new common fixed point results in hyperconvex … · 2018. 3. 24. · 1 new common fixed point...

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NEW COMMON FIXED POINT RESULTS IN HYPERCONVEX

ULTRAMETRIC SPACES

Qazi Aftab Kabir *1

, Rizwana Jamal2, Jyoti Mishra

3, Masroor Mohammad

4,

Ramakant

Bhardwaj5

*1,4Research Scholars, Department of Mathematics Saifia Science College Bhopal, India

2Assistant Professor Department of Mathematics Saifia Science College Bhopal, India

3Department of Mathematics, Gyan Ganga Institute of Technology and Sciences, Jabalpur,

India.

5Department of Mathematics, Technocrats Institute of Technology Bhopal, India

Email*: [email protected]

Abstract: In this paper strong contracting mapping is applied and establish some new fixed

point results are established in complete hyperconvex ultrametric space. We prove common

fixed point theorems for a single-valued and multi-valued maps, which in turn extend, unify

and improve several results in the related literature.

Keywords: Ultrametric spaces, hyperconvex spaces, contractive mappings, fixed points.

𝟏. INTRODUCTION

In 1956, Aronszajn and Panitchpakdi [2] introduced the term hyperconvex spaces. These

spaces are also called injective metric spaces. The results of sine [12] and soardi [13]proved

that nonexpansive mappings holds the fixed point property in hyperconvex spaces. Jawhari et

al. [5] proved that the fixed point theorems of Sine and Soardi are equivalent to the classical

Tarski’s theorem through the concept of generalised metric spaces. Using a recent fixed point

result of Agarwal et al. [1] we are able to establish a new and very general fixed point

theorems for multivalued maps in hyperconvex spaces.The concept of hyperconvex

ultrametric spaces introduced by Bouamama and Misane [3] who proved the strictly

contracting mappings for hyperconvex ultrametric space and successfully used it in fixed

point theorem. In 1977, Rhodes [11] listed contractive type mappings which were

generalizations of Banach contraction principle.Generalized ultrametric spaces are a common

generalization of partially ordered sets and ordinary ultrametric spaces. In this paper, we

establish a unique common fixed point theorem for a single-valued and the multi-valued

International Journal of Pure and Applied MathematicsVolume 119 No. 10 2018, 1261-1271ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

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maps involving some strong contractive type mappings inhyperconvex complete ultrametric

spaces which at the generalisations of the results of Bouamama and Misane [3] andRhodes

[11]

𝟐.PRELIMINARIES

Definition 2.1.Let 𝑍 be a set and let (Γ,≤) be a complete lattice with a least element 0 and a

greatest element 1. A mapping 𝑑:𝑍 × 𝑍 → Γ is said to be an hyperconvex ultrametric

distance on Z if for all 𝑥,𝑦, 𝑧 ∈ 𝑍,

1 𝑑 𝑥, 𝑦 = 0 ⇔ 𝑦 = 𝑥

(2)𝑑 𝑥,𝑦 = 𝑑 𝑦, 𝑥

(3)𝑑 𝑥,𝑦 ≤ max 𝑑 𝑥, 𝑧 ,𝑑 𝑧,𝑦 .

Definition 2.2. A quasi-pseudo-metric space 𝑋,𝑑 is called hyperconvex that for each

family (𝑥𝑖)𝑖∈𝐼 of points in 𝑋 and families (𝑟𝑖)𝑖∈𝐼 and (𝑠𝑖)𝑖∈𝐼 of nonnegative real numbers

satisfying 𝑑 𝑥𝑗 , 𝑥𝑗 ≤ 𝑟𝑖 + 𝑠𝑗 whenever 𝑖, 𝑗 ∈ 𝐼, the following condition holds:

(

𝑖∈𝐼

𝐶𝑑(𝑥𝑖 , 𝑟𝑖) ∩ 𝐶𝑑𝑡(𝑥𝑖 , 𝑠𝑖)) ≠ ∅

Definition 2.3. An ultrametric space (𝑍,𝑑) is said to be spherically complete if every

shrinking collection of balls in Z has a nonempty intersection.

