A topological property of the set of fixed points of a multivalued contraction with convex values

Biagio Ricceri

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti (1987)

  • Volume: 81, Issue: 3, page 283-286
  • ISSN: 0392-7881

Abstract

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In this Note we first establish a result on the structure of the set of fixed points of a multi-valued contraction with convex values. As a consequence of this result, we then obtain the following theorem: Let ( U , ∥ ⋅ ∥ U ) , ( V , ∥ ⋅ ∥ V ) be two real Banach spaces and let Φ be a continuous linear operator from U onto V . Put: α = sup ⁡ { inf ⁡ { ∥ u ∥ U : u ∈ Φ - 1 ⁢ ( v ) } : v ∈ V , ∥ v ∥ V ≤ 1 } . Then, for every v ∈ V and every lipschitzian operator Ψ : U → V , with Lipschitz constant L such that α ⁢ L < 1 , the set { u ∈ U : Φ ⁢ ( u ) + Ψ ⁢ ( u ) = v } is non-empty and arc wise connected.

How to cite

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Ricceri, Biagio. "Une propriété topologique de l'ensemble des points fixes d'une contraction multivoque à valeurs convexes." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 81.3 (1987): 283-286. <http://eudml.org/doc/289321>.

@article{Ricceri1987,
author = {Ricceri, Biagio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
keywords = {Multi-valued contraction; Fixed point; Absolute extensor},
language = {fre},
month = {9},
number = {3},
pages = {283-286},
publisher = {Accademia Nazionale dei Lincei},
title = {Une propriété topologique de l'ensemble des points fixes d'une contraction multivoque à valeurs convexes},
url = {http://eudml.org/doc/289321},
volume = {81},
year = {1987},
}

TY - JOUR
AU - Ricceri, Biagio
TI - Une propriété topologique de l'ensemble des points fixes d'une contraction multivoque à valeurs convexes
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1987/9//
PB - Accademia Nazionale dei Lincei
VL - 81
IS - 3
SP - 283
EP - 286
LA - fre
KW - Multi-valued contraction; Fixed point; Absolute extensor
UR - http://eudml.org/doc/289321
ER -

References

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  1. MICHAEL, E. (1953) - «Pacific J. Math.», 3, 789-806. Zbl0052.11502MR59541
  2. SMITHSON, R.E. (1972) - «Nieuw Arch. Wisk.», 20, 32-53. Zbl0236.54013MR305338
  3. MICHAEL, E. (1956) - «Ann. of Math.», 63, 361-382. Zbl0071.15902MR77107
  4. AUBIN, J.-P. et CELLINA, A. (1984) - «Differential inclusions», Springer-Verlag. MR755330DOI10.1007/978-3-642-69512-4
  5. COVITZ, H. et NADLER, S.B. (1970) - «Israel J. Math.», 8, 5-11. Zbl0192.59802MR263062
  6. ROBINSON, S.M. (1972) - «Trans. Amer. Math. Soc.», 174, 127-140. Zbl0264.47018MR313769
  7. SAINT RAYMOND, J. (1984) - «C.R. Acad. Sc. Paris», 298, série I, 71-74. Zbl0561.54042MR740940

Citations in EuDML Documents

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  1. Alessandro Margheri, Pietro Zecca, Solution sets of multivalued Sturm-Liouville problems in Banach spaces
  2. Paolo Cubiotti, Beatrice Di Bella, A generalization of the Schauder fixed point theorem via multivalued contractions
  3. Salvatore A. Marano, Fixed points of multivalued contractions with nonclosed, nonconvex values
  4. Grzegor Gabor, Some results on existence and structure of solution sets to differential inclusions on the halfline
  5. Evgenios P. Avgerinos, Nikolaos S. Papageorgiou, Topological properties of the solution set of integrodifferential inclusions
  6. Lech Górniewicz, Topological structure of solution sets: current results

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