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A generalization of Ekeland’s variational principle and application to the study of the relation between the weak P. S. condition and coercivity. (English) Zbl 0912.49021
The paper gives a generalization of the celebrated variational principle formulated by Ekeland in 1974. The main hypotheses are the same, excepting the contribution of a weight function \(h:[0,+\infty)\rightarrow [0,+\infty)\) which models the null map in the sense that \(h\) is continuous, nondecreasing and \(\int_0^\infty (1/(1+h(r))dr=+\infty\). The proof follows inductively and uses the same steps as in the classical case. The abstract result is illustrated by several examples, under the additional assumption that the functional satisfies the weak Palais-Smale condition, in the variant used by Cerami. It is refound the well-known result which asserts that a lower semicontinuous, proper, Gâteaux-differentiable bounded from below functional achieves its minimum, provided that the weak Palais-Smale condition is fulfilled. It is also analysed in the paper the link between this compactness condition and the notion of coercivity.

MSC:
49K27 Optimality conditions for problems in abstract spaces
49J45 Methods involving semicontinuity and convergence; relaxation
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E20 Harmonic maps, etc.
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