# A Novel Picture Fuzzyn-Banach Space with Some New Contractive Conditions and Their Fixed Point Results

Authors: Awais Asif; Hassen Aydi; Muhammad Arshad; Zeeshan Ali
Journal: Journal of Function Spaces
 (2020)

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## Abstract

A picture fuzzyn-normed linear space (NPF), a mixture of a picture fuzzy set and an n-normed linear space, is a proficient concept to cope with uncertain and unpredictable real-life problems. The purpose of this manuscript is to present some novel contractive conditions based on NPF. By using these contractive conditions, we explore some fixed point theorems in a picture fuzzy n-Banach space (BPF). The discussed modified results are more general than those in the existing literature which are based on an intuitionistic fuzzy n-Banach space (BIF) and a fuzzy n-Banach space. To express the reliability and effectiveness of the main results, we present several examples to support our main theorems.


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## Body

## 1. Introduction

In various real-life problems, for a suitable mapping, the existence of a solution and existence of a fixed point (FP) are equivalent. Thus, the existence of a FP is a proficient technique to cope with awkward and difficult problems in real-life issues. Various scholars have utilized such results in the environment of many fields [1, 2]. The extensive useful techniques capable with both algebraic and topological properties are those of a normed linear space (NLS), but the continuous maps are more proficient in the sense of NLS. Moreover, in a metric space, every contractive map is uniformly continuous. One of the fundamental applications of Banach’s contraction principle is the “Picard’s theorem,” which is the basic theorem for the existence and uniqueness of solution to the ordinary differential equations. Various scholars have utilized this application in the environment of a partial differential equation [3], in the Gauss-Seidel method for evaluating systems of linear equations [4], in the proof of the inverse function theorem [5], etc.The theory of a fuzzy set (FS) was investigated by Zadeh [6], characterized by only positive grades restricted to 0,1. FS has achieved more success due to its ability to cope with complications and troubles. However, in some practice cases, the concept of FS cannot cope with complications and uncertainty because of lack of knowledge of the problem. Therefore, Atanassov [7] investigated the intuitionistic FS (IFS) containing both positive and negative grades, whose sum is bounded to 0,1. IFS is regarded as a more improved way to cope with complex and awkward information. Further, Cường [8] investigated the picture FS (PFS) including positive, abstinence, and negative grades, whose sum is bounded to 0,1. PFS is regarded as a more improved way to deal with even more complex information. For more related works, we may refer to References [9–16].Keeping the advantages of the PFS, the objective of this manuscript is summarized in the following ways:(1)
To present some novel contractive conditions, we usedNPF as a basis(2)
By using these contractive conditions, some fixed point theorems are explored for a picture fuzzyn-Banach space (BPF). These results are more modified and more general than the existing results which are based on an intuitionistic fuzzy n-Banach space (BIF) and a fuzzy n-Banach space(3)
To express the reliability and effectiveness of the explored approaches, we explain examples in support of the main resultsThe rest of this manuscript is summarized in the following ways: In Section2, we review some basic notions like NIF and their related properties used in the presented work. In Section 3, we describe the notion of NPF and their fundamental properties. In Section 4, we present some novel contractive conditions based on NPF. By using these contractive conditions, we instigate some fixed point theorems for a picture fuzzy n-Banach space (BPF). Finally, the conclusion of this manuscript is discussed in Section 5.


