# Solving Nonlinear Fractional Differential Equations for Contractive and Weakly Compatible Mappings in Neutrosophic Metric Spaces

Authors: Usman Ali; Haifa A. Alyousef; Umar Ishtiaq; Khalil Ahmed; Shajib Ali
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we aim to prove various unique fixed point results for contractive and weakly compatible mappings in the sense of neutrosophic metric spaces. Several nontrivial examples are also imparted. To support main result, uniqueness of solution of nonlinear fractional differential equations is examined.


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## Body

## 1. Introduction and Preliminaries

Uncertainty and fuzziness are prevalent in many applications in today’s complicated world. For capturing the ambiguity and fuzziness of information, Zadeh [1] first developed the concept of fuzzy sets (FSs). Many extensions of FSs, such as intuitionistic fuzzy sets (IFSs), picture FSs, q-rung orthopair FSs, and neutrosophic sets (NSs), have been proposed since its inception to better convey complicated information. Researchers and scholars have gathered a large number of research findings related to their decision-making theories and approaches.As a generalization of FSs, Atanassove [2] proposed and explored IFSs. With the use of continuous t-norms (CTNs) and continuous t-conorms (CTCNs), Park [3] established the concept of intuitionistic fuzzy metric space (IFMS) in 2004. Alaca et al. [4] proposed IFMS with the help of CTNs and CTCNs, a generalization of fuzzy metric space according to Kramosil and Michalek [5] in 2006, using the idea of IFSs and proved various fixed point theorems for contraction mappings. Hu [6] did nice work for different type contractions. Smarandache [7] proposed NSs, which is a generalization of IFSs. Kirişci and Simşek [8] proposed the concept of neutrosophic metric space (NMS), based on the concept of NSs and proved several theorems in the proposed space. Ishtiaq et al. [9] proposed the notion of orthogonal NMS and proved some fixed point results in the sense of complete orthogonal NMS. Several fixed point results for generalized contractions in NMS were demonstrated by Sowndrara et al. [10]. Gulzar et al. [11] used the notion of FSs in subgroups and proved numerous nice results. Gulzar et al. [12] used the notion of FSs in the structure of field and did an amazing work. Several fixed point results for weakly compatible mappings were proved by Sharma et al. [13] in the structure of IFMS. Davvaz et al. [14] did exquisite work by using IFSs. Simsek and Kirişci [15] used the notion of NMS and proved various fixed point theorems.We aim to establish a number of unique fixed point results for contractive and weakly compatible mappings in the context of NMS in this paper. A number of nontrivial examples are also presented. The uniqueness of solutions to nonlinear fractional differential equations is investigated to support the main result.First, we present definitions of CTN, CTCN, IFMS, NSs, and NMS, contraction mapping, and weekly contractive mapping that are helpful for this study.I=0,1 is used in this study.Definition 1 (see [2]).
A binary operation∗:I×I⟶I is called a CTN ifS1.Þ∗ϖ=ϖ∗Þ,for allÞ,ϖ∈I;S2.∗ is continuous;S3.Þ∗1=Þ,for allÞ∈I;S4.Þ∗ϖ∗z=Þ∗ϖ∗z,for allÞ,ϖ,z∈I;S5.Þ≤c and ϖ≤d, with Þ,ϖ,c,d∈I, then Þ∗ϖ≤c∗d.Definition 2 (see [2]).
A binary operation○:I×I⟶I is called a CTCN ifC1.Þ○ϖ=ϖ○Þ,for allÞ,ϖ∈I;C2.○ is continuous;C3.Þ○0=0,for allÞ∈I;C4.Þ○ϖ○z=Þ○ϖ○z,for allÞ,ϖ,z∈I;C5.Þ≤c and ϖ≤d, with Þ,ϖ,c,d∈I, then Þ○ϖ≤c○d.Definition 3 (see [4]).
LetX be nonempty and ∗ be a CTN and ○ be a CTCN. Let MandN be FSs on X2×0,+∞, if the following conditions are satisfied:FS1.MÞ,ϖ,Θ>0;FS2.MÞ,ϖ,Θ+NÞ,ϖ,Θ≤1;FS3.MÞ,ϖ,Θ=1for allΘ>0,if and only ifÞ=ϖ;FS4.MÞ,ϖ,Θ=Mϖ,Þ,Θ;FS5.MÞ,z,Θ+∂≥MÞ,ϖ,Θ∗Mϖ,z,∂;FS6.MÞ,ϖ,⋅:0,+∞⟶0,1 is continuous and limΘ⟶+∞MÞ,ϖ,Θ=1;FS7.NÞ,ϖ,Θ>0;FS8.NÞ,ϖ,Θ=0for allΘ>0,if and only ifÞ=ϖ;FS9.NÞ,ϖ,Θ=Nϖ,Þ,Θ;FS10.NÞ,z,Θ+∂≤NÞ,ϖ,Θ○Nϖ,z,∂;FS11.NÞ,ϖ,⋅:0,+∞⟶0,1 is continuous and limΘ⟶+∞NÞ,ϖ,Θ=0.
Then,X,M,N,∗,○ is called an IFMS.Definition 4 (see [7]).
Let a setX be nonempty and Þ∈Ε. A NS G in X is classified by a truth-membership function, MGÞ, an indeterminacy-membership function NGÞ, and a falsity-membership function OGÞ. The functions MGÞ, NGÞ, and OGÞ are subsets of ]0−,1+[, that is, MGÞ:X⟶]0−,1+[, NGÞ:X⟶]0−,1+[, and OGÞ:X⟶]0−,1+[. So,(1)0−≤supMGÞ+supNGÞ+supOGÞ≤3+.Definition 5 (see [8]).
LetX be nonempty and ∗ be a CTN and ○ be a CTCN. M,N,andO are NSs on X×X×0,+∞, then X,M,N,O is named neutrosophic metric on X, if for all Þ,ϖ,z∈X, the below circumstances are satisfied:(nm1).MÞ,ϖ,Θ+NÞ,ϖ,Θ+OÞ,ϖ,Θ≤3;(nm2).MÞ,ϖ,Θ>0;(nm3).MÞ,ϖ,Θ=1for allΘ>0,if and only ifÞ=ϖ;(nm4).MÞ,ϖ,Θ=Mϖ,Þ,Θ;(nm5).MÞ,z,Θ+∂≥MÞ,ϖ,Θ∗Mϖ,z,∂;(nm6).MÞ,ϖ,⋅:0,+∞⟶0,1 is continuous and limΘ⟶+∞MÞ,ϖ,Θ=1;(nm7).NÞ,ϖ,Θ<1;(nm8).NÞ,ϖ,Θ=0for allΘ>0,if and only ifÞ=ϖ;(nm9).NÞ,ϖ,Θ=Nϖ,Þ,Θ;(nm10).NÞ,z,Θ+∂≤NÞ,ϖ,Θ○Nϖ,z,∂;(nm11).NÞ,ϖ,⋅:0,+∞⟶0,1 is continuous and limΘ⟶+∞NÞ,ϖ,Θ=0;(nm12).OÞ,ϖ,Θ<1;(nm13).OÞ,ϖ,Θ=0for allΘ>0,if and only ifÞ=ϖ;(nm14).OÞ,ϖ,Θ=Oϖ,Þ,Θ;(nm15).OÞ,z,Θ+∂≤OÞ,ϖ,Θ○Oϖ,z,∂;(nm16).OÞ,ϖ,⋅:0,+∞⟶0,1 is continuous and limΘ⟶+∞OÞ,ϖ,Θ=0;(nm17). IfΘ≤0, then MÞ,ϖ,Θ=0,NÞ,ϖ,Θ=1andOÞ,ϖ,Θ=1.
