# Hybrid Initial Value Problems of Fractional Order withq-Derivatives

Authors: Roja Mahmoudi Matankolae; Bahram Agheli; Sirous Ghobadi; Rahmat Darzi; Mohammad Adabitabar Firozja
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the present article, it has been tried to extend the theory ofq-fractional hybrid differential equations that entails Riemann–Liouville q-differential operators. We have also attempted to prove an existence theorem for FHDEs under Lipschitz and Caratheodory conditions. We have also presented some fundamental fractional differential inequalities that can be employed to indicate that some external solutions exist for such problems. The required tools are taken into consideration and the principle for comparison is offered which can be used to conduct further research work regarding qualitative behavior of solutions.


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## Body

## 1. Introduction

A long history is often attributed to Fractional calculus that can trace back to the emergence of the classical calculus. Even though a number of research have been accomplished during the last few decades, only recently, this novel trend in calculus plus dynamic equation have received more attention. There are several categories of noninteger derivative, yet the most commonly used definitions include Caputo derivatives and Riemann–Liouville derivatives. While the first one has an abstract mathematical nature, the second one is most extensively employed among engineers.Investigations into differential equations featuring fractional derivative has appreciably improved within current years, the fact that signifies the crucial importance of the calculus featuring fractional derivative and fractional integral in engineering, sciences, and technologies. There are some more books related to fractional calculus for interested readers [1, 2].During the last decade, having the solution of liner initial FDEs regarding special functions was explored in such sorts of problems, which often investigated solutions (or positive solution) applying Leray–Schoder theory and FPT [3, 4].To put it in other words, there have been the similar necessary conditions for boundary conditions and initial conditions [5, 6].In the last few years, researchers’ attention has often been drawn towards perturbations of various types on nonlinear differential equation in particular hybrid differential equation.Dhage and Lakshmikantham [7], for instance, took the next HDE under study comprising linear perturbations of first kind:(1)ddtxtmt,xt=ht,xt,h∈Cw×ℝ,ℝ,m∈Cw×ℝ,ℝ−0,xt0=x0∈ℝ.While employing a FPT in Banach algebra (BA), Zhao et al. [6] examined the existence of solution based on the noninteger derivative version of the above initial value problem (IVP), i.e.,(2)Dtαxtmt,xt=ht,xt,0<α<1,x0=0.The possibility of taking solutions for the following FHDE including supremum(3)Dtαxtmt,xt,max0≤t≤τxτ=ht,xt,x0=0,in which α∈0,1, h∈Cw×ℝ,ℝ, and m∈Cw×ℝ×ℝ,ℝ−0 was investigated by Caballero et al. [8], whose major tool was the approach of measures of noncompactness in the Banach space.In the present article, we will tackle with the HDE via fractionalq-derivatives mentioned below:(4)tDqαxtmt,xt,max0≤t≤τxτ=ht,xt,t∈0,1,(5)x0=0,in which tDqα is the Riemann–Liouville q-derivative which α∈0,1, m∈Cw×ℝ×ℝ,ℝ−0, w=0,1, for each x∈ℝ,h is measurable map and continuous for each t∈w.Through utilizing a FPT in BA due to Bai and Lü [5], with respect to mixed Lipschitz and Caratheodory condition (LCC) and by supposing the following hypotheses, we have been able to obtain the existence of solution to the FHDEs.Accordingly, a number of basicq-differential inequalities have been determined, which are employed to show the existence of minimal and maximal solutions. Following that, a number of comparison theorems have been proved.Hypotheses:(i)
The functiont⟶xt/mt,xt,max0≤τ≤txτ is increasing in ℝ almost everywhere for t∈0,1.(ii)
There exists a constantL>0 such that(6)mt,x1,x2−mt,y1,y2≤Lmaxx1−y1,x2−y2,for allt∈0,1 and x1,x2,y1,y2∈ℝ.(iii)
There exists a functiong∈L10,1,ℝ such that(7)ft,x≤gt,in whicht∈0,1, for all x∈ℝ.


