# On Unique and Nonunique Fixed Points in Metric Spaces and Application to Chemical Sciences

Authors: Meena Joshi; Anita Tomar
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We introduce the notions of a generalizedΘ-contraction, a generalized ΘE-weak contraction, a ΨE-weak JS-contraction, an integral-type ΘE-weak contraction, and an integral-type ΨE-weak JS-contraction to establish the fixed point, fixed ellipse, and fixed elliptic disc theorems. Further, we verify these by illustrative examples with geometric interpretations to demonstrate the authenticity of the postulates. The motivation of this work is the fact that the set of nonunique fixed points may include a geometric figure like a circle, an ellipse, a disc, or an elliptic disc. Towards the end, we provide an application of Θ-contraction to chemical sciences.


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## Body

## 1. Introduction and Preliminaries

The study of the geometry of the set of nonunique fixed points of a map is a significant area of research. There are numerous examples of a map where the set of nonunique fixed points of the self-map includes some geometric shapes. For example, consider a self-mapM on the metric space U,d with the usual metric defined on the two-dimensional plane ℝ2 as(1)Mu,v=u,v,u,v∈u2+v2=1,1,0,otherwise.Noticeably, the set of nonunique fixed pointscosnθ,sinnθ:n∈ℤ,θ∈0,2π includes the circle C0,0,1 centered at 0,0 having radius 1; that is, C0,0,1 is a fixed circle of M. It is significant to mention that there exist maps that map the circle Cu0,r to itself but do not fix all the points of the circle Cu0,r. For example, let M be a self-map on the two-dimensional plane ℝ2 defined by(2)Mu,v=uu2+v2,vu2+v2,u,v∈ℝ.Then,MC0,1=C0,1, but map M fixes only two points 1,0 and −1,0 of the circle C0,1. Noticeably, M does not fix all the points of C0,1. For details on this work, one may refer to [1–23] and the references therein. A geometric figure (a circle, a disc, an ellipse, and so on) included in the set of nonunique fixed points is called a fixed figure (a fixed circle, a fixed disc, a fixed ellipse, and so on) of the self-map [15].The aim of the present work is to introduce notions of a generalizedΘ-contraction, a generalized ΘE-weak contraction, a ΨE-weak JS-contraction, a generalized integral-type ΘE-weak contraction, and an integral-type ΨE-weak JS-contraction to study the geometry of nonunique fixed points. In the sequel, we establish the fixed point, fixed ellipse, and fixed elliptic disc theorems. Further, we verify these by illustrative examples to demonstrate the authenticity of the postulates. Further, we provide an application of Ćirić-type Θ-contraction to obtain an application to chemical sciences. To be specific, we solve a boundary value problem arising when a diffusing material is kept in an absorbing medium between parallel walls of specified concentrations.Definition 1 [24].
A metric on a nonempty setU is a function d:U×U⟶ℝ+ satisfying(i)
du,v=0 iff u=v(ii)
du,v=dv,u(iii)
du,v≤du,w+dw,v,u,v,w∈UDefinition 2 [5].
An ellipse having foci atc1 and c2 in a metric space U,d is defined as(3)Ec1,c2,a=u∈U:dc1,u+dc2,u=a,c1,c2∈U,a∈0,∞.The midpointC of a line c1c2 is known as a center of an ellipse. Here, the segment of length a on line c1c2 is the major axis, the line perpendicular to it through the center is the minor axis, and a/2 is the length of a semimajor axis of an ellipse. The distance f=1/2dc1,c2 is the linear eccentricity, and the ratio of linear eccentricity and semimajor axis is the eccentricity; that is, e=dc1,c2/a. Visibly, the circles are the ellipses of vanishing eccentricity in which both the focal points are the same; that is, dc1,c2=0. Actually, an ellipse is a compressed circle. Generally, eccentricity is the measure of the deviation of the curve from the circularity of the particular shape.Example 1.
LetU=ℝ and a metric d:U×U⟶ℝ+ be defined as du,v=u−v,u,v∈U; then,(4)E5,10,8=u∈M:d5,u+d10,u=8=u∈U:5−u+10−u=8=3.5,11.5.
That is, an ellipse centered at 7.5 having foci at 5 and 10 is3.5,11.5.Definition 3 [25].
LetΩ symbolize the class of functions Θ:0,∞⟶1,∞ such that the subsequent conditions hold:
Θ1: Θ is nondecreasing;
Θ2: for every sequence un⊆0,∞, limn⟶∞Θun=1⇔limn⟶∞un=0+;
Θ3: there exist α∈0,1 and β∈0,∞ such that limu⟶0+Θu−1/uα=β.Definition 4 [5].
LetEc1,c2,a be an ellipse having foci at c1 and c2 in a metric space U,d. Then, Ec1,c2,a is said to be a fixed ellipse of M:U⟶U if Mu=u,u∈Ec1,c2,a,a∈0,∞.


