# On a Coupled System of Fractional Differential Equations via the Generalized Proportional Fractional Derivatives

Authors: M. I. Abbas; M. Ghaderi; Sh. Rezapour; S. T. M. Thabet
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This work investigates the existence and uniqueness of solutions for a coupled system of fractional differential equations with three-point generalized fractional integral boundary conditions within generalized proportional fractional derivatives of the Riemann-Liouville type. By using the Schauder and Banach fixed point theorems, we study the existence and uniqueness of solutions for the aforesaid system. Finally, we present an example to validate our theoretical outcomes.


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## Body

## 1. Introduction

The theory of fractional calculus has become an attractive area of research for mathematicians and physicians because of its fertile aspects in many applications in natural science [1, 2], engineering [3], and many other fields. Moreover, the fractional differential equations have been employed successfully in the modeling of many biological problems, for example, human liver [4], hepatitis B [5–7], mumps virus [8] and methanol detoxification in the human body [9], and other differential models in thermodynamic and physics such as thermostat [10], pantograph [11], diffusion-wave system [12], and dynamical systems [13]. For additional specifics about the theory of fractional calculus and applications, we suggest the books of Kilbas et al. [14], Podlubny [15], and Samko et al. [16]. During the last years, there have exhibited several concepts about fractional derivatives. Here, we point out the most famous kinds including Riemann-Liouville, Liouville-Caputo, generalized Caputo [17], and Hadamard derivatives [18]. This has lead researchers to numerous research papers concerning several fractional operators which were conducted that one can see, for example, in complex plain [19, 20], extended Riemann-Liouville [21], the Mittag-Leer type function [22], the q-derivative [23], the local fractional derivative [24], and in stability result [25–27].More recently, Jarad et al. [28] constructed a new generalized fractional derivative which is called the generalized proportional fractional derivative. This new fractional operator has the advantage of being well-behaved as it is considered to be a generalization of many of the previously known and widely used fractional operators such as Liouville-Caputo and Riemann-Liouville fractional operators. In detail, fractional differential equations with generalized proportional derivatives have seen significant contributions from an interested researcher. For instance, we refer to works of Abbas and Ragusa and Hristova and Abbas [29, 30] and Khaminsou et al. [31, 32], and the references existing therein.At the same time, coupled systems of differential equations of fractional order with different boundary conditions have been the focus of many mathematicians. The literature on the topic involves the existence, uniqueness, and stability results. Ahmad and Luca [33] studied a system of nonlinear Caputo fractional differential equations with coupled boundary conditions involving Riemann-Liouville fractional integrals. Baitiche et al. [34] discussed the existence and uniqueness of solutions to some nonlinear fractional differential equations involving the ψ-Caputo fractional derivative with multipoint boundary conditions. Mahmudov et al. [35] investigated existence and uniqueness results for a coupled system of Caputo fractional differential equations with integral boundary conditions.Some existing frameworks mentioned above encourage us to study the following coupled system of fractional differential equations:(1)DaRα,ρut=ψ1t,ut,vt,DaRβ,ρvt=ψ2t,ut,vt,t∈J≔a,b,equipped with the generalized fractional integral boundary conditions:(2)uδ1=0,ub=Iaγ1,ρuμ1,vδ2=0,vb=Iaγ2,ρuμ2,whereρ∈0,1, DaRα,ρ and DaRβ,ρ denote the generalized proportional fractional derivatives of Riemann-Liouville type of order α,β∈1,2, Iaγ1,ρ and Iaγ2,ρ denote the generalized proportional fractional integrals of order γ1,γ2∈0,1, and δ1,δ2,μ1,μ2∈a,b and ψ1,ψ2:J×ℝ×ℝ⟶ℝ are continuous functions. In the current work, we establish the existence and uniqueness of solutions of the coupled system (1) and (2) by means of Schauder’s and Banach’s fixed point theorems.To the best of our knowledge, there are no contributions considering a coupled system of generalized proportional fractional differential equations with generalized fractional integral boundary conditions.The paper structure is designed as follows: in Section2, we collect some essential definitions and lemmas relevant to the generalized proportional fractional derivatives and integrals; in Section 3, we establish the Green function associated with the linear issue of the coupled system (1) and (2), while in Section 4, we prove the main existence and uniqueness results in the current paper; in Section 5, an example is given to validate our theoretical outcomes.


