Nonvanishing Preservers and Compact Weighted Composition Operators between Spaces of Lipschitz Functions

Abstract

We will give the 𝛼-Lipschitz version of the Banach-Stone type theorems for lattice-valued 𝛼-Lipschitz functions on some metric spaces. In particular, when 𝑋 and 𝑌 are bounded metric spaces, if 𝑇∶Lip(𝑋)→Lip(𝑌) is a nonvanishing preserver, then 𝑇 is a weighted composition operator 𝑇𝑓=ℎ⋅𝑓∘𝜑, where 𝜑∶𝑌→𝑋 is a Lipschitz homeomorphism. We also characterize the compact weighted composition operators between spaces of Lipschitz functions.

1. Introduction

The classical Banach-Stone theorem tells us that, when 𝑋 and 𝑌 are compact Hausdorff spaces, every linear surjective isometry from 𝐶(𝑋) onto 𝐶(𝑌) can be written as a weighted composition operator; that is, it is of the form 𝑇𝑓𝑦=𝐽𝑦𝑓𝜑𝑦, (1) where 𝜑 is a homeomorphism from 𝑌 onto 𝑋 and 𝐽∈𝐶(𝑌) with |𝐽(𝑦)|=1 for all 𝑦∈𝑌. The theorem has many variable extensions concerning isometries, algebra isomorphisms, and disjointness preserving mappings between continuous function spaces; and we refer the surveys [1, 2] for more history about Banach-Stone theorems. Moreover, Kamowitz [3] gave the representation and spectrum of the compact weighted composition operators on the continuous functions.

Cao et al. stated a lattice version of the classical Banach-Stone theorem in [4]. Later, Chen et al. [5], Ercan and Önal [6, 7], and Miao et al. [8] generalized this result. When 𝑋, 𝑌 are compact Hausdorff spaces and 𝐸, 𝐹 are Banach lattices, by the main results of [5, 7], we can see that every vector lattice isomorphism 𝑇 from 𝐶(𝑋,𝐸) onto 𝐶(𝑌,𝐹) preserving the nonvanishing functions must be a weighted composition operator.

Garrido and Jaramillo [9, 10] and Weaver [11] tackled the Banach-Stone type theorem for lattices of real Lipschitz functions. Later, Jiménez-Vargas and Villegas-Vallecillos [12] proved that two little Lipschitz algebras are order isomorphic if and only if the corresponding compact metric spaces are Lipschitz homeomorphic. Recently, Jiménez-Vargas et al. [13] presented a Lipschitz version of the result in [5], in which the underlying spaces should be compact.

Our first goal of this paper is to prove the Banach-Stone type theorem in the setting of lattice-valued 𝛼-Lipschitz functions. Section 2 is devoted to the preliminaries about vector lattices and 𝛼-Lipschitz functions. Then we will give the 𝛼-Lipschitz version of Banach-Stone theorem in Section 3. In particular, when 𝑋, 𝑌 are bounded metric spaces, if 𝑇∶Lip(𝑋)→Lip(𝑌) is a nonvanishing preserver, then we will show that 𝑇 is a weighted composition operator 𝑇𝑓=ℎ⋅𝑓∘𝜑, where 𝜑∶𝑌→𝑋 is a Lipschitz homeomorphism. Our second aim is to give the characterization of compact weighted composition operators on the 𝛼-Lipschitz functions.

2. Preliminaries

An ordered vector space 𝐸 is said to be a vector lattice if max{𝑥,𝑦} exists for any 𝑥, 𝑦 in 𝐸. A vector lattice 𝐸 is said to be a Banach lattice if it is complete under its norm ‖⋅‖ and satisfies the Riesz law: |𝑥|≤𝑦⟹‖𝑥‖≤𝑦, (2) where |𝑥|=max{𝑥,−𝑥}.

Let (𝑋,𝑑) be a metric space and 𝐸 a Banach space; if 0<𝛼≤1, a function 𝑓 from 𝑋 to 𝐸 is said to be 𝛼-Lipschitz if 𝐿_𝛼𝑓𝑓(𝑥)𝑦=sup−𝑓𝑑^𝛼𝑥,𝑦∶𝑥,𝑦∈𝑋,𝑥≠𝑦<∞. (3) The 𝛼-Lipschitz function space  Lip_𝛼(𝑋,𝐸) is the space of all 𝐸-valued 𝛼-Lipschitz functions on 𝑋. Lip^𝑏_𝛼(𝑋,𝐸) is the Banach space of all bounded 𝛼-Lipschitz functions 𝑓∶𝑋→𝐸 with the 𝛼-Lipschitz norm 𝑓=max𝐿_𝛼𝑓,𝑓_∞, (4) where ‖𝑓‖_∞=sup{‖𝑓(𝑥)‖∶𝑥∈𝑋}. Furthermore, the little Lipschitz space  lip_𝛼(𝑋,𝐸) is then defined to be the closed subspace of Lip_𝛼(𝑋,𝐸) of these functions 𝑓 with the following property: for every 𝜀>0, there exists 𝛿>0 such that ‖𝑓(𝑥₁)−𝑓(𝑥₂)‖<𝜀𝑑^𝛼(𝑥₁,𝑥₂) whenever 𝑑(𝑥₁,𝑥₂)≤𝛿. lip^𝑏_𝛼(𝑋,𝐸) is the subspace of lip_𝛼(𝑋,𝐸) consisting of all bounded functions. Notice that when 𝛼=1, Lip_𝛼(𝑋,𝐸) is just Lipschitz space Lip(𝑋,𝐸). Moreover, if 𝐸 is a Banach lattice, then Lip_𝛼(𝑋,𝐸) is a vector lattice with the usual pointwise order 𝑓≤𝑔⟺𝑓(𝑥)≤𝑔(𝑥),∀𝑥∈𝑋. (5) However, Lip^𝑏_𝛼(𝑋,𝐸) is not a Banach lattice since ‖⋅‖ does not satisfy the Riesz law in general.

