The Representation and Continuity of a Generalized Metric Projection onto a Closed Hyperplane in Banach Spaces

Abstract

Let 𝐶 be a closed bounded convex subset of a real Banach space 𝑋 with 0 as its interior and 𝑝_𝐶 the Minkowski functional generated by the set 𝐶. For a nonempty set 𝐺 in 𝑋 and 𝑥∈𝑋, 𝑔₀∈𝐺 is called the generalized best approximation to 𝑥 from 𝐺 if 𝑝_𝐶(𝑔₀−𝑥)≤𝑝_𝐶(𝑔−𝑥) for all 𝑔∈𝐺. In this paper, we will give a distance formula under 𝑝_𝐶 from a point to a closed hyperplane 𝐻(𝑥^∗,𝛼) in 𝑋 determined by a nonzero continuous linear functional 𝑥^∗ in 𝑋 and a real number α, a representation of the generalized metric projection onto 𝐻(𝑥^∗,𝛼), and investigate the continuity of this generalized metric projection, extending corresponding results for the case of norm.

1. Introduction

Throughout this paper, (𝑋,‖⋅‖) is a real Banach space with the closed unit ball 𝐵(𝑋), and 𝑋^∗ is its the topological dual. For a nonempty subset 𝐴 of 𝑋, as usual, int 𝐴 and bd 𝐴 stand for the interior and the boundary of 𝐴, respectively. Let 𝐶 be a bounded closed convex subset of 𝑋 with 0∈ int 𝐶. Recall that the Minkowski function 𝑝_𝐶∶𝑋→ℝ with respect to the set 𝐶 is defined by 𝑝_𝐶(𝑥)∶=inf{𝑡>0∶𝑥∈𝑡𝐶},∀𝑥∈𝑋. (1) Let 𝐺 be a nonempty subset of 𝑋 and 𝑥∈𝑋. If there exists 𝑔₀∈𝐺 such that 𝑝_𝐶𝑔₀−𝑥=𝜏_𝐶(𝑥,𝐺), (2) where 𝜏_𝐶(𝑥,𝐺)∶=inf𝑝_𝐶𝑔−𝑥∶𝑔∈𝐺 (3) is the distance from the point 𝑥 to the set 𝐺, then following [1] 𝑔₀ is called the generalized best approximation to 𝑥 from 𝐺. The set of all generalized best approximations to 𝑥 from 𝐺 is denoted by 𝑃^𝐶_𝐺(𝑥); that is, 𝑃^𝐶_𝐺(𝑥)=𝑔₀∈𝐺∶𝑝_𝐶𝑔₀−𝑥=𝜏_𝐶(𝑥,𝐺), (4) which is called the generalized metric projection onto 𝐺.

When 𝑝_𝐶 is the norm of 𝑋, the generalized best approximation is reduced to the classical best approximation, which has been studied deeply and extensively since the late 1950s; see [2–4] and references therein. Thus, natural problems are that whether we can extend results in the classical approximation theory to the setting of the generalized approximation. In this direction, some meaning results, such as existences, characterizations, and well-posedness of this kind of approximation, have been established recently; see [1, 5–7]. In this paper, we will consider the problem of representation of generalized metric projection 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} onto a closed hyperplane 𝐻(𝑥^∗,𝛼) and one of continuity of 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}, where 0≠𝑥^∗∈𝑋^∗,   𝛼∈𝐑, and 𝐻𝑥^∗,𝛼=𝑥∈𝑋∶𝑥^∗(𝑥)=𝛼. (5) When 𝑝_𝐶 is the norm of 𝑋 or, equivalently, 𝐶 is the closed unit ball 𝐵(𝑋) of 𝑋, this problem has been studied by a few authors; see [8–12]. In particular, when 𝑋 is reflexive, Wang and Yu have given in [10] the representation of 𝑃^𝐵(𝑋)_{𝐻(𝑥∗,𝛼)}, which was further extended by Ni in [8] to the case of nonreflexive Banach spaces. When 𝑋 is nearly strictly convex, Wang has shown in [11] that 𝑃^𝐵(𝑋)_{𝐻(𝑥∗,𝛼)} is norm-to-weak upper semicontinuous on 𝑋, while, when 𝑋 is arbitrary Banach space, Zhang and Shi have given in [12] the pointwise continuity of 𝑃^𝐵(𝑋)_{𝐻(𝑥∗,𝛼)} under an additional condition.

