# On System of Mixed Fractional Hybrid Differential Equations

Authors: Muath Awadalla; Nazim I. Mahmudov
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we find the necessary conditions for the existence and uniqueness of solutions to a system of hybrid equations that contain mixed fractional derivatives (Caputo and Riemann-Liouville). We also verify the stability of these solutions using the Ulam-Hyers (U-H) technique. Finally, this study ends with applied examples that show how to proceed and verify the conditions of our theoretical results.


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## Body

## 1. Introduction

Although the concept of fractional calculus was established 300 years ago, interest in this type of derivative appeared for a short period. So that it is no secret to anyone that the most important use of fractional derivatives is to find analytical solutions to differential equations if possible, or by using numerical analysis methods to find an approximation to these solutions. In this study, we will focus on the idea of studying theories that investigate the existence of a solution to a system of hybrid fractional equations that contain mixed fractional derivatives with boundary conditions attached to them.As mentioned before, fractional calculus as a concept is not very recent. It is worth mentioning here the great names who have given a lot to this science, such as A.V. Letnikov, J. Hadamard, J. Liouville, B. Riemann M., and Caputo L. worked in this field. These names must be mentioned by way of example. To get acquainted with some of the names of scientists who have made great contributions to fractional calculus in the modern world, we ask the reader to look at [1].Fractional derivatives have played a very important role in mathematical modeling in many diverse applied sciences, see [2, 3]. For example, the authors in [4] employed the fractional derivative of the Psi-Caputo type in modeling the logistic population equation, through which they were able to show that the model with the fractional derivative led to a better approximation of the variables than the classical model. In addition, the authors in [5] employed the fractional derivative of the Psi-Caputo type and used the kernel Rayleigh, to improve the model again in modeling the logistic population equation.As a final example, the authors in [6] employed the fractional derivatives of the Caputo and Caputo-Fabrizio type by modeling the equation that gives the relationship between atmospheric pressure and altitude, and they were also able to show that the fractional equation gave less error in estimating atmospheric pressure at a certain altitude. There are many scientific papers in the literature that prove the superiority of fractional derivatives over classical ones.There are a large number of manuscripts published in the literature that investigate the issue of the existence of a solution to fractional differential equations, whether they are sequential equations of type or nonsequential equations [7–14].In 2012, the authors in [7] studied a nonlinear three-point boundary value problem of sequential fractional differential equations. Green’s function of the associated problem involving the classical gamma function is obtained. Existence results are obtained using Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem.
(1)DCqD+λχτ=wτ,χτ,τ∈0,1q∈1,2,χ0=0,χ′0=0,χ1=δχη,η∈0,1.Here,D is the ordinary derivative, ψ:0,1×ℝ⟶ℝ,λ∈ℝ+,δ is a real number such that δ≠λ+e−λ−1/λη+e−λη−1.In 2019, Ahmad et al. [15] developed the existence theory for a new kind of nonlocal three-point boundary value problems for differential equations involving both Caputo and Riemann–Liouville fractional derivatives. The existence of solutions for the multivalued problem concerning the upper semicontinuous and Lipschitz cases is proved by applying nonlinear alternative for Kakutani maps and Covitz and Nadler fixed point theorem.
(2)CD1−qDRL0+pωτ=ϑτ,ωτ,1<q≤2,0<p≤1,ω0=ω′0=0,ω1=δωζ,whereφ:0,1×ℝ⟶ℝ,δ∈ℝ,ζ∈0,1.It is known that fractional calculus and FDEs are used in different fields such as physics, signal and image processing, control theory, robotics, economics, biology, and metallurgy, see for example [16, 17] and references therein. On the other hand, recently, many researchers have paid much attention to hybrid differential equations of fractional order. This is because of the development and new advanced applications of fractional calculus. The fractional hybrid modeling is of great significance in different engineering fields, and it can be a unique idea for future combined research between various applied sciences, for example, see [18] in which fractional hybrid modeling of a thermostat is simulated, for some recent results on hybrid.For FDEs, we refer to [19, 20]. Freshly, some authors have studied different characteristics of hybrid FDEs including the existence of solutions, see for some detail [21–29], and some go further and studied Hyers-Ulam stability for FDEs by different mathematical theories, see for some detail [26].Zhao et al. [29] investigated the existence result for the fractional hybrid differential equations with Riemann–Liouville fractional derivatives given by
(3)DRL0+rxtft,xt=gt,xt,t∈0,T,r∈0,1,x0=0,whereDRLr is Riemann–Liouville fractional derivative, f:0,T×ℝ⟶ℝ/0,g:0,1×ℝ⟶ℝ are assumed to be continuous.Hilal and Kajouni [30] studied the Caputo hybrid BVP of the form
(4)DC0+rxtft,xt=gt,xt,t∈0,L,r∈0,1,a1x0f0,x0+a2xLfL,xL=d,in whichf:0,T×ℝ⟶ℝ/0, g:0,1×ℝ⟶ℝ are assumed to be continuous and a1+a2≠0.In [31], the authors have considered the following coupled hybrid system. A new generalization of Darbo’s theorem associated with measures of noncompactness is the main tool in their approach:
(5)DCqχτℏτ,χτ,ϑτ=ψτ,χτ,ϑτ,τ∈0,10<q≤1,DRLpϑtƛτ,χτ,ϑτ=φτ,χτ,ϑτ,1<p≤2,supplemented with nonlocal hybrid boundary conditions.Inspired by the aforementioned studies, the following sequential hybrid BVP is considered for investigating the existence of the solution and for the stability of its solution via the U-H sense(6)DC1−qDRL0+rχτℏτ,χτ,ϑτ=ψτ,χτ,ϑτ,τ∈0,11<q≤2,0<r≤1,DC1−qRLD0+pϑtƛτ,χτ,ϑτ=φτ,χτ,ϑτ,0<p≤1,χ0=χ′0=0,χ1=δχζ,δ∈ℝ,ζ∈0,1,ϑ0=ϑ′0=0,ϑ1=εϑξ,ε∈ℝ,ξ∈0,1.After this introductory section of this work, the manuscript is organized as the following hierarchical structure: Section 2 delivers the basic elements of fractional calculus definitions, Section 3 introduces the main results of the work, Section 4 introduces the (U-H) stability result for our problem, and the last section is arranged for a numerical example to support the theoretical results.


