Startpoints and (𝛼,𝛾)-Contractions in Quasi-Pseudometric Spaces

Abstract

We introduce the concept of startpoint and endpoint for multivalued maps defined on a quasi-pseudometric space. We investigate the relation between these new concepts and the existence of fixed points for these set valued maps.

Dedicated to my beloved Clémence on the occasion of her 25th birthday

1. Introduction

In the last few years there has been a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces (see, e.g., [1]), because such a theory provides an important tool and a convenient framework in the study of several problems in theoretical computer science, applied physics, approximation theory, and convex analysis. Many works on general topology have been done in order to extend the well-known results of the classical theory. In particular, various types of completeness are studied in [2], showing, for instance, that the classical concept of Cauchy sequences can be accordingly modified. In the same reference, which uses an approach based on uniformities, the bicompletion of a 𝑇₀-quasi-pseudometric has been explored. It is worth mentioning that, in the fixed point theory, completeness is a key element, since most of the constructed sequences will be assumed to have a Cauchy type property.

It is the aim of this paper to continue the study of quasi-pseudometric spaces by proving some fixed point results and investigating a bit more the behaviour of set-valued mappings. Thus, in Section 3 a suitable notion of (𝛼,𝛾)-contractive mapping is given for self-mappings defined on quasi-pseudometric spaces and some fixed point results are discussed. In Sections 4 and 5, the notions of startpoint and endpoint for set-valued mappings are introduced and different variants of such concepts, as well as their connections with the fixed point of a multivalued map, are characterized.

For recent results in the theory of asymmetric spaces, the reader is referred to [3–8].

2. Preliminaries

Definition 1. Let 𝑋 be a nonempty set. A function 𝑑∶𝑋×𝑋→[0,∞) is called a quasi-pseudometric on 𝑋 if (i)𝑑(𝑥,𝑥)=0 for all 𝑥∈𝑋,(ii)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋.Moreover, if 𝑑(𝑥,𝑦)=0=𝑑(𝑦,𝑥)⇒𝑥=𝑦, then 𝑑 is said to be a 𝑇₀-quasi-pseudometric. The latter condition is referred to as the 𝑇₀-condition.

Remark 2. (i) Let 𝑑 be a quasi-pseudometric on 𝑋; then the map 𝑑⁻¹ defined by 𝑑⁻¹(𝑥,𝑦)=𝑑(𝑦,𝑥) whenever 𝑥,𝑦∈𝑋 is also a quasi-pseudometric on 𝑋, called the conjugate of 𝑑. In the literature, 𝑑⁻¹ is also denoted by 𝑑^𝑡 or 𝑑.
(ii) It is easy to verify that the function 𝑑^𝑠 defined by 𝑑^𝑠∶=𝑑∨𝑑⁻¹, that is, 𝑑^𝑠(𝑥,𝑦)=max{𝑑(𝑥,𝑦),𝑑(𝑦,𝑥)}, defines a metric on 𝑋 whenever 𝑑 is a 𝑇₀-quasi-pseudometric on 𝑋.

Let (𝑋,𝑑) be a quasi-pseudometric space. For 𝑥∈𝑋 and 𝜀>0, 𝐵_𝑑(𝑥,𝜀)=𝑦∈𝑋∶𝑑𝑥,𝑦<𝜀 (1) denotes the open 𝜀-ball at 𝑥. The collection of all such balls yields a base for the topology 𝜏(𝑑) induced by 𝑑 on 𝑋. Hence, for any 𝐴∈𝑋, we will, respectively, denote by int_𝜏(𝑑)𝐴 and cl_𝜏(𝑑)𝐴 the interior and the closure of the set 𝐴 with respect to the topology 𝜏(𝑑).

Similarly, for 𝑥∈𝑋 and 𝜀≥0, 𝐶_𝑑(𝑥,𝜀)=𝑦∈𝑋∶𝑑𝑥,𝑦≤𝜀 (2) denotes the closed 𝜀-ball at 𝑥. We will say that a subset 𝐸⊂𝑋 is join-closed if it is 𝜏(𝑑^𝑠)-closed, that is, closed with respect to the topology generated by 𝑑^𝑠. The topology 𝜏(𝑑^𝑠) is finer than the topologies 𝜏(𝑑) and 𝜏(𝑑⁻¹).

Definition 3. Let (𝑋,𝑑) be a quasi-pseudometric space. The convergence of a sequence (𝑥_𝑛) to 𝑥 with respect to 𝜏(𝑑), called 𝑑-convergence or left-convergence and denoted by 𝑥→−_𝑛^𝑑𝑥, is defined in the following way: 𝑥→−_𝑛^𝑑𝑥⟺𝑑𝑥,𝑥_𝑛⟶0. (3)
Similarly, the convergence of a sequence (𝑥_𝑛) to 𝑥 with respect to 𝜏(𝑑⁻¹), called 𝑑⁻¹-convergence or right-convergence and denoted by 𝑥−−−→_𝑛^𝑑−1𝑥, is defined in the following way: 𝑥−−−→_𝑛^𝑑−1𝑥⟺𝑑𝑥_𝑛,𝑥⟶0. (4)
Finally, in a quasi-pseudometric space (𝑋,𝑑), we will say that a sequence (𝑥_𝑛)𝑑^𝑠-converges to 𝑥 if it is both left and right convergent to 𝑥, and we denote it by 𝑥−−→_𝑛^𝑑𝑠𝑥 or 𝑥_𝑛→𝑥 when there is no confusion. Hence, 𝑥−−→_𝑛^𝑑𝑠𝑥⟺𝑥→−_𝑛^𝑑𝑥,𝑥−−−→_𝑛^𝑑−1𝑥. (5)

Definition 4. A sequence (𝑥_𝑛) in a quasi-pseudometric (𝑋,𝑑) is called(a) left 𝑑-Cauchy if for every 𝜖>0, there exist 𝑥∈𝑋 and 𝑛₀∈ℕ such that ∀𝑛≥𝑛₀𝑑𝑥,𝑥_𝑛<𝜖; (6)(b) left 𝐾-Cauchy if for every 𝜖>0, there exists 𝑛₀∈ℕ such that ∀𝑛,𝑘∶𝑛₀≤𝑘≤𝑛𝑑𝑥_𝑘,𝑥_𝑛<𝜖; (7)(c) 𝑑^𝑠-Cauchy if for every 𝜖>0, there exists 𝑛₀∈ℕ such that ∀𝑛,𝑘≥𝑛₀𝑑𝑥_𝑛,𝑥_𝑘<𝜖. (8)
Dually, we define, in the same way, right 𝑑-Cauchy and right 𝐾-Cauchy sequences.

