# Someα−ϕ-Fuzzy Cone Contraction Results with Integral Type Application

Authors: Saif Ur Rehman; Shamoona Jabeen; Sami Ullah Khan; Mohammed M. M. Jaradat
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we defineα-admissible and α-ϕ-fuzzy cone contraction in fuzzy cone metric space to prove some fixed point theorems. Some related sequences with contraction mappings have been discussed. Ultimately, our theoretical results have been utilized to show the existence of the solution to a nonlinear integral equation. This application is also illustrative of how fuzzy metric spaces can be used in other integral type operators.


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## Body

## 1. Introduction

The concept of fuzzy metric space (FM space) was first introduced by Kramosil and Michale [1] while George and Veeramani [2] illustrated some well-known FM space properties. In the sense of Kramosil and Michale [1], George and Sapena [3] and Grabice [4] introduced the idea of fuzzy contraction of complete FM spaces and developed some fixed point (FP) results. Some more related results can be found in [5–8]. Samet et al. [9] proposed the concept of α-ψ-contraction in complete metric spaces in 2012. Later, Gopal and Vetro [10] presented the concepts of α−ϕ and β−ψ fuzzy contractive mappings, as well as several novel FP theorems in FM spaces. More FP results in the context of FM spaces can be found in references (see, for example, [11–15]). Mohammadi et al. [16] proved some generalized contraction results in FM spaces with application in integral equations. Recently, the rational type fuzzy contraction concept in complete FM space is given by Rehman et al. [17], and they proved some FP results with an application.Oner et al. [18] introduced the idea of fuzzy cone metric space (FCM space), proved some basic properties, and developed the first version of “Banach contraction principle for fixed point” in FCM spaces which is stated as follows: “let U,M¨α,∗ be a complete FCM space in which fuzzy cone contractive sequences are Cauchy and let h:U⟶U be a fuzzy cone contractive mapping being a∈0,1 the contractive constant. Then, h has a unique fixed point.” Ur Rehman et al. [19] presented some extended “fuzzy cone Banach contraction results” in FCM spaces for some weaker conditions. In topology and analysis, the definition of FCM spaces with different contractive conditions has been commonly used. For further reading, refer to [20–28].The definition ofα admissibility has been applied to certain directions by several writers. For a pair of functions, some authors expanded the concept of α admissibility. We may advise readers to look for more work in the field of α admissibility, as well as references (see [29–33]). Recently, Islam et al. [34] established some FP results in cone b2-metric space by using generalized a-admissible Hardy–Rogers’ contractions over Banachs algebras with application.In this paper, we define that a mappingh is α-admissible with respect to η and α−ϕ fuzzy cone contraction in FCM space. By using the concept of α-admissibility with respect to η under mapping h, we establish some FP theorems under the α−ϕ fuzzy cone contraction conditions in FCM space with an example. In support of our work, we present an integral type application. By using this concept, one can prove different contractive type FP results for nonlinear mappings with different types of applications in the context of FM spaces. The paper is organized as follows. In Section 2, we introduce the preliminary concepts to support our work. In Section 3, we present some FP results by using different types of contraction conditions in FCM spaces with an illustrative example. In Section 4, we present an integral equation application to validate the concept defined in the paper. Finally, in Section 5, we discuss the conclusion.


## 2. Preliminaries

The continuoust-norm is defined by by Schweizer and Sklar [35].Definition 1 (see [35]).
An operation∗:0,12⟶0,1 is called continuous t′-norm if it satisfies the following conditions:(i)
∗ is commutative, associative, and continuous(ii)
1∗ϱ1=ϱ1 and ϱ1∗ϱ2≤ϱ3∗ϱ4, whenever, ϱ1≤ϱ3 and ϱ2≤ϱ4, for each ϱ1,ϱ2,ϱ3,ϱ4∈0,1
The basict′-norms, i.e., product, minimum, and L Lukasiewicz continuous t′-norms are defined as follows (see [35]):(i)
ϱ1∗ϱ2=ϱ1ϱ2(ii)
ϱ1∗ϱ2=minϱ1,ϱ2(iii)
ϱ1∗ϱ2=maxϱ1+ϱ2−1,0Definition 2 (see [36]).
A subsetC of a real Banach space E is called cone if(i)
C≠∅, closed, and C≠θ(ii)
0≤ϱ1,ϱ2<∞ and μ,ω∈C, then ϱ1μ+ϱ2ω∈C(iii)
−μ,μ∈C, then μ=θ
A coneC⊂E and ⪯¯ is a partial ordering on E via C which is defined by μ⪯¯ω iff ω−μ∈C. μ≺ω stands for μ⪯¯ω and μ≠ω, while μ≪ω stands for ω−μ∈intC. All the cones in this paper have a nonempty interior.Definition 3 (see [18]).