## 2. Preliminaries

The purpose of this section is to review some existing notions, likeNIF and their related properties. Throughout this section, the symbols RRn+, RRn, NNn, ӾUNI, Μm, Νn, ∗ct, and °ctc represent the positive real numbers, real numbers, natural numbers, universal set, supporting grade, supporting against, continuous t-norm, and continuous t-conorm, respectively.Definition 1.
[9]. A NIF is stated by ӾUNI,Μm,Αa,Νn,∗ct,°ctc, where Μm,Αa,Νn is defined on ӿ1,ӿ2,⋯,ӿk,p∈ӾUNIk×0,∞, where the following conditions hold:
(i)
Μmӿ1,ӿ2,⋯,ӿk,p+Αaӿ1,ӿ2,⋯,ӿk,p+Νnӿ1,ӿ2,⋯,ӿk,p≤1(ii)
Μmӿ1,ӿ2,⋯,ӿk,p>0(iii)
Μmӿ1,ӿ2,⋯,ӿk,p=1 iff ӿ1,ӿ2,⋯,ӿk are linearly dependent(iv)
Μmӿ1,ӿ2,⋯,ӿk,p is invariant under any permutation of ӿ1,ӿ2,⋯,ӿk(v)
Μmӿ1,ӿ2,⋯,αӿk,p=Μmӿ1,ӿ2,⋯,ӿk,p/∣α∣ if α≠0∈ℝRn(vi)
Μmӿ1,ӿ2,⋯,ӿk+ӿk′,p+q≥minΜmӿ1,ӿ2,⋯,ӿk,p,Μmӿ1,ӿ2,⋯,ӿk′,q(vii)
Μmӿ1,ӿ2,⋯,ӿk+ӿk,. is a nondecreasing function of ℝRn.+ and logp→∞Μmӿ1,ӿ2,⋯,ӿk+ӿk,p=1(viii)
Νnӿ1,ӿ2,⋯,ӿk,p<1(ix)
Νnӿ1,ӿ2,⋯,ӿk,p=0 iff ӿ1,ӿ2,⋯,ӿk are linearly dependent(x)
Νnӿ1,ӿ2,⋯,ӿk,p is invariant under any permutation of ӿ1,ӿ2,⋯,ӿk(xi)
Νnӿ1,ӿ2,⋯,αӿk,p=Νnӿ1,ӿ2,⋯,ӿk,p/∣α∣ if α≠0∈ℝRn(xii)
Νnӿ1,ӿ2,⋯,ӿk+ӿk′,p+q≥maxΝnӿ1,ӿ2,⋯,ӿk,p,Νnӿ1,ӿ2,⋯,ӿk′,q(xiii)
Νnӿ1,ӿ2,⋯,ӿk+ӿk,. is a nonincreasing function of ℝRn+ and logp→∞Νnӿ1,ӿ2,⋯,ӿk+ӿk,p=0(xiv)
Further,Mnӿ1,ӿ2,⋯,ӿk,p>0 and Νnӿ1,ӿ2,⋯,ӿk,p<1 imply ӿ=0,∀p>0(xv)
Forp≠0, Mnӿ1,ӿ2,⋯,ӿk,. and Νnӿ1,ӿ2,⋯,ӿk,. are continuous functions of ℝRn+ and are strictly increasing and strictly decreasing, respectively, on the subset p:0<Μmӿ1,ӿ2,⋯,ӿk,p,Νnӿ1,ӿ2,⋯,ӿk,p<1 of ℝRn+Moreover, we explain some important theories based on convergent and Cauchy convergent sequences.Definition 2.
[9]. Consider NIFӾUNI,Μm,Νn,∗ct,°ctc; then, the sequence ӿ=ӿf in ӾUNI is convergent to ℊ∈ӾUNI based on the intuitionistic fuzzy n-norm Μm,Νnk if for every ϵ,p>0 and ӳ1,ӳ2,⋯,ӳk−1∈ӾUNI, there exists ὧ0∈ℕNn such that
(1)Mnӳ1,ӳ2,⋯,ӳk−1,ӿὧ−ℊ,p>1−ϵ,Nnӳ1,ӳ2,⋯,ӳk−1,ӿὧ−ℊ,p<ϵ,for all ὧ≥ὧ0 and it is represented by Μm,Νnk−limӿὧ=ℊ.Definition 3.
[9]. Let NIFӾUNI,Μm,Νn,∗ct,°ctc; then, the sequence ӿ=ӿf in ӾUNI is Cauchy convergent based on the intuitionistic fuzzy n-norm Μm,Νnk if for every ϵ,p>0 and ӳ1,ӳ2,⋯,ӳk−1∈ӾUNI, there exists ὧ0∈ℕNn such that
(2)Mnӳ1,ӳ2,⋯,ӳk−1,ӿὧ−ӿὛ,p>1−ϵ,Nnӳ1,ӳ2,⋯,ӳk−1,ӿὧ−ӿὛ,p<ϵ,for all ὧ,Ὓ≥ὧ0 and it is represented by Μm,Νnk−limӿὧ=ӿὛ.


