Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 175453, 35 pages
doi:10.1155/2010/175453
Research Article

Maximality Principle and General Results of Ekeland and Caristi Types without Lower Semicontinuity Assumptions in Cone Uniform Spaces with Generalized Pseudodistances

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 31 December 2009; Accepted 8 March 2010

Academic Editor: Tomonari Suzuki

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

Our aim is twofold: first, we want to introduce a partial quasiordering in cone uniform spaces with generalized pseudodistances for giving the general maximality principle in these spaces. Second, we want to show how this maximality principle can be used to obtain new and general results of Ekeland and Caristi types without lower semicontinuity assumptions, which was not done in the previous publications on this subject.

1. Introduction

The famous Banach contraction principle [1], fundamental in fixed point theory, has been extended in many different directions. Among these extensions, Caristi's fixed point theorem [2] concerning dissipative maps with lower semicontinuous entropies, equivalent to celebrated Ekeland's variational principle [3] providing approximate solutions of nonconvex minimization problems concerning lower semicontinuous maps, may be the most valuable one.

These results are very useful, simple, and important tools for investigating various problems in nonlinear analysis, mathematical programming, control theory, abstract economy, global analysis, and others. They have many generalizations and extensive applications in many fields of mathematics and applied mathematics.

In the literature, the several generalizations of the variational principle of Ekeland type, for lower semicontinuous maps and fixed point and endpoint theorem of Caristi type for dissipative single-valued and set-valued dynamic systems with lower semicontinuous entropies in metric and uniform spaces are given, and various techniques and methods of investigations (notably based on maximality principle) are presented. However, in all these papers the restrictive assumptions about lower semicontinuity are essential. For details see [4–29] and references therein. It is not our purpose to give a complete list of related papers here.

A long time ago, we did not know how to define the distances in metric, uniform, or cone uniform spaces, which generalize metrics, pseudometrics, or cone pseudometrics, which are connected with metrics, pseudometrics, or cone pseudometrics, respectively, and which have applications to obtaining the solutions of several new important problems in nonlinear analysis. The pioneering effort in this direction is papersof Tataru [30] in Banach spaces,Kada et al.[31], Suzuki [32], and Lin and Du [33] in metric spaces, and Vályi [34] in uniform spaces. In these papers, among other things, various distances are introduced, and relations between Tataru [30], and Kada et al. [31] distances and distances of Suzuki [32] and Lin and Du [33] are established. For many applications of these distances, see the papers [30–48] where, among other things, in metric and uniform spaces with generalized distances [30–34], the new fixed point theorems of Caristi' type for dissipative maps with lower semicontinuous entropies and variational principles of Ekeland type for lower semicontinuous maps are given.

In this paper, in cone uniform spaces [49, 50], the families of generalized pseudodistances are introduced (see Section 2), a partial quasiordering is defined and the general maximality principle is formulated and proved (see Section 3). As applications, in cone uniform spaces with the families of generalized pseudodistances, the general variational principle of Ekeland type for not necessarily lower semicontinuous maps and a fixed point and endpoint theorem of Caristi type for dissipative set-valued dynamic systems with not necessarily lower semicontinuous entropies are established (see Section 4). Special cases are discussed and examples and comparisons show a fundamental difference between our results and the well-known ones in the literature where the standard lower semicontinuity assumptions are essential (see Section 5). Relations between our generalized pseudodistances and generalized distances are described (see Section 6; the aim of this section is to prove that each generalized distance [30–34] is a generalized pseudodistance and we construct the examples which show that the converse is not true). The definitions, the results, the ideas and the methods presented here are new for set-valued and single-valued dynamic systems in cone uniform, cone locally convex and cone metric spaces and even in uniform, locally convex, and metric spaces.

2. 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 .

Let 𝐿 be a real normed space. 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 } , and ( 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 .

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 ≪ 𝑏 to indicate that 𝑏 − 𝑎 ∈ i n t ( 𝐻 ) .

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 that ‖ 𝑎 ‖ ⩽ 𝑀 ‖ 𝑏 ‖ . The number 𝑀 satisfying the above is called the normal constant of 𝐻 .

The following terminologies will be much used.

Definition 2.1 (see [49, 50]). 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 } , 𝒜 -index set, 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 a 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.

