Metric Subregularity for Subsmooth Generalized Constraint Equations in Banach Spaces

Abstract

This paper is devoted to metric subregularity of a kind of generalized constraint equations. In particular, in terms of coderivatives and normal cones, we provide some necessary and sufficient conditions for subsmooth generalized constraint equations to be metrically subregular and strongly metrically subregular in general Banach spaces and Asplund spaces, respectively.

1. Introduction

Let 𝑋 be a Banach space and 𝑓∶𝑋→𝑅 be a function. Consider the following inequality: 𝑓(𝑥)≤0. (1.1) Let 𝑆∶={𝑥∈𝑋∶𝑓(𝑥)≤0}. Recall that (1.1) has a local error bound at 𝑎∈𝑆 if there exist 𝜏,𝛿∈(0,+∞) such that 𝑑(𝑥,𝑆)≤𝜏𝑓(𝑥)₊∀𝑥∈𝐵(𝑎,𝛿), (1.2) where [𝑓(𝑥)]₊∶=max{𝑓(𝑥),0} and 𝐵(𝑎,𝛿) denotes the open ball of center 𝑎 and radius 𝛿. The error bound has been studied by many authors (see [1–3] and the references therein).

Let 𝑌 be another Banach space, 𝑏∈𝑌, and let 𝐹∶𝑋⇉𝑌 be a closed multifunction. The following generalized equation: 𝑏∈𝐹(𝑥) (GE) concludes most of systems in optimization and was investigated by many researchers (see [4–9] and the references therein). Let 𝑥∈𝑋 and 𝑏∈𝐹(𝑎). Recall that (GE) is metrically subregular at (𝑎,𝑏) if there exist 𝜏,𝛿∈(0,∞) such that 𝑑𝑥,𝐹⁻¹(𝑏)≤𝜏𝑑(𝑏,𝐹(𝑥))∀𝑥∈𝐵(𝑎,𝛿) (1.3) (see [4–6] and the references therein). This property provides an estimate how far for an element 𝑥 near 𝑎 can be from the solution set of (GE). A stronger notion is the metric regularity: a multifunction 𝐹 is metrically regular at (𝑎,𝑏) if there exist 𝜏,𝛿∈(0,+∞) such that 𝑑𝑥,𝐹⁻¹𝑦≤𝜏𝑑𝑦,𝐹(𝑥)∀𝑥,𝑦∈𝐵((𝑎,𝑏),𝛿). (1.4) There exists a wide literature on this topic. We refer the interested readers to [3, 7–11] and to the references contained therein. Let 𝐴 be a closed subset of 𝑋. Consider the generalized constraint equation as follows: 𝑏∈𝐹(𝑥)subjectto𝑥∈𝐴. (GCE)

Let 𝑆 denote the solution set of (GCE), that is, 𝑆={𝑥∈𝐴∶𝑏∈𝐹(𝑥)}. We say that (GCE) is metrically subregular at 𝑎∈𝑆 if there exist 𝜏,𝛿∈(0,∞) such that 𝑑(𝑥,𝑆)≤𝜏(𝑑(𝑏,𝐹(𝑥))+𝑑(𝑥,𝐴))∀𝑥∈𝐵(𝑎,𝛿). (1.5)

When 𝐴=𝑋, (GCE) reduces (GE) and (1.5) means that (GE) is metrically subregular at (𝑎,𝑏). When 𝐹(𝑥)=[𝑓(𝑥),+∞),𝑏=0 and 𝐴=𝑋, (GCE) reduces the inequality (1.1) and (1.5) means that this inequality has a local error bound at 𝑎. Error bounds, metric subregularity and regularity have important applications in mathematical programming and have been extensively studied (see [1–12] and the references therein). The Authors [13] introduced the notions of primal smoothness and investigated the properties of primal smooth functions. Under proper conditions, the distance function is primal smooth. Differentiability of the distance function was discussed in [14]. As extension of primal smoothness and convexity, the notion of subsmoothness was introduced and some functional characterizations were provided in [15]. Recently, by variational analysis techniques (for more details, see [16–19]), Zheng and Ng [6] investigated metric subregularity of (GE) under the subsmooth assumption. In this paper, in terms of normal cones and coderivatives, we devote to metric subregularity of generalized constraint equation (GCE) under the subsmooth assumption. We will build some new necessary and sufficient conditions for (GCE) to be metrically subregular and strongly metrically subregular.

2. Preliminaries

Let 𝑋 be a Banach space. We denote by 𝐵_𝑋 and 𝑋^∗ the closed unit ball and the dual space of 𝑋, respectively. Let 𝐴 be a nonempty subset of 𝑋, int(𝐴) and bd(𝐴), respectively, denote the interior and the boundary of 𝐴. For 𝑎∈𝑋 and 𝛿>0, let 𝐵(𝑎,𝛿) denote the open ball with center 𝑎 and radius 𝛿.

We introduce some notions of variations and derivatives needed to state our results.

For a closed subset 𝐴 of 𝑋 and 𝑎∈𝐴, let 𝑇_𝑐(𝐴,𝑎) and 𝑇(𝐴,𝑎), respectively, denote the Clarke tangent cone and contingent (Bouligand) cone of 𝐴 at 𝑎 defined by 𝑇_𝑐(𝐴,𝑎)∶=liminf_𝑥𝐴−→_{𝑎,𝑡→0}⁺𝐴−𝑥𝑡,𝑇(𝐴,𝑎)∶=limsup_𝑡→0+𝐴−𝑎𝑡, (2.1) where 𝑥−→^𝐴𝑎 means that 𝑥→𝑎 with 𝑥∈𝐴. It is easy to verify that 𝑣∈𝑇_𝑐(𝐴,𝑎) if and only if for each sequence {𝑎_𝑛} in 𝐴 converging to 𝑎 and each sequence {𝑡_𝑛} in (0, ∞) decreasing to 0, there exists a sequence {𝑣_𝑛} in 𝑋 converging to 𝑣 such that 𝑎_𝑛+𝑡_𝑛𝑣_𝑛∈𝐴 for each natural number 𝑛; while 𝑣∈𝑇(𝐴,𝑎) if and only if there exists a sequence {𝑣_𝑛} in 𝑋 converging to 𝑣 and a sequence {𝑡_𝑛} in (0,∞) decreasing to 0 such that 𝑎+𝑡_𝑛𝑣_𝑛∈𝐴 for all 𝑛.