## 3. Picture Fuzzyn-Normed Linear Space

The purpose of this section is to explore some new approaches likeNPF and their related properties, which are extensively efficient for the proof of our main work in the next section. Throughout this section, the symbols ӾUNI, Μm, Αa, Νn, ∗ct, and °ctc represented the universal set, supporting grade, abstinence grade, supporting against, continuous t-norm, and continuous t-conorm, respectively.Definition 4.
ANPF is stated as ӾUNI,Μm,Αa,Νn,∗ct,°ctc, where Μm,Αa,Νn is defined on ӿ1,ӿ2,⋯,ӿk,p∈ӾUNIk×0,∞, where the following conditions hold:
(i)
Μmӿ1,ӿ2,⋯,ӿk,p+Αaӿ1,ӿ2,⋯,ӿk,p+Νnӿ1,ӿ2,⋯,ӿk,p≤1(ii)
Μmӿ1,ӿ2,⋯,ӿk,p>0(iii)
Μmӿ1,ӿ2,⋯,ӿk,p=1 iff ӿ1,ӿ2,⋯,ӿk are linearly dependent(iv)
Μmӿ1,ӿ2,⋯,ӿk,p is invariant under any permutation of ӿ1,ӿ2,⋯,ӿk(v)
Μmӿ1,ӿ2,⋯,αӿk,p=Μmӿ1,ӿ2,⋯,ӿk,p/∣α∣ if α≠0∈ℝRn(vi)
Μmӿ1,ӿ2,⋯,ӿk+ӿk′,p+q≥minΜmӿ1,ӿ2,⋯,ӿk,p,Μmӿ1,ӿ2,⋯,ӿk′,q(vii)
Μmӿ1,ӿ2,⋯,ӿk+ӿk,. is a nondecreasing function of ℝRn+ and logp→∞Μmӿ1,ӿ2,⋯,ӿk+ӿk,p=1(viii)
Αaӿ1,ӿ2,⋯,ӿk,p<1(ix)
Αaӿ1,ӿ2,⋯,ӿk,p=0 iff ӿ1,ӿ2,⋯,ӿk are linearly dependent(x)
Αaӿ1,ӿ2,⋯,ӿk,p is invariant under any permutation of ӿ1,ӿ2,⋯,ӿk(xi)
Αaӿ1,ӿ2,⋯,αӿk,p=Αaӿ1,ӿ2,⋯,ӿk,p/∣α∣ if α≠0∈ℝRn(xii)
Αaӿ1,ӿ2,⋯,ӿk+ӿk′,p+q≥maxΑaӿ1,ӿ2,⋯,ӿk,p,Αaӿ1,ӿ2,⋯,ӿk′,q(xiii)
Αaӿ1,ӿ2,⋯,ӿk+ӿk,. is a nonincreasing function of ℝRn+ and logp→∞Αaӿ1,ӿ2,⋯,ӿk+ӿk,p=0(xiv)
Νnӿ1,ӿ2,⋯,ӿk,p<1(xv)
Νnӿ1,ӿ2,⋯,ӿk,p=0 iff ӿ1,ӿ2,⋯,ӿk are linearly dependent(xvi)
Νnӿ1,ӿ2,⋯,ӿk,p is invariant under any permutation of ӿ1,ӿ2,⋯,ӿk(xvii)
Νnӿ1,ӿ2,⋯,αӿk,p=Νnӿ1,ӿ2,⋯,ӿk,p/∣α∣ if α≠0∈ℝRn(xviii)
Νnӿ1,ӿ2,⋯,ӿk+ӿk′,p+q≥maxΝnӿ1,ӿ2,⋯,ӿk,p,Νnӿ1,ӿ2,⋯,ӿk′,q(xix)
ӿ1,ӿ2,⋯,ӿk+ӿk,. is a nonincreasing function of ℝRn+ and logp→∞Νnӿ1,ӿ2,⋯,ӿk+ӿk,p=0(xx)
Further,Mnӿ1,ӿ2,⋯,ӿk,p>0,Anӿ1,ӿ2,⋯,ӿk,p<1 and Νnӿ1,ӿ2,⋯,ӿk,p<1; then, ӿ=0,∀p>0(xxi)
Forp≠0, Mnӿ1,ӿ2,⋯,ӿk,., Anӿ1,ӿ2,⋯,ӿk,., and Νnӿ1,ӿ2,⋯,ӿk,. are continuous functions of ℝRn+ and also strictly increasing and strictly decreasing, respectively, on the subset p:0<Μmӿ1,ӿ2,⋯,ӿk,p,Αaӿ1,ӿ2,⋯,ӿk,p,Νnӿ1,ӿ2,⋯,ӿk,p<1 of ℝRn+Moreover, we explain some important theories based on convergent and Cauchy convergent sequences.Definition 5.
For aNPFӾUNI,Μm,Αa,Νn,∗ct,°ctc, the sequence ӿ=ӿf in ӾUNI is convergent to ℊ∈ӾUNI based on the picture fuzzy n-norm Μm,Αa,Νnk if for every ϵ,p>0 and ӳ1,ӳ2,⋯,ӳk−1∈ӾUNI, there exists ὧ0∈ℕNn such that
(3)Mnӳ1,ӳ2,⋯,ӳk−1,ӿὧ−ℊ,p>1−ϵ,Anӳ1,ӳ2,⋯,ӳk−1,ӿὧ−ℊ,p<ϵ,Nnӳ1,ӳ2,⋯,ӳk−1,ӿὧ−ℊ,p<ϵ,for all ὧ≥ὧ0 and it is represented by Μm,Αa,Νnk−limӿὧ=ℊ.Definition 6.
For aNPFӾUNI,Μm,Αa,Νn,∗ct,°ctc, the sequence ӿ=ӿf in ӾUNI is Cauchy convergent based on the picture fuzzy n-norm Μm,Αa,Νnk if for every ϵ,p>0 and ӳ1,ӳ2,⋯,ӳk−1∈ӾUNI, there exists ὧ0∈ℕNn such that
(4)Mnӳ1,ӳ2,⋯,ӳk−1,ӿὧ−ӿὛ,p>1−ϵ,Anӳ1,ӳ2,⋯,ӳk−1,ӿὧ−ӿὛ,p<ϵ,Nnӳ1,ӳ2,⋯,ӳk−1,ӿὧ−ӿὛ,p<ϵ,for all ὧ,Ὓ≥ὧ0 and it is represented by Μm,Αa,Νnk−limӿὧ=ӿὛ.Remark 7.
The following assumptions are important for our main results.(1)
SupposeȘm−1 is the set of functions Ψm−1:0,+∞⟶0,+∞ such that
(i)
Ψm−1 is continuous and nondecreasing(ii)
Ψm−1p=0⇔p=0(2)
SupposeȘa−2,Șn−3 is the set of functions Ψa−2, Ψn−3:0,+∞⟶0,+∞ such that
(i)
Ψa−2,Ψn−3 is continuous and nonincreasing(ii)
Ψa−2p,Ψn−3p=0⇔p=0(3)
Suppose₸m−1 is the set of functions Θ1:0,+∞⟶0,+∞ such that
(i)
Θ1 is continuous and strictly increasing(ii)
Θ1p=0⇔p=0(4)
Suppose₸a−2,₸n−3 is the set of functions with Θ2,Θ3:0,+∞⟶0,+∞ such that
(i)
Θ2,Θ3 is continuous and strictly decreasing(ii)
Θ2p,Θ3p=0⇔p=0


