Tripled Fixed Points of Multivalued Nonlinear Contraction Mappings in Partially Ordered Metric Spaces

Abstract

Berinde and Borcut (2011), introduced the concept of tripled fixed point for single mappings in partially ordered metric spaces. Samet and Vetro (2011) established some coupled fixed point theorems for multivalued nonlinear contraction mappings in partially ordered metric spaces. In this paper, we obtain existence of tripled fixed point of multivalued nonlinear contraction mappings in the framework of partially ordered metric spaces. Also, we give an example.

1. Introduction and Preliminaries

Let (𝑋,𝑑) be a metric space. Consistent with [1], we denote by 𝐶𝐵(𝑋) the family of all nonempty closed bounded and nonempty closed subsets of 𝑋. Let CL(𝑋)={𝐴⊂𝑋∶𝐴≠∅and𝐴=𝐴}, where 𝐴 denotes the closure of 𝐴 in 𝑋. For 𝐴,𝐵∈𝐶𝐵(𝑋), and 𝑥∈𝑋, set 𝐷(𝑥,𝐴)∶=inf{𝑑(𝑥,𝑎)∶𝑎∈𝐴}. We define a Hausdorff metric 𝐻 on 𝐶𝐵(𝑋) by𝐻(𝐴,𝐵)∶=maxsup_𝑎∈𝐴𝐷(𝑎,𝐵),sup_𝑏∈𝐵𝐷(𝑏,𝐴).(1.1) A point 𝑥∈𝐾 is called a fixed point of 𝑇 if 𝑥∈𝑇𝑥.

The study of fixed points for multivalued contractions and nonexpansive maps using the Hausdorff metric was initiated by Markin [2]. Later, an interesting and rich fixed point theory for such maps was developed. Several authors studied the problem of existence of fixed point of multivalued mappings satisfying different contractive conditions (see, e.g., [3–10]). The theory of multivalued maps has application in control theory, convex optimization, differential equations, and economics.

Existence of fixed points in ordered metric spaces has been initiated in 2004 by Ran and Reurings [11], further studied by Nieto and Rodríguez-López [12]. Samet and Vetro [13] introduced the notion of fixed point of 𝑁 order in case of single-valued mappings. In particular for 𝑁=3 (tripled case), we have the following definition.

Definition 1.1 (see, e.g., [13]). An element (𝑥,𝑦,𝑧)∈𝑋³ is called a tripled fixed point of a mapping 𝐹∶𝑋³→𝑋 if and only if 𝑥=𝐹(𝑥,𝑦,𝑧),𝑦=𝐹(𝑦,𝑧,𝑥),𝑧=𝐹(𝑧,𝑥,𝑦).(1.2)

Recently, Berinde and Borcut [14] established the existence of tripled fixed point of single-valued mappings in partially ordered metric spaces. The aim of this paper is to initiate the study of tripled fixed point of multivalued mappings in the framework of partially ordered metric spaces which in turn extend and strengthen various known results [5, 15].

2. Tripled Fixed Point Results for Multivalued Mappings

First, we introduce the following concepts.

Definition 2.1. An element (𝑥,𝑦,𝑧)∈𝑋³ is called a tripled fixed point of 𝐹∶𝑋³→CL(𝑋) if 𝑥∈𝐹(𝑥,𝑦,𝑧),𝑦∈𝐹(𝑦,𝑧,𝑥),𝑧∈𝐹(𝑧,𝑥,𝑦).(2.1)

Definition 2.2. A mapping 𝑓∶𝑋³→ℝ is called lower semicontinuous if, for any sequences {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛} in 𝑋 and (𝑥,𝑦,𝑧)∈𝑋³, one has lim_𝑛→∞𝑥_𝑛,𝑦_𝑛,𝑧_𝑛=(𝑥,𝑦,𝑧)⟹𝑓(𝑥,𝑦,𝑧)≤liminf_𝑛→∞𝑥_𝑛,𝑦_𝑛,𝑧_𝑛.(2.2)

Let (𝑋,𝑑) be a metric space endowed with a partial order ⪯ and 𝑇∶𝑋→𝑋. Define the set Ψ⊂𝑋³ by Ψ=(𝑥,𝑦,𝑧)∈𝑋³∶𝑇(𝑥)⪯𝑇(𝑦)⪯𝑇(𝑧).(2.3)

Definition 2.3. A mapping 𝐹∶𝑋³→𝑋 is said to have a Ψ-property if (𝑥,𝑦,𝑧)∈Ψ⟹𝐹(𝑥,𝑦,𝑧)×𝐹(𝑦,𝑧,𝑥)×𝐹(𝑧,𝑦,𝑥)⊂Ψ.(2.4)

We give some examples to illustrate Definition 2.3.

