On the range of the sum of monotone operators in general Banach spaces

Riahi, Hassan

doi:10.1090/S0002-9939-96-03314-X

On the range of the sum of monotone operators in general Banach spaces
HTML articles powered by AMS MathViewer

by Hassan Riahi PDF

Proc. Amer. Math. Soc. 124 (1996), 3333-3338 Request permission

Abstract:

The purpose of this paper is to generalize the Brézis-Haraux theorem on the range of the sum of monotone operators from a Hilbert space to general Banach spaces. The result obtained provides that the range $\mathcal R(\overline {A+B}{}^\tau )$ is topologically almost equal to the sum $\mathcal R(A)+\mathcal R(B)$ where $\tau$ is a compatible topology in $X^{**}\times X^*$ as proposed by Gossez. To illustrate the main result we consider some basic properties of densely maximal monotone operators.

References

Hédy Attouch and Haïm Brezis, Duality for the sum of convex functions in general Banach spaces, Aspects of mathematics and its applications, North-Holland Math. Library, vol. 34, North-Holland, Amsterdam, 1986, pp. 125–133. MR 849549, DOI 10.1016/S0924-6509(09)70252-1
Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753
Haïm Brezis and Alain Haraux, Image d’une somme d’opérateurs monotones et applications, Israel J. Math. 23 (1976), no. 2, 165–186. MR 399965, DOI 10.1007/BF02756796
H. Brézis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 225–326. MR 513090
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
Felix E. Browder, Image d’un opérateur maximal monotone et principe de Landesman-Lazer, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 9, A715–A718 (French, with English summary). MR 515673
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
Jean-Pierre Gossez, Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs, J. Math. Anal. Appl. 34 (1971), 371–395 (French). MR 313890, DOI 10.1016/0022-247X(71)90119-3
Jean-Pierre Gossez, On the range of a coercive maximal monotone operator in a nonreflexive Banach space, Proc. Amer. Math. Soc. 35 (1972), 88–92. MR 298492, DOI 10.1090/S0002-9939-1972-0298492-7
Jean-Pierre Gossez, On the extensions to the bidual of a maximal monotone operator, Proc. Amer. Math. Soc. 62 (1976), no. 1, 67–71 (1977). MR 428121, DOI 10.1090/S0002-9939-1977-0428121-5
Chaitan P. Gupta and Peter Hess, Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems, J. Differential Equations 22 (1976), no. 2, 305–313. MR 473942, DOI 10.1016/0022-0396(76)90030-9
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
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
R. R. Phelps, Lectures on maximal monotone operators, Lectures given at Prague-Paseky Summer School, 1993.
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
R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216. MR 262827, DOI 10.2140/pjm.1970.33.209
Eberhard Zeidler, Nonlinear functional analysis and its applications. II/B, Springer-Verlag, New York, 1990. Nonlinear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033498, DOI 10.1007/978-1-4612-0985-0