Common Fixed Point Theorems of Multivalued Maps in Fuzzy Ultrametric Spaces

Abstract

In the setting of fuzzy ultrametric spaces, we study common fixed point theorems of multivalued maps. Our results unify, extend, and generalize some related common fixed point theorems of the literature for both ultrametric spaces (Wang and Song (2013), Gajić (2002) and (2001)) and fuzzy metric spaces (Vijayaraju and Sajath (2011)).

1. Introduction

In 1965, Zadeh [1] introduced the theory of fuzzy sets. Many authors introduced the notion of fuzzy metric space in different ways. George and Veeramani [2] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [3] and defined Hausdorff topology in fuzzy metric space. Several authors [4–10] studied and developed the concept in different directions and proved fixed point theorems in fuzzy metric spaces. Vijayaraju and Sajath [11] extended some previous results and proved some common fixed points theorems for hybrid pair of single and multivalued maps under hybrid contractive conditions. Wang and Song [12] established some results on coincidence and common fixed point for two pairs of multivalued and single-valued maps in ultrametric spaces. In 2009, Savchenko and Zarichnyi [13] introduced the concept of fuzzy ultrametric space. Sedghi and Shobe [14] proved common fixed point theorems for self-maps satisfying contractive conditions on spherically complete fuzzy ultrametric spaces. In this paper, in the setting of fuzzy ultrametric spaces, we study common fixed point theorems of multivalued maps. Our results unify, extend, and generalize some related common fixed point theorems of the literature for both ultrametric spaces [12, 15, 16] and fuzzy metric spaces [11].

2. Preliminaries and Notations

Definition 1 (see [1]). Let𝑋be any nonempty set. A fuzzy set𝐴in𝑋is a function with domain𝑋and values in [0,1].

Definition 2 (see [17]). A binary operation ∗∶[0,1]²→[0,1] is called a continuous triangular norm (shortly t-norm) if it satisfies the following conditions:(1)∗ is associative and commutative,(2)∗ is continuous,(3)𝑎∗1=𝑎 for all 𝑎,𝑏,𝑐,𝑑∈[0,1],(4)𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for all 𝑎,𝑏,𝑐,𝑑∈[0,1].

Definition 3 (see [2]). The 3-tuple (X, M, *) is called a fuzzy metric space if𝑋is an arbitrary (nonempty) set, ∗ is a continuous t-norm, and M is a fuzzy set on 𝑋²×[0,∞) satisfying the following conditions, for all 𝑥,𝑦,𝑧∈𝑋 and each 𝑡 and 𝑠>0:(1)𝑀(𝑥,𝑦,𝑡)=0,(2)𝑀(𝑥,𝑦,𝑡)=1 if and only if 𝑥=𝑦,(3)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡),(4)𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑡)≤𝑀(𝑥,𝑧,𝑡+𝑠),(5)𝑀(𝑥,𝑦,⋅)∶(0,∞)→[0,1] is continuous.
𝑀is called a fuzzy metric on𝑋. The functions 𝑀(𝑥,𝑦,𝑡) denote the degree of nearness between x and y with respect to𝑡, respectively.
Let (𝑋,𝑀, *) be a fuzzy metric space. For 𝑡>0, the open ball 𝐵(𝑥,𝑟,𝑡) with center 𝑥∈𝑋 and radius 0<𝑟<1 is defined by 𝐵(𝑥,𝑟,𝑡)=𝑦∈𝑋∶𝑀𝑥,𝑦,𝑡>1−𝑟. (1)
A subset 𝐴⊂𝑋 is called open if, for each 𝑥∈𝐴, there exist𝑡>0and 0<𝑟<1 such that 𝐵(𝑥,𝑟,𝑡)⊂𝐴. Let 𝜏 denote the family of all open subsets of𝑋. Then 𝜏 is a topology on𝑋 induced by the fuzzy metric𝑀. This topology is Hausdorff and first countable.

