# L-Simulation Functions over b-Metric-Like Spaces and Fractional Hybrid Differential Equations

Authors: Shimaa I. Moustafa; Ayman Shehata
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we establish some fixed point results forαqsp-admissible mappings embedded in L-simulation functions in the context of b-metric-like spaces. As an application, we discuss the existence of a unique solution for fractional hybrid differential equation with multipoint boundary conditions via Caputo fractional derivative of order 1<α≤2. Some examples and corollaries are also considered to illustrate the obtained results.


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## Body

## 1. Introduction

Fixed point theory has received much attention due to its applications in pure mathematics and applied sciences. Generalization of this theory depends on generalizing the metric type space or the contractive type mapping. The concept of metric spaces has been extended in various directions by reducing or modifying the metric axioms. Since, losing or weakening some of the metric axioms causes loss of some topological properties, hence bringing obstacles in proving some fixed point theorems. These obstacles force researchers to develop new techniques in the development of fixed point theory in order to resolve more real concrete applications.In 1989, Bakhtin [1] (and also Czerwik [2]) introduced the concept of b−metric spaces and presented a generalization of Banach contraction principle. Amini-Harandi [3] introduced the notion of metric-like spaces which play an important role in topology and logical programming. In 2013, Alghamdi et al. [4] generalized the notions of b−metric and metric-like spaces by introducing a new space called b−metric-like space and proved some related fixed point results. Recently, many results of fixed point of mappings under certain contractive conditions in such spaces have been obtained (see [5–8]).Zoto et al. [9] introduced the concept of αqsp−admissible mappings and provided some fixed point theorems for these mappings under some new conditions of contractivity in the setting of b−metric-like spaces. Recently, Cho [10] proposed the notion of θL−contractions and confine his fixed point results for such contractions to generalized metric spaces. Aydi et al. [11] proved that those results are also valid in partial metric spaces.Fractional calculus is a field of mathematics that deals with the derivatives and integrals of arbitrary order. Indeed, it is found to be more realistic in describing and modeling several natural phenomena than the classical one. In recent years, many researchers have focused on joining fixed point theory with fractional calculus, see for example [12–15].The study of differential equations with fractional order has attracted many authors because of its intensive development of fractional calculus itself and its applications in various fields of science and engineering, see [16–19].On the other hand, hybrid differential equations have attracted much attention after the pioneering works appeared in [20, 21] which discussed main aspects about first-order hybrid differential equations with perturbations of 1st and 2nd types, respectively.Fractional hybrid differential equations (in short, FHDEs) have been studied using Riemann-Liouville and Caputo fractional derivatives of orderα>0 in many literatures, see [22–27].In [23], Derbazi et al. applied Dhage hybrid fixed point theorem [28] to provide sufficient conditions that guarantee the existence only of solutions for a class of FHDEs with three-point boundary conditions due to Caputo fractional derivative of order 1<α≤2 in Banach algebra spaces.Inspired by the above works, we investigate the existence of a unique fixed point forαqsp-admissible mapping via L-simulation function ξ:1,∞×1,∞→ℝ and control function θ:0,∞→1,∞ in more general setting (b-metric-like space) than partial metric, b−metric and metric-like spaces. Also, as an application, we provide appropriate conditions that guarantee the existence of a unique solution to the following FHDE.
(1)cDαxt−ft,xtgt,xt=ht,xt,Iqxt,t∈J=0,T,ζ1xt−ft,xtgt,xtt=0+ζ2xt−ft,xtgt,xtt=T=λ1,∑i=3mζicDβxt−ft,xtgt,xtt=ηi+ζm+1cDβxt−ft,xtgt,xtt=T=λ2,where cDα and cDβ denote the Caputo fractional derivatives of orders α and β, respectively, 0<α≤2, q>0, 0<β≤1, 0<ηi<T, ζi, i=1,2,3,⋯,m+1, m∈ℕ are real constants such that
(2)ζ1+ζ2≠0,∑i=3mζiηi1−β+ζm+1T1−β≠0,g∈CJ×ℝ,ℝ\0andf,h∈CJ×ℝ,ℝ.


