# On Unique and Nonunique Fixed Points and Fixed Circles inMvb-Metric Space and Application to Cantilever Beam Problem

Authors: Meena Joshi; Anita Tomar; Hossam A. Nabwey; Reny George
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We introduceMvb-metric to generalize and improve Mv-metric and unify numerous existing distance notions. Further, we define topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology on Mvb-metric space and to create an environment for the survival of a unique fixed point. Also, we introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map and establish fixed circle and fixed disc theorems. Further, we verify all these results by illustrative examples to demonstrate the authenticity of the postulates. Towards the end, we solve a fourth order differential equation arising in the bending of an elastic beam.


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## Body

## 1. Introduction

A Greek mathematician Euclid of Alexandria (323-283 BC) was the first to communicate the notion of distance. Indeed, Euclidean distance is a noteworthy measure of closeness among two quantities, which is one of the initial conceptions appreciated by mankind. Fréchet [1] deliberated the general and more axiomatic form of distance as “L-space.” Hausdorff [2] reexamined it as a metric space in the setting of points which has been refined, discussed, and generalized in numerous ways. Bakhtin [3], Branciari [4], Asadi et al. [5], George et al. [6], Mitrović and Radenović [7], Özgür et al. [8], Karahan and Isik [9], and Asim et al. [10] introduced the notions of a b-metric, a rectangular metric and a v-generalized metric, an M-metric, a rectangular b-metric, a bvs-metric, a rectangular M-metric, a generalized pvb-partial metric, and an Mv-metric, respectively. Further, one may also refer to Fernandez et al. [11] and Kanwal et al. [12, 13] for work in Nb-cone metric spaces over Banach algebra, orthogonal F-metric spaces, and weak partial b-metric spaces, respectively. One may allude to Kirk and Shahzed [14], to study in detail about the generalizations of the metric notion.The aim of this work is to introduce a novel distance structure called anMvb-metric space which is an improvement and generalization of an Mv-metric space. Also, we introduce topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology and to create an environment for the survival of a unique fixed point in an Mvb-metric space. Further, we demonstrate that the collection of open balls, which forms a basis on Mvb-metric space, generates a T0 topology on it. In the sequel, with the help of examples and remarks, we demonstrate that an Mvb-metric space unifies and combines numerous distance conceptions and marks supremacy over all those spaces wherein the continuity of a map is required for the survival of a fixed point. We also introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map in an Mvb-metric space and establish fixed circle and greatest fixed disc theorems. We verify these results by illustrative examples with geometric interpretations to establish the authenticity of the postulates. Towards the end, we solve fourth order differential equation arising in the Cantilever beam problem.


## 2. Preliminaries

In this section, we use notationsmvu,w=minmvu,u,mvw,w and Mvu,w=maxmvu,u,mvw,w.In 2017, Mitrović and Radenović [7] introduced bvs-metric space.Definition 1.
Abvs-metric on a nonempty set M with s≥1 is a map bvs:M×M⟶ℝ+ satisfying
(1)bvsibvsu,v=0iffu=v,bvsiibvsu,v≥0,bvsiiibvsu,w=bvsw,u,bvsivbvsu,w≤sbvsu,z1+bvsz1,z2+⋯+bvszv,w,
u,z1,z2,⋯,zv,w∈M and are distinct. A pair M,bvs is a bvs-metric space.Remark 2.
Abvs-metric may be reduced to a v-generalized metric [4] for s=1, a rectangular metric [4] for v=2 and s=1, a rectangular b-metric [6] for v=2, a b-metric [3] for v=1, and a usual metric [1] for v=s=1.In 2018, Karahan and Isik [9] introduced the pvb-partial metric.Definition 3.
A partialbvs-metric on a nonempty set M with s≥1 is a map pbvs:M×M⟶ℝ+ satisfying
(2)pbvsipbvsu,u=pbvsw,w=pbvsu,wiffu=w,pbvsiipbvsu,u≤pbvsu,w,pbvsiiipbvsu,w=pbvsw,u,pbvsivpbvsu,w≤spbvsu,z1+pbvsz1,z2+⋯+pbvszv,w−Σi=1vpbvszi,zi,
u,z1,z2,⋯,zv,w∈M and are distinct. A pair M,pbvs is a partial bvs-metric space.Remark 4.
A partialbvs-metric may be reduced to a rectangular partial metric [15] for v=2 and s=1, a rectangular partial b-metric for v=2, a partial b-metric [16] for v=1, and a partial metric [17] for v=s=1.
In 2019, Asim et al. [10] have familiarized the notion of an Mv-metric space.Definition 5.
AnMv-metric on a nonempty set M is a map mv:M×M⟶ℝ+satisfying(3)mvimvu,u=mvw,w=mvu,wiffu=w,mviimvu,w≤mvu,w,mviiimvu,w=mvw,u,mvivmvu,w−mvu,w≤mvu,z1−mvu,z1+mvz1,z2−mvz1,z2+⋯+mvzv,w−mvzv,w,
u,z1,z2,⋯,zv,w∈M and are distinct. A pair M,mv is an Mv-metric space.Example 6 (see [10]).
LetM=ℝ. Define mv:M×M⟶ℝ+bymvu,w=u+w/2,u,w∈M, thenM,mvis anMv-metric space.Remark 7.
In the above example, ifv=1, mv will be an M-metric [5]. In this case, for 1,2,3∈M,(4)mv1,3−mv1,3>mv1,2−mv1,2+mv2,3−mv2,3−mv2,2.Ifv=2, it will be a rectangular M-metric [18]. Again, in this case, for 1,2,3,4∈M,
(5)mv1,4−mv1,4>mv1,2−mv1,2+mv2,3−mv2,3+mv3,4−mv3,4−mv2,2−mv3,3.This suggests that if in anMv-metric, we subtract the term Σi=1vmvzi,zi, then the function mv will no longer be a metric. However, the terms like Σi=1vmvzi,zi are indispensable in the analogous generalizations of a partial metric as a self-distance of a point in this space is not essentially zero.For instance, ifmvu,w=3−n, then for a finite n and u=w,mvu,u is not equal to zero.Definition 8 (see [19]).
A functionψ:0,∞⟶0,∞ is a subadditive altering distance function if
(i)
ψ is an altering distance function (i.e., ψ is continuous, strictly increasing, and ψt=0 iff t=0)(ii)
ψu+w≤ψu+ψw,u,w∈0,∞


