Previous |  Up |  Next

Article

Keywords:
Fréchet spaces; row-finite systems; one-sided estimates; row-finite matrices
Summary:
We prove an existence and uniqueness theorem for row-finite initial value problems. The right-hand side of the differential equation is supposed to satisfy a one-sided matrix Lipschitz condition with a quasimonotone row-finite matrix which has an at most countable spectrum.
References:
[1] Bogachev V. I.: Deterministic and stochastic differential equations in infinite dimensional spaces. Acta Appl. Math. 40 (1995), 25-93. DOI 10.1007/BF01518365 | MR 1338443 | Zbl 0829.34050
[2] Deimling K.: Ordinary Differential Equations in Banach Spaces. Lecture Notes in Mathematics 596, Springer, 1977. MR 0463601 | Zbl 0361.34050
[3] Feldstein A., Iserles A., Levin D.: Embedding of delay equations into an infinite-dimensional ODE system. J. Differential Equations 117(1995), 127-150. DOI 10.1006/jdeq.1995.1050 | MR 1320185 | Zbl 0817.34045
[4] Herzog G.: Über gewöhnliche Differentialgleichungen in Fréchetraumen. Dissertation, Univ. Karlsruhe, 1992.
[5] Herzog G.: On ordinary linear differential equations in $C^J$. Demonstratio Math. 28 (1995), 383-398. MR 1346844
[6] Herzog G.: On linear time-dependent row-finite systems of differential equations. Travaux Math. Sem. Math. Luxemb. 8 (1996), 167-176. MR 1476744 | Zbl 0978.34042
[7] Herzog G.: On existence and uniqueness conditions for ordinary differential equations in Fréchet spaces. Studia Sci. Math. Hungar. 23 (1996), 367-375. MR 1432178 | Zbl 0880.34066
[8] Herzog G.: One-sided estimates for ordinary differential equations in Fréchet spaces. Ricerche Mat. 46 (1997), 479-487. MR 1760389 | Zbl 0947.34048
[9] Kaplansky I.: Infinite Abelian Groups. The University of Michigan Press, 4th printing, 1962. MR 0065561
[10] Körber K.H.: Das Spektrum zeilenfiniter Matrizen. Math. Ann. 181 (1969), 8-34. DOI 10.1007/BF01351175 | MR 0262863
[11] Lemmert R., Weckbach Ä.: Charakterisierungen zeilenendlicher Matrizen mit abzählbarem Spektrum. Math. Z. 188 (1984), 119-124. DOI 10.1007/BF01163880 | MR 0767369 | Zbl 0554.47015
[12] Lemmert R.: On ordinary differential equations in locally convex spaces. Nonlinear Analysis 10 (1986), 1385-1390. DOI 10.1016/0362-546X(86)90109-4 | MR 0869547 | Zbl 0612.34056
[13] Lobanov S. G., Smolyanov O. G.: Ordinary differential equations in locally convex spaces. Russ. Math. Surv. 49 (1994), 97-175. DOI 10.1070/RM1994v049n03ABEH002258 | MR 1289388 | Zbl 0834.34076
[14] Martin R. H.: Nonlinear Operators and Differential Equations in Banach Spaces. Robert E. Krieger Publishing Company, Malabar, Florida, 1987. MR 0887947
[15] Moszyński K., Pokrzywa A.: Sur les systèmes infinis d'équations différentielles ordinaires dans certains espaces de Fréchet. Dissertationes Math. 115 (1974), 1-37. MR 0390415 | Zbl 0291.34049
[16] Polewczak J.: Ordinary differential equations on closed subsets of Fréchet spaces with applications to fixed point theorems. Proc. Amer. Math. Soc. 107 (1989), 1005-1012. MR 1019755 | Zbl 0684.34062
[17] Polewczak J.: Ordinary differential equations on closed subsets of locally convex spaces with applications to fixed point theorems. J. Math. Anal. Appl. 151 (1990), 208-225. DOI 10.1016/0022-247X(90)90252-B | MR 1069457 | Zbl 0721.34074
[18] Schmidt S.: Existenzsätze für gewöhnliche Differentialgleichungen in Banachräumen. Funkcial. Ekvac. 35 (1992), 199-222. MR 1189893 | Zbl 0785.34045
[19] Tychonov A.N.: Ein Fixpunktsatz. Math. Ann. 111 (1935), 767-776. MR 1513031
[20] Vim H.: Elementarteilertheorie unendlicher Matrizen. Math. Ann. 114 (1937), 493-505. DOI 10.1007/BF01594190 | MR 1513152
[21] Volkmann P.: Ein Existenzsatz für gewöhnliche Differentialgleichungen in Banachräumen. Proc. Amer. Math. Soc. 80 (1980), 297-300. MR 0577763 | Zbl 0506.34051
[22] Wagner E.: Über ein abzählbares System gewöhnlicher linearer Differentialgleichungen von Bandstruktur. Demonstratio Math. 23 (1990), 753-773. MR 1101540 | Zbl 0734.34016
[23] Williamson J. H.: Spectral representation of linear transformations in $\omega$. Proc. Cambridge Philos. Soc. 41 (1951), 461-472. MR 0042615 | Zbl 0042.12201
Partner of
EuDML logo