A Unique Common Triple Fixed Point Theorem for Hybrid Pair of Maps

Abstract

We obtain a unique common triple fixed point theorem for hybrid pair of mappings in metric spaces. Our result extends the recent results of B. Samet and C. Vetro (2011). We also introduced a suitable example supporting our result.

1. Introduction

The study of fixed points for multivalued contraction mappings using the Hausdorff metric was initiated by Nadler [1].

Let (𝑋,𝑑) be a metric space. We denote 𝐶𝐵(𝑋) the family of all nonempty closed and bounded subsets of 𝑋 and 𝐶𝐿(𝑋) the set of all nonempty closed subsets of 𝑋. For 𝐴,𝐵∈𝐶𝐵(𝑋) and 𝑥∈𝑋, we denote 𝐷(𝑥,𝐴)=inf{𝑑(𝑥,𝑎)∶𝑎∈𝐴}. Let 𝐻 be the Hausdorff metric induced by the metric 𝑑 on 𝑋, that is, 𝐻(𝐴,𝐵)=maxsup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑𝑦,𝐴, (1.1) for every 𝐴,𝐵∈𝐶𝐵(𝑋).

It is clear that for 𝐴,𝐵∈𝐶𝐵(𝑋) and 𝑎∈𝐴, we have 𝑑(𝑎,𝐵)≤𝐻(𝐴,𝐵).

Definition 1.1. An element 𝑥∈𝑋 is said to be a fixed point of a set-valued mapping 𝑇∶𝑋→𝐶𝐵(𝑋) if and only if 𝑥∈𝑇𝑥.

In 1969, Nadler [1] extended the famous Banach contraction principle [2] from single-valued mapping to multivalued mapping and proved the following fixed point theorem for the multivalued contraction.

Theorem 1.2 (see, Nadler [1]). Let (𝑋,𝑑) be a complete metric space and let 𝑇 be a mapping from 𝑋 into 𝐶𝐵(𝑋). Assume that there exists 𝑐∈[0,1) such that 𝐻𝑇𝑥,𝑇𝑦≤𝑐𝑑𝑥,𝑦, (1.2) for all 𝑥,𝑦∈𝑋. Then, 𝑇 has a fixed point.

Lemma 1.3   (see, Nadler [1]). Let 𝐴,𝐵∈𝐶𝐵(𝑋) and 𝛼>1. Then for every 𝑎∈𝐴, there exists 𝑏∈𝐵 such that 𝑑(𝑎,𝑏)≤𝛼𝐻(𝐴,𝐵).

Lemma 1.4   (see, Nadler [1]). Let 𝛼>0. If 𝐴,𝐵∈𝐶𝐵(𝑋) with 𝐻(𝐴,𝐵)≤𝛼, then for each 𝑎∈𝐴, there exists 𝑏∈𝐵 such that 𝑑(𝑎,𝑏)≤𝛼.

Lemma 1.5   (see, Nadler [1]). Let {𝐴_𝑛} be a sequence in 𝐶𝐵(𝑋) with lim_𝑛→+∞𝐻(𝐴_𝑛,𝐴)=0, for 𝐴∈𝐶𝐵(𝑋). If 𝑥_𝑛∈𝐴_𝑛 and lim_𝑛→+∞𝑑(𝑥_𝑛,𝑥)=0, then 𝑥∈𝐴.

The existence of fixed points for various multivalued contractive mappings has been studied by many authors under different conditions. For details, we refer the reader to [1, 3–11] and the references therein.

The concept of coupled fixed point for multivalued mapping was introduced by Samet and Vetro [12], and later several authors, namely, Hussain and Alotaibi [13], Aydi et al. [14], and Abbas et al. [15], proved coupled coincidence point theorems in partially ordered metric spaces.