Definition 4 (see [18]). Let (𝑋,𝑑) be a metric space. If the metric𝑑satisfies strong triangle inequality 𝑑𝑥,𝑦≤max𝑑(𝑥,𝑧),𝑑𝑧,𝑦,∀𝑥,𝑦,𝑧∈𝑋, (2) then𝑑is called an ultrametric on𝑋and the pair (𝑋,𝑑) is called an ultrametric space.

Definition 5 (see [18]). An ultrametric space (𝑋,𝑑) is said to be spherically complete if every shrinking collection of balls in𝑋has a nonempty intersection.

Definition 6 (see [14]). Let (𝑋,𝑀, *) be a fuzzy metric space. If the fuzzy metric𝑀satisfies strong triangle inequality 𝑀𝑥,𝑦,𝑡≥min𝑀(𝑥,𝑧,𝑡),𝑀𝑧,𝑦,𝑡,∀𝑥,𝑦,𝑧∈𝑋,𝑡>0, (3) then𝑀is called a fuzzy ultrametric on𝑋and the 3-tuple (𝑋,𝑀, *) is called a fuzzy ultrametric space.

Definition 7 (see [14]). A fuzzy ultrametric space (𝑋,𝑀, *) is said to be spherically complete if every shrinking collection of balls in𝑋has a nonempty intersection.

Remark 8 (see [14]). (i) Let𝑑be an ultrametric on𝑋and 𝑎∗𝑏=𝑎⋅𝑏 for all 𝑎,𝑏∈[0,1]. For each 𝑡∈(0,∞), define 𝑀=1𝑥,𝑦,𝑡1+𝑑𝑥,𝑦^𝜓(𝑡), (4)for all 𝑥,𝑦,𝑧∈𝑋,𝑡>0, and 𝜓∈Ψ. Then fuzzy metric𝑀is also a fuzzy ultrametric.
(ii) Let an ultrametric space (𝑋,𝑑) be spherically complete and 𝑎∗𝑏=𝑎⋅𝑏 for all 𝑎,𝑏∈[0,1]. For each 𝑡∈(0,∞), define 𝑀=1𝑥,𝑦,𝑡1+𝑑𝑥,𝑦^𝜓(𝑡), (5) for all 𝑥,𝑦,𝑧∈𝑋,𝑡>0, and 𝜓∈Ψ. Then fuzzy metric space (𝑋,𝑀, *) is also spherically complete.

3. Main Results

Let (𝑋,𝑀, *) be a fuzzy ultrametric space. Let CB(𝑋) denote the set of all nonempty closed bounded subsets of𝑋. For every 𝐴,𝐵,𝐶∈CB(𝑋) and 𝑡>0, we define 𝑀_∇(𝐴,𝐵,𝑡)=min_𝑎∈𝐴𝑀^∇(𝑎,𝐵,𝑡),_𝑏∈𝐵𝑀^∇(𝑏,𝐴,𝑡), (6) where 𝑀^∇(𝑥,𝐶,𝑡)=⋁_𝑐∈𝐶𝑀(𝑥,𝑐,𝑡).

Remark 9 (see [8]). It is obvious that, for all 𝐴,𝐵∈CB(𝑋) and 𝑡>0, we have(i)𝑀_∇(𝐴,𝐵,𝑡)≤𝑀^∇(𝑎,𝐵,𝑡), whenever 𝑎∈𝐴,(ii)𝑀_∇(𝐴,𝐵,𝑡)=1 if and only if 𝐴=𝐵.
It is easy to prove that (CB(𝑋),𝑀_∇,^∗) is a Hausdorff ultrametric space.

Definition 10. Let (𝑋,𝑀, *) be a fuzzy ultrametric space. An element 𝑥∈𝑋 is said to be a coincidence point of a multivalued map 𝑇∶𝑋→CB(𝑋) and a single-valued map 𝑓∶𝑋→𝑋 if 𝑓𝑥∈𝑇𝑋. We denote 𝐶𝑓,𝑇=𝑥∈𝑋∣𝑓𝑥∈𝑇𝑥 (7) the set of coincidence points of a multivalued map𝑇and single-valued map𝑓.