## 3. Main Results

We use notationsmvu,wb=minmvbu,u,mvbw,w and Mvu,wb=maxmvbu,u,mvbw,w. First, we introduce an Mvb-metric space.Definition 9.
AnMvb-metric on a nonempty set M with s≥1 is a map mvb:M×M⟶ℝ+ satisfying
(6)mvbimvbu,u=mvbw,w=mvbu,wiffu=w,mvbiimvu,wb≤mvbu,w,mvbiiimvbu,w=mvbw,u,mvbivmvbu,w−mvu,wb≤smvbu,z1−mvu,z1b+mvbz1,z2−mvz1,z2b+⋯+mvbzv,w−mvzv,wb−Σi=1vmvbzi,zi,
u,z1,z2,⋯,zv,w∈M and are distinct. A pair M,mvb is called an Mvb-metric space.Example 10.
LetM=ℝ+ and mvb:M×M⟶0,∞ be defined by mvbu,w=u−wα+maxu,wα,α>1. By routine calculation, we verify that M,mvb is an Mvb-metric space with s≥3α−1. But M,mvb is not an Mv-metric space. Since, for u=1,w=n and z1=2,z2=3,⋯,zv=n−1, we attain
(7)mvb1,n−mv1,nb=1−nα+max1,nα−1α=1+nα−1=nα,mvb1,2−mv1,2b=1−2α+max1,2α−1α=1+2α−1=2α,mvb2,3−mv2,3b=2−3α+max2,3α−2α=1+3α−2α,⋮mvbn−2,n−1−mvn−2,n−1b=n−2−n+1α+maxn−2,n−1α−n−2α=1+n−1α−n−2α.
Therefore,mvb1,n−mv1,nb>mvb1,2−mv1,2b+mvb2,3−mv2,3b+⋯+mvbn−2,n−1−mvn−2,n−1b.Also, ifs=1, M,mvb is an improvement and extension of an Mv-metric space [10]. Inclusion of the terms containing nonzero self-distances demonstrate that it is a proper generalization of a notion of an Mv-metric space and consequently partial metric space as well. In particular, M,mvb is an Mv-metric [18] for v=1 and rectangular Mb-metric space [20] for v=2.Theorem 11.
Letmvb be an Mvb-metric on a set M. Let pbvs∗u,w=mvbu,w+Mvu,wb−mvu,wb, then pbvs∗ is a generalized partial bvs-metric on M.Proof.
Letu,w∈M:
pbvsi
Letpbvs∗u,u=pbvs∗w,w=pbvs∗u,w⇔mvbu,u=mvbw,w=pbvs∗u,w.
So,pbvs∗u,w=mvbu,w, i.e., mvbu,u=mvbw,w=mvbu,w⟹u=w.
Hence,pbvs∗u,u=pbvs∗v,v=pbvs∗u,v⇔u=v.
pbvsii For u,w∈M,pbvs∗u,u=mvbu,u+Mvu,ub−mvu,ub=mvbu,u and mvbu,w−mvu,wb≥0.
Now,pbvs∗u,u=mvbu,u≤Mvu,wb≤Mvu,wb+mvbu,w−mvu,wb=pbvs∗u,w.
pbvsiii Symmetric condition is trivially satisfied.
pbvsiv For u,z1,z2,⋯zv,∈M,
(8)pbvs∗u,w=mvbu,w+Mvu,wb−mvu,wb≤smvbu,z1−mvu,z1b+mvbz1,z2−mvz1,z2b+⋯+mvbzv,w−mvzv,wb−Σi=1vmvbzi,zi=smvbu,z1−mvu,z1b+mvbz1,z2−mvz1,z2b+⋯+mvbzv,w−mvzv,wb−Σi=1vmvbzi,zi+Mvu,wb≤smvbu,z1−mvu,z1b+mvbz1,z2−mvz1,z2b+⋯+mvbzv,w−mvzv,wb+Mvu,z1b+Mvz1,z2b+⋯+Mvzv,wb−Σi=1vmvbzi,zi≤smvbu,z1−mvu,z1b+mvbz1,z2−mvz1,z2b+⋯+mvbzv,w−mvzv,wb+sMvu,z1b+Mvz1,z2b+⋯+Mvzv,wb−Σi=1vpbvs∗zi,zi=smvbu,z1+Mvu,z1b−mvu,z1b+mvbz1,z2+Mvz1,z2b−mvz1,z2b+⋯+mvbzv,w+Mvzv,wb−mvzv,wb−Σi=1vpbvs∗zi,zi=spbvs∗u,z1+pbvs∗z1,z2+⋯+pbvs∗zv,w−Σi=1vpbvs∗zi,zi.
Thus,M,pbvs∗ is a generalized partial bvs-metric space.Theorem 12.
Letmvb be an Mvb-metric on a set M. Let bvs∗u,w=mvbu,w−mvu,wb, then bvs∗ is a bvs-metric on M.Proof.
bvsi For u,w∈M,(9)letbvs∗u,w=0⇔mvbu,w=mvu,wb⇔mvbu,w=mvbu,u=mvbw,w⇔u=w.
bvsii Since, mvbu,w≥mvu,wb,mvbu,w−mvu,wb≥0, i.e., bvs∗u,w≥0,u,w∈M.
bvsiii Symmetric condition is trivially satisfied.
bvsiv For u,z1,z2,⋯zv,w∈M,
(10)bvs∗u,w=mvbu,w−mvu,wb,≤smvbu,z1−mvu,z1b+mvbz1,z2−mvz1,z2b+⋯+mvbzv,w−mvzv,wb−Σi=1vmvbzi,zi,≤smvbu,z1−mvu,z1b+mvbz1,z2−mvz1,z2b+⋯+mvbzv,w−mvzv,wb=sbvs∗u,z1+bvs∗z1,z2+⋯+bvs∗zv,w.
Thus,M,bvs∗ is a bvs-metric.Remark 13.
Letpbvs be a partial bvs-metric on a set M, then bvs1∗=pbvsu,w−pbvsu,u−pbvsw,w is bvs-metric space.
To discuss the topology corresponding toMvb-metric, the open ball with centre at u and radius r∈0,∞ is defined as Umvbu,ε=w∈M:mvbu,w<mvu,wb+ε/s.
Similarly, the closed ball with centre atu and radius r∈0,∞ is defined as Umvbu,ε=w∈M:mvbu,w≤mvu,wb+ε/s.Lemma 14.
In anMvb-metric space M,mvb, every open ball is an open set.Proof.
Letw0∈Umvbu,r, then mvbu,w0<mvu,w0b+r/s. Choose ε/s=mvu,w0b+r/s−mvbu,w0>0.
Again, letw1∈Umvbw0,ε, so mvbw1,w0<mvw1,w0b+ε/s and choose ε1/s=mvw1,w0b+ε/s−mvbw1,w0>0.
