# On Impulsive Boundary Value Problem with Riemann-Liouville Fractional Order Derivative

Authors: Zareen A. Khan; Rozi Gul; Kamal Shah
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Our manuscript is devoted to investigating a class of impulsive boundary value problems under the concept of the Riemann-Liouville fractional order derivative. The subject problem is of implicit type. We develop some adequate conditions for the existence and puniness of a solution to the proposed problem. For our required results, we utilize the classical fixed point theorems from Banach and Scheafer. It is to be noted that the impulsive boundary value problem under the fractional order derivative of the Riemann-Liouville type has been very rarely considered in literature. Finally, to demonstrate the obtained results, we provide some pertinent examples.


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## 1. Introduction

The fractional order differential equations (abbreviated as FODEs) are the generalization of the ordinary differential equations of the integer order. In the 17th century (1665), the great mathematicians Newton, Leibnitz and L’Hospital introduced for the first time the idea of fractional order differential equations (FODEs). Later on, in 1823, another mathematician by the name of Lacroix, introduced the fractional derivative [1] of simple power function. Furthermore, this area has been studied by many researchers because it has significant applications in various fields of science and technology in mathematical modeling of different fields of Science and Technology. For instance, some phenomena including the diffusion process [2], some chemical processes of electrochemistry [3], infectious disease in biology [4], signal and image processing [5], dynamic processes [6], and systems control theory [7] can be excellently described by using FODEs instead of the ordinary derivative. For further applications of FODEs, we refer to [8–13] and the references therein.On the other hand, an interesting and important branch recently got warm attention known as impulsive differential equations (IDEs). In recent times, the said area has been increasingly used to model many physical and social phenomena in social sciences in a very interesting way. Currently in the said area, significant contribution has been done by various researchers like Simeonov and Bainov [14], Benchohra et al. [15], Lakshmikantham et al. [16], and Samoilenko and Perestyuk [17]. Benchohra and Slimani [18] has initiated the study of FODEs under impulsive conditions by using fractional derivatives of the Caputo and Riemann-Liouville type with order α∈0,1. In addition, some applications of IDEs have been studied in various scientific disciplines such as biology, geography, engineering, dynamics, physics, geology, and management sciences. In terms of the important applications of IDEs, due to important applications of IDEs this field has a lot of significance and concentration (see [19–22]). For general research and significance, we refer some more important publications [23–26]. Many researchers have recently studied nonlinear FODEs with different kinds of boundary and initial conditions. Boundary value problems have significant applications in various fields of dynamics and fluid mechanics as well as engineering disciplines. Here, it is remarkable that problems under integral boundary conditions have some important applications in fluid mechanics, chemical engineering, thermoelasticity, flow of groundwater, population dynamics, and more (see [27–29]). Furthermore, we also refer some significance of FODEs under integral boundary conditions as discussed in [27–31].Recently, due to increasing applications of FODEs to model real world problems more comprehensively, researchers are taking keen interest to investigate different areas of fractional calculus. In particular, the use of FODEs in mathematical modeling of infectious diseases and other biological phenomena have got more attention. Various researchers have studied the fractional predator-prey pathogen model, then-predator-prey model with herd behavior, etc. (see [32–34]). Further, there is also modeling of the interaction between tumor growth and the immune system, edge-detecting techniques, an infectious diesease on a