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Transactions of the American Mathematical Society

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Ranges of perturbed maximal monotone and $ m$-accretive operators in Banach spaces


Authors: Zhengyuan Guan and Athanassios G. Kartsatos
Journal: Trans. Amer. Math. Soc. 347 (1995), 2403-2435
MSC: Primary 47H05; Secondary 47H06, 47H11, 47H15
DOI: https://doi.org/10.1090/S0002-9947-1995-1297527-2
MathSciNet review: 1297527
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Abstract: A more comprehensive and unified theory is developed for the solvability of the inclusions $ S \subset \overline {R(A + B)} $, int $ S \subset R(A + B)$, where $ A:X \supset D(A) \to {2^Y}$, $ B:X \supset D(B) \to Y$ and $ S \subset X$. Here, $ X$ is a real Banach space and $ Y = X$ or $ Y = {X^*}$. Mainly, $ A$ is either maximal monotone or maccretive, and $ B$ is either pseudo-monotone or compact. Cases are also considered where $ A$ has compact resolvents and $ B$ is continuous and bounded. These results are then used to obtain more concrete sets in the ranges of sums of such operators $ A$ and $ B$. Various results of Browder, Calvert and Gupta, Gupta, Gupta and Hess, and Kartsatos are improved and/or extended. The methods involve the application of a basic result of Browder, concerning pseudo-monotone perturbation of maximal monotone operators, and the Leray-Schauder degree theory.


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  • [1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, The Netherlands, 1975. MR 0390843 (52:11666)
  • [2] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Math. Studies $ 5$, North-Holland, Amsterdam, 1973.
  • [3] -, Monotone operators, nonlinear semigroups and applications, Proc. Internat. Congress of Mathematicians, Canadian Mathematical Congress, Vancouver, 1974.
  • [4] H. Brézis and A. Haraux, Image d'une somme d'opérateurs monotones et applications, Israel J. Math. 23 (1976), 165-186. MR 0399965 (53:3803)
  • [5] H. Brézis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa 5 (1978), 225-326. MR 0513090 (58:23813)
  • [6] F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math., vol 18, Amer. Math. Soc., Providence, RI, 1976. MR 0405188 (53:8982)
  • [7] -, On a principle of H. Brézis and its applications, J. Funct. Anal. 25 (1977), 356-365. MR 0445344 (56:3686)
  • [8] B. D. Calvent and C. P. Gupta, Nonlinear elliptic boundary value problems in $ {L^p}$ spaces and sums of ranges of accretive operators, Nonlinear Anal. TMA 2 (1978), 1-26. MR 512651 (80i:35083)
  • [9] I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer, Boston, 1990. MR 1079061 (91m:46021)
  • [10] Z. Guan, Solvability of semilinear equations with compact perturbations of operators of monotone type, Proc. Amer. Math. Soc. 121 (1994), 93-102. MR 1174492 (94g:47080)
  • [11] -, Existence of solutions of nonlinear equations involving compact perturbations of monotone operators, Proc. First World Congress of Nonlinear Analysts, Tampa, Florida, 1992, De Gruyter (to appear).
  • [12] Z. Guan and A. G. Kartsatos, Solvability of nonlinear equations with coercivity generated by compact perturbations of $ m$-accretive operators in Banach spaces, Houston J. Math. (to appear). MR 1331252 (96c:47099)
  • [13] C. P. Gupta, Sums of ranges of operators and applications, Nonlinear Systems and Applications, Academic Press, New York, 1977. MR 0454758 (56:13006)
  • [14] C. P. Gupta and P. Hess, Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems, J. Differential Equations 22 (1976), 305-313. MR 0473942 (57:13600)
  • [15] A. G. Kartsatos, Mapping theorems involving ranges of sums of nonlinear operators, Nonlinear Anal. TMA 6 (1982), 271-278. MR 654318 (83k:47036)
  • [16] -, Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces, Proc. First World Congress of Nonlinear Analysts, Tampa, Florida, 1992, De Gruyter (to appear). MR 1389246
  • [17] -, On compact perturbations and compact resolvents of nonlinear $ m$-accretive operators in Banach spaces, Proc. Amer. Math. Soc. 119 (1993), 1189-1199. MR 1216817 (94c:47076)
  • [18] -, Sets in the ranges of sums for perturbations of nonlinear $ m$-accretive operators in Banach spaces, Proc Amer. Math. Soc. 123 (1995), 145-156. MR 1213863 (95c:47072)
  • [19] -, Sets in the ranges of nonlinear accretive operators in Banach spaces, Studia Math. (to appear). MR 1338831 (97h:47053)
  • [20] N. Kenmochi, Pseudomonotone operators and nonlinear elliptic boundary value problems, J. Math. Soc. Japan 27 (1975), 121-149. MR 0372419 (51:8628)
  • [21] V. Lakshmikantham and S. Leela, Nonlinear differential equations in abstract spaces, Pergamon Press, Oxford, 1981. MR 616449 (82i:34072)
  • [22] M. Nagumo, Degree of mapping in convex linear topological spaces, Amer. J. Math. 73 (1951), 497-511. MR 0042697 (13:150b)
  • [23] D. Pascali and Sburlan, Nonlinear mappings of monotone type, Sijthoff and Noordhoof, Bucharest, 1978.
  • [24] J. Prüß, A characterization of uniform convexity and applications to accretive operators, Hiroshima Math. J. 11 (1981), 229-234. MR 620534 (82g:46040)
  • [25] E. Zeidler, Nonlinear functional analysis and its applications, II/B, Springer-Verlag, New York, 1990. MR 1033498 (91b:47002)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1297527-2
Keywords: Maximal monotone operator, pseudo-monotone operator, $ m$-accretive operator, compact perturbation, compact resolvent, range of sums
Article copyright: © Copyright 1995 American Mathematical Society

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