PERTURBATION ANALYSIS OF METRIC SUBREGULARITY FOR MULTIFUNCTIONS

Considering a closed multifunction \Psi between two Banach spaces, it is known that metric regularity and strong metric subregularity of \Psi are, respectively, stable with respect to ``small Lipschitz perturbations"" and ``small calm perturbations,"" but the corresponding results are no longer true for metric subregularity of \Psi . This paper further deals with the stability issues of metric subregularity with respect to these two kinds of perturbations. We prove that either metric regularity or strong metric subregularity of \Psi at (x=\, y=\) is sufficient for the stability of metric subregularity of \Psi at (x=\, y=\) with respect to small calm subsmooth perturbations and that, under the convexity assumption on \Psi , it is also necessary for the stability of metric subregularity of \Psi at (x=\, y=\) with respect to small calm subsmooth (or Lipschitz) perturbations. Moreover, in terms of the coderivative of \Psi , we provide some sufficient and necessary conditions for metric subregularity of \Psi to be stable with respect to small calm perturbations. Some results obtained in this paper improve and generalize the corresponding results for error bounds in the literature.