Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 745769, 13 pages
doi:10.1155/2010/745769
Research Article

Fixed Points of Discontinuous Multivalued Operators in Ordered Spaces with Applications

Shihuang H. Hong and Zheyong Qiu

Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China

Received 24 September 2009; Accepted 3 March 2010

Academic Editor: T. Dominguez Benavides

Copyright © 2010 Shihuang H. Hong and Zheyong Qiu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Existence theorems of fixed points for multivalued increasing operators in partially ordered spaces are presented. Here neither the continuity nor compactness is assumed for multivalued operators. As an application, we lead to the existence principles for integral inclusions of Hammerstein type multivalued maps.

1. Introduction

The influence of fixed point theorems for contractive and nonexpansive mappings (see [1, 2]) on fixed point theory is so huge that there are many results dealing with fixed points of mappings satisfying various types of contractive and nonexpansive conditions. On the other hand, it is also huge that well-known Brouwer's and Schauder's fixed point theorems for set-contractive mappings exert an influence on this theory. However, if a mapping is not completely continuous, in general, it is difficult to verify that the mapping satisfies the set-contractive condition. In 1980, Mönch [3] has obtained the following important fixed point theorem which avoids the above mentioned difficulty.

Theorem 1.1. Let 𝐸 be a Banach space, 𝐾 ⊂ 𝐸 a closed convex subset. Suppose that (single) operator 𝐹 ∶ 𝐾 → 𝐾 is continuous and satisfies that (i)there exists 𝑥 ∈ 𝐾 such that if 𝐶 ⊂ 𝐾 ∩ c o ( { 𝑥 } ∪ 𝐹 ( 𝐶 ) ) is countable, then 𝐶 is relatively compact,then 𝐹 has a fixed point in 𝐾 .

It has been observed that continuity is an ideal and important property in the above cited works, while in some applications the mapping under consideration may not be continuous, yet at the same time it may be “not very discontinuous”. this idea has motivated many authors to study corresponding problems, for instance, the stability of Brouwer's fixed point theorem [4], similar result for nonexpansive mappings [5], and existence and approximation of the synthetic approaches to fixed point theorems [6]. Recently, fixed point theory for discontinuous multivalued mappings has attracted much attention and many authors studied the existence of fixed points for such mappings. We refer to [7–11]. For example, Hong [8] has extended Mönch [3] to discontinuous multivalued operators in ordered Banach spaces by using a quite weak compactness condition; that is, assuming the following condition is satisfied.

(H)If 𝐶 = { 𝑥 𝑛 } is a countable totally ordered set and 𝐶 ⊂ w c l ( { 𝑥 1 } ∪ 𝐴 ( 𝐶 ) ) , then 𝐶 is weakly relatively compact. Here 𝐴 is a multivalued operator and w c l ( 𝐵 ) denotes the weak closure of the set 𝐵 .

The purpose of this paper is to present some results on fixed point theorems of Mönch type of multivalued increasing operators for which neither the continuity nor the compactness is assumed in ordered topological spaces. However, we will use the following hypothesis.

(H1)If 𝐶 = { 𝑥 𝑛 } ⊂ 𝐾 is a countable totally ordered set and 𝐶 ⊂ c l ( { 𝑥 1 } ∪ 𝐴 ( 𝐶 ) ) , then 𝐶 has a supremum.

𝐸 is a topological vector space endowed with partial ordering “ ≤ ”, c l ( 𝐵 ) stands for the closure of the set 𝐵 , and 𝐾 = { 𝑥 ∈ 𝐸 ∣ 𝑥 ≥ 𝑢 0 } with 𝑢 0 ∈ 𝐸 is a given ordered set of 𝐸 .

This paper is organized as follows. In Section 2, we introduce some definitions and preliminary facts from partially ordered theory and multivalued analysis which are used later. In especial, we introduce a new partial ordering of sets which forms a basis to our main results. In Section 3, we state and prove existence of fixed points, also, maximal and minimal fixed point theorem is presented for discontinuous multivalued increasing operators which are our main results. To illustrate the applicability of our theory, in Section 4, we discuss the existence of solutions to the Hammerstein integral inclusions of the form

𝑢 ( 𝑡 ) ∈ 𝑇 0 [ ] . 𝑘 ( 𝑡 , 𝑠 ) 𝐺 ( 𝑠 , 𝑢 ( 𝑠 ) ) 𝑑 𝑠 a . e . o n 0 , 𝑇 ( 1 . 1 )

2. Preliminaries

Let ( 𝐸 , ≤ ) be a partially ordered topological vector space. By the notation “ 𝑥 < 𝑦 ” we always mean that 𝑥 ≤ 𝑦 and 𝑥 ≠ 𝑦 . Let 2 𝐸 stand for the collection of all nonempty subsets of 𝐸 . Take 𝑢 0 ∈ 𝐸 and let 𝐾 𝑢 0 = { 𝑥 ∈ 𝐸 ∣ 𝑥 ≥ 𝑢 0 } be a given ordered set of 𝐸 . The ordered interval of 𝐸 is written as [ 𝑢 , 𝑣 ] = { 𝑥 ∈ 𝐸 ∶ 𝑢 ≤ 𝑥 ≤ 𝑣 } .

