# Fixed Point of Rational Contractions and Its Application for Secure Dynamic Routing in Wireless Sensor Networks

Authors: Awais Asif; Ekrem Savas; Hussain AlSalman; Muahammad Arshad; Abdu Gumaei; Abdul Rehman
Journal: Security and Communication Networks
 (2021)

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## Abstract

Security is one of the major concerns for data communication over wireless sensor networks (WSNs). Dynamic routing algorithms can provide small similarity paths of data delivery between two consecutive transmitted packets, improving data security without adding extra information or control messages. This article illustrates the iteration of the fixed point (FP) of rational contractions and generalized Banach contractions (BC) in the setting ofF-metric space (F-MS). It also describes an FP of the said mappings, while restricting the imposition of the contraction only to a subset of the F-MS, the closed ball, rather than executing it on the entire F-MS. The results have been verified and supported by concise examples. Further, the application of the functional equation proved results with randomization is given to find a solution for secure dynamic routing of data transmission in WSNs. The application is a tool to analyze and model a network structure in which sensors can be deployed with high security and low risk in a greater region (sensor field), thus boosting the accuracy.


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## Body

## 1. Introduction and Preliminaries

The idea of metric space is extended and generalized by many authors in different ways (see [1–8]). Getting motivation from Wardowski’s F-contraction, an interesting generalization was put forwarded by Jleli [8], which he named as F-metric space (F-MS). He proved fixed point results of Banach contractions (BC) in the frame of F-MS and defined the topological properties in the new given setting. Following this conception, various authors (see [8–13]) furthered this idea by presenting their fixed point (FP) models in F-MS, and a new and more discussion was initiated on the topic.In continuation to this, our paper focuses on the advancement of the said F-MS, and we prove certain FP and typical FP results in F-MS. Also, following the concept of the closed ball in generalized metric spaces (for discussion on closed ball, see [13–17]), we will present both FP and typical FP results for rational contractions and Banach contractions, which are imposed only on a subset that is closed ball for the whole F-MS. With the help of examples, the difference between locally contractive and globally contractive mappings and their outcomes will be explained.Section 1 of this manuscript defines the basic notions and definitions that will be used or referred to in our paper. Section 2 of the article is dedicated to the Banach as well rational contractions and its investigation for FP in F-MS, while Section 3 of the paper revises the same results for the structure of the closed ball. Section 4 presents the application of our results. The role of the functional equation in solving many real-world problems is inevitable. Besides