## 2. Preliminaries

In this part, we present some basic elements and definitions needed to find solutions to the main mathematical problem presented in this study.Definition 1 (see [3]).
The Riemann-Liouville (RL) fractional integral is defined by(7)IRL0+δϑϖ≔1Γδ∫0ϖϖ−tδ−1ϑtdt,ϖ>0,Reδ>0.Definition 2 (see [3]).
The Caputo fractional derivative of orderν of a function ϑ:ℝ+⟶ℝ is given by
(8)DC0+νϑτ=∫0ττ−zp−ν−1ϑpzΓp−νdz,p−1<ν<p,p=ν+1.Theorem 3 (see [3], Banach’s contraction mapping principle).
LetS,d be a complete metric space; H:S⟶S is a contraction then
(i)
H has a unique fixed point s∈S; that, is Hs=s(ii)
∀s0∈S, we have limn⟶∞Hnu0=uTheorem 4 (see [3], nonlinear alternative of Leray-Schauder type).
Assume thatV is an open subset of a Banach space U, 0∈V, and F:V¯⟶U be a contraction such that FV¯ is bounded then either
(i)
F has a fixed point in V¯, or(ii)
∃μ∈0,1 and v∈∂V such that v=μFv holdsTheorem 5 (see [2], Arzela-Ascoli theorem).
F⊂CU,ℝ is compact if and only if it is closed, bounded, and equicontinuous.


