# Common Fixed-Point Theorems in the Partialb-Metric Spaces and an Application to the System of Boundary Value Problems

Authors: Muhammad Nazam; Zahida Hamid; Hamed Al Sulami; Aftab Hussain
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we investigate the conditions for the existence of the common fixed points of generalized contractions in the partialb-metric spaces endowed with an arbitrary binary relation. We establish some unique common fixed-point theorems. The obtained results may generalize and improve earlier fixed-point results. We provide examples to illustrate our findings. As an application, we discuss the common solution to the system of boundary value problems.


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## Body

## 1. Introduction, Preliminaries, and Motivations

Theb-metric space was introduced by Czerwik [1]. It is obtained by modifying the triangle property of the metric space. Every metric is a b-metric, but the converse is not true. Almost all the fixed-point theorems in the metric spaces have been proved true in the b-metric spaces; for example, see [2–10] and references therein.Matthews [11] introduced the notion of the partial metric space as a part of the study of denotational semantics of the dataflow network. In this space, the usual metric is replaced by a partial metric having a property that the self-distance of any point of the space may not be zero. Every metric is a partial metric, but the converse is not true. Matthews [11] also initiated the fixed-point theory in the partial metric space. He proved the Banach contraction principle in this space to be applied in program verification. We can find so many fixed-point theorems in the metric spaces which have been proved in the partial metric spaces by many fixed-point theorists ([12, 26] and references therein).Shukla [13] introduced the concept of partial b-metric by modifying the triangle property of the partial metric and investigated fixed points of Banach contraction and Kannan contraction in the partial b-metric spaces. Mustafa et al. [14] modified the triangle property of partial b-metric and established a convergence criterion and some working rules in partial b-metric spaces. Moreover, Mustafa et al. [14] investigated common fixed-point results for ϕ,ψ-weakly contractive mappings. Dolicanin-Ðekic [15] obtained the fixed-point theorems for Ciric-type contractions in the partial b-metric spaces. Singh et al. [16] investigated some conditions to show the existence of the common fixed points of power graphic F,ψ-contractions defined on the partial b-metric space endowed with directed graphs. More results on F-contractions can be seen in [8, 17, 18].LetX be a nonempty set, then the nonempty binary relation R is a subset of X2. The set X2 itself is known as universal relation, and the empty set is known as an empty relation; both are trivial relations. If any two elements α,β∈X are related with respect to R, then we shall write α,β∈R. We shall use the notation α,β∈R if either α,β∈R or β,α∈R. R is reflexive if α,α∈R, for all α∈X.R is symmetric if α,β∈R implies β,α∈R, for all α,β∈X.R is antisymmetric if α,β∈R and β,α∈R implies α=β, for all α,β∈X.R is transitive if α,β∈R and β,γ∈R implies α,γ∈R, for all α,β∈X. The inverse, transpose, or dual of binary relation R is denoted by R−1 and defined as follows: R−1=α,β∈X∣β,α∈R. Let Rs=R∪R−1, then it is easy to prove that α,β∈Rs if and only if α,β∈R.Definition 1 (see [19]).
LetT be a self-mapping on a nonempty set X. A binary relation R on X is said to be T-closed if for all α,β∈X,
(1)α,β∈R⇒Tα,Tβ∈R.Definition 2 (see [19]).
LetR be a binary relation on X. A path in R from αtoβ is a sequence α0,α1,α2,α3,⋯,αn⊆X such that
(1)
α0=αandαn=β(2)
αj,αj+1∈R for all j∈0,1,2,⋯⋯⋯,n−1
The set of all paths fromα to β in R is denoted by Γα,β,R. The path of length n involves n+1 element of X.Definition 3 (see [19]).
A metric spaceX,d equipped with the binary relation R is called R-regular (or d-self-closed) if for each sequence αninX, whenever αn,αn+1∈R and αn⟶dα, we have αn,α∈R, for all n∈ℕ∪0.Alam and Imdad [19] used nonempty binary relation on the nonempty set X to obtain the following relation-theoretic contraction principle.Theorem 4 (see [19]).
LetX,d be a complete metric space and R be a binary relation on X. Let T be a self-mapping defined on X,d satisfying the following conditions:
(a)
There existsα0∈X such that α0,Tα0∈R and R is T-closed(b)
EitherT is continuous or X,d is R−regular(c)
There existsk∈0,1 such that dTα,Tβ≤kdα,β for α,β∈X with α,β∈RThenT admits a fixed point in X. Moreover, if Γα,β,Rs is a nonempty set for all α,β∈X, then the fixed point is unique.al-Sulami et al. [20] generalized Theorem 4 by replacing Banach contraction with θ-contraction as follows.Theorem 5 (see [20]).
LetX,d be a complete metric space and R be a binary relation on X. Let T be a self-mapping defined on X,d satisfying the following conditions:
(a)
There existsα0∈X such that α0,Tα0∈R and R is T-closed(b)
EitherT is continuous or X,dis R−regular(c)
There existsk∈0,1 such that θdTα,Tβ≤θdα,βk for α,β∈X with α,β∈RThenT admits a fixed point in X. Moreover, if Γα,β,Rs is a nonempty set for all α,β∈X, then the fixed point is unique.Definition 6 (see [21]).
LetT and S be two self-mappings on a nonempty set X. A binary relation R on X is said to be T,S-closed if for all α,β∈X,
(2)α,β∈R⇒Tα,Sβ∈RorSα,Tβ∈R.Zada and Sarwar [21] generalized Theorem 4 by replacing Banach contraction with F-contraction as follows.Theorem 7 (see [21]).
LetX,d be a complete metric space and R be a binary relation on X. If the self-mappings T and S defined on X,d satisfy the following conditions:
(a)
There existsα0∈X such that α0,Tα0∈R and R is T,S-closed(b)
EitherT,S are continuous or X,d is R−regular(c)
There existsτ>0, such that for all α,β∈R with dTα,Sβ>0,(3)τ+FdTα,Sβ≤Fdα,β+dα,Sβdβ,Tα1+dα,βThenT and S have a unique common fixed point in X. Moreover, if Γα,β,Rs is nonempty for all α,β∈X, then the common fixed point is unique.Liu et al. [22] introduced the D,C-contractions where the mapping D maps positive real numbers to positive real numbers and satisfies the conditions D1−D3:(D1) D is nondecreasing(D2) limn⟶∞Dtn=0⟺limn⟶∞tn=0, for each positive sequence tn(D3) D is continuousC:0,∞⟶0,∞ is a comparison function; that is, it satisfies the following conditions:
(i)
C is monotone increasing, that is,(4)α<β⟹Cα<Cβ(ii)
limn⟶∞Cnt=0 for all t>0, where Cn stands for the nth iterate of CLetD=D:0,∞⟶0,∞DsatisfiesD1−D3. If D is defined by Dt=t;Dt=lnt, then D belongs to D.Note that ifC is a comparison function, then Ct<t, for every t>0. The mappings Ct=αt,0<α<1,t>0,andCt=t/1+t,t>0, are comparison functions.Definition 8 ([22], D,C-contraction).
LetT be a self-mapping defined on the metric space X,d. Let
(5)I=α,β∈X2:dTα,Tβ>0.
The mappingT is called D,C-contraction if it satisfies the following condition:
(6)DdTα,Tβ≤CDdα,β,forallα,β∈I.Definition 9 ([22], generalized D,C-contraction).
LetT be a self-mapping defined on the metric space X,d. If the mapping T satisfies the condition DdTα,Tβ≤CDMα,β,forallα,β∈I, where Mα,β is defined by Mα,β=maxdα,β,dα,Tα,dβ,Tβ,dα,Tβ+dβ,Tα/2. Then it is called generalized D,C-contraction.Liu et al. established the following theorem forD,C-contractions.Theorem 10 (see [22]).
Every generalizedD,C-contraction has a unique fixed point in a complete metric space X,d.In this paper, in Section3, we investigate common fixed-point results for generalized contractions in the partial b-metric spaces endowed with binary relation R. The obtained results generalize Theorems 4, 5, 7, 10. We support the results with a nontrivial example and counter the remarks given in [23].