It should be noted that, when one uses a nonnegative convex function 𝜑 on the Euclidean space 𝐑^𝑛 satisfying 𝜑(0)=0 and 𝜑(𝜆𝑥)=𝜆𝜑(𝑥) for all 𝑥∈𝐑^𝑛 and 𝜆≥0 as a metric on 𝐑^𝑛 (i.e., the distance from a point 𝑥 to a subset 𝐺 of 𝐑^𝑛 is defined as 𝑑(𝑥,𝐺)=inf_𝑔∈𝐺𝜑(𝑔−𝑥)), Ferreia and Nemeth have investigated in [13] the problem of the best approximation in 𝐑^𝑛 and, in particular, given some properties of corresponding metric projections on a hyperplane in 𝐑^𝑛.

The organization of the present paper is as follows. In Section 2, we define the notions of near strict convexity and weak near strict convexity for the underlying set 𝐶, which are, respectively, natural extensions of corresponding notions in norm context, and provide an example of a real Banach space 𝑋 for which 𝐵(𝑋) is weakly nearly strictly convex but not nearly strictly convex. In Section 3, under 𝑝_𝐶, we give a distance formula from a point in 𝑋 to a hyperplane 𝐻(𝑥^∗,𝛼) in 𝑋 and a representation of the generalized metric projection 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} and consider the continuity of 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}. Results obtained in the present paper extend classical Ascoli Theorem (i.e., the distance formula under the case of norm from a point to a closed hyperplane in a Banach space) and main results in [8, 10–12] from the setting of norm to that of the Minkowski functional.

2. Preliminaries and an Example

Recall that (𝑋,‖⋅‖) is a real Banach space with the topological dual 𝑋^∗, 𝐶 is be a closed bounded convex subset of 𝑋 with 0∈int𝐶, and 𝑝_𝐶 is the Minkowski function given by (1). Define the polar 𝐶^∘ of the set 𝐶 by 𝐶^∘∶=𝑥^∗∈𝑋^∗∶𝑥^∗(𝑥)≤1,∀𝑥∈𝐶. (6) Then 𝐶^∘ is a nonempty weakly^∗ compact convex subset of 𝑋^∗ with 0∈ int 𝐶^∘.

We first list some useful properties of the Minkowski function 𝑝_𝐶 which can be proved easily by the definition.

Proposition 1. Let 𝑥,𝑦∈𝑋 and 𝑥^∗∈𝑋^∗. Then(i)𝑝_𝐶(𝑥)≥0 and 𝑝_𝐶(𝑥)=0⇔𝑥=0;(ii)𝑥∈𝐶⇔𝑝_𝐶(𝑥)≤1 and 𝑥∈bd𝐶⇔𝑝_𝐶(𝑥)=1;(iii)𝑝(𝑥+𝑦)≤𝑝(𝑥)+𝑝(𝑦) and 𝑝_𝐶(𝑡𝑥)=𝑡𝑝_𝐶(𝑥) for each 𝑡≥0;(iv)𝑝_𝐶(𝑥)=sup_{𝑦∗∈𝐶}^∘𝑦^∗(𝑥) and 𝑝_𝐶∘(𝑥^∗)=sup_𝑥∈𝐶𝑥^∗(𝑥);(v)𝑥^∗(𝑥)≤𝑝_𝐶(𝑥)𝑝_𝐶∘(𝑥^∗);(vi)there exist positive numbers 𝑚₁ and 𝑚₂ such that 𝑚₁‖𝑥‖≤𝑝_𝐶(𝑥)≤𝑚₂‖𝑥‖. (7)

We then give the following definitions which will be used in the rest of this paper.

Definition 2. Let 𝑇 be a set-valued mapping from 𝑋 into 2^𝑋, where 2^𝑋 is the set of all subsets of 𝑋.(i)Let 𝑥∈𝑋 with 𝑇(𝑥)≠∅. Then 𝑇 is said to be norm-to-norm (resp., norm-to-weak) upper semicontinuous at 𝑥 if, for each open set (resp., weakly open set) 𝑊⊇𝑇(𝑥) there exists an open neighborhood 𝑉 of 𝑥 such that 𝑇(𝑦)⊆𝑊 whenever 𝑦∈𝑉.(ii)𝑇 is said to be norm-to-norm (resp., norm-to-weak) upper semicontinuous on 𝑋 if, for each 𝑥∈𝑋, 𝑇(𝑥)≠∅ and 𝑇 is norm-to-norm (resp., norm-to-weak) upper semicontinuous at 𝑥.(iii)𝑇 is said to be norm-to-norm continuous on 𝑋 if, for each 𝑥∈𝑋, 𝑇(𝑥) is single valued and 𝑇 is norm-to-norm upper semicontinuous at 𝑥.

Definition 3. The set 𝐶 is said to be strictly convex (resp., nearly strictly convex and weakly nearly strictly convex) if each convex subset of bd𝐶 is a singleton (resp., relatively compact and relatively weakly compact).