## 2. Preliminaries

Here, we review some definitions of the generalized proportional fractional derivatives and integrals; see [28, 30, 36].Definition 1 (see [37]).
Takeρ∈0,1, let the functions ε0,ε1:0,1×ℝ⟶0,∞ be continuous such that for all t∈ℝ we have limρ⟶0+ε1ρ,t=1, limρ⟶0+ε0ρ,t=0, limρ⟶1−ε1ρ,t=0, limρ⟶1−ε0ρ,t=1, ε1ρ,t=0 for ρ∈0,1, and ε0ρ,t=0 for ρ∈0,1. Then, the amended conformable derivative of order ρ is defined by
(3)Dρυt=ε1ρ,tυt+ε0ρ,tυ′t.
The above amended conformable derivative (3) is said to be a proportional derivative (see [37]). When ε1ρ,t=1−ρ and ε0ρ,t=ρ, (3) takes the form
(4)Dρυt=1−ρυt+ρυ′t.Note that,limρ⟶0+Dρυt=υt and limρ⟶1−Dρυt=υ′t.Remark 2.
By using (4) for the function υt=et and any arbitrary order ρ, it can be easily concluded that Dρυt=et.Example 1.
Ifυt=sint, then Dρυt=1−ρsint+ρcost. One can find the graphs of Dρsint for different value of ρ=0.1,0.5,0.9,1, in Figure 1. As can be seen from Figure 1, in some points, the value of conformable derivative in this case is independent of ρ, and this can be one of the interesting properties of fractional calculus.Definition 3 (see [28, 30]).
Takeρ∈0,1, α≥0, we define the left generalized proportional fractional integral of the function υ∈L1J by Ia0,ρυt=υt and
(5)Iaα,ρυt=1ραΓα∫ateρ−1/ρt−st−sα−1υsds,t∈J.Definition 4 (see [28, 30]).
Takeρ∈0,1, α≥0, we define the left generalized Caputo-proportional fractional derivative of the function υ∈CnJ by DaC0,ρυt=υt and
(6)DaCα,ρυt=Ian−α,ρDn,ρυt=1ρn−αΓn−α∫ateρ−1/ρt−st−sn−α−1Dn,ρυsds,where n−1<α≤n,n∈ℕ, D1,ρυt=Dρυt=1−ρυt+ρυ′t, and Dn,ρυt=DρDρ⋯Dρ⏟ntimesυt.Definition 5 (see [28, 30]).
Takeρ∈0,1, α≥0, we define the left generalized proportional fractional derivative of Riemann-Liouville type of the function υ by Da0,ρυt=υt and
(7)DaRα,ρυt=Dn,ρIan−α,ρυt=Dn,ρρn−αΓn−α∫ateρ−1/ρt−st−sn−α−1υsds,where n−1<α≤n,n∈ℕ.Lemma 6 (see [36]).
Ifρ∈0,1,β>0, and α>0 with n−1<α≤n and υ∈L1J, we have the following statements:
(8)Iaα,ρeρ−1/ρττ−aβ−1t=ΓβραΓβ+αeρ−1/ρtt−aα+β−1,(9)DaRα,ρeρ−1/ρττ−aβ−1t=ραΓβΓβ−αeρ−1/ρtt−aβ−1−α,(10)Iaα,ρIaβ,ρυt=Iaβ,ρIaα,ρυt=Iaα+β,ρυt,(11)DaRη,ρIaα,ρυt=Iaα−η,ρυt,0<η<α,(12)DaRα,ρIaα,ρυt=υt,(13)Iaα,ρDaRα,ρυt=υt−∑k=1ndkeρ−1/ρt−at−aα−k,where dk=aIk−α,ρυa/ρα−kΓα−k+1.Theorem 7 (Schauder’s fixed point theorem) [38].
LetU be a closed, convex, and nonempty subset of a Banach space C; let H:U⟶U be a continuous mapping such that HU is a relatively compact subset of C. Then, H has at least one fixed point in U.Figure 1
The graph ofDρsint.


