Limit Superior of Subdifferentials of Uniformly Convergent Functions

In this paper we show that the – subdifferential of a lower semicontinuous function is contained in the limit superior of the – subdifferential of lower semicontinuous uniformly convergent family to this function. It happens that this result is equivalent to the corresponding normal cones formulas for family of sets which converges in the sense of the bounded Hausdorff distance. These results extend to the infinite dimensional case those of Ioffe for – functions and of Benoist for Clarkes normal cone. As an application we characterize the subdifferential of any function which is bounded from below by a negative quadratic form in terms of its Moreau–Yosida proximal approximation.     

A differential equation approach to implicit sweeping processes

In this paper, we study an implicit version of the sweeping process. Based on methods of convex analysis, we prove the equivalence of the implicit sweeping process with a differential equation, which enables us to show the existence and uniqueness of the solution to the implicit sweeping process in a very general framework. Moreover, this equivalence allows us to give a characterization of nonsmooth Lyapunov pairs and invariance for implicit sweeping processes. The results of the paper are illustrated with two applications to quasistatic evolution variational inequalities and electrical circuits.     

An isothermal (30 °C) study of heterogeneous equilibria in the sucrose (C12H22O11)-calcium hydroxide (Ca(OH)2)-water system

Solid-liquid equilibria in the C₁₂H₂₂O₁₁-Ca(OH)₂-H₂O system have been determined at 30 °C. Two phases C₁₂H₂₂O₁₁·3Ca(OH)₂ and C₁₂H₂₂O₁₁·2Ca (OH)₂ have been evidenced, and solubility (liquidus) curves have been determined. It is thus shown that these ternary phases exhibit incongruent solubilities at 30 °C. These solid phases were characterized using Infrared Spectroscopy (IR), Scanning Electron Microscopy (SEM) and Inductively Coupled Plasma Atomic Emission Spectroscopy (ICP-AES).     

Qualification conditions for multivalued functions in Banach spaces with applications to nonsmooth vector optimization problems

In this paper we introduce qualification conditions for multivalued functions in Banach spaces involving the A-approximate subdifferential, and we show that these conditions guarantee metric regularity of multivalued functions. The results are then applied for deriving Lagrange multipliers of Fritz—John type and Kuhn—Tucker type for infinite non-smooth vector optimization problems.     

Sufficient Conditions for Error Bounds and Applications

Our aim in this paper is to present sufficient conditions for error bounds in terms of Fréchet and limiting Fréchet subdifferentials in general Banach spaces. This allows us to develop sufficient conditions in terms of the approximate subdifferential for systems of the form (x, y) C × D, g(x, y, u) = 0, where g takes values in an infinite-dimensional space and u plays the role of a parameter. This symmetric structure offers us the choice of imposing conditions either on C or D. We use these results to prove the nonemptiness and weak-star compactness of Fritz–John and Karush–Kuhn–Tucker multiplier sets, to establish the Lipschitz continuity of the value function and to compute its subdifferential and finally to obtain results on local controllability in control problems of nonconvex unbounded differential inclusions.     

Metric regularity and subdifferential calculus in Banach spaces

In this paper we give verifiable conditions in terms of limiting Fréchet subdifferentials ensuring the metric regularity of a multivalued functionF(x)=–g(x)+D. We apply our results to the study of the limiting Fréchet subdifferential of a composite function defined on a Banach space.     

A general metric regularity in asplund banach spaces

This paper establishes a simple and easily-applied criterion for determining whether a multivalued mapping is metrically regular relatively to a subset in the range space.     

Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems

In this paper, we present several constraint qualifications, and we show that these conditions guarantee the nonvacuity and the boundedness of the Lagrange multiplier sets for general nondifferentiable programming problems. The relationships with various constraint qualifications are investigated.The author gratefully acknowledges the comments made by the two referees.     

The positiveness of lower limits of the Hoffman constant in parametric polyhedral programs

If K(t) are sets of admissible solutions in parametric programs then it is natural to ask about the Lipschitz-like property and the lower semi-continuity of the multifunction. Answers to this question are related to the problem of the continuity or Lipschitz continuity of the value function, namely having the lower semi-continuity of K(·) we get the upper semi-continuity of the function easily and the Lipschitz-like property of K(·) leads to the Lipschitz-continuity of it. Herein sufficient conditions to get these properties of the polyhedral multifunction of admissible solutions are given in terms of the lower limit of the Hoffman constant. It is shown that the multifunction is Lipschitz-like at these parameters at which the lower limit of the Hoffman constant are positive.