Clearly, the notions of near strict convexity and weak near strict convexity for the set 𝐶 are extensions of corresponding notions for the unit ball 𝐵(𝑋), which were, respectively, posed by Banaś in [14] and by Wang in [11]. In the following we will provide an example to show that the near strict convexity for 𝐵(𝑋) is strictly stronger than the weak near strict convexity for 𝐵(𝑋).

Example 4. Let 𝑋=𝑙₂ be the space of all square convergent real sequences, endowed with the norm by ‖𝑥‖₀‖𝑥‖=max²√2,‖𝑥‖_∞,∀𝑥∈𝑋, (8) where ‖⋅‖₂ and ‖⋅‖_∞ are the 𝑙₂-norm and the supremum norm on 𝑋, respectively. Then ‖⋅‖₀ is equivalent to the 𝑙₂-norm on 𝑋 because ‖𝑥‖₂√2≤‖𝑥‖₀≤‖𝑥‖₂,∀𝑥∈𝑋. (9) Hence, (𝑋,‖⋅‖₀) is reflexive. It implies that each convex subset of bd 𝐵(𝑋) is relatively weakly compact, and consequently 𝐵(𝑋) is weakly nearly strictly convex. Below we show that 𝐵(𝑋) is not nearly strictly convex. To this end, let {𝑒_𝑛} be the natural basis of 𝑙₂, where the 𝑛th coordinate of 𝑒_𝑛 is 1 and the other coordinates are 0. Furthermore, let 𝑥₁=𝑒₁ and 𝑥_𝑛=𝑒₁+𝑒_𝑛 for each 𝑛≥2. We claim that co{𝑥_𝑛}_𝑛≥1⊆bd𝐵(𝑋). Indeed, let 𝑦∈co{𝑥_𝑛}_𝑛≥1. Then there exist a positive integer 𝑛 and a sequence {𝜆_𝑖}^𝑛_𝑖=1 with {𝜆_𝑖}^𝑛_𝑖=1⊆[0,1] satisfying ∑^𝑛_𝑖=1𝜆_𝑖=1 such that 𝑦=∑^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖. Since 𝑦=𝑒₁+∑^𝑛_𝑖=2𝜆_𝑖𝑒_𝑖, one has that ‖𝑦‖_∞=1 and 𝑦₂=1+^𝑛_𝑖=2𝜆²_𝑖^1/2≤1+^𝑛_𝑖=2𝜆_𝑖^21/2=1+1−𝜆₁^21/2≤√2. (10) It follows that ‖𝑦‖₀=1 and the claim is proved. Since {𝑥_𝑛}^∞_𝑛=1 has no convergent subsequences, one sees that co{𝑥_𝑛}_𝑛≥1 is not relatively compact. This shows that 𝐵(𝑋) is not nearly strictly convex.

3. The Representation and Continuity of Metric Projection onto a Hyperplane

Let 𝑥∈𝑋 and define 𝜎(𝑥)∶=𝑥^∗∈𝑋^∗∶𝑥^∗(𝑥)=𝑝_𝐶(𝑥)𝑝_𝐶∘𝑥^∗=𝑝_𝐶(𝑥)²=𝑝_𝐶∘𝑥^∗2, (11) which is analogous to the dual mapping in Banach spaces. Then, for 𝑥^∗∈𝑋^∗, one obtains from the definition that 𝜎⁻¹(𝑥^∗)={𝑥∈𝑋∶𝑥^∗∈𝜎(𝑥)} and 𝜎⁻¹𝜆𝑥^∗=𝜆𝜎⁻¹𝑥^∗,∀𝜆∈𝐑. (12) Hence, for 0≠𝑥^∗∈𝑋^∗ and 𝑥∈𝑋 (noting that 𝑝_𝐶∘(𝑥^∗)≠0), one has that 𝑥∈𝜎⁻¹𝑥^∗⟺𝑥^∗(𝑥)=𝑝_𝐶(𝑥)𝑝_𝐶∘𝑥^∗=𝑝_𝐶∘𝑥^∗2 (13) and so 𝑥∈𝜎⁻¹𝑥^∗𝑝_𝐶∘(𝑥^∗)⟺𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)=𝑝_𝐶(𝑥)=1. (14)

Recall that the hyperplane 𝐻(𝑥^∗,𝛼) determined by 𝑥^∗∈𝑋^∗⧵{0} and 𝛼∈𝐑 is given by (5) and also that 𝜏_𝐶(𝑥,𝐻(𝑥^∗,𝛼)) is the distance from the point 𝑥 to 𝐻(𝑥^∗,𝛼) defined by (3). The following result is an extension of the classical Ascoli Theorem for the distance formula under the case of norm from a pint to a hyperplane in a Banach space; see [3, Lemma 1.2, p. 24].

