Applications of ordertheoretic fixed point theorems to discontinuous quasiequilibrium problems
 Congjun Zhang^{1} and
 Yuehu Wang^{2}Email author
https://doi.org/10.1186/s1366301503065
© Zhang and Wang; licensee Springer. 2015
Received: 28 December 2014
Accepted: 2 April 2015
Published: 22 April 2015
Abstract
In this paper, we apply ordertheoretic fixed point theorems and isotone selection theorems to study quasiequilibrium problems. Some existence theorems of solutions to quasiequilibrium problems are obtained on Hilbert lattices, chaincomplete lattices and chaincomplete posets, respectively. In contrast to many papers on equilibrium problems, our approach is ordertheoretic and all results obtained in this paper do not involve any topological continuity with respect to the considered mappings.
Keywords
1 Introduction
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, FKKM theorem, Ekeland’s variational principles, topological fixed point theorems, auxiliary principle and many others (see, e.g., [5–18]). Among these methods, a variety of continuity or Brezistype monotonicity conditions of f are commonly necessary. For instance, Cubiotti [10] studies the lower semicontinuous quasiequilibrium problems in topological vector spaces. AlHomidan and Ansari [15] consider the systems of generalized vector quasiequilibrium problems on topological semilattice spaces and establish some existence results for solutions of systems of generalized vector quasiequilibrium 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 [19], Chitra and Subrahmanyam [20] and Borwein and Dempster [21] have adopted an ordertheoretic approach for studying the nonlinear complementarity problems, where the mappings only need to satisfy some orderpreserving properties. Along this line, Nishimura and Ok [22] have extended these results to the case of (generalized) variational inequalities on Hilbert lattices. Very recently, Li and Ok [23] used the variational characterization and orderpreservation properties of a generalized metric projection operator to study the (generalized) variational inequalities on Banach lattices.
Motivated and inspired by the work of Li and Ok [23], Cubiotti [10], AlHomidan and Ansari [15] et al., in this paper we aim to study the quasiequilibrium problems by using ordertheoretic methods, where we do not require f to be continuous and semicontinuous. Furthermore, it is worthy to mention that some of ordertheoretic methods and techniques developed in variational inequality problems are not suitable for studying quasiequilibrium 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 selfcorrespondence \(\pi_{C}\circ (\mathrm{id}_{X}\Gamma)\), where \(\mathrm{id}_{X}\) is the identity mapping on X and Γ is the involved mapping and \(\pi_{C}\) is the metric projection operator onto C. To guarantee that \(\pi_{C}\circ (\mathrm{id}_{X}\Gamma)\) has fixed points, we always need the metric projection operator to be orderpreserving. That is, the (generalized) metric projection operators play crucial roles in dealing with the variational inequalities. Unfortunately, there is no (generalized) metric projection operator in quasiequilibrium problems. Therefore, it is necessary to provide some other techniques to circumvent this difficulty. In this paper, we use the ordertheoretic fixed point theorem and isotone selection theorems to obtain some existence theorems for discontinuous quasiequilibrium problems.
The rest of the present paper can be summarized as follows. Section 2 is devoted to some basic concepts on posets as well as the orderpreserving properties of correspondences. In Section 3, based on the ordertheoretic fixed point theorem of Nishimura and Ok, we study the existence of solutions for quasiequilibrium problems and equilibrium problems on Hilbert lattices. In Section 4, we apply the isotone selection theorems and the ordertheoretic fixed point theorem introduced by Tarski to establish some existence results for quasiequilibrium problems and equilibrium problems on chaincomplete lattices, which are equipped with neither an algebraic structure nor a topological structure. Furthermore, we also apply the ordertheoretic fixed point theorems introduced by Li to establish some existence results for quasiequilibrium problems and equilibrium problems on chaincomplete posets.
2 Preliminaries
