Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 864536, 32 pages
doi:10.1155/2010/864536
Research Article

Periodic Point, Endpoint, and Convergence Theorems for Dissipative Set-Valued Dynamic Systems with Generalized Pseudodistances in Cone Uniform and Uniform Spaces

Kazimierz Włodarczyk and Robert Plebaniak

Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland

Received 29 September 2009; Accepted 17 November 2009

Academic Editor: Mohamed A. Khamsi

Copyright © 2010 Kazimierz Włodarczyk and Robert Plebaniak. 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

In cone uniform and uniform spaces, we introduce the three kinds of dissipative set-valued dynamic systems with generalized pseudodistances and not necessarily lower semicontinuous entropies, we study the convergence of dynamic processes and generalized sequences of iterations of these dissipative dynamic systems, and we establish conditions guaranteeing the existence of periodic points and endpoints of these dissipative dynamic systems and the convergence to these periodic points and endpoints of dynamic processes and generalized sequences of iterations of these dissipative dynamic systems. The paper includes examples.

1. Introduction

A set-valued dynamic system is defined as a pair ( 𝑋 , 𝑇 ) , where 𝑋 is a certain space and 𝑇 is a set-valued map 𝑇 ∶ 𝑋 → 2 𝑋 ; in particular, a set-valued dynamic system includes the usual dynamic system where 𝑇 is a single-valued map. Here 2 𝑋 denotes the family of all nonempty subsets of a space 𝑋 .

Let ( 𝑋 , 𝑇 ) be a dynamic system. By F i x ( 𝑇 ) , P e r ( 𝑇 ) , and E n d ( 𝑇 ) we denote the sets of all fixed points, periodic points, and endpoints of 𝑇 , respectively, that is, F i x ( 𝑇 ) = { 𝑤 ∈ 𝑋 ∶ 𝑤 ∈ 𝑇 ( 𝑤 ) } , P e r ( 𝑇 ) = { 𝑤 ∈ 𝑋 ∶ 𝑤 ∈ 𝑇 [ 𝑞 ] ( 𝑤 ) for some 𝑞 ∈ ℕ } and E n d ( 𝑇 ) = { 𝑤 ∈ 𝑋 ∶ { 𝑤 } = 𝑇 ( 𝑤 ) } . For each 𝑥 ∈ 𝑋 , a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) such that

∀ 𝑚 ∈ { 0 } ∪ ℕ 𝑤 𝑚 + 1 𝑤 ∈ 𝑇 𝑚 , 𝑤 0 = 𝑥 , ( 1 . 1 ) is called a dynamic process or a trajectory starting at 𝑤 0 = 𝑥 of the system ( 𝑋 , 𝑇 ) (for details see Aubin and Siegel [1], Aubin and Ekeland [2], and Aubin and Frankowska [3]). For each 𝑥 ∈ 𝑋 , a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) such that

∀ 𝑚 ∈ { 0 } ∪ ℕ 𝑤 𝑚 + 1 ∈ 𝑇 [ 𝑚 + 1 ] ( 𝑥 ) , 𝑤 0 = 𝑥 , ( 1 . 2 ) 𝑇 [ 𝑚 ] = 𝑇 ∘ 𝑇 ∘ ⋯ ∘ 𝑇 ( 𝑚 -times), 𝑚 ∈ ℕ , is called a generalized sequence of iterations starting at 𝑤 0 = 𝑥 of the system ( 𝑋 , 𝑇 ) (for details see Yuan [4, page 557], Tarafdar and Vyborny [5] and Tarafdar and Yuan [6]). Each dynamic process starting from 𝑤 0 is a generalized sequence of iterations starting from 𝑤 0 , but the converse may not be true; the set 𝑇 [ 𝑚 ] ( 𝑤 0 ) is, in general, bigger than 𝑇 ( 𝑤 𝑚 − 1 ) . If ( 𝑋 , 𝑇 ) is single valued, then, for each 𝑥 ∈ 𝑋 , a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) such that

∀ 𝑚 ∈ { 0 } ∪ ℕ 𝑤 𝑚 + 1 = 𝑇 [ 𝑚 + 1 ] ( 𝑥 ) , 𝑤 0 = 𝑥 , ( 1 . 3 ) is called a Picard iteration starting at 𝑤 0 = 𝑥 of the system ( 𝑋 , 𝑇 ) . If ( 𝑋 , 𝑇 ) is single valued, then (1.1)–(1.3) are identical.

The notion of Banach’s contraction belongs to the most fundamental mathematical ideas. Caristi [7], Ekeland [8], Aubin and Siegel [1], Yuan [4], and Kirk [9] extended this notion to several directions (dissipative single-valued maps with lower semicontinuous entropies, variational inequlities for lower semicontinuous maps, dissipative set-valued dynamic systems with not necessarily lower semicontinuous entropies, generalized contractions and asymptotic contractions, resp.). It is not our purpose to give a complete list of related papers here.

