Convergence Theorem for Multivalued Almost Type Contractions via

Generalized Simulation Functions

Abstract: -The purpose of this work is to introduce the concepts of generalized multivalued almost type Z-

contraction along with C-class functions and generalized Suzuki multivalued almost type Z-contraction along

with C-class functions for a pair of mappings, as well as to show that common fixed point theorems for such

mappings in complete metric spaces. The results of this study generalize and expand on some established fixed

point findings in the literature. We derive several corollaries from our core results and offer examples to support

our results.

Key-Words: Multivalued mapping, C-class functions, Z-contraction, Almost type, Suzuki type

Received: May 19, 2022. Revised: September 9, 2022. Accepted: October 12, 2022. Published: November 4, 2022.

1 Introduction

The origins of fixed point theory can be traced back

to the last quarter of the nineteenth century, when

repeated approximations were used to establish the

existence and uniqueness of solutions to differential

equations. It is worth noting that the Banach contrac-

tion principle, which was developed by Banach [1].

This solution has been expanded for single and mul-

tivalued cases on a metric space in a variety of ways.

Nadler [2] developed the concept of multivalued con-

traction mapping in 1969 and established that it had

a fixed point in the entire metric space. Several fixed

point theorems were then established by various writ-

ers as a generalization of Nadler’s theory (see [3], [4],

[5], [6], [7], [8]).

Let (Υ, d)be a metric space and CB(Υ) denote the

collection of all nonempty closed and bounded subset

of Υ. For ω∈Υand A, B ∈ CB(Υ), we have

d(A, B) = inf{d(a, b) : ρ∈Aand ρ∈B},

D(ω, A) = inf {d(ω, ρ) : ρ∈A}

and

H(A, B) = max sup

ω∈A

D(ω, B),sup

ρ∈B

D(ρ, A).

The function His a Hausdorff metric induced by the

metric d. It is a metric on CB(Υ).

Let (Υ, d)be a complete metric space and Ω :

Υ→ CB(Υ) be a contraction mapping such that

H(Ωω, Ωρ)≤δd(ω, ρ)

for all ω, ρ ∈Υand for some δ∈[0,1)]. It’s a typical

Banach contraction, [1].

Berinde [9] extended the Zamfirescu fixed point

theorem [10] to almost contractions, a class of con-

tractive type mappings, for δ∈[0,1) and L ≥ 0such

that

d(Ωω, Ωρ)≤δd(ω, ρ)+Ld(ω, Ωρ)for all ω, ρ ∈Υ.

(1)

Khojasteh et al. [11] defined Z-contraction with

respect to ζ, which generalizes the Banach contrac-

tion principle and integrates various kinds of contrac-

tion. Olgun et al. [12] achieved fixed point solutions

for generalized Z-contractions.

Later, Chandok et al. [13] expanded the conclu-

sions of [11], [12] by combining the concept of sim-

ulation functions with C-class functions and proving

the existence and uniqueness of point of coincidence.

Motivated and inspired by almost contractions

in (1), Definition 2.3, Definition 2.4 and work of

[13], we introduce the notion of extended multi-

valued almost type Z-contraction with C-class func-

tions and extended multivalued Suzuki almost type

Z-contraction with C-class functions for metric space

mapping pair.

2 Preliminaries

Definition 2.1. [11] Let ζ: [0,∞)×[0,∞)→Rbe

a mapping. Then ζis called a simulation function if

it satisfies the following conditions:

(ζ1): ζ(0,0) = 0;

(ζ2): ζ(v, u)< u −vfor all u, v > 0;

(ζ2): if {vn},{un}are sequence in (0,∞)such

that lim

n→∞ vn=lim

n→∞ un>0,then

lim sup

n→∞

ζ(vn, un)<0.

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1CHUANPIT MUNGKALA, 2PHEERACHATE BUNPATCHARACHAROEN

Department of Mathematics, Faculty of Science and Technology

Rambhai Barni Rajabhat University, Chanthaburi 22000, THAILAND

1chuanpit.t@rbru.ac.th, 2pheerachate.b@rbru.ac.th

Argoubi et al. [14] applying innovative the sim-

ulation function definition by omitting the condition

(ζ1).

