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 Lincei. Matematica e Applicazioni (1987)

  • Volume: 81, Issue: 3, page 283-286
  • ISSN: 1120-6330

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 Lincei. Matematica e Applicazioni 81.3 (1987): 283-286. <http://eudml.org/doc/287233>.

@article{Ricceri1987,
author = {Ricceri, Biagio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Multi-valued contraction; Fixed point; Absolute extensor; absolute extensor; structure of the set of fixed points of a multi-valued contraction with convex values; lipschitzian operator; Lipschitz constant},
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/287233},
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 Lincei. Matematica e Applicazioni
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; absolute extensor; structure of the set of fixed points of a multi-valued contraction with convex values; lipschitzian operator; Lipschitz constant
UR - http://eudml.org/doc/287233
ER -

References

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  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

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