Let 𝑋 be a metric space with metric 𝑑 and let ( 𝑋 , 𝑇 ) be a single-valued dynamic system. Racall that if

∃ 𝜆 ∈ [ 0 , 1 ) ∀ 𝑥 , 𝑦 ∈ 𝑋 { 𝑑 ( 𝑇 ( 𝑥 ) , 𝑇 ( 𝑦 ) ) ⩽ 𝜆 𝑑 ( 𝑥 , 𝑦 ) } , ( 1 . 4 ) then ( 𝑋 , 𝑇 ) is called a Banach's contraction (Banach [10]). ( 𝑋 , 𝑇 ) is called contractive if ∀ 𝑥 , 𝑦 ∈ 𝑋 { 0 < 𝑑 ( 𝑥 , 𝑦 ) ⇒ 𝑑 ( 𝑇 ( 𝑥 ) , 𝑇 ( 𝑦 ) ) < 𝑑 ( 𝑥 , 𝑦 ) } . If ∃ 𝜖 > 0 ∀ 𝑥 , 𝑦 ∈ 𝑋 { 0 < 𝑑 ( 𝑥 , 𝑦 ) < 𝜖 ⇒ 𝑑 ( 𝑇 ( 𝑥 ) , 𝑇 ( 𝑦 ) ) < 𝑑 ( 𝑥 , 𝑦 ) } , then ( 𝑋 , 𝑇 ) is called 𝜖 -contractive (Edelstein [11]). Contractive and 𝜖 -contractive maps are some modifications of Banach’s contractions.

If ( 𝑋 , 𝑇 ) is single valued and ∀ 𝑥 ∈ 𝑋 { 𝑑 ( 𝑥 , 𝑇 ( 𝑥 ) ) ⩽ 𝜔 ( 𝑥 ) − 𝜔 ( 𝑇 ( 𝑥 ) ) } ( 1 . 5 ) for some 𝜔 ∶ 𝑋 → [ 0 , + ∞ ) , then 𝑇 is called Caristi's map (Caristi [7]). Caristi’s maps (1.5) generalize Banach's contractions (1.4) (for details see Kirk and Saliga [12, page 2766]). Banach’s contraction principle and Caristi's fixed point theorem are essentially different: in complete metric space, Banach’s contraction is continuous, each Picard iteration of this contraction is convergent to a fixed point and this fixed point is unique (Banach [10]) while Caristi’s map is not necessarily continuous and if 𝜔 in (1.5) is lower semicontinuous, then each Picard iteration of this map is convergent to a fixed point and this fixed point is not necessarily unique (Caristi [7]). Recall that Ekeland’s [8] variational principle concerning lower semicontinuous maps and Caristi's fixed point theorem are equivalent.

A map 𝜔 ∶ 𝑋 → [ 0 , + ∞ ) is called a weak entropy or entropy of a set-valued dynamic system ( 𝑋 , 𝑇 ) if

∀ 𝑥 ∈ 𝑋 ∃ 𝑦 ∈ 𝑇 ( 𝑥 ) { 𝑑 ( 𝑥 , 𝑦 ) ⩽ 𝜔 ( 𝑥 ) − 𝜔 ( 𝑦 ) } ( 1 . 6 ) or

∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑇 ( 𝑥 ) { 𝑑 ( 𝑥 , 𝑦 ) ⩽ 𝜔 ( 𝑥 ) − 𝜔 ( 𝑦 ) } , ( 1 . 7 ) respectively, and ( 𝑋 , 𝑇 ) is called weak dissipative or dissipative if it has a weak entropy or an entropy, respectively; here 𝜔 is not necessarily lower semicontinuous. These two kinds of dissipative maps were introduced and studied by Aubin and Siegel [1]. If ( 𝑋 , 𝑇 ) is single valued, then (1.5)–(1.7) are identical.

Various periodic, fixed point, convergence, and invariant set theorems for contractive and 𝜖 -contractive single-valued and set-valued dynamic systems have been obtained by Edelstein [11], Ding and Nadler [13], and Nadler [14]. Investigations concerning the existence of fixed points and endpoints and convergence of dynamic processes or generalized sequences of iterations to fixed points or endpoints of single-valued and set-valued generalized contractions (Yuan [4], Tarafdar and Yuan [6, 15], Tarafdar and Chowdhury [16], Tarafdar and Vyborny [5]) and dissipative dynamic systems when entropy 𝜔 is not necessarily lower semicontinuous (Aubin and Siegel [1]) have been conducted by a number of authors in different contexts; for example, see Kirk and Saliga [12], Willems [17], Zangwill [18], Justman [19], Maschler and Peleg [20] and Petruşel, Sîntămărian [21].

