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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 596971, 9 pages
http://dx.doi.org/10.1155/2011/596971
Research Article

Approximate Best Proximity Pairs in Metric Space

1Faculty of Mathematics, Valiasr Rafsanjan University, Rafsanjan, Iran
2Faculty of Mathematics, Yazd University, Yazd, Iran

Received 8 January 2011; Accepted 12 February 2011

Academic Editor: NorimichiΒ Hirano

Copyright Β© 2011 S. A. M. Mohsenalhosseini et al. 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

Let 𝐴 and 𝐡 be nonempty subsets of a metric space 𝑋 and also π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 and 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄. We are going to consider element π‘₯∈𝐴 such that 𝑑(π‘₯,𝑇π‘₯)≀𝑑(𝐴,𝐡)+πœ– for some πœ–>0. We call pair (𝐴,𝐡) an approximate best proximity pair. In this paper, definitions of approximate best proximity pair for a map and two maps, their diameters, 𝑇-minimizing a sequence are given in a metric space.

1. Introduction

Let 𝑋 be a metric space and 𝐴 and 𝐡 nonempty subsets of 𝑋, and 𝑑(𝐴,𝐡) is distance of 𝐴 and 𝐡. If 𝑑(π‘₯0,𝑦0)=𝑑(𝐴,𝐡), then the pair (π‘₯0,𝑦0) is called a best proximity pair for 𝐴 and 𝐡 and put prox(𝐴,𝐡)∢={(π‘₯,𝑦)βˆˆπ΄Γ—π΅βˆΆπ‘‘(π‘₯,𝑦)=𝑑(𝐴,𝐡)}(1.1) as the set of all best proximity pair (𝐴,𝐡). Best proximity pair evolves as a generalization of the concept of best approximation. That reader can find some important result of it in [1–4].

Now, as in [5] (see also [4, 6–11]), we can find the best proximity points of the sets 𝐴 and 𝐡, by considering a map π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 such that 𝑇(𝐴)βŠ†π΅ and 𝑇(𝐡)βŠ†π΄. Best proximity pair also evolves as a generalization of the concept of fixed point of mappings. Because if π΄βˆ©π΅β‰ βˆ…, every best proximity point is a fixed point of 𝑇.

We say that the point π‘₯∈𝐴βˆͺ𝐡 is an approximate best proximity point of the pair (𝐴,𝐡), if 𝑑(π‘₯,𝑇π‘₯)≀𝑑(𝐴,𝐡)+πœ–, for some πœ–>0.

In the following, we introduce a concept of approximate proximity pair that is stronger than proximity pair.

Definition 1.1. Let 𝐴 and 𝐡 be nonempty subsets of a metric space 𝑋 and π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 a map such that 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄. put π‘ƒπ‘Žπ‘‡(𝐴,𝐡)={π‘₯∈𝐴βˆͺπ΅βˆΆπ‘‘(π‘₯,𝑇π‘₯)≀𝑑(𝐴,𝐡)+πœ–forsomeπœ–>0}.(1.2) We say that the pair (𝐴,𝐡) is an approximate best proximity pair if π‘ƒπ‘Žπ‘‡(𝐴,𝐡)β‰ βˆ….

Example 1.2. Suppose that 𝑋=𝐑2, 𝐴={(π‘₯,𝑦)βˆˆπ‘‹βˆΆ(π‘₯βˆ’π‘¦)2+𝑦2≀1}, and 𝐡={(x,𝑦)βˆˆπ‘‹βˆΆ(π‘₯+𝑦)2+𝑦2≀1} with 𝑇(π‘₯,𝑦)=(βˆ’π‘₯,𝑦) for (π‘₯,𝑦)βˆˆπ‘‹. Then 𝑑((π‘₯,𝑦),𝑇(π‘₯,𝑦))≀𝑑(𝐴,𝐡)+πœ– for some πœ–>0. Hence π‘ƒπ‘Žπ‘‡(𝐴,𝐡)β‰ βˆ….