Remark 5. Consider the following:
(i) 𝑑^𝑠-Cauchy ⇒ left 𝐾-Cauchy ⇒ left 𝑑-Cauchy. The same implications hold for the corresponding right notions. None of the above implications is reversible.
(ii) A sequence is left 𝑑-Cauchy with respect to 𝑑 if and only if it is right 𝐾-Cauchy with respect to 𝑑⁻¹.
(iii) A sequence is left 𝐾-Cauchy with respect to 𝑑 if and only if it is right 𝐾-Cauchy with respect to 𝑑⁻¹.
(iv) A sequence is 𝑑^𝑠-Cauchy if and only if it is both left and right 𝐾-Cauchy.

Definition 6. A quasi-pseudometric space (𝑋,𝑑) is called(i)left-𝐾-complete provided that any left 𝐾-Cauchy sequence is 𝑑-convergent,(ii)left Smyth sequentially complete if any left 𝐾-Cauchy sequence is 𝑑^𝑠-convergent.The dual notions of right-completeness are easily derived from the above definition.

Definition 7. A 𝑇₀-quasi-pseudometric space (𝑋,𝑑) is called bicomplete provided that the metric 𝑑^𝑠 on 𝑋 is complete.

As usual, a subset 𝐴 of a quasi-pseudometric space (𝑋,𝑑) will be called bounded provided that there exists a positive real constant 𝑀 such that 𝑑(𝑥,𝑦)<𝑀 whenever 𝑥,𝑦∈𝐴. This is equivalent to saying that there exist 𝑥₀∈𝑋, 𝑟,𝑠≥0 such that 𝐴⊆𝐶_𝑑(𝑥₀,𝑟)∩𝐶_𝑑⁻¹(𝑥₀,𝑟).

We also define the diameter 𝛿(𝐴) of 𝐴 by 𝛿(𝐴)∶=sup{𝑑(𝑥,𝑦)∶𝑥,𝑦∈𝐴}. Hence, 𝐴 is bounded if and only if 𝛿(𝐴)<∞. It is not difficult to see that this definition coincides with that of a bounded set in a metric space.

Let (𝑋,𝑑) be a quasi-pseudometric space. We set 𝒫₀(𝑋)∶=2^𝑋⧵{∅} where 2^𝑋 denotes the power set of 𝑋. For 𝑥∈𝑋 and 𝐴,𝐵∈𝒫₀(𝑋), we define 𝑑(){𝑑()},𝑑(){𝑑()},𝑥,𝐴=inf𝑥,𝑎,𝑎∈𝐴𝐴,𝑥=inf𝑎,𝑥,𝑎∈𝐴 (9) and we define 𝐻(𝐴,𝐵) by 𝐻(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝐴,𝑏). (10)

Then 𝐻 is an extended quasi-pseudometric on 𝒫₀(𝑋). Moreover, we know from [9] that the restriction of 𝐻 to 𝑆_cl(𝑋)={𝐴⊆𝑋∶𝐴=(cl_𝜏(𝑑)𝐴)∩(cl_𝜏(𝑑⁻¹₎𝐴)} is an extended 𝑇₀-quasi-pseudometric. We will denote by 𝐶𝐵(𝑋) the collection of all nonempty bounded and 𝜏(𝑑^𝑠)-closed subsets of 𝑋.

We complete this section by the following lemma.

Lemma 8. Let (𝑋,𝑑) be a quasi-pseudometric space. For every fixed 𝑥∈𝑋, the mapping 𝑦↦𝑑(𝑥,𝑦) is 𝜏(𝑑)-upper semicontinuous (𝜏(𝑑)-usc in short) and 𝜏(𝑑⁻¹)-lower semicontinuous (𝜏(𝑑⁻¹)-lsc in short). For every fixed 𝑦∈𝑋, the mapping 𝑥↦𝑑(𝑥,𝑦) is 𝜏(𝑑)-lsc and 𝜏(𝑑⁻¹)-usc.

Proof. To prove that 𝑑(𝑥,⋅) is 𝜏(𝑑)-usc and 𝜏(𝑑⁻¹)-lsc, we have to show that the set {𝑦∈𝑋∶𝑑(𝑥,𝑦)<𝛼} is 𝜏(𝑑)-open and {𝑦∈𝑋∶𝑑(𝑥,𝑦)>𝛼} is 𝜏(𝑑⁻¹)-open, for every 𝛼∈ℝ, properties that are easy to check.
Indeed, for 𝑦∈𝑋 such that 𝑑(𝑥,𝑦)<𝛼, let 𝑟∶=𝛼−𝑑(𝑥,𝑦)>0. If 𝑧∈𝑋is such that 𝑑(𝑦,𝑧)<𝑟, then 𝑑(𝑥,𝑧)≤𝑑𝑥,𝑦+𝑦,𝑧<𝑑𝑥,𝑦+𝑟=𝛼, (11) showing that 𝐵_𝑑(𝑦,𝑟)⊂{𝑦∈𝑋∶𝑑(𝑥,𝑦)<𝛼}.
Similarly, for 𝑦∈𝑋 with 𝑑(𝑥,𝑦)>𝛼 take 𝑟∶=𝑑(𝑥,𝑦)−𝛼>0. If 𝑧∈𝑋 satisfies 𝑑(𝑧,𝑦)=𝑑⁻¹(𝑦,𝑧)<𝑟, then 𝑑𝑥,𝑦≤𝑑(𝑥,𝑧)+𝑧,𝑦<𝑑(𝑥,𝑧)+𝑟, (12) so that 𝑑(𝑥,𝑧)>𝑑(𝑥,𝑦)−𝑟=𝛼. Consequently, 𝐵_𝑑⁻¹(𝑦,𝑟)⊂{𝑦∈𝑋∶𝑑(𝑥,𝑦)>𝛼}.