Definition 2.2. [14] A simulation function is a map-

ping ζ: [0,∞)×[0,∞)→Rsatisfying the following

conditions:

(ζ2): ζ(v, u)< u −v,u, v > 0;

(ζ′

3): if {vn},{un}are sequence in (0,∞)such that

lim

n→∞ vn=lim

n→∞ un>0,and vn< unthen

lim supn→∞ ζ(vn, un)<0.

Definition 2.3. [12] Let (Υ, d)be a metric space, Ω :

Υ→Υa mapping and ζ∈Z. Then Ωis called a

generalized Z-contraction with respect to ζif

ζ(d(Ωω, Ωρ),Θ(ω, ρ)) ≥0for all ω, ρ ∈Υ,

where

Θ(ω, ρ) = max d(ω, ρ), d(ω, Ωω), d(ρ, Ωρ),

d(ω, Ωρ) + d(ρ, Ωω)

2.

Padcharoen et al. [15] on the other hand de-

fined generalized Suzuki type Z-contraction on met-

ric spaces as follows.

Definition 2.4. [15] Let (Υ, d)be a metric space, Ω :

Υ→Υa mapping and ζ∈Z. Then Ωis called a

generalized Suzuki type Z-contraction with respect to

ζif

2(d(ω, Ωω)< d(ω, ρ)⇒ζ(d(Ωω, Ωρ),Θ(ω, ρ)) ≥0

for all distinct ω, ρ ∈Υ, where

Θ(ω, ρ) = max d(ω, ρ), d(ω, Ωω), d(ρ, Ωρ),

d(ω, Ωρ) + d(ρ, Ωω)

2.

Definition 2.5. [16] A mapping G: [0,∞)2→Rhas

the property CG, if there exists CG≥0such that

(G1): G(u, v)>CGimplies u > v;

(G2): G(u, v)≤ CGfor all v∈[0,∞).

Definition 2.6. [17] ACGsimulation function is a

mapping G: [0,∞)×[0,∞)→Rsatisfying the fol-

lowing conditions:

(i): ζ(v, u)<G(u, v)for all v, u > 0,where G:

[0,∞)[0,∞)×[0,∞)→Ris a C-class function;

(ii): if {vn},{un}are sequence in (0,∞)such that

lim

n→∞ vn=lim

n→∞ un>0, and vn< un,then

lim sup

n→∞

ζ(vn, un)<CG.

Lemma 2.7. [18] Let (Υ, d)be a metric space and let

{ωn}be a sequence in Υsuch that

lim

n→∞ d(ω2n, ω2n+1) = 0.

If {xn}is not a Cauchy sequence in Υ, then there ex-

ists ϵ > 0and two sequence ωm(k)and ωn(k)of pos-

itive integers such that ωn(k)> ωm(k)> k and the

following sequence tend to ϵwhen k→ ∞:

d(ωm(k), ωn(k)), d(ωm(k), ωn(k)+1), d(ωm(k)−1, ωn(k)),

d(ωm(k)−1, ωn(k)+1), d(ωm(k)+1, ωn(k)+1).

For a non-empty set Υ, let P(Υ) denotes the power

set of Υ. If (Υ, d)is a metric space, then let

N(Υ) = P(Υ) − {∅},

CB(Υ) = {A∈N(Υ) : Ais closed and bounded},

K(Υ) = {A∈N(Υ) : Ais compact}.

Definition 2.8. [19] Let Υbe a non empty set, Ω :

Υ→N(Υ) and α: Υ ×Υ→[0,∞)be two map-

pings. Then Ωis said to be an α-admissible whenever

for each ω∈Υand ρ∈Ωω,

α(ω, ρ)≥1⇒α(ρ, η)≥1for all η∈Ωρ.

Definition 2.9. [20] Let Υbe a nonempty set, Ω :

Υ→N(Υ) and α: Υ ×Υ→[0,∞)be two map-

pings. Then Ωis said to be triangular α-admissible if

Ωis α-admissible and

α(ω, ρ)≥1and α(ρ, η)≥1

⇒α(ω, η)≥1for all η∈Ωρ.

Lemma 2.10. [20] Let Ω : Υ →N(Υ) be a triangu-

lar α-admissible mapping. Assume that there exists

ω0∈Υand ω1∈Ωω0such that α(ω0, ω1)≥1.

Then for a sequence {ωn}such that ωn+1 ∈Ωωn,we

have α(ωn, ωm)≥1for all m, n ∈Nwith n < m.

