# Edge Theoretic Extended Contractions and Their Applications

Authors: R. Rajagopalan; Ekta Tamrakar; Fahad. S. Alshammari; H. K. Pathak; Reny George
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

Edge theoretic extended contractions are introduced and coincidence point theorems and common fixed-point theorems are proved for such contraction mappings in a metric space endowed with a graph. As further applications, we have proved the existence of a solution of a nonlinear integral equation of Volterra type and given a suitable example in support of our result.


---

## Body

## 1. Introduction and Preliminaries

The celebrated Banach contraction principle is a motivation for many fixed-point theorems. It guarantees the existence and uniqueness of solution of various equations arising in mathematics. The initial generalizations of Banach’s result came up in the form of Kannan’s contraction, Chatterjea’s contraction, Reich’s contraction, Ciric’s contraction, Hardy-Roger’s contraction, and Ciric’s quasicontraction. Among these, Ciric’s quasicontraction is the most general form in the sense that any mapping which does not satisfy Ciric’s quasicontraction does not satisfy any of the previously mentioned contractions. Further, these results have been widely investigated and many interesting applications have been found by many authors (see [1–7]). F-contraction and fixed-point theorem for F-contraction mappings were introduced by Wardowski [8] as a generalisation of the Banach contraction principle.Definition 1 (see [8]).
Consider the collection of functionsF:0,∞⟶ℝ satisfying the following:
F1F is strictly increasing
F2 If αn⊂0,∞ is a sequence, then limn⟶∞αn=0 iff limn⟶∞Fαn=−∞
F2 There exists k∈0,1 such that limγ⟶0+γkFγ=0
An operatorT:Xi,di⟶Xi is an F-contraction if we can find τ>0 such that
(1)∀xi,yi∈Xi,diTxi,Tyi>0⟹τ+FdiTxi,Tyi≤Fdixi,yi.Later, the concept ofF-weak contraction and ordered F-contractions was introduced by Wardowski and Van Dung [9] and Durmaz et al. [10], respectively. In 2016, Sawangsup et al. [11] extended the F-contraction using a relation theoretic approach which was later generalised by Imdad et al. [12] and Alfaqih et al. [13]. Espinola and Kirk [14] introduced graph theory in fixed-point theory, and Jachymski [15] continued this idea by using different views thereby introducing the G-contraction and proved fixed-point theorem for a G-contraction mapping. These ideas were further extended and generalised by [16–24].It is interesting to note that all these contraction conditions ensure the existence of a unique fixed point or common fixed point of the mappings under consideration. However, it is observed that a mapping which possesses nonunique fixed points does not satisfy the above contractions, for ifxi and yi are any two fixed points of a self-map Ti of a metric space Xi,di, then
(2)diTixi,Tiyi=dixi,yi=maxdixi,yi,dixi,Tixi,diyi,Tiyi,dixi,Tiyi+diyi,Tixi2,diTixi,Tiyi=dixi,yi=maxdixi,yi,dixi,Tixi,diyi,Tiyi,dixi,Tiyi,diyi,Tixi,and thus, we see that Ti does not satisfy Ciric’s quasicontraction, Wardowski’s F-contraction, and Wardowski and Van Dung’s F-weak contraction. Thus, these contraction conditions cannot be used to prove the existence of nonunique fixed points of a function defined in a metric space. On the other hand, many equations obtained by modeling various problems of engineering and science need not necessarily have a unique solution. Thus, it becomes meaningful to obtain extended forms of above contractions which will ensure the existence of nonunique fixed points of self-maps defined in a metric space.Motivated by this fact, in this paper, we have introduced extendedJW-contraction (Jungck-Wardowski contraction), extended CW-contraction (Ciric-Wardowski contraction), and extended CWQ-contraction (Ciric-Wardowski quasicontraction) and established fixed-point theorems which will ensure the existence of nonunique fixed points of a self-map and coincidence points of a pair of self-maps, respectively, in a metric space endowed with a graph. As an application of our result, we have also proven the existence of solution of a nonlinear integral equation of Volterra type.Throughout this paper, we consider the metric spaceXj,dj to be endowed with the graph G=VG,EG, VG=Xj, and Δ⊆EG; Δ=xj,xj:xj∈Xj.Definition 2 (see [15]).
