Survey of Numerical Methods for Shock Physics Applications

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21 but, instead of the functional F (r) = max09:S1 f (r (t)), it is considered the perturbed functional I (r) = max (f (r (t)) 099 + 'ljJ (r (t))), where 'ljJ (x) = max{O, c2 - cd (x, G n fe)}. 22 assuming a variant of (PS)C,e condition in the sense of Cerami. 13. The differentiable functional f : X --+ R, satisfies (PSC)C,e condition around a set G at the level c if every sequence (Xj)j in X such that: (1) limj d (Xj, G) = 0, (2) limj f (Xj) = c, (3) limj(l + Ilxjll) 11/ (Xj)11 = 0, has a convergent subsequence.

1985;60:142149. [RaO] Rabinowitz P. A note on nonlinear eigenvalue problems for a class of differential equations. J. Diff. , 1971;9:536-548. [Ral] Rabinowitz P. The mountain-pass theorem: Theme and variations. : Springer-Verlag, 1982;237-269. [Ra2] Rabinowitz P. Minimax methods in Critical Point Theory and Applications to Differential Equations. CBMS Reg. Conf. , 1986. [Ram] Ramos, Miguel. Teoremas de Enlace na Teoria dos Pontos Criticos. Universidade de Lisboa, Faculdade de Ciencias, 1993. , Tsachev T.

5, consider V : X\K --+ E, a locally Lipschitz mapping such that IIV (x)11 ::; 2. (x) I ::; (/ (x), V (x)), III Let 9 (x) = X (x) V (x) and a (t, x) be the solution of Cauchy problem { o-(t)=g(a(t)), a (0) = x. Since IIg (u)1I ::; 2 the solution a (t, x) of the above problem is defined for every t E R+. We have Iia (t, x) - xii::; lot 110- (s)11 ds ::; lot IIV (a (s))11 ds ::; 2t, and : / (a (t, x)) = (/ (a (t, x)), X (a (t, x)) V (a (t, x))) = X (a (t,x)) (/ (a (t,x)), V (a (t,x))) ~ o. Therefore, for each x E X\K, the mapping t --+ a (t,x) is f-increasing.

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