Stability of Singular Solutions of Nonlinear Equations with Restricted Smoothness Assumptions

Abstract

This work is concerned with conditions ensuring stability of a given solution of a system of nonlinear equations with respect to large (not asymptotically thin) classes of right-hand side perturbations. Our main focus is on those solutions that are in a sense singular, and hence, their stability properties are not guaranteed by “standard” inverse function-type theorems. In the twice differentiable case, these issues have received some attention in the existing literature. Moreover, a few results in this direction are known in the case when the first derivative is merely B-differentiable. Here, we further elaborate on a similar setting, but the main attention is paid to the case of piecewise smooth equations. Specifically, we study the effect of singularity of a solution for some active smooth selection on the overall stability properties, and we provide sufficient conditions ensuring the needed stability properties in the cases when such smooth selections may exist. Finally, an application to a piecewise smooth reformulation of complementarity problems is given.

1 Introduction

In this paper, we are concerned with stability issues for a given solution of a finite system of nonlinear equations in a finite number of variables. When the system is smooth, and the solution in question is nonsingular, by which we mean that the Jacobian of the system at this solution has full row rank, those stability issues are addressed in full by classical covering theorems; see [13] for a recent exposition of these results and their far-reaching extensions. In particular, such solutions are stable subject to any sufficiently small right-hand side perturbations, with the distance to solutions of the perturbed system being of the order of perturbation.

Elementary examples demonstrate that for singular solutions (and in particular for nonisolated solutions when the number of equations coincides with the number of variables), one cannot expect stability with respect to right-hand side perturbations restricted only by their size. Nevertheless, a natural question in this context is about conditions ensuring stability of a given solution with respect to large (not asymptotically thin) classes of right-hand side perturbations. In the twice differentiable case, these issues have received some attention in the existing literature: see, e.g., [20], where stability properties were established for critical solutions, a special type of singular solutions. Moreover, some stability results of this kind are known even in the absence of twice differentiability, in the case when the first derivative is merely B-differentiable [22]. Here, we further elaborate on the latter setting as well, but the main attention is paid to the case of piecewise smooth equations. Specifically, we study the effect of singularity of a solution for some active smooth selection of a piecewise smooth mapping on the overall stability properties for the equation in question. We provide sufficient conditions ensuring the desired stability properties with respect to large classes of perturbations in the cases when such smooth selections may exist.

In Sect. 2, we start with the problem setting and some preliminaries. Among others, we develop the needed directional tools for dealing with active smooth selections and a stability result for constrained equations in Theorem 2.1. The latter is not concerned with any singularity of solutions; its specificity is in restricted smoothness assumptions appropriate for the use of this result in the subsequent analysis. Section 3 contains a discussion of relations between error bounds, criticality of singular solutions for active smooth selections, and stability issues for piecewise smooth equations. In particular, employing Theorem 2.1, we specify some cases when nonsingularity of a solution with respect to some active smooth selection ensures stability with respect to large classes of perturbations. In Sect. 4, we develop stability properties of singular solutions for mappings whose first derivative fulfills some additional smoothness requirements, while Sect. 5 extends these results to the piecewise smooth case. An application of the obtained sufficient conditions for stability to piecewise smooth reformulations of complementarity problems is given in Sect. 6.

Some words about our notation and terminology. By $\mathcal {I}$, we denote the identity matrix of a size always clear from the context. For a given index set J, we write $u_J$ for the subvector of a vector u, with components $u_j$, $j\in J$. Let $\ker M$ and $\textrm{im}M$ stand for the null space and the range space of a matrix (linear operator) M, respectively. By $\textrm{cone}U$, we denote the conic hull of a set $U\subset \mathbb {R}^p$, i.e., the smallest convex cone containing U. A set U is called starlike with respect to a point $\bar{u}\in U$ if $tu+(1-t)\bar{u}\in U$ for all $u\in U$ and all $t\in [0,\, 1]$. For such a set, $v\in \mathbb {R}^p$ is referred to as an excluded direction if $\bar{u}+tv\not \in U$ for all $t > 0$.

Moreover, let the inner product $\langle \cdot ,\, \cdot \rangle$ and the norm $\Vert \cdot \Vert$ be Euclidean. We will use the same symbol $\Vert \cdot \Vert$ for a consistent matrix norm. We further define the open ball $B(\bar{u},\, \varepsilon ):= \{ u\in \mathbb {R}^p\mid \Vert u-\bar{u}\Vert < \varepsilon \}$ centered at $\bar{u}\in \mathbb {R}^p$ with radius $\varepsilon > 0$. For some fixed $\bar{v}\in \mathbb {R}^p$ and $\delta >0$, we define the sets

$$\begin{aligned} C_{\delta }(\bar{v}):= & {} \{ v\in \mathbb {R}^p\mid \Vert \Vert \bar{v}\Vert v -\Vert v\Vert \bar{v}\Vert \le \delta \Vert v\Vert \Vert \bar{v}\Vert \}, \end{aligned}$$

(1)

$$\begin{aligned} K_{\varepsilon ,\, \delta }(\bar{u},\,\bar{v}):= & {} (\bar{u}+C_{\delta }(\bar{v}))\cap B(\bar{u},\, \varepsilon ). \end{aligned}$$

(2)

In particular, $C_{\delta }(\bar{v})$ is a closed convex cone, while the set $K_{\varepsilon ,\, \delta }(\bar{u},\,\bar{v})$ is star-shaped at $\bar{u}$. It results from shifting $C_{\delta }(\bar{v})$ by $\bar{u}$ and by intersecting the shifted set with the ball $B(\bar{u},\, \varepsilon )$. Observe that $C_{\delta }(0)=\mathbb {R}^p$ and $K_{\varepsilon ,\, \delta }(\bar{u},\, 0) = B(\bar{u},\, \varepsilon )$. Let $\textrm{dist}(u,\, U):= \inf _{v\in U} \Vert u-v\Vert$ stand for the distance from u to $U\subset \mathbb {R}^p$ (with the convention that $\textrm{dist}(u,\, \emptyset ):= +\infty$), and

$$\begin{aligned} \mathcal {H}(U_1,\, U_2):= \max \left\{ \sup _{u\in U_1}\textrm{dist}(u,\, U_2),\, \sup _{u\in U_2}\textrm{dist}(u,\, U_1)\right\} \end{aligned}$$

denote the Hausdorff distance between sets $U_1,\, U_2\subset \mathbb {R}^p$.

The directional derivative of a mapping $\varPhi :\mathbb {R}^p\rightarrow \mathbb {R}^q$ at $\bar{u}\in \mathbb {R}^p$ in a direction $v\in \mathbb {R}^p$ is understood in a standard way:

$$\begin{aligned} \varPhi '(\bar{u};\, v) = \lim _{t\rightarrow 0+} \frac{1}{t} (\varPhi (\bar{u}+tv)-\varPhi (\bar{u})). \end{aligned}$$

2 Problem Setting and Preliminaries

We consider the equation

$$\begin{aligned} \varPhi (u) = 0 \end{aligned}$$

(3)

with a mapping $\varPhi :\mathbb {R}^p\rightarrow \mathbb {R}^q$ assumed to be piecewise smooth near a point $\bar{u}\in \mathbb {R}^p$. By this, we mean that there exists a finite collection of smooth (near $\bar{u}$) selection mappings $\varPhi ^1,\, \ldots ,\, \varPhi ^s:\mathbb {R}^p\rightarrow \mathbb {R}^q$ such that

$$\begin{aligned} \varPhi (u)\in \{ \varPhi ^1(u),\, \ldots ,\, \varPhi ^s(u)\} \quad \forall \, u\in \mathbb {R}^p, \end{aligned}$$

and $\varPhi$ is continuous near $\bar{u}$. Here, a mapping is called smooth if it is continuously differentiable. Some results presented below will require stronger smoothness assumptions on the selection mappings. That said, one new feature of this work is that we would want to avoid invoking second derivatives of selection mappings, thus covering, in particular, the equations with merely locally Lipschitzian and directionally differentiable first derivatives [21, 22, 25].

In general, the collection $\{ \varPhi ^1(u),\, \ldots ,\, \varPhi ^s(u)\}$ of smooth selections associated with a piecewise smooth mapping $\varPhi$, and even the number s of these selections, is not uniquely defined. However, from this point on, we will assume that this collection is fixed. In other words, every piecewise smooth mapping is characterized by a given collection of smooth selections, and by a given rule defining selections active at every point.

Our main interest in this work is in characterization of those $w\in \mathbb {R}^q$ for which one can guarantee that the equation

$$\begin{aligned} \varPhi (u) = w \end{aligned}$$

(4)

has a solution close to a given solution $\bar{u}$ of the unperturbed equation (3).

When dealing with piecewise smooth mappings, the key role is played by the set

$$\begin{aligned} \mathcal {A}(u) := \{ j\in \{ 1,\, \ldots ,\, s\} \mid \varPhi (u) = \varPhi ^j(u)\} \end{aligned}$$

(5)

that, for a given $u\in \mathbb {R}^p$, stands for the set of indices of all selection mappings active at u. By the continuity of $\varPhi$ and its smooth selection mappings near $\bar{u}$, the set-valued mapping $\mathcal {A}(\cdot )$ is evidently outer semicontinuous at $\bar{u}$, i.e., $\mathcal {A}(u)\subset \mathcal {A}(\bar{u})$ holds for any $u\in \mathbb {R}^p$ close enough to $\bar{u}$.

According to [15, Lemma 4.6.1] and under the stated assumptions, $\varPhi$ is Lipschitz-continuous near $\bar{u}$ and B-differentiable at $\bar{u}$. The latter concept stems from [28] and implies that $\varPhi$ is directionally differentiable at $\bar{u}$ in any direction $v\in \mathbb {R}^p$ and (see [15, Proposition 3.1.3])

$$\begin{aligned} \varPhi (\bar{u}+v) = \varPhi (\bar{u})+\varPhi '(\bar{u};\, v)+o(\Vert v\Vert ) \end{aligned}$$

(6)

as $v\rightarrow 0$. Moreover, $\varPhi '(\bar{u};\, \cdot )$ is everywhere continuous, and

$$\begin{aligned} \varPhi '(\bar{u};\, v)\in \{ (\varPhi ^j)'(\bar{u})v\mid j\in \mathcal {A}(\bar{u})\} \quad \forall \, v\in \mathbb {R}^p, \end{aligned}$$

(7)

where $(\varPhi ^j)'(\bar{u})$ denotes the usual Jacobian of the smooth mapping $\varPhi ^j$ at $\bar{u}$, for $j\in \{ 1,\, \ldots ,\, s\}$.

Many considerations below will be directional by nature. To that end, for any $\bar{u}\in \mathbb {R}^p$ and $v\in \mathbb {R}^p$, we define the index set

$$\begin{aligned} \mathcal {A}(\bar{u},\, v) := \{ j\in \mathcal {A}(\bar{u})\mid \varPhi '(\bar{u};\, v) = (\varPhi ^j)'(\bar{u})v\}, \end{aligned}$$

(8)

which is nonempty by (7). Observe that $\mathcal {A}(\bar{u},\, 0) = \mathcal {A}(\bar{u})$.

The following facts can be considered as generalizations of the results in [16, Proposition 2.1, Corollary 2.1]; they will serve as our key tools for dealing with piecewise smoothness.

Proposition 2.1

Let $\varPhi :\mathbb {R}^p\rightarrow \mathbb {R}^q$ be piecewise smooth near $\bar{u}\in \mathbb {R}^p$.

Then, for any $\bar{v}\in \mathbb {R}^p$, there exist $\varepsilon >0$ and $\delta >0$ such that

$$\begin{aligned} \mathcal {A}(u) \subset \mathcal {A}(\bar{u},\, \bar{v})\quad \forall \, u\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}). \end{aligned}$$

(9)

Proof

For $\bar{v}= 0$, (9) is evident by outer semicontinuity of $\mathcal {A}(\cdot )$. Let $\bar{v}\not = 0$. We argue by contradiction: suppose that there exist sequences $\{ \varepsilon _k\} \rightarrow 0+$, $\{ \delta _k\} \rightarrow 0+$, $\{ u^k\} \subset \mathbb {R}^p$ such that, for all k, it holds that $u^k\in K_{\varepsilon _k,\, \delta _k}(\bar{u},\, \bar{v})$ and there exists $j^k\in \mathcal {A}(u^k) {\setminus } \mathcal {A}(\bar{u},\, \bar{v})$. From (1)–(2), it then follows that $\{ u^k\} \rightarrow \bar{u}$ and $\{ (u^k-\bar{u})/\Vert u^k-\bar{u}\Vert \} \rightarrow \bar{v}/\Vert \bar{v}\Vert$.