Definition 11. Let (𝑋,𝑀, *) be a fuzzy ultrametric space, 𝑇∶𝑋→CB(𝑋) a multivalued map, and 𝑓∶𝑋→𝑋 if 𝑓𝑥∈𝑇𝑋 is a single-valued map. 𝑇and𝑓are said to be coincidentally computing at 𝑧∈𝑋 if 𝑓𝑧∈𝑇𝑧⇒𝑓𝑇𝑧⊆𝑇𝑓𝑧.

Definition 12. Let (𝑋,𝑀, *) be a fuzzy ultrametric space. An element 𝑥∈𝑋 is said to be a common fixed point of two multivalued maps 𝑆,𝑇∶𝑋→CB(𝑋) and a single-valued map 𝑓∶𝑋→𝑋 if 𝑥=𝑓𝑥∈𝑆𝑥∩𝑇𝑥. (8)

Theorem 13. Let (𝑋,𝑀, *) be a fuzzy ultrametric space. Let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be a pair of multivalued maps and 𝑓,𝑔∶𝑋→𝑋 a pair of single-valued maps satisfying the following conditions:(a)𝑓𝑔(𝑥) is spherically complete; (b)𝑀_∇(𝑆𝑥,𝑇𝑦,𝑡)>min{𝑀(𝑓𝑥,𝑔𝑦,𝑡),𝑀^∇(𝑓𝑥,𝑆𝑥,𝑡),𝑀^∇(𝑔𝑦,𝑇𝑦,𝑡)},  for all 𝑥,𝑦∈𝑋, with 𝑓𝑥≠𝑔𝑦, and for 𝑡>0; (c)𝑓𝑆=𝑆𝑔,𝑓𝑔=𝑔𝑓,𝑓𝑇=𝑇𝑓,𝑔𝑆=𝑆𝑔,𝑔𝑇=𝑇𝑔,𝑆𝑇=𝑇𝑆; (d)If 𝑆(𝑋)⊆𝑓(𝑋),𝑇(𝑋)⊆𝑔(𝑋), then there exist points𝑢and𝑣in𝑋, such that 𝑓𝑢∈𝑆𝑢,𝑔𝑣∈𝑇𝑣,𝑓𝑢=𝑔𝑣,𝑆𝑢=𝑇𝑣. (9)

