Research Article A Triple Fixed Point Theorem for...

Hindawi Publishing CorporationJournal of MathematicsVolume 2013, Article ID 812153, 6 pageshttp://dx.doi.org/10.1155/2013/812153

Research ArticleA Triple Fixed Point Theorem for Multimap ina Hausdorff Fuzzy Metric Space

K. P. R. Rao1 and K. R. K. Rao2

1 Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar 522 510, India2Department of Mathematics, Vignana Bharathi Institute of Technology, Aushapur, Ghatkesar, Hyderabad 501 301, India

Correspondence should be addressed to K. P. R. Rao; [email protected]

Received 25 September 2012; Revised 30 November 2012; Accepted 5 December 2012

Academic Editor: Pierpaolo D’Urso

Copyright © 2013 K. P. R. Rao and K. R. K. Rao.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.

We obtain two triple fixed point theorems for a multimap in a Hausdorff fuzzy metric space.

1. Introduction and Preliminaries

The concept of fuzzy sets was introduced by Zadeh [1] in1965 as a mathematical tool to represent vagueness in everyday life. Since then, it was developed extensively by manyauthors, which also include interesting applications of thistheory in diverse areas. To use this concept in topology andanalysis, several researchers have defined fuzzy metric spacesin several ways (e.g., [2–4]). George and Veeramani [2] havemodified the concept of fuzzy metric space introduced byKramosil and Michalek [3] and also have succeeded ininducing a Hausdroff topology on such fuzzy metric spacewhich is often used in current research these days. LaterGrabiec [5] proved the contraction principle in the setting offuzzy metric spaces introduced in [2]. Fuzzy metric spaceshave many applications, for example, the various concepts offuzzy topology have already been found in vital applicationsin quantum particle physics particularly in connection withboth string and 𝜖(∞) theory which were studied and formu-lated by El Naschie [6] and also most recently Gregori etal. [7] have furnished several interesting examples of fuzzymetrics in the sense of George and Veeramani [2] and havealso utilized such fuzzy metrics to color image processing.For fixed point theorems in fuzzy metric spaces some of theinteresting references are in [2, 5, 8–16].

The theory of set valued maps has applications in con-trol theory, convex optimization, differential inclusions, and

economics. The study of fixed points for multivalued con-traction mappings using the Hausdorff metric was initiatedby Nadler [17]. In 2004, Rodrıguez-Lopez and Romaguera[18] introduced Hausdorff ’s fuzzy metric on the set of thenonempty compact subsets of a given fuzzy metric space.Later several authors proved some fixed point theorems formultivalued maps in fuzzy metric spaces (e.g., [19–22]). Theexistence of fixed points for various multivalued contractivemappings has been studied by many authors under differentconditions. For details, we refer the reader to [3, 17, 23–29] and the references therein. In 2006, Gnana Bhaskar andLakshmikantham [30] introduced the notion of a coupledfixed point in partially ordered metric spaces, also discussedsome problems of the uniqueness of a coupled fixed point,and applied their results to the problems of the existenceand uniqueness of a solution for the periodic boundary valueproblems. In 2011, Samet and Vetro [31] extended the coupledfixed point theorems for a multivalued mapping and laterseveral authors, namely, Hussain and Alotaibi [32], Aydi etal. [33], and Abbas et al. [34] proved coupled coincidencepoint theorems in partially orderedmetric spaces. Borcut [35]observed that the coupled fixed points technique cannot solvea system with the following form: 𝑥2 + 2𝑦𝑧− 6𝑥+ 3 = 0, 𝑦2 +2𝑧𝑥−6𝑦+3 = 0, 𝑧

2+2𝑥𝑦−6𝑧+3 = 0 and hence Berinde and

Borcut [36] introduced the concept of triple fixed points andobtained a tripled fixed point theorem for a single valuedmapin partially ordered metric spaces. Moreover, these results

2 Journal of Mathematics

could be used to study the existence of solutions of periodicboundary value problem involving 𝑦11 = 𝑓(𝑡, 𝑦, 𝑦1).

In this paper, we obtain a triple fixed point theorem fora multimap in a Hausdorff fuzzy metric space and using it,we obtain a common triple fixed point for a multi- and singlevalued maps.

In the sequel, we need the following.

