Download PDF - Sets in the ranges of nonlinear accretive operators in Banach spacesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Sets in the ranges of nonlinear accretive operators in Banach spaces

Authors 1

Affiliations

  1. Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700, U.S.A.

Abstract

Let X be a real Banach space and G ⊂ X open and bounded. Assume that one of the following conditions is satisfied: (i) X* is uniformly convex and T:Ḡ→ X is demicontinuous and accretive; (ii) T:Ḡ→ X is continuous and accretive; (iii) T:X ⊃ D(T)→ X is m-accretive and Ḡ ⊂ D(T). Assume, further, that M ⊂ X is pathwise connected and such that M ∩ TG ≠ ∅ and M∩T(∂G)¯=∅. Then M⊂TG¯. If, moreover, Case (i) or (ii) holds and T is of type (S1), or Case (iii) holds and T is of type (S2), then M ⊂ TG. Various results of Morales, Reich and Torrejón, and the author are improved and/or extended.

Keywords

ENG
accretive operator, m-accretive operator, compact perturbations, compact resolvents, Leray-Schauder boundary condition, mapping theorems

Bibliography

  1. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1975.
  2. F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math. 18, Part 2, Amer. Math. Soc., Providence, 1976.
  3. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Boston, 1990.
  4. K. Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974), 365-374.
  5. J. Gatica and W. A. Kirk, Fixed point theorems for Lipschitzian pseudo-contractive mappings, Proc. Amer. Math. Soc. 36 (1972), 111-115.
  6. J. Gatica and W. A. Kirk, Fixed point theorems for contraction mappings with applications to nonexpansive and pseudo-contractive mappings, Rocky Mountain J. Math. 4 (1974), 69-79.
  7. D. R. Kaplan and A. G. Kartsatos, Ranges of sums and the control of nonlinear evolutions with pre-assigned responses, J. Optim. Theory Appl. 81 (1994), 121-141.
  8. A. G. Kartsatos, Some mapping theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 82 (1981), 169-183.
  9. A. G. Kartsatos, Zeros of demicontinuous accretive operators in reflexive Banach spaces, J. Integral Equations 8 (1985), 175-184.
  10. A. G. Kartsatos, On the solvability of abstract operator equations involving compact perturbations of m-accretive operators, Nonlinear Anal. 11 (1987), 997-1004.
  11. 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.
  12. A. G. Kartsatos, Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces, in: Proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida, 1992, Walter de Gruyter, New York, to appear.
  13. A. G. Kartsatos, On the construction of methods of lines for functional evolutions in general Banach spaces, Nonlinear Anal., to appear.
  14. A. G. Kartsatos, Existence of zeros and asymptotic behaviour of resolvents of maximal monotone operators in reflexive Banach spaces, to appear.
  15. A. G. Kartsatos and R. D. Mabry, Controlling the space with pre-assigned responses, J. Optim. Appl. Theory 54 (1987), 517-540.
  16. W. A. Kirk, Fixed point theorems for nonexpansive mappings satisfying certain boundary conditions, Proc. Amer. Math. Soc. 50 (1975), 143-149.
  17. W. A. Kirk and R. Schöneberg, Some results on pseudo-contractive mappings, Pacific J. Math. 71 (1977), 89-100.
  18. W. A. Kirk and R. Schöneberg, Zeros of m-accretive operators in Banach spaces, Israel J. Math. 35 (1980), 1-8.
  19. V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, Oxford, 1981.
  20. C. Morales, Nonlinear equations involving m-accretive operators, J. Math. Anal. Appl. 97 (1983), 329-336.
  21. C. Morales, Existence theorems for demicontinuous accretive operators in Banach spaces, Houston J. Math. 10 (1984), 535-543.
  22. C. Morales, Zeros for accretive operators satisfying certain boundary conditions, J. Math. Anal. Appl. 105 (1985), 167-175.
  23. M. Nagumo, Degree of mapping in convex linear topological spaces, Amer. J. Math. 73 (1951), 497-511.
  24. S. Reich and R. Torrejón, Zeros of accretive operators, Comment. Math. Univ. Carolin. 21 (1980), 619-625.
  25. R. Torrejón, Some remarks on nonlinear functional equations, in: Contemp. Math. 18, Amer. Math. Soc., 1983, 217-246.
Studia Mathematica
1995/114/3
Export to PBN, Opens in new tab
Pages:
261-273
Main language of publication
English
Received
1994-03-30
Published
1995
Exact and natural sciences