Definition 2.11. [21] Let (Υ, d)be a metric space, α:

Υ×Υ→[0,∞)and Ω : Υ →K(Υ) mappings. Then

Ωis said to be an α-continuous multivalued mapping

on (K(Υ),H),if for all sequences {ωn}with ωn→

ω∈Υas n→ ∞,and α(ωn, ωn+1)≥1for all

n∈N,we have Ωωn→Ωωas n→ ∞,that is,

lim

n→∞ d(ωn, ω) = 0 and α(ωn, ωn+1)≥1

for all n∈N⇒lim

n→∞ H(Ωωn,Ωω) = 0.

Definition 2.12. [22] Let (Υ, d)be a metric space,

α: Υ ×Υ→[0,∞).The metric space (Υ, d)is said

to be α-complete if and only if every Cauchy sequence

{ωn}with α(ωn, ωn+1)≥1for all n∈Nconverges

in Υ.

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3 Main Result

Now we state our main results.

Definition 3.1. Let (Υ, d)be a metric space and

Ω,Λ : Υ →K(Υ) and α: Υ ×Υ→[0,1) be a

function. We say Ωis Z(α,G)multivalued almost type

contraction with respect to ζsuch that

ζ(α(ω, ρ)H(Ωω, Λρ), β(₩)₩)≥ CG(2)

for all ω, ρ ∈Υwith ω=ρand L ≥ 0,where

₩= Θ(ω, ρ) + LΨ(ω, ρ)

with

Θ(ω, ρ) = max d(ω, ρ), D(ω, Ωω), D(ρ, Λρ),

D(ω, Λρ) + D(ρ, Ωω)

2

and

Ψ(ω, ρ) = min D(ω, Ωω), D(ρ, Λρ),

D(ω, Λρ), D(ρ, Ωω).

Definition 3.2. Let (Υ, d)be a metric space and

Ω,Λ : Υ →K(Υ) and α: Υ ×Υ→[0,1) be a

function. We say Ωis Z(α,G)Suzuki multivalued al-

most type contraction with respect to ζif

2min{D(ω, Ωω), D(ρ, Λρ)}< d(ω, ρ)

⇒ζ(₤, β(₩)₩)≥ CG

(3)

for all ω, ρ ∈Υwith Ωω= Λρand L ≥ 0,where

₤=α(ω, ρ)H(Ωω, Λρ),

₩= Θ(ω, ρ) + LΨ(ω, ρ)

with

Θ(ω, ρ) = max d(ω, ρ), D(ω, Ωω), D(ρ, Λρ),

D(ω, Λρ) + D(ρ, Ωω)

2

and

Ψ(ω, ρ) = min D(ω, Ωω), D(ρ, Λρ),

D(ω, Λρ), D(ρ, Ωω).

Theorem 3.3. Let (Υ, d)be a metric space and Ω,Λ :

Υ→K(Υ) be Z(α,G)Suzuki almost type multivalued

contraction satisfying:

(i) (Υ, d)is an α-complete metric space;

(ii) Ω,Λare triangular α-admissible;

(iii) Ω,Λare an α-continuous multivalued mapping.

Then Ωand Λhave a common fixed point.

Proof. Let ω0∈Υ. Choose ω1∈Ωω0. Then by the

definition of Hausdorff metric there exists ω2∈Λω1

such that

0< d(ω1, ω2)

=D(ω1,Λω1)

≤α(ω0, ω1)H(Ωω0,Λω1).

(4)

Assume that D(ω0,Ωω0)>0and D(ω1,Λω1)>0

then

2min {D(ω0,Ωω0), D(ω1,Λω1)}< d(ω0, ω1).

Therefore from (3), we have

2min{D(ω0,Ωω0), D(ω1,Λω1)}< d(ω0, ω1)

⇒ζ(₤0, β(₩0)₩0)≥ CG,

where ₤0=α(ω0, ω1)H(Ωω0,Λω1)and ₩0=

Θ(ω0, ω1) + LΨ(ω0, ω1).

Consider

CG≤ζ(₤0, β(₩0)₩0)

<G(β(₩0)₩0,₤0).(5)

Consequently, we get

d(ω1, ω2)≤₤0< β(₩0)₩0,(6)

where

Θ(ω0, ω1)

=max d(ω, ω1), D(ω0,Ωω0), D(ω1,Λω1),

D(ω0,Λω1) + D(ω1,Ωω0)

2

≤max d(ω0, ω1), d(ω0, ω1), d(ω1, ω2),

d(ω0, ω2) + d(ω1, ω1)

2

=max d(ω0, ω1), d(ω1, ω2),d(ω0, ω2)

2.

