^{1}

^{2}

^{1}

^{3}

^{1}

^{2}

^{3}

We consider the global error bound for the generalized nonlinear complementarity problem over a polyhedral cone (GNCP). By a new technique, we establish an easier computed global error bound for the GNCP under weaker conditions, which improves the result obtained by Sun and Wang (2013) for GNCP.

Let

The GNCP is a direct generalization of the classical nonlinear complementarity problem and a special case of the general variational inequalities problem [

This paper is a follow-up to [

To end this section, we give some notations used in this paper. Vectors considered in this paper are taken in Euclidean space

First, we give the following definition used in the subsequent.

The mapping

Based on this definition,

Now, we give some assumptions for our analysis based on Definition

For mappings

mapping

matrix

In the following, we give the conclusion established in [

A point

From Theorem

For the ease of description, let

In the following, we give the error bound for a single quadratic function to reach our aims.

Let

For any

It is easy to verify that

To establish a global error bound for GNCP, we also give the following result from [

For polyhedral cone

Before proceeding, we present the following definition introduced in [

The mapping

Now, we are at the position to state our main results in this paper.

Suppose that

Using Lemma

Furthermore,

For any

Firstly, from remark of Definition

In the following, we also present an example to compare the condition in Theorem

When

It is easy to see that the solution set of the LCP

However, letting

Secondly, if

In the end of this paper, we will consider a special case of GNCP which was discussed in [

When

Suppose that the hypotheses of Theorem

It is clear that if

In this paper, we established a global error bound on the generalized nonlinear complementarity problems over a polyhedral cone, which improves the result obtained for variational inequalities and the GNCP in [

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

The authors wish to give their sincere thanks to the associated editor and two anonymous referees for their valuable suggestions and helpful comments which improve the presentation of the paper. This work was supported by the Natural Science Foundation of China (nos. 11171180, 11101303, 11171362, and 11271226), the Specialized Research Fund for the Doctoral Program of Chinese Higher Education (20113705110002, and 20120191110031), the Shandong Provincial Natural Science Foundation (ZR2010AL005), the Shandong Province Science and Technology Development Projects (2013GGA13034), the Domestic Visiting Scholar Project for the Outstanding Young Teacher of Shandong Province Universities (2013), and the Applied Mathematics Enhancement Program of Linyi University.