2. Approximate Best Proximity

In this section, we will consider the existence of approximate best proximity points for the map π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡, such that 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄, and its diameter.

Theorem 2.1. Let 𝐴 and 𝐡 be nonempty subsets of a metric space 𝑋. Suppose that the mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is satisfying 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄, and limπ‘›β†’βˆžπ‘‘ξ€·π‘‡π‘›π‘₯,𝑇𝑛+1π‘₯ξ€Έ=𝑑(𝐴,𝐡)forsomeπ‘₯∈𝐴βˆͺ𝐡.(2.1) Then the pair (𝐴,𝐡) is an approximate best proximity pair.

Proof. Let πœ–>0 be given and π‘₯∈𝐴βˆͺ𝐡 such that limπ‘›β†’βˆžπ‘‘(𝑇𝑛π‘₯,𝑇𝑛+1π‘₯)=𝑑(𝐴,𝐡); then there exists 𝑁0>0 such that βˆ€π‘›β‰₯𝑁0ξ€·π‘‡βˆΆπ‘‘π‘›π‘₯,𝑇𝑛+1π‘₯ξ€Έ<𝑑(𝐴,𝐡)+πœ–.(2.2) If 𝑛=𝑁0, then 𝑑(𝑇𝑁0(π‘₯),𝑇(𝑇𝑁0(π‘₯)))<𝑑(𝐴,𝐡)+πœ–, and 𝑇𝑁0(π‘₯)βˆˆπ‘ƒπ‘Žπ‘‡(𝐴,𝐡) and π‘ƒπ‘Žπ‘‡(𝐴,𝐡)β‰ βˆ….

Theorem 2.2. Let 𝐴 and 𝐡 be nonempty subsets of a metric space 𝑋. Suppose that the mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is satisfying 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄ and 𝑑[𝑑](𝑇π‘₯,𝑇𝑦)≀𝛼𝑑(π‘₯,𝑦)+𝛽(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦)+𝛾𝑑(𝐴,𝐡)(2.3) for all π‘₯,π‘¦βˆˆπ΄βˆͺ𝐡, where 𝛼,𝛽,𝛾β‰₯0 and 𝛼+2𝛽+𝛾<1. Then the pair (𝐴,𝐡) is an approximate best proximity pair.

Proof. If π‘₯∈𝐴βˆͺ𝐡, then 𝑑𝑇π‘₯,𝑇2π‘₯≀𝛼𝑑(π‘₯,𝑇π‘₯)+𝛽𝑑(π‘₯,𝑇π‘₯)+𝑑𝑇π‘₯,𝑇2π‘₯ξ€Έξ€»+𝛾𝑑(𝐴,𝐡).(2.4) Therefore, 𝑑𝑇π‘₯,𝑇2π‘₯≀𝛼+𝛽𝛾1βˆ’π›½π‘‘(π‘₯,𝑇π‘₯)+1βˆ’π›½π‘‘(𝐴,𝐡).(2.5) Now if π‘˜=(𝛼+𝛽)/(1βˆ’π›½), then 𝑑𝑇π‘₯,𝑇2π‘₯ξ€Έβ‰€π‘˜π‘‘(π‘₯,𝑇π‘₯)+(1βˆ’π‘˜)𝑑(𝐴,𝐡)(2.6) also 𝑑𝑇2π‘₯,𝑇3π‘₯ξ€Έβ‰€π‘˜2𝑑(π‘₯,𝑇π‘₯)+1βˆ’π‘˜2𝑑(𝐴,𝐡).(2.7) Therefore, 𝑑𝑇𝑛π‘₯,𝑇𝑛+1π‘₯ξ€Έβ‰€π‘˜π‘›π‘‘(π‘₯,𝑇π‘₯)+(1βˆ’π‘˜π‘›)𝑑(𝐴,𝐡),(2.8) and so 𝑑𝑇𝑛π‘₯,𝑇𝑛+1π‘₯ξ€ΈβŸΆπ‘‘(𝐴,𝐡),asπ‘›βŸΆβˆž.(2.9) Therefore, by Theorem 2.1, π‘ƒπ‘Žπ‘‡(𝐴,𝐡)β‰ βˆ…; then pair (𝐴,𝐡) is an approximate best proximity pair.

