CUBO A Mathematical Journal

Vol.10, N
o
¯ 04, (119–135). December 2008

Fixed Point Results for Set-Valued and Single-Valued

Mappings in Ordered Spaces

Seppo Heikkilä

Department of Mathematical Sciences,

University of Oulu,

BOX 3000, FIN-90014 University of Oulu, Finland

email: sheikki@cc.oulu.fi

ABSTRACT

In this article we use a recursion principle and generalized iteration methods to prove

existence and comparison results for fixed points of set- and single-valued mappings in

ordered spaces.

RESUMEN

En este art́ıculo usamos el principio de recurrencia y métodos de iteración generalizados

para provar resultados de exitencia y comparación para puntos fijos de aplicaciones

conjunto (uni-)valoradas en espacios ordenados.

Key words and phrases: Poset, set-valued mapping, fixed point, solution, maximal, minimal,

sup-center, inf-center, recursion principle, generalized iteration methods, order compact, chain

complete.

Math. Subj. Class.: 47H04, 47H07, 47H10.



120 Seppo Heikkilä CUBO
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1 Introduction

Let P be a nonempty partially ordered set (poset). As an introductory result we show that a

set-valued mapping F from P to the set 2P \∅ of nonempty subsets of P has minimal and maximal

fixed points, that is, the set Fix F = {x ∈ P | x ∈ F(x)} has minimal and maximal elements, if

the following conditions hold.

(c1) sup{c, y} ∈ P for some c ∈ P and for every y ∈ P .

(c2) If x ≤ y in P , then for every z ∈ F(x) there exists a w ∈ F(y) such that z ≤ w, and for

every w ∈ F(y) there exists a z ∈ F(x) such that z ≤ w.

(c3) Strictly monotone sequences of F[P ] =
⋃

{F(x) : x ∈ P } are finite.

As for the proof, denote x0 = c, and choose y0 from F(x0). If y0 6≤ x0, then x0 < x1 := sup{c, y0}.

Apply then condition (c2) to choose y1 from F(x1) such that y0 ≤ y1. If y0 = y1, then stop.

Otherwise, y0 < y1, whence x1 = sup{c, y0} ≤ x2 := sup{c, y1}, and apply again condition (c2) to

choose y2 from F(x2) such that y1 ≤ y2. Continuing in a similar way, condition (c3) ensures that

after a finite number of choices we get the situation, where yn−1 = yn ∈ F(xn). In view of the

above construction we then have xn := sup{c, yn−1} = sup{c, yn}.

Denoting z0 := xn and w0 := yn then w0 ∈ F(z0) and w0 ≤ sup{c, w0} = z0. If w0 = z0, then

z0 is a fixed point of F. Otherwise, denoting z1 := w0, we have z1 < z0. In view of condition (c2)

there exists a w1 ∈ F(z1) such that w1 ≤ w0. If equality holds, then z1 = w0 = w1 ∈ F(z1), so

that z1 is a fixed point of F. Otherwise, w1 < w0, denote z2 := w1, and choose by (c2) such a

w2 ∈ F(z2) that w2 ≤ w1, and so on. Condition (c3) implies that a finite number of steps yields

the situation zm := wm−1 = wm ∈ F(zm). Thus zm belongs to Fix F. Being a subset of F[P ],

strictly monotone sequences of Fix F are finite by condition (c3). This property implies in turn

that Fix F has minimal and maximal elements.

The above described result will be generalized in Section 3. For instance, we show that F has

minimal and maximal fixed points when the above condition (c1) holds, condition (c2) is replaced

by a stronger monotonicity condition, and (c3) is replaced by a chain completeness of the order

closure of the range F[P ]. Applications to single-valued mappings are also given. Fixed points of

a concrete mapping are approximated by using an algorithmic method developed from the above

described reasoning.

The obtained results are used in Section 4 to derive fixed point results in ordered normed

spaces and in ordered topological spaces. Existence proofs require several consecutive applications

of a recursion principle and generalized iteration methods introduced in [4, 6] and presented in

Section 2.



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Fixed Point Results for Set-Valued ... 121

2 Recursions and iterations in posets

Given a nonempty set P , a relation x < y in P × P is called a partial ordering, if x < y implies

y 6< x, and if x < y and y < z imply x < z. Defining x ≤ y if and only if x < y or x = y, we say

that P = (P, ≤) is a partially ordered set (poset).

An element b of a poset P is called an upper bound of a subset A of P if x ≤ b for each x ∈ A.

If b ∈ A, we say that b is the greatest element of A, and denote b = max A. A lower bound of A

and the least element, min A, of A are defined similarly, replacing x ≤ b above by b ≤ x. If the set

of all upper bounds of A has the least element, we call it a supremum of A and denote it by sup A.

We say that y is a maximal element of A if y ∈ A, and if z ∈ A and y ≤ z imply that y = z. An

infimum of A, inf A, and a minimal element of A are defined similarly. We say that a poset P is

a lattice if inf{x, y} and sup{x, y} exist for all x, y ∈ P . W is called a chain if x ≤ y or y ≤ x

for all x, y ∈ W . We say that W is well-ordered if nonempty subsets of W have least elements,

and inversely well-ordered if nonempty subsets of W have greatest elements. In both cases W is a

chain.

A basis to our considerations is the following recursion principle (cf. [6], Lemma 1.1.1).

Lemma 2.1. Given a nonempty poset P , a subset D of 2P = {A : A ⊆ P } with ∅ ∈ D and a

mapping f : D → P , there is a unique well-ordered chain C in P such that

x ∈ C if and only if x = f (C<x), where C<x = {y ∈ C : y < x}. (2.1)

If C ∈ D, then f (C) is not a strict upper bound of C.

