Inductive properties of fixed point sets of mappings on posets and on partially ordered topological spaces
 Jinlu Li^{1}Email author
Received: 15 August 2015
Accepted: 6 November 2015
Published: 18 November 2015
Abstract
In this paper, we prove several fixed point theorems on both posets and partially ordered topological spaces for setvalued mappings. We also provide the inductive properties of the sets of fixed points for some orderincreasing mappings.
Keywords
MSC
1 Introduction
Fixed point theory has been developed for more than a century and has become an important branch in mathematics. It has been widely applied to many branches in pure mathematics and applied mathematics, such as: differential equations; integral equations; optimization theory; game theory; economics theory (see [1, 2]). Fixed point theorems have played important roles to prove the solvability of equations, the existence of solutions for some optimization problems and the existence of Pareto and Nash equilibriums in game theory.
Under similar circumstances, fixed point theorems on posets have become very significant tools in solving some problems with partially ordered underlying spaces. For example, they have been applied to solve equilibrium problems for strategic games with incomplete preferences (see [3]), ordered optimization problems and ordered variational inequality problems (see [4]), nonlinear Hammerstein integral equations (see [5]) and ordinal differential equations (see [1] and [2]).
In the ordinary fixed point theory, the underlying spaces are topological spaces. The considered mappings are required to hold some continuity properties, with respect to the given topologies. In the real world, there are many problems in economics, military affairs, ecology, etc. for which the utilities cannot be totally ordered. The underlying spaces may not be equipped with a topological structure or an algebraic structure. In these cases, the traditional techniques used in totally ordered spaces endowed with topology structure cannot be appropriately applied. It leads many authors to develop some new analysis techniques in nontotally ordered spaces (see [1, 3–5]). So the first important task is to develop some fixed point theorem on ordered spaces without using topological properties.
In 1955, Tarski proved a fixed point theorem on chaincomplete lattices for singlevalued mappings.
Tarski fixed point theorem
[6]
Let \((P, \succeq )\) be a chaincomplete lattice and let \(F: P \rightarrow P\) be an orderincreasing singlevalued mapping. If there is an \(x^{*}\) in P with \(F(x^{*}) \succeq x^{*}\), then F has a fixed point.
It was extended to chaincomplete posets in 1961.
AbianBrown fixed point theorem
[7]
Let \((P, \succeq )\) be a chaincomplete poset and let \(F: P \rightarrow P\) be an orderincreasing singlevalued mapping. If there is an \(x^{*}\) in P with \(F(x^{*}) \succeq x^{*}\), then F has an ⪰maximum fixed point.
In 1984, Fujimoto extended Tarski fixed point theorem from singlevalued mappings to setvalued mappings:
Fujimoto fixed point theorem
[8]
 A1.
F is isotone (F is orderincreasing upward).
 A2.
The set \(\{z \in P: z \preceq u\textit{ for some }u \in F(x)\}\) is an inductively ordered set for each \(x \in P\).
The present author generalized the Fujimoto fixed point theorem to chaincomplete posets in 2014.
Theorem 1.1
([3], Theorems 2.2)
 A1.
F is orderincreasing upward.
 A2.
\((F(x), \succeq )\) is inductive with a finite number of maximal elements, for every \(x \in P\).
 A3.
There is an element y in P with \(y \preceq v\), for some \(v \in F(y)\).
In [3] and [4], several fixed point theorems are proved for setvalued mappings on chinacomplete posets and some applications have been provided. In the theorems in [3] and [4], the condition A2 requires that the values of the considered mapping must be inductive with a finite number of maximal elements. In this paper, we first show that this condition is necessary for the considered mappings to have a fixed point. Then we will show that the set of fixed points of some orderpreserving setvalued mappings is inductive, which provides some useful properties for applications. We will also develop more fixed point theorems on both partially ordered sets and partially ordered topological spaces. Moreover, we will investigate some conditions for the considered mappings to substitute the chaincompleteness of the underlying spaces.
2 Preliminaries
In this section, we recall some concepts and properties of posets. The notations in order theory used in this paper closely follow that from [1, 3, 4, 7–9], and [10].