## 2. Basic Notions in the Partialb-Metric Spaces

LetX be a nonempty set, and the mapping P:X×X⟶⟶0,∞ satisfies the following axioms:
(1)
x=y⇔Px,x=Px,y=Py,y,∀x,y∈X(2)
Px,x≤Px,y∀x,y∈X(3)
Px,y=Py,x∀x,y∈X(4)
Px,z≤Px,y+Py,z−Py,y∀x,y,z∈X(5)
There exists a real numbers≥1 such that(7)Px,z≤sPx,y+Py,z−Py,y∀x,y,z∈XAccording to Matthews [11], if the mapping P satisfies axioms (1-4), we say that it is a partial metric on the set X and X,P is called partial metric space. According to Shukla [13], if P satisfies axioms (1, 2, 3, and 5), then it is a partialb-metric on the set X and X,Pb is called partialb-metric space. For convenience, we denote partial b-metric by Pb.Every partialb-metric Pb induces a b-metric dPb:X×X⟶0,∞ defined by
(8)dPbx,y=2Pbx,y−Pbx,x−Pby,y∀x,y∈X.It is calledinducedb-metric on X.LetBPbx,ϵ=y∈X:Pbx,y<ϵ+Pbx,x, then BPbx,ϵ:x∈X,ϵ>0 is a collection of Pb-balls which forms a base for partialb-metric topology.The following relation can be observed.Remark 11.
(1)
InX,Pb, Pbx,y=0,⇒x=y, ∀x,y∈X, but the converse is not true (in this case, X,Pb reduces to a b-metric space) Figure 1.Figure 1Example 1 (see [13]).
LetX=0,∞,l>1, be a constant and Pb:X×X⟶0,∞ be defined by
(9)Pbx,y=maxx,yl+x−ylforallx,y∈X.
ThenX,Pb is a partial b-metric space with coefficient s=2l>1, but it is neither a b-metric space nor a partial metric space.Example 2 (see [13]).
LetP:X×X⟶0,∞ and d∗:X×X⟶0,∞ be the partial metric and b-metric on X, respectively. Then the mapping Pb:X×X⟶0,∞ defined by Pbx,y=Px,y+d∗x,y for all x,y∈X defines a partial b-metric on X.Example 3 (see [13]).
LetP:X×X⟶0,∞ be a partial metric. Then the mapping Pb:X×X⟶0,∞ defined by Pbx,y=Px,yl for all x,y∈Xandl≥1 is a partial b-metric on X provided s=2l−1.Definition 12 (see [13]).
A sequencexnn∈N in the partial b-metric space X,Pb,s is called a convergent sequence if there exists x∈X such that
(10)limPbn⟶∞xn,x=Pbx,x.The uniqueness of the limit of a convergent sequence may not be guaranteed in the partialb-metric spaces (see [23]).Definition 13 (see [13]).
A sequencexnn∈N in a partial b-metric space X,Pb,s is called the Cauchy sequence if
(11)limPbxn,m,n⟶∞xm=Pbx,x.
The partialb-metric space X,Pb,s is said to be complete if every Cauchy sequence xnn∈N in X converges to a point x∈X.Lemma 14 (see [14]).
(1)
Every Cauchy sequence in theb-metric space is also Cauchy in the partial b-metric space and vice versa(2)
The partialb-metric space is complete if and only if b-metric space (induced b-metric space) is complete(3)
For any sequencexnn∈N in X, limn⟶∞dPbx∗,xn=0 if and only if(12)limn⟶∞Pbx∗,xn=Pbx∗,x∗=limPbn,m⟶∞xn,xm


## 3. Common Fixed-Point Theorems in the Partialb-Metric Spaces

This section is the main part of this paper. It contains some new common fixed-point theorems in the partialb-metric spaces. The existence theorems given in [12, 15, 19–22, 24, 27] can be seen as a special case of the results proved in this section.The results in this paper are based on the following contractive condition.Definition 15.
LetT and S be two self-mappings on the partial b-metric space X,Pb,s and R be a binary relation on X. Let
(13)I=x,y∈R:PbTx,Sy>0.The mappingsT and S form a DC-contraction if there exists a continuous comparison functionCandD∈D such that
(14)Ds2PbTx,Sy≤CDPbx,y,forallx,y∈I.In [23], it was remarked that some contraction conditions on partial b-metric spaces imply contraction conditions on b-metric spaces (see Theorem 2.6 in [23]). In the following example, we show that the contraction condition (14) is independent of these remarks.Example 4.
LetX=0,∞ and R=X2. Let Pb:X×X⟶0,∞ be defined by
(15)Pbx,y=maxx,y2+x−y2forallx,y∈X.ThenX,Pb is a partial b-metric space with coefficient s=4. The associated b-metric is given by
(16)dPbx,y=2maxx,y2+x−y2−x2−y2.DefineT≡S:0,1⟶0,1 by Tx=x/5ifx∈0,1 and T1=0. Consider
(17)Ds2dPbT1,T56≤CDdPb1,56.This implies,(18)D1636≤CD1336<D1336,a contradiction to the definition of mappingD∈D. On the other hand, for partial b-metric, we have
(19)Ds2PbT1,T56=D3236≤CDPb1,56<D3736.Note that we have taken1,5/6∈R. Similarly, it can be shown that the above conclusion holds for all other cases.Since, in general,b-metric is discontinuous mapping (see [5]), so by Example 2, the partial b-metric is not continuous in general. The following lemma is necessary for the upcoming results.Lemma 16 (see [14]).
LetX,Pb,s be a partial b-metric space. If there exists a xn in X,Pb,s and x∗,y∗ such that limn⟶∞xn=x∗. Then
(20)1sPbx∗,y∗≤limn⟶∞infPbxn,y∗≤limn⟶∞supPbxn,y∗≤sPbx∗,y∗.