A sequencexnj⊆Xj is edge-preserving if xjn,xjn+1∈EG for all n∈ℕ0.Definition 3.
Letg:Xj⟶Xj. A sequence xnj⊆Xj is g-edge-preserving if gxjn,gxjn+1∈EG for all n∈ℕ0.Definition 4.
T:Xj⟶Xj is edge-preserving if xj,yj∈EG implies Txj,Tyj∈EG.Definition 5.
T,g:Xj⟶Xj is g-edge-preserving if for all xj,yj∈X, gxj,gyj∈EG implies Txj,Tyj∈EG.Definition 6 (see [15]).
Xj,dj is edge-complete if every edge-preserving Cauchy sequence in Xj converges to some point in Xj.Definition 7 (see [15]).
T:Xj⟶Xj is edge-continuous at xj if xjn⟶xj implies Txjn⟶Txj for any edge-preserving sequence xjn⊆Xj. If T is edge-continuous at all xj∈Xj, then T is an edge-continuous mapping.Definition 8.
LetT,g:Xj⟶Xj and xj∈Xj. We say that T is g-edge continuous at xj if gxjn⟶gxj implies Txjn⟶Txj for any edge-preserving sequence xjn⊆Xj. If T is g-edge continuous at all xj∈Xj, then T is an g-edge continuous mapping.Definition 9.
T,g is edge-compatible if and only if for any sequence T and g edge-preserving sequence xjn⊆X, limn⟶∞gxjn=limn⟶∞Txjn=x∈Xj implies limn⟶∞djgTxjn,Tgxjn=0.We will use the following lemmas taken from [25, 26]:Lemma 10. (see [25]).
LetM be a nonempty set and g:M⟶M. Then, there exists a subset S⊆M such that gs=gM and g:S⟶S is one-one.Lemma 11 (see [26]).
Letxnj be a sequence in metric space Xj,dj such that limn⟶+∞djxnj,xn+1j=0. If xnj is not Cauchy in Xj,dj, then there exist ξ>0 and sequences nk and pk in ℕ such that nk>pk>k, and the sequences
(3)djxnkj,xpkj,djxnk+1j,xpkj,djxnkj,xpk−1j,djxnk+1j,xpk−1j,djxnk+1j,xpk+1j,
tend to beξ+, as k⟶+∞.


## 2. Edge Theoretic Extended Contractions

LetF be the collection of all nondecreasing continuous functions F:0,∞⟶ℝ.Example 1.
Some examples of function belonging to the classF are
(4)Fy=y2,Fy=lny,Fy=y−1y,Fy=lny3+siny.
LetA⊂0,∞ and Ξ be the collection of all continuous functions ξ:A×A⟶0,∞ satisfying the following:
(i)
α=0 or β=0 implies ξα,β=0(ii)
α>0 and β>0 implies ξα,β>0(5)supα,β∈Aξα,β=ζ>0.Some examples of functionξ are as follows:Example 2.
(i)
ξα,β=k.αβ, for some k>0(ii)
ξα,β=minα,β(iii)
ξα,β=α/1+lnβ(iv)
ξα,β=α+β/1+lnαβ(v)
ξα,β=αβα+β(vi)
ξα,β=αβ/1+αβ(vii)
ξα,β=ln1+K.minα,β
LetΘ be the family of all functions θ:0,∞⟶R which satisfy the following conditions:
θ1θ is strictly increasing
θ2θt=0 iff t=0
θ3supt>0θt=λ for some λ>0Example 3.
Some examples of elements ofΘ are
(6)θt=t1+t,θt=ln1+t1+t,θt=t1+ln1+t.Definition 12.
