^{1}

^{2}

^{2}

^{3}

^{1}

^{2}

^{3}

We will give the

The classical Banach-Stone theorem tells us that, when

Cao et al. stated a lattice version of the classical Banach-Stone theorem in [

Garrido and Jaramillo [

Our first goal of this paper is to prove the Banach-Stone type theorem in the setting of lattice-valued

An ordered vector space

Let

A mapping

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

This means that when we refer to

Suppose that

Let

When

It is easy to prove the following lemma.

In the above contexts,

In the Contexts

We are going to use the Closed Graph Theorem to prove this lemma. Suppose that the sequence of functions

In order to prove that

For any fixed element

By Theorem

Suppose on the contrary that there exists a sequence

Suppose that

We can define the linear map

Assume that

Suppose that

For any

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 [

Let

For completeness, we will sketch the proof. Observe that

Suppose that

From the previous paragraph, if

Suppose that

By Lemma

In Theorem

Let

Suppose that

Suppose that

Since

Let

On the other hand, for any

On the other hand, suppose on the contrary that

Suppose that

Let

Since

We will show that

Hence we derive that

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

Let

If

If

Also here, the result of [

Suppose that

When

Suppose that

In the following part of this section we have

Let

Suppose that

Assume that

When

When

On the other hand, for each

From the assumption

Given any

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.