In this paper, inspired by these results, we introduce in cone uniform and uniform spaces the three kinds of dissipative set-valued dynamic systems with generalized pseudodistances and with not necessarily lower semicontinuous entropies and we present the methods which are useful for establishing general conditions guaranteeing the existence of periodic points and endpoints of these set-valued dynamic systems and conditions that for each starting point the dynamic processes or generalized sequences of iterations converge and the limit is a periodic point or endpoint (see Sections 3–6). The presented definitions and results are more general and different from those given in the literature and are new even for single-valued and set-valued dynamic systems in metric spaces. For details, see Section 7 where examples, remarks, and some comparisons are included. This paper is a continuation of [22, 23].

2. Dissipative Set-Valued Dynamic Systems with Generalized Pseudodistances in Cone Uniform Spaces

We define a real normed space to be a pair ( 𝐿 , ‖ ⋅ ‖ ) , with the understanding that a vector space 𝐿 over ℝ carries the topology generated by the metric ( 𝑎 , 𝑏 ) → ‖ 𝑎 − 𝑏 ‖ , 𝑎 , 𝑏 ∈ 𝐿 .

A nonempty closed convex set 𝐻 ⊂ 𝐿 is called a cone in 𝐿 if it satisfies: ( H 1 ) ∀ 𝑠 ∈ ( 0 , ∞ ) { 𝑠 𝐻 ⊂ 𝐻 } ; ( H 2 ) 𝐻 ∩ ( − 𝐻 ) = { 0 } ; ( H 3 ) 𝐻 ≠ { 0 } .

It is clear that each cone 𝐻 ⊂ 𝐿 defines, by virtue of “ 𝑎 ⪯ 𝐻 𝑏 if and only if 𝑏 − 𝑎 ∈ 𝐻 ,” an order of 𝐿 under which 𝐿 is an ordered normed space with cone 𝐻 . We will write 𝑎 ≺ 𝐻 𝑏 to indicate that 𝑎 ⪯ 𝐻 𝑏 but 𝑎 ≠ 𝑏 .

The following terminologies will be much used.

Definition 2.1 (see [22]). Let 𝑋 be a nonempty set and let 𝐿 be an ordered normed space with cone 𝐻 . (i)The family 𝒫 = { p 𝛼 ∶ X × X → L , 𝛼 ∈ 𝒜 } is said to be a 𝒫 -family of cone pseudometrics on 𝑋 ( 𝒫 -family, for short ) if the following three conditions hold:( 𝒫 1 ) ∀ 𝛼 ∈ 𝒜 ∀ 𝑥 , 𝑦 ∈ 𝑋 { 0 ⪯ 𝐻 𝑝 𝛼 ⋀ ( 𝑥 , 𝑦 ) 𝑥 = 𝑦 ⇒ 𝑝 𝛼 ( 𝑥 , 𝑦 ) = 0 } ;( 𝒫 2 ) ∀ 𝛼 ∈ 𝒜 ∀ 𝑥 , 𝑦 ∈ 𝑋 { 𝑝 𝛼 ( 𝑥 , 𝑦 ) = 𝑝 𝛼 ( 𝑦 , 𝑥 ) } ;( 𝒫 3 ) ∀ 𝛼 ∈ 𝒜 ∀ 𝑥 , 𝑦 , 𝑧 ∈ 𝑋 { 𝑝 𝛼 ( 𝑥 , 𝑧 ) ⪯ 𝐻 𝑝 𝛼 ( 𝑥 , 𝑦 ) + 𝑝 𝛼 ( 𝑦 , 𝑧 ) } . (ii)If 𝒫 is 𝒫 -family, then the pair ( 𝑋 , 𝒫 ) is called a cone uniform space.(iii)A 𝒫 -family 𝒫 is said to be separating if( 𝒫 4 ) ∀ 𝑥 , 𝑦 ∈ 𝑋 { 𝑥 ≠ 𝑦 ⇒ ∃ 𝛼 ∈ 𝒜 { 0 ≺ 𝐻 𝑝 𝛼 ( 𝑥 , 𝑦 ) } } . (iv)If a 𝒫 -family 𝒫 is separating, then the pair ( 𝑋 , 𝒫 ) is called a Hausdorff cone uniform space.

A cone 𝐻 is said to be solid if i n t ( 𝐻 ) ≠ ∅ ; i n t ( 𝐻 ) denotes the interior of 𝐻 . We will write 𝑎 ≪ 𝐻 𝑏 to indicate that 𝑏 − 𝑎 ∈ i n t ( 𝐻 ) .

