
AIMS Mathematics, 2019, 4(3): 626647. doi: 10.3934/math.2019.3.626
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Nonlinear Langevin equations and inclusions involving mixed fractional order derivatives and variable coefficient with fractional nonlocalterminal 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
Received: , Accepted: , Published:
Special Issues: Initial and Boundary Value Problems for Differential Equations
References
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), 14401452.
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: 110.
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), 599606.
4.B. Ahmad, S. K. Ntouyas and A. Alsaedi, New existence results for nonlinear fractional differential equations with threepoint integral boundary conditions, Adv. Differ. Equations, 2011 (2011), Article ID 107384: 111.
5.B. Ahmad, S. K. Ntouyas, A. Alsaedi, et al. Fractionalorder multivalued problems with nonseparated integralflux boundary conditions, Electron. J. Qual. Theory Differ. Equations, (2015), 117.
6.B. Ahmad, A. Alsaedi, S. K. Ntouyas, et al. HadamardType 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), 13401350.
8.C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, BerlinHeidelberg: SpringerVerlag, 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), 511.
11.K. Deimling, Multivalued Differential Equations, Berlin: De Gruyter, 1992.
12.K. Diethelm, The Analysis of Fractional Differential Equations, Berlin, Heidelberg: Springerverlag, 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), 235246.
14.A. Granas and J. Dugundji, Fixed Point Theory, New York: SpringerVerlag, 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), 123127.
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 KakutaniKy Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys., 13 (1965), 781786.
21.J. S. Leszczynski and T. Blaszczyk, Modeling the transition between stable and unstable operation while emptying a silo, Granular Matter, 13 (2011), 429438.
22.S. C. Lim, M. Li and L. P. Teo, Langevin equation with two fractional orders, Phys. Lett. A, 372 (2008), 63096320.
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 semilinear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions, Fract. Calc. Appl. Anal., 19 (2016), 463479.
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 HadamardCaputo type fractional derivatives with nonlocal fractional integral conditions, Adv. Math. Phys., 2014 (2014), Article ID 372749: 115.
28.M. Uranagase and T. Munakata, Generalized Langevin equation revisited: Mechanical random force and selfconsistent 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), 6775.
30. C. Yu and G. Gao, Existence of fractional differential equations, J. Math. Anal. Appl., 310 (2005), 2629.
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