Proceeding as above, letwv∈Umvbwv−1,εv, so mvbwv,wv−1<mvwv,wv−1b+εv−1/s, choose εv/s=mvwv,wv−1b+εv−1−mvbu,w1>0.
Now, foru,w0,w1,⋯,wv,(11)mvbu,wv−mvu,wv≤smvbu,w0−mvu,w0+mvbw0,w1−mvw0,w1+⋯+mvbwv−1,wv−mvwv−1,wv−mvbw1,w1−mvbw2,w2−⋯−mvbwv−1,wv−1≤smvbu,w0−mvu,w0+mvbw0,w1−mvw0,w1+⋯+mvbwv−1,wv−mvwv−1,wv=srs−εs+εs−ε1s+⋯+εv−1s−εvs=r−εv.
Hence,Umvbw0,ε⊆Umvbu,r.Theorem 15.
IfM,mbv is an Mvb-metric space and τvb a topology generated by the open balls Umvbu,r, then M,τvb is a T0-space.Proof.
LetM,τvb be an Mvb-metric space and u,w∈M are two distinct points. Then,
(12)mvu,wb≤mvbu,w⟹minmvbu,u,mvbw,w≤mvbu,w,
i.e.,mvbu,u≤mvbu,w or mvbw,w≤mvbu,w.
Firstly, assume thatmvbu,u=mvbu,w,(13)thenmvu,wb=mvbu,u=mvbw,w<mvbu,w⟹mvu,wb−mvbu,w=mvu,wb−mvbu,u<0.
Fors≥1, if we chose ε>0 such that mvbu,w=mvu,wb+ε/s, so w∉Umvbu,ε.
Next, assume thatmvbu,u<mvbw,w, then mvu,wb=mvbu,u<mvbu,w:
(14)⟹mvu,wb−mvu,wb=mvu,wb−mvbu,u<0.
Again, fors≥1, if we chose ε1>0 such that mvbu,w=mvu,wb+ε/s, so w∉Umvbu,ε1.
Similarly, formvbw,w<mvbu,u, one may easily find an open ball so that u∈Umvbu,ε1 and w∉Umvbu,ε1, i.e., for two distinct points u,w∈M, there is a ball including the point u but not including the other point w. Thus, M,mvb is a T0-space.Now, we discuss the convergence of the sequence and introduce definitions related to it.Definition 16.
(i)
A sequenceun in M,mvb is mvb-convergent to u∈M iff limn⟶∞mvbun,u−mvun,u=0
In other words, a sequenceun in a topological space M,τvb converges to a point u in M if for each open ball Umvbu,ε containing u there exists k so that for each n>k,un∈Umvbu,ε.
(ii)
A sequenceun in M,mvb is mvb-Cauchy sequence iff limn,m⟶∞mvbun,um−mvun,umb and limn,m⟶∞Mvun,umb−mvun,umb exist and are finite(iii)
AnMvb-metric space is mvb-complete if each mvb-Cauchy sequence un converges to a point u∈M so that limn,m⟶∞mvbun,u−mvun,ub=0 and limn,m⟶∞Mvun,ub−mvun,ub=0Lemma 17.
LetM,mvb be an Mvb-metric space and A∶M⟶M a self-map on M so that there exists η∈0,1/s, satisfying
(15)mvbAu,Aw≤ηmvbu,w.
Consider the sequenceun defined by un+1=Aun. If un⟶u as n⟶∞, then Aun⟶Au as n⟶∞.Proof.
IfmvbAun,Au=0,mvAun,Aub≤mvbAun,Au=0⟹mvbAun,Au−mvAun,Aub⟶0, i.e., Aun⟶Aun as n⟶∞.
Now, ifmvbAun,Au>0, then by (15),
(16)mvbAun,Au≤ηmvbun,u.There are two cases.Case 1.
Ifmvbu,u≤mvbun,un, then mvbun,un⟶0 as n⟶∞⟹mvbu,u=0.
SincemvbAu,Au≤mvbu,u, we have mvbAu,Au=0. Again, un⟶u implies that limn⟶∞mvbun,u−mvun,ub=0.
Since,mvun,ub=minmvbun,un,mvbu,u=0, so limn⟶∞mvbun,u=0.
Now, from equation (16),
(17)limn⟶∞mvbAun,Au=0,i.e.,Aun⟶Auasn⟶∞.Case 2.
Ifmvbun,un≤mvbu,u, then mvbun,un⟶0 as n⟶∞.
So, as abovelimn⟶∞mvbAun,Au=0,i.e.,Aun⟶Au as n⟶∞.
Now, we prove the first main result for a Kannan type contraction.Theorem 18.
LetM,mvb be an Mvb-complete metric space with coefficient s≥1. Suppose, a self-map A∶M⟶M satisfies
(18)mvbAu,Aw≤μmvbu,Au+mvbw,Aw,μ<12s,u,w∈M.
Then,A has a unique fixed point u∗ so that mvbu∗,u∗=0 and the sequence of iterates Anu0⊆M converges to u∗∈M.Proof.
Starting from the given elementu0∈M, form the sequence un, where un=Aun−1,n∈ℕ. If mvbun,un+1=0,n≥0, then Aun=un+1=un and mvbun,un=0, and this completes the proof.
Further, takemvbun,un+1>0, n≥0. For u=un,w=un+1, utilizing condition (18),
(19)mvbun+1,un+2=mvbAun,Aun+1≤μmvbun,Aun+mvbun+1,Aun+1=μmvbun,un+1+mvbun+1,un+2,i.e.,1−μmvbun+1,un+2≤μmvbun,un+1,i.e.,mvbun+1,un+2≤μ1−μmvbun,un+1.
Let,ξvbn=mvbun,un+1,n≥0,λ=μ/1−μ<1, then ξvbn+1≤λξvbn.
On repeating these steps, we getξvbn+1≤λnξvb0⟶0,n⟶∞.
Now,(20)mvbun+1,un+1=mvbAun,Aun≤μmvbun,Aun+mvbun,Aun=2μmvbun,un+1=2μξvbn≤2λn−1ξvb0⟶0,asn⟶∞.
First, we show thatun≠um, for n≠m. Suppose, un=um, for n>m, then Aun=un+1=Aum=um+1.