## 3. The Equivalent Integral Equations

LetC=CJ,ℝ be the Banach space of all continuous functions from J into ℝ with the norm
(14)uC=maxt∈Jut,andC=C×C be the product Banach space with the norm
(15)u,vC=uC+vC.Definition 8.
By a solution of the coupled system (1) and (2), we mean a coupled ordered pair of continuous functions u,v∈C that satisfy (1) and (2).Lemma 9.
Letρ∈0,1, Λ1Λ4=Λ2Λ3, and ω:J⟶ℝ. Then, the solution of
(16)DaRα,ρut=ωt,t∈J,α∈1,2,uδ1=0,ub=Iaγ1,ρuμ1,δ1,μ1∈a,b,γ1∈0,1,
is equivalent to the integral equation(17)ut=Iaα,ρωt+Λ6t−aα−1eρ−1/ρt−aΛ2Iaα,ρωb−Λ2Iaα+γ1,ρωμ1−Λ4Iaα,ρωδ1+Λ5t−aα−2eρ−1/ρt−aΛ1Iaα,ρωb−Λ1Iaα+γ1,ρωμ1−Λ3Iaα,ρωδ1,
where(18)Λ1=δ1−aα−1eρ−1/ρδ1−a,Λ2=δ1−aα−2eρ−1/ρδ1−a,Λ3=b−aα−1eρ−1/ρb−a−Γαργ1Γα+γ1μ1−aα+γ1−1eρ−1/ρμ1−a,Λ4=b−aα−2eρ−1/ρb−a−Γα−1ργ1Γα+γ1−1μ1−aα+γ1−2eρ−1/ρμ1−a,Λ5=Λ2Λ3−Λ1Λ4−1,andΛ6=Λ1Λ4−Λ2Λ3−1.Proof.
By applying the generalized fractional proportional integralIaα,ρ· to both sides of the first equation (16) and using (13) in Lemma 6, one has
(19)ut=Iaα,ρωt+d1eρ−1/ρt−at−aα−1+d2eρ−1/ρt−at−aα−2.
Using the boundary conditionuδ1=0 and (19), one has
(20)δ1−aα−1eρ−1/ρδ1−ad1+δ1−aα−2eρ−1/ρδ1−ad2=−Iaα,ρωδ1.
Using (18), the above equation becomes
(21)Λ1d1+Λ2d2=−Iaα,ρωδ1.
In the light of (8) and (10) in Lemma 6, the boundary condition ub=Iaγ1,ρuμ1 and (19) give
(22)Iaα,ρωb+d1eρ−1/ρb−ab−aα−1+d2eρ−1/ρb−ab−aα−2=Iaα+γ1,ρωμ1+d1Γαργ1Γα+γ1μ1−aα+γ1−1eρ−1/ρμ1−a+d2Γα−1ργ1Γα+γ1−1μ1−aα+γ1−2eρ−1/ρμ1−a.
Again, using (18), the above equation takes the form
(23)Λ3d1+Λ4d2=−Iaα,ρωb+Iaα+γ1,ρωμ1.
Therefore, by merging equations (21) and (23), using (18), we get
(24)d1=Λ6Λ2Iaα,ρωb−Λ2Iaα+γ1,ρωμ1−Λ4Iaα,ρωδ1,d2=Λ5Λ1Iaα,ρωb−Λ1Iaα+γ1,ρωμ1−Λ3Iaα,ρωδ1.
Thus, by inserting the values ofd1 and d2 in (19), we obtain (17). The proof is finished.By hint of Lemma9, the solution u,v∈C of the system (1) and (2) is given by
(25)ut=Iaα,ρψ1t,ut,vt+Λ6t−aα−1eρ−1/ρt−aΛ2Iaα,ρψ1b,ub,vb−Λ2Iaα+γ1,ρψ1μ1,uμ1,vμ1−Λ4Iaα,ρψ1δ1,uδ1,vδ1+Λ5t−aα−2eρ−1/ρt−aΛ1Iaα,ρψ1b,ub,vb−Λ1Iaα+γ1,ρψ1μ1,uμ1,vμ1−Λ3Iaα,ρψ1δ1,uδ1,vδ1,t∈J,vt=Iaβ,ρψ2t,ut,vt+Λ6′t−aβ−1eρ−1/ρt−aΛ2′Iaβ,ρψ2b,ub,vb−Λ2′Iaβ+γ2,ρψ2μ2,uμ2,vμ2−Λ4′Iaβ,ρψ2δ2,uδ2,vδ2+Λ5′t−aβ−2eρ−1/ρt−aΛ1′Iaβ,ρψ2b,ub,vb−Λ1′Iaβ+γ2,ρψ2μ2,uμ2,vμ2−Λ3′Iaβ,ρψ2δ2,uδ2,vδ2,t∈J,where(26)Λ1′=δ2−aβ−1eρ−1/ρδ2−a,Λ2′=δ2−aβ−2eρ−1/ρδ2−a,Λ3′=b−aβ−1eρ−1/ρb−a−Γβργ2Γβ+γ2μ2−aβ+γ2−1eρ−1/ρμ2−a,Λ4′=b−aβ−2eρ−1/ρb−a−Γβ−1ργ2Γβ+γ2−1μ2−aβ+γ2−2eρ−1/ρμ2−a,Λ5′=Λ2′Λ3′−Λ1′Λ4′−1,andΛ6′=Λ1′Λ4′−Λ2′Λ3′−1,Λ1′Λ4′=Λ2′Λ3′.