A pair of mappingsT,g:Xj⟶Xj is an ξ-extended JW-contraction pair if we can find τ>0, F∈F, ξ∈Ξ, and L≥0 such that for all xj,yj∈Xj,
(7)djTxj,Tyj>0⟹τ+FdjTxj,Tyj≤Fdjgxj,gyj+Lξdjgyj,Txj,djgxj,Tyj,Definition 13.
A pair of mappingsT,g:Xj⟶Xj is an ξ-extended CW-contraction pair if we can find τ>0, F∈F, ξ∈Ξ, and L≥0 such that for all xj,yj∈Xj,
(8)djTxj,Tyj>0⟹τ+FdjTxj,Tyj≤FMjxj,yj+Lξdjgyj,Txj,djgxj,Tyj,where
(9)Mjxj,yj=maxdjgxj,gyj,djgxj,Txj,djgyj,Tyj,djgxj,Tyj+djgyj,Txj2.Definition 14.
A pair of mappingsT,g:Xj⟶Xj is an ξ-extended CWQ-contraction pair provided that there is a τ>0, F∈F, ξ∈Ξ, and L≥0 such that for all xj,yj∈Xj,
(10)djTxj,Tyj>0⟹τ+FdjTxj,Tyj≤FMj∗xj,yj+Lξdjgyj,Txj,djgxj,Tyj,where
(11)Mj∗xj,yj=maxdjgxj,gyj,djgxj,Txj,djgyj,Tyj,djgxj,Tyj,djgyj,Txj.Definition 15.
In Definitions12, 13, and 14, if conditions (7), (8), and (10) are satisfied only for all xj,yj∈Xj with xj,yj∈EG, then the pair T,g is an ξ-extended JW-edge contraction, ξ-extended CW-edge contraction, and ξ-extended CWQ-edge contraction, respectively.Definition 16.
T,g:Xj⟶Xj is a θ-extended JW-edge contraction if we can find τ>0, F∈F, and θ∈Θ such that for all xj,yj∈Xj with gxj,gyj∈EG,
(12)djTxj,Tyj>0⟹τ+FdjTxj,Tyj≤Fdjgxj,gyj+Lθdjgyj,Txj.Definition 17.
A pair of mappingsT,g:Xj⟶Xj is a θ-extended CW-edge contraction if we can find τ>0, F∈F, and θ∈Θ such that
(13)djTxj,Tyj>0⟹τ+FdjTxj,Tyj≤FMjxj,yj+Lθdjgyj,Txj,for all xj,yj∈Xj with gxj,gyj∈EG and Mjxj,yj, is as in (9).Ifg=I in the above definitions, then T is an ξ-extended F-contraction mapping, ξ-extended CW-contraction mapping, θ-extended JW-edge contraction mapping, and θ-extended CW-edge contraction mapping, respectively.Property (∗). The space Xj,dj is said to have property∗ if for any edge-preserving sequence xjn∈X such that xjn⟶x; there exists a subsequence xnkj of xjn such that xnkj,x∈EGX for all k∈ℕ0Example 4.
LetX=0,1⋃2, djxi,yj=xi−yj, and Txi=xi4/8 for all xi∈X. Then, at xi=0 and yj=2, T does not satisfy the conditions of Ciric’s quasicontraction, Wardowski’s F-contraction, and Wardowski and Van Dung’s F- weak contraction. However, T is an ξ-extended F-contraction with τ=ln2, as shown below:
LetF:0,∞⟶ℝ be defined by
(14)Ft=lnt,and ξα,β=ln1+K.minα,β.Case 1.
xi,yj∈0,1. Clearly,
(15)djTxi,Tyj=18xi4−yj4≤18xi−yjxi+yjxi2+yj2≤14xi−yjxi+yj<12xi−yj≤12djxj,yj.