Passing to subsequences, if necessary, we may suppose that $j^k = j$ is the same for all k. Outer semicontinuity of $\mathcal {A}(\cdot )$ at $\bar{u}$ then yields that $j\in \mathcal {A}(\bar{u})$. Taking into account continuity of $\varPhi '(\bar{u};\, \cdot )$ and (6), we now obtain that

$$\begin{aligned} \varPhi '\left( \bar{u};\, \frac{\bar{v}}{\Vert \bar{v}\Vert } \right)= & {} \lim _{k\rightarrow \infty } \varPhi '\left( \bar{u};\, \frac{u^k-\bar{u}}{\Vert u^k-\bar{u}\Vert }\right) \\= & {} \lim _{k\rightarrow \infty } \frac{\varPhi (u^k) -\varPhi (\bar{u})}{\Vert u^k-\bar{u}\Vert } \\= & {} \lim _{k\rightarrow \infty } \frac{\varPhi ^j(u^k) -\varPhi ^j(\bar{u})}{\Vert u^k-\bar{u}\Vert } \\= & {} \lim _{k\rightarrow \infty } (\varPhi ^j)'(\bar{u})\frac{u^k-\bar{u}}{\Vert u^k-\bar{u}\Vert } \\= & {} (\varPhi ^j)'(\bar{u})\frac{\bar{v}}{\Vert \bar{v}\Vert }. \end{aligned}$$

According to (8), this implies that $j\in \mathcal {A}(\bar{u},\, \bar{v})$, a contradiction. $\square$

Corollary 2.1

Under the assumptions of Proposition 2.1, let $\mathcal {A}(\bar{u},\, \bar{v})$ consist of a single index $\widehat{\jmath }$.

Then, there exist $\varepsilon >0$ and $\delta >0$ such that

$$\begin{aligned} \mathcal {A}(u) = \{ \widehat{\jmath }\} \quad \forall \, u\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}). \end{aligned}$$

(10)

Some particular cases of Proposition 2.1 and Corollary 2.1 were used in [16] in order to demonstrate local attraction of piecewise Newton-type methods to solutions that are critical with respect to some active smooth selections. The idea here is to employ Proposition 2.1 and Corollary 2.1 for establishing stability properties of a solution $\bar{u}$ of (3) from the corresponding properties for smooth selections $\varPhi ^j$, $j\in \mathcal {A}(\bar{u},\, \bar{v})$, for a given $\bar{v}$.

We complete this section with a covering result that is not directly concerned with piecewise smoothness, but will be used below for deriving some conclusions concerning stability of (in some sense) nonsingular solutions of a piecewise smooth equation. The main specificity of this result is that the smoothness assumptions in it are still rather weak: $\varPhi$ is supposed to be differentiable only at the solution $\bar{u}$ of interest, and only with respect to a given set $U\in \mathbb {R}^p$ containing $\bar{u}$. According to the terminology in [6, Appendix II], this property means the existence of a linear operator $\mathcal {J}:\mathbb {R}^p\rightarrow \mathbb {R}^q$ such that

$$\begin{aligned} \varPhi (\bar{u}+v) = \varPhi (\bar{u})+\mathcal {J} v+o(\Vert v\Vert ) \end{aligned}$$

as $v\rightarrow 0$ in such a way that $\bar{u}+v$ stays in U. An interesting observation (not used in this paper though) is that this kind of differentiability allows $\varPhi$ to be defined only on U rather than on an entire neighborhood of $\bar{u}$. If U is convex, and $\textrm{int}U\not = \emptyset$, there can be no more than one linear operator $\mathcal {J}$ with the specified properties, in which case we denote it in a standard way, as $\varPhi '(\bar{u})$. This is the only reason for the assumption $\textrm{int}U\not = \emptyset$ in the following theorem.

Theorem 2.1

Let $U\subset \mathbb {R}^p$ be a convex set with $\textrm{int}U\not = \emptyset$, and let a continuous mapping $\varPhi :U\rightarrow \mathbb {R}^q$ be differentiable at some solution $\bar{u}\in U$ of Eq. (3) with respect to U.

Then, for any $\bar{w}\in \textrm{int}\varPhi '(\bar{u})(U-\bar{u})$, there exist $N >0$, $\widetilde{\varepsilon } > 0$, and $\widetilde{\delta } > 0$, such that, for every $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, \bar{w})$, Eq. (4) has a solution $u(w)\in U$ satisfying the estimate

$$\begin{aligned} \Vert u(w)-\bar{u}\Vert \le N\Vert w\Vert . \end{aligned}$$

(11)

Proof

According to [3, Lemma 2.2], there exist a closed convex (actually, even polyhedral) cone $K\subset \mathbb {R}^p$ and $\varepsilon > 0$ such that $K\cap B(0,\, \varepsilon )\subset (U-\bar{u})$ and $\bar{w}\in \textrm{int}\varPhi '(\bar{u})K$. (The assumptions in [3, Lemma 2.2] require the set U to be closed, but this requirement is never used in the proof of that lemma.) Then, according to [3, Lemma 2.1], there exist $R > 0$, $\delta > 0$, and a continuous mapping $v(\cdot ):C_{\delta }(\bar{w})\rightarrow K$ such that

$$\begin{aligned} \varPhi '(\bar{u})v(\chi ) = \chi ,\quad \Vert v(\chi )\Vert \le R\Vert \chi \Vert \quad \forall \, \chi \in C_{\delta }(\bar{w}). \end{aligned}$$

(12)

Fix any $\widetilde{\delta } \in (0,\, \min \{ 2,\, \delta /2\} ]$. According to (2) and to the inequality in (12), for any $\rho \in (0,\, \varepsilon /R]$, it then holds that for every $\chi \in K_{\rho ,\, \delta }(0,\, \bar{w})$, the value $v(\chi )$ is well-defined, $\bar{u}+ v(\chi )\in U$, and by differentiability of $\varPhi$ at $\bar{u}$ with respect to U, taking $\rho$ small enough ensures

$$\begin{aligned} \Vert \varPhi (\bar{u}+v(\chi ))-\varPhi '(\bar{u})v(\chi ) \Vert \le \frac{\widetilde{\delta } }{4R} \Vert v(\chi )\Vert \le \frac{\widetilde{\delta } }{4} \Vert \chi \Vert \quad \forall \, \chi \in K_{\rho ,\, \delta }(0,\, \bar{w}). \end{aligned}$$

(13)

Now fix any $\widetilde{\varepsilon } \in (0,\, \rho /2]$ and $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, \bar{w})$, and define the mapping $\mathcal {S}_w: K_{\rho ,\, \delta }(0,\, \bar{w})\rightarrow \mathbb {R}^q$,

$$\begin{aligned} \mathcal {S}_w(\chi ) = \chi -\varPhi (\bar{u}+v(\chi ))+w. \end{aligned}$$

According to the reasoning above, this mapping is well-defined and continuous. Moreover, for any $\chi \in C_{\delta }(\bar{w})$ satisfying $\Vert \chi \Vert \le 2\Vert w\Vert$ we then have $\chi \in K_{\rho ,\, \delta }(0,\, \bar{w})$, and hence, employing the equality in (12), (13), and the definition of $\widetilde{\delta }$,

$$\begin{aligned} \Vert \mathcal {S}_w(\chi )-w\Vert\le & {} \Vert \chi -\varPhi '(\bar{u})v(\chi )\Vert + \Vert \varPhi (\bar{u}+v(\chi ))-\varPhi '(\bar{u})v(\chi ) \Vert \nonumber \\\le & {} \frac{\widetilde{\delta } }{4} \Vert \chi \Vert \nonumber \\\le & {} \frac{\widetilde{\delta } }{2} \Vert w\Vert \nonumber \\\le & {} \Vert w\Vert \end{aligned}$$

(14)

implying, in particular, that $\Vert \mathcal {S}_w(\chi )\Vert \le 2\Vert w\Vert$. Furthermore, employing (1)–(2), the next-to-the-last inequality in (14), and again the definition of $\widetilde{\delta }$,

$$\begin{aligned} \Vert \Vert \bar{w} \Vert \mathcal {S}_w(\chi )-\Vert \mathcal {S}_w(\chi )\Vert \bar{w} \Vert\le & {} \Vert \Vert \bar{w} \Vert w-\Vert w\Vert \bar{w}\Vert \\{} & {} +\Vert \bar{w} \Vert \Vert \mathcal {S}_w(\chi )-w\Vert +\left| \Vert \mathcal {S}_w(\chi )\Vert - \Vert w\Vert \right| \Vert \bar{w}\Vert \\\le & {} \widetilde{\delta } \Vert w\Vert \Vert \bar{w}\Vert + \frac{1}{2} \widetilde{\delta } \Vert w\Vert \Vert \bar{w}\Vert + \frac{1}{2} \widetilde{\delta } \Vert w\Vert \Vert \bar{w}\Vert \\= & {} 2\widetilde{\delta } \Vert w\Vert \Vert \bar{w}\Vert \\\le & {} \delta \Vert w\Vert \Vert \bar{w}\Vert , \end{aligned}$$

implying that $\mathcal {S}_w(\chi )\in C_{\delta }(\bar{w})$.

We have thus demonstrated that $\mathcal {S}_w$ continuously maps a convex compact set $\{ \chi \in C_{\delta }(\bar{w})\mid \Vert \chi \Vert \le 2\Vert w\Vert \}$ into itself. Therefore, by Brouwer’s fixed-point theorem, there exists $\chi (w)$ in this set, satisfying $\chi (w) = \mathcal {S}_w(\chi (w))$, and hence,

$$\begin{aligned} \varPhi (\bar{u}+v(\chi (w))) = w. \end{aligned}$$

Moreover, according to the inequality in (12), and since $\Vert \chi (w)\Vert \le 2\Vert w\Vert$, we have

$$\begin{aligned} \Vert v(\chi (w))\Vert \le R\Vert \chi (w)\Vert \le 2R\Vert w\Vert . \end{aligned}$$

This gives the needed conclusion with $u(w) = \bar{u}+v(\chi (w))$ and $N = 2R$. $\square$

Remark 2.1

Examining the proof in [3, Lemma 2.1], and employing [2, Corollary 3], we see that $N > 0$ in Theorem 2.1 can be chosen independently of U in the following sense. Consider two sets $U_1,\, U_2\subset \mathbb {R}^p$. If the assumptions of Theorem 2.1 are satisfied with $U = U_1$ and with $U = U_2$, then the assertion of Theorem 2.1 holds for both these cases with the same $N > 0$. This further implies that in Propositions 3.1, 3.2, and 6.1, $N > 0$ can actually be chosen independently of $\varepsilon > 0$ and $\delta > 0$.

Properties involving estimates to the given solution, like (11), appear in the literature under different names, e.g., covering property in [22], hemiregularity in [24, Examlpe 3.58], semiregularity in [10], etc.¹ Note that a solution u(w) in Theorem 2.1 and in other results below need not be unique. In particular, the assertion of Theorem 2.1 can be stated as follows: for any $\bar{w}\in \textrm{int}\varPhi '(\bar{u})(U-\bar{u})$, there exist $N >0$, $\widetilde{\varepsilon } > 0$, and $\widetilde{\delta } > 0$, such that, for every $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, \bar{w})$,

$$\begin{aligned} \varPhi ^{-1}(w)\cap U\cap B(\bar{u},\, N\Vert w\Vert ) \not = \emptyset \end{aligned}$$

is satisfied.

The proof of Theorem 2.1 is essentially finite-dimensional: the number of equations in (3) must be finite. For this finite-dimensional setting, Theorem 2.1 generalizes various known results, including the classical covering result following from Robinson’s stability theorem (see [27, Theorem 1], or [9, Theorem 2.89, comments on p. 71]) under Robinson’s regularity condition $0\in \textrm{int}\varPhi '(\bar{u})(U-\bar{u})$ allowing to apply Theorem 2.1 with $\bar{w} = 0$. The results beyond Robinson’s regularity generalized by Theorem 2.1 are those in [26, Theorem 3], [4, Corollary 2.1, Remark 2.1], [5, Corollary 6.1, Remark 6.2]. In particular, Theorem 2.1 generalizes the mentioned results in a sense that they all assume strict differentiability, while here we only assume differentiability at a given point with respect to a set.

The presented result is strongly related to the one in [3, Thereom 3.1] where, however, the smoothness requirements are also stronger than those employed here.

Finally, we mention that Theorem 2.1 also generalizes the finite-dimensional result in [17, Theorem G], where differentiability is assumed only at a given point, but with respect to the entire space, and the derivative is assumed to be surjective, again allowing to take $\bar{w} = 0$.

We complete the section by mentioning that if $\textrm{int}\varPhi '(\bar{u})(U-\bar{u}) = \emptyset$, Theorem 2.1 is vacuous (says nothing). This is the case when the first-order analysis does not allow to establish covering of non-thin sets. Then, some other tools relying on higher-order (generalized) derivatives have to be involved, as it is done, for example, in Sects. 4 and 5.

3 Error Bounds, Criticality, and Stability Issues

According to [20, Theorem 2], in the case of a smooth mapping $\varPhi$, criticality of a solution $\bar{u}$ of (3) can be understood as the violation of the local Lipschitzian error bound

$$\begin{aligned} \textrm{dist}(u,\, \varPhi ^{-1}(0)) = O(\Vert \varPhi (u)\Vert ) \end{aligned}$$

as $u\in \mathbb {R}^p$ tends to $\bar{u}$. Moreover, as demonstrated in [20, Proposition1 and Theorem 4], critical solutions possess special stability properties that are necessarily missing for noncritical singular solutions.