In this section, we recall some basic concepts about Hilbert lattices as well as several useful lemmas. For more details, the readers are referred to [24–26].
2.1 Some concepts in poset
A poset is an ordered pair \((X,\succcurlyeq)\), where X is a nonempty set and ≽ denotes the partial order defined on X. For each \(x\in(X,\succcurlyeq)\), we define \(x^{\uparrow}=\{y\in (X,\succcurlyeq):y\succcurlyeq x \}\) and \(x_{\downarrow}=\{y\in (X,\succcurlyeq): x\succcurlyeq y \}\). In turn, for any nonempty subset S, we define \(S^{\uparrow}=\bigcup\{x^{\uparrow}:x\in S\}\) and \(S^{\downarrow}=\bigcup\{x^{\downarrow}:x\in S\}\). We say that an element x of \((X,\succcurlyeq)\) is an ≽upper bound for S if \(x\succcurlyeq S\), that is, \(x\succcurlyeq y\) for each \(y\in S\). The notation \(S\succcurlyeq x\) is similarly understood. We say that S is ≽bounded from above if \(x\succcurlyeq S\) for some \(x\in(X,\succcurlyeq)\) and ≽bounded from below if \(S\succcurlyeq x\) for some \(x\in(X,\succcurlyeq)\). In turn, S is said to be ≽bounded if it is ≽bounded from above and below. Particularly, if \(x\in S\) and x is an ≽upper bound for S, then we say that x is the ≽maximum in S. The ≽minimum element of S is similarly defined. We say that x is the ≽maximal element of S if \(x\in S\) and \(y\succcurlyeq x\) does not hold for any \(y\in S\setminus\{x\}\). Similarly, x is said to be the ≽minimal element of S if \(x\in S\) and \(x\succcurlyeq y\) does not hold for any \(y\in S\setminus\{x\}\). A nonempty subset S of X is said to be a ≽chain in X if either \(x\succcurlyeq y\) or \(y\succcurlyeq x\) hold for each \(x,y\in S\).
The ≽supremum of S is the ≽minimum of the set of all ≽upper bounds for S, and is denoted by \(\bigvee_{X}S\). The ≽infimum of S which is denoted by \(\bigwedge_{X}S\) is defined similarly. As is conventional, we denote \(\bigvee_{X}\{x,y\}\) as \(x\vee y\) and \(\bigwedge_{X}\{x,y\}\) as \(x\wedge y\) for any x, y in \((X,\succcurlyeq)\). If \(x\vee y\) and \(x\wedge y\) exist for every x and y in \((X,\succcurlyeq)\), then we say that \((X,\succcurlyeq)\) is a lattice, and if \(\bigvee_{X}S\) and \(\bigwedge_{X}S\) exist for every nonempty ≽bounded \(S\subseteq(X,\succcurlyeq)\), then we say that \((X,\succcurlyeq)\) is a Dedekind complete lattice. If Y is a nonempty subset of \((X,\succcurlyeq)\) which contains \(\bigvee_{X}\{x,y\}\) and \(\bigwedge_{X}\{x,y\}\) for every \(x,y\in Y\), then it is said to be a ≽sublattice of \((X,\succcurlyeq)\). In turn, if Y contains \(\bigvee_{X}S\) and \(\bigwedge_{X}S\) for every nonempty \(S\subseteq Y\), then Y is said to be a complete ≽sublattice of \((X,\succcurlyeq)\).
Let A be a nonempty subset of X, A is said to be inductive if every chain in A has an upper bound in A. Moreover, A is said to be chaincomplete if every chain C in A possesses its supremum in A.
2.2 Orderpreservation for correspondences
Definition 2.1
Let \(\Gamma: X\rightarrow2^{Y}\setminus\{ \emptyset\}\) be a setvalued mapping. A selection for Γ is a singlevalued function \(F:X\rightarrow Y\) such that \(F(x)\in\Gamma(x)\) for each \(x\in X\). An isotone selection is a selection which is orderpreserving.
Lemma 2.1
(see [27])
Let \((X,\succcurlyeq)\) be a poset, and let \(\Gamma:X\rightarrow2^{X}\setminus\{\emptyset\}\) be a setvalued mapping. If Γ is upper ≽preserving and has upper bound ≽closed values, then the singlevalued mapping \(F:X\rightarrow X\) defined by \(F(x)=\bigvee_{X}\Gamma(x)\) is an isotone selection for Γ.
Lemma 2.2
(see [27])
Let \((X,\succcurlyeq)\) be a poset, and let \(\Gamma:X\rightarrow2^{X}\setminus\{\emptyset\}\) be a setvalued mapping. If Γ is lower ≽preserving and has lower bound ≽closed values, then the singlevalued mapping \(F:X\rightarrow X\) defined by \(F(x)=\bigwedge_{X}\Gamma(x)\) is an isotone selection for Γ.
2.3 Hilbert lattice and several notations
 (i)
\((X,\succcurlyeq)\) is a lattice.
 (ii)
The mapping \(\alpha\mathrm{id}_{X}+z\) is a ≽preserving selfmapping on X for every \(z\in X\) and positive number α, where \(\mathrm{id}_{X}\) denotes the identical mapping on X.
 (iii)
The norm \(\\cdot\\) on X is compatible with the partial order ≽, that is, \(x\succcurlyeqy\) implies \(\x\\geqslant\ y\\), where \(z=(z\vee\mathbf{0})+(z\vee\mathbf{0})\) for every \(z\in X\) and 0 denotes the origin of X.