Definition 2.4. A self-mapping𝑇 of an ultrametric space 𝑍 is said to be contractive (or,

strictly contractive ) mapping if:

𝑑 𝐺𝑥,𝐺𝑦 < 𝑑 𝑥,𝑦 ∀ 𝑥,𝑦 ∈ 𝑍 with 𝑥 ≠ 𝑦

Definition 2.5. An element 𝑧 ∈ 𝑍 is called a coincidence point of 𝐺:𝑍 → 𝑍 and 𝐻:𝑍 →

2𝑍(where 2𝑍 is the space of all nonempty compact subsets in 𝑍)

if 𝐺𝑧 ∈ 𝐻𝑧.

Definition 2.6. let 𝐺:𝑍 → 𝑍 and 𝐻:𝑍 → 2𝑍 . The mappings 𝐺 and 𝐻 are called coincidently

communting at 𝑧 ∈ 𝑍 if 𝐺𝐻𝑧 ⊆ 𝐻𝐺𝑧 whenever 𝐺𝑧 ∈ 𝐻𝑧.

Definition 2.7. Let 𝑍,𝑑, Γ be an hyperconvex ultrametric space if it satisfies the following

two properties:

International Journal of Pure and Applied Mathematics Special Issue

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1 For any family {𝐵(𝑧𝑖 ;𝑢𝑖)}𝑖∈𝐼 of balls

𝐵 𝑧𝑖 ;𝑢𝑖 ∩ 𝐵 𝑧𝑗 ;𝑢𝑗 ≠ ∅∀ 𝑖, 𝑗 ∈ 𝐼 ⇒ 𝐵

𝑖∈𝐼

𝑧𝑖 ;𝑢𝑗 ≠ ∅.

(2) For all 𝑧,𝑢 ∈ 𝑍 and 𝑢1 ,𝑢2 ∈ Γ

𝑑 𝑧,𝑢 ≤ sup 𝑢1,𝑢2 ⇒ ∃ 𝑤 ∈ 𝑍

Such that 𝑑(𝑧,𝑤) ≤ 𝑢1 and 𝑑(𝑤,𝑢) ≤ 𝑢2

We first observe that in the classical setting the second condition is redundant, indeed. In any

classical hyperconvex ultrametric space,

𝑑(𝑧,𝑢) ≤ max{𝑢1, 𝑢2} ⇔ 𝐵(𝑧;𝑢1)⋂𝐵(𝑢;𝑢2) ≠ ∅

We will generalize and proceede in the similar way for the intersection of two balls and so

one for an arbitrary family of balls.

Let (𝑍, 𝑑, Γ) be an ultrametric space and 𝑇:𝑍 → 𝑍 a mapping. 𝑇 is called set-valued or

contracting if 𝑑 𝑇 𝑥 ,𝑇 𝑦 ≤ 𝑑(𝑥,𝑦) for all 𝑥,𝑦 ∈ 𝑍. If for all

𝑥,𝑦 ∈ 𝑍, 𝑥 ≠ 𝑦,𝑑 𝑇 𝑥 ,𝑇 𝑦 < 𝑑 𝑥,𝑦 then 𝑇 is strictly contracting.

Definition 2.8. For 𝐴,𝐵 ∈ 𝐵 𝑍 , (𝐵(𝑍) is the space of all nonempty bounded subsets of 𝑍),

the Hausdorff metric is defined as:

𝐻 𝐴,𝐵 = 𝑚𝑎𝑥 sup𝑥∈𝐵 𝑑 𝑥,𝐴 , sup𝑦∈𝐴

𝑑(𝑥,𝐵) ;

Where 𝑑 𝑥,𝐴 = inf 𝑑 𝑥,𝑎 :𝑎 ∈ 𝐴 .

3. MAIN RESULTS

In this section, we apply strong contractive type mappings on the results of Bouamama and

Misane [3] and Rhodes [11].and established some new fixed point results in Hyperconvex

ultrametric spaces for multi-valued maps.

Theorem 3.1. Let 𝑍,𝑑, Γ be a complete hyperconvex ultrametric space and 𝐺,𝐻: 𝑍 →Z be

two self-maps on 𝑍 which satisfies the following conditions:

1 𝐺𝑍 ⊂ 𝐻𝑍;


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2 𝑑 𝐺𝑥,𝐺𝑦 < max 𝑑 𝐻𝑥,𝐻𝑦 ,𝑑 𝐻𝑥,𝐺𝑥 ,𝑑 𝐻𝑦,𝐺𝑦 ,𝑑 𝐻𝑥,𝐺𝑦 ,𝑑(𝐻𝑦,𝐺𝑥) ∀ 𝑥,𝑦

∈ 𝑍;

3 𝐻𝑍 is spherically complete.