3. Some First Results

We begin by recalling the following.

Definition 9. A function 𝛾∶[0,∞)→[0,∞) is called a (𝑐)-comparison function if (𝛾₁)𝛾 is nondecreasing;(𝛾₂)∑^∞_𝑛=1𝛾^𝑛(𝑡)<∞ for all 𝑡>0, where 𝛾^𝑛 is the 𝑛th iterate of 𝛾.We will denote by Γ the set of such functions. Note that for any 𝛾∈Γ,   𝛾(𝑡)<𝑡 for any 𝑡>0.

We then introduce the following definitions.

Definition 10. Let (𝑋,𝑑) be a quasi-pseudometric type space. A function 𝑇∶𝑋→𝑋 is called 𝑑-sequentially continuous or left-sequentially continuous if for any 𝑑-convergent sequence (𝑥_𝑛) with 𝑥→−_𝑛^𝑑𝑥, the sequence (𝑇𝑥_𝑛)𝑑-converges to 𝑇𝑥; that is, 𝑇𝑥→−_𝑛^𝑑𝑇𝑥.
Similarly, we define a 𝑑⁻¹-sequentially continuous or right-sequentially continuous function.

Definition 11. Let (𝑋,𝑑) be a quasi-pseudometric space, and let 𝑇∶𝑋→𝑋 and 𝛼∶𝑋×𝑋→[0,∞) be mappings. We say that 𝑇 is 𝛼-admissible if 𝛼𝑥,𝑦≥1⟹𝛼𝑇𝑥,𝑇𝑦≥1, (13) whenever 𝑥,𝑦∈𝑋.

Definition 12. Let (𝑋,𝑑) be a quasi-pseudometric space and let 𝑇∶𝑋→𝑋 be a mapping. We say that 𝑇 is an (𝛼,𝛾)-contractive mapping if there exist two functions 𝛼∶𝑋×𝑋→[0,∞) and 𝛾∈Γ such that 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝛾𝑑𝑥,𝑦, (14) whenever 𝑥,𝑦∈𝑋.

We now state the first fixed point theorem.

Theorem 13. Let (𝑋,𝑑) be a Hausdorff left 𝐾-complete 𝑇₀-quasi-pseudometric space. Suppose that 𝑇∶𝑋→𝑋 is an (𝛼,𝛾)-contractive mapping which satisfies the following conditions: (i)𝑇 is 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1;(iii)𝑇 is 𝑑-sequentially continuous.Then 𝑇 has a fixed point.

Proof. By (ii), there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1. Let us define the sequence (𝑥_𝑛) in 𝑋 by 𝑥_𝑛+1=𝑇𝑥_𝑛 for all 𝑛=0,1,2,…. Without loss of generality, we can always assume that 𝑥_𝑛≠𝑥_𝑛+1 for all 𝑛∈ℕ, since if 𝑥_𝑛0=𝑥_𝑛0+1 for some 𝑛₀∈ℕ, the proof is complete.

From assumption (i), we derive 𝛼𝑥₀,𝑇𝑥₀=𝛼𝑥₀,𝑥₁≥1⟹𝛼𝑇𝑥₀,𝑇𝑥₁=𝛼𝑥₁,𝑥₂≥1. (15) Recursively, we get 𝛼𝑥_𝑛,𝑥_𝑛+1≥1∀𝑛=0,1,2,…. (16) Since 𝑇 is (𝛼,𝛾)-contractive, we can write 𝑑𝑥_𝑛,𝑥_𝑛+1=𝑑𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝛼𝑥_𝑛−1,𝑥_𝑛𝑑𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝛾𝑑𝑥_𝑛−1,𝑥_𝑛, (17) for all 𝑛=0,1,2,…. Inductively, we obtain 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝛾^𝑛𝑑𝑥₀,𝑥₁,𝑛=1,2,…. (18) Therefore, for any 𝑘≥1, using the triangle inequality, we get 𝑑𝑥_𝑛,𝑥_𝑛+𝑘≤𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑑𝑥_{𝑛+𝑘−1},𝑥_𝑛+𝑘≤^{𝑛+𝑘−1}_𝑖=𝑛𝛾^𝑖𝑑𝑥₀,𝑥₁≤^∞_𝑖=𝑛𝛾^𝑖𝑑𝑥₀,𝑥₁. (19) Letting 𝑛→∞, we derive that 𝑑(𝑥_𝑛,𝑥_𝑛+𝑘)→0. Hence, (𝑥_𝑛) is a left 𝐾-Cauchy sequence. Since (𝑋,𝑑) is left 𝐾-complete and 𝑇𝑑-sequentially continuous, there exists 𝑥^∗ such that 𝑥→−_𝑛^𝑑𝑥^∗ and 𝑥^𝑑_𝑛+1→−𝑇𝑥^∗. Since 𝑋 is Hausdorff, we have that 𝑥^∗=𝑇𝑥^∗.

Corollary 14. Let (𝑋,𝑑) be a Hausdorff right 𝐾-complete 𝑇₀-quasi-pseudometric space. Suppose that 𝑇∶𝑋→𝑋 is an (𝛼,𝛾)-contractive mapping which satisfies the following conditions: (i)𝑇 is 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑇𝑥₀,𝑥₀)≥1;(iii)𝑇 is 𝑑⁻¹-sequentially continuous.Then 𝑇 has a fixed point.

Corollary 15. Let (𝑋,𝑑) be a bicomplete quasi-pseudometric space. Suppose that 𝑇∶𝑋→𝑋 is an (𝛼,𝛾)-contractive mapping which satisfies the following conditions: (i)𝑇 is 𝛼-admissible and the function 𝛼 is symmetric; that is, 𝛼(𝑎,𝑏)=𝛼(𝑏,𝑎) for any 𝑎,𝑏∈𝑋;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑇𝑥₀,𝑥₀)≥1;(iii)𝑇 is 𝑑^𝑠-sequentially continuous.Then 𝑇 has a fixed point.