A mapping 𝜑 from 𝑌 to 𝑋 is said to be a 𝛼-Lipschitz homeomorphism if it is bijective and 𝜑 and 𝜑⁻¹ are both 𝛼-Lipschitz. If 𝑓 is in Lip_𝛼(𝑋) and 𝑒 is a vector in 𝐸, denote by 𝑓⊗𝑒 the function 𝑥↦𝑓(𝑥)𝑒 in Lip_𝛼(𝑋,𝐸). In particular, 1⊗𝑒 denotes the constant function 𝑥↦𝑒 on 𝑋. For any function 𝑓 in Lip_𝛼(𝑋,𝐸), the zero set {𝑥∈𝑋∶𝑓(𝑥)=0} of 𝑓 is denoted by 𝑧(𝑓) and its cozero set {𝑥∈𝑋∶𝑓(𝑥)≠0} is coz(𝑓), and 𝑓 is said to be nonvanishing if 𝑧(𝑓)=∅. An operator 𝑇∶Lip_𝛼(𝑋,𝐸)→Lip_𝛼(𝑌,𝐹) is said to be a nonvanishing preserver if 𝑧𝑓=∅⟺𝑧𝑇𝑓=∅,∀𝑓∈Lip_𝛼(𝑋,𝐸). (6)𝑇 is said to be a Riesz isomorphism if 𝑇(𝑓∨𝑔)=𝑇𝑓∨𝑇𝑔 and 𝑇(𝑓∧𝑔)=𝑇𝑓∧𝑇𝑔 for any 𝑓,𝑔∈Lip_𝛼(𝑋,𝐸).

3. Nonvanishing Preservers on Lipschitz Functions

In this section, our results will be valid (with the same proof) for different kinds of spaces. For this reason we first consider several situations to work in. Throughout this section we will assume that 0<𝛼≤1, 𝑋, 𝑌 are metric spaces and 𝐸, 𝐹 are Banach lattices.

Context 1. 𝐴(𝑋,𝐸)=Lip^𝑏_𝛼(𝑋,𝐸), 𝐴(𝑌,𝐹)=Lip^𝑏_𝛼(𝑌,𝐹).

Context 2 (0<𝛼<1). 𝐴(𝑋,𝐸)=lip^𝑏_𝛼(𝑋,𝐸), 𝐴(𝑌,𝐹)=lip^𝑏_𝛼(𝑌,𝐹).

This means that when we refer to 𝑋, 𝑌, 𝐴(𝑋,𝐸), 𝐴(𝑌,𝐹), we assume that all of them are included at the same time in one of the above two contexts.

Suppose that 𝑋 is a metric space and 0<𝛼≤1, for any 𝑥₁≠𝑥₂∈𝑋, the function 𝑓(𝑥)𝑑=max0,1−^𝛼𝑥,𝑥₂𝑑^𝛼𝑥₁,𝑥₂ (7) belongs to Lip^𝑏_𝛼(𝑋). Moreover, if 0<𝛼<1, then we can find 𝛽>0 such that 𝛼<𝛽<1, and the function 𝑓(𝑥)𝑑=max0,1−^𝛽𝑥,𝑥₂𝑑^𝛽𝑥₁,𝑥₂ (8) belongs to lip^𝑏_𝛼(𝑋). The function 𝑓 defined in (7) or (8) has the property: 0≤𝑓≤1, 𝑓(𝑥₁)=0, 𝑓(𝑥₂)=1 and 𝑧(𝑓)={𝑥∈𝑋∶𝑑(𝑥,𝑥₂)≥𝑑(𝑥₁,𝑥₂)}.

Theorem 1. Let 𝑇∶𝐴(𝑋,𝐸)→𝐴(𝑌,𝐹) be a Riesz isomorphism preserving nonvanishing functions. Then 𝑇 carries the form 𝑇𝑓𝑦=𝐽𝑦𝑓𝜑𝑦,∀𝑓∈𝐴(𝑋,𝐸),𝑦∈𝑌. (9) Here, 𝜑 is a homeomorphism from 𝑌 onto 𝑋 and all fiber linear maps 𝐽𝑦∶𝐸→𝐹 are isomorphisms.

Remark 2. When 𝛼=1, the previous theorem is not valid for the little Lipschitz space lip₁(𝑋,𝐸), where 𝑋 is a connected Banach and 𝐸 is a Banach lattice. Note that if 𝑋 is a connected Banach spaces, we have that lip^𝑏₁(𝑋,𝐸)=lip₁(𝑋,𝐸) consisting of all 𝐸-valued constant functions defined on 𝑋. Let 𝜑 be any map from ℝ² to ℝ and 𝑇∶lip₁(ℝ,𝐸)→lip₁(ℝ²,𝐸) a linear bijection operator defined by 𝑇𝑓𝑦=𝑓𝜑𝑦,∀𝑦∈ℝ². (10) It is obvious that the operator 𝑇 is a Riesz isomorphism preserving nonvanishing functions with a weighted composition representation, but ℝ and ℝ² are not homeomorphic.
It is easy to prove the following lemma.

Lemma 3. 𝑇 preserves common zeros, that is, ^𝑛_𝑖=1𝑧𝑓_𝑖≠∅⟺^𝑛_𝑖=1𝑧𝑇𝑓_𝑖≠∅ (11) for any 𝑓₁,…,𝑓_𝑛∈𝐴(𝑋,𝐸) and 𝑛∈ℕ.

Proof of Theorem 1. In the above contexts, 𝐴(𝑋,𝐸) and 𝐴(𝑌,𝐹) contain constant functions, so 𝐸_𝑦=𝐸 and 𝐹_𝑦=𝐹, where 𝐸_𝑦, 𝐹_𝑦 are defined in [14, Definition 3.8]. Therefore, by [14, Theorem 3.1] we can derive the result.

Lemma 4. In the Contexts 1 and 2, 𝑇 is automatically continuous.

Proof. We are going to use the Closed Graph Theorem to prove this lemma. Suppose that the sequence of functions {𝑓_𝑛} converges to 𝑓₀ in 𝐴(𝑋,𝐸) and {𝑇𝑓_𝑛} converges to 𝑔₀ in 𝐴(𝑌,𝐹); then for any 𝑥∈𝑋 and 𝑦∈𝑌, we have that {𝑓_𝑛(𝑥)} converges to 𝑓₀(𝑥) in 𝐸 and {(𝑇𝑓_𝑛)(𝑦)} converges to 𝑔₀(𝑦) in 𝐹, respectively. Notice that, for any 𝑥∈𝑋, 𝐽𝜑⁻¹(𝑥)∶𝐸→𝐹 is continuous; then one can derive that 𝑇𝑓_𝑛𝜑⁻¹(𝑥)=𝐽𝜑⁻¹(𝑥)𝑓_𝑛(𝑥)⟶𝐽𝜑⁻¹(𝑥)𝑓₀(𝑥)=𝑇𝑓₀𝜑⁻¹(𝑥) (12) for all 𝑥 in 𝑋. Since 𝜑 is a bijection from 𝑌 onto 𝑋, we get that the sequence {(𝑇𝑓_𝑛)(𝑦)} converges to (𝑇𝑓₀)(𝑦) for all 𝑦 in 𝑌, and hence 𝑔₀=𝑇𝑓₀. This means that 𝑇 is a closed operator from 𝐴(𝑋,𝐸) to 𝐴(𝑌,𝐹), and then 𝑇 is continuous.

