×

zbMATH — the first resource for mathematics

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)

MSC:
34C23 Bifurcation theory for ordinary differential equations
34A40 Differential inequalities involving functions of a single real variable
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] J. P. Aubin, A. Cellina: Differential Inclusions. Springer-Verlag, Berlin, 1984.
[2] M. Degiovanni, A. Marino: Non-smooth variational bifurcation. Atti Acc. Lincei Rend. Fis. (8) LXXXI (1987), 259-270. · Zbl 0671.58029
[3] J. Eisner, M. Kučera: Hopf bifurcation and ordinary differential inequalities. · Zbl 0848.34020
[4] M. Kučera: Bifurcation of periodic solutions to ordinary differential inequalities. · Zbl 0809.34021
[5] M. Kučera: Bifurcation points of variational inequalities. Czechoslovak Math. J. 32 (107) (1982), 208-226. · Zbl 0621.49006
[6] E. Miersemann: Über höhere Verzweigungspunkte nichtlinearen Variationsungleichungen. Math. Nachr. 85 (1978), 195-213. · Zbl 0324.49036
[7] M. Pazy: Semi-groups of nonlinear contractions in Hilbert space. Problems in Nonlinear Analysis (C.I.M.E., IV Ciclo, Varenna, 1970), Edizioni Cremonese, Rome, 1971, pp. 343-430. · Zbl 0228.47038
[8] P. Quittner: Spectral analysis of variational inequalities. Comment. Math. Univ. Carol. 27 (1986), 605-629.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.