## 2.q-Calculus

We now will give preliminaries ofq-calculus, definitions, and lemmas that will be used in the remainder of our article. The presentation here can be gained in [9].Letq∈0,1 and define(8)cq=1−qc1−q,c∈ℝ.Theq-analogue of the function c−dp featuring p∈ℕ0≔0,1,2,3,… can be described by(9)c−dn=∏k=0m−1c−dqk,m∈ℕ,d,c∈ℝ,in which c−d0=1.In more general terms, ifc≠0 and α∈ℝ, thus,(10)c−dα=cα∏k=0∞c−dqkc−dqα+k.It is obvious that, asd=0, therefore, cα=cα. The q-gamma function can be described via(11)Γqt=1−qt−11−qt−1,t∈−0,−1,−2,…,and satisfies Γqt+1=tqΓqt.Theq-derivative of a function m can be described wherein(12)Dqmt=mqt−rt1−qt,Dqm0=limt⟶0Dqmt,which is the q-derivative of higher order due to(13)Dqmt=mt,Dqnmt=DqDqn−1mt,n∈ℕ.Theq-integral of a function m, is resulted from(14)Iqmt=∫0tmsdqs=t1−q∑k=0∞mtqkqk,t∈0,d,which is defined in interval 0,d.Theq-integral of a function m, from c to d is resulted from(15)∫cdmsdqs=∫0dmsdqs−∫0cmsdqs,in which c∈0,d and m is defined on 0,d.In the same way as to that for derivatives, an operatorIqn for n∈ℕ is resulted from(16)Iqnmt=IqIqn−1mt,and for n=0, is Iq0mt=mt.The basic theorem of calculus is applicable in such operators asIq and Dq, i,e.,(17)Dqmt=mt,where m|t=0 is continuous, hence,(18)DqIqmt=mt−m0.The formulas mentioned below will be employed in the rest of the article, i.e., the integration by parts formula:(19)∫0tmsDqhsdqs=mshss=0s=t−∫0tDqhshqsdqs,and(20)ct−xα=cαt−xα,(21)tDqt−xα=αqt−xα−1,(22)sDqt−sα=−αqt−qxα−1,(23)tDq∫0tmt,sdqst=∫0tDtqmt,sdqs+mqt,t,in which tDq stands for the derivative.Definition 1.
The fractionalq-integral of Riemann–Liouville of order α and t∈0,1 in which m is a function on 0,1 for α>0, is described by(24)Iqαmt=1Γqα∫0tt−qsα−1msdqs,and Iq0mt=mt, for α=0.Definition 2.
The fractionalq-derivative of Riemann–Liouville of order α≥0 is described via Dq0mt=mt and(25)Dqαmt=DqαIqα−αmt,α>0.Lemma 1 (see [9]).
Assume thatc≤d≤t and α≥0, consequently, t−cα=t−dα.Lemma 2 (see [9]).
Assume thatm be a function which is defined on 0,1. Following that two properties can be stipulated:(1)
IqβIqαmt=Iqα+βmt,(2)
DqαIqαmt=mt,in which α,β≥0.Lemma 3 (see [9]).
Assume thatα∈ℝ+ and k∈Z+. Accordingly, the next equality holds:(26)IqαDqkgt=DqkIqαgt−∑p=0k−1tα−k+pΓqα+p−k+1Dqpg0.Lemma 4 (see [10]).
Letα∈ℝ+ and m:0,c⟶ℂ. If ∈ Lq10,c, then, Iqαm∈Lq10,c and Iqαm≤cαm1/Γqα+1.