prepredator model, etc., (we refer to [35–40]).As stated earlier, the area devoted to IDEs with a fractional order has many applications. These differential equations can be modeled to those evolutionary processes which are subjected to abrupt changes in their states. Recently, some authors have used IDEs for the mathematical modeling of certain biological events. It is remarkable that impulsive differential equations are using in mathematical models which give rise to some important dual-layered impulsive systems. The said systems will open new doors in the future to develop a general mathematical theory for the said systems. For instance, the author of [41] has obtained very interesting results in this regard for various kinds of biological models of infectious diseases. Here, we remark that a very basic and important qualitative problem in the investigation of IDEs with a fractional order concerns the existence theory of solutions. For these purposes, researchers have used the classical fixed point theory and some tools of nonlinear analysis. For instance, in [42], the authors have applied fixed point results to develop the corresponding existence theory of solutions by using the Caputo derivative of the fractional order. In the same line, in [43], the authors have used the Picard-type analysis to investigate the stochastic-type IDEs of a fractional order by using the Caputo operator. In all these papers, the Caputo operator has been increasingly used. It is to be noted that the fractional order derivative of the Riemann-Liouville type has been very rarely used in IDEs.Authors [44] have established existence theory for fractional order IDEs with initial conditions by using the fixed point theory. The authors in [45] investigated the following problem of IDEs under the fractional order derivative of the Riemann-Liouville type as
(1)DRLαkrz=fz,rz,z∈0,T,z≠zm,1<αm≤2,z∈J,z≠zm,Δrzm=ψmrzm,m=1,2,3,⋯,q,Δ∗rzm=ψm∗rzm,m=1,2,3,⋯,q,r0=0,Dα0−1r0=β,where β∈R and f:0,T×R⟶R is a continuous function. They developed sufficient conditions for the existence of at least one solution to the considered problem by using a fixed point approach.Motivated from the said work given in (1), we are interested in studying a class of nonlinear implicit fractional order IDEs under the Riemann-Liouville derivative with the Riemann-Liouville-type integral boundary conditions as
(2)DRLαkrz=fz,rz,DRLαrz,1<α≤2,z∈J,z≠zm,Δrzm=ψmrzm,m=1,2,3,⋯,q,Δ∗rzm=ψm∗rzm,m=1,2,3,⋯,q,I1−αr0=0,I2−αr1=0,where DRL is denoted as the Riemann-Liouville fractional order derivative, J=0,1, f:J×R×R⟶R is a continuous function. Furthermore, ψm,ψm∗:R⟶R are continuous functions for m=1,2,⋯,q and Δrzm=Izm1−αrzm+−Izm1−αrzm−, Δ∗rzm=Izm2−αrzm+−Izm2−αrzm− with rzm+=limh⟶0+rzm+h, rzm−=limh⟶0−rzm+h, m=1,2,.⋯,q, for 0=z0<z1<z2⋯<zq+1=1. And also, where I1−α, I2−α are denoted as the Riemann-Liouville integral of fractional order 1−α<0, 2−α>0 on J, respectively. To establish the required results, we utilize the Scheafer fixed point theorem to investigate sufficient conditions for the existence of at least one solution to the problem under consideration (2). Furthermore, the criterion of uniqueness is derived by using the Banach contraction theorem. For the demonstration of our results, we provide some concrete examples.


## 2. Preliminaries

In this section, we provide some important results, basic definitions, and lemmas from the literature of fractional calculus [1, 3, 10, 11], which are needed in this manuscript.LetJ=0,1, J0=z0,z1, and Jm=zm,zm+1 for m=1,2,3⋯q. Suppose that PCJ,R=r:J⟶R;r∈Czm,zm+1,R,m=0,1,2,⋯,q+1 and rzm+ and rzm− exist with rzm−=rzm,m=1,2,⋯,q. Note that PCJ,R is a Banach space of piece-wise continuous function with norm r=supz∈Jrz.Definition 1.
The integral of the Riemann-Liouville fractional orderβ>0 of a continuous function f:0,∞⟶R is defined by
(3)Iβfz=∫0zz−sβ−1Γβfsds,z∈0,1.
Therefore, the right side is point-wise defined on0,∞, where Γ is the symbol of gamma function defined as Γβ=∫0∞e−ssβ−1ds.Definition 2.
The derivative of the Riemann-Liouville fractional orderβ>0, of a continuous function f∈C0,1∩L0,1, b>0 is defined by
(4)DRLβfz=ddzn∫0zz−sn−β−1Γn−βfsds,n−1<β<n,z∈0,1,where n=β+1, β represent the whole part of the real number β; therefore, the right side is point-wise defined on 0,∞.Definition 3.