Definition 2.2 (see [49, Definition 2 . 3 ]). 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 𝛼 ∈ 𝐿 , 0 ≪ 𝑐 𝛼 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝑐 𝛼 ) ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 𝑝 𝛼 𝑤 𝑚 , 𝑤 ≪ 𝑐 𝛼 . ( 2 . 1 ) (ii)We say that a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is a 𝒫 -Cauchy sequence in 𝑋 , if ∀ 𝛼 ∈ 𝒜 ∀ 𝑐 𝛼 ∈ 𝐿 , 0 ≪ 𝑐 𝛼 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝑐 𝛼 ) ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 < 𝑛 𝑝 𝛼 𝑤 𝑚 , 𝑤 𝑛 ≪ 𝑐 𝛼 . ( 2 . 2 ) (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.

The following holds.

Theorem 2.3 (see [49, Theorem 2 . 1 ]). Let 𝐿 be an ordered normed space with normal solid cone 𝐻 and let ( 𝑋 , 𝒫 ) be a Hausdorff cone uniform space with cone 𝐻 . (a)Let ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) be a sequence in 𝑋 and let 𝑤 ∈ 𝑋 . The sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) is 𝒫 -convergent to 𝑤 if and only if ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ‖ ‖ 𝑝 𝛼 𝑤 𝑚 ‖ ‖ , 𝑤 < 𝜀 𝛼 . ( 2 . 3 ) (b)Let ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) be a sequence in 𝑋 . The sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) is a 𝒫 -Cauchy sequence if and only if ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 < 𝑛 ‖ ‖ 𝑝 𝛼 𝑤 𝑚 , 𝑤 𝑛 ‖ ‖ < 𝜀 𝛼 . ( 2 . 4 ) (c)Each 𝒫 -convergent sequence is a 𝒫 -Cauchy sequence.

Definition 2.4. Let 𝐿 be an ordered normed space with solid cone 𝐻 . The cone 𝐻 is called regular if for every increasing (decreasing) 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 2.5. Every regular cone is normal; see [51].

Definition 2.6. Let 𝐿 be an ordered normed space with normal solid cone 𝐻 and let ( 𝑋 , 𝒫 ) be a Hausdorff 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 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ⩽ 𝑛 ‖ ‖ 𝐽 𝛼 𝑤 𝑚 , 𝑤 𝑛 ‖ ‖ < 𝜀 𝛼 , ( 2 . 5 ) if there exists a sequence ( 𝑣 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 satisfying ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ‖ ‖ 𝐽 𝛼 𝑤 𝑚 , 𝑣 𝑚 ‖ ‖ < 𝜀 𝛼 , ( 2 . 6 ) then ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ‖ ‖ 𝑝 𝛼 𝑤 𝑚 , 𝑣 𝑚 ‖ ‖ < 𝜀 𝛼 . ( 2 . 7 ) (ii)Let the family 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a 𝒥 -family on 𝑋 . One says that a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is a 𝒥 -Cauchy sequence in 𝑋 if (2.5) holds.

Remark 2.7. Each 𝒫 -family is a 𝒥 -family.

The following result is useful.

Proposition 2.8. Let ( 𝑋 , 𝒫 ) be a Hausdorff cone uniform space with cone 𝐻 . Let the 𝒥 -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 ⋀ 𝐽 ( 𝑥 , 𝑦 ) = 0 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 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ⩽ 𝑛 { ‖ 𝐽 𝛼 ( 𝑤 𝑚 , 𝑤 𝑛 ) ‖ < 𝜀 𝛼 } and ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 { ‖ 𝐽 𝛼 ( 𝑤 𝑚 , 𝑣 𝑚 ) ‖ < 𝜀 𝛼 } where 𝑤 𝑚 = 𝑥 , 𝑣 𝑚 = 𝑦 , and 𝑚 ∈ ℕ , and, by ( 𝒥 3), ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 { ‖ 𝑝 𝛼 ( 𝑤 𝑚 , 𝑣 𝑚 ) ‖ < 𝜀 𝛼 } , that is, ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 { ‖ 𝑝 𝛼 ( 𝑥 , 𝑦 ) ‖ < 𝜀 𝛼 } . Hence, ∀ 𝛼 ∈ 𝒜 { 𝑝 𝛼 ( 𝑥 , 𝑦 ) = 0 } which, according to ( 𝒫 4), implies that 𝑥 = 𝑦 .

