Applications of order-theoretic fixed point theorems to discontinuous quasi-equilibrium problems

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R E S E A R C H

Open Access

Applications of order-theoretic ﬁxed point

theorems to discontinuous quasi-equilibrium

problems

Congjun Zhang

1

_{and Yuehu Wang}

2*

*_{Correspondence:} [email protected]

2_{School of Mathematical Sciences,} Anhui University, Hefei, 230601, P.R. China

Full list of author information is available at the end of the article

Abstract

In this paper, we apply order-theoretic ﬁxed point theorems and isotone selection theorems to study quasi-equilibrium problems. Some existence theorems of solutions to quasi-equilibrium problems are obtained on Hilbert lattices, chain-complete lattices and chain-complete posets, respectively. In contrast to many papers on equilibrium problems, our approach is order-theoretic and all results obtained in this paper do not involve any topological continuity with respect to the considered mappings.

Keywords: order-theoretic ﬁxed points; quasi-equilibrium problems; Hilbert lattices; posets; discontinuous; existence

1 Introduction

LetXbe a given set andCbe a subset ofX. LetT:C→C_{\ {∅}}_{be a set-valued mapping.}

Letf:C×C→Rbe a given function. The quasi-equilibrium problem (shortly QEP) is to ﬁndx∗∈T(x∗) such that

fx∗,y≥ for ally∈Tx∗. (.)

This problem was introduced by Noor and Oettli in []. In particular, ifT(x) =Cfor anyx∈ C, then the quasi-equilibrium problem is reduced to the following classical equilibrium problem (shortly EP), which is to ﬁndx∗∈Csuch that

fx∗,y≥ for ally∈C. (.)

EP was initially introduced by Blum and Oettli []. It is well known that EP include varia-tional inequality problems, ﬁxed point problems, complementarity problems, saddle point problems and Nash equilibrium problems as special cases (see,e.g., [–]).

If there is at least one solution to QEP (EP), then we say QEP (EP) is solvable. For studying the existence of solutions to QEP and EP, various methods have been developed, for instance, KKM theorem, F-KKM theorem, Ekeland’s variational principles, topolog-ical ﬁxed point theorems, auxiliary principle and many others (see,e.g., [–]). Among these methods, a variety of continuity or Brezis-type monotonicity conditions off are

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commonly necessary. For instance, Cubiotti [] studies the lower semicontinuous quasi-equilibrium problems in topological vector spaces. Al-Homidan and Ansari [] consider the systems of generalized vector quasi-equilibrium problems on topological semilattice spaces and establish some existence results for solutions of systems of generalized vector quasi-equilibrium problems and their special cases, where they require the considered mappings to be upper semicontinuous or lower semicontinuous. However, this approach may fail when the topological continuity of f is unknown. On the other hand, we note that Fujimoto [], Chitra and Subrahmanyam [] and Borwein and Dempster [] have adopted an order-theoretic approach for studying the nonlinear complementarity prob-lems, where the mappings only need to satisfy some order-preserving properties. Along this line, Nishimura and Ok [] have extended these results to the case of (generalized) variational inequalities on Hilbert lattices. Very recently, Li and Ok [] used the varia-tional characterization and order-preservation properties of a generalized metric projec-tion operator to study the (generalized) variaprojec-tional inequalities on Banach lattices.

Motivated and inspired by the work of Li and Ok [], Cubiotti [], Al-Homidan and Ansari []et al., in this paper we aim to study the quasi-equilibrium problems by using order-theoretic methods, where we do not requiref to be continuous and semicontinu-ous. Furthermore, it is worthy to mention that some of order-theoretic methods and tech-niques developed in variational inequality problems are not suitable for studying quasi-equilibrium problems. Actually, the existence results for variational inequalities are based on the fact that the solutions to variational inequalities coincide with the ﬁxed points of the self-correspondenceπC◦(idX–), where idX is the identity mapping onXand

is the involved mapping andπC is the metric projection operator ontoC. To guarantee

thatπC◦(idX–) has ﬁxed points, we always need the metric projection operator to be

order-preserving. That is, the (generalized) metric projection operators play crucial roles in dealing with the variational inequalities. Unfortunately, there is no (generalized) met-ric projection operator in quasi-equilibrium problems. Therefore, it is necessary to pro-vide some other techniques to circumvent this diﬃculty. In this paper, we use the order-theoretic ﬁxed point theorem and isotone selection theorems to obtain some existence theorems for discontinuous quasi-equilibrium problems.