As an application of Lemma 2.1 we get the following result (cf. [4], Lemma 2).

Lemma 2.2. Given G : P → P and c ∈ P , there exists a unique well-ordered chain C = C(G) in

P , called a w-o chain of cG-iterations, satisfying

x ∈ C if and only if x = sup{c, G[C<x]}. (2.2)

Proof. Denote D = {W ⊆ P : W is well-ordered and sup{c, G[W ]} exists}. Defining f (W ) =

sup{c, G[W ]}, W ∈ D, we get a mapping f : D → P , and (2.1) is reduced to (2.2). Thus the

assertion follows from Lemma 2.1.

A subset W of a chain C is called an initial segment of C if x ∈ W and y < x imply y ∈ W .

The following result is also used in the sequel.

Lemma 2.3. Given F : P → 2P \ ∅, denote by G the set of all selections from F, i.e.,

G := {G : P → P : G(x) ∈ F(x) for all x ∈ P }. (2.3)

For every G : P → P denote by CG the longest such an initial segment of the w-o chain C(G)

of cG-iterations that the restriction G|CG of G to CG is increasing (i.e., G(x) ≤ G(y) whenever

x ≤ y in CG). Define a partial ordering ≺ on G as follows.



122 Seppo Heikkilä CUBO
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(O) F ≺ G if and only if CF is a proper initial segment of CG and G|CF = F |CF .

Then (G, �) has a maximal element.

Proof. Let C be a chain in G. The definition (O) of ≺ implies that the sets CF , F ∈ C, form a nested

family of well-ordered sets of P . Thus the set C := ∪{CF : F ∈ C} is well-ordered. Moreover, it

follows from (O) that the functions F |CF , F ∈ C, considered as relations in P × P , are nested.

This ensures that g := ∪{F |CF : F ∈ C} is a function from C to P . Since each F ∈ C is increasing

in CF , then g is increasing, and g(x) ∈ F(x) for each x ∈ C. Let G be such a selection from F

that G|C = g. Then G ∈ G, and G is increasing on C. If x ∈ C, then x ∈ CF for some F ∈ C.

The definitions of C and the partial ordering ≺ imply that CF is C or its initial segment, whence

C<x
F

= C<x. Because F |CF = g|CF = G|CF , then

x = sup{c, F [C<xF ]} = sup{c, G[C
<x

]}. (2.4)

This result implies (cf. the proof of Lemma 2.1) that C is C(G) or its proper initial segment. Since

G is increasing on C, then C is CG or its proper initial segment. Consequently, G is an upper

bound of C in G. This result implies by Zorn’s Lemma that G has a maximal element.

Let X = (X, ≤) be a poset. When z, w ∈ X, denote

[z) = {x ∈ X : z ≤ x}, (w] = {x ∈ X : x ≤ w} and [z, w] = [z) ∩ (w].

A subset A of X is called a causal set (causet) if [z, w] ∩ A is finite for all z, w ∈ A. We say that

A is bounded from above if A ⊆ (z], bounded from below if A ⊆ [z), and order bounded if A ⊆ [z, w]

for some z, w ∈ X.

We say that X, equipped with a topology is an ordered topological space if the order intervals

[z) and (z] are closed for each z ∈ X. If the topology of X is induced by a metric, we say that X

is an ordered metric space.

Next we define some concepts for sequences and set-valued functions.

Definition 2.1. A sequence (zn)
∞
n=0 of a poset is called increasing if zn ≤ zm whenever n ≤ m,

decreasing if zm ≤ zn whenever n ≤ m, and monotone if it is increasing or decreasing. If the

above inequalities are strict, the sequence (zn)
∞
n=0 is called strictly increasing, strictly decreasing

or strictly monotone, respectively.

Definition 2.2. Given posets X and P , we say F : X → 2P \ ∅ is increasing upward if x ≤ y in

X and z ∈ F(x) imply that [z) ∩ F(y) is nonempty. F is increasing downward if x ≤ y in X and

w ∈ F(y) imply that (w] ∩ F(x) is nonempty. If F is increasing upward and downward, we say

that F is increasing.

Definition 2.3. Given posets P and X and a set-valued function F : X → 2P \ ∅, consider chains

of the form G[C], where C is a nonempty chain in X and G is an increasing selection from F|C. F



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Fixed Point Results for Set-Valued ... 123

is called chain complete upward if such chains G[C] have supremums whenever C is well-ordered,

chain complete downward if such chains G[C] have infimums whenever C is inversely well-ordered,

and chain complete if both these conditions hold. If every such a chain G[C] has an upper bound in

F(x) whenever x is an upper bound of C in X, we say that F is called strongly increasing upward.

If this condition holds with upper bounds replaced by lower bounds, we say that F is strongly

increasing downward. If both these conditions hold, then F is said to be strongly increasing.

The following Corollary is an easy consequence of Definitions 2.2 and 2.3.

Corollary 2.1. (a) F : X → 2P \ ∅ is chain complete upward (respectively downward) if every

nonempty chain of F[X] has a supremum (respectively an infimum).

(b) If F is strongly increasing upward (respectively downward), then it is increasing upward (re-

spectively downward).

(c) If F is increasing upward (respectively downward), and if max F(x) (respectively min F(x))

exists for every x ∈ X, then F is strongly increasing upward (respectively downward).

(d) An increasing mapping F is strongly increasing, if chains of X are causets, or if chains of

F[X] are finite, or if the values of F are finite sets.

In the case when P is an ordered topological space we have the following results.

Lemma 2.4. Let X be a poset, P an ordered topological space, and F : X → 2P \ ∅.

(a) If F is increasing upward (respectively downward), and if its values are compact, then F is

strongly increasing upward (respectively downward).