We denote by 𝑁_𝑐(𝐴,𝑎) the Clarke normal cone of 𝐴 at 𝑎, that is, 𝑁_𝑐(𝐴,𝑎)∶=𝑥^∗∈𝑋^∗∶𝑥^∗,ℎ≤0∀ℎ∈𝑇_𝑐(𝐴,𝑎). (2.2) For 𝜀≥0 and 𝑎∈𝐴, the nonempty set 𝑁_𝜀(𝐴,𝑎)∶=𝑥^∗∈𝑋^∗∶limsup_𝑥𝐴−→_𝑎𝑥^∗,𝑥−𝑎‖𝑥−𝑎‖≤𝜀 (2.3) is called the set of Fréchet 𝜀-normals of 𝐴 at 𝑎. When 𝜀=0, 𝑁_𝜀(𝐴,𝑎) is a convex cone which is called the Fréchet normal cone of 𝐴 at 𝑎 and is denoted by 𝑁(𝐴,𝑎). Let 𝑁(𝐴,𝑎) denote the Mordukhovich limiting or basic normal cone of 𝐴 at 𝑎, that is, 𝑁(𝐴,𝑎)∶=limsup_𝑥𝐴−→_{𝑎,𝜀→0}⁺𝑁_𝜀(𝐴,𝑥), (2.4) that is, 𝑥^∗∈𝑁(𝐴,𝑎) if and only if there exist sequences {(𝑥_𝑛,𝜀_𝑛,𝑥^∗_𝑛)} in 𝐴×𝑅₊×𝑋^∗ such that (𝑥_𝑛,𝜀_𝑛)→(𝑎,0),𝑥−−→^∗_𝑛^𝑤∗𝑥^∗ and 𝑥^∗_𝑛∈𝑁_𝜀𝑛(𝐴,𝑥_𝑛) for each natural number 𝑛. It is known that 𝑁(𝐴,𝑎)⊆𝑁(𝐴,𝑎)⊆𝑁_𝑐(𝐴,𝑎), (2.5) (see [4, 9, 16, 18, 19] and the references contained therein). If 𝐴 is convex, then 𝑇(𝐴,𝑎)=𝑇_𝑐(),𝑁()()𝐴,𝑎𝐴,𝑎=𝑁𝐴,𝑎=𝑁_𝑐(𝐴,𝑎)=𝑥^∗∈𝑋^∗∶𝑥^∗,𝑥−𝑎≤0∀𝑥∈𝐴. (2.6) Recall that a Banach space 𝑋 is called an Asplund space if every continuous convex function on 𝑋 is Fréchet differentiable at each point of a dense subset of 𝑋 (for other definitions and their equivalents, see [19]). It is well known that 𝑋 is an Asplund space if and only if every separable subspace of 𝑋 has a separable dual space. In particular, every reflexive Banach space is an Asplund space. When X is an Asplund space, it is well known that 𝑁_𝑐(𝐴,𝑎)=cl^∗(co(𝑁(𝐴,𝑎))),𝑁(𝐴,𝑎)=limsup_𝑥𝐴−→_𝑎𝑁(𝐴,𝑥), (2.7) where cl^∗(⋅) denotes the closure with respect to the weak^∗ topology, see [9, 19]. Recently, Zheng and Ng [5] established an approximate projection result for a closed subset of 𝑋, which will play a key role in the proofs of our main results.

Lemma 2.1. Let 𝐴 be a closed nonempty subset of a Banach space 𝑋 and let 𝛽∈(0,1). Then for any 𝑥∉𝐴 there exist 𝑎∈bd(𝐴) and 𝑎^∗∈𝑁_𝑐(𝐴,𝑎) with ‖𝑎^∗‖=1 such that 𝛽‖𝑥−𝑎‖<min𝑑(𝑥,𝐴),𝑎^∗,𝑥−𝑎. (2.8) If 𝑋 is an Asplund space, then 𝑁_𝑐(𝐴,𝑎) can be replaced by 𝑁(𝐴,𝑎).

Let 𝐹∶𝑋⇉𝑌 be a multifunction and let Gr(𝐹) denote the graph of 𝐹, that is, Gr(𝐹)∶=𝑥,𝑦∈𝑋×𝑌∶𝑦∈𝐹(𝑥). (2.9) As usual, 𝐹 is said to be closed (resp., convex) if Gr(𝐹) is a closed (resp., convex) subset of 𝑋×𝑌. Let (𝑥,𝑦)∈Gr(𝐹). The Clarke tangent and contingent derivatives 𝐷_𝑐𝐹(𝑥,𝑦),𝐷𝐹(𝑥,𝑦) of 𝐹 at (𝑥,𝑦) are defined by 𝐷_𝑐𝐹𝑥,𝑦(𝑢)∶=𝑣∈𝑌∶(𝑢,𝑣)∈𝑇_𝑐(𝐹),𝑢)()(𝐹),Gr𝑥,𝑦∀𝑢∈𝑋,𝐷𝐹𝑥,𝑦(∶=𝑣∈𝑌∶𝑢,𝑣∈𝑇Gr𝑥,𝑦∀𝑢∈𝑋, (2.10) respectively. Let 𝐷^∗𝐹(𝑥,𝑦),𝐷^∗𝐹(𝑥,𝑦), and 𝐷^∗_𝑐𝐹(𝑥,𝑦) denote the coderivatives of 𝐹 at (𝑥,𝑦) associated with the Fréchet, Mordukhovich, and Clarke normal structures, respectively. They are defined by the following: 𝐷^∗𝐹𝑥,𝑦𝑦^∗∶=𝑥^∗∈𝑋^∗∶𝑥^∗,−𝑦^∗∈𝑁Gr(𝐹),𝑥,𝑦∀𝑦^∗∈𝑌^∗,𝐷^∗𝐹𝑥,𝑦𝑦^∗∶=𝑥^∗∈𝑋^∗∶𝑥^∗,−𝑦^∗∈𝑁Gr(𝐹),𝑥,𝑦∀𝑦^∗∈𝑌^∗,𝐷^∗_𝑐𝐹𝑥,𝑦𝑦^∗∶=𝑥^∗∈𝑋^∗∶𝑥^∗,−𝑦^∗∈𝑁_𝑐Gr(𝐹),𝑥,𝑦∀𝑦^∗∈𝑌^∗. (2.11) The more details of the coderivatives can be found in [9, 18, 19] and the references therein.