If the functionf:c,d⟶R is at least n-times differentiable, then the Caputo fractional derivative of order β>0 is defined as
(5)DCcβfz=∫czz−sn−β−1Γn−βfnsds,n−1<β<n,z∈c,d,where n=β+1.Lemma 4 [11].
Supposeβ>0, and r∈Cb,d∩Lb,d. Then, FODE
(6)DRLβrz=0,has a unique solution given by
(7)rz=d1z−bβ−1+d2z−bβ−2+⋯+dnz−bβ−n,where di∈R, i=1,2,⋯,n, and n−1<β<n.Lemma 5 [11].
In particular,β>0 and r∈C0,1∩L0,1. We have
(8)IβDβrz=rz+a1zβ−1+a2zβ−2+⋯+anzβ−n,where ak∈R, k=1,2,⋯,n, and n−1<β<n.


## 3. Main Works

To convert our considered problem in to an impulsive fractional integral equation, the given Lemma is provided.Lemma 6.
The solution of the given linear IDE of the fractional order with the Riemann-Liouville derivative(9)DRLαrz=σz,1<α≤2,z∈J,z≠zm,Δrzm=ψmrzm,m=1,2,3,⋯,q,Δ∗rzm=ψm∗rzm,m=1,2,3,⋯q,I1−αr0=0,I2−αr1=0,is given by
(10)rz=∫0zz−sα−1Γασsds+zm∫zm11−s2σsds+1−zm∫0zmzm−sσsds−ψmrzm+∫0zmzm−s2σsds−ψm∗rzmzα−2,z∊0,z1,∫zmzz−sα−1Γασsds−z−zmα−1+z−zmα−1+zmz−zmα−21−zm×∫zm−1zmzm−sσsds−ψmrzm−z−zmα−2+z−zmα−1+zmz−zmα−2×∫zm−1zmzm−s2σsds−ψm∗rzm−z−zmα−1+zmz−zmα−2∫zm11−s2σsds,z∊zm,zm+1.Proof.
Supposerz is a solution to Problem (9); then, taking the Riemann-Liouville integral on both sides to using Lemma 5, there exist some constants c0,c1∈R such that
(11)rz=∫0zz−sα−1Γασsds−c0zα−1−c1zα−2,z∈0,z1.
Again taking the Riemann-Liouville integral to using Lemma (9), for some constant d0,d1∈R, we have
(12)rz=∫z1zz−sα−1Γασsds−d0z−z1α−1−d1z−z1α−2,z∈z1,z2.
Now, by using the impulsive conditions, we haveΔrz1=Iz11−αrz1+−Iz11−αrz1−=ψ1rz1 and Δ∗rz1=Iz12−αrz1+−Iz12−αrz1−=ψ1∗rz1, and we find that
(13)−d0=∫0z1z1−sσsds−c0−c1z1−1+ψ1rz1,−d1=∫0z1z1−s2σsds−c0z1−c1+ψ1∗rz1.
Thus, putting the values in (12), we have
(14)rz=∫z1zz−sα−1Γασsds−z−z1α−1∫0z1z1−sσsds−ψ1rz1−z−z1α−2∫0z1z1−s2σsds−ψ1∗rz1−c0z−z1α−1+z1z−z1α−2−c1z1−1z−z1α−1+z−z1α−2,z∈z1,z2.
The above process can be repeated in this way until we obtain the solutionrz for z∈zm,zm+1 as
(15)rz=∫zmzz−sα−1Γασsds−z−zmα−1∫zm−1zmzm−sσsds−ψmrzm−z−zmα−2∫zm−1zmzm−s2σsds−ψm∗rzm−c0z−zmα−1+zmz−zmα−2−c1zm−1z−zmα−1+z−zmα−2,z∈zm,zm+1.