Definition 2.3. Let 𝐴 and 𝐡 be nonempty subsets of a metric space 𝑋. Suppose that the mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is satisfying 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄. We say that the sequence {𝑧𝑛}βŠ†π΄βˆͺ𝐡 is T-minimizing if limπ‘›β†’βˆžπ‘‘ξ€·π‘§π‘›,𝑇𝑧𝑛=𝑑(𝐴,𝐡).(2.10)

Theorem 2.4. Let 𝐴 and 𝐡 be nonempty subsets of a metric space 𝑋, suppose that the mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is satisfying 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄. If {𝑇𝑛π‘₯} is a T-minimizing for some π‘₯∈𝐴βˆͺ𝐡, then (𝐴,𝐡) is an approximate best pair proximity.

Proof. Since limπ‘›β†’βˆžπ‘‘ξ€·π‘‡π‘›π‘₯,𝑇𝑛+1π‘₯ξ€Έ=𝑑(𝐴,𝐡)forsomeπ‘₯∈𝐴βˆͺ𝐡,(2.11) therefore, by Theorem 2.1, π‘ƒπ‘Žπ‘‡(𝐴,𝐡)β‰ βˆ…; then pair (𝐴,𝐡) is an approximate best proximity pair.

Theorem 2.5. Let 𝐴 and 𝐡 be nonempty subsets of a normed space 𝑋 such that 𝐴βˆͺ𝐡 is compact. Suppose that the mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is satisfying 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄, 𝑇 is continuous and ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–,(2.12) where (π‘₯,𝑦)βˆˆπ΄Γ—π΅. Then π‘ƒπ‘Žπ‘‡(𝐴,𝐡) is nonempty and compact.

Proof. Since 𝐴βˆͺ𝐡 compact, there exists a 𝑧0∈𝐴βˆͺ𝐡 such that ‖‖𝑧0βˆ’π‘‡π‘§0β€–β€–=infπ‘§βˆˆπ΄βˆͺπ΅β€–π‘§βˆ’π‘‡π‘§β€–β‹…(βˆ—) If ‖𝑧0βˆ’π‘‡π‘§0β€–>𝑑(𝐴,𝐡), then ‖𝑇𝑧0βˆ’π‘‡2𝑧0β€–<‖𝑧0βˆ’π‘‡π‘§0β€– which contradict to the definition of 𝑧0, (𝑇𝑧0∈𝐴βˆͺ𝐡 and by (*) ‖𝑇𝑧0βˆ’π‘‡(𝑇𝑧0)β€–β‰₯‖𝑧0βˆ’π‘‡π‘§0β€–). Therefore, ‖𝑧0βˆ’π‘‡π‘§0β€–=𝑑(𝐴,𝐡)≀𝑑(𝐴,𝐡)+πœ– for some πœ–>0 and 𝑧0βˆˆπ‘ƒπ‘Žπ‘‡(𝐴,𝐡). Therefore, π‘ƒπ‘Žπ‘‡(𝐴,𝐡) is nonempty.
Also, if {𝑧𝑛}βŠ†π‘ƒπœ–π‘‡(𝐴,𝐡), then β€–π‘§π‘›βˆ’π‘‡π‘§π‘›β€–<𝑑(𝐴,𝐡)+πœ–, for some πœ–>0, and by compactness of 𝐴βˆͺ𝐡, there exists a subsequence π‘§π‘›π‘˜ and a 𝑧0∈𝐴βˆͺ𝐡 such that π‘§π‘›π‘˜β†’π‘§0 and so ‖‖𝑧0βˆ’π‘‡π‘§0β€–β€–=limπ‘˜β†’βˆžβ€–β€–π‘§π‘›π‘˜βˆ’π‘‡π‘§π‘›π‘˜β€–β€–<𝑑(𝐴,𝐡)+πœ–(2.13) for some πœ–>0, hence π‘ƒπ‘Žπ‘‡(𝐴,𝐡) is compact.