3. Subsmooth Generalized Constraint Equation

Let 𝐴 be a closed subset of 𝑋. Recall (see [13, 14]) that 𝐴 is said to be prox-regular at 𝑎∈𝐴 if there exist 𝜏,𝛿>0 such that 𝑥^∗−𝑢^∗,𝑥−𝑢≥−𝜏‖𝑥−𝑢‖² (3.1) whenever 𝑥,𝑢∈𝐴∩𝐵(𝑎,𝛿),𝑥^∗∈𝑁_𝑐(𝐴,𝑥)∩𝐵_𝑋^∗, and 𝑢^∗∈𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗.

As a generalization of the prox-regularity, Aussel et al. [15] introduced and studied the subsmoothness. 𝐴 is said to be subsmooth at 𝑎∈𝐴 if for any 𝜀>0 there exist 𝜏,𝛿>0 such that 𝑥^∗−𝑢^∗,𝑥−𝑢≥−𝜀‖𝑥−𝑢‖, (3.2) whenever 𝑥,𝑢∈𝐴∩𝐵(𝑎,𝛿),𝑥^∗∈𝑁_𝑐(𝐴,𝑥)∩𝐵_𝑋^∗, and 𝑢^∗∈𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗.

It is easy to verify that 𝐴 is subsmooth at 𝑎∈𝐴 if and only if for any 𝜀>0, there exists 𝛿>0 such that ⟨𝑢^∗,𝑥−𝑢⟩≤𝜀‖𝑥−𝑢‖ (3.3) whenever 𝑥,𝑢∈𝐴∩𝐵(𝑎,𝛿) and 𝑢^∗∈𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗.

Let 𝐹∶𝑋⇉𝑌 be a closed multifunction, 𝑏∈𝑌 and 𝑎∈𝐹⁻¹(𝑏). Zheng and Ng [6] introduce the concept of the L-subsmoothness of 𝐹 at 𝑎 for 𝑏: 𝐹 is called to be L-subsmooth at 𝑎 for 𝑏 if for any 𝜀>0 there exists 𝛿>0 such that 𝑢^∗,𝑥−𝑎+𝑣^∗,𝑦−𝑣≤𝜀‖𝑥−𝑎‖+𝑦−𝑣, (3.4) whenever 𝑣∈𝐹(𝑎)∩𝐵(𝑏,𝛿),(𝑢^∗,𝑣^∗)∈𝑁_𝑐(Gr(𝐹),(𝑎,𝑣))∩𝐵_𝑋^∗_×𝑌^∗and(𝑥,𝑦)∈Gr(𝐹)with‖𝑥−𝑎‖+‖𝑦−𝑏‖<𝛿. Next, we introduce the concept of the subsmoothness of generalized constraint equation (GCE) which will be useful in our discussion.

Definition 3.1. Generalized equation (GCE) is subsmooth at 𝑎∈𝑆 if for any 𝜀>0, there exists 𝛿>0 such that 𝑢^∗,𝑥−𝑢−𝑣^∗,𝑦−𝑏≤𝜀‖𝑥−𝑢‖+𝑦−𝑏,𝑤^∗,𝑥^′−𝑢≤𝜀𝑥^′−𝑢, (3.5) whenever 𝑥∈𝐵(𝑎,𝛿),𝑥^′∈𝐴∩𝐵(𝑎,𝛿),𝑢∈𝑆∩𝐵(𝑎,𝛿),𝑦∈𝐹(𝑥)∩𝐵(𝑏,𝛿),𝑣^∗∈𝐵_𝑌^∗,𝑢^∗∈𝐷^∗_𝑐𝐹(𝑢,𝑏)(𝑣^∗)∩𝐵_𝑋^∗,and𝑤^∗∈𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗.

Remark 3.2. The subsmoothness of (GCE) at 𝑎 means the subsmoothness of 𝐹 at 𝑎 for 𝑏 when 𝐴=𝑋, while the subsmoothness of (GCE) at 𝑎 means the subsmoothness of 𝐴 at 𝑎 when 𝐹(𝑥)=𝑏 for all 𝑥∈𝑋. If 𝐴=𝑋 and Gr(𝐹) is prox-regular at (𝑎,𝑏), then generalized equation (GCE) is subsmooth at 𝑎. If 𝐴 and Gr(𝐹) are convex, then 𝐹 is also subsmooth at 𝑎. Finally when 𝐴 is prox-regular and 𝐹 is single-valued and smooth, (GCE) is subsmooth at 𝑎, too. Hence, Definition 3.1 extends notions of smoothness, convexity and prox-regularity.

Proposition 3.3. Suppose that Generalized equation (GCE) is subsmooth at 𝑎∈𝑆. Then for any 𝜀>0 there exists 𝛿>0 such that 𝑢^∗,𝑥−𝑢≤(2+𝜀)𝑑(𝑏,𝐹(𝑥))+𝜀‖𝑥−𝑢‖,𝑤^∗,𝑥−𝑢≤𝑑(𝑥,𝐴)+𝜀‖𝑥−𝑢‖, (3.6)(3.7) whenever 𝑥∈𝐵(𝑎,𝛿),𝑢∈𝑆∩𝐵(𝑎,𝛿), 𝑢^∗∈𝐷^∗_𝑐𝐹(𝑢,𝑏)(𝐵_𝑌^∗)∩𝐵_𝑋^∗, and 𝑤^∗∈𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗.