## 2. Main Results

In this section, we are dealing with maps satisfying some novel contractions which fix one element of the space or more than one element of the space under suitable conditions and a set of nonunique fixed points, including some geometrical shapes, may be either an ellipse or an elliptic disc. First, we define a generalizedΘ-contraction to establish a unique fixed point by giving a short and simple proof.Definition 5.
LetΘ:0,∞⟶1,∞∈Ω, and the map M:U⟶U of a metric space U,d is said to be a generalized Θ-contraction with u≠v if(5)dMu,Mv>0⇒ΘdMu,Mv≤ΘLu,vα,where Lu,v=maxdu,v,γdu,Mu+1−γdv,Mv,1−γdu,Mu+γdv,Mv,γdu,Mv+1−γdv,Mu,1−γdu,Mv+γdv,Mu,γ∈0,1,α∈0,1,u,v∈U.Remark 6.
In the above contraction, ifγ=0, then M:U⟶U is a Ćirić-type Θ-contraction.Theorem 7.
LetU,d be a complete metric space and map M:U⟶U be a continuous generalized Θ-contraction. Then, M has a unique fixed point. Also, the sequence of iterates Mnu converges to a fixed point of M in U.Proof.
Define a Picard sequenceun⊆U, un+1=Mun, n∈ℕ0, with initial point u0∈U. If for some n∈ℕ, Mnu=Mn+1u, then Mnu is a fixed point of M and the proof is complete. So, presume that for each n, dMnu,Mn+1u>0; then,(6)ΘdMnu,Mn+1u≤ΘLMn−1u,Mnu,where(7)LMn−1u,Mnu=maxdMn−1u,Mnu,γdMn−1u,Mnu+1−γdMnu,Mn+1u,1−γdMn−1u,Mnu+γdMnu,Mn+1u,γdMn−1u,Mn+1u+1−γdMnu,Mnu,1−γdMn−1u,Mn+1u+γdMnu,Mnu=maxdMn−1u,Mnu,γdMn−1u,Mnu+1−γdMnu,Mn+1u,1−γdMn−1u,Mnu+γdMnu,Mn+1u,γdMn−1u,Mn+1u,1−γdMn−1u,Mn+1u.Case 1.
IfdMn−1u,Mnu≤dMnu,Mn+1u, then(8)LMn−1u,Mnu=dMnu,Mn+1u.
That is,ΘdMnu,Mn+1u≤ΘdMnu,Mn+1uα,α∈0,1, a contradiction.Case 2.
IfdMn−1u,Mnu≥dMnu,Mn+1u, then(9)LMn−1u,Mnu=dMn−1u,Mnu.
That is,ΘdMnu,Mn+1u≤ΘdMn−1u,Mnuα.Following a similar pattern,(10)ΘdMnu,Mn+1u≤ΘdMn−1u,Mnuα⋯≤Θdu,Muαn⟶1,asn⟶∞.UsingΘ2, limn⟶∞du,Mu=0.UsingΘ3, there exist β∈0,∞ such that limn⟶∞ΘdMnu,Mn+1u−1/dMnu,Mn+1uα=β.Ifβ∈0,∞, then for ε1=β/4>0, there exists N1>0 such that(11)ΘdMnu,Mn+1u−1dMnu,Mn+1uα−β<ε,n≥N1,⇒ΘdMnu,Mn+1u−1dMnu,Mn+1uα>β−ε1=34β>ε1,n≥N1.That is,dMnu,Mn+1uα<1/ε1ΘdMnu,Mn+1u−1,n≥N1.Ifβ=∞, then for any ε2>0, there exists N2>0 such that(12)ΘdMnu,Mn+1u−1dMnu,Mn+1uα>ε2,n≥N2.That is,dMnu,Mn+1uα<1/ε2ΘdMnu,Mn+1u−1,n>N2.Thus, for allβ∈0,∞ and μ=max1/ε1,1/ε2, there exists N=maxN1,N2 such that(13)dMnu,Mn+1uα<μΘdMnu,Mn+1u−1,n>N,≤μΘdu,Mu−1⟶0,asn⟶∞usingΘ2.That is,limn⟶∞dMnu,Mn+1uα=0 implies that there exists n≥N such that(14)dMnu,Mn+1u≤1n1/α,n≥N.Ifn>m,(15)dMmu,Mnu≤dMmu,Mm+1u+dMm+1u,Mm+2u+⋯+dMn−1u,Mnu≤1m1/α+1m+11/α+⋯+1n−11/α≤∑i=n∞1i1/α.Sinceα∈0,1, series Σi=n∞1/i1/α is convergent and limn,m⟶∞dMmu,Mnu exists and is finite; that is, Mnu is a Cauchy sequence.SinceU is complete, Mnu converges to u∗∈U. Since M is continuous, Mnu⟶u∗⇒Mn+1u⟶Mu∗. By definition of limit Mu∗=u∗, that is, u∗ is a fixed point of M.Letw∗ be another fixed point of U. So dMu∗,Mw∗=du∗,w∗>0. Now,(16)ΘdMu∗,Mw∗≤ΘLu∗,w∗α, where (17)Lu∗,w∗=maxdu∗,w∗,γdu∗,Mu∗+1−γdw∗,Mw∗,1−γdu∗,Mu∗+γdw∗,Mw∗,γdu∗,Mw∗+1−γdw∗,Mu∗,1−γdu∗,Mw∗+γdw∗,Mu∗=maxdu∗,w∗,γdu∗,u∗+1−γdw∗,w∗,1−γdu∗,u∗+γdw∗,w∗,γdu∗,w∗+1−γdw∗,u∗,1−γdu∗,w∗+γdw∗,u∗=maxdu∗,w∗,γdu∗,w∗+1−γdw∗,u∗,1−γdu∗,w∗+γdw∗,u∗=du∗,w∗.That is,ΘdMu∗,Mw∗≤Θdu∗,w∗α≤Θdu∗,w∗.That is,Θdu∗,w∗≤Θdu∗,w∗, a contradiction.Hence,M has a unique fixed point in U.Theorem 8.