Now, using inequality (18), for u=un and w=un+1,
(21)ξvbm=mvbum,um+1=mvbun,un+1=ξvbn≤λξvbn−1≤λ2ξvbn−2≤⋯≤λn−mξvbm<ξvbm,
a contradiction. Thus,un≠um, for n≠m.Now, we assert thatun is a Cauchy sequence in M,mvb. We discuss two cases.Case 1.
First, letl be odd, i.e., l=2m+1, for n,m∈ℕ. Now, by using mvbiv for n≤v≤n+l,(22)mvbun,un+l=mvbun,un+2m+1≤smvbun,un+1+mvbun+1,un+2+⋯+mvbun+v−1,un+v+mvbun+v,un+2m+1−mvbun+1,un+1−mvbun+2,un+2−⋯−mvbun+v,un+v≤sλn−1+λn+⋯λn+v−2ξvb0−2λn+λn+1+⋯+λn+v−1ξvb0+smvbun+v,un+2m+1=sλn−11−λv1−λξvb0−2λn1−λv1−λξvb0+smvbun+v,un+2m+1≤sλn−11−λv1−λξvb0−λn1−λv1−λξvb0+s2mvbun+v,un+v+1+mvbun+v+1,un+v+2+⋯+mvbun+2v−1,un+2v+mvbun+2v,un+2m+1−smvbun+v+1,un+v+1+mvbun+v+2,un+v+2+⋯+mvbun+2v,un+2v≤sλn−11−λv1−λξvb0−λn1−λv1−λξvb0+s2λn+v−1+λn+v+⋯+λn+2v−2ξvb0+s2mvbun+2v,un+2m+1−sλn+v+λn+v+1+⋯+λn+2v−1ξvb0≤sλn−11−λv1−λξvb0−λn1−λv1−λ+s2λn+v−11−λv1−λξvb0−sλn+v1−λv1−λξvb0⋯+s2m/v−1mvbun+2m−v,un+2m+1≤sλn−11−λv1−λξvb0−λn1−λv1−λξvb0+s2λn+v−11−λv1−λξvb0−sλn+v1−λv1−λξvb0+⋯+s2m/vmvbun+2m−v,un+2m−v+1+mvbun+2m−v+1,un+2m−v+2+⋯+mvbun+2m,un+2m+1−s2m/v−1mvbun+2m−v+1,un+2m−v+1+⋯+mvbun+2m,un+2m≤sλn−11−λv1−λξvb0−λn1−λv1−λξvb0+s2λn+v−11−λv1−λξvb0−sλn+v1−λv1−λξvb0+⋯+s2m/vλn+2m−v−11−λv1−λξvb0−s2m/v−1λn+2m−v1−λv1−λξvb0⟶0,asn⟶∞,i.e., limn,m⟶∞mvbun,un+2m+1=0.Case 2.
Now, letl is even, i.e., l=2m for n,m∈ℕ. Now, by using mvbiv for n≤v≤n+l,(23)mvbun,un+l=mvbun,un+2m≤smvbun,un+1+mvbun+1,un+2+⋯+mvbun+v−1,un+v+mvbun+v,un+2m−mvbun+1,un+1−mvbun+2,un+2−⋯−mvbun+v,un+v≤sλn−1+λn+⋯+λn+v−2ξvb0−λn+λn+1+⋯+λn+v−1ξvb0+smvbun+v,un+2m=sλn−11−λv1−λξvb0−λn1−λv1−λξvb0+smvbun+v,un+2m≤sλn−11−λv1−λξvb0−λn1−λv1−λξvb0+s2mvbun+v,un+v+1+mvbun+v+1,un+v+2+⋯+mvbun+2v−1,un+2v+mvbun+2v,un+2m+1−smvbun+v+1,un+v+1+mvbun+v+2,un+v+2+⋯+mvbun+2v,un+2v≤sλn−11−λv1−λξvb0−λn1−λv1−λξvb0+s2λn+v−1+λn+v+⋯+λn+2v−2mvb0+s2mvbun+2v,un+2m−sλn+v+λn+v+1+⋯+λn+2v−1ξvb0≤sλn−11−λv1−λξvb0−λn1−λv1−λξvb0+s2λn+v−11−λv1−λξvb0−sλn+v1−λv1−λξvb0+⋯+s2m/v−1mvbun+2m−1−v,un+2m≤sλn−11−λv1−λξvb0−λn1−λv1−λξvb0+s2λn+v−11−λv1−λξvb0−sλn+v1−λv1−λξvb0⋯+s2m/vmvbun+2m−1−v,un+2m−v+mvbun+2m−v,un+2m−v+1+⋯+mvbun+2m−1,un+2m−s2m/v−1mvbun+2m−v,un+2m−v+⋯+mvbun+2m−1,un+2m−1≤sλn−11−λv1−λξvb0−λn1−λv1−λξvb0+s2λn+v−11−λv1−λξvb0−sλn+v1−μv1−λξvb0+⋯+s2m/vλn+2m−v−21−λv1−λξvb0−s2m/v−1λn+2m−v−11−λv1−λξvb0⟶0,asn⟶∞,i.e., limn,m⟶∞mvbun,un+2m=0.
So,limn,m⟶∞mvbun,um−mvun,umb=0.
LetMvbun,um=mvbun,un. Now,Mvbun,um−mvbun,um≤Mvbun,um=mvbun,un≤2λn−1ξvb0⟶0, as n⟶∞.
Hence,limn,m⟶∞Mvbun,um−mvbun,um=0.
Therefore, the sequenceun is mvb-Cauchy in M.
Since,M is mvb-complete, there exists u∗∈U so that un⟶u∗. Now, we assert that Au∗=u∗:
(24)limn⟶∞mvbun,u∗−mvun,u∗b=0⟹limn⟶∞mvbun+1,u∗−mvun+1,u∗b=0⟹limn⟶∞mvbAun,u∗−mvAun,u∗b=0⟹mvbAu∗,u∗−mvAu∗,u∗b=0usingLemma17,i.e., mvbAu∗,u∗=minmvbAu∗,Au∗,mvbu,u∗⟹mvbAu∗,u∗=mvbAu∗,Au∗ or mvbAu∗,u∗=mvbu∗,u∗.
Hence,Au∗=u∗, i.e., u∗ is a fixed point of A.
Now, we assert thatmvbu∗,u∗=0:
(25)mvbu∗,u∗=mvbMu∗,Mu∗≤μmvbu∗,Au∗+mvbu∗,Au∗=2μmvbu∗,Au∗=2μmvbu∗,u∗<mvbu∗,u∗,a contradiction. Hence, mvbu∗,u∗=0.
Suppose,u∗ and w∗ are two different fixed points of A, so
(26)mvbu∗,w∗=mvbAu∗,Aw∗≤μmvbu∗,Au∗+mvbw∗,Aw∗=μmvbu∗,u∗+mvbw∗,w∗=0⟹mvbu∗,w∗=0.
Hence,u∗=w∗.Now, we furnish two examples (one of continuous and another of discontinuous self-map) to validate Theorem18 and demonstrate the fact that the continuity of a self-map is not indispensable for the survival of a unique fixed point in an Mvb-metric space for the Kannan type contraction. Also, the self-distance of a fixed point is zero, and the sequence of iterates Anu0⊆M converges to a fixed point.Example 19.
LetM=−10,10 and an Mvb-metric mvb:M×M⟶ℝ+ be defined by mvbu,w=∣u∣+∣w∣/22,u,w∈M. Then, M,mvb is a complete Mvb-metric space for s=3. Define a self-map A on M as Au=3/19u,u∈M. Observe that, for all u,w∈M, we obtain