Example 2.4. Let 𝑋=ℝ be endowed with the usual order ≤, and 𝑇∶𝑋→𝑋. Define 𝐹∶𝑋³→CL(𝑋) by 𝐹(𝑥,𝑦,𝑧)={𝑥}∀𝑥,𝑦,𝑧∈ℝ.(2.5) Obviously, 𝐹 has the Ψ-property.

Example 2.5. Let 𝑋=ℝ⁺ be endowed with the usual order ≤, and let 𝑇∶𝑋→𝑋 be defined by 𝑇𝑥=exp(𝑥). Define 𝐹∶𝑋³→CL(𝑋) by 𝐹(𝑥,𝑦,𝑧)={𝑥+𝑧}∀𝑥,𝑦,𝑧∈ℝ⁺.(2.6) We have Ψ={(𝑥,𝑦,𝑧)∈𝑋³,exp(𝑥)≤exp(𝑦)≤exp(𝑧)}. Moreover, 𝐹 has the Ψ-property.

Now, we prove the following theorem.

Theorem 2.6. Let (𝑋,𝑑) be a complete metric space endowed with a partial order ⪯ and Ψ≠∅; that is, there exists (𝑥₀,𝑦₀,𝑧₀)∈Ψ. Suppose that 𝐹∶𝑋³→CL(𝑋) has a Ψ-property such that 𝑓∶𝑋³→[0,∞) given by 𝑓(𝑥,𝑦,𝑧)=𝐷(𝑥,𝐹(𝑥,𝑦,𝑧))+𝐷(𝑦,𝐹(𝑦,𝑧,𝑥))+𝐷(𝑧,𝐹(𝑧,𝑥,𝑦))∀𝑥,𝑦,𝑧∈𝑋(2.7) is lower semicontinuous and there exists a function 𝜙∶[0,∞)→[𝑀,1),0<𝑀<1, satisfying limsup_𝑟→𝑡+𝜙(𝑟)<1foreach𝑡∈[0,∞).(2.8) If for any (𝑥,𝑦,𝑧)∈Ψ there exist 𝑢∈𝐹(𝑥,𝑦,𝑧),𝑣∈𝐹(𝑦,𝑧,𝑥), and 𝑤∈𝐹(𝑧,𝑦,𝑥) with √𝜙(𝑓(𝑥,𝑦,𝑧))[𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)+𝑑(𝑧,𝑤)]≤𝑓(𝑥,𝑦,𝑧)(2.9) such that 𝑓(𝑢,𝑣,𝑤)≤𝜙(𝑓(𝑥,𝑦,𝑧))[𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)+𝑑(𝑧,𝑤)],(2.10) then 𝐹 has a tripled fixed point.

