Opuscula Math. 35, no. 4 (2015), 427-443

Opuscula Mathematica

More on the behaviors of fixed points sets of multifunctions and applications

Boualem Alleche
Khadra Nachi

Abstract. In this paper, we study the behaviors of fixed points sets of non necessarily pseudo-contractive multifunctions. Rather than comparing the images of the involved multifunctions, we make use of some conditions on the fixed points sets to establish general results on their stability and continuous dependence. We illustrate our results by applications to differential inclusions and give stability results of fixed points sets of non necessarily pseudo-contractive multifunctions with respect to the bounded proximal convergence.

Keywords: multifunction, fixed point, Pompeiu-Hausdorff metric, bounded proximal convergence, differential inclusion.

Mathematics Subject Classification: 26E25, 34A12, 34A60, 54C60.

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  1. R.P. Agarwal, D. O'Regan, D.R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, 2009.
  2. B. Alleche, On hemicontinuity of bifunctions for solving equilibrium problems, Adv. Nonlinear Anal. 3 (2014) 2, 69-80.
  3. B. Alleche, V.D. Rădulescu, Equilibrium problem techniques in the qualitative analysis of quasi-hemivariational inequalities, Optimization (2014), DOI 10.1080/02331934.2014.917307. http://dx.doi.org/10.1080/02331934.2014.917307.
  4. H. Attouch, J.-P. Penot, H. Riahi, The continuation method and variational convergence, Fixed Point Theory and Applications, [in:] M. Théra, J.-B. Baillon (eds.), vol. 252, Pitman Research Notes in Mathematics, Longman, London, 1991, 9-32.
  5. H. Attouch, R. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), 695-730.
  6. J.-P. Aubin, A. Cellina, Differential Inclusion, Set-Valued Map and Viability Theory, Springer-Verlag, 1984.
  7. J.-P. Aubin, H. Frankowska, On inverse function theorems for set-valued maps, J. Maths. Pures Appl. (1987), 71-89.
  8. D. Azé, J.-P. Penot, On the dependence of fixed point sets of pseudo-contractive multifunctions. Application to differential inclusions, Nonlinear Dyn. Syst. Theory 6 (2006) 1, 31-47.
  9. L. Barbet, K. Nachi, Sequences of contractions and convergence of fixed points, Monogr. Semin. Mat. Garcia Galdeano 33 (2006), 51-58.
  10. G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, 1993.
  11. G. Beer, S. Levi, Gap, excess and bornological convergence, Set-Valued Var. Anal. 16 (2008), 489-506.
  12. G. Beer, R. Lucchetti, Weak topologies for the closed subsets of a metrizable space, Trans. Amer. Math. Soc. 335 (1993), 805-822.
  13. G. Beer, R. Lucchetti, Well-posed optimization problems and a new topology for the closed subsets of a metric space, Rocky Mountain J. Math. 23 (1993), 1197-1220.
  14. S. Benahmed, On differential inclusions with unbounded right-hand side, Serdica Math. J. 37 (2011) 1, 1-8.
  15. S. Benahmed, D. Azé, On fixed points of generalized set-valued contractions,Bull. Aust. Math. Soc. 81 (2010) 1, 16-22.
  16. V. Berinde, M. Pacurar, The role of the Pompeiu-Hausdorff metric in fixed point theory, Creat. Math. Inform. 22 (2013) 2, 35-42.
  17. M. Bianchi, G. Kassay, R. Pini, An inverse map result and some applications to sensitivity of generalized equations, J. Math. Anal. Appl. 399 (2013), 279-290.
  18. M. Bianchi, G. Kassay, R. Pini, Stability results of variational systems under openness with respect to fixed sets, J. Optim. Theory. Appl. (2014), DOI 10.1007/s10957-014-0560-4. http://dx.doi.org/10.1007/s10957-014-0560-4.
  19. K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, New York, 1992.
  20. A.L. Dontchev, W.W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994) 2, 481-489.
  21. A.F. Filippov, Classical solutions of differential equations with multivalued right-hand side, SIAM J. Control 5 (1967), 609-621.
  22. M.H. Geoffroy, G. Pascaline, Generalized differentiation and fixed points sets behaviors with respect to Fisher convergence, J. Math. Anal. Appl. 387 (2012), 464-474.
  23. A. Granas, J. Dugundji, Fixed Point Theory, Springer, 2003.
  24. L. Holá, D. Holý, A weakening of the Attouch-Wets topology on function spaces, Tatra Mount. Math. Publ. 2 (1993), 105-121.
  25. S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis, vol. I, Theory, Kluwer Academics Publishers, 1997.
  26. A.D. Ioffe, Existence and relaxation theorems for unbounded differential inclusions, J. Convex Anal. 13 (2006), 353-362.
  27. T.-C. Lim, On fixed point stability for set-valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985), 436-441.
  28. R. Lucchetti, Convexity and Well-Posed Problems, Springer, 2006.
  29. J.T. Markin, Continuous dependence of fixed point sets, Proc. Amer. Math. Soc. 38 (1973) 3, 436-441.
  30. K. Nachi, J.-P. Penot, Inversion of multifunctions and differential inclusions, Control Cybernet. 34 (2005) 3, 871-901.
  31. D. Pai, P. Shunmugaraj, On stability of approximate solutions of minimization problems, Indian Inst. Thec., Bombay, India, 1990.
  32. N.S. Papageorgiou, Convergence theorems for fixed points of multifunctions and solutions of differential inclusions in Banach spaces, Glas. Mat. Ser. III 23 (1988) 2, 247-257.
  33. J.-P. Penot, The cosmic Hausdorff topology, the bounded Hausdorff topology, and continuity of polarity, Proc. Amer. Math. Soc. 113 (1991), 275-286.
  34. J.-P. Penot, C. Zalinescu, Bounded (Hausdorff) convergence: basic facts and applications, Proc. Amer. Math. Soc. (2005).
  35. R.T. Rockafellar, R.J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, Heidelberg, 2009.
  36. Y. Sonntag, C. Zalinescu, Set convergence. An attempt of classification, Trans. Amer. Math. Soc. 340 (1993) 1, 199-226.
  37. Q.J. Zhu, On the solution set of differential inclusions in Banach spaces, J. Differential Equations 93 (1991) 1, 213-237.
  • Boualem Alleche
  • Laboratoire de Mécanique, Physique et Modélisation Mathématique, Université de Médéa, Algeria
  • Khadra Nachi
  • Laboratoire de Mathématiques et ses Applications, Université d'Oran, Algeria
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2014-04-20.
  • Revised: 2014-10-25.
  • Accepted: 2014-10-28.
  • Published online: 2015-02-06.
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Cite this article as:
Boualem Alleche, Khadra Nachi, More on the behaviors of fixed points sets of multifunctions and applications, Opuscula Math. 35, no. 4 (2015), 427-443, http://dx.doi.org/10.7494/OpMath.2015.35.4.427

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