Let \((X, \succeq ^{X})\), \((U, \succeq ^{U})\) be posets and \(F: X \rightarrow 2^{U}\backslash\{\emptyset\}\) a setvalued mapping. F is said to be isotone, or to be orderincreasing upward, if \(x \preceq ^{X} y\) in X, then for any \(z \in F(x)\), there is a \(w \in F(y)\) such that \(z \preceq ^{U} w\). F is said to be orderincreasing downward, if \(x \preceq ^{X} y\) in X, then for any \(w \in F(y)\), there is a \(z \in F(x)\) such that \(z \preceq ^{U} w\). If F is both orderincreasing upward and orderincreasing downward, then F is said to be orderincreasing.
As a special case, a singlevalued mapping F from a poset \((X, \succeq ^{X})\) to a poset \((U, \succeq ^{U})\) is said to be orderincreasing whenever \(x \preceq ^{X} y\) implies \(F(x) \preceq ^{U} F(y)\). An increasing mapping \(F: X \rightarrow U\) is said to be strictly orderincreasing whenever \(x \prec ^{X} y\) implies \(F(x) \prec ^{U} F(y)\).
A nonempty subset A of a poset \((P, \succeq )\) is said to be inductive whenever any given chain \(\{x_{\alpha}\}\subseteq A\) has an upper cover in A. A is said to be reinductive whenever any given chain \(\{x_{\alpha}\}\subseteq A\) has a lower bound in A. A is said to be biinductive if and only if A is both inductive and reinductive. From Zorn’s lemma and its dual version, every biinductive subset of a poset \((P, \succeq )\) contains an ⪰minimal element and an ⪰maximal element (in A). The following concept was introduced in [4] (Definition 3.1 therein).
A nonempty subset A of a poset \((P, \succeq )\) is said to be universally inductive in P if, for any given chain \(\{x_{\alpha}\}\subseteq P\) satisfying the requirement that if every element \(x_{\beta}\in\{x_{\alpha}\}\) has an upper cover in A, then the chain \(\{x_{\alpha}\}\) has an upper bound in A. By applying the concept of universally inductive posets, a fixed point theorem is proved in [4].
Theorem 2.1
([4], Theorem 3.4)
 A1.
F is orderincreasing upward.
 A2.
\((F(x), \succeq )\) is universally inductive, for every \(x \in P\).
 A3.
There is an element \(y_{*}\) in P and \(v_{*} \in F(y_{*})\) with \(y_{*} \preceq v_{*}\).
The concept of universally inductive posets is more general than the concept of inductive posets with a finite number of maximal elements. It has been proved in [4].
Lemma 2.2
([4], Lemma 3.2)
Every inductive subset A in a chaincomplete poset with a finite number of maximal elements is universally inductive.
From Lemma 3.2 in [4], Theorems 2.2 in [3] can be immediately obtained as a corollary of Theorem 3.4 in [4]. In Theorems 2.2 in [3] listed above, the conditions A1 and A2 for the given mapping F look very natural; and the condition that the poset \((P, \succeq )\) is chaincomplete and condition A2 for F seem to be very strong and ‘superfluous’. In fact, these conditions cannot be deleted in this theorem. We provide some counter examples below to show that in condition A2 of Theorems 2.2 [3], the condition that, for every \(x \in P\), the set of the maximal elements of the inductive set \((F(x), \succeq )\) is finite is necessary for ensuring F to have a fixed point.
Example 2.3
Let S be the closed set in \(R_{2} = \{(u, v): u, v \in R\}\) enclosed by the triangle with vertices \((1, 0)\), \((0, 1)\) and \((2, 2)\); and let L be the closed segment in \(R_{2}\) with ending points \((2, 2)\) and \((4, 0)\). Take P to be the union of S and L and let P be equipped with the componentwise ordering in \(R_{2}\), which is denoted by ⪰^{2}.
For any two distinct points \((u_{1}, v_{1}), (u_{2}, v_{2}) \in R_{2}\), let \(((u_{1}, v_{1}), (u_{2}, v_{2}))\) denote the open interval in \(R_{2}\) with (not included) ending points \((u_{1}, v_{1})\) and \((u_{2}, v_{2})\); and let \(((u_{1}, v_{1}), (u_{2}, v_{2})]\) denote the half open interval in \(R_{2}\) not including the ending point \((u_{1}, v_{1})\) but including \((u_{2}, v_{2})\).