## 4. Contractive Mappings Based on the Picture Fuzzyn-Banach Space

Based on the definitions introduced in Section3, we describe some contractive mappings using the BPF named as picture fuzzy n-normed contractive mapping (NCM) and verify it with the help of numerical examples.Definition 8.
For aNPFӾUNI,Μm,Αa,Νn,∗ct,°ctc, the mapping Ț:ӾUNI⟶ӾUNI is called NCM, if
(5)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≤Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,pAnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,pNnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p,for all ӿ1,ӿ2,⋯,ӿk−1∈ӾUNI,ӿ,ӳ∈ӾUNI,p>0.Further, based on equation (5) and using Remark 7, we explore the following results, which are very helpful for future work.Theorem 9.
For aNPFӾUNI,Μm,Αa,Νn,∗ct,°ctc, we define NCM,Ț:ӾUNI⟶ӾUNI such that
(6)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥α⟹Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p−Ψm−1p≥αAnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α⟹Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p−Ψa−2p<1−αNnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α⟹Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p−Ψn−3p<1−α,where Ψm−1∈Șm−1, Ψa−2∈Șa−2, and Ψn−3∈Șn−3, for all ӿ1,ӿ2,⋯,ӿk−1∈ӾUNI,ӿ,ӳ∈ӾUNI,p>0 with α∈0,1. Then, Ț possesses a unique fixed point in ӾUNI.Proof.
Letӿ0∈ӾUNI with ӿk+1=Țӿk∀k∈ℕNn. By using Remark 7 and inequality (6), we get
(7)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≤Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p−Ψm−1p,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p−Ψa−2p,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p−Ψn−3p.
Further, we write the above equations as(8)Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p≤Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p−Ψm−1p≤Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p,Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p≥Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p−Ψa−2p≥Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p,Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p≥Nnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p−Ψn−3p≤Nnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p.
It is clear from the above analysis thatMnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p is a bounded nondecreasing sequence while Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p and Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p are bounded nonincreasing sequences. Then, the limit of these equations exists. Hence,
(9)Mnӿ2,ӿ3,⋯,ӿk−1,ӿ1−ӿ0,p+Ψm−1p≤Mnӿ2,ӿ3,⋯,ӿk,ӿ2−ӿ1,p+Ψm−1p−Ψm−1p+Ψm−1p≤Mnӿ2,ӿ3,⋯,ӿk,ӿ2−ӿ1,p,Anӿ2,ӿ3,⋯,ӿk−1,ӿ1−ӿ0,p+Ψa−1p≥Anӿ2,ӿ3,⋯,ӿk,ӿ2−ӿ1,p,Nnӿ2,ӿ3,⋯,ӿk−1,ӿ1−ӿ0,p+Ψn−1p≥Nnӿ2,ӿ3,⋯,ӿk,ӿ2−ӿ1,p.
By using the induction onk, we have
(10)Mnӿ2,ӿ3,⋯,ӿk−1,ӿ1−ӿ0,p+Ψm−1p≤Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p,Anӿ2,ӿ3,⋯,ӿk−1,ӿ1−ӿ0,p+Ψa−1p≥Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p,Nnӿ2,ӿ3,⋯,ӿk−1,ӿ1−ӿ0,p+Ψn−1p≥Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p.
Ask⟶∞, we have
(11)limk→∞Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p=1,limk→∞Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p=0,limk→∞Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p=0.
Supposingp,ϵ>0, we have
(12)Mnӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,p2≥1−ϵ,Mnӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,Ψm−1p2≥1−ϵ.
Similarly, from abstinence and falsity grades, we have(13)Anӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,Ψa−2p2<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,Ψn−3p2<ϵ.
By using the above analysis, we write, ifMnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿK,p/2≥1−ϵ and Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿK,p/2,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿK,p/2<ϵ, then
(14)Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿK,p2≥minMnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ȚӿK,p2−Ψm−1p2,Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿK,Ψm−1p2≥minMnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿK,p2,Mnӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,Ψm−1p2≥1−ϵ.
Similarly, solving the grades of abstinence and falsity, we have(15)Anӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,Ψa−2p2<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,Ψn−3p2<ϵ.
Therefore,(16)Mnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿK,p2≥1−ϵ.
Similarly, dealing with the grades of abstinence and falsity, we have(17)Anӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿK,p2<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿK,p2<ϵ.
Then, for allk≥ℕNn,
(18)Mnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿὛ,p≥minMnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿK,p,Mnӿ1,ӿ2,⋯,ӿk−1,ӿὛ−ӿK,p≥1−ϵ,Mnӿ1,ӿ2,⋯,ӿk−1,ӿὛ−ӿK,p≥1−ϵ.
Also, we find(19)Anӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿὛ,p<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿὛ,p<ϵ.
Sinceϵ is arbitrary and the sequence ӿf is Cauchy, hence they are convergent. Therefore, limӿf=ӿ.
Supposep,ϵ>0; then, there exists k0∈ℕNn such that
(20)Mnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿ,p2≥1−ϵ,Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿk,Ψm−1p2≥1−ϵ.
