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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Boundary value problems on infinite intervals


Authors: Jan Andres, Grzegorz Gabor and Lech Górniewicz
Journal: Trans. Amer. Math. Soc. 351 (1999), 4861-4903
MSC (1991): Primary 34A60, 34B15, 47H04, 54C60
Published electronically: August 30, 1999
MathSciNet review: 1603870
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Abstract: We present two methods, both based on topological ideas, to the solvability of boundary value problems for differential equations and inclusions on infinite intervals. In the first one, related to the rich family of asymptotic problems, we generalize and extend some statements due to the Florence group of mathematicians Anichini, Cecchi, Conti, Furi, Marini, Pera, and Zecca. Thus, their conclusions for differential systems are as well true for inclusions; all under weaker assumptions (for example, the convexity restrictions in the Schauder linearization device can be avoided). In the second, dealing with the existence of bounded solutions on the positive ray, we follow and develop the ideas of Andres, Górniewicz, and Lewicka, who considered periodic problems. A special case of these results was previously announced by Andres. Besides that, the structure of solution sets is investigated. The case of l.s.c. right hand sides of differential inclusions and the implicit differential equations are also considered. The large list of references also includes some where different techniques (like the Conley index approach) have been applied for the same goal, allowing us to envision the full range of recent attacks on the problem stated in the title.


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Additional Information

Jan Andres
Affiliation: Department of Mathematic al Analysis, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc-Hejčín, Czech Republic

Grzegorz Gabor
Affiliation: Faculty of Mathematics and Informatics, N. Copernicus University, Chopina 12/18, 87–100 Toruń, Poland
Email: ggabor@mat.uni.torun.pl

Lech Górniewicz
Email: gorn@mat.uni.torun.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02297-7
PII: S 0002-9947(99)02297-7
Keywords: Asymptotic boundary value problems, differential equations and inclusions, topological methods, Fr spaces
Received by editor(s): September 10, 1996
Received by editor(s) in revised form: June 11, 1997
Published electronically: August 30, 1999
Additional Notes: The first author was supported by grant no. 0053/1996 of FRVŠ
The second author was supported by grant no. 335-M of N. Copernicus University
The third author was supported by Polish scientific grant KBN no. 2 P03A 06109.
Article copyright: © Copyright 1999 American Mathematical Society