Abstract
This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x(t) and ẋ(t). In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C 1(T,H). Also we examine the lsc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C 1(T,H) to the solutions of the original convex problem (strong relaxation). An example of a nonlinear hyperbolic optimal control problem is also discussed.
Similar content being viewed by others
References
Ahmed Kerbal S., N. U.: Optimal control of nonlinear second order evolution equations. J. Appl. Math. Stoch. Anal., 6, 123–136 (1993)
Bian W.: Existence results for second order nonlinear evolution inclusions. Indian J. Pure Appl. Math., 11, 1177–1193 (1998)
Gasinski, L. Smolka, M.: An existence theorem for wave–type hyperbolic hemivariational inequalities. Math. Nachr., 242, 79–90 (2002)
Kartsatos, A. Markov, L.: An L 2–approach to second order nonlinear functional evolutions involving maccretive operators in Banach spaces. Diff. Integral Equations, 14, 833–866 (2001)
Migorski, S.: Existence, variational and optimal control problems for nonlinear second order evolution inclusions. Dynam. Syst. and Appl., 4, 513–528 (1995)
Kartsatos, A., Liu, X.: On the construction and the convergence of the method of lines for quasi–linear functional evolutions in general Banach spaces. Nonlin. Anal., 29, 385–414 (1997)
Kartsatos, A., Shin, K. Y.: The method of lines and the approximation of zeros of m–accretive operators in general Banach spaces. J. Diff. Eqns., 113, 128–149 (1994)
Ding, Z. Kartsatos, A.: Nonresonance problems for differential inclusions in separable Banach spaces. Proc. AMS, 124, 2357–2365 (1996)
Papageorgiou, N. S., Papalini, F., Renzacci, F.: Existence of solutions and periodic solutions for nonlinear evolution inclusions. Rend. Circolo Mat. di Palermo, 48, 341–364 (1999)
Papageorgiou, N. S., Papalini, F., Yannakakis N.: Nonmonotone, nonlinear evolution inclusions, in Nonlinear Operator Theory, special issue in Math. and Computer Modelling, 32, 1345–1366 (2000)
Papageorgiou, N. S., Yannakakis, N.: Second–order nonlinear–evolution inclusions II: Properties of the solution set, submitted
Hu, S., Papageorgiou, N. S.: Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, The Netherlands, 1997
Hu, S., Papageorgiou, N. S.: Handbook of Multivalued Analysis, Volume II: Applications, Kluwer, Dordrecht, The Netherlands, 2000
Zeidler, E.: Nonlinear Functional Analysis and its Applications II, Springer–Verlag, New York, 1990
Lions, J. L.: Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969
Browder, F. E., Hess, P.: Nonlinear Mappings of Monotone Type in Banach spaces. J. Funct. Anal., 11, 251–294 (1972)
Guan, Z., Kartsatos, A.: Ranges of generalized pseudomonotone perturbations of maximal monotone operators in reflexive Banach spaces. Contemporary Math, 207, 107–123 (1997)
Papageorgiou, N. S.: On the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities. J. Math. Anal. Appl., 205, 434–453 (1997)
Pachpatte, B. G.: A note on the Gronwall–Bellman inequality. J. Math. Anal. Appl., 44, 758–762 (1973)
De Blasi, F. S. Pianigiani, G.: Nonconvex valued differential inclusions in Banach spaces. J. Math. Anal. Appl., 157, 469–494 (1991)
De Blasi, F. S. Pianigiani, G.: On the density of extremal solutions of differerntial inclusions. Annales Polonici Math. LVI, 133–142 (1992)
De Blasi, F. S. Pianigiani, G.: Topological properties of nonconvex differential inclusions. Nonlin. Anal., 23, 669–681 (1994)
Brezis H.: Operateurs Maximaux Monotones, North–Holland, Amsterdam, 1973
Gossez, J. P., Mustonen, V.: Pseudo–monotonicity and the Leray–Lions condition. Diff. and Integral Eqns., 6, 37–45 (1993)
Balder, E.: Necessary and sufficient conditions for L 1 strong–weak lower semicontinuity of integral functionals. Nonlin. Anal–TMA, 11, 1399–1404 (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Papageorgiou, N.S., Yannakakis, N. Second Order Nonlinear Evolution Inclusions I: Existence and Relaxation Results. Acta Math Sinica 21, 977–996 (2005). https://doi.org/10.1007/s10114-004-0508-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-004-0508-y
Keywords
- Evolution triple
- Pseudomonotone and demicontinuous operator
- Coercive operator
- L–pseudomonotonicity
- Upper semicontinuous and lower semicontinuous multifunction
- Solution set
- Integration by parts formula
- Compact embedding
- Extremal solutions
- Strong relaxation
- Hyperbolic control system
- Surjective operator