For two subsets 𝑃 , 𝑄 of 𝐸 , we write 𝑃 ≤ 𝑄 (or 𝑄 ≥ 𝑃 ) if

∀ 𝑝 ∈ 𝑃 , ∃ 𝑞 ∈ 𝑄 s u c h t h a t 𝑝 ≤ 𝑞 . ( 2 . 1 )

Given a nonempty subsets Ω of 𝐸 we say that 𝐴 ∶ Ω ⊸ 2 𝐸 is increasing upwards if 𝑢 , 𝑣 ∈ Ω , 𝑢 ≤ 𝑣 , and 𝑥 ∈ 𝐴 ( 𝑢 ) imply that there exists 𝑦 ∈ 𝐴 ( 𝑣 ) such that 𝑥 ≤ 𝑦 . 𝐴 is increasing downwards if 𝑢 , 𝑣 ∈ Ω , 𝑢 ≤ 𝑣 , and 𝑦 ∈ 𝐴 ( 𝑣 ) imply an existence of 𝑥 ∈ 𝐴 ( 𝑢 ) such that 𝑥 ≤ 𝑦 . If 𝐴 is increasing upwards and downwards we say that 𝐴 is increasing.

Let Γ ⊂ 𝐸 be nonempty. The element 𝑦 ∈ 𝐸 is called an upper (lower) bound of Γ if 𝑥 ≤ 𝑦 ( 𝑥 ≥ 𝑦 ) whenever 𝑥 ∈ Γ . Γ is called upper (lower) bounded with respect to the ordering if its upper (lower) bounds exist. The element 𝑧 ∈ 𝐸 is called a supremum of Γ , written as 𝑧 = s u p Γ , if 𝑧 is an upper bound and 𝑧 ≤ 𝑦 as long as 𝑦 is another upper bound of Γ . Similarly, we can define the infimum i n f Γ of Γ .

Throughout this paper, unless otherwise mentioned, the partial ordering of 𝐸 always introduced by a closed cone if 𝐸 is a Banach space. The following lemmas will be used in after sections.

Lemma 2.1 (see [12]). Let 𝐸 be an ordered Banach space and 𝐵 a totally ordered and weakly relatively compact subset of 𝐸 , then there exists 𝑥 ∗ ∈ w c l ( 𝐵 ) such that 𝑥 ≤ 𝑥 ∗ foe all 𝑥 ∈ 𝐵 .

An ordered topological vector space 𝐸 is said to have the limit ordinal property if 𝑥 𝑛 , 𝑦 𝑛 ∈ 𝐸 with 𝑥 𝑛 ≤ 𝑦 𝑛 for 𝑛 = 1 , 2 , … , and 𝑥 𝑛 → 𝑥 ∗ , 𝑦 𝑛 → 𝑦 ∗ for 𝑛 → ∞ imply 𝑥 ∗ ≤ 𝑦 ∗ . By an analogy of the proof of Lemma 1.1.2 in [12], we have the following.

Lemma 2.2. If 𝐸 has the limit ordinal property and { 𝑥 𝑛 } is a relatively compact monotone sequence of 𝐸 , then { 𝑥 𝑛 } is convergent. Moreover, 𝑥 𝑛 ≤ 𝑥 ∗ if { 𝑥 𝑛 } is increasing and 𝑥 ∗ ≤ 𝑥 𝑛 if { 𝑥 𝑛 } is decreasing for 𝑛 = 1 , 2 , … . Here l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑥 ∗ .

Remark 2.3. Under the assumptions of Lemma 2.2, it is evident that 𝑥 ∗ is the supremum (infimum) of increasing (decreasing) sequence { 𝑥 𝑛 } .

Lemma 2.4. Let the increasing sequence { 𝑥 𝑛 } have the supremum 𝑧 . If { 𝑥 𝑛 𝑖 } is a infinity subsequence of { 𝑥 𝑛 } , then { 𝑥 𝑛 𝑖 } has the supremum 𝑧 , too.

Proof. Evidently, 𝑧 is an upper bound of { 𝑥 𝑛 𝑖 } . Let 𝑦 be the other one, then 𝑥 𝑛 𝑖 ≤ 𝑦 for 𝑖 = 1 , 2 , … . For any given 𝑛 , since { 𝑥 𝑛 𝑖 } is infinity, there exists 𝑖 0 such that 𝑥 𝑛 ≤ 𝑥 𝑛 𝑖 0 , which implies that 𝑥 𝑛 ≤ 𝑦 for all 𝑛 ≥ 1 . From the definition of supremums it follows that 𝑧 ≤ 𝑦 , that is, 𝑧 is the supremum of { 𝑥 𝑛 𝑖 } .

Lemma 2.5. Suppose that every countable totally ordered subset of the partially ordered set 𝑌 has a supremum in 𝑌 . Let the operator 𝐹 ∶ 𝑌 → 𝑌 satisfy 𝐹 ( 𝑥 ) ≥ 𝑥 for all 𝑥 ∈ 𝑌 , then there exists 𝑥 0 ∈ 𝑌 such that 𝐹 ( 𝑥 0 ) = 𝑥 0 .