 (1)For any point \((u,  u+ 1 + a) \in S_{a}\), with a given number \(a \in [0, 3)\), let$$F(u,  u + 1 + a) = (1+a/3, 1+a/3). $$
 (2)For any point \((u,  u + 4)\) on the interval \(((2, 2), (4, 0)]\), define$$F(u, u + 4) = (2, 2),\quad \mbox{for all }2 < u \leq 4. $$
 (3)
\(F(2, 2) = ((2, 2), (4, 0))\).
Furthermore, by modifying the above Example 2.3, we provide counterexamples for Theorems 2.2 [3] below with chaincomplete lattices.
Example 2.4
Let S, L be defined as in Example 2.3; and let T be the closed triangle in \(R_{2}\) with vertices \((2, 2)\), \((4, 0)\) and \((4, 2)\) (note that L is the hypotenuse of the right triangle T). Take P to be the union of S and T and let P be equipped with the componentwise ordering ⪰^{2} on \(R_{2}\). One can show that \((P, \succeq ^{2})\) is a chaincomplete lattice!
 1.For any point \((u,  u+ 1 + a) \in S_{a}\), with a given number \(a \in [0, 3)\), let$$F(u,  u + 1 + a) = (1+a/3, 1+a/3). $$
 2.For any point \((u,  u + 4)\) on the open interval \(((2, 2), (4, 0))\) (where we must have \(2 < u < 4\)), define$$F(u,  u + 4) = (2, 2). $$
 3.
\(F(2, 2) = ((2, 2), (4, 0))\).
 4.For any point \((b, v) \in T\backslash L\), with a number \(b \in (2, 4]\), let$$F(b, v) = (1+b/2, 2),\quad \mbox{for any } {}b + 4 \leq v \leq 2. $$
Furthermore, we will next provide a counterexample to demonstrate that if the poset \((P, \succeq )\) is not chaincomplete, then a mapping F satisfying conditions A1, A2, and A3 in Theorem 2.2 [3] may not have a fixed point.
Example 2.5
3 Inductive properties of fixed point sets
In this section, we apply fixed point Theorem 3.4 in [4] to go one step further to study the inductive properties of fixed point sets of some mappings on chaincomplete posets. For a setvalued mapping F on a poset, let \(\mathscr{F}(F)\) denote the set of fixed points of F.
Theorem 3.1
 A1.
F is orderincreasing upward.
 A2.
\((F(x), \succeq )\) is universally inductive, for every \(x \in P\).
 A3.
There is an element \(y_{*}\) in P and \(v_{*} \in F(y_{*})\) with \(y_{*} \preceq v_{*}\).
 (i)
\((\mathscr{F}(F), \succeq )\) is a nonempty inductive poset.
 (ii)
\((\mathscr{F}(F)\cap[y_{*}), \succeq )\) is a nonempty inductive poset; and F has an ⪰maximal fixed point \(x^{*}\) with \(x^{*} \succeq y_{*}\).
Proof
1. Suppose \([x)\cap A = \{x\}\). From the above proof, the ⪰maximal point \(v \in F(x)\) satisfies \(x \preceq v\). From the orderincreasing upward property of F, for the given \(v \in F(x)\) with \(x \preceq v\), there is \(w \in F(v)\) such that \(v \preceq w\). It implies that \(v \in A\). From the assumption \([x)\cap A = \{x\}\) (x is an ⪰maximal element in \((A, \succeq )\)) and \(x \preceq v\), it follows that \(x = v \in F(x)\). Hence \(x \in \mathscr{F}(F)\), and, therefore, x is an upper bound of the chain \(\{x_{\alpha}\}\) in \(\mathscr{F}(F)\).
On the other hand, from the orderincreasing upward property of F, from \(y \preceq u \in F(y)\), there is \(w \in F(u)\) such that \(u \preceq w\). So we have \(u \in A\). From \(x \preceq y \preceq u \preceq w\), it implies that \(u \in [x)\cap A\). Since \(\{x_{\beta}\}\) is a maximal chain in \(([x)\cap A, \succeq )\) and \(y = \bigvee(\{x_{\alpha}\}\cup \{x_{\beta}\}) \in \{x_{\beta}\}\), by \(y \preceq u \in [x)\cap A\), it follows that \(u = y\). It implies that \(y \in \mathscr{F}(F)\). From \(y = \bigvee(\{x_{\alpha}\}\cup \{x_{\beta}\})\), it shows that y is an upper bound of the chain \(\{x_{\alpha}\}\) in \(\mathscr{F}(F)\).
