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A principle of linearization in theory of stability of solutions of variational inequalities

Jiří Neustupa (1995)

Mathematica Bohemica

It is shown that the uniform exponential stability and the uniform stability at permanently acting disturbances of a sufficiently smooth but not necessarily steady-state solution of a general variational inequality is a consequence of the uniform exponential stability of a zero solution of another (so called linearized) variational inequality.

A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization

Robert Feßler (1995)

Annales Polonici Mathematici

The following problem of Markus and Yamabe is answered affirmatively: Let f be a local diffeomorphism of the euclidean plane whose jacobian matrix has negative trace everywhere. If f(0) = 0, is it true that 0 is a global attractor of the ODE dx/dt = f(x)? An old result of Olech states that this is equivalent to the question if such an f is injective. Here the problem is treated in the latter form by means of an investigation of the behaviour of f near infinity.

Additive groups connected with asymptotic stability of some differential equations

Árpád Elbert (1998)

Archivum Mathematicum

The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient λ 2 q ( s ) , s [ s 0 , ) is investigated, where λ and q ( s ) is a nondecreasing step function tending to as s . Let S denote the set of those λ ’s for which the corresponding differential equation has a solution not tending to 0. It is proved that S is an additive group. Four examples are given with S = { 0 } , S = , S = 𝔻 (i.e. the set of dyadic numbers), and S .

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