# On a fixed point theorem for weakly sequentially continuous mapping

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

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

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topIreneusz 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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- [9] R. E. Edwards, Functional Analysis, Holt Rinehart and Winston New York 1965. Zbl0182.16101
- [10] G. Emanuelle, Measure of weak noncompactness and fixed points theorems Bull. Math. Soc. Sci. R. S. Roumanie 25 (1981), 353-358.
- [11] W. J. Knight, Solutions of differential equations in B-spaces, Duke Math. J. 41 (1974), 437-442. Zbl0288.34063
- [12] M. A. Krasnoselski, B. N. Sadovskii (ed), Measures of Noncompactness and Condensing Operators, Novosibirsk 1986 (in Russian).
- [13] I. Kubiaczyk, Kneser type theorems for ordinary differential equations in Banach spaces, J. Differential Equations 45 (1982), 139-146. Zbl0505.34048
- [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] 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] B. N. Sadovskii A fixed point principle, Functional Analysis and its Applications 1 (1967), 151-153 (in Russian).
- [17] A. Szep Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar. 6 (1971), 197-203.

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