Abstract
In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation Au = b. We prove that if A is a quasi-uniformly monotone and hemi-continuous operator, then A−1 is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence of Galerkin approximations to the solution of steady-state electromagnetic p-curl systems.
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References
S. Antontsev, F. Miranda, L. Santos: A class of electromagnetic p-curl systems: Blow-up and finite time extinction. Nonlinear Anal., Theory Methods Appl., Ser. A 75 (2012), 3916–3929.
S. Antontsev, F. Miranda, L. Santos: Blow-up and finite time extinction for p(x,t)-curl systems arising in electromagnetism. J. Math. Anal. Appl. 440 (2016), 300–322.
J. Aramaki: Lp theory for the div-curl system. Int. J. Math. Anal., Ruse 8 (2014), 259–271.
A. Bahrouni, D. Repovš: Existence and nonexistence of solutions for p(x)-curl systems arising in electromagnetism. Complex Var. Elliptic Equ. 63 (2018), 292–301.
V. Barbu: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics. Springer, Berlin, 2010.
I. Cimrák, R. Van Keer: Level set method for the inverse elliptic problem in nonlinear electromagnetism. J. Comput. Phys. 229 (2010), 9269–9283.
N. Dunford, J. T. Schwartz: Linear Operators. I. General Theory. Pure and Applied Mathematics 7. Interscience Publishers, New York, 1958.
J. Franců: Monotone operators: A survey directed to applications to differential equations. Appl. Math. 35 (1990), 257–301.
H. Gajewski, K. Gröger, K. Zacharias: Nichtlineare Operatorgleichungen und Operator-differentialgleichungen. Mathematische Lehrbücher und Monographien. II. Abteilung. Band 38. Akademie, Berlin, 1974. (In German.)
J.-F. Gerbeau, C. Le Bris, T. Lelièvre: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2006.
E. Janíková, M. Slodička: Fully discrete linear approximation scheme for electric field diffusion in type-II superconductors. J. Comput. Appl. Math. 234 (2010), 2054–2061.
S. C. László: The Theory of Monotone Operators with Applications. Babes-Bolyai University, Budapest, 2011.
M. Xiang, F. Wang, B. Zhang: Existence and multiplicity for p(x)-curl systems arising in electromagnetism. J. Math. Anal. Appl. 448 (2017), 1600–1617.
E. Zeidler: Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators. Springer, New York, 1990.
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The authors would like to thank the anonymous referee for his or her very constructive comments and suggestions that helped us to improve the quality of the original manuscript and make it more readable.
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Song, CH., Ri, YG. & Sin, C. Properties of a quasi-uniformly monotone operator and its application to the electromagnetic p-curl systems. Appl Math 67, 431–444 (2022). https://doi.org/10.21136/AM.2021.0365-20
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DOI: https://doi.org/10.21136/AM.2021.0365-20