Then,X,M,N,O,∗,○ is called an NMS.
Here, functionsMÞ,ϖ,Θ,NÞ,ϖ,Θ, and OÞ,ϖ,Θ, respectively, denote the degrees of nearness, nonnearness, and indeterminacy. In X,M,N,O,∗,○, the following conditions hold for all Þ,ϖ∈X,andΘ>0.(i)
limΘ⟶+∞MÞ,ϖ,Θ=1,limΘ⟶+∞NÞ,ϖ,Θ=0,andlimΘ⟶+∞OÞ,ϖ,Θ=0.(ii)
IfÞn⟶Þ,ϖn⟶ϖandΘn⟶Θ, then NÞn,ϖn,Θn=NÞ,ϖ,ϖ.(iii)
A sequenceÞn converges to Þ∈X if the following limits exist:(2)limΘ⟶+∞MÞn,ϖ,Θ=1,limΘ⟶+∞NÞn,ϖ,Θ=0andlimΘ⟶+∞OÞn,ϖ,Θ=0.(iv)
A sequenceÞn∈X is a Cauchy sequence if and only if for each ε∈0,1,there existsn0∈ℕ such that(3)limΘ⟶+∞MÞn+σ,Þn,Θ=1,limΘ⟶+∞NÞn+σ,Þn,Θ=0,(v)
and(4)limΘ⟶+∞OÞn+σ,Þn,Θ=0for allσ>0andΘ>0.(vi)
Every Cauchy sequence is convergent inX if and only if NMS is complete.(vii)
Every sequence contains a convergent subsequence if and only if NMS is compact.(viii)
A mappingT:X⟶X is a neutrosophic contractive mapping if there exists k∈0,1,for allÞ,ϖ∈X,andΘ>0, such that(5)1MTÞ,Tϖ,Θ≤k1MÞ,ϖ,Θ−1,NTÞ,Tϖ,Θ≤kNÞ,ϖ,Θ,(ix)
and(6)OTÞ,Tϖ,Θ≤kOÞ,ϖ,Θ.
Now, we state several useful definitions from [13].(i)
Letsup0<Θ<1ΔΘ,Θ=1. A CTN Δ is said to be H-type if ΔmΘm=1+∞ is the family of functions and is equicontinuous at Θ=1, where(7)Δ1Θ=ΘΔΘ,Δm+1Θ=ΘΔΔmΘ,m=1,2,…,Θ∈0,1.(ii)
For anyλ∈0,1, clearly that Δ is a CTN of H-type if and only if, there existsδλ∈0,1 such that ΔmΘ<λfor allm∈ℕ,Θ<δ.(iii)
ConsiderФ=∅|∅:ℝ+⟶ℝ+ each ∅∈Ф fulfill the below assertions:(iv)
∅ is nondecreasing;(v)
∅ is an upper semicontinuous from the right;(vi)
∑n=0+∞∅nΘ<+∞,for allΘ>0 where ∅n+1Θ=∅(∅nΘ,n∈ℕ, and also∅Θ<Θfor allΘ>0.(vii)
A coupled fixed point is an elementÞ,ϖ∈X×X of the mapping T:X×X⟶X if TÞ,ϖ=Þ and Tϖ,Þ=ϖ.(viii)
A coupled coincidence point is an elementÞ,ϖ∈X×X of the mapping T:X×X⟶X and G:X⟶X if TÞ,ϖ=GÞ and Tϖ,Þ=Gϖ.(ix)
A common coupled fixed point is an elementÞ,ϖ∈X×X of the mapping T:X×X⟶X and G:X⟶X if TÞ,ϖ=GÞ=Þ and Tϖ,Þ=Gϖ=ϖ.(x)
A common fixed point is an elementÞ∈X of the mapping T:X×X⟶X and G:X⟶X if TÞ,Þ=GÞ=Þ.(xi)
The mappingsT:X×X⟶XandG:X×X⟶X are said to be weakly compatible mapping if TÞ,ϖ=GÞ,Tϖ,Þ=Gϖ implies that(8)GTÞ,ϖ=TGÞ,Gϖ,GTϖ,Þ=TGϖ,GÞfor allÞ,ϖ∈X.(xii)
IfTandG are weakly compatible, then maybe they are not compatible, but the converse is true.


## 2. Main Results

This section contains several results for contraction mappings.Definition 6.
LetX,M,N,O,∗,○ be a NMS. The mapping T:X×X⟶X and G:X⟶X are called compatible if the following limits exist:(9)limn⟶+∞MGTÞn,ϖn,TGÞn,Gϖn,Θ=1,limn⟶+∞MGTϖn,Þn,TGϖn,GÞn,Θ=1,limn⟶+∞NGTÞn,ϖn,TGÞn,Gϖn,Θ=0,limn⟶+∞NGTϖn,Þn,TGϖn,GÞn,Θ=0.
For allΘ>0 and(10)limn⟶+∞OGTÞn,ϖn,TGÞn,Gϖn,Θ=0,limn⟶+∞OGTϖn,Þn,TGϖn,GÞn,Θ=0.
WheneverÞn and ϖn are sequence in X, such that(11)limn⟶+∞TÞn,ϖn=limn⟶+∞GÞn=Þandlimn⟶+∞Tϖn,Þn=limn⟶+∞Gϖn=ϖfor allÞ,ϖ∈X.Lemma 1.
LetX,M,N,O,∗,○ be a NMS, where ∗and○ are CTN and CTCN of H-type, if there exists ∅∈Ф, such that(12)MÞ,ϖ,∅Θ≥MÞ,ϖ,Θ,NÞ,ϖ,∅Θ≤NÞ,ϖ,Θ,and(13)OÞ,ϖ,∅Θ≤OÞ,ϖ,Θfor allΘ>0,thenÞ=ϖ.By the help of [13], we have the following definition.Definition 7.
