Abstract
Our aim is to study a new class of differential variational inequalities involving fractional derivatives. Using the fixed point approach, the existence of decay solutions to the mentioned problem is proved.
References
[1] J.-P. Aubin, A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer- Verlag, Berlin (1984).10.1007/978-3-642-69512-4Search in Google Scholar
[2] R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina, B.N. Sadovskii, Measures of Noncompactness and Condensing Operators. Birkh¨auser, Boston-Basel-Berlin (1992).10.1007/978-3-0348-5727-7Search in Google Scholar
[3] 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, No 3 (2010), 973-1033.Search in Google Scholar
[4] E.P. Avgerinos, N.S. Papageorgiou, Differential variational inequalities in RN. Indian J. Pure Appl. Math. 28, No 9 (1997), 1267-1287.Search in Google Scholar
[5] J. Banas, L. Olszowy, On a class of measures of noncompactness in banach algebras and their application to nonlinear integral equations. J. Anal. Appl. 28 (2009), 475-498.Search in Google Scholar
[6] K. Balachandran, Yong Zhou, J. Kokila, Relative controllability of fractional dynamical systems with distributed delays in control. Comput. Math. Appl. 64 (2012), 3201-3209.Search in Google Scholar
[7] K.Balachandran, J.Y. Park, J.J.Trujillo, Controllability of nonlinear fractional dynamical systems. Nonlinear Anal. 75 (2012), 1919-1926.Search in Google Scholar
[8] K. Balachandran, V. Govindaraj, M. Rivero, J.A. Tenreiro Machado, J.J. Trujillo, Observability of nonlinear fractional dynamical systems. Abstr. Appl. Anal. (2013), Art. ID 346041.Search in Google Scholar
[9] D. Bothe, Multivalued perturbations of m-accretive differential inclusions. Israel J. Math. 108 (1998), 109-138.Search in Google Scholar
[10] J. Diestel, W.M. Ruess, W. Schachermayer, Weak compactness in Ll(μ,X). Proc. Amer. Math. Soc. 118 (1993), 447-453.Search in Google Scholar
[11] I. Ekeland, R. Temam, Convex Analysis and Variational Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia - PA (1999).Search in Google Scholar
[12] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, # 194, Springer- Verlag, New York (2000).Search in Google Scholar
[13] Tian Liang Guo, The necessary conditions of fractional optimal control in the sense of Caputo. J. Optim. Theory Appl. 156 (2013), 115-126.Search in Google Scholar
[14] J. Gwinner, On differential variational inequalities and projected dynamical systems - equivalence and a stability result. Discrete Contin. Dynam. Syst. (Dynamical Systems and Differential Equations. Proc. of the 6th AIMS Internat. Conference), Suppl. (2007), 467-476.Search in Google Scholar
[15] J. Gwinner, A note on linear differential variational inequalities in hilbert spaces. Syst. Model. Optim. 391 (2013), 85-91.Search in Google Scholar
[16] H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-Leffler functions and their applications. J. Appl. Math. 2011 (2011), Art ID 298628, 51 pages.10.1155/2011/298628Search in Google Scholar
[17] J.K. Hale, S.M. Verduyn Lunel, Theory of Functional Differential Equations. Springer-Verlag, New York (1993).10.1007/978-1-4612-4342-7Search in Google Scholar
[18] M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Walter de Gruyter, Berlin - New York (2001).10.1515/9783110870893Search in Google Scholar
[19] T.D. Ke, V. Obukhovskii, N.C.Wong, J.C. Yao, On a class of fractional order differential inclusions with infinite delays. Appl. Anal. 92 (2013), 115-137.Search in Google Scholar
[20] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar
[21] V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Res. Notes in Math. Ser. #301, Longman Sci. and Techn., Harlow & John Wiley, New York (1994).Search in Google Scholar
[22] Z. Liu, N.V. Loi, V. Obukhovskii, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities. Int. J. Bifur. Chaos. 23, No 7 (2013), # 1350125.10.1142/S0218127413501253Search in Google Scholar
[23] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Intersci. Publ., John Wiley & Sons, Inc., New York (1993).Search in Google Scholar
[24] V. Obukhovskii and J.-C. Yao, Some existence results for fractional functional differential equations. Fixed Point Theory 11, No 1 (2010), 85-96.Search in Google Scholar
[25] J.-S. Pang, D.E. Steward, Differential variational inequalities. Math. Program. Ser. A 113 (2008), 345-424.10.1007/s10107-006-0052-xSearch in Google Scholar
[26] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Math. in Science and Engin. # 198, Academic Press, San Diego - CA (1999).Search in Google Scholar
[27] T.I. Seidman, Invariance of the reachable set under nonlinear perturbations. SIAM J. Control Optim. 25 (1987), 1173-1191.10.1137/0325064Search in Google Scholar
[28] R.-N. Wang, D.-H. Chena, T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators. J. Differential Equations 252 (2012), 202-235.Search in Google Scholar
[29] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations. Comp. Math. Appl. 59 (2010), 1063-1077. Search in Google Scholar
© Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Fcaa Related News, Events And Books (Fcaa-Volume 18-3-2015)
- Decay solutions for a class of fractional differential variational inequalities
- A biomathematical view on the fractional dynamics of cellulose degradation
- The spreading property for a prey-predator reaction-diffusion system with fractional diffusion
- Fractional variation of Hölderian functions
- Periodic disturbance rejection for fractional-order dynamical systems
- Successive approximation: A survey on stable manifold of fractional differential systems
- When do fractional differential equations have solutions that are bounded by the Mittag--Leffler function ?
- On explicit stability conditions for a linear fractional difference system
- Fractional differential inclusions in the Almgren sense
- Time-optimal control of fractional-order linear systems
- Analytical solutions for the multi-term time-space fractional reaction-diffusion equations on an infinite domain
- Nonexistence results for a class of evolution equations in the Heisenberg group
- High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)
- Dyadic nonlocal diffusions in metric measure spaces
- Fractional derivative anomalous diffusion equation modeling prime number distribution
- Time-fractional diffusion equation in the fractional Sobolev spaces
- Continuous time random walk models associated with distributed order diffusion equations
Articles in the same Issue
- Frontmatter
- Fcaa Related News, Events And Books (Fcaa-Volume 18-3-2015)
- Decay solutions for a class of fractional differential variational inequalities
- A biomathematical view on the fractional dynamics of cellulose degradation
- The spreading property for a prey-predator reaction-diffusion system with fractional diffusion
- Fractional variation of Hölderian functions
- Periodic disturbance rejection for fractional-order dynamical systems
- Successive approximation: A survey on stable manifold of fractional differential systems
- When do fractional differential equations have solutions that are bounded by the Mittag--Leffler function ?
- On explicit stability conditions for a linear fractional difference system
- Fractional differential inclusions in the Almgren sense
- Time-optimal control of fractional-order linear systems
- Analytical solutions for the multi-term time-space fractional reaction-diffusion equations on an infinite domain
- Nonexistence results for a class of evolution equations in the Heisenberg group
- High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)
- Dyadic nonlocal diffusions in metric measure spaces
- Fractional derivative anomalous diffusion equation modeling prime number distribution
- Time-fractional diffusion equation in the fractional Sobolev spaces
- Continuous time random walk models associated with distributed order diffusion equations