### 3.1. Main Results

We state our main results which describe the conditions for the existence of the common fixed points ofDC-contraction in the partial b-metric spaces.Theorem 17.
LetX,Pb be a complete partial b-metric space and R be a transitive binary relation on X. Let T and S form a DC-contraction. Then T and S have a common fixed point in X, if the following conditions are satisfied.
(a)
There existsα0∈X such that α0,Tα0∈R(b)
RisT,S-closed(c)
TandS are continuousProof.
By assumption (a), there existsα0∈X such that α0,Tα0∈R. Taking α0∈X as the initial point, we define the sequence αn in X by
(21)α1=Tα0,α2=Sα1,continuingwiththesamepattern,wehaveα2n+1=Tα2n,α2n+2=Sα2n+1,wheren∈ℕ∪0.☐Moreover, by assumptions (a) and (b), we have(22)α1,α2=Tα0,Sα1∈R,α2,α3=Sα1,Tα2∈R,α3,α4=Tα2,Sα3∈R,α4,α5=Sα3,Tα4∈R.In general, we haveα2n,α2n+1=Sα2n−1,Tα2n∈R and α2n+1,α2n+2=Tα2n,Sα2n+1∈R.Case 1.
Ifα2n∗=α2n∗+1, for some n∗, then
(23)α2n∗+1=α2n∗+2.Indeed, on the contrary, ifα2n∗+1≠α2n∗+2, then α2n∗+1,α2n∗+2∈I, and by contractive condition (14), we have
(24)DPbα2n∗+1,α2n∗+2≤Ds2PbTα2n∗,Sα2n∗+1≤CDPbα2n∗,α2n∗+1.SinceCt<t, for every t>0, we obtain
(25)DPbα2n∗+1,α2n∗+2<DPbα2n∗,α2n∗+1.Since the functionD is nondecreasing, so Pbα2n∗+1,α2n∗+2<Pbα2n∗,α2n∗+1. This contradicts the second condition of partial b-metric spaces Pbx,x≤Pbx,y∀x,y∈X. Hence, α2n∗=α2n∗+1 implies α2n∗+1=α2n∗+2. Consequently, α2n∗ is a common fixed point of T, and that is α2n∗=Tα2n∗=Sα2n∗+1=Sα2n∗.Case 2.
Ifα2n≠α2n+1 for all n∈ℕ. We have PbTα2n,Sα2n−1>0 for all n∈ℕ. Since α2n,α2n−1∈R, so α2n,α2n−1∈I. Setting α=α2n and β=α2n−1 in (14), we get
(26)DPbα2n+1,α2n≤Ds2Pbα2n+1,α2n=Ds2PbTα2n,Sα2n−1≤CDPbα2n,α2n−1,
for alln∈ℕ.Similarly, settingα=α2n and β=α2n+1 in (14), we get
(27)DPbα2n+1,α2n+2≤Ds2PbTα2n,Sα2n+1≤CDPbα2n,α2n+1.In general, for allhn∈ℕ, either even or odd, we have
(28)DPbαhn,αhn+1≤CDPbαhn−1,αhn≤C2DPbαhn−2,αhn−1⋮≤ChnDPbα0,α1.Taking limitn⟶∞ in the above inequality, we get
(29)0≤limn⟶∞DPbαhn,αhn+1≤limn⟶∞ChnDPbα0,α1=0.This implieslimn⟶∞DPbαhn,αhn+1=0, and by D2, we havelimn⟶∞Pbαhn,αhn+1=0. This implies (by (8)) that
(30)limn⟶∞dPbαhn,αhn+1=0.By axiom (2), we havelimn⟶∞Pbαhn,αhn≤limn⟶∞Pbαhn,αhn+1=0. Thus, for all n,m≥1, we have
(31)limn,m⟶∞dPbαhm,αhn=2limn,m⟶∞Pbαhm,αhn.We claim thatαn is a Cauchy sequence in X,dPb, for this is sufficient to prove that α2n is Cauchy sequence. On the contrary, if α2n is not Cauchy, then for some subsequences α2nkk=1∞ and α2mkk=1∞, there exist ∈>0, and a positive integer k∈, such that for all nk,mk>k, we have dPbα2mk,α2nk≥ε and dPbα2mk,α2nk−2<ε; thus,
(32)ε≤dPbα2mk,α2nk≤sdPbα2mk,α2mk+1+sdPbα2mk+1,α2nk.Ask⟶∞ in the above inequality, we have
(33)εs≤limk⟶∞supdPbα2mk+1,α2nk.By using triangular inequality (axiom (5)), we get(34)dPbα2mk,α2nk−1≤sdPbα2mk,α2nk−2+sdPbα2nk−2,α2nk−1.Taking limitk⟶∞, we have
(35)limk⟶∞supdPbα2mk,α2nk−1≤sε.Also, we have the following information:(36)dPbα2mk,α2nk≤sdPbα2mk,α2nk−2+sdPbα2nk−2,α2nk≤sdPbα2mk,α2nk−2+s2dPbα2nk−2,α2nk−1+s2dPbα2nk−1,α2nk.Taking limitk⟶∞, we have
(37)limk⟶∞supdPbα2mk,α2nk≤sε.By axiom (5), we have(38)dPbα2mk+1,α2nk−1≤sdPbα2mk+1,α2mk+sdPbα2mk,α2nk−1.Taking limitk⟶∞ and using (35), we have
(39)limk⟶∞supdPbα2mk+1,α2nk−1≤s2ε.By using (31), we have the following information from (33), (35), (37), and (39):
(40)ε2s≤limk⟶∞supPbα2mk+1,α2nk,(41)limk⟶∞supPbα2mk,α2nk−1≤sε2,(42)limk⟶∞supPbα2mk,α2nk≤sε2,(43)limk⟶∞supPbα2mk+1,α2nk−1≤s2ε2,Sinceα2mk,α2nk−1∈I, by (14), we have
(44)Ds∈2=Ds2·s∈2≤Ds2limk⟶∞supPbα2mk+1,α2nk=limk⟶∞supDs2PbTα2mk,Sα2nk−1≤limk⟶∞supCDPbα2mk,α2nk−1=CDlimk⟶∞supPbα2mk,α2nk−1≤CDs∈2<Ds∈2.This is a contradiction to the definition of functionD. Thus, αn is a Cauchy sequence in X,dPb. By Lemma 14 (1), αn is a Cauchy sequence in X,Pb. Since X,Pb is a complete Partial b-metric space, so by Lemma 14 (2), X,dpb is also a complete metric space. Thus, there exists α∗∈X such that αn⟶α∗, that is, limn⟶∞dpbαn,α∗=0. By Lemma 14 (3), we get
(45)limn⟶∞Pbαn,α∗=Pbα∗,α∗=limn,m⟶∞Pbαn,αm.Sincelimn,m⟶∞Pbαn,αm=0, so that Pbα∗,α∗=0. Thus, αn converges to α∗ in X,Pb.Now, we claim thatTα∗=Sα∗=α∗. By (40), we have
(46)limn⟶∞Pbα2n+1,α∗=0,limn⟶∞Pbα2n+2,α∗=0.SinceT and S are continuous, we have
(47)limn⟶∞PbTα2n,Tα∗=0,limn⟶∞PbSα2n+1,Sα∗=0.By Lemma16, we have
(48)1sPbα∗,Tα∗≤limn⟶∞infPbα2n+1,Tα∗=liminfn⟶∞PbTα2n,Tα∗=0.Thus,PbTα∗,α∗=Pbα∗,α∗=PbTα∗,Tα∗.This impliesTα∗=α∗. Similar arguments lead us to have Sα∗=α∗.Hence,Tα∗=Sα∗=α∗; that is, T and S have a common fixed point α∗∈X.IfΓα,β,R≠∅, then we have the following theorem.Theorem 18.