Proof. Take 𝑧 0 ∈ 𝑌 any fixed and let 𝑧 𝑖 + 1 = 𝐹 ( 𝑧 𝑖 ) for 𝑖 = 0 , 1 , … , then 𝑧 𝑖 + 1 ≥ 𝑧 𝑖 that is, { 𝑧 𝑖 } is increasing. From our assumption it follows that { 𝑧 𝑖 } has a supremum denoted by 𝑧 1 0 = s u p 𝑧 𝑖 . Let Γ 1 = 𝑧 0 , 𝑧 1 ∪ 𝑧 , … 1 0 . ( 2 . 2 ) If 𝑧 1 0 = 𝐹 ( 𝑧 1 0 ) , then the conclusion of the lemma is proved. Otherwise, take 𝑧 1 𝑖 = 𝐹 ( 𝑧 1 𝑖 − 1 ) for 𝑖 = 1 , 2 , … . Again, the set { 𝑧 1 0 , 𝑧 1 1 , … } has the supremum 𝑧 2 0 = s u p 𝑧 1 𝑖 . Denote Γ 2 = { 𝑧 1 0 , 𝑧 1 1 , … } ∪ { 𝑧 2 0 } . If 𝑧 2 0 = 𝐹 ( 𝑧 2 0 ) , then the conclusion of the lemma is proved. Otherwise, take 𝑧 2 𝑖 = 𝐹 ( 𝑧 2 𝑖 − 1 ) for 𝑖 = 1 , 2 , … , and let Γ 3 = { 𝑧 2 0 , 𝑧 2 1 , … } ∪ { 𝑧 3 0 } with 𝑧 3 0 = s u p 𝑧 2 𝑖 . In general, having defined Γ 𝑘 = { 𝑧 0 𝑘 − 1 , 𝑧 1 𝑘 − 1 , … } ∪ { 𝑧 𝑘 0 } with 𝑧 𝑖 𝑘 − 1 = 𝐹 ( 𝑧 𝑖 𝑘 − 1 ) and 𝑧 𝑘 0 = s u p 𝑧 𝑖 𝑘 − 1 , where 𝑧 0 𝑖 = 𝑧 𝑖 and 𝑘 , 𝑖 = 1 , 2 , … , if 𝑧 𝑘 0 = 𝐹 ( 𝑧 𝑘 0 ) , which completes the proof. Otherwise, repeating this process, either the conclusion of the lemma is proved, or we can obtain a set sequence Γ 1 , Γ 2 , … satisfying (i) Γ 𝑘 = { 𝑧 1 𝑘 − 1 , 𝑧 2 𝑘 − 1 , … } ∪ { 𝑧 𝑘 0 } with 𝑧 𝑘 0 = s u p 𝑧 𝑖 𝑘 − 1 and 𝑧 𝑘 𝑖 = 𝐹 ( 𝑧 𝑘 𝑖 − 1 ) , 𝑖 , 𝑘 = 1 , 2 , … ; (ii) 𝑧 𝑘 𝑖 − 1 ≤ 𝑧 𝑘 𝑖 for 𝑖 , 𝑘 = 1 , 2 , … ; (iii) 𝑧 𝑗 𝑘 − 1 ≤ 𝑧 𝑘 𝑡 𝑗 , 𝑡 = 0 , 1 , 2 , … , and 𝑧 0 𝑖 = 𝑧 𝑖 . Let ⋃ Γ = ∞ 𝑘 = 1 Γ 𝑘 , then Γ is a countable subset and 𝑧 0 ≤ 𝑥 ∀ 𝑥 ∈ Γ . ( 2 . 3 ) We claim that 𝐹 ( Γ ) ⊂ Γ . ( 2 . 4 ) In fact, for any 𝑦 ∈ 𝐹 ( Γ ) , there exists 𝑥 ∈ Γ such that 𝑦 = 𝐹 ( 𝑥 ) . There exists Γ 𝑘 such that 𝑥 ∈ Γ 𝑘 . If 𝑥 = 𝑧 𝑖 𝑘 − 1 for some nature number 𝑖 , then 𝑦 = 𝐹 ( 𝑥 ) = 𝑧 𝑘 − 1 𝑖 + 1 ∈ Γ 𝑘 which yields 𝑦 ∈ Γ . Otherwise, we have 𝑥 = s u p 𝑧 𝑖 𝑘 − 1 = 𝑧 𝑘 0 ∈ Γ 𝑘 + 1 . This implies that 𝑦 = 𝐹 ( 𝑥 ) = 𝐹 ( 𝑘 0 ) = 𝑧 𝑘 1 ∈ Γ 𝑘 + 1 . Consequently, 𝑦 ∈ Γ . From the arbitrariness of 𝑦 it follows that (2.4) is satisfied.
Finally, combining ( i i ) and ( i i i ) we see easily that Γ is totally ordered. Our hypothesis guarantees that Γ has a supremum, written as 𝑥 ∗ = s u p Γ . Note that (2.4) guarantees 𝐹 ( 𝑥 ∗ ) ∈ Γ , we have 𝐹 ( 𝑥 ∗ ) ≤ 𝑥 ∗ . On the other hand, the definition of 𝐹 ensures that 𝐹 ( 𝑥 ∗ ) ≥ 𝑥 ∗ . Hence 𝐹 ( 𝑥 ∗ ) = 𝑥 ∗ . This proof is completed.