LetU,d be a complete metric space and map M:U⟶U be a continuous Ćirić-type Θ-contraction. Then, M has a unique fixed point. Also, the sequence of iterates Mnu converges to a fixed point of M in U.Proof.
The proof follows the pattern of Theorem7 on taking γ=0.The subsequent example appreciates that Theorem8 gives assurance of the uniqueness of the fixed point.Example 2.
LetU=un=2n−1:n∈ℕ and a metric d:U×U⟶ℝ+ be defined as du,v=u−v. Then, U,d is a complete metric space.
LetΘt=etet∈Ω,γ=0.
Define a self-mapM:U⟶U as(18)Mu=u1,u=u1,un−1,u=un,n≥2.
Then,(19)Lun,u1=maxdun,u1,dun,Mun,du1,Mu1,du1,Mun,dun,Mu1=maxd2n−1,1,d2n−1,2n−3,0,d1,2n−3,d2n−1,1=max2n−2,2,2n−4=2n−2,n≥2.
Now,(20)limn⟶∞dMun,Mu1Lun,u1=limn⟶∞Mun−Mu1Lun,u1=limn⟶∞2n−3−12n−2=1,n≥2.
Clearly,M is neither a Ćirić-type contraction [26] nor a Banach contraction [27].
Now, we claim thatM satisfies Ćirić-type Θ-contraction; that is,(21)dMun,Mum≠0⇒edMun,MumedMun,Mum≤eαdun,umedun,um,α∈0,1⇒dMun,MumedMun,Mum≤α2dun,umedun,um,α∈0,1⇒dMun,MumedMun,Mumdun,umedun,um≤α2,α∈0,1.Case 1.
Whenn=1 and m≥2,(22)dMun,MumedMun,Mumdun,umedun,um=4−2me∣4−2m∣2−2me∣2−2m∣=2m−4e2m−42m−2e2m−2≤e−2.Case 2.
Whenn>m>1,(23)dMun,MumedMun,Mumdun,umedun,um=2n−2m−6e∣2n−2m−6∣2n−2m−2e∣2n−2m−2∣=2n−2m−6e2n−2m−62n−2m−2e2n−2m−2≤e−4.Thus,M is a Ćirić-type Θ-contraction with α=maxe−4,e−2=e−2 and has a unique fixed point 1. Further, limn⟶∞Mnu1=1.Remark 9.
Theorems7 and 8 are improvements, extensions, and generalizations of Banach [27], Ćirić [26], Jleli and Samet [25], and references therein. Further, on taking Θt=et, t>0, in these results, we obtain some novel results which are generalizations of existing results in the literature.Next, following Joshi et al. [5], we define an elliptic disc and a fixed elliptic disc to study the geometry of nonunique fixed points in a metric space.Definition 10.
An elliptic disc having foci atc1 and c2 in a metric space U,d is defined as EDc1,c2,a=u∈U:dc1,u+dc2,u≤a,c1,c2∈M,a∈0,∞.Remark 11.
For defining an ellipse or elliptic disca≥dc1,c2.Example 3.
LetU=ℝ2 and a metric d:U×U⟶ℝ+ be defined as du,v=u1−v12+u2−v22,u≪u1,u2,v≪v1,v2∈U; then,(24)ED−5,0,5,0,12=u∈U:d−5,0,u1,u2+d5,0,u1,u2≤12=u∈U:u1+52+u22+u1−52+u22≤12,which is shown by the blue shaded portion in Figure 1.Definition 12.