Hence in either case, we proved that the arbitrary chain \(\{x_{\alpha}\} \subseteq \mathscr{F}(F)\) has an upper bound in \((\mathscr{F}(F), \succeq )\). It follows that \((\mathscr{F}(F), \succeq )\) is inductive.
Then we prove part (ii). From condition A3, \(v_{*} \in [y_{*})\), and, therefore, \([y_{*}) \neq \emptyset\). It is clear that \(y_{*} \in [y_{*})\cap A\). By the increasing upward property of F, we can show that \(v_{*} \in [y_{*})\cap A\). From the Hausdorff maximality theorem, \(([y_{*})\cap A, \succeq )\) contains a maximal chain \(\{x_{\gamma}\}\) containing \(y_{*}\) as its smallest element. Since \((P, \succeq )\) is chain complete, there exists \(z = \bigvee \{x_{\gamma}\}\). Similarly to the proof of step 2 above, we can show that \(z \in \mathscr{F}(F)\). From \(z = \bigvee \{x_{\gamma}\} \succeq y_{*}\), it follows that \(z \in \mathscr{F}(F)\cap[y_{*})\). Hence \(\mathscr{F}(F)\cap[y_{*}) \neq \emptyset\). From part (i), \(\mathscr{F}(F)\) is inductive and as an orderinterval of a chaincomplete poset, [\(y_{*}\)) is also inductive. It implies that \(\mathscr{F}(F)\cap[y_{*})\) is a nonempty inductive subset of \((P, \succeq )\). □
By Lemma 3.2 in [4], as a consequence of Theorem 3.1, we have the following result. Since it is important for application, we state it with full details as a corollary.
Corollary 3.2
 A1.
F is orderincreasing upward.
 A2.
\((F(x), \succeq )\) is inductive with a finite number of maximal elements, for every \(x \in P\).
 A3.
There is an element \(y_{*}\) in P and \(v_{*} \in F(y_{*})\) with \(y_{*} \preceq v_{*}\).
 (i)
\((\mathscr{F}(F), \succeq )\) is a nonempty inductive poset.
 (ii)
\((\mathscr{F}(F)\cap[y_{*}), \succeq )\) is a nonempty inductive poset, and F has an ⪰maximal fixed point \(x^{*}\) with \(x^{*} \succeq y_{*}\).
Theorem 3.3
 A1.
F is ⪰increasing upward.
 A2.
\((F(x), \succeq )\) is universally inductive, for every \(x \in P\).
 A3.
There is an element \(y_{*}\) in P and \(v_{*} \in F(y_{*})\) with \(y_{*} \preceq v_{*}\).
 A4.
The range \((F(P), \succeq )\) is a chaincomplete poset.
 (i)
\((\mathscr{F}(F), \succeq )\) is a nonempty inductive poset.
 (ii)
\((\mathscr{F}(F)\cap[y_{*}), \succeq )\) is a nonempty inductive poset, and F has an ⪰maximal fixed point \(x^{*}\) with \(x^{*} \succeq y_{*}\).
Proof
 1.
\((\mathscr{F}(F_{R}), \succeq )\) is a nonempty inductive poset, and, therefore, F has an ⪰maximal fixed point.
 2.
\((\mathscr{F}(F_{R})\cap[y_{*})\cap F(P), \succeq )\) is a nonempty inductive poset, and, therefore, F has an ⪰maximal fixed point \(x^{*}\) with \(x^{*} \succeq y_{*}\).
Similarly to Corollary 3.2, we can apply Lemma 3.2 in [4] to get the following corollary of Theorem 3.3.
Corollary 3.4
 A1.
F is ⪰increasing upward.
 A2.
\((F(x), \succeq )\) is inductive with a finite number of maximal elements, for every \(x \in P\).
 A3.
There is an element \(y_{*}\) in P and \(v_{*} \in F(y_{*})\) with \(y_{*} \preceq v_{*}\).
 A4.
The range \((F(P), \succeq )\) is a chaincomplete poset.
