On a fixed point theorem for weakly sequentially continuous mapping

• Autorzy: Ireneusz Kubiaczyk
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

fixed point; differential equations in abstract spaces

Kategorie tematyczne

• 47H10: Fixed-point theorems
• 34G20: Nonlinear equations

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 1995
• Tom: 15
• Numer: 1
• Strony: 15-20

Daty

wydano

1995

Twórcy

autor

Ireneusz Kubiaczyk

• Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland

Bibliografia

• [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.
• [2] J. M. Ball, Properties of mappings and semigroups, Proc. Royal Soc. Edinburgh Sect. A 72 (1973/74), 275-280.
• [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.
• [4] J Banaś, J. Rivero, On measure of weak noncompactness, Ann. Mat. Pura Appl. 125 (1987), 213-224.
• [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] M. Cichoń, On a fixed point theorem of Sadowskii, (to appear).
• [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.
• [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.
• [9] R. E. Edwards, Functional Analysis, Holt Rinehart and Winston New York 1965.
• [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.
• [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.
• [14] I. Kubiaczyk, On the existence of solutions of differential equations in Banach spaces, Bull. Polon. Acad. Sci. Math. 33 (1985), 607-614.
• [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.