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nonlinear operators; maximal monotone operators; range of maximal monotone operator; an approximation method of maximal monotone operators
It is shown that every maximal monotone operator on a real Banach space with relatively compact range is of type NI. Moreover, if the space has a separable dual space then every maximally monotone operator $T$ can be approximated by a sequence of maximal monotone operators of type NI, which converge to $T$ in a reasonable sense (in the sense of Kuratowski-Painleve convergence).
[1] Debrunner, H., Flor, P.: Ein Erweiterungssatz für monotone Mengen. Arch. Math. 15 (1964), 445-447 German. DOI 10.1007/BF01589229 | MR 0170189 | Zbl 0129.09203
[2] Fitzpatrick, S. P., Phelps, R. R.: Bounded approximants to monotone operators on Banach spaces. Ann. Inst. Henri Poincaré 9 (1992), 573-595. MR 1191009 | Zbl 0818.47052
[3] Holmes, R. B.: Geometric Functional Analysis and its Applications. Springer New York (1975). MR 0410335 | Zbl 0336.46001
[4] Phelps, R. R.: Lecture on maximal monotone operators. Lecture given at Prague/Paseky, Summer school, arXiv:math/9302209v1 [math.FA] (1993). MR 1627478
[5] Phelps, R. R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics 1364. Springer Berlin (1989). MR 0984602
[6] Rockafellar, R. T.: On the virtual convexity of the domain and range of a nonlinear maximal monotone operator. Math. Ann. 185 (1970), 81-90. DOI 10.1007/BF01359698 | MR 0259697 | Zbl 0181.42202
[7] Rudin, W.: Functional Analysis (2nd edition). McGraw-Hill New York (1991). MR 1157815
[8] Simons, S.: From Hahn-Banach to Monotonicity. Lecture Notes in Mathematics 1693 (2nd expanded ed.). Springer Berlin (2008). MR 2386931
[9] Simons, S.: Minimax and Monotonicity. Lecture Notes in Mathematics 1693. Springer Berlin (1998). MR 1723737
[10] Zagrodny, D.: The closure of the domain and the range of a maximal monotone multifunction of type NI. Set-Valued Anal. 16 (2008), 759-783. DOI 10.1007/s11228-008-0087-7 | MR 2465516 | Zbl 1173.47031
[11] Zeidler, E.: Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators. Springer Berlin (1990). MR 1033498 | Zbl 0684.47029
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