Then, we havelndjTxi,Tyj<ln1/2djxj,yj or
(16)ln2+lndjTxi,Tyj<lndjxj,yj+Lξdjgyj,Txj,djgxj,Tyj.Case 2.
xi∈0,1 and yj=2. Note that in this case, djxj,yj≥1.
(17)djTxi,Tyj=xi48−2≤12+2minxi−2,2−xi48⟹djTxi,Tyj≤12djxj,yj1+8minxi−2,2−xi48⟹lndjTxi,Tyj≤−ln2+lndjxj,yj+ln1+8minxi−2,2−xi48⟹ln2+lndjTxi,Tyj≤lndjxj,yj+ln1+8minxi−2,2−xi48⟹ln2+FdjTxi,Tyj≤Fdjxj,yj+ξ1+8mindjxj,Tyj,djyj,Txj.Example 5.
LetXj=0,∞, djxj,yj=xj−yj, EG=n,n,n,n+1:n=0,1,2,3,⋯, and T,g:Xj⟶Xj be given by
(18)Txj=0,if0≤xj≤1,xj−1,ifxj≥1,gxj=xj+n+1−xjxj−n,whenevern≤xj≤n+1.
LetF:0,∞⟶ℝ be defined by
(19)Ft=t−1t,and θ∈Θ be defined by θt=t/t+1. Then,
(20)τ+FdjTn,Tn+1≤Fdjgn,gn+1+Lθdjgn+1,Tn⟹τ+Fdjn−1,n≤Fdjn,n+1+Lθdjn+1,n−1⟹τ≤F1−F1+Lθ2⟹τ≤Lθ2.Hence, for any0<τ<2/3 and L=1, (13) is satisfied and thus T,g is a θ-extended WJ-edge contraction and θ-extended WC-edge contraction. However, the pair (T,g) is neither an ξ-extended JW-edge contraction pair nor an ξ-extended CW-contraction pair. If we take g to be the identity mapping, then T is a θ-extended JW-edge contraction mapping and θ-extended CW-edge contraction mapping. However, again T is none of Wardowski’s F-contraction, Wardowski and Van Dung’s F- weak contraction, and Ciric’s quasicontraction.


## 3. Main Results

We start by proving the following main theorems:Theorem 18.
SupposeXj,dj be endowed with a graph G satisfying transitivity property, and the following conditions hold for T,g:Xj⟶Xj.
agx0j,Tx0j∈EG for some x0j∈Xj
bT is g-edge preserving
cT,g is an θ-extended CW-edge contraction pair of mappings
dd1 There exists an edge-complete subset Mj of Xj for which TXj⊆Mj⊆gXj
d2 One of the following conditions holds:
(i)
T is g-edge continuous(ii)
T and g are continuous(iii)
EGXj satisfies property∗
Then, the pairT,g has a coincidence point.Proof.