Then 𝐺 and 𝐻 have a common point 𝑐∗ ∈ 𝑍 and if 𝐺 and 𝐻 are coincidently commuting at 𝑐∗

i.e. 𝐺𝐻𝑐∗ = 𝐻𝐺𝑐∗ then 𝐺𝑐∗ = 𝐻𝑐∗ = 𝑐∗.

Proof. Let 𝐶𝑟 = 𝐻𝑟, 𝑑 𝐻𝑟,𝐺𝑟 ∩ 𝐻𝑍 is a closed sphere in 𝐻𝑍 whose centre is Hr for all 𝑟

in Z with radii 𝑑 𝐻𝑟,𝐺𝑟 . Let ℱ is the collection of all such spheres on which the partial

order is defined like 𝐶𝑟𝐶𝑠 𝑖𝑓 𝐶𝑠 ⊆ 𝐶𝑟 . Let ℱ1 is totally ordered subfamily of ℱ. As 𝐻𝑍 is

spherically complete

𝐶𝑟𝐶𝑟∈ℱ1

= 𝐶 ≠ ∅.

For 𝐻𝑠 ∈ 𝐶 ⟹ 𝐻𝑠 ∈ 𝐶𝑟 as 𝐶𝑟 ∈ ℱ1, hence 𝑑 𝐻𝑟,𝐻𝑠 ≤ 𝑑 𝐻𝑟,𝐺𝑟 . If 𝐻𝑟 = 𝐻𝑠 then

𝐶𝑟 = 𝐶𝑠. Assume that 𝑟 ≠ 𝑠, and let 𝑥 ∈ 𝐶𝑠 .

Then

𝑑(𝑥,𝐻𝑠) ≤ 𝑑(𝐻𝑠,𝐺𝑠) ≤ max 𝑑 𝐻𝑠,𝐻𝑟 ,𝑑 𝐻𝑟,𝐺𝑟 ,𝑑 𝐺𝑟,𝐺𝑠

≤ max 𝑑 𝐻𝑠,𝐻𝑟 ,𝑑 𝐻𝑟,𝐺𝑟 𝑚𝑎𝑥 𝑑 𝐻𝑟,𝐻𝑠 ,𝑑 𝐻𝑠,𝐺𝑠 ,𝑑 𝐻𝑟,𝐺𝑟 ,𝑑 𝐻𝑟,𝐺𝑠 ,𝑑(𝐻𝑠,𝐺𝑟) .

As𝑑 𝐻𝑟,𝐺𝑠 ≤ max 𝑑 𝐻𝑟,𝐻𝑠 ,𝑑 𝐻𝑠,𝐺𝑠 ,

And𝑑 𝐻𝑠,𝐺𝑟 ≤ max 𝑑 𝐻𝑠,𝐻𝑟 ,𝑑 𝐻𝑟,𝐺𝑟 .

Then

𝑑 𝑥,𝐻𝑠 ≤ max{𝑑 𝐻𝑠,𝐻𝑟 ,𝑑 𝐻𝑟,𝐺𝑟 max 𝑑 𝐻𝑟,𝐻𝑠 ,𝑑 𝐻𝑠,𝐺𝑠 ,𝑑 𝐻𝑟,𝐺𝑟 ,

max{𝑑 𝐻𝑟,𝐻𝑠 ,𝑑(𝐻𝑠,𝐺𝑠)}},

max 𝑑 𝐻𝑠,𝐻𝑟 ,𝑑 𝐻𝑟,𝐺𝑟 = 𝑑 𝐻𝑟,𝐺𝑟 .

Now

𝑑 𝑥,𝐻𝑟 ≤ max 𝑑 𝑥,𝐻𝑠 ,𝑑 𝐻𝑠,𝐻𝑟 ≤ 𝑑 𝐻𝑟,𝐺𝑟 .Implies 𝑥 ∈ 𝐶𝑟 so 𝐶𝑠 ⊆ 𝐶𝑟 for all

𝐶𝑟 ∈ ℱ1. Hence 𝐶𝑠 is the upper bound of ℱ for the family ℱ1, hence by thezorn’s lemma ℱ

has a maximal element 𝐶𝑐∗ for some 𝑐∗ ∈ 𝑍. We are going to prove that 𝐺𝑐∗ = 𝐻𝑐∗. Suppose

𝐺𝑐∗ ≠ 𝐻𝑐∗, as


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𝐺𝑍 ⊆ 𝐻𝑍 there is 𝑤 ∈ 𝑍 such that 𝐺𝑐∗ = 𝐻𝑤 and 𝑐∗ ≠ 𝑤.

