^{1}

^{2}

^{1}

^{2}

The class

In 1929, Knaster, Kurnatoaski, and Mazurkiewicz proved the well-known

Let

Recently, many scholars (see [

In this paper, we first introduce the new generalized class

Let

The multivalued mapping

Let

Then

Let

for each compact subset

Let

the mapping

for each compact subset

for each compact subset

Let

Suppose that

Let

Then

Define

Theorem

Let

for each compact subset

there exists a nonempty compact subset

Suppose that the conclusion is not true; then

We claim that

Theorem

Let

for any compact subset

there exist a nonempty compact convex subset

Then

Assume that

Theorem

In order to apply the above theorem to show the fixed-point theorems, we first establish the following generalization of the Ky Fan’s matching theorem.

Let

Then there exists

Assume that, for any

Theorem

Let

Then for any

Let

If

In the sequel, we give the famous Fan-Browder type fixed-point theorem. We first give the following conclusion.

Suppose that

for any

there exists a transfer compactly open values mapping

there exist a nonempty compact convex subset

Then there exists a finite subset

Define

(i) Suppose that

(ii) Suppose that

Theorem

For Theorem

Suppose that

for any

there exists a transfer compactly open values mapping

there exist a nonempty compact convex subset

Then there exists a finite subset

The proof is similar to Theorem

Suppose that

for any

there exists a transfer compactly open values mapping

Then there exists a finite subset

Let

for any

Then there is an

Let

If

In this section, we will introduce some definitions and conclusions and show the existence of solutions to the generalized vector equilibrium problems.

Let

Let

for each

for each

for each

Then

If the conclusion does not hold, then there exists

Let

for each

for each

for each

Then

There exists

To avoid the structure of the space, Lemmas

Let

Let

Let

Then the mapping

Let

for each

for each

for each

for each compact subset

setting

Then there exists

By condition (1),

Let

for each

for each

for each

for each compact subset

setting

Then there exists

By condition (1),

Let

for each

for each

for each

for each

setting

Then there exists

By condition (1) and Proposition

Let

for each

for each

for each

for each

setting

Then there exists

By condition (1) and Proposition

The following result is a simplicity version of Theorem 1 in [

Let

If, for each fixed

If, for each fixed

Let

for each

for each

for each

for each

setting

Then there exists

Define set-valued mappings

For the above results, Theorems

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

The authors are grateful to the referees’ suggestion for improving the paper. The research is supported by the Natural Science Foundation of Hunan Province (no. 2014JJ4044).