Perturbations of maximal monotone random operators

Let X be a Banach space, X¹ its dual, and Ω a measurable space. We study the solvability of nonlinear random equations involving operators of the form L + T, where L is a maximal monotone random operator from Ω × X into X¹ and T : Ω × X → X¹ a random operator of monotone type.     

Cyclical monotonicity of maximal monotone step operators

Summary Maximal monotone operatorsT:X→2 ^y such that {Tx} _xεx is a finite family of sets are shown to be cyclically monotone.     

On the convergence of maximal monotone operators

We study the convergence of maximal monotone operators with the help of representations by convex functions. In particular, we prove the convergence of a sequence of sums of maximal monotone operators under a general qualification condition of the Attouch-Brezis type.

     

Non-convex sweeping processes involving maximal monotone operators

By using a regularization method, we study in this paper the global existence and uniqueness property of a new variant of non-convex sweeping processes involving maximal monotone operators. The system can be considered as a maximal monotone differential inclusion under a control term of normal cone type forcing the trajectory to be always contained in the desired moving set. When the set is fixed, one can show that the unique solution is right-differentiable everywhere and its right-derivative is right-continuous. Non-smooth Lyapunov pairs for this system are also analysed.     

Differential inclusions with nonstationary maximal monotone operators

Institute for Control Problems. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 4, pp. 14–24, October–December, 1990.     

Some eigenvalue results for maximal monotone operators

We study the eigenvalue problem of the form

0∈Tx−λCx,     

Maximality of sums of two maximal monotone operators

We use methods from convex analysis, relying on an ingenious function of Simon Fitzpatrick, to prove maximality of the sum of two maximal monotone operators on reflexive Banach space under weak transversality conditions.

     

Existence via compactness for maximal monotone elliptic operators

In this Note we propose a new method of proving the existence of solutions to $?\mathrm{div}A(x,\mathrm{?}u)?f$, when $A(x,\mathrm{?}u)$ has x-dependent maximal monotone graph. The idea is based on the theory of Young measures and on the method of compensated compactness. Alternative approaches were proposed elsewhere. However, our method allows us to obtain also the strong convergence of approximate solutions. To cite this article: P. Gwiazda, A. Zatorska-Goldstein, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On maximal monotone operators with relatively compact range

It is shown that every maximal monotone operator on a real Banach space with relatively compact range is of type NI. Moreover, if the space has a separable dual space then every maximally monotone operator T can be approximated by a sequence of maximal monotone operators of type NI, which converge to T in a reasonable sense (in the sense of Kuratowski-Painleve convergence).     

New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators

The purpose of this article is to prove the strong convergence theorems for hemi-relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems for hemi-relatively nonexpansive mappings, a new monotone hybrid iteration algorithm is presented and is used to approximate the fixed point of hemi-relatively nonexpansive mappings. Noting that, the general hybrid iteration algorithm can be used for relatively nonexpansive mappings but it can not be used for hemi-relatively nonexpansive mappings. However, this new monotone hybrid algorithm can be used for hemi-relatively nonexpansive mappings. In addition, a new method of proof has been used in this article. That is, by using this new monotone hybrid algorithm, we firstly claim that, the iterative sequence is a Cauchy sequence. The results of this paper modify and improve the results of Matsushita and Takahashi, and some others.     

A characterization of boundary conditions yielding maximal monotone operators

We provide a characterization for maximal monotone realizations for a certain class of (nonlinear) operators in terms of their corresponding boundary data spaces. The operators under consideration naturally arise in the study of evolutionary problems in mathematical physics. We apply our abstract characterization result to Port–Hamiltonian systems and a class of frictional boundary conditions in the theory of contact problems in visco-elasticity.     

Parallel algorithms for maximal monotone operators of local type

Summary. This paper presents general algorithms for the parallel solution of finite element problems associated with maximal monotone operators of local type. The latter concept, which is also introduced here, is well suited to capture the idea that the given operator is the discretization of a differential operator that may involve nonlinearities and/or constraints as long as those are of a local nature. Our algorithms are obtained as a combination of known algorithms for possibly multi-valued maximal monotone operators with appropriate decompositions of the domain. This work extends a method due to two of the authors in the single-valued and linear case. Received April 25, 1994     

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X?D(T) → 2 ^X * is a maximal monotone multi-valued operator and C: X?D(C) → X* is a generalized pseudomonotone quasibounded operator with L ? D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T _x + λ C _x , with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.     

A new proof for Rockafellar's characterization of maximal monotone operators

We provide a new and short proof for Rockafellar's characterization of maximal monotone operators in reflexive Banach spaces based on S. Fitzpatrick's function and a technique used by R. S. Burachik and B. F. Svaiter for proving their result on the representation of a maximal monotone operator by convex functions.