New Characterization for Nonlinear Weighted Best Simultaneous Approximation

Abstract

This paper is concerned with the problem of a wide class of weighted best simultaneous approximation in normed linear spaces, and it establishes a new characterization result for the class of approximation by virtue of the notion of simultaneous regular point.

1. Introduction

The problem of best simultaneous approximation has a long history and continues to generate much interest. The problem of approximating simultaneously two continuous functions on a finite closed interval was first studied by Dunham [1]. Since then, such problems have been extended extensively, see, for example, [1–7] and references therein. In particular, characterization and uniqueness results were obtained in [7] for a wide class of best simultaneously approximating problem, which includes early results as special cases.

The setting for the problems considered here is as follows. Let 𝐑^∞ be a real linear space consisting of some sequences in the field of real numbers 𝐑 and 𝐞^𝑖=(𝛿_𝑖𝑗)∈𝐑^∞ for each 𝑖∈𝐍, where 𝛿_𝑖𝑗=1 if 𝑗=𝑖, and 0 otherwise. We endow a norm ‖⋅‖_𝐴 on 𝐑^∞ such that the norm is monotonic; that is, for (𝑎_𝑣)∈𝐑^∞ and a real sequence (𝑏_𝑣), the condition that |𝑏_𝑖|≤|𝑎_𝑖| for each 𝑖=1,2,…, implies that (𝑏₁,𝑏₂,…)∈𝐑^∞ and ‖(𝑏_𝑣)‖_𝐴≤‖(𝑎_𝑣)‖_𝐴. Let {𝜆_𝑖}∈𝐑^∞ be a fixed sequence of positive numbers. Let 𝑋 be a complex normed linear space with the norm ‖⋅‖ and 𝐺⊂𝑋. The problem considered here is, for a sequence (𝑥_𝑖) in 𝑋 with (𝜆_𝑖‖𝑥_𝑖‖)∈𝐑^∞, finding 𝑔₀∈𝐺 such that

‖(𝜆_𝑖‖𝑥_𝑖−𝑔₀‖)‖_𝐴≤‖(𝜆_𝑖‖𝑥_𝑖−𝑔‖)‖_𝐴,∀𝑔∈𝐺. (1.1) Any element 𝑔₀ satisfying (1.1) is called a best simultaneous approximation to 𝑥 from 𝐺. The set of all best simultaneous approximations to 𝑥 from 𝐺 is denoted by 𝑃_𝐺(𝑥).

In order to characterize restricted Chebyshev centers of a set in normed linear spaces, the work in [8] introduced the concept of simultaneous regular point of a set. In this paper, we propose a same notion and the notion of simultaneous strongly regular point of a set for studying best simultaneous approximation to a sequence from the set in 𝑋, and establish new characterization results for this class of approximation problem. It should be remarked that our results are new even in the case when 𝑋 is real (noting that results obtained in this paper is valid for real normed linear spaces) and when the approximated sequence is finite.

2. Preliminaries

Let 𝐑^∞ be as in Section 1 with the monotonic norm ‖⋅‖_𝐴 and let (𝜆_𝑖)∈𝐑^∞ be a fixed sequence of positive numbers satisfying

lim_𝑖→∞‖(0,…,0,𝜆_𝑖,𝜆_𝑖+1,…)‖_𝐴=0 (2.1) (noting that such a sequence (𝜆_𝑖) satisfying (2.1) exists, see [7]), which plays a fundamental role in the present paper. Let 𝑋 be a complex normed linear space with the norm ‖⋅‖. We use (𝐑^∞)^∗ and 𝑋^∗ to denote the duals of 𝐑^∞ and 𝑋, respectively. The inner product between 𝐑^∞ and (𝐑^∞)^∗ is denoted by ⟨⋅,⋅⟩ while 𝑓(𝑥) stands for the inner product of 𝑥∈𝑋 and 𝑓∈𝑋^∗. Also, we denote by 𝑉 and 𝑊 the closed unit ball of (𝐑^∞)^∗ and 𝑋^∗, respectively. For a set 𝐴 in the dual of a Banach space, let 𝐴^∗ signify the weak∗ closure of the set 𝐴 and be endowed with the weak∗ topology. Let Ω=𝑉×𝑊×𝑊×⋯, and let Ω be endowed with the product topology. Then Ω is a compact Hausdorff space.