(b) Assume that (yn) converges whenever yn ∈ F(xn) for every n and both (xn) and (yn) are in-

creasing (respectively decreasing). If P is second countable or metrizable, then F is chain complete

upward (respectively downward).

Proof. Consider a chain W = G[C], where C is a chain in X and G is an increasing selection from

F|C.

(a) Assume that x ∈ X is an upper bound of C. If F is increasing upward, then to every

y ∈ W there corresponds a z ∈ [y) ∩ F(x). Because W is a chain, then the sets [y) ∩ F(x), y ∈ W ,

satisfy the finite intersection property. Thus their intersection is nonempty if F(x) is compact,

and every element from that intersection is an upper bound of W in F(x). Similarly one can prove

that if x ∈ X is a lower bound of C, if F is increasing downward, and if F(x) is compact, then

F(x) contains a lower bound of W .

(b) If C is well-ordered, and (yn) is an increasing sequence of W = G[C], then yn = G(xn),

where xn = min{x ∈ C : G(x) = yn} for every n, and (xn) is increasing. Thus (yn) converges by

a hypothesis of (b). If P is second countable, then every subset of W is separable. It then follows

from [6], Lemma 1.1.7 that W contains an increasing sequence which converges to sup W . This

result follows from [6], Proposition 1.1.5 if P is an ordered metric space. These results and their

duals imply the conclusions of (b).



124 Seppo Heikkilä CUBO
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3 Fixed point results in posets

In this section we prove existence and comparison results for fixed points of a set-valued and

single-valued functions in a poset P .

3.1 Fixed point results for set-valued functions

As an application of Lemma 2.1 we obtain the following result.

Proposition 3.1. Assume that F : P → 2P \ ∅ is strongly increasing upward and chain complete

upward, and that the set S+ = {x ∈ P : [x) ∩ F(x) 6= ∅} is nonempty. Then F has a maximal fixed

point, which is also a maximal element of S+.

Proof. Denote D = {W ⊂ S+ : W is well-ordered and has a strict upper bound in S+}. Because

S+ is nonempty, then ∅ ∈ D. Let f : D → P be a function which assigns to each W ∈ D an

element y = f (W ) ∈ [x) ∩ F(x), where x is a fixed strict upper bound of W in S+. By Lemma

2.1 there exists exactly one well ordered chain W in P satisfying (2.1). By the above construction

and (2.1) each element y of W belongs to [x) ∩F(x), where x is a fixed strict upper bound of W <y

in S+. Because F is increasing upward and x ≤ y ∈ F(x), then [y) ∩ F(y) 6= ∅, so that y ∈ S+.

It is easy to verify that the set C of these elements x form a well ordered chain in S+, that the

correspondence x 7→ y defines an increasing selection G : C → S+ from F|C, and that W = G[C].

Because F is chain complete upward, then x = sup W exists in P . The above construction implies

that x is also an upper bound of C. Since F is strongly increasing upward, then W has an upper

bound y in F(x). Because x = sup W , then x ≤ y, so that y ∈ [x) ∩ F(x), and thus x = sup W

belongs to S+. x = max W , for otherwise f (W ) would exist, and being a strict upper bound of W ,

would contradict the last conclusion of Lemma 2.1. By the same reason x is a maximal element of

S+.

Because x ≤ y ∈ F (x), then [y) ∩ F(y) 6= ∅, or equivalently, y ∈ S+, since F is increasing

upward. Because x is a maximal element of S+, then x = y ∈ F(x), so that x is a fixed point of

F. If z is a fixed point of F and x ≤ z, then z ∈ S+, whence x = z. Thus x is a maximal fixed

point of F.

The next result is the dual of Proposition 3.1.

Proposition 3.2. Assume that F : P → 2P \∅ is strongly increasing downward and chain complete

downward, and that the set S− = {x ∈ P : (x] ∩ F(x) 6= ∅} is nonempty. Then F has a minimal

fixed point, which is also a minimal element of S−.

If F[P ] has an upper bound (respectively a lower bound) in P , it belongs to S− (respectively

to S+).

Next we derive other conditions under which the set S− or the set S+ is nonempty.



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Fixed Point Results for Set-Valued ... 125

Definition 3.1. Let A be a subset of a poset P . The set ocl(A) of all possible supremums and

infimums of chains of A is called an order closure of A. If A = ocl(A), then A is order closed. We

say that a subset A of poset P has a sup-center c in P if c ∈ P and sup{c, x} exists in P for each

x ∈ A. If inf{c, x} exists in P for each x ∈ A, we say that c is an inf-center of A in P . If c has

both these properties it is called an order center of A in P . Phrase ”in P ” is omitted if A = P .

If P is an ordered topological space, then the order closure ocl(A) of A is contained in the

topological closure of A. If c is the greatest element (respectively the least element) of P , then c

is an inf-center, (respectively a sup-center) of P . If P is a lattice, then its every point is an order

center of P . If P is a subset of R2, ordered coordinatewise, a necessary and sufficient condition

for a point c = (c1, c2) of P to be a sup-center of a subset A of P in P is that whenever a point

y = (y1, y2) of A and c are unordered, then (y1, c2) ∈ P if y2 < c2 and (c1, y2) ∈ P if y1 < c1. No

conditions are imposed on other points of A.

The following result is an application of Lemma 2.3.

Proposition 3.3. Let F : P → 2P \ ∅ be chain complete upward and strongly increasing upward.

If ocl(F[P ]) has a sup-center in P , then the set S− = {x ∈ P : (x] ∩ F(x) 6= ∅} is nonempty.