Definition 2.2. Let 𝐿 be an ordered normed space with solid cone 𝐻 and let ( 𝑋 , 𝒫 ) be a cone uniform space with cone 𝐻 .(i)We say that a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is a 𝒫 -convergent in 𝑋 , if there exists 𝑤 ∈ 𝑋 such that ∀ 𝛼 ∈ 𝒜 ∀ 𝑐 ∈ 𝐿 , 0 ≪ 𝐻 𝑐 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 { 𝑝 𝛼 ( 𝑤 𝑚 , 𝑤 ) ≪ 𝐻 𝑐 } .(ii)We say that a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is a 𝒫 -Cauchy sequence in 𝑋 , if ∀ 𝛼 ∈ 𝒜 ∀ 𝑐 ∈ 𝐿 , 0 ≪ 𝐻 𝑐 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 < 𝑛 { 𝑝 𝛼 ( 𝑤 𝑚 , 𝑤 𝑛 ) ≪ 𝐻 𝑐 } .(iii)If every 𝒫 -Cauchy sequence in 𝑋 is 𝒫 -convergent in 𝑋 , then ( 𝑋 , 𝒫 ) is called a 𝒫 -sequentially complete cone uniform space.(iv)The set-valued dynamic system ( 𝑋 , 𝑇 ) is called a cone closed set-valued dynamic system in 𝑋 if whenever ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) is a sequence in X 𝒫 -converging to 𝑤 ∈ 𝑋 and ( 𝑣 𝑚 ∶ 𝑚 ∈ ℕ ) is a sequence 𝒫 -converging to 𝑣 ∈ 𝑋 such that 𝑣 𝑚 ∈ 𝑇 ( 𝑤 𝑚 ) for all 𝑚 ∈ ℕ , then 𝑣 ∈ 𝑇 ( 𝑤 ) .(v)Let ( 𝑋 , 𝒫 ) be a 𝒫 -sequentially complete cone uniform space. For an arbitrary subset 𝐸 of 𝑋 , the cone closure of 𝐸 , denoted by c l ( 𝐸 ) , is defined as the set c l ( 𝐸 ) = { 𝑤 ∈ 𝑋 ∶ ∃ ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) ⊂ 𝐸 ∀ 𝛼 ∈ 𝒜 ∀ 𝑐 ∈ 𝐿 , 0 ≪ 𝐻 𝑐 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 { 𝑝 𝛼 ( 𝑤 𝑚 , 𝑤 ) ≪ 𝐻 𝑐 } } . The subset 𝐸 of 𝑋 is said to be a cone closed subset in 𝑋 if c l ( 𝐸 ) = 𝐸 .

The cone 𝐻 is normal if a real number 𝑀 > 0 exists such that for each 𝑎 , 𝑏 ∈ 𝐻 , 0 ⪯ 𝐻 𝑎 ⪯ 𝐻 𝑏 implies ‖ 𝑎 ‖ ⩽ 𝑀 ‖ 𝑏 ‖ . The number 𝑀 satisfying the above is called the normal constant of 𝐻 .

The following holds.

Theorem 2.3 (see [22]). Let 𝐿 be an ordered normed space with normal solid cone 𝐻 and let ( 𝑋 , 𝒫 ) be a cone uniform space with cone 𝐻 . (a)Let ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) be a sequence in 𝑋 and let 𝑤 ∈ 𝑋 . Then the sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) is 𝒫 -convergent to 𝑤 if and only if ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ‖ ‖ 𝑝 𝛼 𝑤 𝑚 ‖ ‖ , 𝑤 < 𝜀 . ( 2 . 1 ) (b)Let ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) be a sequence in 𝑋 . Then the sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) is a 𝒫 -Cauchy sequence if and only if ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 < 𝑛 ‖ ‖ 𝑝 𝛼 𝑤 𝑚 , 𝑤 𝑛 ‖ ‖ < 𝜀 . ( 2 . 2 ) (c) Each 𝒫 -convergent sequence is a 𝒫 -Cauchy sequence.

Definition 2.4. Let 𝐿 be an ordered normed space with normal solid cone 𝐻 and let ( 𝑋 , 𝒫 ) be a cone uniform space with cone 𝐻 .(i)The family 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } is said to be a 𝒥 -family of cone pseudodistances on 𝑋 ( 𝒥 -family on 𝑋 , for short ) if the following three conditions hold:( 𝒥 1 ) ∀ 𝛼 ∈ 𝒜 ∀ 𝑥 , 𝑦 ∈ 𝑋 { 0 ⪯ 𝐻 𝐽 𝛼 ( 𝑥 , 𝑦 ) } ;( 𝒥 2 ) ∀ 𝛼 ∈ 𝒜 ∀ 𝑥 , 𝑦 , 𝑧 ∈ 𝑋 { 𝐽 𝛼 ( 𝑥 , 𝑧 ) ⪯ 𝐻 𝐽 𝛼 ( 𝑥 , 𝑦 ) + 𝐽 𝛼 ( 𝑦 , 𝑧 ) } ;( 𝒥 3 )for any sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 such that ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ⩽ 𝑛 ‖ ‖ 𝐽 𝛼 𝑤 𝑚 , 𝑤 𝑛 ‖ ‖ < 𝜀 , ( 2 . 3 ) if there exists a sequence ( 𝑣 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 satisfying ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ‖ ‖ 𝐽 𝛼 𝑤 𝑚 , 𝑣 𝑚 ‖ ‖ < 𝜀 , ( 2 . 4 ) then ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ‖ ‖ 𝑝 𝛼 𝑤 𝑚 , 𝑣 𝑚 ‖ ‖ < 𝜀 . ( 2 . 5 ) (ii)Let the family 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a 𝒥 -family on 𝑋 . We say that a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is a 𝒥 -Cauchy sequence in 𝑋 if (2.3) holds.

