In this paper we study the Darboux problem in some class of Banach spaces. The right-hand side of this problem is a Pettis-integrable function satisfying some conditions expressed in terms of measures of weak noncompactness. We prove that the set of all local pseudo-solutions of our problem is nonempty, compact and connected in the space of continuous functions equipped with the weak topology.
@article{119242,
author = {Mieczys\l aw Cicho\'n and Ireneusz Kubiaczyk},
title = {Kneser-type theorem for the Darboux problem in Banach spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {42},
year = {2001},
pages = {267-279},
zbl = {1115.35141},
mrnumber = {1832146},
language = {en},
url = {http://dml.mathdoc.fr/item/119242}
}
Cichoń, Mieczysław; Kubiaczyk, Ireneusz. Kneser-type theorem for the Darboux problem in Banach spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 267-279. http://gdmltest.u-ga.fr/item/119242/
Some remarks on the existence and uniqueness of solutions of the hyperbolic equation, Studia Math. 15 156-160 (1956). (1956) | MR 0079711 | Zbl 0070.09204
Weak continuity properties of mappings and semi-groups, Proc. Royal Soc. Edinbourgh Sect.A 72 275-280 (1979). (1979) | MR 0397495
On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 259-262 (1977). (1977) | MR 0482402
On the structure of the set of solutions of the Darboux problem for hyperbolic equations, Proc. Edinbourgh Math. Soc. Ser.2 29 17-23 (1986). (1986) | MR 0829175
Kneser's theorem for weak solutions of the Darboux problem in Banach spaces, Nonlinear Analysis T.M.A. 20 169-173 (1993). (1993) | MR 1200387 | Zbl 0776.34048
Weak solutions of differential equations in Banach spaces, Disc. Math. Differential Inclusions 15 5-14 (1995). (1995) | MR 1344523
On the set of solutions of the Cauchy problem in Banach spaces, Arch. Math. 63 251-257 (1994). (1994) | MR 1287254
Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces, Annales Polon. Math. 62 13-21 (1995). (1995) | MR 1348215 | Zbl 0836.34062
On bounded solutions of hyperbolic differential inclusion in Banach spaces, Demonstratio Math. 25 153-159 (1992). (1992) | MR 1170678 | Zbl 0780.35120
Solution Sets of Differential Equations in Abstract Spaces, Pitman Research Notes in Mathematics Series 342, Longman, 1996. | MR 1427944 | Zbl 0847.34004
Characterizations of Banach Spaces Not Containing $l^1$, CWI Tract, Amsterdam, 1989. | MR 1002733
Pettis integration, Proc. Amer. Math. Soc. 82 81-86 (1981). (1981) | MR 0603606 | Zbl 0506.28007
On the set of solutions of the Darboux problem for some hyperbolic equations, Bull. Acad. Polon. Sci. Math. 28 279-286 (1980). (1980) | MR 0620202
An application of the topological degree theory to the study of the Darboux problem for hyperbolic equations, J. Math. Anal. Appl. 76 107-115 (1980). (1980) | MR 0586649
Solutions of differential equations in B-spaces, Duke Math. J. 41 437-442 (1974). (1974) | MR 0344624 | Zbl 0288.34063
On a fixed point theorem for weakly sequentially continuous mapping, Disc. Math. Differential Inclusions 15 15-20 (1995). (1995) | MR 1344524
On the Fubini theorem for the Pettis integral for bounded functions, Bull. Polish Sci. Math. 49 (1) (2001), in press. (2001) | MR 1824153 | Zbl 0995.46026
An existence theorem for weak solutions of differential equations in Banach spaces, in Nonlinear Equations in Abstract Spaces, ed. by V. Laksmikantham, 1978, pp.387-404. | MR 0502554 | Zbl 0452.34054
Sul problema di Darboux negli spazi di Banach, Boll. U.M.I. (5) 17-A 201-215 (1956). (1956)
Fixed point theory for weakly sequentially continuous mappings, to appear.
On integration in vector spaces, Trans. Amer. Math. Soc. 44 277-304 (1938). (1938) | MR 1501970 | Zbl 0019.41603
Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Bull. Acad. Polon. Sci. Math. 26 407-413 (1978). (1978)