At the end of this section, we introduce a notation, which will be used frequently in the following sections. Let X be a given set and let C be a subset of X. Let \(T:C\rightarrow2^{C}\setminus\{\emptyset\} \) be a setvalued mapping. Throughout the paper, the set \(\{x\in C: x\in T(x)\}\) is always denoted by E.
3 Solvability of quasiequilibrium problems on Hilbert lattices
For studying generalized variational inequalities, Nishimura and Ok introduced the following ordertheoretic fixed point theorem on Hilbert lattices.
Lemma 3.1
(see [22])
Let \((X,\succcurlyeq)\) be a separable Hilbert lattice and C be a weakly compact and convex ≽sublattice of X. Then every upper ≽preserving and compactvalued correspondence \(F:C\rightarrow 2^{C}\setminus\{\emptyset\}\) has a fixed point.
Based on Lemma 3.1, we can get an existence theorem for quasiequilibrium problems as follows.
Theorem 3.1
 (i)
\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)
T is upper ≽preserving and compactvalued. \(x \nsucc C\setminus E\) for any \(x\in E\).
 (iii)The setvalued mapping \(\Phi: C\rightarrow2^{C}\) defined by settingis upper ≽preserving and compactvalued.$$\Phi(x)=\bigl\{ y\in T(x): f(x,y)< 0 \bigr\} $$
Proof
Step 1. Show that Ψ is upper ≽preserving.
 Case I.
If \(x_{1}, x_{2}\in E\), then the upper ≽preservation of Ψ is equivalent to the upper ≽preservation of Φ. Thus, we only need to prove that Φ is upper ≽preserving. From assumption (iii), it is obvious.
 Case II.
If \(x_{1}, x_{2}\in C\setminus E\), then the upper ≽preservation of Ψ is reduced to the upper ≽preservation of T. It follows immediately from assumption (ii).
 Case III.
If \(x_{1}\in C\setminus E\) and \(x_{2}\in E\), then \(y_{2}\in \Psi(x_{2})\) is reduced to \(y_{2}\in\Phi(x_{2})\), which implies that \(y_{2}\in T(x_{2})\). Again, since T is upper ≽preserving, there exists \(y_{1}\in T(x_{1})=\Psi(x_{1})\) such that \(y_{1}\succcurlyeq y_{2}\).
Furthermore, since \(x \nsucc C\setminus E\) for any \(x\in E\), \(x_{1}\succcurlyeq x_{2}\) does not hold for any \(x_{1}\in E\) and \(x_{2}\in C\setminus E\). Above all, from Case I, Case II and Case III, we conclude that Ψ is upper ≽preserving on C.
Step 2. Prove that Ψ has a fixed point.
Since C is a weakly compact and convex ≽sublattice of X and Ψ is upper ≽preserving and compactvalued, Ψ has a fixed point by Lemma 3.1. Denote this fixed point by \(\bar{x}\). Noting that \(\{x\in C: x\in\Psi (x)\}\subseteq E\), we get \(\bar{x}\in E\cap\Phi(\bar{x})\), and hence we have \(f(\bar{x},\bar{x})<0\), which contradicts with (i). Therefore, there exists \(x^{*}\in E\) such that \(\Phi(x^{*})=\emptyset\). That is, \(x^{*}\in T(x^{*})\) and \(f(x^{*},y)\geqslant0\) for all \(y\in T(x^{*})\). □
Example 3.1
In Theorem 3.1, the new mapping Φ related to the considered mapping f is involved in assumption (iii). It is a kind of burdensome for the applications of Theorem 3.1. Hence, to whittle down the nuisance caused by Φ, it is desirable to find some different conditions only on f such that Φ is still upper ≽preserving. Therefore, the following result is obtained.
Theorem 3.2
 (i)
\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)
T is upper ≽preserving and compactvalued. \(x \nsucc C\setminus E\) for any \(x\in E\).
 (iii)
\(f(\cdot,y)\) is orderreversing for any \(y\in C\) and \(f(x,\cdot )\) is orderreversing for any \(x\in C\).
Moreover, the set \(\{y\in C: f(x,y)<0 \}\) is closed for any \(x\in C\).
Proof
On the other hand, since \(\{y\in C: f(x,y)<0 \}\) is closed for any \(x\in C\) by assumption (iii) and \(T(x)\) is a compact subset of C for any \(x\in C\) by assumption (ii), we conclude that Φ is compactvalued. Therefore, Φ satisfies the assumption (iii) of Theorem 3.1, and then quasiequilibrium problem (1.1) has a solution. □
In particular, if \(T(x)=C\) for any \(x\in C\), then we can deduce the following results from Theorem 3.1 and Theorem 3.2.
Corollary 3.1
 (i)
\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)The setvalued mapping \(\Phi: C\rightarrow2^{C}\) defined by settingis upper ≽preserving and compactvalued.$$\Phi(x)=\bigl\{ y\in C: f(x,y)< 0 \bigr\} $$
Corollary 3.2
 (i)
\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)
\(f(\cdot,y)\) is orderreversing for any \(y\in C\) and \(f(x,\cdot)\) is orderreversing for any \(x\in C\).
Moreover, the set \(\{y\in C: f(x,y)<0 \}\) is compact for any \(x\in C\).
In fact, the assumption (ii) of Corollary 3.2 can be weakened as follows.
Corollary 3.3
 (i):

