Semimonotone Operators Research Articles

Monotone type operators in nonreflexive Banach spaces

Let E be a real Banach space, be the dual space of E, be the dual space of . Let be a monotone type mapping. In this paper, first, we introduce the special case when T is the weak* sub-differential of a convex function ϕ and obtain a surjective result for the mapping , where . Second, we show the existence of solutions of the variational inequality problems for strictly quasi-monotone operators and semi-monotone operators. Finally, we construct a degree theory for mappings of the class and then construct a generalized degree for the weak* sub-differential of a convex function.

On a generalization of the method of monotone operators

For a family of operator equations of the first kind with nonlinear nonmonotone hemicontinuous operators in a reflexive Banach space, we prove a theorem on the solvability and a uniform estimate of the solution in the norm of the space. Our approach is related to the method of semimonotone operators but has some essential differences from the latter. In a specific example, we show that our theorem can be used to prove the total (with respect to the set of admissible controls) preservation of solvability for distributed control systems.

Regularity results for semimonotone operators

We introduce the concept of ρ-semimonotone point-to-set operators in Hilbert spaces. This notion is symmetrical with respect to the graph of T, as is the case for monotonicity, but not for other related notions, like e.g. hypomonotonicity, of which our new class is a relaxation. We give a necessary condition for ρ-semimonotonicity of T in terms of Lispchitz continuity of [T + ρ-11]-1 and a sufficient condition related to expansivity of T. We also establish surjectivity results for maximal ρ-semimonotone operators.

Existence Results for System of Variational Inequality Problems with Semimonotone Operators

We introduce the system of variational inequality problems for semimonotone operators in reflexive Banach space. Using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain some existence results for system of variational inequality problems for semimonotone with finite-dimensional continuous operators in real reflexive Banach spaces. The results presented in this paper extend and improve the corresponding results for variational inequality problems studied in recent years.

Vector Variational Inequalities with Semi-monotone Operators

In this paper, the concept of a vector semi-monotone operator is introduced. This concept is applied to establish several existence results of vector variational inequalities. These extend some results of others.

Random equations for weakly semimonotone operators of type (S) and semi-J-monotone operators of type (J-S)

Random equations for weakly semimonotone operators of type (S) and semi-J-monotone operators of type (J-S)

Nonlinear variational inequalities for random weakly semimonotone operators

Nonlinear variational inequalities for random weakly semimonotone operators

On the Semi-monotone Operator Theory and Applications

In this paper,Eis a real Banach space,A:E**×E**→E* is a semi-monotone operator. We first study the variational inequality problem: Findu∈K, such that (A(u,u),v−u)≥0, whereK⊂E** is a closed convex subset; then the operator equationA(u,u)=p*, and we also construct a degree theory for demi-continuous semi-monotone operators in reflexive Banach spaces.

On solvability of variational inequalities with discontinuous semimonotone operators

By using the method of monotone operators, a theorem on the existence of the solution with a special property is obtained for an elliptic variational inequality with discontinuous semimonotone operator; this theorem is then used to prove the existence of a semicorrect solution of a variational inequality with a differential semilinear high-order operator of elliptic type with a nonsymmetric linear part and discontinuous nonlinearity.

Nonlinear elliptic functional equations in nonreflexive Banach spaces

If the coefficient functions Aa have at most polynomial growth in u and its derivatives, we have shown in [ l ] how existence theorems for solutions of variational boundary value problems for the equation A (u) —f may be obtained from monotonicity assumptions on the nonlinear Dirichlet form of A, with extensions to weaker ellipticity assumptions in [3], [4], [9], and [15]. These results extend earlier theorems of Visik [18] obtained by other arguments, but (despite an apparent remark to the contrary in [l5]) do not extend automatically to the case of rapidly increasing coefficients treated by Visik in [ l9] . The crucial point for this more general case is that the Banach spaces in which the problems are appropriately formulated are derived from nonreflexive Orlicz spaces [14] and are themselves nonreflexive and nonseparable. On the other hand, the basic treatments of monotone and semimonotone operator equations have been carried out in reflexive Banach spaces (and in [15], it is even essential to consider only separable spaces). I t is our object in the present note to outline a treatment of these more general problems in nonreflexive spaces, using the remark already made by the writer in the final section of [4] on the useful properties of the weak* topology on a conjugate space. We formulate this treatment in a very general context of functional rather than simply differential equations, and apply to the resulting operator equations a new theorem on perturbation of semimonotone operators (Theorem 2 below) which is particularly efficacious in this general context. This result is extended and applied in more detail in [ l l ] to include all the elliptic results of [ l ] , [3], [4], [9], [IS], [18], and [ l9] , and a corresponding extension to unbounded operators and to monotone operators on convex sets gives a significant extension of