## 2. Basic Concepts

In order to fix the framework needed to state our main results, we recall the following notions.Definition 1 [1].
LetX be a nonempty set and s≥1 be a given real number. A function d:X×X→0,∞ is a b−metric if for all x,y,z∈X, the following conditions are satisfied.
(b1) dx,y=0⇔x=y
(b2) dx,y=dy,x
(b3) dx,y≤sdx,z+dz,y
The pairX,d is called a b−metric space, and s is the coefficient of it.Note that, every metric space is ab−metric space with coefficient s=1.Definition 2 [3].
A metric-like space on a nonempty setX is a function σ:X×X→0,∞ such that for all x,y,z∈X:
(σ1) σx,y=0⇒x=y
(σ2) σx,y=σy,x
(σ3) σx,y≤σx,z+σz,y
Then, the pairX,σ is called a metric-like space.It should be noted thatσ satisfies all of the conditions of a metric except that σx,x may be positive for x∈X.Definition 3 [4].
A functionσb:X×X→0,∞ on a nonempty set X is called b−metric-like if for any x,y,z∈X, the following conditions hold true.
(σb1) σbx,y=0⇒x=y
(σb2) σbx,y=σby,x
(σb3) σbx,y≤sσbx,z+σbz,y
The pairX,σb is called a b−metric-like space.Remark 4.
The class ofb−metric-like spaces is considerably larger than both b−metric spaces and metric-like spaces. Since, every b−metric is a b−metric-like with same coefficient and zero self-distance. Also, every metric-like is a b−metric-like with s=1. However, the converse implications do not hold (see for example, [1, 4]).Example 5.
LetX=R and σb:X×X→0,∞ be defined as
(3)σbx,y=∣x∣+∣y∣p,∀x,y∈Xandp∈ℕ,then, X,σb is a b−metric-like space with parameter s=2p−1.Example 6.
LetX⊆ℝ and CbX=f:X→R:supx∈X∣fx∣<∞. The function σb:X×X→0,∞, defined by
(4)σbf,g=supx∈X∣fx∣,∣gx∣nn,∀f,g∈CbX,n∈ℕ,is a b−metric-like with constant s=2n−1n, and so, X,σb is a b−metric-like space.Definition 7.
LetX,σb be a b−metric-like space and xn be a sequence in X, and x∈X. Then,
(1)
The set(5)Bx,r=y∈X:σbx,y−σbx,x<r,is called an open ball with center x and radius r. Also, the family
(6)Bx,r,∀x∈X,r>0,forms a base of the topology τσb generated by σb on X.
(2)
xn is said to converge to x w.r.t. τσb if(7)limn→∞σbxn,x=σbx,x.(3)
xn is said to be σb−Cauchy if(8)limn,m→∞σbxn,xm,exists and is finite.
(4)
X,σb is said to be complete if for every Cauchy sequence xn in X, there exists x∈X such that(9)limn,m→∞σbxn,xm=limn→∞σbxn,x=σbx,x.Lemma 8.
LetX,σb be a b−metric-like space with parameter s≥1 and xn be a convergent sequence in X such that
(10)limn,m→∞σbxn,xm=limn→∞σbxn,x=σbx,x=0,x∈X.Then, every subsequencexnk with nk≥k∈ℕ converges to the same limit x∈X.Proof.
Sincexn→x and limn→∞σbxn,x=0, then for a given ε>0(11)∃n0∈ℕ:n>n0⇒σbxn,x<ε.
Fromσb3 and (10), we have
(12)σbxnk,x<sσbxnk,xk+σbxk,x→0asnk≥k→∞.
Therefore,(13)limk→∞σbxnk,x=0=σbx,x.Definition 9 [10].
LetL be the set of all L−simulation functions ξ:1,∞×1,∞→ℝ that fulfil:
(ξ1) ξ1,1=1
(ξ2) ξu,v<v/u,∀u,v>1
(ξ3) For any two sequences un,vn∈1,∞ with un<vn,∀n∈ℕ(14)limn→∞un=limn→∞vn>1⇒limsupn→∞ξun,vn<1.In [10], authors used the function θ:0,∞→1,∞ defined by Jleli and Samet in [29] to propose the following result.Definition 10 [29].
LetΘ be the set of all functions θ:0,∞→1,∞ that fulfil:
(θ1) θ is non-decreasing
(θ2) For any sequence un∈0,∞(15)limn→∞θun=1⇒limn→∞un=0.Theorem 11 [10].
LetX,d be a complete generalized metric space and T:X→X satisfy
(16)ξθdTx,Ty,θdx,y≥1,∀x,y∈XwithdTx,Ty>0,ξ∈L,θ∈Θ.Then,T has a unique fixed point, and for every initial point x0∈X, the Picard sequence Tnx0 converges to that fixed point.Definition 12 [9].
LetX,σb be a b−metric-like space with parameter s≥1, α:X×X→0,∞ be a function, and q≥1 and p≥2 be arbitrary constants. A mapping T:X→X is αqsp−admissible if
(17)αx,Tx≥qsp⇒αTx,T2x≥qsp,∀x∈X.In addition,T is said to be triangular αqsp−admissible if it is αqsp−admissible and
(18)αx,yandαy,Ty≥qsp⇒αx,Ty≥qsp,∀x,y∈X.Definition 13 [18, 19].
The Riemann-Liouville fractional integral of orderα>0 of a function x:0,∞→ℝ is given by
(19)Iαxt=1Γα∫0tt−sα−1xsds.The Caputo fractional derivative of orderα of x is given by
(20)cDαxt=1Γn−α∫0tt−sn−α−1xnsds,where n=α+1 and Γ denote the gamma function, provided that the right side is point-wise defined on 0,∞.Lemma 14 [23].
Let0<β≤α and x∈Cn0,T. Then, for all t∈0,T, we have:
(1)
IαIβxt=Iα+βxt and cDβIαxt=Iα−βft(2)
IαcDαxt=xt+∑i=0n−1citi, for some c0,⋯,cn−1∈ℝ(3)
Iα:C0,T→C0,T