LetEDc1,c2,a be an elliptic disc having foci at c1 and c2 in a metric space U,d. Then, EDc1,c2,a is said to be a fixed elliptic disc of map M:U⟶U if Mu=u,u∈EDc1,c2,a,a∈0,∞.Figure 1
The elliptic discED−5,0,5,0,12 centered at 0,0 having foci at −5,0 and 5,0 is shown in this figure.We now introduce and exploit a generalizedΘE-weak contraction to demonstrate that the set of nonunique fixed points of a map includes an ellipse or an elliptic disc.Definition 13.
LetΘ:0,∞⟶1,∞ be an increasing function. A map M:U⟶U of a metric space U,d is said to be a generalized ΘE-weak contraction with u≠v, if(25)du,Mu>0⇒Θdu,Mu≤ΘLu,vα,where Lu,v=maxdu,v,γdu,Mu+1−γdv,Mv,1−γdu,Mu+γdv,Mv,γdu,Mv+1−γdv,Mu,1−γdu,Mv+γdv,Mu,γ∈0,1,α∈0,1,u,v∈U.Remark 14.
In the above contraction, ifγ=0, then M:U⟶U is said to be a Ćirić-type ΘE-weak contraction.Theorem 15.
LetEc1,c2,a be an ellipse in a metric space U,d and a=1/2infdu,Mu:u≠Mu. If map M:U⟶U is a generalized ΘE-weak contraction with c1,c2∈U and dc1,Mu+dc2,Mu=a,u∈Ec1,c2,a, then Ec1,c2,a is a fixed ellipse of M.Proof.
Letu∈Ec1,c2,a be any arbitrary point and Mu≠u. From the definition of a,du,Mu≥2a, suppose Mc1≠c1 and Mc2≠c2, so we have dc1,Mc1>0, dc2,Mc2>0, and(26)Θdc1,Mc1≤ΘLc1,c1α=Θmaxdc1,c1,γdc1,Mc1+1−γdc1,Mc1,1−γdc1,Mc1+γdc1,Mc1γdc1,Mc1+1−γdc1,Mc1,1−γdc1,Mc1+γdc1,Mc1α=Θmax0,dc1,Mc1α=Θdc1,Mc1α<Θdc1,Mc1,α∈0,1,acontradiction.
SoMc1=c1. Similarly, Mc2=c2.
Again, sincedu,Mu>0,(27)Θdu,Mu≤ΘLu,c1α=Θmaxdu,c1,γdu,Mu+1−γdc1,Mc1,1−γdu,Mu+γdc1,Mc1γdu,Mc1+1−γdc1,Mu,1−γdu,Mc1+γdc1,Muα<Θmax2a,γdu,Mu,1−γdu,Mu,γdu,c1+1−γdc1,Mu,1−γdu,c1+γdc1,Muα<Θmax2a,γdu,Mu,1−γdu,Mu,γdu,c1+1−γdc1,Mu,1−γdu,c1+γdc1,Mu,α∈0,1.Case 1.
Ifmax2a,γdu,Mu,1−γdu,Mu,1−γdu,Mu,γdu,c1+1−γdc1,Mu,1−γdu,c1+γdc1,Mu=2a, then Θdu,Mu<Θ2a.
By definition ofa and Θ, Θ2a≤Θdu,Mu<Θ2a, a contradiction.Case 2.
Ifmax2a,γdu,Mu,1−γdu,Mu,1−γdu,Mu,γdu,c1+1−γdc1,Mu,1−γdu,c1+γdc1,Mu=γdu,Mu, then Θdu,Mu<Θγdu,Mu.
Ifγ=0, Θdu,Mu<Θ0, a contradiction.
Ifγ∈0,1, Θdu,Mu<Θγdu,Mu<Θdu,Mu, a contradiction.Case 3.
Ifmax2a,γdu,Mu,1−γdu,Mu,1−γdu,Mu,γdu,c1+1−γdc1,Mu,1−γdu,c1+γdc1,Mu=1−γdu,Mu, then Θdu,Mu<Θ1−γdu,Mu≤Θdu,Mu, a contradiction.Case 4.
Ifmax2a,γdu,Mu,1−γdu,Mu,1−γdu,Mu,γdu,c1+1−γdc1,Mu,1−γdu,c1+γdc1,Mu=γdu,c1+1−γdc1,Mu, then(28)Θdu,Mu<Θγdu,c1+1−γdc1,Mu<Θγa+1−γa=Θa.
By definition ofa and Θ, Θ2a≤Θdu,Mu<Θa, a contradiction.Case 5.
Ifmax2a,γdu,Mu,1−γdu,Mu,1−γdu,Mu,γdu,c1+1−γdc1,Mu,1−γdu,c1+γdc1,Mu=1−γdu,c1+γdc1,Mu, then(29)Θdu,Mu<Θ1−γdu,c1+γdc1,Mu<Θγa+1−γa=Θa.
By definition ofa and Θ, Θ2a≤Θdu,Mu<Θa, a contradiction.Similarly, we can prove forc2∈U.Hence,Mu=u,u∈Ec1,c2,a; that is, Ec1,c2,a is a fixed ellipse of M.Theorem 16.