3. Maximality (Minimality) Principle in Cone Uniform Spaces with Generalized Pseudodistances

We start with the following result.

Proposition 3.1. Let 𝐿 be an ordered Banach space with normal solid cone 𝐻 , let ( 𝑋 , 𝒫 ) be a Hausdorff cone uniform space with cone 𝐻 and let 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a 𝒥 -family on 𝑋 . Every 𝒥 -Cauchy sequence in 𝑋 is 𝒫 -Cauchy sequence in 𝑋 .

Proof. Indeed, assume that a sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is 𝒥 -Cauchy, that is, by Definition 2.6(ii), assume that ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ⩽ 𝑛 ‖ ‖ 𝐽 𝛼 𝑤 𝑚 , 𝑤 𝑛 ‖ ‖ < 𝜀 𝛼 . ( 3 . 1 ) Hence ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ∀ 𝑞 ∈ { 0 } ∪ ℕ { ‖ 𝐽 𝛼 ( 𝑤 𝑚 , 𝑤 𝑞 + 𝑚 ) ‖ < 𝜀 𝛼 } , and if 𝑖 0 ∈ ℕ , 𝑗 0 ∈ { 0 } ∪ ℕ , 𝑖 0 > 𝑗 0 , and 𝑢 𝑚 = 𝑤 𝑖 0 + 𝑚 , 𝑣 𝑚 = 𝑤 𝑗 0 + 𝑚 f o r 𝑚 ∈ ℕ , ( 3 . 2 ) then ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ‖ ‖ 𝐽 𝛼 𝑤 𝑚 , 𝑢 𝑚 ‖ ‖ < 𝜀 𝛼 ∧ ‖ ‖ 𝐽 𝛼 𝑤 𝑚 , 𝑣 𝑚 ‖ ‖ < 𝜀 𝛼 . ( 3 . 3 ) By ( 𝒥 3), (3.1) and (3.3), ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ‖ ‖ 𝑝 𝛼 𝑤 𝑚 , 𝑢 𝑚 ‖ ‖ < 𝜀 𝛼 ∧ ‖ ‖ 𝑝 𝛼 𝑤 𝑚 , 𝑣 𝑚 ‖ ‖ < 𝜀 𝛼 . ( 3 . 4 ) If 𝑀 is a normal constant of 𝐻 , then (3.2) and (3.4) give ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 ‖ ‖ 𝑝 𝛼 𝑤 𝑚 , 𝑤 𝑖 0 + 𝑚 ‖ ‖ < 𝜀 𝛼 ∧ ‖ ‖ 𝑝 2 𝑀 𝛼 𝑤 𝑚 , 𝑤 𝑗 0 + 𝑚 ‖ ‖ < 𝜀 𝛼 . 2 𝑀 ( 3 . 5 )
Let 𝛼 ∈ 𝒜 and 𝜀 𝛼 > 0 be arbitrary and fixed and let 𝑚 , 𝑛 ∈ ℕ satisfy 𝑛 0 ⩽ 𝑚 < 𝑛 . We may suppose that 𝑛 = 𝑖 0 + 𝑛 0 and 𝑚 = 𝑗 0 + 𝑛 0 for some 𝑖 0 ∈ ℕ and 𝑗 0 ∈ { 0 } ∪ ℕ such that 𝑖 0 > 𝑗 0 . Then, by ( 𝒫 1)–( 𝒫 3), ∀ 𝛼 ∈ 𝒜 { 0 ⪯ 𝐻 𝑝 𝛼 ( 𝑤 𝑚 , w 𝑛 ) = 𝑝 𝛼 ( 𝑤 𝑗 0 + 𝑛 0 , 𝑤 𝑖 0 + 𝑛 0 ) ⪯ 𝐻 𝑝 𝛼 ( 𝑤 𝑛 0 , 𝑤 𝑗 0 + 𝑛 0 ) + 𝑝 𝛼 ( 𝑤 𝑛 0 , 𝑤 𝑖 0 + 𝑛 0 ) } . Hence, using (3.5), ∀ 𝛼 ∈ 𝒜 { ‖ 𝑝 𝛼 ( 𝑤 𝑚 , 𝑤 𝑛 ) ‖ ⩽ 𝑀 ‖ 𝑝 𝛼 ( 𝑤 𝑛 0 , 𝑤 𝑗 0 + 𝑛 0 ) ‖ + 𝑀 ‖ 𝑝 𝛼 ( 𝑤 𝑛 0 , 𝑤 𝑖 0 + 𝑛 0 ) ‖ < 𝜀 𝛼 } and, consequently, ∀ 𝛼 ∈ 𝒜 ∀ 𝜀 𝛼 > 0 ∃ 𝑛 0 = 𝑛 0 ( 𝛼 , 𝜀 𝛼 ) ∈ ℕ ∀ 𝑚 , 𝑛 ∈ ℕ ; 𝑛 0 ⩽ 𝑚 < 𝑛 { ‖ 𝑝 𝛼 ( 𝑤 𝑚 , 𝑤 𝑛 ) ‖ < 𝜀 𝛼 } . Therefore, by Theorem 2.3(b), the sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) is 𝒫 -Cauchy.