Proof. Suppose that (GCE) is subsmooth at 𝑎∈𝑆. Then for any 𝜀>0, there exists 𝛿>0 such that 𝑢^∗,𝑥−𝑢−𝑣^∗≤𝜀,𝑦−𝑏2‖𝑥−𝑢‖+𝑦−𝑏,𝑤^∗,𝑥^′≤𝜀−𝑢2𝑥^′−𝑢, (3.8) whenever 𝑥∈𝐵(𝑎,2𝛿),𝑥^′∈𝐴∩𝐵(𝑎,2𝛿),𝑢∈𝑆∩𝐵(𝑎,2𝛿),𝑦∈𝐹(𝑥)∩𝐵(𝑏,2𝛿),𝑣^∗∈𝐵_𝑌^∗,𝑢^∗∈𝐷^∗_𝑐𝐹(𝑢,𝑏)(𝑣^∗𝐵)∩_𝑋∗,and𝑤^∗∈𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗.
Let 𝑥∈𝐵(𝑎,𝛿),𝑢∈𝑆∩𝐵(𝑎,𝛿),𝑣^∗∈𝐵_𝑌^∗,𝑢^∗∈𝐷^∗_𝑐𝐹(𝑢,𝑏)(𝑣^∗)∩𝐵_𝑋^∗,and𝑤^∗∈𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗. If 𝐹(𝑥)∩𝐵(𝑏,𝛿)=∅, then 𝑢^∗,𝑥−𝑢≤‖𝑥−𝑢‖≤‖𝑥−𝑎‖+‖𝑎−𝑢‖≤2𝛿,𝑑(𝑏,𝐹(𝑥))≥𝛿. (3.9) Thus, (3.6) holds. Otherwise, one has 𝑢^∗,𝑥−𝑢≤𝑣^∗+𝜀,𝑦−𝑏2‖𝑥−𝑢‖+𝑦−𝑏≤(1+𝜀)𝑦−𝑏+𝜀‖𝑥−𝑢‖, (3.10) whenever 𝑦∈𝐹(𝑥)∩𝐵(𝑏,𝛿). Noting that 𝑑(𝑏,𝐹(𝑥))=𝑑(𝑏,𝐹(𝑥)∩𝐵(𝑏,𝛿)) (sine 𝐹(𝑥)∩𝐵(𝑏,𝛿)≠∅), it follows that 𝑢^∗,𝑥−𝑢≤(1+𝜀)𝑑(𝑏,𝐹(𝑥))+𝜀‖𝑥−𝑢‖. (3.11)
It remains to show that (3.7) holds. Since 𝑤^∗,𝑥−𝑢=𝑤^∗,𝑥−𝑥^′+𝑤^∗,𝑥^′−𝑢≤𝑥−𝑥^′+𝜀2𝑥^′≤𝜀−𝑢1+2𝑥−𝑥^′+𝜀2‖𝑥−𝑢‖, (3.12) whenever 𝑥^′∈𝐴∩𝐵(𝑎,𝛿). One has 𝑤^∗()+𝜀,𝑥−𝑢≤𝑑𝑥,𝐴2(𝑑(𝑥,𝐴)+‖𝑥−𝑢‖). (3.13) Noting that 𝑢∈𝐴, it follows that 𝑑(𝑥,𝐴)≤‖𝑥−𝑢‖ which implies that (3.7) holds and completes the proof.

4. Main Results

This section is devoted to metric subregularity of generalized equation (GCE). We divide our discussion into two subsections addressing the necessary conditions and the sufficient conditions for metric subregularity.

4.1. Necessary Conditions for Metric Subregularity

There are two results in this subsection: one is on the Banach space setting and the other on the Asplund space setting.

Theorem 4.1. Suppose that 𝑋,𝑌 are Banach spaces and that generalized equation (GCE) is metrically subregular at 𝑎∈𝑆. Then there exist 𝜏,𝛿∈(0,+∞) such that 𝑁(𝑆,𝑢)∩𝐵_𝑋^∗⊆𝜏𝐷^∗_𝑐𝐹(𝑢,𝑏)𝐵_𝑌^∗+𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗∀𝑢∈𝑆∩𝐵(𝑎,𝛿). (4.1)

Proof. Let 𝛿_Gr(𝐹) denote the indicator function of Gr(𝐹) and 𝛿>0 such that (1.5) holds. Then (1.5) can be rewritten as 𝑑(𝑥,𝑆)≤𝛿_Gr(𝐹)𝑥,𝑦+𝜏𝑦−𝑏+𝑑(𝑥,𝐴)∀𝑥,𝑦∈𝐵(𝑎,𝛿)×𝑌. (4.2) Let 𝑢∈𝑆∩𝐵(𝑎,𝛿) and 𝑢^∗∈𝑁(𝑆,𝑢)∩𝐵_𝑋^∗. Noting (cf. [9, Corollary 1.96]) that 𝑁(𝑆,𝑢)∩𝐵_𝑋^∗=𝜕𝑑(⋅,𝑆)(𝑢), one gets that for any natural number 𝑛, there exists 𝑟∈(0,𝛿) such that 𝐵(𝑢,𝑟)⊆𝐵(𝑎,𝛿) and 𝑢^∗()+1,𝑥−𝑢≤𝑑𝑥,𝑆𝑛‖𝑥−𝑢‖∀𝑥∈𝐵(𝑢,𝑟). (4.3) Hence, by (4.2), it follows that 𝑢^∗,𝑥−𝑢≤𝛿_Gr(𝐹)()+1𝑥,𝑦+𝜏𝑦−𝑏+𝜏𝑑𝑥,𝐴𝑛‖𝑥−𝑢‖∀𝑥,𝑦∈𝐵(𝑢,𝑟)×𝑌, (4.4) that is, (𝑢,𝑏) is a local minimizer of 𝜙 defined by 𝜙𝑥,𝑦∶=−𝑢^∗,𝑥−𝑢+𝛿_Gr(𝐹)()+1𝑥,𝑦+𝜏𝑦−𝑏+𝜏𝑑𝑥,𝐴𝑛‖𝑥−𝑢‖∀𝑥,𝑦∈𝑋×𝑌. (4.5) Hence, (0,0)∈𝜕_𝑐𝜙(𝑢,𝑏). It follows from [16] that (0,0)∈−𝑢^∗,0+𝑁_𝑐(Gr(𝐹),(𝑢,𝑏))+{0}×𝜏𝐵_𝑌^∗+𝜏𝜕_𝑐𝑑(𝑢)×{0}+1⋅,𝐴)(𝑛𝐵_𝑋^∗×{0}, (4.6) that is, 1𝜏𝑢^∗+1𝑥𝜏𝑛^∗𝑛,−𝑦^∗_𝑛∈𝑁_𝑐(Gr(𝐹),(𝑢,𝑏))+𝜕_𝑐𝑑(⋅,𝐴)(𝑢)×{0}, (4.7) for some 𝑥^∗_𝑛∈𝐵_𝑋^∗ and 𝑦^∗_𝑛∈𝐵_𝑌^∗. Since 𝐵_𝑋^∗ and 𝐵_𝑌^∗ are weak^∗ compact, without loss of generality (otherwise take a generalized subsequence), we can assume 𝑥−−→^∗_𝑛^𝑤∗𝑥^∗,𝑦−−→^∗_𝑛^𝑤∗𝑦^∗ for some 𝑥^∗∈𝐵_𝑋^∗ and 𝑦^∗∈𝐵_𝑌^∗ as 𝑛→∞. Noting that 𝑁_𝑐(Gr(𝐹),(𝑢,𝑏))+𝜕_𝑐𝑑(⋅,𝐴)(𝑢)×{0} (4.8) is weak^∗ closed (since 𝑁_𝑐(Gr(𝐹),(𝑢,𝑏)) is weak^∗ closed and 𝜕_𝑐𝑑(⋅,𝐴)(𝑢)×{0} is weak^∗ compact), one has 𝑢^∗𝜏,−𝑦^∗∈𝑁_𝑐(Gr(𝐹),(𝑢,𝑏))+𝜕_𝑐𝑑(⋅,𝐴)(𝑢)×{0}. (4.9) This implies that 𝑢^∗∈𝜏𝐷^∗_𝑐𝐹(𝑢,𝑏)𝐵_𝑌^∗+𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗. (4.10) This shows that (4.1) holds true. The proof is completed.