The rest of the present paper can be summarized as follows. Section  is devoted to some basic concepts on posets as well as the order-preserving properties of correspon-dences. In Section , based on the order-theoretic fixed point theorem of Nishimura and Ok, we study the existence of solutions for quasi-equilibrium problems and equilibrium problems on Hilbert lattices. In Section , we apply the isotone selection theorems and the order-theoretic fixed point theorem introduced by Tarski to establish some existence results for quasi-equilibrium problems and equilibrium problems on chain-complete lat-tices, which are equipped with neither an algebraic structure nor a topological structure. Furthermore, we also apply the order-theoretic fixed point theorems introduced by Li to establish some existence results for quasi-equilibrium problems and equilibrium problems on chain-complete posets.

2 Preliminaries

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2.1 Some concepts in poset

A poset is an ordered pair (X,), whereXis a nonempty set anddenotes the partial order deﬁned onX. For eachx∈(X,), we deﬁnex↑={y∈(X,) :yx}andx↓={y∈

(X,) :xy}. In turn, for any nonempty subsetS, we deﬁneS↑={x↑:x∈S}andS↓=

{x↓:x∈S}. We say that an element xof (X,) is an-upper bound forS if xS, that is,xyfor eachy∈S. The notationSxis similarly understood. We say thatSis -bounded from aboveifxSfor somex∈(X,) and-bounded from belowifSx

for somex∈(X,). In turn,Sis said to be-boundedif it is-bounded from above and below. Particularly, ifx∈Sandxis an-upper bound forS, then we say thatxis the -maximuminS. The-minimumelement ofSis similarly deﬁned. We say thatxis the -maximalelement ofSifx∈Sandyxdoes not hold for anyy∈S\ {x}. Similarly,xis said to be the-minimalelement ofSifx∈Sandxydoes not hold for anyy∈S\ {x}. A nonempty subsetSofXis said to be a-chaininXif eitherxyoryxhold for each

x,y∈S.

The-supremumofSis the-minimum of the set of all-upper bounds forS, and is denoted by_XS. The-inﬁmumofSwhich is denoted by_XSis deﬁned similarly. As is conventional, we denote_X{x,y}asx∨yand_X{x,y}asx∧yfor anyx,yin (X,). If

x∨yandx∧yexist for everyxandyin (X,), then we say that (X,) is alattice, and if

XSand

XSexist for every nonempty-boundedS⊆(X,), then we say that (X,) is

aDedekind complete lattice. IfYis a nonempty subset of (X,) which contains_X{x,y}

and_X{x,y}for every x,y∈Y, then it is said to be a-sublatticeof (X,). In turn, if

Y contains_XS and_XS for every nonemptyS⊆Y, thenY is said to be acomplete

-sublatticeof (X,).

LetAbe a nonempty subset ofX,Ais said to beinductiveif every chain inAhas an upper bound inA. Moreover,Ais said to bechain-completeif every chainCinApossesses its supremum inA.

2.2 Order-preservation for correspondences

For any lattices (X,X) and (Y,Y), we say that a mapF:X→Y isorder-preserving if

xXyimpliesF(x)YF(y) for everyx,y∈X. In turn, if:X→Y is a set-valued

map-ping, we say thatisupper order-preservingifxXyimplies that(y) =∅, or for every

y∈(y) there isx∈(x) such thatxY y. Upper order-reversing maps are deﬁned

dually. Similarly,islower order-preservingifxXyimplies that(x) =∅, or for every

x∈(x) there isy∈(y) such thatxYy.isorder-preservingif it is both upper and

lower order-preserving. If (X,X) and (Y,Y) are subposets of a given poset (Z,), then

we use the phrase-preservinginstead of order-preserving.is said to beupper-bound

if there existsy∗∈Ysuch that_Y(x) exists and

y∗

Y

(x). (.)

is said to haveupper bound-closed valuesifx∈Ximplies(x) =∅or

Y

(x)∈(x). (.)

is said to belower-boundif there existsy∗∈Y such thatY(x) exists and

Y

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is said to havelower bound-closed valuesifx∈Ximplies(x) =∅or

Y

(x)∈(x). (.)