## 3. Main Results

Lemma 6.
Ifh∈C0,1,ℝ, and
(9)DC1−qDRL0+rχτℏτ,χτ,ϑτ=wτ,τ∈0,1,1<q≤2,0<r≤1,χ0=χ′0=0,χ1=δχζ,δ∈ℝ,ζ∈0,1,then the solution to the problem mentioned above is given by
(10)χτ=ℏτ,χτ,ϑτ×1Γr∫0ττ−zr−1I1−qwzdz+τr+11−δζr+1Γrδ∫0ζζ−zr−1I1−qwzdz−∫011−zr−1I1−qwzdz.Proof.
TakingIRL1−q to DC1−qDRL0+rχτ/ℏτ,χτ,ϑτ=wτ, then take IRL0+r to the resulting equation, we get
(11)χτℏτ,χτ,ϑτ=IRL0+rIRL1−qwτ+a0+a1t+a2τr−1=IRL0+rIRL1−qwτ+a0τrΓr+1+a1τr+1Γr+2+a2τr−1.
Substitution ofχ0=0 and χ′0=0 in Equation (11) gives a2=0 and a0=0, respectively, and consequently, Equation (6) becomes
(12)χτℏτ,χτ,ϑτ=IRL0+rIRL1−qwτ+a1τr+1Γr+2.Use of the condition χ1=δχζ in Equation (12) yields
(13)a1=Γr+21−δζr+1δIRL0+rIRL1−qwζ−IRL0+rIRL1−qw1.
Insertinga1 in Equation (12) gives
(14)χτℏτ,χτ,ϑτ=IRL0+rIRL1−qwτ+Γr+21−δζr+1δIRL0+rIRL1−qwζ−IRL0+rIRL1−qw1τr+1Γr+2.
Alternatively, we have(15)χτ=ℏτ,χτ,ϑτ×IRL0+rIRL1−qwτ+Γr+21−δζr+1δIRL0+rIRL1−qwζ−IRL0+rIRL1−qw1τr+1Γr+2,τ∈0,1.
Equation (15) is equivalent to Equation (10), which makes the proof done.Denote the Banach space byC=C0,1 with the norm h=sup0≤t≤1ht. Then, the product space C×C,χ,ϑ with the norm χ,ϑ=χ+ϑ, ∀x,y∈C×C is indeed a Banach space too. We define an operator ϒ:C×C⟶C×C as
(16)ϒχ,ϑτ=ϒ1χ,ϑτϒ2χ,ϑτ,where
(17)ϒ1χ,ϑτ=ℏτ,χτ,ϑτ×1Γr∫0ττ−zr−1IRL1−qψz,χz,ϑzdz+τr+11−δζr+1Γr×δ∫0ζζ−zr−1IRL1−qψz,χz,ϑzdz−∫011−zr−1IRL1−qψz,χz,ϑzdz,ϒ2χ,ϑτ=ƛτ,χτ,ϑτ×1Γp∫0ττ−zp−1IRL1−qφz,χz,ϑzdz+τp+11−εξp+1Γp×ε∫0ξξ−zp−1IRL1−qφz,χz,ϑzdz−∫011−zp−1IRL1−qφz,χz,ϑzdz.To construct the necessary conditions for the results of uniqueness and existence of the problem (6), let us consider the following hypotheses.(C1) Let the functions f and g are assumed to be continuous and bounded; that is, ∃λf,λg>0 such that