Let

𝒳={𝑥=(𝑥_𝑖)∶(𝜆_𝑖‖𝑥_𝑖‖)∈𝐑^∞} (2.2) with the norm ‖𝑥‖_𝒳=‖(𝜆_𝑖‖𝑥_𝑖‖)‖_𝐴 for each 𝑥∈𝒳. Then 𝑋⊂𝒳, where 𝑥 is viewed as an element(𝑥,𝑥,…) in 𝒳 for each 𝑥∈𝑋. For simplicity, we write 𝑓 for (𝑓_𝑖). Thus (𝑎^∗,𝑓)∈Ω means that (𝑎^∗,𝑓₁,𝑓₂,…)∈Ω. Let 𝑥=(𝑥_𝑖)∈𝒳. Define the function 𝜙(𝑥) on Ω by

𝜙(𝑥)(𝜔)=⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑥_𝑖))⟩,∀𝜔=(𝑎^∗,𝑓)∈Ω. (2.3) Furthermore, define

𝜙(𝑥)⁺(𝜔)=inf_{𝑂∈𝑁𝜔}sup_{𝜔′∈𝑂}𝜙(𝑥)(𝜔^′), (2.4) where 𝑁_𝜔 denotes the family of all open neighborhoods of 𝜔 in Ω. Then, by (2.1) and [9, Remark  1,  2,  4], we have the following proposition (see also [7, Proposition  2.1]).

Proposition 2.1. Let 𝑥=(𝑥_𝑖)∈ℱ. Then the following assertions hold: (i)𝜙(𝑥)⁺ is upper semicontinuous on Ω,(ii)𝜙(𝑥) is continuous on   Ω   for all 𝑥∈𝑋,(iii) For each 𝑥∈𝑋,𝜙(𝑥−𝑥)⁺=𝜙(𝑥)⁺−𝜙(𝑥),max_𝜔∈Ω𝜙(𝑥−𝑥)⁺(𝜔)=max_𝜔∈Ω[𝜙(𝑥)⁺(𝜔)−𝜙(𝑥)(𝜔)]=sup_𝜔∈Ω𝜙(𝑥−𝑥)(𝜔)=‖𝑥−𝑥‖_𝒳. (2.5)(2.6)

In what follows, we always assume that 𝐺 is a nonempty subset of 𝑋. Let 𝑥∈𝒳 and 𝑔∈𝐺. Write

Ω𝑥_−𝑔={𝜔∈Ω∶𝜙(𝑥)⁺(𝜔)−𝜙(𝑔)(𝜔)=‖𝑥−𝑔‖_𝒳}. (2.7) Then Ω𝑥⁺_−𝑔 is a nonempty compact subset of Ω since 𝜙(𝑥)⁺−𝜙(𝑔) is upper-semicontinuous on the compact set Ω.

The following concept is an extension of the notion of local best approximation given in [10] to the case of best simultaneous approximation.

Definition 2.2. Let 𝑔₀∈𝐺 and 𝑥∈𝒳. 𝑔₀ is called a local best simultaneous approximation to 𝑥 from 𝐺 if there exists an open neighborhood 𝑈(𝑔₀) (i.e., an open ball with center 𝑔₀) of 𝑔₀ such that 𝑔₀∈𝑃_{𝐺∩𝑈(𝑔0)}(𝑥).

The notion of suns, introduced by Efimov and Stečhkin in [11], has proved to be very important in nonlinear approximation theory in normed linear spaces. The following definition is an extension of the notion of suns to the case of simultaneous approximations, see [4].