Because

d(ω0, ω2)

2≤d(ω0, ω1) + d(ω2, ω1)

≤max {d(ω0, ω1), d(ω1, ω2)}.

Thus,

Θ(ω0, ω1)≤max {d(ω0, ω1), d(ω1, ω2)}

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and

Ψ(ω0, ω1)

=min{D(ω0,Ωω0), D(ω1,Λω1),

D(ω0,Λω1), D(ω1,Ωω0)}

=min{d(ω0, ω1), d(ω1, ω2), d(ω0, ω2),

d(ω1, ω1)}

= 0.

If max {d(ω0, ω1), d(ω1, ω2)}=d(ω1, ω2)and

Ψ(ω0, ω1) = 0,then (6) becomes

d(ω1, ω2)≤α(ω0, ω1)H(Ωω0,Λω1)

< β(d(ω1, ω2))d(ω1, ω2),(7)

obtain that

d(ω1, ω2)≤α(ω0, ω1)H(Ωω0,Λω1)< d(ω1, ω2),

which is a contradiction. Thus we conclude that

max {d(ω0, ω1), d(ω1, ω2)}=d(ω0, ω1).

By (6) we get

d(ω1, ω2)< d(ω0, ω1).

Similarly, for ω2∈Λω1and ω3∈Ωω2we have

d(ω2, ω3)≤α(ω1, ω2)H(Λω1,Ωω2)< d(ω1, ω2).

This implies

d(ω2, ω3)< d(ω1, ω2).

By continuing in this manner, we construct a sequence

{ωn}in Υsuch that ω2n+1 ∈Ωω2nand ω2n+2 ∈

Λω2n+1, n = 0,1,2, ... such that

0< d(ω2+1, ω2n+2)

=D(ω2n+1,Λω2n+1)

≤α(ω2n, ω2n+1)H(Ωω2n,Λω2n+1)

and

2min {D(ω2n,Ωω2n), D(ω2n+1,Λω2n+1)}

< d(ω2n, ω2n+1).

Hence from (3), we have

2min {D(ω2n,Ωω2n), D(ω2n+1,Λω2n+1)}

< d(ω2n, ω2n+1)⇒ζ(₤2n, β(₩2n)₩2n)≥ CG,

where ₤2n=α(ω2n, ω2n+1)H(Ωω2n,Λω2n+1)and

₩2n= Θ(ω2n, ω2n+1) + LΨ(ω2n, ω2n+1).

Consider

CG≤ζ(₤2n, β(₩2n)₩2n)

<G(β(₩2n)₩2n,₤2n).(8)

Consequently, we get

d(ω2n+1, ω2n+2)≤₤2n< β(₩2n)₩2n,(9)

where

Θ(ω2n, ω2n+1)

=max d(ω2n, ω2n+1), D(ω2n,Ωω2n),

D(ω2n+1,Λω2n+1),

D(ω2n,Λω2n+1) + D(ω2n+1,Ωω2n)

2

≤max d(ω2n, ω2n+1), d(ω2n, ω2n+1),

d(ω2n+1, ω2n+2),

d(ω2n, ω2n+2) + d(ω2n+1, ω2n+1)

2

=max d(ω2n, ω2n+1), d(ω2n+1, ω2n+2),

d(ω2n, ω2n+2)

2.

Because

d(ω2n, ω2n+2)

≤d(ω2n, ω2n+1) + d(ω2n+2, ω2n+1)

≤max {d(ω2n, ω2n+1), d(ω2n+1, ω2n+2)}.

Thus,

Θ(ω2n, ω2n+1)

≤max {d(ω2n, ω2n+1), d(ω2n+1, ω2n+2)}

and

Ψ(ω2n, ω2n+1)

=min D(ω2n,Ωω2n), D(ω2n+1,Λω2n+1),

D(ω2n,Λω2n+1), D(ω2n+1,Ωω2n)

=min d(ω2n, ω2n+1), d(ω2n+1, ω2n+2),

d(ω2n, ω2n+2), d(ω2n+1, ω2n+1)

= 0.

