Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces

Mieczysław Cichoń ; Ireneusz Kubiaczyk

Annales Polonici Mathematici, Tome 62 (1995), p. 13-21 / Harvested from The Polish Digital Mathematics Library

Résumé

We investigate the structure of the set of solutions of the Cauchy problem x’ = f(t,x), x(0) = x₀ in Banach spaces. If f satisfies a compactness condition expressed in terms of measures of weak noncompactness, and f is Pettis-integrable, then the set of pseudo-solutions of this problem is a continuum in $C_{w}(I,E)$, the space of all continuous functions from I to E endowed with the weak topology. Under some additional assumptions these solutions are, in fact, weak solutions or strong Carathéodory solutions, so we also obtain Kneser-type theorems for these classes of solutions.

Détail

Publié le : 1995-01-01

Zbl 0836.34062

EUDML-ID : urn:eudml:doc:262264

Bibliographie

[000] [1] O. Arino, S. Gautier and J. P. Penot, A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkc. Ekvac. 27 (1984), 273-279. | Zbl 0599.34008

[001] [2] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math. 60, Marcel Dekker, New York, 1980. | Zbl 0441.47056

[002] [3] J. Banaś and J. Rivero, On measure of weak noncompactness, Ann. Mat. Pura Appl. 125 (1987), 213-224. | Zbl 0653.47035

[003] [4] M. Cichoń, Weak solutions of differential equations in Banach spaces, Discuss. Math. 15 (1994) (in press).

[004] [5] E. Cramer, V. Lakshmikantham and A. R. Mitchell, On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal. 2 (1978), 169-177. | Zbl 0379.34041

[005] [6] S. J. Daher, On a fixed point principle of Sadovskii, Nonlinear Anal., 643-645. | Zbl 0377.47038

[006] [7] 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. | Zbl 0365.46015

[007] [8] J. Diestel and J. J. Uhl Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.

[008] [9] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Side, Nauka, Moscow, 1985 (in Russian). | Zbl 0571.34001

[009] [10] W. J. Knight, Solutions of differential equations in B-spaces, Duke Math. J. 41 (1974), 437-442. | Zbl 0288.34063

[010] [11] I. Kubiaczyk, Kneser type theorems for ordinary differential equations in Banach spaces, J. Differential Equations 45 (1982), 139-146. | Zbl 0505.34048

[011] [12] I. Kubiaczyk and S. Szufla, Kneser's theorem for weak solutions of ordinary differential equations in Banach spaces, Publ. Inst. Math. (Beograd) 32 (1982), 99-103. | Zbl 0516.34058

[012] [13] A. R. Mitchell and Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, in: Nonlinear Equations in Abstract Spaces, V. Lakshmikantham (ed.), 1978, 387-404.

[013] [14] N. S. Papageorgiou, Kneser's theorems for differential equations in Banach spaces, Bull. Austral. Math. Soc. 33 (1986), 419-434. | Zbl 0579.34046

[014] [15] B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304. | Zbl 0019.41603

[015] [16] A. Szep, Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar. 6 (1971), 197-203. | Zbl 0238.34100

[016] [17] S. Szufla, Some remarks on ordinary differential equations in Banach spaces, Bull. Acad. Polon. Sci. Math. 16 (1968), 795-800. | Zbl 0177.18902

[017] [18] S. Szufla, Kneser's theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Bull. Acad. Polon. Sci. Math. 26 (1978), 407-413. | Zbl 0384.34039