Proposition 5. Let 𝑥^∗∈𝑋^∗⧵{0}, 𝛼∈𝐑, and 𝑥∈𝑋. Then 𝜏_𝐶𝑥,𝐻𝑥^∗,𝛼=𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)sign𝛼−𝑥^∗(𝑥). (15)

Proof. Without loss of generality, we assume that 𝛼>𝑥^∗(𝑥). Let 𝑦∈𝐻(𝑥^∗,𝛼). Then 𝑥^∗(𝑦)=𝛼; hence 𝛼−𝑥^∗(𝑥)=𝑥^∗𝑦−𝑥≤𝑝_𝐶𝑦−𝑥𝑝_𝐶∘𝑥^∗ (16) by Proposition 1(v). This implies that (𝛼−𝑥^∗(𝑥))/𝑝_𝐶∘(𝑥^∗)≤𝑝_𝐶(𝑦−𝑥) (noting that 𝑝_𝐶∘(𝑥^∗)≠∅), and therefore 𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)≤𝜏_𝐶𝑥,𝐻𝑥^∗,𝛼 (17) because 𝑦∈𝐻(𝑥^∗,𝛼) is arbitrary.
To show the converse inequality, let 𝜖∈(0,𝑝_𝐶∘(𝑥^∗)). Then, by Proposition 1(iv), there is 𝑧∈𝐶 such that 𝑥^∗(𝑧)>𝑝_𝐶∘𝑥^∗−𝜖. (18) Multiplying two sides of (18) by (𝛼−𝑥^∗(𝑥))/𝑥^∗(𝑧)(𝑝_𝐶∘(𝑥^∗)−𝜖), one has that 𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)>−𝜖𝛼−𝑥^∗(𝑥)𝑥^∗(𝑧). (19) Now let 𝑦=𝑥+((𝛼−𝑥^∗(𝑥))/𝑥^∗(𝑧))𝑧. Then, 𝑦∈𝐻(𝑥^∗,𝛼), and 𝑝_𝐶𝑦−𝑥=𝛼−𝑥^∗(𝑥)𝑥^∗(𝑧)𝑝_𝐶(𝑧)≤𝛼−𝑥^∗(𝑥)𝑥^∗(𝑧) (20) because 𝑝_𝐶(𝑧)≤1 by Proposition 1(ii) (noting that 𝑧∈𝐶). It follows from (19) and (20) that (𝛼−𝑥^∗(𝑥))/(𝑝_𝐶∘(𝑥^∗)−𝜖)>𝑝(𝑦−𝑥). Hence,   (𝛼−𝑥^∗(𝑥))/(𝑝_𝐶∘(𝑥^∗)−𝜖)>𝜏_𝐶(𝑥,𝐻(𝑥^∗,𝛼)). Letting 𝜖→0 in this inequality gives 𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)≥𝜏_𝐶𝑥,𝐻𝑥^∗,𝛼. (21) Thus the converse inequality of (17) follows. The proof is complete.

The first main result of this section is as follows, which gives a presentation of the generalized metric projection onto a closed hyperplane in 𝑋.

Theorem 6. Let 𝑥^∗∈𝑋^∗⧵{0}, 𝛼∈𝐑, and 𝑥∈𝑋. Then the following assertion holds: 𝑃^𝐶_{𝐻(𝑥∗),𝛼}(𝑥)=𝑥+𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)²𝜎⁻¹𝑥^∗,if𝜎⁻¹𝑥^∗≠∅,𝑥^∗(𝑥)𝑥<𝛼,𝑥+^∗(𝑥)−𝛼𝑝_𝐶∘(𝑥^∗)²𝜎⁻¹−𝑥^∗,if𝜎⁻¹−𝑥^∗≠∅,𝑥^∗(𝑥)>𝛼. (22)