𝑑 𝐻𝑤,𝐺𝑤 = 𝑑 𝐺𝑐∗,𝐺𝑤

< max 𝑑 𝐻𝑐∗,𝐻𝑤 ,𝑑 𝐻𝑐∗,𝐺𝑐∗ ,𝑑 𝐻𝑤,𝐺𝑤 ,𝑑 𝐻𝑐∗,𝐺𝑤 ,𝑑(𝐻𝑤,𝐺𝑤)

= max 𝑑 𝐻𝑐∗,𝐻𝑤 ,𝑑 𝐻𝑤,𝐺𝑤

≤ max 𝑑 𝐻𝑐∗,𝐻𝑤 ,𝑑 𝐻𝑤,𝐺𝑤 ,𝑑 𝐻𝑤,𝐻𝑤 𝑑 𝐻𝑐∗,𝐻𝑤 ,𝑑(𝐻𝑤,𝐺𝑤)

= 𝑑 𝐻𝑐∗,𝐻𝑤 .

This means that

𝑑 𝐻𝑤,𝐺𝑤 < 𝑑 𝐻𝑐∗,𝐻𝑤 .

Let 𝑦 ∈ 𝐶𝑤 . Then

𝑑 𝑦,𝐻𝑤 ≤ 𝑑 𝐻𝑤,𝐺𝑤 < 𝑑 𝐻𝑐∗,𝐺𝑐∗ ,

Implies

𝑑 𝑦,𝐻𝑤 < 𝑑 𝐻𝑐∗,𝐺𝑐∗ .

As

𝑑 𝑦,𝐻𝑐∗ ≤ max 𝑑 𝑦,𝐻𝑤 ,𝑑 𝐻𝑤,𝐻𝑐∗ = 𝑑 𝐻𝑐∗,𝐺𝑐∗ ,

𝑦 ∈ 𝐶𝑐∗ so 𝐶𝑤 ⊆ 𝐶𝑐∗. As 𝐻𝑐∗ ∉ 𝐶𝑤 ⇒ 𝐶𝑤𝐶𝑐∗ . It is contradiction to the maximality of 𝐶𝑐∗ ,

hence 𝐺𝑐∗ = 𝐻𝑐∗. Suppose 𝐻 𝑎𝑛𝑑 𝐺 are coincidently communting at 𝑐∗ ∈ 𝑍,

then

𝐻2𝑐∗ = 𝐻 𝐻𝑐∗ = 𝐻𝐺𝑐∗ = 𝐺𝐻𝑐∗ = 𝐺2𝑐∗.

Let 𝐻𝑐∗ ≠ 𝑐∗. Now

𝑑 𝐺𝐻𝑐∗,𝐺𝑐∗

< max 𝑑 𝐻2𝑐∗,𝐻𝑐∗ ,𝑑 𝐻2𝑐∗,𝐺𝑐∗ ,𝑑 𝐻𝑐∗,𝐺𝑐∗ ,𝑑 𝐻2𝑐∗,𝐺𝑐∗ ,𝑑 𝐻𝑐∗,𝐺𝐻𝑐∗

≤ max 𝑑 𝐻2𝑐∗,𝐻𝑐∗ ,𝑑 𝐻𝑐∗,𝐺𝐻𝑐∗

= 𝑑 𝐺𝐻𝑐∗,𝐺𝑐∗ ⟹ 𝑑 𝐺𝐻𝑐∗,𝐺𝑐∗ < 𝑑 𝐺𝐻𝑐∗,𝐺𝑐∗ .

Which is contradiction, hence 𝐺𝑐∗ = 𝐻𝑐∗ = 𝑐∗.

For the uniqueness, let 𝑤∗ be another fixed point. 𝐺𝑤∗ = 𝐻𝑤∗ = 𝑤∗.


