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Additional info for Equadiff 10: Czechoslovak international conference on differential equations and their applications, Prague, August 27-31, 2001 : papers

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The existence condition includes the case of a function Φ − G∗ of constant sign. In particular, the result applies if max Φ < G+ or min Φ > G+ . (4) That situation has been treated by Krasnosel’skii and Mawhin [9] for perturbations of a linear oscillator. Arguing as in [9], it can be shown that, when g has a sublinear primitive, G+ = G+ = 0. Hence, if one considers, for instance, the equation x + αx+ − βx− + sin(x) = p(t) , where x+ = max{x, 0}, x− = max{−x, 0}, and α, β satisfy condition (2) with k = 1, or the equation x − 1 (x + 1) + + sin(x) = p(t) , 4(x + 1)3 4 for both of which g(x) = sin(x), it results from Theorem 1 that these equations have at least one 2π-periodic solution if the number of zeros (supposed to be simple), in [0, 2π), of the function Φ defined by (3), is different from 2.

If there exists a weak nontrivial solution of BVP −∆p u = λ|u|p−2 u in Ω, u = 0 on ∂Ω. Now we define the even functional I(u) := |∇u|p dx : W01,p (Ω) \ {0} → R, p Ω |u| dx Ω and for any k ∈ N we consider set Fk := A ⊂ {u ∈ W01,p (Ω) : u Lp (Ω) = 1} : there exists a continuous odd surjection h : S k → A , where S k represents the unit sphere in Rk . Pavel Drábek and Stephen B. Robinson proved in 1999 that for any k ∈ N the number λk := inf sup I(u) A∈Fk u∈A is an eigenvalue of −∆p . This situation is very interesting, because it is not known if this represents a complete list of eigenvalues 1 but it is known that: 1 Nobody knows how to obtain all eigenvalues of −∆p ; we only know that we have complete list of eigenvalues if N = 1 or p = 2.

S. Melnik and J. Valero theory of global attractors is not applicable. Hence, a different approach has been considered [4,5,6,9]. A new difficulty appears if the solution corresponding to each initial state can be non-unique. The classical results on attractors in the autonomous and nonautonomous cases are generalized to the multivalued case in [7] and [8], respectively, with applications to evolution inclusions. In [1,2,3] the study of multivalued dynamical systems is extended to the stochastic case, generalizing in this way the results of [4,5].

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