A 3-tupleU,M¨α,∗ is called FCM space if C is a cone of E, U is an arbitrary set, ∗ is a continuous t-norm, and a mapping M¨α:U×U×intC⟶0,1 satisfies the following axioms:(i)
M¨αμ,ω,t′>0 and M¨αμ,ω,t′=1 iff μ=ω(ii)
M¨αμ,ω,t′=M¨αω,μ,t′(iii)
M¨αμ,ρ,s′∗M¨αρ,ω,t′≤M¨αμ,ω,s′+t′(iv)
M¨αμ,ω,.:intC⟶0,1 is continuous
∀
μ
,
ω
,
ρ
∈
U and s′,t′∈intC.Definition 4 (see [18]).
LetU,M¨α,∗ be a FCM space, and μ∈U and μℓ be a sequence in U. Then,(i)
The sequenceμi converges to μ, if t′≫θ, 0<r′<1 and ∃i1∈N so that M¨αμℓ,μ,t′>1−r′, ∀ℓ≥ℓ1. We denote this by limℓ⟶∞μℓ=μ or μℓ⟶μ as ℓ⟶∞.(ii)
The sequenceμℓ is Cauchy sequence if 0<r′<1, t′≫θ, and ∃ℓ1∈N so that M¨αμℓ,μj,t′>1−r′, ∀ℓ,j≥ℓ1.(iii)
The sequenceμℓ is G-C auchy sequence if 0<r′<1, t′≫θ and ∃ℓ1∈N so that M¨αμℓ,μj,t′=1, ∀ℓ,j≥ℓ1.(iv)
U,M¨α,∗ is G-complete if every G-C auchy sequence is convergent in U.(v)
μℓ is said to be fuzzy cone contractive if ∃0<ϱ<1 so that(1)1M¨αμℓ,μℓ+1,t′−1≤ϱ1M¨αμℓ−1,μℓ,t′−1,fort′≫θℓ≥1.Definition 5 (see [19]).
Let a FCMM¨α be triangular in FCM space U,M¨α,∗ if(2)1M¨αμ,ω,t′−1≤1M¨αμ,ρ,t′−1+1M¨αρ,ω,t′−1,∀μ,ω,ρ∈U and t′≫θ.Lemma 1 (see [18]).
Letμ∈U in a FCM space U,M¨α,∗ and μℓ be a sequence in U. Then, μℓ⟶μ⇔M¨αμℓ,μ,t′⟶1 as ℓ⟶∞, for t′≫θ.Definition 6 (see [18]).
LetU,M¨α,∗ be a FCM space and A:U⟶U. Then, T is called fuzzy cone contractive if ∃0<ϱ<1, so that(3)1M¨αAμ,Aω,t′−1≤ϱ1M¨αμ,ω,t′−1,∀μ,ω∈U,t′≫θ.


## 3. Main Result

Definition 7.
LetU,M¨α,∗ be a FCM space, and let α,η:U×U×intC⟶0,∞ be two functions. We say that h:U⟶U is α-admissible w.r.t η if(4)αμ,ω,t′≥ημ,ω,t′⇒αhμ,hω,t′≥ηhμ,hω,t′,∀μ,ω∈U.
Note that in a special case, if we takeημ,ω,t′=1, then Definition 7 is reduced to the α-admissible mapping (i.e., Definition 3.4 in the study by Gopal and Vetro [10]), and also if we take αμ,ω,t′=1, then we can say that h is a η-subadmissible mapping.
In the following,Φ denotes the family of all right continuous functions ϕ:0,+∞⟶0,+∞ with ϕτ<τ,∀τ>0.Definition 8.
LetU,M¨α,∗ be a FCM space, and the mapping h:U⟶U is called α-ϕ-fuzzy cone contractive if there exist three functions α,η:U×U×intC⟶0,∞ and ϕ∈Φ, so that(5)αμ,ω,t′≥ημ,ω,t′⇒1M¨αhμ,hω,t′−1≤ϕ1Ωh,μ,ω,t′−1,where(6)Ωh,μ,ω,t′=minM¨αμ,ω,t′,M¨αμ,hμ,t′,M¨αω,hμ,t′,M¨αω,hω,t′,∀μ,ω∈U,t′≫θ.Theorem 1.
Let a FMM¨α be triangular in a G-complete FCM space U,M¨α,∗ and let h:U⟶U be an α-ϕ-fuzzy cone contractive if the following axioms hold:(1)
h is an α-admissible w.r.t η(2)
∃μ0∈U such that αμ0,hμ0,t′≥ημ0,hμ0,t′, for t′≫θ(3)
The sequenceμℓ in U with αμℓ,μℓ+1,t′≥ημℓ,μℓ+1,t′,∀ℓ∈N,t′≫θ and μℓ⟶z as ℓ⟶+∞, then αμℓ,z,t′≥ημℓ,z,t′, ∀ℓ∈Nand,t′≫θ
Then,h has a FP z∈U such that hz=z.Proof.
Letμ0∈U, such that(7)αμ0,hμ0,t′≥ημ0,hμ0,t′,fort′≫θ.