Proof. Let c be a sup-center of ocl(F[P ]) in P , let G be defined by (2.3), and let the partial

ordering ≺ be defined by (O). By Lemma 2.3 (G, �) has a maximal element G. Let C(G) be the

w-o chain of cG-iterations, and let C = CG be the longest initial segment of C(G) on which G is

increasing. Thus C is well-ordered and G is an increasing selection from F|C. Since F is chain

complete upward, then w = sup G[C] exists. Moreover, x = sup{c, w} exists in P by the choice of

c, and it is easy to see that x = sup{c, G[C]}. This result and (2.4) imply that for each x ∈ C,

x = sup{c, G[C<x]} ≤ sup{c, G[C]} = x.

This proves that x is an upper bound of C. Since F is strongly increasing upward, then W = G[C]

has an upper bound z in F(x), and w = sup G[C] ≤ z. To show that x = max C, assume on the

contrary that x is a strict upper bound of C. Let F be a selection from F whose restriction to

C ∪ {x} is G|C ∪ {(x, z)}. Since G is increasing on C and F (x) = G(x) ≤ w ≤ z = F (x) for each

x ∈ C, then F is increasing on C ∪ {x}. Moreover,

x = sup{c, G[C]} = sup{c, F [C]} = sup{c, F [{y ∈ C ∪ {x} : y < x}]},

whence C ∪ {x} is a subset of the longest initial segment CF of the w-o chain of cF -iterations

where F is increasing. Thus C = CG is a proper subset of CF , and F |CG = F |CF . This means by

(O) that G ≺ F . But this is impossible because G is a maximal element of (G, �). Consequently,

x = max C. Since G is increasing on C, then x = sup{c, G[C]} = sup{c, G(x)}. In particular,

F(x) ∋ G(x) ≤ x, whence G(x) belongs to the set (x] ∩ F(x).

As a consequence of Propositions 3.1, 3.2 and 3.3 we obtain the following result.



126 Seppo Heikkilä CUBO
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Theorem 3.1. Assume that F : P → 2P \∅ is strongly increasing and chain complete. If ocl(F[P ])

has a sup-center or an inf-center in P , then F has minimal and maximal fixed points.

Proof. We shall give the proof in the case when ocl(F[P ]) has a sup-center in P , the proof in the

case of an inf-center being similar. The hypotheses of Proposition 3.3 are then valid, whence there

exists a x ∈ P such that (x] ∩ F(x) 6= ∅. Thus the hypotheses of Proposition 3.2 hold, whence F

has by Proposition 3.2 a minimal fixed point x−. In particular [x−) ∩ F(x−) 6= ∅. The hypotheses

of Proposition 3.1 are then valid, whence F has also a maximal fixed point.

Example 3.1. Assume that Rm is ordered as follows. For all x = (x1, . . . , xm), y = (y1, . . . , ym) ∈

R
m

,

x ≤ y if and only if xi ≤ yi, i = 1, . . . , j, and xi ≥ yi, i = j + 1, . . . , m, (3.1)

where j ∈ {0, . . . , m}. Show that if F : Rm → 2R
m

\ ∅ is increasing, and its values are closed

subsets of R
m

, and if F[Rm] is contained in

B
p
R
(c) = {(x1, . . . , xm) ∈ R

m
:

m
∑

i=1

|xi − ci|
p ≤ Rp}

for some p, R ∈ (0, ∞) and c = (c1, . . . , cm) ∈ R
m

, then F has minimal and maximal fixed points.

Solution. Let x = (x1, . . . , xm) ∈ B
p
R
(c) be given. Since | max{ci, xi} − ci| ≤ |xi − ci| and

| min{ci, xi} − ci| ≤ |xi − ci| for each i = 1, . . . , m, it follows that sup{c, x} and inf{c, x} belong to

B
p
R
(c) for all x ∈ B

p
R
(c). Moreover, every B

p
R
(c) is a closed and bounded subset of Rm, whence its

monotone sequences converge in B
p
R
(c) with respect to the Euclidean metric of Rm. These results,

Lemma 2.4 and the given hypotheses imply that F is chain complete, and strongly increasing, and

that c is an order center of ocl(F[Rm]). Thus the hypotheses of Theorem 3.1 hold, whence F has

minimal and maximal fixed points.

3.2 Fixed point results for single-valued functions

Next we present existence and comparison results for fixed points of single-valued functions. In

the proofs we use the following consequence of Proposition 3.3.

Proposition 3.4. Assume that G : P → P is increasing, that ocl(G[P ]) has a sup-center c in

P , and that sup G[C] exists whenever C is a nonempty chain in P . If C is the w-o chain of

cG-iterations, then x = max C exists, x = sup{c, G(x)} = sup{c, G[C]} and

x = min{z ∈ P : sup{c, G(z)} ≤ z}. (3.2)

Moreover, x is the least solution of the equation x = sup{c, G(x)} and is increasing in G.



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Fixed Point Results for Set-Valued ... 127

Proof. When F : P → 2P \ ∅ is single-valued, it coincides with its unique selection function

G : P → P . Moreover, F is strongly increasing upward if and only if G is increasing, in which

case C in Lemma 2.3 is the w-o chain of cG-iterations. The hypotheses given for G imply also that

G = F is chain complete upward, and that c is a sup-center of ocl(F[P ]) in P . As a single valued

mapping it is also strongly increasing. Thus the proof of Proposition 3.3 implies that x = max C ex-

ists and x = sup{c, G(x)} = sup{c, G[C]}. To prove (3.2), let z ∈ P satisfy sup{c, G(z)} ≤ z. Then

c = min C ≤ z. If x ∈ C and sup{c, G(y)} ≤ z for each y ∈ C<x, then x = sup{c, G[C<x]} ≤ z.

This implies by transfinite induction that x ≤ z for each x ∈ C. In particular x = max C ≤ z. This

result and the fact that x = sup{c, G(x)} imply that x = x is the least solution of the equation

x = sup{c, G(x)}, and that (3.2) holds. The last assertion is an immediate consequence of (3.2).