Definition 1.6   (see, Samet and Vetro [12]). Let 𝐹∶𝑋×𝑋→𝐶𝐿(𝑋) be a given mapping. We say that (𝑥,𝑦)∈𝑋×𝑋 is a coupled fixed point of 𝐹 if and only if𝑥∈𝐹𝑥,𝑦,𝑦∈𝐹𝑦,𝑥. (1.3)

Definition 1.7   (see, Hussain and Alotaibi [13]). Let the mappings 𝐹∶𝑋×𝑋→𝐶𝐵(𝑋) and 𝑔∶𝑋→𝑋 be given. An element (𝑥,𝑦)∈𝑋×𝑋 is called(1)a coupled coincidence point of a pair {𝐹,𝑔} if 𝑔𝑥∈𝐹(𝑥,𝑦) and 𝑔𝑦∈𝐹(𝑦,𝑥);(2)a coupled common fixed point of a pair {𝐹,𝑔} if 𝑥=𝑔𝑥∈𝐹(𝑥,𝑦) and 𝑦=𝑔𝑦∈𝐹(𝑦,𝑥).
Berinde and Borcut [16] introduced the concept of triple fixed points and obtained a tripled fixed point theorem for single valued map.
Now we give the following.

Definition 1.8. Let 𝑋 be a nonempty set, 𝑇∶𝑋×𝑋×𝑋→2^𝑋 (collection of all nonempty subsets of 𝑋). 𝑓∶𝑋→𝑋. (i)The point (𝑥,𝑦,𝑧)∈𝑋×𝑋×𝑋 is called a tripled fixed point of 𝑇 if 𝑥∈𝑇𝑥,𝑦,𝑧,𝑦∈𝑇𝑦,𝑥,𝑦,𝑧∈𝑇𝑧,𝑦,𝑥. (1.4)(ii)The point (𝑥,𝑦,𝑧)∈𝑋×𝑋×𝑋 is called a tripled coincident point of 𝑇 and 𝑓 if 𝑓𝑥∈𝑇𝑥,𝑦,𝑧,𝑓𝑦∈𝑇𝑦,𝑥,𝑦,𝑓𝑧∈𝑇𝑧,𝑦,𝑥. (1.5)(iii)The point (𝑥,𝑦,𝑧)∈𝑋×𝑋×𝑋 is called a tripled common fixed point of 𝑇 and 𝑓 if 𝑥=𝑓𝑥∈𝑇𝑥,𝑦,𝑧,𝑦=𝑓𝑦∈𝑇𝑦,𝑥,𝑦,𝑧=𝑓𝑧∈𝑇𝑧,𝑦,𝑥. (1.6)

Definition 1.9. Let 𝑇∶𝑋×𝑋×𝑋→2^𝑋 be a multivalued map and 𝑓 be a self map on 𝑋. The Hybrid pair {𝑇,𝑓} is called 𝑤-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 Results

Theorem 2.1. Let (𝑋,𝑑) be a metric space and let 𝑇∶𝑋×𝑋×𝑋→𝐶𝐵(𝑋) and 𝑓∶𝑋→𝑋 mappings satisfying (2.1.1)𝐻(𝑇(𝑥,𝑦,𝑧),𝑇(𝑢,𝑣,𝑤))≤𝑗𝑑(𝑓𝑥,𝑓𝑦)+𝑘𝑑(𝑓𝑦,𝑓𝑣)+𝑙𝑑(𝑓𝑧,𝑓𝑤), for all 𝑥,𝑦,𝑧,𝑢,𝑣,𝑤∈𝑋 and 𝑗,𝑘,𝑙∈[0,1) with 𝑗+𝑘+𝑙≤ℎ<1, where ℎ is a fixed number, (2.1.2)𝑇(𝑋×𝑋×𝑋)⊆𝑓(𝑋) and 𝑓(𝑋) is a complete subspace of 𝑋.
Then the maps 𝑇 and 𝑓 have a tripled coincidence point.
Further, 𝑇 and 𝑓 have a tripled common fixed point if one of the following conditions holds.(2.1.3) (a) {𝑇,𝑓} is 𝑤-compatible, there exist 𝑢,𝑣,𝑤∈𝑋 such that lim_𝑛→∞𝑓^𝑛𝑥=𝑢,lim_𝑛→∞𝑓^𝑛𝑦=𝑣 and lim_𝑛→∞𝑓^𝑛𝑧=𝑤, whenever (𝑥,𝑦,𝑧) is a tripled coincidence point of {𝑇,𝑓} and 𝑓 is continuous at 𝑢,𝑣,𝑤. (b) There exist 𝑢,𝑣,𝑤∈𝑋 such that lim_𝑛→∞𝑓^𝑛𝑢=𝑥,  lim_𝑛→∞𝑓^𝑛𝑣=𝑦 and lim_𝑛→∞𝑓^𝑛𝑤=𝑧 whenever (𝑥,𝑦,𝑧) is a tripled coincidence point of {𝑇,𝑓} and 𝑓 is continuous at 𝑥,𝑦, and 𝑧.

