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Variational study of bifurcations in von Kármán equations
Rongrong JIN, Guangcun LU
Front. Math. China. 2019, 14 (3): 567-590.
https://doi.org/10.1007/s11464-019-0766-8
For a class of nonlinear elliptic boundary value problems including the von Kármán equations considered by D. M. Duc, N. L. Luc, L. Q. Nam, and T. T. Tuyen [Nonlinear Anal., 2003, 55: 951{968], we give a new proof of a corresponding theorem of three solutions via Morse theory instead of topological degree theory. Several bifurcation results for this class of boundary value problems are also obtained with Morse theory methods. In addition, for the von Kármán equations studied by A. Borisovich and J. Janczewska [Abstr. Appl. Anal., 2005, 8: 889{899], we prove a few of bifurcation results under Dirichlet boundary conditions based on the second named author's recent work about parameterized splitting theorems and bifurcations for potential operators.
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Absence of eigenvalues for quasiperiodic Schrödinger type operators
Jiahao XU, Xin ZHAO
Front. Math. China. 2019, 14 (3): 645-659.
https://doi.org/10.1007/s11464-019-0773-9
We obtain the matrix-valued Schrödinger-type operators [Hα,θ] with Lipschitz potentials having no eigenvalues on the set {E: L(E)<δC,d(α,θ)}, where δ is an explicit function depending on the sampling function C(θ), dimension d, phase θ, and frequency α, and L(E) is the Lyapunov exponent.
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Estimations on upper and lower bounds of solutions to a class of tensor complementarity problems
Yang XU, Weizhe GU, Zheng-Hai HUANG
Front. Math. China. 2019, 14 (3): 661-671.
https://doi.org/10.1007/s11464-019-0770-z
We introduce a class of structured tensors, called generalized row strictly diagonally dominant tensors, and discuss some relationships between it and several classes of structured tensors, including nonnegative tensors, B-tensors, and strictly copositive tensors. In particular, we give estimations on upper and lower bounds of solutions to the tensor complementarity problem (TCP) when the involved tensor is a generalized row strictly diagonally dominant tensor with all positive diagonal entries. The main advantage of the results obtained in this paper is that both bounds we obtained depend only on the tensor and constant vector involved in the TCP; and hence, they are very easy to calculate.
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