If in the above theoremdc1,Mu+dc2,Mu≤a, then EDc1,c2,a is a fixed elliptic disc of M.Proof.
Now, to showEDc1,c2,a is a fixed elliptic disc of M, it is sufficient to demonstrate that M fixes an ellipse Ec1,c2,b with b⊲a. Since M is a generalized ΘE-weak contraction, then proceeding as in Theorem 15, Ec1,c2,b is a fixed ellipse of M as dc1,Mu+dc2,Mu=b≤a; that is, Mu=u,∀u∈EDc1,c2,b. Hence, EDc1,c2,a is a fixed elliptic disc of M.Theorem 17.
Theorem15 remains true if we substitute Ćirić-type ΘE-weak contraction in place of generalized ΘE-weak contraction.Proof.
The proof follows the pattern of Theorem15 on taking γ=0.Theorem 18.
Theorem16 remains true if we substitute Ćirić-type ΘE-weak contraction in place of generalized ΘE-weak contraction.Proof.
The proof follows the pattern of Theorem16 on taking γ=0.The subsequent examples elucidate Theorems17 and 18.Example 4.
LetU=5,∞ and a metric d:U×U⟶ℝ+ be defined as du,v=u−v.
LetΘt=et, c1=−2, c2=3, a=6, γ=0, and α=6/7.
The ellipse(30)E−2,3,6=u∈U:d−2,u+d3,u=6=u∈U:−2−u+3−u=6=−2.5,3.5.
Define a self-mapM:U⟶U as Mu=u,u∈−5,5u+12,otherwise.
Since foru∈−5,5, du,Mu=0, and for u∈5,∞, du,Mu=12>0.Case 1.
Foru>5 and c1=−2,(31)Lu,−2=maxdu,−2,d−2,M−2,du,Mu,d−2,Mu,du,M−2=maxdu,−2,d−2,−2,du,Mu,d−2,Mu,du,−2=maxdu,−2,0,12,d−2,u+12=maxu+2,12,u+14=u+14>19,and Θdu,Mu=Θ12=e12<e12/13∣u+8∣=eLu,−212/19=ΘLu,−212/19.Case 2.
Foru>5 and c2=3,(32)Lu,3=maxdu,3,d3,M3,du,Mu,d3,Mu,du,M3=maxdu,3,d3,3,du,Mu,d3,Mu,du,3=maxdu,3,0,12,d3,u+12=maxu−3,12,u+9=u+9>14,and Θdu,Mu=Θ12=e12<e12/14u+9=eLu,312/14=ΘLu,312/14.
That is,M is a Ćirić-type ΘE-weak contraction with c1=−2, c2=3, and α=max12/19,12/14=12/14. Hence, E−2,3,6=−2.5,3.5 is a fixed ellipse and ED−2,3,6=−2.5,3.5 is a fixed elliptic disc of M. One may verify that d−2,u+d3,u≤6,u∈ED−2,3,6.Example 5.
LetU=ℝ2 and a metric d:U×U⟶ℝ+ be defined as du,v=u1−v12+u2−v22, where u=u1,u2 and v=v1,v2.
LetΘt=1+t, c1=3+23,−1, c2=3−23,−1, a=8, γ=0, and α=6/7.
The ellipse(33)Ec1,c2,8=u∈U:dc1,u+dc2,u=8=u∈U:u1−3−232+u2+12+u1−3+232+u2+12=8=u∈U:u1−3216+u2+124=1,which is shown by the blue line in Figure 2.
Further, the elliptic discEDc1,c2,8=u∈U:u1−32/16+u2+12/4≤1, which is shown as the blue shaded portion in Figure 2.
Define a self-mapM:U⟶U as Mu=u,u∈3+6cosθ,−1+6sinθu+82,82,otherwise.
Since foru∈3+6cosθ,−1+6sinθ, du,Mu=0, and for u∈ℝ2\3+6cosθ,−1+6sinθ, du,Mu=16>0.Case 1.
Foru∈ℝ2\3+6cosθ,−1+6sinθ and c1=3+23,−1,(34)Lu,3+23,−1=maxdu,3+23,−1,d3+23,−1,M3+23,−1,du,Mu,d3+23,−1,Mu,du,M3+23,−1=maxdu,3+23,−1,d3+23,−1,3+23,−1,du,Mu,d3+23,−1,Mu,du,3+23,−1=maxdu,3+23,−1,0,16,d3+23,−1,u+82,82=maxu1−3−232+u2+12,16,u1+82−3−232+u2+82+12>16,and Θdu,Mu=Θ16=e16<e16/17Lu,3+23,−1∣=ΘLu,3+23,−116/17.Case 2.