Definition 2.3. Let 𝑔₀∈𝐺. 𝑔₀ is called simultaneous solar point of 𝐺 if for each 𝑥∈𝒳, 𝑔₀∈𝑃_𝐺(𝑥) implies that 𝑔₀∈𝑃_𝐺(𝑥_𝛼) for each 𝛼>0, here and in the sequel, 𝑥_𝛼=𝑔₀+𝛼(𝑥−𝑔₀). 𝐺 is called a simultaneous sun if each point of 𝐺 is a simultaneous solar point of 𝐺.

The following notion stated in Definition 2.4 (i) is similar to the notion of simultaneous regular point in [8], which was used to characterize restricted Chebyshev centers of a set in a normed linear space.

Definition 2.4. Let 𝑔₀∈𝐺. Then 𝑔₀ is called (i)simultaneous regular point of 𝐺 if, for each 𝑥∈𝒳, 𝑔∈𝐺 and closed set 𝐴 satisfying Ω𝑥_−𝑔0⊂𝐴⊂Ω and min_(𝑎∗,𝑓_)∈𝐴⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔−𝑔₀))⟩>0, (2.8) there exists {𝑔_𝑛}⊂𝐺 such that ‖𝑔_𝑛−𝑔₀‖→0 and 𝜙(𝑔_𝑛−𝑔₀)(𝑎^∗,𝑓)>𝜙(𝑥−𝑔₀)⁺(𝑎^∗,𝑓)−‖𝑥−𝑔₀‖_𝒳,∀(𝑎^∗,𝑓)∈𝐴,𝑛∈𝐍, (2.9)(ii)simultaneous strongly regular point of 𝐺 if, for each 𝑥∈𝒳, 𝑔∈𝐺 and closed set 𝐴 satisfying Ω𝑥_−𝑔0⊂𝐴⊂Ω and (2.8), there exists {𝑔_𝑛}⊂𝐺 such that ‖𝑔_𝑛−𝑔₀‖→0 and 𝜙(𝑔_𝑛−𝑔₀)(𝑎^∗,𝑓)>0,∀(𝑎^∗,𝑓)∈𝐴,𝑛∈𝐍, (2.10)(iii)𝐺 is called a simultaneous regular set (resp., simultaneous strongly regular set ) of 𝑋 if each point of 𝐺 is a simultaneous regular point (resp., simultaneous strongly regular point) of 𝐺.

The following notions are respectively analogues to Kolmogorov Condition (cf. [10, 12]) and Papini Condition (cf. [12]) in nonlinear approximation theory in normed linear spaces.

Definition 2.5. Let 𝑔₀∈𝐺 and 𝑥∈𝒳. Then 𝑔₀ is said to satisfy(i)simultaneous Kolmogorov Condition if max{⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔₀−𝑔))⟩∶(𝑎^∗,𝑓)∈Ω𝑥_−𝑔0}≥0,∀𝑔∈𝐺, (2.11)(ii)simultaneous Papini Condition if max{⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔−𝑔₀))⟩∶(𝑎^∗,𝑓)∈Ω𝑥^−𝑔}≤0,∀𝑔∈𝐺. (2.12)

Proposition 2.6. Let 𝑔₀∈𝐺 and 𝑥∈𝒳. (i)If (𝑥,𝑔₀) satisfies simultaneous Kolmogorov Condition, then 𝑔₀∈𝑃_𝐺(𝑥).(ii)If 𝑔₀∈𝑃_𝐺(𝑥), then (𝑥,𝑔₀) satisfies simultaneous Papini Condition.