Proof. Let 𝑥₀,𝑦₀,𝑧₀∈𝑋. From (2.1.2), there exist sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} in 𝑋 such that 𝑓𝑥_𝑛+1∈𝑇(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛), 𝑓𝑦_𝑛+1∈𝑇(𝑦_𝑛,𝑥_𝑛,𝑦_𝑛) and 𝑓𝑧_𝑛+1∈𝑇(𝑧_𝑛,𝑦_𝑛,𝑥_𝑛), 𝑛=0,1,2,3,….
For simplification, denote 𝑑^𝑥_𝑛=𝑑𝑓𝑥_𝑛−1,𝑓𝑥_𝑛,𝑑^𝑦_𝑛=𝑑𝑓𝑦_𝑛−1,𝑓𝑦_𝑛,𝑑^𝑧_𝑛=𝑑𝑓𝑧_𝑛−1,𝑓𝑧_𝑛. (2.1) From (2.1.1), we obtain 𝑑^𝑥₂=𝑑𝑓𝑥₁,𝑓𝑥₂≤𝐻𝑇𝑥₀,𝑦₀,𝑧₀,𝑇𝑥₁,𝑦₁,𝑧₁+ℎ≤𝑗𝑑𝑓𝑥₀,𝑓𝑥₁+𝑘𝑑𝑓𝑦₀,𝑓𝑦₁+𝑙𝑑𝑓𝑧₀,𝑓𝑧₁+ℎ=𝑗𝑑^𝑥₁+𝑘𝑑^𝑦₁+𝑙𝑑^𝑧₁+ℎ,𝑑^𝑦₂=𝑑𝑓𝑦₁,𝑓𝑦₂≤𝐻𝑇𝑦₀,𝑥₀,𝑦₀,𝑇𝑦₁,𝑥₁,𝑦₁+ℎ≤𝑗𝑑𝑓𝑦₀,𝑓𝑦₁+𝑘𝑑𝑓𝑥₀,𝑓𝑥₁+𝑙𝑑𝑓𝑦₀,𝑓𝑦₁+ℎ=𝑘𝑑^𝑥₁+𝑗+𝑙𝑑^𝑦₁+ℎ,𝑑^𝑧₂=𝑑𝑓𝑧₁,𝑓𝑧₂≤𝐻𝑇𝑧₀,𝑦₀,𝑥₀,𝑇𝑧₁,𝑦₁,𝑥₁+ℎ≤𝑗𝑑𝑓𝑧₀,𝑓𝑧₁+𝑘𝑑𝑓𝑦₀,𝑓𝑦₁+𝑙𝑑𝑓𝑥₀,𝑓𝑥₁+ℎ=𝑙𝑑^𝑥₁+𝑘𝑑^𝑦₁+𝑗𝑑^𝑧₁+ℎ,𝑑^𝑥₃=𝑑𝑓𝑥₂,𝑓𝑥₃≤𝐻𝑇𝑥₁,𝑦₁,𝑧₁,𝑇𝑥₂,𝑦₂,𝑧₂+ℎ²≤𝑗𝑑𝑓𝑥₁,𝑓𝑥₂+𝑘𝑑𝑓𝑦₁,𝑓𝑦₂+𝑙𝑑𝑓𝑧₁,𝑓𝑧₂+ℎ²=𝑗𝑑^𝑥₂+𝑘𝑑^𝑦₂+𝑙𝑑^𝑧₂+ℎ²≤𝑗𝑗𝑑^𝑥₁+𝑘𝑑^𝑦₁+𝑙𝑑^𝑧₁+ℎ+𝑘𝑘𝑑^𝑥₁+𝑗+𝑙𝑑^𝑦₁+ℎ+𝑙𝑙𝑑^𝑥₁+𝑘𝑑^𝑦₁+𝑗𝑑^𝑧₁+ℎ+ℎ²=𝑗²+𝑘²+𝑙²𝑑^𝑥₁+2𝑗𝑘+2𝑙𝑘𝑑^𝑦₁+2𝑗𝑙𝑑^𝑧₁+ℎ²+ℎ=𝑗𝑗+𝑘+𝑙₂+𝑘²+𝑙²𝑑^𝑥₁+2𝑗𝑘+2𝑙𝑘𝑑^𝑦₁+2𝑗𝑙𝑑^𝑧₁+2ℎ²,𝑑^𝑦₃=𝑑𝑓𝑦₂,𝑓𝑦₃≤𝐻𝑇𝑦₁,𝑥₁,𝑦₁,𝑇𝑦₂,𝑥₂,𝑦₂+ℎ²≤𝑗𝑑𝑓𝑦₁,𝑓𝑦₂+𝑘𝑑𝑓𝑥₁,𝑓𝑥₂+𝑙𝑑𝑓𝑦₁,𝑓𝑦₂+ℎ²=𝑘𝑑^𝑥₂+𝑗+𝑙𝑑^𝑦₂+ℎ²≤𝑘𝑗𝑑^𝑥₁+𝑘𝑑^𝑦₁+𝑙𝑑^𝑧₁+ℎ+𝑗+𝑙𝑘𝑑^𝑥₁+𝑗+𝑙𝑑^𝑦₁+ℎ+ℎ²=2𝑗𝑘+𝑙𝑘𝑑^𝑥₁+𝑗+𝑙²+𝑘²𝑑^