Moreover, doing the same process to abstinence and falsity grades, we obtain(21)Anӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿ,Ψa−2p2<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿ,Ψn−3p2<ϵ,for all k≥k0. Hence,
(22)Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p≥minMnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿk+1,p−Ψm−1p,Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿ,Ψm−1p≥minMnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿk,p,Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿ,Ψm−1p≥1−ϵ,Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p<ϵ,for all k≥k0. Therefore,
(23)Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p=1,Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p=0,Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p=0,for all p>0. Hence, Țӿ=ӿ; that is, Ț has a fixed point in ӾUNI. Next, we prove its uniqueness. For this, we suppose ӳ is another fixed point of Ț in ӾUNI; then,
(24)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p+kΨm−1p≥minMnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p=Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p+kΨa−2p=Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p+kΨn−3p=Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p,for all k∈ℕNn and k⟶∞; then,
(25)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=1,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=0,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=0,for all p>0. Hence, ӿ=ӳ. Thus, Ț has a unique fixed point in ӾUNI.Example 10.
For a Banach spaceӾUNI,., we define a mapping Ț:ӾUNI⟶ӾUNI such that for all ӿ,ӳ∈ӾUNI,
(26)Țӿ−Țӳ≤ӿ−ӳ−Ψm−1ӿ−ӳ,Țӿ−Țӳ≥ӿ−ӳ−Ψa−2ӿ−ӳ,Țӿ−Țӳ≥ӿ−ӳ−Ψn−3ӿ−ӳ.We know thatΨm−1∈Șm−1, Ψa−2∈Șa−2, and Ψn−3∈Șn−3. We consider that Ψm−1βp≤βΨm−1p, Ψa−2βp≥βΨa−2p, and Ψn−3βp≥βΨn−3p, where p>0 and β∈0,1. Now, we describe the picture fuzzy n-norm Mn, An, and Nn:
(27)Mnӿ1,ӿ2,⋯,ӿk−1,ӿk,p=pӿ1,ӿ2,⋯,ӿk−1,ӿk,0<p≤ӿ1,ӿ2,⋯,ӿk−1,ӿk,1,ӿ1,ӿ2,⋯,ӿk−1,ӿk<p,0,p≤0,Anӿ1,ӿ2,⋯,ӿk−1,ӿk,p=1−pӿ1,ӿ2,⋯,ӿk−1,ӿk,0<p≤ӿ1,ӿ2,⋯,ӿk−1,ӿk,0,ӿ1,ӿ2,⋯,ӿk−1,ӿk<p,1,p≤0,Nnӿ1,ӿ2,⋯,ӿk−1,ӿk,p=1−pӿ1,ӿ2,⋯,ӿk−1,ӿk,0<p≤ӿ1,ӿ2,⋯,ӿk−1,ӿk,0,ӿ1,ӿ2,⋯,ӿk−1,ӿk<p,1,p≤0.We consider that(28)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥α,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α.The first three parts are discussed for the truth grade. We have the following cases:Case 1.
Suppose0<p≤ӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p; then,
(29)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=pӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥α,p≥αӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p.Further, we write(30)αӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ≤αӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ−αΨm−1ӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ≤αӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ−Ψm−1αӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ≤p−Ψm−1p.Therefore, we get(31)Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p−Ψm−1p=p−Ψm−1pӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ≥α.Case 2.
Supposeӿ1,ӿ2,⋯,ӿk−1,ӿk<p; then,
(32)ӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ≤ӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ−Ψm−1ӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ≤p−Ψm−1p.Therefore, we get(33)Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p−Ψm−1p=1≥α.Case 3.
Supposep≤0 and Mnӿ1,ӿ2,⋯,ӿk−1,ӿk,p=0; then,
(34)Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,p−Ψm−1p=1≥α.Similarly, we can prove these conditions for abstinence and falsity grades. Hence, the solution is completed. Further, we instigate more results based onBPF to show the proficiency of the discussed results.Theorem 11.
For aBPFӾUNI,Μm,Αa,Νn,∗ct,°ctc, the grade of truth, abstinence, and falsity satisfies the conditions of Definition 4. Now, we define the decreasing mapping Γ1 and increasing mappings Γ2 and Γ3, such that Γ1:0,+∞⟶0,1 and Γ2:0,+∞⟶0,1,Γ3:0,+∞⟶0,1 with Ț:ӾUNI⟶ӾUNI, such that
(35)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥α⟹Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ1−1Γ1PΘ1P≥αAnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α⟹Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ2−1Γ2PΘ2P<1−αNnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α⟹Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ3−1Γ3PΘ3P<1−α,where Θ1∈₸m−1, Θ2∈₸a−2, and Θ3∈₸n−3, for all ӿ1,ӿ2,⋯,ӿk−1∈ӾUNI,ӿ,ӳ∈ӾUNI,p>0 with α∈0,1. Then, Ț has a unique fixed point in ӾUNI.Proof.
Letӿ0∈ӾUNI with ӿk+1=Țӿk∀k∈ℕNn. By using Remark 7 and inequality (35), we get
(36)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≤Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ1−1Γ1PΘ1P,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ2−1Γ2PΘ2P,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ3−1Γ3PΘ3P.
Further, we write the above equations as(37)Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p≤Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ1−1Γ1PΘ1P≤Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p,Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p≥Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ2−1Γ2PΘ2P≥Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p,Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p≥Nnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ3−1Γ3PΘ3P≤Nnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p.