In order to prove that 𝜑 is a 𝛼-Lipschitz map from 𝑌 onto 𝑋, we need the following lemma, and some idea of the proof comes from [15, Lemma 5.8].

Lemma 5. For any fixed element 𝑒∈𝐸 with ‖𝑒‖=1, we have that inf_𝑦∈𝑌𝐽𝑦(𝑒)=inf_𝑦∈𝑌𝑇(1⊗𝑒)𝑦>0. (13)

Proof. By Theorem 1 we can also find a map 𝐽̃ from 𝑋 to 𝐈𝐬𝐨(𝐹,𝐸) (which is the set of all linear isomorphisms from 𝐹 to 𝐸) and a bijection 𝜑 from 𝑋 onto 𝑌 such that 𝑇⁻¹𝑔(𝑥)=𝐽𝑥̃𝑔𝜑(𝑥) (14) for all 𝑥∈𝑋 and 𝑔∈𝐴(𝑌,𝐹). From the definition of 𝜓, 𝜑, and 𝜑, we can see that 𝜑⁻¹=𝜑=𝜓.
Suppose on the contrary that there exists a sequence {𝑦_𝑛}⊂𝑌 such that ‖𝑇(1⊗𝑒)(𝑦_𝑛)‖=‖(𝐽𝑦_𝑛)(𝑒)‖≤2^−2𝑛 for all 𝑛∈ℕ. If {𝑦_𝑛} has a limit point 𝑦^′ in 𝑌, notice that 𝑇 preserves nonvanishing functions, then we can see that (𝐽𝑦^′)(𝑒)=0 and hence 𝑒=0. This leads to a contradiction. On the other hand, if there exists a positive scalar 𝜏>0 such that 𝑑^𝛼(𝑦_𝑛,𝑦_𝑚)≥𝜏 for any 𝑛,𝑚∈ℕ with 𝑛≠𝑚, when we take the norm one element 𝑏_𝑛=𝑇(1⊗𝑒)𝑦_𝑛𝑇(1⊗𝑒)𝑦_𝑛, (15) then we can derive that 𝑇⁻¹(𝑇(1⊗𝑒))𝜑𝑦_𝑛=(1⊗𝑒)𝜑𝑦_𝑛𝑇=𝑒,₋₁(𝑇(1⊗𝑒))𝜑𝑦_𝑛=𝐽𝜑̃𝑦_𝑛𝑇(1⊗𝑒)𝑦_𝑛 (16) for all 𝑛∈ℕ. Therefore, for any 𝑛∈ℕ, we know that 𝐽𝜑̃𝑦_𝑛𝑏_𝑛=𝐽𝜑̃𝑦_𝑛𝑇(1⊗𝑒)𝑦_𝑛𝑇(1⊗𝑒)𝑦_𝑛=1𝑇(1⊗𝑒)𝑦_𝑛≥2^2𝑛. (17) Moreover, for any 𝑛∈ℕ, by the similar manner of (7) and (8) we can define the function 𝜓_𝑛(𝑦)∈𝐴^𝑏(𝑌) such that 0≤𝜓_𝑛≤1, 𝐿_𝛼(𝜓_𝑛)≤𝑚 for some 𝑚>0, 𝜓_𝑛(𝑦_𝑛)=1 and 𝜓_𝑛(𝑦)=0 for all 𝑦 such that 𝑑(𝑦,𝑦_𝑛)≥𝜏/2. When put ℎ₀=^∞_𝑛=2𝜓_𝑛⊗𝑏_𝑛2^𝑛, (18) we can see that ℎ₀ belongs to 𝐴^𝑏(𝑌,𝐹) and ℎ₀(𝑦_𝑛)=𝑏_𝑛/2^𝑛 for 𝑛>1. Then one can conclude that 𝑇⁻¹ℎ₀𝜑𝑦_𝑛=𝐽𝜑̃𝑦_𝑛ℎ₀𝜑𝜑𝑦_𝑛=𝐽𝜑̃𝑦_𝑛ℎ₀𝑦_𝑛=12^𝑛𝐽𝜑̃𝑦_𝑛𝑏_𝑛, (19) and hence 𝑇⁻¹ℎ₀𝜑𝑦_𝑛=𝐽𝜑̃𝑦_𝑛𝑏_𝑛2^𝑛>2^𝑛,∀𝑛≥2. (20) This is a contradiction in Contexts 1 and 2 since ‖𝑇⁻¹ℎ₀‖_∞<∞.

Theorem 6. Suppose that 𝑋, 𝑌 are bounded metric spaces in the Contexts 1 and 2; then 𝜑 is a 𝛼-Lipschitz map from 𝑌 onto 𝑋.