## 3. A Set of Fixed Point Results

Our first main result is the following theorem.Theorem 15.
LetX,σb be a complete b−metric-like space with parameter s≥1. Suppose that T:X→X is a triangular αqsp−admissible mapping and satisfy
(21)ξαx,yθσbTx,Ty,θσbx,y≥1,∀x,y∈X,ξ∈L,θ∈Θ.Consider that the following properties hold true(a)
Ifxn is a sequence in X such that xn→x∈X as n→∞ and αxn,xn+1≥qsp, then αxn,x≥qsp,∀n∈ℕ(b)
For allx,y∈FixT, we have αx,y≥qsp, where FixT denotes the set of fixed points of TMoreover, if there existsx0∈X such that αx0,Tx0≥qsp, then T has a unique fixed point.Proof.
Starting with that pointx0∈X:αx0,Tx0≥qsp. We define a sequence xn⊂X by
(22)xn+1=Txn,∀n∈ℕ0=ℕ∪0.
Regarding thatT is an αqsp−admissible, then by induction, we get
(23)αxn,xn+1≥qsp,∀n∈ℕ0.
Ifσbxn,xn+1=0 for some n, then xn=xn+1, that is, xn is a fixed point of T, and the proof is completed. So, we assume that
(24)σbxn,xn+1>0,∀n.
From (23) and (24), we apply (21) at x=xn−1 and y=xn to get
(25)1≤ξαxn−1,xnθσbTxn−1,Txn,θσbxn−1,xn<θσbxn−1,xnαxn−1,xnθσbxn,xn+1,⇒θσbxn,xn+1≤αxn−1,xnθσbxn,xn+1<θσbxn−1,xn.
Hence, the sequenceθσbxn,xn+1 is monotone decreasing and bounded below by 1. Therefore, there exists m≥1 such that
(26)limn→∞θσbxn,xn+1=m.
To prove thatm=1, suppose the contrary that m>1 and obtain a contradiction. From (25), (26), and ξ3, we have
(27)1≤limsupn→∞ξαxn−1,xnθσbTxn−1,Txn,θσbxn−1,xn<1,that is all we need. Thus, m=1 and
(28)limn→∞θσbxn,xn+1=1.
Also,θ2 implies
(29)limn→∞σbxn,xn+1=0.
Now, we show that(30)limn,m→∞σbxn,xm=0.
Consider the sequence(31)Rk=supθσbxn,xm:m≥n≥k,∀k=1,2,3,⋯.
It is easy to verify that(32)limk→∞Rk=1⇒limn,m→∞θσbxn,xm=1,1≤⋯≤Rk+1≤Rk≤⋯≤R1.
Hence, the sequenceRk is decreasing and bounded below by 1. Consequently, there exists r≥1 such that
(33)limk→∞Rk=r.
Assume thatr>1, then from (31), we conclude that
(34)∀k=1,2,3,⋯1k>0,∃mk≥nk≥k:Rk−1k<θσbxnk,xmk<Rk.
Taking limit ask→∞, together with (33), implies
(35)limk→∞θσbxnk,xmk=r.
Also, we have(36)θσbxnk,xmk−σbxnk,xnk−1−σbxmk−1,xmk≤θσbxnk−1,xmk−1<RkorRk−1.
Again, taking limit ask→∞, together with (33), (35) implies