Let Ω be a nonempty subset of 𝐾 𝑢 0 . In this section we impose the following hypotheses on the increasing upwards multivalued operator 𝐴 ∶ Ω ⊸ 2 𝐸 . Set

ℛ = { 𝑥 ∈ Ω ∣ t h e r e e x i s t s 𝑢 ∈ 𝐴 𝑥 s u c h t h a t 𝑥 ≤ 𝑢 } ( 2 . 5 ) and for any 𝑥 ∈ ℛ define that

𝐶 ( 𝑥 ) = 𝑥 , 𝑢 1 , 𝑢 2 , … , 𝑢 𝑛 , … , 𝐷 ( 𝑥 ) = 𝐶 ( 𝑥 ) ∪ { 𝑤 ( 𝑥 ) } , ( 2 . 6 ) where, 𝑤 ( 𝑥 ) = s u p 𝐶 ( 𝑥 ) and 𝑢 𝑖 ( 𝑖 = 1 , 2 , … ) is given as follows: since 𝑥 ∈ ℛ , there exists 𝑢 1 ∈ 𝐴 𝑥 such that 𝑥 ≤ 𝑢 1 . In virtue of the fact that 𝐴 is increasing upwards, there exists 𝑢 2 ∈ 𝐴 𝑢 1 such that 𝑢 1 ≤ 𝑢 2 . On the analogy of this process, there exists 𝑢 𝑛 ∈ 𝐴 𝑢 𝑛 − 1 such that 𝑢 𝑛 − 1 ≤ 𝑢 𝑛 for 𝑛 = 2 , 3 , … , Obviously, 𝐶 ( 𝑥 ) ⊂ c l ( { 𝑥 } ∪ 𝐴 ( 𝐶 ) ) , thus, the condition (H1) guarantees that the supremum 𝑤 ( 𝑡 ) of 𝐶 ( 𝑥 ) exists.

Remark 2.6. In general, the sequences of these kinds, { 𝑢 𝑛 } , may not be unique, that is, every { 𝑢 𝑛 } corresponds to 𝐶 ( 𝑥 ) , moreover, corresponds to 𝐷 ( 𝑥 ) . For given 𝑥 ∈ ℛ , we denote with 𝒞 ( 𝑥 ) and 𝒟 ( 𝑥 ) the families of 𝐶 ( 𝑥 ) and 𝐷 ( 𝑥 ) as above, respectively.
In addition, if 𝐸 has the limit ordinal property, 𝐷 ( 𝑥 ) is a closed set for any 𝑥 ∈ ℛ . In fact, let { 𝑢 𝑛 𝑖 } be any infinity subsequence of 𝐷 for which
𝑢 𝑛 𝑖 ⟶ 𝑥 ∗ f o r 𝑖 ⟶ ∞ . ( 2 . 7 ) observing that { 𝑢 𝑛 𝑖 } is increasing, by Lemma 2.2 we get that 𝑥 ∗ is a supremum of { 𝑢 𝑛 𝑖 } and by Lemma 2.4 we get 𝑤 ( 𝑥 ) = 𝑥 ∗ .

Definition 2.7. A set Γ is said to be sup-closed if the supremum of each countable subset of Γ (provided that it exists) belongs to Γ . A multivalued operator 𝐴 ∶ Ω ⊸ 2 𝐸 is said to have sup-closed values if 𝐴 𝑥 is sup-closed for each 𝑥 ∈ Ω .

Defining

𝑋 ( 𝑥 ) = { 𝑢 ∶ t h e r e e x i s t s 𝐷 ( 𝑥 ) ∈ 𝒟 ( 𝑥 ) s u c h t h a t 𝑢 ∈ 𝐷 ( 𝑥 ) } . ( 2 . 8 )

Lemma 2.8. Let 𝐸 be an ordered topological space, Ω a nonempty subset of 𝐾 𝑢 0 with 𝑢 0 ∈ 𝐸 ; let 𝐴 ∶ Ω ⊸ 2 𝐸 have sup-closed values and satisfy hypothesis (H1). Moreover, assume that (H2) 𝐴 is increasing upwards and satisfies 𝑢 0 ≤ 𝐴 𝑢 0 , then for any 𝐶 ( 𝑥 ) ∈ 𝒞 ( 𝑥 ) , 𝐶 ( 𝑥 ) has the supremum 𝑤 ( 𝑥 ) which belongs to ℛ , that is, 𝑤 ( 𝑥 ) ≤ 𝑥 ∗ f o r s o m e 𝑥 ∗ ∈ 𝐴 ( 𝑤 ( 𝑥 ) ) . ( 2 . 9 )

Proof. It is clear that 𝐶 ( 𝑥 ) has the supremum 𝑤 ( 𝑥 ) ∈ 𝐸 . For any 𝑢 𝑖 ∈ 𝐶 ( 𝑥 ) / { 𝑥 } , from 𝑢 𝑖 ∈ 𝐴 ( 𝑢 𝑖 − 1 ) and 𝑢 𝑖 − 1 ≤ 𝑤 ( 𝑥 ) there exists 𝑥 𝑖 ∈ 𝐴 ( 𝑤 ( 𝑥 ) ) such that 𝑢 𝑖 ≤ 𝑥 𝑖 . We can assume that the sequence { 𝑥 𝑖 } is increasing. Indeed, if 𝑥 𝑖 ≤ 𝑥 𝑖 + 1 for 𝑖 = 1 , 2 , … , our purpose is reached. Otherwise, there exists 𝑖 0 such that 𝑥 𝑖 0 ≰ 𝑥 𝑖 0 + 1 , then we take 𝑥 𝑖 0 + 1 instead of 𝑥 𝑖 0 . Let 𝑀 = { 𝑤 ( 𝑥 ) , 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 , … } , then 𝑀 ⊂ c l ( { 𝑤 ( 𝑥 ) } ∪ 𝐴 ( 𝑀 ) ) . Condition (H1) guarantees that 𝑀 has a supremum 𝑥 ∗ = s u p 𝑀 . Clearly, 𝑤 ( 𝑥 ) ≤ 𝑥 ∗ . By virtue of the fact that 𝐴 has sup-closed values, we have 𝑥 ∗ ∈ 𝐴 ( 𝑤 ( 𝑥 ) ) . This proof is complete.