For other families of cone pseudodistances in cone uniform spaces and various applications, see [22, 23]. The following is a consequence of Definition 2.4(i).

Proposition 2.5. Let ( 𝑋 , 𝒫 ) be a Hausdorff cone uniform space with cone 𝐻 . Let the family 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a 𝒥 -family. If  ∀ 𝛼 ∈ 𝒜 { 𝐽 𝛼 ⋀ 𝐽 ( 𝑥 , 𝑦 ) = 0 𝛼 ( 𝑦 , 𝑥 ) = 0 } , then 𝑥 = 𝑦 .

Proof. Let 𝑥 , 𝑦 ∈ 𝑋 be such that ∀ 𝛼 ∈ 𝒜 { 𝐽 𝛼 ( 𝑥 , 𝑦 ) = 0 ⋀ 𝐽 𝛼 ( 𝑦 , 𝑥 ) = 0 } . By ( 𝒥 2), ∀ 𝛼 ∈ 𝒜 { 𝐽 𝛼 ( 𝑥 , 𝑥 ) ⪯ 𝐻 𝐽 𝛼 ( 𝑥 , 𝑦 ) + 𝐽 𝛼 ( 𝑦 , 𝑥 ) } . By ( 𝒥 1), this gives ∀ 𝛼 ∈ 𝒜 { 𝐽 𝛼 ( 𝑥 , 𝑥 ) = 0 } . Thus, we get ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ⩽ 𝑛 { ‖ 𝐽 𝛼 ( 𝑤 𝑚 , 𝑤 𝑛 ) ‖ < 𝜀 } and ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 { ‖ 𝐽 𝛼 ( 𝑤 𝑚 , 𝑣 𝑚 ) ‖ < 𝜀 } where 𝑤 𝑚 = 𝑥 , 𝑣 𝑚 = 𝑦 , 𝑚 ∈ ℕ , and, by ( 𝒥 3), ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 { ‖ 𝑝 𝛼 ( 𝑤 𝑚 , 𝑣 𝑚 ) ‖ < 𝜀 } , that is, ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 { ‖ 𝑝 𝛼 ( 𝑥 , 𝑦 ) ‖ < 𝜀 } . Hence, ∀ 𝛼 ∈ 𝒜 { 𝑝 𝛼 ( 𝑥 , 𝑦 ) = 0 } which, according to ( 𝒫 4), implies that 𝑥 = 𝑦 .

Now we introduce the following three kinds of dissipative set-valued dynamic systems with generalized pseudodistances in cone uniform spaces (conditions (ii)–(iv) below).

Definition 2.6. Let ( 𝑋 , 𝒫 ) be a Hausdorff cone uniform space and let ( 𝑋 , 𝑇 ) be a set-valued dynamic system. Let 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a 𝒥 -family on 𝑋 and let Ω = { 𝜔 𝛼 ∶ 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a family of maps such that ∀ 𝛼 ∈ 𝒜 ∀ 𝑥 ∈ 𝑋 0 ⪯ 𝐻 𝜔 𝛼 ( 𝑥 ) . ( 2 . 6 ) (i)We say that a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) in 𝑋 is ( 𝒥 , Ω ) -admissible if ∀ 𝛼 ∈ 𝒜 ∀ 𝑚 ∈ { 0 } ∪ ℕ 𝐽 𝛼 𝑤 𝑚 , 𝑤 𝑚 + 1 ⪯ 𝐻 𝜔 𝛼 𝑤 𝑚 − 𝜔 𝛼 𝑤 𝑚 + 1 . ( 2 . 7 ) (ii)If the following conditions are satisfied: ( C 1 ) ∅ ≠ 𝑋 0 ⊂ 𝑋 ; and(C2) 𝑥 ∈ 𝑋 0 if and only if there exists a ( 𝒥 , Ω ) -admissible dynamic process ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) starting at 𝑤 0 = 𝑥 of the system ( 𝑋 , 𝑇 ) ,
then we say that 𝑇 is weak ( 𝒥 , Ω ; 𝑋 0 ) -dissipative on 𝑋
(iii)We say that 𝑇 is ( 𝒥 , Ω ) -dissipative on 𝑋 if, for each 𝑥 ∈ 𝑋 , each dynamic process ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) starting at 𝑤 0 = 𝑥 of the system ( 𝑋 , 𝑇 ) is ( 𝒥 , Ω ) -admissible.(iv)We say that 𝑇 is strictly ( 𝒥 , Ω ) -dissipative on 𝑋 if, for each 𝑥 ∈ 𝑋 , each generalized sequence of iterations ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) starting at 𝑤 0 = 𝑥 of the system ( 𝑋 , 𝑇 ) is ( 𝒥 , Ω ) -admissible.
If one of the conditions (ii)–(iv) holds, then we say that ( 𝑋 , 𝑇 ) is a dissipative set-valued dynamic system with respect to ( 𝒥 , Ω ) (dissipative set-valued dynamic system, for short).