 (i)
\((\mathscr{F}(F), \succeq )\) is a nonempty inductive poset.
 (ii)
\((\mathscr{F}(F)\cap[y_{*}), \succeq )\) is a nonempty inductive poset, and F has an ⪰maximal fixed point \(x^{*}\) with \(x^{*} \succeq y_{*}\).
Let \((P, \succeq )\) be a chaincomplete poset and let \(F: P \rightarrow 2^{P}\backslash\{\emptyset\}\) be a setvalued mapping. We provide a counter example below to show that the two conditions A1 and A2 for F given in Theorem 3.1 does not guarantee the range \((F(P), \succeq )\) to be a chaincomplete poset contained in \((P, \succeq )\).
Example 3.5
4 The reversed orders of partial orders
 1.
\((P, \succeq )\) is reinductive, if and only if \((P, \succeq ^{})\) is inductive.
 2.
\((P, \succeq )\) is rechain complete if and only if \((P, \succeq ^{})\) is chain complete.
 3.
A mapping \(F: P \rightarrow 2^{P}\backslash\{\emptyset\}\) is ⪰increasing downward, if and only if it is ⪰^{−}increasing upward.
The reversal version of Definition 3.1 in [4] is that a nonempty subset A of a poset \((P, \succeq )\) is said to be universally reinductive in P if, for any given chain \(\{x_{\alpha}\}\subseteq P\) satisfying the requirement that every element \(x_{\beta}\in\{x_{\alpha}\}\) has a lower bound in A, the chain \(\{x_{\alpha}\}\) has a lower bound in A.
The dual version of Lemma 3.2 in [4] is that every inductive subset A in a poset satisfying that A has a finite number of minimal elements is universally reinductive. Under the above definitions, the results in the previous section can easily be adapted to the cases of the reversed order with some suitable modifications. We list the dual versions of Theorem 3.1 as theorems below.
Theorem 4.1
 A1.
F is orderincreasing downward.
 A2.
\((F(x), \succeq )\) is universally reinductive, for every \(x \in P\).
 A3.
There is an element \(y^{*}\) in P and \(v^{*} \in F(y^{*})\) with \(v^{*} \preceq y^{*}\).
 (i)
\((\mathscr{F}(F), \succeq )\) is a nonempty reinductive poset.
 (ii)
\((\mathscr{F}(F)\cap(y^{*}], \succeq )\) is nonempty and reinductive, and F has an ⪰minimal fixed point \(x^{*}\) with \(x^{*} \preceq y^{*}\).
The inductive property and the ordermaximal of the set of fixed points are studied in Theorem 3.1. By applying the reversed order, the reinductive property and the orderminimal points of the set of fixed points are studied in Theorem 4.1. By combining these two theorems, we can derive the following result dealing with biinductive properties of the set of fixed points.
Theorem 4.2
 A1.
F is ⪰increasing.
 A2.
\((F(x), \succeq )\) is both universally inductive and universally reinductive, for every \(x \in P\).
 A3.
There are elements \(y^{*}\), \(z^{*}\) in P with \(y^{*} \preceq z^{*}\) satisfying \(F(y^{*}) \cap[y^{*}) \neq \emptyset\) and \(F(z^{*}) \cap(z^{*}] \neq \emptyset\).
 (i)
\((\mathscr{F}(F), \succeq )\) is a nonempty biinductive poset.
 (ii)
\((\mathscr{F}(F)\cap[y^{*}), \succeq )\) is a nonempty inductive poset.
 (iii)
\((\mathscr{F}(F)\cap(z^{*}], \succeq )\) is a nonempty reinductive poset.
 A3′.:

There are elements \(y^{*}\), \(z^{*}\) in P with \(y^{*} \preceq z^{*}\) satisfying \(F(y^{*}) \subseteq [y^{*}, z^{*}]\) and \(F(z^{*}) \subseteq [y^{*}, z^{*}]\).
 (iv)
\((\mathscr{F}(F)\cap[y^{*}, z^{*}], \succeq )\) is a nonempty biinductive poset; and F has a fixed point \(x^{*}\) with \(y^{*} \preceq x^{*} \preceq z^{*}\).