(27)mvbAu,Aw=Au+Aw22=3/19u+3/19w22=9361u+w22,mvbu,Au+mvbw,Aw=u+Au22+w+Aw22=u+3/19u22+w+3/19w22=484361u22+484361w22≤484361u+w22,i.e., mvbAu,Aw≤μmvbu,Au+mvbw,Aw,μ=9/484<1/2s.
Thus, all the postulates of Theorem18 are verified and A has a unique fixed point 0 in M and clearly, mvb0,0=0. Further, there exists a sequence Anu0⊆M,Anu0=−n/n2+n,n∈ℕ, which mvb-converges to 0.Example 20.
ConsiderM=0,1 and an Mvb-metric mvb:M×M⟶ℝ+ defined by mvbu,w=u+w/22,u,w∈M. Then, M,mvb is a complete Mvb-metric space for s=3. Define a self-map A on M as
(28)Au=u3,u∈0,119,u=1,u∈M.Observe that, for allu,w∈M, we obtain the following.Case 21.
Whenu,w∈0,1,(29)mvbAu,Aw=Au+Aw22=u/3+w/322=u+w62,mvbu,Au+mvbw,Aw=u+Au22+w+Aw22=u+u/322+w+w/322=4u62+4w62≤49u+w2.Case 22.
Whenu=w=1,(30)mvbAu,Aw=1/9+1/922=181,mvbu,Au+mvbw,Aw=1+1/922+1+1/922=5081.Case 23.
Whenu=0,w=1,(31)mvbAu,Aw=0+1/922=1324,mvbu,Au+mvbw,Aw=0+022+1+1/922=2581,i.e., mvbAu,Aw≤μmvbu,Au+mvbw,Aw,μ=1/16<1/2s.
Thus, all the postulates of Theorem18 are verified and A has a unique fixed point 0 in M, and clearly, mvb0,0=0. Further, there exists a sequence Anu0⊆M,Anu0=1/3n2+2n,n∈ℕ, which mvb-converges to 0.Remark 24.
Examples19 and 20 demonstrate that Theorem 18 is an extension, generalization, and improvement of Asim et al. [10], Banach [21], Branciari [4], Kannan [22], Karahan and Isik [9], Mitrović and Radenović [7], Özgür et al. [8], Shukla [15, 16], and references therein to an Mvb-metric space. It is fascinating to see that the continuity of a self-map is not an indispensable requirement for the survival of a unique fixed point satisfying (18) in this novel space.Our next result extends the result of Reich [23] to Mvb-metric space.Theorem 25.
LetM,mvb be an Mvb-complete metric space with coefficient s≥1. Suppose a self-map A∶M⟶M satisfies
(32)mvbAu,Aw≤αmvbu,Au+βmvbw,Aw+γmvbu,w,α,β,γ≥0,α+β+γ<12s+1,u,w∈M.
Then,A has a unique fixed point u∗ so that mvbu∗,u∗=0 and the sequence of iterates Anu0⊆M converges to u∗∈M.Proof.
It follows the pattern of Theorem18.
Theorem25 is an extension, improvement, and generalization of Banach [21] (α=β=0), Kannan [22] (α=β,γ=0), Reich [23], and Asim et al. [10] (α=β=0) to an Mvb-metric space.
Now, we prove the result for a Hardy and Rogers type contraction [24], which includes all the results stated above as a special case.Theorem 26.
LetM,mvb be an Mvb-complete metric space. Suppose, a self-map A∶M⟶M satisfies
(33)mvbAu,Aw≤αmvbu,Aw+βmvbw,Au+γmvbu,w+δmvbu,Au+κmvbw,Aw,α,β,γ,δ,κ≥0,α+β+γ+δ+κ<12s+1,u,w∈M.
Then,A has a unique fixed point u∗ so that mvbu∗,u∗=0 and the sequence of iterates Anu0⊆M converges to u∗∈M.Proof.
It follows the pattern of Theorem18.Theorem26 is an extension, improvement, and generalization of Banach [21] (α=β=δ=κ=0), Kannan [22] (α=β=γ=0,δ=κ), Chatterjee [25] (γ=δ=κ=0,α=β), Reich [23] (α=β=0), Hardy and Rogers [24] (s=1), Asim et al. [10] (α=β=δ=κ=0), and references therein to an Mvb-metric space.The following result is more fascinating as it is proved by altering the distances between the points, exploiting subadditive altering distance function [19].Theorem 27.
LetM,mvb be a Mvb-complete metric space. Suppose, we have a self-map A∶M⟶M satisfying
(34)ψmvbAu,Aw≤μψmvbu,w+ψmvbu,Au+ψmvbw,Aw,μ<12s+1,u,w∈M.
Then,A has a unique fixed point u∗ so that mvbu∗,u∗=0, and the sequence of iterates Anu0⊆M converges to u∗∈M and for κ=2μ/1−μ<1,(35)MvbAn+1u∗,Anu∗≤κnMvbu∗,Au∗,n=0,1,2,⋯.Proof.
For an arbitraryu∈M, let w=Au. Then,
