Sets in the ranges of nonlinear accretive operators in Banach spaces

• Autorzy: Athanassios G. Kartsatos
• Języki publikacji: EN

Abstrakty

EN

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 $M ∩ \overline{T(∂ G)} = ∅$. Then $M ⊂ \overline{TG}$. If, moreover, Case (i) or (ii) holds and T is of type $(S_1)$, or Case (iii) holds and T is of type $(S_2)$, then M ⊂ TG. Various results of Morales, Reich and Torrejón, and the author are improved and/or extended.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 47H05: Monotone operators and generalizations
• 47H10: Fixed-point theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1995
• Tom: 114
• Numer: 3
• Strony: 261-273

Daty

wydano

1995

otrzymano

1994-03-30

Twórcy

autor

Athanassios G. Kartsatos

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

Bibliografia

• [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.