## 4. Existence and Uniqueness Results

Define the operatorH:C⟶C by
(27)Hu,vt=H1u,vtH2u,vt,where(28)H1u,vt=aIα,ρψ1t,ut,vt+Λ6t−aα−1eρ−1/ρt−aΛ2Iaα,ρψ1b,ub,vb−Λ2Iaα+γ1,ρψ1μ1,uμ1,vμ1−Λ4Iaα,ρψ1δ1,uδ1,vδ1+Λ5t−aα−2eρ−1/ρt−aΛ1Iaα,ρψ1b,ub,vb−Λ1Iaα+γ1,ρψ1μ1,uμ1,vμ1−Λ3Iaα,ρψ1δ1,uδ1,vδ1,t∈J,(29)H2u,vt=aIβ,ρψ2t,ut,vt+Λ6′t−aβ−1eρ−1/ρt−aΛ2′Iaβ,ρψ2b,ub,vb−Λ2′Iaβ+γ2,ρψ2μ2,uμ2,vμ2−Λ4′Iaβ,ρψ2δ2,uδ2,vδ2+Λ5′t−aβ−2eρ−1/ρt−aΛ1′Iaβ,ρψ2b,ub,vb−Λ1′Iaβ+γ2,ρψ2μ2,uμ2,vμ2−Λ3′Iaβ,ρψ2δ2,uδ2,vδ2,t∈J.According to Lemma9, the solution u,v∈C of the coupled system (1) and (2) conforms with the fixed point operator H.For fulfillment the main results, the following assumptions will be imposed.(A1) The functionsψ1,ψ2:J×ℝ×ℝ⟶ℝ are continuous(A2) There exist nonnegative constantsL1 and L2 such that
(30)ψit,u1,v1−ψit,u2,v2≤Liu1−u2+v1−v2,for each t∈J and ui,vi∈ℝ,i=1,2Further, we setψi∗=maxt∈Jψit,0,0,i=1,2.The following notations will be introduced:(31)Δ1≔b−aαραΓα+1+Λ6b−aα−1Λ2b−aαραΓα+1+Λ2μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ4δ1−aαραΓα+1+Λ5b−aα−2Λ1b−aαραΓα+1+Λ1μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ3δ1−aαραΓα+1,(32)Δ2≔b−aβρβΓβ+1+Λ6′b−aβ−1Λ2′b−aβρβΓβ+1+Λ2′μ2−aβ+γ2ρβ+γ2Γβ+γ2+1+Λ4′δ2−aβρβΓβ+1+Λ5′b−aβ−2Λ1′b−aβρβΓβ+1+Λ1′μ2−aβ+γ2ρβ+γ2Γβ+γ2+1+Λ3′δ2−aβρβΓβ+1.Theorem 10.
Assume that the assumptions (A1) and (A2) are satisfied. Then, the coupled system (1) and (2) has at least one solution on J.Proof.
Consider the operatorH:C⟶C as defined in (27). Let us introduce the ball
(33)ζℓ=u,v∈C:u,vC≤ℓ,
whereℓ is a positive real number such that
(34)ℓ≥Δ1ψ1∗+Δ2ψ2∗1−Δ1L1+Δ2L2,Δ1L1+Δ2L2<1.
It is obvious thatζℓ is a closed, bounded, and convex subset of the Banach space C. We shall show that H achieves the hypothesis of Schauder’s fixed point theorem in four steps.
Step 1.H maps bounded sets into bounded sets in C.
By virtue of (A2) and sinceeρ−1/ρt−a<1,∀t>a, then for each t∈J and u,v∈ζℓ, one has
