Second-order differentiability of generalized perturbation maps

The second-order contingent derivative of generalized perturbation maps

“…Then, we modify main results concerning the sum rule of second-order contingent derivatives for set-valued maps and its application to the sensitivity analysis of generalized perturbation maps. The obtained results are new and better than those in [1]. Some examples are proposed to illustrate our results.…”

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In the paper, we give some remarks on [1]. Then, we modify main results concerning the sum rule of second-order contingent derivatives for set-valued maps and its application to the sensitivity analysis of generalized perturbation maps.

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Calculus of tangent sets and derivatives of set-valued maps under metric subregularity conditions

“…One speaks here about conditions concerning several types of generalized compactness for the graphs of underlying set-valued maps and such requirements are known to be quite strong in infinite dimensional spaces. Notice as well that the second-order conditions used in [11] are also of the same nature. It is true that there is a difference between the conditions used in [13] (working on finite dimensional vector spaces for first order objects) and [11] (working on general Banach spaces for second order objects) as the authors of the later paper emphasize, but, however, the techniques and the arguments are not so different.…”

“…Notice as well that the second-order conditions used in [11] are also of the same nature. It is true that there is a difference between the conditions used in [13] (working on finite dimensional vector spaces for first order objects) and [11] (working on general Banach spaces for second order objects) as the authors of the later paper emphasize, but, however, the techniques and the arguments are not so different. Another remark is that in all these papers the approach inherits the main line of arguments and techniques from [10,Chapter 3].…”

“…Once one has a concept of tangent set to an arbitrary set in a normed vector space, then one can construct a corresponding derivative for set-valued maps, and after that the question of calculus rules for these objects arises naturally. Several recent papers in literature are devoted to the study of such calculus for Bouligand and Ursescu first and second order tangent sets: we cite here [13], [12], [11] and the references therein. The starting point of this paper is the remark that, in general, the quoted papers employ quite strong conditions on the initial data in order to get the desired calculus.…”