Deﬁnition . Let:X→Y_{\ {∅}}_{be a set-valued mapping. A selection for}_{is a}

single-valued functionF:X→Ysuch thatF(x)∈(x) for eachx∈X. An isotone selection is a selection which is order-preserving.

Lemma .(see []) Let(X,)be a poset,and let:X→X_{\ {∅}}_{be a set-valued}

map-ping. If is upper-preserving and has upper bound-closed values,then the single-valued mapping F:X→X deﬁned by F(x) =_X(x)is an isotone selection for. Lemma .(see []) Let(X,)be a poset,and let:X→X_{\ {∅}}_{be a set-valued}

map-ping.Ifis lower-preserving and has lower bound-closed values,then the single-valued mapping F:X→X deﬁned by F(x) =_X(x)is an isotone selection for.

2.3 Hilbert lattice and several notations

We say that (X,) is a Hilbert lattice ifXis a Hilbert space with the inner product·,· and with the induced norm · andXis also a poset with the partial ordersatisfying the following conditions:

(i) (X,)is a lattice.

(ii) The mappingαidX+zis a-preserving self-mapping onXfor everyz∈Xand

positive numberα, whereidXdenotes the identical mapping onX.

(iii) The norm · onXis compatible with the partial order, that is,|x||y|implies x ≥ y, where|z|= (z∨) + (–z∨)for everyz∈Xanddenotes the origin ofX.

At the end of this section, we introduce a notation, which will be used frequently in the following sections. LetXbe a given set and letCbe a subset ofX. LetT:C→C\ {∅}be a set-valued mapping. Throughout the paper, the set{x∈C:x∈T(x)}is always denoted byE.

3 Solvability of quasi-equilibrium problems on Hilbert lattices

For studying generalized variational inequalities, Nishimura and Ok introduced the fol-lowing order-theoretic ﬁxed point theorem on Hilbert lattices.

Lemma .(see []) Let(X,)be a separable Hilbert lattice and C be a weakly com-pact and convex-sublattice of X.Then every upper-preserving and compact-valued correspondence F:C→C_{\ {∅}}_{has a ﬁxed point}_.

Based on Lemma ., we can get an existence theorem for quasi-equilibrium problems as follows.

Theorem . Let(X,)be a separable Hilbert lattice,and let C be a weakly compact convex-sublattice of X.Let f :C×C→R be a bifunction and T :C→C_{\ {∅}}_{be a}

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(ii) Tis upper-preserving and compact-valued.xC\Efor anyx∈E. (iii) The set-valued mapping:C→C_{deﬁned by setting}

(x) = y∈T(x) :f(x,y) < 

is upper-preserving and compact-valued.

Then QEP(.)is solvable.

Proof We claim that there existsx∗∈E such that(x∗) =∅. Arguing by contradiction, assume(x)=∅for allx∈E, then we can deﬁne a set-valued mapping:C→C_{\ {∅}}

by

(x) =

⎧ ⎨ ⎩

(x) ifx∈E,

T(x) ifx∈C\E.

Next, we divide the rest of the proof into two steps.

Step. Show thatis upper-preserving.

Take anyx,x∈Cwithxxand pick an arbitraryy∈(x). We wish to ﬁndy∈

(x) such thatyy. To this end, we consider the following three cases.

Case I. Ifx,x∈E, then the upper-preservation ofis equivalent to the upper -preservation of. Thus, we only need to prove thatis upper

-preserving. From assumption (iii), it is obvious.

Case II. Ifx,x∈C\E, then the upper-preservation ofis reduced to the upper

-preservation ofT. It follows immediately from assumption (ii). Case III. Ifx∈C\Eandx∈E, theny∈(x)is reduced toy∈(x), which

implies thaty∈T(x). Again, sinceTis upper-preserving, there exists

y∈T(x) =(x)such thatyy.

