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Weak solutions of differential equations in Banach spaces

Published online by Cambridge University Press:  17 April 2009

Nikolaos S. Papageorgiou
Nikolaos S. Papageorgiou
Affiliation:
University of Illinois, Department of Mathematics, 1409 W. Green St., Urbana, Illinois 61801, U.S.A.
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Abstract

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We consider the Cauchy problem x (t) = f (t,x (t)), x (0) = x0 in a nonreflexive Banach space X and for f: T × X → X a weakly continuous vector field. Using a compactness hypothesis involving a weak measure of noncompactness we prove an existence result that generalizes earlier theorems by Chow-Shur, Kato and Cramer-Lakshmikantham-Mitchell.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 33 , Issue 3 , June 1986 , pp. 407 - 418
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Ambrosette, A., Unteorema di esistenza per le equazioni differenziali nagli spazi di Banach, Rend, Sem. Univ. Padova, 39 (1967), 349–360.Google Scholar
[2]Banas, J., Goebel, K., Measures of Noncompactness in Banach Spaces. (Marcel Dekker, Inc., New York 1980).Google Scholar
[3]Boudourides, M., An existence theorem for ordinary differential equations in Banach spaces, Bull. Austral. Math. Soc. 22 (1980), 457–463.CrossRefGoogle Scholar
[4]Chow, S. N., Shur, J., An existence theorem for ordinary differential equations in Banach spaces, Bull. Amer. Math. Soc. 77 (1971), 1018–1020.CrossRefGoogle Scholar
[5]Cramer, E., Lakshmikantham, V., Mitchell, A., On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal. 2 (1976), 169–177.CrossRefGoogle Scholar
[6]DeBlasi, F., On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 21 (1977), 259–262.Google Scholar
[7]Deimling, K., Ordinary Differential Equations in Banach Spaces. Lecture Notes in Mathematice, Vol. 596, Springer, Berlin (1977).CrossRefGoogle Scholar
[8]Dinculeanu, N., Vector Measures, Pergamon Press, London (1966).Google Scholar
[9]Kato, S., On existence and uniqueness conditions for nonlinear ordinary differential equations in Banach spaces, Funkcial. Ekvac. 19 (1976), 239–245.Google Scholar
[10]Pianigiani, G.,Existence of solutions for ordinary differential equations in Banach spaces, Bull, Acad. Polan. Sci. Ser. Sci. Math. Astronom. Phys. 23 (1975), 853–857.Google Scholar
[11]Szep, A., Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar. 6 (1971), 197–203.Google Scholar
[12]Szufla, S., On the existence of solutions of differential equations in Banach spaces, Bull. Acad. Polan. Sci. Ser. Sci. Math. 30 (1982), 507–514.Google Scholar