When 𝑋 and 𝑌 are Asplund spaces, the conclusion in Theorem 4.1 can be strengthened with 𝐷^∗_𝑐𝐹(𝑢,𝑏)(𝐵_𝑌^∗) and 𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗) replaced by 𝐷^∗𝐹(𝑢,𝑏)(𝐵_𝑌^∗) and 𝑁(𝐴,𝑢)∩𝐵_𝑋^∗), respectively. Its proof is similar to that of Theorem 4.1.

Theorem 4.2. Suppose that 𝑋 and 𝑌 are Asplund spaces and that generalized equation (GCE) is metrically subregular at 𝑎∈𝑆. Then there exist 𝜏,𝛿∈(0,+∞) such that 𝑁(𝑆,𝑢)∩𝐵_𝑋^∗⊆𝜏𝐷^∗𝐹(𝑢,𝑏)𝐵_𝑌^∗+𝑁(𝐴,𝑢)∩𝐵_𝑋^∗∀𝑢∈𝑆∩𝐵(𝑎,𝛿). (4.11)

4.2. Sufficient Conditions for Metric Subregularity

Under the subsmooth assumption, we will show in the next result that some conditions similar to (4.1) turns out to be sufficient conditions for metric subregularity.

Theorem 4.3. Let 𝑋 and 𝑌 be Banach spaces. Suppose that generalized constraint equation (GCE) is subsmooth at 𝑎 and that there exist 𝜏,𝛿∈(0,+∞) such that 𝑁_𝑐(𝑆,𝑢)∩𝐵_𝑋^∗⊆𝜏𝐷^∗_𝑐𝐹(𝑢,𝑏)𝐵_𝑌^∗+𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗, (4.12) whenever 𝑢∈bd(𝑆)∩𝐵(𝑎,𝛿). Then (GCE) is metrically subregular at 𝑎 and, more precisely, for any 𝜀∈(0,1/(2(1+𝜏))) there exists 𝛿_𝜀∈(0,𝛿/2) such that 𝑑()≤()𝑥,𝑆1+𝜏)(2+𝜀()𝜀(𝑑((𝑥(1−21+𝜏𝑏,𝐹))+𝑑𝑥,𝐴))∀𝑥∈𝐵𝑎,𝛿_𝜀. (4.13)

Proof. Let 𝜀∈(0,1/(2(1+𝜏))). Then, by subsmooth assmption of (GCE) at 𝑎 and Proposition 3.3, there exists 𝛿^′∈(0,𝛿/2) such that 𝑢^∗₁≤()𝑑((𝑥‖‖,𝑢,𝑥−𝑢2+𝜀𝑏,𝐹))+𝜀𝑥−𝑢_∗2,𝑥−𝑢≤𝑑(𝑥,𝐴)+𝜀‖𝑥−𝑢‖, (4.14) whenever 𝑥∈𝐵(𝑎,𝛿^′),𝑢∈𝑆∩𝐵(𝑎,𝛿^′),𝑢^∗₁∈𝐷^∗_𝑐𝐹(𝑢,𝑏)(𝐵_𝑌^∗)∩𝐵_𝑋^∗and𝑢^∗₂∈𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗.
Let 𝛿_𝜀∈(0,𝛿^′/2) and 𝑥∈𝐵(𝑎,𝛿_𝜀)⧵𝑆. Now we need only show (4.13).(i)If 𝐹(𝑥)∩𝐵(𝑏,𝛿^′)=∅, then 𝑑(𝑥,𝑆)≤‖𝑥−𝑎‖<𝛿_𝜀, 𝑑(𝑏,𝐹(𝑥))≥𝛿^′. Hence (4.13) holds.(ii)Suppose 𝐹(𝑥)∩𝐵(𝑏,𝛿^′)≠∅ and let 𝑑()𝛽∈max𝑥,𝑆𝛿_𝜀()1,21+𝜏𝜀,2,1. (4.15)By Lemma 2.1 there exist 𝑢₀∈bd(𝑆) and 𝑢^∗∈𝑁_𝑐(𝑆,𝑢₀) with ‖𝑢^∗‖=1 such that 𝛽𝑥−𝑢₀≤min𝑢^∗,𝑥−𝑢₀,𝑑(𝑥,𝑆). (4.16) Thus, ‖𝑥−𝑢₀‖≤(𝑑(𝑥,𝑆)/𝛽)<𝛿_𝜀. Hence, 𝑢₀−𝑎≤𝑢₀−𝑥+‖𝑥−𝑎‖<2𝛿_𝜀<𝛿^′<𝛿. (4.17) By (4.12) there exist 𝑦^∗₁∈𝐵_𝑌^∗,𝑥^∗₁∈𝐷^∗_𝑐𝐹(𝑢₀,𝑏)(𝑦^∗₁), and 𝑥^∗₂∈𝑁_𝑐(𝐴,𝑢₀)∩𝐵_𝑋^∗ such that 𝑢^∗=𝜏(𝑥^∗₁+𝑥^∗₂). Applying (4.14) with ((𝜏/(1+𝜏))𝑥^∗₁,(𝜏/(1+𝜏))𝑥^∗₂,𝑢₀) in place of (𝑢^∗₁,𝑢^∗₂,𝑢), it follows that 𝑢^∗,𝑥−𝑢₀=𝜏𝑥^∗₁,𝑥−𝑢₀+𝑥^∗₂,𝑥−𝑢₀≤(1+𝜏)(2+𝜀)(𝑑(𝑏,𝐹(𝑥))+𝑑(𝑥,𝐴))+2(1+𝜏)𝜀𝑥−𝑢₀. (4.18) This and (4.16) imply that 𝑑(𝑥,𝑆)≤𝑥−𝑢₀≤(1+𝜏)(2+𝜀)()𝜀(𝑑((𝑥(.𝛽−21+𝜏𝑏,𝐹))+𝑑𝑥,𝐴)) (4.19) Letting 𝛽→1, it follows that (4.13) holds. The proof is completed.