Let∅:0,1×0,1⟶−1/2,1/2 be a mapping fulfilling the below assertions:(i)
∅1,1=0,∅0,0=0,(ii)
∅Θ,∂<1/∂+1−1/Θ+1,(iii)
IfΘn,∂n are any two sequences in [0, 1] such that limn⟶+∞Θn=limn⟶+∞∂n<1,thenlimn⟶+∞∅Θn,∂n<0.Definition 8.
LetX,M,N,O,∗,○ be a NMS. A mapping T:X⟶X is said to be a contractive mapping (CM) with respect to ∅ if it fulfills the below assertions:(14)∅MTÞ,Tϖ,Θ,MÞ,ϖ,Θ≥0for allÞ,ϖ∈X,∅NTÞ,Tϖ,Θ,NÞ,ϖ,Θ≤0for allÞ,ϖ∈X,∅OTÞ,Tϖ,Θ,OÞ,ϖ,Θ≤0.for allÞ,ϖ∈X.Remark 1.
From the definition of∅, it is clear that for all ∂>Θ,∅Θ,∂≤0, also ∅Θ,∂>0for all∂<Θ. If it is CM w.r.t. ∅, then(15)MTÞ,Tϖ,Θ>MÞ,ϖ,Θ,NTÞ,Tϖ,Θ<NÞ,ϖ,Θ,OTÞ,Tϖ,Θ<OÞ,ϖ,Θ.Lemma 2.
SupposeX,M,N,O,∗,○ is a NMS and T is a contraction with respect to ∅, then the fixed point of TinX is unique, if exists.Proof.
SupposeÞ∈X and TÞ=Þ. Let ϖ∈X be another fixed point of T distinct from Þ,i.e.,TÞ=ÞandÞ≠ϖ, then by the definition, we have(16)0≤∅MTÞ,Tϖ,Θ=MÞ,ϖ,Θ,for allÞ,ϖ∈X,0≥∅NTÞ,Tϖ,Θ=NÞ,ϖ,Θ,for allÞ,ϖ∈X,0≥∅OTÞ,Tϖ,Θ=OÞ,ϖ,Θ,for allÞ,ϖ∈X.
This is contradiction, henceÞ=ϖ.Theorem 1.
SupposeX,M,N,O,∗,○ is a complete NMS and T is a contraction w.r.t. ∅, then Thas a unique fixed point inX.Proof.
AssumeÞ0∈X is a point and suppose Þnis a sequence inX such that Þn=TÞn−1for alln∈ℕ. Now, we suppose that (without the loss of generality), Þn≠Þn+1for alln∈ℕ; also, if there existsn0 such that Þn=Þn0+1,thenÞn=Þn0+1=TÞn0. This deduces that TÞn0=Þn0.
Currently, by contradiction, we will examine thatlimn⟶+∞MÞn,Þn+1,Θ=1,limn⟶+∞NÞn,Þn+1,Θ=0,andOÞn,Þn+1,Θ=0for allΘ>0. Suppose that there exist some Θ0 such that limn⟶+∞MÞn,Þn+1,Θ<1,limn⟶+∞NÞn,Þn+1,Θ>0,andlimn⟶+∞OÞn,Þn+1,Θ>0for alln∈ℕ. Then, we have(17)0≤∅MTÞ,Tϖ,Θ,MÞ,ϖ,Θ<1MÞn−1,Þn,Θ0−1MÞn,Þn+1,Θ0,0≥∅NTÞ,Tϖ,Θ,NÞ,ϖ,Θ<1NÞn−1,Þn,Θ0−1NÞn,Þn+1,Θ0,and(18)0≥∅OTÞ,Tϖ,Θ,OÞ,ϖ,Θ<1OÞn−1,Þn,Θ0−1OÞn,Þn+1,Θ0for alln∈ℕ.
Now, sinceMÞn−1,Þn,Θ0 is nondecreasing sequence and NÞn−1,Þn,Θ0 and OÞn−1,Þn,Θ0 are nonincreasing sequences of ℝ+, then there exist l≤1,σ≥0andh≥0 such that(19)limn⟶+∞MÞn−1,Þn,Θ0=l,limn⟶+∞NÞn−1,Þn,Θ0=σ,limn⟶+∞OÞn−1,Þn,Θ0=h.
Now, by contradiction, we will prove thatl=1,σ=0,andh=0. Using (iii), let l<1,Θn=MÞn,Þn+1,Θ0,and∂n=MÞn−1,Þn,Θ0. So, it concluded that(20)0≤∅MÞn,Þn+1,Θ0,MÞn−1,Þn,Θ0<0.
This is a contradiction, hencel=1,i.e.,limn⟶+∞MÞn,Þn+1,Θ0=1.
Letσ>0andΘn=NÞn,Þn+1,Θ0,and∂n=NÞn−1,Þn,Θ0. So, it concluded that(21)0≥∅NÞn,Þn+1,Θ0,NÞn−1,Þn,Θ0>0.
This is a contradiction, henceσ=0,i.e.,limn⟶+∞NÞn,Þn+1,Θ0=0.
Letu>0andΘn=OÞn,Þn+1,Θ0and∂n=OÞn−1,Þn,Θ0. So, it concluded that(22)0≥∅OÞn,Þn+1,Θ0,OÞn−1,Þn,Θ0>0.
This is a contradiction, henceh=0,i.e.,limn⟶+∞OÞn,Þn+1,Θ0=0.
Now, we show thatÞn is a Cauchy sequence in X. Let us assume that Þn is not a Cauchy sequence in X, then there existε∈0,1 and two subsequences ÞmkandÞnkofÞn such that nk>mk≥k, then(23)MÞmk,Þnk,Θ0≤1−ε,NÞmk,Þnk,Θ0≥εandOÞmk,Þnk,Θ0≥ε,and(24)MÞmk,Þnk−1,Θ0≥1−ε,NÞmk,Þnk−1,Θ0≤εandOÞmk,Þnk−1,Θ0≤ε.
Using triangular inequalities (23) and (24), we have(25)1−ε≥MÞmk,Þnk,Θ0≥MÞmk,Þnk−1,Θ0∗MÞnk−1,Þnk,Θ0≥1−ε∗MÞnk−1,Þnk,Θ0.
Therefore,k⟶+∞1−ε≥limk⟶+∞MÞmk,Þnk,Θ0≥1−ε, also k⟶+∞,(26)ε≤NÞmk,Þnk,Θ0≤NÞmk,Þnk−1,Θ0○NÞnk−1,Þnk,Θ0≤ε○NÞnk−1,Þnk,Θ0ε≤limk⟶+∞NÞmk,Þnk,Θ0≤ε,and(27)ε≤OÞmk,Þnk,Θ0≤OÞmk,Þnk−1,Θ0○OÞnk−1,Þnk,Θ0≤ε○OÞnk−1,Þnk,Θ0ε≤limk⟶+∞OÞmk,Þnk,Θ0≤ε.
Hence,(28)limk⟶+∞MÞmk,Þnk,Θ0=1−ε,limk⟶+∞NÞmk,Þnk,Θ0=ε,limk⟶+∞OÞmk,Þnk,Θ0=ε.
