Abstract
We use topological methods to develop an existence theory for nonlinear operator equations of Hammerstein type in Banach spaces. In particular our theory yields existence results to initial and boundary value problems for functional-differential equations in abstract spaces.
Authors
Donal O’Regan
Department of Mathematics National University of Ireland Galway, Ireland
Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Keywords
Abstract Hammerstein equation; Nonlinear integral equation; Integro-differential equation; Evolution equation.
Paper coordinates
D. O’Regan, R. Precup, Existence theory for nonlinear operator equations of Hammerstein type in Banach spaces, Dynamic Systems Appl. 14 (2005), 121-134.
About this paper
Print ISSN
Online ISSN
1056-2176
MR2128316, Zbl 1086.47034. journal website: https://www.acadsol.eu/dsa/
google scholar link
[1] Agarwal, R.P. and O’Regan, D., Existence criteria for operator inclusions in abstract spaces, J. Comput. Appl. Math. 113 (2000), 183-193.
[2] Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei–Noordhoff International Publishing, Bucuresti–Leyden, 1976.
[3] Brezis, H. and Browder, F.E., Existence theorems for nonlinear integral equations of Hammerstein type, Bull. Amer. Math. Soc. 81 (1975), 73- 78.
[4] Couchouron, J-F. and Kamenski, M., An abstract topological point of view and a general averaging principle in the theory of differential inclusions, Nonlinear Anal. 42 (2000), 1101-1129.
[5] Couchouron, J-F. and Precup, R., Existence principles for inclusions of Hammerstein type involving noncompact acyclic multivalued maps, Electron. J. Differential Equations 2002 (2002), No. 04, 1-21.
[6] Deimling, K., Nonlinear Functional Analysis, Springer-Verlag, Berlin– Heidelberg–New York–Tokyo, 1985.
[7] Diestel, J., Ruess, W.M. and Schachermayer, W., Weak compactness in L 1 (µ, X), Proc. Amer. Math. Soc. 118 (1993), 447-453.
[8] Guo, D., Lakshmikantham, V. and Liu, X., Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, Dordrecht–Boston–London, 1996.
[9] Mitidieri, E. and Vrabie, I.I., Existence for nonlinear functional differential equations, Hiroshima Math. J. 17 (1987), 627-649.
[10] Monch, H., Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal.4 (1980), 985-999.
[11] O’Regan, D., Volterra and Urysohn integral equations in Banach spaces, J. Appl. Math. Stochastic Anal. 11 (1998), 449-464.
[12] O’Regan, D. and Meehan, M., Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer Academic Publishers, Dordrecht–Boston–London, 1998.
[13] O’Regan, D. and Precup, R., Existence criteria for integral equations in Banach spaces, J. Inequal. Appl. 6 (2001), 77-97.
[14] O’Regan, D. and Precup, R., Theorems of Leray-Schauder Type and Applications, Gordon and Breach Science Publishers, Amsterdam, 2001.
[15] O’Regan, D. and Precup, R., Integrable solutions of Hammerstein integral inclusions in Banach spaces, Dynamics Cont., Discrete Impulsive Systems 9 (2002), 165-176.
[16] Precup, R., Methods in Nonlinear Integral Equations, Kluwer Academic Publishers, Dordrecht–Boston–London, 2002.