Definition 1 (see [37]). A binary operation∗ : [0, 1]×[0, 1] →[0, 1] is a continuous 𝑡-norm if it satisfies the followingconditions:

(1) ∗ is associative and commutative,

(2) ∗ is continuous,

(3) 𝑎 ∗ 1 = 𝑎 for all 𝑎 ∈ [0, 1],

(4) 𝑎 ∗ 𝑏 ≤ 𝑐 ∗ 𝑑 whenever 𝑎 ≤ 𝑐 and 𝑏 ≤ 𝑑, for each𝑎, 𝑏, 𝑐, 𝑑 ∈ [0, 1].

Two typical examples of continuous 𝑡-norm are 𝑎∗𝑏 = 𝑎𝑏and 𝑎 ∗ 𝑏 = min{𝑎, 𝑏}.

Definition 2 (see [2]). A 3-tuple (𝑋,𝑀, ∗) is called a fuzzymetric space if 𝑋 is an arbitrary (nonempty) set, ∗ is acontinuous 𝑡-norm, and 𝑀 is a fuzzy set on 𝑋2 × (0,∞),satisfying the following conditions for each 𝑥, 𝑦, 𝑧 ∈ 𝑋 andeach 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, 𝑠) ≤ 𝑀(𝑥, 𝑧, 𝑡 + 𝑠),

(5) 𝑀(𝑥, 𝑦, ⋅) : (0,∞) → [0, 1] is continuous.

Let (𝑋,𝑀, ∗) be a fuzzy metric space. For 𝑡 > 0, the openball 𝐵(𝑥, 𝑟, 𝑡) with centre 𝑥 ∈ 𝑋 and radius 0 < 𝑟 < 1 isdefined by 𝐵(𝑥, 𝑟, 𝑡) = {𝑦 ∈ 𝑋 : 𝑀(𝑥, 𝑦, 𝑡) > 1 − 𝑟}.

A subset 𝐴 ⊂ 𝑋 is called open if for each 𝑥 ∈ 𝐴, thereexist 𝑡 > 0 and 0 < 𝑟 < 1 such that 𝐵(𝑥, 𝑟, 𝑡) ⊂ 𝐴. Let 𝜏denote the family of all open subsets of𝑋.Then 𝜏 is called thetopology on𝑋 induced by the fuzzy metric𝑀. This topologyis Hausdorff and first countable. A subset 𝐴 of 𝑋 is said tobe F-bounded if there exist 𝑡 > 0 and 0 < 𝑟 < 1 such that𝑀(𝑥, 𝑦, 𝑡) > 1 − 𝑟 for all 𝑥, 𝑦 ∈ 𝐴.

Grabiec [5] obtained the following important lemma.

Lemma 3 (see [5]). Let (𝑋,𝑀, ∗) be a fuzzy metric space.Then𝑀(𝑥, 𝑦, 𝑡) is nondecreasing with respect to 𝑡, for all 𝑥, 𝑦in𝑋.

Rodrıguez-Lopez and Romaguera [18] defined the conti-nuity of fuzzy metric 𝑀 and obtained the following lemmarelating to the continuity of𝑀.

Definition 4. Let (𝑋,𝑀, ∗) be a fuzzy metric space.𝑀 is saidto be continuous on𝑋2 × (0,∞) if

lim𝑛→∞

𝑀(𝑥𝑛, 𝑦𝑛, 𝑡𝑛) = 𝑀(𝑥, 𝑦, 𝑡) , (1)

whenever a sequence {(𝑥𝑛, 𝑦𝑛, 𝑡𝑛)} in 𝑋2× (0,∞) converges

to a point (𝑥, 𝑦, 𝑡) ∈ 𝑋2 × (0,∞), that is, whenever

lim𝑛→∞

𝑀(𝑥𝑛, 𝑥, 𝑡) = lim𝑛→∞

𝑀(𝑦𝑛, 𝑦, 𝑡) = 1,

lim𝑛→∞

𝑀(𝑥, 𝑦, 𝑡𝑛) = 𝑀(𝑥, 𝑦, 𝑡) .

(2)

Lemma 5 (see [18]). Let (𝑋,𝑀, ∗) be a fuzzy metric space.Then𝑀 is a continuous function on𝑋2 × (0,∞).

