Gritsans, Armand and Sadyrbaev, Felix: A twopoint boundary value problem for third order asymptotically linear systems. In: Electronic journal of qualitative theory of differential equations 28. pp. 124. (2019)

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Abstract
We consider a third order system x 000 = f(x) with the twopoint boundary conditions x(0) = 0, x 0 (0) = 0, x(1) = 0, where f(0) = 0 and the vector field f ∈ C 1 (Rn , Rn ) is asymptotically linear with the derivative at infinity f 0 (∞). We introduce an asymptotically linear vector field φ such that its singular points (zeros) are in a onetoone correspondence with the solutions of the boundary value problem. Using the vector field rotation theory, we prove that under the nonresonance conditions for the linearized problems at zero and infinity the indices of φ at zero and infinity can be expressed in the terms of the eigenvalues of the matrices f 0 (0) and f 0 (∞), respectively. This proof constitutes an essential part of our article. If these indices are different, then standard arguments of the vector field rotation theory ensure the existence of at least one nontrivial solution to the boundary value problem. At the end of the article we consider the consequences for the scalar case.
Item Type:  Article 

Journal or Publication Title:  Electronic journal of qualitative theory of differential equations 
Date:  2019 
Number:  28 
ISSN:  14173875 
Page Range:  pp. 124 
DOI:  https://doi.org/10.14232/ejqtde.2019.1.28 
Uncontrolled Keywords:  Határérték probléma  differenciálegyenletek 
Additional Information:  Bibliogr.: p. 2224. ; összefoglalás angol nyelven 
Date Deposited:  2019. Sep. 27. 11:54 
Last Modified:  2019. Sep. 27. 13:21 
URI:  http://acta.bibl.uszeged.hu/id/eprint/62106 
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