Proof
We only need to prove part (iv). It is clear that \(([y^{*}, z^{*}], \succeq )\) is also a chaincomplete and rechaincomplete poset. First, we show that, for any \(x \in [y^{*}, z^{*}]\), \(F(x) \subseteq [y^{*}, z^{*}]\). To this end, take an arbitrary \(w \in F(x)\), \(y^{*} \preceq x\) and \(x \succeq z^{*}\), the ⪰increasing property of F implies that there are elements \(v \in F(y^{*})\) and \(u \in F(z^{*})\) such that \(v \preceq w \preceq u\). Since \(u, v \in [y^{*}, z^{*}]\), it follows that \(w \in [y^{*}, z^{*}]\). It implies that the restriction of \(F: [y^{*}, z^{*}] \rightarrow 2^{[y^{*}, z^{*}]}\backslash \{\emptyset\}\) is well defined. Then this theorem immediately follows from Theorems 3.1 and 3.3. □
Theorems 4.1 and 4.2 still hold if condition A2 is modified as given in the following theorem.
Theorem 4.3
 A1.
F is ⪰increasing.
 A2.
\((F(x), \succeq )\) is biinductive with a finite number of maximal elements and a finite number of minimal elements, for every \(x \in P\).
 A3.
There are elements \(y^{*}\), \(z^{*}\) in P with \(y^{*} \preceq z^{*}\) satisfying \(F(y^{*}) \cap[y^{*}) \neq \emptyset\) and \(F(z^{*}) \cap(z^{*}] \neq \emptyset\).
 (i)
\((\mathscr{F}(F), \succeq )\) is a nonempty biinductive poset.
 (ii)
\((\mathscr{F}(F)\cap[y^{*}), \succeq )\) is a nonempty inductive poset.
 (iii)
\((\mathscr{F}(F)\cap(z^{*}], \succeq )\) is a nonempty reinductive poset.
5 Several fixed point theorems on partially ordered topological spaces
In many application problems, the underlying spaces are partially ordered topological spaces, which are equipped with both topology structures and ordering relations, such as Banach lattices, Riesz spaces (see [1, 6, 10]). In this section, we particularly examine the inductive properties of mappings on partially ordered topological spaces and partially ordered topological vector spaces.
Remark 5.1
In this paper, a topological space equipped with a partial order is called a partially ordered topological space, if the topology on this space is natural with respect to the given partial order. Particularly, a Banach space equipped with a partial order is called a partially ordered Banach space if the norm of this space induces a natural topology with respect to the given partial order, which satisfies the orderlinearity properties (3). For example, it is well known that the norm topology of a Banach lattice is always natural with respect to the given lattice order. Hence any Banach lattice can be considered as a special case of partially ordered Banach spaces.
Notice that the topology τ on a topology space X is natural with respect to a partial order ⪰ on X if and only if τ is natural with respect to the reversed order ⪰^{−} on X. Then Theorem 2.3 in [4] can be immediately extended as follows.
Lemma 5.2
Let \((X, \tau, \succeq )\) be a partially ordered Hausdorff topological space. Then every nonempty compact subset of X is both chain complete and rechain complete.
In the proof of Theorem 3.1, the condition that the values of the considered mapping are universally inductive is important for the existence of fixed point. As examples, we showed that every nonempty inductive subset with a finite number of maximal elements in a poset is universally inductive. Similarly to Lemma 5.2, we extend Lemma 3.6 in [4] to provide more examples of universally inductive, which are useful in order theory with partially ordered Hausdorff topological spaces.
Lemma 5.3
Every nonempty compact subset of a partially ordered Hausdorff topological space is both universally inductive and universally reinductive.
As a consequence of Lemma 5.3, we have the result below, which is more practical in partially ordered reflexive Banach spaces.
Lemma 5.4
Every nonempty bounded closed and convex subset of a partially ordered reflexive Banach space is both universally inductive and universally reinductive.
Proof
Since every nonempty bounded closed and convex subset of a reflexive Banach space is weakly compact, and in a partially ordered Banach space, the norm topology is natural with respect to the given partial order if and only if the weak topology is natural with respect to the same partial order, this corollary follows from Lemma 5.3 immediately. □
We provide the following example to demonstrate that the concept of universally inductive is indeed broader than the concept of inductive with a finite number of maximal elements.