Proof. We can define the linear map 𝑇 from 𝐴(𝑋) to 𝐴(𝑌) by 𝑇𝑓𝑦=𝑓𝜑𝑦,∀𝑦∈𝑌. (21) We have to show that 𝑇 is well defined at first. For any fixed element 𝑒∈𝐸 with ‖𝑒‖=1, from Lemma 5 we can choose a positive scalar 𝜈 such that ‖𝑇(1⊗𝑒)(𝑦)‖≥𝜈>0 for all 𝑦 in 𝑌, and then it is easy to see that the function ℎ which maps 𝑦 to 1/‖𝑇(1⊗𝑒)(𝑦)‖ belongs to 𝐴(𝑌).
Assume that 𝑓 is a positive function in 𝐴(𝑋); one can get that, for any 𝑦₁,𝑦₂∈𝑌, 𝑇𝑓𝑦₁𝑇(1⊗𝑒)𝑦₁−𝑇𝑓𝑦₂𝑇(1⊗𝑒)𝑦₂=𝐽𝑦₁𝑓𝜑𝑦₁𝑒−𝐽𝑦₂𝑓𝜑𝑦₂𝑒≤𝐽𝑦₁𝑓𝜑𝑦₁𝑒−𝐽𝑦₂𝑓𝜑𝑦₂𝑒=𝑇𝑓⊗𝑒𝑦₁−𝑇𝑓⊗𝑒𝑦₂≤𝐿_𝛼𝑇𝑓⊗𝑒𝑑^𝛼𝑦₁,𝑦₂≤𝑇𝑓⊗𝑒𝑑^𝛼𝑦₁,𝑦₂; (22) that is, (𝑇𝑓)(𝑦)‖𝑇(1⊗𝑒)(𝑦)‖ is a bounded 𝛼-Lipschitz function. Moreover, in Context 2 we can derive that (𝑇𝑓)(𝑦)‖𝑇(1⊗𝑒)(𝑦)‖ is also a little Lipschitz function. This means that the function (𝑇𝑓)(𝑦)=(𝑇𝑓)(𝑦)‖𝑇(1⊗𝑒)(𝑦)‖ℎ(𝑦) belongs to 𝐴(𝑌). Therefore, 𝑇 is a well-defined bijective linear operator from 𝐴(𝑋) onto 𝐴(𝑌), and 𝑇 is also a nonvanishing preserver.
Suppose that {𝑓_𝑛} is a sequence which converges to 0 in 𝐴(𝑋) and the sequence {𝑇𝑓_𝑛} converges to 𝑔₀ in 𝐴(𝑌). For any 𝑛∈ℕ and 𝑦∈𝑌, (𝑇𝑓_𝑛)(𝑦)=𝑓_𝑛(𝜑(𝑦)), and hence we have that {𝑓_𝑛(𝜑(𝑦))} converges to 𝑔₀(𝑦) for all 𝑦∈𝑌. Notice that {𝑓_𝑛} converges to 0; one can conclude that {𝑓_𝑛(𝑥)} converges to 0 for all 𝑥∈𝑋, and, since 𝜑 is a bijective map from 𝑌 onto 𝑋, we have that 𝑔₀(𝑦)=0 for any 𝑦 in 𝑌. Therefore, 𝑇 is a closed operator and hence 𝑇 is continuous.
For any 𝑦₁ and 𝑦₂ in 𝑌, there exists a function 𝑓₀∈𝐴(𝑋) such that ‖𝑓₀‖≤𝐷(𝑋)+𝐷(𝑋)^1−𝛼 and 𝑓₀(𝜑(𝑦₁))=𝑑(𝜑(𝑦₁),𝜑(𝑦₂)) and 𝑓₀(𝜑(𝑦₂))=0 (in fact, 𝑓₀(𝑥)=𝑑(𝑥,𝜑(𝑦₂)) has the properties that we need). Here 𝐷(𝑋) denotes the diameter of 𝑋. Then we can derive that 𝑇𝑓₀𝑦₁−𝑇𝑓₀𝑦₂≤𝐿_𝛼𝑇𝑓₀𝑑^𝛼𝑦₁,𝑦₂≤𝑇𝑓₀𝑑^𝛼𝑦₁,𝑦₂. (23) Furthermore, we have that 𝑑𝜑𝑦₁,𝜑𝑦₂=𝑓₀𝜑𝑦₁−𝑓₀𝜑𝑦₂=𝑇𝑓₀𝑦₁−𝑇𝑓₀𝑦₂≤𝑇𝐷(𝑋)+𝐷(𝑋)^1−𝛼𝑑^𝛼𝑦₁,𝑦₂, (24) and this means that 𝜑 is a 𝛼-Lipschitz mapping from 𝑌 onto 𝑋. Similarly, we can see that 𝜑⁻¹ is also 𝛼-Lipschitz, and then 𝜑 is a 𝛼-Lipschitz homeomorphism.

For the spaces of scalar-valued Lipschitz functions, we give a complete characterization of nonvanishing preservers. But at first we need to recall a special case of [16, Lemma 25].

Lemma 7. Let 𝐴(𝑋), 𝐴(𝑌) be in Contexts 1 and 2. Suppose that 𝑇∶𝐴(𝑋)→𝐴(𝑌) is a linear nonvanishing preserver; then the map 𝑆∶𝐴(𝑋)→𝐴(𝑌) given by 𝑆𝑓=𝑇𝑓⋅𝑇1|𝑇1| (25) is a Riesz isomorphism preserving nonvanishing functions.

Proof. For completeness, we will sketch the proof. Observe that 𝑇1 is never vanishing. If 𝑓∈𝐴(𝑋) and 𝜆∈ℝ, then 𝜆∈range𝑓 if and only if 0∈range(𝑓−𝜆) if and only if 0∈range(𝑇𝑓−𝜆𝑇1) if and only if 𝜆∈range𝑇𝑓/𝑇1. In particular, if 𝑓≥0, then 𝑇𝑓/𝑇1≥0. Let 𝑌₊={𝑦∈𝑌∶(𝑇1)(𝑦)>0} and 𝑌₋={𝑦∈𝑌∶(𝑇1)(𝑦)<0}. Then 𝑌₊∪𝑌₋ is a partition of 𝑌 into two open sets.
Suppose that 𝑓∈𝐴(𝑋) and 𝑓≥0. Then 𝑇𝑓≥0 on 𝑌₊ and 𝑇𝑓≤0 on 𝑌₋. Hence 𝑇𝑓⋅𝑇1/|𝑇1|=|𝑇𝑓|∈𝐴(𝑌). For any 𝑓∈𝐴(𝑋), we have that 𝑓₊,𝑓₋∈𝐴(𝑋), and |𝑇(𝑓₊)|=𝑇(𝑓₊)⋅𝑇1/|𝑇1| and |𝑇(𝑓₋)|=𝑇(𝑓₋)⋅𝑇1/|𝑇1|. Then we can derive that 𝑆𝑓=𝑇𝑓⋅𝑇1||=𝑇𝑓𝑇1⁺−𝑇𝑓₋⋅𝑇1||=𝑇𝑓𝑇1₊−𝑇𝑓₋∈𝐴(𝑌). (26) This means that 𝑆 is well defined. Moreover, it is easy to check that 𝑆 is bijective.
From the previous paragraph, if 0≤𝑓∈𝐴(𝑋), then 𝑆𝑓=|𝑇𝑓|≥0. If 𝑓∈𝐴(𝑋) and 𝑔=𝑆𝑓≥0, then by the above, 0≤𝑆𝑓=𝑇𝑓⋅𝑇1_𝑋𝑇1_𝑋=𝑇𝑓₊−𝑇𝑓₋. (27) By [17, Lemma 2.3], 𝑇 is biseparating, and hence 𝑇(𝑓₊)⋅𝑇(𝑓₋)=0. It follows that 𝑇(𝑓₋)=0 and thus 𝑓₋=0. Therefore, 𝑓≥0. Thus 𝑆 is a Riesz isomorphism. It is trivial to check that 0∈range𝑓 if 0∈range𝑆𝑓 for any 𝑓∈𝐴(𝑋).