Furthermore, sincexC\E for anyx∈E,xx does not hold for anyx∈Eand x ∈C\E. Above all, from Case I, Case II and Case III, we conclude that is upper -preserving onC.

Step . Prove thathas a ﬁxed point.

SinceCis a weakly compact and convex-sublattice ofXandis upper-preserving and compact-valued, has a ﬁxed point by Lemma .. Denote this ﬁxed point byx¯. Noting that{x∈C:x∈(x)} ⊆E, we getx¯∈E∩(¯x), and hence we havef(¯x,x¯) < , which contradicts with (i). Therefore, there exists x∗∈E such that(x∗) =∅. That is,

x∗∈T(x∗) andf(x∗,y)≥ for ally∈T(x∗).

Example . It is easy to construct examples of a set-valued mappingT that satisﬁes the assumption (ii) of Theorem .. For instance, take X= (R,≥) andC= [, ]. Deﬁne

T: [, ]→[,]_{by setting}

T(x) =

⎧ ⎨ ⎩

[, ] ifx∈[, ],

{} ifx∈(, ]. (.)

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In Theorem ., the new mappingrelated to the considered mappingf is involved in assumption (iii). It is a kind of burdensome for the applications of Theorem .. Hence, to whittle down the nuisance caused by , it is desirable to ﬁnd some diﬀerent condi-tions only onf such thatis still upper-preserving. Therefore, the following result is obtained.

Theorem . Let(X,)be a separable Hilbert lattice, and let C be a weakly compact convex-sublattice of X.Let f :C×C→R be a bifunction and T :C→C_{\ {∅}}_{be a}

set-valued mapping.Assume that the following conditions hold: (i) f(x,x)≥for any(x,x)∈C×C.

(ii) Tis upper-preserving and compact-valued.xC\Efor anyx∈E.

(iii) f(·,y)is order-reversing for anyy∈Candf(x,·)is order-reversing for anyx∈C. Moreover,the set{y∈C:f(x,y) < }is closed for anyx∈C.

Proof Deﬁne a set-valued mapping:C→Cby setting (x) = y∈T(x) :f(x,y) < .

For applying Theorem ., we only need to prove that is upper -preserving and compact-valued. In fact, take anyx,x∈Cwithxxand pick an arbitraryy∈(x). We wish to ﬁndy∈(x) such thatyy.

Sincey∈(x), we have

y∈T(x) and f(x,y) < . (.)

SinceTis upper-preserving by assumption (ii), there existsy∈T(x) such thatyy. Noting thatxxand thatf(·,y) is order-reversing by assumption (iii), we have

f(x,y)≥f(x,y). (.)

Again, sinceyyandf(x,·) is order-reversing by assumption (iii), we get

f(x,y)≥f(x,y). (.)

Combining (.), (.) and (.), we obtain

f(x,y) < . (.)

Hence,is upper-preserving onC.

On the other hand, since{y∈C:f(x,y) < }is closed for anyx∈Cby assumption (iii) andT(x) is a compact subset ofCfor anyx∈Cby assumption (ii), we conclude that is compact-valued. Therefore,satisﬁes the assumption (iii) of Theorem ., and then

quasi-equilibrium problem (.) has a solution.

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Corollary . Let(X,)be a separable Hilbert lattice,and let C be a compact convex

-sublattice of X.Let f :C×C→R be a bifunction.Assume that the following conditions hold:

(i) f(x,x)≥for any(x,x)∈C×C.

(ii) The set-valued mapping:C→Cdeﬁned by setting (x) = y∈C:f(x,y) < 

is upper-preserving and compact-valued.

Then EP(.)is solvable.

Corollary . Let(X,)be a separable Hilbert lattice,and let C be a compact convex

(ii) f(·,y)is order-reversing for anyy∈Candf(x,·)is order-reversing for anyx∈C. Moreover,the set{y∈C:f(x,y) < }is compact for anyx∈C.

In fact, the assumption (ii) of Corollary . can be weakened as follows.