Proof. Similar to the proof of Proposition 5, we assume that 𝜎⁻¹(𝑥^∗)≠∅ and 𝑥^∗(𝑥)<𝛼. Let 𝑦∈𝜎⁻¹(𝑥^∗). Then 𝑥^∗∈𝜎(𝑦); hence 𝑥^∗𝑦=𝑝_𝐶𝑦𝑝_𝐶∘𝑥^∗=𝑝_𝐶𝑦²=𝑝_𝐶∘𝑥^∗2. (23) Since 𝑝_𝐶∘(𝑥^∗)≠0, one has that 𝑝_𝐶𝑦=𝑝_𝐶∘𝑥^∗. (24) Let 𝑧=𝑥+((𝛼−𝑥^∗(𝑥))/𝑝_𝐶∘(𝑥^∗)²)𝑦. We then have from (23) that 𝑧∈𝐻(𝑥^∗,𝛼) and from (24) and Proposition 5 that 𝑝_𝐶(𝑧−𝑥)=𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)²𝑝_𝐶𝑦=𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)=𝜏_𝐶𝑥,𝐻𝑥^∗,𝛼. (25) This implies that 𝑧∈𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥), and further 𝑃^𝐶_{𝐻(𝑥∗),𝛼}(𝑥)⊇𝑥+𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)²𝜎⁻¹𝑥^∗. (26)
To show the reverse inclusion, let 𝑧∈𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥). Then, 𝑧∈𝐻(𝑥^∗,𝛼), and 𝑝_𝐶(𝑧−𝑥)=𝜏_𝐶𝑥,𝐻𝑥^∗,𝛼=𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗) (27) thanks to Proposition 5. Noting that 𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)=𝑥^∗(𝑧−𝑥)𝑝_𝐶∘(𝑥^∗)≤𝑝_𝐶(𝑧−𝑥), (28) we get from (27) that 𝛼−𝑥^∗(𝑥)=𝑥^∗(𝑧−𝑥)=𝑝_𝐶(𝑧−𝑥)𝑝_𝐶∘𝑥^∗. (29) Now let 𝑦=(𝑝_𝐶∘(𝑥^∗)²/(𝛼−𝑥^∗(𝑥)))(𝑧−𝑥). It follows from (29) that 𝑥^∗𝑦=𝑝_𝐶∘𝑥^∗2𝛼−𝑥^∗(𝑥)𝑥^∗(𝑧−𝑥)=𝑝_𝐶∘𝑥^∗2,𝑝_𝐶𝑦=𝑝_𝐶∘𝑥^∗2𝛼−𝑥^∗(𝑥)𝑝_𝐶(𝑧−𝑥)=𝑝_𝐶∘𝑥^∗. (30) Hence, 𝑥^∗𝑦=𝑝_𝐶𝑦𝑝_𝐶∘𝑥^∗=𝑝_𝐶∘𝑥^∗2. (31) This means that 𝑦∈𝜎⁻¹(𝑥^∗) by (13), and so 𝑧=𝑥+𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)²𝑦∈𝑥+𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)²𝜎⁻¹𝑥^∗. (32) Consequently, 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥) is contained in the right-hand side of (26). The proof is complete.

The following result gives necessary and sufficient conditions for 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥)≠∅.

Proposition 7. Let 𝑥^∗∈𝑋^∗⧵{0}, 𝛼∈𝐑, and 𝑥∈𝑋 satisfy that 𝑥^∗(𝑥)<𝛼 (𝑟𝑒𝑠𝑝.,𝑥^∗(𝑥)>𝛼). Then 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥)≠∅ if and only if 𝑥^∗(𝑟𝑒𝑠𝑝.,−𝑥^∗) attains its supremum 𝑝_𝐶∘(𝑥^∗)  (𝑟𝑒𝑠𝑝.,𝑝_𝐶∘(−𝑥^∗)) on bd𝐶.

Proof. Let 𝑥^∗,𝛼, and 𝑥 be as in Proposition 7, and let 𝑥^∗(𝑥)<𝛼. Suppose that 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥)≠∅. Take 𝑦∈𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥). Then 𝑥^∗(𝑦)=𝛼 and 𝑝_𝐶(𝑦−𝑥)=𝜏_𝐶(𝑥,𝐻(𝑥^∗,𝛼)). It follows from Propositions 5 and 1(v) that 𝑝_𝐶𝑦−𝑥=𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)=𝑥^∗𝑦−𝑥𝑝_𝐶∘(𝑥^∗)≤𝑝_𝐶𝑦−𝑥. (33) Hence, 𝑥^∗ attains its supremum 𝑝_𝐶∘(𝑥^∗) at (𝑦−𝑥)/(𝑝_𝐶(𝑦−𝑥))∈bd𝐶.
Conversely, suppose that 𝑥^∗ attains its supremum 𝑝_𝐶∘(𝑥^∗) at 𝑥₀∈bd𝐶. Then 𝑥^∗(𝑥₀)=𝑝_𝐶∘(𝑥^∗) and 𝑝_𝐶(𝑥₀)=1; hence, 𝑥^∗𝑝_𝐶∘𝑥^∗𝑥₀=𝑝_𝐶∘𝑥^∗2=𝑝_𝐶𝑝_𝐶∘𝑥^∗𝑥₀𝑝_𝐶∘𝑥^∗. (34) This together with (13) implies that 𝑝_𝐶∘(𝑥^∗)𝑥₀∈𝜎⁻¹(𝑥^∗), and therefore 𝜎⁻¹(𝑥^∗)≠∅. By Theorem 6, one sees that 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥)≠∅. Similarly, we can prove another assertion for the case of 𝑥^∗(𝑥)>𝛼.

The second main result of this section is as follows, which describes the continuity of the generalized metric projection 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} onto the hyperplane 𝐻(𝑥^∗,𝛼) under the condition that the set 𝐶 is weakly nearly strictly convex.