Theorem 8. Suppose that 𝑋, 𝑌 are bounded metric spaces and 𝑇 is a nonvanishing preserver between the following function spaces: (i)0<𝛼≤1 and 𝑇∶𝐿𝑖𝑝_𝛼(𝑋)→𝐿𝑖𝑝_𝛼(𝑌);(ii)0<𝛼<1 and 𝑇∶𝐿𝑖𝑝_𝛼(𝑋)→𝐿𝑖𝑝_𝛼(𝑌). Then 𝑇 is a weighted composition operator of the form 𝑇𝑓𝑦=ℎ𝑦𝑓𝜑𝑦. (28) Here ℎ=𝑇1 and 𝜑∶𝑌→𝑋 is a 𝛼-Lipschitz map.

Proof. By Lemma 7 we have that 𝑇 is a Riesz isomorphism. Then by Theorem 6 we can derive the conclusion.

In Theorem 8, the boundedness of the metric spaces can not be dropped.

Example 9. Let ℕ₁ be the positive integers with the discrete metric, and we can derive that ℕ₁ is not Lipchitz homeomorphic to ℕ. By [18, Example 1.6.4] we can derive that Lip^𝑏(ℕ)=Lip^𝑏(ℕ₁)=ℓ^∞, and then the identity map 𝐼∶Lip^𝑏(ℕ)→Lip^𝑏(ℕ₁) is a nonvanishing preserver. However, the underlying metric spaces are not Lipschitz homeomorphic.

4. Compact Weighted Composition Operators on Lipschitz Spaces

Suppose that 𝑋, 𝑌 are metric spaces, 0<𝛼≤1, and 𝑇∶Lip^𝑏_𝛼(𝑋)→Lip^𝑏_𝛼(𝑌) is a weighted composition operator, that is, 𝑇𝑓𝑦=ℎ𝑦𝑓𝜑𝑦,∀𝑦∈𝑌,𝑓∈Lip^𝑏_𝛼(𝑋). (29) Here ℎ=𝑇1 and 𝜑∶𝑌→𝑋 is a 𝛼-Lipschitz mapping. Put 𝑌₀={𝑦∈𝑌∶ℎ(𝑦)=0}. Recall that 𝜑∶𝑌→𝑋 is supercontractive on 𝑌^′⊂𝑌 if for each 𝜀>0 there exists 𝛿>0 such that 𝑑(𝜑(𝑦₁),𝜑(𝑦₂))<𝜀𝑑(𝑦₁,𝑦₂) whenever 𝑦₁,𝑦₂∈𝑌^′ and 0<𝑑(𝑦₁,𝑦₂)<𝛿. In this section, we will characterize the compact weighted composition operator 𝑇 and consider its spectrum.

Theorem 10. Suppose that 𝑇 is compact. For any 𝑦₀∈𝑌⧵𝑌₀, there is an open neighborhood 𝑈₀ of 𝑦₀ such that 𝜑 is supercontractive on 𝑈₀ and 𝜑(𝑈₀) is totally bounded.

Proof. Since ℎ(𝑦₀)≠0, we can find an open neighborhood 𝑈₀ of 𝑦₀ such that |ℎ(𝑦)|≥|ℎ(𝑦₀)|/2>0 for all 𝑦∈𝑈₀. Suppose on the contrary that there exist {𝑥_𝑛},{𝑦_𝑛}⊂𝑈₀ such that 𝑑(𝑥_𝑛,𝑦_𝑛)→0 and 𝑑𝜑𝑥_𝑛,𝜑𝑦_𝑛𝑑𝑥_𝑛,𝑦_𝑛>𝜀₀ (30) for some 𝜀₀>0. Without loss of generality we can assume that 𝑑^𝛼2(𝑥_𝑛,𝑦_𝑛)<1/𝑛.
Let 𝑓_𝑛(𝑥)=1−𝑒^{−𝑛𝑑𝛼(𝑥,𝜑(𝑦𝑛))}𝑛; (31) we can derive that ‖𝑓_𝑛‖_∞≤1/𝑛 and |𝑓_𝑛(𝑥₁)−𝑓_𝑛(𝑥₂)|≤𝑑^𝛼(𝑥₁,𝑥₂) for any 𝑥₁,𝑥₂∈𝑋. This implies that {𝑓_𝑛} is a bounded sequence in Lip^𝑏_𝛼(𝑋). If 𝑇 is compact, then there exists a subsequence {𝑓_𝑛𝑘} such that 𝑇𝑓_𝑛𝑘→𝑔₀∈Lip^𝑏_𝛼(𝑌). Since 𝑓_𝑛→0 uniformly, for any 𝑦∈𝑌, we have that 𝑇𝑓_𝑛𝑘𝑦=ℎ𝑦𝑓_𝑛𝑘𝜑𝑦≤‖ℎ‖_∞𝑓_𝑛𝑘𝜑𝑦⟶0, (32) and then 𝑔₀=0. This means that 𝑇𝑓_𝑛𝑘→0 in Lip^𝑏_𝛼(𝑌).
On the other hand, for any 𝑛∈ℕ, by the Mean Value Theorem we have that 𝑇𝑓_𝑛𝑥_𝑛−𝑇𝑓_𝑛𝑦_𝑛𝑑^𝛼𝑥_𝑛,𝑦_𝑛=ℎ𝑥_𝑛𝑓_𝑛𝜑𝑥_𝑛𝑑^𝛼𝑥_𝑛,𝑦_𝑛≥ℎ𝑦₀21−𝑒^{−𝑛𝑑𝛼(𝜑(𝑥𝑛),𝜑(𝑦𝑛))}𝑛𝑑^𝛼𝑥_𝑛,𝑦_𝑛=ℎ𝑦₀2𝑒^{−𝑛𝜉𝑛}𝑑^𝛼𝜑𝑥_𝑛,𝜑𝑦_𝑛𝑑^𝛼𝑥_𝑛,𝑦_𝑛≥ℎ𝑦₀2𝑒^{−𝐿𝛼𝛼(𝜑)}𝜀^𝛼₀. (33) Here 0<𝜉_𝑛<𝑑^𝛼(𝜑(𝑥_𝑛),𝜑(𝑦_𝑛))≤𝐿^𝛼_𝛼(𝜑)𝑑^𝛼2(𝑥_𝑛,𝑦_𝑛)<𝐿^𝛼_𝛼(𝜑)/𝑛. Therefore, we can derive that 𝐿_𝛼(𝑇𝑓_𝑛)↛0, and this is a contradiction.
On the other hand, suppose on the contrary that 𝜑(𝑈₀) is not totally bounded, then there exist a constant 𝜏>0 and 𝑧_𝑛=𝜑(𝑢_𝑛)∈𝜑(𝑈₀) such that 𝑑^𝛼(𝑧_𝑛,𝑧_𝑚)>𝜏 whenever 𝑛≠𝑚. Let 𝑓_𝑛(𝑥)=1−𝑒^{−𝑑𝛼(𝑥,𝑧𝑛)},∀𝑥∈𝑋; (34) then it is easy to see that 𝑓_𝑛(𝑧_𝑛)=0 and ‖𝑓_𝑛‖≤1. Moreover, for any 𝑛≠𝑚, we can derive that 𝑇𝑓_𝑛−𝑇𝑓_𝑚∞≥ℎ𝑢_𝑛𝑓_𝑚𝑧_𝑛≥ℎ𝑦₀21−𝑒^{−𝑑𝛼(𝑧𝑛,𝑧𝑚)}≥ℎ𝑦₀21−𝑒^−𝜏. (35) Therefore, {𝑇𝑓_𝑛} has no Cauchy subsequence, and hence 𝑇 is not compact. This leads to a contradiction.

