In this paper we prove a theorem for the existence of pseudo-solutions to the Cauchy problem, x' = f(t,x), x(0) = x₀ in Banach spaces. The function f will be assumed Pettis-integrable, but this assumption is not sufficient for the existence of solutions. We impose a weak compactness type condition expressed in terms of measures of weak noncompactness. We show that under some additionally assumptions our solutions are, in fact, weak solutions or even strong solutions. Thus, our theorem is an essential generalization of previous results.
@article{bwmeta1.element.bwnjournal-article-div15i1n1bwm,
author = {Mieczys\l aw Cicho\'n},
title = {Weak solutions of differential equations in Banach spaces},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {15},
year = {1995},
pages = {5-14},
zbl = {0829.34051},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-div15i1n1bwm}
}
Mieczysław Cichoń. Weak solutions of differential equations in Banach spaces. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 15 (1995) pp. 5-14. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-div15i1n1bwm/
[000] [1] A. Alexiewicz, Functional Analysis, Monografie Matematyczne 49, Polish Scientific Publishers, Warsaw 1968 (in Polish).
[001] [2] O. Arino, S. Gautier, J.P.Penot, A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkcialaj Ekvac. 27 (1984), 273-279. | Zbl 0599.34008
[002] [3] J. M. Ball, Weak continuity properties of mappings and semi-groups, Proc. Royal Soc. Edinbourgh Sect. A 72 (1979), 275-280.
[003] [4] J. Banas, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, Marcel Dekker, New York - Basel 1980. | Zbl 0441.47056
[004] [5] J. Banas, J. Rivero, On measures of weak noncompactness, Ann. Mat. Pura Appl. 125 (1987), 213-224. | Zbl 0653.47035
[005] [6] M. Cichoń, On measures of weak noncompactness, Publicationes, Math. Debrecen 45 (1994), 93-102. | Zbl 0829.47042
[006] [7] E. Cramer, V. Lakshmikantham and A.R.Mitchell, On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlin. Anal. TMA 2 (1978), 169-177. | Zbl 0379.34041
[007] [8] F. S. DeBlasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R. S. Roumanie 21 (1977), 259-262.
[008] [9] G. A. Edgar, Geometry and the Pettis integral, Indiana Univ. Math. J. 26 (1977), 663-677.
[009] [10] G. A. Edgar, Geometry and the Pettis integral II, ibid. 28 (1979), 559-579. | Zbl 0418.46034
[010] [11] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Side, Moscow 1985 (in Russian). | Zbl 0571.34001
[011] [12] R. F. Geitz, Pettis integration, Proc. Amer. Math. Soc. 82 (1981), 81-86. | Zbl 0506.28007
[012] [13] R. F. Geitz, Geometry and the Pettis integral, Trans. Amer. Math. Soc. 269 (1982), 535-548. | Zbl 0498.28005
[013] [14] E. Hille, R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, Rhode-Island 1957. | Zbl 0078.10004
[014] [15] W. J. Knight, Solutions of differential equations in B-spaces, Duke Math. J. 41 (1974), 437-442. | Zbl 0288.34063
[015] [16] I. Kubiaczyk, On the existence of solutions of differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 33 (1985), 607-614. | Zbl 0607.34055
[016] [17] I. Kubiaczyk, S. Szufla, Kneser's theorem for weak solutions of ordinary differential equations in Banach spaces, Publ. Inst. Mat. Beograd 32 (1982), 99-103. | Zbl 0516.34058
[017] [18] A. R. Mitchell, Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, pp. 387-404 in: Nonlinear Equations in Abstract Spaces, ed. by V. Lakshmikantham 1978.
[018] [19] B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304. | Zbl 0019.41603
[019] [20] 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
[020] [21] M. Talagrand, Pettis integral and measure theory, Memoires Amer. Math. Soc. 307 Vol. 51, Amer. Math. Soc., Providence, Rhode-Island 1984.