Remark 2.7. It is worth noticing that if a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) in 𝑋 is ( 𝒥 , Ω ) -admissible, then, for each 𝑘 ∈ ℕ , a sequence ( 𝑤 𝑚 + 𝑘 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) is ( 𝒥 , Ω ) -admissible. Consequently, if 𝑇 is weak ( 𝒥 , Ω ; 𝑋 0 ) -dissipative on 𝑋 , 𝑥 ∈ 𝑋 0 , and ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) is a dynamic process starting at 𝑤 0 = 𝑥 of the system ( 𝑋 , 𝑇 ) which is ( 𝒥 , Ω ) -admissible, then ∀ 𝑚 ∈ ℕ { 𝑤 𝑚 ∈ 𝑋 0 } ; in general, 𝑇 ( 𝑋 0 ) ≠ 𝑋 0 (see Example 7.3).

Now we can give the following conclusion.

Proposition 2.8. Let ( 𝑋 , 𝒫 ) be a Hausdorff cone uniform space and let ( 𝑋 , 𝑇 ) be a set-valued dynamic system. (a)If 𝑇 is weak ( 𝒥 , Ω ; 𝑋 0 ) -dissipative on 𝑋 , then ( 𝑋 0 , 𝕂 𝒥 ; 𝑇 ) is a set-valued dynamic system where, for each 𝑥 ∈ 𝑋 0 , 𝕂 𝒥 ; 𝑇 𝑤 ( 𝑥 ) = 0 , 𝑤 1 , 𝑤 2 ∶ 𝑤 , … 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ∈ 𝒦 𝒥 𝒦 ( 𝑇 , 𝑥 ) , ( 2 . 8 ) 𝒥 𝑤 ( 𝑇 , 𝑥 ) = 𝑚 , 𝑤 ∶ 𝑚 ∈ { 0 } ∪ ℕ 0 = 𝑥 ∶ ∀ 𝑚 ∈ { 0 } ∪ ℕ 𝑤 𝑚 + 1 𝑤 ∈ 𝑇 𝑚 ∧ ∀ 𝛼 ∈ 𝒜 𝐽 𝛼 𝑤 𝑚 , 𝑤 𝑚 + 1 ⪯ 𝐻 𝜔 𝛼 𝑤 𝑚 − 𝜔 𝛼 𝑤 𝑚 + 1 . ( 2 . 9 ) (b)If 𝑇 is ( 𝒥 , Ω ) -dissipative on 𝑋 , then ( 𝑋 , 𝕎 𝒥 ; 𝑇 ) is a set-valued dynamic system where, for each 𝑥 ∈ 𝑋 , 𝕎 𝒥 ; 𝑇 𝑤 ( 𝑥 ) = 0 , 𝑤 1 , 𝑤 2 ∶ 𝑤 , … 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ∈ 𝒲 𝒥 𝒲 ( 𝑇 , 𝑥 ) , ( 2 . 1 0 ) 𝒥 𝑤 ( 𝑇 , 𝑥 ) = 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ , 𝑤 0 = 𝑥 ∶ ∀ 𝑚 ∈ { 0 } ∪ ℕ 𝑤 𝑚 + 1 𝑤 ∈ 𝑇 𝑚 . ( 2 . 1 1 ) (c)If 𝑇 is strictly ( 𝒥 , Ω ) -dissipative on 𝑋 , then ( 𝑋 , 𝕊 𝒥 ; 𝑇 ) is a set-valued dynamic system where, for each 𝑥 ∈ 𝑋 , 𝕊 𝒥 ; 𝑇 𝑤 ( 𝑥 ) = 0 , 𝑤 1 , 𝑤 2 ∶ 𝑤 , … 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ∈ 𝒮 𝒥 𝒮 ( 𝑇 , 𝑥 ) , ( 2 . 1 2 ) 𝒥 ( 𝑤 𝑇 , 𝑥 ) = 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ , 𝑤 0 = 𝑥 ∶ ∀ 𝑚 ∈ { 0 } ∪ ℕ 𝑤 𝑚 + 1 ∈ 𝑇 [ 𝑚 + 1 ] 𝑤 0 . ( 2 . 1 3 )

Proof. The fact that 𝕂 𝒥 ; 𝑇 ∶ 𝑋 0 ⟶ 2 𝑋 0 , 𝕎 𝒥 ; 𝑇 ∶ 𝑋 ⟶ 2 𝑋 , 𝕊 𝒥 ; 𝑇 ∶ 𝑋 ⟶ 2 𝑋 ( 2 . 1 4 ) follows from (1.1), (1.2), Definition 2.6, Remark 2.7, and (2.8)–(2.13).