Proof. By our assumption, 𝜙(𝑓(𝑥,𝑦,𝑧))<1 for each (𝑥,𝑦,𝑧)∈𝑋³. Hence, for any (𝑥,𝑦,𝑧)∈𝑋³, there exist 𝑢∈𝐹(𝑥,𝑦,𝑧),𝑣∈𝐹(𝑦,𝑧,𝑥), and 𝑤∈𝐹(𝑧,𝑥,𝑦) satisfying √√𝜙(𝑓(𝑥,𝑦,𝑧))𝑑(𝑥,𝑢)≤𝐷(𝑥,𝐹(𝑥,𝑦,𝑧)),√𝜙(𝑓(𝑥,𝑦,𝑧))𝑑(𝑦,𝑣)≤𝐷(𝑦,𝐹(𝑦,𝑧,𝑥)),𝜙(𝑓(𝑥,𝑦,𝑧))𝑑(z,𝑤)≤𝐷(𝑧,𝐹(𝑧,𝑥,𝑦)).(2.11) Let (𝑥₀,𝑦₀,𝑧₀) be an arbitrary point in Ψ. By (2.9) and (2.10), we can choose 𝑥₁∈𝐹(𝑥₀,𝑦₀,𝑧₀), 𝑦₁∈𝐹(𝑦₀,𝑧₀,𝑥₀), and 𝑧₁∈𝐹(𝑧₀,𝑥₀,𝑦₀) satisfying 𝜙𝑓𝑥₀,𝑦₀,𝑧₀𝑑𝑥₀,𝑥₁+𝑑𝑦₀,𝑦₁+𝑑𝑧₀,𝑧₁≤𝑓𝑥₀,𝑦₀,𝑧₀(2.12) such that 𝑓𝑥₁,𝑦₁,𝑧₁≤𝜙𝑓𝑥₀,𝑦₀,𝑧₀𝑑𝑥₀,𝑥₁+𝑑𝑦₀,𝑦₁+𝑑𝑧₀,𝑧₁.(2.13) By (2.12) and (2.13), we obtain 𝑓𝑥₁,𝑦₁,𝑧₁≤≤𝜙𝑓𝑥₀,𝑦₀,𝑧₀𝑑𝑥₀,𝑥₁+𝑑𝑦₀,𝑦₁+𝑑𝑧₀,𝑧₁≤𝜙𝑓𝑥₀,𝑦₀,𝑧₀𝜙𝑓𝑥₀,𝑦₀,𝑧₀𝑑𝑥₀,𝑥₁+𝑑𝑦₀,𝑦₁+𝑑𝑧₀,𝑧₁𝜙𝑓𝑥₀,𝑦₀,𝑧₀𝑓𝑥₀,𝑦₀,𝑧₀.(2.14) Thus, 𝑓𝑥₁,𝑦₁,𝑧₁≤𝜙𝑓𝑥₀,𝑦₀,𝑧₀𝑓𝑥₀,𝑦₀,𝑧₀.(2.15) Since 𝐹 has a Ψ-property and (𝑥₀,𝑦₀,𝑧₀)∈Ψ, so we have 𝐹𝑥₀,𝑦₀,𝑧₀×𝐹𝑦₀,𝑧₀,𝑥₀×𝐹𝑧₀,𝑥₀,𝑦₀⊂Ψ(2.16) which implies that (𝑥₁,𝑦₁,𝑧₁)∈Ψ.
Again by (2.9) and (2.10), we can choose 𝑥₂∈𝐹(𝑥₁,𝑦₁,𝑧₁), 𝑦₂∈𝐹(𝑦₁,𝑧₁,𝑥₁), and 𝑧₂∈𝐹(𝑧₁,𝑥₁,𝑦₁) satisfying 𝜙𝑓𝑥₁,𝑦₁,𝑧₁𝑑𝑥₁,𝑥₂+𝑑𝑦₁,𝑦₂+𝑑𝑧₁,𝑧₂≤𝑓𝑥₁,𝑦₁,𝑧₁(2.17) such that 𝑓𝑥₂,𝑦₂,𝑧₂≤𝜙𝑓𝑥₁,𝑦₁,𝑧₁𝑑𝑥₁,𝑥₂+𝑑𝑦₁,𝑦₂+𝑑𝑧₁,𝑧₂.(2.18) Thus, we have 𝑓𝑥₂,𝑦₂,𝑧₂≤𝜙𝑓𝑥₁,𝑦₁,𝑧₁𝑓𝑥₁,𝑦₁,𝑧₁(2.19) and (𝑥₂,𝑦₂,𝑧₂)∈Ψ.
Continuing this process, we can choose sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} in 𝑋 such that for each 𝑛∈ℕ with (𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)∈Ψ.
𝑥_𝑛+1∈𝐹(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛),𝑦_𝑛+1∈𝐹(𝑦_𝑛,𝑧_𝑛,𝑥_𝑛) and 𝑧_𝑛+1∈𝐹(𝑧_𝑛,𝑥_𝑛,𝑦_𝑛) satisfying 𝜙𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑦_𝑛,𝑦_𝑛+1+𝑑𝑧_𝑛,𝑧_𝑛+1≤𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛(2.20) such that 𝑓𝑥_𝑛+1,𝑦_𝑛+1,𝑧_𝑛+1≤𝜙𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑦_𝑛,𝑦_𝑛+1+𝑑𝑧_𝑛,𝑧_𝑛+1.(2.21) Hence, we obtain 𝑓𝑥_𝑛+1,𝑦_𝑛+1,𝑧_𝑛+1≤𝜙𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛(2.22) with (𝑥_𝑛+1,𝑦_𝑛+1,𝑧_𝑛+1)∈Ψ. We claim that 𝑓(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)→0 as 𝑛→∞. If 𝑓(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)=0 for some 𝑛∈ℕ, then 𝐷(𝑥_𝑛,𝐹(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛))=0 implies that 𝑥_𝑛∈𝐹(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)=𝐹(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛). Analogously, 𝐷(𝑦_𝑛,𝐹(𝑦_𝑛,𝑧_𝑛,𝑥_𝑛))=0 implies that 𝑦_𝑛∈𝐹(𝑦_𝑛,𝑧_𝑛,𝑥_𝑛) and 𝐷(𝑧_𝑛,𝐹(𝑧_𝑛,𝑦_𝑛,𝑥_𝑛))=0 implies that 𝑧_𝑛∈𝐹(𝑧_𝑛,𝑦_𝑛,𝑥_𝑛). Hence, (𝑥_𝑛,𝑦_𝑛,𝑧_𝑛) becomes a tripled fixed point of 𝐹 for such 𝑛 and the result follows. Suppose that 𝑓(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)>0 for all 𝑛∈ℕ.