The results presented in the next proposition are dual to those of Lemma 2.2 and Proposition

3.4.

Proposition 3.5. Given G : P → P and c ∈ P there exists exactly one inversely well-ordered

chain D in P , called an inversely well-ordered (i.w-o) chain of cG- iterations, satisfying

x ∈ D if and only if x = inf{c, G[{y ∈ C : x < y}]}. (3.3)

Assume that G is increasing, that ocl(G[P ]) has an inf-center c in P , and that inf G[C] exists

whenever C is a nonempty chain in P . If D is the i.w-o chain of cG-iterations, then x = min D

exists, x = inf{c, G(x)} = inf{c, G[D]} and

x = max{z ∈ P : z ≤ inf{c, G(z)}}. (3.4)

Moreover, x is the greatest solution of the equation x = inf{c, G(x)} and is increasing in G.

Our first fixed point result is a consequence of Propositions 3.4 and 3.5.

Lemma 3.1. Let P be a poset and G : P → P an increasing mapping.

(a) If P ∋ x ≤ G(x), and if sup G[C] exists whenever C is a chain in [x), then the w-o chain C of

xG-iterations has a maximum x∗ and

x∗ = max C = sup G[C] = min{y ∈ [x) : G(y) ≤ y}. (3.5)

Moreover, x∗ is the least fixed point of G in [x) and is increasing in G.

(b) If G(x) ≤ x ∈ P , and if inf G[C] exists whenever C is a chain (x], then the i.w-o chain D of

xG-iterations has a minimum x∗ and

x
∗

= min D = inf G[D] = max{y ∈ (x] : y ≤ G(y)}. (3.6)

Moreover, x∗ is the greatest fixed point of G in (x] and is increasing in G.

Proof. (a) Since G is increasing and x ≤ G(x), then G[[x)] ⊂ [x). Thus the conclusions of (a) are

immediate consequences of the conclusion of Proposition 3.4 when c = x and G is replaced by its

restriction to [x).



128 Seppo Heikkilä CUBO
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The proof of (b) is dual to that of (a).

As an application of Propositions 3.4 and 3.5 and Lemma 3.1 we get the following fixed point

results.

Theorem 3.2. Assume that G : P → P is increasing and that c ∈ P .

(a) If c is a sup-center of ocl(G[P ]) in P , and if sup G[C] and inf G[C] exist whenever C is a chain

in P , then G has minimal and maximal fixed points. Moreover, G has the greatest fixed point x∗ in

(x], where x is the least solution of the equation x = sup{c, G(x)}. Both x and x∗ are increasing

with respect to G.

(b) If c is an inf-center of ocl(G[P ]) in P , and if sup G[C] and inf G[C] exist whenever C is a

chain in P , then G has minimal and maximal fixed points. Moreover, G has the least fixed point

x∗ in [x), where x is the greatest solution of the equation x = inf{c, G(x)}. Both x and x∗ are

increasing with respect to G.

Lemma 3.1, Proposition 3.1 and Proposition 3.2 imply the following results.

Proposition 3.6. Assume that G : P → P is increasing.

(a) If sup G[C] exists whenever C is a well-ordered chain in P , and if G[P ] has a lower bound in

P , then G has the least fixed point and a maximal fixed point.

(b) If sup G[D] exists whenever D is an inversely well-ordered chain in P , and if G[P ] has an

upper bound in P , then G has the greatest and a minimal fixed point.

Example 3.2. Let Rm be ordered coordinatewise, and assume that G : Rm → Rm maps increasing

sequences of R
m

to bounded and increasing sequences of R
m
+

, where R+ is the set of nonnegative

reals. Show that G has the least fixed point and a maximal fixed point.

Solution. Let C be a well-ordered chain in Rm. Since G is increasing, by definition, then G[C] is a

well-ordered chain in R
m
+

. If (yn) is an increasing sequence in G[C], and xn = min{x ∈ C : G(x) =

yn}, then the sequence (xn) is increasing and yn = G(xn) for every n. Thus (yn) is bounded,

by definition of G, and hence converges with respect to the Euclidean metric of Rm. This result

implies by Lemma 2.4 that sup G[C] exists. Moreover, the origin is a lower bound of G[Rm]. Thus

the assertions follow from Proposition 3.6.

3.3 Algorithmic methods

It can be shown that the first elements of the w-o chain C of cG-iterations are: x0 = c, xn+1 =

sup{c, Gxn}, n = 0, 1, . . . , as long as xn+1 exists and xn < xn+1. Assuming that strictly monotone

sequences of G[P ] are finite, then C is a finite strictly increasing sequence (xn)
m
n=0. If sup{c, x}

exists for every x ∈ G[P ], then x = sup{c, G[C]} = max C = xm is the least solution of the

equation x = sup{c, G(x)} by the proof of Proposition 3.4. In particular, Gx ≤ x. If G(x) < x,

then first elements of the i.w-o chain D of xG-iterations of x are y0 = x = xm, yj+1 = Gyj , as long



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Fixed Point Results for Set-Valued ... 129

as yj+1 < yj . Since strictly monotone sequences of G[P ] are finite, D is a finite strictly decreasing

sequence (yj )
k
j=0, and x

∗
= inf G[D] = yk is the greatest fixed point of G in (x] by the proof of

Lemma 3.1. This reasoning and its dual imply the following results.