It is clear from the above analysis thatMnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p is a bounded nondecreasing sequence and Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p and Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p are the bounded nonincreasing sequences. Then, the limit of these equations exists. We suppose that
(38)limk→∞Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p<β1<1,limk→∞Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p>β2>1,limk→∞Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p>β3>1.
Therefore, we have(39)Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q≥Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q,Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q≤Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q,Nnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q≤Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q,0<p≤ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β1≤ӿ2,ӿ3,⋯,ӿk−1,ӿk+1−ӿkβ1,0<p≤ӿ2,ӿ3,⋯,ӿk−1,ӿk+1−ӿkβ2≤ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β2,0<p≤ӿ2,ӿ3,⋯,ӿk−1,ӿk+1−ӿkβ3≤ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β3.
Then, the limit of these equations also exists. We have(40)limk→∞ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1=b1,limk→∞ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ2=b2,limk→∞ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ3=b3.
IfMnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q≥β1, Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q<β2, and Nnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q<β3, then
(41)Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ1−1Γ1PΘ1P≥Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q≥β1,Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ2−1Γ2PΘ2P≤Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q<β2,Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ3−1Γ3PΘ3P≤Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q<β3.
Therefore,(42)ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β1≤Θ1−1Γ1PΘ1P,Θ1ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β1≤Γ1PΘ1P≤Γ1ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1Θ1P≤Γ1b1Θ1P.
Similarly, we can find that(43)Θ2ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β2≥Θ2−1Γ2PΘ2P≥Γ2b2Θ2P,Θ3ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β3≥Θ3−1Γ3PΘ3P≥Γ3b3Θ3P.
And it is clear thatp⟶ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1,ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ2,ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ3; then,
(44)Θ1ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β1≤Γ1b1Θ1ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1.
Again,(45)Θ2ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β2≥Γ2b2Θ2ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ2,Θ3ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β3≥Γ3b3Θ3ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ3.
Thus, we get(46)Θ1b1≤Γ1b1Θ1b1⟹Γ1b1≥1,Θ2b2≥Γ2b2Θ2b2⟹Γ2b2≤1,Θ3b3≥Γ3b3Θ3b3⟹Γ3b3≤1,which is a contradiction; hence,
(47)limk→∞Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p=1,limk→∞Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p=0,limk→∞Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p=0.
Supposep,ϵ>0. We have
(48)Mnӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,p2−Θ1−1Γ1P2Θ1P2≥1−ϵ.
Similarly, for abstinence and falsity grades, we have(49)Anӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,p2−Θ2−1Γ2P2Θ2P2<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,p2−Θ3−1Γ3P2Θ3P2<ϵ.
By using the above analysis, we get, ifMnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿK,p/2≥1−ϵ and Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿK,p/2,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿK,p/2<ϵ, then
(50)Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿK,p2≥minMnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ȚӿK,p2−Θ1−1Γ1P2Θ1P2,Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿK,Θ1−1Γ1P2Θ1P2≥minMnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿK,p2,Mnӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,Θ1−1Γ1P2Θ1P2≥1−ϵ.
Similarly, resolving the grades of abstinence and falsity, we have(51)Anӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,Θ2−1Γ2P2Θ2P2<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,ӿK+1−ӿK,Θ3−1Γ3P2Θ3P2<ϵ.
Therefore,(52)Mnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿK,p2≥1−ϵ.
Also, we note(53)Anӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿK,p2<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿK,p2<ϵ.
Then, for allk≥ℕNn,
(54)Mnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿὛ,p≥minMnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿK,p,Mnӿ1,ӿ2,⋯,ӿk−1,ӿὛ−ӿK,p≥1−ϵ,Mnӿ1,ӿ2,⋯,ӿk−1,ӿὛ−ӿK,p≥1−ϵ.
Further, we find(55)Anӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿὛ,p<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿὛ,p<ϵ.
Sinceϵ is arbitrary and the sequence ӿf is Cauchy, hence they are convergent. Therefore, limӿf=ӿ.