Proof. Following the proof of Theorem 13, it is clear that the sequence (𝑥_𝑛) in 𝑋 defined by 𝑥_𝑛+1=𝑇𝑥_𝑛 for all 𝑛=0,1,2,… is 𝑑^𝑠-Cauchy. Since (𝑋,𝑑^𝑠) is complete and 𝑇 sequentially continuous, there exists 𝑥^∗ such that 𝑥−−→_𝑛^𝑑𝑠𝑥^∗ and 𝑥−−→^𝑑_𝑛+1^𝑠𝑇𝑥^∗. Since (𝑋,𝑑^𝑠) is Hausdorff, we have that 𝑥^∗=𝑇𝑥^∗.

Remark 16. In fact, we do not need 𝛼 to be symmetric. It is enough, for the result to be true, to have a point 𝑥₀∈𝑋 for which 𝛼(𝑥₀,𝑇𝑥₀)≥1 and 𝛼(𝑇𝑥₀,𝑥₀)≥1.

We conclude this section by the following results which are in fact consequences of Theorem 13.

Theorem 17. Let (𝑋,𝑑) be a Hausdorff left 𝐾-complete 𝑇₀-quasi-pseudometric space. Suppose that 𝑇∶𝑋→𝑋 is an (𝛼,𝛾)-contractive mapping which satisfies the following conditions:(i)𝑇 is 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1;(iii)if (𝑥_𝑛) is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛=1,2,… and 𝑥→−_𝑛^𝑑𝑥, then there exists a subsequence (𝑥_𝑛(𝑘)) of (𝑥_𝑛) such that 𝛼(𝑥_𝑛(𝑘),𝑥)≥1 for all 𝑘.Then 𝑇 has a fixed point.

Proof. Following the proof of Theorem 13, we know that the sequence (𝑥_𝑛) defined by 𝑥_𝑛+1=𝑇𝑥_𝑛 for all 𝑛=0,1,2,…𝑑-converges to some 𝑥^∗ and satisfies 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for 𝑛≥1. From condition (iii), we know that there exists a subsequence (𝑥_𝑛(𝑘)) of (𝑥_𝑛) such that 𝛼(𝑥_𝑛(𝑘),𝑥^∗)≥1 for all 𝑘. Since 𝑇 is an (𝛼,𝛾)-contractive mapping, we get 𝑑𝑥_𝑛(𝑘)+1,𝑇𝑥^∗=𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑥^∗≤𝛼𝑥_𝑛(𝑘),𝑥^∗𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑥^∗≤𝛾𝑑𝑥_𝑛(𝑘),𝑥^∗. (20) Letting 𝑘→∞, we obtain 𝑑(𝑥_𝑛(𝑘)+1,𝑇𝑥^∗)→0. Since 𝑋 is Hausdorff, we have that 𝑇𝑥^∗=𝑥^∗.
This completes the proof.

Corollary 18. Let (𝑋,𝑑) be a Hausdorff right 𝐾-complete 𝑇₀-quasi-pseudometric space. Suppose that 𝑇∶𝑋→𝑋 is an (𝛼,𝛾)-contractive mapping which satisfies the following conditions:(i)𝑇 is 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑇𝑥₀,𝑥₀)≥1;(iii)if (𝑥_𝑛) is a sequence in 𝑋 such that 𝛼(𝑥_𝑛+1,𝑥_𝑛)≥1 for all 𝑛=1,2,… and 𝑥−−−→_𝑛^𝑑−1𝑥, then there exists a subsequence (𝑥_𝑛(𝑘)) of (𝑥_𝑛) such that 𝛼(𝑥,𝑥_𝑛(𝑘))≥1 for all 𝑘.Then 𝑇 has a fixed point.

Corollary 19. Let (𝑋,𝑑) be a bicomplete quasi-pseudometric space. Suppose that 𝑇∶𝑋→𝑋 is an (𝛼,𝛾)-contractive mapping which satisfies the following conditions:(i)𝑇 is 𝛼-admissible and the function 𝛼 is symmetric;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑇𝑥₀,𝑥₀)≥1;(iii)if (𝑥_𝑛) is a sequence in 𝑋 such that 𝛼(𝑥_𝑚,𝑥_𝑛)≥1 for all 𝑛,𝑚∈ℕ and 𝑥−−→_𝑛^𝑑𝑠𝑥, then there exists a subsequence (𝑥_𝑛(𝑘)) of (𝑥_𝑛) such that 𝛼(𝑥,𝑥_𝑛(𝑘))≥1 for all 𝑘.Then 𝑇 has a fixed point.

4. Startpoint Theory

It is important to mention that there are a variety of endpoint concepts in the literature (see, e.g., [10]), each of them corresponding to a specified setting. Here we introduce a similar notion for set-valued maps defined on quasi-pseudometric spaces. Let (𝑋,𝑑) be a 𝑇₀-quasi-pseudometric space.

Definition 20. Let 𝐹∶𝑋→2^𝑋 be a set-valued map. An element 𝑥∈𝑋 is said to be (i)a fixed point of 𝐹 if 𝑥∈𝐹𝑥,(ii)a startpoint of 𝐹 if 𝐻({𝑥},𝐹𝑥)=0,(iii)an endpoint of 𝐹 if 𝐻(𝐹𝑥,{𝑥})=0,(iv)an 𝜀-startpoint of 𝐹 for some 𝜀∈(0,1) if 𝐻({𝑥},𝐹𝑥)<𝜀,(v)an 𝜀-endpoint of 𝐹 for some 𝜀∈(0,1) if 𝐻(𝐹𝑥,{𝑥})<𝜀.

Remark 21. It is therefore obvious that if 𝑥 is both a startpoint of 𝐹 and an endpoint of 𝐹, then 𝑥 is a fixed point of 𝐹. In fact, 𝐹𝑥 is a singleton. But a fixed point need not be a startpoint nor an endpoint.
Indeed, consider the 𝑇₀-quasi-pseudometric space (𝑋,𝑑), where 𝑋={0,1} and 𝑑 is defined by 𝑑(0,1)=0, 𝑑(1,0)=1, and 𝑑(𝑥,𝑥) for 𝑥=0,1. We define on 𝑋 the set-valued map 𝐹∶𝑋→2^𝑋 by 𝐹𝑥=𝑋. Obviously, 1 is a fixed point, but 𝐻({1},𝐹1)=𝐻({1},𝑋)=max{𝑑(1,1),𝑑(1,0)}=1≠0.

