On Some Fixed Point Theorems for 1-Set Weakly Contractive Multi-Valued Mappings with Weakly Sequentially Closed Graph

Afif Ben Amar, Aneta Sikorska-Nowak

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DOI: 10.4236/apm.2011.14030                

Abstract

In this paper we prove Leray-Schauder and Furi-Pera types fixed point theorems for a class of multi-valued mappings with weakly sequentially closed graph. Our results improve and extend previous results for weakly sequentially closed maps and are very important in applications, mainly for the investigating of boundary value problems on noncompact intervals.

Keywords

Measures of Weak Noncompactness, Weakly Condensing, Weakly Nonexpansive

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	O. Arino, S. Gautier and J. P. Penot, “A Fixed Point Theorem for Sequentially Continuous Mapping with Application to Or-dinary Differential Equations,” Functional Ekvac, Vol. 27, No. 3, 1984, pp. 273-279.
[2]	D. Averna and S.A. Marano, “Ex-istence of Solutions for Operator Inclusions: a Unified Ap-proach,” Rendiconti del Seminario Matematico della Universit di Padova, Vol. 102, 1999, pp. 285-303.
[3]	A. Ben Amar, A. Jeribi and M. Mnif, “Some Fixed Point Theorems and Applica-tion to Biological Model,” Numerical Functional Analysis and Optimization, Vol. 29, No. 1, 2008, pp. 1-23. HHUUdoi:10.1080/01630560701749482UU
[4]	A. Ben Amar and M. Mnif, “Leray-Schauder Alternatives for Weakly Sequentially Continuous Mappings and Application to Transport Equation,” Mathematical Methods in The Applied Sciences, Vol. 33, No. 1, 2010, pp. 80-90.
[5]	G. Bonanno and S. A. Marano, “Positive Solutions of Elliptic Equations with Discontinuous Nonlineari-ties”, Topological Methods in Nonlinear Analysis, Vol. 8, 1996, pp. 263-273.
[6]	F. S. DeBlasi, “On a Property of the Unit Sphere in Banach Space,” Bulletin Mathématiques de la Société des Sciences Mathématiques de Roumanie, Vol. 21, 1977, pp. 259-262.
[7]	R. E. Edwards, “Functional Analysis, Theory and Applications,” Reinhart and Winston, New York, 1965.
[8]	N. Dunford and J. T. Schwartz, “Linear Operators: Part I” Intersciences, 1958.
[9]	K. Floret, “Weakly Compact Sets,” Lecture Notes in Mathematics.
[10]	M. Furi and P. Pera, “A Continuation Method on Locally Convex Spaces and Ap-plications to Ordinary Differential Equations on Noncompact Intervals,” Annales Polonici Math-Ematici, Vol. 47, 1987, pp. 331-346.
[11]	L. Gorniewicz, “Topological Fixed Point The-ory of Multivalued Mappings,” 2nd edition, Springer, New York, 2006.
[12]	J. Himmelberg, “Fixed Points of Multifunc-tions,” Journal of Mathematical Analysis and Applications, Vol. 38, No. 1, 1972, pp. 205-207. HHUUdoi:10.1016/0022-247X(72)90128-XUU
[13]	I. M. James, “Topological and Uniform Spaces,” Spring- er-Verlag, New York, 1987.
[14]	G. J. O. Jameson, “An Ele-mentary Proof of the Arens and Borusk Extension Theorems,” Journal of the London Mathematical Society, Vol. 14, No. 2, 1976, pp. 364-368. HHUUdoi:10.1112/jlms/s2-14.2.364UU
[15]	D. O'Regan, “A Continua-tion Method for Weakly Condensing Operators,” Zeitschrift fr Analysis und ihre Anwendungen, Vol. 15, 1996, pp. 565-578.
[16]	D. O'Regan, “Fixed-Point Theory for Weakly Sequen-tially Continuous Mapping,” Mathematical and Computer Modelling, Vol. 27, No. 5, 1998, pp. 1-14.
[17]	D. O'Regan, “Fixed Point Theorems for Weakly Sequentially Closed Maps,” Archivum Mathematicum (Brno) Tomus, Vol. 36, 2000, pp. 61-70.
[18]	M. Palmucci and F. Papalini, “Periodic and Boundary Value Problems for Second Order Differential Inclu-sions,” Journal of Applied Mathematics and Stochastic Analy-sis, Vol. 14, No.2, 2001, pp. 161-182. HHUUdoi:10.1155/S1048953301000120UU
[19]	H. H. Schaefer, “Topological Vector Spaces,” Macmillan Company, New York, 1966.
[20]	M. A. Taoudi, “Krasnosel'skii Type Fixed Point Theorems under Weak Topology Features,” Nonlinear Analysis, Vol. 72, No. 1, 2010, pp. 478-482. HHUUdoi:10.1016/j.na.2009.06.086UU
[21]	E. Zeidler, “Nonlinear Functional Analysis and Its Applications,” Vol. I, Springer, New York, 1986.