From now onwards, (𝐴)will denote the following condi-tion:

lim𝑡→∞

𝑀(𝑥, 𝑦, 𝑡) = 1, ∀𝑥, 𝑦 ∈ 𝑋. (𝐴)

In 1994,Mishra et al. [13] proved the following lemma relatingto Cauchy sequences in fuzzy metric spaces.

Lemma 6 (see [13]). Let {𝑦𝑛} be a sequence in fuzzy metricspace (𝑋,𝑀, ∗) satisfying (𝐴). If there exists a positive number𝑘 < 1 such that

𝑀(𝑦𝑛, 𝑦𝑛+1, 𝑘𝑡) ≥ 𝑀(𝑦𝑛−1, 𝑦𝑛, 𝑡) , 𝑡 > 0, 𝑛 = 1, 2, . . . ,

(3)

then {𝑦𝑛} is a Cauchy sequence in𝑋.

Definition 7 ([18], Definition 2.2). Let 𝐵 be a nonemptysubset of a fuzzy metric space (𝑋,𝑀, ∗). For 𝑎 ∈ 𝑋 and 𝑡 > 0,define𝑀(𝑎, 𝐵, 𝑡) = sup{𝑀(𝑎, 𝑏, 𝑡)/𝑏 ∈ 𝐵}.

Throughout the paper, let 𝐾(𝑋) denote the class of allnonempty compact subsets of𝑋.

Lemma 8 ([18], Lemma 1). Let (𝑋,𝑀, ∗) be a fuzzy metricspace. Then for each 𝑎 ∈ 𝑋, 𝐵 ∈ 𝐾(𝑋) and 𝑡 > 0, there exists𝑏 ∈ 𝐵 such that𝑀(𝑎, 𝐵, 𝑡) = 𝑀(𝑎, 𝑏, 𝑡).

Definition 9 (see [18]). Let (𝑋,𝑀, ∗) be a fuzzy metric space.For each 𝐴, 𝐵 ∈ 𝐾(𝑋) and 𝑡 > 0, set

𝐻𝑀 (𝐴, 𝐵, 𝑡) = min{inf𝑥∈𝐴𝑀(𝑥, 𝐵, 𝑡) , inf

𝑦∈𝐵𝑀(𝐴, 𝑦, 𝑡)} . (4)

The 3-tuple (𝐾(𝑋),𝐻𝑀, ∗) is called a Hausdorff fuzzy metricspace.

We use the following lemma proved by Haghi et al. [38]in Theorem 15 in Section 2.

Lemma 10 (see [38]). Let 𝑋 be a nonempty set and 𝑔 : 𝑋 →

𝑋 be a mapping. Then there exists a subset 𝐸 ⊆ 𝑋 such that𝑔(𝐸) = 𝑔(𝑋) and 𝑔 : 𝐸 → 𝑋 is one one.

Now, we give the following definitions for a hybrid pair ofmappings (see also [39]).

Journal of Mathematics 3

Definition 11. Let 𝑋 be a nonempty set, 𝑇 : 𝑋 × 𝑋 × 𝑋 →

2𝑋 (collection of all nonempty subsets of𝑋) and𝑓 : 𝑋 → 𝑋.

(i) The point (𝑥, 𝑦, 𝑧) ∈ 𝑋 × 𝑋 × 𝑋 is called a tripledfixed of 𝑇 if

𝑥 ∈ 𝑇 (𝑥, 𝑦, 𝑧) ,

𝑦 ∈ 𝑇 (𝑦, 𝑥, 𝑦) ,

𝑧 ∈ 𝑇 (𝑧, 𝑦, 𝑥) .

(5)

(ii) The point (𝑥, 𝑦, 𝑧) ∈ 𝑋 × 𝑋 × 𝑋 is called a tripledcoincident point of 𝑇 and 𝑓 if

𝑓𝑥 ∈ 𝑇 (𝑥, 𝑦, 𝑧) ,

𝑓𝑦 ∈ 𝑇 (𝑦, 𝑥, 𝑦) ,

𝑓𝑧 ∈ 𝑇 (𝑧, 𝑦, 𝑥) .

(6)

(iii) The point (𝑥, 𝑦, 𝑧) ∈ 𝑋 × 𝑋 × 𝑋 is called a tripledcommon fixed point of 𝑇 and 𝑓 if

𝑥 = 𝑓𝑥 ∈ 𝑇 (𝑥, 𝑦, 𝑧) ,

𝑦 = 𝑓𝑦 ∈ 𝑇 (𝑦, 𝑥, 𝑦) ,

𝑧 = 𝑓𝑧 ∈ 𝑇 (𝑧, 𝑦, 𝑥) .