Lemma 22. Let (𝑋,𝑑) be a 𝑇₀-quasi-pseudometric space and let 𝐹∶𝑋→2^𝑋 be a set-valued map. An element 𝑥∈𝑋 is a startpoint of 𝐹 if and only if it is an 𝜀-startpoint of 𝐹 for every 𝜀∈(0,1).

Lemma 23. Let (𝑋,𝑑) be a 𝑇₀-quasi-pseudometric space and let 𝐹∶𝑋→2^𝑋 be a set-valued map. An element 𝑥∈𝑋 is an endpoint of 𝐹 if and only if it is an 𝜀-endpoint of 𝐹 for every 𝜀∈(0,1).

Definition 24. Let (𝑋,𝑑) be a 𝑇₀-quasi-pseudometric space. We say that a set-valued map 𝐹∶𝑋→2^𝑋 has the approximate startpoint property (resp., approximate endpoint property) if inf_𝑥∈𝑋sup_{𝑦∈𝐹𝑥}𝑑𝑥,𝑦=0resp.,inf_𝑥∈𝑋sup_{𝑦∈𝐹𝑥}𝑑𝑦,𝑥=0. (21)

Definition 25. Let (𝑋,𝑑) be a 𝑇₀-quasi-pseudometric space. We say that a set-valued map 𝐹∶𝑋→2^𝑋 has the approximate mix-point property if inf_𝑥∈𝑋sup_{𝑦∈𝐹𝑥}𝑑^𝑠𝑥,𝑦=0. (22) Here, it is also very clear that 𝐹 has approximate mix-point property if and only if 𝐹 has both the approximate startpoint and the approximate endpoint properties.

We are therefore naturally led to this definition.

Definition 26. Let 𝑓∶𝑋→𝑋 be a single-valued map on a 𝑇₀-quasi-pseudometric space (𝑋,𝑑). Then 𝑓 has the approximate startpoint property (resp., approximate endpoint property) if and only if inf_𝑥∈𝑋𝑑𝑥,𝑓𝑥=0resp.,inf_𝑥∈𝑋𝑑𝑓𝑥,𝑥=0. (23)

We motivate our coming results by the following examples. We basically show that the concepts defined above are independent and do not necessarily coincide. The list of examples presented is not exhaustive and more can be constructed, showing the connection between the notions defined above.

Example 27. Let 𝑋={0,1,2}. The map 𝑑∶𝑋×𝑋→[0,∞) defined by 𝑑(0,1)=𝑑(0,2)=0, 𝑑(1,0)=𝑑(1,2)=1, 𝑑(2,0)=𝑑(2,1)=2, and 𝑑(𝑥,𝑥)=0 for all 𝑥∈𝑋 is a 𝑇₀-quasi-pseudometric on 𝑋. Let 𝐹∶𝑋→2^𝑋 be the set mapping defined by 𝐹𝑎=𝑋⧵{𝑎} for any 𝑎∈𝑋. By definition, 𝐹 does not have any fixed point. Nevertheless, a simple computation shows that 𝐻({0},𝐹0)=0, and hence 0 is a startpoint and it is the only one. Also there is no endpoint. Again, with a direct computation, we have inf_𝑥∈𝑋sup_{𝑦∈𝐹𝑥}𝑑(𝑥,𝑦)=0, showing that 𝐹 has the approximate startpoint property, but inf_𝑥∈𝑋sup_{𝑦∈𝐹𝑥}𝑑(𝑦,𝑥)=1, showing that 𝐹 does not have the approximate endpoint property.

Example 28. Let 𝑋={1/𝑛,𝑛=1,2,…}. The map 𝑑∶𝑋×𝑋→[0,∞) defined by 𝑑(1/𝑛,1/𝑚)=max{1/𝑛−1/𝑚,0} is a 𝑇₀-quasi-pseudometric on 𝑋. Let 𝐹∶𝑋→2^𝑋 be the set-valued mapping defined by 𝐹𝑎=𝑋⧵{𝑎} for any 𝑎∈𝑋. By definition, 𝐹 does not have any fixed point.
For a fixed 𝑛₀∈ℕ, 𝑑1𝑛₀,1𝑚=0if𝑚≤𝑛₀,1𝑛₀−1𝑚if𝑚>𝑛₀. (24) Similarly for a fixed 𝑛₀∈ℕ, 𝑑1𝑚,1𝑛₀=0if𝑚≥𝑛₀,1𝑚−1𝑛₀if𝑚<𝑛₀. (25) Hence, 𝐹 does not have any startpoint nor endpoint (which also implies that 𝐹 does not have any fixed point).

But for a given 𝜀∈(0,1), there exists 𝑘₀∈ℕ such that 1/𝑘₀<𝜀. We also know from definition that 𝐻({1/𝑘₀},𝐹(1/𝑘₀))=1/𝑘₀, so, 1/𝑘₀ is an 𝜀-startpoint of 𝐹. More generally, for a given 𝜀∈(0,1), there exists 𝑘_𝜀∈ℕ such that 1/𝑘_𝜀 is an 𝜀-startpoint of 𝐹. Moreover, for any 𝑘≥𝑘_𝜀, 1/𝑘 is an 𝜀-startpoint of 𝐹.

Similarly, we can show that 𝐹 admits an 𝜀-endpoint.

We can now state our first result.

Theorem 29. Let (𝑋,𝑑) be a bicomplete quasi-pseudometric space. Let 𝐹∶𝑋→𝐶𝐵(𝑋) be a set-valued map that satisfies 𝐻𝐹𝑥,𝐹𝑦≤𝜓𝑑𝑥,𝑦,𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑥,𝑦∈𝑋, (26) where 𝜓∶[0,∞)→[0,∞) is upper semicontinuous, 𝜓(𝑡)<𝑡 for each 𝑡>0, and liminf_𝑡→∞(𝑡−𝜓(𝑡))>0. Then there exists a unique 𝑥₀∈𝑋 which is both a startpoint and an endpoint of 𝐹 if and only if 𝐹 has the approximate mix-point property.