Corollary . Let(X,)be a separable Hilbert lattice,and let C be a compact convex

(ii) f(·,y)is order-reversing for anyy∈C,and the set{y∈C:f(x,y) < }is compact for any

x∈C.

Proof Deﬁne a set-valued mapping:C→C_{by setting}

(x) = y∈C:f(x,y) < .

For applying Theorem ., we only need to prove thatis upper-preserving. In fact, take anyx,x∈Cwithxxand pick an arbitraryy∈(x). Sincey∈(x), we have

y∈Candf(x,y) < . Again, sincef(·,y) is order-reversing, we getf(x,y)≥f(x,y).

Choosey=y, theny∈Candf(x,y) < , which implies thaty∈(x). Hence,is upper-preserving. By Theorem ., EP (.) is solvable.

Remark . Actually, based on the dual version of Zorn’s lemma and Lemma ., a ﬁxed point theorem for lower-preserving correspondence can be obtained. Applying this new ﬁxed point theorem, we can explore some existence theorems for quasi-equilibrium prob-lems and equilibrium probprob-lems under the condition of lower-preservation.

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for example. Let (X,) = (R,≥) andC= [, ]⊆R. Denote byDthe set{(x,y)∈[, ]× [, ] :x–y≥}. Deﬁne a mappingf : [, ]×[, ]→Rby

z=f(x,y) =

⎧ ⎨ ⎩  y–

 x+



, (x,y)∈[, ]×[, ]\D, y–x, (x,y)∈D,

and deﬁne a set-valued mappingT:C→C_{\ {∅}}_by

T(x) =

⎧ ⎨ ⎩

[,_], x∈[,_], {

}, x∈(  , ].

We can check thatf andTsatisfy all the conditions (including assumption (iii)) in Theo-rem ., butf is discontinuous. Furthermore, if we takexˆ=_, then we have

f

 ,y

≥ for ally∈T

 

,

which impliesxˆ=_ is a solution to a quasi-equilibrium problem. Similarly, examples for Theorem . can also be given.

4 Solvability of quasi-equilibrium problems on chain-complete lattices and chain-complete posets

In this section, we explore several existence theorems on chain-complete lattices and chain-complete posets, on which there is neither a topological structure nor an algebraic structure.

Firstly, let us recall the following order-theoretic ﬁxed point theorem, which was intro-duced by Tarski [] in .

Lemma . (see []) Let(X,)be a chain-complete lattice, and let F:X→X be an order-preserving single-valued mapping.If there isxˆ∈X with F(xˆ)xˆ,then F has a ﬁxed point.

Theorem . Let(X,)be a poset and let C be a chain-complete-sublattice of X.Let f :C×C→R be a bifunction and T:C→C_{\ {∅}}_{be a set-valued mapping}_._{Assume that}

the following conditions hold:

(ii) Tis upper-preserving and has upper bound-closed values.xC\Efor any

x∈E.

(iii) The set-valued mapping:C→Cdeﬁned by setting

(x) = y∈T(x) :f(x,y) < 

is upper-preserving and has upper bound-closed values.

(iv) There isxˆ∈C\Esuch that_CT(xˆ)xˆ,or there isxˆ∈Esuch that_C(xˆ)xˆ.

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Proof We claim that there existsx∗∈E such that(x∗) =∅. Arguing by contradiction, assume(x)=∅for allx∈E. By the same argument as that in Theorem ., we can deﬁne the set-valued mapping:C→C_{\ {∅}}_{and prove that}_{is upper}_{-preserving. From}

assumption (ii) and assumption (iii), it follows that is upper bound-closed. From Lemma ., there is an isotone selectionψfor. From assumption (iv), there existsxˆ∈C

such thatψ(ˆx)xˆ. SinceCis a chain-complete-sublattice ofX, therefore, by Lemma ., there existsx¯inCsuch thatx¯=ψ(x¯)∈(x¯). Since{x∈C:x∈(x)} ⊆E, we getx¯∈E∩

(x¯). In particular, we havef(x¯,x¯) < , which contradicts with assumption (i). Therefore, there existsx∗∈E such that(x∗) =∅. That is,x∗∈T(x∗) andf(x∗,y)≥ for all y∈

T(x∗).