LetX,Pb be a complete partial b-metric space and R be a transitive binary relation on X. Let T and S form a DC-contraction. Suppose that Γα,β,R≠∅ and statement of Theorem 17 holds, then the mappings T and S admit a unique common fixed point in X.Proof.
We have proved the existence of a common fixed point in Theorem17. On the contrary, suppose that v and v∗ are two distinct common fixed points of T and S in X. Then the class of paths of finite length ℓ in R from v to v∗ is Γv,v∗,R. Let one of the paths be A0,A1,A2,⋯,Aℓ in X from v to v∗ with
(49)A0=v,Aℓ=v∗,Aj,Aj+1∈R;j=0,1,2,3⋯⋯⋯⋯ℓ−1.
By transitivity ofR, we have
(50)v,A1∈R,A1,A2∈R,⋯,Aℓ−1,v∗∈R⇒v,v∗∈R.
It is given thatT and S form a DC-contraction, that is,
(51)Ds2PbTv,Sv∗≤CDPbv,v∗.
This impliesDs2Pbv,v∗≤CDPbv,v∗<DPbv,v∗. This is a contradiction to the definition of D. Hence, v=v∗. This shows that v is a unique common fixed point of T and S.☐Remark 19.
If the mappingsT and S are discontinuous, then we have the following theorem.Theorem 20.
LetX,Pb be an R−regular complete partial b-metric space. Let T and S form a DC-contraction. Suppose that R is an antisymmetric relation, then T and S admit a common fixed point in X if they meet the conditions (a) and (b):
(a)
There existsα0∈X such that α0,Tα0∈R(b)
RisT,S-closedProof.
By Theorem17, we know that αn,αn+1∈R and αn⟶α∗ as n⟶∞. It is given that X,Pb is R−regular, so αn,α∗∈R, for all n∈ℕ. There are two possible cases.☐Case 1.
If the sequenceαn is constant. Let αn=α∗ for each n∈ℕ so that α2n=α∗ and Tα∗=Tα2n=α2n+1. Since X,Pb is R−regular, so α2n+1,α∗=Tα∗,α∗∈R. We know that α2n,α2n+1∈R; thus, α∗,Tα∗∈R. As R is an antisymmetric relation, so α∗=Tα∗, by the same arguments we have α∗=Sα∗ as required.Case 2.
Ifαn is not constant and arbitrary, we claim that Pbα∗,Sα∗=0. Let Pbα∗,Sα∗>0. It is proved in Theorem 17 that limi⟶∞α2i+1=α∗, so there must be an integer n0>0, such that
(52)Pbα2i+1,Sα∗>0,Pbα2i,α∗<Pbα∗,Sα∗2,foralli≥n0.It is assumed thatX,Pb is R−regular, and by Theorem 17, we know that α2i⟶α∗asi⟶∞;thus,α2i,α∗∈R. By contractive condition (2.1), monotonicity of D, and Lemma 16, we have
(53)Dpbα∗,Sα∗≤Dslimi⟶∞infPbα2i+1,Sα∗≤Ds2limi⟶∞infPbα2i+1,Sα∗=limi⟶∞infDs2PbTα2i,Sα∗≤limi⟶∞infCDPbα2i,α∗<limi⟶∞infCDPbα∗,Sα∗2<DPbα∗,Sα∗2.This is a contradiction to the definition of mappingD. Thus, Pbα∗,Sα∗=0. Also, we have the following information:
(54)PbSα∗,Sα∗=0=Pbα∗,α∗.Thus,α∗=Sα∗. By interchanging roles of S and T, we have α∗=Tα∗.Hence,Tα∗=Sα∗=α∗; that is, α∗ is a common fixed point of T and S in X.The following is the most general theorem of this section.Theorem 21.
LetX,Pb be an R−regular complete partial b-metric space and R be a transitive and antisymmetric binary relation on X. Let T and S form a DC-contraction. Suppose that Γα,β,R≠∅, and assumptions (a) and (b) in Theorem 17 hold. Then the mappings T and S admit a unique common fixed point in X.Proof.
See the proofs of Theorems17, 18, and 20, respectively.☐Remark 22.
(1)
The results in this section are independent of the observation made in [23], and hence, our results are a real generalization of the related results in literature (see [12, 19–22])(2)
Theorem21 remains true if Pbα,β is replaced by Mα,β
The following example explains the main results.Example 5.
LetX=an=nn+1/2:n∈ℕ.Define the partial b-metric function Pb:X×X⟶0,∞ by
(55)Pbα,β=maxα,β2,forallα,β∈X.Then (X, Pb,2) is a complete partial b-metric space. Define D∶0,∞⟶0,∞ by Da=aea for each a>0, then D∈D. Let the function C:0,∞⟶0,∞ be defined by Cr=r/2 for all r∈0,∞. Then C is continuous comparison. Define the binary relation R on X by
(56)R=an,am:an+am≥2foreachm≥n.Define the mappingsT,S∶X⟶X by
(57)Tan=a1,n=1,nn−12,n≥2,Sam=a1,m∈1,2,m−1m−22,m≥3,m∈ℕ.We observe that there existsa1∈X such that a1,Ta1∈R(sincea1+Ta1=2 by definition of R, so assumption (a) is satisfied in Theorem 17. Let an,am∈R, then we have Tan+Sam≥2 for each m≥n, so Tan,Sam∈R. Thus, R is T,S-closed (this verifies assumption (b) of Theorem 17. Also, T,S are continuous (assumption (c) is satisfied). Now, we show that TandS form DC-contraction. It is noted that the mappings T, S do not form Banach contraction in the partial b-metric sense. Indeed,
(58)limn⟶∞PbTan,Sa1Pban,a1=limn⟶∞n2−n2n2+n2=1.We noticed thatPbTan,Sam>0 for each m≥n. Thus, an,am∈I.Consider
(59)4PbTan,Same4PbTan,Sam≤12Pban,amePban,am.This implies(60)8PbTan,SamPban,am≤ePban,am−4PbTan,Sam.Forn=1 and m=2, the inequality (41) reduces to e5≥8/9. Thus, (41) holds for this case. For n=2 and m=3, the inequality (41) gets the form e32≥2/9. Similarly, for each m≥n, (41) holds true. Thus, we have
(61)Ds2PbTα,Sβ≤CDPbα,β,forallα,β∈X.We note thata1=Ta1=Sa1.