Remark 2.9. By Proposition 2.8 and Definition 2.6, we get:(i)If 𝑇 is ( 𝒥 , Ω ) -dissipative on 𝑋 , then 𝑇 is weak ( 𝒥 , Ω ; 𝑋 0 ) -dissipative on 𝑋 for 𝑋 0 = 𝑋 and ∀ 𝑥 ∈ 𝑋 0 { 𝕂 𝒥 ; 𝑇 ( 𝑥 ) = 𝕎 𝒥 ; 𝑇 ( 𝑥 ) } .(ii)If 𝑇 is strictly ( 𝒥 , Ω ) -dissipative on 𝑋 , then 𝑇 is ( 𝒥 , Ω ) -dissipative on 𝑋 and ∀ 𝑥 ∈ 𝑋 { 𝕎 𝒥 ; 𝑇 ( 𝑥 ) ⊂ 𝕊 𝒥 ; 𝑇 ( 𝑥 ) } .

Definition 2.10. Let 𝐿 be an ordered normed space with solid cone 𝐻 . The cone 𝐻 is called regular if for every incresing (decresing) sequence which is bounded from above (below), that is, if for each sequence ( 𝑐 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝐿 such that 𝑐 1 ⪯ 𝐻 𝑐 2 ⪯ 𝐻 ⋯ ⪯ 𝐻 𝑐 𝑚 ⪯ 𝐻 ⋯ ⪯ 𝐻 𝑏 ( 𝑏 ⪯ 𝐻 ⋯ ⪯ 𝐻 𝑐 𝑚 ⪯ 𝐻 ⋯ ⪯ 𝐻 𝑐 2 ⪯ 𝐻 𝑐 1 ) for some 𝑏 ∈ 𝐿 , there exists 𝑐 ∈ 𝐿 such that l i m 𝑚 → ∞ ‖ 𝑐 𝑚 − 𝑐 ‖ = 0 .

Remark 2.11. Every regular cone is normal; see [24].

3. Periodic Point and Convergence Theorem for Weak ( 𝒥 , Ω ; 𝑋 0 ) -Dissipative, ( 𝒥 , Ω ) -Dissipative, and Strictly ( 𝒥 , Ω ) -Dissipative Set-Valued Dynamic Systems in Cone Uniform Spaces

Our main result of this section is the following.

Theorem 3.1. Let 𝐿 be an ordered Banach space with a regular solid cone 𝐻 and let ( 𝑋 , 𝒫 ) be a Hausdorff sequentially complete cone uniform space with cone 𝐻 . Let 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a 𝒥 -family on 𝑋 and let Ω = { 𝜔 𝛼 ∶ 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a family of maps such that ∀ 𝛼 ∈ 𝒜 ∀ 𝑥 ∈ 𝑋 { 0 ⪯ 𝐻 𝜔 𝛼 ( 𝑥 ) } . Let ( 𝑋 , 𝑇 ) be a set-valued dynamic system. The following hold. (a)If 𝑇 is weak ( 𝒥 , Ω ; 𝑋 0 ) -dissipative on 𝑋 , then, for each 𝑥 ∈ 𝑋 0 and for each dynamic process ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) ∈ 𝒦 𝒥 ( 𝑇 , 𝑥 ) , there exists 𝑤 ∈ c l ( 𝑋 0 ) such that ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) is 𝒫 -convergent to 𝑤 . If, in addition, the map 𝑇 [ 𝑞 ] is cone closed in 𝑋 for some 𝑞 ∈ ℕ , then 𝑤 ∈ 𝑇 [ 𝑞 ] ( 𝑤 ) .(b)If 𝑇 is ( 𝒥 , Ω ) -dissipative on 𝑋 , then, for each 𝑥 ∈ 𝑋 and for each dynamic process ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) ∈ 𝒲 𝒥 ( 𝑇 , 𝑥 ) , there exists 𝑤 ∈ 𝑋 such that ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) is 𝒫 -convergent to 𝑤 . If, in addition, the map 𝑇 [ 𝑞 ] is cone closed in 𝑋 for some 𝑞 ∈ ℕ , then 𝑤 ∈ 𝑇 [ 𝑞 ] ( 𝑤 ) .(c)If 𝑇 is strictly ( 𝒥 , Ω ) -dissipative on 𝑋 , then, for each 𝑥 ∈ 𝑋 and for each generalized sequence of iterations ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) ∈ 𝒮 𝒥 ( 𝑇 , 𝑥 ) , there exists 𝑤 ∈ 𝑋 such that ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) is 𝒫 -convergent to 𝑤 . If, in addition, the map 𝑇 [ 𝑞 ] is cone closed in 𝑋 for some 𝑞 ∈ ℕ , then, for each 𝑥 ∈ 𝑋 , there exists a generalized sequence of iterations ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) ∈ 𝒮 𝒥 ( 𝑇 , 𝑥 ) and 𝑤 ∈ 𝑋 such that ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) is 𝒫 -convergent to 𝑤 and 𝑤 ∈ 𝑇 [ 𝑞 ] ( 𝑤 ) .