We choose a sequence(8)μℓ∈U,i.e.μℓ=hμℓ−1=hℓμ0,∀ℓ∈N.
Ifμℓ+1=uℓ for some ℓ∈N, then μℓ=μ is a FP of h in U. Otherwise, we assume that μℓ+1≠μℓ,∀ℓ∈N. Since the mapping h is an α-admissible w.r.t η and αμ0,hμ0,t′≥ημ0,hμ0,t′, for t′≫θ, we have to find that(9)αμ1,μ2,t′=αhμ0,h2μ0,t′≥ηhμ0,h2μ0,t′=ημ1,μ2,t′,⇒αμ1,μ2,t′≥ημ1,μ2,t′,fort′≫θ.
Continuing this process∀ℓ∈N, we get(10)αμℓ,μℓ+1,t′≥ημℓ,μℓ+1,t′,fort′≫θ.
From (5), we have(11)1M¨αμℓ,μℓ+1,t′−1=1M¨αhμℓ−1,hμℓ,t′−1,≤ϕ1Ωh,μℓ−1,μℓ,t′−1,where(12)1Ωh,μℓ−1,μℓ,t′−1=1minM¨αμℓ−1,μℓ,t′,M¨αμℓ−1,hμℓ−1,t′,M¨αμℓ,hμℓ−1,t′,M¨αμℓ,hμℓ,t′−1,=1minM¨αμℓ−1,μℓ,t′,M¨αμℓ−1,μℓ,t′,M¨αμℓ,μℓ+1,t′−1=1minM¨αμℓ−1,μℓ,t′,M¨αμℓ,μℓ+1,t′−1.
Now, from (11), for t′≫θ, we have(13)1M¨αμℓ,μℓ+1,t′−1≤ϕ1minM¨αμℓ−1,μℓ,t′,M¨αμℓ,μℓ+1,t′−1.
IfM¨αμℓ,μℓ+1,t′ is minimum for some ℓ∈N, then by (13), we obtain(14)1M¨αμℓ,μℓ+1,t′−1≤ϕ1M¨αμℓ,μℓ+1,t′−1,<1M¨αμℓ,μℓ+1,t′−1,which is not possible. Hence, ∀ℓ∈N and t′≫θ, we get(15)1M¨αμℓ,μℓ+1,t′−1≤ϕ1M¨αμℓ−1,μℓ,t′−1<1M¨αμℓ−1,μℓ,t′−1.
This implies that(16)M¨αμℓ,μℓ+1,t′>M¨αμℓ−1,μℓ,t′.
Thus,M¨αμℓ−1,μℓ,t′ is an increasing sequence in 0,1. Let m1t′=limℓ⟶∞M¨αμℓ−1,μℓ,t′, we have to show that m1t′=1, for t′≫θ. Let ∃t′∗≫θ∋mt′∗<1.(17)1M¨αμℓ,μℓ+1,t′∗−1<1M¨αμℓ−1,μℓ,t′∗−1.
Using the right side continuity of a functionϕ and let the limit ℓ⟶+∞, we obtained the contradiction as follows:(18)1mt′∗−1≤ϕ1mt′∗−1<1mt′∗−1.
This implies thatlimℓ⟶∞M¨αμℓ−1,μℓ,t′=1, for t′≫θ. Let j>ℓ, where ℓ,j∈N and j=ℓ+p, for a fixed p∈N,(19)1M¨αμℓ,μj,t′−1=1M¨αμℓ,μℓ+p,t′−1,≤1M¨αμℓ,μℓ+1,t′/p∗M¨αμℓ+1,μℓ+2,t′/p∗⋯∗M¨αμℓ+p−1,μℓ+p,t′/p−1⟶θ,asℓ⟶∞.
This implies thatlimℓ⟶∞M¨αμℓ,μj,t′=1. It is proved that the sequence μℓ is G-C auchy. Since U,M¨α,∗ is G-complete, then ∃z∈U such that μℓ⟶z as ℓ⟶+∞, i.e.,(20)limℓ⟶∞M¨αμℓ,z,t′=1,fort′≫θ.
Now, by the view of Definition4 (iii),(21)αμℓ,z,t′≥ημℓ,z,t′,∀ℓ∈N,t′≫θ.
Ifhz≠z, i.e., M¨αz,hz,t′<1, then t′≫θ. Since M¨α is triangular, we have that(22)1M¨αz,hz,t′−1≤1M¨αz,μℓ+1,t′−1+1M¨αμℓ+1,hz,t′−1,fort′≫θ.
Then, from (5) and (21), we have(23)1M¨αμℓ+1,hz,t′−1=1M¨αhμℓ,hz,t′−1≤ϕ1Ωh,μℓ,z,t′−1,where(24)1Ωh,μℓ,z,t′−11minM¨αμℓ,z,t′,M¨αμℓ,hμℓ,t′,M¨αz,hμℓ,t′,M¨αz,hz,t′−1,=1minM¨αμℓ,z,t′,M¨αμℓ,μℓ+1,t′,M¨αz,μℓ+1,t′,M¨αz,hz,t′−1⟶1min1,1,1,M¨αz,hz,t′−1,asℓ⟶∞,=1M¨αz,hz,t′−1.