\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)′:

\(f(\cdot,y)\) is orderreversing for any \(y\in C\), and the set \(\{y\in C: f(x,y)<0 \}\) is compact for any \(x\in C\).
Proof
Remark 3.1
Actually, based on the dual version of Zorn’s lemma and Lemma 3.1, a fixed point theorem for lower ≽preserving correspondence can be obtained. Applying this new fixed point theorem, we can explore some existence theorems for quasiequilibrium problems and equilibrium problems under the condition of lower ≽preservation.
Remark 3.2
4 Solvability of quasiequilibrium problems on chaincomplete lattices and chaincomplete posets
In this section, we explore several existence theorems on chaincomplete lattices and chaincomplete posets, on which there is neither a topological structure nor an algebraic structure.
Firstly, let us recall the following ordertheoretic fixed point theorem, which was introduced by Tarski [28] in 1955.
Lemma 4.1
(see [28])
Let \((X,\succcurlyeq)\) be a chaincomplete lattice, and let \(F:X\rightarrow X\) be an orderpreserving singlevalued mapping. If there is \(\hat{x}\in X\) with \(F(\hat{x})\succcurlyeq\hat {x}\), then F has a fixed point.
Theorem 4.1
 (i)
\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)
T is upper ≽preserving and has upper bound ≽closed values. \(x \nsucc C\setminus E\) for any \(x\in E\).
 (iii)The setvalued mapping \(\Phi: C\rightarrow2^{C}\) defined by settingis upper ≽preserving and has upper bound ≽closed values.$$\Phi(x)=\bigl\{ y\in T(x): f(x,y)< 0 \bigr\} $$
 (iv)
There is \(\hat{x}\in C\setminus E\) such that \(\bigvee_{C}T(\hat {x})\succcurlyeq\hat{x}\), or there is \(\hat{x}\in E\) such that \(\bigvee_{C}\Phi(\hat{x})\succcurlyeq\hat{x}\).
Proof
We claim that there exists \(x^{*}\in E\) such that \(\Phi (x^{*})=\emptyset\). Arguing by contradiction, assume \(\Phi(x)\neq \emptyset\) for all \(x\in E\). By the same argument as that in Theorem 3.1, we can define the setvalued mapping \(\Psi: C\rightarrow 2^{C}\setminus\{\emptyset\}\) and prove that Ψ is upper ≽preserving. From assumption (ii) and assumption (iii), it follows that Ψ is upper bound ≽closed. From Lemma 2.1, there is an isotone selection ψ for Ψ. From assumption (iv), there exists \(\hat{x}\in C\) such that \(\psi(\hat{x})\succcurlyeq \hat{x}\). Since C is a chaincomplete ≽sublattice of X, therefore, by Lemma 4.1, there exists \(\bar{x}\) in C such that \(\bar {x}=\psi(\bar{x})\in\Psi(\bar{x})\). Since \(\{x\in C: x\in\Psi(x) \} \subseteq E\), we get \(\bar{x}\in E\cap\Phi(\bar{x})\). In particular, we have \(f(\bar{x},\bar{x})<0\), which contradicts with assumption (i). Therefore, there exists \(x^{*}\in E\) such that \(\Psi(x^{*})=\emptyset\). That is, \(x^{*}\in T(x^{*})\) and \(f(x^{*},y)\geqslant0\) for all \(y\in T(x^{*})\). □
Replacing the assumption (iii) of Theorem 4.1 by some conditions on f, we can obtain the following results.
Theorem 4.2
 (i):