(36)ψmvbw,Aw=ψmvbAu,Aw≤μψmvbu,w+ψmvbu,Au+ψmvbw,Aw=μψmvbu,Au+ψmvbu,Au+ψmvbAu,Aw1−μψmvbAu,Aw≤2μψmvbu,AuψmvbAu,Aw≤2μ1−μψmvbu,AuψmvbAu,Aw≤κψmvbu,w,whereκ=2μ1−μ<1,⟹mvbAu,Aw≤κmvbu,w.
Now, starting from the given elementu0∈M, form the sequence un, where un=Aun−1,n∈ℕ. If mvbun,un+1=0,n≥0, then Aun=un+1=un and mvbun,un=0, and this completes the proof.Further, takemvbun,un+1>0, n≥0. For u=un,w=un+1, utilizing condition (36),
(37)mvbun+1,un+2=mvbAun,Aun+1≤κmvbun,un+1≤κnmvbu0,u1⟶0,as,n⟶∞.Also,(38)mvbun+1,un+1=mvbAun,Aun≤κmvbun,un≤κnmvbu0,u0⟶0,asn⟶∞.So, following a similar pattern as that of Theorem18, we may easily conclude that A has a unique fixed point u∈M, where mvbu,u=0.Finally, by takingu=An−1u, from (36),
(39)mvbAnu,An+1u≤κmvbAn−1u,Anu≤κ2mvbAn−2u,An−1u⋯≤κnmvbu,Au,n=1,2,3,⋯.Now, we furnish two examples of a discontinuous self-map to validate Theorem27.Example 1.
LetM=ℝ and an Mvb-metric mvb:M×M⟶ℝ+ be defined as mvbu,w=maxu2,w2+u−w2,u,w∈M with s=3.
Define a self-mapA∶M×M⟶ℝ by
(40)Au=u9,u∈−9,9,3u5,otherwise,and ψ:0,∞⟶0,∞ as ψu=u/7, then observe that, for all u,w∈M, we obtain
(41)ψmvbAu,Aw=ψmaxAu2,Aw2+Au−Aw2≤ψ925maxu2,w2+u−w2<925ψmvbu,w+ψmvbu,Au+ψmvbw,Aw.
Thus, all the postulates of Theorem27 are verified and A has a unique fixed point 0 in M, and clearly, mvb0,0=0. Further, there exists a sequence Anu0⊆M,Anu0=n/n2+1,n∈ℕ, which mvb-converges to 0.Example 2.
LetM=−2,2 and an Mvb-metricmvb∶M×M⟶ℝ+ be defined as
(42)mvbu,w=u+w22,u,w∈Mwiths=3.
Define a self-mapA∶M×M⟶ℝ by
(43)Au=0,u∈−2,0,−1,otherwise,and ψ∶0,∞⟶0,∞ as ψu=u/2, then observe that, for all u,w∈M, we obtain
(44)ψmvbAu,Aw<μψmvbu,w+ψmvbu,Au+ψmvbw,Aw,μ=211<12s+1.
Thus, all the postulates of Theorem27 are verified and A has a unique fixed point 0 in M, and clearly, mvb0,0=0. Further, there exists a sequence Anu0⊆M,Anu0=−3n/n2+n,n∈ℕ, which mvb-converges to 0.Remark 28.
Examples1 and 2 demonstrate that Theorem 27 is an extension, generalization, and improvement of Asim et al. [10], Banach [21], Kannan [22], Hardy and Rogers [24], Karahan and Isik [9], Khan et al. [19], Mlaiki et al. [18], Mitrović and Radenović [7], Reich [23], and references therein to Mvb-metric space. It is fascinating to see that continuity of a self-map is not indispensable for the existence of a fixed point satisfying (34) in this novel space.Corollary 29.
The conclusion of Theorem27 is true even if we replace (34) by
(45)mvbAu,Aw≤μmvbu,v+mvbu,Au+mvbv,Av,μ<12s+1,u,w∈M.Proof.
The proof follows, if we take,ψu=u in (34).Corollary 30.
The conclusion of Theorem27 is true even if we replace (34) by
(46)ψmvbAju,Ajw≤μψmvbu,w+ψmvbu,Aju+ψmvbw,Ajw,μ<12s+1,u,w∈M.Proof.
Applying Theorem27 to the self-map T=Aj, we get T has a unique fixed point, say u, i.e., Tu=Aju=u.
Now,(47)TAu=AjAu=Aj+1u=AAju=Au,so Au is a unique fixed point of T.Corollary 31.
The conclusion of Theorem27 remains true even if we replace (34) by
(48)1+mvbAu,Aw1/μ≤1+mvbu,w1+mvbu,Au1+mvbw,Aw,μ<12s+1,u,w∈M.Proof.
The proof follows the pattern of Theorem27 if we take ψu=ln1+u in (34).Theorem 32.
LetM,mvbbe anMvb-complete metric space with coefficient sβ<1. Suppose A∶M⟶M satisfies
(49)ψmvbAu,Aw≤αψmvbu,Au+βψmvbw,Aw+γψmvbu,w,α+β+γ<1,u,w∈M.
Then,A has a unique fixed point u∗ so that mvbu∗,u∗=0, and the sequence of iterates Anu0⊆M converges to u∗∈M and for δ=α+β/1−γ,(50)MvbAn+1u,Anu≤δnMvbu,Au,n=0,1,2,⋯.Proof.
The proof follows the pattern of Theorem27.Theorem 33.
The conclusion of Theorem27 remains true even if we replace inequality (34) by
(51)ψmvbAu,Aw≤μψmvbw,Aw+ψmvbu,Aw,μ<12s,u,w∈M.Proof.
The proof follows the pattern of Theorem27.
We note that whenψu=u, the above theorem is the same as Theorem 18.


