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A companion preorder to G-majorization and a Tarski type fixed-point theorem section: convex analysis

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Abstract

In this paper, a companion preorder \( \prec _G^\textrm{comp}\,\) to G-majorization \( \prec _G \) of Eaton type is introduced and studied. Attention is paid to the case of effective groups G. A criterion for G-majorization inequalities to hold is established by utilizing that companion preorder. A characterization of Gateaux differentiable \( \prec _G^\textrm{comp}\,\)-increasing functions is provided using their gradients. Next, some G-majorization relations are derived for gradients of some functions. New classes of c-strongly (weakly) \( \prec _G \)-increasing functions and c-strongly (weakly) \( \prec _G^\textrm{comp}\,\)-increasing functions are introduced. A Tarski like theorem is established on fixed points of the gradients maps of weakly \( \prec _G^\textrm{comp}\,\)-increasing functions. Some interpretations for the weak-majorization preorder and singular values of matrices are shown.

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Acknowledgements

The author would like to thank anonymous referees for giving some suggestions and pointing out the references [7, 16].

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  1. Institute of Mathematics, Pedagogical University of Cracow, Podchora̧żych 2, 30-084, Kraków, Poland

    Marek Niezgoda

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  1. Marek Niezgoda
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Correspondence to Marek Niezgoda.

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Marek Niezgoda is retired from Pedagogical University of Cracow, Kraków, Poland.

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Niezgoda, M. A companion preorder to G-majorization and a Tarski type fixed-point theorem section: convex analysis. J. Fixed Point Theory Appl. 25, 49 (2023). https://doi.org/10.1007/s11784-023-01053-z

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