The aim of this lecture is to present a new compactness method for operator inclusions in general, and for Hammerstein like inclusions, in particular. This method applies to acyclic multivalued maps which satisfy a generalized compactness condition of Monch type.
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Multivalued map; acyclic map; Hammerstein operator; operator inclusion; compactness; fixed point.
R. Precup, Fixed point theorems for acyclic multivalued maps and inclusions of Hammerstein type, Seminar on Fixed Point Theory Cluj-Napoca, 3 (2002), 327-334.
Seminar on Fixed Point Theory Cluj-Napoca, Volume 3, 2002, 327-334
University ”Babeș-Bolyai”, Cluj-Napoca, Romania
MR 1929778, Zbl 1043.47037.
google scholar link
 J.-F. Couchouron and R. Precup, Existence principles for inclusions of Hammerstein type involving noncompact acyclic multivalued maps, to appear.
 S. Eilenberg and D. Montgomery, Fixed point theorems for multivalued transformations, Amer. J. Math. 68 (1946), 214-222.
 D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, Dordrecht, 1996.
 S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer Academic Publishers, Dordrecht-Boston-London, 1997.
 H. Monch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), 985-999.
 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.
 D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Gordon and Breach Science Publishers, 2001.
 D. O’Regan and R. Precup, Integrable solutions of Hammerstein integral inclusions in Banach spaces, Dynam. Contin. Discrete Impuls. Systems, to appear.
 R. Precup, A Monch type generalization of the Eilenberg-Montgomery fixed point theorem, Seminar on Fixed Point Theory Cluj-Napoca 1 (2000), 69-71.
 R. Precup, On the Palais-Smale condition for Hammerstein integral equations, Nonlinear Anal. 47, no 2 (2001), 1233-1244.
 R. Precup, Inequalities and compactness, to appear