Zhong, Cheng-Kui On Ekeland’s variational principle and a minimax theorem. (English) Zbl 0870.49015 J. Math. Anal. Appl. 205, No. 1, 239-250 (1997). Summary: Ekeland’s variational principle states that if a Gâteaux differentiable function \(f\) has a finite lower bound (although it need not attain it), then for every \(\varepsilon>0\), there exists some point \(x_\varepsilon\) such that \(|f'(x_\varepsilon)|\leq\varepsilon\). This paper shows that there exists some point \(z_\varepsilon\) such that \(|f'(z_\varepsilon)|\leq\varepsilon/(1+ h(|z_\varepsilon|))\), where \(h:[0,+\infty)\to [0,+\infty)\) is a continuous nondecreasing function such that \(\int^{+\infty}_0(1/(1+h(r)))dr= +\infty\). As an application, a minimax theorem is proved under a weak “compactness condition”. Cited in 1 ReviewCited in 33 Documents MSC: 49K35 Optimality conditions for minimax problems 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 49J35 Existence of solutions for minimax problems 49J50 Fréchet and Gateaux differentiability in optimization Keywords:Gâteaux differentiable function; minimax theorem PDFBibTeX XMLCite \textit{C.-K. Zhong}, J. Math. Anal. Appl. 205, No. 1, 239--250 (1997; Zbl 0870.49015) Full Text: DOI References: [1] Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063 [2] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal., 7, 939-1012 (1983) · Zbl 0522.58012 [3] Brezis, H.; Nirenberg, L., Remarks on finding critical points, Comm. Pure Appl. Math., 64, 939-963 (1991) · Zbl 0751.58006 [4] Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-357 (1974) · Zbl 0286.49015 [5] Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0676.58017 [6] Shi, S. Z., Ekeland’s variational principle and the mountain pass lemma, Acta Math. Sinica (N.S.), 1, 348-358 (1985) · Zbl 0605.58017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.