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𝑑 𝐺𝑤∗,𝐺𝑐∗ < max 𝑑 𝐻𝑤∗,𝐻𝑐∗ ,𝑑 𝐻𝑤∗,𝐺𝑤∗ ,𝑑 𝐻𝑐∗,𝐺𝑐∗ ,𝑑 𝐻𝑤∗,𝐺𝑐∗ ,𝑑 𝐻𝑐∗,𝐺𝑤∗

< 𝑑 𝐺𝑤∗,𝐺𝑐∗ ,

Which is contradiction.

Theorem 3.2. Let 𝑍,𝑑, Γ be an hyperconvex ultrametric space. Let 𝐺:𝑍 → 𝑍 and 𝐻:𝑍 → 2𝑍

be maps satisfying.

𝑇 𝐻𝑥,𝐻𝑦 < max 𝑑 𝐺𝑥,𝐺𝑦 ,𝑑 𝐺𝑥,𝐻𝑥 ,𝑑 𝐺𝑦,𝐻𝑦 ,𝑑 𝐺𝑥,𝐻𝑦 ,𝑑(𝐺𝑦,𝐻𝑥) (2.1)

For all 𝑥,𝑦 ∈ 𝑍, such that 𝑥 ≠ 𝑦 suppose that

i. 𝐺𝑍 is spherically complete

Then there exist 𝑢 ∈ 𝑍 such that 𝐺𝑢 ∈ 𝐻𝑢. Assume in addition that

ii. 𝐺 and 𝐻 are coincidentally at 𝑢;

iii. 𝑑 𝐺𝑥,𝐺𝑦 ≤ 𝑑 𝑦,𝐻𝑥 ∀ 𝑥,𝑦 ∈ 𝑍

Then 𝐺𝑢 is the unique common fixed point of 𝐺 and 𝐻, that is

𝐺 𝐺𝑢 = 𝐺𝑢 ∈ 𝐻 𝐺𝑢

Proof : Assume that 𝑑 𝐺𝑥,𝐻𝑥 = 𝑖𝑛𝑓𝑤∈𝐻𝑥𝑑 𝐺𝑥,𝑊 > 0 ∀ 𝑥 ∈ 𝑍

Let 𝐶𝑟 = 𝐶 𝐺𝑟 ,𝑑 𝐺𝑟 ,𝐻𝑟 ∩ 𝐺𝑍 denote the closed ball centred at 𝐺𝑟 with radius 𝑑 𝐺𝑟,𝐻𝑟 >

0 ∀ 𝑟 ∈ 𝑍 and let ℱ be the collection of these balls. We define ℱ the following partial order

𝐶𝑟 ≼ 𝐶𝑠 ⇔ 𝐶𝑠 ⊆ 𝐶𝑟

Let ℱ1 be a totally ordered subfamily of ℱ. We shall prove that ℱ1 has an upper

bound. By condition 1 ,𝐺𝑍 is spherically complete. It follows that

𝐶𝑟𝐶𝑟∈ℱ

= 𝐶 ≠ ∅

Let 𝐺𝑠 ∈ 𝐶. This implies that 𝐺𝑠 ∈ 𝐶𝑟 , as 𝐶𝑟 ∈ ℱ1 so 𝑑 𝐺𝑠,𝐺𝑟 ≤ 𝑑 𝐺𝑟,𝐻𝑟 . Since 𝐻𝑟 is

nonempty compact set, then there exist 𝑝 ∈ 𝐻𝑟 such that 𝑑 𝐺𝑟,𝑝 = 𝑑(𝐺𝑟,𝐻𝑟) from (2.1)

and by the strong triangle inequality, we get

𝑑(𝐺𝑠,𝐻𝑠) ≤ max 𝑑 𝐺𝑠,𝐺𝑟 ,𝑑 𝐺𝑟, 𝑝 ,𝑑 𝑝,𝐻𝑠

≤ max 𝑑 𝐺𝑟,𝐻𝑟 ,𝑇 𝐻𝑟,𝐻𝑠

< max 𝑑 𝐺𝑟,𝐻𝑟 ,𝑑 𝐺𝑟,𝐺𝑠 ,𝑑 𝐺𝑟,𝐻𝑟 ,𝑑 𝐺𝑠,𝐻𝑠 ,𝑑 𝐺𝑟,𝐻𝑠 ,𝑑(𝐺𝑠,𝐻𝑟)