Proof. (i) Let 𝑔∈𝐺⧵{𝑔₀}. Then by the assumption, there exists (𝑎^∗,𝑓)∈Ω𝑥_−𝑔0 such that ⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔₀−𝑔))⟩≥0. It follows from (2.6) that
‖𝑥−𝑔‖_𝒳≥⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑥_𝑖−𝑔))⟩=⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑥_𝑖−𝑔₀))⟩+⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔₀−𝑔))⟩≥‖𝑥−𝑔₀‖_𝒳. (2.13) This means that 𝑔₀∈𝑃_𝐺(𝑥).
(ii) Let 𝑔₀∈𝑃_𝐺(𝑥) and 𝑔∈𝐺. Then for each (𝑎^∗,𝑓)∈Ω𝑥^−𝑔, one has that ⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔−𝑔₀))⟩=⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔−𝑥_𝑖))⟩+⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑥_𝑖−𝑔₀))⟩≤−‖𝑥−𝑔‖_𝒳+‖𝑥−𝑔₀‖_𝒳≤0. (2.14) Hence (𝑥,𝑔₀) satisfies simultaneous Papini Condition. The proof is complete.

3. Characterizations for Best Simultaneous Approximations

The relationship of best simultaneous approximations and local best simultaneous approximations is as follows.

Theorem 3.1. Let 𝑔₀∈𝐺. Suppose that 𝑔₀ is a simultaneous solar point of 𝐺. Then for each 𝑥∈𝒳, the condition that 𝑔₀ is a local best simultaneous approximation to 𝑥 from 𝐺 implies that 𝑔₀∈𝑃_𝐺(𝑥).

Proof. Let 𝑥∈𝒳 and 𝑔₀ is a local best simultaneous approximation to 𝑥 from 𝐺. Then there is an open neighborhood 𝑈(𝑔₀,𝛿) of 𝑔₀ such that ‖𝑥−𝑔₀‖_𝒳≤‖𝑥−𝑔‖_𝒳,∀𝑔∈𝑈(𝑔₀,𝛿)𝐺. (3.1) Clearly, we may assume that 𝑥≠(𝑔₀). Let 𝛼=min{1,𝛿‖(𝜆_𝑖)‖_𝐴(2‖𝑥−𝑔₀‖_𝒳)}. (3.2) We assert that 𝑔₀∈𝑃_𝐺(𝑥_𝛼). In fact, let 𝑔∈𝐺⧵𝑈(𝑔₀,𝛿). Then ‖𝑔−𝑔₀‖≥𝛿. It follows from (3.2) that ‖𝑥_𝛼−𝑔₀‖_𝒳≥‖𝑔₀−𝑔‖_𝒳−‖𝑥_𝛼−𝑔₀‖_𝒳=‖(𝜆_𝑖)‖_𝐴‖𝑔₀−𝑔‖−𝛼‖𝑥−𝑔₀‖_𝒳≥𝛿‖(𝜆_𝑖)‖_𝐴−𝛿2‖(𝜆_𝑖)‖_𝐴≥𝛼‖𝑥−𝑔₀‖_𝒳=‖𝑥_𝛼−𝑔₀‖_𝒳. (3.3) On the other hand, let 𝑔∈𝐺∩𝑈(𝑔₀,𝛿) be arbitrary. Suppose on the contrary that ‖𝑥−𝑔‖_𝒳<‖𝑥−𝑔₀‖_𝒳. Then ‖𝑥−𝑔‖_𝒳≤‖𝑥−𝑥_𝛼‖_𝒳+‖𝑥_𝛼−𝑔₀‖_𝒳<‖𝑥−𝑥_𝛼‖_𝒳+‖𝑥_𝛼−𝑔₀‖_𝒳=(1−𝛼)‖𝑥−𝑔₀‖_𝒳+𝛼‖𝑥−𝑔₀‖_𝒳=‖𝑥−𝑔₀‖_𝒳, (3.4) which contradicts (3.1). This completes the proof of the assertion. Since 𝑔₀ is a simultaneous solar point of 𝐺, one has that 𝑔₀∈𝑃_𝐺(𝑔₀+(1/𝛼)(𝑥_𝛼−𝑔₀)). Noting that 𝑥=𝑔₀+(1/𝛼)(𝑥_𝛼−𝑔₀), we obtain that 𝑔₀∈𝑃_𝐺(𝑥).