(35)H1u,vt≤1ραΓα∫att−sα−1ψ1s,us,vs−ψ1s,0,0+ψ1s,0,0ds+Λ6t−aα−1Λ2ραΓα∫abb−sα−1ψ1s,us,vs−ψ1s,0,0+ψ1s,0,0ds+Λ2ρα+γ1Γα+γ1∫aμ1μ1−sα+γ1−1ψ1s,us,vs−ψ1s,0,0+ψ1s,0,0ds+Λ4ραΓα∫aδ1δ1−sα−1ψ1s,us,vs−ψ1s,0,0+ψ1s,0,0ds+Λ5t−aα−2Λ1ραΓα∫abb−sα−1ψ1s,us,vs−ψ1s,0,0+ψ1s,0,0ds+Λ1ρα+γ1Γα+γ1∫aμ1μ1−sα+γ1−1ψ1s,us,vs−ψ1s,0,0+ψ1s,0,0ds+Λ3ραΓα∫aδ1δ1−sα−1ψ1s,us,vs−ψ1s,0,0+ψ1s,0,0ds≤b−aαραΓα+1+Λ6b−aα−1Λ2b−aαραΓα+1+Λ2μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ4δ1−aαραΓα+1+Λ5b−aα−2Λ1b−aαραΓα+1+Λ1μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ3δ1−aαραΓα+1×L1uC+vC+ψ1∗.
Thus, by using (31), we get
(36)H1u,vC≤Δ1L1ℓ+ψ1∗.
Similarly, we obtain that(37)H2u,vC≤Δ2L2ℓ+ψ2∗,where Δ2 is defined in (32). Hence, we conclude that
(38)H1u,vC+H2u,vC≤Δ1L1+Δ2L2ℓ+Δ1ψ1∗+Δ2ψ2∗≤ℓ,which implies that H:ζℓ⟶ζℓ
Step 2.H is continuous.
In view of the assumption (A1), we conclude thatH1 and H2 are continuous on J. Thus, the operator H is also continuous
Step 3.Hζℓ is equicontinuous.
Setmaxt∈Jψit,ut,vt≔Mi<∞,i=1,2. For t1,t2∈J, with t1<t2 and u,v∈ζℓ, we have
(39)H1u,vt2−H1u,vt1≤1ραΓα∫at2eρ−1/ρt2−st2−sα−1ψ1s,us,vsds−∫at1eρ−1/ρt1−st1−sα−1ψ1s,us,vsds+Λ6eρ−1/ρt2−at2−aα−1−eρ−1/ρt1−at1−aα−1Λ2ραΓα∫abb−sα−1ψ1s,us,vsds+Λ2ρα+γ1Γα+γ1∫aμ1μ1−sα+γ1−1ψ1s,us,vsds+Λ4ραΓα∫aδ1δ1−sα−1ψ1s,us,vsds+Λ5eρ−1/ρt2−at2−aα−2−eρ−1/ρt1−at1−aα−2Λ1ραΓα∫abb−sα−1ψ1s,us,vsds+Λ1ρα+γ1Γα+γ1∫aμ1μ1−sα+γ1−1ψ1s,us,vsds+Λ3ραΓα∫aδ1δ1−sα−1ψ1s,us,vsds≤M1ραΓα∫at2eρ−1/ρt2−st2−sα−1−t1−sα−1ds+∫at2eρ−1/ρt2−s−eρ−1/ρt1−st1−sα−1ds+∫t1t2eρ−1/ρt1−st1−sα−1ds+Λ6M1eρ−1/ρt2−at2−aα−1−t1−aα−1+eρ−1/ρt2−a−eρ−1/ρt1−at1−aα−1×Λ2b−aαραΓα+1+Λ2μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ4δ1−aαραΓα+1+Λ5M1eρ−1/ρt2−at2−aα−2−t1−aα−2+eρ−1/ρt2−a−eρ−1/ρt1−at1−aα−2×Λ1b−aαραΓα+1+Λ1μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ3δ1−aαραΓα+1≤M1ραΓα∫at2t2−sα−1−t1−sα−1ds+∫at2ρ−1ρt2−t1eρ−1/ρξ1−st1−sα−1ds+∫t1t2t1−sα−1ds+Λ6M1t2−aα−1−t1−aα−1+ρ−1ρt2−t1eρ−1/ρξ2−st1−aα−1×Λ2b−aαραΓα+1+Λ2μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ4δ1−aαραΓα+1+Λ5M1t2−aα−2−t1−aα−2+ρ−1ρt2−t1eρ−1/ρξ2−st1−aα−2×Λ1b−aαραΓα+1+Λ1μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ3δ1−aαραΓα+1,where the mean value theorem is used on the function eρ−1/ρt with ξ1,ξ2∈t1,t2.
Thus, we get(40)H1u,vt2−H1u,vt1≤M1ραΓα+1t2−aα−t1−aα+t2−t1t1−aα+Λ6M1t2−aα−1−t1−aα−1+t2−t1t1−aα−1×Λ2b−aαραΓα+1+Λ2μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ4δ1−aαραΓα+1+Λ5M1t2−aα−2−t1−aα−2+t2−t1t1−aα−2×Λ1b−aαραΓα+1+Λ1μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ3δ1−aαραΓα+1.