(7)

Definition 12. Let𝑇 : 𝑋×𝑋×𝑋 → 2𝑋 be amultivaluedmap

and 𝑓 be a self-map on 𝑋. The Hybrid pair {𝑇, 𝑓} is called𝑤-compatible if 𝑓(𝑇(𝑥, 𝑦, 𝑧)) ⊆ 𝑇(𝑓𝑥, 𝑓𝑦, 𝑓𝑧) whenever(𝑥, 𝑦, 𝑧) is a tripled coincidence point of 𝑇 and 𝑓.

2. Main Result

First we prove a slightly different result from Lemma 6 whichis essential in proving our main result.

Lemma 13. Let {𝑥𝑛}, {𝑦𝑛}, and {𝑧𝑛} be sequences in fuzzymetric space (𝑋,𝑀, ∗) satisfying (𝐴). If there exists a positivenumber 𝑘 < 1 such that

min {𝑀 (𝑥𝑛, 𝑥𝑛+1, 𝑘𝑡) ,𝑀 (𝑦𝑛, 𝑦𝑛+1, 𝑘𝑡) ,𝑀 (𝑧𝑛, 𝑧𝑛+1, 𝑘𝑡)}

≥ min {𝑀 (𝑥𝑛−1, 𝑥𝑛, 𝑡) ,𝑀 (𝑦𝑛−1, 𝑦𝑛, 𝑡) ,𝑀 (𝑧𝑛−1, 𝑧𝑛, 𝑡)} ,

(8)

for all 𝑡 > 0, 𝑛 = 1, 2, . . ., then {𝑥𝑛}, {𝑦𝑛} and {𝑧𝑛} are Cauchysequences in𝑋.

Proof. We have

min {𝑀 (𝑥𝑛, 𝑥𝑛+1, 𝑘𝑡) ,

𝑀 (𝑦𝑛, 𝑦𝑛+1, 𝑘𝑡) ,𝑀 (𝑧𝑛, 𝑧𝑛+1, 𝑘𝑡)}

≥ min {𝑀 (𝑥𝑛−1, 𝑥𝑛, 𝑡) ,

𝑀 (𝑦𝑛−1, 𝑦𝑛, 𝑡) ,𝑀 (𝑧𝑛−1, 𝑧𝑛, 𝑡)}

≥ min {𝑀(𝑥𝑛−2, 𝑥𝑛−1,𝑡

𝑘2) ,

𝑀(𝑦𝑛−2, 𝑦𝑛−1,𝑡

𝑘2) ,𝑀(𝑧𝑛−2, 𝑧𝑛−1,

𝑡

𝑘2)}

...

≥ min { 𝑀(𝑥0, 𝑥1,𝑡

𝑘𝑛) ,𝑀(𝑦0, 𝑦1,

𝑡

𝑘𝑛) ,

𝑀(𝑧0, 𝑧1,𝑡

𝑘𝑛)} .

(9)

Hence,

𝑀(𝑥𝑛, 𝑥𝑛+1, 𝑡)

≥ min {𝑀(𝑥0, 𝑥1,𝑡

𝑘𝑛) ,𝑀(𝑦0, 𝑦1,

𝑡

𝑘𝑛) ,


𝑘𝑛)} .

(10)

Now, for any positive integer 𝑝,

𝑀(𝑥𝑛, 𝑥𝑛+𝑝, 𝑡)

≥ 𝑀(𝑥𝑛, 𝑥𝑛+1,𝑡

𝑝

) ∗𝑀(𝑥𝑛+1, 𝑥𝑛+2,𝑡

𝑝

)

∗ ⋅ ⋅ ⋅ ∗ 𝑀(𝑥𝑛+𝑝−1, 𝑥𝑛+𝑝,𝑡

𝑝

)

≥ min{𝑀(𝑥0, 𝑥1,𝑡

𝑝𝑘𝑛) ,

𝑀(𝑦0, 𝑦1,𝑡

𝑝𝑘𝑛) ,𝑀(𝑧0, 𝑧1,

𝑡

𝑝𝑘𝑛)}

∗min{𝑀(𝑥0, 𝑥1,𝑡

𝑝𝑘𝑛+1) ,


𝑝𝑘𝑛+1) ,𝑀(𝑧0, 𝑧1,

𝑡

𝑝𝑘𝑛+1)}


∗ ⋅ ⋅ ⋅ ∗min{𝑀(𝑥0, 𝑥1,𝑡

𝑝𝑘𝑛+𝑝−1

) ,



) ,



)} .