Let ( Λ , ≤ Λ ) denote a directed set whose elements will be indicated by the letters 𝜆 , 𝜂 , and 𝜇 . In the sequel, 𝜆 < Λ 𝜂 will stand for 𝜆 ≤ Λ 𝜂 and 𝜆 ≠ 𝜂 .

The relation ≤ 𝑋 on 𝑋 which is reflexive (i.e., for all 𝑥 ∈ 𝑋 the condition 𝑥 ≤ 𝑋 𝑥 holds) and transitive (i.e., for all 𝑥 , 𝑦 , 𝑧 ∈ 𝑋 the conditions 𝑥 ≤ 𝑋 𝑦 and 𝑦 ≤ 𝑋 𝑧 imply that 𝑥 ≤ 𝑋 𝑧 ) is called a quasiordering on 𝑋 and the pair ( 𝑋 , ≤ 𝑋 ) is called a quasiordering space. If, additionally, relation ≤ 𝑋 satisfies, for all 𝑥 , 𝑦 ∈ 𝑋 , the conditions: 𝑥 ≤ 𝑋 𝑦 and 𝑦 ≤ 𝑋 𝑥 which imply that 𝑥 = 𝑦 , then it is called a partial quasiordering on 𝑋 and the pair ( 𝑋 , ≤ 𝑋 ) is called a partial quasiordering space. In the sequel, 𝑢 < 𝑋 𝑣 will stand for 𝑢 ≤ 𝑋 𝑣 and 𝑢 ≠ 𝑣 .