𝑦₁+𝑘𝑙𝑑^𝑧₁+𝑗+𝑘+𝑙ℎ+ℎ²≤2𝑗𝑘+𝑙𝑘𝑑^𝑥₁+𝑗+𝑙²+𝑘²𝑑^𝑦₁+𝑘𝑙𝑑^𝑧₁+2ℎ²,𝑑^𝑧₃=𝑑𝑓𝑧₂,𝑓𝑧₃≤𝐻𝑇𝑧₁,𝑦₁,𝑥₁,𝑇𝑧₂,𝑦₂,𝑥₂+ℎ²≤𝑗𝑑𝑓𝑧₁,𝑓𝑧₂+𝑘𝑑𝑓𝑦₁,𝑓𝑦₂+𝑙𝑑𝑓𝑥₁,𝑓𝑥₂+ℎ²=𝑗𝑑^𝑧₂+𝑘𝑑^𝑦₂+𝑙𝑑^𝑥₂+ℎ²=𝑙𝑑^𝑥₂+𝑘𝑑^𝑦₂+𝑗𝑑^𝑧₂+ℎ²≤𝑙𝑗𝑑^𝑥₁+𝑘𝑑^𝑦₁+𝑙𝑑^𝑧₁+ℎ+𝑘𝑘𝑑^𝑥₁+𝑗+𝑙𝑑^𝑦₁+ℎ+𝑗𝑙𝑑^𝑥₁+𝑘𝑑^𝑦₁+𝑗𝑑^𝑧₁+ℎ+ℎ²=2𝑗𝑙+𝑘²𝑑^𝑥₁+2𝑗𝑘+𝑙𝑘𝑑^𝑦₁+𝑗²+𝑙²𝑑^𝑧₁+𝑗+𝑘+𝑙ℎ+ℎ²≤2𝑗𝑙+𝑘²𝑑^𝑥₁+2𝑗𝑘+𝑙𝑘𝑑^𝑦₁+𝑗²+𝑙²𝑑^𝑧₁+2ℎ². (i)(ii)(iii)(iv)(v)(vi) Let 𝐴=^{𝑗𝑘𝑙𝑘𝑗+𝑙0}_𝑙𝑘𝑗  denoted by  𝑎₁𝑏₁𝑐₁𝑑₁𝑒₁𝑓₁𝑔₁𝑏₁ℎ₁.
Clearly, 𝑎₁+𝑏₁+𝑐₁=𝑑₁+𝑒₁+𝑓₁=𝑔₁+𝑏₁+ℎ₁=(𝑗+𝑘+𝑙)≤ℎ<1.
Then, 𝐴²=𝑗²+𝑘²+𝑙²2𝑗𝑘+2𝑙𝑘2𝑗𝑙2𝑗𝑘+𝑙𝑘𝑗+𝑙²+𝑘²𝑘𝑙2𝑗𝑙+𝑘²2𝑗𝑘+2𝑙𝑘𝑗²+𝑙²denote𝐴²𝑎by²𝑏₂𝑐₂𝑑₂𝑒₂𝑓₂𝑔₂𝑏₂ℎ₂. (2.2) It is clear that 𝑎₂+𝑏₂+𝑐₂=𝑑₂+𝑒₂+𝑓₂=𝑔₂+𝑏₂+ℎ₂=(𝑗+𝑘+𝑙)²≤ℎ²<1.
Now we prove by induction that 𝐴^𝑛=𝑎_𝑛𝑏_𝑛𝑐_𝑛𝑑_𝑛𝑒_𝑛𝑓_𝑛𝑔_𝑛𝑏_𝑛ℎ_𝑛, (2.3) where 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛=𝑑_𝑛+𝑒_𝑛+𝑓_𝑛=𝑔_𝑛+𝑏_𝑛+ℎ_𝑛=𝑗+𝑘+𝑙^𝑛≤ℎ^𝑛<1. (2.4) Equation (2.3) is true for 𝑛=1,2.
Assume that (2.3) is true for some 𝑛. Consider 𝐴^𝑛+1=𝐴^𝑛𝑎⋅𝐴=^𝑛𝑏_𝑛𝑐_𝑛𝑑_𝑛𝑒_𝑛𝑓_𝑛𝑔_𝑛𝑏_𝑛ℎ_𝑛=𝑗𝑘𝑙𝑘𝑗+𝑙0𝑙𝑘𝑗𝑗𝑎_𝑛+𝑘𝑏_𝑛+𝑙𝑐_𝑛𝑘𝑎_𝑛+𝑗+𝑙𝑏_𝑛+𝑘𝑐_𝑛𝑙𝑎_𝑛+𝑗𝑐_𝑛𝑗𝑑_𝑛+𝑘𝑒_𝑛+𝑙𝑓_𝑛𝑘𝑑_𝑛+𝑗+𝑙𝑒_𝑛+𝑘𝑓_𝑛𝑙𝑑_𝑛+𝑗𝑓_𝑛𝑗𝑔_𝑛+𝑘𝑏_𝑛+𝑙ℎ_𝑛𝑘𝑔_𝑛+𝑗+𝑙𝑏_𝑛+𝑘ℎ_𝑛𝑙𝑔_𝑛+𝑗ℎ_𝑛. (2.5) We