(11)

Letting 𝑛 → ∞ and using (𝐴), we have

lim𝑛→∞

𝑀(𝑥𝑛, 𝑥𝑛+𝑝, 𝑡) ≥ 1 ∗ 1 ∗ ⋅ ⋅ ⋅ ∗ 1 = 1.

Hence, lim𝑛→∞

𝑀(𝑥𝑛, 𝑥𝑛+𝑝, 𝑡) = 1.

(12)

Thus {𝑥𝑛} is a Cauchy sequence in 𝑋. Similarly, we can showthat {𝑦𝑛} and {𝑧𝑛} are also Cauchy sequences in𝑋.

Now, we are ready to prove our first main result.

Theorem 14. Let (𝑋,𝑀, ∗) be a complete fuzzy metric spacesatisfying condition (𝐴) and 𝐹 : 𝑋 ×𝑋 ×𝑋 → K(𝑋) be a setvalued mapping satisfying

𝐻𝑀 (𝐹 (𝑥, 𝑦, 𝑧) , 𝐹 (𝑢, 𝑣, 𝑤) , 𝑘𝑡)

≥ min {𝑀 (𝑥, 𝑢, 𝑡) ,𝑀 (𝑦, 𝑣, 𝑡) ,𝑀 (𝑧, 𝑤, 𝑡)}

(13)

for each 𝑥, 𝑦, 𝑧, 𝑢, 𝑣, 𝑤 ∈ 𝑋, 𝑡 > 0, where 0 < 𝑘 < 1.Then 𝐹 has a tripled fixed point.

Proof. Let 𝑥0, 𝑦0, 𝑧0 ∈ 𝑋.Choose 𝑥1 ∈ 𝐹(𝑥0, 𝑦0, 𝑧0), 𝑦1 ∈ 𝐹(𝑦0, 𝑥0, 𝑦0), 𝑧1 ∈

𝐹(𝑧0, 𝑦0, 𝑥0).Since 𝐹 is compact valued, by Lemma 8, there exists 𝑥2 ∈

𝐹(𝑥1, 𝑦1, 𝑧1) such that

𝑀(𝑥1, 𝑥2, 𝑘𝑡)

= sup𝑥∈𝐹(𝑥1 ,𝑦1,𝑧1)

𝑀 (𝑥1, 𝑥, 𝑘𝑡)

≥ 𝐻𝑀 (𝐹 (𝑥0, 𝑦0, 𝑧0) , 𝐹 (𝑥1, 𝑦1, 𝑧1) , 𝑘𝑡)

≥ min {𝑀 (𝑥0, 𝑥1, 𝑡) ,𝑀 (𝑦0, 𝑦1, 𝑡) ,𝑀 (𝑧0, 𝑧1, 𝑡)} .

(14)

Since 𝐹 is compact valued, by Lemma 8, there exists 𝑦2 ∈𝐹(𝑦1, 𝑥1, 𝑦1) such that

𝑀(𝑦1, 𝑦2, 𝑘𝑡)

= sup𝑦∈𝐹(𝑦1 ,𝑥1,𝑦1)

𝑀 (𝑦1, 𝑦, 𝑘𝑡)

≥ 𝐻𝑀 (𝐹 (𝑦0, 𝑥0, 𝑦0) , 𝐹 (𝑦1, 𝑥1, 𝑦1) , 𝑘𝑡)

≥ min {𝑀 (𝑦0, 𝑦1, 𝑡) ,𝑀 (𝑥0, 𝑥1, 𝑡) ,𝑀 (𝑦0, 𝑦1, 𝑡)}


(15)

Since 𝐹 is compact valued, by Lemma 8, there exists 𝑧2 ∈𝐹(𝑧1, 𝑦1, 𝑥1) such that

𝑀(𝑧1, 𝑧2, 𝑘𝑡)