Example 2.6. If 𝐴=[βˆ’3,βˆ’1],𝐡=[1,3], and π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 such that ⎧βŽͺ⎨βŽͺβŽ©π‘‡(π‘₯)=1βˆ’π‘₯2,π‘₯∈𝐴,βˆ’1βˆ’π‘₯2,π‘₯∈𝐡,(2.14) then π‘ƒπ‘Žπ‘‡(𝐴,𝐡) is compact, and we have π‘ƒπ‘Žπ‘‡(𝐴,𝐡)={π‘₯∈𝐴βˆͺπ΅βˆΆπ‘‘(π‘₯,𝑇π‘₯)<𝑑(𝐴,𝐡)+πœ–forsomeπœ–>0}={π‘₯∈𝐴βˆͺπ΅βˆΆπ‘‘(π‘₯,𝑇π‘₯)<2+πœ–forsomeπœ–>0}={1,βˆ’1}.(2.15) That is compact.

In the following, by diam(π‘ƒπ‘Žπ‘‡(𝐴,𝐡)) for a set π‘ƒπ‘Žπ‘‡(𝐴,𝐡)β‰ βˆ…, we will understand the diameter of the set π‘ƒπ‘Žπ‘‡(𝐴,𝐡).

Definition 2.7. Let π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 be a continuous map such that 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄ and πœ–>0. We define diameter π‘ƒπ‘Žπ‘‡(𝐴,𝐡) by 𝑃diamπ‘Žπ‘‡ξ€Έξ€½(𝐴,𝐡)=sup𝑑(π‘₯,𝑦)∢π‘₯,π‘¦βˆˆπ‘ƒπ‘Žπ‘‡ξ€Ύ.(𝐴,𝐡)(2.16)

Theorem 2.8. Let π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡, such that 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄ and πœ–>0. If there exists an π›Όβˆˆ[0,1] such that for all (π‘₯,𝑦)βˆˆπ΄Γ—π΅π‘‘(𝑇π‘₯,𝑇𝑦)≀𝛼𝑑(π‘₯,𝑦),(2.17) then 𝑃diamπ‘Žπ‘‡ξ€Έβ‰€(𝐴,𝐡)2πœ–+1βˆ’π›Ό2𝑑(𝐴,𝐡).1βˆ’π›Ό(2.18)

Proof. If π‘₯,π‘¦βˆˆπ‘ƒπ‘Žπ‘‡(𝐴,𝐡), then 𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑇π‘₯,𝑇𝑦)+𝑑(𝑇𝑦,𝑦)β‰€πœ–1+𝛼𝑑(π‘₯,𝑦)+2𝑑(𝐴,𝐡)+πœ–2.(2.19) Put πœ–=Max{πœ–1,πœ–2}, therefore, 𝑑(π‘₯,𝑦)≀2πœ–/(1βˆ’π›Ό)+(2𝑑(𝐴,𝐡))/(1βˆ’π›Ό). Hence diam(π‘ƒπ‘Žπ‘‡(𝐴,𝐡))≀2πœ–/(1βˆ’π›Ό)+(2𝑑(𝐴,𝐡))/(1βˆ’π›Ό).

3. Approximate Best Proximity for Two Maps

In this section, we will consider the existence of approximate best proximity points for two maps π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 and π‘†βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡, and its diameter.