For the sake of convenience, in this paper, by 𝑤 ( 𝑥 ) we always stand for the supremum of 𝐶 ( 𝑥 ) . For given 𝑥 ∈ ℛ , let 𝒲 ( 𝑥 ) be a set consisting of all 𝑤 ( 𝑥 ) given as in Lemma 2.8, then 𝒲 is an increasing map. Now for any 𝑢 𝑛 ∈ 𝐶 ( 𝑥 ) Lemma 2.4 shows 𝐷 ( 𝑢 𝑛 ) ⊂ 𝐷 ( 𝑥 ) , thus, 𝒲 ( 𝑢 𝑛 ) ⊂ 𝒲 ( 𝑥 ) . Define

𝑍 = { 𝐷 ( 𝑥 ) ∶ 𝑥 ∈ ℛ } . ( 2 . 1 0 ) It is obvious that 𝐷 ( 𝑢 0 ) ∈ 𝑍 . Hence, 𝑍 is nonempty. A relation ” ≤ 1 ” on Z is defined as follows (it is easy to see that ( 𝑍 , ≤ 1 ) is a partially ordered set):

𝐷 ( 𝑥 ) = 𝐷 ( 𝑦 ) ⇔ 𝑥 = 𝑦 , 𝑤 ( 𝑥 ) = 𝑤 ( 𝑦 ) ; 𝐷 ( 𝑥 ) < 1 𝐷 ( 𝑦 ) ⇔ 𝑥 < 𝑦 and 𝑤 ( 𝑥 ) ≤ 𝑤 ( 𝑦 ) .

Remark 2.9. It is clear we may assume that, for any 𝑢 ∈ 𝐷 ( 𝑥 ) , there exists 𝑣 ∈ 𝐷 ( 𝑦 ) such that 𝑢 ≤ 𝑣 if 𝐷 ( 𝑥 ) < 1 𝐷 ( 𝑦 ) .
Let us assume that there exists some 𝑢 0 ∈ ℛ such that
(H3) 𝒲 ( 𝑢 0 ) ⊂ c l ( 𝐴 ( 𝑋 ( 𝑢 0 ) ) ) .
Define
𝑆 = { 𝒟 ( 𝑥 ) ∶ 𝑥 ∈ ℛ , 𝒲 ( 𝑥 ) ⊂ c l ( 𝐴 ( 𝑋 ( 𝑥 ) ) ) } . ( 2 . 1 1 ) Obviously, 𝒟 ( 𝑢 0 ) ∈ 𝑆 , that is, 𝑆 is nonempty if 𝐴 is increasing upwards. Now we denote ≤ 2 as a relation on 𝑆 defined by, for any 𝒟 ( 𝑥 ) , 𝒟 ( 𝑦 ) ∈ 𝑆 , (I) 𝒟 ( 𝑥 ) = 𝒟 ( 𝑦 ) ⇔ 𝑥 = 𝑦 ; (II) 𝒟 ( 𝑥 ) < 2 𝒟 ( 𝑦 ) ⇔ ( 𝑎 ) f o r a l l 𝐷 ( 𝑥 ) ∈ 𝒟 ( 𝑥 ) , there exists 𝐷 ( 𝑦 ) ∈ 𝒟 ( 𝑦 ) such that 𝐷 ( 𝑥 ) < 1 𝐷 ( 𝑦 ) and (b)there exists a countable at most and totally ordered subset 𝑄 ⊂ ℛ such that ( b 1 ) 𝑥 < 𝑞 < 𝑦 for any 𝑞 ∈ 𝑄 . ( b 2 ) There exists 𝐷 ( 𝑥 ) ∈ 𝒟 ( 𝑥 ) such that 𝑦 ∈ c l { 𝑤 ( 𝑥 ) } ∪ 𝑞 ∈ 𝑄 𝒲 ( 𝑞 ) , { 𝑦 } ≥ 𝒲 ( 𝑞 ) , 𝑤 ( 𝑥 ) ≤ 𝑞 ( ∀ 𝑞 ∈ 𝑄 ) ; ( 2 . 1 2 ) ( b 3 ) ⋃ 𝑞 ∈ 𝑄 𝑋 ( 𝑞 ) is a totally ordered set and satisfies ⋃ 𝑞 ∈ 𝑄 ⋃ 𝑋 ( 𝑞 ) ⊂ c l ( 𝒲 ( 𝑥 ) ∪ 𝐴 ( 𝑞 ∈ 𝑄 𝑋 ( 𝑞 ) ) ) . 𝑄 is called a link of linking 𝒟 ( 𝑥 ) with 𝒟 ( 𝑦 ) .

Remark 2.10. ( b 2 ) may be satisfied. In fact, we can take empty set as a link of linking 𝒟 ( 𝑥 ) and 𝒟 ( 𝑦 ) . Thus, 𝒟 ( 𝑥 ) < 2 𝒟 ( 𝑦 ) implies that for any 𝐷 ( 𝑥 ) ∈ 𝒟 ( 𝑥 ) we can find 𝐷 ( 𝑦 ) ∈ 𝒟 ( 𝑦 ) such that 𝐷 ( 𝑥 ) < 1 𝐷 ( 𝑦 ) . In this case, we take 𝑤 ( 𝑥 ) = 𝑦 . Besides, 𝑄 can be a finite set, for example, 𝑄 = { 𝑞 1 , 𝑞 2 , … , 𝑞 𝑚 } with 𝑞 1 < 𝑞 2 < ⋯ < 𝑞 𝑚 , then 𝑞 1 = i n f 𝒲 ( 𝑥 ) , 𝑦 = s u p 𝒲 ( 𝑞 𝑚 ) . ( b 3 ) and the condition (H1) ensure ⋃ 𝑞 ∈ 𝑄 𝑋 ( 𝑞 ) to exist the supremum, so, from Lemma 2.2 the element 𝑦 satisfying ( b 2 ) exists.

