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We apply the extensions of the Abian-Brown fixed point theorem for set-valued mappings on chain-complete posets to examine the existence of generalized and extended saddle points of bifunctions defined on posets. We also study the generalized and extended equilibrium problems and the solvability of ordered variational inequalities on posets, which are equipped with a partial order relation and have neither an algebraic structure nor a topological structure.

Let

In optimization theory, the standard tools for dealing with the existence of saddle points and equilibria of a bifunction

The concepts defined in

In economic theory and social sciences, there are some examples that both of the income and outcome spaces of mappings are posets, which are equipped with neither topological structure nor algebraic structure. Then, in these circumstances, the optimization problems will be order-optimization problems, which are not normal optimization problems (with respect to real valued functions) and they cannot be solved by using the standard methods. The saddle points and equilibria of bifunction must be generalized from real valued functions to functions with values in ordered sets. In this case, more fixed point theorems on ordered sets must be acquired and some new analysis techniques dealing with mappings on ordered sets must be developed.

In [

On the other hand, in [

References [

The order-increasing property of mappings is important for the considered mappings to have a fixed point. In this section, we recall the notations of the order-increasing property of mappings, which are used in [

Let

Let

Similar to the well-known Kakutani contribution that extended the Brouwer fixed point theorem from single valued mappings to set-valued mappings in topological spaces, the main results of [

Let

F is order-increasing upward;

the set

there is a y in P with

Let

The generalized saddle points of bifunctions on Banach lattices were studied in [

Let

The proof of the following lemma is straight forward and is omitted.

Let

Let

Let

order-negative with respect to

order-positive with respect to

Let

for every fixed

For every fixed

There is an element

From Lemma

Next we show that

That is,

We claim that an element

From condition

Let

Let

Let

for every fixed

for every fixed

there is an element

Define

From the proof of Theorem

Let

We prove the following theorem for the existence of generalized equilibrium. The proof is similar to the proofs of Theorems

Let

for every fixed

there is an element

Taking the mapping

Next we show that

Let

It is clear to see that any generalized equilibrium of a mapping is an extended equilibrium of this mapping. For a given mapping

Let

for every fixed

there is an element

Taking the mapping

Next we show that

Let

Let

Let

Let

for every fixed

there is an element

Define a mapping

Let

Let

Let

for every fixed

there is an element

Define a mapping

In Sections

It is clearly seen that, from different fixed point theorems, one can obtain various results for solving some optimization problems. Note that all extensions of the Abian-Brown fixed point theorem listed in Section

For any positive integer

So if we are able to get some fixed point theorem on conditionally chain-complete posets, then we can study some optimization problems under more general underlying spaces: conditionally chain-complete posets, which include all problems studied in Sections

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

The authors greatly appreciate the anonymous referees’ comments and suggestions, which improved the presentation of this paper.