## 3. Existence Results

In the present part of the article, it has been attempted to show the existence outcomes for theq-FHIVP equations (4) and (5) both on the bounded and closed on 0,1 with LCCs on the nonlinearities included in it. We have posited the q-FHIVP equations (4) and (5) in the space Cw,ℝ of continuous functions with real-value considered on w. In Cw,ℝ, a supremum norm. is defined based on(27)f=supt∈wft,and a multiplied in Cw,ℝ based on(28)fgt=ftgt,for f,g∈Cw,ℝ. Considering the above norm and multiplication in it, it is evident that Cw,ℝ is a BA. Based on L1w,ℝ denotes the space of Lebesgue which is an integrable function with real-value in interval w=0,1 related to the .L1 defined by(29)fL1=∫01fsds.What follows is considered a FPT in BA according to Dhage [11], which is a fundamental theorem that confirms our major result.Theorem 1 (see [11]).
Assume thatS be a closed, convex, nonempty, and bounded subcategory of the BA on T and assume L and H be two operators in a manner wherein(i)
L:T⟶T and H:S⟶T.(ii)
A is Lipschitz and γ is a Lipschitz constant.(iii)
B is complete and continuous.(iv)
x=LxHx⟹x∈S,∀x∈S,(v)
Bγ<1, in which B=Hs=supHx,x∈S. Meanwhile, for the operator equation LxHx=x has a solution in S.The lemma given below is useful for the following sections.Lemma 5.
Presume that hypothesisH0 holds. Thus, for α∈0,1 and any h∈L10,1, the function x∈C0,1,ℝ is a solution of the following IVP:(30)Dqαxtmt,xt,max0≤t≤τxτ=ht,(31)x0=0,if and only if x fulfills the following hybrid integral equation:(32)xt=φ∫0thst−qs1−αdqs,in which φ=mt,xt,max0≤t≤τxτ/Γqα.Proof.
see [6].Theorem 2.
Assume that hypothesesH0−H2 hold. Furthermore, if LhL1/Γqα+1<1, afterward for q-FHIVP equations (4) and (5) has a solution defined on 0,1.Proof.
SetG=C0,1×ℝ×ℝ,ℝ−0, and consider a subset:(33)S=g∈G,g≤kLTα,q1−LTα,q,where k=supmt,0,0 and is Tα,q=LhL1/Γqα+1.
It can certainly be claimed that,S is a convex, closed, and bounded subcategory of the Banach space G. With Lemma 5, equations (4) and (5) is equivalent to the equation below:(34)xt=mt,xt,max0≤t≤τxτ1Γqα∫0tt−qs1−αgs,xsdqs=mt,xt,max0≤t≤τxτIqαgs,xs.
Assume two operatorsL:T⟶T, H:S⟶T by Lxt=mt,xt,max0≤τ≤txτ and Hxt=Iqαgs,xs. Then, equation (34) can be converted to the operator equation as LxtHxt=xt, t∈w.
In [6], we showed which of the two operators L and H fulfill all the requirements of Theorem 1 and consequently, for q-FHIVP equations (4) and (5),1.1−1.2 we can get to a solution defined on 0,1.


## 4. An Illustrative Example

Now, assume the FHIVP given below:(35)Dqαxt1/2+1/4tan−1max0≤τ≤txτ=cosx,x0=0.Here, α=q=1/2,ht,xt=cosx,mt,xt,max0≤τ≤txτ=1/2+1/4tan−1max0≤τ≤txτ and ht=1. We have(36)mt,x1,y1−mt,x2,y2=≤14maxx1−x2,y1−y2.Sincetan−1tt,t0. Also, ht,xt=ht=cost≤1.Then, hypothesisH1 and H2 hold also for q=α=1/2, we have(37)Γqα+1=1−1/21/21−1/21/2=0.93201317,M≤hL1Γqα+1=1410.93201317=1.07294529,Mγ≤141.53647204.Therefore, theq-FHIVP equation (35)4.1 has a solution.


