Abstract
A real function φ of a real variable which is bounded below on the real line needs not to have a minimum, as it is clear from the example of the exponential function. If we call minimizing sequence for φ any sequence (a k ) such that
as k → ∞, a necessary condition for the real number a to be such that
is that φ has a minimizing sequence which converges to a (take a k = a for all integers k). Without suitable continuity assumptions on φ this condition will not be sufficient, as shown by the example of the function φ defined by φ(x) = |x| for x ≠ 0 and φ(0) = 1, which does not achieve its infimum 0 although all its minimizing sequences converge to zero. In order that the limit a of a convergent minimizing sequence be such that φ(a) = inf φ, we have to impose that
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© 1989 Springer Science+Business Media New York
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Mawhin, J., Willem, M. (1989). The Direct Method of the Calculus of Variations. In: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, vol 74. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2061-7_1
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DOI: https://doi.org/10.1007/978-1-4757-2061-7_1
Publisher Name: Springer, New York, NY
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