Lemma 2.11. The relation “ ≤ 2 ” satisfies that (i) 𝒟 ( 𝑥 ) ≤ 2 𝒟 ( 𝑥 ) ;(ii) 𝒟 ( 𝑥 ) ≤ 2 𝒟 ( 𝑦 ) and 𝒟 ( 𝑦 ) ≤ 2 𝒟 ( 𝑥 ) implies 𝒟 ( 𝑥 ) = 𝒟 ( 𝑦 ) ;(iii) 𝒟 ( 𝑥 ) ≤ 2 𝒟 ( 𝑦 ) , 𝐷 ( 𝑦 ) ≤ 2 𝒟 ( 𝑧 ) implies 𝒟 ( 𝑥 ) ≤ 2 𝒟 ( 𝑧 ) .Therefore, ( 𝑆 , ≤ 2 ) is a partially ordered set.

Proof. (i) and (ii) are satisfied. Trivial by (I) and (II)(a). To prove (iii), for any given 𝐷 ( 𝑥 ) ∈ 𝒟 ( 𝑥 ) we take 𝐷 ( 𝑦 ) ∈ 𝒟 ( 𝑦 ) such that 𝐷 ( 𝑥 ) ≤ 1 𝐷 ( 𝑦 ) and we can find 𝐷 ( 𝑧 ) ∈ 𝒟 ( 𝑧 ) such that 𝐷 ( 𝑦 ) ≤ 1 𝐷 ( 𝑧 ) . It is sufficient to assume that at least one of the above equalities does not hold. The definition of < 1 guarantees that 𝑥 ≤ 𝑦 ≤ 𝑧 , ( 2 . 1 3 ) 𝑤 ( 𝑥 ) ≤ 𝑤 ( 𝑦 ) ≤ 𝑤 ( 𝑧 ) , ( 2 . 1 4 ) and at least one strictly inequality in (2.13) holds. The links linking 𝒟 ( 𝑥 ) with 𝒟 ( 𝑦 ) and linking 𝒟 ( 𝑦 ) with 𝒟 ( 𝑧 ) are written, respectively, as 𝑄 ′ and 𝑄 . Let 𝑄 = 𝑄 ′ ∪ 𝑄 ∪ { 𝑦 } , for any 𝑞 ′ ∈ 𝑄 ′ , 𝑞 ∈ 𝑄 , if none of equalities in (2.13) holds, then by (b 1 ) we have 𝑥 < 𝑞 ′ < 𝑦 , 𝑦 < 𝑞 < 𝑧 . ( 2 . 1 5 ) If at least one equality in (2.13) holds, for instance, 𝑥 = 𝑦 , then (b 1 ) and (b 2 ) show that 𝑤 ( 𝑥 ) ≤ 𝑞 ′ ≤ 𝑤 ( 𝑦 ) , 𝑤 ( 𝑦 ) ≤ 𝑞 < 𝑧 . ( 2 . 1 6 ) Hence, 𝑄 ⊂ ℛ is a countable totally ordered subset and satisfies (b 1 ).
Next we will prove that 𝑄 satisfies (b 2 ). It is clear that the following consequences are true, that is, 𝑧 ≥ 𝑤 ( 𝑦 ) ≥ 𝑤 ( 𝑥 ) and ⋃ 𝑧 ∈ c l ( { 𝑤 ( 𝑦 ) } ∪ 𝑞 ′ ′ ∈ 𝑄 ′ ′ 𝒲 ( 𝑞 ⋃ ) ) ⊂ c l ( { 𝑤 ( 𝑥 ) } ∪ 𝑞 ∈ 𝑄 𝒲 ( 𝑞 ) ) . It is easy to see that { 𝑧 } ≥ 𝒲 ( 𝑞 ) for all 𝑞 ∈ 𝑄 by { 𝑧 } ≥ 𝒲 ( 𝑞 ) and 𝒲 ( 𝑞 ′ ′ ) ≥ 𝒲 ( 𝑞 ′ ) for all 𝑞 ′ ∈ 𝑄 ′ . Also, 𝑤 ( 𝑥 ) ≤ 𝑞 for all 𝑞 ∈ 𝑄 .
Finally, we prove that 𝑄 satisfies (b 3 ). F o r a l l 𝑥 1 , 𝑥 2 ∈ ⋃ 𝑞 ∈ 𝑄 𝑋 ( 𝑞 ) , there exist 𝑞 ′ , 𝑞 ∈ 𝑄 with 𝑞 ′ ≤ 𝑞 (because 𝑄 is totally ordered) and 𝐷 ( 𝑞 ′ ) ∈ 𝒟 ( 𝑞 ′ ) , 𝐷 ( 𝑞 ) ∈ 𝒟 ( 𝑞 ) such that 𝑥 1 ∈ 𝐷 ( 𝑞 ′ ) , 𝑥 2 ∈ 𝐷 ( 𝑞 ) . If 𝑞 ′ , 𝑞 ∈ 𝑄 ′ (or 𝑞 ′ , 𝑞 ∈ 𝑄 ), then 𝑥 1 and 𝑥 2 are ordered by (b 3 ). If 𝑞 ′ ∈ 𝑄 ′ , 𝑞 ∈ 𝑄 , from (b 1 ) and (b 2 ) it follows that 𝐷 ( 𝑞 ′ ) < 1 𝐷 ( 𝑞 ) , which shows that 𝑥 1 ≤ 𝑤 ( 𝑞 ′ ) ≤ 𝑞 ≤ 𝑥 2 . To conclude, ⋃ 𝑞 ∈ 𝑄 𝑋 ( 𝑞 ) is totally ordered. Noting that both 𝑄 ′ and 𝑄 have supremums, by the definition of 𝑆 , we have
𝒲 ( 𝑦 ) ⊂ c l ( 𝐴 ( 𝑋 ( 𝑦 ) ) ) . ( 2 . 1 7 ) Therefore, 𝑞 ∈ 𝑄 𝑋 ( 𝑞 ) = 𝑞 ∈ 𝑄 𝑋 ⎛ ⎜ ⎜ ⎝ ( 𝑞 ) 𝑞 ∈ 𝑄 ′ ′ 𝑋 ⎞ ⎟ ⎟ ⎠ 𝑋 ( 𝑞 ) ( 𝑦 ) ⊂ c l 𝒲 ( 𝑥 ) ∪ 𝐴 𝑞 ∈ 𝑄 ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ 𝑋 ( 𝑞 ) c l 𝑤 ( 𝑦 ) ∪ 𝐴 𝑞 ∈ 𝑄 ′ ′ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ 𝒲 𝑋 ( 𝑞 ) 𝑋 ( 𝑦 ) ⊂ c l ( 𝑥 ) ∪ 𝐴 𝑞 ∈ 𝑄 𝑋 ⎛ ⎜ ⎜ ⎝ 𝒲 ⎛ ⎜ ⎜ ⎝ ( 𝑞 ) ( 𝑦 ) ∪ 𝐴 𝑞 ∈ 𝑄 ′ ′ 𝑋 ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ⊂ ( 𝑞 ) c l ( 𝐴 ( 𝑋 ( 𝑦 ) ) ) 𝒲 ( 𝑥 ) ∪ 𝐴 𝑞 ∈ 𝑄 . 𝑋 ( 𝑞 ) ( 2 . 1 8 ) This shows that 𝑄 satisfies (b 3 ). Consequently, 𝒟 ( 𝑥 ) < 2 𝒟 ( 𝑧 ) , which completes this proof.

