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On the maximal monotonicity and the range of the sum of nonlinear maximal monotone operators

Published online by Cambridge University Press:  20 January 2009

J. R. L. Webb  and
Weiyu Zhao
J. R. L. Webb
Affiliation:
Department of MathematicsUniversity of GlasgowGlasgow G12 8QW
Weiyu Zhao
Affiliation:
Department of MathematicsUniversity of GlasgowGlasgow G12 8QW
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Abstract

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Conditions are given on two maximal monotone (multivalued) operators A and B which ensure that A + B is also maximal. One condition used is that ∥Bx∥≦k(∥x∥)Ax| +d|(A + B)x| + c(∥x∥) for every x∈D(A)⊆D(B), where 0≦k(r)<1, and c(r)≧0 are nondecreasing functions, and 0≦d≦1 is a constant. Here, for a set C, |C| denotes inf{∥y∥:y∈C}. This extends the well known result which has d = 0 (and is used in the proof here). The second part of the paper uses similar hypotheses to give conditions under which the range of the sum, R(A + B), has the same interior and same closure as the sum of the ranges, R(A) + R(B).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 1 , February 1991 , pp. 143 - 153
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

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