When 𝑋 and 𝑌 are Asplund spaces, the assumption in Theorem 4.3 can be weakened with 𝑁_𝑐(𝑆,𝑢) replaced by 𝑁(𝑆,𝑢).

Theorem 4.4. Suppose 𝑋 and 𝑌 are Asplund spaces. Suppose that generalized constraint equation (GCE) is subsmooth at 𝑎 and that there exist 𝜏,𝛿∈(0,+∞) such that 𝑁(𝑆,𝑢)∩𝐵_𝑋^∗⊆𝜏𝐷^∗_𝑐𝐹(𝑢,𝑏)𝐵_𝑌^∗+𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗, (4.20) whenever 𝑢∈bd(𝑆)∩𝐵(𝑎,𝛿). Then for any 𝜀>0 there exists 𝛿_𝜀>0 such that (4.13) holds.

With the Asplund space version of Lemma 2.1 applied in place of the Banach space version, similar to the proof of Theorem 4.3, it is easy to verify Theorem 4.4.

In general, (GCE) is not necessarily metrically subregular at 𝑎 if (GCE) only has that 𝑁_𝑐(𝑆,𝑎)∩𝐵_𝑋^∗⊆𝜏(𝐷^∗_𝑐𝐹(𝑎,𝑏)(𝐵_𝑌^∗)+𝑁_𝑐(𝐴,𝑎)∩𝐵_𝑋^∗).

Finally, we end this subsection with a sufficient and necessary condition for the Clarke tangent derivative mapping 𝐷_𝑐𝐹(𝑎,𝑏) to be metrically subregular at 0 for 0 over the Clerke tangent cone 𝑇_𝑐(𝐴,𝑎).

Let 𝜏(𝐹,𝑎,𝑏;𝐴)∶=inf{𝜏>0∶thereexists𝛿>0suchthat(1.5)holds}. (4.21) For 𝑢∈𝑆, let 𝛾(𝐹,𝑢,𝑏;𝐴)∶=inf𝜏>0∶𝑁_𝑐(𝑆,𝑢)∩𝐵_𝑋^∗⊆𝜏𝐷^∗_𝑐𝐹(𝑎,𝑏)𝐵_𝑌^∗+𝑁_𝑐(𝐴,𝑢)∩𝐵_𝑋^∗. (4.22) The following lemma is known ([5, Theorem 3.2]) and useful for us in the sequel.

Lemma 4.5. Assume that 𝐹∶𝑋⇉𝑌 is a closed convex multifunction, 𝐴 is a closed convex subset of 𝑋, and 𝑎∈𝑆. And suppose that there exist a cone 𝐶 and a neighborhood 𝑉 of 𝑎 such that 𝑆∩𝑉=(𝑎+𝐶)∩𝑉. Then, 𝜏(𝐹,𝑎,𝑏;𝐴)=𝛾(𝐹,𝑎,𝑏;𝐴). (4.23) Consequently, (GCE) is metrically subregular at 𝑎 if and only if 𝛾(𝐹,𝑎,𝑏;𝐴)<+∞.

Theorem 4.6. Let 𝑎∈𝑆 and 𝜏∶=inf𝜏>0∶𝑑ℎ,𝑇_𝑐(𝑆,𝑎)≤𝜏𝑑0,𝐷_𝑐𝐹(𝑎,𝑏)(ℎ)+𝑑ℎ,𝑇_𝑐(𝐴,𝑎)∀ℎ∈𝑋. (4.24) Suppose that 𝑇_𝑐(𝑆,𝑎)⊆𝑇_𝑐(𝐴,𝑎)∩𝐷_𝑐𝐹(𝑎,𝑏)⁻¹(0). (4.25) Then, 𝜏=𝛾(𝐹,𝑎,𝑏;𝐴). (4.26) If, in addition, 𝜏<+∞, then 𝑇_𝑐(𝑆,𝑎)=𝑇_𝑐(𝐴,𝑎)∩𝐷_𝑐𝐹(𝑎,𝑏)⁻¹(0). (4.27) Consequently, 𝐷_𝑐𝐹(𝑎,𝑏) is metrically subregular at (0,0) over 𝑇_𝑐(𝐴,𝑎) if and only if 𝛾(𝐹,𝑎,𝑏;𝐴)<+∞.