3. Main Results

Now we can state and prove our main results.

Definition 3.1. 𝑢 ∈ 𝐸 is said to be a fixed point of the multivalued operator 𝐴 if 𝑢 ∈ 𝐴 ( 𝑢 ) . The fixed point 𝑥 ∗ of 𝐴 is said to be a maximal fixed point of 𝐴 if 𝑢 = 𝑥 ∗ whenever 𝑢 ∈ 𝐴 ( 𝑢 ) and 𝑥 ∗ ≤ 𝑢 . If 𝑥 ∗ is a fixed point and if 𝑥 ∗ = 𝑢 whenever 𝑢 ∈ 𝐴 ( 𝑢 ) and 𝑢 ≤ 𝑥 ∗ , we say that 𝑥 ∗ is a minimal fixed point of 𝐴 .

Theorem 3.2. Assume that 𝐸 is an ordered topological space. Let 𝑢 0 ∈ 𝐸 , Ω ⊂ 𝐾 𝑢 0 be nonempty and the multivalued operator 𝐴 ∶ Ω ⊸ 2 𝐸 have sup-closed values such that hypotheses (H1)–(H3) hold. Then 𝐴 admits at least one fixed point in 𝐾 𝑢 0 .

Proof. If 𝑆 has a maximal element 𝒟 ( 𝑥 ∗ ) , then 𝑥 ∗ is a fixed point of 𝐴 . In fact, since 𝑥 ∗ ∈ ℛ , we can find 𝑢 ∈ 𝐴 ( 𝑥 ∗ ) such that 𝑥 ∗ ≤ 𝑢 . From the definition of 𝐶 ( 𝑥 ∗ ) we can let 𝑢 ∈ 𝐷 ( 𝑥 ∗ ) ∈ 𝒟 ( 𝑥 ∗ ) . This implies 𝑢 ≤ 𝑤 ( 𝑥 ∗ ) . We claim that 𝑥 ∗ = 𝑢 . Suppose that 𝑥 ∗ < 𝑢 , then 𝑥 ∗ < 𝑤 ( 𝑥 ∗ ) and 𝐷 ( 𝑥 ∗ ) < 1 𝐷 ( 𝑤 ( 𝑥 ∗ ) ) . Take empty set as a link of linking 𝒟 ( 𝑥 ∗ ) with 𝒟 ( 𝑤 ( 𝑥 ∗ ) ) , we have 𝒟 ( 𝑥 ∗ ) < 2 𝒟 ( 𝑤 ( 𝑥 ∗ ) ) , which contradicts the definition of maximal element.
To prove the existence of maximal element of 𝑆 , by Zorn's lemma, is thus sufficient to show that every totally ordered subset of 𝑆 has an upper bound. Let 𝑀 be any such a subset of 𝑆 . To this purpose, we consider the set ⋃ 𝑁 = 𝒟 ( 𝑥 ) ∈ 𝑀 𝑋 ( 𝑥 ) . Obviously, 𝑁 ⊂ 𝐾 𝑢 0 . We claim that 𝑁 is totally ordered. Indeed, for any 𝑦 1 , 𝑦 2 ∈ 𝑁 , there exist 𝒟 ( 𝑥 1 ) , 𝒟 ( 𝑥 2 ) ∈ 𝑀 and 𝐷 ( 𝑥 1 ) ∈ 𝒟 ( 𝑥 1 ) , 𝐷 ( 𝑥 2 ) ∈ 𝒟 ( 𝑥 2 ) such that 𝑦 1 ∈ 𝐷 ( 𝑥 1 ) , 𝑦 2 ∈ 𝐷 ( 𝑥 2 ) . If 𝐷 ( 𝑥 1 ) = 𝐷 ( 𝑥 2 ) , then 𝑦 1 , 𝑦 2 is ordered. Otherwise, we can assume that 𝐷 ( 𝑥 1 ) < 1 𝐷 ( 𝑥 2 ) , thus, from Lemma 2.8 and (b 2 ) it follows that 𝑦 1 ≤ 𝑤 ( 𝑥 1 ) ≤ 𝑥 2 ≤ 𝑦 2 . Conclusively, 𝑁 is a totally ordered subset.