## 4. Existence of a Fixed Circle/Disc

Following Özgür and Tas [26], we first familiarize fixed circle in an Mvb-metric space to establish fixed circle theorems. Next, we exploit mvb-metric version (mvb-Caristi map) of the classical Caristi map [27] to establish that the set of nonunique fixed points of a map includes a circle. Also, following Aydi et al. [28], we familiarize fixed disc to establish the greatest fixed disc in an Mvb-metric space.Definition 34.
We define a circle centred atu0 and radius r in an Mvb-metric space M,mvb as
(52)Cu0,r=u∈M:mvbu0,u=r+mvbu0,u0,u0∈M,r∈0,∞.Definition 35.
We define a disc centred atu0 and radius r in an Mvb-metric space M,mvb as
(53)Du0,r=u∈M:mvbu0,u≤r+mvbu0,u0,u0∈M,r∈0,∞.Geometrically, it is not necessary that a circle/disc defined in aMvb-metric space is the same as the circle/disc in a Euclidean space.Definition 36.
LetCu0,r/Du0,r be a circle/disc centred at u0 and radius r in an Mvb-metric space M,mvb. For a self-map A:M⟶M in an Mvb-metric space M,mvb, if mvbu,Au=mvbu,u=mvbAu,Au,u∈Cu0,r/u∈Du0,r, then Cu0,r/Du0,r is called the fixed circle/fixed disc of A.Theorem 37.
LetCu0,r be a circle in an Mvb-metric space M,mvb. Define ς:M⟶0,∞ as
(54)ςu=mvbu,u0+mvbu,u,u∈M.
If there exists a self-mapA:M⟶M so that
(55)imvbu,Au≤ςAu−ςu+mvbu,u,iimvbAu,u0≤r+mvbu0,u0,iiimvbAu,Au≤mvbu,u,(iv)
IfmvbAu,Aw≤μmvbu,w,u∈Cu0,r,w∈M\Cu0,r,μ∈0,1, then Cu0,r is a unique fixed circle of MProof.
Letu∈Cu0,r be any arbitrary point. Using (i) and equation (54),
(56)mvbu,Au≤mvbAu,u0+mvbAu,Au−mvbu,u0−mvbu,u+mvbu,u=mvbAu,u0+mvbAu,Au−mvbu,u0≤r+mvbu0,u0+mvbAu,Au−r−mvbu0,u0usingii,i.e.,mvbu,Au≤mvbAu,Au.
But by the definition of anMvb-metric space, mvbAu,Au≤mvbu,Au. So, mvbu,Au=mvbAu,Au. Using (iii) mvbu,Au≤mvbu,u. Again, by the definition of an Mvb-metric space, mvbu,u≤mvbu,Au. Hence,
(57)mvbu,Au=mvbAu,Au=mvbu,u,i.e., u is a fixed point of A,∀u∈Cu0,r. So, the self-map A fixes the circle Cu0,r, i.e., the set of nonunique fixed points of a map A includes a circle.
Let there exist two fixed circles,Cu0,r0 and Cu1,r1,r≠r1 of A, i.e., A satisfies all the conditions (i) to (iii) for each of the circles Cu0,r0 and Cu1,r1. Let u∈Cu0,r0 and w∈Cu1,r1. Using (iv), mvbu,w=mvbAu,Aw≤μmvbu,w, a contradiction.Hence,Cu0,r0 is a unique fixed circle of A.The following example illustrates Theorem37.Example 38.
LetM=ℝ and an Mvb-metric mvb:M×M⟶ℝ+ be defined as mvbu,w=maxuα,wα+u−wα,α>1,u,w∈M with s=3α−1. The circle
(58)C−2,6=u∈M:mvb−2,u=6+mvb−2,−2=u∈M:max−2α,uα+∣−2−uα=6+−2α=u∈M:max2α,uα+∣2+uα=6+2α.
Now, we discuss circles for different values ofα:
(i)
Whenα=2,max4,u2+2+u2=6+4, then(59)either4+2+u2=10oru2+2+u2=10,2+u2=6oru2+4+u2+4u=10,2+u=±6or2u2+4u−6=0,u=−2±6oru=−3,u=1(ii)
Whenα=3,max4,∣u3+2+u3=6+8, then(60)either8+2+u3=14oru3+2+u3=14±2+u3=6or±u3±2+u3=14,2+u=±61/3oru=−2.40629,0.6097,−1±2,u=−2±61/3oru=−2.40629,0.6097,−1±2
Now, define a self-mapA:M⟶M as
(61)Au=u,u∈C−2,6,−1+2,u∉C−2,6.
Then, the self-mapA verifies all the postulates of Theorem 37 except (iv) and fixes the circle C−2,6. Clearly, the set of fixed points of A contains a circle C−2,6. However, one may notice that there are more than one circle corresponding to each value of α. For instance, C2,6 is also a fixed circle of A.
It is obvious that geometrically (i) implies thatAu is in the exterior of a circle and (ii) implies that Au is in the interior of a circle. It means ACu0,r⊆Cu0,r (see Example 38).Theorem 39.
The conclusion of Theorem37 is true even if we substitute (i) by (i)′ and (ii) by (ii)′:
(62)i′mvbu,Au≤ςu−ςAu+mvbAu,Au,ii′mvbAu,u0≥r+mvbu0,u0.Proof.
Letu∈Cu0,r be any arbitrary point. Using (i)′ and equation (54)
(63)mvbu,Au≤mvbu,u0+mvbu,u−mvbAu,u0−mvbAu,Au+mvbAu,Au=r+mvbu0,u0+mvbu,u−mvbAu,u0≤r+mvbu0,u0+mvbu,u−r−mvbu0,u0usingii′,(64)i.e.,mvbu,Au≤mvbu,u.
But by the definition of anMvb-metric space, mvbu,u≤mvbu,Au.
So,mvbu,Au=mvbu,u. Using (63) and (iii), mvbu,Au≤mvbAu,Au. Also, by definition of an Mvb-metric mvbAu,Au≤mvbu,Au. So, mvbu,Au=mvbAu,Au, i.e., mvbu,Au=mvbAu,Au=mvbu,u⟹u is a fixed point of A,∀u∈Cu0,r, i.e., the self-map A fixes the circle Cu0,r, i.e., the set of fixed points of a map A includes a circle.
Uniqueness of a fixed circle may be proved as in Theorem37.Example 40.
LetM=ℝ and an Mvb-metric mvb:M×M⟶ℝ+ be defined as mvbu,w=u+w/22+u−w2,u,w∈M with s=4:(65)C−4,11=u∈M:mvb−4,u=11+mvb−4,−4=u∈M:4+u22+−4−u2=11+4+422,⟹4+u22+4+u2=11+16,⟹4+u2+44+u2=108.
There arises two cases:(i)
4+u2+44+u2=108,4+u2=108/5=21.6,4+u=±4.65⟹u=0.65,u=−8.65(ii)
4−u2+44+u2=1084+u2−8u+416+u2+8u=108,5u2+24u+68=108,5u2+24u−40=0⟹u=−6.1,u=−1.3
Define a self-mapA:M⟶M as
(66)Au=u,u=C−4,11,0.65,u≠C−4,11.
Then, mapA verifies all the postulates of Theorem 39 and fixes the unique circle C−4,11, i.e., the set of nonunique fixed points of A=−8.65,−6.1,−1.3,0.65contains a unique circle C−4,11.
It is clear that geometrically (i)′ implies that Au is in the interior of a circle and (ii)′ implies that Au is in the exterior of a circle. It means ACu0,r⊆Cu0,r (see Example 40).
In the above theorems, we have assumed Banach contraction to demonstrate the uniqueness of fixed circle. So, it is significant to establish the uniqueness of the fixed circle using different contractive conditions. In the following, we establish uniqueness using a more general contractive condition.Theorem 41.
LetM,mvbbe anMvb-metric space andCu0,rbe a circle onM.LetA:M⟶Mbe a self-map satisfying (i)-(iii), (i)′and (ii)′of Theorem37or39along with the contraction condition(67)mvbMu,Mw≤μmaxmvbu,w,mvbu,Aw,mvbw,Mu,12mvbu,Mu+mvbw,Mw,12mvbu,Mw+mvbw,Mu,u∈Cu0,r,w∈M\Cu0,r, where ς:ℝ+⟶ℝ+ is a continuous nondecreasing function and ςt<t, then Cu0,r is a unique fixed circle of M.Proof.
LetCu0,r0 and Cu1,r1 be two fixed circles of M, i.e., M satisfies the conditions (i)-(iii), (i)′ and (ii)′ (Theorem 37 and 39) for both the circles Cu0,r0 and Cu1,r1. Let u∈Cu0,r0 and w∈Cu1,r1. Using inequality (67),
(68)mvbu,w=mvbMu,Mw≤μmvbu,w,mvbu,w,mvbu,w,12mvbu,w+mvbu,w,12mvbu,w+mvbu,w≤μmvbu,w<mvbu,w,a contradiction. Hence, Cu0,r0 is a unique fixed circle of M.
Next, we establish the existence of a greatest fixed disc.Theorem 42.
LetDu0,r be a disc in an Mvb-metric spaceM,mvb. Define ς:M⟶0,∞ as in (54). If there exists a self-map A:M⟶M so that
(69)mvbAu,u0≤r+mvbu0,u0,mvbAu,Au≤mvbu,u,(i)
mvbu,Au≤ςAu+ςAu−2r−2mvbu0,u0, then Du0,r is a fixed disc of M(ii)
IfmvbAu,Aw≤μmvbu,w,u∈Du0,r,w∈M\Du0,r,μ∈0,1, then Du0,r is a fixed disc of maximum radius r, i.e., there is no fixed disc Du0,r′ of A having a radius greater than rProof.
Letu∈Du0,r be any arbitrary point:
(70)mvbu,Au≤mvbu,u0+mvbu,u+mvbAu,u0+mvbAu,Au−2r−2mvbu0,u0≤r+mvbu0,u0+mvbu,u+r+mvbu0,u0−2r−2mvbu0,u0=mvbu,u,(71)i.e.,mvbu,Au≤mvbu,u.
But by the definition of anMvb-metric space, mvbu,u≤mvbu,Au.
So,mvbu,Au=mvbu,u. Using (70) and (iii), mvbu,Au≤mvbAu,Au. Also, by definition of an Mvb-metric mvbAu,Au≤mvbu,Au. So, mvbu,Au=mvbAu,Au, i.e., mvbu,Au=mvbAu,Au=mvbu,u⟹u is a fixed point of A,∀u∈Du0,r, i.e., the self-map A fixes the disc Du0,r, i.e., the set of fixed points of a map A includes a disc.
Let there exist two fixed discsDu0,r and Du1,r′,r<r′, of A, i.e., A satisfies the conditions (i) to (iii) for each of the discs Du0,r and Du1,r′.
Letu∈Du0,r and w∈Du1,r′ such that w∉Du0,r.
Using (iv),mvbu,w=mvbAu,Aw≤μmvbu,w, a contradiction. Hence, Du0,r0 is a fixed disc of A having maximum a radius r.Remark 43.
Following the pattern of Theorem41, we may prove the existence of a greatest fixed disc of A having a maximum radius.Remark 44.
For more work on the set of nonunique fixed points forming a circle or a disc or an ellipse, one may refer to [26, 28–31] and references therein.


