A note on spherical maxima sharing the same Lagrange multiplier

Abstract

In this paper, we establish a general result on spherical maxima sharing the same Lagrange multiplier of which the following is a particular consequence: Let $X$ be a real Hilbert space. For each $r>0$, let $S_r=\{x\in X:\Vert x{\Vert }^2=r\}$. Let $J:X\to \mathbf{R}$ be a sequentially weakly upper semicontinuous functional which is Gâteaux differentiable in $X\setminus \{0\}$. Assume that $$\underset{x\to 0}{\mathrm{limsup}}\frac{J(x)}{\Vert x{\Vert }^2}=+\infty .$$ Then, for each $\rho >0$, there exists an open interval $I\subseteq ]0,+\infty [$ and an increasing function $\phi :I\to ]0,\rho [$ such that, for each $\lambda \in I$, one has $$\varnothing \ne \left\{x\in S_{\phi (\lambda )}:J(x)=\underset{S_{\phi (\lambda )}}{\sup }J\right\}\subseteq \{x\in X:x=\lambda J^{\prime }(x)\}.$$

References in corpus (1)

• Uniqueness properties of functionals with Lipschitzian derivative