In this paper, some algebraic fixed point theorems for multi-valued discontinuous operators on ordered spaces are proved. These theorems improve the earlier fixed point theorems of Dhage (1988, 1991) Dhage and Regan (2002) and Heikkilä and Hu (1993) under weaker conditions. The main fixed point theorems are applied to the first order discontinuous differential inclusions for proving the existence of the solutions under certain monotonicity condition of multi-functions.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1063,
author = {Bupurao C. Dhage},
title = {Some algebraic fixed point theorems for multi-valued mappings with applications},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {26},
year = {2006},
pages = {5-55},
zbl = {1228.47049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1063}
}
Bupurao C. Dhage. Some algebraic fixed point theorems for multi-valued mappings with applications. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 26 (2006) pp. 5-55. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1063/
[000] [1] R.P. Agarwal, B.C. Dhage and D. O'Regan, The method of upper and lower solution for differential inclusions via a lattice fixed point theorem, Dynamic Systems & Appl. 12 (2003), 1-7.
[001] [2] H. Amann, Order Structures and Fixed Points, ATTI 2⁰ Sem Anal. Funz. Appl. Italy 1977.
[002] [3] J. Appell, H.T. Nguyen and P. Zabreiko, Multivalued superposition operators in ideal spaces of vector functions III, Indag. Math. 3 (1992), 1-9. | Zbl 0770.47029
[003] [4] J. Aubin and A. Cellina, Differential Inclusions, Springer Verlag, New York 1984.
[004] [5] M. Benchohra, Upper and lower solutions method for second order differential inclusions, Dynam. Systems Appl. 11 (2002), 13-20. | Zbl 1023.34008
[005] [6] H. Covitz and S.B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5-11. | Zbl 0192.59802
[006] [7] K. Deimling, Multi-valued Differential Equations, De Gruyter, Berlin 1998.
[007] [8] B.C. Dhage, On some variants of Schauder's fixed point principle and applications to nonlinear integral equations, J. Math. Phy. Sci. 25 (1988), 603-611. | Zbl 0673.47043
[008] [9] B.C. Dhage, A lattice fixed point theorem for multi-valued mappings and applications, Chinese Journal of Math. 19 (1991), 11-22.
[009] [10] B.C. Dhage, Some fixed point theorems in ordered Banach spaces and applications, The Math. Student 63 (1-4) (1992), 81-88. | Zbl 0882.47031
[010] [11] B.C. Dhage, Fixed point theorems in ordered Banach algebras and applications, PanAmer. Math. J. 9 (4) (1999), 93-102. | Zbl 0964.47026
[011] [12] B.C. Dhage, A functional integral inclusion involving Carathéodories, Electronic J. Qualitative Theory Diff. Equ. 14 (2003), 1-18. | Zbl 1056.47040
[012] [13] B.C. Dhage, A functional integral inclusion involving discontinuities, Fixed Point Theory 5 (2004), 53-64. | Zbl 1057.47063
[013] [14] B.C. Dhage, Multi-valued operators and fixed point theorems in Banach algebras I, Taiwanese J. Math. 10 (2006), 1025-1045. | Zbl 1144.47321
[014] [15] B.C. Dhage, A fixed point theorem in Banach algebras involving three operators with applications, Kyungpook Math. J. 44 (2004), 145-155. | Zbl 1057.47062
[015] [16] B.C. Dhage, On existence of extremal solutions of nonlinear functional integral equations in Banach algebras, J. Appl. Math. Stoch. Anal. 2004 (2004), 271-282. | Zbl 1088.47051
[016] [17] B.C. Dhage, A fixed point theorem for multi-valued mappings in ordered Banach spaces with applications I, Nonlinear Anal. Forum 10 (2005), 105-126.
[017] [18] B.C. Dhage and D. O'Regan, A lattice fixed point theorem and multi-valued differential equations, Functional Diff. Equations 9 (2002), 109-115. | Zbl 1095.34510
[018] [19] B.C. Dhage and S.M. Kang, Upper and lower solutions method for first order discontinuous differential inclusions, Math. Sci. Res. J. 6 (11) (2002), 527-533. | Zbl 1028.34012
[019] [20] Dajun Guo and V. Lakshmikanthm, Nonlinear Problems in Abstract Cones, Academic Press, New York-London 1988.
[020] [21] J. Dugundji and A. Granas, Fixed Point Theory, Springer Verlag 2003.
[021] [22] N. Halidias and N. Papageorgiou, Second order multi-valued boundary value problems, Arch. Math. (Brno) 34 (1998), 267-284. | Zbl 0915.34021
[022] [23] S. Heikkilä, On chain method used in fixed point theory, Nonlinear Studies 6 (2) (1999), 171-180. | Zbl 0954.47044
[023] [24] S. Heikkilä and S. Hu, On fixed points of multi-functions in ordered spaces, Appl. Anal. 53 (1993), 115-127. | Zbl 0839.54033
[024] [25] S. Heikkilä and V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinues Differential Equations, Marcel Dekker Inc., New York 1994. | Zbl 0804.34001
[025] [26] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer Academic Publishers Dordrechet-Boston-London 1997. | Zbl 0887.47001
[026] [27] M. Kisielewicz, Differential inclusions and optimal control, Kluwer Acad. Publ., Dordrecht 1991.
[027] [28] M.A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press 1964.
[028] [29] A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phy. 13 (1965), 781-786. | Zbl 0151.10703
[029] [30] A. Tarski, A lattice theoretical fixed-point theorem and its applications, Pacific J. Math. 5 (1955), 285-309. | Zbl 0064.26004
[030] [31] D.M. Wagner, Survey of measurable selection theorems, SIAM Jour. Control & Optim. 15 (1975), 859-903. | Zbl 0407.28006
[031] [32] E. Zeidler, Nonlinear Functional Analysis and Its Applications:~Part I, Springer Verlag 1985. | Zbl 0583.47051