Foru∈ℝ2\3+6cosθ,−1+6sinθ and c1=3−23,−1,(35)Lu,3−23,−1=maxdu,3−23,−1,d3−23,−1,M3−23,−1,du,Mu,d3−23,−1,Mu,du,M3−23,−1=maxdu,3−23,−1,d3−23,−1,3−23,−1,du,Mu,d3−23,−1,Mu,du,3−23,−1=maxdu,3−23,−1,0,16,d3−23,−1,u+82,82=maxu1−3+232+u2+12,16,u1+82−3+232+u2+82+12>21,(36)Θdu,Mu=Θ16=e16<e16/21Lu,3−23,−1=ΘLu,3−23,−116/21.
That is,M is a Ćirić-type ΘE-weak contraction with c1=3+23,−1, c2=3−23,−1, and α=max16/17,16/21=16/17. Hence, Ec1,c2,8 is a fixed ellipse and EDc1,c2,8 is a fixed elliptic disc of M. One may verify that dc1,u+dc2,u≤8,u∈EDc1,c2,8.Figure 2
The blue line and its interior demonstrate the ellipseEc1,c2,8 and elliptic disc EDc1,c2,8, respectively, which is fixed by map M.Now, we introduceΨE-weak JS-contraction to study the geometry of nonunique fixed points.Definition 19.
LetΨ:0,∞⟶1,∞ be an increasing function with Ψ0=1; then, a map M:U⟶U of a metric space U,d is said to be a ΨE-weak JS-contraction with u≠v, if(37)du,Mu>0⇒Ψdu,Mu≤Ψdu,vaΨdu,MubΘdv,MvcΨdu,MveΨdv,Muf,where a, b, c, e, and f are nonnegative and a+b+c+e+f∈0,1,u,v∈U.Theorem 20.
LetEc1,c2,a be an ellipse in a metric space U,d and a=1/2infdu,Mu:u≠Mu. If map M:U⟶U is a ΨE-weak JS-contraction with c1,c2∈U and dc1,Mu+dc2,Mu=a,u∈Ec1,c2,a, then Ec1,c2,a is a fixed ellipse of M.Proof.
Letu∈Ec1,c2,a be any arbitrary point and Mu≠u. From the definition of a,du,Mu≥2a, since M is ΨE-weak JS-contraction for c1,c2∈U, suppose Mc1≠c1 and Mc2≠c2, so we have dc1,Mc1>0, dc2,Mc2>0, and(38)Ψdc1,Mc1≤Ψdc1,c1aΨdc1,Mc1bΨdc1,Mc1cΨdc1,Mc1eΨdc1,Mc1f=Ψ0aΨdc1,Mc1b+c+e+f=Ψdc1,Mc11−a<Ψdc1,Mc1,acontradiction.
SoMc1=c1. Similarly, Mc2=c2.
Again, sincedu,Mu>0, so(39)Ψdu,Mu≤Ψdu,c1aΨdu,MubΨdc1,Mc1cΨdu,Mc1eΨdc1,Muf<ΨaaΨ2abΨdc1,c1cΨaeΨaf<Θ2aaΨ2abΘ0cΨ2aeΨ2afasΨisincreasing=Ψ2aa+b+e+f<Ψ2a1−c<Ψ2a.
Sincedu,Mu≥2a and Ψ is increasing, Ψ2a≤Ψdu,Mu<Ψ2a, a contradiction.
Similarly, we can prove forc2∈U.
Hence,Mu=u,u∈Ec1,c2,a; that is, Ec1,c2,a is a fixed ellipse of M.Theorem 21.
If in Theorem20, dc1,Mu+dc2,Mu≤a, then EDc1,c2,a is a fixed elliptic disc of M.Proof.
The proof follows the pattern of Theorem16.The subsequent example elucidates Theorems20 and 21.Example 6.
LetU=−2,0,1/2ln6/e,1/2ln15/e,1/2ln18/e,1/2ln21/e,1/2ln24/e,1/2ln27/e,1/2ln30/e,1/2ln6e,1/2ln9e,1/2ln12e,1/2ln15e,ln2,ln3,ln5 and a metric d:U×U⟶ℝ+ be defined as du,v=u−v. Let Ψt=et, c1=ln3, c2=ln5, a=1, γ=0, and α=3/4.
The ellipse(40)Eln3,ln5,1=u∈U:dln3,u+dln5,u=1=u∈U:ln3−u+ln5−u=1=12ln15e,12ln15e.
Define a self-mapM:U⟶U as(41)Mu=0,u=−2,−2,u=0,u,otherwise.
Then,du,Mu=2,u∈−2,00,otherwise.
Then,du,Mu=2>0.Case 1.