have 𝑎_𝑛+1+𝑏_𝑛+1+𝑐_𝑛+1=𝑗+𝑘+𝑙𝑎_𝑛+𝑏_𝑛+𝑐_𝑛=𝑗+𝑘+𝑙^𝑛+1≤ℎ^𝑛+1<1. (2.6) Similarly, we have 𝑑_𝑛+1+𝑒_𝑛+1+𝑓_𝑛+1=𝑔_𝑛+1+𝑏_𝑛+1+ℎ_𝑛+1=𝑗+𝑘+𝑙^𝑛+1≤ℎ^𝑛+1<1. (2.7) Thus (2.3) is true for all +^ve integer values of 𝑛.
Now from (i)–(vi) and continuing this process, we get 𝑑^𝑥_𝑛+1𝑑^𝑦_𝑛+1𝑑^𝑧_𝑛+1≤𝑎_𝑛𝑏_𝑛𝑐_𝑛𝑑_𝑛𝑒_𝑛𝑓_𝑛𝑔_𝑛𝑏_𝑛ℎ_𝑛𝑑^𝑥₁𝑑^𝑦₁𝑑^𝑧₁+𝑛ℎ^𝑛𝑛ℎ^𝑛𝑛ℎ^𝑛, (2.8) for all 𝑛=1,2,3,…. That is, 𝑑^𝑥_𝑛+1≤𝑎_𝑛𝑑^𝑥₁+𝑏_𝑛𝑑^𝑦₁+𝑐_𝑛𝑑^𝑧₁+𝑛ℎ^𝑛,𝑑^𝑦_𝑛+1≤𝑑_𝑛𝑑^𝑥₁+𝑒_𝑛𝑑^𝑦₁+𝑓_𝑛𝑑^𝑧₁+𝑛ℎ^𝑛,𝑑^𝑧_𝑛+1≤𝑔_𝑛𝑑^𝑥₁+𝑏_𝑛𝑑^𝑦₁+ℎ_𝑛𝑑^𝑧₁+𝑛ℎ^𝑛,∀𝑛=1,2,3,…. (2.9)For 𝑚>𝑛, we have𝑑𝑓𝑥_𝑚,𝑓𝑥_𝑛≤𝑑𝑓𝑥_𝑚,𝑓𝑥_𝑚−1+𝑑𝑓𝑥_𝑚−1,𝑓𝑥_𝑚−2+⋯+𝑑𝑓𝑥_𝑛+2,𝑓𝑥_𝑛+1+𝑑𝑓𝑥_𝑛+1,𝑓𝑥_𝑛=𝑑^𝑥_𝑚+𝑑^𝑥_𝑚−1+⋯+𝑑^𝑥_𝑛+2+𝑑^𝑥_𝑛+1≤𝑎_𝑚−1𝑑^𝑥₁+𝑏_𝑚−1𝑑^𝑦₁+𝑐_𝑚−1𝑑^𝑧₁+(𝑚−1)ℎ^𝑚−1+𝑎_𝑚−2𝑑^𝑥₁+𝑏_𝑚−2𝑑^𝑦₁+𝑐_𝑚−2𝑑^𝑧₁+(𝑚−2)ℎ^𝑚−2+⋯+𝑎_𝑛+1𝑑^𝑥₁+𝑏_𝑛+1𝑑^𝑦₁+𝑐_𝑛+1𝑑^𝑧₁+(𝑛+1)ℎ^𝑛+1+𝑎_𝑛𝑑^𝑥₁+𝑏_𝑛𝑑^𝑦₁+𝑐_𝑛𝑑^𝑧₁+𝑛ℎ^𝑛≤𝑎_𝑚−1+𝑎_𝑚−2+⋯+𝑎_𝑛+1+𝑎_𝑛𝑑^𝑥₁+𝑏_𝑚−1+𝑏_𝑚−2+⋯+𝑏_𝑛+1+𝑏_𝑛𝑑^𝑦₁+𝑐_𝑚−1+𝑐_𝑚−2+⋯+𝑐_𝑛+1+𝑐_𝑛𝑑^𝑧₁+(𝑚−1)ℎ^𝑚−1+(𝑚−2)ℎ^𝑚−2+⋯+(𝑛+1)ℎ^𝑛+1+𝑛ℎ^𝑛≤ℎ^𝑚−1+ℎ^𝑚−2+⋯+ℎ^𝑛+1+ℎ^𝑛𝑑^𝑥₁+𝑑^𝑦₁+𝑑^𝑧₁+^𝑚−1_𝑗=𝑛𝑗ℎ^𝑗≤ℎ^𝑛𝑑1−ℎ^𝑥1+𝑑^𝑦₁+𝑑^𝑧₁+^𝑚−1_𝑗=𝑛𝑗ℎ^𝑗→0as𝑛→∞,because0≤ℎ<1. (2.10) Hence {𝑓𝑥_𝑛} is a Cauchy. Similarly, we can show that {𝑓𝑦_𝑛} and {𝑓𝑧_𝑛} are Cauchy.