Using (2.22) and 𝜙(𝑡)<1, we conclude that {𝑓(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)} is a decreasing sequence of positive real numbers. Thus, there exists a 𝛿≥0 such that lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛=𝛿.(2.23) We will show that 𝛿=0. Assume on contrary that 𝛿>0. Letting 𝑛→∞ in (2.22) and by assumption (2.8), we obtain 𝛿≤limsup_{𝑓(𝑥𝑛,𝑦𝑛,𝑧𝑛)→𝛿+}𝜙𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛𝛿<𝛿,(2.24) a contradiction. Hence, lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛=0⁺.(2.25) Now, we prove that {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}⊂𝑋 are Cauchy sequences in (𝑋,𝑑). Assume that 𝛼=limsup_{𝑓(𝑥𝑛,𝑦𝑛,𝑧𝑛)→0+}𝜙𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛.(2.26) By (2.8), we conclude that 𝛼<1. Let 𝑘 be a real number such that 𝛼<𝑘<1. Thus, there exists 𝑛₀∈ℕ such that 𝜙𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛<𝑘foreach𝑛≥𝑛₀.(2.27) Using (2.22), we obtain 𝑓𝑥_𝑛+1,𝑦_𝑛+1,𝑧_𝑛+1<𝑘𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛foreach𝑛≥𝑛₀.(2.28) By mathematical induction, 𝑓𝑥_𝑛+1,𝑦_𝑛+1,𝑧_𝑛+1<𝑘^{𝑛+1−𝑛0}𝑓𝑥_𝑛0,𝑦_𝑛0,𝑧_𝑛0foreach𝑛≥𝑛₀.(2.29) Since 𝜙(𝑡)≥𝑀>0 for all 𝑡≥0 so (2.20), and (2.29) gives that 𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑦_𝑛,𝑦_𝑛+1+𝑑𝑧_𝑛,𝑧_𝑛+1<𝑘^𝑛−𝑛0√𝑀𝑓𝑥^𝑛0,𝑦_𝑛0,𝑧_𝑛0foreach𝑛≥𝑛₀(2.30) which yields that {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}⊂𝑋 are Cauchy sequences in 𝑋. Since 𝑋 is complete, there exists (𝑎,𝑏,𝑐)∈𝑋³ such that lim_𝑛→∞𝑥_𝑛=𝑎,lim_𝑛→∞𝑦_𝑛=𝑏,lim_𝑛→∞𝑧_𝑛=c.(2.31) Finally, we show that (𝑎,𝑏,𝑐)∈𝑋³ is tripled fixed point of 𝐹. As 𝑓 is lower semicontinuous, (2.25) implies that 0≤𝑓(𝑎,𝑏,𝑐)=𝐷(𝑎,𝐹(𝑎,𝑏,𝑐))+𝐷(𝑏,𝐹(𝑏,𝑐,𝑎))+𝐷(𝑐,𝐹(𝑐,𝑎,𝑏))≤liminf_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛=0.(2.32) Hence, 𝐷(𝑎,𝐹(𝑎,𝑏,𝑐))=𝐷(𝑏,𝐹(𝑏,𝑐,𝑎))=𝐷(𝑐,𝐹(𝑐,𝑎,𝑏))=0 gives that (𝑎,𝑏,𝑐) is a tripled fixed point of 𝐹.