= sup𝑧∈𝐹(𝑧1 ,𝑦1,𝑥1)

𝑀 (𝑧1, 𝑧, 𝑘𝑡)

≥ 𝐻𝑀 (𝐹 (𝑧0, 𝑦0, 𝑥0) , 𝐹 (𝑧1, 𝑦1, 𝑥1) , 𝑘𝑡)

≥ min {𝑀 (𝑧0, 𝑧1, 𝑡) ,𝑀 (𝑦0, 𝑦1, 𝑡) ,𝑀 (𝑥0, 𝑥1, 𝑡)}

= min {𝑀 (𝑥0, 𝑥1, 𝑡) ,𝑀 (𝑧0, 𝑧1, 𝑡) ,𝑀 (𝑦0, 𝑦1, 𝑡)} .

(16)

Thus,

min {𝑀 (𝑥1, 𝑥2, 𝑘𝑡) ,𝑀 (𝑦1, 𝑦2, 𝑘𝑡) ,𝑀 (𝑧1, 𝑧2, 𝑘𝑡)}


(17)

Continuing in this way, we can obtain sequences {𝑥𝑛}, {𝑦𝑛}and {𝑧𝑛} in 𝑋 such that 𝑥𝑛+1 ∈ 𝐹(𝑥𝑛, 𝑦𝑛, 𝑧𝑛), 𝑦𝑛+1 ∈

𝐹(𝑦𝑛, 𝑥𝑛, 𝑦𝑛) and 𝑧𝑛+1 ∈ 𝐹(𝑧𝑛, 𝑦𝑛, 𝑥𝑛) such that

min {𝑀 (𝑥𝑛, 𝑥𝑛+1, 𝑘𝑡)𝑀 (𝑦𝑛, 𝑦𝑛+1, 𝑘𝑡)𝑀 (𝑧𝑛, 𝑧𝑛+1, 𝑘𝑡)}

≥ min {𝑀 (𝑥𝑛−1, 𝑥𝑛, 𝑡) ,𝑀 (𝑦𝑛−1, 𝑦𝑛, 𝑡) ,𝑀 (𝑧𝑛−1, 𝑧𝑛, 𝑡)} .

(18)

Hence, by Lemma 13, {𝑥𝑛} {𝑦𝑛} and {𝑧𝑛} are Cauchy sequencesin𝑋.

Since 𝑋 is complete, there exist 𝑥, 𝑦, 𝑧 ∈ 𝑋 suchthat lim𝑛→∞{𝑥𝑛} = 𝑥, lim𝑛→∞{𝑦𝑛} = 𝑦 and lim𝑛→∞{𝑧𝑛} =𝑧. Consider

𝐻𝑀 (𝐹 (𝑥𝑛, 𝑦𝑛, 𝑧𝑛) , 𝐹 (𝑥, 𝑦, 𝑧) , 𝑘𝑡)

≥ min {𝑀 (𝑥𝑛, 𝑥, 𝑡) ,𝑀 (𝑦𝑛, 𝑦, 𝑡) ,𝑀 (𝑧𝑛, 𝑧, 𝑡)} .

(19)

Letting 𝑛 → ∞, we get

lim𝑛→∞

𝐻𝑀 (𝐹 (𝑥𝑛, 𝑦𝑛, 𝑧𝑛) , 𝐹 (𝑥, 𝑦, 𝑧) , 𝑘𝑡) = 1 so that

lim𝑛→∞

𝐻𝑀 (𝐹 (𝑥𝑛, 𝑦𝑛, 𝑧𝑛) , 𝐹 (𝑥, 𝑦, 𝑧) , 𝑡) = 1.

(20)

Similarly, we can show that

lim𝑛→∞

𝐻𝑀 (𝐹 (𝑦𝑛, 𝑥𝑛, 𝑦𝑛) , 𝐹 (𝑦, 𝑥, 𝑦) , 𝑡) = 1,

lim𝑛→∞

𝐻𝑀 (𝐹 (𝑧𝑛, 𝑦𝑛, 𝑥𝑛) , 𝐹 (𝑧, 𝑦, 𝑥) , 𝑡) = 1.