However, even in the simplest nonsmooth cases (see Example 3.1), and even for the particular case of piecewise smooth reformulations of complementarity systems (see [4]), solutions violating strict complementarity (i.e., those with $\mathcal {A}(\bar{u})$ not being a singleton) can be stable subject to wide classes of perturbations, even if the local Lipschitzian error bound is satisfied. This suggests that the absence of the error bound is perhaps not an adequate understanding of criticality when $\mathcal {A}(\bar{u})$ is not a singleton.

Moreover, if we think of critical solutions as those possessing special stability properties, and/or attraction properties for Newton-type methods [19], perhaps the criticality concept should be left for smooth equations altogether, while for piecewise smooth ones, its impact should be regarded as arising through active smooth selections. To begin with, the discussion below suggests to consider a solution of the piecewise smooth equation as singular if it is singular for at least one active selection.

Suppose that for some sequences $\{ w^k\} \subset \mathbb {R}^q {\setminus } \{ 0\}$ and $\{ u^k\} \subset \mathbb {R}^p$ it holds that $\{ w^k\} \rightarrow 0$, $\{ u^k\} \rightarrow \bar{u}$, and $u^k$ solves (4) with $w = w^k$ for all k. Then, there exists $j\in \mathcal {A}(\bar{u})$ such that $u^k$ solves the equation

$$\begin{aligned} \varPhi ^j(u) = w \end{aligned}$$

(15)

with $w = w^k$ for infinitely many k, and passing to subsequences, we may suppose that it holds for all k. For every k, let $\widehat{u}^k$ stand for a projection of $u^k$ onto $(\varPhi ^j)^{-1}(0)$. Then, assuming that $\varPhi ^j$ is strictly differentiable at $\bar{u}$ with respect to $(\varPhi ^j)^{-1}(0)$ (as defined in [20]), and that $\bar{u}$ is a noncritical solution of the equation

$$\begin{aligned} \varPhi ^j(u) = 0, \end{aligned}$$

(16)

similarly to [20, Proposition 1] we find that the sequence $\{ (w^k,\, u^k-\widehat{u}^k)/\Vert w^k\Vert \}$ is bounded, and any accumulation point $(d,\, v)$ of this sequence satisfies $d\not = 0$ and the equality

$$\begin{aligned} (\varPhi ^j)'(\bar{u})v = d. \end{aligned}$$

If $\bar{u}$ is a singular solution of Eq. (16), i.e., $\textrm{rank}(\varPhi ^j)'(\bar{u}) < q$, this may only hold for special $d\not = 0$, namely, for those in $\textrm{im}(\varPhi ^j)'(\bar{u})$. Therefore, an active smooth selection with the specified properties can give rise to stability of the solution $\bar{u}$ of Eq. (3) subject to the right-hand side perturbations approaching 0 tangentially to the thin subset $\textrm{im}(\varPhi ^j)'(\bar{u})$ of $\mathbb {R}^q$ only. In particular, if $\bar{u}$ is a singular noncritical solution of Eq. (16) for all $j\in \mathcal {A}(\bar{u})$, then the solution $\bar{u}$ can be stable subject to special right-hand side perturbations of Eq. (3) only, forming an asymptotically thin subset of $\mathbb {R}^q$.

Therefore, when attempting to characterize the lack of stability, the case of interest is when $\bar{u}$ is a singular solution of (16) for at least one $j\in \mathcal {A}(\bar{u})$. It is evident that, say, this always holds if $\bar{u}$ is a nonisolated solution of (3) and $p=q$. However, even in the presence of such j, $\bar{u}$ can be a nonsingular (and hence noncritical) solution of (16) for some other $j\in \mathcal {A}(\bar{u})$, and the existence of such selections may of course have a positive impact on stability properties.

Example 3.1

Let $p = q = 1$, $\varPhi (u) = \min \{ u,\, 0\}$. This $\varPhi$ is piecewise smooth, with smooth selection functions $\varPhi ^1(u) = u$ and $\varPhi ^2(\cdot ) \equiv 0$, and

$$\begin{aligned} \varPhi (u) = \left\{ \begin{array}{ll} \varPhi ^1(u),&{}\text{ if } u\le 0,\\ \varPhi ^2(u),&{}\text{ if } u > 0. \end{array}\right. \end{aligned}$$

Then, every $\bar{u}\ge 0$ is a solution of Eq. (3), with $\mathcal {A}(\bar{u}) = \{ 2\}$ provided $\bar{u}> 0$, and every $\bar{u}$ is a singular noncritical solution of Eq. (16) with $j = 2$. Evidently, all solutions $\bar{u}> 0$ are unstable: Eq. (4) does not possess solutions in some neighborhood of any such $\bar{u}$ for any $w\not = 0$.

As for the remaining solution $\bar{u}= 0$ with $\mathcal {A}(0) = \{ 1,\, 2\}$, this solution is nonsingular for Eq. (16) with $j = 1$, and Eq. (4) has the unique solution $u(w) = w$ for every $w < 0$, and no solutions for $w > 0$.

One may ask whether the existence of $j\in \mathcal {A}(\bar{u})$ such that $\bar{u}$ is a nonsingular (and hence noncritical) solution of (16) necessarily implies stability with respect to large classes of perturbations. Without further assumptions on a smooth selection in question, the answer is evidently negative, as this selection can be inactive at any nearby point, and hence, locally redundant, i.e., can be removed from the list of smooth selections without changing $\varPhi$ near $\bar{u}$.

Moreover, even in the absence of redundant selections, the existence of a “nonsingular” active selection may have no strong positive effect on stability properties.

Example 3.2

Let $p = q = 2$,

$$\begin{aligned} \varPhi ^1(u)= & {} \left( \begin{array}{c} u_1\\ u_2 \end{array} \right) ,\quad \varPhi ^2(u) = \left( \begin{array}{c} -u_1\\ u_2 \end{array} \right) ,\quad \varPhi ^3(u) = \left( \begin{array}{c} u_2^2\\ u_2 \end{array} \right) ,\\ \varPhi (u)= & {} \left\{ \begin{array}{ll} \varPhi ^1(u),&{}\text{ if } 0\le u_1\le u_2^2,\\ \varPhi ^2(u),&{}\text{ if } 0 < -u_1 \le u_2^2,\\ \varPhi ^3(u),&{}\text{ if } |u_1| > u_2^2. \end{array}\right. \end{aligned}$$

This $\varPhi$ is piecewise smooth, with the specified smooth selection functions $\varPhi ^1$, $\varPhi ^2$, and $\varPhi ^3$. Any solution $\bar{u}$ of (3) fulfills $\bar{u}_2 = 0$, where $\mathcal {A}(\bar{u}) = \{ 3\}$ provided $\bar{u}\not = 0$. Moreover, every solution is a singular noncritical solution of Eq. (16) with $j = 3$. Evidently, all nonzero solutions are unstable in a sense that Eq. (4) possess solutions close to any such $\bar{u}$ only provided $w_1 = w_2^2$.

For the remaining solution $\bar{u}= 0$ we have $\mathcal {A}(0) = \{ 1,\, 2,\, 3\}$, and this solution is nonsingular for Eq. (16) with $j = 1,\, 2$. Nevertheless, Eq. (4) has solutions only for w satisfying $0\le w_1\le w_2^2$, i.e., w belongs to a set that is asymptotically thin at 0.

One case when “nonsingular” active selection gives rise to stability with respect to large classes of perturbations is specified in the following result.

Proposition 3.1

Let $\varPhi :\mathbb {R}^p\rightarrow \mathbb {R}^q$ be piecewise smooth near a given solution $\bar{u}$ of (3). Let $\bar{v}\in \mathbb {R}^p$ be such that $\mathcal {A}(\bar{u},\, \bar{v}) = \{ \widehat{\jmath } \}$, and $\textrm{rank}(\varPhi ^{\widehat{\jmath } })'(\bar{u}) = q$.

Then, for any $\varepsilon > 0$ and $\delta > 0$, there exist $N > 0$, $\widetilde{\varepsilon } > 0$, and $\widetilde{\delta } > 0$ such that, for any $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPhi ^{\widehat{\jmath } })'(\bar{u})\bar{v})$, Eq. (4) has a solution $u(w)\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ satisfying estimate (11).

Proof

Without loss of generality, we may suppose that $\varepsilon > 0$ and $\delta > 0$ are taken as in the assertion of Corollary 2.1 (as the smaller are those quantities, the more restrictive is the assertions of the proposition being proven). Then, (10) implies that $\varPhi (K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})) = \varPhi ^{\widehat{\jmath } }(K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}))$. Observe that since $\textrm{rank}(\varPhi ^{\widehat{\jmath } })'(\bar{u}) = q$, it holds that $(\varPhi ^{\widehat{\jmath } })'(\bar{u})\textrm{int}K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})\subset \textrm{int}(\varPhi ^{\widehat{\jmath } })'(\bar{u})K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$. The needed result now follows, e.g., from [4, Corollary 2.1, Remark 2.1] applied with $\varPhi := \varPhi ^{\widehat{\jmath } }$ and $K:= C_{\delta }(\bar{v})$, or from Theorem 2.1, applied with the same $\varPhi$, with $U:= K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$, and with $\bar{w} = (\varPhi ^{\widehat{\jmath } })'(\bar{u})\bar{v}$. $\square$

Example 3.3

Let $p = q = 1$, $\varPhi (u) = \min \{ u,\, u^2\}$. This $\varPhi$ is piecewise smooth, with smooth selection functions $\varPhi ^1(u) = u$ and $\varPhi ^2(u) = u^2$, and

$$\begin{aligned} \varPhi (u) = \left\{ \begin{array}{ll} \varPhi ^1(u),&{}\text{ if } u\le 0 \text{ or } u\ge 1,\\ \varPhi ^2(u),&{}\text{ if } 0< u < 1. \end{array} \right. \end{aligned}$$

Then, $\bar{u}= 0$ is the unique solution of Eq. (3), with $\mathcal {A}(0) = \{ 1,\, 2\}$. Moreover, this $\bar{u}$ is the unique nonsingular solution of Eq. (16) with $j = 1$. Thus, Proposition 3.1 is applicable with $\bar{v}= -1$ and $\widehat{\jmath } = 1$ and yields the existence of a solution u(w) of Eq. (4) with $w \le 0$ close enough to 0, satisfying the estimate (11). And indeed, for every $w \le 0$, Eq. (4) has the unique solution $u(w) = w$. In other words, the “nonsingular” selection $\varPhi ^1$ allows to cover all $w \le 0$, and with the specified estimate.

Observe that at the same time, the “singular” selection $\varPhi ^2$ allows to cover all $w\in [0,\, 1]$, and the latter is justified (at least for $w\ge 0$ small enough) by Proposition 5.1 (b), applied with $\bar{v}= 1$ and $\widehat{\jmath } = 2$.

Proposition 3.1 is also applicable at the solution $\bar{u}= 0$ in Example 3.1, with $\bar{v}= -1$ and $\widehat{\jmath } = 1$. At the same time, it is not applicable at the solution $\bar{u}= 0$ in Example 3.2. Indeed, for directions $\bar{v}$ with $\bar{v}_1\not = 0$, it holds that $\mathcal {A}(\bar{u},\, \bar{v}) = \{ 3\}$, and $(\varPhi ^3)'(0)$ is singular. For directions $\bar{v}$ with $\bar{v}_1 = 0$, it holds that $\mathcal {A}(\bar{u},\, \bar{v}) = \{ 1,\, 2,\, 3\}$, and not only it is not a singleton, but it also involves a “singular” selection $\varPhi ^3$.

If $p > q$, the assumptions of Proposition 3.1 allow for the case $\varPhi '(\bar{u};\, \bar{v}) = 0$, and in this case $K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPhi ^{\widehat{\jmath } })'(\bar{u})\bar{v}) = B(0,\, \widetilde{\varepsilon } )$, thus yielding covering of an entire neighborhood of 0 in $\mathbb {R}^q$. This also agrees with Proposition 5.1 (a) and Remark 4.2.

We next discuss the case when $\mathcal {A}(\bar{u},\, \cdot )$ is not necessarily a singleton. From continuity of $\varPhi '(\bar{u};\, \cdot )$ it follows that the set-valued mapping $\mathcal {A}(\bar{u},\, \cdot )$ is everywhere outer semicontinuous, i.e., $\mathcal {A}(\bar{u},\, v)\subset \mathcal {A}(\bar{u},\, \bar{v})$ holds for any $v\in \mathbb {R}^p$ close enough to a given $\bar{v}\in \mathbb {R}^p$.