Replacing the assumption (iii) of Theorem . by some conditions onf, we can obtain the following results.

Theorem . Let(X,)be a poset and let C be a chain-complete-sublattice of X.Let f :C×C→R be a bifunction and T:C→C_{\ {∅}}_{be a set-valued mapping}_._{Assume that}

the following conditions hold:

(ii) Tis upper-preserving and has upper bound-closed values.xC\Efor anyx∈E. (iii) f(·,y)is order-reversing for anyy∈Candf(x,·)is order-reversing for anyx∈C.

{y∈T(x) :f(x,y) < }is a complete-sublattice ofCfor anyx∈C.

(iv) There isxˆ∈C\Esuch that_CT(ˆx)xˆ,or there isxˆ∈Esuch that_C(ˆx)xˆ.

Proof Deﬁne a set-valued mapping:C→C_{by setting}

(x) = y∈T(x) :f(x,y) < .

We only need to prove thatsatisﬁes the assumption (iii) of Theorem .. By the same ar-gument as that in Theorem .,is upper-preserving. On the other hand, from the def-inition of, we have(x)⊆T(x) for eachx∈C. Since{y∈T(x) :f(x,y) < }is a complete -sublattice ofCfor anyx∈C, we have_C{y∈T(x) :f(x,y) < } ∈ {y∈T(x) :f(x,y) < }, that is,_C(x)∈(x), which implies thathas upper bound-closed values. There-fore,satisﬁes the assumption (iii) of Theorem .. From Theorem ., there is a solution

for quasi-equilibrium problem (.).

If_CC∈CandT(x) =Cfor eachx∈C, then we can deduce the following existence theorems for equilibrium problem (.) from Theorem . and Theorem ..

Corollary . Let(X,)be a poset and let C be a chain-complete-sublattice of X such that_CC∈C.Let f :C×C→R be a bifunction.Assume that the following conditions hold:

(ii) The set-valued mapping:C→C_{deﬁned by setting}

(x) = y∈C:f(x,y) < 

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(iii) There isxˆ∈Esuch that_C(ˆx)xˆ.

Corollary . Let(X,)be a poset and let C be a chain-complete-sublattice of X such that_CC∈C.Let f :C×C→R be a bifunction.Assume that the following conditions hold:

(ii) f(·,y)is order-reversing for anyy∈Candf(x,·)is order-reversing for anyx∈C. {y∈C:f(x,y) < }is a complete-sublattice ofCfor anyx∈C.

(iii) There isxˆ∈Csuch that_C{y∈C:f(ˆx,y) < }xˆ.

Remark . In Theorem ., Theorem ., Corollary . and Corollary . above, we al-ways assume thatTis upper-preserving andf(·,y) is order-reversing for eachy∈Cand

f(x,·) is order-reversing for eachx∈C. In fact, all of these conditions are used to guaran-tee thatis upper-preserving, so that we can obtain an isotone selection by Lemma .. However, it is worthy to mention that these conditions related to upper-preservation are suﬃcient. That is, for getting an isotone selection, the set-valued mapping need not be up-per order-preserving or lower-preserving. To see this, takeX= (R,≥) andC= [, ]. Deﬁne a set-valued mapping: [, ]→[,]_{by setting}

(x) = y∈R:sin(x) + ≥y≥sin(x) + . (.)

It is easy to check thatis neither upper≥-preserving nor lower≥-preserving; however, we can ﬁnd an isotone selectionFfor, whereFis deﬁned by setting

F(x) = 

x+ . (.)

Remark . We can also consider the case whenT is lower-preserving andf(x,·) is order-preserving for eachx∈Candf(·,y) is order-preserving for eachy∈C. Applying Lemma ., we can explore some existence theorems similar to Theorem .etc.

Now we consider the quasi-equilibrium problems on chain-complete posets. To this end, we need some order-theoretic ﬁxed point theorems on chain-complete posets. The following result is usually called Abian-Brown ﬁxed point theorem, which extends Lemma . from chain-complete lattices to chain-complete posets.