Thus, to avoid the contradiction withϕc<c, for c>0,(25)limsupℓ⟶∞1M¨αμℓ+1,hz,t′−1≤1M¨αz,hz,t′−1,
Thus, together with (20) and (22), we conclude that M¨αz,hz,t′=1⇒hz=z.Corollary 1.
LetU,M¨α,∗ be a G-complete FCM space in which M¨α is triangular, and let h:U⟶U be an α-admissible. Assume that ∃ϕ∈Φ, so that(26)αμ,ω,t′≥1⇒1M¨αhμ,hω,t′−1≤ϕ1Ωh,μ,ω,t′−1,where(27)Ωh,μ,ω,t′=minM¨αμ,ω,t′,M¨αμ,hμ,t′,M¨αω,hμ,t′,M¨αω,hω,t′,∀μ,ω∈U,t′≫θ.
Assume that the following assertions hold:(i)
∃μ0∈U such that αμ0,hμ0,t′≥1, for t′≫θ(ii)
Any sequenceμℓ in U with αμℓ,μℓ+1,t′≥1, ∀ℓ∈N,t′≫θ and μℓ⟶z as ℓ⟶+∞, then αμℓ,z,t′≥1, ∀ℓ∈Nand,t′≫θ
Then,h has a FP in U.Corollary 2.
Let a FMM¨α be triangular in a G-complete FCM space U,M¨α,∗, and let h:U⟶U be an η-admissible. Assume that ∃ϕ∈Φ, so that(28)ημ,ω,t′≤1⇒1M¨αhμ,hω,t′−1≤ϕ1Ωh,μ,ω,t′−1,where(29)Ωh,μ,ω,t′=minM¨αμ,ω,t′,M¨αμ,hμ,t′,M¨αω,hμ,t′,M¨αω,hω,t′,∀μ,ω∈U,t′≫θ.
Suppose that the following axioms hold:(i)
∃μ0∈U such that ημ0,hμ0,t′≤1, for t′≫θ(ii)
Any sequenceμℓ in U with αμℓ,μℓ+1,t′≥1, ∀ℓ∈N,t′≫θ and μℓ⟶z as ℓ⟶+∞, then αμℓ,z,t′≥1, ∀ℓ∈Nandt′≫θ
Then.h has a FP in U.
Now, to establish the unique FP of anα-ϕ-fuzzy cone contraction map, let the hypothesis (H) is given as follows:(H)
For allμ,ω∈U, t′≫θ, ∃ρ∈U such that(30)αμ,ρ,t′≥ημ,ρ,t′,αω,ρ,t′≥ηω,ρ,t′,limℓ⟶∞M¨αhℓ−1ρ,hℓρ,t′=1.Theorem 2.
Adding the hypothesis (H) in Theorem1, we obtain the uniqueness of a FP of h provided the function ϕ∈Φ is nondecreasing.Proof.
Assume thatz and ρ be the two FPs of h in U. If αz,ρ,t′≥ηz,ρ,t′, then by (5), we get z=ρ. Suppose αz,ρ,t′<ηz,ρ,t′, then from (H), ∃y∈U, so that(31)αz,y,t′≥ηz,y,t′,andαρ,y,t′≥ηρ,y,t′,fort′≫θ.
Sinceh is an α-admissible w.r.t η, then we get(32)αz,hℓy,t′≥ηz,hℓy,t′,∀ℓ∈Nand,t′≫θ.
Now, we have to show thatM¨αz,hℓy,t′⟶1, as ℓ⟶∞, for t′≫θ. Since M¨α is triangular, then from (5) and (31),(33)1M¨αz,hℓy,t′−1≤1M¨αz,hz,t′−1+1M¨αhz,hℓy,t′−1=1M¨αz,hz,t′−1+1M¨αhz,hhℓ−1y,t′−1≤ϕ1Ωh,z,hℓ−1y,t′−1,where z is the FP of h and for t′≫θ,(34)1Ωh,z,hℓ−1y,t′−1=1minM¨αz,hℓ−1y,t′,M¨αz,hz,t′,M¨αhℓ−1y,hz,t′,M¨αhℓ−1y,hℓy,t′−1,=1minM¨αz,hℓ−1y,t′,1,M¨αhℓ−1y,z,t′,M¨αhℓ−1y,hℓy,t′−1=1minM¨αz,hℓ−1y,t′,M¨αhℓ−1y,hℓy,t′−1.
Letℓ0∈N such that M¨αz,hℓ−1y,t′≤M¨αhℓ−1y,hℓy,t′, ∀ℓ≥ℓ0 and by the hypothesis H. Thus,(35)1M¨αz,hℓy,t′−1≤ϕ1M¨αz,hℓ−1y,t′−1,fort′≫θ.