The argument below is related to the one used in [16, Section 2.3] for somewhat different purposes. Suppose that there exist $j_1,\, j_2\in \mathcal {A}(\bar{u},\, \bar{v})$ such that $(\varPhi ^{j_1})'(\bar{u})\not = (\varPhi ^{j_2})'(\bar{u})$, and pick up any $\widehat{v}\in \mathbb {R}^p$ satisfying $(\varPhi ^{j_1})'(\bar{u})\widehat{v}\not = (\varPhi ^{j_2})'(\bar{u})\widehat{v}$. Then, for any real t close enough to 0, it holds that $(\varPhi ^{j_1})'(\bar{u})(\bar{v}+t\widehat{v})\not = (\varPhi ^{j_2})'(\bar{u})(\bar{v}+t\widehat{v})$, and $\mathcal {A}(\bar{u},\, \bar{v}+t\widehat{v})\subset \mathcal {A}(\bar{u},\, \bar{v})$, implying, in particular, that $\mathcal {A}(\bar{u},\, \bar{v}+t\widehat{v})$ cannot contain both indices $j_1$ and $j_2$ simultaneously. Continuing this procedure with $\bar{v}$ replaced by $\bar{v}+t\widehat{v}$, we end up with some $\bar{v}$ such that either $\mathcal {A}(\bar{u},\, \bar{v})$ is a singleton, or $(\varPhi ^j)'(\bar{u})$ coincide with the same matrix $\mathcal {J}$ for all $j\in \mathcal {A}(\bar{u},\, \bar{v})$. In the latter case, by (1)–(2), (6), and by outer semicontinuity of $\mathcal {A}(\bar{u},\, \cdot )$ at $\bar{v}$, we conclude that for any $\varepsilon > 0$ and a sufficiently small $\delta > 0$, the mapping $\varPhi$ is differentiable at $\bar{u}$ with respect to the set $K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$, with the derivative $\varPhi '(\bar{u}) = \mathcal {J}$. Moreover, this $\bar{v}$ can be taken arbitrarily close to the original one, and if, say, for the original $\bar{v}$ it holds that $\textrm{rank}(\varPhi ^j)'(\bar{u}) = q$ for all $j\in \mathcal {A}(\bar{u},\, \bar{v})$, then in the former case, the assumptions of Proposition 3.1 are satisfied with the newly constructed $\bar{v}$ and some $\widehat{\jmath }$, while in the latter case, $\varPhi '(\bar{u})$ is nonsingular, and then Theorem 2.1 is applicable with $U:= K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$, and with $\bar{w} = \varPhi '(\bar{u})\bar{v}$, yielding the same assertion as the one of Proposition 3.1. Therefore, the following generalization of Proposition 3.1 holds.

Proposition 3.2

Let $\varPhi :\mathbb {R}^p\rightarrow \mathbb {R}^q$ be piecewise smooth near a given solution $\bar{u}$ of (3). Let $\bar{v}\in \mathbb {R}^p$ be such that for some $q\times p$ matrix $\mathcal {J}$ it holds that $\textrm{rank}\mathcal {J} = q$ and $(\varPhi ^j)'(\bar{u}) = \mathcal {J}$ for all $j\in \mathcal {A}(\bar{u},\, \bar{v})$.

Then, for any $\varepsilon > 0$ and $\delta > 0$, there exist $N > 0$, $\widetilde{\varepsilon } > 0$, and $\widetilde{\delta } > 0$ such that for any $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, \mathcal {J} \bar{v})$, Eq. (4) has a solution $u(w)\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ satisfying estimate (11).

Considerations above lead, in particular, to the following conclusion: if $\bar{u}$ is a nonsingular solution of (16) for all $j\in \mathcal {A}(\bar{u})$, then it is stable subject to large classes of perturbations. If $p = q$, the assumption that $\bar{u}$ is a nonsingular solution of (16) for all $j\in \mathcal {A}(\bar{u})$ (combined with some additional requirements, including the requirement that the signs of all determinants of $(\varPhi ^j)'(\bar{u})$, $j\in \mathcal {A}(\bar{u})$, are the same) appears in [15, Theorem 4.6.5 (b)] providing necessary and sufficient conditions for $\varPhi$ to be a local Lipschitzian homeomorphism. However, here we are interested in much weaker covering properties, not concerned with local injectivity of $\varPhi$ (e.g., like for $\varPhi (u) = |u|$).

4 Stability of Singular Solutions in the Case of B-Differentiable Derivatives

In this section, we consider a single smooth selection. To that end, we skip the index $j\in \{ 1,\, \ldots ,\, s\}$ and consider the mapping $\varPhi$ satisfying the following assumptions for a given $\bar{u}\in \mathbb {R}^p$:

Assumption 1

$\varPhi$ is differentiable near $\bar{u}$, with its Jacobian $\varPhi '$ being continuous at $\bar{u}$.

Assumption 2

$\varPi$ is the projector in $\mathbb {R}^q$ onto some complementary linear subspace of $\textrm{im}\varPhi '(\bar{u})$, parallel to $\textrm{im}\varPhi '(\bar{u})$.

Assumption 3

The mapping $\varPi \varPhi '$ is Lipschitz-continuous near $\bar{u}$ and directionally differentiable at $\bar{u}$ in every direction.

If we were to restrict ourselves in Sect. 5 to piecewise smooth mappings with twice differentiable smooth selections, then instead of Theorems 4.1 and 4.2 presented in the current section, we might refer in Sect. 5 to some earlier results of this kind known for the twice differentiable case [8, 20]. However, the setting adopted here allows for more generality, and Theorem 4.2 is a new result of independent interest and importance.

Observe that Assumption 3 certainly holds if $\varPhi '$ is itself Lipschitz-continuous near $\bar{u}$ and directionally differentiable at $\bar{u}$ in every direction. As demonstrated in [29], Assumption 3 implies that the mapping $\varPi \varPhi '$ is B-differentiable at $\bar{u}$. Taking into account Assumption 2, the latter means that

$$\begin{aligned} \varPi \varPhi '(\bar{u}+v) = (\varPi \varPhi ')'(\bar{u};\, v)+o(\Vert v\Vert ) \end{aligned}$$

as $v\rightarrow 0$.

For any $v\in \mathbb {R}^p$, define the matrix (linear operator)

$$\begin{aligned} \varPsi (\bar{u},\, \varPi ;\, v) := \varPhi '(\bar{u})+(\varPi \varPhi ')'(\bar{u};\, v). \end{aligned}$$

(17)

Following [21, 22], we will be saying that $\varPhi$ is 2-regular at $\bar{u}$ in the direction $\bar{v}\in \mathbb {R}^p$ if

$$\begin{aligned} \textrm{rank}\varPsi (\bar{u},\, \varPi ;\, \bar{v}) = q. \end{aligned}$$

(18)

One can easily see that 2-regularity is indeed a directional condition, i.e., if it holds with some $\bar{v}$, then it also holds with $\bar{v}$ replaced by $t\bar{v}$ for every $t> 0$. Observe that condition (18) holds with any $\bar{v}$ (including $\bar{v}= 0$) provided the regularity condition

$$\begin{aligned} \textrm{rank}\varPhi '(\bar{u}) = q \end{aligned}$$

(19)

holds. However, here we are mostly interested in the cases when $\bar{u}$ is a singular solution of Eq. (3), by which in the setting of this section we mean precisely that (19) is violated.

In the twice differentiable case, the 2-regularity construction dates back at least to [7]; see also [1] for a systematic use of this construction in nonlinear analysis and optimization theory.

In the results presented below in this section, it will be assumed that $\bar{v}\in \ker \varPhi '(\bar{u})$. Similarly to [19, Proposition 1], employing [22, Theorem 2.1 (a)], it can be seen that in that case, $\bar{v}\in \ker \varPsi (\bar{u},\, \varPi ;\, \bar{v})$ provided $\bar{u}$ is a noncritical solution in a sense of [20, Definition 1], and hence, if $p = q$, the 2-regularity condition (18) can only hold if the solution $\bar{u}$ is either nonsingular or critical. Moreover, the next result can only be applicable for singular $\bar{u}$ when $p > q$.

Theorem 4.1

Let $\varPhi :\mathbb {R}^p\rightarrow \mathbb {R}^q$ be satisfying Assumptions 1–3 with some solution $\bar{u}$ of Eq. (3). Let $\bar{v}\in \ker \varPhi '(\bar{u})$ be such that $(\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}= 0$, and let $\varPhi$ be 2-regular at $\bar{u}$ in the direction $\bar{v}$, i.e., (18) holds.

Then, for any $\varepsilon > 0$ and $\delta > 0$, there exist $N > 0$ and $\widetilde{\varepsilon } > 0$ such that

$$\begin{aligned} \textrm{dist}(\bar{u},\, \varPhi ^{-1}(w)\cap K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}))\le N(\Vert (\mathcal {I}-\varPi )w\Vert +\Vert \varPi w\Vert ^{1/2})\quad \forall \, w\in B(0,\, \widetilde{\varepsilon } ). \end{aligned}$$

(20)

This result is essentially [22, Theorem 4.1], but with one improvement, important, in particular, for tackling the piecewise smooth case in Sect. 5: According to (20), solutions of the perturbed equation (4) are restricted to $K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$. To obtain Theorem 4.1, a slight modification of the proof in [22, Theorem 4.1] is sufficient, similar to taking $\gamma > 0$ small enough in the proof of Theorem 4.2.

Example 4.1

Let $p = 2$, $q = 1$, $\varPhi (u) = 2u_1u_2-(\min \{ 0,\, u_1+u_2\} )^2$. This $\varPhi$ is a complementarity function (dating back at least to [14], and studied in [23]), and in particular, the solution set of Eq. (3) consists of $\bar{u}\in \mathbb {R}^2$ satisfying $\bar{u}_1\ge 0$, $\bar{u}_2\ge 0$, $\bar{u}_1\bar{u}_2 = 0$. Except for $\bar{u}= 0$, all these solutions are nonsingular, and hence stable subject to small but otherwise arbitrary right-hand side perturbations, with the distance from solutions of (4) estimated from above by O(|w|) as $w\rightarrow 0$.

As for $\bar{u}= 0$, we have that $\varPhi '(0) = 0$,

$$\begin{aligned} (\varPhi ')'(0;\, v) = 2(v_2-\min \{ 0,\, v_1+v_2\},\, v_1-\min \{ 0,\, v_1+v_2\} ), \end{aligned}$$

implying, in particular, that $(\varPhi ')'(0;\, v)v = 0$ for, say, $v = (1,\, 0)$ or $v = (0,\, 1)$, and $\varPhi$ is 2-regular at 0 in any such direction v. Therefore, Theorem 4.1 is applicable, ensuring that for every w close enough to zero, Eq. (4) has a solution whose distance to 0 is estimated from above as $O(|w|^{1/2})$ as $w\rightarrow 0$. And indeed, for every w, Eq. (4) has a solution

$$\begin{aligned} u(w) = \left\{ \begin{array}{ll} ((w/2)^{1/2},\, (w/2)^{1/2}),&{} \text{ if } w \ge 0,\\ ((-w/2)^{1/2},\, -(-w/2)^{1/2}),&{} \text{ if } w < 0. \end{array}\right. \end{aligned}$$

Observe that the assumptions of Theorem 4.1 are only sufficient but not necessary for its assertion to hold. To see this, consider, e.g., $p = 2$, $q = 1$, $\varPhi (u) = u_1u_2$, and $\bar{u}= 0$. Then, $\varPhi$ is not 2-regular at $\bar{u}$ in the direction $\bar{v}= 0$, while the assertion of Theorem 4.1 is valid with this $\bar{v}$.

The next theorem is new, though [20, Theorem 5] can be regarded as its predecessor in the twice differentiable case. This result complements Theorem 4.1 in the following sense: the assumptions are the same, except that $(\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}$ is assumed to be not equal to 0. In this case, one cannot expect $\varPhi$ to cover an entire neighborhood of 0 (see Example 4.2), but the set being covered is still large, and the case $p = q$ is allowed even when $\bar{u}$ is a singular solution.

Theorem 4.2

Let $\varPhi :\mathbb {R}^p\rightarrow \mathbb {R}^q$ be satisfying Assumptions 1–3 with some solution $\bar{u}$ of Eq. (3). Let $\bar{v}\in \ker \varPhi '(\bar{u})$ be such that $(\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\not = 0$, and let $\varPhi$ be 2-regular at $\bar{u}$ in the direction $\bar{v}$, i.e., (18) holds.