Theorem 2.7. Let (𝑋,𝑑) be a complete metric space endowed with a partial order ⪯ and Ψ≠∅; that is, there exists (𝑥₀,𝑦₀,𝑧₀)∈Ψ. Suppose that 𝐹∶𝑋³→CL(𝑋) has a Ψ-property such that function 𝑓∶𝑋³→[0,∞) defined by 𝑓(𝑥,𝑦,𝑧)=𝐷(𝑥,𝐹(𝑥,𝑦,𝑧))+𝐷(𝑦,𝐹(𝑦,𝑧,𝑥))+𝐷(𝑧,𝐹(𝑧,𝑥,𝑦))∀𝑥,𝑦,𝑧∈𝑋,(2.33) is lower semicontinuous and there exists a function 𝜙∶[0,∞)→[𝑀,1),0<𝑀<1, satisfying limsup_𝑟→𝑡+𝜙(𝑟)<1foreacht∈[0,∞).(2.34) If for any (𝑥,𝑦,𝑧)∈Ψ there exist 𝑢∈𝐹(𝑥,𝑦,𝑧),𝑣∈𝐹(𝑦,𝑧,𝑥), and 𝑤∈𝐹(𝑧,𝑦,𝑥) satisfying √𝜙(Δ)Δ≤𝐷(𝑥,𝑓(𝑥,𝑦,𝑧))+𝐷(𝑦,f(𝑦,𝑧,𝑥))+𝐷(𝑧,𝑓(𝑧,𝑥,𝑦))(2.35) such that 𝐷(𝑢,𝑓(𝑢,𝑣,𝑤))+𝐷(𝑣,𝑓(𝑣,𝑤,𝑢))+𝐷(𝑤,𝑓(𝑤,𝑢,𝑣))≤𝜙(Δ)Δ,(2.36) where Δ=Δ((𝑥,𝑦,𝑧),(𝑢,𝑣,𝑤))=[𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)+𝑑(𝑧,𝑤)], then 𝐹 admits a tripled fixed point.

