This paper studies the existence of the random attractor for a Klein-Gordon-Schödinger system under a small ε-random perturbation on a high dimensional infinite lattice. Firstly, we prove the asymptotic compactness of the random dynamical system and obtain the random attractor. Then, by comparing to the case without random perturbation (ε = 0), we show the upper semicontinuity of the attractors.
We present and analyze some variants of Cauchy's methods free from second derivative for obtaining simple roots of nonlinear equations. The convergence analysis of the methods is discussed. It is established that the methods have convergence order three. Per iteration the new methods require two function and one first derivative evaluations. Numerical examples show that the new methods are comparable with the well-known existing methods and give better numerical results in many aspects.
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