= max 𝑑 𝐺𝑟,𝐻𝑟 ,𝑑 𝐺𝑠,𝐻𝑠 ,𝑑 𝐺𝑟,𝐻𝑠 ,𝑑 𝐺𝑠,𝐻𝑟


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As 𝑑 𝐺𝑟,𝐻𝑠 ≤ max 𝑑 𝐺𝑟,𝐺𝑠 ,𝑑 𝐺𝑠,𝐻𝑠

and

𝑑 𝐺𝑠,𝐻𝑟 ≤ max 𝑑 𝐺𝑠,𝐺𝑟 ,𝑑 𝐺𝑟,𝐻𝑟 , Then

𝑑 𝐺𝑠,𝐻𝑠 < max 𝑑 𝐺𝑟,𝐻𝑟 ,𝑑 𝐺𝑠,𝐻𝑠 .

Necessarily, we have 𝑑 𝐺𝑠,𝐻𝑠 < 𝑑 𝐺𝑟,𝐻𝑟 .

For 𝑥 ∈ 𝐶𝑠, we have

𝑑 𝐺𝑠, 𝑥 ≤ 𝑑 𝐺𝑠,𝐻𝑠 < 𝑑 𝐺𝑟,𝐻𝑟 .

Then

𝑑 𝐺𝑟, 𝑥 ≤ max 𝑑 𝐺𝑟,𝐺𝑠 ,𝑑 𝐺𝑠, 𝑥 ≼ 𝑑 𝐺𝑟,𝐻𝑟 .

It follows that 𝑥 ∈ 𝐶𝑟 and so 𝐶𝑠 ⊆ 𝐶𝑟 . Thus 𝐶𝑟 ≼ 𝐶𝑠 for all 𝐶𝑟 ∈ ℱ1. Hence 𝐶𝑠 is an upper

bound in ℱ for the family ℱ1. By zorn’s lemma, there exist a maximalelement in ℱ, say 𝐶𝑢 .

We claim that 𝐺𝑢 ∈ 𝐻𝑢. We argue by contradiction, that is, 𝐺𝑢 ∉ 𝐻𝑢. Since 𝐻𝑢 is nonempty

compact set, there exists 𝐺𝑣 ∈ 𝐻𝑢 such that

𝑑 𝐺𝑣,𝐺𝑢 = 𝑑(𝐺𝑢,𝐻𝑢) and 𝐺𝑣 ≠ 𝐺𝑢. We shall prove that 𝐶𝑣 ⊆ 𝐶𝑢 . We have

𝑑(𝐺𝑣,𝐻𝑣) ≤ 𝑇(𝐻𝑢,𝐻𝑣)

< max 𝑑 𝐺𝑢,𝐺𝑣 ,𝑑 𝐺𝑢,𝐻𝑢 ,𝑑 𝐺𝑣,𝐻𝑣 ,𝑑 𝐺𝑤,𝐻𝑣 ,𝑑(𝐺𝑣,𝐻𝑢)

< max 𝑑 𝐺𝑢,𝐻𝑢 ,𝑑 𝐺𝑣,𝐻𝑣 ,𝑑 𝐺𝑢,𝐺𝑣 ,𝑑 𝐺𝑣,𝐻𝑣 ,𝑑 𝐺𝑣,𝐺𝑢 ,𝑑(𝐺𝑢,𝐻𝑢)

= max 𝑑 𝐺𝑢,𝐻𝑢 ,𝑑 𝐺𝑣,𝐻𝑣 .

Then 𝑑 𝐺𝑣,𝐻𝑣 < 𝑑 𝐺𝑢,𝐻𝑢 . Now, for𝑥 ∈ 𝐶𝑣 , we have

𝑑 𝐺𝑣, 𝑥 ≤ 𝑑 𝐺𝑣,𝐻𝑣 < 𝑑 𝐺𝑢,𝐻𝑢

It follows that

𝑑 𝐺𝑢, 𝑥 ≤ max 𝑑 𝐺𝑢,𝐺𝑣 ,𝑑 𝐺𝑣, 𝑥 = 𝑑(𝐺𝑢,𝐻𝑢)

Hence 𝑥 ∈ 𝐶𝑢 and so 𝐶𝑣 ⊆ 𝐶𝑢 . Moreover, 𝐺𝑢 ∈ 𝐶𝑢 but 𝐺𝑢 ∈ 𝐶𝑣 , because 𝑑 𝐺𝑣,𝐺𝑢 =

𝑑 𝐺𝑢,𝐻𝑢 > 𝑑 𝐺𝑣,𝐻𝑣 . Then 𝐶𝑣 ⊆ 𝐶𝑢 , which is a contradiction to the maximality of 𝐶𝑢 .

