Implicit differential inclusions in reflexive smooth Banach spaces
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Abstract:
In this paper we prove the existence of solutions of some types of implicit differential inclusions in reflexive smooth Banach spaces. A positive answer is given to a question of Ding [On a class of implicit differential inclusions, Proc. Amer. Math. Soc., 124 (1996), no. 3, 745–749.]References
- Yakov Alber and Irina Ryazantseva, Nonlinear ill-posed problems of monotone type, Springer, Dordrecht, 2006. MR 2213033
- Ya. I. Al′ber, Generalized projection operators in Banach spaces: properties and applications, Functional-differential equations, Funct. Differential Equations Israel Sem., vol. 1, Coll. Judea Samaria, Ariel, 1993, pp. 1–21. MR 1297965
- Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., vol. 178, Dekker, New York, 1996, pp. 15–50. MR 1386667
- Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
- Claudio Baiocchi and António Capelo, Variational and quasivariational inequalities, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. Applications to free boundary problems; Translated from the Italian by Lakshmi Jayakar. MR 745619
- Alain Bensoussan and Jacques-Louis Lions, Impulse control and quasivariational inequalities, $\mu$, Gauthier-Villars, Montrouge; Heyden & Son, Inc., Philadelphia, PA, 1984. Translated from the French by J. M. Cole. MR 756234
- Messaoud Bounkhel, Existence and uniqueness of some variants of nonconvex sweeping processes, J. Nonlinear Convex Anal. 8 (2007), no. 2, 311–323. MR 2351911
- Messaoud Bounkhel and Lionel Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal. 6 (2005), no. 2, 359–374. MR 2159846
- Messaoud Bounkhel and Rabab Al-Yusof, First and second order convex sweeping processes in reflexive smooth Banach spaces, Set-Valued Var. Anal. 18 (2010), no. 2, 151–182. MR 2645247, DOI 10.1007/s11228-010-0134-z
- J. M. Borwein and H. M. Strójwas, Proximal analysis and boundaries of closed sets in Banach space. I. Theory, Canad. J. Math. 38 (1986), no. 2, 431–452. MR 833578, DOI 10.4153/CJM-1986-022-4
- Robert Wayne Carroll and Ralph E. Showalter, Singular and degenerate Cauchy problems, Mathematics in Science and Engineering, Vol. 127, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0460842
- C. Castaing, Équation différentielle multivoque avec contrainte sur l’état dans les espaces de Banach, Travaux Sém. Anal. Convexe 8 (1978), no. 2, exp. no. 13, 20 (French). MR 521156
- Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
- F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth analysis and control theory, Graduate Texts in Mathematics, vol. 178, Springer-Verlag, New York, 1998. MR 1488695
- Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
- Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
- Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, and Václav Zizler, Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001. MR 1831176, DOI 10.1007/978-1-4757-3480-5
- A. D. Ioffe, Proximal analysis and approximate subdifferentials, J. London Math. Soc. (2) 41 (1990), no. 1, 175–192. MR 1063554, DOI 10.1112/jlms/s2-41.1.175
- Kenneth Kuttler, Non-degenerate implicit evolution inclusions, Electron. J. Differential Equations (2000), No. 34, 20. MR 1764715
- Jinlu Li, The generalized projection operator on reflexive Banach spaces and its applications, J. Math. Anal. Appl. 306 (2005), no. 1, 55–71. MR 2132888, DOI 10.1016/j.jmaa.2004.11.007
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- P. Manchanda and A. H. Siddiqi, A rate-independent evolution quasivariational inequalities and state-dependent sweeping processes, Adv. Nonlinear Var. Inequal. 5 (2002), no. 1, 17–28. MR 1881205
- Manuel D. P. Monteiro Marques, Differential inclusions in nonsmooth mechanical problems, Progress in Nonlinear Differential Equations and their Applications, vol. 9, Birkhäuser Verlag, Basel, 1993. Shocks and dry friction. MR 1231975, DOI 10.1007/978-3-0348-7614-8
- Jean-Jacques Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations 26 (1977), no. 3, 347–374. MR 508661, DOI 10.1016/0022-0396(77)90085-7
- Z. Naniewicz and P. D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 188, Marcel Dekker, Inc., New York, 1995. MR 1304257
- Jean-Paul Penot and Robert Ratsimahalo, Characterizations of metric projections in Banach spaces and applications, Abstr. Appl. Anal. 3 (1998), no. 1-2, 85–103. MR 1700278, DOI 10.1155/S1085337598000451
- A. H. Siddiqi and P. Manchanda, Variants of Moreau’s sweeping process, Adv. Nonlinear Var. Inequal. 5 (2002), no. 1, 1–16. MR 1881204
- Wataru Takahashi, Nonlinear functional analysis, Yokohama Publishers, Yokohama, 2000. Fixed point theory and its applications. MR 1864294
- G. Wenzel, On a class of implicit differential inclusions, J. Differential Equations 63 (1986), no. 2, 162–182. MR 848266, DOI 10.1016/0022-0396(86)90046-X
- Zouhua Ding, On a class of implicit differential inclusions, Proc. Amer. Math. Soc. 124 (1996), no. 3, 745–749. MR 1328345, DOI 10.1090/S0002-9939-96-03402-8
- M. M. Vainberg, Variational methods and method of monotone operators in the theory of nonlinear equations, John Wiley, New York, 1973.
Additional Information
- Messaoud Bounkhel
- Affiliation: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Riyadh, Saudi-Arabia
- Email: bounkhel@ksu.edu.sa
- Received by editor(s): November 9, 2009
- Received by editor(s) in revised form: March 6, 2010, and March 7, 2010
- Published electronically: December 7, 2011
- Communicated by: Yingfei Yi
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 2767-2782
- MSC (2010): Primary 34A60, 49J53
- DOI: https://doi.org/10.1090/S0002-9939-2011-11122-5
- MathSciNet review: 2910764