## 5. Fractional Hybridq-Differential Inequalities

In the next section, we dissert a fundamental relevance to strict inequalities for the hybrid differential equation with fractionalq-derivative.Lemma 6.
Presume thatm:ℝ+⟶ℝ be a locally Holder continuous in a way which for any t1∈0,+∞, the result for 0≤t≤b is(38)mt≤0andmb=0.Then, it follows that(39)Dqαmb≥0.Proof.
We know that(40)Dqαmt=1Γq1−αddtq∫0tt−qsp−1msdqs,where p=1−α. Let Ht=∫0tt−qsp−1msdqs.
Forh>0, we have(41)Hb−Hb−h=∫0b−hb−qsp−1−b−h−qsp−1msdqs+∫b−hbb−qsp−1msdqs=I1+I2.
Sinceb−qsp−1−b−h−qsp−1<0, for 0≤s≤b−h and ms≤0 by hypothesis, we have I1≥0. For by,(42)Hb−Hb−h≥∫0b−hb−qsp−1msdqs=I2.
In as much asmt is a locally Holder continuous and mb, such that, for t1−h≤s≤b+h, there exists a constant kb>0:(43)−kbb−qsλ≤ms≤kbb−qsλ,where λ>0 is such that λ+p−1>0 and 0<λ<0. As a result, we get to(44)I2≥−kb∫b−hbb−qsp−λ+1dqs=kbΓqp+λhp+λ.
Hence,Hb−Hb−h−kb/Γqp+λhp+λ≥0 for small enough h>0.
Countingb−h=bq, we have b1−q=h.
Then,Hb−Hqb/1−qt1−kt1hp+λ−1/Γqp+λ≥0 for small enough h>0. Now, it is simple to show that Dqαmb=1/ΓqpDqHb>0 and we have the full proof.Theorem 3.
Presume that hypothesisH0 hold. Also, assume that there exists the function μ,γ:0,T⟶ℝ that is a locally Holder continuous in a manner that(45)Dqαμtmt,μt,max0≤τ≤tμτ≤ht,μt,t∈w,and(46)Dqαγtmt,γt,max0≤τ≤tγτ≥ht,γt,t∈w,in case of one of the inequalities being strict. Thence,(47)μ0<γ0,implies for all t∈0,1(48)μt<γt.Proof.
Assume that inequality equation (46) is strict. Take the claim is not true. Thus there exists a b∈w, b>0 in a way that μb=γb and μt<γt for 0≤t≤b.
DefineY=μt/mt,μt,max0≤τ≤tμτ and Z=γt/mt,γt,max0≤τ≤tγτ.
Then, we haveYb=Zb and by virtue of hypothesis H0, we take Yt<Zt for all 0≤t≤b. For 0≤t≤b, putting mt=Yt−Zt, for 0≤t≤b, we can gain mt≤0, and mb=0. Thence, by Lemma 6,(49)gb,μb≥DqαYb≥DqαZb>gt1,γb.
This repugn viaμb=γb. Accordingly, equation (48) is a valid conclusion and we have the full proof.
The result which comes next is related to strict fractionalq-differential inequalities that is regarded a sort of one way Lipschitz condition.Theorem 4.
Assume that the requirement of Theorem3 hold via q-inequalities equations (45) and (46). Presume that there exists J∈ℝ+ in such a way that for all θ,ϑ∈ℝ via θ≥ϑ.(50)ht,θ−ht,ϑ≤J1+tαθmt,θ,max0≤τ≤tθτ−ϑmt,ϑ,max0≤τ≤tϑτ,t∈w.Therefore,μ0≤ρ0 implies,(51)μt≤ρt,for allt∈0,1, provided J≤1/Γq1−α.Proof.
For smallε>0, we put(52)ρεtmt,ρεt,max0≤τ≤tρετ=ρtmt,ρt,max0≤τ≤tρτ+ε1+tα.
Thence, it follows that(53)ρεtmtρεt,max0≤τ≤tρετ>ztmt,ρt,max0≤τ≤tρτ⟹ρεt>ρt.
Letzεt=ρεt/mmax0≤t≤τρετ.
Fort∈0,1, so that(54)ρt=ρtmt,ρt,max0≤t≤τρτ.
Since(55)ht,ρε−ht,ρ≤J1+tαρεtmt,ρεt,max0≤τ≤tρετ−ρtmt,ρt,max0≤τ≤tρτ,for all t∈0,1 and J≤1/Γq1−α, one has(56)Dqαρεt=Dqαρt+εDqα1+tα≥ht,ρt+εDqα1+Dqαtα>ht,ρεt−ρεtmt,ρt,max0≤τ≤tρετ−ρtmt,ρt,max0≤τ≤tρτJ1+tα+εt−αΓq1−α+Γqα+1>ht,ρεt−εJ+ε1Γq1−α>ht,ρεt.
Likewise, we haveρε0>ρ0≥μ0. Consequently, applying Theorem 3 with ρ=ρε results in μt<ρεt for all t∈w. By assuming ε>0, considering the limits as ε⟶0, we have μt≤ρt for all t∈0,1, which gives us the full proof.Remark 1.
Lethx,t≡1 and gx,t=t. We can facilely determine that h and g fulfill the condition equation (50).