(21)

Since 𝑥𝑛+1 ∈ 𝐹(𝑥𝑛, 𝑦𝑛, 𝑧𝑛), 𝑦𝑛+1 ∈ 𝐹(𝑦𝑛, 𝑥𝑛, 𝑦𝑛), and 𝑧𝑛+1 ∈𝐹(𝑧𝑛, 𝑦𝑛, 𝑥𝑛), from (20) and (21), we have

lim𝑛→∞

sup𝑎∈𝐹(𝑥,𝑦,𝑧)

𝑀 (𝑥𝑛+1, 𝑎, 𝑡) = 1,

lim𝑛→∞

sup𝑏∈𝐹(𝑦,𝑥,𝑦)

𝑀 (𝑦𝑛+1, 𝑏, 𝑡) = 1,

lim𝑛→∞

sup𝑐∈𝐹(𝑧,𝑦,𝑥)

𝑀 (𝑧𝑛+1, 𝑐, 𝑡) = 1.

(22)

Journal of Mathematics 5

Hence there exist sequences 𝑝𝑛 ∈ 𝐹(𝑥, 𝑦, 𝑧), 𝑞𝑛 ∈ 𝐹(𝑦, 𝑥, 𝑦)and 𝑟𝑛 ∈ 𝐹(𝑧, 𝑦, 𝑥) such that

lim𝑛→∞

𝑀(𝑥𝑛+1, 𝑝𝑛, 𝑡) = 1,

lim𝑛→∞

𝑀(𝑦𝑛+1, 𝑞𝑛, 𝑡) = 1,

lim𝑛→∞

𝑀(𝑧𝑛+1, 𝑟𝑛, 𝑡) = 1,

for each 𝑡 > 0.

(23)

Now, for each 𝑛 ∈N, we have

𝑀(𝑝𝑛, 𝑥, 𝑡) ≥ 𝑀(𝑝𝑛, 𝑥𝑛+1,𝑡

2

) ∗𝑀(𝑥𝑛+1, 𝑥,𝑡

2

) . (24)

Letting 𝑛 → ∞, we obtain

lim𝑛→∞

𝑀(𝑝𝑛, 𝑥, 𝑡) = 1 so that lim𝑛→∞

𝑝𝑛 = 𝑥. (25)

Similarly, we can show that

lim𝑛→∞

𝑞𝑛 = 𝑦, lim𝑛→∞

𝑟𝑛 = 𝑧. (26)

Since 𝐹(𝑥, 𝑦, 𝑧), 𝐹(𝑦, 𝑥, 𝑦), and 𝐹(𝑧, 𝑦, 𝑥) are compact, wehave 𝑥 ∈ 𝐹(𝑥, 𝑦, 𝑧),𝑦 ∈ 𝐹(𝑦, 𝑥, 𝑦), and 𝑧 ∈ 𝐹(𝑧, 𝑦, 𝑥).

Thus, (𝑥, 𝑦, 𝑧) is a tripled fixed point of 𝐹.

Using the above theorem, we now prove a tripled coinci-dence and common fixed point theorem for a hybrid pair ofmultivalued and single-valued mapping.

Theorem 15. Let (𝑋,𝑀, ∗) be a complete fuzzy metric spacesatisfying condition (𝐴) and 𝐹 : 𝑋 × 𝑋 × 𝑋 → K(𝑋) and𝑔 : 𝑋 → 𝑋 be a mappings satisfying

𝐻𝑀 (𝐹 (𝑥, 𝑦, 𝑧) , 𝐹 (𝑢, 𝑣, 𝑤) , 𝑘𝑡)

≥ min {𝑀 (𝑔𝑥, 𝑔𝑢, 𝑡) ,𝑀 (𝑔𝑦, 𝑔𝑣, 𝑡) ,𝑀 (𝑔𝑧, 𝑔𝑤, 𝑡)} ,

(27)

for all 𝑥, 𝑦, 𝑧, 𝑢, 𝑣, 𝑤 ∈ 𝑋, 𝑡 > 0 and 0 < 𝑘 < 1. Furtherassume that 𝐹(𝑋×𝑋×𝑋) ⊆ 𝑔(𝑋), then 𝐹 and 𝑔 have a tripledcoincidence point. Moreover, 𝐹 and 𝑔 have a tripled commonfixed point if one of the following conditions holds.

