On an Extension of Minty's Theorem

The notion of monotone operator in a Hilbert space is extended, and a considerably broader class of possibly multivalued operators is introduced. It is shown that well-known Minty's theorems on maximal monotone operators in Hilbert spaces can be extended to the cases of operators belonging to the class. Typical examples of partial differential operators are given to illustrate that the results can be applied to nonlinear equations, which involve nonmonotone differential operators. In 1962 Minty [4] gave the following two important results: ,., A monotone operator A in a Hilbert space H is maximal if W and only if R(I + XA) = H for some X > 0. ,jTs If a monotone operator A in a Hilbert space H is defined on all of H and is continuous, then A is maximal monotone. These results have been applied to show the existence of solutions of a variety of partial differential equations, especially nonlinear elliptic equations. Since then these results have been generalized in various directions to Banach spaces (see for instance Barbu [1]). However, to the best of the author's knowledge, it seems that Minty's theorems have not been extended to the case of nonmonotone operators even though they are linear. In this paper we extend the above results to a class of nonmonotone operators in Hilbert spaces. This class, denoted herein by Jf(a), 0 < a < 1, includes monotone operators as a special class. Our arguments are based on the method exploited in Kömura [3] (cf. [2]). In §1 we introduce classes of multivalued operators, Jf(a), a G [0, 1), and state our main theorems. In §2 we give the proofs after preparing two lemmas. Section 3 contains several remarks on perturbations as well as an example of a linear partial differential operator that is elliptic but not strongly elliptic. 1. Definitions and main results Let H be a complex Hilbert space with norm | • | and inner-product (•, •). By a multivalued operator in H we mean an operator that assigns to each Received by the editors January 4, 1990 and, in revised form, August 28, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 47H15, 47H05.