## 6. Existence of Minimal and Maximal Solutions

In the present part, the existence of minimal and maximal solutions is confirmed in order to the FHDE equation (30) in interval w=0,1. Accordingly, the definition below is required for the rest of the article.Definition 3.
Lett∈w. A solution r of the q-FHDE equation (20)2.1 be maximal, if γt≤rt, where on 0,1, γ is any solution of the q-FHDE equation (30).Definition 4.
Lett∈w. A solution r of the q-FHDE equation (20)2.1 be minimal, if rt≤γt, where on 0,1, γ is any solution of the q-FHDE equation (30).
We only tussle the condition of minimal solution, whereas the condition of maximal solution is equivalent and may be achieved via some arguments with some suitable variations, presuming a small numberε∈ℝ+.
Now assume the following IVP ofq-FHDE:(57)Dqαxtmt,xt,max0≤τ≤txτ=ht,xt+ε,t∈w,x0=0,in which m∈Cw×ℝ×ℝ,ℝ−0, 0<α<1 and h∈Cw×ℝ,ℝ.
Now, we give an existence theorem for theq-FHDE equation (71) given below:Theorem 5.
Assume that hypothesesH0−H2 hold. Presume that inequality in Theorem 2 holds. Therefore, for per small number ε∈ℝ+, the FHDE equation (57) has a solution defined on 0,1.Proof.
By a hypothesis as long as(58)Lh1Γqα+1<1,there exists an ε0∈ℝ+, in a way that(59)Lh1+εΓqα+1<1,for all 0<ε≤ε0. Now, the remaining part of the proof just like Theorem 4.Theorem 6.
Assume that hypothesesH0−H2 hold. Moreover, if the condition of Theorem 2 is valid, then, on 0,1 for the q-FHDE equations (4) and (5), we can find a maximal defined (definite) solution.Proof.
Assumeεn0∞ be a declining sequence of real numbers and positive in such a manner that limn⟶∞εn=0, in which ε0 is a real number and positive fulfilling the next inequality:(60)Lh1+ε0Γqα+1<1.
The numberε0 exists in view of inequality LhL1/Γqα+1<1. With the help of Theorem 5, therefore, there exists a solution rt,εn defined on 0,1 of the FHDE(61)Dqαxtmt,xt,max0≤τ≤txτ=gt,xt+εna.e.t∈w,x0=0.