Then, we obtain(29)1−ε≥MÞmk,Þnk,Θ0≥MÞmk,Þmk−1,Θ0∗MÞmk−1,Þnk−1,Θ0∗MÞnk−1,Þnk,Θ0,MÞmk−1,Þmk−1,Θ0≥MÞmk−1,Þmk,Θ0∗MÞmk,Þnk,Θ0∗MÞnk,Þnk−1,Θ0.
Ask⟶+∞, we get(30)limk⟶+∞MÞmk−1,Þnk−1,Θ0=1−ε,∅1−ε,1−ε<0,and(31)ε≤NÞmk,Þnk,Θ0≤NÞmk,Þmk−1,Θ0○NÞmk−1,Þnk−1,Θ0○NÞnk−1,Þnk,Θ0,NÞmk−1,Þmk−1,Θ0≤NÞmk−1,Þmk,Θ0○NÞmk,Þnk,Θ0○NÞnk,Þnk−1,Θ0.
Ask⟶+∞, we get(32)limk⟶+∞NÞmk−1,Þnk−1,Θ0=ε,limk⟶+∞sup∅NÞmk,Þnk,Θ0,NÞmk−1,Þnk−1,Θ0>0,∅ε,ε>0,also(33)ε≤OÞmk,Þnk,Θ0≤OÞmk,Þmk−1,Θ0○OÞmk−1,Þnk−1,Θ0○OÞnk−1,Þnk,Θ0,OÞmk−1,Þmk−1,Θ0≤OÞmk−1,Þmk,Θ0○OÞmk,Þnk,Θ0○OÞnk,Þnk−1,Θ0.
Ask⟶+∞, we get(34)limk⟶+∞OÞmk−1,Þnk−1,Θ0=ε,limk⟶+∞sup∅OÞmk,Þnk,Θ0,OÞmk−1,Þnk−1,Θ0>0,∅ε,ε>0.
These inequalities are obviously untrue andÞn is a Cauchy sequence in X.X,M,N,O,∗,○ is complete, so Þn converges to some point u∈X, i.e.,(35)limn⟶+∞MÞn,u,Θ=1,limn⟶+∞NÞn,u,Θ=0andlimn⟶+∞OÞn,u,Θ=0for allΘ>0.
Now, we examine thatu is a fixed point of T. Let Tu≠u, then(36)Mu,Tu,Θ<1,Nu,Tu,Θ>0andlimn⟶+∞OÞn,u,Θ=0,0≤limn⟶+∞sup∅MTÞn,Tu,Θ,MÞn,u,Θ,0≤limn⟶+∞sup1MÞn,u,Θ−1MTÞn,Tu,Θ,0≤limn⟶+∞sup1MÞn,u,Θ−1MTÞn,Tu,Θ,0≤1−1Mu,Tu,Θ,1≤Mu,Tu,Θ,and(37)0≥limn⟶+∞sup∅NTÞn,Tu,Θ,NÞn,u,Θ,0≥limn⟶+∞sup1NÞn,u,Θ−1NTÞn,Tu,Θ,0≥limn⟶+∞sup1NÞn,u,Θ−1NTÞn+1,Tu,Θ,0≥10−1Nu,Tu,Θ,0≥Nu,Tu,Θ,also(38)0≥limn⟶+∞sup∅OTÞn,Tu,Θ,OÞn,u,Θ,0≥limn⟶+∞sup1OÞn,u,Θ−1OTÞn,Tu,Θ,0≥limn⟶+∞sup1OÞn,u,Θ−1OTÞn+1,Tu,Θ,0≥10−1Ou,Tu,Θ,0≥Ou,Tu,Θ.
From (36)–(38), this is a contradiction, and therefore, we examine(39)Mu,Tu,Θ=1,Nu,Tu,Θ=0andOu,Tu,Θ=0.
Thus,Tu=u.
Using the idea of NSs and NMS with CTN and CTCN, we generalize the theorems in [6] in the context of NMS using the following convention:(40)MÞ,ϖ,Θn=MÞ,ϖ,Θ∗MÞ,ϖ,Θ∗…∗MÞ,ϖ,Θ︸n,NÞ,ϖ,Θn=NÞ,ϖ,Θ○NÞ,ϖ,Θ○…○NÞ,ϖ,Θ︸n,OÞ,ϖ,Θn=OÞ,ϖ,Θ○OÞ,ϖ,Θ○…○OÞ,ϖ,Θ︸nfor alln∈ℕ.Theorem 2.
LetX,M,N,O,∗,○ be a complete NMS with CTN ‘∗’ and CTCN ‘○’ of H-type. Let T:X×X⟶XandG:X⟶X be two mappings and there exist∅∈Ф such that(41)MTÞ,ϖ,Tu,v,∅Θ≥MGÞ,Gu,Θ∗MGϖ,Gv,Θ,NTÞ,ϖ,Tu,v,∅Θ≤NGÞ,Gu,Θ○NGϖ,Gv,Θ,and(42)OTÞ,ϖ,Tu,v,∅Θ≤OGÞ,Gu,Θ○OGϖ,Gv,Θfor allÞ,ϖ,u,v∈X,Θ>0.IfT⊆G,G is continuous, TandG are compatible, then TandG have a unique fixed point.Theorem 3.
LetX,M,N,O,∗,○ be a complete NMS with CTN ‘∗’ and CTCN ‘○’ of H-type. Let T:X×X⟶XandG:X⟶X be two weakly compatible mappings and theirexist∅∈Ф if TX×X⊆GXandTX×XorGX is complete, then TandG have a unique common fixed point in X.Proof.
We assume two pointsÞ0,ϖ0∈X. Since, TX×X⊆GX, we have Þ1,ϖ1∈X such that GÞ1=TÞ0,ϖ0andGϖ1=Tϖ0,Þ0 and two sequences ÞnandϖninX can be constructed(43)GÞn+1=TÞn,ϖnandGϖn+1=Tϖn,Þnfor alln≥0.
We shall prove thatGÞnandGϖn are Cauchy sequences. Therefore, for any λ>0 there exist μ>0 and following conditions are hold for all k∈ℕ(44)1−μ∗1−μ∗…∗1−μ︸k≥1−λμ○μ…○μ≤λ︸kμ○μ…○μ≤λ︸k.
Since,MÞ,ϖ,•,NÞ,ϖ,•andOÞ,ϖ,• are continuous and limn⟶+∞MÞ,ϖ,Θ=1,limn⟶+∞NÞ,ϖ,Θ=0andlimn⟶+∞OÞ,ϖ,Θ=0forallÞ,ϖ∈XtheirexistΘ0>0, such that (45)MGÞ0,GÞ1,Θ0≥1−λ,MGϖ0,Gϖ1,Θ0≥1−λNGÞ0,GÞ1,Θ0≤μ,NGϖ0,Gϖ1,Θ0≤μOGÞ0,GÞ1,Θ0≤μ,OGϖ0,Gϖ1,Θ0≤μ.