## 5. Application to Cantilever Beam

As an application of Theorem18, we solve fourth-order differential equation arising in two point boundary value problem of a bending of an elastic beam. This problem is employed in the distortion of an elastic beam in equilibrium, of which one end is free whereas another is fixed. Let I=0,1 and M=CI,ℝ be the set of all continuous functions on 0,1. Define Mvb-metric, mvb:M×M⟶ℝ as mvbu,w=∣u∣+∣w∣/22,u,w∈M and s=4.Theorem 45.
We consider the Cantilever beam problem(72)d4udt=ψt,u,dudt,d2udt,d3udt,u0=dudt0=d2udt21=d3udt31=0,t∈0,1,where ψ:0,1×ℝ4⟶0,1 is a continuous function. If
(73)ψξ,u,dudt<μuξ,ξ∈0,1,μ∈0,18,then Problem (72) has a unique solution.Proof.
The Cantilever beam problem (72) may be rewritten as
(74)ut=∫01Gt,ξψξ,uξ,dudtξ,u∈M,where the Green function
(75)Gt,ξ=16ξ23t−ξ,0≤ξ≤t≤1,16t23ξ−t,0≤t≤ξ≤1.
Define a mapA:M⟶M by
(76)Aut=∫01Gt,ξψξ,uξ,dudtξdξ.
Now,u∈M is a solution of (72) iff it is a fixed point of A:(77)mvbAut,Awt=Aut+Awt22=14∫01Gt,ξψξ,uξ,dudtξdξ+∫01Gt,ξψξ,wξ,dwdtξdξ2≤14μut∫01Gt,ξdξ+μwt∫01Gt,ξdξ2=μ4ut+wt2∫01Gt,ξdξ2=μ4ut+wt2∫0t16ξ23t−ξdξ+∫t116t23ξ−tdξ2=μ4ut+wt218t4+1123t2−2t3−t42≤μ4ut+wt218+6122=25μ256ut+wt2,(78)mvbu,Aut+mvbw,Awt=ut+Aut22+wt+Awt22=ut+∫01Gt,ξψξ,uξ,du/dtξdξ22+∣ut∣+∣∫01Gt,ξψξ,wξ,dw/dtξdξ∣22=ut+μut∫01Gt,ξdξ22+wt+μwt∫01Gt,ξdξ22≤14ut+58μut2+wt+58μwt2=148+5μ264ut2+8+5μ264wt2=8+5μ2256ut2+∣wt2<8+5μ2256u+wt2.
Equations (77) and (78) imply that
(79)mvbAut,Awt≤μmvbut,Aut+mvbwt,Awt,i.e., A satisfies Theorem 18, and hence, A has a unique fixed point, i.e., a Cantilever beam problem has a unique solution.


## 6. Conclusion

We have introducedMvb-metric as an improvement and generalization of an Mv-metric which need not be continuous and defined topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of a space to discuss the topology of Mvb-metric and to create an environment for the survival of a unique fixed point in an Mvb-metric space. Further, we have demonstrated that the collection of open ball, which forms a basis on Mvb-metric space and generates a T0 topology on it. Also, we have introduced a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map. Our results are sharpened versions of the well-known results, wherein continuity of a self-map is not indispensable for the survival of a unique fixed point. Examples and applications to solve a Cantilever beam problem employed in distortion of an elastic beam in equilibrium substantiate the utility of these improvements and extensions. It is interesting to mention that the Cantilever structures permit overhanging constructions deprived of peripheral bracing.



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*Source: 6681044-2021-03-08.xml*