Proof. Let 𝐵_𝑎=(𝑓𝑔𝑎;min{𝑀^∇(𝑓𝑔𝑎,𝑆𝑔𝑎,𝑡),𝑀^∇(𝑓𝑔𝑎,𝑇𝑓𝑎,𝑡)}) denote the closed ball with centered 𝑓𝑔𝑎 and radius min𝑀^∇𝑓𝑔𝑎,𝑆𝑔𝑎,𝑡,𝑀^∇𝑓𝑔𝑎,𝑇𝑓𝑎,𝑡. (10) Let 𝐀 be the collection of all the spheres for all 𝑎∈𝑓𝑔(𝑋). Then the relation 𝐵_𝑎≤𝐵_𝑏if𝐵_𝑏⊆𝐵_𝑎 (11) is a partial order on 𝐀.
Consider a totally ordered subfamily 𝐀₁ of 𝐀. Since 𝑓𝑔(𝑋) is spherically complete, we have _{𝐵𝑎∈𝐀1}𝐵_𝑎=𝐵≠𝜙. (12)
Let 𝑓𝑔𝑏∈𝐵, where 𝑏∈𝑓𝑔(𝑋) and 𝐵_𝑎∈𝐀₁. Then 𝑓𝑔𝑏∈𝐵_𝑎. Hence 𝑀𝑀𝑓𝑔𝑏,𝑓𝑔𝑎,𝑡≥min^∇𝑓𝑔𝑎,𝑆𝑔𝑎,𝑡,𝑀^∇𝑓𝑔𝑎,𝑇𝑓𝑎,𝑡. (13) If 𝑎=𝑏, then 𝐵_𝑎=𝐵_𝑏. Assume that 𝑎≠𝑏. Let, for all 𝑥∈𝐵_𝑏, 𝑀𝑀𝑥,𝑓𝑔𝑏,𝑡≥min^∇𝑓𝑔𝑏,𝑆𝑔𝑏,𝑡,𝑀^∇𝑓𝑔𝑏,𝑇𝑓𝑏,𝑡. (14) Since𝑆𝑔𝑎is a nonempty closed bounded set, then there exists an element 𝑝∈𝑆𝑔𝑎, such that 𝑀^∇𝑓𝑔𝑎,𝑆𝑔𝑎,𝑡=𝑀𝑓𝑔𝑎,𝑝,𝑡. (15)𝑇𝑓𝑎 is a nonempty closed bounded set; then there exists an element 𝑞∈𝑇𝑓𝑎 such that 𝑀^∇,𝑀𝑓𝑔𝑎,𝑇𝑓𝑎,𝑡=𝑀𝑓𝑔𝑎,𝑞,𝑡min^∇𝑓𝑔𝑏,𝑆𝑔𝑏,𝑡,𝑀^∇𝑓𝑔𝑏,𝑇𝑓𝑏,𝑡=min_{𝑐∈𝑆𝑔𝑏}𝑀𝑓𝑔𝑏,𝑐,𝑡,_{𝑑∈𝑇𝑔𝑏}𝑀𝑀,𝑓𝑔𝑏,𝑑,𝑡≥min𝑓𝑔𝑏,𝑓𝑔𝑎,𝑡,𝑀𝑓𝑔𝑎,𝑞,𝑡_{𝑐∈𝑆𝑔𝑏}𝑀,𝑀,𝑞,𝑐,𝑡,𝑀𝑓𝑔𝑏,𝑓𝑔𝑎,𝑡𝑓𝑔𝑎,𝑝,𝑡_{𝑑∈𝑇𝑓𝑏}𝑀𝑀𝑝,𝑑,𝑡≥min𝑓𝑔𝑏,𝑓𝑔𝑎,𝑡,𝑀^∇,𝑀𝑓𝑔𝑎,𝑇𝑓𝑎,𝑡_∇,𝑀𝑇𝑓𝑎,𝑆𝑔𝑏,𝑡,𝑀𝑓𝑔𝑏,𝑓𝑔𝑎,𝑡_∇𝑓𝑔𝑎,𝑆𝑔𝑎,𝑡,𝑀_∇𝑀𝑆𝑔𝑎,𝑇𝑓𝑏,𝑡>min𝑓𝑔𝑏,𝑓𝑔𝑎,𝑡,𝑀^∇,𝑀𝑓𝑔𝑎,𝑇𝑓𝑎,𝑡_∇,𝑀𝑓𝑔𝑎,𝑆𝑔𝑎,𝑡min𝑓𝑔𝑏,𝑓𝑔𝑎,𝑡,𝑀^∇,𝑀𝑓𝑔𝑎,𝑇𝑓𝑎,𝑡_∇,𝑀𝑓𝑔𝑏,𝑆𝑔𝑏,𝑡min𝑓𝑔𝑎,𝑓𝑔𝑏,𝑡,𝑀^∇,𝑀𝑓𝑔𝑎,𝑆𝑔𝑎,𝑡_∇𝑀𝑔𝑓𝑏,𝑇𝑓𝑏,𝑡≥min^∇𝑓𝑔𝑎,𝑇𝑓𝑎,𝑡,𝑀^∇𝑓𝑔𝑎,𝑆𝑔𝑎,𝑡. (16)
From the two conditions (a),(b) and (13), we now have 𝑀𝑀𝑀𝑥,𝑓𝑔𝑎,𝑡≥min𝑥,𝑓𝑔𝑏,𝑡,𝑀𝑓𝑔𝑏,𝑓𝑔𝑎,𝑡≥min^∇𝑓𝑔𝑎,𝑆𝑔𝑎,𝑡,𝑀^∇𝑓𝑔𝑎,𝑇𝑓𝑎,𝑡. (17)
So 𝑥∈𝐵_𝑎; we have just proved that 𝐵_𝑏⊆𝐵_𝑎, for every 𝐵_𝑎∈𝐀₁. Thus 𝐵_𝑏 is an upper bound in 𝐀 for the family 𝐀₁, and hence by Zorn’s lemma, there is a maximal element in 𝐀, say 𝐵_𝑍,𝑧∈𝑓𝑔(𝑋). There exists an element 𝑤∈𝑋 such that 𝑧=𝑓𝑔𝑤.
Suppose 𝑓,𝑔.𝑔𝑓𝑔𝑤∉𝑆𝑔𝑓𝑔𝑤𝑓𝑓𝑔𝑤∉𝑇𝑓𝑓𝑔𝑤 (18)
Since 𝑆𝑔𝑓𝑔𝑤 and 𝑇𝑓𝑓𝑔𝑤 are nonempty closed bounded sets, then there exist 𝑘∈𝑆𝑔𝑓𝑔𝑤 and 𝑙∈𝑇𝑓𝑓𝑔𝑤 such that 𝑀^∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡=𝑀𝑓𝑔𝑓𝑔𝑤,𝑘,𝑡_∇𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡=𝑀𝑓𝑔𝑓𝑔𝑤,𝑙,𝑡. (19)(20)
From the two conditions (b), (c) and (19), we have 𝑀^∇=𝑆𝑔𝑓𝑔𝑤,𝑇𝑆𝑓𝑔𝑤,𝑡_{𝑒∈𝑇𝑆𝑓𝑔𝑤}𝑀𝑀𝑔𝑆𝑓𝑔𝑤,𝑒,𝑡≥min^∇,𝑀,𝑆𝑔𝑓𝑔𝑤,𝑓𝑔𝑓𝑔𝑤,𝑡𝑓𝑔𝑓𝑔𝑤,𝑘,𝑡_{𝑒∈𝑇𝑆𝑓𝑔𝑤}𝑀()𝑀𝑘,𝑒,𝑡≥min^∇,𝑀𝑆𝑔𝑓𝑔𝑤,𝑓𝑔𝑓𝑔𝑤,𝑡_∇𝑀𝑆𝑔𝑓𝑔𝑤,𝑇𝑆𝑓𝑔𝑤,𝑡>min^∇,𝑀,𝑀𝑆𝑔𝑓𝑔𝑤,𝑓𝑔𝑓𝑔𝑤,𝑡min𝑓𝑔𝑓𝑔𝑤,𝑓𝑆𝑓𝑔𝑤,𝑡_∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡_∇𝑔𝑆𝑓𝑔𝑤,𝑇𝑆𝑓𝑔𝑤,𝑡=𝑀^∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡_∇=𝑇𝑓𝑓𝑔𝑤,𝑆𝑇𝑓𝑔𝑤,𝑡_{ℎ∈𝑆𝑇𝑓𝑔𝑤}𝑀𝑀𝑓𝑇𝑓𝑔𝑤,ℎ,𝑡≥min^∇,𝑀𝑇𝑓𝑓𝑔𝑤,𝑓𝑔𝑓𝑔𝑤,𝑡_∇𝑀𝑇𝑓𝑓𝑔𝑤,𝑆𝑇𝑓𝑔𝑤,𝑡>min^∇,,𝑀𝑇𝑓𝑓𝑔𝑤,𝑔𝑓𝑓𝑔𝑤,𝑡min𝑓𝑇𝑓𝑔𝑤,𝑔𝑓𝑓𝑔𝑤,𝑡_∇,𝑀𝑓𝑇𝑓𝑔𝑤,𝑆𝑇𝑓𝑔𝑤,𝑡_∇𝑔𝑓𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡=𝑀^∇𝑇𝑓𝑓𝑔𝑤,𝑓𝑔𝑓𝑔𝑤,𝑡. (21)(22)
