Summary
Consider the Ordinary Differential System
and a subset ω of Ω. It is known that the consequent mappingK is upper semicontinuous at any point P ∈ Ω at whichK is defined and moreoverK(P) is a continuum in ∂ω. Here we study the topological properties of the setK(P) in the case where P is a singular point ofK, i.e. there exists a solution of (E) through P which stays (right) asymptotic in ω. As an application we get an existence result of a general boundary value problem concerning (E) and we also prove that the second-order BVP
has a solution.
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References
J. Bebernes -J. Schuur,Investigations in the topological method of Ważewski, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.,49 (1970), pp. 39–42.
C. Berge,Espaces Topologiques. Functions Multivoques, Dunod, Paris (1966).
L. Jackson -G. Klaasen,A variation of Ważewski's topological method, SIAM J. Appl. Math.,20 (1971), pp. 124–130.
J.Kaplan - A.Lasota - J.Yorke,An application of the Ważewski retract method to boundary value problems, Zeszyty Nauk. Univ. Jagiellon. Prace Mat., to appear.
K. Kuratowski,Topology II, Academic Press, New York, 1968.
S. Bernfeld -V. Lakshmikantham,An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York (1974).
P. Palamides,Kneser's type properties for Caratheodory differential equations, Funkcial. Ekvac.,23 (1980), pp. 25–37.
P. Palamides,Kneser's type properties at the extreme points, Technical Report 46 (1980), Univ. of Ioannina, Greece.
P.Palamides,A new topological method for the study of asymptotic behavior of Caratheodory systems, Math. Nachr., to appear.
P. Palamides -Y. Sficas -V. Staikos,Asymptotic behavior of Caratheodory systems. Anti-Lyapunov method, J. Differential Equations,36 (1980), pp. 442–457.
P. Palamides -Y. Sficas -V. Staikos,Ważewski's topological method for Caratheodory systems, Technical Report 36 (1980), Univ. of Ioannina, Greece.
V. Staikos,On the asymptotic relationship at infinity between the solutions of two differential systems, Bull. Soc. Math. Grèce (N-S),13 (1972), pp. 1–13.
T. Wazewski,Sur un principle topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Math. Polon.,20 (1947), pp. 279–313.
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Palamides, P.K. Singular points of the consequent mapping. Annali di Matematica pura ed applicata 129, 383–395 (1981). https://doi.org/10.1007/BF01762151
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DOI: https://doi.org/10.1007/BF01762151