Suppose 𝑓(𝑋) is complete, the sequences {𝑓𝑥_𝑛},  {𝑓𝑦_𝑛}, and {𝑓𝑧_𝑛} are convergent to some 𝛼,𝛽,𝛾 in 𝑓(𝑋), respectively. There exist 𝑥,𝑦,𝑧∈𝑋 such that 𝛼=𝑓𝑥,  𝛽=𝑓𝑦, and 𝛾=𝑓𝑧.
Now, we have 𝑑𝑇𝑥,𝑦,𝑧,𝛼≤𝑑𝑇𝑥,𝑦,𝑧,𝑓𝑥_𝑛+1+𝑑𝑓𝑥_𝑛+1𝑇𝑥,𝛼≤𝐻𝑥,𝑦,𝑧,𝑇_𝑛,𝑦_𝑛,𝑧_𝑛+𝑑𝑓𝑥_𝑛+1,𝛼≤𝑗𝑑𝑓𝑥,𝑓𝑥_𝑛+𝑘𝑑𝑓𝑦,𝑓𝑦_𝑛+𝑙𝑑𝑓𝑧,𝑓𝑧_𝑛+𝑑𝑓𝑥_𝑛+1,𝛼=𝑗𝑑𝛼,𝑓𝑥_𝑛+𝑘𝑑𝛽,𝑓𝑦_𝑛+𝑙𝑑𝛾,𝑓𝑧_𝑛+𝑑𝑓𝑥_𝑛+1,𝛼. (2.11) Letting 𝑛→∞, we get 𝑑(𝑇(𝑥,𝑦,𝑧),𝛼)≤0 so that 𝛼∈𝑇(𝑥,𝑦,𝑧). That is, 𝑓𝑥∈𝑇(𝑥,𝑦,𝑧). Similarly, we can show that 𝑓𝑦∈𝑇(𝑦,𝑥,𝑦) and 𝑓𝑧∈𝑇(𝑧,𝑦,𝑥). Thus (𝑥,𝑦,𝑧) is a tripled coincidence point of 𝑇 and 𝑓. Suppose (2.1.3) (a) holds.
Since (𝑥,𝑦,𝑧) is a tripled coincidence point of 𝑇 and 𝑓, there exist 𝑢,𝑣,𝑤∈𝑋 such that lim_𝑛→∞𝑓^𝑛𝑥=𝑢, lim_𝑛→∞𝑓^𝑛𝑦=𝑣 and lim_𝑛→∞𝑓^𝑛𝑧=𝑤.
Since 𝑓 is continuous at 𝑢,𝑣 and 𝑤, we have 𝑓𝑢=𝑢,  𝑓𝑣=𝑣 and 𝑓𝑤=𝑤.
Since 𝑓𝑥∈𝑇(𝑥,𝑦,𝑧), we have 𝑓²𝑥∈𝑓(𝑇(𝑥,𝑦,𝑧))⊆𝑇(𝑓𝑥,𝑓𝑦,𝑓𝑧).
Since 𝑓𝑦∈𝑇(𝑦,𝑥,𝑦), we have 𝑓²𝑦∈𝑓(𝑇(𝑦,𝑥,𝑦))⊆𝑇(𝑓𝑦,𝑓𝑥,𝑓𝑦).
Since 𝑓𝑧∈𝑇(𝑧,𝑦,𝑥), we have 𝑓²𝑧∈𝑓(𝑇(𝑧,𝑦,𝑥))⊆𝑇(𝑓𝑧,𝑓𝑦,𝑓𝑥).
