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Keywords:
ordered Banach space; hybrid fixed point theorem; neutral functional differential inclusion and existence theorem
ordered Banach space; hybrid fixed point theorem; neutral functional differential inclusion and existence theorem
Summary:
In this paper, some hybrid fixed point theorems for the right monotone increasing multi-valued mappings in ordered Banach spaces are proved via measure of noncompactness and they are further applied to the neutral functional nonconvex differential inclusions involving discontinuous multi-functions for proving the existence results under mixed Lipschitz, compactness and right monotonicity conditions. Our results improve the multi-valued hybrid fixed point theorems of Dhage (Dhage, B. C., A fixed point theorem for multivalued mappings on ordered Banach spaces with applications I, Nonlinear Anal. Forum 10 (2005), 105–126.) under weaker convexity conditions.
References:
[1] Agarwal R. P., Dhage B. C., O’Regan D.: The method of upper and lower solution for differential inclusions via a lattice fixed point theorem. Dynamic Systems Appl. 12 (2003), 1–7. MR 1989018
[2] Akhmerov P. P., Kamenskii M. I., Potapov A. S., Sadovskii B. N.: Measures of Noncompactness and Condensing Operators. Birkhäuser 1992. MR 1153247
[3] Andres J., Górniewicz L.: Topological Fixed Point Principles for Boundary Value Problems. Kluwer, 2003. MR 1998968
[4] Banas J., Lecko M.: Fixed points of the product of operators in Banach algebras. PanAmer. Math. J. 12 (2002), 101–109. MR 1895774
[5] Covitz H., Nadler S. B., Jr.: Multivalued contraction mappings in generalized metric spaces. Israel J. Math. 8 (1970), 5–11. MR 0263062
[6] Deimling K.: Multi-valued Differential Equations. De Gruyter, Berlin 1998.
[7] Dhage B. C.: Multi-valued operators and fixed point theorems in Banach algebras I. Taiwanese J. Math. 10 (4), (2006), 1025–1045. MR 2229639 | Zbl 1144.47321
[8] Dhage B. C.: Multi-valued mappings and fixed points I. Nonlinear Funct. Anal. Appl. 10 (3), (2005), 359–378. MR 2194603 | Zbl 1100.47040
[9] Dhage B. C.: Hybrid fixed point theory for strictly monotone increasing multi-valued mappings with applications. Comput. Math. Appl. 53 (2007), 803–824. MR 2327635 | Zbl 1144.47041
[10] Dhage B. C.: A fixed point theorem for multivalued mappings on ordered Banach spaces with applications I. Nonlinear Anal. Forum 10 (2005), 105–126. MR 2162344
[11] Dhage B. C.: A general multi-valued hybrid fixed point theorem and perturbed differential inclusions. Nonlinear Anal. 64 (2006), 2747–2772. MR 2218544 | Zbl 1100.47045
[12] Dhage B. C.: Some algebraic fixed point theorems for multi-valued operators with applications. DISC. Math. Differential inclusions, Control & Optimization 26 (2006), 5–55. MR 2330779
[13] Dhage B. C., Ntouyas S. K.: Existence results for neutral functional differential inclusions. Fixed Point Theory 5 (2005), 235–248. MR 2117335 | Zbl 1080.34063
[14] Dajun Guo, Lakshmikanthm V.: Nonlinear Problems in Abstract Cones. Academic Press, New York–London, 1988.
[15] Hale J. K.: Theory of Functional Differential Equations. Springer, New York 1977. MR 0508721 | Zbl 0352.34001
[16] Heikkilä S., Lakshmikantham V.: Monotone Iterative Technique for Nonlinear Discontinues Differential Equations. Marcel Dekker Inc., New York, 1994. MR 1280028
[17] Heikkilä S., Hu S.: On fixed points of multi-functions in ordered spaces. Appl. Anal. 53 (1993), 115–127.
[18] Hu S., Papageorgiou N. S.,: Handbook of Multivalued Analysis, Vol. I: Theory. Kluwer Academic Publishers, Dordrechet–Boston–London 1997. MR 1485775
[19] Lasota A., Opial Z.: An application of the Kakutani- Ky Fan theorem in the theory of ordinary differential equations. Bull. Polish Acad. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781–786. MR 0196178 | Zbl 0151.10703
[20] Ntouyas S. K.: Initial and boundary value problems for functional differential equations via topological transversality method : A Survey. Bull. Greek Math. Soc. 40 (1998), 3–41. MR 1671786
[21] Petruşel A.: Operatorial Inclusions. House of the Book of Science, Cluj-Napoca, 2002. MR 1939244 | Zbl 1057.47004
[22] Zeidler E.: Nonlinear Functional Analysis and Its Applications: Part I. Springer Verlag, 1985. MR 0768749
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