Foru=−2,0 and c1=ln3,(42)Ψdu,ln3aΨdu,MubΨdln3,Mln3cΨdu,Mln3eΨdln3,Muf=Ψ∣u−ln3∣aΨ∣u−Mu∣bΨ∣ln3−Mln3∣cΨ∣u−Mln3∣eΨ∣ln3−Mu∣f=Ψ∣u−ln3∣aΨ2bΨ∣ln3−ln3∣cΨ∣u−ln3∣eΨ∣ln3−Mu∣f=Ψ∣u−ln3∣a+eΨ2bΨ∣ln3−Mu∣f=Ψ∣u−ln3∣a+eΨ2bΨln3−Muf,Ψu−ln3a+eΨ2bΨln3−Muf=Ψln3a+eΨ2bΨln3+2f,ifu=0Ψ2+ln3a+eΨ2bΨln3f,ifu=−2=Ψln3a+eΨ2bΨln3e2,ifu=0Ψln3e2a+eΨ2bΨln3f,ifu=−2=3a+ee2b3e2f,ifu=03e2a+ee2b3f,ifu=−2>e2=Ψdu,Mu,for a=e=1/4, b=1/4, c=0, and f=1/3, satisfying a+b+c+e+f<1; that is,(43)Ψdu,Mu<Ψdc1,c1aΨdc1,Mc1bΨdc1,Mc1cΨdc1,Mc1eΨdc1,Mc1f.Case 2.
Foru∈−2,0 and c2=ln5,(44)Ψdu,ln5aΨdu,MubΨdln5,Mln5cΨdu,Mln5eΨdln5,Muf=Ψu−ln5aΨu−MubΨln5−Mln5cΨu−Mln5eΨln5−Muf=Ψu−ln5aΨ2bΨln5−ln5cΨu−ln5eΨln5−Muf=Ψu−ln5a+eΨ2bΨln5−Muf,Ψu−ln5a+eΨ2bΨln5−Muf=Ψln5a+eΨ2bΨln5+2f,ifu=0Ψ2+ln5a+eΨ2bΨln5f,ifu=−2=Ψln5a+eΨ2bΨln5e2f,ifu=0Ψln5e2a+eΨ2bΨln5f,ifu=−2=5a+ee2b5e2f,ifu=05e2a+ee2b5f,ifu=2>e2=Ψdu,Mu,for a=e=1/4, b=1/4, c=0, and f=1/3, satisfying a+b+c+e+f<1; that is,(45)Ψdu,Mu<Ψdc1,c1aΨdc1,Mc1bΨdc1,Mc1cΨdc1,Mc1eΨdc1,Mc1f.
That is,M is a Ψ-weak JS-contraction with c1=ln3, c2=ln5, and a=e=1/4, b=1/4, c=0, and f=1/3. Hence, Eln3,ln5,1=1/2ln15/e,1/2ln15e is a fixed ellipse and EDln3,ln5,1=U\−2,0 is a fixed elliptic disc of M. One may verify that dln3,u+dln5,u≤1,u∈EDln3,ln5,1. Noticeably, a fixed ellipse or a fixed elliptic disc is not essentially unique, as Eln3,ln5,1 is also a fixed ellipse and EDln3,ln5,1 is also a fixed elliptic disc of M.Now, we prove the fixed ellipse and fixed elliptic disc conclusions of the integra-type. LetΦ:0,∞⟶0,∞ be a locally integrable function so that, for each t>0, ∫01Φtdt>0.Definition 22.
LetΘ:0,∞⟶1,∞ be an increasing function. A map M:U⟶U of a metric space U,d is said to be a generalized integral-type ΘE-weak contraction with u≠v, if(46)du,Mu>0⇒∫0Θdu,MuΦtdt≤∫0ΘLu,vαΦtdt,where Lu,v=maxdu,v,γdu,Mu+1−γdv,Mv,1−γdu,Mu+γdv,Mv,γdu,Mv+1−γdv,Mu,1−γdu,Mv+γdv,Mu,γ∈0,1,α∈0,1,u,v∈U.Theorem 23.
LetEc1,c2,a be an ellipse in a metric space U,d and a=1/2infdu,Mu:u≠Mu. If map M:U⟶U is a generalized integral-type ΘE-weak contraction with c1,c2∈U and dc1,Mu+dc2,Mu=a,u∈Ec1,c2,a, then Ec1,c2,a is a fixed ellipse of M.Proof.
The proof follows the pattern of Theorem15.Theorem 24.
If in Theorem23, dc1,Mu+dc2,Mu≤a, then Ec1,c2,a is a fixed elliptic disc of M.Proof.
The proof follows the pattern of Theorem16.Next, we introduce an integral-typeΨE-weak JS-contraction.Definition 25.
LetΨ:0,∞⟶1,∞ be an increasing function with Ψ0=1; then, map M:U⟶U of a metric space U,d is said to be an integral-type ΨE-weak JS-contraction with u≠v, if du,Mu>0 implies that(47)∫0Ψdu,MuΦtdt≤∫0Ψdu,vaΨdu,MubΘdv,MvcΨdu,MveΨdv,MufΦtdt,where a, b, c, e, and f are nonnegative and a+b+c+e+f∈0,1,u,v∈U.Theorem 26.