Lemma .(see []) Let(X,)be a chain-complete poset,and let F:X→X be an order-preserving single-valued mapping.If there isx¯∈X with F(x¯)x¯,then F has a ﬁxed point.

By using Lemma . and the methodology given in Theorem ., we can prove the fol-lowing result, which extends Theorem . from chain-complete lattices to chain-complete posets.

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(ii) Tis upper-preserving and has upper bound-closed values.xC\Efor any

x∈E.

(iii) f(·,y)is order-reversing for anyy∈Candf(x,·)is order-reversing for anyx∈C. {y∈T(x) :f(x,y) < }is a complete-sublattice ofCfor anyx∈C.

(iv) There isxˆ∈C\Esuch that_CT(ˆx)xˆ,or there isxˆ∈Esuch that

C{y∈T(ˆx) :f(ˆx,y) < }xˆ.

Remark . In a similar way, Theorem ., Corollary . and Corollary . can be ex-tended to chain-complete posets by using Lemma ..

Very recently, Li [] proved several extensions of Lemma . from single-valued map-pings to set-valued mapmap-pings on chain-complete posets.

Lemma . (see []) Let(X,)be a chain-complete poset and F :X→X_{\ {∅}}_{be a}

set-valued mapping.Assume that:

A. Fis upper order-preserving.

A. There isyinXwithuyfor someu∈F(y).

In addition,one of the following assumptions holds:

A. SF={z∈X:uzfor someu∈F(x)}is an inductive poset for eachx∈X. A. (F(x),)is inductive with a ﬁnite number of maximal elements for everyx∈X. A. F(x)has a maximum element for everyx∈X.

A. F(x)is a chain-complete lattice for eachx∈X.

Then F has a ﬁxed point.

By using conditions A, A and A in Lemma . and the methodology given in Theo-rem ., we can obtain an existence theoTheo-rem of solutions to quasi-equilibrium problems on chain-complete posets, where the isotone selection is dropped.

Theorem . Let(X,)be a poset and let C be a chain-complete subset of X.Let f:C× C→R be a bifunction and T:C→C_{\ {∅}}_{be a set-valued mapping}_._{Assume that the}

following conditions hold:

(ii) Tis upper-preserving andT(x)has a maximum element for everyx∈C.

xC\Efor anyx∈E.

(iii) f(·,y)is order-reversing for anyy∈Candf(x,·)is order-reversing for anyx∈C,and

f(x,_CT(x)) < for eachx∈E.

(iv) There isy∈C\Esuch thatyufor someu∈T(y),or there isy∈Esuch thatyu

for someu∈T(y)andf(y,u) < .

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Theorem . Let(X,)be a poset and let C be a chain-complete subset of X.Let f:C× C→R be a bifunction and T:C→C_{\ {∅}}_{be a set-valued mapping}_._{Assume that the}

following conditions hold:

(ii) Tis upper-preserving andT(x)is a chain-complete sublattice ofCfor every

x∈C.xC\Efor anyx∈E.

(iii) f(·,y)is order-reversing for anyy∈Candf(x,·)is order-reversing for anyx∈C. (iv) There isy∈C\Esuch thatyufor someu∈T(y),or there isy∈Esuch thatyu

for someu∈T(y)andf(y,u) < .

Remark . Based on Theorem . and Theorem ., it is easy to deduce some corollaries for EP (.).

Remark . In the same manner, we can also consider the case when T is lower -preserving andf(x,·) is order-preserving for eachx∈Candf(·,y) is order-preserving for eachy∈C.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210023, P.R. China.}2_{School of} Mathematical Sciences, Anhui University, Hefei, 230601, P.R. China.

Acknowledgements

The authors sincerely thank the anonymous reviewers and editor for their valuable suggestions, which improved the presentation of this paper. The ﬁrst author was supported ﬁnancially by the National Natural Science Foundation of China (11071109). This work was also supported by the National Natural Science Foundation of China (11401296) and Jiangsu Provincial Natural Science Foundation of China (BK20141008).

Received: 28 December 2014 Accepted: 2 April 2015 References

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