(a) The pair (𝐹, 𝑔) is𝑤-compatible and there exist 𝛼, 𝛽, 𝛾 ∈𝑋 such that lim𝑛→∞𝑔𝑛𝑥 = 𝛼, lim𝑛→∞𝑔𝑛𝑦 = 𝛽,and lim𝑛→∞𝑔

𝑛𝑧 = 𝛾, whenever (𝑥, 𝑦, 𝑧) is a tripled

coincidence point of 𝐹 and 𝑔 and 𝑔 is continuous at𝛼, 𝛽, 𝛾.

(b) There exist 𝛼, 𝛽, 𝛾 ∈ 𝑋 such that lim𝑛→∞𝑔𝑛𝛼 = 𝑥,lim𝑛→∞𝑔

𝑛𝛽 = 𝑦 and lim𝑛→∞𝑔

𝑛𝛾 = 𝑧, whenever

(𝑥, 𝑦, 𝑧) is a tripled coincidence point of 𝐹 and 𝑔 and𝑔 is continuous at 𝑥, 𝑦, 𝑧.

Proof. By Lemma 10, There exists 𝐸 ⊆ 𝑋 such that 𝑔 : 𝐸 →𝑋 is one to one and 𝑔(𝐸) = 𝑔(𝑋).

Now, define 𝐺 : 𝑔(𝐸) × 𝑔(𝐸) × 𝑔(𝐸) → K(𝑋) by 𝐺(𝑔𝑥,𝑔𝑦, 𝑔𝑧) = 𝐹(𝑥, 𝑦, 𝑧) for all 𝑔𝑥, 𝑔𝑦, 𝑔𝑧 ∈ 𝑔(𝐸). Since 𝑔 is one-one on 𝐸, 𝐺 is well defined.

Now,

𝐻𝑀 (𝐺 (𝑔𝑥, 𝑔𝑦, 𝑔𝑧) , 𝐺 (𝑔𝑢, 𝑔𝑣, 𝑔𝑤) , 𝑘𝑡)

= 𝐻𝑀 (𝐹 (𝑥, 𝑦, 𝑧) , 𝐹 (𝑢, 𝑣, 𝑤) , 𝑘𝑡)

≥ min {𝑀 (𝑔𝑥, 𝑔𝑢, 𝑡) ,𝑀 (𝑔𝑦, 𝑔𝑣, 𝑡) ,𝑀 (𝑔𝑧, 𝑔𝑤, 𝑡)} .

(28)

Hence 𝐺 satisfies (13) and all the conditions of Theorem 14.By Theorem 14, 𝐺 has a tripled fixed point (𝑢, 𝑣, 𝑤) ∈

𝑔(𝐸) × 𝑔(𝐸) × 𝑔(𝐸). Thus,

𝑢 ∈ 𝐺 (𝑢, 𝑣, 𝑤) ,

𝑣 ∈ 𝐺 (𝑣, 𝑢, 𝑣) ,

𝑤 ∈ 𝐺 (𝑤, 𝑣, 𝑢) .

(29)

Since 𝐹(𝑋 × 𝑋 × 𝑋) ⊆ 𝑔(𝑋), there exist 𝑢, 𝑣, 𝑤 ∈ 𝑋 × 𝑋 × 𝑋such that 𝑔𝑢1 = 𝑢, 𝑔𝑣1 = 𝑣, and 𝑔𝑤1 = 𝑤. So from (29), wehave

𝑔𝑢1 ∈ 𝐺 (𝑔𝑢1, 𝑔𝑣1, 𝑔𝑤1)

= 𝐹 (𝑢1, 𝑣1, 𝑤1) ,

𝑔𝑣1 ∈ 𝐺 (𝑔𝑣1, 𝑔𝑢1, 𝑔𝑣1)

= 𝐹 (𝑣1, 𝑢1, 𝑣1) ,

𝑔𝑤1 ∈ 𝐺 (𝑔𝑤1, 𝑔𝑣1, 𝑔𝑢1)

= 𝐹 (𝑤1, 𝑣1, 𝑢1) .

(30)

This implies that (𝑢1, 𝑣1, 𝑤1) ∈ 𝑋 × 𝑋 × 𝑋 is a tripledcoincidence point of 𝐹 and 𝑔.

Now, following as in [39] except from the inequalitiessatisfied by 𝑀 we can show that 𝐹 and 𝑔 have a tripledcommon fixed point.

Acknowledgment

The authors are thankful to the referee for his valuable sug-gestions.

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