Proof. First, we assume that 𝜏<+∞. By the definition of 𝜏, we have 𝑑𝑥,𝑇_𝑐(𝑆,𝑎)≤𝜏𝑑0,𝐷_𝑐𝐹(𝑎,𝑏)(𝑥)+𝑑𝑥,𝑇_𝑐(𝐴,𝑎)∀𝑥∈𝑋. (4.28) This implies that 𝑇_𝑐(𝐴,𝑎)∩𝐷_𝑐𝐹(𝑎,𝑏)⁻¹(0)⊆𝑇_𝑐(𝑆,𝑎). (4.29) This and (4.25) imply that (4.27) holds.
We consider the following constraint equation: 0∈𝐷_𝑐𝐹(𝑎,𝑏)(𝑥)subjectto𝑥∈𝑇_𝑐(𝐴,𝑎). GCE^′ Let 𝑆^′ denote the solution set of GCE^′. Then, 𝑆^′=𝑇_𝑐(𝐴,𝑎)∩𝐷_𝑐𝐹(𝑎,𝑏)⁻¹(0). (4.30) Noting that 𝑁_𝑐𝑆^′,0=𝑁_𝑐𝑇_𝑐(𝑆,𝑎),0=𝑁_𝑐(),𝑁𝑆,𝑎_𝑐(Gr(𝐹),(𝑎,𝑏))=𝑁_𝑐Gr𝐷_𝑐𝐹(𝑎,𝑏),(0,0), (4.31) it is straightforward to verify that 𝜏=𝜏𝐷_𝑐𝐹(𝑎,𝑏),0,0;𝑇_𝑐(𝐴,𝑎),𝛾(𝐹,𝑎,𝑏;𝐴)=𝛾𝐷_𝑐𝐹(𝑎,𝑏),0,0;𝑇_𝑐(𝐴,𝑎). (4.32) On the other hand, since 𝐷_𝑐𝐹(𝑎,𝑏) is a closed convex multifunction from 𝑋 to 𝑌 and 𝑇_𝑐(𝐴,𝑎) is a closed convex cone, Lemma 4.5 implies that 𝜏𝐷_𝑐𝐹(𝑎,𝑏),0,0;𝑇_𝑐(𝐴,𝑎)=𝛾𝐷_𝑐𝐹(𝑎,𝑏),0,0;𝑇_𝑐(𝐴,𝑎). (4.33) This gives us 𝜏=𝛾(𝐹,𝑎,𝑏;𝐴).
It remains to show that 𝛾(𝐹,𝑎,𝑏;𝐴)=+∞ when 𝜏=+∞. Suppose that 𝛾(𝐹,𝑎,𝑏;𝐴)<+∞. (4.34) We need only show that 𝜏<+∞. Let 𝑥∈𝑋⧵𝑇_𝑐(𝑆,𝑎) and 𝛽∈(0,1). By Lemma 2.1 there exist 𝑢∈𝑇_𝑐(𝑆,𝑎) and 𝑥^∗∈𝑁_𝑐(𝑇_𝑐(𝑆,𝑎),𝑢) with ‖𝑥^∗‖=1 such that 𝛽‖𝑥−𝑢‖≤𝑥^∗,𝑥−𝑢. (4.35) Noting that 𝑇_𝑐(𝑆,𝑎) is a convex cone, it is easy to verify that 𝑥^∗∈𝑁_𝑐𝑇_𝑐(𝑆,𝑎),0=𝑁_𝑐(𝑆,𝑎),𝑥^∗,𝑢=0. (4.36) Take a fixed 𝑟 in (𝛾(𝐹,𝑎,𝑏;𝐴),+∞). Then there exist 𝑦^∗∈𝑟𝐵_𝑌^∗,𝑥^∗₁∈𝐷^∗_𝑐𝐹(𝑎,𝑏)(𝑦^∗) and 𝑥^∗₂∈𝑁_𝑐(𝐴,𝑎)∩𝑟𝐵_𝑋^∗ such that 𝑥^∗=𝑥^∗₁+𝑥^∗₂. (4.37) We equip the product space 𝑋×𝑌 with norm 𝑥,𝑦_𝑟𝑟∶=‖𝑥‖+𝑦∀1+𝑟𝑥,𝑦∈𝑋×𝑌. (4.38) Noting that the unit ball of the dual space of (𝑋×𝑌,‖⋅‖_𝑟) is (((1+𝑟)/𝑟)𝐵_𝑋^∗)×𝐵_𝑌^∗, it follows from the convexity of 𝐷_𝑐𝐹(𝑥,𝑦) and 𝑇_𝑐(𝐴,𝑎) that 1𝑟𝑥^∗₁,−𝑦^∗∈𝑁_𝑐(Gr(𝐹),(𝑎,𝑏))∩1+𝑟𝑟𝐵_𝑋^∗×𝐵_𝑌^∗=𝑁_𝑐Gr𝐷_𝑐𝐹(𝑎,𝑏),(0,0)∩1+𝑟𝑟𝐵_𝑋^∗×𝐵_𝑌^∗=𝜕_𝑐𝑑_{‖⋅‖𝑟}⋅,Gr𝐷_𝑐𝐹()(),1𝑎,𝑏0,0𝑟𝑥^∗₂∈𝑁_𝑐(𝐴,𝑎)∩𝐵_𝑋^∗=𝑁_𝑐𝑇_𝑐(𝐴,𝑎),0∩𝐵_𝑋^∗=𝜕_𝑐𝑑⋅,𝑇_𝑐(𝐴,𝑎)(0). (4.39) Hence, 1𝑟𝑥^∗₁,𝑥≤𝑑_{‖⋅‖𝑟}(𝑥,0),Gr𝐷_𝑐𝐹(𝑎,𝑏)≤𝑑0,𝐷_𝑐𝐹(𝑥,1𝑎,𝑏)()𝑟𝑥^∗₂,𝑥≤𝑑𝑥,𝑇_𝑐(𝐴,𝑎), (4.40) whenever 𝑥∈𝑋. Noting that ⟨𝑥^∗,𝑢⟩=0, it follows from (4.35) that 𝛽‖𝑥−𝑢‖𝑟≤𝑑0,𝐷_𝑐𝐹(𝑎,𝑏)(𝑥)+𝑑𝑥,𝑇_𝑐(𝐴,𝑎). (4.41) Therefore, 𝛽𝑑𝑥,𝑇_𝑐(𝑆,𝑎)𝑟≤𝑑0,𝐷_𝑐𝐹(𝑎,𝑏)(𝑥)+𝑑𝑥,𝑇_𝑐(𝐴,𝑎). (4.42) Letting 𝛽→1, one has 𝑑𝑥,𝑇_𝑐(𝑆,𝑎)≤𝑟𝑑0,𝐷_𝑐𝐹(𝑎,𝑏)(𝑥)+𝑑𝑥,𝑇_𝑐(𝐴,𝑎). (4.43) This contradicts with 𝜏=+∞. The proof is completed.

4.3. Strongly Metric Subregularity

Let 𝐹∶𝑋⇉𝑌 be a multifunction and 𝑏∈𝐹(𝑎). Recall that 𝐹 is strongly subregular at 𝑎 if there exist 𝜏∈(0,+∞), neighborhoods 𝑈 of 𝑎, and 𝑉 of 𝑏 such that ‖𝑥−𝑎‖≤𝜏𝑑(𝑏,𝐹(𝑥)∩𝑉)∀𝑥∈𝑈. (4.44) It is clear that this definition is equivalent to the next one when 𝐴=𝑋.

Definition 4.7. One says that generalized constraint equation (GCE) is strongly metrically subregular at 𝑎 if there exists 𝜏,𝛿∈(0,∞) such that ‖𝑥−𝑎‖≤𝜏(𝑑(𝑏,𝐹(𝑥))+𝑑(𝑥,𝐴))∀𝑥∈𝐵(𝑎,𝛿). (4.45) It is clear that (GCE) is strongly metrically subregular at 𝑎 if and only if 𝑎 is an isolated point of 𝑆 (i.e., 𝑆∩𝐵(𝑎,𝑟)={𝑎} for some 𝑟>0) and it is metrically subregular at 𝑎. Thus, if (GCE) is strongly metrically subregular at 𝑎, Then 𝑁_𝑐(𝑆,𝑎)=𝑋^∗. We immediately have the following Corollary 4.8 from Theorem 4.1.

Corollary 4.8. Suppose that there exists 𝜏,𝛿∈(0,∞) such that (4.45) holds. Then, 𝐵_𝑋^∗⊆𝜏𝐷^∗_𝑐𝐹(𝑎,𝑏)𝐵_𝑌^∗+𝑁_𝑐(𝐴,𝑎)∩𝐵_𝑋^∗. (4.46)

Applying Theorem 4.3, one obtains a sufficient condition for (GCE) to be strongly metrically subregular at 𝑎.