the application discussed in the abstract, the functional equation approach may be executed to construct the algorithm that finds the solution point in no more than steps equal to N−1 for a network havingN nodes (see [18]). Also, the functional equations enable multistage decision-making and lead to water security sustainability evaluation (see [19]). Therefore, the importance and contribution of our work are not limited to what we have discussed in this Section 4 of our article. It can apply to stochastic processes, economics, classical mechanics, dynamic programming, computer graphics, game theory, neural networks, artificial intelligence, fuzzy set theory, decision theory, digital image processing, multivalued logic, and many other fields. At last, the work done in this paper is concluded in the Conclusion section. Following are related and relevant to the main study of our work.Definition 1 (see [8]).
SupposeG is a family of mappings g:0,+∞⟶R with the following constraints:(F1)g be a nondecreasing mapping, i.e., 0<q<s⟹gq≤gs(F2) For every sequenceun⊂0,+∞, we have(1)limn⟶∞un=0⟺limn⟶∞gun=−∞.Definition 2 (see [8]).
Suppose a setE≠∅ and a map d:E×E⟶0,∞. as well as suppose there is some g,σ∈G×0,+∞ in whichDistance (d1)q,s∈E×E,dq,s=0⟺q=sandq,s∈E×E,dq,s=ds,qDistance (d2)q,s∈E×E,dq,s=ds,qDistance (d3) For eachq,s∈E×E, and for all N∋′N′≥2, as well as for each uii=1n⊂E with u1,uN′=q,s, we have(2)q,s>0⟹gdq,s≤g∑i=1N′−1dui,ui+1+σ.
Then,E,d is called an F-metric space (F-MS)Example 1 (see [8]).
LetE=N and d:E×E⟶0,∞ be given as(3)dq,s=q−s2ifq,s∈0,3×0,3,q−sifq,s∉0,3×0,3.
For eachq,s∈E×E, then d is an F-metric space (F-MS).Example 2 (see [8]).
AssumingE=N and distance d:E×E⟶0,∞ is given as(4)dq,s=0,ifq=s,eq−s,ifq≠s.
For eachq,s∈E×E, then d is an F-metric space (F-MS).Definition 3 (see [8]).
Letqn∈E. If(i)
limn⟶∞dqn,q=0 for some q∈E, at this point qn will be F-convergent to q(ii)
limn,m⟶∞dqn,qm=0, then the sequence qnF-Cauchy(iii)
E,d is said to be F-complete if for every qn⊂E⟹qnDefinition 4 (see [8]).
LetE,d an F-metric space (F-MS). A set O⊂E is an F-metric open (F-MO) if and only if for each q∈O, there is a number σ>0 provided that Bq,σ⊂O, and(5)Bq,σ=s∈E:dq,s<σ.
While a subsetC from the set E is an F-metric closed (F-MC) whenever EorC is an F-open.Definition 5 (see [8]).
Letϕ≠B⊂E , and distance (d) be an F-metric space (F-MS); then, the subsequent rules are equivalent:(i)
Bis anF-MC(ii)
For everyqn⊂B, we have(6)limn⟶∞dqn,q=0,q∈E⟹q∈B.
For convenience, we write onlyE rather than E,d for an F-MS. For a complete F-MS, we will write CF-MS.Theorem 1 (see [8]).
Leth be a self-mapping on a CF-MS E and k∈0,1 such that(7)dhq,hs≤kdq,s,q,s∈E×E.Then,hq∗=q∗ for a unique q∗∈E. Further, for any q0∈E, the sequence qn⊂E defined by qn+1  =  hqn,n∈N, is F-convergent to q∗.Theorem 2 (see [8]).
LetE,d is a complete MS and h is a self-mapping on E such that(8)dhq,hs≤αdq,s+βdq,hs+γdq,hs.For allqands∈E, where α,β,andγ are nonnegative numbers satisfying α+β/1−γ<1, then h has a unique fixed point in E.Lemma 1 (see [8]).
LetBW,⋅be a Banach space and distance (d) is stated by(9)dJ,h=J−h=maxq∈WJq−hq,J,h∈BW.Then,BW,⋅ will be an F-MS.