Theorem 8. Let the set 𝐶 be weakly nearly strictly convex, 𝑥^∗∈𝑋^∗⧵{0}, and 𝛼∈𝐑. Then the following assertions hold.(i)Suppose that 𝑥∈𝑋 satisfies that 𝑥^∗(𝑥)<𝛼  (𝑟𝑒𝑠𝑝.,𝑥^∗(𝑥)>𝛼) and that 𝑥^∗(𝑟𝑒𝑠𝑝.,−𝑥^∗) attains its supremum on bd𝐶; then 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} is norm-to-weak upper semicontinuous at 𝑥.(ii)If 𝑥^∗ and −𝑥^∗ attain their supremum on bd𝐶, then 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} is norm-to-weak upper semicontinuous on 𝑋. Furthermore, 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} is norm-to-norm upper semicontinuous at each point of 𝐻(𝑥^∗,𝛼).

Proof. (i) Without loss of generality, we assume that 𝑥^∗∈𝑋^∗⧵{0}, 𝑥∈𝑋, and 𝛼∈𝐑 satisfy 𝑥^∗(𝑥)<𝛼 and that 𝑥^∗ attains its supremum on bd𝐶. We first show that 𝜎⁻¹(𝑥^∗) is convex. To do this, let 𝑦₁,𝑦₂∈𝜎⁻¹(𝑥^∗) and 𝜆∈[0,1]. Then we obtain from (13) that 𝑥^∗𝑦₁=𝑝_𝐶𝑦₁𝑝_𝐶∘𝑥^∗=𝑝_𝐶∘𝑥^∗2=𝑝_𝐶𝑦₂𝑝_𝐶∘𝑥^∗=𝑥^∗𝑦₂. (35) This, together with Proposition 1(v) and (iii), implies that 𝑝_𝐶∘𝑥^∗2=𝑥^∗𝜆𝑦₁+(1−𝜆)𝑦₂≤𝑝_𝐶𝜆𝑦₁+(1−𝜆)𝑦₂𝑝_𝐶∘𝑥^∗≤𝜆𝑝_𝐶𝑦₁+(1−𝜆)𝑝_𝐶𝑦₂𝑝_𝐶∘𝑥^∗=𝑝_𝐶∘𝑥^∗2. (36) Hence, 𝜆𝑦₁+(1−𝜆)𝑦₂∈𝜎⁻¹(𝑥^∗) by (13), and 𝜎⁻¹(𝑥^∗) is convex.
We then show that 𝜎⁻¹(𝑥^∗) is weakly compact. Since 𝜎⁻¹(𝑥^∗/𝑝_𝐶∘(𝑥^∗))⊆bd𝐶 by (14) and since 𝜎⁻¹(𝑥^∗/𝑝_𝐶∘(𝑥^∗))=(1/𝑝_𝐶∘(𝑥^∗))𝜎⁻¹(𝑥^∗) by (12), one sees that 𝜎⁻¹(𝑥^∗/𝑝_𝐶∘(𝑥^∗)) is a convex subset of bd𝐶. It follows that 𝜎⁻¹(𝑥^∗/𝑝_𝐶∘(𝑥^∗)) is relatively weakly compact because 𝐶 is weakly nearly strictly convex; hence, 𝜎⁻¹(𝑥^∗) is relatively weakly compact. Thus, to complete the proof, it suffices to show that 𝜎⁻¹(𝑥^∗) is weakly closed. To do this, let {𝑥_𝛿} be a net in 𝜎⁻¹(𝑥^∗) convergent weakly to some 𝑥∈𝑋. Since 𝑥^∗𝑥_𝛿=𝑝_𝐶𝑥_𝛿𝑝_𝐶∘𝑥^∗=𝑝_𝐶∘𝑥^∗2∀𝛿 (37) and since 𝑝_𝐶 is weakly lower semicontinuous by [15, Theorem 2.2.1, page 60], we have that 𝑝_𝐶∘𝑥^∗2=𝑥^∗(𝑥)=lim_𝛿𝑥^∗𝑥_𝛿=lim_𝛿𝑝_𝐶𝑥_𝛿𝑝_𝐶∘𝑥^∗≥𝑝_𝐶(𝑥)𝑝_𝐶∘𝑥^∗. (38) Noting that 𝑥^∗(𝑥)≤𝑝_𝐶(𝑥)𝑝_𝐶∘(𝑥^∗), we get that 𝑥^∗(𝑥)=𝑝_𝐶(𝑥)𝑝_𝐶∘𝑥^∗=𝑝_𝐶∘𝑥^∗2. (39) Hence, 𝑥∈𝜎⁻¹(𝑥^∗) by (13), and the weak closedness of 𝜎⁻¹(𝑥^∗) is proved.