The first main result of this paper is as follows.

Theorem 3.2. Let 𝑔₀∈𝐺. Then the following statements are equivalent: (i)𝑔₀ is a simultaneous solar point of 𝐺,(ii)For each 𝑥∈𝒳, 𝑔₀∈𝑃_𝐺(𝑥) if and only if (𝑥,𝑔₀) satisfies simultaneous Kolmogorov Condition,(iii)𝑔₀ is a simultaneous regular point of 𝐺.

Proof. The equivalence of (i)⇔(ii) is exactly [7, Theorem  3.1].
(ii) ⇒(iii) For each 𝑥∈𝒳, 𝑔∈𝐺 and closed set 𝐴 satisfying Ω𝑥_−𝑔0⊂𝐴⊂Ω and (2.8), we obtain from (2.8) that
max_(𝑎∗,𝑓_)∈Ω𝑥_0−𝑔⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔₀−𝑔))⟩<0. (3.5) Hence 𝑔₀∉𝑃_𝐺(𝑥) by (ii). Using the equivalence of (i) and (ii) as well as Theorem 3.1, 𝑔₀ is not local best simultaneous approximation to 𝑥 from 𝐺. Thus there exists a sequence {𝑔_𝑛}⊂𝐺 such that ‖𝑔_𝑛−𝑔₀‖→0 and ‖𝑥−𝑔_𝑛‖_𝒳<‖𝑥−𝑔₀‖_𝒳,∀𝑛∈𝐍. (3.6) Let (𝑎^∗,𝑓)∈𝐴 be arbitrary. Then by (2.5) and (2.6), one has that ‖𝑥−𝑔_𝑛‖_𝒳≥𝜙(𝑥−𝑔_𝑛)⁺(𝑎^∗,𝑓)=𝜙(𝑥−𝑔₀)⁺(𝑎^∗,𝑓)−𝜙(𝑔_𝑛−𝑔₀)(𝑎^∗,𝑓). (3.7) This together with (3.6) implies that (2.9) holds, which shows that 𝑔₀ is a simultaneous regular point of 𝐺.
(iii)⇒(ii) Let (iii) hold. By Proposition 2.6, it suffices to prove that (𝑥,𝑔₀) satisfies simultaneous Kolmogorov Condition for each 𝑥∈𝒳 with 𝑔₀∈𝑃_𝐺(𝑥). For this end, suppose on the contrary that there exists 𝑥∈𝒳 with 𝑔₀∈𝑃_𝐺(𝑥) such that (𝑥,𝑔₀) does not satisfy simultaneous Kolmogorov Condition. Then there exist 𝑔∈𝐺 and 𝜖>0 such that