(37)limk→∞θσbxnk−1,xmk−1=r.
According to (23) and the fact that T is a triangular αqsp−admissible, we derive
(38)αxnk−1,xmk−1≥1.
On account of the above observations, we apply condition (21) and then (ξ3) to obtain
(39)1≤ξαxnk−1,xmk−1θσbxnk,xmk,θσbxnk−1,xmk−1<θσbxnk−1,xmk−1αxnk−1,xmk−1θσbxnk,xmk,⇒θσbxnk,xmk≤αxnk−1,xmk−1θσbxnk,xmk<θσbxnk−1,xmk−1.
Therefore, we have(40)limk→∞αxnk−1,xmk−1θσbxnk,xmk=r.
And hence(41)1≤limsupk→∞ξαxnk−1,xmk−1θσbxnk,xmk,θσbxnk−1,xmk−1<1,which is a contradiction, then r=1 and
(42)limn,m→∞θσbxn,xm=1.
Thus, (30) holds true, and the sequence xn is σb−Cauchy. By the completeness of X,σb, there exists x∈X such that
(43)limn→∞σbxn,x=σbx,x=limn,m→∞σbxn,xm=0.
Now, consider the subsequencexnk of the sequence xn such that
(44)σbxnk,x>0andσbxnk+1,Tx>0,∀nk≥k∈ℕ.
Lemma8, together with (43), imply that
(45)limk→∞σbxnk,x=0.
Apply (21), we obtain
(46)1≤ξαxnk,xθσbxnk+1,Tx,θσbxnk,x<θσbxnk,xαxnk,xθσbxnk+1,Tx.⇒θσbxnk+1,Tx≤αxnk,xθσbxnk+1,Tx<θσbxnk,x,0≤σbxnk+1,Tx≤σbxnk,x→0ask→∞.
Hence,(47)limk→∞σbxnk+1,Tx=0.
From (45), (47), and σ3, we conclude that
(48)σbx,Tx≤sσbx,xnk+σbxnk,Tx→0ask→∞,⇒σbx,Tx=0⇒x=Tx.
To see that this fixed point in unique, suppose thaty∈X is another fixed point of T and apply (21) to get the opposite.
(49)1≤ξαx,yθσbTx,Ty,θσbx,y<θσbx,yαx,yθσbTx,Ty,⇒θσbx,y≤αx,yθσbx,y<θσbx,y.
This is impossible, sox=y, and the fixed point is unique.
LetR denote the class of β:0,∞→0,1 which satisfies the condition
(50)limn→∞tn>0⇒limn→∞βtn<1.Remark 16.
Sinceb−metric-like space is a proper extension of partial metric, metric-like, and b−metric spaces. Then, we can derive our main results in the setting of these spaces.Corollary 17.
LetX,σ be a complete metric-like space and T:X→X be a mapping such that
(51)σTx,Ty≤βσx,yσx,y,∀x,y∈X,β∈R.Then,T has a unique fixed point.Proof.
For allx,y∈X with x≠y and Tx≠Ty, condition (51) can be written as
(52)eσTx,Ty≤eσx,yβlneσx,y.
Therefore, Corollary17 follows from Theorem 15 by taking
(53)q=s=1,;αx,y=1,θt=etandξu,v=vβlnvu,for all x,y∈X, t∈0,∞, and u,v∈1,∞.