Corollary 3.1. Conclusions of Theorem 3.2 hold if G : P → P is increasing and strictly monotone

sequences of G[P ] are finite, and if sup{c, x} and inf{c, x} exist for every x ∈ G[P ]. Moreover, x∗

is the last element of the finite sequence determined by the following algorithm:

(i) x0 = c. For n from 0 while xn 6= Gxn do: xn+1 = Gxn if Gxn < xn else xn+1 =

sup{c, Gxn},

and x∗ is the last element of the finite sequence determined by the following algorithm:

(ii) x0 = c. For n from 0 while xn 6= Gxn do: xn+1 = Gxn if Gxn > xn else xn+1 =

inf{c, Gxn}.

Let G : P → P satisfy the hypotheses of Theorem 3.2. The result Corollary 3.1 can be applied

to approximate the fixed points x∗ and x∗ of G introduced in Theorem 3.2 in the following manner.

Assume that G, G : P → P satisfy the hypotheses given for G in Corollary 3.1, and that

G(x) ≤ G(x) ≤ G(x) for all x ∈ P. (3.7)

Since x∗ is increasing with respect to G, it follows from (3.7) that x∗ ≤ x∗ ≤ x∗, where x∗ and x∗

are obtained by algorithm (i) of Corollary 3.1 with G replaced by G and G, respectively.

Since partial ordering is the only structure needed in the proofs, the above results can be

applied to problems where only ordinal scales are available. On the other hand, these results have

some practical value also in real analysis. We shall demonstrate this by an example where the

above described method is applied to a system of the form

xi = Gi(x1, . . . , xm), i = 1, . . . , m, (3.8)

where the functions Gi are real valued functions of m real variables.

Example 3.3. In this example we approximate a solution x∗ = (x1, y1) of the system

x = G1(x, y) :=
N1(x, y)

2 − |N1(x, y)|
, y = G2(x, y) :=

N2(x, y)

3 − |N2(x, y)|
, (3.9)

where

N1(x, y) =
11

12
x +

12

13
y +

1

234
and N2(x, y) =

15

16
x +

14

15
y −

7

345
, (3.10)

by calculating such upper and lower estimates of (x1, y1) whose corresponding coordinates differ

less than 10
−100

.

The mapping G = (G1, G2), defined by (3.9), (3.10) maps the set P = {(x, y) ∈ R
2

: |x|+|y| ≤
1

2
} into P , and is increasing on P . It follows from Example 2.1 that c = (0, 0) is an order center

of P , and that P is chain complete. Thus the results of Theorem 3.2 are valid.



130 Seppo Heikkilä CUBO
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Upper and lower estimates to the fixed point x∗ = (x1, y1) of G, and hence to a solution

(x1, y1) of system (3.9), (3.10), can be obtained by applying the algorithm (i) given in Corollary

3.1 to operators G and G, defined by

{

G(x, y) = (10−101ceil(10101G1(x, y)), 10
−101

ceil(10
101G2(x, y)),

G(x, y) = (10−101floor(10101G1(x, y)), 10
−101

floor(10
101G2(x, y)),

(3.11)

where ceil(x) is the least integer ≥ x and floor(x) is the greatest integer ≤ x. The so defined

operators G, G are increasing and map the set P = {(x, y) ∈ R2 : |x| + |y| ≤ 1
2
} into finite subsets

of P , and (3.7) holds. We are going to show that the required upper and lower estimates are

obtained by algorithm (i) of Corollary 3.1 with G replaced by G and G, respectively. The following

Maple program is used in calculations of the upper estimate x∗ = (x1, y1).

(N1, N2):=(11/12*x+12/13*y+1/234,15/16*x+14/15*y-7/345):

(z,w):=(N1/(2-abs(N1)),N2/(3-abs(N2))):(G1,G2):=(ceil(10
101

z)/10
101

,ceil(10
101

w)/10
101

):

(x0,y0):=(0,0);x:=x0:y:=y0:u:=G1:v:=G2:b[0]:=[x,y]:

for k from 1 while abs(u-x)+abs(v-y)> 0 do:

if u<=x and v<=y then (x,y):=(u,v) else (x,y):=(max(x,u),max(y,v)):fi:

u:=G1:v:=G2:b[k]:=[x,y]:od:n:=k-1: x1:=x;y1=y;

The above program yields the following results (n=1246).























x1 = −0.00775318684978081165491069304103701961947143138774717254950456999535

626408273278584836718225237250043,

y1 : −0.01359961542461090148983671991312928002452425440128992737588059916178

38548683927620135569441397855721

In particular, (x1, y1) is the fixed point x∗ of G.

Replacing ’ceil’ by ’floor’ in the above program we obtain components of the fixed point

x∗ = (x2, y2) of G (n:=1248).























x2 = −0.00775318684978081165491069304103701961947143138774717254950456999535

62640827327858483671822523725005,

y2 = −0.01359961542461090148983671991312928002452425440128992737588059916178

385486839276201355694413978557215

The above calculated components of x∗ and x∗ are exact, and their differences are < 10−100. Ac-

cording to the above reasoning the exact fixed point x∗ of G belongs order interval [x∗, x∗]. In

particular, both (x1, y1) and (x2, y2) approximate an exact solution (x1, y1) of system (3.9), (3.10)

with the required precision. Moreover, x1 ≤ x1 ≤ y1 and x2 ≤ y1 ≤ y2.

Remarks 3.1. The results of Lemma 3.1 and its dual and Corollary 3.1 could be combined to obtain

upper and lower estimates also to fixed points of set-valued functions.



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Fixed Point Results for Set-Valued ... 131

4 Special cases

In this section we shall first present existence and comparison results for equations and inclusions

in ordered normed spaces. Next we formulate in ordered topological spaces some existence and

comparison results derived in section 3.