max_(𝑎∗,𝑓_)∈Ω𝑥_0−𝑔⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔₀−𝑔))⟩=−𝜖<0. (3.8) Let 𝑈={(𝑎^∗,𝑓)∈Ω∶⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔₀𝜖−𝑔))⟩<−2}. (3.9) Then 𝑈 is an open subset of Ω because the function 𝜔=(𝑎^∗,𝑓)↦𝜙(𝑔₀−𝑔)(𝜔)=⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔−𝑔₀))⟩ (3.10) is continuous on   Ω by Proposition 2.1(ii). Let 𝐴=𝑈. Then 𝐴 is closed and Ω𝑥_−𝑔0⊂𝐴⊂Ω. Furthermore, min_(𝑎∗,𝑓_)∈𝐴⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔−𝑔₀𝜖))⟩≥2. (3.11) Since 𝑔₀ is a simultaneous regular point of 𝐺, one has that {𝑔_𝑛}⊂𝐺 such that ‖𝑔_𝑛−𝑔₀‖→0 and (2.9) holds. It follows from (2.9) and Proposition 2.1 that 𝜙(𝑥−𝑔_𝑛)⁺(𝑎^∗,𝑓)<‖𝑥−𝑔₀‖_𝒳,∀(𝑎^∗,𝑥)∈𝐴,𝑛∈𝐍. (3.12) Therefore, max_(𝑎∗,𝑥_)∈𝐴𝜙(𝑥−𝑔_𝑛)⁺(𝑎^∗,𝑓)<‖𝑥−𝑔₀‖_𝒳,∀𝑛∈𝐍. (3.13) On the other hand, let 𝐾=Ω⧵𝑈. Then 𝐾 is a compact subset of Ω and 𝐾∩Ω𝑥^−𝑔₀=∅. Thus there is 𝛿>0 such that max_(𝑎∗,𝑥_)∈𝐾𝜙(𝑥−𝑔₀)⁺(𝑎^∗,𝑓)<‖𝑥−𝑔₀‖_𝒳−𝛿. (3.14) For each (𝑎^∗,𝑓)∈Ω, since ⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑥_𝑖−𝑔_𝑛))⟩=⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑥_𝑖−𝑔₀))⟩+⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔₀−𝑔_𝑛))⟩≤⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑥_𝑖−𝑔₀))⟩+‖(𝜆_𝑖)‖_𝐴‖𝑔₀−𝑔_𝑛‖, (3.15) one has that 𝜙(𝑥−𝑔_𝑛)⁺(𝑎^∗,𝑓)≤𝜙(𝑥−𝑔₀)⁺(𝑎^∗,𝑓)+‖(𝜆_𝑖)‖_𝐴‖𝑔₀−𝑔_𝑛‖. (3.16) Thus when 𝑛 is large enough, we obtain from (3.16) and (3.14) that max_(𝑎∗,𝑥_)∈𝐾𝜙(𝑥−𝑔_𝑛)⁺(𝑎^∗,𝑓)≤max_(𝑎∗,𝑥_)∈𝐾𝜙(𝑥−𝑔_𝑛)⁺(𝑎^∗,𝑓)+‖(𝜆_𝑖)‖_𝐴‖𝑔₀−𝑔_𝑛‖<‖𝑥−𝑔₀‖_𝒳. (3.17) This together with (2.6) and (3.13) implies that ‖𝑥−𝑔_𝑛‖_𝒳=max_(𝑎∗,𝑥_)∈Ω𝜙(𝑥−𝑔_𝑛)⁺(𝑎^∗,𝑓)<‖𝑥−𝑔₀‖_𝒳, (3.18) which contradicts that 𝑔₀∈𝑃_𝐺(𝑥). The proof is complete.