If max {d(ω2n, ω2n+1), d(ω2n+1, ω2n+2)}=

d(ω2n+1, ω2n+2)and Ψ(ω2n, ω2n+1)=0,then

(9) becomes

d(ω2n+1, ω2n+2)

≤α(ω2n, ω2n+1)H(Ωω2n,Λω2n+1)

< β(d(ω2n+1, ω2n+2))d(ω2n+1, ω2n+2),

(10)

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obtain that

d(ω2n+1, ω2n+2)≤α(ω2n, ω2n+1)H(Ωω2n,Λω2n+1)

< d(ω2n+1, ω2n+2),

which is a contradiction. Thus we conclude that

max {d(ω2n, ω2n+1), d(ω2n+1, ω2n+2)}

=d(ω2n, ω2n+1).

By (10) we get

d(ω2n+1, ω2n+2)< d(ω2n, ω2n+1).

Then from (10) we have

d(ω2n+2, ω2n+3)

≤α(ω2n+1, ω2n+2)H(Ωω2n+1,Λω2n+2)

< d(ω2n+1, ω2n+2).

This implies

d(ω2n+2, ω2n+3)< d(ω2n+1, ω2n+2).(11)

Thus d(ωn+1, ωn+2)< d(ωn, ωn+1)for all n. Hence

{d(ωn, ωn+1)}is a strictly decreasing sequence of

non-negative real numbers. Thus there exists Z≥0

such that lim

n→∞ d(ωn, ωn+1) = Z.

Assume that Z > 0.So by inequality (8) we obtain,

lim

n→∞ ₤2n=Z(12)

and

lim

n→∞ β(₩2n)₩2n=Z. (13)

Using (2) and (G2) of Definition 2.5, get

CG≤lim sup

n→∞

ζ(₤2n, β(₩2n)₩2n)

=lim sup

n→∞

ζ(₤2n, β(d(ω2n, ω2n+1))d(ω2n, ω2n+1))

<CG,

which is a contradiction and nence z= 0, i.e.,

lim

n→∞ d(ωn, ωn+1) = 0.(14)

We now show that {ωn}is a Cauchy sequence. As-

sume, however, that it is not a Cauchy sequence. We

suppose that ϵ > 0exists, as well as two sequences of

positive integers, {n(k)}and {m(k)}such that

n(k)> m(k)> k, d(ωn(k), ωm(k))≥ϵ,

d(ωn(k)−1, ωm(k))< ϵ. (15)

We obtain using the triangular inequality

ϵ≤d(ωn(k), ωm(k))

≤d(ωn(k), ωm(k)−1) + d(ωn(k)−1, ωm(k))

< d(ωn(k), ωn(k)−1) + ϵ.

Taking the limit as k→ ∞ and applying (14), we get

that lim

k→∞ d(ωn(k), ωm(k)) = ϵ. (16)

Using the triangle inequlity, we have

ϵ≤d(ωn(k), ωm(k))

≤d(ωn(k), ωm(k)+1) + d(ωn(k)+1, ωm(k))

and

d(ωn(k), ωm(k)+1)

≤d(ωn(k), ωm(k)) + d(ωm(k), ωm(k)+1).

Again, by taking the limit as k→ ∞ and using (11),

(12) and (13), we get

lim

k→∞ d(ωn(k), ωm(k)+1) = ϵ. (17)

Similarly, we obtain

lim

k→∞ d(ωn(k)+1, ωm(k)) = ϵ. (18)

Also, we observe that

d(ωn(k)+1, ωm(k)+1)

≤d(ωn(k)+1, ωm(k)) + d(ωm(k), ωm(k)+1)

and

d(ωn(k)+1, ωm(k)+1)

≤d(ωn(k)+1, ωm(k)+1) + d(ωm(k), ωm(k)).

By taking the limit k→ ∞ and using (12), (13), (14)

and (16), we get

lim

n→∞ d(ωn(k)+1, ωm(k)+1) = ϵ. (19)

From (14) and (15) we can choose a positive integer

n0≥1such that

2D(ωn(k),Ωωn(k)), D(ωm(k),Λωm(k))<ϵ

< d(ωn(k), ωm(k))

and consequently,

lim

k→∞ Θ(ωm(k), ωn(k)) = ϵ. (20)

Since α(ω0,Ωω0)≥1and Ω,Λare α-admissile, we

get

α(ω0, ω1) = α(ω0,Ωω0)≥1.