In view of the assumption (a), we havegx0j,Tx0j∈EG. If Tx0j=gx0j, then x0 is a coincidence point of T,g, i.e., CoinT,g≠ϕ, and there is nothing to prove. Assume that Tx0j≠gx0j; then, since TXj⊆gXj, there exists xj1∈Xj such that gx1j=Tx0j.Similarly, there isxj2∈Xj such that gx2j=Tx1j with gx1j,gx2j∈EG and consequently Tx0j,Tx1j∈EG. Inductively, one can construct a sequence xjn⊆Xj such that
(21)gxn+1j=Txjn,foralln∈ℕ0,with
(22)gxjn,gxn+1j∈EGforalln∈ℕ0,and consequently, asT is g-edge preserving,
(23)Txjn,Txn+1j∈EG.Now, ifTxn0j=Txn0j for some n0∈ℕ0, then xn0 is a coincidence point T,g and we are done. Assume that Txjn≠Txn+1j, for all n∈ℕ0. On using (21), (22), (23), and condition c, we have
(24)τ+Fdgxjn,gxn+1j=τ+FdTxn−1j,Txjn≤FMxn−1j,xjn+Lθdgxjn,Txn−1j.Now,(25)Mxn−1j,xjn=maxdjgxjn−1,gxjn,djgxn−1j,Txn−1j,djgxjn,Txjn,djgxn−1j,Txjn+djgxjn,Txn−1j2=maxdjgxn−1j,gxjn,djgxjn,gxjn+1,θdgxjn,Txn−1j=θdgxjn,gxjn=0.Thus, we get(26)τ+Fdgxjn,gxn+1j≤Fmaxdjgxn−1j,gxjn,djgxjn,gxjn+1,i.e.,(27)Fdgxjn,gxn+1j<τ+Fdgxjn,gxn+1j≤Fdgxn−1j,gxjn.SinceF is nondecreasing, we get dgxjn,gxn+1j<dgxn−1j,gxjn. This further means that djxnj,xn+1j⟶δ≥0 as n⟶+∞. If δ>0, we obtain from (27) that
(28)Fδ+≤τ+Fδ+≤Fδ+,which is a contradiction. Hence, limn⟶+∞djxnj,xn+1j=0. Suppose the sequence gxnj is not a Cauchy sequence. By Lemma 11, there exist ξ>0 and sequences nk and pk in ℕ such that nk>pk>k, such that the sequences djxnkj,xpkj and djxnk+1j,xpk+1j tend to be ξ+, as k⟶+∞. By (27) we get
(29)τ+Fξ++≤Fξ++,which is a contradiction. So sequencegxnj is a Cauchy sequence.By (21) and (22), gxjn is an edge-preserving Cauchy sequence in TXj⊂Mj, and since Mj is edge-complete, there exists yj∈Mj such that gxjn⟶yj. As Mj⊆gXj, there exists uj∈Xj such that yj=guj. Hence, on using (21), we obtain
(30)limn⟶∞gxjn=limn⟶∞Txjn=guj.Now, suppose conditiond2i is true. Using (22) and (30), we obtain
(31)limn⟶∞Txjn=Tuj.By (30) and (31), we have
(32)Tuj=guj.Suppose condition (d2ii) is true. By Lemma 10, there is S⊆Xj for which gS=gXj and g:S⟶S is one-one. Consider the function f:gS⟶gXj given by
(33)fgs=Tsgs∈gS,s∈S.Asg:S⟶Xj is one-one and TXj⊆gXj, f is well-defined. Since T and g are continuous, f is also continuous by condition d1 of the hypothesis TXj⊆Mj⊆gS. Thus, we have xjn⊆S and uj∈S. Therefore,
(34)Tuj=fguj=flimn⟶∞gxjn=limn⟶∞fgxjn=limn⟶∞Txjn=guj.Suppose condition (d2iii) is true; that is, EGXj satisfied Property∗. Since gxjn⊆X, it follows that gxjn is EGXj-preserving (due to (22)) and gxjn⟶guj (by (30)) and so we have a subsequence gxnkj⊆gxjn such that
(35)gxnkj,guj∈EGX,forallk∈ℕ0.Using (35) and condition b of the hypothesis, we have
(36)Txnkj,Tuj∈EGXj⊆S,forallk∈ℕ0.Now, letPj=k∈ℕ:Txnkj=Tu.IfPj is finite, then Txnkj has a subsequence Txnkij such that Txnkij≠Tu for all i∈ℕ. Also, gxnkij,guj∈EGX⊂EG. Thus, we have
(37)τ+FdTxnkij,Tuj≤FMxnkij,uj+Lθdguj,Txnkij,Mxnkij,uj=maxdjgxjnki,guj,djgxjnki,Txjnki,djguj,Tuj,djgxjnki,Tuj+djguj,Tgxjnki2.Lettingi⟶∞, we obtain Mxnkij,uj=djguj,Tuj and θdguj,Txnkij=0. Thus, we get
(38)τ+Fdjguj,Tuj≤Fdguj,Tuj,which is a contradiction. Hence,Pj is not finite. Thus, Pj is infinite and so Txnkj has a subsequence Txnkij such that Txnkij=Tuj for all i∈ℕ. Thus, limi⟶∞Txnkij=Tuj. As limn⟶∞Txjn=guj (by (30)), we get Tuj=guj.Theorem 19.