Since,∅∈Ф, therefore by (iii), we have ∑n=1+∞∅nΘ0<+∞, then for any Θ>0,theirexistn0∈ℕ, such that(46)Θ>∑n=1+∞∅nΘ0.
From Theorem2, we have(47)MGÞ1,GÞ2,∅Θ0=MTÞ0,ϖ0,TÞ1,ϖ1,∅Θ0,≥MGÞ0,GÞ1,Θ0∗MGϖ0,Gϖ1,Θ0,MGϖ1,Gϖ2,∅Θ0=MTϖ0,Þ0,Tϖ1,Þ1,∅Θ0,≥MGϖ0,Gϖ1,Θ0∗MGÞ0,GÞ1,Θ0,and(48)NGÞ1,GÞ2,∅Θ0=NTÞ0,ϖ0,NÞ1,ϖ1,∅Θ0,≤NGÞ0,GÞ1,Θ0○NGϖ0,Gϖ1,Θ0,NGϖ1,Gϖ2,∅Θ0=NTϖ0,Þ0,Tϖ1,Þ1,∅Θ0,≤NGϖ0,Gϖ1,Θ0○NGÞ0,GÞ1,Θ0,also(49)OGÞ1,GÞ2,∅Θ0=OTÞ0,ϖ0,OÞ1,ϖ1,∅Θ0,≤OGÞ0,GÞ1,Θ0○OGϖ0,Gϖ1,Θ0,OGϖ1,Gϖ2,∅Θ0=OTϖ0,Þ0,Tϖ1,Þ1,∅Θ0,≤OGϖ0,Gϖ1,Θ0○OGÞ0,GÞ1,Θ0.
Similarly,(50)MGÞ2,GÞ3,∅2Θ0=MTÞ1,ϖ1,TÞ2,ϖ2,∅2Θ0,≥MGÞ1,GÞ2,Θ0∗MGϖ1,Gϖ2,Θ0,≥MGÞ0,GÞ1,Θ02∗MGϖ0,Gϖ1,Θ02,MGϖ2,Gϖ3,∅2Θ0=MTϖ1,Þ1,Tϖ2,Þ2,∅2Θ0,≥MGϖ1,Gϖ2,Θ0∗MGÞ1,GÞ2,Θ0,≥MGϖ0,Gϖ1,Θ02∗MGÞ0,GÞ1,Θ02,and(51)NGÞ2,GÞ3,∅2Θ0=NTÞ1,ϖ1,TÞ2,ϖ2,∅2Θ0,≤NGÞ1,GÞ2,Θ0○NGϖ1,Gϖ2,Θ0,≤NGÞ0,GÞ1,Θ02○NGϖ0,Gϖ1,Θ02,NGϖ2,Gϖ3,∅2Θ0=NTϖ1,Þ1,Tϖ2,Þ2,∅2Θ0,≤NGϖ1,Gϖ2,Θ0○NGÞ1,GÞ2,Θ0,≤NGϖ0,Gϖ1,Θ02○NGÞ0,GÞ1,Θ02,also(52)OGÞ2,GÞ3,∅2Θ0=OTÞ1,ϖ1,TÞ2,ϖ2,∅2Θ0,≤OGÞ1,GÞ2,Θ0○OGϖ1,Gϖ2,Θ0,≤OGÞ0,GÞ1,Θ02○OGϖ0,Gϖ1,Θ02,OGϖ2,Gϖ3,∅2Θ0=OTϖ1,Þ1,Tϖ2,Þ2,∅2Θ0,≤OGϖ1,Gϖ2,Θ0○OGÞ1,GÞ2,Θ0,≤OGϖ0,Gϖ1,Θ02○OGÞ0,GÞ1,Θ02.
By induction, we deduce(53)MGÞn,GÞn+1,∅nΘ0≥MGÞ0,GÞ1,Θ02n−1∗MGϖ0,Gϖ1,Θ02n−1,MGϖn,Gϖn+1,∅nΘ0≥MGϖ0,Gϖ1,Θ02n−1∗MGÞ0,GÞ1,Θ02n−1,MGϖn,Gϖn+1,∅nΘ0≥MGϖ0,Gϖ1,Θ02n−1∗MGÞ0,GÞ1,Θ02n−1,NGϖn,Gϖn+1,∅nΘ0≤NGϖ0,Gϖ1,Θ02n−1○NGÞ0,GÞ1,Θ02n−1and(54)OGÞn,GÞn+1,∅nΘ0≤OGÞ0,GÞ1,Θ02n−1○OGϖ0,Gϖ1,Θ02n−1,OGϖn,Gϖn+1,∅nΘ0≤OGϖ0,Gϖ1,Θ02n−1○OGÞ0,GÞ1,Θ02n−1.
From (45) and (46) for m>n≥n0,(55)MGÞn,GÞm,Θ≥MGÞn,GÞm,∑k=n0+∞∅kΘ0≥MGÞn,GÞm,∑k=nm−1∅kΘ0,≥MGÞn,GÞn+1,∅nΘ0∗MGÞn+1,GÞn+2,∅nΘ0∗…∗MGÞm−1,GÞm,∅m−1Θ0,≥MGϖ0,Gϖ1,Θ02n−1∗MGÞ0,GÞ1,Θ02n−1∗MGϖ0,Gϖ1,Θ02n∗MGÞ0,GÞ1,Θ02n∗…∗MGÞm−1,GÞm,∅m−1Θ0,=MGϖ0,Gϖ1,Θ02m−1−2n−1∗MGÞ0,GÞ1,Θ02m−1−2n−1,≥1−μ∗1−μ∗…∗1−μ︸2m−2n≥1−λ.and(56)NGÞn,GÞm,Θ≤NGÞn,GÞm,∑k=n0+∞∅kΘ0≤NGÞn,GÞm,∑k=nm−1∅kΘ0,≤NGÞn,GÞn+1,∅nΘ0○NGÞn+1,GÞn+2,∅nΘ0○…○NGÞm−1,GÞm,∅m−1Θ0,≤NGϖ0,Gϖ1,Θ02n−1○NGÞ0,GÞ1,Θ02n−1○NGϖ0,Gϖ1,Θ02n,○NGÞ0,GÞ1,Θ02n○…○NGÞm−1,GÞm,∅m−1Θ0,=NGϖ0,Gϖ1,Θ02m−1−2n−1○NGÞ0,GÞ1,Θ02m−1−2n−1,≤μ○μ○…○μ︸2m−2n≤λ,also(57)OGÞn,GÞm,Θ≤OGÞn,GÞm,∑k=n0+∞∅kΘ0≤OGÞn,GÞm,∑k=nm−1∅kΘ0,≤OGÞn,GÞn+1,∅nΘ0○OGÞn+1,GÞn+2,∅nΘ0○…○OGÞm−1,GÞm,∅m−1Θ0,≤OGϖ0,Gϖ1,Θ02n−1○OGÞ0,GÞ1,Θ02n−1○OGϖ0,Gϖ1,Θ02n,○OGÞ0,GÞ1,Θ02n○…○OGÞm−1,GÞm,∅m−1Θ0,=OGϖ0,Gϖ1,Θ02m−1−2n−1○OGÞ0,GÞ1,Θ02m−1−2n−1,≤μ○μ○…○μ︸2m−2n≤λ.