By triangular α-admissile, we get

α(Ωω0,Λω1) = α(ω1, ω2)≥1

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and

α(ΛΩω0,ΩΛω1) = α(ω2, ω3)≥1.

By proceeding the above process, we conclude that

α(ωn, ωn+1)≥1for all nNow, we prove that

α(ωn, ωn+1)≥1, for all m, n ∈Nwith n<m.

Since α(ωn, ωn+1)≥1,

α(ωn+1, ωn+2)≥1,

then, we have

α(ωn, ωn+2)≥1.

Again, since

α(ωn, ωn+2)≥1,

α(ωn+2, ωn+3)≥1,

we deduce that

α(ωn, ωn+3)≥1.

By proceeding this process, we have

α(ωn, ωm)≥1

for all m, n ∈Nwith m > n. Let ω=ωm(k), ρ =

ωn(k). from above we obtain α(ωn, ωm)≥1. Then

by 2.1,

CG≤ζ(₤m(k), β(₩m(k))₩m(k))

<G(β(₩m(k))₩m(k),₤m(k)),

where ₤m(k)=α(ωm(k), ωn(k))H(Ωωm(k),Λωn(k))

and ₩m(k)= Θ(ωm(k), ωn(k)) + LΨ(ωm(k), ωn(k)).

Here Θ(ωm(k), ωn(k)) = d(ωm(k), ωn(k)), by (G1),

we get

d(ωm(k), ωn(k))

≤₤m(k)

< β(₩m(k))₩m(k)

<₩m(k)

=d(ωm(k), ωn(k)) + LΨ(ωm(k), ωn(k)).

(21)

Using (16), (15) and limn→∞ Ψ(ωm(k), ωn(k)) = 0 in

(21), we get

lim

k→∞ α(ωm(k), ωn(k))H(Ωωm(k),Λωn(k)) = ϵ,

and lim

k→∞ β(₩m(k))₩m(k)=ϵ,

where ₩m(k)= Θ(ωm(k), ωn(k)) +

LΨ(ωm(k), ωn(k)). Therefore using (3.1)

and (ζ2) of Definition 2.2, putting ₤m(k)=

α(ωm(k), ωn(k))H(Ωωm(k),Λωn(k))and ₩m(k)=

Θ(ωm(k), ωn(k)) + LΨ(ωm(k), ωn(k)), we get

CG≤ζ(₤m(k), β(₩m(k))₩m(k))<CG,

which is a contradiction. As a result, {ωn}is a

Cauchy sequence. Because Υis complete, we can

guarantee that {ωn}convergence to some ω∗∈Υ,

i.e.,

lim

n→∞ d(ωn, ω∗) = 0

and so

lim

n→∞ d(ωn, ω∗) = lim

n→∞ d(ω2n, ω∗)

=lim

n→∞ d(ω2n+1, ω∗) = 0.(22)

We now assert that

2min {D(ωn,Ωωn), D(ω∗,Λω∗)}< d(ωn, ω∗)

2min {D(ω∗,Ωω∗), D(ωn+1,Λωn+1)}

< d(ω∗, ωn+1)

(23)

for all n∈N. Suppose that it is not the case. Then

there exist m∈Nsuch that

2min {D(ωm,Ωωm), D(ω∗,Λω∗)} ≥ d(ωm, ω∗)

(24)

and

2min {D(ω∗,Ωω∗), D(ωm+1,Λωm+1)}

≥d(ω∗, ωm+1).

(25)

Therefore

2d(ωm, ω∗)

≤min {D(ωm,Ωωm), D(ω∗,Λω∗)}

≤min {d(ωm, ω∗) + D(ω∗,Ωωm), D(ω∗,Λω∗)}

≤d(ωm, ω∗) + D(ω∗,Ωωm)

≤d(ωm, ω∗) + d(ω∗, ωm+1),

which implies that

d(ωm, ω∗)≤d(ω∗, dωm+1).(26)

From (23) and (24)

d(ωm, ω∗)

≤d(ωm+1, ω∗)

≤1

2min {D(ω∗,Ωω∗), D(ωm+1,Λωm+1)}.