## 4. Fractional Hybrid Differential Equations

Here, we place our considered problem (1) in the space
(54)X=CJ,ℝ=x:J→ℝ;xiscontinuousonJ,with a mapping σb:X×X→ℝ+∪0 defined on it as:
(55)σbx,y=supt∈J∣xt∣+∣yt∣p,∀x,y∈X,t∈J,p>1.It is evident thatX,σb is a complete b−metric-like space with coefficient s=2p−1.For convenience, we define the following functionsρi:J→ℝ, i=1,2,3,4:
(56)ρ1t=ζ2Γ2−βTζ1+ζ2∑i=3mζiηi1−β+ζm+1T1−β−Γ2−βt∑i=3mζiηi1−β+ζm+1T1−β,ρ2t=ζ2ζm+1Γ2−βTζ1+ζ2∑i=3mζiηi1−β+ζm+1T1−β−ζm+1Γ2−βt∑i=3mζiηi1−β+ζm+1T1−β,ρ3t=−ζ2ζ1+ζ2,ρ4t=−ζ1Γ2−βTλ2ζ1+ζ2∑i=3mζiηi1−β+ζm+1T1−β+λ1ζ1+ζ2+Γ2−βtλ2∑i=3mζiηi1−β+ζm+1T1−β.Lemma 18.
Leth∈CJ, then the integral representation of the boundary value problem
(57)cDαxt−ft,xtgt,xt=ht,∀t∈J,(58)ζ1xt−ft,xtgt,xtt=0+ζ2xt−ft,xtgt,xtt=T=λ1,∑i=3mζicDβxt−ft,xtgt,xtt=ηi+ζm+1cDβxt−ft,xtgt,xtt=T=λ2,where 1<α≤2, 0<β≤1, 0<ηi<T and ζi, i=1,2,3,⋯,m+1, m∈ℕ are real constants such that
(59)ζ1+ζ2≠0,∑i=3mζiηi1−β+ζm+1T1−β≠0,is given by
(60)xt=gt,xt∫0TGt,shsds+ρ4t+ft,xt,where Gt,s is the Green function and is given by
(61)Gt,s=t−sα−1Γα+ρ1tΓα−β∑j=imζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1,ifs<¯¯t,ηi−1<¯¯s<¯¯ηi,=ρ1tΓα−β∑j=imζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1,ifs≥t,ηi−1<¯¯s<¯¯ηi.Proof.
Applying the operatorIα on both sides of (57) and using Lemma 14, we have
(62)xt−ft,xtgt,xt=Iαht−c0−c1t.
Using the boundary conditions (58), we get
(63)−ζ1c0+ζ2Iα−c0−c1T=λ1,(64)∑i=3mζiIα−βhηi−c1ηi1−βΓ2−β+ζm+1Iα−βhT−c1T1−βΓ2−β=λ2.
Note that,(65)cDβc0=0andcDβt=t1−βΓ2−β.
Solving (63) and (64) for c0,c1 yields
(66)c0=−ζ2TΓ2−β∑i=3mζiIα−βhηi+ζm+1Iα−βhT−λ2ζ1+ζ2∑i=3mζiηi1−β+ζm+1T1−β+ζ2ζ1+ζ2IαhT−λ1ζ1+ζ2,c1=Γ2−β∑i=3mζiIα−βhηi+ζm+1Iα−βhT−λ2∑i=3mζiηi1−β+ζm+1T1−β.
Substituting the values ofc0 and c1 into (62), we get
(67)xt−ft,xtgt,xt=∫0tt−sα−1Γαhsds+ζ2Γ2−βTζ1+ζ2∑i=3mζiηi1−β+ζm+1T1−β−Γ2−βt∑i=3mζiηi1−β+ζm+1T1−β.∑i=3mζi∫0ηiηi−sα−β−1Γα−βhsds+ζ2ζm+1Γ2−βTζ1+ζ2∑i=3mζiηi1−β+ζm+1T1−β−ζm+1Γ2−βt∑i=3mζiηi1−β+ζm+1T1−β.∫0TT−sα−β−1Γα−βhsds−ζ2ζ1+ζ2.∫0TT−sα−1Γαhsds+−ζ1Γ2−βTλ2ζ1+ζ2∑i=3mζiηi1−β+ζm+1T1−β+λ1ζ1+ζ2+Γ2−βtλ2∑i=3mζiηi1−β+ζm+1T1−β.