Now, applying boundary conditionI1−αr0=0 and I2−αr1=0 to get the values of constant c0 and c1, we have
(16)c0=0,c1=zm∫zm11−s2σsds−zm1−zm∫0zmzm−sσsds−ψmrzm−zm∫0zmzm−s2σsds−ψm∗rzm.
The values ofc0,c1 putting in (11) and (15), one can obtain (10). On the contrary, suppose rz is a solution of the impulsive fractional integral equation (10). Following the direct calculation, we see that (10) satisfies the problem (9).For simplification, we use the following notations:(17)σ1=supz∈0,1z−zmα−1+z−zmα−1+zmz−zmα−21−zm,σ2=supz∈0,1z−zmα−2+z−zmα−1+zmz−zmα−2,σ3=supz∈0,1z−zmα−1+zmz−zmα−2.For the existence and uniqueness of the solution, we use some fixed point theorems. To transform the considered Problem (2) to a fixed point problem, we need to define the operator by M:PCJ,R⟶PCJ,R as
(18)Mrz=∫zmzz−sα−1Γασsds−z−zmα−1+z−zmα−1+zmz−zmα−21−zm×∫zm−1zmzm−sσsds−ψmrzm−z−zmα−2+z−zmα−1+zmz−zmα−2×∫zm−1zmzm−s2σsds−ψm∗rzm−z−zmα−1+zmz−zmα−2∫zm11−s2σsds.By using Lemma6 with rz=fz,rz,DRLαrz, Problem (2) is reduced to a fixed point problem Mrz=rz, where the operator M is given by (18). Therefore, Problem (2) has a solution if and only if operator M has a fixed point, where τz=fz,rz,τz and τz=DRLαrz. The following hypotheses are satisfied:(H1) The function f:J×R×R⟶R is continuous(H2) There exist some constants K∗>0 and 0<L∗<1, such that
(19)fz,rz,τz−fz,r¯z,τ¯z≤K∗rz−r¯z+L∗τz−τ¯z,for any r,r¯∈PCJ,R, τ,τ¯∈R, and z∈J.(H3) There exists a constant N1∗>0, such that
(20)ψmrz−ψmr¯z≤N1∗rz−r¯z,for each r,r¯∈PCJ,R and m=1,2,3,⋯,q.(H4) There exists a constant N2∗>0, such that
(21)ψm∗rz−ψm∗r¯z≤N2∗rz−r¯z,for every r,r¯∈PCJ,R and m=1,2,3,⋯,q.We use Banach fixed point theorem to prove that problem (2) has unique solution.Theorem 7.
Under hypotheses (H1)–(H4) and if the following condition holds
(22)K∗1−L∗1Γα+1+σ12+σ23+σ33+σ1N1∗+σ2N2∗<1,then, there exists a unique solution for Problem (2) on J.Proof.
Letr,r¯∈PCJ,R for some z∈J, we have
(23)Mrz−Mr¯z≤∫zmzz−sα−1Γαfs,rs,τs−fs,r¯s,τ¯sds+z−zmα−1+z−zmα−1+zmz−zmα−21−zm×∫zm−1zmzm−sfs,rs,τs−fs,r¯s,τ¯sds+ψmrz−ψmr¯z+z−zmα−2+z−zmα−1+zmz−zmα−2×∫zm−1zmzm−s2fs,rs,τs−fs,r¯s,τ¯sds+ψm∗rz−ψm∗r¯z+z−zmα−1+zmz−zmα−2×∫zm11−s2fs,rs,τs−fs,r¯s,τ¯sds,which further gives
(24)Mrz−Mr¯z≤∫zmzz−sα−1Γατs−τ¯sds+z−zmα−1+z−zmα−1+zmz−zmα−21−zm×∫zm−1zmzm−sτs−τ¯s+ψmrz−ψmr¯z+z−zmα−2+z−zmα−1+zmz−zmα−2×∫zm−1zmzm−s2τs−τ¯sds+ψm∗rz−ψm∗r¯z+z−zmα−1+zmz−zmα−2∫zm11−s2τs−τ¯sds,where τ,τ¯∈CJ,R are given by
(25)τz=fz,rz,τz,τ¯z=fz,r¯z,τ¯z.