Proof. The proof will be broken into three steps.Step 1. Let ( i ) 𝑥 ∈ 𝑋 0 and ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) ∈ 𝒦 𝒥 ( 𝑇 , 𝑥 ) ; or ( i i ) 𝑥 ∈ 𝑋 and ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) ∈ 𝒲 𝒥 ( 𝑇 , 𝑥 ) ∪ 𝒮 𝒥 ( 𝑇 , 𝑥 ) . We show that ( 𝑤 𝑚 ∶ 𝑚 ∈ { 0 } ∪ ℕ ) is 𝒥 -Cauchy and 𝒫 -Cauchy, that is, ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ⩽ 𝑛 ‖ ‖ 𝐽 𝛼 𝑤 𝑚 , 𝑤 𝑛 ‖ ‖ , ∀ < 𝜀 ( 3 . 1 ) 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 < 𝑛 ‖ ‖ 𝑝 𝛼 𝑤 𝑚 , 𝑤 𝑛 ‖ ‖ , < 𝜀 ( 3 . 2 ) respectively; see Definitions 2.4(ii) and 2.2(ii) and Theorem 2.3(b).Indeed, since ⪯ 𝐿 is transitive, by (2.9), (2.11), (2.13), Definition 2.6(ii)–(iv) and ( 𝒥 1), we get that ∀ 𝛼 ∈ 𝒜 ∀ 𝑚 ∈ { 0 } ∪ ℕ { 𝜔 𝛼 ( 𝑤 𝑚 + 1 ) ⪯ 𝐻 𝜔 𝛼 ( 𝑤 𝑚 ) } . According to (2.6), for each 𝛼 ∈ 𝒜 , the sequence ( 𝜔 𝛼 ( 𝑤 𝑚 ) ∶ 𝑚 ∈ { 0 } ∪ ℕ ) is contained in 𝐻 , bounded from below and, by the above, nonincreasing. Since 𝐻 is a closed and regular cone, it follows that ∀ 𝛼 ∈ 𝒜 ∃ 𝑢 𝛼 ∈ 𝐻 l i m 𝑚 → ∞ ‖ ‖ 𝜔 𝛼 𝑤 𝑚 − 𝑢 𝛼 ‖ ‖ = 0 . ( 3 . 3 ) Let now 𝛼 0 ∈ 𝒜 and 𝜀 0 > 0 be arbitrary and fixed. By (3.3), ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 ; 𝑛 0 ⩽ 𝑚 ‖ ‖ 𝜔 𝛼 0 𝑤 𝑚 − 𝑢 𝛼 0 ‖ ‖ < 𝜀 0 2 𝑀 , ( 3 . 4 ) where 𝑀 is a normal constant of 𝐻 (see Remark 2.11). Furthermore, for 𝑛 0 < 𝑚 ⩽ 𝑛 , using ( 𝒥 1), ( 𝒥 2), and (2.7), 0 ⪯ 𝐻 𝐽 𝛼 0 ( 𝑤 𝑚 , 𝑤 𝑛 ) ⪯ 𝐻 ∑ 𝑛 − 1 𝑘 = 𝑚 𝐽 𝛼 0 ( 𝑤 𝑘 , 𝑤 𝑘 + 1 ⪯ 𝐻 𝜔 𝛼 0 ( 𝑤 𝑚 ) − 𝜔 𝛼 0 ( 𝑤 𝑛 ) and next, by (3.4) and the fact that 𝐻 is normal (see Remark 2.11), ‖ 𝐽 𝛼 0 ( 𝑤 𝑚 , 𝑤 𝑛 ) ‖ ⩽ 𝑀 ‖ 𝜔 𝛼 0 ( 𝑤 𝑚 ) − 𝜔 𝛼 0 ( 𝑤 𝑛 ) ‖ = 𝑀 ‖ 𝜔 𝛼 0 ( 𝑤 𝑚 ) − 𝑢 𝛼 0 − 𝜔 𝛼 0 ( 𝑤 𝑛 ) + 𝑢 𝛼 0 ‖ ⩽ 𝑀 ‖ 𝜔 𝛼 0 ( 𝑤 𝑚 ) − 𝑢 𝛼 0 ‖ + 𝑀 ‖ 𝜔 𝛼 0 ( 𝑤 𝑛 ) − 𝑢 𝛼 0 ‖ ⩽ 𝜀 0 / 2 + 𝜀 0 / 2 = 𝜀 0 . Therefore, (3.1) holds.
Also we can show that (3.2) holds. Indeed, by (3.1), ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 > 0 ∃ 𝑛 0 ∈ ℕ ∀ 𝑚 ⩾ 𝑛 0 ∀ 𝑘 ∈