Therefore, achieving (obtaining) (attaining) any solution of theq-FHDE equations (4) and (5) fulfills(62)Dqαxtmt,xt,max0≤τ≤txτ≤ht,xt,and any solution of supportive problem equation (73) fulfills(63)Dqαzt,εnmt,zt,εn,max0≤t≤τzτ,εn≤ht,zt,εn+εn>gt,zt,εn,in which x0=0≤εn=z0,εn. With the help of Theorem 4, we deduce that(64)xt≤zt,εn,for all t∈0,1 and n∈ℕ0.
Sinceε2=z0,ε2≤z0,ε1=ε1, by Theorem 5, we deduce that zt,ε2≤zt,ε1. Hence, zt,εn is a declining sequence of real numbers and positive, the limit(65)zt=limn⟶∞zt,εn,exists. We can indicate that the convergence in equation (65) is uniform on w. To put it to an end, what we need is to demonstrate that the sequence zt,εn is equicontinuous in cw,ℝ. Assume a,b∈w with a<b be arbitrary. Thus,(66)za,εn−zt2,εn=|ma,zt1,εn,max0≤τ≤azτ,εn1Γqα∫0aa−qsα−1gs,zs,εn+εndqs≤|ma,zt1,εn,max0≤τ≤azτ,εn1Γqα∫0aa−qsα−1gs,zs,εn+εndqs−mt2,zt2,εn,max0≤τ≤bzτ,εn1Γqα∫0ab−qsα−1gs,zs,εn+εndqs|+mb,zb,εn,max0≤τ≤bzτ,εn1Γqα∫abb−qsα−1gs,zs,εn+εndqs≤A1−A2h1+εnΓqα+1+Fh1+εnΓqα+1bα−aα−b−aα+b−aα,where F=supt,x,y∈w×−N×N×0,Nmt,xt,max0≤τ≤txτ. Because m is continuous on the compact set w×−N×N×0,N, it is uniformly continuous there. Hereupon, for all n∈ℕ,(67)ma,za,εn,max0≤τ≤azτ,εn−mb,rb,εn,max0≤τ≤bzτ,εn⟶0,as a⟶b uniformly.
Accordingly, regarding the inequality mentioned above, it can be concluded thatza,εn−zb,εn⟶0 as a⟶b uniformly for all n∈ℕ. Afterwards, for all t∈0,1zt,εn⟶zt as n⟶∞.
In later section, we indicate that the functionzt is a solution of q-FHDE equations (4) and (5) defined on 0,1. In as much as zt,εn is a solution of the q-FHDE equation (61), we will get to(68)zt,εn=mt,zt,εn,max0≤τ≤tzτ,εn×1Γqα∫0tt−qsα−1gs,zs,εn+εndqs,for all t∈w. Imagining the limit as n⟶∞ in the equation (48)5.6 leads to(69)zt=mt,zt,max0≤τ≤trτ1Γqα∫0tt−qsα−1gs,zsdqs,for all t∈0,1. Therefore, on 0,1, the function z is a solution of q-FHDE equations (4) and (5). Ultimately, from inequality equation (64), it can be concluded that xt≤zt for all t∈w. Thus, for the q-FHDE equations (4) and (5), we can get to a maximal solution on 0,1, which results in the full proof