Theorem 11. Suppose that 𝜑 is supercontractive on 𝑌⧵𝑌₀ and 𝜑(𝑌⧵𝑌₀) is totally bounded; then the weighted composition operator defined by (29) is compact.

Proof. Let {𝑓_𝑛}⊂Lip^𝑏_𝛼(𝑋) be a bounded sequence, that is, ‖𝑓_𝑛‖≤𝑀 for some 𝑀>0. Since 𝜑(𝑌⧵𝑌₀) is totally bounded, there exists a subsequence of {𝑓_𝑛}, which is also denoted by {𝑓_𝑛}, such that {𝑓_𝑛} is convergent uniformly in 𝜑(𝑌⧵𝑌₀). Denote the limit by 𝑓₀(𝑥) for all 𝑥∈𝜑(𝑌⧵𝑌₀). It is easy to verify that 𝑓₀ is a bounded Lipschitz function in 𝜑(𝑌⧵𝑌₀). By the similar argument of [18, Theorem 1.5.6] we can extend 𝑓₀ to be a bounded Lipschitz function in Lip^𝑏_𝛼(𝑋), which is also denoted by 𝑓₀. It suffices to show that {𝑇𝑓_𝑛} converges to 𝑇𝑓₀ in Lip^𝑏_𝛼(𝑌).
Since 𝑇 is a weighted composition operator, it is easy to see that {𝑇𝑓_𝑛} converges to 𝑇𝑓₀ uniformly on 𝑌. Let 𝜀>0 be given. Since 𝜑 is supercontractive on 𝑌⧵𝑌₀, there exists 𝛿>0 such that 𝑑𝜑𝑦₁,𝜑𝑦₂𝑑𝑦₁,𝑦₂<𝜀 (36) whenever 𝑦₁,𝑦₂∈𝑌⧵𝑌₀ and 0<𝑑(𝑦₁,𝑦₂)<𝛿.
We will show that 𝐿_𝛼(𝑇𝑓_𝑛−𝑇𝑓₀)→0 by dividing into four cases as the following arguments. For any 𝑦₁,𝑦₂∈𝑌 with 𝑦₁≠𝑦₂.
Case  1. If 𝑦₁,𝑦₂∈𝑌₀, we have that (𝑇𝑓_𝑛)(𝑦_𝑖)=(𝑇𝑓₀)(𝑦_𝑖)=0 for 𝑖=1,2.
 Case  2. If 𝑦₁,𝑦₂∈𝑌⧵𝑌₀ and 0<𝑑(𝑦₁,𝑦₂)<𝛿, we have that 𝑇𝑓_𝑛−𝑓₀𝑦₁−𝑇𝑓_𝑛−𝑓₀𝑦₂≤ℎ𝑦₁−ℎ𝑦₂𝑓_𝑛−𝑓₀𝜑𝑦₁+ℎ𝑦₂𝑓_𝑛−𝑓₀𝜑𝑦₁−𝑓_𝑛−𝑓₀𝜑𝑦₂≤𝐿_𝛼(ℎ)𝑑^𝛼𝑦₁,𝑦₂𝑓_𝑛−𝑓₀𝜑𝑦₁+‖ℎ‖_∞𝐿_𝛼𝑓_𝑛+𝐿_𝛼𝑓₀𝑑^𝛼𝜑𝑦₁,𝜑𝑦₂. (37) Moreover, by (36) we can derive that 𝑑^𝛼𝜑𝑦₁,𝜑𝑦₂=𝑑^𝛼𝜑𝑦₁,𝜑𝑦₂𝑑^𝛼𝑦₁,𝑦₂𝑑^𝛼𝑦₁,𝑦₂≤𝜀^𝛼𝑑^𝛼𝑦₁,𝑦₂. (38)
Case  3. If 𝑦₁,𝑦₂∈𝑌⧵𝑌₀ and 𝑑(𝑦₁,𝑦₂)>𝛿, we have that 𝑇𝑓_𝑛−𝑓₀𝑦₁−𝑇𝑓_𝑛−𝑓₀𝑦₂𝑑^𝛼𝑦₁,𝑦₂≤2𝑇𝑓_𝑛−𝑇𝑓_0∞𝛿^𝛼. (39)
Case  4. If 𝑦₁∈𝑌⧵𝑌₀ and 𝑦₂∈𝑌₀, we have that ℎ(𝑦₂)=0 and then 𝑇𝑓_𝑛−𝑓₀𝑦₁−𝑇𝑓_𝑛−𝑓₀𝑦₂=ℎ𝑦₁𝑓_𝑛−𝑓₀𝜑𝑦₁=ℎ𝑦₁−ℎ𝑦₂⋅𝑓_𝑛−𝑓₀𝜑𝑦₁≤𝐿_𝛼(ℎ)𝑑^𝛼𝑦₁,𝑦₂𝑓_𝑛−𝑓₀𝜑𝑦₁. (40)
Hence we derive that 𝐿_𝛼(𝑇𝑓_𝑛−𝑇𝑓₀)→0 and then 𝑇𝑓_𝑛→𝑇𝑓₀. This means that 𝑇 is a compact operator.