Proof. It is clear that if 𝐹 admits a point which is both a startpoint and an endpoint, then 𝐹 has the approximate startpoint property and the approximate endpoint property. Just observe that 𝐻({𝑥},𝐹𝑥)=sup_{𝑦∈𝐹𝑥}𝑑(𝑥,𝑦) and 𝐻(𝐹𝑥,{𝑥})=sup_{𝑦∈𝐹𝑥}𝑑(𝑦,𝑥). Conversely, suppose 𝐹 has the approximate mix-point property. Then 𝐶_𝑛=𝑥∈𝑋∶sup_{𝑦∈𝐹𝑥}𝑑^𝑠≤1𝑥,𝑦𝑛≠∅, (27) for each 𝑛∈ℕ. Also it is clear that for each 𝑛∈ℕ, 𝐶_𝑛+1⊆𝐶_𝑛. The map 𝑥↦sup_{𝑦∈𝐹𝑥}𝑑^𝑠(𝑥,𝑦) is 𝜏(𝑑^𝑠)-lower semicontinuous (as supremum of 𝜏(𝑑^𝑠)-continuous mappings); we have that 𝐶_𝑛 is 𝜏(𝑑^𝑠)-closed.
Next we prove that, for each 𝑛∈ℕ,𝐶_𝑛 is bounded.
Assume by the way of contradiction that 𝛿(𝐶_𝑛)=∞ for each 𝑛∈ℕ. Then there exist 𝑥_𝑛,𝑦_𝑛∈𝐶_𝑛 such that 𝑑(𝑥_𝑛,𝑦_𝑛)≥𝑛. From (26), we obtain that 𝑑{𝑥},𝑦𝑥})𝑦=2𝑥,𝑦=𝐻≤𝐻({,𝐹𝑥+𝐻𝐹𝑥,𝐹𝑦+𝐻𝐹𝑦,𝑛+𝜓𝑑𝑥,𝑦, (28) whenever 𝑥,𝑦∈𝐶_𝑛.
Therefore, 𝑑𝑑≤2𝑥,𝑦−𝜓𝑥,𝑦𝑛, (29) whenever 𝑥,𝑦∈𝐶_𝑛. Hence, lim_𝑛→∞𝑑𝑥_𝑛,𝑦_𝑛−𝜓𝑑𝑥_𝑛,𝑦_𝑛=0,lim_𝑛→∞𝑑𝑥_𝑛,𝑦_𝑛=∞. (30) This contradicts our assumption. Now we show that lim_𝑛→∞𝛿(𝐶_𝑛)=0. On the contrary, assume lim_𝑛→∞𝛿(𝐶_𝑛)=𝑟₀>0 (note that the sequence (𝛿(𝐶_𝑛)) is nonincreasing and bounded below and then has a limit). Let 𝛽=inf_𝑛∈ℕinfliminf_𝑘→∞𝑟_𝑛,𝑘−𝜓𝑟_𝑛,𝑘∶𝑥_𝑛,𝑘,𝑦_𝑛,𝑘∈𝐶_𝑛,𝑟_𝑛,𝑘=𝑑𝑥_𝑛,𝑘,𝑦_𝑛,𝑘⟶𝛿𝐶_𝑛as𝑘⟶∞. (31) Now we show that 𝛽>0 (notice 𝛽≥0). Arguing by contradiction, we assume 𝛽=0; then by the definition of 𝛽, there exists a sequence 𝑟_𝑛 such that 𝑟_𝑛→𝑟₀ and lim_𝑛→∞(𝑟_𝑛−𝜓(𝑟_𝑛))=0. Then lim_𝑛→∞𝜓(𝑟_𝑛)=𝑟₀. But since 𝜓 is upper semicontinuous and 𝑟₀>0, then 𝑟₀=lim_𝑛→∞𝜓𝑟_𝑛≤𝜓𝑟₀<𝑟₀. (32) This contradiction shows that 𝛽>0. For each 𝑛∈𝑁, let (𝑥_𝑘,𝑦_𝑘)∈𝐶_𝑛 be a sequence such that 𝑑(𝑥_𝑘,𝑦_𝑘)→𝛿(𝐶_𝑛), as 𝑘→∞. Then from (29) we get 𝛽≤liminf_𝑘→∞𝑑𝑥_𝑘,𝑦_𝑘−𝜓𝑑𝑥_𝑘,𝑦_𝑘≤2𝑛,foreach𝑛∈𝑁. (33) Hence, 𝛽=0. This contradiction shows that lim_𝑛→∞𝛿(𝐶_𝑛)=0. It follows from the Cantor intersection theorem that ⋂_𝑛∈𝑁𝐶_𝑛={𝑥₀}.
Thus, 𝐻({𝑥₀},𝐹𝑥₀)=sup_{𝑦∈𝐹𝑥0}𝑑(𝑥,𝑦)=0=sup_{𝑦∈𝐹𝑥0}𝑑(𝑦,𝑥)=𝐻(𝐹𝑥₀,{𝑥₀}). For uniqueness, if 𝑧₀ is an arbitrary startpoint of 𝐹, then 𝐻({𝑧₀},𝐹𝑧₀)=0=𝐻(𝐹𝑧₀,{𝑧₀}), and so 𝑧₀∈⋂_𝑛∈𝑁𝐶_𝑛={𝑥₀}.
This completes the proof.

Corollary 30. Let (𝑋,𝑑) be a bicomplete quasi-pseudometric space. Let 𝐹∶𝑋→𝐶𝐵(𝑋) be a set-valued map that satisfies 𝐻𝐹𝑥,𝐹𝑦≤𝜓𝑑𝑥,𝑦,𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑥,𝑦∈𝑋, (34) where 𝜓∶[0,∞)→[0,∞) is upper semicontinuous, 𝜓(𝑡)<𝑡 for each 𝑡>0, and liminf_𝑡→∞(𝑡−𝜓(𝑡))>0. If 𝐹 has the approximate mix-point property then 𝐹 has a fixed point.

Proof. From Theorem 29, we conclude that there exists 𝑥₀ which is both a startpoint and an endpoint; that is, 𝐻({𝑥₀},𝐹𝑥₀)=0=𝐻(𝐹𝑥₀,{𝑥₀}). The 𝑇₀-condition therefore guarantees the desired result.