From (45) and (46), we have(58)MGÞn,GÞm,Θ>1−λ,NGÞn,GÞm,Θ<λandOGÞn,GÞm,Θ<λ,for allm,n∈ℕwithm>n>n0andΘ>0 and it proved that GÞn is a Cauchy sequence. Similarly, Gϖn is also a Cauchy sequence. Now, we prove that GandT have a coupled coincidence point. Without loss of generality, GX is complete, and hence, there exists ā,ē∈XÞ,ϖ∈GX, then(59)limn⟶+∞GÞn=limn⟶+∞TÞn,ϖn=Gā=Þlimn⟶+∞Gϖn=limn⟶+∞Tϖn,Þn=Gē=ϖ.
Using Theorem2,(60)MTÞn,ϖn,Tā,ē,∅Θ≥MGÞn,Gā,Θ∗MGϖn,Gē,Θ,NTÞn,ϖn,Tā,ē,∅Θ≤NGÞn,Gā,Θ○NGϖn,Gē,Θ,and(61)OTÞn,ϖn,Tā,ē,∅Θ≤OGÞn,Gā,Θ○OGϖn,Gē,Θ.
SinceM,N,andO are continuous, therefore, as n⟶+∞, we get(62)MGā,Tā,ē,∅Θ=1,NGā,Tā,ē,∅Θ=0andOGā,Tā,ē,∅Θ=0.
This implies thatTā,ē=Gā=Þ and similarly Tē,ā=Gē=ϖ.
SinceTandG are weekly compatible, therefore GTā,ē=TGā,GēandGTē,ā=TGē,Gā,thenGÞ=TÞ,ϖandGϖ=Tϖ,Þ, so GÞ=ϖandGϖ=Þ, then for any λ>0,thereexistμ>0 such that(63)1−μ∗1−μ∗…∗1−μ︸k≥1−λ,and(64)μ○μ○…○μ︸k≤λfor allk∈ℕ.
SinceMÞ,ϖ,•,NÞ,ϖ,•,andOÞ,ϖ,• are continuous and(65)limΘ⟶+∞MÞ,ϖ,Θ=1,limΘ⟶+∞NÞ,ϖ,Θ=0andlimΘ⟶+∞OÞ,ϖ,Θ=0for allÞ,ϖ∈X,there exist Θ0>0 such that(66)MGÞ,ϖ,Θ0≥1−μ,MGϖ,Þ,Θ0≥1−μ,NGÞ,ϖ,Θ0≤μ,NGϖ,Þ,Θ0≤μ,and(67)OGÞ,ϖ,Θ0≤μ,OGϖ,Þ,Θ0≤μ.
Also, since∅∈Ф, therefore, we have ∑n=1+∞∅nΘ0<+∞. Thus, for any Θ>0,thereexistn0∈ℕ such that Θ>∑k=n0+∞∅kΘ0,(68)MGÞ,Gϖn+1,∅Θ0=MTÞ,ϖ,Tϖn,Þn,∅Θ0,≥MGÞ,Gϖn,Θ0∗MGϖ,GÞn,Θ0,NGÞ,Gϖn+1,∅Θ0=NTÞ,ϖ,Tϖn,Þn,∅Θ0,≤NGÞ,Gϖn,Θ0○NGϖ,GÞn,Θ0,and(69)OGÞ,Gϖn+1,∅Θ0=OTÞ,ϖ,Tϖn,Þn,∅Θ0,≤OGÞ,Gϖn,Θ0○OGϖ,GÞn,Θ0.
Asn⟶+∞, we get(70)MGÞ,ϖ,∅Θ0≥MGÞ,ϖ,Θ0∗MGϖ,Þ,Θ0NGÞ,ϖ,∅Θ0≤NGÞ,ϖ,Θ0○NGϖ,Þ,Θ0OGÞ,ϖ,∅Θ0≤OGÞ,ϖ,Θ0○OGϖ,Þ,Θ0.
Similarly,(71)MGϖ,Þ,∅Θ0≥MGϖ,Þ,Θ0∗MGÞ,ϖ,Θ0NGϖ,Þ,∅Θ0≤NGÞ,ϖ,Θ0○NGϖ,Þ,Θ0OGϖ,Þ,∅Θ0≤OGÞ,ϖ,Θ0○OGϖ,Þ,Θ0.
From (70) and (71), we obtain(72)MGÞ,ϖ,∅Θ0∗MGϖ,Þ,∅Θ0≥MGÞ,ϖ,Θ02∗MGϖ,Þ,Θ02,NGÞ,ϖ,∅Θ0○NGϖ,Þ,∅Θ0≤NGÞ,ϖ,Θ02○NGϖ,Þ,Θ02,and(73)OGÞ,ϖ,∅Θ0○OGϖ,Þ,∅Θ0≤OGÞ,ϖ,Θ02○OGϖ,Þ,Θ02.
From these inequalities, we obtain(74)MGÞ,ϖ,∅nΘ0∗MGϖ,Þ,∅nΘ0≥M(GÞ,ϖ,∅n−1Θ0)2∗M(Gϖ,Þ,∅n−1Θ0)2,≥MGÞ,ϖ,Θ02n∗MGϖ,Þ,Θ02n,NGÞ,ϖ,∅nΘ0○NGϖ,Þ,∅nΘ0≤N(GÞ,ϖ,∅n−1Θ0)2○N(Gϖ,Þ,∅n−1Θ0)2,≤NGÞ,ϖ,Θ02n○NGϖ,Þ,Θ02n,OGÞ,ϖ,∅nΘ0○OGϖ,Þ,∅nΘ0≤O(GÞ,ϖ,∅n−1Θ0)2○O(Gϖ,Þ,∅n−1Θ0)2,≤OGÞ,ϖ,Θ02n○OGϖ,Þ,Θ02n,for alln∈ℕ.