## 2. Fixed Points of Generalized Contractions inF−MS

This portion of the article illustrates FP and typical FP results of rational type and Banach type contraction in the setting ofF-MS. The results have been explained with the help of concise examples, and some essential corollaries are developed.Theorem 3.
Assumeg,α∈G×0,∞ and A,d is a CF-MS, and suppose S,T:A⟶A be self-mappings such that(10)dSs,Ty≤λds,y.Forλ∈0,∞ and for all s,y∈A×A, then there exists a single typical FP of the mappings S and T in A.Proof.
Pick an arbitrary elements0 and iterate a sequence sn as(11)Ss2x=s2x+1,Ts2x+1=s2x+2,x=0,1,2,….
From (12) and (13), we obtain(12)ds2x+1,s2x+2=dSs2x,Ts2x+1≤λds2x,s2x+1.
This implies(13)ds2x+1,s2x+2<λds2x,s2x+1.
Similarly,(14)ds2x,s2x+1=dSs2x−1,Ts2x≤λds2x−1,s2x,i.e.,(15)ds2x,s2x+1≤λds2x−1,s2x.
Generalizing this as follows:(16)dsn,sn+1<λdsn−1,snfor allnϵℕ,which yields(17)dsn,sn+1<λdsn−1,sn<λ2dsn−2,sn−1<⋯<λnds0,s1,n∈ℕ.
Using (16), we can write(18)∑k=nm−1dsk,sk+1=dsn,sn+1+dsn+1,sn+2+⋯+dsm−1,sm<λn1+λ+λ2+⋯+λm−n−1ds0,s1≤λn1−λds0,s1,m>n.i.e.,(19)∑k=nm−1dsk,sk+1≤λn1−λds0,s1,m>n.
Sincelimn⟶∞λn/1−λds0,s1=0, for any δ>0, there are some n′∈ℕ in which (20)0<λn1−λds0,s1<δ,n≥n′.
Further, supposeg,α∈G×0,∞ meets (d3) and ϵ>0 is fixed. By F2, there exists some δ>0 where(21)0<l<δ⟹gl<gϵ−α.
By (20) and (21), we write(22)g∑k=nm−1dsk,sk+1≤gλn1−λds0,s1<gϵ−α,m>n≥n′.
By (d3) and the above equation, we can get(23)dsn,sm>0,m>n>n′⟹gdsn,sm<gϵ.
This shows(24)dsn,sm<ϵ,m>n≥n′.
Thus, it proved that the sequence (sn) is F-Cauchy in A. Now, as A,d is F-complete, so there must be some element c∗ in A and sn converges to c∗ i.e.,(25)limn⟶∞dsn,c∗=0.
Next, we show thatc∗ is the FP of S. For this,(26)dSc∗,s2x+2=dSc∗,Ts2x+1≤λdc∗,s2x+1.
If limitx⟶∞, we obtain(27)dSc∗,c∗≤λ.dc∗,c∗.
Thus,dSc∗,c∗=0,i.e., Sc∗=c∗.
Similarly, we obtainTc∗=c∗. Hence, Tc∗=Sc∗=c∗.
Uniqueness: letc∗∗ is also a typical FP of S and T, andc∗≠c∗∗.Then,(28)dc∗,c∗∗=dSc∗,Tc∗∗≤λdc∗,c∗∗,which is a contradiction. Thus, c∗=c∗∗.
TakingS=T, the following outcome is achieved.Corollary 1.
Supposeg,α∈G×0,∞ and A,d is an F-complete F-MS and S:A⟶A is a self-mapping such that(29)dSs,Sy≤λds,y.Forλ∈0,∞ and each s,y∈A×A, then there exists a single FP of the mappings S in A.Theorem 4.
Assumeg,α∈G×0,∞, A,d is a CF-MS, and S,T:A⟶A are self-maps such that(30)dSs,Ty≤αds,y+β.p+ds,Ss.dy,Typ+ds,y+γds,Ss+dy,Ty.Forα,β,γ,andp∈0,∞ such that α+γ/1−β−γ<1 and for all s,y∈A×A, then there exists a single typical FP of the mappings S and T in A.Proof.
