Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

New results in the perturbation theory of
maximal monotone and $M$-accretive
operators in Banach spaces


Author: Athanassios G. Kartsatos
Journal: Trans. Amer. Math. Soc. 348 (1996), 1663-1707
MSC (1991): Primary 47H17; Secondary 47B44, 47H09, 47H10
DOI: https://doi.org/10.1090/S0002-9947-96-01654-6
MathSciNet review: 1357397
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $X$ be a real Banach space and $G$ a bounded, open and convex subset of $X.$ The solvability of the fixed point problem $(*)~Tx+Cx \owns x$ in $D(T)\cap \overline{G}$ is considered, where $T:X\supset D(T)\to 2^{X}$ is a possibly discontinuous $m$-dissipative operator and $C: \overline{G}\to X$ is completely continuous. It is assumed that $X$ is uniformly convex, $D(T)\cap G \not = \emptyset $ and $(T+C)(D(T)\cap \partial G)\subset \overline{G}.$ A result of Browder, concerning single-valued operators $T$ that are either uniformly continuous or continuous with $X^{*}$ uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let $\Gamma = \{\beta :\mathcal{R}_{+}\to \mathcal{R}_{+}~;~\beta (r)\to 0\text{ as }r\to \infty \}.$ The effect of a weak boundary condition of the type $\langle u+Cx,x\rangle \ge -\beta (\|x\|)\|x\|^{2}$ on the range of operators $T+C$ is studied for $m$-accretive and maximal monotone operators $T.$ Here, $\beta \in \Gamma ,~x\in D(T)$ with sufficiently large norm and $u\in Tx.$ Various new eigenvalue results are given involving the solvability of $Tx+ \lambda Cx\owns 0$ with respect to $(\lambda ,x)\in (0,\infty )\times D(T).$ Several results do not require the continuity of the operator $C.$ Four open problems are also given, the solution of which would improve upon certain results of the paper.


References [Enhancements On Off] (What's this?)

