It is shown that a set-valued map M : R ⇒ R is maximal monotone if and only if the following five conditions are satisfied: (i) M is monotone; (ii) M has a nearly convex domain; (iii) M is convex-valued; (iv) the recession cone of the values M(x) equals the normal cone to the closure of the domain of M at x; (v) M has a closed graph. We also show that the conditions (iii) and (v) can be replace...

where T is a maximal monotone multi-valued operator and the operator C satisfies condition (S+) or (S̃+). In a regularization method by the duality operator, we use the degree theories of Kartsatos and Skrypnik upon conditions of C as well as Browder’s degree. There are two cases to consider: One is that C is demicontinuous and bounded with condition (S+); and the other is that C is quasibounded...

An important and perhaps interesting topic in nonlinear analysis and convex optimization concerns solving inclusions of the form 0 ∈ A(x), where A is a maximal monotone operator on a Hilbert space H. Its importance in convex optimization is evidenced from the fact that many problems that involve convexity can be formulated as finding zeros of maximal monotone operators. For example, convex mini...

Approximation-solvability of a class of nonlinear implicit variational inequalities involving a class of partially relaxed monotone mappings a computation-oriented class in a Hilbert space settingis presented with some applications. Partially Strongly Monotone Mapping, Partially Relaxed Mono