(27)

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Since 1

2min {D(ωm,Ωωm), D(ω∗,Λω∗)}<

d(ωm, ωm+1), from (2) we have

CG≤ζ(₤m, β(₩m)₩m)

<G(β(₩m)₩m,₤m),

where ₤m=α(ωm, ωm+1)H(Ωωm,Λωm+1)and

₩m= Θ(ωm, ωm+1) + LΨ(ωm, ωm+1).

Consequently, we get

d(ωm+1, ωm+2)≤₤m< β(₩m)₩m<₩m,(28)

where

Θ(ωm, ωm+1)

=max d(ωm, ωm+1)D(ωm,Ωωm),

D(ωm+1,Λωm+1),

D(ωm,Λωm+1) + D(ωm+1,Ωωm)

2

≤max d(ωm, ωm+1), d(ωm, ωm+1),

d(ωm+1, ωm+2),

d(ωm, ωm+2) + d(ωm+1, ωm+1)

2

=max d(ωm, ωm+1), d(ωm+1, ωm+2),

d(ωm, ωm+2)

2.

Since

d(ωm, ωm+2)

2≤d(ωm, ωm+1) + d(ωm+1, ωm+2)

≤max {d(ωm, ωm+1), d(ωm+1, ωm+2)}.

Thus,

Θ(ωm, ωm+1)≤max {d(ωm, ωm+1), d(ωm+1, ωm+2)}.

Also,

Ψ(ωm, ωm+1) = 0.

Suppose that max {d(ωm, ωm+1), d(ωm+1, ωm+2)}=

d(ωm+1, ωm+2),then from (28) we have

d(ωm+1, ωm+2)< d(ωm+1, ωm+2),

which is a contradiction. Thus we conclude that

max {d(ωm, ωm+1), d(ωm+1, ωm+2)}=d(ωm, ωm+1).

From (26) we get that

d(ωm+1, ωm+2)< d(ωm, ωm+1).(29)

From (27), (28) and (29), we get

d(ωm+1, ωm+2)

< d(ωm, ωm+1)

≤d(ωm, ω∗) + d(ω∗, ωm+1)

≤1

2min {D(ω∗,Ωω∗), D(ωm+1,Λωm+1)}

=min {D(ω∗,Ωω∗), D(ωm+1,Λωm+1)}

≤d(ωm+1, ωm+2),

which is a contradiction. Hence (25) holds, i.e., for

every n≥2

2min {D(ωn,Ωωn), D(ω∗,Λω∗)< d(ωn, ω∗)}

holds. Hence from (3)

CG≤ζ(₤n, β(₩n)₩n)

<G(β(₩n)₩n,₤n),(30)

where ₤n=α(ωn, ω∗)H(Ωωn,Λω∗)and ₩n=

Θ(ωn, ω∗) + LΨ(ωn, ω∗).

Consequently, we get

D(ωn+1,Λω∗)≤₤n<₩n,(31)

where

Θ(ωn, ω∗)

=max d(ωm, ω∗)D(ωn,Ωωn), D(ω∗,Λω∗),

D(ωn,Λω∗) + D(ω∗,Ωωn)

2

≤max d(ωn, ω∗), d(ωn, ωn+1), D(ω∗,Λω∗),

D(ωn,Λω∗) + d(ω∗, ωn+1)

2

and

Ψ(ωn, ω∗)

=min{D(ωn,Ωωn), D(ω∗,Λω∗),

D(ωn,Λω∗), D(ω∗,Ωωn)}

=min{d(ωn, ωn+1), D(ω∗,Λω∗),

D(ωn,Λω∗), D(ω∗,Ωωn)}.

Letting n→ ∞ and by using (14) and (22), we obtain

lim

n→∞ Θ(ωn, ω∗) = D(ω∗,Λω∗),

lim

n→∞ Ψ(ωn, ω∗) = 0.(32)

Now we show that ω∗∈Λω∗. Suppose, on the other

hand, that D(ω∗,Λω∗)>0.By allowing n→ ∞ in

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(31), we obtain

D(ω∗,Λω∗)

=lim

n→∞ D(ωn+1,Λω∗)

≤lim

n→∞ α(ωn, ω∗)H(Ωωn,Λω∗)

<lim

n→∞ Θ(ωn, ω∗) + Llim

n→∞ Ψ(ωn, ω∗)

=D(ω∗,Λω∗),

which is a contradiction. Therefore ω∗∈Λω∗.

