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Nonlinear Langevin equations and inclusions involving mixed fractional order derivatives and variable coefficient with fractional nonlocal-terminal conditions

1 Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2 Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Special Issues: Initial and Boundary Value Problems for Differential Equations

In this paper, we discuss the existence and uniqueness of solutions for a new kind of Langevin equation involving Riemann-Liouville as well as Caputo fractional derivatives, and variable coefficient, supplemented with nonlocal-terminal fractional integro-differential conditions. The proposed study is based on modern tools of functional analysis. We also extend our discussion to the associated inclusions problem. For the applicability of the obtained results, several examples are constructed. Some interesting observations are also presented.
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1.R. P. Agarwal, Y. Zhou, J. R. Wang, et al. Fractional functional differential equations with causal operators in Banach spaces, Math. Comput. Modell., 54 (2011), 1440-1452.    

2.B. Ahmad and J. Nieto, Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions, Int. J. Differ. Equations, 2010 (2010), Article ID 649486: 1-10.

3.B. Ahmad, J. J. Nieto, A. Alsaedi, et al. A study of nonlinear Langevin equation involving two fractional orders in different intervals,Nonlinear Anal. Real World Appl., 13 (2012), 599-606.    

4.B. Ahmad, S. K. Ntouyas and A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Differ. Equations, 2011 (2011), Article ID 107384: 1-11.

5.B. Ahmad, S. K. Ntouyas, A. Alsaedi, et al. Fractional-order multivalued problems with non-separated integral-flux boundary conditions, Electron. J. Qual. Theory Differ. Equations, (2015), 1-17.

6.B. Ahmad, A. Alsaedi, S. K. Ntouyas, et al. Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Switzerland: Springer, Cham, 2017.

7.M. Benchohra, J. Henderson, S. K. Ntouyas, et al. Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338 (2008), 1340-1350.    

8.C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Berlin-Heidelberg: Springer-Verlag, 1977.

9.W. T. Coffey, Yu. P. Kalmykov and J. T. Waldron, The Langevin Equation, 2nd Edition., Singapore: World Scientific, 2004.

10.H. Covitz and S. B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces,Isr. J. Math., 8 (1970), 5-11.    

11.K. Deimling, Multivalued Differential Equations, Berlin: De Gruyter, 1992.

12.K. Diethelm, The Analysis of Fractional Differential Equations, Berlin, Heidelberg: Springer-verlag, 2010.

13.H. Ergören and B. Ahmad, Neutral functional fractional differential inclusions with impulses at variable times, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 24 (2017), 235-246.

14.A. Granas and J. Dugundji, Fixed Point Theory, New York: Springer-Verlag, 2003.

15.F. Jiao and Y. Zhou,Existence results for fractional boundary value problem via critical point theory, Int. J. Bifurcation Chaos, 22 (2012), Article ID 1250086.

16.A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier Science B.V., 2006.

17.M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991.

18.M. A. Krasnoselskii, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk, 10 (1955), 123-127.

19.V. Lakshmikantham, S. Leela and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge: Cambridge Academic Publishers, 2009.

20.A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys., 13 (1965), 781-786.

21.J. S. Leszczynski and T. Blaszczyk, Modeling the transition between stable and unstable operation while emptying a silo, Granular Matter, 13 (2011), 429-438.    

22.S. C. Lim, M. Li and L. P. Teo, Langevin equation with two fractional orders, Phys. Lett. A, 372 (2008), 6309-6320.    

23.K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, New York: John Wiley, 1993.

24.I. Podlubny, Fractional Differential Equations, San Diego: Academic Press, 1999.

25.D. Qarout, B. Ahmad and A. Alsaedi, Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions, Fract. Calc. Appl. Anal., 19 (2016), 463-479.

26.S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Yverdon: Gordon and Breach, 1993.

27.J. Tariboon, S. K. Ntouyas and Ch. Thaiprayoon, Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions, Adv. Math. Phys., 2014 (2014), Article ID 372749: 1-15.

28.M. Uranagase and T. Munakata, Generalized Langevin equation revisited: Mechanical random force and self-consistent structure,J. Phys. A Math. Theor., 43 (2010), Article ID 455003.

29.C. Yu and G. Gao, Some results on a class of fractional functional differential equations, Commun. Appl. Nonlinear Anal., 11 (2004), 67-75.

30. C. Yu and G. Gao, Existence of fractional differential equations, J. Math. Anal. Appl., 310 (2005), 26-29.    

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