Abstract : We consider maps defined on the interior of a normal, closed cone in a real Banach space that are nonexpansive with respect to Thompson's metric. With mild compactness assumptions, we prove that the Krasnoselskii iterates of such maps converge to a fixed point when one exists. For maps that are also order-preserving, we give simple necessary and sufficient conditions in terms of upper and lower Collatz-Wielandt numbers for the fixed point set to be nonempty and bounded in Thompson's metric. When the map is also real analytic, these conditions are both necessary and sufficient for the map to have a unique fixed point and for all iterates of the map to converge to the fixed point. We demonstrate how these results apply to certain nonlinear matrix equations on the cone of positive definite Hermitian matrices.
Abstract : In this article, we consider the boundedness for a class of parameterized Littlewood-Paley integrals and their commutators. More precisely, Let $\Omega \in L^{2}\left(\mathrm{~S}^{n-1}\right)$ be a homogeneous function of degree zero, we prove that parameterized Littlewood-Paley area integral $\mu_{\Omega, S}^{\rho}$, $g_{\lambda}^{*}$ function $\mu_{\Omega, \lambda}^{*, \rho}$ are bounded on two weighted Herz spaces with variable exponents. It is worth noting that above operators in two weighted Herz spaces with variable exponents are more complex than operators themselves. Moreover, let $b$ be a BMO function, the boundedness of commutators generated by $b$ and parameterized Littlewood-Paley operators will also be showed.
Abstract : Various methods exist to address the challenges posed by moving domain problems, including the transform method, penalty method, and push-forward and pull-back method. These methods play a crucial role in transforming non-cylindrical domain problems into cylindrical domain problems. In \cite{RINCON2007}, the existence of at least one weak solution for a more general extensible beam was established. However, this paper does not specifically address the uniqueness of solutions and exploring their relationship within any finite-dimensional space. Our primary objective is to formulate approximate solutions in a finite-dimensional space using the penalty method, with a focus on achieving uniqueness. Furthermore, we present an illustrative example to enhance the understanding of the concepts discussed.
Hyungbin Park, Junsu Park
J. Korean Math. Soc. 2024; 61(5): 875-898
https://doi.org/10.4134/JKMS.j230053
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014
Jangwon Ju
J. Korean Math. Soc. 2023; 60(5): 931-957
https://doi.org/10.4134/JKMS.j220231
Yoon Kyung Park
J. Korean Math. Soc. 2023; 60(2): 395-406
https://doi.org/10.4134/JKMS.j220180
Jangwon Ju
J. Korean Math. Soc. 2023; 60(5): 931-957
https://doi.org/10.4134/JKMS.j220231
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014
Hyungbin Park, Junsu Park
J. Korean Math. Soc. 2024; 61(5): 875-898
https://doi.org/10.4134/JKMS.j230053
Xing Yu Song, Ling Wu
J. Korean Math. Soc. 2023; 60(5): 1023-1041
https://doi.org/10.4134/JKMS.j220589
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