Choose an arbitrary elements0 and iterate a sequence sn as(31)Ss2x=s2x+1,Ts2x+1=s2x+2;x=0,1,2,….
From (30) and (31), we obtain(32)ds2x+1,s2x+2=dSs2x,Ts2x+1≤αds2x,s2x+1+β.p+ds2x,Ss2x.ds2x+1,Ts2x+1p+ds2x,s2x+1+γds2x,Ss2x+ds2x+1,Ts2x+1=αds2x,s2x+1+β.p+ds2x,s2x+1.ds2x+1,s2x+2p+ds2x,s2x+1+γds2x,s2x+1+ds2x+1,s2x+2=αds2x,s2x+1+βds2x+1,s2x+2+γds2x,s2x+1+ds2x+1,s2x+2=α+γds2x,s2x+1+β+γds2x+1,s2x+2,or(33)1−β−γdS2x+1,S2x+2≤α+γS2x+1,S2x+2,which implies that(34)ds2x+1,s2x+2≤α+γ1−β−γds2x,s2x+1=λds2x,s2x+1sαyα+γ1−β−γ=λ.
Similarly,(35)ds2x,s2x+1≤λds2x−1,s2x.
Continuing the same way, we obtain(36)dsn,sn+1<λdsn−1,sn.
For allnϵℕ, it yields(37)dsn,sn+1<λdsn−1,sn<λ2dsn−2,sn−1<⋯<λnds0,s1,n∈ℕ.
Using (37), we can write(38)∑k=nm−1dsk,sk+1=dsn,sn+1+dsn+1,sn+2+⋯+dsm−1,sm<λn1+λ+λ2+⋯+λm−n−1ds0,s1≤λn1−λds0,s1,m>n.i.e.,(39)∑k=nm−1dsk,sk+1≤λn1−λds0,s1,m>n.
Sincelimn⟶∞λn/1−λds0,s1=0, for any δ>0, there exists some n′∈ℕ such that(40)0<λn1−λds0,s1<δn≥n′.
Further, supposeg,α∈G×0,∞ satisfies (d3), and ϵ>0 is fixed. By F2, there is some δ>0 such that(41)0<l<δ⟹gl<gϵ−α.
By (40) and (41), we write(42)g∑k=nm−1dsk,sk+1≤gλn1−λds0,s1<gϵ−αm>n≥n′.
By (d3) and the above equation, we obtain(43)dsn,sm>0,m>n>n′⟹gdsn,sm<gϵ.
This shows(44)dsn,sm<ϵ,m>n≥n′.
Thus, it proved that the sequence (sn) is F-Cauchy in A. Now, as A,d is F-complete, so there must be some element c∗ in A and sn converges to c∗ i.e.,(45)limn⟶∞dsn,c∗=0.
Next, we show thatc∗ is the FP of S:(46)dSc∗,s2x+2=dSc∗,Ts2x+1≤αdc∗,s2x+1+β.p+dc∗,Sc∗.ds2x+1,Ts2x+1p+dc∗,s2x+1+γdc∗,Sc∗+ds2x+1,Ts2x+1=αdc∗,s2x+1+β.p+dc∗,s2x+1.ds2x+1,s2x+2p+dc∗,s2x+1+γdc∗,s2x+1+ds2x+1,s2x+2.
Applying limitn⟶∞, we obtain(47)dSc∗,c∗≤α.dc∗,c∗.
Hence,dSc∗,c∗=0, i.e., Sc∗=c∗.
Similarly, we obtainTc∗=c∗. Hence, Tc∗=Sc∗=c∗.
Uniqueness: if there is another distinct typical FPc∗∗S and T, then(48)dc∗,c∗∗=dSc∗,Tc∗∗≤αdc∗,c∗∗+β.p+dc∗,Sc∗.dc∗∗,Tc∗∗p+dc∗,c∗∗+γdc∗,Sc∗+dc∗∗,Tc∗∗=αdc∗,c∗∗+β.p+dc∗,c∗.dc∗∗,c∗∗p+dc∗,c∗∗+γdc∗,c∗+dc∗∗,c∗∗.
Thus,1−αdc∗,c∗∗≤dc∗,c∗∗, and this implies that dc∗,c∗∗, i.e., c∗=c∗∗.Example 3.
SupposeA=Ax6x+2/3,x∈N, dAx,Ak=0ifAx=AkeAx−AkifAx≠Ak, gAx=−1/Ax, and S,T:A⟶A are defined by(49)TAx=A1,ifx=1,2Ax−1,ifx>2,SAx=A1,ifx=1,A2,ifx=2,Ax−2,ifx>4.
One can easily verify thatd is an F-metric and g satisfies F1−F2. Suppose x≠k, α=e−1/2,andβ=γ=0; then,(50)dSAx,TAk=eAx−2−Ak−1=e2x−k−2<e−12.e2x−k=αdAx,Ak<αdAx,Ak+β.p+dAx,SAx.dAk,TAkp+dAx,Ak+γdAk,SAk+dAx,TAx.
Hence, inequality (30) holds. Clearly, A1 is the only typical FP of S and T.


