On a fixed point theorem for weakly sequentially continuous mapping

Ireneusz Kubiaczyk

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (1995)

  • Volume: 15, Issue: 1, page 15-20
  • ISSN: 1509-9407

Abstract

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Let E be a metrizable locally convex topological vector space x ∈ E, and let D be a closed convex subset of E such that x ∈ D. In this paper we prove that the weakly sequentially continuous mapping F: D ∪ D which satisfies V̅ = c̅o̅n̅v̅({x} ∪ F(V))⇒ V is relatively weakly compact, has a fixed point. Employing the above results we prove the existence theorem for the Cauchy problem x'(t) = f(t,x(t)), x(0) = x₀. As compared with the previous results of this type, in this theorem the continuity hypothesis on f is essentially weakened. Our results generalize those of [1,7,15,17].

How to cite

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Ireneusz Kubiaczyk. "On a fixed point theorem for weakly sequentially continuous mapping." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 15.1 (1995): 15-20. <http://eudml.org/doc/276001>.

@article{IreneuszKubiaczyk1995,
abstract = { Let E be a metrizable locally convex topological vector space x ∈ E, and let D be a closed convex subset of E such that x ∈ D. In this paper we prove that the weakly sequentially continuous mapping F: D ∪ D which satisfies V̅ = c̅o̅n̅v̅(\{x\} ∪ F(V))⇒ V is relatively weakly compact, has a fixed point. Employing the above results we prove the existence theorem for the Cauchy problem x'(t) = f(t,x(t)), x(0) = x₀. As compared with the previous results of this type, in this theorem the continuity hypothesis on f is essentially weakened. Our results generalize those of [1,7,15,17]. },
author = {Ireneusz Kubiaczyk},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {fixed point; differential equations in abstract spaces; weakly sequentially continuous mapping; Cauchy problem},
language = {eng},
number = {1},
pages = {15-20},
title = {On a fixed point theorem for weakly sequentially continuous mapping},
url = {http://eudml.org/doc/276001},
volume = {15},
year = {1995},
}

TY - JOUR
AU - Ireneusz Kubiaczyk
TI - On a fixed point theorem for weakly sequentially continuous mapping
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 1995
VL - 15
IS - 1
SP - 15
EP - 20
AB - Let E be a metrizable locally convex topological vector space x ∈ E, and let D be a closed convex subset of E such that x ∈ D. In this paper we prove that the weakly sequentially continuous mapping F: D ∪ D which satisfies V̅ = c̅o̅n̅v̅({x} ∪ F(V))⇒ V is relatively weakly compact, has a fixed point. Employing the above results we prove the existence theorem for the Cauchy problem x'(t) = f(t,x(t)), x(0) = x₀. As compared with the previous results of this type, in this theorem the continuity hypothesis on f is essentially weakened. Our results generalize those of [1,7,15,17].
LA - eng
KW - fixed point; differential equations in abstract spaces; weakly sequentially continuous mapping; Cauchy problem
UR - http://eudml.org/doc/276001
ER -

References

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  1. [1] O. Arino, S. Gautier, J. P. Penot, A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Func. Ekvac. 27 (1984), 273-279. Zbl0599.34008
  2. [2] J. M. Ball, Properties of mappings and semigroups, Proc. Royal Soc. Edinburgh Sect. A 72 (1973/74), 275-280. 
  3. [3] J. Banaś, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics Vol. 60, Marcel Dekker, New York-Basel 1980. Zbl0441.47056
  4. [4] J Banaś, J. Rivero, On measure of weak noncompactness, Ann. Mat. Pura Appl. 125 (1987), 213-224. Zbl0653.47035
  5. [5] M. Cichoń, Application of Measure of Noncompactness in the Theory of Differential Inclusions in Banach Spaces, Ph. D. Thesis Poznań 1992 (in Polish). 
  6. [6] M. Cichoń, On a fixed point theorem of Sadowskii, (to appear). 
  7. [7] E. Cramer, V. Lakshmikantham, A. R. Mitchell, On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlin. Ann. TMA 2 (1978), 169-177. Zbl0379.34041
  8. [8] F. S. De Blasi, On a property of the unit sphere in a Banach space Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259-262. Zbl0365.46015
  9. [9] R. E. Edwards, Functional Analysis, Holt Rinehart and Winston New York 1965. Zbl0182.16101
  10. [10] G. Emanuelle, Measure of weak noncompactness and fixed points theorems Bull. Math. Soc. Sci. R. S. Roumanie 25 (1981), 353-358. 
  11. [11] W. J. Knight, Solutions of differential equations in B-spaces, Duke Math. J. 41 (1974), 437-442. Zbl0288.34063
  12. [12] M. A. Krasnoselski, B. N. Sadovskii (ed), Measures of Noncompactness and Condensing Operators, Novosibirsk 1986 (in Russian). 
  13. [13] I. Kubiaczyk, Kneser type theorems for ordinary differential equations in Banach spaces, J. Differential Equations 45 (1982), 139-146. Zbl0505.34048
  14. [14] I. Kubiaczyk, On the existence of solutions of differential equations in Banach spaces, Bull. Polon. Acad. Sci. Math. 33 (1985), 607-614. Zbl0607.34055
  15. [15] A. R. Mitchell, Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, 387-404 in: Nonlinear Equations in Banach Spaces, ed. V. Lakshmikantham 1978. 
  16. [16] B. N. Sadovskii A fixed point principle, Functional Analysis and its Applications 1 (1967), 151-153 (in Russian). 
  17. [17] A. Szep Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar. 6 (1971), 197-203. 

Citations in EuDML Documents

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  1. Józef Banaś, Afif Ben Amar, Measures of noncompactness in locally convex spaces and fixed point theory for the sum of two operators on unbounded convex sets
  2. Anna Kisiołek, Asymptotic behaviour of solutions of difference equations in Banach spaces
  3. Mieczysław Cichoń, Ireneusz Kubiaczyk, Kneser-type theorem for the Darboux problem in Banach spaces
  4. Bianca Satco, Volterra integral inclusions via Henstock-Kurzweil-Pettis integral
  5. Mieczysław Cichoń, Ireneusz Kubiaczyk, Sikorska-Nowak, Aneta Sikorska-Nowak, Aneta, The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem
  6. A. Sikorska-Nowak, Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals

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