By using hypothesisH2, we have
(26)τz−τ¯z=fz,rz,τz−fz,r¯z,τ¯z≤K∗rz−r¯z+L∗τz−τ¯z.
Repeating this process, we get(27)τz−τ¯z≤K∗1−L∗rz−r¯z.
Therefore, for everyz∈J and from (24), using hypothesis H3, H4, and (27), one has
(28)Mrz−Mr¯z≤K∗1−L∗∫zmzz−sα−1Γαrs−r¯sds+z−zmα−1+z−zmα−1+zmz−zmα−21−zm×K∗1−L∗∫zm−1zmzm−srs−r¯sds+N1∗rz−r¯z+z−zmα−2+z−zmα−1+zmz−zmα−2×K∗1−L∗∫zm−1zmzm−s2rs−r¯sds+N2∗rz−r¯z+z−zmα−1+zmz−zmα−2K∗1−L∗∫zm11−s2rs−r¯sds.
Upon further simplification, (29) yields
(29)Mr−Mr¯≤K∗1−L∗z−zmαΓα+1r−r¯+σ1K∗1−L∗zm−zm−122r−r¯+N1∗r−r¯+σ2K∗1−L∗zm−zm−133r−r¯∣+N2∗r−r¯+σ3K∗1−L∗1−zm33∥r−r¯∥.
Hence, from (29), we have
(30)Mr−Mr¯≤K∗1−L∗1Γα+1+σ12+σ23+σ33+σ1N1∗+σ2N2∗r−r¯.
By (22), operator M is a contraction. Thus, according to Banach’s contraction principle, operator M has a unique fixed point which is the unique solution to Problem (2).Next, we will prove that Problem (2) has at least one solution for this, and we use Schaefer’s fixed point theorem. Let the given hypotheses hold true:(H5) There exist x,y,h∈PCJ,R, with
(31)x∗=supz∈0,1xz,y∗=supz∈0,1yz,h∗=supz∈0,1hz<1,such that
(32)fz,rz,τz≤xz+yzrz+hzτz,for z∈J, r∈PCJ,R, and τ∈R.(H6) The function ψm:PCJ,R⟶R is continuous, and there exist constants A1∗,B1∗>0 such that
(33)ψmrz≤A1∗rz+B1∗,for every r∈PCJ,R, m=1,⋯,q.(H7) The function ψm∗:PCJ,R⟶R is continuous, and there exist constants A2∗,B2∗>0 such that
(34)ψm∗rz≤A2∗rz+B2∗,for every r∈PCJ,R, m=1,⋯,q.Theorem 8.
If the hypotheses (H1), (H2), (H5)-(H7) are satisfied, then Problem (2) has at least one solution.Proof.
The proof is performed in several steps.(Step 1)
The operatorM is continuous
Assumern be a sequence such that rn⟶r on PCJ,R.
Forz∈J, we have
(35)Mrnz−Mrz≤∫zmzz−sα−1Γατns−τsds+z−zmα−1+z−zmα−1+zmz−zmα−21−zm×∫zm−1zmzm−sτns−τsds+ψmrnz−ψmrz+z−zmα−2+z−zmα−1+zmz−zmα−2×∫zm−1zmzm−s2τns−τsds+ψm∗rnz−ψm∗rz+z−zmα−1+zmz−zmα−2∫zm11−s2τns−τsds,where τnz, τz∈PCJ,R are given by
(36)τnz=fz,rnz,τnz,τz=fz,rz,τz.
Now, from assumptionH2, we have
(37)τnz−τz=fz,rnz,τnz−fz,rz,τz≤K∗rn−r+L∗τnz−τz.
Repeating this process, we get(38)τnz−τz≤K∗1−L∗rn−r.