If, in addition to hypothesisa-d of Theorem 18, we assume the following:
(i)
For alluj,vj∈CoinT,g,(39)djTuj,Tvj>0⟹τ+FdjTuj,Tvj≤FMjuj,vj+Lθdjgui,Tui,(ii)
One ofT or g is one-one(iii)
T and g are weakly compatible
thenT,g has a unique common fixed point.Proof.
In view of Theorem18, the set CoinT,g is nonempty. Let uj,vj∈CoinT,g. If djTuj,vj=0, then we have Tuj=guj=gvj=Tvj, and hence, uj=vj as one of T and g is one-one. Otherwise, using condition (39), we obtain
(40)τ+FdTuj,Tvj≤Fdguj,gvj+Lθdguj,Tuj,=FdTuj,Tvj,
which is a contradiction. So the coincidence point ofT and g is unique.
Letuj be the unique coincidence point of T and g, and let zj∈X such that zj=Tuj=guj. As T and g are weakly compatible, we have Tzj=Tguj=gTuj=gzj. Thus, zj is a coincidence point of T and g. By the uniqueness of the coincidence point, we conclude uj=zj; that is, u is a common fixed point of the pair T,g which is indeed unique. as the coincidence point of T and g is unique.Remark 20.
If we replace conditiond of Theorem 18 with the following alternate condition:
d∗d1∗ There exists a subset Yj of Xj such that TXj⊆gXj⊆Yj and Yj is edge-complete
d2∗T,g is an edge-compatible pair
d3∗T and g are edge-continuous
the conclusions of Theorems18 and 19 still hold.Proof.
Clearly,gxjn is an edge-preserving Cauchy sequence in Yj, and by edge-completeness of Y, we get vj∈Yj such that
(41)limn⟶∞gxjn=vj,and then, by (21), we have
(42)limn⟶∞Txjn=vj.
Using the edge continuity ofg and T, we also have
(43)limn⟶∞Tgxjn=Tlimn⟶∞gxjn=Tvj,(44)limn⟶∞gTxjn=glimn⟶∞Txjn=gvj.
Then, by edge-compatibility ofg and T, we get
(45)limn⟶∞dgTxjn,Tgxjn=0.
Finally from (44), (45), and (43), we get
(46)dgvj,Tvj=dlimn⟶∞gTxjn,limn⟶∞Tgxjn=limn⟶∞gTxjn,Tgxjn=0.
Hence,vj is a coincidence point of the pair T,g.Remark 21.
Since everyξ-extended contraction mapping is a θ-extended contraction, the conclusions of Theorems 18 and 19 remain true for an edge theoretic ξ-extended CW-contraction pair of mappings also.
On settingg=I in Theorem 18, we deduce the following corresponding fixed-point result.Theorem 22.
LetM,d be a metric space endowed with a directed graph G and T:M⟶M. Assume that the following conditions are fulfilled:
(a)
There existsx0∈M such that x0,Tx0j∈EG(b)
T is edge-preserving(c)
T is a θ-extended CW-edge contraction mapping(d)
d1 There exists a subset X of M such that TM⊆X and X is edge-complete
d2 One of the following conditions is satisfied:
(i)
T is edge-continuous(ii)
EGX satisfies Property∗
Then,T has a fixed point.Example 6.