By the similar argument, one can conclude the following results for the scalar-valued little Lipschitz function spaces.

Theorem 12. Let 𝛼∈(0,1). Suppose that 𝑇∶𝐿𝑖𝑝^𝑏_𝛼(𝑋)→𝐿𝑖𝑝^𝑏_𝛼(𝑌) is a nonzero weighted composition operator of the form (29).(1)If 𝑇 is compact, then, for any 𝑦₀∈𝑌⧵𝑌₀, there is an open neighborhood 𝑈₀ of 𝑦₀ such that 𝜑 is supercontractive on 𝑈₀ and 𝜑(𝑈₀) is totally bounded.(2)If 𝜑 is supercontractive on 𝑌⧵𝑌₀ and 𝜑(𝑌⧵𝑌₀) is totally bounded, then 𝑇 is compact.

Also here, the result of [19] also refers to the case where 𝑇 is a composition operator.

Corollary 13. Suppose that 𝑋, 𝑌 are compact metric spaces, and 𝑇 is a weighted composition operator of the form (29) between the following function spaces: (i)0<𝛼≤1 and 𝑇∶𝐿𝑖𝑝^𝑏_𝛼(𝑋)→𝐿𝑖𝑝^𝑏_𝛼(𝑌);(ii)0<𝛼<1 and 𝑇∶𝐿𝑖𝑝^𝑏_𝛼(𝑋)→𝐿𝑖𝑝^𝑏_𝛼(𝑌).Then 𝑇 is compact if and only if 𝜑 is supercontractive on 𝑌⧵𝑌₀.

When 𝑇 is a composition operator, that is, ℎ=𝑇1=1 in the form (29), then 𝑌₀=∅ and we can establish the following results in [20, Theorem 1.1].

Corollary 14. Suppose that 𝑋, 𝑌 are metric spaces and 𝑇∶𝐿𝑖𝑝^𝑏_𝛼(𝑋)→𝐿𝑖𝑝^𝑏_𝛼(𝑌) is a composition operator; then 𝑇 is compact if and only if 𝜑 is supercontractive and 𝜑(𝑌) is totally bounded.

In the following part of this section we have 𝑋=𝑌. Define 𝜑₀(𝑥)=𝑥 and 𝜑_𝑛(𝑥)=𝜑(𝜑_𝑛−1(𝑥)) for all 𝑥∈𝑋 by induction. A point 𝑥₀∈𝑋 is said to be the fixed point of 𝜑 of order 𝑛, 𝑛∈ℕ, if 𝜑_𝑛(𝑥₀)=𝑥₀ and 𝜑_𝑖(𝑥₀)≠𝑥₀ for any 𝑖=0,1,…,𝑛−1.

Theorem 15. Let 𝑋 be a complete metric space and 𝑇∶𝐿𝑖𝑝^𝑏(𝑋)→𝐿𝑖𝑝^𝑏(𝑋) a weighted composition operator of form (29) satisfying: 𝜑 is supercontractive on 𝑋⧵𝑋₀ and 𝜑(𝑋⧵𝑋₀) is totally bounded. Then we can derive that 𝜎(𝑇)={0}∪𝒮, where 𝒮=𝜆∶𝜆^𝑛=ℎ𝑥₀ℎ𝜑𝑥₀⋯ℎ𝜑_𝑛−1𝑥₀,𝑥₀𝑖𝑠𝑎𝑓𝑖𝑥𝑒𝑑𝑝𝑜𝑖𝑛𝑡𝑜𝑓𝜑𝑜𝑓𝑜𝑟𝑑𝑒𝑟𝑛. (41)