Provided the functionsρit,i=1,2,3,4 are defined as in Eq. (56), we obtain
(68)xt−ft,xtgt,xt=∫0tt−sα−1Γαhsds+ρ1tΓα−β∑i=3mζi∫0ηiηi−sα−β−1hsds+ρ2tΓα−β∫0TT−sα−β−1hsds+ρ3tΓα∫0TT−sα−1hsds+ρ4t.
One can easily verify that(69)∑i=3mζi∫0ηiηi−sα−β−1hsds=∫0η3∑j=3mζjηj−sα−β−1hsds+∑i=4m∫ηi−1ηi∑j=imζjηj−sα−β−1hsds.
Thus, for0≤t≤η3(70)xt−ft,xtgt,xt=∫0tt−sα−1Γα+ρ1tΓα−β∑j=3mζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∫tη3ρ1tΓα−β∑j=3mζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∑i=4m∫ηi−1ηiρ1tΓα−β∑j=imζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∫ηmTρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+ρ4t.
Forηk−1≤t≤ηk, we have
(71)xt−ft,xtgt,xt=∫0η3t−sα−1Γα+ρ1tΓα−β∑j=3mζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∑i=4k−1∫ηi−1ηit−sα−1Γα+ρ1tΓα−β∑j=imζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∫ηk−1tt−sα−1Γα+ρ1tΓα−β∑j=kmζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∫tηk−1ρ1tΓα−β∑j=kmζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∑i=k+1m∫ηi−1ηiρ1tΓα−β∑j=1mζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∫ηmTρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+ρ4t.
Forηm≤t≤T, we get
(72)xt−ft,xtgt,xt=∫0η3t−sα−1Γα+ρ1tΓα−β∑j=3mζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∑i=4m∫ηi−1ηit−sα−1Γα+ρ1tΓα−β∑j=imζjηj−sα−β−1+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∫ηmtt−sα−1Γα+ρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+∫tTρ2tΓα−βT−sα−β−1+ρ3tΓαT−sα−1hsds+ρ4t.
Joining the previous three cases together, we get (60).
Define the operatorT:CJ,ℝ→CJ,ℝ by
(73)Txt=gt,xt∫0TGt,shs,xs,Iqxsds+ρ4t+ft,xt.
In view of Lemma18, fixed points of T are solutions of the FHDE (1). Now, we assume the following conditions which allow us to establish the existence and uniqueness results for the solution of the multipoint boundary value problem (1) by applying Theorem 15.
(C1) The functions g:J×ℝ→ℝ\0 and f,h:J×ℝ→ℝ are continuous
(C2) There exist two functions φ1,φ2:J→ℝ with bounds φ1 and φ2, respectively, such that
(74)ft,x+ft,y≤φ1tx+y,gt,x+gt,y≤φ2tx+y,∀t,x,y∈J,ℝ2.
(C3) There exist a functions p∈L∞J,ℝ and a continuous nondecreasing function ψ:0,∞→0,∞ such that