SinceΘ>∑k=n0+∞∅kΘ0, then(75)MGÞ,ϖ,Θ0∗MGϖ,Þ,Θ0≥MGÞ,ϖ,∑k=n0+∞∅kΘ0∗MGϖ,Þ,∑k=n0+∞∅kΘ0,≥MGÞ,ϖ,∅n0Θ0∗MGϖ,Þ,∅n0Θ0,≥MGÞ,ϖ,Θ02n0∗MGϖ,Þ,Θ02n0,≥1−μ∗1−μ∗…∗1−μ︸22n0≥1−λ,NGÞ,ϖ,Θ0○NGϖ,Þ,Θ0≤NGÞ,ϖ,∑k=n0+∞∅kΘ0○NGϖ,Þ,∑k=n0+∞∅kΘ0,≤NGÞ,ϖ,∅n0Θ0○NGϖ,Þ,∅n0Θ0,≤NGÞ,ϖ,Θ02n0○NGϖ,Þ,Θ02n0,≤μ○μ○…○μ︸22n0≤λ,and(76)OGÞ,ϖ,Θ0○OGϖ,Þ,Θ0≤OGÞ,ϖ,∑k=n0+∞∅kΘ0○OGϖ,Þ,∑k=n0+∞∅kΘ0,≤OGÞ,ϖ,∅n0Θ0○OGϖ,Þ,∅n0Θ0,≤OGÞ,ϖ,Θ02n0○OGϖ,Þ,Θ02n0,≤μ○μ○…○μ︸22n0≤λ.
For anyλ>0, we obtain(77)MGÞ,ϖ,Θ∗MGϖ,Þ,Θ≥1−λNGÞ,ϖ,Θ○NGϖ,Þ,Θ≤λOGÞ,ϖ,Θ○OGϖ,Þ,Θ≤λ.
For allΘ>0, hence, we conclude that GÞ=ϖandGϖ=Þ.
We prove thatÞ=ϖ. From Theorem 2, we have(78)M(G(Þn+1),G(ϖn+1),∅(Θ0))=M(T(Þn,ϖn),T(ϖn,Þn),∅(Θ0),≥MGÞn,Gϖn,Θ0∗MGϖn,GÞ,Θ0,N(G(Þn+1),G(ϖn+1),∅(Θ0))=N(T(Þn,ϖn),T(ϖn,Þn),∅(Θ0),≤NGÞn,Gϖn,Θ0○NGϖn,GÞ,Θ0,and(79)O(G(Þn+1),G(ϖn+1),∅(Θ0))=O(T(Þn,ϖn),T(ϖn,Þn),∅(Θ0),≤OGÞn,Gϖn,Θ0○OGϖn,GÞ,Θ0.
Asn⟶+∞, we obtain(80)MÞ,ϖ,∅Θ0≥MÞ,ϖ,Θ0∗Mϖ,Þ,Θ0,NÞ,ϖ,∅Θ0≤NÞ,ϖ,Θ0○Nϖ,Þ,Θ0,OÞ,ϖ,∅Θ0≤OÞ,ϖ,Θ0○Oϖ,Þ,Θ0.
Thus, we obtain(81)MÞ,ϖ,Θ≥MÞ,ϖ,∑k=n0+∞∅kΘ0≥MÞ,ϖ,∅n0Θ0,≥MÞ,ϖ,Θ02n0−1∗Mϖ,Þ,Θ02n0−1,≥1−μ∗1−μ∗…∗1−μ︸22n0−2≥1−λ,NÞ,ϖ,Θ≤NÞ,ϖ,∑k=n0+∞∅kΘ0≤NÞ,ϖ,∅n0Θ0,≤NÞ,ϖ,Θ02n0−1○Nϖ,Þ,Θ02n0−1,≤μ○μ○…○μ︸22n0−2≤λ,and(82)OÞ,ϖ,Θ≤OÞ,ϖ,∑k=n0+∞∅kΘ0≤OÞ,ϖ,∅n0Θ0,≤OÞ,ϖ,Θ02n0−1○Oϖ,Þ,Θ02n0−1,≤μ○μ○…○μ︸22n0−2≤λ.
Hence,Þ=ϖ, it is clear that TandG have a common fixed point.Example 1.
SupposeX,M,N,O,∗,○ is a complete NMS with ā∗ē=āē and ā○ē=min1,ā+ē. Let X=0,10 with the metric dÞ,ϖ=Þ−ϖfor allÞ,ϖ∈X, and(83)MÞ,ϖ,Θ=ΘΘ+dÞ,ϖ,NÞ,ϖ,Θ=dÞ,ϖΘ+dÞ,ϖ,OÞ,ϖ,Θ=dÞ,ϖΘ.
The mapT,S:X⟶X is defined by TÞ=3+Þ/4andSÞ=Þ. Let Þn=1−1/n.(84)limn⟶+∞MTSÞn,STÞn,Θ=limn⟶+∞MTÞn,S3+Þn4,Θ,limn⟶+∞M3+Þn4,3+Þn4,Θ=1,limn⟶+∞NTSÞn,STÞn,Θ=limn⟶+∞NTÞn,S3+Þn4,Θlimn⟶+∞N3+Þn4,3+Þn4,Θ=0,and(85)limn⟶+∞OTSÞn,STÞn,Θ=limn⟶+∞OTÞn,S3+Þn4,Θ,limn⟶+∞O3+Þn4,3+Þn4,Θ=0,limn⟶+∞TÞn=limn⟶+∞3+Þn4=limn⟶+∞3+1−1/n4=1,limn⟶+∞SÞn=limn⟶+∞Þn=limn⟶+∞1−1n=1.
Therefore,TandS are weakly compatible mappings. We define a map α:0,1⟶0,1byα∂=2∂/∂+1 for each ∂∈0,1andα∈Ф.(86)MTÞ,Tϖ,Θ≥αMSÞ,Sϖ,Θ,M3+Þ4,3+ϖ4,Θ≥αMÞ,ϖ,Θ,ΘΘ+d3+Þ/4,3+ϖ/4≥2Θ/Θ+dÞ,ϖΘ/Θ+dÞ,ϖ+1,ΘΘ+3+Þ/4−3+ϖ/4≥2Θ/Θ+Þ−ϖΘ/Θ+Þ−ϖ+1,ΘΘ+Þ−ϖ/4≥ΘΘ+Þ−ϖ/2,Θ+Þ−ϖ2≥Θ+Þ−ϖ4,⇒4>2.
A mapη:0,1⟶0,1byηr=r/2−r for each r∈0,1andη∈Ф(87)NTÞ,Tϖ,Θ≤ηNSÞ,Sϖ,Θ,N3+Þ4,3+ϖ4,Θ≤ηNÞ,ϖ,Θ,d3+Þ/4,3+ϖ/4Θ+d3+Þ/4,3+ϖ/4≤dÞ,ϖ/Θ+dÞ,ϖ2−dÞ,ϖ/Θ+dÞ,ϖ,2Θ+Þ−ϖ≤4Θ+Þ−ϖ,⇒2<4.
A mapγ:0,1⟶0,1byγw=1ifw=1,w/2otherwise, for each w∈0,1andγ∈Ф,(88)OTÞ,Tϖ,Θ≤γOSÞ,Sϖ,Θ,N3+Þ4,3+ϖ4,Θ≤γOÞ,ϖ,Θ,d3+ϖ/4,3+ϖ/4Θ≤dÞ,ϖ2Θ,Þ−ϖ4Θ≤Þ−ϖ2Θ,⇒2<4.