## 3. Fixed Points of Rational Type Contractions (RTC) on Metric Closed (F-MC) Ball

In this section of the manuscript, we explore similar results in the domain of an metric closed (F-MC) ball rather than in the wholeF-MS. We will show that the FP of the above contraction can be iterated even if the contractive condition is imposed only on the metric closed (F-MC) ball. The example in this section gives a comparative analysis of the results in this section to those in the previous section.Definition 6.
LetA,d be a CF-MS and S,T:A⟶A be self-mappings. Suppose forα,β,γ,andp∈0,∞withα+γ/1−β−γ<1; then, the map T is called an RTC on Bs0,μ⊆A if(51)dSs,Ty≤αds,y+βp+ds,Ss.dy,Typ+ds,y+γds,Ss+dy,Ty,for each s,y∈Bs0,μ⊆A.Theorem 5.
Assumeg,α∈G×0,∞, A,d is a CF-MS, and Tbe an RTC on Bs0,μ. Supposefor  somes0∈A and μ>0, and the next conditions can hold:(a)
Bs0,μ is an F-MC(b)
ds0,s1≤1−λμ, for s1∈A and λ=α+γ/1−β−γ(c)
There exist0<ϵ<μ such as g1−λk+1μ≤gϵ−α, where k∈ℕ
Then, there exists a single typical FP of the mappingsS and T in Bs0,μ.Proof:.
Choose an arbitrary elements0 and iterate a sequence sn as(52)Ts2x=s2x+1,Ss2x+1=s2x+2,x=0,1,2,….
Before proceeding to our proof, first we show thatsn is in Bs0,μ for every n∈ℕ. We do this by mathematical induction. Using (b), we write(53)ds0,s1<μ
Hence,s1∈Bs0,μ. Let s2,…,sk∈Bs0,μ for some k∈ℕ. Next, if s2x+1≤sk, then by (51), we can write(54)ds2x,s2x+1=dSs2x−1,Ts2x≤αds2x−1,s2x+β.p+ds2x−1,Ss2x−1.ds2x,Ts2xp+ds2x−1,s2x+γds2x−1,Ss2x−1+ds2x,Ts2x=αds2x−1,s2x+β.p+ds2x−1,s2x.ds2x,s2x+1p+ds2x−1,s2x+γds2x−1,s2x+ds2x,s2x+1=αds2x−1,s2x+β.ds2x,s2x+1+γds2x−1,s2x+ds2x,s2x+1=α+γds2x−1,s2x+β+γds2x,s2x+1,or(55)1−β−γds2x,s2x+1≤α+γds2x−1,s2x,which implies that(56)ds2x,s2x+1≤α+γ1−β−γds2x−1,s2x=λds2x−1,s2xsαyα+γ1−β−γ=λ.
Similarly,(57)ds2x−1,s2x≤λds2x−2,s2x−1.
Similarly, ifs2x≤sk,(58)ds2x−1,s2x<α+γ1−β−γds2x−2,s2x−1=λds2x−2,s2x−1.
Therefore, from inequality (55) and (56), we write(59)ds2x,s2x+1<λds2x−1,s2x<…<λ2xds0,s1,and(60)ds2x−1,s2x<λds2x−2,s2x−1<⋯<λ2x−1ds0,s1.
From (59) and (60), we write(61)dsk,sk+1≤λkds0,s1for somekɛN.
Now, using (61), we have(62)gds0,sk+1≤g∑i=1k+1dsi−1,si+α=gds0,s1+⋯+dsk,sk+1+α≤g1+λ+λ2+⋯+λkds0,s1+α=g1−λk+11−λds0,s1+α.
By (b) and (c), we can write(63)gds0,sk+1≤g1−λk+1μ+α≤gϵ<gμ.
Hence, by (F1), we deduce that(64)sk+1∈Bs0,μ.