### 3.2. Discussions

In this part of the current section, we state some corollaries which are themselves prominent fixed-point theorems in the literature.The following corollary generalizes the results presented by Jleli and Samet [6] and al-Sulami et al. [20].Corollary 23.
LetX,Pb be a complete partial b-metric space and R be a transitive and antisymmetric binary relation on X. If the self-mappings T and S defined on X,Pb satisfy the following conditions:
(a)
Γα,β,R is nonempty for all α,β∈X(b)
There existsα0∈X such that α0,Tα0∈R and RisT,S-closed(c)
EitherT,S are continuous or X,Pb is R−regular(d)
There exists a functionθ∈Θandk∈0,1, such that for all α,β∈I,(62)θs2PbTα,Sβ≤θPbα,βk
Then the mappingsT and S admit a unique common fixed point.Proof.
SettingCt=lnkt and Dt=θs2t in Theorem 17 and following the proofs of Theorems 17, 18, and 20 respectively, we obtain the required result.☐The following corollary generalizes and improves the results presented by Zada and Sarwar [21] and Wardowski [25].Corollary 24.
Suppose that the self-mappingsT and S defined on the complete partial b-metric space X,Pb satisfy the following conditions:
(a)
Γα,β,R is nonempty for all α,β∈X(b)
There existsα0∈X such that α0,Tα0∈R and RisT,S-closed(c)
EitherT,S are continuous or X,Pb is R−regular(d)
There existsF∈Fandτ>0,such that for all α,β∈I,(63)τ+Fs2PbTα,Sβ≤FPbα,β
IfR is a transitive and antisymmetric binary relation on X, then the mappings T,S admit a unique common fixed point.Proof.
SettingCt=e−τt and Dt=es2Ft in Theorem 17 and following the proofs of Theorems 17, 18, and 20, respectively, we obtain the required result.☐Corollary 25 (see [21]).
LetX,Pb be a complete partial b-metric space and R be a transitive and antisymmetric binary relation on X. If the self-mappings T and S defined on X,Pb satisfy the following conditions:
(a)
Γα,β,R is nonempty for all α,β∈X(b)
There existsα0∈X such that α0,Tα0∈R and RisT,S-closed(c)
EitherT,S are continuous or X,Pb is R−regular(d)
There existsF∈Fandτ>0, such that for all α,β∈I,(64)τ+FsPbTα,Sβ≤FPbα,β+Pbα,SβPbβ,Tα1+Pbα,β
ThenT and S have a unique common fixed point in X.Proof.
This proof follows the proof of Corollary24.
The following corollary improves the fixed-point results presented by Geraghty [24].☐Corollary 26.
LetX,Pb be a complete partial b-metric space and R be a transitive and antisymmetric binary relation on X. If the self-mappings T and S defined on X,Pb satisfy the following conditions:
(a)
Γα,β,R is nonempty for all α,β∈X(b)
There existsα0∈X such that α0,Tα0∈R and RisT,S-closed(c)
EitherT,S are continuous or X,Pb is R−regular(d)
For allα,β∈X and α,β∈R(65)s2PbTα,Sβ≤γPbα,βPbα,β,where γ:0,∞⟶0,∞ such that limr⟶t+γr<1/s, for each t∈0,∞Proof.
By definingCt=tγt and Dt=s2tin Theorem 17 and following the proofs of Theorems 17, 18, and 20, respectively, we obtain the required result.☐Remark 27.
(1)
Fors=1, Theorems 17, 18, and 20 establish criteria for the existence of common fixed points of Jc-contractions in the partial metric spaces [12] and correspondingly for Corollaries 23, 24, 25, and 26(2)
For the zero self-distance (Pbα,β=0forallα,β) and for the zero self-distance with s=1, the results stated in Remark 27 (1) hold in the b-metric spaces and metric spaces, respectively


