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

Mieczysław Cichoń; Ireneusz Kubiaczyk

Annales Polonici Mathematici (1995)

- Volume: 62, Issue: 1, page 13-21
- ISSN: 0066-2216

## Access Full Article

top## Abstract

top## How to cite

topMieczysław Cichoń, and Ireneusz Kubiaczyk. "Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces." Annales Polonici Mathematici 62.1 (1995): 13-21. <http://eudml.org/doc/262264>.

@article{MieczysławCichoń1995,

abstract = {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.},

author = {Mieczysław Cichoń, Ireneusz Kubiaczyk},

journal = {Annales Polonici Mathematici},

keywords = {set of solutions; pseudo-solutions; measures of weak noncompactness; Pettis integral; Cauchy problem; Banach space},

language = {eng},

number = {1},

pages = {13-21},

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

url = {http://eudml.org/doc/262264},

volume = {62},

year = {1995},

}

TY - JOUR

AU - Mieczysław Cichoń

AU - Ireneusz Kubiaczyk

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

JO - Annales Polonici Mathematici

PY - 1995

VL - 62

IS - 1

SP - 13

EP - 21

AB - 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.

LA - eng

KW - set of solutions; pseudo-solutions; measures of weak noncompactness; Pettis integral; Cauchy problem; Banach space

UR - http://eudml.org/doc/262264

ER -

## References

top- [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. Zbl0599.34008
- [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. Zbl0441.47056
- [3] J. Banaś and J. Rivero, On measure of weak noncompactness, Ann. Mat. Pura Appl. 125 (1987), 213-224. Zbl0653.47035
- [4] M. Cichoń, Weak solutions of differential equations in Banach spaces, Discuss. Math. 15 (1994) (in press).
- [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. Zbl0379.34041
- [6] S. J. Daher, On a fixed point principle of Sadovskii, Nonlinear Anal., 643-645. Zbl0377.47038
- [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. Zbl0365.46015
- [8] J. Diestel and J. J. Uhl Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.
- [9] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Side, Nauka, Moscow, 1985 (in Russian). Zbl0571.34001
- [10] W. J. Knight, Solutions of differential equations in B-spaces, Duke Math. J. 41 (1974), 437-442. Zbl0288.34063
- [11] I. Kubiaczyk, Kneser type theorems for ordinary differential equations in Banach spaces, J. Differential Equations 45 (1982), 139-146. Zbl0505.34048
- [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. Zbl0516.34058
- [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.
- [14] N. S. Papageorgiou, Kneser's theorems for differential equations in Banach spaces, Bull. Austral. Math. Soc. 33 (1986), 419-434. Zbl0579.34046
- [15] B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304. Zbl0019.41603
- [16] A. Szep, Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar. 6 (1971), 197-203. Zbl0238.34100
- [17] S. Szufla, Some remarks on ordinary differential equations in Banach spaces, Bull. Acad. Polon. Sci. Math. 16 (1968), 795-800. Zbl0177.18902
- [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. Zbl0384.34039

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.