Thus,sn∈Bs0,μ for all n∈ℕ. By (30), we have the following equations:(65)ds2i+1,s2i+2=dSs2i,Ts2i+1≤αds2i,s2i+1+β.p+ds2i,Ss2i.ds2i+1,Ts2i+1p+ds2i,s2i+1+γds2i,Ss2i+ds2i+1,Ts2i+1=αds2i,s2i+1+β.p+ds2i,s2i+1.ds2i+1,s2i+2p+ds2i,s2i+1+γds2i,s2i+1+ds2i+1,s2i+2.
Now, proceeding in a similar way as in Theorem4 and using (a), we find that sn converges to some c∗ in Bs0,μ.Also, we prove c∗ as the single common FP of S and T by following the method of Theorem 4.
Putα=0 in Theorem 5, and the following results are given.Corollary 2.
Assumeg,α∈G×0,∞, A,d is an F-complete F-MS, S,T:A⟶A are self-mappings, and β/1−α−β with α,β,p∈0,∞. Let, fors0∈A and μ>0, the following conditions hold:(i)
Bs0,μ⊆A is F-closed(ii)
dSs,Ty≤α.p+ds,Ss.dy,Ty/p+ds,y+βds,Ss+dy,Ty,for allsandy∈Bs0,μ(iii)
ds0,s1≤1−λμ, for s1∈A and λ=β/1−α−β(iv)
There exists0<ϵ<μ such as g1−λk+1μ≤gϵ−α, where k∈ℕ.Then, there exists a single typical FP of the mappingsS and Tin Bs0,μ.If we takeS=T in Theorem 5, the following results are developed.Corollary 3.
Letg,α∈G×0,∞, A,d is an F-complete F-MS, and T:A⟶A is a self-mapping, and assume that α+γ/1−β−γ with α,β,γ,andp∈0,∞. Assume that, fors0∈A and μ>0, the following conditions hold:(i)
Bs0,μ⊆A is F-closed(ii)
dTs,Ty≤αds,y+β.p+ds,Ts.dy,Ty/p+ds,y+γds,Ts+dy,Ty, for every sandy∈Bs0,μ(iii)
ds0,s1≤1−λμ, for s1∈A and λ=α+γ/1−β−γ(iv)
There exist0<ϵ<μ such as g1−λk+1μ≤gϵ−α, where k∈ℕ
Then, there is a single FP of the mappingT in Bs0,μ.Example 4.
LetA=0,∞ and gs=−1/s. Define T:A⟶A by(66)Ts=s3,ifs∈0,2,s3,ifs∈2,∞,and define dβy:(67)ds,y=s−y2,ifs,y∈0,2×0,2,s−y,ifs,y∉0,2×0,2.
Observe thatd is an F-metric space (F-MS) and the mapping g fulfills F1-F2. Choose s0=μ=1; then, Bs0,μ=0,2. Notice that Bs0,μ is ℱ-closed, so condition (a) of Corollary 2 is fulfilled. Next, if α=3/4andβ=γ=0, then λ=α and(68)ds0,s1=ds0,Ts0=12−162=19<1−34.1=1−λμ.
Thus, condition (b) is obeyed. Further, as the functiong is increasing and λ<1, therefore, for every k∈ℕ, we can locate some ϵ<μ and α∈0,∞ such that g1−λk+1μ=gε−α is satisfied. i.e., condition (c) is obeyed.
Now, whens,y∈Bs0,μ×Bs0,μ, then(69)dTs,Ty=s3−y32=19s−y2<34s−y2=αds,y=αds,y+β.p+ds,Ts.dy,Typ+ds,y+γds,Ts+dy,Ty.
Hence, (d) holds for alls,y∈Bs0,μ×Bs0,μ. But, if s,y∉Bs0,μ×Bs0,μ e.g., s=3 and y=4, then(70)dTs,Ty=33−43>343−4=αds,y=αds,y+β.p+ds,Ts.dy,Typ+ds,y+γds,Ts+dy,Ty.
Hence, it is now verified that the condition (b) holds true only for and not on the whole spaceA. Finally, 0∈Bs0,μ is the FP of T.