Again by the triangular property ofM¨α and the view of (5) and (31),(36)1M¨αz,hℓy,t′−1≤ϕ1M¨αz,hℓ−1y,t′−1,≤ϕ1M¨αz,hz,t′−1+1M¨αhz,hℓ−1y,t′−1=ϕ1M¨αz,hz,t′−1+1M¨αhz,hhℓ−2y,t′−1≤ϕ21Ωh,z,hℓ−1y,t′−1,where z is the FP of h and for t′≫θ,(37)1Ωh,z,hℓ−2y,t′−1=1minM¨αz,hℓ−2y,t′,M¨αz,hz,t′,M¨αhℓ−2y,hz,t′,M¨αhℓ−2y,hℓ−1y,t′−1,=1minM¨αz,hℓ−2y,t′,1,M¨αhℓ−2y,z,t′,M¨αhℓ−2y,hℓ−1y,t′−1=1minM¨αz,hℓ−2y,t′,M¨αhℓ−2y,hℓ−1y,t′−1.
Letℓ0∈N such that M¨αz,hℓ−2y,t′≤M¨αhℓ−2y,hℓ−1y,t′, ∀ℓ≥ℓ0, and by the hypothesis H. Thus,(38)1M¨αz,hℓy,t′−1≤ϕ21M¨αz,hℓ−2y,t′−1,fort′≫θ.
By continuing the same argument, we obtain(39)1M¨αz,hℓy,t′−1≤ϕℓ−ℓ01M¨αz,hℓ0y,t′−1,fort′≫θ.
Then, by taking the limitℓ⟶∞, we get that(40)hℓy⟶z.
Similarly, we can prove that(41)hℓy⟶ρ,asℓ⟶∞.
Now, from (40) and (41), we get the uniqueness, i.e., z=ρ. Subsequently, we use the following classes of functions in our results without M¨α triangularity condition. Suppose that(42)Ψ=ψ:0,+∞⟶0,+∞,where ψ is nondecreasing and continuous, and(43)Φ0=ϕ:0,+∞⟶0,+∞,where ϕ is lower semicontinuous, where ψc=ϕc=0⇔c=0.Theorem 3.
LetU,M¨α,∗ be a G-complete FCM space, and let h:U⟶U be an α-admissible w.r.t η. Suppose that ∃ψ∈Ψ and ϕ∈Φ0, so that(44)αμ,hμ,t′αω,hω,t′≥ημ,hμ,t′ηω,hω,t′,⇒ψ1M¨αhμ,hω,t′−1≤ψ1M¨αμ,ω,t′−1−ϕ1M¨αμ,ω,t′−1,∀μ,ω∈U and t′≫θ. Let the following axioms hold:(i)
∃μ0∈U such that αμ0,hμ0,t′≥ημ0,hμ0,t′, for t′≫θ(ii)
Any sequenceμℓ in U with αμℓ,μℓ+1,t′≥ημℓ,μℓ+1,t′, ∀ℓ∈N,t′≫θ and μℓ⟶z∈U, as ℓ⟶+∞, then αμℓ,z,t′≥ημℓ,z,t′ and αz,hz,t′≥ηz,hz,t′, ∀ℓ∈Nandt′≫θ
Then,h has an FP in U.Proof.
Letμ0∈U such that(45)αμ0,hμ0,t′≥ημ0,hμ0,t′,fort′≫θ.
We define a sequenceμℓ in U such that μℓ=hμℓ−1=hℓμ0,∀ℓ∈N. If μℓ+1=uℓ for some ℓ∈N, then μℓ=μ is an FP of h in U.
Otherwise, we assume thatμℓ+1≠μℓ∀ℓ∈N. However, the mapping h is an α-admissible w.r.t η and αμ0,hμ0,t′≥ημ0,hμ0,t′. Now, we have to deduce that(46)αμ1,μ2,t′=αhμ0,h2μ0,t′≥ηhμ0,h2μ0,t′=ημ1,μ2,t′,⇒αμ1,μ2,t′≥ημ1,μ2,t′,fort′≫θ.
Continuing this process∀ℓ∈N, we may obtain(47)αμℓ,μℓ+1,t′≥ημℓ,μℓ+1,t′,fort′≫θ.
Clearly,(48)αμℓ−1,hμℓ−1,t′αμℓ,hμℓ,t′≥ημℓ−1,hμℓ−1,t′ημℓ,hμℓ,t′,fort′≫θ.
Now, by (44), for t′≫θ,(49)ψ1M¨αμℓ,μℓ+1,t′−1=ψ1M¨αhμℓ−1,hμℓ,t′−1≤ψ1M¨αμℓ−1,μℓ,t′−1−ϕ1M¨αμℓ−1,μℓ,t′−1.
