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Bifurcation of periodic solutions to differential inequalities in \(\mathbb{R}^ 3\). (English) Zbl 0794.34031
The authors consider the inequality \((*)\) \(U(t)\in K\), \(\forall t\in [0,T)\), \((\dot U(t)- A_ \lambda U(t)- G(\lambda,U(t)),\;v-U(t))\geq 0\) for all \(v\in K\), a.a. \(t\in [0,T)\), where \(K\) is a closed convex cone with its vertex at the origin in \(\mathbb{R}^ 3\), \(A_ \lambda\) is a real \(3\times 3\) matrix depending continuously on a real parameter \(\lambda\), \(G:\mathbb{R}\times\mathbb{R}^ 3\to\mathbb{R}^ 3\) is a continuous mapping locally Lipschitzian in the variable \(u\) and satisfying the usual condition \(\lim_{u\to 0} {G(\lambda,u)\over | u|}=0\) uniformly on compact \(\lambda\)-intervals. Under certain assumptions concerning the eigenvalues of \(A_ \lambda\) and a relation of the cone \(K\) to the eigenvectors of \(A_ \lambda\) they prove the existence of a bifurcation point \(\lambda_ I\) at which periodic solutions to the inequality \((*)\) bifurcate from the branch of trivial solutions.
Reviewer: A.Martynyuk (Kiev)

34C23 Bifurcation theory for ordinary differential equations
34A40 Differential inequalities involving functions of a single real variable
Full Text: EuDML
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