Then (𝑓𝑥,𝑓𝑦,𝑓𝑧) is tripled coincidence point of 𝑇 and 𝑓.
Similarly, we can show that (𝑓^𝑛𝑥,𝑓^𝑛𝑦,𝑓^𝑛𝑧) is a tripled coincidence point of 𝑇 and 𝑓.
Also it is clear that 𝑓^𝑛𝑥∈𝑇𝑓^𝑛−1𝑥,𝑓^𝑛−1𝑦,𝑓^𝑛−1𝑧,𝑓^𝑛𝑦∈𝑇𝑓^𝑛−1𝑦,𝑓^𝑛−1𝑥,𝑓^𝑛−1𝑦,𝑓^𝑛𝑧∈𝑇𝑓^𝑛−1𝑧,𝑓^𝑛−1𝑦,𝑓^𝑛−1𝑥. (2.12) From (2.1.1), we have 𝑑𝑓𝑢,𝑇(𝑢,𝑣,𝑤)≤𝑑𝑓𝑢,𝑓^𝑛𝑥+𝑑𝑓^𝑛𝑥,𝑇(𝑢,𝑣,𝑤)≤𝑑𝑓𝑢,𝑓^𝑛𝑥+𝐻𝑇𝑓^𝑛−1𝑥,𝑓^𝑛−1𝑦,𝑓^𝑛−1𝑧,𝑇(𝑢,𝑣,𝑤)≤𝑑𝑓𝑢,𝑓^𝑛𝑥+𝑗𝑑𝑓^𝑛𝑥,𝑓𝑢+𝑘𝑑𝑓^𝑛𝑦,𝑓𝑣+𝑙𝑑𝑓^𝑛𝑧,𝑓𝑤. (2.13) Letting 𝑛→∞, we obtain 𝑑𝑓𝑢,𝑇(𝑢,𝑣,𝑤)≤0, (2.14) which implies that 𝑓𝑢∈𝑇(𝑢,𝑣,𝑤). (2.15) Thus 𝑢=𝑓𝑢∈𝑇(𝑢,𝑣,𝑤). Similarly, we can show that 𝑣=𝑓𝑣∈𝑇(𝑣,𝑢,𝑣) and 𝑤=𝑓𝑤∈𝑇(𝑤,𝑣,𝑢). Thus (𝑢,𝑣,𝑤) is a tripled common fixed point of 𝑇 and 𝑓. Suppose (2.1.3) (b) holds.
Since (𝑥,𝑦,𝑧) is a tripled coincidence point of {𝑇,𝑓}, there exist 𝑢,𝑣,𝑤∈𝑋 such that lim_𝑛→∞𝑓^𝑛𝑢=𝑥, lim_𝑛→∞𝑓^𝑛𝑣=𝑦 and lim_𝑛→∞𝑓^𝑛𝑤=𝑧.
Since 𝑓 is continuous at 𝑥,𝑦 and 𝑧, we have 𝑓𝑥=𝑥,  𝑓𝑦=𝑦 and 𝑓𝑧=𝑧. Thus 𝑥=𝑓𝑥∈𝑇(𝑥,𝑦,𝑧), 𝑦=𝑓𝑦∈𝑇(𝑦,𝑥,𝑦) and 𝑧=𝑓𝑧∈𝑇(𝑧,𝑦,𝑥). Hence (𝑥,𝑦,𝑧) is a tripled common fixed point of {𝑇,𝑓}.
The following example illustrates Theorem 2.1.