Then, for any $\varepsilon > 0$ and $\delta > 0$, there exist $N > 0$, $\widetilde{\varepsilon } > 0$, and $\widetilde{\delta } > 0$, such that

$$\begin{aligned} \textrm{dist}(\bar{u},\, \varPhi ^{-1}(w)\cap K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}))\le N\Vert w\Vert ^{1/2}\quad \forall \, w\in K_{\widetilde{\varepsilon } ,\, \widetilde{\delta } }(0,\, (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}). \end{aligned}$$

(21)

Proof

Let $\varepsilon > 0$ and $\delta > 0$ be fixed. Set

$$\begin{aligned} R = \sup _{w\in \mathbb {R}^q,\; \Vert w\Vert = 1} \textrm{dist}(0,\, (\varPsi (\bar{u},\, \varPi ;\, \bar{v}))^{-1}(w)). \end{aligned}$$

Condition (18) implies that $R < \infty$. Fix an arbitrary $\theta \in (0,\, 1)$. From [22, Lemma 3.1] we have the existence of $\tau >0$ and $\gamma >0$ such that for all $t\in (0,\, \tau )$ and all $u^1,\, u^2\in B(0,\, \gamma t)$

$$\begin{aligned}{} & {} \Vert (\mathcal {I}-\varPi )(\varPhi (\bar{u}+t\bar{v}+u^1)-\varPhi (\bar{u}+t\bar{v}+u^2))-\varPhi '(\bar{u})(u^1-u^2)\Vert \le \frac{\theta }{2R} \Vert u^1-u^2\Vert , \nonumber \\ \end{aligned}$$

(22)

$$\begin{aligned}{} & {} \Vert \varPi (\varPhi (\bar{u}+t\bar{v}+u^1)-\varPhi (\bar{u}+t\bar{v}+u^2))- (\varPi \varPhi ')'(\bar{u};\, t\bar{v})(u^1-u^2)\Vert \le \frac{\theta }{2R} t\Vert u^1-u^2\Vert .\nonumber \\ \end{aligned}$$

(23)

Evidently, we can further reduce $\gamma > 0$, if necessary, to ensure the inclusion

$$\begin{aligned} B(\bar{v},\, 2\gamma )\subset C_\delta (\bar{v}). \end{aligned}$$

(24)

Taking into account that $(\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\in \ker (\mathcal {I} -\varPi )$, and employing (1)–(2), for any $\widetilde{\varepsilon } > 0$ and $\widetilde{\delta } > 0$, and any $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v})$, we have

$$\begin{aligned} \Vert (\mathcal {I}-\varPi )w\Vert= & {} \left\| (\mathcal {I}-\varPi )\left( w -\frac{\Vert w\Vert }{\Vert (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\Vert } (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v})\right) \right\| \nonumber \\\le & {} \widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert \Vert w\Vert , \end{aligned}$$

(25)

and hence,

$$\begin{aligned} \Vert w\Vert \le \Vert (\mathcal {I}-\varPi )w\Vert +\Vert \varPi w\Vert \le \widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert \Vert w\Vert +\Vert \varPi w\Vert \end{aligned}$$

implying that

$$\begin{aligned} \Vert w\Vert \le \frac{1}{1-\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert } \Vert \varPi w\Vert \end{aligned}$$

(26)

provided $\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert < 1$. On the other hand, by (25),

$$\begin{aligned} \Vert \varPi w\Vert = \Vert w -(\mathcal {I}-\varPi )w\Vert \le \Vert w\Vert +\Vert (\mathcal {I}-\varPi )w\Vert ) \le (1+\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert )\Vert w\Vert . \end{aligned}$$

(27)

Define the function $\rho :\mathbb {R}^q\rightarrow \mathbb {R}_+$,

$$\begin{aligned} \rho (w) = \left( \frac{2\Vert \varPi w\Vert }{\Vert (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\Vert }\right) ^{1/2}. \end{aligned}$$

(28)

Take $\widetilde{\varepsilon } > 0$ and $\widetilde{\delta } > 0$ small enough so that

$$\begin{aligned}{} & {} (\Vert \bar{v}\Vert +2\gamma )\left( \frac{2\Vert \varPi \Vert \widetilde{\varepsilon } }{\Vert (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\Vert }\right) ^{1/2} < \varepsilon , \end{aligned}$$

(29)

$$\begin{aligned}{} & {} \widetilde{\varepsilon } < \tau ^2\frac{\Vert (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\Vert }{2\Vert \varPi \Vert }, \end{aligned}$$

(30)

$$\begin{aligned}{} & {} \left( \frac{\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert }{1-\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert } \right) ^2\Vert \varPi \Vert \widetilde{\varepsilon } \le \left( \frac{1-\theta }{4R} \gamma \right) ^2\frac{2}{\Vert (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\Vert }, \end{aligned}$$

(31)

and

$$\begin{aligned} \widetilde{\delta } < \frac{1-\theta }{2R} \frac{\gamma }{\Vert (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\Vert } . \end{aligned}$$

(32)

Note that (26), (28), and (30) imply the inequalities $0< \rho (w) < \tau$ for all nonzero $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v})$. Then, for the set-valued mapping $\mathcal {S}_w: B(0,\, \gamma \tau )\rightarrow 2^{\mathbb {R}^p}$,

$$\begin{aligned} \mathcal {S}_w(u) = u-(\varPsi (\bar{u},\, \varPi ;\, \rho (w)\bar{v}))^{-1}(\varPhi (\bar{u}+\rho (w)\bar{v}+u)-w), \end{aligned}$$

again employing (18), we have that

$$\begin{aligned} \mathcal {S}_w(u)\not =\emptyset \quad \forall \, w \in K_{\widetilde{\varepsilon } ,\, \widetilde{\delta } }(0,\, (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v})\setminus \{ 0\} ,\; \forall \, u\in B(0,\, \gamma \tau ). \end{aligned}$$

(33)

According to [22, Lemma 3.2], and recalling $\bar{v}\in \ker \varPhi '(\bar{u})$, by further reducing $\widetilde{\varepsilon } > 0$ if necessary, we can ensure that for all $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v})$ it holds that

$$\begin{aligned}{} & {} \Vert (\mathcal {I}-\varPi )\varPhi (\bar{u}+\rho (w)\bar{v})\Vert =\Vert (\mathcal {I}-\varPi )\varPhi (\bar{u}+\rho (w)\bar{v})-\varPhi '(\bar{u})(\rho (w)\bar{v})\Vert \nonumber \\{} & {} \le \frac{1-\theta }{4R} \gamma \rho (w), \end{aligned}$$

(34)

$$\begin{aligned}{} & {} \left\| \varPi \varPhi (\bar{u}+\rho (w)\bar{v}) -\frac{1}{2} (\varPi \varPhi ')'(\bar{u};\, \rho (w)\bar{v})(\rho (w)\bar{v})\right\| \le \frac{1-\theta }{4R} \gamma (\rho (w))^2. \end{aligned}$$

(35)

Moreover, from (25), (26), (28), and (31), we have

$$\begin{aligned} \Vert (\mathcal {I}-\varPi )w\Vert\le & {} \frac{\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert }{1-\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert } \Vert \varPi w\Vert \nonumber \\= & {} \frac{\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert }{1-\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert } \Vert \varPi w\Vert ^{1/2}\Vert \varPi w\Vert ^{1/2} \nonumber \\\le & {} \frac{\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert }{1-\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert } (\Vert \varPi \Vert \widetilde{\varepsilon } )^{1/2} \left( \frac{1}{2} \Vert (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\Vert \right) ^{1/2} \rho (w) \nonumber \\\le & {} \frac{1-\theta }{4R} \gamma \rho (w), \end{aligned}$$

(36)

while additionally employing (32), for $w\not = 0$ we get

$$\begin{aligned} \varPi w-\frac{1}{2} (\varPi \varPhi ')'(\bar{u};\, \rho (w)\bar{v})(\rho (w)\bar{v})= & {} \frac{1}{2} \left( \frac{\Vert (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\Vert }{\Vert \varPi w\Vert } \varPi w \right. \\{} & {} -(\varPi \varPhi ')'(\bar{u};\, \bar{v})(\bar{v})\Bigg ) (\rho (w))^2 \end{aligned}$$

and hence

$$\begin{aligned} \left\| \varPi w-\frac{1}{2} (\varPi \varPhi ')'(\bar{u};\, \rho (w)\bar{v})(\rho (w)\bar{v})\right\|\le & {} \frac{1}{2} \widetilde{\delta } \Vert (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\Vert (\rho (w))^2\nonumber \\< & {} \frac{1-\theta }{4R} \gamma (\rho (w))^2. \end{aligned}$$

(37)

Defining $r(w)=\gamma \rho (w)$, and making use of (22) and (23), for all nonzero elements $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v})$, and all $u^1,\, u^2\in B(0,\, r(w))$, since $\mathcal {S}_w(u^1)$ and $\mathcal {S}_w(u^2)$ are parallel affine sets, we obtain that

$$\begin{aligned} \mathcal {H} (\mathcal {S}_w(u^1),\, \mathcal {S}_w(u^2))= & {} \inf \{\Vert v^1-v^2\Vert \mid v^i\in \mathcal {S}_w(u^i),\; i = 1,\, 2\} \nonumber \\= & {} \inf \{ \Vert v\Vert \mid v\in \mathbb {R}^p,\,\varPsi (\bar{u},\, \varPi ;\, \bar{v})v = \varPsi (\bar{u},\, \varPi ;\, \bar{v})(u^1-u^2) \nonumber \\{} & {} -(\mathcal {I}-\varPi )(\varPhi (\bar{u}+\rho (w)\bar{v}+u^1)-\varPhi (\bar{u}+\rho (w)\bar{v}+u^2))\nonumber \\{} & {} -(\rho (w))^{-1}\varPi (\varPhi (\bar{u}+\rho (w)\bar{v}+u^1)-\varPhi (\bar{u}+\rho (w)\bar{v}+u^2))\} \nonumber \\\le & {} R(\Vert (\mathcal {I}-\varPi )(\varPhi (\bar{u}+\rho (w)\bar{v}+u^1)-\varPhi (\bar{u}+\rho (w)\bar{v}+u^2)) \nonumber \\{} & {} -\varPhi '(\bar{u})(u^1-u^2)\Vert \nonumber \\{} & {} +(\rho (w))^{-1}\Vert \varPi (\varPhi (\bar{u}+\rho (w)\bar{v}+u^1)-\varPhi (\bar{u}+\rho (w)\bar{v}+u^2)) \nonumber \\{} & {} -(\varPi \varPhi ')'(\bar{u};\, \rho (w)\bar{v})(u^1-u^2)\Vert )\nonumber \\\le & {} \theta \Vert u^1-u^2\Vert . \end{aligned}$$

(38)

Furthermore, by (28) and (34)–(37), for all nonzero $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v})$ we have

$$\begin{aligned} \textrm{dist}(0,\, \mathcal {S}_w(0))= & {} \inf \{ \Vert v\Vert \mid v\in \mathbb {R}^p, \varPsi (\bar{u},\, \varPi ;\, \bar{v})v = (\mathcal {I}-\varPi )(\varPhi (\bar{u}+\rho (w)\bar{v})-w) \nonumber \\{} & {} +(\rho (w))^{-1}\varPi (\varPhi (\bar{u}+\rho (w)\bar{v})-w)\} \nonumber \\\le & {} R(\Vert (\mathcal {I}-\varPi )(\varPhi (\bar{u}+\rho (w)\bar{v})-w)\Vert \nonumber \\{} & {} +(\rho (w))^{-1}\Vert \varPi (\varPhi (\bar{u}+\rho (w)\bar{v})-w)\Vert ) \nonumber \\\le & {} R\Big (\Vert (\mathcal {I}-\varPi )(\varPhi (\bar{u}+\rho (w)\bar{v})\Vert \nonumber \\{} & {} +(\rho (w))^{-1}\left\| \varPi \varPhi (\bar{u}+\rho (w)\bar{v}) -\frac{1}{2} (\varPi \varPhi ')'(\bar{u};\, \rho (w)\bar{v})(\rho (w)\bar{v})\right\| \nonumber \\{} & {} +\Vert (\mathcal {I}-\varPi )w\Vert +\nonumber \\{} & {} \left. (\rho (w))^{-1}\left\| \varPi w-\frac{1}{2} (\varPi \varPhi ')'(\bar{u};\, \rho (w)\bar{v})(\rho (w)\bar{v})\right\| \right) \nonumber \\< & {} (1-\theta )\gamma \rho (w) \nonumber \\= & {} (1-\theta ) r(w). \end{aligned}$$

(39)

For every nonzero $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v})$, relations (33), (38), and (39) constitute the assumptions of the contraction multimapping principle as it appears in [18, Lemma 1, pp. 31–32] for $\mathcal {S}_w$ in the ball $B(0,\, r(w))$, and therefore, there exists $u(w)\in B(0,\, r(w))$ such that

$$\begin{aligned} u(w)\in \mathcal {S}_w(u(w)),\quad \Vert u(w)\Vert \le \frac{2}{1-\theta } \textrm{dist}(0,\, \mathcal {S}_w(0)). \end{aligned}$$

(40)

By the definition of $\mathcal {S}_w$, the inclusion in (40) implies that

$$\begin{aligned} \varPhi (\bar{u}+\rho (w)\bar{v}+u(w)) = w. \end{aligned}$$

(41)

Employing (24), (29), the next-to-the-last inequality in (39), and the estimate in (40), we further obtain that

$$\begin{aligned} \bar{u}+\rho (w)\bar{v}+u(w)\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}) \end{aligned}$$

and

$$\begin{aligned} \Vert \rho (w)\bar{v}+u(w)\Vert \le (\Vert \bar{v}\Vert +2\gamma )\rho (w). \end{aligned}$$

Combining these relations with (41), and recalling (27), (28), we obtain (21) with

$$\begin{aligned} N = (\Vert \bar{v}\Vert +2\gamma )\left( \frac{2 (1+\widetilde{\delta } \Vert \mathcal {I}-\varPi \Vert )}{\Vert (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\Vert } \right) ^{1/2}, \end{aligned}$$

(42)

and with the specified $\widetilde{\varepsilon } > 0$ and $\widetilde{\delta } > 0$. $\square$

Example 4.2

Let $p = q = 1$, $\varPhi (u) = (\max \{ 0,\, u\} )^2$. Then, every $\bar{u}\le 0$ is a solution of Eq. (3), all these solutions are singular, and all are noncritical except for $\bar{u}= 0$. Evidently, all noncritical solutions are unstable: Eq. (4) does not possess solutions in some neighborhood of any $\bar{u}< 0$ for any $w\not = 0$.