From the two conditions (b), (c) and (19)–(22), we have 𝑀^∇=𝑓𝑔𝑔𝑆𝑤,𝑆𝑔𝑔𝑆𝑤,𝑡_{𝑚∈𝑆𝑔𝑔𝑆𝑤}𝑀𝑀𝑓𝑔𝑔𝑆𝑤,𝑚,𝑡≥min^∇,𝑀,𝑀𝑓𝑔𝑔𝑆𝑤,𝑇𝑆𝑓𝑔𝑤,𝑡𝑇𝑆𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡_∇,𝑀,𝑇𝑓𝑓𝑔𝑤,𝑓𝑔𝑓𝑔𝑤,𝑡𝑔𝑓𝑓𝑔𝑤,𝑙,𝑡_{𝑚∈𝑆𝑔𝑔𝑆𝑤}𝑀()𝑀𝑙,𝑚,𝑡≥min^∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡_∇,𝑀,𝑀𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡min𝑓𝑔𝑔𝑠𝑤,𝑔𝑓𝑓𝑔𝑤,𝑡_∇,𝑀𝑓𝑔𝑔𝑆𝑤,𝑆𝑔𝑔𝑆𝑤,𝑡_∇𝑀𝑔𝑓𝑓𝑔𝑆𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡≥min^∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡_∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡_∇=𝑓𝑔𝑓𝑇𝑤,𝑇𝑓𝑓𝑇𝑤,𝑡_{𝑛∈𝑇𝑓𝑓𝑇𝑤}𝑀𝑀𝑓𝑔𝑓𝑇𝑤,𝑛,𝑡≥min^∇,𝑀,𝑀𝑓𝑔𝑓𝑇𝑤,𝑆𝑇𝑓𝑔𝑤,𝑡𝑆𝑇𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡_∇,𝑀,𝑆𝑔𝑓𝑔𝑤,𝑓𝑔𝑓𝑔𝑤,𝑡𝑓𝑔𝑓𝑔𝑤,𝑘,𝑡_{𝑛∈𝑇𝑓𝑓𝑇𝑤}𝑀()𝑀𝑘,𝑛,𝑡≥min^∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡_∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡_∇𝑀𝑆𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑇𝑤,𝑡≥min^∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡_∇,𝑀,𝑀𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡min𝑓𝑔𝑓𝑔𝑤,𝑔𝑓𝑓𝑇𝑤,𝑡_∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡_∇𝑀𝑔𝑓𝑓𝑇𝑤,𝑇𝑓𝑓𝑇𝑤,𝑡≥min^∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡_∇𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡. (23)
From (21) and (23), we have min𝑀𝑆𝑔𝑓𝑔𝑤,𝑇𝑆𝑓𝑔𝑤,𝑡,𝑀^∇𝑀𝑓𝑔𝑔𝑆𝑤,𝑆𝑔𝑔𝑆𝑤,𝑡>min^∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡_∇𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡. (24)
From (22) and (3.7), we have min𝑀𝑆𝑇𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡,𝑀^∇𝑀𝑓𝑔𝑓𝑇𝑤,𝑇𝑓𝑓𝑇𝑤,𝑡>min^∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡_∇𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡. (25)
If min𝑀^∇𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡,𝑀^∇𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡=𝑀^∇𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡, (26) then from (24), we have 𝑓𝑔𝑓𝑔𝑤∉𝐵_𝑔𝑆𝑤⇒𝑓𝑔𝑧∉𝐵_𝑔𝑆𝑤.
Hence 𝐵_𝑧⊂𝐵̸_𝑔𝑆𝑤. It is a contradiction to the maximality of 𝐵_𝑧 in 𝐀, since 𝑔𝑆𝑤⊆𝑔𝑓(𝑋)=𝑓𝑔(𝑋).
If min𝑀^∇𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡,𝑀^∇𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡=𝑀^∇𝑓𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡, (27) then from (25), we have 𝑓𝑔𝑓𝑔𝑤∉𝐵_𝑓𝑇𝑤⇒𝑓𝑔𝑧∉𝐵_𝑓𝑇𝑤.
Hence 𝐵_𝑧⊂𝐵̸_𝑓𝑇𝑤. It is a contradiction to the maximality of 𝐵_𝑧 in 𝐀, since 𝑓𝑇𝑤⊆𝑓𝑔(𝑋),
So 𝑓𝑔𝑓𝑔𝑤∈𝑆𝑓𝑓𝑔𝑤,𝑔𝑔𝑓𝑔𝑤∈𝑇𝑓𝑓𝑔𝑤. (28) In addition, 𝑓(𝑔𝑓𝑔𝑤)=𝑔(𝑓𝑓𝑔𝑤).
Using the two conditions (b), (c) and (28), we obtain 𝑀_∇𝑀𝑆𝑔𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡>min𝑓𝑔𝑓𝑔𝑤,𝑔𝑓𝑓𝑔𝑤,𝑡,𝑀^∇,𝑀𝑓𝑔𝑓𝑔𝑤,𝑆𝑔𝑓𝑔𝑤,𝑡_∇𝑔𝑓𝑓𝑔𝑤,𝑇𝑓𝑓𝑔𝑤,𝑡=1. (29)
Hence 𝑆(𝑔𝑓𝑔𝑤)=𝑇(𝑓𝑓𝑔𝑤). Then the proof is completed.