Proof. By replacing 𝜙(𝑓(𝑥,𝑦,𝑧)) with 𝜙([𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)+𝑑(𝑧,𝑤)]) in the proof of Theorem 2.6, we obtain sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}⊂𝑋 such that for each 𝑛∈ℕ with 𝑥_𝑛,𝑦_𝑛,𝑧_𝑛∈Ψ,𝑥_𝑛+1∈𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝑦_𝑛+1∈𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛,𝑧_𝑛+1∈𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛,(2.37) such that 𝐷𝜙𝑥Δ_𝑛Δ_𝑛≤𝐷𝑥_𝑛,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝐷𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛,(2.38)_𝑛+1≤,𝐹𝑥_𝑛+1,𝑦_𝑛+1,𝑧_𝑛+1+𝐷𝑦_𝑛+1,𝐹𝑦_𝑛+1,𝑧_𝑛+1,𝑥_𝑛+1+𝐷𝑧_𝑛+1,𝐹𝑧_𝑛+1,𝑥_𝑛+1,𝑦_𝑛+1𝜙Δ_𝑛𝐷𝑥_𝑛,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝐷𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛,(2.39) where Δ_𝑛=Δ_𝑛𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+1,𝑦_𝑛+1,𝑧_𝑛+1=𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑦_𝑛,𝑦_𝑛+1+𝑑𝑧_𝑛,𝑧_𝑛+1.(2.40) Again, following arguments similar to those given in the proof of Theorem 2.6, we deduce that 𝐷𝑥_𝑛,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝐷𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛(2.41) is a decreasing sequence of real numbers. Thus, there exists a 𝛿>0 such that lim_𝑛→∞𝐷𝑥_𝑛,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛=𝛿.(2.42) Now, we need to prove that {Δ_𝑛} admits a subsequence converging to certain 𝜂⁺ for some 𝜂≥0. Since 𝜙(𝑡)≥𝑀>0, using (2.38), we obtain Δ_𝑛≤1√𝑎𝐷𝑥^𝑛,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝐷𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛.(2.43) From (2.42) and (2.43), it is clear that the sequence 𝐷𝑥_𝑛,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝐷𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛(2.44) is bounded. Therefore, there is some 𝜃≥0 such that liminf_𝑛→+∞Δ_𝑛=𝜃.(2.45) From (2.37), we have 𝑥_𝑛+1∈𝐹(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛),𝑦_𝑛+1∈𝐹(𝑦_𝑛,𝑧_𝑛,𝑥_𝑛), and 𝑧_𝑛+1∈𝐹(𝑧_𝑛,𝑥_𝑛,𝑦_𝑛), Δ_𝑛≥𝐷𝑥_𝑛,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝐷𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛foreach𝑛≥0.