Consider the Cauchy problem for an ordinary diierential equation _ x = g(t; x); x(0) = x; t 2 0; T]: (1:1) When g is continuous, the local existence of solutions is provided by Peano's theorem. Several existence and uniqueness results are known also in the case of a discontinuous right hand side 7]. We recall here the classical theorem of Carath eodory 8]: Theorem A. Let g : 0; T] IR n 7 ! IR n...

In this paper, we study the following Duffing-type equation: x′′ + cx′ + g(t, x) = h(t), where g(t, x) is a 2π-periodic continuous function in t and concave-convex in x, and h(t) is a small continuous 2π-periodic function. The exact multiplicity and stability of periodic solutions are obtained.

It is well known that if G is a countable amenable group and G y (Y ,ν) factors onto G y (X ,μ), then the entropy of the first action must be at least the entropy of the second action. In particular, if G y (X ,μ) has infinite entropy, then the action G y (Y ,ν) does not admit any finite generating partition. On the other hand, we prove that if G is a countable nonamenable group then there exis...

The international tokamak ITER is progressing towards assembly completion and first-plasma operation, which will be a physics engineering challenge for the fusion community. In preparation experimental scenarios, non-linear MHD simulations are playing an essential role to actively understand predict behaviour stability of plasmas in future power plant. development codes like JOREK key aspect th...

Conditions are found upon satisfaction of which the differential equation x(n)(t)− λx(n)(t− σ) + f(t, x(g(t))) = 0 has solutions which are asymptotically equivalent to the solutions of the equation x(n)(t)− λx(n)(t− σ) = 0. § 0. Introduction We consider the neutral functional differential equation x(n)(t)− λx(n)(t− σ) + f(t, x(g(t))) = 0 (A) under the assumptions that (i) n ≥ 1 is an integer; λ...

Let δ be a coaction of a locally compact group G on a C*-algebra A. We show that if I is a δ-invariant ideal in A, then 0! I¬δ I G!A¬δ G! (A}I )¬δI G! 0 for full crossed products, as Landstad et al. have done for spatial crossed products by coactions. We prove that for suitable coactions, the crossed products of C ! (X )-algebras are again C ! (X )-algebras, and the crossed products of continuo...