Proof. Suppose that 𝑥₀ is a fixed point of 𝜑 of order 𝑛. If ℎ(𝜑_𝑘(𝑥₀))=0 for some 𝑘, we can see that 𝑇 is not surjective and hence 0∈𝜎(𝑇).
Assume that ℎ(𝜑_𝑘(𝑥₀))≠0 for any 𝑘=0,1,2,…,𝑛−1 and 𝜆^𝑛=ℎ(𝑥₀)⋯ℎ(𝜑_𝑛−1(𝑥₀)).
When 𝑛=1, we have that 𝜆=ℎ(𝑥₀) and 𝜑(𝑥₀)=𝑥₀. There exists 𝑔∈Lip^𝑏(𝑋) such that 𝑔(𝑥₀)=1. There is no 𝑓∈Lip^𝑏(𝑋) such that (𝜆−𝑇)𝑓=𝑔. Indeed, if such 𝑓 exists, we can derive that 0=𝜆𝑓𝑥₀−ℎ𝑥₀𝑓𝑥₀=𝜆𝑓𝑥₀−ℎ𝑥₀𝑓𝜑𝑥₀=𝑔𝑥₀=1, (42) and this is impossible. This means that 𝜆∈𝜎(𝑇).
When 𝑛≥2, let 𝛿∶=min{𝑑(𝜑_𝑖(𝑥₀),𝜑_𝑗(𝑥₀))∶0≤𝑖≠𝑗≤𝑛−1}, and define 1𝑔=𝑛^𝑛−1_𝑖=01𝜆^{𝑛−𝑖−1}ℎ𝑥₀⋯ℎ𝜑_𝑖−1𝑥₀𝑑×max0,1−𝑥,𝜑_𝑖𝑥₀𝛿. (43) Here ℎ(𝜑₋₁(𝑥₀))∶=1. Then, similar to the argument of [3, Proposition 3], we can derive that 𝜆∈𝜎(𝑇).
On the other hand, for each 𝑓∈Lip^𝑏(𝑋) with 𝜆𝑓=𝑇𝑓, for some 𝜆∉{0}∪𝒮, we will prove that 𝑓=0. This implies that 𝜆∉𝜎(𝑇) and completes the proof.
From the assumption 𝜆𝑓=𝑇𝑓=ℎ⋅𝑓∘𝜑, for all 𝑥∈𝑋 and 𝑛∈ℕ, we derive that 𝜆^𝑛𝑓(𝑥)=ℎ(𝑥)ℎ𝜑(𝑥)⋯ℎ𝜑_𝑛−1(𝑥)𝑓𝜑_𝑛(𝑥). (44)
Given any 𝑧∈𝑋, let ℱ={𝜑_𝑛(𝑧)∶𝑛∈ℕ∪{0}} and 𝒩={𝑛∈ℕ∶|ℎ(𝜑_𝑛(𝑧))|≥𝛿₀}; here 𝛿₀ is any fixed number with 0<𝛿₀<|𝜆|. We provide that 𝑓(𝑧)=0, which implies that 𝑓=0, by dividing into the following cases.
Case I (ℱ∩𝑋₀≠∅). If there exists 𝑖₀ such that ℎ(𝜑_𝑖0(𝑧))=0, by (44) we can see that 𝜆^𝑖0+1𝑓(𝑧)=ℎ(𝑧)ℎ𝜑(𝑧)⋯ℎ𝜑_𝑖0(𝑧)𝑓𝜑_𝑖0+1(𝑧)=0. (45) This implies that 𝑓(𝑧)=0.
Case II (ℱ⊂𝑋⧵𝑋₀ and ℱ is finite). Let ℱ={𝑧,𝜑(𝑧),…,𝜑_𝑛0(𝑧)}. Then there exists 0≤𝑘≤𝑛₀ such that 𝜑_𝑛0+1(𝑧)=𝜑_𝑘(𝑧). This means that 𝜑_𝑘(𝑧) is a fixed point of 𝜑 of order 𝑛₀−𝑘+1. By (44), we have that 𝜆^{𝑛0−𝑘+1}𝑓𝜑_𝑘(𝑧)=ℎ𝜑_𝑘(𝑧)ℎ𝜑_𝑘+1(𝑧)⋯ℎ𝜑_𝑛0(𝑧)𝑓𝜑_𝑛0+1(𝑧), (46) and 𝑓(𝜑_𝑘(𝑧))=0 since 𝜆∉{0}∪𝒮. Once again, by (44), we derive that 𝜆^𝑘𝑓(𝑧)=ℎ(𝑧)ℎ𝜑(𝑧)⋯ℎ𝜑_𝑘−1(𝑧)𝑓𝜑_𝑘(𝑧)=0, (47) and 𝑓(𝑧)=0.
Case III (ℱ⊂𝑋⧵𝑋₀, ℱ is infinite and 𝒩 is infinite). Notice that {𝜑_𝑛(𝑧)∶𝑛∈𝒩}⊂(𝑋⧵𝑋₀)∩𝜑(𝑋⧵𝑋₀). Since 𝜑(𝑋⧵𝑋₀) is totally bounded, we can derive that {𝜑_𝑛(𝑧)∶𝑛∈𝒩} converges to a point 𝑥∈𝑋. Moreover, 𝜑_𝑛(𝑧)→𝑥 since 𝜑 is supercontractive. Then we have that |ℎ(𝑥)|≥𝛿₀ and 𝜑(𝑥)=𝑥. By (44) we can see that 𝑓(𝑥)=0. Since 𝜑 is supercontractive, there exists 𝛿₁>0 such that 𝑑𝜑𝑧₁,𝜑𝑧₂<|𝜆|2‖ℎ‖_∞𝑑𝑧₁,𝑧₂when0<𝑑𝑧₁,𝑧₂<𝛿₁. (48) Choose 𝑁∈ℕ such that 𝑑(𝜑_𝑛(𝑧),𝑥)<𝛿₁ for all 𝑛≥𝑁, and we have, for any 𝑛∈ℕ, that 𝜆^𝑛𝑓𝜑_𝑁(𝑧)=ℎ𝜑_𝑁(𝑧)ℎ𝜑_𝑁+1(𝑧⋯ℎ𝜑_{𝑁+𝑛−1})(𝑧)𝑓𝜑_𝑛+𝑁(𝑧)≤‖ℎ‖^𝑛_∞𝑓𝜑_𝑛+𝑁(𝑧)=‖ℎ‖^𝑛_∞𝑓𝜑_𝑛+𝑁(𝑧)−𝑓(𝑥≤‖ℎ‖_𝑛∞)𝐿𝑓𝑑𝜑_𝑛+𝑁(𝑧),𝑥≤‖ℎ‖^𝑛_∞𝐿𝑓|𝜆|2‖ℎ‖_∞𝑑𝜑_{𝑛+𝑁−1}(𝑧),𝑥≤⋯≤‖ℎ‖^𝑛_∞𝐿𝑓|𝜆|2‖ℎ‖_∞^𝑛𝑑𝜑_𝑁(𝑧),𝑥𝑓|𝜆|=𝐿2^𝑛𝑑𝜑_𝑁(𝑧),𝑥. (49) That is, 𝑓𝜑_𝑁(𝑧≤1)2^𝑛𝐿𝑓𝑑𝜑_𝑁(𝑧),𝑥. (50) Since 𝑛 is arbitrary, we can derive that 𝑓(𝜑_𝑁(𝑧))=0, and then 𝑓(𝑧)=0 since (44).
Case IV (ℱ⊂𝑋⧵𝑋₀, ℱ is infinite and 𝒩 is finite). We can choose 𝑁₀∈ℕ such that |ℎ(𝜑_𝑛(𝑧))|<𝛿₀ for 𝑛>𝑁₀. From (44), we have that 𝜆^𝑛𝑓(𝑧)=ℎ(𝑧)ℎ𝜑(𝑧)⋯ℎ𝜑_𝑛−1(𝑧)𝑓𝜑_𝑛(𝑧), (51) and then 𝑓(𝑧)≤‖ℎ‖^𝑁0_∞𝛿^−𝑁0₀𝛿₀𝜆^𝑛𝑓_∞, (52) for all 𝑛>𝑁₀. This implies that 𝑓(𝑧)=0 as 𝛿₀<|𝜆|.

Acknowledgments

The authors would like to express their thanks to the referees for several helpful comments which improved the presentation of this paper. Research of the second author was partially supported by NSF of China (11301285, 11371201). The fourth author was supported by Department of Applied Mathematics and the Research Group for Nonlinear Analysis and Optimization, post-doctoral fellowship at the National Sun Yat-sen University when this work was started, and this research is supported in part by Taiwan NSC grant 102-2115-M-033-006.