Hence


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𝐺𝑢 ∈ 𝐻𝑢. Let 𝑤∗ = 𝐺𝑢. We claim that 𝑤∗ is a common fixed point of 𝐺 and 𝐻.

By condition 4 , we have

𝑑 𝑤∗,𝐺𝑤∗ = 𝑑 𝐺𝑢,𝐺 𝐺𝑢 ≤ 𝑑 𝐺𝑢,𝐻𝑢 = 0,

Because 𝐺𝑢 ∈ 𝐻𝑢, which implies that 𝑤∗ = 𝐺𝑤∗. Further, as 𝐺𝑢 ∈ 𝐻𝑢, by condition (2), we

get 𝐺 𝐺𝑢 ∈ 𝐺𝐻𝑢 ⊆ 𝐻𝐺𝑢. Then 𝑤∗ = 𝐺𝑤∗ ∈ 𝐻𝑤∗. Hence 𝑤∗ is a common fixed point of 𝐺

and 𝐻.

Let 𝑤∗∗ another common fixed point of 𝐺 and 𝐻. Suppose that 𝑤∗ ≠ 𝑤∗∗. Using the

condition (3), from (2.1), we have

0 < 𝑑 𝑤∗,𝑤∗∗ = 𝑑(𝐺𝑤∗,𝐺𝑤∗∗)

≤ 𝑑 𝑤∗∗,𝐻𝑤∗

≤ 𝑇 𝐻𝑤∗∗,𝐻𝑤∗

< max 𝑑 𝐺𝑤∗,𝐺𝑤∗∗ ,𝑑 𝐺𝑤∗,𝐻𝑤∗ ,𝑑 𝐺𝑤∗∗,𝐻𝑤∗∗

= max{𝑑 𝑤∗,𝑤∗∗ ,𝑑(𝑤∗,𝐻𝑤∗∗)}

≤ max 𝑑 𝑤∗,𝑤∗∗ ,𝑑 𝑤∗∗,𝐻𝑤∗∗ ,𝑑 𝑤∗∗,𝑤∗ ,𝑑 𝑤∗,𝐻𝑤∗

= 𝑑 𝑤∗,𝑤∗∗ ,

Which is a contradiction. Hence 𝑤∗ = 𝑤∗∗.

If there exists 𝑥 ∈ 𝑍 such that 𝑑 𝐺𝑥,𝐻𝑥 = 𝑑 𝐻𝑥,𝐺𝑥 = 0, then 𝐺𝑥 ∈ 𝐻𝑥. Similarly , we

prove that 𝐺𝑥 is the unique common fixed point of 𝐺 and 𝐻 and this completes the proof.

Corollary 3.3.Let 𝑍,𝑑, Γ be an hyperconvex ultrametric space. Let 𝐺:𝑍 → 𝑍 and 𝐻:𝑍 →

2𝑍 be maps satisfying.

𝑇 𝐻𝑥,𝐻𝑦 < max 𝑑 𝐺𝑥,𝐺𝑦 ,𝑑 𝐺𝑥,𝐻𝑥 ,𝑑 𝐺𝑦,𝐻𝑦 ,𝑑 𝐺𝑥,𝐻𝑦 ,𝑑(𝐺𝑦,𝐻𝑥) (2.1)

For all 𝑥,𝑦 ∈ 𝑍, such that 𝑥 ≠ 𝑦 suppose that

i. 𝐺𝑍 is spherically complete

Then there exist 𝑢 ∈ 𝑍 such that 𝐺𝑢 ∈ 𝐻𝑢. Assume in addition that

ii. 𝐺 and 𝐻 are coincidentally at 𝑢;

iii. 𝑑 𝐺𝑥,𝐺𝑦 ≤ 𝑑 𝑦,𝐻𝑥 ∀ 𝑥,𝑦 ∈ 𝑍


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Then 𝐺𝑢 is the unique common fixed point of 𝐺 and 𝐻, that is

𝐺 𝐺𝑢 = 𝐺𝑢 ∈ 𝐻 𝐺𝑢

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