Finally, we show that 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} is norm-to-weak upper semicontinuous at 𝑥. Otherwise, there exist a weakly open set 𝑊⊇𝑃^𝐶_{𝐻(𝑥∗),𝛼}(𝑥) (40) and a sequence {𝑥_𝑛}⊆𝑋 with ‖𝑥_𝑛−𝑥‖→0 such that 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥_𝑛)⊆𝑊̸. Since 𝑥^∗(𝑥)<𝛼 and ‖𝑥_𝑛−𝑥‖→0, we may assume that each 𝑥^∗(𝑥_𝑛)<𝛼. Now take 𝑦_𝑛∈𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥_𝑛)⧵𝑊 for each 𝑛. By Theorem 6, there exists 𝑧_𝑛∈𝜎⁻¹(𝑥^∗) such that 𝑦_𝑛=𝑥_𝑛+((𝛼−𝑥^∗(𝑥_𝑛))/𝑝_𝐶∘(𝑥^∗))𝑧_𝑛 for all 𝑛. Using the weak compactness of 𝜎⁻¹(𝑥^∗), one has a subsequence {𝑧_𝑛𝑘} of {𝑧_𝑛} such that lim_𝑘𝑧_𝑛𝑘=𝑧 weakly for some 𝑧∈𝜎⁻¹(𝑥^∗). Therefore, weak−lim_𝑘𝑦_𝑛𝑘=𝑦∶=𝑥+𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)𝑧∈𝑃^𝐶_{𝐻(𝑥∗),𝛼}(𝑥). (41) This and (40) imply that 𝑦∈𝑊. Since 𝑊 is weakly open, one has that 𝑦_𝑛𝑘∈𝑊 for sufficiently large 𝑘, which contradicts the choice of 𝑦_𝑛𝑘 and the proof of assertion (i) is complete.
(ii) Let 𝑥∈𝐻(𝑥^∗,𝛼). Note that the norm-to-norm upper semicontinuity of 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} at 𝑥 implies the norm-to-weak upper semicontinuity of 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} at 𝑥. It suffices to verify that 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} is norm-to-norm upper semicontinuous at 𝑥. To this end, let 𝑊 be an open neighborhood of 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥)=𝑥. Then there exists a positive number 𝛿₀ such that 𝐵(𝑥,𝛿₀)⊆𝑊, where 𝐵(𝑥,𝛿₀) denotes the closed ball with center 𝑥 and radius 𝛿₀. Below we show that there is 𝛿∈(0,𝛿₀] such that 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑦)⊆𝐵(𝑥,𝛿₀) whenever ‖𝑦−𝑥‖<𝛿 or, equivalently (noting that if 𝛿∈(0,𝛿₀], one always has that 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑦)=𝑦∈𝐵(𝑥,𝛿₀) whenever 𝑦∈𝐻(𝑥^∗,𝛼) and 𝑦∈𝐵(𝑥,𝛿)), 𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)²‖𝑧‖≤𝛿₀foreach𝑧∈𝜎⁻¹𝑥^∗if𝑥^∗𝑦𝑥<𝛼,_∗(𝑥)−𝛼𝑝_𝐶∘(𝑥^∗)²‖𝑧‖≤𝛿₀foreach𝑧∈𝜎⁻¹−𝑥^∗if𝑥^∗𝑦>𝛼 (42)(43) whenever ‖𝑦−𝑥‖<𝛿 due to Theorem 6.
To proceed, we first verify ‖𝑧‖≤𝑝_𝐶∘𝑥^∗𝑚₁,∀𝑧∈𝜎⁻¹𝑥^∗, (44) where the positive number 𝑚₁ is as in Proposition 1(vi). In fact, let 𝑧∈𝜎⁻¹(𝑥^∗). Then 𝑝_𝐶(𝑧)=𝑝_𝐶∘(𝑥^∗) by (13). This, together with Proposition 1(vi), implies that ‖𝑧‖≤(1/𝑚₁)𝑝_𝐶(𝑧)=(𝑝_𝐶∘(𝑥^∗)/𝑚₁), and (44) is proved. Next, take 𝑚𝛿=min¹𝛿₀‖𝑥^∗‖𝑝_𝐶∘𝑥^∗,𝑚₁𝛿₀‖𝑥^∗‖𝑝_𝐶∘−𝑥^∗,𝛿₀. (45) Then when ‖𝑦−𝑥‖<𝛿 and 𝑥^∗(𝑦)<𝛼, one has, for each 𝑧∈𝜎⁻¹(𝑥^∗), that 𝛼−𝑥^∗(𝑥)𝑝_𝐶∘(𝑥^∗)²‖𝑧‖=𝑥^∗𝑥−𝑦𝑝_𝐶∘(𝑥^∗)²‖𝑧‖≤𝑥^∗𝑥−𝑦𝑝_𝐶∘(𝑥^∗)²‖𝑧‖<𝑥^∗𝛿𝑝_𝐶∘(𝑥^∗)²𝑝_𝐶∘𝑥^∗𝑚₁≤𝛿₀; (46) hence, (42) holds. While when ‖𝑦−𝑥‖<𝛿 and 𝑥^∗(𝑦)>𝛼, we can similarly show that (43) is true. Thus, the proof of (ii) is complete.