Definition 3.2. Let 𝐿 be an ordered normed space with solid cone 𝐻 , let ( 𝑋 , 𝒫 ) be a Hausdorff cone uniform space with cone 𝐻 and let 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a 𝒥 -family on 𝑋 .(i)One says that the net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) in 𝑋 is 𝒥 -Cauchy ( 𝒫 -Cauchy) in 𝑋 if ∀ 𝛼 ∈ 𝒜 ∀ 𝑐 𝛼 ∈ 𝐿 , 0 ≪ 𝑐 𝛼 ∃ 𝜋 0 ∈ Λ ∀ 𝜂 , 𝜇 ∈ Λ ; 𝜋 0 ≤ Λ 𝜂 ≤ Λ 𝜇 { 𝐽 𝛼 ( 𝑤 𝜂 , 𝑤 𝜇 ) ≪ 𝑐 𝛼 } ( ∀ 𝛼 ∈ 𝒜 ∀ 𝑐 𝛼 ∈ 𝐿 , 0 ≪ 𝑐 𝛼 ∃ 𝜋 0 ∈ Λ ∀ 𝜂 , 𝜇 ∈ Λ ; 𝜋 0 ≤ Λ 𝜂 < Λ 𝜇 { 𝑝 𝛼 ( 𝑤 𝜂 , 𝑤 𝜇 ) ≪ 𝑐 𝛼 } ).(ii)One says that the net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) in 𝑋 is 𝒥 -convergent ( 𝒫 -convergent ) in 𝑋 , if there exists 𝑤 ∈ 𝑋 such that ∀ 𝛼 ∈ 𝒜 ∀ 𝑐 𝛼 ∈ 𝐿 , 0 ≪ 𝑐 𝛼 ∃ 𝜋 0 ∈ Λ ∀ 𝜂 ∈ Λ ; 𝜋 0 ≤ Λ 𝜂 { 𝐽 𝛼 ( 𝑤 𝜂 , 𝑤 ) ≪ 𝑐 𝛼 } ( ∀ 𝛼 ∈ 𝒜 ∀ 𝑐 𝛼 ∈ 𝐿 , 0 ≪ 𝑐 𝛼 ∃ 𝜋 0 ∈ Λ ∀ 𝜂 ∈ Λ ; 𝜋 0 ≤ Λ 𝜂 { 𝑝 𝛼 ( 𝑤 𝜂 , 𝑤 ) ≪ 𝑐 𝛼 } ) .(iii)One says that ( 𝑋 , 𝒫 ) is complete, if every 𝒫 -Cauchy net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) in 𝑋 is 𝒫 -convergent in 𝑋 .(iv)Let ( 𝑋 , 𝒫 ) be complete. For an arbitrary subset 𝐸 of 𝑋 , the closure of 𝐸 , denoted by c l ( 𝐸 ) , is defined as the set c l ( 𝐸 ) = { 𝑤 ∈ 𝑋 ∶ ∃ ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) ⊂ 𝐸 ∀ 𝛼 ∈ 𝒜 ∀ 𝑐 𝛼 ∈ 𝐿 , 0 ≪ 𝑐 𝛼 ∃ 𝜋 0 ∈ Λ ∀ 𝜂 ∈ Λ ; 𝜋 0 ≤ Λ 𝜂 { 𝑝 𝛼 ( 𝑤 𝜂 , 𝑤 ) ≪ 𝑐 𝛼 } } . The subset 𝐸 of 𝑋 is said to be a closed subset in 𝑋 if c l ( 𝐸 ) = 𝐸 .(v)Let ( 𝑋 , ≤ 𝑋 ) be a partial quasiordering space. One says that the net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) in ( 𝑋 , ≤ 𝑋 ) is increasing ( 𝑑 𝑒 𝑐 𝑟 𝑒 𝑎 𝑠 𝑖 𝑛 𝑔 ) with respect to ≤ 𝑋 if ∀ 𝜂 , 𝜇 ∈ Λ { 𝜂 < Λ 𝜇 ⇒ 𝑤 𝜂 ≤ 𝑋 𝑤 𝜇 } ( ∀ 𝜂 , 𝜇 ∈ Λ { 𝜂 < Λ 𝜇 ⇒ 𝑤 𝜇 ≤ 𝑋 𝑤 𝜂 } ) .

Of course, each 𝒫 -convergent net is a 𝒫 -Cauchy net. Also we show the following

Proposition 3.3. Let 𝐿 be an ordered Banach space with a solid cone 𝐻 and let ( 𝑋 , 𝒫 ) be a Hausdorff cone uniform space with cone 𝐻 . Let 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a 𝒥 -family on 𝑋 and let ( 𝑋 , ≤ 𝑋 ) be a partial quasiordering space. (a)Assume that each increasing sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is 𝒥 -Cauchy ( 𝒫 -Cauchy ) . Then each increasing net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) in 𝑋 is 𝒥 -Cauchy ( 𝒫 -Cauchy ) .(b)Assume that each decreasing sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is 𝒥 -Cauchy ( 𝒫 -Cauchy ) . Then each decreasing net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) in 𝑋 is 𝒥 -Cauchy ( 𝒫 -Cauchy ) .