Corollary 4.9. Let 𝑋,𝑌 be Banach spaces. Suppose that generalized constraint equation (GCE) is subsmooth at 𝑎 and that there exists 𝜏∈(0,+∞) such that 𝐵_𝑋^∗⊆𝜏𝐷^∗_𝑐𝐹(𝑎,𝑏)𝐵_𝑌^∗+𝑁_𝑐(𝐴,𝑎)∩𝐵_𝑋^∗. (4.47) Then (GCE) is strongly metrically subregular at 𝑎 and, more precisely, for any 𝜀∈(0,1/(1+2𝜏)) there exists 𝛿_𝜀>0 such that ‖‖≤()𝑥−𝑎1+𝜏)(2+𝜀()𝜀(𝑑((𝑥(1−21+𝜏𝑏,𝐹))+𝑑𝑥,𝐴))∀𝑥∈𝐵𝑎,𝛿_𝜀. (4.48)

Proof. From Theorem 4.3, we need only show that 𝑆∩𝐵(𝑎,𝛿)={𝑎} for some 𝛿>0. Since the assumption that (GCE) is subsmooth at 𝑎, by Proposition 3.3, for any 𝜀∈(0,1/2(1+𝜏)), there exists 𝛿>0 such that 𝑎^∗₁,𝑥−𝑎≤(2+𝜀)𝑑(𝑏,𝐹(𝑥))+𝜀‖𝑥−𝑢‖,𝑎^∗₂,𝑥−𝑎≤𝑑(𝑥,𝐴)+𝜀‖𝑥−𝑢‖, (4.49)(4.50) whenever 𝑥∈𝐵(𝑎,𝛿),𝑎^∗₁∈𝐷^∗_𝑐𝐹(𝑎,𝑏)(𝐵_𝑌^∗)∩𝐵_𝑋^∗and𝑎^∗₂∈𝑁_𝑐(𝐴,𝑎)∩𝐵_𝑋^∗.
Take an arbitrary 𝑥^∗∈𝐵_𝑋^∗. By (4.47), there exist 𝑥^∗₁∈𝐷^∗_𝑐𝐹(𝑎,𝑏)(𝐵_𝑌^∗), 𝑥^∗₂∈𝑁_𝑐(𝐴,𝑎)∩𝐵_𝑋^∗ such that 𝑥^∗=𝜏(𝑥^∗₁+𝑥^∗₂). Let 𝑥∈𝑆∩𝐵(𝑎,𝛿). Applying (4.49) with (𝜏/(1+𝜏))𝑥^∗₁ in place of 𝑎^∗₁, it follows from this and (4.50) that we have 𝜏𝑥^∗₁,𝑥−𝑎≤(1+𝜏)𝜀‖𝑥−𝑎‖,𝑥^∗₂,𝑥−𝑎≤𝜀‖𝑥−𝑎‖. (4.51) Then, 𝑥^∗,𝑥−𝑎=𝜏𝑥^∗₁,𝑥−𝑎+𝑥^∗₂≤()𝜀‖‖.,𝑥−𝑎1+2𝜏𝑥−𝑎 (4.52) And so, ‖𝑥−𝑎‖=sup_{𝑥∗∈𝐵𝑋∗}𝑥^∗,𝑥−𝑎≤(1+2𝜏)𝜀‖𝑥−𝑎‖. (4.53) This shows that 𝑆∩𝐵(𝑎,𝛿)={𝑎}. The proof is completed.

From Corollaries 4.8 and 4.9, we also have the following equivalent results.

Corollary 4.10. Suppose that generalized constraint equation (GCE) is subsmooth at 𝑎. Then the following statements are equivalent:(i)(GCE) is strongly metrically subregular at 𝑎;(ii)there exists 𝜏∈(0,∞) such that 𝐵_𝑋^∗⊆𝜏(𝐷^∗_𝑐𝐹(𝑎,𝑏)(𝐵_𝑌^∗)+𝑁_𝑐(𝐴,𝑎)∩𝐵_𝑋^∗));(iii)0∈int(𝐷^∗_𝑐𝐹(𝑎,𝑏)(𝑌^∗)+𝑁_𝑐(𝐴,𝑎));(iv)𝑋^∗=𝐷^∗_𝑐𝐹(𝑎,𝑏)(𝑌^∗)+𝑁_𝑐(𝐴,𝑎);(v)𝐷_𝑐𝐹(𝑎,𝑏) is strongly metrically subregular at 0 for 0 over 𝑇_𝑐(𝐴,𝑎).

Proof. First, by Corollaries 4.8 and 4.9, it is clear that (i)⇔(ii). Noting that 𝐷^∗_𝑐(𝐷_𝑐𝐹(𝑎,𝑏))(0,0)(𝐵_𝑌^∗)=𝐷^∗_𝑐𝐹(𝑎,𝑏)(𝐵_𝑌^∗), (ii)⇔(v) is immediate from (i)⇔(ii).
It is clear that (ii)⇔(iii). Noting that 𝑁_𝑐(𝐴,𝑎) and 𝐷^∗_𝑐𝐹(𝑎,𝑏)(𝑌^∗) are cones, hence, 𝐷^∗_𝑐𝐹(𝑎,𝑏)𝑌^∗+𝑁_𝑐(𝐴,𝑎)=^∞_𝑛=1𝐷^∗_𝑐𝐹(𝑎,𝑏)𝑛𝐵_𝑌^∗+𝑁_𝑐(𝐴,𝑎)∩𝑛𝐵_𝑋^∗. (4.54) This shows that (ii)⇒(iv).
It remains to show that (iv)⇒(ii). Suppose that (iv) holds, by the Alaogu theorem, for each 𝑛, the set 𝐷^∗_𝑐𝐹(𝑎,𝑏)(𝑛𝐵_𝑌^∗)+𝑁_𝑐(𝐴,𝑎)∩𝑛𝐵_𝑋^∗ is weakly star-closed, it follows from the well-known Baire category theorem and (iv) that 0∈int𝐷^∗_𝑐𝐹(𝑎,𝑏)𝐵_𝑌^∗+𝑁_𝑐(𝐴,𝑎)∩𝐵_𝑋^∗. (4.55) Hence, (ii) holds. The proof is completed.

Acknowledgment

This paper was supported by the National Natural Science Foundations, China, (Grants nos. 11061039, 11061038, and 11261067), and IRTSTYN.