As a result, all of the assertions of Theorem3 are met and a unique fixed point is 1. Hence, TandS have the unique fixed point in X.Corollary 1.
LetX,M,N,O,ē,∗,○ be complete neutrosophic b-metric space (NBMS), i.e., ē≥1, multiplying in the right sides of triangle inequalities of NMS definition, such that(89)limΘ⟶++∞MÞ,ϖ,Θ=1,limΘ⟶+∞NÞ,ϖ,Θ=0,limΘ⟶+∞OÞ,ϖ,Θ=0for allÞ,ϖ∈X.
LetT:X⟶X be a mapping satisfying(90)MTÞ,Tϖ,kΘ≥MÞ,ϖ,Θ,NTÞ,Tϖ,kΘ≤NÞ,ϖ,Θ,OTÞ,Tϖ,kΘ≤OÞ,ϖ,Θ,for all Þ,ϖ∈X,k∈0,1/e. Then, T has a unique fixed point.


## 3. Solution of Nonlinear Fractional Differential Equations: A Fixed Point Technique

The main goal of this section is to apply Corollary1 to examine the existence and uniqueness of solution to a nonlinear fractional differential equation (NFDE),(91)D0+σϖΘ=GΘ,ϖΘ,0<Θ<1,with the boundary condition(92)ϖ0+ϖ′0=0,ϖ1+ϖ′1=0,where 1<σ≤2 is a number, D0+σ is the Caputo fractional derivative, and G:[0,1]×[0,+∞)⟶[0,+∞) is a continuous function. Let X=C0,1,ℝ denote the space of all continuous functions on 0,1 with CTN ‘∗’c∗d=c.d and CTCN c○d=maxc,d for all c,d∈0,1 and specify the complete NBMS on X as follows:(93)Mϖ,w,Θ=αΘαΘ+γSupΘ∈0,1ϖΘ−wΘ6,Nϖ,w,Θ=SupΘ∈0,1ϖΘ−wΘ6αΘ+SupΘ∈0,1ϖΘ−wΘ6,and(94)Nϖ,w,Θ=SupΘ∈0,1ϖΘ−wΘ6αΘ.For allΘ>0 and ϖ,w∈X. For ϖ∈X, we have the following integral equation:(95)ϖΘ=1Гσ∫011−∂σ−11−ΘG∂,ϖ∂d∂+1Гσ−1∫011−∂σ−21−ΘG∂,ϖ∂d∂+1Гσ∫0ΘΘ−∂σ−1G∂,ϖ∂d∂.Theorem 4.
The integral operatorT:X⟶X is given by(96)TϖΘ=1Гσ∫011−∂σ−11−ΘG∂,ϖ∂d∂+1Гσ−1∫011−∂σ−21−ΘG∂,ϖ∂d∂+1Гσ∫0ΘΘ−∂σ−1G∂,ϖ∂d∂,where G:[0,1]×[0,+∞)⟶[0,+∞) fulfils the following criteria:(97)G∂,ϖ∂−G∂,w∂≤14ϖ∂−w∂,for allϖ,w∈X,supΘ∈0,1140961−ΘГσ+1+1−ΘГσ+ΘσГσ+16≤η<1.Then, NFDE has a uniqueX.Proof.
We have(98)TϖΘ−TwΘ6=1−ΘГσ∫011−∂σ−1G∂,ϖ∂−G∂,w∂d∂+1−ΘГσ−1∫011−∂σ−2G∂,ϖ∂−G∂,w∂d∂+1Гσ∫0ΘΘ−∂σ−1G∂,ϖ∂−G∂,w∂d∂6,≤1−ΘГσ∫011−∂σ−1G∂,ϖ∂−G∂,w∂d∂+1−ΘГσ−1∫011−∂σ−2G∂,ϖ∂−G∂,w∂d∂+1Гσ∫0ΘΘ−∂σ−1|G∂,ϖ∂−G∂,w∂|6,≤1−ΘГσ∫011−∂σ−1ϖ∂−w∂4d∂+1−ΘГσ−1∫011−∂σ−2ϖ∂−w∂4d∂+1Гσ∫0ΘΘ−∂σ−1ϖ∂−w∂4d∂6,≤146sup|ϖΘ−wΘ|6Θ∈0,11−ΘГσ∫011−∂σ−1d∂+1−ΘГσ−1∫011−∂σ−2d∂+1Гσ∫0ΘΘ−∂σ−1d∂6,≤146supΘ∈0,1ϖΘ−wΘ61−ΘГσ+1+1−ΘГσ+ΘσГσ+16,=ηsupΘ∈0,1|ϖΘ−wΘ|6,where(99)η=supΘ∈0,1140961−ΘГσ+1+1−ΘГσ+ΘσГσ+16.
Therefore, the above inequality(100)supΘ∈0,1TϖΘ−TwΘ6≤ηsupΘ∈0,1ϖΘ−w(Θ6,⇒αΘ+γηsupΘ∈0,1TϖΘ−TwΘ6≤αΘ+γsupΘ∈0,1ϖΘ−wΘ6,⇒αηΘαηΘ+γsupΘ∈0,1|TϖΘ−TwΘ|6≥αΘαΘ+γsupΘ∈0,1|ϖΘ−wΘ|6,⇒MTϖ,Tw,ηΘ≥Mϖ,w,Θ.
Similarly, we can deduce(101)γsupΘ∈0,1|TϖΘ−TwΘ|6αηΘ+γsupΘ∈0,1|TϖΘ−TwΘ|6≤γsupΘ∈0,1|ϖΘ−wΘ|6αΘ+γsupΘ∈0,1|ϖΘ−wΘ|6,⇒NTϖ,Tw,ηΘ≤Nϖ,w,Θ,and(102)γsupΘ∈0,1|TϖΘ−TwΘ|6αηΘ≤γsupΘ∈0,1|ϖΘ−wΘ|6αΘ,⇒OTϖ,Tw,ηΘ≤Oϖ,w,Θ.for some α,γ>0. As a result, we can conclude that Corollary 1 assumptions are met. Hence, T has a unique fixed point and NFDE has a unique solution.


## 4. Conclusion

In the presented study, various fixed point results for contraction and weakly compatible mappings are proved. As known that fixed point theory has a wide range of applications in economics, engineering, and computer science, we proved nonlinear fractional differential equations application via neutrosophic metric space and unique solution exists. This work can be extended in several structures, such as neutrosophic b-metric spaces and orthogonal neutrosophic metric spaces. Our techniques may help many researchers working in the field of plasma physics. We only remember plasma physics as an example due to its richness by several differential equations that are used to describe many nonlinear phenomena that can propagate in different plasma models [16–18].



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