LetEc1,c2,a be an ellipse in a metric space U,d and a=1/2infdu,Mu:u≠Mu. If map M:U⟶U is an integral-type ΨE-weak JS-contraction with c1,c2∈U and dc1,Mu+dc2,Mv=a,u∈Ec1,c2,a, then Ec1,c2,a is a fixed ellipse of M.Proof.
The proof follows the pattern of Theorem15.Theorem 27.
If in Theorem26, dc1,Mu+dc2,Mu≤a, then Ec1,c2,a is a fixed elliptic disc of M.Proof.
The proof follows the pattern of Theorem16.Remark 28.
(1)
It is interesting to mention here that if a self-map fixes an elliptic disc, then it also fixes an ellipse (see Examples4, 5, and 6). However, the reverse may not be true(2)
The ellipses (elliptic discs) in metric spaces change their shapes on changing the center, lengths of a semiminor or semimajor axis, or metric under consideration. Also, it is not necessary that an ellipse or an elliptic disc defined in a metric space be the same as an ellipse or an elliptic disc in a Euclidean space. Noticeably, ellipses discussed in Examples1, 4, and 6 and elliptic discs in Examples 4 and 6 are different from the ellipse and elliptic disc in a Euclidean space(3)
If there exists a self-map that maps the ellipse (elliptic disc) to itself, then that ellipse (elliptic disc) may not be a fixed ellipse (elliptic disc); that is, the self-map may not fix all the points of the ellipse (elliptic disc)(4)
All the elliptic discs inside a fixed elliptic disc are also fixed elliptic discs of a self-map. Further, a fixed ellipse is not essentially unique (see Example6)(5)
If both the focuses coincide, then fixed ellipse results, as well as fixed elliptic disc results, reduce to analogous fixed circle and fixed disc results, respectively. Noticeably, ifc1=c2=u0 (say), then Ec1,c2,a=Cu0,a/2 and EDc1,c2,a=Du0,a/2, with center u0 and radius a/2(6)
For work on the set of nonunique fixed points forming a fixed figure, one may refer to [1–23] and references therein


## 3. Applications

Inspired by the fact that the fixed point theory is applicable in many real-world problems, we solve a boundary value problem arising when a diffusing matter is kept in an absorbing medium between parallel walls of fixed concentrationsγ and δ. The concentration ut of the substance at time t is given by(48)−d2udt2+ftu=ξt,u0=γ,u1=δ,where ft is the absorption coefficient and ξt is the source density.The Green function associated with an initial value problem (48) is(49)Gt,s=t1−s,0≤t≤s≤1,s1−t,0≤s≤s≤1.Problem (48) is equivalent to(50)ut=γ+δ−γt+∫01Gt,sξs−fsusds,s∈0,1.LetU be a set of Riemann integrable functions on 0,1; that is, U=R0,1. Define d:U×U⟶ℝ+ by(51)du,v=u−v∞,u,v∈U,where u∞=supt∈0,1ut. Clearly, U,d is a complete metric space.Theorem 29.
Consider the boundary value problem (48). Let M:U×U⟶ℝ be a self-map in a complete metric space U,d, satisfying(52)ut−vt∞>0⇒ftut−ftvt∞≤e−λut−vt∞,λ>0.
Then, the boundary value problem (48) has a solution.Proof.
Define a mapM:U⟶U by(53)Mut=γ+δ−γt+∫01Gt,sξs−fsusds,s∈0,1.
Clearly, a fixed point of mapM is a solution to the problem (48).
Sinceut−ut∞>0,(54)dMu,Mv=γ+δ−γt+∫01Gt,sξs−fsusds−γ−δ−γt−∫01Gt,sξs−fsvsds=∫01Gt,sξs−fsusds−∫01Gt,sξs−fsvsds=∫01Gt,sfsus−fsvsds<fsus−fsvs∞supt∈0,1∫01Gt,sds<e−λus−vs∞supt∈0,1∫01Gt,sds<18e−λus−vs∞=18e−λdu,v.
IfΘt=et,t∈0,∞, then(55)ΘdMu,Mv=edMu,Mv<e1/8e−λdu,v=edu,v1/8e−λ=Θdu,vα≤Θmaxdu,v,dv,Mv,du,Mu,dv,Mu,du,Mvα,where α=1/8e−λ and α∈0,1. Therefore, all the conditions of Theorem 8 are verified. Hence, M has a unique fixed point, which is indeed a unique solution to a boundary value problem (48).


## 4. Conclusion

We have explored new directions as a fixed ellipse and a fixed elliptic disc to the geometry of the set of nonunique fixed points of a map on a metric space via novel contractions. Further, we have utilized a generalizedΘ-contraction and Ćirić-type Θ-contraction to establish a unique fixed point. Also, we have verified established results by illustrative examples to demonstrate the authenticity of the postulates and substantiated utility of our results by solving a boundary value problem of chemical sciences. The study of a set of unique and nonunique fixed points in the current context would be an interesting area for future study.



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*Source: 5525472-2021-05-07.xml*