Abstract
We apply Monch type fixed point theorems for acyclic multivalued maps to the solvability of inclusions of Hammerstein type in Banach spaces. Our approach makes possible to unify and improve the existence theories for nonlinear evolution problems and abstract integral inclusions of Volterra and Fredholm type.
Authors
Jean-François Couchouron
France
Radu Precup
Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania
Keywords
Fixed point, multivalued map, acyclic set, integral inclusion, Hammerstein equation, evolution equation, boundary value problem.
Paper cordinates
J.-F. Couchouron, R. Precup, Existence principles for inclusions of Hammerstein type involving noncompact acyclic multivalued maps, Electron. J. Differential Equations. 2002 (2002), no.4, 1-21.
About this paper
Journal
Electron. Journal Differential Equations
Publisher Name
Print ISSN
Not available yet.
Online ISSN
Not available yet.
Google Scholar Profile
MR: 1872799, Zbl 0991.47050
References
[1] A.R. Aftabizadeh, S. Aizicovici and N.H. Pavel, On a class of second-order anti-periodic boundary value problems, J. Math. Anal. Appl. 171 (1992), 301-320.
[2] R. Bader, On the semilinear multi-valued flow under constraints and the periodic problem, Comment. Math. Univ. Carolin. 41 (2000), 719-734.
[3] V. Barbu, “Nonlinear Semigroups and Differential Equations in Banach Spaces”, Ed. Academiei & Noordhoff International Publishing, Bucure¸stiLeyden, 1976.
[4] J.-F. Couchouron, Probleme de Cauchy non autonome pour des equations d’evolution, Potential Anal. 13 (2000), 213-248.
[5] J.-F. Couchouron and M. Kamenski, A unified topological point of view for integro-differential inclusions, in “Differential Inclusions and Optimal Control”, Lecture Notes in Nonlinear Analysis, Vol. 2, 1998, 123-137.
[6] J.-F. Couchouron and M. Kamenski, An abstract topological point of view and a general averaging principle in the theory of differential inclusions, Nonlinear Anal. 42 (2000), 1101-1129.
[7] K. Deimling, “Multivalued Differential Equations”, Walter de Gruyter, Berlin-New York, 1992.
[8] J. Diestel, W.M. Ruess and W. Schachermayer, Weak compactness in L1(µ, X), Proc. Amer. Math. Soc. 118 (1993), 447-453.
[9] S. Eilenberg and D. Montgomery, Fixed point theorems for multivalued transformations, Amer. J. Math. 68 (1946), 214-222.
[10] P.M. Fitzpatrick and W.V. Petryshyn, Fixed point theorems for multivalued noncompact acyclic mappings, Pacific J. Math. 54 (1974), 17-23.
[11] M. Frigon, Theoremes d’existence de solutions d’inclusions differentielles, in “Topological Methods in Differential Equations and Inclusions” (A. Granas and M. Frigon eds.), NATO ASI Series C, Vol. 472, Kluwer Academic Publishers, Dordrecht-Boston-London, 1995, 51-87.
[12] L. Gorniewicz, “Homological Methods in Fixed Point Theory of Multivalued Maps”, Dissertationes Math. 129, Polish Scientific Publishers, Warsaw, 1976.
[13] D. Guo, V. Lakshmikantham and X. Liu, “Nonlinear Integral Equations in Abstract Spaces”, Kluwer Academic Publishers, Dordrecht-Boston-London, 1996.
[14] S. Gutman, Existence theorems for nonlinear evolution equations, Nonlinear Anal. 11 (1987), 1193-1206.
[15] A. Haraux, Anti-periodic solutions of some nonlinear evolution equations, Manuscripta Math. 63 (1989), 479-505.
[16] S. Hu and N.S. Papageorgiou, “Handbook of Multivalued Analysis, Vol. I: Theory”, Kluwer Academic Publishers, Dordrecht-Boston-London, 1997.
[17] D. O’Regan and R. Precup, Fixed point theorems for set-valued maps and existence principles for integral inclusions, J. Math. Anal. Appl. 245 (2000), 594-612.
[18] D. O’Regan and R. Precup, Integrable solutions of Hammerstein integral inclusions in Banach spaces, Dynam. Contin. Discrete Impuls. Systems, to appear.
[19] R. Precup, A Monch type generalization of the Eilenberg-Montgomery fixed point theorem, Seminar on Fixed Point Theory Cluj-Napoca 1 (2000), 69-71.
[20] I.I. Vrabie, “Compactness Methods for Nonlinear Evolutions”, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 32, Longman Scientific & Technical, 1987.