Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces

The asymptotic behavior of resolvents of a proper convex lower semicontinuous function is studied in the various settings of spaces. In this paper, we consider the asymptotic behavior of the resolvents of a sequence of functions defined in a complete geodesic space. To obtain the result, we assume the Mosco convergence of the sets of minimizers of these functions.

$L^{p}-$Variational solutions of multivalued backward stochastic differential equations

We prove the existence and uniqueness of the $L^{p}-$variational solution, with $p>1,$ of the fo\-llo\-wing multivalued backward stochastic differential equation with $p-$integrable data: \[ \left\{ \begin{array}[c]{l} -dY_{t}+\partial_{y}\Psi(t,Y_{t})dQ_{t}\ni H(t,Y_{t},Z_{t})dQ_{t}-Z_{t}dB_{t},\;0\leq t<\tau,\\[0.2cm] Y_{\tau}=\eta, \end{array} \right. \] where $\tau$ is a stopping time, $Q$ is a progressively measurable increasing continuous stochastic process and $\partial_{y}\Psi$ is the subdifferential of the convex lower semicontinuous function $y\mapsto\Psi(t,y).$

Higher Connectedness Properties of Support Points and Functionals of Convex Sets

We prove that the set of all support points of a nonempty closed convex bounded set C in a real infinite-dimensional Banach space X is AR(σ-compact) and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of C and for the domain, the graph, and the range of the subdifferential map of a proper convex lower semicontinuous function on X.

Minimizing non-convex multiple integrals: a density result

We consider variational problems of the form where Ω is a bounded open set in RN, f : RN → R is a possibly non-convex lower semicontinuous function with p-growth at infinity for some 1 < p < ∞, and the boundary datum u0 is any function in W1, p (Ω). Assuming that the convex envelope f** of f is affine on each connected component of the set {f** < f}, we prove the existence of solutions to ( P) for every continuous function g such that (i) g has no strict local minima and (ii) every convergent sequence of extremum points of g eventually belongs to an interval where g is constant, thus showing that the set of continuous functions g that yield existence to (P) is dense in the space of continuous functions on R.

AN EXISTENCE RESULT FOR VIBRATIONS WITH UNILATERAL CONSTRAINTS

We are motivated by the study of dynamical systems with a finite number of degrees of freedom, subject to unilateral convex constraints without loss of energy at impacts. If we denote the set of constraints by K, the motion is described by [Formula: see text], where ψK is the indicatrix function of K. More generally we consider dynamical systems with a convex potential described by [Formula: see text], where φ is a proper, convex, lower semicontinuous function. We prove that these systems possess a solution whose kinetic energy is conserved through impact in the first case, or more generally, whose energy [Formula: see text] is a continuous function of time in the second case.

Regularisations of convex functions and slicewise suprema

For a number of years, there has been interest in the regularisation of a given proper convex lower semicontinuous function on a Banach space, defined to be the episum (=inf-convolution) of the function with a scalar multiple of the norm. There is an obvious geometric way of characterising this regularisation as the lower envelope of cones lying above the graph of the original function. In this paper, we consider the more interesting problem of characterising the regularisation in terms of approximations from below, expressing the regularisation as the upper envelope of certain subtangents to the graph of the original function. We shall show that such an approximation is sometimes (but not always) valid. Further, we shall give an extension of the whole procedure in which the scalar multiple of the norm is replaced by a more general sublinear functional. As a by-product of our analysis, we are led to the consideration of two senses stronger than the pointwise sense in which a function on a Banach space can be expressed as the upper envelope of a family of functions. These new senses of suprema lead to some questions in Banach space theorey.

Subdifferentials are locally maximal monotone

In a recent paper, Fitzpatrick and Phelps introduced a new class of operators on a Banach space, the locally maximal monotone operators, and showed that this kind of operator can be approximated by a sequence of nicer maximal monotone operators. We give here an affirmative answer to a question posed in this paper: is the subdifferential of a proper convex lower semicontinuous function necessarily locally maximal monotone? Since a locally maximal operator is maximal monotone, our result represents a strengthening of Rockafellar's maximal monotonicity theorem.