  • 1. Y. I. Alber, On the solution of nonlinear equations with monotone operators in a Banach space, Siberian Math. J. 16 (1975), 3-11. MR 51:6512
  • 2. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff Int. Publ., Leyden (The Netherlands), 1975. MR 52:11666
  • 3. P. Bénilan, Équations d'Évolution dans un Espace de Banach Quelconque et Applications, Thése, Orsay, 1972.
  • 4. H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl Math. 23 (1970), 123-144. MR 41:2454
  • 5. F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Symp. Pure Appl. Math., 18, Part 2, Amer. Math. Soc., Providence, 1976. MR 53:8982
  • 6. B. D. Calvert and C. P. Gupta, Nonlinear elliptic boundary value problems in $L^{p}-$spaces and sums of ranges of accretive operators, Nonlinear Anal. 2 (1978), 1-26. MR 80i:35083
  • 7. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Acad. Publ., Boston, 1990. MR 91m:46021
  • 8. K. Deimling, Zeros of accretive operators, Manuscr. Math. 13 (1974), 365-374. MR 50:3030
  • 9. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. MR 86j:47001
  • 10. Z. Ding and A. G. Kartsatos, Nonzero solutions of nonlinear equations involving compact perturbations of accretive operators in Banach spaces, Nonlinear Anal. 25 (1995), 1333--1342.
  • 11. Z. Ding and A. G. Kartsatos, P-Regular mappings and alternative results for perturbations of $m$-accretive operators in Banach spaces, Topol. Meth. Nonl. Anal. (to appear).
  • 12. P. M. Fitzpatrick and W. V. Petryshyn, On the nonlinear eigenvalue problem $T(u)=\lambda C(u)$ involving noncompact abstract and differential operators, Boll. Un. Mat. Ital. B (5) 15 (1978), 80-107. MR 80b:47083
  • 13. Z. Guan, Ranges of operators of monotone type in Banach spaces, J. Math. Anal. Appl. 174 (1993), 256-264. MR 95b:47068
  • 14. Z. Guan, Solvability of semilinear equations with compact perturbations of operators of monotone type, Proc. Amer. Math. Soc. 121 (1994), 93-102. MR 94g:47080
  • 15. 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. 21 (1995), 149-188. CMP 95:12
  • 16. Z. Guan and A. G. Kartsatos, On the eigenvalue problem for perturbations of nonlinear accretive and monotone operators in Banach spaces, Nonlinear Anal. (to appear).
  • 17. Z. Guan and A. G. Kartsatos, Ranges of perturbed maximal monotone and $m$-accretive operators in Banach spaces, Trans. Amer. Math. Soc. 347 (1995), 2403-2435. MR 95i:47096
  • 18. C. P. Gupta and P. Hess, Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems, J. Diff. Eq. 22 (1976), 305-313. MR 57:13600
  • 19. N. Hirano and A. K. Kalinde, On perturbations of $m$-accretive operators in Banach spaces, Proc. Amer. Math. Soc. (to appear).
  • 20. D. R. Kaplan and A. G. Kartsatos, Ranges of sums and control of nonlinear evolutions with preassigned responses, J. Optim. Theory Appl. 81 (1994), 121-141. CMP 94:12
  • 21. A. G. Kartsatos, Zeros of demicontinuous accretive operators in reflexive Banach spaces, Integral Equations 8 (1985), 175-184. MR 86j:47078
  • 22. A. G. Kartsatos, On compact perturbations and compact resolvents of nonlinear m-accretive operators in Banach spaces, Proc. Amer. Math. Soc. 119 (1993), 1189-1199. MR 94c:47076
  • 23. A. G. Kartsatos, Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces, Proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida, 1992, Walter De Gruyter, New York, (1995), pp. 2197-2222.
  • 24. A. G. Kartsatos, 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 95c:47072
  • 25. A. G. Kartsatos, On the construction of methods of lines for functional evolutions in general Banach spaces, Nonlinear Anal. 25 (1995), 1321-1331.
  • 26. A. G. Kartsatos, A compact evolution operator generated by a time-dependent $m$-accretive operator in a general Banach space, Math. Ann. 302 (1995), 473-487. CMP 95:15
  • 27. A. G. Kartsatos, Sets in the ranges of nonlinear accretive operators in Banach spaces, Studia Math. 114 (1995), 261-273. CMP 95:14
  • 28. A. G. Kartsatos, On the perturbation theory of $m$-accretive operators in Banach spaces, Proc. Amer. Math. Soc. (to appear).
  • 29. A. G. Kartsatos and X. Liu, Nonlinear equations involving compact perturbations of $m$-accretive operators in Banach spaces, Nonlinear Anal. 24 (1995), 469-492. CMP 95:7
  • 30. A. G. Kartsatos and R. D. Mabry, Controlling the space with pre-assigned responses, J. Optim. Theory Appl. 54 (1987), 517-540. MR 88j:49024
  • 31. V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, Oxford, 1981. MR 82i:34072
  • 32. N. G. Lloyd, Degree Theory, Cambridge Univ. Press, Cambridge, 1978. MR 58:12558
  • 33. I. Massabo and C. A. Stuart, Positive eigenvectors of $k$-set contractions, Nonlinear Anal. 3 (1979), 35-44. MR 80b:47073
  • 34. M. Nagumo, Degree of mapping in convex linear topological spaces, Amer. J. Math. 73 (1951), 497-511. MR 13:150b
  • 35. D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Sijthoff and Noordhoff Intern. Publ., Alphen aan den Rijn, 1978. MR 80g:47056
  • 36. S. Reich, Extension problems for accretive sets in Banach spaces, J. Funct. Anal. 26 (1977), 378-395. MR 57:17393
  • 37. E. H. Rothe, Introduction to Various Aspects of Degree Theory in Banach Spaces, Math. Surveys and Monographs, 23, A.M.S., Providence, 1986. MR 87m:47145
  • 38. I. V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, Transl. Math. Monographs, 139, A.M.S., Providence, 1994. MR 95i:35109
  • 39. M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden-Day, Inc., San Francisco, 1964. MR 31:638
  • 40. I. I. Vrabie, Compactness Methods in Nonlinear Evolutions, Longman Sci. Tech., London, 1987. MR 90f:47101
  • 41. E. Zeidler, Nonlinear Functional Analysis and its Applications, II/B, Springer-Verlag, New York, 1990. MR 91b:47002

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47H17, 47B44, 47H09, 47H10

Retrieve articles in all journals with MSC (1991): 47H17, 47B44, 47H09, 47H10


Additional Information

Athanassios G. Kartsatos
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
Email: hermes@gauss.math.usf.edu

DOI: https://doi.org/10.1090/S0002-9947-96-01654-6
Keywords: $m$-accretive operator, maximal monotone operator, compact perturbation, compact resolvent, eigenvalues for nonlinear operators, fixed point theory, degree theory
Received by editor(s): February 7, 1995
Additional Notes: The results of this paper were announced in a lecture at the International Conference on Nonlinear Differential Equations, Kiev, Ukraine, August 21-27, 1995.
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society