## 4. Application

This section assures a solution for dynamic programming (DP) by using the fixed point technique. Thus, in turn, it renders a solution to dynamic routing with randomization for secure wireless sensor networks.A DP is two states process: a state space (SS) and a decision space (DC). One can further divide SS into (a) initial state, (b) transitional state, and (c) action state. On the contrary, DS comprises the procedure and steps adopted to iterate the solution of the particular problem. Such algorithms are mainly used in computer programming and optimization.There used to be hundreds or thousands of sensor nodes in a wireless sensor. They sometimes communicate among themselves and sometimes directly to a base station. If there are more sensors in a network, they can sense a large region with more considerable accuracy, minimizing the risk. The sensor nodes are often dispersed on a broader sensor field, where they share the quality data/information.All such sensor nodes tend to gather and route the data back to the base station or to among themselves. Sending the traffic of a network can be done by the process of routing. For a high communication capacity, efficient tools are needed to select a network that can respond sharply to changes in the communication link. For this purpose, many algorithms have already been developed for routing protocols in a network.Bellman [20], in 1958, using the approach of functional equations, designed an algorithm that converges in N−1 (or lesser) iterations to perform dynamic routing with randomization in a sensor network of N nodes and supporting its security in terms of data transmission. One can formulate the problem of DP in the structure of a function equation as follows:(71)ps=maxt∈TFs,t+f1s,t,pηs,tfors∈S.(72)xs=maxt∈TFs,t+f2s,t,pηs,tfors∈S,where Y and Z are Banach spaces such as S⊆Y and T⊆Z, and(73)η:S×T⟶S,F:S×T⟶R,f1,f2:S×T×R⟶R.SupposeS and T are the DS and SS, respectively. We tend to investigate a similar solution point occurrence for both (71) and (72). Let us represent the family of real-valued bounded mappings on S by WS. Suppose an arbitrary element j∈WS, and j=maxs∈Sjs. Then, WS,. is Banach space endowed with d defined as(74)dj,k=maxs∈Sjs−ks.Suppose the below speculations hold:(C1): F,f1,andf2 are bounded.(C2): For s∈S and j∈WS, define Q,Z:WS⟶WS by(75)Qju=maxt∈TFs,t+f1s,t,jηs,tfors∈S,Zju=maxt∈TFs,t+f2s,t,jηs,tfors∈S.Observe that the mappingsF, f1, and f2 are all bounded.(C3) : Fors,t∈S×T, jandk∈WS and l∈S, and we write(76)f1s,t,jl−f1s,t,kl≤Mj,k,where(77)Mj,k=αdj,k+βp+dj,Qj.dk,Zkp+dj,k+γdj,Qj+dk,Zk,for α,β,andγ∈0,∞ and α+2β+2γ<1.Now, we develop the following theorem.Theorem 6.
Suppose conditions (C1)-(C3) hold; then, at most one identical bounded solution for both (71) and (72).Proof:.
By Lemma1, it is evident that WS,d is an F-complete F-MS. d is defined by (74), and from (C1), we deduce that S and T are self-mappings on WS. Let ω be an arbitrary positive number and j1andj2∈WS. Take s∈S and t1andt2∈T such as(78)Qjx<Fs,tx+f1s,tx,jxηs,tx+ω,(79)Zjx<Fs,tx+f2s,tx,jxηs,tx+ω,and(80)Qj1≥Fs,t2+f1s,t2,j1ηs,t2,(81)Zj2≥Fs,t1+f1s,t,j2ηs,t1.
Then, using (78) and (81), we obtain(82)Qj1s−Zj2s<f1s,t1,j1ηs,t1−f1s,t1,j2ηs,t1+ω≤f1s,t1,j1ηs,t1−f1s,t1,j2ηs,t1+ω≤Mj1s,j2s+ω.
Also, from (78) and (79), we get(83)Zj2s−Qj1s<Mj1s,j2s+ω.
Merging the above two inequalities, we write(84)Qj1s−Zj2s<Mj1s,j2s+ω,for all ω>0. Thus,(85)dQj1s,Zj2s≤Mj1s,j2s,i.e.,(86)dSj1,Tj2≤Mj1,j2,for every s∈S. All the requirements of Theorem 4 are fulfilled. Therefore, using Theorem 4, S and T have a unique bounded and standard solution for equations (70) and (71).


## 5. Conclusion

This research work has highlighted the essentialness and usefulness of the newly-introduced F-MS by establishing interesting FP theorems of some contractions. It is obtained that the FP and typical FP of a contractive mapping is beneficial even if the contraction condition is not imposed on the whole F-MS and is shrunk only to a closed ball inside it. The two different examples are indeed a comparative analysis of the outcome of assessing the contraction locally and globally. Few practical corollaries have been developed from the proven results. We will extend this idea to the frame of fuzzy metric space and picture fuzzy metric space in the future. The effects will be investigated in the mentioned setting, and its application in multistage optimization will be discussed. Finally, there is a discussion on the applications side for development and establishing a single solution of the functional equation, which leads to a dynamic routing with randomization and improving data security tasks in a wireless sensor network.



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*Source: 9939546-2021-10-08.xml*