Example 2.2. Let 𝑋=[0,1],𝑇∶𝑋×𝑋×𝑋→𝐶𝐵(𝑋) and 𝑓∶𝑋→𝑋 defined as 𝑇(𝑥,𝑦,𝑧)=[0,(1/8)sin𝑥+(1/4)sin𝑦+(1/3)sin𝑧] and 𝑓𝑥=(7/8)𝑥. Then 𝐻𝑇(=1𝑥,𝑦,𝑧,𝑇𝑢,𝑣,𝑤)81sin𝑥+41sin𝑦+3−1sin𝑧81sin𝑢+41sin𝑣+3≤1sin𝑤8||+1sin𝑥−sin𝑢4+1sin𝑦−sin𝑣3||≤1sin𝑧−sin𝑤8||+1𝑥−𝑢4+1𝑦−𝑣3||=1𝑧−𝑤7787𝑥−8𝑢+27787𝑦−8𝑣+872187𝑧−8𝑤=17𝑑+2𝑓𝑥,𝑓𝑢7𝑑+8𝑓𝑦,𝑓𝑣𝑑.21𝑓𝑧,𝑓𝑤 (2.16) It is clear that all conditions of Theorem 2.1 are satisfied and (0,0,0) is the tripled common fixed point of 𝑇 and 𝑓.
The following example shows that 𝑇 and 𝑓 have no tripled common fixed point if (2.1.3) (a) or (2.1.3) (b) is not satisfied.

Example 2.3. Let 𝑋=[0,4], 𝑇(𝑥,𝑦,𝑧)=[1.5,2] and 𝑓𝑥=2−(1/2)𝑥. Then (0,1/2,1) is a tripled coincidence point of 𝑇 and 𝑓. Clearly 𝑇 and 𝑓 have no tripled common fixed point.