Proof. (a)Suppose that there exists an increasing net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) in 𝑋 which is not 𝒥 -Cauchy, that is, which satisfies ∀ 𝜂 , 𝜇 ∈ Λ { 𝜂 < Λ 𝜇 ⇒ 𝑤 𝜂 ≤ 𝑋 𝑤 𝜇 } and ∃ 𝛼 0 ∈ 𝒜 ∃ 𝑐 𝛼 0 ∈ 𝐿 , 0 ≪ 𝑐 𝛼 0 ∀ 𝜋 ∈ Λ ∃ 𝜂 , 𝜇 ∈ Λ ; 𝜋 ≤ Λ 𝜂 ≤ Λ 𝜇 𝐽 𝛼 0 𝑤 𝜂 , 𝑤 𝜇 − 𝑐 𝛼 0 . ∉ i n t ( 𝐻 ) ( 3 . 6 ) Assume that 𝜋 1 ∈ Λ is arbitrary and fixed. By (3.6), there exist 𝜂 1 , 𝜇 1 ∈ Λ , 𝜋 1 ≤ Λ 𝜂 1 ≤ Λ 𝜇 1 , such that 𝐽 𝛼 0 ( 𝑤 𝜂 1 , 𝑤 𝜇 1 ) − 𝑐 𝛼 0 ∉ i n t ( 𝐻 ) and define 𝑣 1 = 𝑤 𝜂 1 and 𝑣 2 = 𝑤 𝜇 1 . Next, for 𝜋 2 = 𝜇 1 , by (3.6), there exist 𝜂 2 , 𝜇 2 ∈ Λ , 𝜋 2 ≤ Λ 𝜂 2 ≤ Λ 𝜇 2 , such that 𝐽 𝛼 0 ( 𝑤 𝜂 2 , 𝑤 𝜇 2 ) − 𝑐 𝛼 0 ∉ i n t ( 𝐻 ) and define 𝑣 3 = 𝑤 𝜂 2 and 𝑣 4 = 𝑤 𝜇 2 . Now, if 𝑣 𝑘 are defined for 𝑘 = 1 , … , 2 ( 𝑛 − 1 ) and if 𝜋 𝑛 = 𝜇 𝑛 − 1 , then, by (3.6), there exist 𝜂 𝑛 , 𝜇 𝑛 ∈ Λ , 𝜋 𝑛 ≤ Λ 𝜂 𝑛 ≤ Λ 𝜇 𝑛 , such that 𝐽 𝛼 0 ( 𝑤 𝜂 𝑛 , 𝑤 𝜇 𝑛 ) − 𝑐 𝛼 0 ∉ i n t ( 𝐻 ) and define 𝑣 2 𝑛 − 1 = 𝑤 𝜂 𝑛 and 𝑣 2 𝑛 = 𝑤 𝜇 𝑛 . By induction, this gives ∀ 𝑚 ∈ ℕ { 𝑣 𝑚 ≤ 𝑋 𝑣 𝑚 + 1 } and ∃ 𝛼 0 ∈ 𝒜 ∃ 𝑐 𝛼 0 ∈ 𝐿 , 0 ≪ 𝑐 𝛼 0 ∀ 𝑛 ∈ ℕ ∃ 𝑚 0 , 𝑛 0 ∈ ℕ ; 𝑛 ⩽ 𝑚 0 ⩽ 𝑛 0 { 𝐽 𝛼 0 ( 𝑣 𝑚 0 , 𝑣 𝑛 0 ) − 𝑐 𝛼 0 ∉ i n t ( 𝐻 ) } . Consequently, there exists an increasing sequence ( 𝑣 𝑛 ∶ 𝑛 ∈ ℕ ) in 𝑋 which is not 𝒥 -Cauchy.
By Remark 2.7, we get the claim.
(b)We use a similar argument as in ( a ).

Let ( 𝑋 , ≤ 𝑋 ) be a partial quasiordering space. Set 𝐸 ⊂ 𝑋 which is called a chain in 𝑋 if any two elements of 𝐸 are comparable, that is, 𝑥 ≤ 𝑋 𝑦 or 𝑦 ≤ 𝑋 𝑥 for all 𝑥 , 𝑦 ∈ 𝐸 . The Zorn lemma says that every partially ordered set in which every chain has an upper (lower) bound contains at least one maximal (minimal) element.

The main result of this section is the following maximality (minimality) principle.

Theorem 3.4. Let 𝐿 be an ordered Banach space with a normal solid cone 𝐻 and let ( 𝑋 , 𝒫 ) be a Hausdorff cone uniform space with cone 𝐻 . Let 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 , 𝛼 ∈ 𝒜 } be a 𝒥 -family on 𝑋 and let ( 𝑋 , ≤ 𝑋 ) be a partial quasiordering space. (A)Assume that ( a 1 ) for each 𝑥 ∈ 𝑋 , the set { 𝑦 ∈ 𝑋 ∶ 𝑥 ≤ 𝑋 𝑦 } is complete, and ( a 2 ) each increasing sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is 𝒥 -Cauchy. Then 𝑋 contains at least one maximal element.(B)Assume that ( b 1 ) for each 𝑥 ∈ 𝑋 , the set { 𝑦 ∈ 𝑋 ∶ 𝑦 ≤ 𝑋 𝑥 } is complete, and ( b 2 ) each decreasing sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is 𝒥 -Cauchy. Then 𝑋 contains at least one minimal element.