Theorem 31. Let (𝑋,𝑑) be a bicomplete quasi-pseudometric space. Let 𝐹∶𝑋→𝐶𝐵(𝑋) be a set-valued map that satisfies 𝐻𝐹𝑥,𝐹𝑦≤𝑘𝑑𝑥,𝑦,𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑥,𝑦∈𝑋, (35) where 0≤𝑘<1. Then there exists a unique 𝑥₀∈𝑋 which is both a startpoint and an endpoint of 𝐹 if and only if 𝐹 has the approximate mix-point property.

Proof. Take 𝜓(𝑡)=𝑘𝑡 in Theorem 29.

We then deduce the following result for single-valued maps.

Theorem 32. Let (𝑋,𝑑) be a bicomplete quasi-pseudometric space. Let 𝑓∶𝑋→𝑋 be a map that satisfies 𝑑𝑓𝑥,𝑓𝑦≤𝜓𝑑𝑥,𝑦,𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑥,𝑦∈𝑋, (36) where 𝜓∶[0,∞)→[0,∞) is upper semicontinuous, 𝜓(𝑡)<𝑡 for each 𝑡>0, and liminf_𝑡→∞(𝑡−𝜓(𝑡))>0. Then 𝑓 has the approximate startpoint property.

Proof. By the way of contradiction, suppose that inf_𝑥∈𝑋𝑑(𝑥,𝑓𝑥)>0. Then inf_𝑥∈𝑋𝑑𝑥,𝑓𝑥≤inf_{(𝑋)𝑦∈𝑓}𝑑𝑦,𝑓𝑦=inf_𝑥∈𝑋𝑑𝑓𝑥,𝑓²𝑥≤inf_𝑥∈𝑋𝜓𝑑𝑥,𝑓𝑥. (37) Since 𝜓(𝑑(𝑥,𝑦))≤𝑑(𝑥,𝑦), then inf_𝑥∈𝑋𝜓(𝑑(𝑥,𝑓𝑥))≤inf_𝑥∈𝑋𝑑(𝑥,𝑓𝑥).
Now, on the contrary, suppose again that inf_𝑥∈𝑋𝜓𝑑𝑥,𝑓𝑥=inf_𝑥∈𝑋𝑑𝑥,𝑓𝑥. (38) Let (𝑥_𝑛)⊂𝑋 be a sequence such that lim_𝑛→∞𝑑(𝑥_𝑛,𝑓(𝑥_𝑛))=inf_𝑥∈𝑋𝑑(𝑥,𝑓𝑥). By passing to subsequences if necessary, we may assume that lim_𝑛→∞𝜓(𝑑(𝑥_𝑛,𝑓(𝑥_𝑛))) exists. Then from (37) we have inf_𝑥∈𝑋𝑑𝑥,𝑓𝑥≤inf_𝑥∈𝑋𝜓𝑑𝑥,𝑓𝑥≤lim_𝑛→∞𝜓𝑑𝑥_𝑛,𝑓𝑥_𝑛≤𝜓inf_𝑥∈𝑋𝑑𝑥,𝑓𝑥<inf_𝑥∈𝑋𝑑𝑥,𝑓𝑥. (39)
We get a contradiction, so inf_𝑥∈𝑋𝜓(𝑑(𝑥,𝑓𝑥))<inf_𝑥∈𝑋𝑑(𝑥,𝑓𝑥) which again contradicts (37).
This completes the proof.

Corollary 33. Let (𝑋,𝑑) be a bicomplete quasi-pseudometric space. Let 𝑓∶𝑋→𝑋 be a map that satisfies 𝑑𝑓𝑥,𝑓𝑦≤𝜓𝑑𝑥,𝑦,𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑥,𝑦∈𝑋, (40) where 𝜓∶[0,∞)→[0,∞) is upper semicontinuous, 𝜓(𝑡)<𝑡 for each 𝑡>0, and liminf_𝑡→∞(𝑡−𝜓(𝑡))>0. Then 𝑓 has the approximate endpoint property.

We finish this section by the following fixed point result.

Corollary 34. Let (𝑋,𝑑) be a bicomplete quasi-pseudometric space. Let 𝑓∶𝑋→𝑋 be a map that satisfies 𝑑𝑓𝑥,𝑓𝑦≤𝜓𝑑𝑥,𝑦,𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑥,𝑦∈𝑋, (41) where 𝜓∶[0,∞)→[0,∞) is upper semicontinuous, 𝜓(𝑡)<𝑡 for each 𝑡>0, and liminf_𝑡→∞(𝑡−𝜓(𝑡))>0. Then 𝑓 has a fixed point.

Proof. From Theorem 32 and Corollary 33, we conclude that 𝑓 has the approximate mix-point property. Hence, by Corollary 30, we have the desired result.

5. More Results

The following theorem is the main result of this section.

Theorem 35. Let (𝑋,𝑑) be a left 𝐾-complete quasi-pseudometric space. Let 𝐹∶𝑋→𝐶𝐵(𝑋) be a set-valued map and 𝑓∶𝑋→ℝ as 𝑓(𝑥)=𝐻({𝑥},𝐹𝑥). If there exists 𝑐∈(0,1) such that for all 𝑥∈𝑋 there exists 𝑦∈𝐹𝑥 satisfying 𝐻𝑦,𝐹𝑦≤𝑐𝑑𝑥,𝑦, (42) then 𝑇 has a startpoint.