Similarly, we can show that ω∗∈Ωω∗. Thus Ωand

Λhave a common fixed point.

Corollary 3.4. Let (Υ, d)be a complete metric space

and Ω:Υ→ CB(Υ) be a generalized multivalued

Suzuki type Z-contraction with respect to ζ, i.e.,

2min{D(ω, Ωω), D(ρ, Λρ)}< d(ω, ρ)

⇒ζ(H(Ωω, Λρ),Θ(ω, ρ)) ≥0for all ω, ρ ∈Υ,

where

Θ(ω, ρ) = max d(ω, ρ), D(ω, Ωω), D(ρ, Λρ),

D(ω, Λρ) + D(ρ, Ωω)

2.

Then Ωand Λhave a common fixed point.

Proof. The proof follows from Theorem 3.3 by taking

α(ω, ρ) = 1, β(v) = vand Ψ(ω, ρ) = 0.

Example 3.5. Let Υ = {0,3,5}be endowed with the

usual metric. Let Ω,Λ : Υ → CB(Υ) be defined by

Ωω=ω

7if ω ∈ {0,5}

0,1

7if ω = 3,

and Λω=ω

5for all ω∈Υ.

We now define ζ: [0,∞)×[0,∞)→Rby ζ(v, u) =

7u−vfor all u, v ∈[0,∞)]. We can now confirm the

inequality (2) for all ω, ρ ∈Υwith Ωω= Λρ. Note

that for all ω, ρ ∈Υwith Ωω= Λρthe inequality

2min {D(ω, Ωω), D(ω, Λω)}< d(ω, ρ)gives

(ω, ρ)∈ {(0,3),(3,0),(0,5),(5,0),(3,5),(5,3)}.

Then from (2), we have

ζ(H(Ωω, Λρ),Θ(ω, ρ)) = 6

7Θ(ω, ρ)− H(Ωω, Λρ)≥0.

That implies that

H(Ωω, Λρ)≤6

7Θ(ω, ρ).

Case (i) for ω= 0, ρ = 3;

H(Ω0,Λ3) = H({0},3

5) = 3

5≤6

7Θ(0,3).

Case (ii) for ω= 3, ρ = 0;

H(Ω3,Λ0) = H(0,1

7,{0}) = 1

7≤6

7Θ(3,0).

Case (iii) for ω= 0, ρ = 5;

H(Ω0,Λ5) = H({0},{1}) = 1 ≤6

7Θ(0,5).

Case (iv) for ω= 5, ρ = 0;

H(Ω5,Λ0) = H(5

7,{0}) = 5

7≤6

7Θ(5,0).

Case (v) for ω= 3, ρ = 5;

H(Ω3,Λ5) = H(0,1

7,{1}) = 1 ≤6

7Θ(3,5).

Case (v) for ω= 5, ρ = 3;

H(Ω5,Λ3) = H(5

7,3

5) = 5

7≤6

7Θ(5,3).

That all of the hypotheses in Corollary 3.4 are met. As

a result, 0is a common fixed point owned by Ωand

Λ.

Corollary 3.6. Let (Υ, d)be a complete metric space

and Ω:Υ→ CB(Υ) be a generalized multivalued

Suzuki type Z-contraction with respect to ζ, i.e.,

2D(ω, Ωω)< d(ω, ρ)

⇒ζ(H(Ωω, Ωρ),Θ(ω, ρ)) ≥0,

(33)

for all ω, ρ ∈Υwith ω=ρ, where

Θ(ω, ρ) = max d(ω, ρ), D(ω, Ωω), D(ρ, Ωρ),

D(ω, Ωρ) + D(ρ, Ωω)

2.

Then Ωhas a fixed point ω∗∈Υand for ω∈Υthe

sequence {Ωnω}convergences to ω∗.

Proof. The proof follows from Theorem 3.3 by taking

Ω = Λ.

4 Conclusion

Despite its novel applications, the search for fixed

point theorems involving contraction type conditions

has received much interest in recent decades. In this

context, we analyzed convergence point results for

such mappings and illustrative for support theorem

based on the new idea of Suzuki type Z-contraction

mappings obeying an admissibility type condition

in generalized metric spaces via the concept of C-

functions.

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Acknowledgment

This project was supported by the Research and De-

velopment Institute, Rambhai Barni Rajabhat Univer-

sity (Grant no.2214/2565).

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