Supposep,ϵ>0; then, there exists k0∈ℕNn such that
(56)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿk,p−Θ1−1Γ1pΘ1p≥1−ϵ.
Similarly, observing for abstinence and falsity grades, we have(57)Anӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿ,p−Θ2−1Γ2pΘ2p<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,ӿk−ӿ,p−Θ3−1Γ3pΘ3p<ϵ,for all k≥k0. Hence,
(58)Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p≥minMnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿk+1,p−Θ1−1Γ1pΘ1p,Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿ,Θ1−1Γ1pΘ1p≥minMnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӿk,p,Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿ,Θ1−1Γ1pΘ1p≥1−ϵ,Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p<ϵ,Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p<ϵ,for all k≥k0. Therefore,
(59)Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p=1,Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p=0,Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−ӿ,p=0,for all p>0. Hence, Țӿ=ӿ; that is, Ț has a fixed point in ӾUNI. Next, we prove the uniqueness of the fixed point. For this, we suppose ӳ is another fixed point Ț in ӾUNI; then,
(60)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥minMnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ1−1Γ1pΘ1p=Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p.
Hence,p=Θ1−1Γ1pΘ1p⟹Θ1p=Γ1pΘ1p,
(61)p=Θ2−1Γ2pΘ2p⟹Θ2p=Γ2pΘ2p,p=Θ3−1Γ3pΘ3p⟹Θ3p=Γ3pΘ3p.
Therefore,Γ1p=1, Γ2p=1, and Γ3p=1. It is a contradiction; thus, for all k∈ℕNn and k⟶∞, we obtain
(62)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=1,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=0,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=0,for all p>0. Hence, ӿ=ӳ. Hence, Ț has a unique fixed point in ӾUNI.Example 12.
For a Banach spaceӾUNI,., we define the decreasing mapping Γ1 and increasing mappings Γ2 and Γ3, such that Γ1:0,+∞⟶0,1 and Γ2:0,+∞⟶0,1,Γ3:0,+∞⟶0,1, and Ț:ӾUNI⟶ӾUNI are such that for all ӿ,ӳ∈ӾUNI,
(63)Θ1Țӿ−Țӳ≤Γ1ӿ−ӳ−Θ1ӿ−ӳ,Θ2Țӿ−Țӳ≥Γ2ӿ−ӳ−Θ2ӿ−ӳ,Θ3Țӿ−Țӳ≥Γ3ӿ−ӳ−Θ3ӿ−ӳ,where Θ1∈₸m−1, Θ2∈₸a−2, and Θ3∈₸n−3. Suppose that Γ1Θ1 is nondecreasing and Γ2Θ2,Γ3Θ3 are nonincreasing functions with
(64)β1Θ1−1Γ1pΘ1p≤Θ1−1Γ1β1pΘ1β1p,β2Θ2−1Γ2pΘ2p≥Θ2−1Γ2β2pΘ2β2p,β3Θ3−1Γ3pΘ3p≥Θ3−1Γ3β3pΘ3β3p,for all p∈0,+∞,β1,β2,β3∈0,1. Further, define picture fuzzy n-norm Mn,An,Nn as in Example 10. Consider that
(65)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥α,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α.By using the three cases of Example10 and using Theorem 11, we explore that the function Ț has a unique fixed point in XUNI. Hence, the solution is completed. Further, we have utilized more results based on BPF to show the proficiency of the proven approaches.Theorem 13.
LetBPFӾUNI,Μm,Αa,Νn,∗ct,°ctc. Let the grade of truth, abstinence, and falsity satisfy the conditions of Definition 4. Now, we define the mapping Ț:ӾUNI⟶ӾUNI, such that
(66)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥α⟹Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ1−1Θ1P−Θ1′P≥αAnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α⟹Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ2−1Θ2P−Θ2′P<1−αNnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α⟹Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ3−1Θ3P−Θ3′P<1−α,where Θ1P,Θ1′P∈₸m−1, Θ2P,Θ2′P∈₸a−2, and Θ3P−Θ3′P∈₸n−3, for all ӿ1,ӿ2,⋯,ӿk−1∈ӾUNI,ӿ,ӳ∈ӾUNI,p>0 with α∈0,1 and Θ1P≥Θ1′P,Θ2P≥Θ2′P,Θ3P≥Θ3′P. Then, Ț has a unique fixed point in ӾUNI.Proof.
Letӿ0∈ӾUNI with ӿk+1=Țӿk∀k∈ℕNn. By using Remark 7 and equation (66), we get
(67)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≤Mnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ1−1Θ1P−Θ1′P,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥Anӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ2−1Θ2P−Θ2′P,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥Nnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ3−1Θ3P−Θ3′P.
Further, from the above equations, we obtain(68)Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p≤Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ1−1Θ1P−Θ1′P≤Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p,Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p≥Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ2−1Θ2P−Θ2′P≥Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p,Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p≥Nnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ3−1Θ3P−Θ3′P≤Nnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,p.
It is clear from the above analysis thatMnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p is a bounded nondecreasing sequence and the sequences Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p and Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p are bounded and nonincreasing. Then, the limit of these equations exists. We suppose that
(69)limk→∞Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p<β1<1,limk→∞Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p>β2>1,limk→∞Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p>β3>1.