4.1 Equations and inclusions in ordered normed spaces

Definition 4.1. A closed subset E+ of a normed space E is called an order cone if E+ + E+ ⊆ E+,

E+ ∩ (−E+) = {0} and cE+ ⊆ E+ for each c ≥ 0. The space E, equipped with an order relation

’≤’, defined by

x ≤ y if and only if y − x ∈ E+

is called an ordered normed space.

It is easy to see that the above defined order relation ≤ is a partial ordering in E.

Lemma 4.1. Let C be a chain in an ordered normed space E, and assume that each monotone

sequence of C has a weak limit in E. Then C contains an increasing sequence which converges

weakly to sup C and a decreasing sequence which converges weakly to inf C. This result holds also

when weak convergence is replaced by strong convergence.

Proof. C has by [6], Lemma 1.1.2 a well-ordered cofinal subchain W . Since all increasing se-

quences of W have weak limits, there is by [2], Lemma A.3.1 an increasing sequence (xn) in W

which converges weakly to x = sup W = sup C. Noticing that −C is a chain whose increas-

ing sequences have weak limits, there exists an increasing sequence (xn) of −C which converges

weakly to sup(−C) = − inf C. Denoting yn = −xn, we obtain a decreasing sequence (yn) of C

which converges weakly to inf C. In the case of strong convergence the conclusion follows from [6],

Proposition 1.1.5.

In what follows, E is an ordered normed space having some of the following properties.

(E0) Bounded and monotone sequences of E have weak limits.

(E1) x+ = sup{0, x} exists, and ‖x+‖ ≤ ‖x‖ for every x ∈ E.

When c ∈ E and R ∈ [0, ∞), denote BR(c) := {x ∈ E : ‖x − c‖ ≤ R}. Recall (cf. e.g., [11])

that if a sequence (xn) of a normed space E converges weakly to x, then (xn) is bounded, i.e.

supn ‖xn‖ < ∞, and

‖x‖ ≤ lim inf
n→∞

‖xn‖. (4.1)

The next auxiliary result is needed in the sequel.



132 Seppo Heikkilä CUBO
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Lemma 4.2. Let E be an ordered normed space with properties (E0) and (E1). If c ∈ E and

R ∈ (0, ∞), then c is an order center of BR(c), and for every chain C of BR(c) both sup C and

inf C exist and belong to BR(c).

Proof. Since

sup{c, x} = (x − c)+ − c and inf{c, x} = c − (c − x)+, for all x ∈ E, (4.2)

then (E1) and (4.2) imply that

‖ sup{c, x} − c‖ = ‖ inf{c, x} − c‖ = ‖(x − c)+‖ ≤ ‖x − c‖ ≤ R for every x ∈ BR(c).

Thus both sup{c, x} and inf{c, x} belong to BR(c).

Let C be a chain in BR(c). Since C is bounded there is by (E0) and Lemma 4.1 an increasing

sequence (xn) in C which converges weakly to x = sup C. Since ‖xn − c‖ ≤ R for each n, it follows

from (4.1) that

‖x − c‖ ≤ lim inf
n→∞

‖xn − c‖ ≤ R.

Thus x = sup C exists and belongs to BR(c). Similarly one can show that inf G[C] exists in E and

belongs to BR(c).

Applying Theorem 3.2 and Lemmas 4.1 and 4.2 we obtain the following fixed point results.

Theorem 4.1. Given a subset P of E, assume that G : P → P is increasing, and that G[P ] ⊆

BR(c) ⊆ P for some c ∈ E and R ∈ (0, ∞). Then G has

(a) minimal and maximal fixed points;

(b) least and greatest fixed points x∗ and x
∗ in the order interval [x, x] of P , where x is the greatest

solution of x = inf{c, G(x)}, and x is the least solution of x = sup{c, G(x)}.

Moreover, x∗, x∗, x and x are all increasing with respect to G.

Proof. Let C be a chain in P . Since G[C] is a chain in BR(c), then both sup G[C] and inf G[C]

exist in E and belongs to BR(c) ⊆ P by Lemma 4.2. Because c is an order center of BR(c) and

ocl(G[P ]) ⊆ G[P ] ⊆ BR(c) ⊆ P , then c is an order center of ocl(G[P ]) in P .

The above proof shows that the hypotheses of Theorem 3.2 are valid.

In the set-valued case we have the following consequence of Theorem 3.1.

Theorem 4.2. Assume that P is a subset of E which contains BR(c) for some c ∈ E and R ∈

(0, ∞). Let F : P → 2P \ ∅ be an increasing mapping whose values are weakly compact in E, and

whose range F[P ] is contained in BR(c). Then F has minimal and maximal fixed points.

Remarks 4.1. Each of the following spaces has properties (E0) and (E1) (as for the proofs, see,

e.g. [1, 2, 3, 5, 6, 7, 8, 10]):



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Fixed Point Results for Set-Valued ... 133

(a) A Sobolev space W 1,p(Ω) or W
1,p
0

(Ω), 1 < p < ∞, ordered a.e. pointwise, where Ω is a bounded

domain in R
N

.

(b) A finite-dimensional normed space ordered by a cone generated by a basis.

(c) lp, 1 ≤ p ≤ ∞, normed by p-norm and ordered coordinatewise.

(d) Lp(Ω), 1 ≤ p ≤ ∞, normed by p-norm and ordered a.e. pointwise, where Ω is a σ-finite measure

space.

(e) A separable Hilbert space whose order cone is generated by an orthonormal basis.

(f) A weakly complete Banach lattice or a UMB-lattice (cf.[1]).

(g) Lp(Ω, Y ), 1 ≤ p ≤ ∞, normed by p-norm and ordered a.e. pointwise, where Ω is a σ-finite

measure space and Y is any of the spaces (b)–(f).

(h) Newtonian spaces N 1,p(Y ), 1 < p < ∞, ordered a.e. pointwise, where Y is a metric measure

space.