\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii):

T is upper ≽preserving and has upper bound ≽closed values. \(x \nsucc C\setminus E\) for any \(x\in E\).
 (iii)′:

\(f(\cdot,y)\) is orderreversing for any \(y\in C\) and \(f(x,\cdot)\) is orderreversing for any \(x\in C\).
\(\{y\in T(x): f(x,y)<0\}\) is a complete ≽sublattice of C for any \(x\in C\).
 (iv):

There is \(\hat{x}\in C\setminus E\) such that \(\bigvee_{C}T(\hat {x})\succcurlyeq\hat{x}\), or there is \(\hat{x}\in E\) such that \(\bigvee_{C}\Phi(\hat{x})\succcurlyeq\hat{x}\).
Proof
If \(\bigvee_{C}C\in C\) and \(T(x)=C\) for each \(x\in C\), then we can deduce the following existence theorems for equilibrium problem (1.2) from Theorem 4.1 and Theorem 4.2.
Corollary 4.1
 (i)
\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)The setvalued mapping \(\Phi: C\rightarrow2^{C}\) defined by settingis upper ≽preserving and has upper bound ≽closed values.$$\Phi(x)=\bigl\{ y\in C: f(x,y)< 0 \bigr\} $$
 (iii)
There is \(\hat{x}\in E\) such that \(\bigvee_{C}\Phi(\hat {x})\succcurlyeq\hat{x}\).
Corollary 4.2
 (i)
\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)
\(f(\cdot,y)\) is orderreversing for any \(y\in C\) and \(f(x,\cdot)\) is orderreversing for any \(x\in C\).
\(\{y\in C: f(x,y)<0\}\) is a complete ≽sublattice of C for any \(x\in C\).
 (iii)
There is \(\hat{x}\in C\) such that \(\bigvee_{C}\{y\in C: f(\hat {x},y)<0\}\succcurlyeq\hat{x}\).
Remark 4.1
Remark 4.2
We can also consider the case when T is lower ≽preserving and \(f(x,\cdot)\) is orderpreserving for each \(x\in C\) and \(f(\cdot,y)\) is orderpreserving for each \(y\in C\). Applying Lemma 2.2, we can explore some existence theorems similar to Theorem 4.1 etc.
Now we consider the quasiequilibrium problems on chaincomplete posets. To this end, we need some ordertheoretic fixed point theorems on chaincomplete posets. The following result is usually called AbianBrown fixed point theorem, which extends Lemma 4.1 from chaincomplete lattices to chaincomplete posets.
Lemma 4.2
(see [25])
Let \((X,\succcurlyeq)\) be a chaincomplete poset, and let \(F:X\rightarrow X\) be an orderpreserving singlevalued mapping. If there is \(\bar{x}\in X\) with \(F(\bar {x})\succcurlyeq\bar{x}\), then F has a fixed point.
By using Lemma 4.2 and the methodology given in Theorem 3.1, we can prove the following result, which extends Theorem 4.2 from chaincomplete lattices to chaincomplete posets.
Theorem 4.3
 (i)
\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)
T is upper ≽preserving and has upper bound ≽closed values. \(x \nsucc C\setminus E\) for any \(x\in E\).
 (iii)
\(f(\cdot,y)\) is orderreversing for any \(y\in C\) and \(f(x,\cdot )\) is orderreversing for any \(x\in C\).
\(\{y\in T(x): f(x,y)<0\}\) is a complete ≽sublattice of C for any \(x\in C\).
 (iv)
There is \(\hat{x}\in C\setminus E\) such that \(\bigvee_{C}T(\hat {x})\succcurlyeq\hat{x}\), or there is \(\hat{x}\in E\) such that \(\bigvee_{C}\{y\in T(\hat{x}): f(\hat{x},y)<0\}\succcurlyeq\hat{x}\).
Remark 4.3
In a similar way, Theorem 4.1, Corollary 4.1 and Corollary 4.2 can be extended to chaincomplete posets by using Lemma 4.2.
Very recently, Li [29] proved several extensions of Lemma 4.2 from singlevalued mappings to setvalued mappings on chaincomplete posets.
Lemma 4.3
(see [29])
 A1.:

F is upper orderpreserving.
 A2.:

There is y in X with \(u\succcurlyeq y\) for some \(u\in F(y)\).
 A3.:

\(SF=\{z\in X: u\succcurlyeq z \textit{ for some }u\in F(x) \}\) is an inductive poset for each \(x\in X\).
 A3′.:

\((F(x),\succcurlyeq)\) is inductive with a finite number of maximal elements for every \(x\in X\).
 A3″.:

\(F(x)\) has a maximum element for every \(x\in X\).
 A3‴.:

\(F(x)\) is a chaincomplete lattice for each \(x\in X\).
By using conditions A1, A2 and A3 in Lemma 4.3 and the methodology given in Theorem 3.2, we can obtain an existence theorem of solutions to quasiequilibrium problems on chaincomplete posets, where the isotone selection is dropped.
Theorem 4.4
 (i)
\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)
T is upper ≽preserving and \(T(x)\) has a maximum element for every \(x\in C\). \(x \nsucc C\setminus E\) for any \(x\in E\).
 (iii)
\(f(\cdot,y)\) is orderreversing for any \(y\in C\) and \(f(x,\cdot )\) is orderreversing for any \(x\in C\), and \(f(x,\bigvee_{C}T(x))<0\) for each \(x\in E\).
 (iv)
There is \(y\in C\setminus E\) such that \(y\preccurlyeq u\) for some \(u\in T(y)\), or there is \(y\in E\) such that \(y\preccurlyeq u\) for some \(u\in T(y)\) and \(f(y,u)<0\).
On the other hand, by using conditions A1, A2 and A3‴ in Lemma 4.3 and the methodology given in Theorem 3.2, we can establish another existence theorem of solutions to quasiequilibrium problems on chaincomplete posets.
Theorem 4.5
 (i)
\(f(x,x)\geqslant0\) for any \((x,x)\in C\times C\).
 (ii)
T is upper ≽preserving and \(T(x)\) is a chaincomplete sublattice of C for every \(x\in C\). \(x \nsucc C\setminus E\) for any \(x\in E\).
 (iii)
\(f(\cdot,y)\) is orderreversing for any \(y\in C\) and \(f(x,\cdot )\) is orderreversing for any \(x\in C\).
 (iv)
There is \(y\in C\setminus E\) such that \(y\preccurlyeq u\) for some \(u\in T(y)\), or there is \(y\in E\) such that \(y\preccurlyeq u\) for some \(u\in T(y)\) and \(f(y,u)<0\).
Remark 4.4
Based on Theorem 4.4 and Theorem 4.5, it is easy to deduce some corollaries for EP (1.2).
Remark 4.5
In the same manner, we can also consider the case when T is lower ≽preserving and \(f(x,\cdot)\) is orderpreserving for each \(x\in C\) and \(f(\cdot,y)\) is orderpreserving for each \(y\in C\).
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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