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On an operator equation involving mappings of monotone type

Author: Chaitan P. Gupta
Journal: Proc. Amer. Math. Soc. 53 (1975), 143-148
MSC: Primary 47H05
MathSciNet review: 0377610
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Abstract: Let $ X$ be a real reflexive Banach space and $ A:X \to {2^{{X^{\ast}}}}$ a maximal monotone mapping such that the graph $ G(A)$ of $ A$ is weakly-closed in $ X \times {X^{\ast}}$ and $ 0\epsilon A(0)$. Also, let $ T:X \to {2^{{X^{\ast}}}}$ be a quasi-bounded coercive mapping of type $ ({\text{M}})$ such that the effective domain $ D(T)$ of $ T$ contains a dense linear subspace $ {X_0}$ of $ X$. Then it is shown that for each $ \omega \epsilon {X^{\ast}}$ there exists a $ u\epsilon X$ such that $ \omega \epsilon Au + Tu$ and the subset $ \{ u\epsilon X\vert\omega \epsilon Au + Tu\} $ is a weakly-compact subset of $ X$. An application to an elliptic nonlinear boundary value problem of Neumann type is given.

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Keywords: Monotone operators, maximal-monotone, generalized pseudo-monotone, type $ ({\text{M)}}$, quasi-bounded, coercive, effective-domain
Article copyright: © Copyright 1975 American Mathematical Society