IfM¨αμℓ−1,μℓ,t′=1, then M¨αμℓ,μℓ+1,t′=1. Otherwise, if M¨αμℓ−1,μℓ,t′<1, then(50)ψ1M¨αμℓ,μℓ+1,t′−1<ψ1M¨αμℓ−1,μℓ,t′−1,fort′≫θ.
Sinceψ is nondecreasing, we may obtain that M¨αμℓ,μℓ+1,t′≥M¨αμℓ−1,μℓ,t′, ∀ℓ∈N and, t′≫θ. Thus, M¨αμℓ−1,μℓ,t′ is nondecreasing sequence in 0,1. Let m1t′=limℓ⟶∞M¨αμℓ−1,μℓ,t′, t′≫θ. Now, we have to show that m1t′=t′, for t′≫θ. If not, then ∃t′≫θ such that m1t′<1. Therefore, by taking the limit ℓ⟶∞ on (50), we get(51)ψ1m1t′−1≤ψ1m1t′−1−ϕ1m1t′−1,which is a contradiction. Thus, we get that(52)limℓ⟶∞M¨αμℓ−1,μℓ,t′=1,fort′≫θ.
Letℓ,j∈N such that j>ℓ and j=ℓ+p, where p∈N. We have that(53)M¨αμℓ,μℓ+p,t′≥M¨αμℓ,μℓ+1,t′p∗M¨αμℓ+1,μℓ+2,t′p∗⋯∗M¨αμℓ+p−1,μℓ+p,t′p⟶1∗1∗⋯∗1=1,asℓ⟶∞,fort′≫θ.
Thus,μℓ is a G-C auchy sequence, and since the space U,M¨α,∗ is G-complete, therefore, μℓ⟶z∈U. Now, for any sequence μℓ in U with αμℓ,μℓ+1,t′≥ημℓ,μℓ+1,t′, ∀ℓ∈N and t′≫θ. By the completeness of U, μℓ⟶z as ℓ⟶∞, αμℓ,z,t′≥ημℓ,z,t′ and αz,hz,t′≥ηz,hz,t′, ∀ℓ∈N and t′≫θ. Then, easily we may obtain(54)αμℓ,μℓ+1,t′αz,hz,t′≥ημℓ,μℓ+1,t′ηz,hz,t′,fort′≫θ.
Now, from (44),(55)ψ1M¨αμℓ+1,hz,t′−1=ψ1M¨αhμℓ,hz,t′−1≤ψ1M¨αμℓ,z,t′−1−ϕ1M¨αμℓ,z,t′−1,fort′≫θ.
IfM¨αμℓ,z,t′=1, then M¨αμℓ+1,hz,t′=1,fort′≫θ. If M¨αμℓ,z,t′<1, for t′≫θ, then we have(56)ψ1M¨αμℓ+1,hz,t′−1<ψ1M¨αμℓ,z,t′−1⇒M¨αμℓ+1,hz,t′≥M¨αμℓ,z,t′⟶1,asℓ⟶∞,for t′≫θ. Thus, we get that hz=z.Example 1.
LetU=0,∞ and t′-norm is a continuous norm. Let a FM M¨α:U×U×0,∞⟶0,1 be defined as(57)M¨αμ,ω,t′=t′t′+μ−ω,∀μ,ω∈U,andt′>0.
Then,U,M¨α,∗ is a G-complete FCM space and a FCM M¨α is triangular. Now, we define a mapping h:U⟶U by(58)hμ=μ3,ifμ∈0,1,3μ,ifμ∈1,∞.
Next, we defineα,η:U×U×intC⟶0,∞ and ϕ,ψ:0,+∞⟶0,+∞, and we have(59)αμ,ω,t′=3,ifμ,ω∈0,1,andt′≫θ,34,otherwise,(60)ημ,ω,t′=2,ifμ,ω∈0,1,andt′≫θ,25,otherwise,(61)ψξ=2ξ3,andϕξ=ξ3,∀ξ∈0,+∞.
However,U,M¨α,∗ is a G-complete FCM space. Now, first, we have to show that h is α-admissible w.r.t η. Since in (59) and (60), αμ,ω,t′≥1 and ημ,ω,t′≥1 for all μ,ω∈0,1 and t′≫θ and αμ,ω,t′≥ημ,ω,t′, which shows that α is admissible w.r.t η and h is an α-admissible w.r.t η, that is, αhμ,hω,t′≥ηhμ,hω,t′ for all μ,ω∈0,1 and t′≫θ. On the other hand, hμ≤1 for all μ∈0,1. If μ∉0,1, then, again by using (59) and (60), we have that αμ,hμ,t′=3/4 and ημ,hμ,t′=2/5 for t′≫θ and so that αμ,hμ,t′αω,hω,t′=9/16<1 and ημ,hμ,t′ηω,hω,t′=4/25<1, for t′≫θ, which is contradiction. Similarly, if ω∉0,1, then again we get that αμ,hμ,t′αω,hω,t′<1 and ημ,hμ,t′ηω,hω,t′<1, for t′≫θ, which is contradiction. Hence, αμ,hμ,t′αω,hω,t′≥1 and ημ,hμ,t′ηω,hω,t′≥1, for t′≫θ. It follows that the mapping h is both α-admissible and η-admissible and αμ,hμ,t′αω,hω,t′≥ημ,hμ,t′ηω,hω,t′, for t′≫θ.
