value problems with nonlinear fractional differential equations. Appl. Anal. 2008, 87: 851–863. 10.1080/00036810802307579MathSciNetView ArticleGoogle Scholar
- 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.MathSciNetView ArticleGoogle Scholar
- 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.MathSciNetGoogle Scholar
- 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.2011.11.065MathSciNetView ArticleGoogle Scholar
- 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.2009.01.073MathSciNetView ArticleGoogle Scholar
- 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)Google Scholar
- 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.2011.04.004MathSciNetView ArticleGoogle Scholar
- 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.2010.09.030MathSciNetView ArticleGoogle Scholar
- 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 7Google Scholar
- 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.2011.04.033MathSciNetView ArticleGoogle Scholar
- Ahmad B, Nieto JJ: Existence of solutions for impulsive anti-periodic boundary value problems of fractional order. Taiwan. J. Math. 2011, 15: 981–993.MathSciNetGoogle Scholar
- 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.2009.09.002MathSciNetView ArticleGoogle Scholar
- 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.MathSciNetGoogle Scholar
- Mophou GM: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 2010, 72: 1604–1615. 10.1016/j.na.2009.08.046MathSciNetView ArticleGoogle Scholar
- 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.2010.01.028MathSciNetView ArticleGoogle Scholar
- 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.2010.05.007MathSciNetView ArticleGoogle Scholar
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