Theorem 14. Let (𝑋,𝑀,*) be a fuzzy ultrametric space. Let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be a pair of multivalued maps and let 𝑓∶𝑋→𝑋 be a single-valued map satisfying the following conditions:(a)𝑓(𝑥) is spherically complete;(b)𝑀_∇(𝑆𝑥,𝑇𝑦,𝑡)>min{𝑀(𝑓𝑥,𝑓𝑦,𝑡),𝑀^∇(𝑓𝑥,𝑆𝑥,𝑡),𝑀^∇(𝑓𝑦,𝑇𝑦,𝑡)},  for all 𝑥,𝑦∈𝑋, with 𝑥≠𝑦, and for 𝑡>0;(c)𝑓𝑆=𝑆𝑓,𝑓𝑇=𝑇𝑓,𝑆𝑇=𝑇𝑆;(d)If 𝑆(𝑋)⊆𝑓(𝑋),𝑇(𝑋)⊆𝑓(𝑋).
Then 𝑓,𝑆, and 𝑇 have a coincidence point in 𝑋. Moreover, if 𝑓 and 𝑆,𝑓 and 𝑇 are coincidentally commuting at 𝑧∈𝐶(𝑓,𝑇) and 𝑓𝑓𝑧=𝑓𝑧, then 𝑓,𝑆, and 𝑇 have a common fixed point in𝑋.