Proof. For any initial 𝑥₀∈𝑋, there exists 𝑥₁∈𝐹𝑥₀⊆𝑋 such that 𝐻𝑥₁,𝐹𝑥₁≤𝑐𝑑𝑥₀,𝑥₁, (43) and for 𝑥₁∈𝑋, there is 𝑥₂∈𝐹𝑥₁⊆𝑋 such that 𝐻𝑥₂,𝐹𝑥₂≤𝑐𝑑𝑥₁,𝑥₂. (44) Continuing this process, we can get an iterative sequence (𝑥_𝑛) where 𝑥_𝑛+1∈𝐹𝑥_𝑛⊆𝑋 and 𝐻𝑥_𝑛+1,𝐹𝑥_𝑛+1≤𝑐𝑑𝑥_𝑛,𝑥_𝑛+1,𝑛=0,1,2,…. (45)
Claim 1. (𝑥_𝑛) is a left 𝐾-Cauchy sequence.
On one hand, 𝐻𝑥_𝑛+1,𝐹𝑥_𝑛+1≤𝑐𝑑𝑥_𝑛,𝑥_𝑛+1,𝑛=0,1,2,…. (46) On the other hand, 𝑥_𝑛+1∈𝐹𝑥_𝑛 implies 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝐻𝑥_𝑛,𝐹𝑥_𝑛,𝑛=0,1,2,…. (47) By the two above inequalities, we have 𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝑐𝑑𝑥_𝑛,𝑥_𝑛+1𝐻𝑥𝑛=0,1,2,…,_𝑛+1,𝐹𝑥_𝑛+1≤𝑐𝐻𝑥_𝑛,𝐹𝑥_𝑛𝑛=0,1,2,…. (48) We then deduce by iteration that 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝑐^𝑛𝑑𝑥₀,𝑥₁𝐻𝑥𝑛=0,1,2,…,_𝑛+1,𝐹𝑥_𝑛+1≤𝑐^𝑛𝐻𝑥₀,𝐹𝑥₀𝑛=0,1,2,…. (49) Then for 𝑛,𝑚∈ℕ, 𝑛<𝑚, 𝑑𝑥_𝑛,𝑥_𝑚≤𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑑𝑥_𝑚−1,𝑥_𝑚≤𝑐^𝑛+𝑐^𝑛+1+⋯+𝑐^𝑚−1𝑑𝑥₀,𝑥₁≤𝑐^𝑛𝑑𝑥1−𝑐⁰,𝑥₁, (50) and since 𝑐^𝑛→0 as 𝑛→∞ we conclude that (𝑥_𝑛) is a left 𝐾-Cauchy sequence. According to the left 𝐾-completeness of (𝑋,𝑑), there exists 𝑥^∗∈𝑋 such that 𝑥→−_𝑛^𝑑𝑥^∗.
Claim 2. 𝑥^∗ is a startpoint of 𝐹.
Observe that the sequence (𝑓𝑥_𝑛)=(𝐻({𝑥_𝑛},𝐹𝑥_𝑛)) is decreasing and hence converges to 0. Since 𝑓 is 𝜏(𝑑)-lower semicontinuous (as supremum of 𝜏(𝑑)-lower semicontinuous functions), we have 0≤𝑓𝑥^∗≤liminf_𝑛→∞𝑓𝑥_𝑛=0. (51) Hence, 𝑓(𝑥^∗)=0; that is, 𝐻({𝑥^∗},𝐹𝑥^∗)=0.
This completes the proof.

Example 36. Let 𝑋={1,1/2,1/3}. The map 𝑑∶𝑋×𝑋→[0,∞) defined by 𝑑(1/𝑛,1/𝑚)=max{1/𝑛−1/𝑚,0} is a 𝑇₀-quasi-pseudometric on 𝑋. Let 𝐹∶𝑋→2^𝑋 be the set mapping defined by 𝐹𝑎=𝑋⧵{𝑎} for any 𝑎∈𝑋. With 𝑐=1/2, the map 𝐹 satisfies the assumptions of our theorem, so it has a startpoint, which in the present case is 1/3.
More generally, if we set 𝑋_𝑛={1/𝑖,𝑖=1,2,…,𝑛} and 𝑑 as defined above, with 𝑐=1/2, the map 𝐹 defined by 𝐹𝑎=𝑋⧵{𝑎} for any 𝑎∈𝑋 satisfies the assumptions of our theorem, so it has a startpoint, which in this case is 1/𝑛.

Corollary 37. Let (𝑋,𝑑) be a right 𝐾-complete quasi-pseudometric space. Let 𝐹∶𝑋→𝐶𝐵(𝑋) be a set-valued map and 𝑓∶𝑋→ℝ defined by 𝑓(𝑥)=𝐻(𝐹𝑥,{𝑥}). If there exists 𝑐∈(0,1) such that for all 𝑥∈𝑋 there exists 𝑦∈𝐹𝑥 satisfying 𝐻𝐹𝑦,𝑦≤𝑐𝑑𝑦,𝑥, (52) then 𝑇 has an endpoint.

Corollary 38. Let (𝑋,𝑑) be a bicomplete quasi-pseudometric space. Let 𝐹∶𝑋→𝐶𝐵(𝑋) be a set-valued map and 𝑓∶𝑋→ℝ defined by 𝑓(𝑥)=𝐻^𝑠(𝐹𝑥,{𝑥})=max{𝐻(𝐹𝑥,{𝑥}),𝐻({𝑥},𝐹𝑥)}. If there exists 𝑐∈(0,1) such that for all 𝑥∈𝑋 there exists 𝑦∈𝐹𝑥 satisfying 𝐻^𝑠𝑦,𝐹𝑦≤𝑐min𝑑𝑦,𝑥,𝑑𝑦,𝑥, (53) then 𝑇 has a fixed point.

Proof. We give here the main idea of the proof.
Observe that inequality (53) guarantees that the sequence (𝑥_𝑛) constructed in the proof of Theorem 35 is a 𝑑^𝑠-Cauchy sequence and hence 𝑑^𝑠-converges to some 𝑥^∗. Using the fact that 𝑓 is 𝜏(𝑑^𝑠)-lower semicontinuous (as supremum of 𝜏(𝑑^𝑠)-continuous functions), we have 0≤𝑓𝑥^∗≤liminf_𝑛→∞𝑓𝑥_𝑛=0. (54) Hence, 𝑓(𝑥^∗)=0; that is, 𝐻({𝑥^∗},𝐹𝑥^∗)=0=𝐻(𝐹𝑥^∗,{𝑥^∗}), and we are done.

Remark 39. All the results given remain true when we replace accordingly the bicomplete quasi-pseudometric space (𝑋,𝑑) with a left Smyth sequentially complete/left 𝐾-complete or a right Smyth sequentially complete/right 𝐾-complete space.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.