Sincern⟶r, τnz⟶τz as n⟶∞ for every z∈J. We know that every convergent sequence is bounded, so for this, let ζ>0 such that for every z∈J, we have τnz≤ζ and ∣τz∣≤ζ. Then, we have
(39)z−sα−1τnz−τz≤z−sα−1τnz+τz≤2ζz−sα−1,zm−sα−1τnz−τz≤zm−sα−1τnz+τz≤2ζzm−sα−1,for every z∈J; the function s⟶2ζz−sα−1 and s⟶2ζzm−sα−1 are integrable on 0,1, upon the use of these facts and the Lebesque-dominated convergence theorem in (35). After using assumptions H5‐H7, we see that
(40)Mrnz−Mrz⟶0,asn⟶∞,and hence, we have
(41)Mrn−Mr⟶0,n⟶∞.
Therefore, operatorM is continuous.
(Step 2)
The operatorM assigns bounded sets to bounded sets on PCJ,R. Just prove it for any ξ∗>0, there exists a positive constant E∗, such that for every r∈D=r∈PCJ,R,r≤ξ∗, we have Mr≤E∗. To derive this result for each z∈J, one has(42)Mrz≤∫zmzz−sα−1Γατsds+z−zmα−1+z−zmα−1+zmz−zmα−21−zm×∫zm−1zmzm−sτsds+ψmrzm+z−zmα−2+z−zmα−1+zmz−zmα−2×∫zm−1zmzm−s2τsds+ψm∗rzm+z−zmα−1+zmz−zmα−2∫zm11−s2τsds,where τz∈CJ,R is given by
(43)τz=fz,rz,τz.
By hypothesis (H5) and for every z∈J, we have
(44)τz=fz,rz,τz≤xz+yzrz+hzτz≤xz+yzr+hzτz≤xz+yzξ∗+hzτz≤x∗+y∗ξ∗+h∗τz.
Thus, we have(45)τz≤x∗+y∗ξ∗1−h∗≔R∗.
Therefore, from (42) by using (45), one has
(46)Mrz≤R∗Γα+1+σ1R∗2+A1∗ξ∗+B1∗+σ2R∗3+A2∗ξ∗+B2∗+σ3R∗3.
Hence, one has(47)Mr≤R∗1Γα+1+σ12+σ23+σ33+A1∗+A2∗ξ∗+B1∗+B2∗≔E∗,Mr≤E∗.
Therefore, the operatorM is bounded.
(Step 3)
The operatorM assigns bounded sets to equicontinuous sets of PCJ,R. Let z1, z2∈J, and z1<z2. D is a bounded set as in Step 2, and let r∈D; then, we have(48)Mrz2−Mrz1≤∫zmz2z2−sα−1Γατsds−∫zmz1z1−sα−1Γατsds+z2−zmα−1+z2−zmα−1+zmz2−zmα−21−zm−z1−zmα−1+z1−zmα−1+zmz1−zmα−21−zm∫zm−1zmzm−sτsds+ψmrzm+z2−zmα−2+z2−zmα−1+zmz2−zmα−2−z1−zmα−2+z1−zmα−1+zmz1−zmα−2∫zm−1zmzm−s2τsds+ψm∗rzm+z2−zmα−1+zmz2−zmα−2−z1−zmα−1+zmz1−zmα−2∫zm11−s2τsds.
Using (45) and hypotheses H6 and H7 in (48), we obtain
(49)Mrz2−Mrz1≤R∗Γα+1z2−zmα−z1−zmα+R∗zm−zm−122+A1∗ξ∗+B1∗z2−zmα−1−z1−zmα−1zm+R∗zm−zm−122+A1∗ξ∗+B1∗z2−zmα−2−z1−zmα−2zm1−zm+R∗zm−zm−133+A2∗ξ∗+B2∗z2−zmα−2−z1−zmα−21+zm+R∗zm−zm−133+A2∗ξ∗+B2∗z2−zmα−1−z1−zmα−1+R∗1−zm33z2−zmα−1−z1−zmα−1+R∗1−zm33z2−zmα−2−z1−zmα−2zm.