## 3.1. Main Results

We state our main results which describe the conditions for the existence of the common fixed points ofDC-contraction in the partial b-metric spaces.Theorem 17.
LetX,Pb be a complete partial b-metric space and R be a transitive binary relation on X. Let T and S form a DC-contraction. Then T and S have a common fixed point in X, if the following conditions are satisfied.
(a)
There existsα0∈X such that α0,Tα0∈R(b)
RisT,S-closed(c)
TandS are continuousProof.
By assumption (a), there existsα0∈X such that α0,Tα0∈R. Taking α0∈X as the initial point, we define the sequence αn in X by
(21)α1=Tα0,α2=Sα1,continuingwiththesamepattern,wehaveα2n+1=Tα2n,α2n+2=Sα2n+1,wheren∈ℕ∪0.☐Moreover, by assumptions (a) and (b), we have(22)α1,α2=Tα0,Sα1∈R,α2,α3=Sα1,Tα2∈R,α3,α4=Tα2,Sα3∈R,α4,α5=Sα3,Tα4∈R.In general, we haveα2n,α2n+1=Sα2n−1,Tα2n∈R and α2n+1,α2n+2=Tα2n,Sα2n+1∈R.Case 1.
Ifα2n∗=α2n∗+1, for some n∗, then
(23)α2n∗+1=α2n∗+2.Indeed, on the contrary, ifα2n∗+1≠α2n∗+2, then α2n∗+1,α2n∗+2∈I, and by contractive condition (14), we have
(24)DPbα2n∗+1,α2n∗+2≤Ds2PbTα2n∗,Sα2n∗+1≤CDPbα2n∗,α2n∗+1.SinceCt<t, for every t>0, we obtain
(25)DPbα2n∗+1,α2n∗+2<DPbα2n∗,α2n∗+1.Since the functionD is nondecreasing, so Pbα2n∗+1,α2n∗+2<Pbα2n∗,α2n∗+1. This contradicts the second condition of partial b-metric spaces Pbx,x≤Pbx,y∀x,y∈X. Hence, α2n∗=α2n∗+1 implies α2n∗+1=α2n∗+2. Consequently, α2n∗ is a common fixed point of T, and that is α2n∗=Tα2n∗=Sα2n∗+1=Sα2n∗.Case 2.
Ifα2n≠α2n+1 for all n∈ℕ. We have PbTα2n,Sα2n−1>0 for all n∈ℕ. Since α2n,α2n−1∈R, so α2n,α2n−1∈I. Setting α=α2n and β=α2n−1 in (14), we get
(26)DPbα2n+1,α2n≤Ds2Pbα2n+1,α2n=Ds2PbTα2n,Sα2n−1≤CDPbα2n,α2n−1,
for alln∈ℕ.Similarly, settingα=α2n and β=α2n+1 in (14), we get
(27)DPbα2n+1,α2n+2≤Ds2PbTα2n,Sα2n+1≤CDPbα2n,α2n+1.In general, for allhn∈ℕ, either even or odd, we have
(28)DPbαhn,αhn+1≤CDPbαhn−1,αhn≤C2DPbαhn−2,αhn−1⋮≤ChnDPbα0,α1.Taking limitn⟶∞ in the above inequality, we get
(29)0≤limn⟶∞DPbαhn,αhn+1≤limn⟶∞ChnDPbα0,α1=0.This implieslimn⟶∞DPbαhn,αhn+1=0, and by D2, we havelimn⟶∞Pbαhn,αhn+1=0. This implies (by (8)) that
(30)limn⟶∞dPbαhn,αhn+1=0.By axiom (2), we havelimn⟶∞Pbαhn,αhn≤limn⟶∞Pbαhn,αhn+1=0. Thus, for all n,m≥1, we have
(31)limn,m⟶∞dPbαhm,αhn=2limn,m⟶∞Pbαhm,αhn.We claim thatαn is a Cauchy sequence in X,dPb, for this is sufficient to prove that α2n is Cauchy sequence. On the contrary, if α2n is not Cauchy, then for some subsequences α2nkk=1∞ and α2mkk=1∞, there exist ∈>0, and a positive integer k∈, such that for all nk,mk>k, we have dPbα2mk,α2nk≥ε and dPbα2mk,α2nk−2<ε; thus,
(32)ε≤dPbα2mk,α2nk≤sdPbα2mk,α2mk+1+sdPbα2mk+1,α2nk.Ask⟶∞ in the above inequality, we have
(33)εs≤limk⟶∞supdPbα2mk+1,α2nk.By using triangular inequality (axiom (5)), we get(34)dPbα2mk,α2nk−1≤sdPbα2mk,α2nk−2+sdPbα2nk−2,α2nk−1.Taking limitk⟶∞, we have
(35)limk⟶∞supdPbα2mk,α2nk−1≤sε.Also, we have the following information:(36)dPbα2mk,α2nk≤sdPbα2mk,α2nk−2+sdPbα2nk−2,α2nk≤sdPbα2mk,α2nk−2+s2dPbα2nk−2,α2nk−1+s2dPbα2nk−1,α2nk.Taking limitk⟶∞, we have
(37)limk⟶∞supdPbα2mk,α2nk≤sε.By axiom (5), we have(38)dPbα2mk+1,α2nk−1≤sdPbα2mk+1,α2mk+sdPbα2mk,α2nk−1.Taking limitk⟶∞ and using (35), we have
(39)limk⟶∞supdPbα2mk+1,α2nk−1≤s2ε.By using (31), we have the following information from (33), (35), (37), and (39):
(40)ε2s≤limk⟶∞supPbα2mk+1,α2nk,(41)limk⟶∞supPbα2mk,α2nk−1≤sε2,(42)limk⟶∞supPbα2mk,α2nk≤sε2,(43)limk⟶∞supPbα2mk+1,α2nk−1≤s2ε2,Sinceα2mk,α2nk−1∈I, by (14), we have
(44)Ds∈2=Ds2·s∈2≤Ds2limk⟶∞supPbα2mk+1,α2nk=limk⟶∞supDs2PbTα2mk,Sα2nk−1≤limk⟶∞supCDPbα2mk,α2nk−1=CDlimk⟶∞supPbα2mk,α2nk−1≤CDs∈2<Ds∈2.This is a contradiction to the definition of functionD. Thus, αn is a Cauchy sequence in X,dPb. By Lemma 14 (1), αn is a Cauchy sequence in X,Pb. Since X,Pb is a complete Partial b-metric space, so by Lemma 14 (2), X,dpb is also a complete metric space. Thus, there exists α∗∈X such that αn⟶α∗, that is, limn⟶∞dpbαn,α∗=0. By Lemma 14 (3), we get
(45)limn⟶∞Pbαn,α∗=Pbα∗,α∗=limn,m⟶∞Pbαn,αm.Sincelimn,m⟶∞Pbαn,αm=0, so that Pbα∗,α∗=0. Thus, αn converges to α∗ in X,Pb.Now, we claim thatTα∗=Sα∗=α∗. By (40), we have
(46)limn⟶∞Pbα2n+1,α∗=0,limn⟶∞Pbα2n+2,α∗=0.SinceT and S are continuous, we have
(47)limn⟶∞PbTα2n,Tα∗=0,limn⟶∞PbSα2n+1,Sα∗=0.By Lemma16, we have
(48)1sPbα∗,Tα∗≤limn⟶∞infPbα2n+1,Tα∗=liminfn⟶∞PbTα2n,Tα∗=0.Thus,PbTα∗,α∗=Pbα∗,α∗=PbTα∗,Tα∗.This impliesTα∗=α∗. Similar arguments lead us to have Sα∗=α∗.Hence,Tα∗=Sα∗=α∗; that is, T and S have a common fixed point α∗∈X.IfΓα,β,R≠∅, then we have the following theorem.Theorem 18.