Proof. (A) The proof will be broken into five steps.Step 1. Suppose ( a 2 ) holds, that is, that each increasing sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is 𝒥 -Cauchy. Then, by Proposition 3.1, each increasing sequence ( 𝑤 𝑚 ∶ 𝑚 ∈ ℕ ) in 𝑋 is 𝒫 -Cauchy and, consequently, Proposition 3.3 ( a ) gives that each increasing net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) in 𝑋 is 𝒫 -Cauchy.Step 2. Let an increasing net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) in 𝑋 be arbitrary and fixed. In view of (a 1 ) and Step 1, ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) is convergent to a 𝑤 ∈ 𝑋 and, since 𝑋 is Hausdorff, 𝑤 is unique.Step 3. Let 𝐸 be a chain in ( 𝑋 , ≤ 𝑋 ) . If ∃ 𝑢 ∈ 𝐸 ∀ 𝑣 ∈ 𝐸 { 𝑣 ≤ 𝑋 𝑢 } , then 𝐸 has an upper bound in 𝑋 .Step 4. Let 𝐸 be a chain in ( 𝑋 , ≤ 𝑋 ) . If ∀ 𝑢 ∈ 𝐸 ∃ 𝑣 ∈ 𝐸 { 𝑢 < 𝑋 𝑣 } , then denoting Λ = 𝐸 and 𝑤 𝜆 = 𝜆 for each 𝜆 ∈ 𝐸 , we can identify 𝐸 with the increasing net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) . Next, using, in particular, Steps 1 and 2, we can show that ∀ 𝜆 ∈ Λ { 𝑤 𝜆 ≤ 𝑋 𝑤 } where 𝑤 is a unique limit of ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) ; which means that 𝑤 is an upper bound of ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ ) . Indeed, let 𝜆 0 ∈ Λ be arbitrary and fixed and define the sets Λ 0 , 𝐸 0 by Λ 0 = { 𝜆 ∈ Λ ∶ 𝜆 0 ≤ Λ 𝜆 } , 𝐸 0 = { 𝑦 ∈ 𝑋 ∶ 𝑤 𝜆 0 ≤ 𝑋 𝑦 } . By assumption ( a 1 ), 𝐸 0 is complete. Clearly, the net ( 𝑤 𝜆 ∶ 𝜆 ∈ Λ 0 ) is increasing in 𝑋 , 𝒫 -Cauchy, convergent to 𝑤 and 𝑤 ∈ 𝐸 0 . This proves that 𝑤 𝜆 0 ≤ 𝑋 𝑤 . Therefore, 𝐸 has an upper bound in 𝑋 .Step 5. Using Steps 3 and 4 and the Zorn lemma, we conclude that 𝑋 contains at least one maximal element.(B) We use a similar argument as in ( A ).

4. Variational Principle of Ekeland Type and Fixed Point and Endpoint Theorem of Caristi Type in Cone Uniform Spaces with Generalized Pseudodistances

Let 2 𝑋 denote the family of all nonempty subsets of a space 𝑋 . Recall that 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.

Let 𝐿 be an ordered Banach space with a cone 𝐻 and let ( 𝑋 , 𝒫 ) be a cone uniform space with cone 𝐻 .

Let an element + ∞ ∉ 𝐿 be such that 𝑎 ⪯ 𝐻 + ∞ for all 𝑎 ∈ 𝐿 . We say that a map 𝐹 ∶ 𝑋 → 𝐿 ∪ { + ∞ } is proper if its effective domain, d o m ( 𝐹 ) = { 𝑥 ∶ 𝜔 ( 𝑥 ) ≠ + ∞ } , is nonempty.

If 𝒥 = { 𝐽 𝛼 ∶ 𝑋 × 𝑋 → 𝐿 ∶