Metric subregularity for subsmooth generalized constraint equations in Banach spaces (Q1952781)

Let \(X\) and \(Y\) be Banach spaces, and let \(F:X \rightrightarrows Y\) be a closed multifunction. Next, let \(b\in Y\) be a given point , and \(A\subset X\) be a closed set. In the paper, the following constraint equation \[ b\in F(x) \eqno (1) \] on the set \(A\), is considred. The equation (1) is called metrically subregular at \(a\in S=\{x\in A:b\in F(x)\}\) if there exist \(\tau,\delta \in (0,\infty)\) such that \[ d(x,S)\leq \tau d(b,F(x))+d(x,A) \; \forall x\in B(a,\delta), \] where \(B(a,\delta)\) denotes the open ball of center \(a\) and radius \(\delta.\) The equation (1) is called strongly subregular at \(a\in S\) if there exist \(\tau,\delta \in (0,\infty)\) such that \[ \|x-a\|\leq \tau d(b,F(x))+d(x,A) \; \forall x\in B(a,\delta). \] In this paper, necessary and sufficient conditions are obtained for subsmooth constraint equations to be metrically and strongly metrically subregular.