As for the unique critical solution $\bar{u}= 0$, we have $\varPhi '(0) = 0$, $(\varPhi ')'(0;\, v) = 2\max \{ 0,\, v\}$, implying, in particular, that $(\varPhi ')'(0;\, v)v = 0$ for every $v\le 0$, but $\varPhi$ is not 2-regular at 0 in any such direction v. At the same time, for $v > 0$ it holds that $(\varPhi ')'(0;\, v)v = v^2 \not = 0$, and $\varPhi$ is 2-regular in every such direction v. Therefore, Theorem 4.1 is not applicable, while Theorem 4.2 is, ensuring that for every $w\ge 0$ small enough, Eq. (4) has a solution whose distance to 0 is estimated from above as $O(w^{1/2})$ as $w\rightarrow 0$. And indeed, Eq. (4) has the unique solution $u(w) = w^{1/2}$ for every $w\ge 0$, and no solutions for $w < 0$.

In addition, for Theorem 4.1, the assumptions of Theorem 4.2 are not necessary for its assertion to hold. This can be seen, e.g., by considering $p = 3$, $q = 2$, $\varPhi (u) = (u_1u_2,\, u_3^2)$, $\bar{u}= 0$, with 2-regularity of $\varPhi$ at $\bar{u}$ violated in the direction $\bar{v}= (0,\, 0,\, 1)$, while the assertion of Theorem 4.2 is valid for this $\bar{v}$.

Remark 4.1

It can be seen from the proof of Theorem 4.2 that constant $N > 0$ in it can actually be chosen independently of $\varepsilon > 0$ and $\delta > 0$. Observe that this is not true for Theorem 4.1, where it may be needed to infinitely increase N as $\delta \rightarrow 0$. This can be demonstrated by considering again $p = 2$, $q = 1$, $\varPhi (u) = u_1u_2$, and $\bar{u}= 0$, but with $\bar{v}= (\pm 1,\, 0)$ or $\bar{v}= (0,\, \pm 1)$.

Remark 4.2

If $\bar{u}$ is a nonsingular solution, i.e., the regularity condition (19) holds, then necessarily $\varPi = 0$, and Theorem 4.1 is applicable with any $\bar{v}\in \ker \varPhi '(\bar{u})$, including $\bar{v}= 0$, with the estimate (20) taking the form

$$\begin{aligned} \textrm{dist}(\bar{u},\, \varPhi ^{-1}(w)\cap K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}))\le N\Vert w\Vert \quad \forall \, w\in B(0,\, \widetilde{\varepsilon } ). \end{aligned}$$

This gives the classical covering (hemiregularity, semiregularity) result for regular mappings, with the additional restriction of solutions to $K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ (making sense when $\bar{v}\not = 0$). The same result follows, for instance, from any of Propositions 3.1 or 3.2, applied with $\mathcal {A}(\bar{u})$ being a singleton. Observe that Theorem 4.2 is not applicable in the nonsingular case.

The other extreme case is when $\varPhi '(\bar{u}) = 0$ (as in both Examples 4.1 and 4.2). Then, necessarily $\varPi = \mathcal {I}$, and the estimate (20) in Theorem 4.1 takes the form

$$\begin{aligned} \textrm{dist}(\bar{u},\, \varPhi ^{-1}(w)\cap K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}))\le N\Vert w\Vert ^{1/2}\quad \forall \, w\in B(0,\, \widetilde{\varepsilon } ), \end{aligned}$$

(43)

while the estimate (21) in Theorem 4.2 takes the form

$$\begin{aligned} \textrm{dist}(\bar{u},\, \varPhi ^{-1}(w)\cap K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}))\le N\Vert w\Vert ^{1/2}\quad \forall \, w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPhi ')'(\bar{u};\, \bar{v})\bar{v}). \end{aligned}$$

The intermediate cases of singularity (when (19) is violated but $\varPhi '(\bar{u}) \not = 0$) give freedom in choosing a complementary subspace in Assumption 2, and hence, in choosing $\varPi$. Observe that different $\varPi$ may give rise to different directions $(\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}$ in Theorem 4.2, and for each of these directions (21) holds with some $N > 0$, $\widetilde{\varepsilon } > 0$, and $\widetilde{\delta } > 0$, also possibly depending on $\varPi$.

That said, if we strengthen Assumption 3 by assuming that $\varPhi '$ is itself directionally differentiable at $\bar{u}$ in every direction, then the norm of $(\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}= \varPi (\varPhi ')'(\bar{u};\, \bar{v})\bar{v}$ is separated from zero by its value for $\varPi$ being the orthogonal projector onto $(\textrm{im}\varPhi '(\bar{u}))^\bot$. Then, from (24) and (42) it can be seen that one can actually take the same $N > 0$ for all $\varPi$, compensating the dependence of N on $\varPi$ by taking $\widetilde{\delta } > 0$ small enough. Furthermore, under the strengthened Assumption 3, condition $(\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}\not = 0$ in Theorem 4.2 can be written in the form $(\varPhi ')'(\bar{u};\, \bar{v})\bar{v}\not \in \textrm{im}\varPhi '(\bar{u})$ independent of $\varPi$, and the union of $(\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v}$ over all possible $\varPi$ coincides with $(\varPhi ')'(\bar{u};\, \bar{v})\bar{v}+\textrm{im}\varPhi '(\bar{u})$. Hence, the latter is contained in the interior of the cone of feasible directions to the corresponding union $W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ of the sets $K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v})$ at 0, and

$$\begin{aligned} \textrm{dist}(\bar{u},\, \varPhi ^{-1}(w)\cap K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}))\le N\Vert w\Vert ^{1/2}\quad \forall \, w\in W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v}). \end{aligned}$$

(44)

In particular, if $\textrm{rank}\varPhi '(\bar{u}) = q-1$, then $\textrm{im}\varPhi '(\bar{u})$ is a hyperplane in $\mathbb {R}^q$, while the set $\textrm{cone}((\varPhi ')'(\bar{u};\, \bar{v})\bar{v}+\textrm{im}\varPhi '(\bar{u}))$ is the entire half-space (the one containing $(\varPhi ')'(\bar{u};\, \bar{v})\bar{v}$) associated with this hyperplane with excluded nonzero points belonging to $\textrm{im}\varPhi '(\bar{u})$. In other words, the set $W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ is starlike with respect to 0, with excluded directions necessarily belonging to the half-space associated with the hyperplane $\textrm{im}\varPhi '(\bar{u})$ (to the one not containing $(\varPhi ')'(\bar{u};\, \bar{v})\bar{v}$). This set that according to (44) is covered by $\varPhi$ on $K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$, is large (half-ball asymptotically), and cannot be expected to be larger, even in the twice differentiable case.

We emphasize that the observations regarding taking the union of sets $K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPi \varPhi ')'(\bar{u};\, \bar{v})\bar{v})$ over possible values of $\varPi$ appear to be new even in the twice differentiable case. This is demonstrated by the following example.

Example 4.3

Let $p = q = 2$, $\varPhi (u) = (u_1,\, u_1^2+u_2^2)$. Then, (3) has the unique solution $\bar{u}= 0$, and it is critical. We have

$$\begin{aligned} \varPhi '(0) = \left( \begin{array}{cc} 1&{}0\\ 0&{}0 \end{array} \right) ,\quad (\varPhi ')'(0;\, v) = \varPhi ''(0)[v] = \left( \begin{array}{cc} 0&{}0\\ 2v_1&{}2v_2 \end{array} \right) , \end{aligned}$$

implying that $\varPhi$ is 2-regular at 0 in any direction $v\in \mathbb {R}^2$ with $v_2\not = 0$, including the directions $\bar{v}= (0,\, \pm 1)$ spanning $\ker \varPhi '(0)$. With any of those $\bar{v}$ we have $(\varPhi ')'(0;\, \bar{v})\bar{v}= \varPhi ''(0)[\bar{v},\, \bar{v}] = (0,\, 2)$, and hence, according to Remark 4.2, for any fixed $\varepsilon > 0$ and $\delta > 0$, the cone of feasible directions to $W_{\varepsilon ,\, \delta }(0,\, \bar{v})$ is the upper half-plane with excluded nonzero points in the horizontal axis, and for any $w\in W_{\varepsilon ,\, \delta }(0,\, \bar{v})$ Eq. (4) has a solution whose distance to 0 is estimated from above as $O(\Vert w\Vert ^{1/2})$. And indeed, Eq. (4) has two solutions $u(w) = (w_1,\, \pm (w_2-w_1^2)^{1/2})$ for every $w\in \mathbb {R}^2$ satisfying $w_2\ge w_1^2$, and no solutions otherwise.

Observe that applying Theorem 4.2 with, say, $\varPi$ fixed as the orthogonal projector onto $(\textrm{im}\varPhi '(\bar{u}))^\bot$ would yield covering only of a cone around $(0,\, 2)$ instead of the entire upper half-plane.

5 Stability of Singular Solutions in the Piecewise Smooth Case

We now get back to the piecewise smooth setting. However, in contrast to Sect. 3, we consider the stability of special singular solutions and apply results from Sect. 4.

Proposition 5.1

Let $\varPhi :\mathbb {R}^p\rightarrow \mathbb {R}^q$ be piecewise smooth near a given solution $\bar{u}$ of (3). Let $\bar{v}\in \mathbb {R}^p$ be such that $\varPhi '(\bar{u};\, \bar{v}) = 0$, $\mathcal {A}(\bar{u},\, \bar{v}) = \{ \widehat{\jmath } \}$, and the selection mapping $\varPhi ^{\widehat{\jmath } }$ satisfies Assumptions 1–3 (with $\varPhi$ substituted by $\varPhi ^{\widehat{\jmath } }$) and is 2-regular at $\bar{u}$ in the direction $\bar{v}$.

Then, the following assertions hold:

• (a) If $(\varPi (\varPhi ^{\widehat{\jmath } })')'(\bar{u};\, \bar{v})\bar{v}= 0$, then, for any $\varepsilon > 0$ and $\delta > 0$, there exist $N > 0$ and $\widetilde{\varepsilon } > 0$ such that, for any $w\in B(0,\, \widetilde{\varepsilon } )$, Eq. (4) has a solution $u(w)\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ satisfying the estimate

  $$\begin{aligned} \Vert u(w)-\bar{u}\Vert \le N(\Vert (\mathcal {I}-\varPi )w\Vert +\Vert \varPi w\Vert ^{1/2}). \end{aligned}$$

• (b) If $(\varPi (\varPhi ^{\widehat{\jmath } })')'(\bar{u};\, \bar{v})\bar{v}\not = 0$, then, for any $\varepsilon > 0$ and $\delta > 0$, there exist $N > 0$, $\widetilde{\varepsilon } > 0$, and $\widetilde{\delta } > 0$, such that, for any $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\varPi (\varPhi ^{\widehat{\jmath } })')'(\bar{u};\, \bar{v})\bar{v})$, Eq. (4) has a solution $u(w )\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ satisfying the estimate

  $$\begin{aligned} \Vert u(w)-\bar{u}\Vert \le N\Vert w\Vert ^{1/2}. \end{aligned}$$

  (45)

  Moreover, if $(\varPhi ^{\widehat{\jmath } })'$ is directionally differentiable at $\bar{u}$ in every direction, then, for any $\varepsilon > 0$ and $\delta > 0$, there exist $N > 0$ and a set $W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})\subset \mathbb {R}^q$ such that $0\in W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$, the set $((\varPhi ^{\widehat{\jmath } })')'(\bar{u};\, \bar{v})\bar{v}+\textrm{im}(\varPhi ^{\widehat{\jmath } })'(\bar{u})$ is contained in the interior of the cone of feasible directions to $W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ at 0, and, for any $w\in W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$, Eq. (4) has a solution $u(w)\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ satisfying the estimate (45).