Example 5.5
From Remark 5.1 and Lemma 5.2, as a consequence of Theorem 3.1, we have the following result on a partially ordered compact Hausdorff topological space.
Theorem 5.6
 A1.
F is ⪰increasing upward.
 A2.
\(F(x)\) is a τcompact subset, for every \(x \in X\).
 A3.
There is an element \(y_{*}\) in X and \(v_{*} \in F(y_{*})\) with \(y_{*} \preceq v_{*}\).
 (i)
\((\mathscr{F}(F), \succeq )\) is a nonempty inductive poset.
 (ii)
\((\mathscr{F}(F)\cap[y_{*}), \succeq )\) is a nonempty inductive poset, and F has an ⪰maximal fixed point \(x^{*}\) with \(x^{*} \succeq y_{*}\).
Remark 5.7
By Theorem 4.2, we can obtain the useful dual versions of Theorem 5.6.
Theorem 5.8
 A1.
F is ⪰increasing.
 A2.
\(F(x)\) is a τcompact subset, for every \(x \in X\).
 A3.
There are element \(y_{*}\), \(z_{*}\) in X and \(v_{*} \in F(y_{*})\), \(u_{*} \in F(z_{*})\) with \(y_{*} \preceq v_{*}\) and \(z_{*} \succeq u_{*}\).
 (i)
\((\mathscr{F}(F), \succeq )\) is a nonempty biinductive poset.
 (ii)
\((\mathscr{F}(F)\cap[y_{*}), \succeq )\) is a nonempty inductive poset.
 (iii)
\((\mathscr{F}(F)\cap(z_{*}], \succeq )\) is a nonempty reinductive poset.
 A3′.:

There are elements \(y^{*}\), \(z^{*}\) in X with \(y^{*} \preceq z^{*}\) satisfying \(F(y^{*}) \subseteq [y^{*}, z^{*}]\) and \(F(z^{*}) \subseteq [y^{*}, z^{*}]\).
 (iv)
\((\mathscr{F}(F)\cap[y^{*}, z^{*}], \succeq )\) is a nonempty biinductive poset; and F has a fixed point \(x^{*}\) with \(y^{*} \preceq x^{*} \preceq z^{*}\).
Similarly to Example 5.5, we provide a counterexample below to show that, for the given mapping F, neither the conditions in Theorem 5.6, nor the conditions in Theorem 5.8, guarantee the set of fixed points of F to be chain complete.
Example 5.9
From Lemmas 5.3 and 5.4, as a consequence of Theorem 5.6, we have the following fixed point theorem on partially ordered reflexive Banach spaces.
Theorem 5.10
 A1.
F is ⪰increasing.
 A2.
\(F(x)\) is a closed and convex subset of X, for every \(x \in X\).
 A3.
There are elements \(y_{*}\), \(z_{*}\) in X and \(v_{*} \in F(y_{*})\), \(u_{*} \in F(z_{*})\) with \(y_{*} \preceq v_{*}\) and \(z_{*} \succeq u_{*}\).
 (i)
\((\mathscr{F}(F), \succeq )\) is a nonempty biinductive poset.
 (ii)
\((\mathscr{F}(F)\cap[y_{*}),\succeq )\) is a nonempty inductive poset.
 (iii)
\((\mathscr{F}(F)\cap(z_{*}], \succeq )\) is a nonempty reinductive poset.
 A3′.:

There are elements \(y^{*}\), \(z^{*}\) in X with \(y^{*} \preceq z^{*}\) satisfying \(F(y^{*}) \subseteq [y^{*}, z^{*}]\) and \(F(z^{*}) \subseteq [y^{*}, z^{*}]\).
 (iv)
\((\mathscr{F}(F)\cap[y^{*}, z^{*}], \succeq )\) is a nonempty biinductive poset; and F has a fixed point \(x^{*}\) with \(y^{*} \preceq x^{*} \preceq z^{*}\).
Recall that, in every Banach lattice, the topology (norm) is always natural with respect to the given lattice order. Hence, in Theorem 5.10, the partially ordered reflexive Banach space \((B, \succeq )\) can be substituted by a reflexive Banach lattice, as it is a special case.
Declarations
Acknowledgements
The author is very grateful to the anonymous reviewers for their valuable suggestions, which improved the presentation of this paper.
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Authors’ Affiliations
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