Sets in the ranges of sums for perturbations of nonlinear 𝑚-accretive operators in Banach spaces

Kartsatos, Athanassios G.

doi:10.1090/S0002-9939-1995-1213863-5

Sets in the ranges of sums for perturbations of nonlinear $m$-accretive operators in Banach spaces
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Proc. Amer. Math. Soc. 123 (1995), 145-156 Request permission

Abstract:

Several results are given involving nonlinear range inclusions of the types $B + D \subset \overline {R(T + C)}$ and $\operatorname {int} (B + D) \subset R(T + C)$, where B, D are subsets of a real Banach space X, the operator $T:X \supset D(T) \to {2^X}$ is at least m-accretive, and the perturbation $C:X \supset D(C) \to X$ is at least compact, or demicontinuous, or m-accretive. Leray-Schauder degree theory is used in most of the results, and extended versions of recent results of Calvert and Gupta, Morales, Reich, and the author are shown to be possible by using mainly homotopies of compact transformations.

References

Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843, DOI 10.1007/978-94-010-1537-0
Felix E. Browder, On a principle of H. Brézis and its applications, J. Functional Analysis 25 (1977), no. 4, 356–365. MR 0445344, DOI 10.1016/0022-1236(77)90044-1
Bruce D. Calvert and Chaitan P. Gupta, Nonlinear elliptic boundary value problems in $L^{p}$-spaces and sums of ranges of accretive operators, Nonlinear Anal. 2 (1978), no. 1, 1–26. MR 512651, DOI 10.1016/0362-546X(78)90038-X
Bruce D. Calvert and Athanassios G. Kartsatos, On the convexity of the interior of the domain of an $m$-accretive operator in a Banach space, Nonlinear Anal. 12 (1988), no. 7, 727–732. MR 947885, DOI 10.1016/0362-546X(88)90025-9
Athanassios G. Kartsatos, Mapping theorems involving ranges of sums of nonlinear operators, Nonlinear Anal. 6 (1982), no. 3, 271–278. MR 654318, DOI 10.1016/0362-546X(82)90094-3
Athanassios G. Kartsatos, On the solvability of abstract operator equations involving compact perturbations of $m$-accretive operators, Nonlinear Anal. 11 (1987), no. 9, 997–1004. MR 907819, DOI 10.1016/0362-546X(87)90080-0
Athanassios G. Kartsatos, Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces, World Congress of Nonlinear Analysts ’92, Vol. I–IV (Tampa, FL, 1992) de Gruyter, Berlin, 1996, pp. 2197–2222. MR 1389246
V. Lakshmikantham and S. Leela, Nonlinear differential equations in abstract spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications, vol. 2, Pergamon Press, Oxford-New York, 1981. MR 616449
N. G. Lloyd, Degree theory, Cambridge Tracts in Mathematics, No. 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978. MR 0493564
Claudio H. Morales, On the range of sums of accretive and continuous operators in Banach spaces, Nonlinear Anal. 19 (1992), no. 1, 1–9. MR 1171607, DOI 10.1016/0362-546X(92)90026-B
Simeon Reich, The range of sums of accretive and monotone operators, J. Math. Anal. Appl. 68 (1979), no. 1, 310–317. MR 531440, DOI 10.1016/0022-247X(79)90117-3
Simeon Reich and Ricardo Torrejón, Zeros of accretive operators, Comment. Math. Univ. Carolin. 21 (1980), no. 3, 619–625. MR 590139
Ricardo M. Torrejón, Remarks on nonlinear functional equations, Nonlinear Anal. 6 (1982), no. 3, 197–207. MR 654312, DOI 10.1016/0362-546X(82)90088-8