Corollary 3.3. The following statements are equivalent: (i)𝐺 is a simultaneous sun,(ii)For each 𝑔₀∈𝐺 and 𝑥∈𝒳, 𝑔₀∈𝑃_𝐺(𝑥) if and only if (𝑥,𝑔₀) satisfies simultaneous Kolmogorov Condition,(iii)𝐺 is a simultaneous regular set.

The second main result of this paper is as follows.

Theorem 3.4. Let 𝑔₀∈𝐺. Consider the following statements: (i)𝑔₀ is a simultaneous strongly regular point of 𝐺,(ii)For each 𝑥∈𝒳, the fact that (𝑥,𝑔₀) satisfies simultaneous Papini Condition implies that (𝑥,𝑔₀) satisfies simultaneous Kolmogorov Condition,(iii)𝑔₀ is a simultaneous solar point of 𝐺.Then (i)⇒(ii)⇒(iii).

Proof. (i)⇒(ii) Let (i) holds. Suppose on the contrary that there exists 𝑥∈𝒳 such that (𝑥,𝑔₀) satisfies simultaneous Papini Condition but does not satisfy simultaneous Kolmogorov Condition. Then there exist 𝑔∈𝐺 and 𝜖>0 such that (3.8) holds. Let 𝑈 and 𝐴 be as in the proof of the implication (iii)⇒(ii) in Theorem 3.2, respectively. Then, 𝐴 is closed, Ω𝑥_−𝑔0⊂𝐴⊂Ω, and (3.11) is valid. In view of the definition of simultaneous strongly regular point, there is a sequence {𝑔_𝑛}⊂𝐺 such that ‖𝑔_𝑛−𝑔₀‖→0 and (2.10) holds. It follows from (2.10) that
max_(𝑎∗,𝑓_)∈𝐴⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔_𝑛−𝑔₀))⟩>0,∀𝑛∈𝐍, (3.19) since 𝐴 is closed. Let 𝐾=Ω⧵𝑈. Then 𝐾 is a closed subset of Ω. Furthermore, let 𝛽=max_(𝑎∗,𝑓_)∈𝐾⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑥_𝑖−𝑔₀))⟩. It is easy to see that 𝛽<‖𝑥−𝑔₀‖_𝒳. Let 𝛽₀=(1/2)(‖𝑥−𝑔₀‖_𝒳−𝛽) and let 𝑛 be sufficiently large such that ‖𝑔_𝑛−𝑔₀‖_𝒳<𝛽₀. Then ‖𝑥−𝑔_𝑛‖_𝒳≥‖𝑥−𝑔₀‖_𝒳−‖𝑔₀−𝑔_𝑛‖_𝒳>‖𝑥−𝑔₀‖_𝒳−𝛽₀. (3.20) It follows that max_(𝑎∗,𝑓_)∈𝐾⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑥_𝑖−𝑔_𝑛))⟩≤max_(𝑎∗,𝑓_)∈𝐾⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑥_𝑖−𝑔₀))⟩+‖𝑔₀−𝑔_𝑛‖_𝒳=𝛽+‖𝑔₀−𝑔_𝑛‖_𝒳=[‖𝑥−𝑔₀‖_𝒳−𝛽₀]−[𝛽₀−‖𝑔₀−𝑔_𝑛‖_𝒳]<‖𝑥−𝑔_𝑛‖_𝒳. (3.21) This shows that 𝐾⊂Ω⧵Ω𝑥^−𝑔_𝑛, and hence Ω𝑥^−𝑔_𝑛⊂𝑈. Moreover, max_(𝑎∗,𝑓_)∈Ω𝑥_𝑛−𝑔⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔_𝑛−𝑔₀))⟩≥inf_(𝑎∗,𝑓_)∈𝑈⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔_𝑛−𝑔₀))⟩=min_(𝑎∗,𝑓_)∈𝐴⟨𝑎^∗,(Re𝜆_𝑖𝑓_𝑖(𝑔_𝑛−𝑔₀))⟩>0 (3.22) thanks to (3.19). This contradicts that (𝑥,𝑔₀) satisfies simultaneous Papini Condition.
(ii)⇒(iii) Let 𝑥∈𝒳 and 𝑔₀∈𝑃_𝐺(𝑥). Then (𝑥,𝑔₀) satisfies simultaneous Papini Condition by Proposition 2.6. Hence (𝑥,𝑔₀) satisfies simultaneous Kolmogorov Condition thanks to (ii). Using Theorem 3.2 and Proposition 2.6, one has that 𝑔₀ is a simultaneous solar point of 𝐺. The proof is complete.

By Theorem 3.4 and Proposition 2.6, we have the following result.

Corollary 3.5. Let 𝐺 be a simultaneous strongly regular set of 𝑋. Let 𝑥∈𝒳 and 𝑔₀∈𝐺. Then 𝑔₀∈𝑃_𝐺(𝑥) if and only if (𝑥,𝑔₀) satisfies simultaneous Papini Condition.

Acknowledgment

The authors would like to thank the support in part by the NNSF of China (Grant no. 90818020) and the NSF of Zhejiang Province of China (Grant no. Y7080235).