LetX,Pb be a complete partial b-metric space and R be a transitive binary relation on X. Let T and S form a DC-contraction. Suppose that Γα,β,R≠∅ and statement of Theorem 17 holds, then the mappings T and S admit a unique common fixed point in X.Proof.
We have proved the existence of a common fixed point in Theorem17. On the contrary, suppose that v and v∗ are two distinct common fixed points of T and S in X. Then the class of paths of finite length ℓ in R from v to v∗ is Γv,v∗,R. Let one of the paths be A0,A1,A2,⋯,Aℓ in X from v to v∗ with
(49)A0=v,Aℓ=v∗,Aj,Aj+1∈R;j=0,1,2,3⋯⋯⋯⋯ℓ−1.
By transitivity ofR, we have
(50)v,A1∈R,A1,A2∈R,⋯,Aℓ−1,v∗∈R⇒v,v∗∈R.
It is given thatT and S form a DC-contraction, that is,
(51)Ds2PbTv,Sv∗≤CDPbv,v∗.
This impliesDs2Pbv,v∗≤CDPbv,v∗<DPbv,v∗. This is a contradiction to the definition of D. Hence, v=v∗. This shows that v is a unique common fixed point of T and S.☐Remark 19.
If the mappingsT and S are discontinuous, then we have the following theorem.Theorem 20.
LetX,Pb be an R−regular complete partial b-metric space. Let T and S form a DC-contraction. Suppose that R is an antisymmetric relation, then T and S admit a common fixed point in X if they meet the conditions (a) and (b):
(a)
There existsα0∈X such that α0,Tα0∈R(b)
RisT,S-closedProof.
By Theorem17, we know that αn,αn+1∈R and αn⟶α∗ as n⟶∞. It is given that X,Pb is R−regular, so αn,α∗∈R, for all n∈ℕ. There are two possible cases.☐Case 1.
If the sequenceαn is constant. Let αn=α∗ for each n∈ℕ so that α2n=α∗ and Tα∗=Tα2n=α2n+1. Since X,Pb is R−regular, so α2n+1,α∗=Tα∗,α∗∈R. We know that α2n,α2n+1∈R; thus, α∗,Tα∗∈R. As R is an antisymmetric relation, so α∗=Tα∗, by the same arguments we have α∗=Sα∗ as required.Case 2.
Ifαn is not constant and arbitrary, we claim that Pbα∗,Sα∗=0. Let Pbα∗,Sα∗>0. It is proved in Theorem 17 that limi⟶∞α2i+1=α∗, so there must be an integer n0>0, such that
(52)Pbα2i+1,Sα∗>0,Pbα2i,α∗<Pbα∗,Sα∗2,foralli≥n0.It is assumed thatX,Pb is R−regular, and by Theorem 17, we know that α2i⟶α∗asi⟶∞;thus,α2i,α∗∈R. By contractive condition (2.1), monotonicity of D, and Lemma 16, we have
(53)Dpbα∗,Sα∗≤Dslimi⟶∞infPbα2i+1,Sα∗≤Ds2limi⟶∞infPbα2i+1,Sα∗=limi⟶∞infDs2PbTα2i,Sα∗≤limi⟶∞infCDPbα2i,α∗<limi⟶∞infCDPbα∗,Sα∗2<DPbα∗,Sα∗2.This is a contradiction to the definition of mappingD. Thus, Pbα∗,Sα∗=0. Also, we have the following information:
(54)PbSα∗,Sα∗=0=Pbα∗,α∗.Thus,α∗=Sα∗. By interchanging roles of S and T, we have α∗=Tα∗.Hence,Tα∗=Sα∗=α∗; that is, α∗ is a common fixed point of T and S in X.The following is the most general theorem of this section.Theorem 21.
LetX,Pb be an R−regular complete partial b-metric space and R be a transitive and antisymmetric binary relation on X. Let T and S form a DC-contraction. Suppose that Γα,β,R≠∅, and assumptions (a) and (b) in Theorem 17 hold. Then the mappings T and S admit a unique common fixed point in X.Proof.
See the proofs of Theorems17, 18, and 20, respectively.☐Remark 22.
(1)
The results in this section are independent of the observation made in [23], and hence, our results are a real generalization of the related results in literature (see [12, 19–22])(2)
Theorem21 remains true if Pbα,β is replaced by Mα,β
The following example explains the main results.Example 5.
LetX=an=nn+1/2:n∈ℕ.Define the partial b-metric function Pb:X×X⟶0,∞ by
(55)Pbα,β=maxα,β2,forallα,β∈X.Then (X, Pb,2) is a complete partial b-metric space. Define D∶0,∞⟶0,∞ by Da=aea for each a>0, then D∈D. Let the function C:0,∞⟶0,∞ be defined by Cr=r/2 for all r∈0,∞. Then C is continuous comparison. Define the binary relation R on X by
(56)R=an,am:an+am≥2foreachm≥n.Define the mappingsT,S∶X⟶X by
(57)Tan=a1,n=1,nn−12,n≥2,Sam=a1,m∈1,2,m−1m−22,m≥3,m∈ℕ.We observe that there existsa1∈X such that a1,Ta1∈R(sincea1+Ta1=2 by definition of R, so assumption (a) is satisfied in Theorem 17. Let an,am∈R, then we have Tan+Sam≥2 for each m≥n, so Tan,Sam∈R. Thus, R is T,S-closed (this verifies assumption (b) of Theorem 17. Also, T,S are continuous (assumption (c) is satisfied). Now, we show that TandS form DC-contraction. It is noted that the mappings T, S do not form Banach contraction in the partial b-metric sense. Indeed,
(58)limn⟶∞PbTan,Sa1Pban,a1=limn⟶∞n2−n2n2+n2=1.We noticed thatPbTan,Sam>0 for each m≥n. Thus, an,am∈I.Consider
(59)4PbTan,Same4PbTan,Sam≤12Pban,amePban,am.This implies(60)8PbTan,SamPban,am≤ePban,am−4PbTan,Sam.Forn=1 and m=2, the inequality (41) reduces to e5≥8/9. Thus, (41) holds for this case. For n=2 and m=3, the inequality (41) gets the form e32≥2/9. Similarly, for each m≥n, (41) holds true. Thus, we have
(61)Ds2PbTα,Sβ≤CDPbα,β,forallα,β∈X.We note thata1=Ta1=Sa1.


