| Title:
|
Kneser-type theorem for the Darboux problem in Banach spaces (English) |
| Author:
|
Cichoń, Mieczysław |
| Author:
|
Kubiaczyk, Ireneusz |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
2 |
| Year:
|
2001 |
| Pages:
|
267-279 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Pettis integral |
| Keyword:
|
Fubini theorem |
| Keyword:
|
Darboux problem |
| Keyword:
|
measure of weak noncompactness |
| MSC:
|
35L90 |
| MSC:
|
35R20 |
| MSC:
|
46G10 |
| idZBL:
|
Zbl 1115.35141 |
| idMR:
|
MR1832146 |
| . |
| Date available:
|
2009-01-08T19:09:47Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119242 |
| . |
| Reference:
|
[1] Alexiewicz A., Orlicz W.: Some remarks on the existence and uniqueness of solutions of the hyperbolic equation.Studia Math. 15 156-160 (1956). Zbl 0070.09204, MR 0079711 |
| Reference:
|
[2] Ball J.M.: Weak continuity properties of mappings and semi-groups.Proc. Royal Soc. Edinbourgh Sect.A 72 275-280 (1979). MR 0397495 |
| Reference:
|
[3] DeBlasi F.: On a property of the unit sphere in a Banach space.Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 259-262 (1977). MR 0482402 |
| Reference:
|
[4] DeBlasi F., Myjak J.: 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). MR 0829175 |
| Reference:
|
[5] Bugajewski D., Szufla S.: Kneser's theorem for weak solutions of the Darboux problem in Banach spaces.Nonlinear Analysis T.M.A. 20 169-173 (1993). Zbl 0776.34048, MR 1200387 |
| Reference:
|
[6] Cichoń M.: Weak solutions of differential equations in Banach spaces.Disc. Math. Differential Inclusions 15 5-14 (1995). MR 1344523 |
| Reference:
|
[7] Cichoń M., Kubiaczyk I.: On the set of solutions of the Cauchy problem in Banach spaces.Arch. Math. 63 251-257 (1994). MR 1287254 |
| Reference:
|
[8] Cichoń M., Kubiaczyk I.: Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces.Annales Polon. Math. 62 13-21 (1995). Zbl 0836.34062, MR 1348215 |
| Reference:
|
[9] Dawidowski M., Kubiaczyk I.: On bounded solutions of hyperbolic differential inclusion in Banach spaces.Demonstratio Math. 25 153-159 (1992). Zbl 0780.35120, MR 1170678 |
| Reference:
|
[10] Dragoni R., Macki J.W., Nistri P., Zecca P.: Solution Sets of Differential Equations in Abstract Spaces.Pitman Research Notes in Mathematics Series 342, Longman, 1996. Zbl 0847.34004, MR 1427944 |
| Reference:
|
[11] van Dulst D.: Characterizations of Banach Spaces Not Containing $l^1$.CWI Tract, Amsterdam, 1989. MR 1002733 |
| Reference:
|
[12] Geitz R.F.: Pettis integration.Proc. Amer. Math. Soc. 82 81-86 (1981). Zbl 0506.28007, MR 0603606 |
| Reference:
|
[13] Górniewicz L., Pruszko T.: On the set of solutions of the Darboux problem for some hyperbolic equations.Bull. Acad. Polon. Sci. Math. 28 279-286 (1980). MR 0620202 |
| Reference:
|
[14] Górniewicz L., Bryszewski J., Pruszko T.: 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). MR 0586649 |
| Reference:
|
[15] Knight W.J.: Solutions of differential equations in B-spaces.Duke Math. J. 41 437-442 (1974). Zbl 0288.34063, MR 0344624 |
| Reference:
|
[16] Kubiaczyk I.: On a fixed point theorem for weakly sequentially continuous mapping.Disc. Math. Differential Inclusions 15 15-20 (1995). MR 1344524 |
| Reference:
|
[17] Michalak A.: On the Fubini theorem for the Pettis integral for bounded functions.Bull. Polish Sci. Math. 49 (1) (2001), in press. Zbl 0995.46026, MR 1824153 |
| Reference:
|
[18] Mitchell A.R., Smith Ch.: 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. Zbl 0452.34054, MR 0502554 |
| Reference:
|
[19] Negrini P.: Sul problema di Darboux negli spazi di Banach.Boll. U.M.I. (5) 17-A 201-215 (1956). |
| Reference:
|
[20] O'Regan D.: Fixed point theory for weakly sequentially continuous mappings.to appear. |
| Reference:
|
[21] Pettis B.J.: On integration in vector spaces.Trans. Amer. Math. Soc. 44 277-304 (1938). Zbl 0019.41603, MR 1501970 |
| Reference:
|
[22] Szep A.: Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces.Bull. Acad. Polon. Sci. Math. 26 407-413 (1978). |
| . |