Proof

Again, without loss of generality, we assume that $\varepsilon > 0$ and $\delta > 0$ are taken as in the assertion of Corollary 2.1. Then, choose $N > 0$ and $\widetilde{\varepsilon } > 0$ according to Theorem 4.1 for assertion (a), and $N > 0$, $\widetilde{\varepsilon } > 0$, and $\widetilde{\delta } > 0$, according to Theorem 4.2 for assertion (b), applied with $\varPhi$ substituted by $\varPhi ^{\widehat{\jmath } }$. With these choices, the needed assertions become evident, taking into account that according to (10), $\varPhi (u) = \varPhi ^{\widehat{\jmath } }(u)$ for all $u\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$, and also employing Remark 4.2. $\square$

Remark 5.1

According to the discussion in Remark 4.2, if in the last assertion of Proposition 5.1$\textrm{rank}(\varPhi ^{\widehat{\jmath } })'(\bar{u}) = q-1$, then the set $\textrm{cone}(((\varPhi ^{\widehat{\jmath } })')'(\bar{u};\, \bar{v})\bar{v}+\textrm{im}(\varPhi ^{\widehat{\jmath } })'(\bar{u})){\setminus } \{ 0\}$ is an open half-space, and this large set is contained in the cone of feasible directions to $W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ at 0.

Examples illustrating Proposition 5.1 can be easily constructed, for instance, on the basis of examples in Sect. 4. We note that all these examples can be modified by adding to functions involved any differentiable terms small of order higher than 2 at 0, without changing the conclusions concerned with application of Theorems 4.1– 4.2 and Proposition  5.1, but making direct explicit solution of Eq. (4) difficult or even impossible, or even making (4) unsolvable for w away from zero.

6 Applications to the Nonlinear Complementarity Problem

In this section we apply the results obtained above to a reformulation of the nonlinear complementarity problem (NCP)

$$\begin{aligned} x\ge 0,\quad F(x)\ge 0,\quad \langle x,\, F(x)\rangle =0, \end{aligned}$$

(46)

with a smooth mapping $F:\mathbb {R}^n\rightarrow \mathbb {R}^n$.

One traditional approach to tackle NCP (46) consists of reformulating it as Eq. (3) with $\varPhi :\mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}^n\times \mathbb {R}^n$ (thus setting $p = q = 2n$) defined by

$$\begin{aligned} \varPhi (u):= (F(x)-y,\, \min \{ x,\, y\} ), \end{aligned}$$

(47)

where $u = (x,\, y)$ with a slack variable y, and the min-operation is applied componentwise. Involving slack variable y in this reformulation is needed exclusively to tackle, by means of Proposition 5.1, additive constant perturbations of F, rather than generic perturbations of $\varPhi$, which would be of unclear meaning.

Let $s:= 2^n$, and fix any one-to-one mapping $j\mapsto I(j)$ from $\{ 1,\, \ldots ,\, s\}$ to the set of all pairwise different subsets of $\{ 1,\, \ldots ,\, n\}$ (including $\emptyset$ and the entire $\{ 1,\, \ldots ,\, n\}$). The mapping $\varPhi$ defined in (47) is piecewise smooth, and the components of the corresponding smooth selection mappings $\varPhi ^j:\mathbb {R}^p\rightarrow \mathbb {R}^p$ for $j\in \{ 1,\, \ldots ,\, s\}$ are given by

$$\begin{aligned} \varPhi ^j_i(u) := F_i(x)-y_i,\quad \varPhi ^j_{n+i}(u) := \left\{ \begin{array}{ll} x_i,&{} \text{ if } i\in I(j),\\ y_i &{} \text{ otherwise }, \end{array} \right. \quad i\in \{ 1,\, \ldots ,\, n\} . \end{aligned}$$

(48)

Then, the set of indices of selection mappings that are active at u, defined according to (5), takes the form

$$\begin{aligned} \mathcal {A}(u) = \{ j\in \{ 1,\, \ldots ,\, s\} \mid I(j) = J\cup I_<(u),\; J\subset I_=(u)\} , \end{aligned}$$

(49)

where

$$\begin{aligned} I_>(u):= & {} \{ i\in \{ 1,\, \ldots ,\, n\} \mid x_i > y_i\},\\ I_=(u):= & {} \{ i\in \{ 1,\, \ldots ,\, n\} \mid x_i = y_i\},\\ I_<(u):= & {} \{ i\in \{ 1,\, \ldots ,\, n\} \mid x_i < y_i\}. \end{aligned}$$

Evidently, any $\bar{x}\in \mathbb {R}^n$ is a solution of NCP (46) if and only if $\bar{u}= (\bar{x},\, F(\bar{x}))$ is a solution of Eq. (3) with $\varPhi$ defined in (47). Set

$$\begin{aligned} I_>:= & {} I_>(\bar{u})=\{ i\in \{ 1,\, \ldots ,\, n\} \mid \bar{x}_i > F_i(\bar{x}) = 0\},\\ I_=:= & {} I_=(\bar{u})=\{ i\in \{ 1,\, \ldots ,\, n\} \mid \bar{x}_i = 0 = F_i(\bar{x})\},\\ I_<:= & {} I_<(\bar{u})=\{ i\in \{ 1,\, \ldots ,\, n\} \mid 0 = \bar{x}_i < F_i(\bar{x})\}. \end{aligned}$$

For any $J\subset I_=$, we will use the notation ${\setminus } J:= I_={\setminus } J$.

For a given $j\in \mathcal {A}(\bar{u})$, and for $J\subset I_=$ associated with j by the equality $I(j) = J\cup I_<$, from (48) we have (after the appropriate re-ordering of rows and columns) that

$$\begin{aligned} (\varPhi ^j)'(\bar{u}) = \left( \begin{array}{cc} F'(\bar{x})&{}-\mathcal {I}\\ \mathcal {I}_{J\cup I_<}&{}0\\ 0&{}\mathcal {I}_{\setminus J\cup I_>} \end{array} \right) . \end{aligned}$$

(50)

This matrix is evidently nonsingular if and only if the matrix

$$\begin{aligned} \frac{\partial F_{\setminus J\cup I_>}}{\partial x_{\setminus J\cup I_>}} (\bar{x}) \end{aligned}$$

is nonsingular.

Furthermore, for any $v = (\xi ,\, \eta )\in \mathbb {R}^n\times \mathbb {R}^n$, from (47) we have (after the appropriate re-ordering of components) that

$$\begin{aligned} \varPhi '(\bar{u};\, v) = (F'(\bar{x})\xi -\eta ,\, (\xi _{I_<},\, \min \{ \xi _{I_=},\, \eta _{I_=}\} ,\, \eta _{I_>})). \end{aligned}$$

(51)

From (50), we then obtain that $\varPhi '(\bar{u};\, v) = (\varPhi ^j)'(\bar{u})v$ holds if and only if

$$\begin{aligned} \min \{ \xi _J,\, \eta _J\} = \xi _J,\quad \min \{ \xi _{\setminus J},\, \eta _{\setminus J}\} = \eta _{\setminus J}, \end{aligned}$$

and therefore, according to (8), (49),

$$\begin{aligned} \mathcal {A}(\bar{u},\, v) = \{ j\in \{ 1,\, \ldots ,\, s\} \mid I(j) = J\cup I_<,\; J\subset I_=,\; \xi _J\le \eta _J,\; \xi _{\setminus J}\ge \eta _{\setminus J}\} . \end{aligned}$$

(52)

We are interested in solutions of the perturbed NCP

$$\begin{aligned} x\ge 0,\quad F(x)-z\ge 0,\quad \langle x,\, F(x)-z\rangle =0, \end{aligned}$$

(53)

where $z\in \mathbb {R}^n$ is a small additive perturbation of F. To that end, we are interested in perturbed equation (4) with $w\in \mathbb {R}^n\times \mathbb {R}^n$ close to the directions of the form $(\zeta ,\, 0)$, $\zeta \in \mathbb {R}^n$. Therefore, it makes sense to consider $\bar{v}= (\bar{\xi },\, \bar{\eta } )\in \mathbb {R}^n\times \mathbb {R}^n$ such that $\varPhi '(\bar{u};\, \bar{v}) = (\bar{\zeta },\, 0)$ with some $\bar{\zeta } \in \mathbb {R}^n$, which according to (51) means that

$$\begin{aligned} \bar{\xi } _{I_<} = 0,\quad \min \{ \bar{\xi } _{I_=},\, \bar{\eta }_{I_=}\} = 0,\quad \bar{\eta }_{I_>} = 0. \end{aligned}$$

(54)

Pick up any $\widehat{J}\subset I_=$ and $\bar{v}= (\bar{\xi },\, \bar{\eta })$ such that

$$\begin{aligned}{} & {} \bar{\xi } _{\widehat{J}\cup I_<} = 0,\quad \bar{\xi } _{\setminus \widehat{J}} > 0, \end{aligned}$$

(55)

$$\begin{aligned}{} & {} \bar{\eta } _{\widehat{J}}> 0,\quad \bar{\eta } _{\setminus \widehat{J}\cup I_>} = 0. \end{aligned}$$

(56)

Any such $\bar{v}$ satisfies (54), and according to (52), it holds that $\mathcal {A}(\bar{u},\, \bar{v}) = \{ \widehat{\jmath } \}$, where $\widehat{\jmath }$ is associated with $\widehat{J}$ by the equality $I(\widehat{\jmath } ) = \widehat{J}\cup I_<$. The latter agrees with discussion at the end of Sect. 3 and the fact that matrices in (50) cannot be the same for two different $J\subset I_=$ (and in particular, Proposition 3.2 cannot be applicable if $\mathcal {A}(\bar{u},\, \bar{v})$ is not a singleton). Furthermore,

$$\begin{aligned} (\varPhi ^{\widehat{\jmath } })'(\bar{u})\bar{v}= (\bar{\zeta },\, 0), \end{aligned}$$

where

$$\begin{aligned} \bar{\zeta } _{\setminus \widehat{J}\cup I_>} = \frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x})\bar{\xi } _{\setminus \widehat{J}\cup I_>},\quad \bar{\zeta }_{\widehat{J}\cup I_<} = \frac{\partial F_{\widehat{J}\cup I_<}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x})\bar{\xi } _{\setminus \widehat{J}\cup I_>}-\bar{\eta } _{\widehat{J}\cup I_<}. \end{aligned}$$

(57)

Proposition 6.1

Let $\bar{x}$ be a solution of NCP (46) with $F:\mathbb {R}^n\rightarrow \mathbb {R}^n$ being continuously differentiable near $\bar{x}$. Let $\widehat{J}\subset I_=$ and $\bar{\xi } \in \mathbb {R}^n$ be satisfying (55), and assume that the matrix

$$\begin{aligned} \frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}) \end{aligned}$$

(58)

is nonsingular.

Then, for any $\varepsilon > 0$ and $\delta > 0$, and any $\bar{\eta } \in \mathbb {R}^n$ satisfying (56), there exist $N > 0$, $\widetilde{\varepsilon } > 0$, and $\widetilde{\delta } > 0$ such that, for any $z\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, \bar{\zeta } )$ with $\bar{\zeta } \in \mathbb {R}^n$ defined by (57), NCP (53) has a solution $x(z)\in K_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } )$ satisfying the estimate

$$\begin{aligned} \Vert x(z)-\bar{x}\Vert \le N\Vert z\Vert . \end{aligned}$$

(59)

Proof

Considerations preceding the statement of the proposition allow to apply Proposition 3.1 with $\varPhi$ defined in (47), and with $\bar{v}= (\bar{\xi },\, \bar{\eta } )$ for any $\bar{\xi }$ and $\bar{\eta }$ with the specified properties, yielding the existence of $N > 0$, $\widetilde{\varepsilon } > 0$, and $\widetilde{\delta } > 0$, such that for any $w\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\bar{\zeta },\, 0))$, Eq. (4) has a solution $u(w)\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ satisfying estimate (11). If we take $w = (z,\, 0)$ with $z\in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, \bar{\zeta } )$, then evidently $w \in K_{\widetilde{\varepsilon },\, \widetilde{\delta } }(0,\, (\bar{\zeta },\, 0))$, and, for any solution $u(w) = (x(z),\, y(z))\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ of Eq. (4), it holds that x(z) is a solution of (53), and (11) evidently implies (59).

Finally, if $\bar{\xi } = 0$, the inclusion $(x(z),\, y(z))\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ clearly implies $x(z)\in B(\bar{x},\, \varepsilon ) = K_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } )$. If $\bar{\xi } \not = 0$, by routine computations, it can be seen that the inclusion $(x(z),\, y(z))\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ yields $x(z)\in K_{\varepsilon ,\, \widehat{\delta }}(\bar{x},\, \bar{\xi } )$, where $\widehat{\delta } = 2\delta \Vert \bar{v}\Vert /\Vert \bar{\xi } \Vert$. Therefore, the needed result can be obtained by repeating the reasoning above with $\delta$ replaced by $\delta \Vert \bar{\xi } \Vert /(2\Vert \bar{v}\Vert )$. $\square$

Assuming nonsingularity of the matrix in (58), the set of $\bar{\zeta }$ satisfying (57) with some $\bar{v}= (\bar{\xi },\, \bar{\eta } )$ satisfying (55)–(56) is characterized by the inequalities

$$\begin{aligned}{} & {} \left( \left( \frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x})\right) ^{-1}\zeta _{\setminus \widehat{J}\cup I_>}\right) _{\setminus \widehat{J}}> 0,\\{} & {} \quad \frac{\partial F_{\widehat{J}}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}) \left( \frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x})\right) ^{-1}\zeta _{\setminus \widehat{J}\cup I_>}-\zeta _{\widehat{J}} > 0, \end{aligned}$$

i.e., it is the (nonempty) interior of a polyhedral cone. This interpretation of Proposition 6.1 essentially recovers [4, Proposition 3.2]. The use of these propositions is demonstrated by a solution $\bar{x}= (0,\, 0)$ in [4, Example 3.3].