Definition 3.1. Let 𝐴 and 𝐡 be nonempty subsets of a metric space (𝑋,𝑑) and let π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺπ΅π‘†βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 two maps such that 𝑇(𝐴)βŠ†π΅, 𝑆(𝐡)βŠ†π΄. A point (π‘₯,𝑦) in 𝐴×𝐡 is said to be an approximate-pair fixed point for (𝑇,𝑆) in 𝑋 if there exists πœ–>0𝑑(𝑇π‘₯,𝑆𝑦)≀𝑑(𝐴,𝐡)+πœ–.(3.1) We say that the pair (𝑇,𝑆) has the approximate-pair fixed property in 𝑋 if π‘ƒπ‘Ž(𝑇,𝑆)(𝐴,𝐡)β‰ βˆ…, where π‘ƒπ‘Ž(𝑇,𝑆)(𝐴,𝐡)={(π‘₯,𝑦)βˆˆπ΄Γ—π΅βˆΆπ‘‘(𝑇π‘₯,𝑆𝑦)≀𝑑(𝐴,𝐡)+πœ–forsomeπœ–>0}.(3.2)

Theorem 3.2. Let 𝐴 and 𝐡 be nonempty subsets of a metric space (𝑋,𝑑) and let π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 and π‘†βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 be two maps such that 𝑇(𝐴)βŠ†π΅, 𝑆(𝐡)βŠ†π΄. If, for every (π‘₯,𝑦)βˆˆπ΄Γ—π΅, 𝑑(𝑇𝑛(π‘₯),𝑆𝑛(𝑦))βŸΆπ‘‘(𝐴,𝐡),(3.3) then (𝑇,𝑆) has the approximate-pair fixed property.

Proof. For πœ–>0, Suppose (π‘₯,𝑦)βˆˆπ΄Γ—π΅. Since 𝑑(𝑇𝑛(π‘₯),𝑆𝑛(𝑦))βŸΆπ‘‘(𝐴,𝐡),βˆƒπ‘›0>0s.t.βˆ€π‘›β‰₯𝑛0βˆΆπ‘‘(𝑇𝑛(π‘₯),𝑆𝑛(𝑦))<𝑑(𝐴,𝐡)+πœ–,(3.4) then 𝑑(𝑇(π‘‡π‘›βˆ’1(π‘₯),𝑆(π‘†π‘›βˆ’1(𝑦))<𝑑(𝐴,𝐡)+πœ– for every 𝑛β‰₯𝑛0. Put π‘₯0=𝑇𝑛0βˆ’1(π‘₯) and 𝑦0=𝑆𝑛0βˆ’1(𝑦)). Hence 𝑑(𝑇(π‘₯0),𝑆(𝑦0))≀𝑑(𝐴,𝐡)+πœ– and π‘ƒπ‘Ž(𝑇,𝑆)(𝐴,𝐡)β‰ βˆ….

Theorem 3.3. Let 𝐴 and 𝐡 be nonempty subsets of a metric space (𝑋,𝑑) and let π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 and π‘†βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 be two maps such that 𝑇(𝐴)βŠ†π΅, 𝑆(𝐡)βŠ†π΄ and, for every (π‘₯,𝑦)βˆˆπ΄Γ—π΅, 𝑑[𝑑](𝑇x,𝑆𝑦)≀𝛼𝑑(π‘₯,𝑦)+𝛽(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑆𝑦)+𝛾𝑑(𝐴,𝐡),(3.5) where 𝛼,𝛽,𝛾β‰₯0 and 𝛼+2𝛽+𝛾<1. Then if π‘₯ is an approximate fixed point for 𝑇, or 𝑦 is an approximate fixed point for 𝑆, then π‘ƒπ‘Ž(𝑇,𝑆)(𝐴,𝐡)β‰ βˆ….