(2.46) So comparing (2.42) to (2.45), we get that 𝜃≥𝛿. Now, we shall show that 𝜃=𝛿. If 𝛿=0, then, by (2.42) and (2.43), we get 𝜃=∶liminf_𝑛→+∞Δ_𝑛=0 and consequently 𝜃=𝛿=0. Suppose that 𝛿>0. Assume on contrary that 𝜃>𝛿. From (2.42) and (2.45), there is a positive integer 𝑛₀ such that 𝐷𝑥_𝑛,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝐷𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛<𝛿+𝜃−𝛿𝛿−𝜃−𝛿4,(2.47)4<Δ^𝑛,(2.48) for all 𝑛≥𝑛₀. We combine (2.38), (2.47) to (2.48) to obtain 𝜙Δ_𝑛𝛿−𝜃−4𝛿≤𝜙Δ_𝑛Δ_𝑛<≤𝐷𝛿+𝑥𝜃_𝑛,−𝐹𝛿𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝐷𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛4,(2.49) for all 𝑛≥𝑛₀. It follows that 𝜙Δ_𝑛≤𝜃3𝜃++3𝛿𝛿∀𝑛≥𝑛⁰.(2.50) By (2.39) and (2.50), we have 𝐷𝑥≤_𝑛+1ℎ,𝐹𝐷𝑥𝑥_𝑛+1𝑛,𝑦_𝑛+1,𝑧_𝑛+1+𝐷𝑦_𝑛+1,𝐹𝑦_𝑛+1,𝑧_𝑛+1,𝑥_𝑛+1+𝐷𝑧_𝑛+1,𝐹𝑧_𝑛+1,𝑥_𝑛+1,𝑦_𝑛+1,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝐷𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛∀𝑛≥𝑛₀,(2.51) where ℎ=(𝜃+3𝛿)/(3𝜃+𝛿). Since 𝜃>𝛿>0, therefore ℎ<1, so proceeding by induction and combining the above inequalities, it follows that 𝛿≤+𝐷𝐷𝑥𝑧_𝑛0+𝑘0,_𝑛0+𝑘0𝐹𝑥_𝑛0+𝑘0,𝑦_𝑛0+𝑘0,𝑧_𝑛0+𝑘0+𝐷𝑦_𝑛0+𝑘0,𝐹𝑦_𝑛0+𝑘0,𝑧_𝑛0+𝑘0,𝑥_𝑛0+𝑘0≤ℎ_𝑘0,𝐹𝑧_𝑛0+𝑘0,𝑥_𝑛0+𝑘0,𝑦_𝑛0+𝑘0𝐷𝑥_𝑛0,𝐹𝑥_𝑛0,𝑦_𝑛0,𝑧_𝑛0+𝐷𝑦_𝑛0,𝐹𝑦_𝑛0,𝑧_𝑛0,𝑥_𝑛0+𝐷𝑧_𝑛0,𝐹𝑧_𝑛0,𝑥_𝑛0,𝑦_𝑛0<𝛿,(2.52) for a positive integer 𝑘₀. Then, we obtain a contradiction, so we must have 𝜃=𝛿.
Now, we shall show that 𝜃=0. Since 𝜃=𝛿≤𝐷𝑥_𝑛,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝐷𝑦_𝑛,𝐹𝑦_𝑛,𝑧_𝑛,𝑥_𝑛+𝐷𝑧_𝑛,𝐹𝑧_𝑛,𝑥_𝑛,𝑦_𝑛≤Δ_𝑛,(2.53) then we rewrite (2.45) as liminf_𝑛→+∞Δ_𝑛=𝜃⁺.(2.54) Hence, there exists a subsequence {Δ_𝑛𝑘} of {Δ_𝑛} such that liminf_𝑘→+∞Δ_𝑛𝑘=𝜃⁺.
By (2.34), we have limsup_{Δ𝑛𝑘→𝜃+}𝜙Δ_𝑛𝑘<1.(2.55) From (2.39), we obtain 𝐷𝑥+_𝑛𝑘+1,𝐷𝐹𝑧𝑥_{𝑛𝑘+1𝑛𝑘+1},𝑦_𝑛𝑘+1,𝑧_𝑛𝑘+1+𝐷𝑦_𝑛𝑘+1,𝐹𝑦_𝑛𝑘+1,𝑧_𝑛𝑘+1,𝑥_𝑛𝑘+1≤,𝐹𝑧_𝑛𝑘+1,𝑥_𝑛𝑘+1,𝑦_𝑛𝑘+1𝜙Δ_𝑛𝑘𝐷𝑥_𝑛𝑘,𝐹𝑥_𝑛𝑘,𝑦_𝑛𝑘,𝑧_𝑛𝑘+𝐷𝑦_𝑛𝑘,𝑓𝑦_𝑛𝑘,𝑧_𝑛𝑘,𝑥_𝑛𝑘+𝐷𝑧_𝑛𝑘,𝐹𝑧_𝑛𝑘,𝑥_𝑛𝑘,𝑦_𝑛𝑘.(2.56) Taking the limit as 𝑘→∞ and using (2.42), we have 𝛿=limsu+p_𝑘→+∞𝐷𝐷𝑧𝑥_{𝑛𝑘+1𝑛𝑘+1},𝐹𝑥_𝑛𝑘+1,𝑦_𝑛𝑘+1,𝑧_𝑛𝑘+1+𝐷𝑦_𝑛𝑘+1,𝐹𝑦_𝑛𝑘+1,𝑧_𝑛𝑘+1,𝑥_𝑛𝑘+1≤limsup_𝑘→+∞,𝑓𝑧_𝑛𝑘+1,𝑥_𝑛𝑘+1,𝑦_𝑛𝑘+1limsup_𝑘→+∞𝜙Δ_𝑛𝑘=l𝐷im𝑥su_𝑛𝑘p_{Δ𝑛𝑘→𝜃+},𝐹𝑥_𝑛𝑘,𝑦_𝑛𝑘,𝑧_𝑛𝑘+𝐷𝑦_𝑛𝑘,𝐹𝑦_𝑛𝑘,𝑧_𝑛𝑘,𝑥_𝑛𝑘+𝐷𝑧_𝑛𝑘,𝐹𝑧_𝑛𝑘,𝑥_𝑛𝑘,𝑦_𝑛𝑘𝜙Δ_𝑛𝑘𝛿.(2.57) Assume that 𝛿>0, then from (2.57) we get that 1≤limsup_{Δ𝑛𝑘→𝜃+}𝜙Δ_𝑛𝑘,(2.58) a contradiction with respect to (2.55), so 𝛿=0. Now, from (2.39), and (2.42) we have 𝛼=limsup_Δ𝑛→0+𝜙Δ_𝑛<1.(2.59) The rest of the proof is similar to the proof of Theorem 2.6, so it is omitted.