We will prove that any countable totally ordered subset of 𝑁 has a supremum. It is enough to prove that any given strictly monotone sequence { 𝑦 𝑛 } of 𝑁 there is a supremum. From the definition of 𝑁 , there exist 𝒟 ( 𝑥 𝑛 ) ∈ 𝑀 and 𝐷 ( 𝑥 𝑛 ) ∈ 𝒟 ( 𝑥 𝑛 ) such that 𝑦 𝑛 ∈ 𝐷 ( 𝑥 𝑛 ) for 𝑛 = 1 , 2 , … . For any 𝑥 ∈ ℛ , from the definition of 𝐷 ( 𝑥 ) , it follows 𝐷 ( 𝑥 ) has a supremum. Moreover, Lemma 2.4 guarantees that { 𝑦 𝑛 } has a supremum if 𝑦 𝑛 ∈ 𝐷 ( 𝑥 𝑚 ) with 𝑛 ≥ 𝑚 for some given 𝑚 . It is suffices to consider the fact that there exists a subsequence of { 𝑦 𝑛 } (without loss of generality, we may assume that it is { 𝑦 𝑛 } itself) such that 𝑦 𝑛 ∉ 𝐷 ( 𝑥 𝑚 ) ( 𝑛 ≠ 𝑚 ) .
Case 1. If { 𝑦 𝑛 } is strictly increasing, then 𝑦 𝑖 < 𝑦 𝑖 + 1 ( 𝑖 = 1 , 2 , … ) . We claim that 𝒟 𝑥 1 < 2 𝒟 𝑥 2 < 2 ⋯ < 2 𝒟 𝑥 𝑛 < 2 ⋯ . ( 3 . 1 ) If it is contrary, there exists some 𝑖 such that 𝒟 ( 𝑥 𝑖 + 1 ) ≤ 2 𝒟 ( 𝑥 𝑖 ) . It is easy to know that 𝒟 ( 𝑥 𝑖 + 1 ) ≠ 𝒟 ( 𝑥 𝑖 ) . (b 2 ) implies that 𝑤 ( 𝑥 𝑖 + 1 ) ≤ 𝑥 𝑖 , therefore, 𝑦 𝑖 + 1 ≤ 𝑤 ( 𝑥 𝑖 + 1 ) ≤ 𝑥 𝑖 ≤ 𝑦 𝑖 . This contradicts { 𝑦 𝑛 } increasing. The claim follows.
Taking 𝑄 𝑖 as the link of linking 𝒟 ( 𝑥 𝑖 + 1 ) with 𝒟 ( 𝑥 𝑖 ) for 𝑖 = 1 , 2 , … . Let
𝐶 = ∞ 𝑖 = 1 ⎛ ⎜ ⎜ ⎝ 𝑋 𝑥 𝑖 ∪ ⎛ ⎜ ⎜ ⎝ 𝑞 ∈ 𝑄 𝑖 𝑋 ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ( 𝑞 ) , ( 3 . 2 ) (b 3 ) shows that 𝐶 is countable totally ordered. For any 𝑧 ∈ 𝐶 ⧵ { 𝑥 1 } , there exists 𝑗 such that 𝑧 ∈ 𝑋 ( 𝑥 𝑗 ⋃ ) ∪ ( 𝑞 ∈ 𝑄 𝑗 𝑋 ( 𝑞 ) ) . If 𝑧 = 𝑥 𝑗 , by means of (b 2 ) and 𝒲 𝑥 𝑗 − 1 𝐴 𝑋 𝑥 ∈ c l 𝑗 − 1 , ( 3 . 3 ) we have ⎛ ⎜ ⎜ ⎝ 𝒲 𝑥 𝑧 ∈ c l 𝑗 − 1 ⎛ ⎜ ⎜ ⎝ ∪ 𝐴 𝑞 ∈ 𝑄 𝑗 − 1 𝑋 ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝐴 𝑋 𝑥 ( 𝑞 ) ⊂ c l 𝑗 − 1 ⎛ ⎜ ⎜ ⎝ ∪ 𝐴 𝑞 ∈ 𝑄 𝑗 − 1 𝑋 ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝐴 ⎛ ⎜ ⎜ ⎝ 𝑋 𝑥 ( 𝑞 ) = c l 𝑗 − 1 ∪ ⎛ ⎜ ⎜ ⎝