A similar proof to that of Theorem 8 yields the following result.

Theorem 9. Let the set 𝐶 be nearly strictly convex, 𝑥^∗∈𝑋^∗⧵{0}, and 𝛼∈𝐑. The the following assertions hold.(i)Suppose that 𝑥∈𝑋 satisfies that 𝑥^∗(𝑥)<𝛼   (resp., 𝑥^∗(𝑥)>𝛼) and that 𝑥^∗ (resp., −𝑥^∗) attains its supremum on bd𝐶; then 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} is norm-to-norm upper semicontinuous at 𝑥.(ii)If 𝑥^∗ and −𝑥^∗ attain their supremum on bd𝐶, then 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} is norm-to-norm upper semicontinuous on 𝑋.

Theorem 10. Suppose that the set 𝐶 is strictly convex and that nonzero continuous linear functional 𝑥^∗ and −𝑥^∗ attain, respectively, their supremum on bd𝐶. Then 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} is norm-to-norm continuous on 𝑋.

Proof. Let 𝑦^∗∈𝑋^∗⧵{0}. We assert that 𝜎⁻¹(𝑦^∗) contains at most one point under the hypothesis made upon the set 𝐶. To do this, let 𝑦₁,𝑦₂∈𝜎⁻¹(𝑦^∗). Then, from the proof of Theorem 8(i), one has that 𝑝_𝐶𝜆𝑦₁+(1−𝜆)𝑦₂=𝑝_𝐶𝑦₁=𝑝_𝐶𝑦₂=𝑝_𝐶∘𝑦^∗,∀𝜆∈[0,1]; (47) hence, [𝑦₁/𝑝_𝐶∘(𝑦^∗),𝑦₂/𝑝_𝐶∘(𝑦^∗)]⊆bd𝐶 by Proposition 1(ii). It follows from the strict convexity of 𝐶 that 𝑦₁/𝑝_𝐶∘(𝑦^∗)=𝑦₂/𝑝_𝐶∘(𝑦^∗), that is, 𝑦₁=𝑦₂, and that the assertion is proved. Applying this conclusion to 𝑥^∗ and −𝑥^∗ (noting that 𝜎⁻¹(𝑥^∗) and 𝜎⁻¹(−𝑥^∗) are nonempty by (13) because 𝑥^∗ and −𝑥^∗ attain their supremum on bd𝐶), one sees that both 𝜎⁻¹(𝑥^∗) and 𝜎⁻¹(−𝑥^∗) are a singleton. Therefore, 𝑃^𝐶_{𝐻(𝑥∗,𝛼)}(𝑥) is single valued for each 𝑥∈𝑋 by Theorem 6, and 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} is norm-to-norm continuous on 𝑋 by Theorem 9, which completes the proof of Theorem 9.

Theorem 11. Suppose that 𝑋 is reflexive and that the set 𝐶 is strictly convex. Then 𝑃^𝐶_{𝐻(𝑥∗,𝛼)} is norm-to-norm continuous on 𝑋 for each 𝑥^∗∈𝑋^∗⧵{0}.

Proof. Let 𝑥^∗∈𝑋^∗⧵{0} be arbitrary. Below we will show that 𝑥^∗ attains its supremum on bd𝐶. Granting this, the conclusion follows from Theorem 10. To this end, we take a sequence {𝑥_𝑛}⊆𝐶 such that lim_𝑛𝑥^∗(𝑥_𝑛)=𝑝_𝐶∘(𝑥^∗). Since the set 𝐶 is weakly compact (noting that 𝑋 is reflexive), there exists a subsequence of {𝑥_𝑛}, denoted still by {𝑥_𝑛}, such that lim_𝑛𝑥_𝑛=𝑥 weakly for some 𝑥∈𝐶. Thus, 𝑝_𝐶∘𝑥^∗=𝑥^∗(𝑥)≤𝑝_𝐶(𝑥)𝑝_𝐶∘𝑥^∗≤𝑝_𝐶∘𝑥^∗. (48) Consequently, 𝑥∈bd𝐶 because 𝑝_𝐶(𝑥)=1, and 𝑥^∗ attains its supremum at 𝑥. The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author was supported in part by the NNSF of China (Grant no. 11271342) and the NSF of Zhejiang Province (Grant no. LY12A01029).