## 7. Comparison Theorems

It can be argued that the major problem related to differential inequalities is to appraise a bound in order to attain the solution set for the differential inequality attributed to theq-FHDE equations (4) and (5).In the following part, we will try to confirm that the minimal and maximal solution function as bounds for the solution of the corresponding differential inequality to theq-FHDE equations (4) and (5) in interval w=0,1.Theorem 7.
Assume that hypothesesH0−H2 and condition LhL1/Γqα+1<1 are valid. Presume that a real number M>0 exists in such a manner that for t∈0,1,(70)ht,a−ht,b≤M1+tαamt,a,max0≤τ≤taτ−bmt,b,max0≤τ≤tbτ,for all a,b∈ℝ with a≥b, where M≤1/Γq1−α. Moreover, in case there exists a function, x∈Cw×ℝ in such a way that(71)Dqαxtmt,x,max0≤τ≤txτ≤ht,xt,x0≤0,in which t∈0,1. Therefore, for all t∈0,1,(72)xt≤γt,in which r is a maximal solution of the q-FHDE equations (4) and (5) on w.Proof.
Presumeε>0 be arbitrarily small. By Theorem 6, γt,ε there is a maximal solution of the q-FHDE equation (57), and that the limit(73)γt=limε⟶0rt,ε,is uniform on 0,1 and the function r is a maximal solution of the FHDE equation (20)2.1 on w. As a result, for t∈0,1, we will get to(74)Dqαγt,εmt,γt,εmax0≤τ≤tγτ,ε=ht,γt,ε+ε,γ0,ε=0.
From the above inequality for∈w, it follows that(75)Dqαγt,εmt,γt,εmax0≤τ≤tγτ,ε>ht,γt,ε,γ0,ε=0.
Accordingly, Theorem4 can be applied to inequalities equations (71) and (75), and one can get to the conclusion that xt<γt,ε for all t∈w. Hence, this in view of limit equation (73) suggests that inequality equation (72) holds true on w, which in consequence leads to make the proof full.Theorem 8.
Assume that hypothesesH0−H2 and condition LhL1/Γqα+1<1 hold. Suppose that a real number K>0 exists in such a manner that for t∈0,1,(76)ht,a−ht,b≤K1+tαamt,a,max0≤τ≤taτ−bmt,b,max0≤τ≤tbτ,for all a,b∈ℝ with a≥b, where K≤1/Γq1−α. Furthermore, if v∈Cw×ℝ exists in a way that(77)Dqαvtmt,vtmax0≤τ≤tvτ≥ht,vt,a.e.t∈wv0>0.Therefore,ϑt≤vt, which ϑ is a minimal solution of the q-FHDE equations (4) and (5) on 0,1.Theorem 9.
Assume that hypothesisH0−H2 and condition LhL1/Γqα+1<1 hold. Presuming that a real number θ>0 exists so that in order to t∈0,1, and for all a,b∈ℝ,(78)ht,a−ht,b≤θ1+tαamt,a,max0≤τ≤taτ−bmt,b,max0≤τ≤tbτ,with a≥b, where θ≤1/Γq1−α. If an identical zero function for t∈0,1 is the only solution of the differential equation,(79)Dqαmt=K1+tαmt,m0=0.Thus, theq-FHDE equations (4) and (5) has a unique solution on 0,1.Proof.
With the assumption that the Theorem2, equations (4) and (5) has a solution defined on 0,1. Assume that there are two solutions x1 and x2 of the q-FHDE equations (4) and (5) on 0,1 with x1>x2. Define a function m:w⟶ℝ, so,(80)ζt=x1tmt,x1,max0≤τ≤tx1τ−x2tmt,x2,max0≤τ≤tx2τ.
With regards to hypothesisH0, it can be concluded that mt>o. Accordingly, we get to(81)Dqαζt=Dqαx1tmt,x1,max0≤τ≤tu1τ−x2tmt,x2,max0≤τ≤tx2τ=gt,x1−gt,x2≤L1+tαx1tmt,x1,max0≤τ≤tx1τ−u2tmt,x2,max0≤τ≤tx2τ=L1+tαζt,for almost everywhere t∈0,1 and that ζ0=0.
Following that, we execute Theorem7 via ft,x≡1 to gain that ζt≤0 for t∈0,1, where an identical zero function is the only solution of differential equation (79). ζt≤0, which is contradictory to ζt>0. Hence, we can gain x1=x2. This completes the proof.


## 8. Conclusion

In this paper, we have developed the theory of the HDE via fractional Riemann–Liouvilleq-derivatives of order 0<α<1. Through utilizing a FPT in BA due to Bai and Lü [5], with respect to mixed Lipschitz and Caratheodory condition (LCC) and by supposing the above hypotheses, we have been able to obtain the existence of a solution to the FHDEs. We proved Lemma 6, which is the main tools to raise new existence results in q-calculus. Some fundamental fractional differential inequalities have been established which are used to prove the existence of extremal solutions. The necessary tools have been considered and the comparison principle has been proved, which will be useful for further study of qualitative behavior of solutions.



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