LetXj,dj, EG, T, and g be as in Example 5. Then, we have the following:
(1)
g0,T0∈EG(2)
T is g-edge-preserving. In fact, we see that gxj,gyj∈EG implies either xj=n,yj=n or xj=n,yj=n+1. If n=0, then T0,T0∈EG and T0,T1∈EG. If n=1, then T1,T1∈EG and T1,T2∈EG. If n=k>1, then Tk,Tk∈EG and Tk,Tk+1=k−1,k∈EG(3)
T,g is a θ-extended CW-edge contraction mapping(4)
TXj⊂gXj(5)
T is g-edge-continuous
Thus, all conditions of Theorem18 are satisfied and 0 is a coincidence point of T and g. Moreover, we see that T and g satisfy conditions (i), (ii) (g is one-one), and (iii) of Theorem 19, and 0 is the unique common fixed point of T and g.Remark 23 (an open problem).
Prove Theorems18, 19, and 22 for ξ-extended CQW-contraction mappings.


## 4. Application to Nonlinear Integral Equations

Consider the Banach spaceM=C0,1,R of all continuous functions x:0,1⟶R equipped with norm
(47)x=maxs∈0,1xs.Define a metricdj on M by djxj,yj=xj−yj for all xj,yj∈M. Then, M,dj is a complete metric space.In this section, we show the applicability of Theorem19 by investigating the existence and uniqueness of a solution for the following nonlinear integral equation of Volterra type:
(48)xjs=∫0μsKs,v,xjηvdv+∫0σsJs,v,xjζvdv+fs,s∈0,1,where K,J:0,1×0,1×R⟶R, f:0,1⟶R, and μ,σ,η,ζ:0,1⟶0,1.Definition 24.
A lower solution for (48) is a function x∈M such that
(49)xjs≤∫0μsKs,v,xjηvdv+∫0σsJs,v,xjζvdv+fs,s∈0,1.Definition 25.
An upper solution for (48) is a function x∈M such that
(50)xjs≥∫0μsKs,v,xjηvdv+∫0σsJs,v,xjζvdv+fs,s∈0,1.
Consider the operatorT:M⟶M defined by
(51)Txjs=∫0μsKs,v,xjηvdv+∫0σsJs,v,xjζvdv+fs,forallx∈M.
Then,xj is a fixed point of the operator T if and only if it is a solution of the integral equation (48).Let(52)M⋄xj,yj=maxxj−yj,xj−Txjs,yj−Tyjs,xj−Tyjs+yj−Txjs2,M⋄xj,yj=maxxj−yj,xj−Txjs,yj−Tyjs,xj−Tyjs+yj−Txjs2.Theorem 26.
Assume thatK and J are nondecreasing in the third variable, μt+σt≤1 for all t∈0,1, and the following conditions hold:
There existsτ>0 such that
(53)Ks,v,gxj−Ks,v,gyj≤M⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1,Js,v,gxj−Js,v,gyj≤M⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1,for all s,v∈0,1, xj,yj∈M with xjs≤yjs and L≥0. If (48) has a lower solution, e.g., xj0s, then a solution exists for the integral equation (48).Proof.
Consider the graphG in M, with edges EG given by
(54)EG=xj,yj∈M×M:xjs≤yjs.
For anyxj,yj∈EG, we have (for all s∈0,1)
(55)Txjs=∫0μsKs,v,xjηvdv+∫0σsJs,v,xjζvdv+fs≤∫0μsKs,v,yjηvdv+∫0σsJs,v,yjζvdv+fs=Tyjs,which shows that Txj,Tyj∈EG. Thus, T is edge-preserving. Now, for all xj,yj∈EG and s∈0,1, we have
(56)Txjs−Tyjs≤∫0sKs,v,xjηv−Ks,v,yjηvdv+∫0sJs,v,xjηv−Js,v,yjηvdv≤∫0μsM⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1dv+∫0σsM⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1dv≤∫0μsmaxs∈0,1M⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1dv+∫0σsmaxs∈0,1M⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1dv≤M⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1∫0μsdv+M⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1∫0σsdv=M⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1μs+σs≤M⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1.
