On the Infimum of Certain Functionals

Biagio Ricceri ()
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Biagio Ricceri: University of Catania, Department of Mathematics

A chapter in Essays in Mathematics and its Applications, 2016, pp 361-367 from Springer

Abstract: Abstract Here is a particular case of our main result: Let X be a real Banach space, $$\varphi: X \rightarrow \mathbf{R}$$ a nonzero continuous linear functional and ψ: X → R a nonconstant Lipschitzian functional with Lipschitz constant equal to $$\|\varphi \|_{X^{{\ast}}}$$ . Then, we have $$\displaystyle\begin{array}{rcl} & & \max \left \{\inf _{x\in X}(\varphi (x) +\psi (x)),\inf _{x\in X}(\varphi (x) -\psi (x))\right \} {}\\ & & \quad =\inf _{x\in X}(\varphi (x) + \vert \psi (x)\vert ) =\liminf _{\|x\|\rightarrow +\infty }(\varphi (x) + \vert \psi (x)\vert ) {}\\ \end{array}$$

Keywords: Dynamical System; Banach Space; Ergodic Theory; Global Analysis; Basic Result (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/978-3-319-31338-2_14

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