Proof. If 𝑓=𝑔 in Theorem 13, we obtain that there exist points 𝑢 and𝑣in𝑋such that 𝑓𝑢∈𝑆𝑢,𝑓𝑣∈𝑇𝑣,𝑓𝑢=𝑓𝑣,𝑆𝑢=𝑇𝑣. (30)
As 𝑢∈𝐶(𝑓,𝑆),𝑓 and 𝑆 are coincidentally commuting at 𝑢 and 𝑓𝑓𝑢=𝑓𝑢.
Writing 𝑤=𝑓𝑢, then 𝑤∈𝑆𝑢,𝑤∈𝑇𝑣. Then we have ()𝑓𝑤=𝑤,𝑤=𝑓𝑤∈𝑓𝑆𝑢⊆𝑆𝑓𝑢=𝑆𝑤. (31)
Now, since also 𝑢∈𝐶(𝑓,𝑇),𝑓 and 𝑇 are coincidentally commuting at 𝑢 and 𝑓𝑓𝑢=𝑓𝑢, so we obtain 𝑤=𝑓𝑤∈𝑓(𝑇𝑣)⊆𝑇𝑓𝑣=𝑇𝑤. (32)
Thus, we have proved that 𝑤=𝑓𝑤∈𝑆𝑤∩𝑇𝑤; that is, 𝑤 is a common fixed point of 𝑓,𝑆, and 𝑇.