In an identical way, we obtain that(41)H2u,vt2−H2u,vt1≤M2ρβΓβ+1t2−aβ−t1−aβ+t2−t1t1−aβ+Λ6′M2t2−aβ−1−t1−aβ−1+t2−t1t1−aβ−1×Λ2′b−aβρβΓβ+1+Λ2′μ2−aβ+γ2ρβ+γ2Γβ+γ2+1+Λ4′δ2−aβρβΓβ+1+Λ5′M1t2−aβ−2−t1−aβ−2+t2−t1t1−aβ−2×Λ1′b−aβρβΓβ+1+Λ1′μ2−aβ+γ2ρβ+γ2Γβ+γ2+1+Λ3′δ2−aβρβΓβ+1.
Ast2⟶t1, the R.H.S. of the last two inequalities ⟶0 independently of u,v∈ζℓ. As a consequence of Steps 1 to 3 and in the light of the Arzelá-Ascoli theorem, we conclude that the operator Hζℓ is relatively compact in C. Hence, in accordance with Schauder’s fixed point theorem (Theorem 7), the operator H has a fixed point and so the coupled system (1) and (2) possesses at least one solution on J. The proof is completed.Theorem 11.
Assume that the assumptions (A1) and (A2) are satisfied. IfΔ1L1+Δ2L2<1, then the coupled system (1) and (2) has a unique solution on J.Proof.
Consider the operatorH as defined in (27). We have to show that H is a contraction mapping.
For eacht∈J and u1,v1,u2,v2∈C, one has
(42)H1u1,v1t−H1u2,v2t≤1ραΓα∫att−sα−1ψ1s,u1s,v1s−ψ1s,u2s,v2sds+Λ6t−aα−1Λ2ραΓα∫abb−sα−1ψ1s,u1s,v1s−ψ1s,u2s,v2sds+Λ2ρα+γ1Γα+γ1∫aμ1μ1−sα+γ1−1ψ1s,u1s,v1s−ψ1s,u2s,v2sds+Λ4ραΓα∫aδ1δ1−sα−1ψ1s,u1s,v1s−ψ1s,u2s,v2sds+Λ5t−aα−2Λ1ραΓα∫abb−sα−1ψ1s,u1s,v1s−ψ1s,u2s,v2sds+Λ1ρα+γ1Γα+γ1∫aμ1μ1−sα+γ1−1ψ1s,u1s,v1s−ψ1s,u2s,v2sds+Λ3ραΓα∫aδ1δ1−sα−1ψ1s,u1s,v1s−ψ1s,u2s,v2sds≤b−aαραΓα+1+Λ6b−aα−1Λ2b−aαραΓα+1+Λ2μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ4δ1−aαραΓα+1+Λ5b−aα−2Λ1b−aαραΓα+1+Λ1μ1−aα+γ1ρα+γ1Γα+γ1+1+Λ3δ1−aαραΓα+1×L1u1−u2C+v1−v2C.
Thus, by (31), we get
(43)H1u1,v1−H1u2,v2C≤Δ1L1u1−u2C+v1−v2C.
In a similar way, using (32), we get
(44)H2u1,v1−H2u2,v2C≤Δ2L2u1−u2C+v1−v2C.
From (43) and (44), we get
(45)Hu1,v1−Hu2,v2C≤Δ1L1+Δ2L2u1,v1−u2,v2C.
By virtue of the conditionΔ1L1+Δ2L2<1, we conclude that H is a contraction mapping.
Hence, with the aid of Banach’s fixed point theorem, we deduce thatH has a unique fixed point, and so the coupled system (1) and (2) possesses a solution on J uniquely. The proof is finished.Example 2.
Consider the following coupled system of fractional differential equations(46)0+RD3/2,1/2ut=ψ1t,ut,vt,0+RD5/4,1/2vt=ψ2t,ut,vt,t∈0,1,