Clearly, in the inequality (49), the right hand side tends to zero as z1⟶z2. Hence, Mrz2−Mrz1⟶0 as z1⟶z2. As a consequence of the passage from Step 1 to Step 3 combined with the Arzellá-Ascolli theorem, we conclude that M:PCJ,R⟶PCJ,R is completely continuous.
(Step 4)
In the last step, we need to show that the setF=rz∈PCJ,R:rz=μMrz,forsome0<μ<1 is bounded. Let r∈F; then, rz=μMrz for some 0<ì<1. Therefore, for every z∈J, we have(50)rz=μMrz=μ∫zmzz−sα−1Γατsds−μz−zmα−1+z−zmα−1+zmz−zmα−21−zm×∫zm−1zmzm−sτsds−ψmrzm−μz−zmα−2+z−zmα−1+zmz−zmα−2×∫zm−1zmzm−s2τsds−ψm∗rzm−μz−zmα−1+zmz−zmα−2∫zm11−s2τsds.
Now, we have(51)rz=μMrz≤∫zmzz−sα−1Γατsds+z−zmα−1+z−zmα−1+zmz−zmα−21−zm×∫zm−1zmzm−sτsds+ψmrzm+z−zmα−2+z−zmα−1+zmz−zmα−2×∫zm−1zmzm−s2τsds+ψm∗rzm+z−zmα−1+zmz−zmα−2∫zm11−s2∣τs∣ds.
Using (45) and hypotheses H6 and H7 in (51), we get
(52)rz≤R∗Γα+1+σ1R∗2+A1∗ξ∗+B1∗+σ2R∗3+A2∗ξ∗+B2∗+σ3R∗3.
Hence, one has from where(53)r≤R∗1Γα+1+σ12+σ23+σ33+A1∗+A2∗ξ∗+B1∗+B2∗≔Z∗,r≤Z∗.
Hence, the given setF is bounded as a result of the Schaefer fixed point theorem, and we conclude that operator M has at least one fixed point. Hence, the corresponding Problem (2) has at least one solution.


## 4. Examples

Here, we provide two pertinent examples to verify the previous results.Example 1.
Consider the following IDE under the Riemann-Liouville-type integral boundary condition and the Riemann-Liouville fractional order derivative(54)DRLαrz=z+cosrz+cosDRLαrz90+z2,1<α≤2,z∈0,1,z≠15,Δr15=ψ1r15=sinr1/530,Δ∗r15=ψ1∗r15=e−r1/515,I1−αr0=0,I2−αr1=0.where α=1/2, we set
(55)fz,rz,τz=z+cosrz+cosτz90+z2,rz∈PCJ,R,τz∈R,andz∈0,1.
Clearlyf is a jointly continuous function.
Now for everyrz, r¯z∈PCJ,R, τz, τ¯z∈R, and z∈0,1, we have
(56)fz,rz,τz−fz,r¯z,τ¯z=z+cosrz+cosτz90+z2−z+cosr¯z+cosτ¯z90+z2=cosrz−cosr¯z+cosτz−cosτ¯z90+z2≤cosrz−cosr¯z90+z2+cosτz−cosτ¯z90+z2,fz,rz,τz−fz,r¯z,τ¯z≤190r−r¯+τ−τ¯.
Which satisfies hypothesisH2 with K∗=L∗=1/90. Now, we set
(57)Δr15=ψ1r15=sinr1/530,r∈PCJ,R.
Then, forrz, r¯z∈PCJ,R, we have
(58)ψ1r15−ψ1r15=sinr1/530−sinr¯1/530≤130r−r¯.
Therefore, withN1∗=1/30, hypothesis H3 is satisfied. Next, we set
(59)Δ∗r15=ψ1∗r15=e−r1/515,r∈PCJ,R.
