Constrained Equilibrium Point of Maximal Monotone Operator Via Variational Inequality (original) (raw)

Characterization of the subdifferentials of convex functions

Pacific Journal of Mathematics, 1966

Each lower semi-continuous proper convex function / on a Banach space E defines a certain multivalued mapping df from E to E* called the subdifferential of /. It is shown here that the mappings arising this way are precisely the ones whose graphs are maximal "cyclically monotone" relations on E X E*, and that each of these is also a maximal monotone relation. Furthermore, it is proved that df determines / uniquely up to an additive constant. These facts generally fail to hold when E is not a Banach space. The proofs depend on establishing a new result which relates the directional derivatives of / to the existence of approximate subgradients.

Maximal Monotonicity, Subdifferentials and Generalizations

Nonconvex Optimization and Its Applications, 2001

In 1966 R.T. Rockafellar [14] exploring the operator of subdifferential of convex function on a Banach space revealed that the operator is maximal monotone i.e. if for every (x, x*) E graph of it holds (x*-q*, x-q) :2: 0, then q* E of(q).

The maximal monotonicity of the subdifferentials of convex functions: Simons' problem

Set-Valued Analysis, 1996

In the paper we deal with the problem when the graph of the subdifferential operator of a convex lower semicontinuous function has a common point with the product of two convex nonempty weak and weak* compact sets, i.e. when graph&j n (Q x Q*) # 0. The results obtained partially solve the problem posed by Simons as well as generalize the Rockafellar Maximal Monotonicity Theorem.

Subdifferential calculus for convex operators

Journal of Mathematical Analysis and Applications, 1981

Well-known scalar results on the subdifferential of composite functions are extended to the framework of ordered topological vector spaces. This is done by using the sandwich theorem for convex operators which is derived from an extension of the Hahn-Banach theorem.

Convex functions, subdifferentiability and renormings

Acta Mathematica Sinica, 1998

This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces.

Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems

Abstract and Applied Analysis, 2016

LetXbe a real locally uniformly convex reflexive Banach space with locally uniformly convex dual spaceX⁎. LetT:X⊇D(T)→2X⁎andA:X⊇D(A)→2X⁎be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved forT+Aunder weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used conditionD(T)∘∩D(A)≠∅and Browder and Hess who used the quasiboundedness ofTand condition0∈D(T)∩D(A). In particular, the maximality ofT+∂ϕis proved provided thatD(T)∘∩D(ϕ)≠∅, whereϕ:X→(-∞,∞]is a proper, convex, and lower semicontinuous function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator.

Subdifferential Characterization of Quasiconvexity and Convexity

Journal of Convex Analysis, 1994

Let f: X! IR f+1g be a lower semicontinuous function on a Banach space X. We show that f is quasiconvex if and only if its Clarke subdi erential@ f is quasimonotone. As an immediate consequence, we get that f is convex if and only if@ f is monotone.