Thus the results of Theorems 4.1–4.2 hold if E is any of the spaces listed in (a)–(h).

4.2 Fixed point results in ordered topological spaces

Let P = (P, ≤) be an ordered topological space, i.e., for each a ∈ P the order intervals [a) = {x ∈

P : a ≤ x} and (a] = {x ∈ P : x ≤ a} are closed in the topology of P . In what follows, we assume

that P has the following property:

(C) Each well-ordered chain C of P whose increasing sequences converge in P contains an in-

creasing sequence which converges to sup C, and each inversely well-ordered chain C of P

whose decreasing sequences converge in P contains a decreasing sequence which converges to

inf C.

Corollary 4.1. The following ordered topological spaces have property (C).

(a) Ordered metric spaces.

(b) Order closed subsets of ordered normed spaces equipped with a norm topology.

(c) Order closed subsets of ordered normed spaces equipped with a weak topology.

(d) Ordered topological spaces which satisfy the second countability axiom.

Proof. (a) and (b) follow from the result of [6], Proposition 1.1.5 and from its dual.

(c) is a consequence of [2], Appendix, Lemma A.3.1 and its dual.

(d) If P is an ordered topological spaces which satisfies the second countability axiom, then

each chain of P is separable, whence P has property (C) by the result of [6], Lemma 1.1.7 and its

dual.

The following result is a consequence of Proposition 3.6.

Proposition 4.1. Given an ordered topological space P with property (C), assume that G : P → P

is an increasing function.



134 Seppo Heikkilä CUBO
10, 4 (2008)

(a) If G[P ] has an upper bound in P , and if G maps decreasing sequences of P to convergent

sequences, then G has greatest and minimal fixed points.

(b) If G[P ] has a lower bound in P , and if G maps increasing sequences of P to convergent

sequences, then G has least and maximal fixed points.

Proof. (a) Let D be an inversely well-ordered chain in P . Since G is increasing, then G[D] is

inversely well-ordered. Every decreasing sequence of G[D] is of the form (G(xn)), where (xn) is a

decreasing sequence in D. Thus the hypotheses of (a) and property (C) imply that x∗ = inf G[D]

exists and belongs to P . It then follows from Proposition 3.6(b) that G has the greatest fixed point

and a minimal fixed point.

The conclusions of (b) is a similar consequence of Proposition 3.6(a).

The next fixed point result is a consequence of Proposition 4.1 and Lemma 4.1.

Corollary 4.2. Let P be an order closed subset of an ordered normed space E whose (order)

bounded and monotone sequences have weak or strong limits, and let G : P → P be increasing.

(a) If G[P ] has an upper bound in P , and if G maps decreasing sequences of P to (order) bounded

sequences, then G has the greatest and a minimal fixed point.

(b) If G[P ] has a lower bound in P , and G maps increasing sequences of P to (order) bounded

sequences, then G has the least and a maximal fixed point.

The next result is a consequence of Theorem 3.2.

Theorem 4.3. Given an ordered topological space P with property (C), assume that G : P → P

is increasing and maps monotone sequences of P to convergent sequences.

(a) If c is a sup-center of ocl(G[P ]) in P , then G has minimal and maximal fixed points. More-

over, G has the greatest fixed point x∗ in (x], where x is the least solution of the equation

x = sup{c, G(x)}. Both x and x∗ are increasing with respect to G.

(b) If c is an inf-center of ocl(G[P ]) in P , then G has minimal and maximal fixed points. Moreover,

G has the least fixed point x∗ in [x), where x is the greatest solution of the equation x = inf{c, G(x)}.

Both x and x∗ are increasing with respect to G.

As a consequence of Propositions 3.1 and 3.2 and Theorem 3.1 we get the following result.

Proposition 4.2. Let P be an ordered topological space with property (C), and let the values of

F : P → 2P \ ∅ be compact.

(a) Assume that following hypothesis holds.

(F+) If (xn) and (yn) are increasing and yn ∈ F(xn) for every n, then (yn) converges.

If the set S+ = {x ∈ P : [x) ∩ F(x) 6= ∅} is nonempty, then F has a maximal fixed point.

(b) Assume that F satisfies the following hypothesis.



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Fixed Point Results for Set-Valued ... 135

(F−) If (xn) and (yn) are decreasing and yn ∈ F(xn) for every n, then (yn) converges.

If the set S− = {x ∈ P : (x] ∩ F(x) 6= ∅} is nonempty, then F has a minimal fixed point.

(b) Assume that the hypotheses (F±) hold. If ocl(F[P ]) has a sup-center or an inf-center in P ,

then F has minimal and maximal fixed points.

Received: April 2008. Revised: April 2008.

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[3] S. Carl and S. Heikkilä, Elliptic problems with lack of compactness via a new fixed point

theorem, J. Differential Equations 186 (2002), 122–140.

[4] S. Heikkilä, A method to solve discontinuous boundary value problems, Nonlinear Anal.,

47: 2387–2394, 2001.

[5] S. Heikkilä, Operator equations in ordered function spaces, in R.P. Agarwal and D. O’Regan

(Eds.), Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80:th Birthday,

Kluwer Acad. Publ., Dordrecht, ISBN 1-4020-1688-3 (2003), 595–616.

[6] S. Heikkilä and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous

Nonlinear Differential Equations, Marcel Dekker, Inc., New York, 1994.

[7] Juha Kinnunen and Olli Martio, Nonlinear potential theory on metric spaces, Illinois J.

Math. 46, 3, 857–883, 2002.

[8] J. Lindenstraus and L. Tzafriri, Classical Banach Spaces II, Function Spaces, Springer-

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[9] H.L. Royden, Real Analysis, The MacMillan Company, London, 1968.

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