The proof of the following corollary is easy, so it is omitted.

Corollary 15. Let (𝑋,𝑀,*) be a spherically complete fuzzy ultrametric space. Let 𝑆,𝑇∶𝑋→𝐶𝐵(𝑋) be a pair of multivalued maps satisfying the following conditions:(a)𝑀_∇(𝑆𝑥,𝑇𝑦,𝑡)>min{𝑀(𝑥,𝑦,𝑡),𝑀^∇(𝑥,𝑆𝑥,𝑡),𝑀^∇(𝑦,𝑇𝑦,𝑡)}, for all 𝑥,𝑦∈𝑋, with 𝑥≠𝑦, and for 𝑡>0;(b)𝑆𝑇=𝑇𝑆.
Then, there exists a point 𝑧 in 𝑋 such that 𝑧∈𝑆𝑧∩𝑇𝑧 and 𝑆𝑧=𝑇𝑧.

Now we give an example to illustrate Theorem 13.

Example 16. Let (𝑋,𝑀,*) be a fuzzy ultrametric space in which 𝑋=[0,1],𝑎∗𝑏=min{𝑎,𝑏} for all 𝑎,𝑏∈[0,1], and 𝑀=𝑡𝑥,𝑦,𝑡𝑡+𝑥−𝑦 (33) for all 𝑡>0.
Define the maps 𝑆,𝑇,𝑓,𝑔 on 𝑋 as follows: 𝑆𝑥=𝑇𝑥=1,𝑓𝑥=𝑥+12,𝑔𝑥=2𝑥+13,∀𝑥∈𝑋. (34)
Define 𝑊𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅=𝑡₁−min𝑡₂,𝑡₃,𝑡₄,𝑡₅^ℎ, (35) where 𝑊 is a function from [0,1]⁵→[−1,1] satisfying the following conditions:(𝑊₁)𝑊(𝑡₁,…,𝑡₅) is increasing in 𝑡₁ and decreasing in 𝑡₂,…,𝑡₅(𝑊₂)𝑊(𝑢,𝑣,𝑣,𝑣,𝑣,)≥0 implies that 𝑢>𝑣, for all 𝑢∈[0,1] and 𝑣∈[0,1).
Then for any ℎ∈(0,1), the following inequality 𝑀_∇>𝑀𝑆𝑥,𝑇𝑦,𝑡min𝑓𝑥,𝑔𝑦,𝑡,𝑀^∇,𝑀𝑓𝑥,𝑆𝑥,𝑡_∇𝑔𝑦,𝑇𝑦,𝑡^ℎ (36) is satisfied for all 𝑥,𝑦∈𝑋, with 𝑥≠𝑦, and for all 𝑡>0 since the LHS of the inequality is 1.
Clearly all conditions in Theorem 13 are satisfied, so 1 is the unique common fixed point of 𝑆,𝑇,𝑓, and 𝑔.

4. Conclusion

In this paper, we get coincidence point theorems and common fixed point theorems for two pairs of multivalued and single-valued maps satisfying different contractive conditions on spherically complete fuzzy ultrametric space, which are generalized results for both ultrametric spaces [12, 15, 16] and fuzzy metric spaces [11].

Acknowledgment

The author is thankful to the referees for their valuable suggestions in preparing this paper.