with the generalized fractional integral boundary conditions:(47)u13=0,u1=aI1/7,1/2u15,v16=0,v1=aI1/9,1/2u18.Here,α=3/2,β=5/4,ρ=1/2,γ1=1/7,γ2=1/9,δ1=1/3,δ2=1/6,μ1=1/5,μ2=1/8, and a,b=0,1. Set ψ1t,u,v=1/25t2+2sin2u+v/1+v and ψ2t,u,v=1/100t2+u+cosv that their graphs show in Figures 2 and 3.Figure 2
The graph ofψ1t,u,v.Figure 3
The graph ofψ2t,u,v.For eacht∈0,1 and u1,v1,u2,v2∈C, one has
(48)ψ1t,u1,v1−ψ1t,u2,v2≤150u1−u2+v1−v2,(49)ψ2t,u1,v1−ψ2t,u2,v2≤1100u1−u2+v1−v2,which implies that the assumption (A2) holds true withL1=1/50 and L2=1/100. We calculate functions in (18), (26), (31), and (32) for ρ=1/4,ρ=1/2, and ρ=3/4 and present their numerical results in Table 1. We have in all three cases:
(50)L1Δ1+L2Δ2<1.Table 1
Numerical results for some functions in Example2.
ρ=14ρ=12ρ=34Λ10.21240.41370.5166Λ20.63721.24111.5499Λ30.03940.05120.0865Λ4-1.2107-1.6677-1.2144Λ50.96801.32721.6257Λ6-0.9680-1.3272-1.6257Λ1′0.32150.54090.8321Λ2′2.78413.24513.9521Λ3′-0.0612-0.09021.5934Λ4′-4.1284-4.9254-5.6418Λ5′0.29960.42170.7294Λ6′-0.2996-0.4217-0.7294Δ16.74518.064510.0121Δ22.74233.12344.0096L1Δ1+L2Δ20.16230.22250.2403By virtue of the above discussion, we infer that all the assumptions of Theorems10 and 11 are satisfied. Consequently, we deduce that the coupled system (46) and (47) has a solution on 0,1 uniquely.


## 5. Conclusion

As you know, there are many events in nature which we know nothing about those. One of the best ways for better understanding these types of phenomena is studying new notions in the fractional calculus field. In this work, we investigated the existence and uniqueness of solutions for a coupled system of fractional differential equations with three-point generalized fractional integral boundary conditions in the frame of the generalized proportional fractional derivatives of the Riemann-Liouville type which was introduced in 2017 by Jarad et al. In this way, we provided some results under some conditions. To better explain the notion, we gave some figures of some functions. Finally, we provided an illustrated example for our main result.



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*Source: 4779213-2022-07-14.xml*