V. I. Burenkov, T. V. Tararykova, “On the spectrum of a nonlinear operator associated with calculation of the norm of a linear vector-functional”, Eurasian Math. J., 2014, Volume 5, Number 2,Pages <nobr>132

Short communications

On the spectrum of a nonlinear operator associated with calculation of the norm of a linear vector-functional

V. I. Burenkov^ab, T. V. Tararykova^ba

^a Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, 2 Mirzoyan St., 010008 Astana, Kazakhstan
^b Cardiff School of Mathematics, Cardiff University, Senghennydd Rd. CF24 4AG Cardiff, UK

Abstract: An explicit formula is presented for the norm if $1\le p\le\infty$ and for the quasi-norm if $0<p<1$ of a linear vector-functional $L\colon H\to l_p$ on a Hilbert space $H$ and the set of all extremal elements is described. All eigenvalues and eigenvectors of a nonlinear homogeneous operator entering the corresponding Euler's equation, are written out explicitly.

Keywords and phrases: continuous linear vector-functional, Riesz theorem, extremal elements, Euler's equation, nonlinear eigenvalue problem.

MSC: 46C99, 47A75

Received: 01.02.2014

Language: English