Now, ifμℓ in a sequence in U such that αμℓ,μℓ+1,t′≥1 and ημℓ,μℓ+1,t′≥1 for all ℓ∈ℕ∪0 and μℓ⟶z as ℓ⟶+∞, then μℓ⊂0,1 and hence z∈0,1. This implies that αμℓ,z,t′≥1 and ημℓ,z,t′≥1 for all ℓ∈ℕ and t′≫θ. Next, we prove that inequality (44) of Theorem 3 is satisfied by using (59)–(61), for all μ,ω∈0,1 and t′≫θ:(62)ψ1M¨αhμ,hω,t′−1=ψhμ−hωt′,=2hμ−hω3t′=2μ−ω9t′≤2μ−ω3t′−μ−ω3t′=ψ1M¨αμ,ω,t′−1−ϕ1M¨αμ,ω,t′−1.
That is,(63)αμ,hμ,t′αω,hω,t′≥ημ,hμ,t′ηω,hω,t′fort′≫θ,⇒ψ1M¨αhμ,hω,t′−1≤ψ1M¨αμ,ω,t′−1−ϕ1M¨αμ,ω,t′−1fort′≫θ.
Hence, the conditions of Theorem3 are satisfied, and the mapping h has a FP in U, i.e., 0.
Note. In special case, by using metric space, the main result of Dutta and Choudhury [37] for the mapping h is not applicable; if we put μ=3 and ω=4, then we have(64)ψdhμ,hω=2>13=ψdμ,ω−ϕdμ,ω.
Note. if we take ημ,ω,t′=1 in Theorem 3, then we obtain the following two corollaries.Corollary 3.
LetU,M¨α,∗ be a G-complete FCM space, and let h:U⟶U be an α-admissible. Suppose that ∃ψ∈Ψ and ϕ∈Φ0, so that(65)αμ,hμ,t′αω,hω,t′≥1⇒ψ1M¨αhμ,hω,t′−1≤ψ1M¨αμ,ω,t′−1−ϕ1M¨αμ,ω,t′−1,∀μ,ω∈U and t′≫θ. Let the following axioms hold:(i)
∃μ0∈U such that αμ0,hμ0,t′≥1, for t′≫θ(ii)
Any sequenceμℓ in U with αμℓ,μℓ+1,t′≥1, ∀ℓ∈N,t′≫θ and μℓ⟶z∈U as ℓ⟶+∞, then αμℓ,z,t′≥1 and αz,hz,t′≥1, ∀ℓ∈Nt′≫θ
Then,h has a FP in U.Corollary 4.
LetU,M¨α,∗ be a G-complete FCM space, and let h:U⟶U be an α-admissible. Suppose that ∃ψ∈Ψ and ϕ∈Φ0, so that(66)αμ,hμ,t′αω,hω,t′≤1⇒ψ1M¨αhμ,hω,t′−1≤ψ1M¨αμ,ω,t′−1−ϕ1M¨αμ,ω,t′−1,∀μ,ω∈U and t′≫θ. Let the following conditions hold:(i)
∃μ0∈U such that αμ0,hμ0,t′≤1, for t′≫θ(ii)
Any sequenceμℓ in U with αμℓ,μℓ+1,t′≤1, ∀ℓ∈N,t′≫θ and μℓ⟶z∈U as ℓ⟶+∞, then αμℓ,z,t′≤1 and αz,hz,t′≤1, ∀ℓ∈Nt′≫θ
Then,h has a FP in U.


## 4. Supportive Application

In this section, we present an integral type application for FP to support our result.Let(67)μr=fτ+∫0rκτ,s′,μs′ds′,∀τ∈0,r=I0,where r>0. Let U=CI0,ℛ be a Banach space of all continuous functions defined on I0. The induced metric is defined by(68)d¨μ,ω=supτ∈I0μτ−ωτ,μ,ω∈CI0,ℛ.Letϱ1∗ϱ2=ϱ1ϱ2,∀ϱ1,ϱ2∈0,1 and consider the fuzzy metric be defined as follows:(69)M¨αμ,ω,t′=t′t′+d¨μ,ω,for t′>0 and μ,ω∈CI,ℛ. The space CI0,ℛ,M¨α,∗3 is G-complete FM space indeed by the Banach space CI0,ℛ. Now, here we discuss an integral type application for FCM space and prove the existing solution for the integral equation (67).Theorem 4.