We improved and corrected the example of Samet and Vetro [13].

Example 2.8. Let 𝑋=[0,1], and let 𝑑∶𝑋×𝑋→[0,∞) be the usual metric. Suppose that 𝑇(𝑥)=𝑀 for all 𝑥∈[0,1] where 𝑀 is a constant in [0,1], and 𝐹∶𝑋³→CL(𝑋) is defined, for all 𝑦,𝑧∈𝑋 as follow: 𝐹(𝑥,𝑦,𝑧)=⎧⎪⎪⎨⎪⎪⎩𝑥²145if𝑥∈0,3152∪1325,1,96,15if𝑥=1325.(2.60) Obviously, 𝐹 has the Ψ-property. Set 𝜙∶[0,∞)→[0,1): 𝜙(𝑡)=⎧⎪⎨⎪⎩111120𝑡if𝑡∈0,23,16if𝑡∈23,∞.(2.61)
Consider the function 𝑓(𝑥,𝑦,𝑧)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑥+𝑦+𝑧−14𝑥²+𝑦²+𝑧²if𝑥,𝑦,𝑧∈0,15𝑥+𝑦−132∪1325,1,4𝑥²+𝑦²+43𝑥−1160if𝑥,𝑦∈0,1352∪1325,1with𝑧=1352,4𝑥²[phone]if𝑥,𝑦∈0,1352∪1325,1with𝑦=𝑧=1352,160if𝑥=𝑦=𝑧=1325(2.62) which is lower semicontinuous. Thus, for all 𝑥,𝑦,𝑧∈𝑋 with 𝑥,𝑦,𝑧≠15/32, there exist 𝑢∈𝐹(𝑥,𝑦,𝑧)={𝑥²/4},𝑣∈𝐹(𝑦,𝑧,𝑥)={𝑦²/4}, and 𝑤∈𝐹(𝑧,𝑦,𝑥)={𝑧²/4} such that 𝐷(𝑢,=𝐹𝑥(𝑢,𝑣,𝑤))+𝐷(𝑣,𝐹(𝑣,𝑤,𝑢))+𝐷(𝑤,𝐹(𝑤,𝑣,𝑢))₂=14−6𝑥4⁴+𝑦4²−6𝑦4⁴+𝑧4²−6𝑧4⁴≤41𝑥+𝑥4²𝑥−𝑥4²+𝑦+𝑦4²𝑦−𝑦4²+𝑧+𝑧4²𝑧−𝑧4²≤14𝑥+𝑥4²𝑑(𝑥,𝑢)+𝑦+𝑦4²𝑑(𝑦,𝑣)+𝑧+𝑧4²𝑑(𝑧,𝑤)<410max𝑥+𝑥4²,𝑦+𝑦4²,𝑧+𝑧4²[𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)+𝑑(𝑧,𝑤)]≤𝜙12(𝑑m(a𝑥x,𝑢)+𝑥𝑑−(𝑦,𝑥4𝑣²)+,𝑑(𝑦𝑧−,𝑤𝑦4)²)[𝑑,(𝑥,𝑧𝑢−)+𝑧4²𝑑(𝑦,𝑣[)𝑑+(𝑥𝑑,𝑢(𝑧),+𝑤𝑑)](.𝑦,𝑣)+𝑑(𝑧,𝑤)](2.63) Hence, for all 𝑥,𝑦,𝑧∈𝑋 with 𝑥,𝑦,𝑧≠15/32, the conditions (2.35) and (2.36) are satisfied. Analogously, one can easily show that conditions (2.35) and (2.36) are satisfied for the cases (𝑥,𝑦∈[0,15/32)∪(15/32,1] with 𝑧=15/32) and (𝑥∈[0,15/32)∪(15/32,1] and 𝑦=𝑧=15/32). For the last case, that is, 𝑥=𝑦=𝑧=15/32, we assume that 𝑢=𝑣=𝑤=15/96, so it follows that [𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)+𝑑(𝑧,𝑤)]=1516>23.(2.64) As a consequence, we get that √𝜙(=𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)+𝑑(𝑧,𝑤))[𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)+𝑑(𝑧,𝑤)]10𝐷(𝑢,𝐹(𝑢,1𝑣6,11𝑤65))<+11𝐷6209(𝑣=,𝐹𝐷(𝑣(𝑥,𝑤,𝐹,𝑢()𝑥),+𝑦,𝐷𝑧)()𝑤+,𝐹𝐷((𝑤𝑦,,𝐹𝑣,(𝑢𝑦),)𝑧=,𝑥3))||||1+5𝐷(𝑧,𝐹(𝑧,𝑥,𝑦)),96−141956²<10||||161156=𝜙(𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)+𝑑(𝑧,𝑤))[𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)+𝑑(𝑧,𝑤)].(2.65) Thus, we conclude that all the conditions of Theorem 2.7 are satisfied and 𝐹 admits a tripled fixed point 𝑎=(0,0,0).

Remark 2.9. If we replace the function 𝜙 with the following, we get the results again: 𝜙(𝑡)=⎧⎪⎨⎪⎩7172𝑡if𝑡∈0,34,16if𝑡∈34,∞.(2.66)