(75)ht,x,y≤ptψx+y,∀t,x∈J,ℝ.
(C4) Let ℓ1 and ℓ2 be equal
(76)ℓ1=8p−1φ2p∫0TGt,spsdsp,ℓ2=1+T/Γq+1,ℓ3=8p−1φ2pρ4p+4p−1φ1p,then,
(77)ℓ1ψℓ2lntp+ℓ3<1,∀t>1.Theorem 19.
IfC1−C4 hold true, then the problem (1) has one solution in X.Proof.
First, we show thatT is θL−contraction on X.
(78)σbTx,Ty=supt∈JTxt+Tytp≤supt∈Jgt,xtψ1+TΓq+1σbx,y∫0TGt,spsds+ρ4t+ft,xt+gt,ytψ1+TΓq+1σbx,y∫0TGt,spsds+ρ4t+ft,ytp≤supt∈Jφ2txt+ytψ1+TΓq+1σbx,y∫0TGt,spsds+ρ4t+φ1txt+ytp≤2p−12p−1φ2pσbx,yψ1+TΓq+1σbx,yp∫0TGt,spsdsp+ρ4p+φ1pσbx,y≤2p−12p−1φ2pψ1+TΓq+1σbx,yp∫0TGt,spsdsp+ρ4p+φ1pσbx,y.
Now, we define the functionsϕ:1,∞→0,1, θ∈Θ, and ξ∈Las
(79)ϕt=ℓ1ψℓ2lntp+ℓ3,
θt=et,
ξs,t=tϕt/s.
Consequently, we obtain(80)αx,yeσbTx,Ty≤eσbx,yϕeσbx,y1≤ξαx,yθσbTx,Ty,θσbx,y,
whereαx,y=2p−1,∀x,y∈X. Also, other hypothesis of Theorem 15 is satisfied. Hence, T has a unique fixed point, and then, FHDE (1) has on solution in X. This completes the proof.Now, we present an example to support our results.Example 20.
Let us consider the following FHDE with five-point boundary conditions.(81)cD3/2xt−ft,xtgt,xt=ht,xt,I5/2xt,t∈J=0,1,xt−ft,xtgt,xtt=0+12xt−ft,xtgt,xtt=1=12,∑i=35ζicD1/2xt−ft,xtgt,xtt=ηi+14cD1/2xt−ft,xtgt,xtt=1=1,
whereν>0. To show the existence of a unique solution of (81), we apply Theorem 19 with
(82)α=32,β=12,q=52,ζ1=1,ζ2=12,ζ3=58,ζ4=512,ζ5=516,ζ6=14,η3=425,η4=925,η5=1625,ft,x=1412x+x2+1−eνt,gt,x=14ln1+x2x,ht,x,y=cos∣x∣+∣y∣,∀t∈0,1andx,y∈ℝ.
By computation, we can show that the hypothesesC1−C4 are satisfied with
(83)p=1,φ1=φ2=14.
Therefore, we conclude that problem (81) has one solution.


## 5. Concluding Remarks and Observations

Our results extend the results of [10, 11] and many others. Indeed, we deal with a class of αqsp-admissible θL-contractions in a larger structure such as a b-metric-like space. Also, the problem we used as an application is different from that one appeared in [23] in the sense that we discussed not only existence but also uniqueness for our considered problem under multipoint boundary conditions via Caputo fractional derivatives.



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