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.
Similar content being viewed by others
References
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).
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).
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.
E.P. Avgerinos, N.S. Papageorgiou, Differential variational inequalities in RN. Indian J. Pure Appl. Math. 28, No 9 (1997), 1267–1287.
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.
K. Balachandran, Yong Zhou, J. Kokila, Relative controllability of fractional dynamical systems with distributed delays in control. Comput. Math. Appl. 64 (2012), 3201–3209.
K. Balachandran, J.Y. Park, J.J. Trujillo, Controllability of nonlinear fractional dynamical systems. Nonlinear Anal. 75 (2012), 1919–1926.
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.
D. Bothe, Multivalued perturbations of m-accretive differential inclusions. Israel J. Math. 108 (1998), 109–138.
J. Diestel, W.M. Ruess, W. Schachermayer, Weak compactness in Ll(μ,X). Proc. Amer. Math. Soc. 118 (1993), 447–453.
I. Ekeland, R. Temam, Convex Analysis and Variational Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia - PA (1999).
K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, # 194, Springer- Verlag, New York (2000).
Tian Liang Guo, The necessary conditions of fractional optimal control in the sense of Caputo. J. Optim. Theory Appl. 156 (2013), 115–126.
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.
J. Gwinner, A note on linear differential variational inequalities in hilbert spaces. Syst. Model. Optim. 391 (2013), 85–91.
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.
J.K. Hale, S.M. Verduyn Lunel, Theory of Functional Differential Equations. Springer-Verlag, New York (1993).
M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Walter de Gruyter, Berlin - New York (2001).
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.
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).
V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Res. Notes in Math. Ser. #301, Longman Sci. and Techn., Harlow & John Wiley, New York (1994).
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.
K.S. Millerand B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Intersci. Publ., John Wiley & Sons, Inc., New York (1993).
V. Obukhovskii and J.-C. Yao, Some existence results for fractional functional differential equations. Fixed Point Theory 11, No 1 (2010), 85–96.
J.-S. Pang, D.E. Steward, Differential variational inequalities. Math. Program. Ser. A 113 (2008), 345–424.
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).
T.I. Seidman, Invariance of the reachable set under nonlinear perturbations. SIAM J. Control Optim. 25 (1987), 1173–1191.
R.-N. Wang, D.-H. Chena, T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators. J. Differential Equations 252 (2012), 202–235.
Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations. Comp. Math. Appl. 59 (2010), 1063–1077.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Ke, T.D., Loi, N.V. & Obukhovskii, V. Decay solutions for a class of fractional differential variational inequalities. FCAA 18, 531–553 (2015). https://doi.org/10.1515/fca-2015-0033
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1515/fca-2015-0033
Keywords
- differential variational inequality
- fixed point
- measure of noncompactness
- MNC estimate
- functional differential inclusion
- condensing operator