Taking the supremum, we get(57)Tx−Ty≤M⋄xj,yjM⋄xj,yjτ−Lyj−Txjs/1+yj−Txjs+1,or
(58)τ+1M⋄xj,yj≤1Tx−Ty+Lyj−Txjs1+yj−Txjs,or
(59)τ−1Tx−Ty≤−1M⋄xj,yj+Lyj−Txjs1+yj−Txjs.
That is,(60)τ−1djTxj,Tyj≤−1Mjxj,yj+Ldjyj,Txjs1+djyj,Txjs.
Thus, inequality (13) is satisfied with Fα=−1/α and θβ=β/1+β, so that λ=supt>0θt=1. Also, by Definition 24, we have xj0,Txj0∈EG. Therefore, all the assumptions of Theorem 22 are satisfied, and thus, problem (48) has a solution.Theorem 27.
Assume thatK is nonincreasing in the third variable and there exists τ>0 such that
(61)Ks,v,gxj−Ks,v,gyj≤gxj−gyjτgxj−gyj+1,for all s,v∈0,1 and x,y∈M. Then, the existence of an upper solution of the integral equation (48) ensures the existence of a solution of (48).Proof.
Define setEG of edges on M by
(62)EG=x,y∈M×M:xs≥ys.
Now, following the steps of the proof of Theorem26 with an analogous procedure, one can check that all the hypotheses of Theorem 22 are validated, and thus, Theorem 22 ensures the existence of a unique solution of the integral equation (48).We now furnish a numerical example to validate the hypothesis of Theorem27.Example 7.
Consider the functionx∈M defined by xs=s2,s∈0,1. We show that this function is an upper solution in M for the following integral equation:
(63)xs=−12s+2s2+arctan12s−3arctan12s2−12s2ln1+14s4+∫0s2/2ln1+xvdv+∫0s/2xv1+xvdv,s∈0,1.Finally, we see thatxus=s2−arctans2/2 is the unique solution of (63).Proof.
Define the operatorT:M⟶M as
(64)Txs=−12s+2s2+arctan12s−3arctan12s2−12s2ln1+14s4+∫0s2/2ln1+xvdv+∫0s/2xv1+xvdv,s∈0,1.
Now, setKs,v,xv=ln1+xv, Js,v,xv=xv/1+xv, μs=1/2s2,σs=1/2s, fs=−1/2s+2s2+arctan1/2s−3arctan1/2s2−1/2s2ln1+1/4s4, and τ≤0.01. We observe the following:
(i)
Both the functionsKs,v,xv=ln1+xv and Js,v,xv=xv/1+xv are nondecreasing in the third variable(ii)
By actual computation, we have(65)∫0s2/2ln1+xvdv=−s2+2arctan12s2+12s2ln1+14s4,s∈0,1,∫0s/2xv1+xvdv=12s−arctan12s,s∈0,1.(iii)
s2≥−1/2s+2s2+arctan1/2s−3arctan1/2s2−1/2s2ln1+1/4s4+∫0s2/2ln1+xvdv+∫0s/2xv/1+xvdv,s∈0,1 so that xs=s2 is an upper solution for (63)(iv)
The following inequalities hold true for allx,y∈0,1 (see Figures 1 and 2):(66)ln1+x−ln1+y≤x−y1+0.01x−y,(67)x1+x−y1+y≤x−y1+x−y≤x−y1+0.01x−y.Figure 1
Inequality in (66).Figure 2
Inequality in (67).Furthermore, using the nondecreasing functions↦s/1+0.01s, we have
(68)ln1+x−ln1+y≤x−y1+0.01x−y≤maxs∈0,1x−y1+0.01maxs∈0,1x−y=x−y1+0.01x−y.Similarly, for allx,y∈0,1, we have
(69)x1+x−y1+y≤x−y1+0.01x−y.Hence, all the conditions of Theorem27 are satisfied. It is evident that the integral equation (63) has a unique solution xu∈M defined by xus=s2−arctans2/2.



---

*Source: 5157708-2021-11-09.xml*