Proof. If (π‘₯,𝑦)βˆˆπ΄Γ—π΅, then 𝑑[𝑑](𝑇π‘₯,𝑆(𝑇π‘₯))≀𝛼𝑑(π‘₯,𝑇π‘₯)+𝛽(π‘₯,𝑇π‘₯)+𝑑(𝑇π‘₯,𝑆(𝑇π‘₯))+𝛾𝑑(𝐴,𝐡).(3.6) Therefore, 𝑑(𝑇π‘₯,𝑆(𝑇π‘₯))≀𝛼+𝛽𝛾1βˆ’π›½π‘‘(π‘₯,𝑇π‘₯)+1βˆ’π›½π‘‘(𝐴,𝐡).(3.7) Now if π‘˜=(𝛼+𝛽)/(1βˆ’π›½), then 𝑑(𝑇π‘₯,𝑆(𝑇π‘₯))β‰€π‘˜π‘‘(π‘₯,𝑇π‘₯)+(1βˆ’π‘˜)𝑑(𝐴,𝐡)(βˆ—) also 𝑑(𝑆𝑦,𝑇(𝑆𝑦))β‰€π‘˜π‘‘(𝑦,𝑆𝑦)+(1βˆ’π‘˜)𝑑(𝐴,𝐡).(βˆ—βˆ—) If π‘₯ is an approximate fixed point for 𝑇, then there exists a πœ–>0 and by (*) 𝑑(𝑇π‘₯,𝑆(𝑇π‘₯))β‰€π‘˜π‘‘(π‘₯,𝑇π‘₯)+(1βˆ’π‘˜)𝑑(𝐴,𝐡)β‰€π‘˜(𝑑(𝐴,𝐡)+πœ–)+(1βˆ’π‘˜)𝑑(𝐴,𝐡)=𝑑(𝐴,𝐡)+π‘˜πœ–<𝑑(𝐴,𝐡)+πœ–.(3.8) And (π‘₯,𝑇π‘₯)βˆˆπ‘ƒπ‘Ž(𝑇,𝑆)(𝐴,𝐡); also if 𝑦 is an approximate fixed point for 𝑆, then there exists a πœ–>0 and by (**) 𝑑(𝑆𝑦,𝑇(𝑆𝑦))β‰€π‘˜π‘‘(𝑦,𝑆𝑦)+(1βˆ’π‘˜)𝑑(𝐴,𝐡)β‰€π‘˜(𝑑(𝐴,𝐡)+πœ–)+(1βˆ’π‘˜)𝑑(𝐴,𝐡)=𝑑(𝐴,𝐡)+π‘˜πœ–<𝑑(𝐴,𝐡)+πœ–.(3.9) And (𝑦,𝑆𝑦)βˆˆπ‘ƒπ‘Ž(𝑇,𝑆)(𝐴,𝐡). Therefore, π‘ƒπ‘Ž(𝑇,𝑆)(𝐴,𝐡)β‰ βˆ….

Theorem 3.4. Let 𝐴 and 𝐡 be nonempty subsets of a metric space (𝑋,𝑑) and let π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 and π‘†βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 be two continuous maps such that 𝑇(𝐴)βŠ†π΅, 𝑆(𝐡)βŠ†π΄. If, for every (π‘₯,𝑦)βˆˆπ΄Γ—π΅, 𝑑(𝑇π‘₯,𝑆𝑦)≀𝛼𝑑(π‘₯,𝑦)+𝛾𝑑(𝐴,𝐡),(3.10) where 𝛼,𝛾β‰₯0 and 𝛼+𝛾=1, also let {π‘₯𝑛} and {𝑦𝑛} be as follows: π‘₯𝑛+1=𝑆𝑦𝑛,𝑦𝑛+1=𝑇π‘₯𝑛π‘₯forsome1,𝑦1ξ€Έβˆˆπ΄Γ—π΅,π‘›βˆˆπ‘.(3.11) If {π‘₯𝑛} has a convergent subsequence in 𝐴, then there exists a π‘₯0∈𝐴 such that 𝑑(π‘₯0,𝑇π‘₯0)=𝑑(𝐴,𝐡).

