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Fixed Point Theorems for Multi-valued Nonexpansive Mappings in Banach Spaces

  • Zineb Bounegab
  • Smaïl DjebaliEmail author
Article
  • 59 Downloads

Abstract

In this paper, we present new fixed point theorems for multivalued nonexpansive mappings. Since Banach space can have any geometric structure, we consider mappings such that their perturbation by the identity operator is expansive. Then we derive some fixed point results including existence theorems for the sum and product of some classes of nonlinear operators. Three illustrating examples for functional and differential inclusions are supplied.

Keywords

Multivalued nonexpansive map \(\psi \)-expansive map sum of operators product of operators banach algebra 

Mathematics Subject Classification

47H09 47H10 47J25 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for his/her careful reading of the original manuscript which led to substantial improvement of the paper.

References

  1. 1.
    Aubin, J., Cellina, A.: Differential inclusions. Set-valued Maps and Viability Theory. Grundlehren der Mathematischen Wissenschaften, vol. 264. Springer, Berlin (1984)Google Scholar
  2. 2.
    Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977)CrossRefGoogle Scholar
  3. 3.
    Chang, T.H., Yen, C.L.: Some fixed point theorems in Banach space. J. Math. Anal. Appl. 138(2), 550–558 (1989)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Covitz, H., Nadler, S.B.: Multi-valued contraction mappings in generalized metric space. Isr. J. Math. 8, 5–11 (1970)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Deimling, K.: Multivalued Differential Equations, Walter de Gruyter Series in Nonlinear Analysis and Applications, vol. 1. Walter de Gruyter Co., Berlin (1992)Google Scholar
  6. 6.
    Dhage, B.C.: Multi-valued operators and fixed point theorems in Banach algebras. I. Taiwan. J. Math. 10(4), 1025–1045 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dhage, B.C.: Multi-valued mappings and fixed points. II. Tamkang J. Math. 37(1), 27–46 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Djebali, S.: Fixed point theory for \(1\)-set contractions: a survey, applied mathematics in Tunisia. In: Proceedings of the International Conference on Advances in Applied Mathematics (ICAAM), Hammamet, Tunisia, December 2013. Series. Springer Proceedings in Mathematics & Statistics, vol. 131, pp. 53–10. Springer-Birkhäuser (2015)Google Scholar
  9. 9.
    Djebali, S., Hammache, K.: Furi-Pera fixed point theorems in Banach algebras with applications. Acta Univ. Palacki. Olomuc Fac. rer. nat. Math. 47, 55–75 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Djebali, S., Hammache, K.: Fixed point theorems for nonexpansive maps in Banach spaces. Nonlinear Anal. 73(10), 3440–3449 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dominguez Benavides, T., Lorenzo Ramirez, P.: Fixed-point theorems for multivalued non-expansive mappings without uniform convexity. Abstr. Appl. Anal. 6, 375–386 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Garcia-Falset, J.: Existence of fixed points for the sum of two operators. Math. Nachr. 283(12), 1736–1757 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83. Marcel Dekker, New York (1984)Google Scholar
  14. 14.
    Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings, Fixed Point Theory and Applications, vol. 4. Springer, Dordrecht (2006)zbMATHGoogle Scholar
  15. 15.
    Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)CrossRefGoogle Scholar
  16. 16.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. I. Theory, Mathematics and its Applications, vol. 419. Kluwer Academic Publishers, Dordrecht (1997)CrossRefGoogle Scholar
  17. 17.
    Kirk, W.A., Massera, S.: Remarks on asymptotic and Chebyshev centers. Houst. J. Math. 16, 357–364 (1990)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lasota, A., Opial, Z.: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13, 781–786 (1965)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lim, T.C.: A fixed point theorem for multivalued nonexpansive mappings in an uniformly convex Banach space. Bull. Am. Math. Soc. 80(6), 1123–1126 (1974)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nussbaum, R.D.: The fixed point index and asymptotic fixed point theorems for \(k\)-set contractions. Bull. Am. Math. Soc. 75, 490–495 (1969)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Petryshyn, W.V.: Remarks on condensing and \(k\)-set-contractive mappings. J. Math. Anal. Appl. 39, 717–741 (1972)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Petryshyn, W.V., Fitzpatrick, P.M.: A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings. Trans. Am. Math. Soc. 194, 1–25 (1974)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rybinski, L.E.: An application of the continuous selection theorem to the study of the fixed points of multivalued mappings. J. Math. Anal. Appl. 153(2), 391–396 (1990)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Xu, H.K.: Multivalued nonexpansive mappings in Banach spaces. Nonlinear Anal. 43(6), 693–706 (2001)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratoire “Théorie du Point Fixe et Applications” ENSAlgiersAlgeria
  2. 2.Al Imam Mohammad Ibn Saud Islamic University (IMSIU)RiyadhSaudi Arabia

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