Therefore, we have(70)Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q≥Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q,Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q≤Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q,Nnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q≤Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q,0<p≤ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β1≤ӿ2,ӿ3,⋯,ӿk−1,ӿk+1−ӿkβ1,0<p≤ӿ2,ӿ3,⋯,ӿk−1,ӿk+1−ӿkβ2≤ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β2,0<p≤ӿ2,ӿ3,⋯,ӿk−1,ӿk+1−ӿkβ3≤ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β3.
Then, the limit of these equations also exists. We have(71)limk→∞ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1=b1,limk→∞ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ2=b2,limk→∞ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ3=b3.
IfMnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q≥β1, Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q<β2, and Nnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,q<β3, then
(72)Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ1−1Θ1P−Θ1′P≥Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q≥β1,Anӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ2−1Θ2P−Θ2′P≤Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q<β2,Mnӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1,Θ3−1Θ3P−Θ3′P≤Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,q<β3.
Therefore,(73)ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β1≤Θ1−1Θ1P−Θ1′P,Θ1ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β1≤Θ1P−Θ1′P.
Similarly, we can find(74)Θ2ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β2≥Θ2P−Θ2′P,Θ3ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β3≥Θ3P−Θ3′P.
Clearly,p⟶ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1,ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ2,ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ3; hence,
(75)Θ1ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β1≤Θ1ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1−Θ1′ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1.
Also, we write(76)Θ2ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β2≥Θ2ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1−Θ2′ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1,Θ3ӿ2,ӿ3,⋯,ӿk,ӿk+2−ӿk+1β3≥Θ3ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1−Θ3′ӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿkβ1.
Thus, we get(77)0≤Θ1p≤Θ1b1≤Θ1b1−Θ1′b1≤Θ1b1.
Similarly,(78)Θ2p≥Θ2b2≥Θ2b2−Θ2′b2≥Θ2b2,Θ3p≥Θ3b3≥Θ3b3−Θ3′b3≥Θ3b3.
It is a contradiction; hence,(79)limk→∞Mnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p=1,limk→∞Anӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p=0,limk→∞Nnӿ1,ӿ2,⋯,ӿk−1,ӿk+1−ӿk,p=0.
The rest of the proof to express for allp>0 the equation Țӿ=ӿ can be obtained using the similar technique of Theorem 9 and Theorem 11; that is, Ț has a fixed point in ӾUNI. Next, we prove the uniqueness of the fixed point. For this, we suppose ӳ is another fixed point Ț in ӾUNI; then,
(80)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥minMnӿ1,ӿ2,⋯,ӿk−1,Țӿ−Țӳ,Θ1−1Θ1P−Θ1′P=Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p.
Hence,p=Θ1−1Θ1P−Θ1′P⟹Θ1p=Θ1P−Θ1′P,
(81)p=Θ2−1Θ2P−Θ2′P⟹Θ2p=Θ2P−Θ2′P,p=Θ3−1Θ3P−Θ3′P⟹Θ3p=Θ3P−Θ3′P.
Therefore,Θ1′p=1, Θ2′p=1, and Θ3′p=1. It is a contradiction; thus, as k⟶∞, we get
(82)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=1,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=0,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p=0,for all p>0. Hence, ӿ=ӳ. Thus, Ț has a unique fixed point in ӾUNI.Example 14.
For a Banach spaceӾUNI,., we define the mapping Ț:ӾUNI⟶ӾUNI such that for all ӿ,ӳ∈ӾUNI,
(83)Θ1Țӿ−Țӳ≤Θ1ӿ−ӳ−Θ1′ӿ−ӳ,Θ2Țӿ−Țӳ≥Θ2ӿ−ӳ−Θ2′ӿ−ӳ,Θ3Țӿ−Țӳ≥Θ3ӿ−ӳ−Θ3′ӿ−ӳ,where Θ1,Θ1′∈₸m−1, Θ2,Θ2′∈₸a−2, and Θ3,Θ3′∈₸n−3. Suppose Θ1−Θ1′ is nondecreasing and Θ2−Θ2′,Θ3−Θ3′ are nonincreasing functions with
(84)β1Θ1−1Θ1−Θ1′≤Θ1−1Θ1β1p−Θ1′β1p,β2Θ2−1Γ2pΘ2p≥Θ2−1Θ2β2p−Θ2′β2p,β3Θ3−1Γ3pΘ3p≥Θ3−1Θ3β3p−Θ3′β3p,for all p∈0,+∞ and β1,β2,β3∈0,1. Further, define picture fuzzy n-norm Mn,An,Nn as in Example 10. Consider that
(85)Mnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p≥α,Anӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α,Nnӿ1,ӿ2,⋯,ӿk−1,ӿ−ӳ,p<1−α.We explore that the functionȚ has a unique fixed point in XUNI. Hence, the solution is completed.


## 5. Conclusion

A picture fuzzy set is more proficient and more capable than an intuitionistic fuzzy set and fuzzy to cope with uncertain and unpredictable information in realistic issues. Keeping the advantages of the picture fuzzy set and an-norm linear space, the manuscript made the following advancements in the existing literature:
(1)
The novel picture fuzzyn-norm linear space and its basic properties are explored(2)
Some novel contractive conditions based onNPF are presented. By using these contractive conditions, we have explored some fixed point theorems for a picture fuzzy n-Banach space (BPF). It was observed that these results are more modified and more general than the existing ones in the literature, which are based on intuitionistic fuzzy n-Banach spaces (BIF) and fuzzy n-Banach spaces(3)
The reliability and effectiveness of the obtained main theorems are expressed, and several examples are presented afterwards



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