Proof. We have 𝑑π‘₯𝑛+1,𝑦𝑛+1ξ€Έξ€·=𝑑𝑇π‘₯𝑛,𝑆𝑦𝑛π‘₯≀𝛼𝑑𝑛,𝑦𝑛+𝛾(𝑑(𝐴,𝐡)≀⋯≀𝛼𝑛+1𝑑π‘₯0,𝑦0ξ€Έ+(1+𝛼+β‹―+𝛼𝑛)𝛾𝑑(𝐴,𝐡).(3.12) If {π‘₯π‘›π‘˜}π‘˜β‰₯1 converges to π‘₯1∈𝐴, that is, π‘₯π‘›π‘˜β†’π‘₯1, then 𝑑π‘₯𝑛𝐾+1,π‘¦π‘›π‘˜+1ξ‚β‰€π›Όπ‘›π‘˜+1𝑑π‘₯0,𝑦0ξ€Έ+ξ€·1+𝛼+β‹―+π›Όπ‘›π‘˜ξ€Έπ›Ύπ‘‘(𝐴,𝐡).(3.13) Since 𝑇 is continuous, then 𝑑π‘₯π‘›π‘˜+1,𝑇π‘₯π‘›π‘˜ξ‚βŸΆπ›Ύ1βˆ’π›Όπ‘‘(𝐴,𝐡)=𝑑(𝐴,𝐡).(3.14) Therefore, 𝑑(π‘₯1,𝑇π‘₯1)=𝑑(𝐴,𝐡).

Definition 3.5. Let π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 and π‘†βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 be continues maps such that 𝑇(𝐴)βŠ†π΅ and 𝑆(𝐡)βŠ†π΄. We define diameter π‘ƒπ‘Ž(𝑇,𝑆)(𝐴,𝐡) by 𝑃diamπ‘Ž(𝑇,𝑆)(𝐴,𝐡)=sup{𝑑(π‘₯,𝑦)βˆΆπ‘‘(𝑇π‘₯,𝑇𝑦)β‰€πœ–+𝑑(𝐴,𝐡)forsomeπœ–>0}.(3.15)

Example 3.6. Suppose 𝐴={(π‘₯,0)∢0≀π‘₯≀1}, 𝐡={(π‘₯,1)∢0≀π‘₯≀1}, 𝑇(π‘₯,0)=𝑇(π‘₯,1)=(1/2,1), and 𝑆(π‘₯,1)=𝑆(π‘₯,0)=(1/2,0). Then 𝑑(𝑇(π‘₯,0),𝑆(𝑦,1))=1 and diam(π‘ƒπ‘Ž(𝑇,𝑆)√(𝐴,𝐡))=diam(𝐴×𝐡)=2.

Theorem 3.7. Let π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 and π‘†βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 be continuous maps such that 𝑇(𝐴)βŠ†π΅, 𝑆(𝐡)βŠ†π΄. If there exists π‘Žβ€‰β€‰π‘˜βˆˆ[0,1], 𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑆𝑦,𝑦)β‰€π‘˜π‘‘(π‘₯,𝑦),(3.16) then 𝑃diamπ‘Ž(𝑇,𝑆)ξ‚β‰€πœ–(𝐴,𝐡)+1βˆ’π‘˜π‘‘(𝐴,𝐡)1βˆ’π‘˜forsomeπœ–>0.(3.17)

Proof. If (π‘₯,𝑦)βˆˆπ‘ƒπ‘Ž(𝑇,𝑆)(𝐴,𝐡), then 𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑇π‘₯,𝑆𝑦)+𝑑(𝑆𝑦,𝑦)β‰€πœ–+π‘˜π‘‘(π‘₯,𝑦)+𝑑(𝐴,𝐡).(3.18) Therefore, 𝑑(π‘₯,𝑦)β‰€πœ–/(1βˆ’π‘˜)+(𝑑(𝐴,𝐡))/(1βˆ’π‘˜). Then diam(π‘ƒπ‘Ž(𝑇,𝑆)(𝐴,𝐡))β‰€πœ–/(1βˆ’π‘˜)+(𝑑(𝐴,𝐡))/(1βˆ’π‘˜).

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