The notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex, lower semicontinuous, and proper function and any (not necessarily symmetric) continuous linear positive operator. It was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing Fitzpatrick functions. In this pa...

In this paper, we analyse theoretical properties of the slice sampler. We nd that the algorithm has extremely robust geometric ergodicity properties. For the case of just one auxiliary variable, we demonstrate that the algorithm is stochasti-cally monotone, and deduce analytic bounds on the total variation distance from stationarity of the method using Foster-Lyapunov drift condition methodology.

we introduce a new concept of general $g$-$eta$-monotone operator generalizing the general $(h,eta)$-monotone operator cite{arvar2, arvar1}, general $h-$ monotone operator cite{xiahuang} in banach spaces, and also generalizing $g$-$eta$-monotone operator cite{zhang}, $(a, eta)$-monotone operator cite{verma2}, $a$-monotone operator cite{verma0}, $(h, eta)$-monotone operator cite{fanghuang}, $h$-...

An acyclic compactum in an orientable, open 3-manifold has arbitrarily close, polyhedral neighborhoods whose components are compact 3-manifolds with a special structure. Frequently, these 3-manifolds have free fundamental groups. These observations and some results from combinatorial group theory are exploited to deduce facts about the homomorphism of fundamental groups induced by an acyclic ma...

The existence of monotone solutions for a second-order functional differential inclusion is obtained in the case when the multifunction that define the inclusion is upper semicontinuous compact valued and contained in the Fréchet subdifferential of a φ-convex function of order two.

For linear and fully non-linear diffusion equations of BellmanIsaacs type, we introduce a class of monotone approximation schemes relying on monotone interpolation. As opposed to classical numerical methods, these schemes converge for degenerate diffusion equations having general nondiagonal dominant coefficient matrices. Such schemes have to have a wide stencil in general. Besides providing a ...

In this paper, the well-known Egoroff’s theorem in classical measure theory is established on monotone non-additive measure spaces. Taylor’s theorem, which concerns almost everywhere convergence of measurable function sequence in classical measure theory, is also generalized. The converse problem of the theorems are discussed, and a necessary and sufficient condition for the Egoroff’s theorem i...

Motivated by a classical result concerning the ε-subdifferential of the sum of two proper, convex and lower semicontinuous functions, we give in this paper a similar result for the enlargement of the sum of two maximal monotone operators defined on a Banach space. This is done by establishing a necessary and sufficient condition for a bivariate inf-convolution formula.

A 0-continuum (9„-continuum) is a compact, connected, metric space that is not separated into infinitely many (more than n) components by any subcontinuum. The following results are among those proved. The first generalizes earlier joint work with E. J. Vought for ^„-continua, and the second generalizes earlier work by Vought for 6,-continua. A 0-continuum X admits a monotone, upper semicontinu...

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 subdiierential @f is quasimonotone. As an immediate consequence, we get that f is convex if and only if @f is monotone.