## 3.2. Discussions

In this part of the current section, we state some corollaries which are themselves prominent fixed-point theorems in the literature.The following corollary generalizes the results presented by Jleli and Samet [6] and al-Sulami et al. [20].Corollary 23.
LetX,Pb be a complete partial b-metric space and R be a transitive and antisymmetric binary relation on X. If the self-mappings T and S defined on X,Pb satisfy the following conditions:
(a)
Γα,β,R is nonempty for all α,β∈X(b)
There existsα0∈X such that α0,Tα0∈R and RisT,S-closed(c)
EitherT,S are continuous or X,Pb is R−regular(d)
There exists a functionθ∈Θandk∈0,1, such that for all α,β∈I,(62)θs2PbTα,Sβ≤θPbα,βk
Then the mappingsT and S admit a unique common fixed point.Proof.
SettingCt=lnkt and Dt=θs2t in Theorem 17 and following the proofs of Theorems 17, 18, and 20 respectively, we obtain the required result.☐The following corollary generalizes and improves the results presented by Zada and Sarwar [21] and Wardowski [25].Corollary 24.
Suppose that the self-mappingsT and S defined on the complete partial b-metric space X,Pb satisfy the following conditions:
(a)
Γα,β,R is nonempty for all α,β∈X(b)
There existsα0∈X such that α0,Tα0∈R and RisT,S-closed(c)
EitherT,S are continuous or X,Pb is R−regular(d)
There existsF∈Fandτ>0,such that for all α,β∈I,(63)τ+Fs2PbTα,Sβ≤FPbα,β
IfR is a transitive and antisymmetric binary relation on X, then the mappings T,S admit a unique common fixed point.Proof.
SettingCt=e−τt and Dt=es2Ft in Theorem 17 and following the proofs of Theorems 17, 18, and 20, respectively, we obtain the required result.☐Corollary 25 (see [21]).
LetX,Pb be a complete partial b-metric space and R be a transitive and antisymmetric binary relation on X. If the self-mappings T and S defined on X,Pb satisfy the following conditions:
(a)
Γα,β,R is nonempty for all α,β∈X(b)
There existsα0∈X such that α0,Tα0∈R and RisT,S-closed(c)
EitherT,S are continuous or X,Pb is R−regular(d)
There existsF∈Fandτ>0, such that for all α,β∈I,(64)τ+FsPbTα,Sβ≤FPbα,β+Pbα,SβPbβ,Tα1+Pbα,β
ThenT and S have a unique common fixed point in X.Proof.
This proof follows the proof of Corollary24.
The following corollary improves the fixed-point results presented by Geraghty [24].☐Corollary 26.
LetX,Pb be a complete partial b-metric space and R be a transitive and antisymmetric binary relation on X. If the self-mappings T and S defined on X,Pb satisfy the following conditions:
(a)
Γα,β,R is nonempty for all α,β∈X(b)
There existsα0∈X such that α0,Tα0∈R and RisT,S-closed(c)
EitherT,S are continuous or X,Pb is R−regular(d)
For allα,β∈X and α,β∈R(65)s2PbTα,Sβ≤γPbα,βPbα,β,where γ:0,∞⟶0,∞ such that limr⟶t+γr<1/s, for each t∈0,∞Proof.
By definingCt=tγt and Dt=s2tin Theorem 17 and following the proofs of Theorems 17, 18, and 20, respectively, we obtain the required result.☐Remark 27.
(1)
Fors=1, Theorems 17, 18, and 20 establish criteria for the existence of common fixed points of Jc-contractions in the partial metric spaces [12] and correspondingly for Corollaries 23, 24, 25, and 26(2)
For the zero self-distance (Pbα,β=0forallα,β) and for the zero self-distance with s=1, the results stated in Remark 27 (1) hold in the b-metric spaces and metric spaces, respectively


## 4. Application to the System of Boundary Value Problems

We will apply Theorem17 to achieve the existence of a common solution to the following system of boundary value problems:
(66)−d2vdt2=Ht,vt;t∈I,v0=v1=0,(67)−d2wdt2=Kt,wt;t∈I,w0=w1=0,where I=0,1,CI represents the set of continuous functions defined on I. The functions H,K:0,1×CI⟶ℝ are continuous and nondecreasing according to ordinates. We define the binary relation N on CI as follows:
(68)N=v,w∈CI×CI:vt≤wt∀t∈I.The associated Green functionℊ:I×I⟶I to (66) and (67) can be defined as follows:
(69)ℊt,b=t1−bif0≤t≤b≤1,b1−tif0≤b≤t≤1.Let the mappingd∗:CI×CI⟶0,∞ be defined by
(70)d∗v,w=v−w2∞=supvt−wt2,∀v,w∈CI,t∈I.It is claimed thatCI,d∗,2 is a complete b-metric space. By integration, we see that (66) and (67) can be written as v=Sv and w=Tw, where S,T:CI⟶CI are defined by
(71)Svt=∫01ℊt,bHb,vbdb,Twt=∫01ℊt,bKb,wbdb.It is remarked that the common solution to (66) and (67) is the common fixed point of the operators S,T. Suppose the following conditions:
(a)
∃k>0 such that for vt≠wt∀t, we have(72)Ht,vt−Kt,wt2≤16e−kvt−wt2∀t∈I(b)
∃v0,w0∈CI such that(73)v0t≤∫01ℊt,bHb,v0bdb,w0t≤∫01ℊt,bKb,w0bdbThe following theorem states the conditions under which equations (66) and (67) have a common solution.Theorem 28.
Let the functionsH,K:0,1×CI⟶ℝ satisfy conditions (a) and (b). Then equations (66) and (67) have a common solution.Proof.
We will apply Theorem17 to show the existence of the common solution to (66) and (67). By condition (b), there exists v0 such that v0,Sv0∈N. Since the functions H,K are continuous, so S,T:CI⟶CI defined above are continuous. Since it is given that H,K are nondecreasing, thus, N is S,T closed. To show that the mappings S,T form DC-contraction, we proceed as follows:
(74)Svt−Twt2=∫01ℊt,bHb,vb−Kb,wbdb2≤∫01ℊt,bHb,vb−Kb,wbdb2≤∫01ℊt,b16e−kvt−wt2db2.☐Sincesup∫01ℊt,bdb2=1/64, for all t∈I, thus, taking supremum on both sides of the above inequality, we have
(75)s2d∗Sv,Tw≤e−kd∗v,w∀v·,w·∈CI.Define theb-metric d∗ on CI by
(76)d∗v,w=Pbv,w,ifv≠w,0,ifv=w.Inequality (75) can be written as
(77)s2PbSv,Tw≤e−kPbv,w∀v·,w·∈CI.Defining the functionsC, F, and D by Ct=e−kt,Ft=lnt, and Dt=eFt, respectively, for all t∈0,∞, we have
(78)k+Fs2PbSv,Tw≤FPbv,w,ek·eFs2PbSv,Tw≤eFPbv,w,eFs2PbSv,Tw≤e−τeFPbv,w,Ds2PbSv,Tw≤CDPbv,w.Hence, applying Theorem17, we say that the boundary value problems (66) and (67) have a common solution in CI.



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*Source: 7777754-2021-09-23.xml*