. Leth:CI0,ℛ⟶CI0,ℛ be an integral operator defined as follows:(70)hμτ=fτ+∫0τκτ,s′,μs′ds′,where τ∈I0, f∈CI0,ℛ, and κ:I0×I0×ℛ⟶ℛ, that is, κ∈CI0×I0×ℛ,ℛ satisfies the following: ∃g:I0∗I0⟶0,+∞ such that ∀r∈I0,gτ,.∈L1I0,ℛ, ∀μ,ω∈CI0,ℛ, and ∀τ,s′∈I0, and we have that(71)κτ,s′,μs′−κτ,s′,ωs′≤gτ,s′Bh,μ,ω,where(72)Bh,μ,ω=minsups′∈I0μs′−ωs′,sups′∈I0μs′−hμs′,sups′∈I0ωs′−hμs′,sups′∈I0ωs′−hωs′,where ∫0τgτ,s′ds′ is bounded on I0 and the supτ∈I0∫0τgτ,s′ds′≤λ<1. Then, the integral equation (67) has a solution μ∗∈CI0,ℛ.Proof.
The integral operatorh is defined in (70). Now, we have to apply Corollary 1, for all μ,ω∈CI0,ℛ, and by the view of (69)–(71), we have(73)1M¨αhμ,hω,t′−1=dhμ,hωt′,=supτ∈I0hμτ−hωτt′=1t′supτ∈I0∫0τκτ,s′,μs′−κτ,s′,ωs′ds′≤1t′supτ∈I0∫0τgτ,s′Bh,μ,ωds′=1t′Bh,μ,ωsupτ∈I0∫0τgτ,s′ds′.
Then, we have the following four cases:(1)
Ifsups′∈I0μs′−ωs′ is the minimum term in (72), then Bh,μ,ω=sups′∈I0μs′−ωs′. Now, by the view of (69), (71), and (73), for t′≫θ, we have(74)1M¨αhμ,hω,t′−1≤1t′sups′∈I0μs′−ωs′supτ∈I0∫0τgτ,s′ds′,≤1t′d¨μ,ωsupτ∈I0∫0τgτ,s′ds′≤λ1M¨αμ,ω,t′−1=λ1Ωh,μ,ω,t′−1,whereΩh,μ,ω,t′=M¨αμ,ω,t′, for t′≫θ.(2)
Ifsups′∈I0μs′−hμs′ is the minimum term in (72), then Bh,μ,ω=sups′∈I0μs′−hμs′. Now, by the view of (69), (71), and (73), for t′≫θ, we have(75)1M¨αhμ,hω,t′−1≤1t′sups′∈I0μs′−hμs′supτ∈I0∫0τgτ,s′ds′,≤1t′d¨μ,hμsupτ∈I0∫0τgτ,s′ds′≤λ1M¨αμ,hμ,t′−1=λ1Ωh,μ,ω,t′−1,whereΩh,μ,ω,t′=M¨αμ,hμ,t′, for t′≫θ.(3)
Ifsups′∈I0ωs′−hμs′ is the minimum term in (72), then Bh,μ,ω=sups′∈I0ωs′−hμs′. Now, by the view of (69), (71), and (73), for t′≫θ, we have(76)1M¨αhμ,hω,t′−1≤1t′sups′∈I0ωs′−hμs′supτ∈I0∫0τgτ,s′ds′,≤1t′d¨ω,hμsupτ∈I0∫0ττ,s′ds′≤λ1M¨αω,hμ,t′−1=λ1Ωh,μ,ω,t′−1,whereΩh,μ,ω,t′=M¨αω,hμ,t′, for t′≫θ.(4)
Ifsups′∈I0ωs′−hωs′ is the minimum term in (72), then Bh,μ,ω=sups′∈I0ωs′−hωs′. Now, by the view of (69), (71), and (73), for t′≫θ, we have(77)1M¨αhμ,hω,t′−1≤1t′sups′∈I0ωs′−hωs′supτ∈I0∫0ττ,s′ds′,≤1t′d¨ω,hωsupτ∈I0∫0τgτ,s′ds′≤λ1M¨αω,hω,t′−1=λ1Ωh,μ,ω,t′−1,whereΩh,μ,ω,t′=M¨αω,hω,t′, for ≫θ.
Hence, in all the cases, it is obvious that (26) holds with ϕc=λc,∀c≥0 and αμ,ω,t′=1,∀μ,ω∈CI0,ℛ and t′≫θ. As we have mentioned above, CI0,ℛ is complete and then the FCM space CI0,ℛ,M¨α,∗3 is G-complete in which M¨α is triangular. The other conditions of Corollary 1 are fulfilled immediately. We deduce that the operator h has a FP μ∗∈CI0,ℛ which is the required solution of the integral equation (67).


## 5. Conclusion

In this paper, we have presented the mappings, andh is α-admissible w.r.t η and α-ϕ-fuzzy cone contraction in FCM spaces. Using this kind of contractions, we proved FP theorems for G-complete FCM space in the sense of George and Veeramani with an illustrative example. Moreover, some extended results in the form of corollaries are discussed. An application is presented to support the concepts defined in the paper. This integral type application is also illustrative of how fuzzy metrics can be used in other integral type operators.



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