Then, forrz, r¯z∈PCJ,R, we have
(60)ψ1∗r15−ψ1∗r15=e−r1/515−e−r¯1/515≤115r−r¯.
Thus, withN2∗=1/15, hypothesis H4 is satisfied. Further, we need to satisfy the given condition of Theorem 7, by
(61)K∗1−L∗1Γα+1+σ12+σ23+σ33+σ1N1∗+σ2N2∗=1901Γα+1+3520+5530+2830+35300+55150<1.
Therefore, all the hypotheses and conditions of Theorem7 are satisfied. Therefore, the considered problem (54) has a unique solution on J.Example 2.
Consider another example of IDE under the Riemann-Liouville-type integral boundary condition and the Riemann-Liouville fractional order derivative:(62)DRLαrz=e−z+e−2zsinrz+sinDRLαrz35+z3,1<α≤2,z∈0,1,z≠17,Δr17=ψ1r17=tanr1/725+tanr1/7,Δ∗r17=ψ1∗r17=erz55+30erz,I1−αr0=0,I2−αr1=0.where α=3/2 and DRLαrz=τz. We set
(63)fz,rz,τz=e−z+e−2zsinrz+sinτz35+z3,rz∈PCJ,R,τz∈R,z∈0,1.
Clearlyf is a jointly continuous function.
Now for everyrz, r¯z∈PCJ,R, τz, τ¯z∈R, and z∈0,1, we have
(64)fz,rz,τz−fz,r¯z,τ¯z=e−z+e−2zsinrz+sinτz35+z3−e−z+e−2zsinr¯z+sinτ¯z35+z3≤e−2zsinrz−sinr¯z35+z3+sinτz−sinτ¯z35+z3≤e−2z35+z3rz−r¯z+135+z3τz−τ¯z,fz,rz,τz−fz,r¯z,τ¯z≤135r−r¯+τ−τ¯.
This satisfies hypothesis (H2), with K∗=L∗=1/35.
Now, another hypotheses for everyrz∈PCJ,R, τz∈R, and z∈0,1, we have
(65)fz,rz,τz=e−z+e−2zsinrz+sinτz35+z3≤e−z35+z3+e−2z35+z3rz+135+z3τz≤e−z35+z3+e−2z35+z3rz+135+z3τz.
Thus, hypothesis (H5) is satisfied with xz=e−z/35+z3, yz=e−2z/35+z3, and hz=1/35+z3. Now, we set
(66)Δr17=ψ1r17=tanr1/725+tanr1/7,r∈PCJ,R.
Then, for everyr∈PCJ,R, we have
(67)ψ1r17=tanr1/725+tanr1/7,≤125r+1.
Therefore, hypothesis (H6) is satisfied with A1∗=1/25 and B1∗=1. Next, we set
(68)Δ∗r17=ψ1∗r17=er1/755+30er1/7,r∈PCJ,R.
Then, for everyr∈PCJ,R, we have
(69)ψ1∗r17=er1/755+30er1/7,≤155r+130.
Thus, hypothesis (H7) is satisfied with A2∗=1/55 and B2∗=1/30. Therefore, all of the hypotheses of Theorem 8 are satisfied, and therefore, the considered problem (62) has at least one solution on J.


## 5. Conclusion

IDEs of the fractional order have received proper attention due to their important applications in various fields of applied sciences. In the past, most studies have been done using the Caputo-type fractional derivatives to handle IDEs. In very few papers, investigating IDEs of the fractional order was done using the Riemann-Liouville derivative. Therefore, we have established successfully some important results devoted to the existence theory of a solution to the considered nonlinear implicit IDE with the Riemann-Liouville-type integral boundary conditions under the Riemann-Liouville fractional order derivative. The corresponding results for the existence and uniqueness of the solution have been archived by utilizing the classical Schiefer and Banach contraction fixed point theorems. For the demonstration of our results, we have enriched the paper by providing two pertinent examples.



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*Source: 8331731-2021-09-30.xml*