For the reformulations of NCP, Proposition 5.1 (a) is not applicable since $p = q$. At the same time, Proposition 5.1 (b) can be applicable.

To that end, we now pick up any $\widehat{J}\subset I_=$ and $\bar{v}= (\bar{\xi },\, \bar{\eta } )$ satisfying (55)–(56) and the equalities

$$\begin{aligned} \frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}) \bar{\xi } _{\setminus \widehat{J}\cup I_>} = 0,\quad \frac{\partial F_{\widehat{J}\cup I_<}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}) \bar{\xi } _{\setminus \widehat{J}\cup I_>}-\bar{\eta } _{\widehat{J}\cup I_<} = 0. \end{aligned}$$

(60)

According to the discussion above (see (51)–(52)), we then have that $\varPhi '(\bar{u};\, \bar{v}) = 0$, and $\mathcal {A}(\bar{u},\, \bar{v}) = \{ \widehat{\jmath } \}$ with $\widehat{\jmath }$ associated with $\widehat{J}$ by the equality $I(\widehat{\jmath } ) = \widehat{J}\cup I_<$.

Furthermore, from (50) we have

$$\begin{aligned}{} & {} \textrm{im}(\varPhi ^{\widehat{\jmath } })'(\bar{u}) = \left\{ (z,\, \omega )\in \mathbb {R}^n\times \mathbb {R}^n\left| \, \begin{array}{l} \displaystyle (z+\omega )_{\setminus \widehat{J}\cup I_>} -\frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\widehat{J}\cup I_<}} (\bar{x}) \omega _{\widehat{J}\cup I_<}\\ \displaystyle \in \textrm{im}\frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}) \end{array} \right. \right\} , \end{aligned}$$

(61)

$$\begin{aligned}{} & {} \ker (\varPhi ^{\widehat{\jmath } })'(\bar{u}) = \left\{ (\xi ,\, \eta ) \in \mathbb {R}^n\times \mathbb {R}^n\left| \, \begin{array}{l} \displaystyle \xi _{\widehat{J}\cup I_<} = 0,\; \frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}) \xi _{\setminus \widehat{J}\cup I_>} = 0,\\ \displaystyle \eta _{\widehat{J}\cup I_<} = \frac{\partial F_{\widehat{J}\cup I_<}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}) \xi _{\setminus \widehat{J}\cup I_>},\\ \eta _ {\setminus \widehat{J}\cup I_>} = 0 \end{array} \right. \right\} .\nonumber \\ \end{aligned}$$

(62)

Assuming that F is twice differentiable at $\bar{x}$, and taking into account (55), from (48) we also derive that

$$\begin{aligned} (\varPhi ^{\widehat{\jmath } })''(\bar{u})[\bar{v} ,\, v] = \left( \frac{\partial ^2F}{\partial x_{\setminus \widehat{J}\cup I_>}^2} (\bar{x}) \left[ \bar{\xi } _{\setminus \widehat{J}\cup I_>},\, \xi _{\setminus \widehat{J}\cup I_>}\right] ,\, 0 \right) \end{aligned}$$

(63)

for any $v = (\xi ,\, \eta )\in \ker (\varPhi ^{\widehat{\jmath } })'(\bar{u})$. Employing (61)–(62), we then readily conclude that 2-regularity of $\varPhi ^{\widehat{\jmath } }$ at $\bar{u}$ in the direction $\bar{v}$ is equivalent to saying that there exists no $\xi \in \mathbb {R}^n$, with $\xi _{\setminus \widehat{J}\cup I_>}\not = 0$, satisfying

$$\begin{aligned} \frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}) \xi _{\setminus \widehat{J}\cup I_>} = 0,\quad \frac{\partial ^2F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}^2} (\bar{x}) \left[ \bar{\xi } _{\setminus \widehat{J}\cup I_>},\, \xi _{\setminus \widehat{J}\cup I_>}\right] \in \textrm{im}\frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}). \end{aligned}$$

(64)

Proposition 6.2

Let $\bar{x}$ be a solution of NCP (46), where $F:\mathbb {R}^n\rightarrow \mathbb {R}^n$ is differentiable near $\bar{x}$, $F'$ is Lipschitz-continuous near $\bar{x}$, and F is twice differentiable at $\bar{x}$. Let $\widehat{J}\subset I_=$ and $\bar{\xi } \in \mathbb {R}^n$ be satisfying (55), the first equality in (60), and

$$\begin{aligned} \frac{\partial F_{\widehat{J}}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}) \bar{\xi } _{\setminus \widehat{J}\cup I_>} > 0, \end{aligned}$$

(65)

and assume that there exists no $\xi \in \mathbb {R}^n$, with $\xi _{\setminus \widehat{J}\cup I_>}\not = 0$, satisfying (64).

Then, for any $\varepsilon > 0$ and $\delta > 0$, there exist $N > 0$ and a set $Z_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } )\subset \mathbb {R}^n$ such that $0\in Z_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } )$, the set $\bar{\zeta } +L$ with $\bar{\zeta } \in \mathbb {R}^n$ and $L \subset \mathbb {R}^n$ defined by

$$\begin{aligned}{} & {} \bar{\zeta } _{\widehat{J}\cup I_<}:= 0,\quad \bar{\zeta } _{\setminus \widehat{J}\cup I_>}:= \frac{\partial ^2F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}^2} (\bar{x}) \left[ \bar{\xi } _{\setminus \widehat{J}\cup I_>},\, \bar{\xi } _{\setminus \widehat{J}\cup I_>}\right] , \end{aligned}$$

(66)

$$\begin{aligned}{} & {} L:= \left\{ z\in \mathbb {R}^n\left| \, z_{\setminus \widehat{J}\cup I_>}\in \textrm{im}\frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}) \right. \right\} , \end{aligned}$$

(67)

is contained in the interior of the cone of feasible directions to $Z_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } )$ at 0, and for any $z\in Z_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } )$, NCP (53) has a solution $x(z)\in K_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } )$ satisfying the estimate

$$\begin{aligned} \Vert x(z)-\bar{x}\Vert \le N\Vert z\Vert ^{1/2}. \end{aligned}$$

(68)

Proof

Observe first that under (65), there exists the unique $\bar{\eta } \in \mathbb {R}^n$ satisfying (56) and the second equality in (60). If $\bar{\xi } _{\setminus \widehat{J}\cup I_>} = 0$, then according to (56) and (60), it holds that $\bar{\eta } = 0$ as well. Then, (55)–(56) are only possible if $\widehat{J} = \setminus \widehat{J} = I_= = \emptyset$, and the assumption that there is no $\xi \in \mathbb {R}^n$, with $\xi _{\setminus \widehat{J}\cup I_>}\not = 0$, satisfying (64), only holds when the matrix

$$\begin{aligned} \frac{\partial F_{I_>}}{\partial x_{I_>}} (\bar{x}) \end{aligned}$$

(69)

is nonsingular. In this case, the needed assertion follows from Proposition 6.1, with $Z_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } ) = B(0,\, \widetilde{\varepsilon } )$ for some $\widetilde{\varepsilon } >0$, and with the estimate (68) replaced by the stronger one in (59).

Suppose now that $\bar{\xi } _{\setminus \widehat{J}\cup I_>}\not = 0$. Considerations preceding the statement of the proposition will allow to apply Proposition 5.1 (b) with $\varPhi$ defined in (47), and with $\bar{v}= (\bar{\xi },\, \bar{\eta } )$, if we show that $\varPi (\varPhi ^{\widehat{\jmath } })''(\bar{u})[\bar{v},\, \bar{v}]\not = 0$ with $\varPi$ defined in Assumption 2 for $\varPhi = \varPhi ^{\widehat{\jmath } }$. According to (61)–(63), this is equivalent to saying that

$$\begin{aligned} \frac{\partial ^2F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}^2} (\bar{x}) \left[ \bar{\xi } _{\setminus \widehat{J}\cup I_>},\, \bar{\xi } _{\setminus \widehat{J}\cup I_>}\right] \not \in \textrm{im}\frac{\partial F_{\setminus \widehat{J}\cup I_>}}{\partial x_{\setminus \widehat{J}\cup I_>}} (\bar{x}). \end{aligned}$$

But if this would have been not the case, taking into account (55) and (60), we would have that $\xi = \bar{\xi }$ satisfies (64), and since $\bar{\xi } _{\setminus \widehat{J}\cup I_>}\not = 0$, this would contradict the assumptions of the proposition.

Applying Proposition 5.1 (b), we now obtain the existence of $N > 0$ and a set $W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})\subset \mathbb {R}^n\times \mathbb {R}^n$ such that $0\in W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$, the set $(\varPhi ^{\widehat{\jmath } })''(\bar{u})[\bar{v},\, \bar{v}]+\textrm{im}(\varPhi ^{\widehat{\jmath } })'(\bar{u})$ is contained in the interior of the cone of feasible directions to $W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ at 0, and for any $w\in W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$, Eq. (4) has a solution $u(w)\in K_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})$ satisfying the estimate (45). Define $Z_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } ) = \{ z\in \mathbb {R}^n\mid (z,\, 0)\in W_{\varepsilon ,\, \delta }(\bar{u},\, \bar{v})\}$. As in the proof of Proposition 6.1, after repeating the argument above with $\delta$ replaced by $\delta \Vert \bar{\xi } \Vert /(2\Vert \bar{v}\Vert )$, we may claim that for any $z\in Z_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } )$, NCP (53) has a solution $x(z)\in K_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } )$ satisfying the estimate (68).

Finally, from (61), (63) and (66)–(67) one can easily see that $(\bar{\zeta } +L)\times \{ 0\} \subset (\varPhi ^{\widehat{\jmath } })''(\bar{u})[\bar{v},\, \bar{v}]+\textrm{im}(\varPhi ^{\widehat{\jmath } })'(\bar{u})$, implying that $\bar{\zeta } +L$ is contained in the interior of the cone of feasible directions to $Z_{\varepsilon ,\, \delta }(\bar{x},\, \bar{v})$ at 0. $\square$

Proposition 6.2 is closely related to [4, Proposition 3.3] where slightly weaker assumptions are employed: some of the strict inequalities in (55) and/or (65) can be replaced by nonstrict ones (under some additional requirements on the structure of the null space of the matrix (58)). However, the assertion of Propositions 6.2 is more exact, as it provides a characterization of the set necessarily contained in the interior of the cone of feasible directions to $Z_{\varepsilon ,\, \delta }(\bar{x},\, \bar{\xi } )$.

Example 6.1

(Example 3.4 in [4]) Let $n = 2$, $F(x) = ((x_1-1)^2+(x_1-1)x_2,\, (x_1-1)^2)$. For a solution $\bar{x}= (1,\, 0)$ of NCP (46) one has $I_> = \{ 1\}$, $I_= = \{ 2\}$, $I_< = \emptyset$, and $F'(\bar{x}) = 0$. This solution can be tackled by means of [4, Proposition 3.3] applied with the branch of the NCP solution set corresponding to $\widehat{J} = \{ 2\}$, and with $\bar{\xi } _1 \not = 0$, $\bar{\xi } _2 = 0$, while Proposition 6.2 is not applicable with such $\widehat{J}$ and $\bar{\xi }$. That said, Proposition 6.2 is applicable with $\widehat{J} = \emptyset$, $\bar{\xi } _1 \not = 0$, $\bar{\xi } _2 > 0$, and with (66)–(67) giving $\bar{\zeta } = (2\bar{\xi } _1(\bar{\xi } _1+\bar{\xi } _2),\, 2\bar{\xi } _1^2)$ and $L = \{ 0\}$. As for [4, Proposition 3.3], it is also applicable in this case, but with a less sharp conclusion regarding the set of perturbations being covered.

7 Conclusions

We presented new results on stability of a given solution of a piecewise smooth equation with respect to large (not asymptotically thin) classes of right-hand side perturbations. Apart from piecewise (rather than true) smoothness, we allow the first derivatives of the smooth selections to be merely B-differentiable and do not require them to be twice differentiable. The main case of interest in this work is when the solution in question is singular for some smooth selection active at it, and hence, its stability properties cannot be tackled by standard analysis tools. Some applications of the results obtained to complementarity problems are also discussed.