(18)ℏτ,χ,ϑ≤λℏ,andƛτ,χ,ϑ≤λƛ,∀τ,χ,ϑ∈0,1×ℝ2.(C2) Let the functions ψ and φ are assumed to be continuous, and ∃υi,ℓi>0,i=1,2 such that
(19)ψτ,χ1,ϑ1−ψτ,χ2,ϑ2≤υ1χ1−χ2+υ2ϑ1−ϑ2,φτ,χ1,ϑ1−φτ,χ2,ϑ2≤ℓ1χ1−χ2+ℓ2ϑ1−ϑ2,∀τ∈0,1,χi,ϑi∈ℝ,i=1,2.(C3) There is positive constants ω0, θ0, and ωi,θi≥0i=1,2 such that
(20)ψτ,χ,ϑ≤ω0+ω1χ+ω2ϑ,(21)φτ,χ,ϑ≤θ0+θ1χ+θ2ϑ,∀τ∈0,1,χi,ϑi∈ℝ,i=1,2.(C4) Let S⊂C×C be a bounded set, then ∃κi>0,i=1,2 such that ψτ,χτ,ϑτ≤κ1,and
(22)φτ,χτ,ϑτ≤κ2,∀χ,ϑ∈S.Observe that(23)1ΓrΓq∫0ττ−zr−1∫s1u−zq−1dudz=∫0ττ−zr−1Γr∫z1u−zq−1Γqdudz=∫0ττ−zr−1Γru−zqΓq+1u=zu=1dz=∫0ττ−zr−1Γru−zqΓq+1u=zu=1dz=∫0ττ−zr−1Γr1−zqΓq+1dz≤∫0ττ−zr−1Γr1Γq+1dz,1−sq≤1,1<q≤2=τrΓr+1Γq+1.To facilitate the calculations below, let us say(24)Λ1=sup0≤τ≤11ΓrΓq∫0ττ−zr−1∫z1u−zq−1dudz+τr+11−δζr+1ΓrΓq×δ∫0ζζ−zr−1∫z1u−zq−1dudz−∫011−zr−1∫z1u−zq−1dudz≤1Γq+1Γr+11+δζr1−δζr+1,Λ2=sup0≤t≤11ΓpΓq∫0ττ−zp−1∫z1u−zq−1dudz+τp+1Γp1−εξp+1Γq×ε∫0ξξ−zp−1∫z1u−zq−1dudz−∫011−zp−1∫s1u−zq−1dudz≤1Γp+1Γq+11+εξp1−εξp+1.Theorem 7.
If both (C1) and (C2) are satisfied, and assume that λℏΛ1υ1+υ2+λƛΛ2ℓ1+ℓ2<1. Then, the system in Equation (6) has a unique solution.Proof.
Define a closed ballBγ¯=χ,ϑ∈C×C:χ,ϑ≤γ with γ≥λℏΛ1Nψ+λƛΛ2Nφ/1−λℏΛ1υ1+υ2+λƛΛ2ℓ1+ℓ2, where Nψ=sup0≤τ≤Tψτ,0,0,Nφ=sup0≤τ≤Tφτ,0,0.
Observe thatψτ,χ,ϑ=ψτ,χ,ϑ−ψτ,0,0+ψτ,0,0≤υ1χ+υ2ϑ+Nψ≤υ1+υ2γ+Nψ.
First, we show thatIB¯γ⊂Bγ¯. For any χ,ϑ∈Bγ¯,τ∈0,1, we have
(25)ϒ1χ,ϑτ=ℏτ,χτ,ϑτ×1Γr∫0ττ−zr−1IRL1−qψz,χz,ϑzdz+τr+11−δζr+1Γr×δ∫0ζζ−zr−1IRL1−qψz,χz,ϑzdz∫011−zr−1IRL1−qψz,χz,ϑzdz≤λℏsup0≤τ≤11ΓrΓq∫0ττ−zr−1∫z1u−zq−1ψz,χz,ϑzdudz+τr+11−δζr+1ΓrΓqδ∫0ζζ−zr−1∫z1u−zq−1ψz,χz,ϑzdudz−∫011−zr−1∫z1u−zq−1ψz,χz,ϑzdudz≤λℏυ1+υ2γ+Nψsup0≤τ≤1∫0ττ−zr−1∫z1u−zq−1dudz+τr+11−δζr+1ΓrΓq×δ∫0ζζ−zr−1∫z1u−zq−1dudz−∫011−zr−1∫z1u−zq−1ψz,χz,ϑzdudz≤λℏΛ1υ1+υ2γ+Nψ,similar to what was done above, we get
(26)ϒ2χ,ϑ≤λƛΛ2ℓ1+ℓ2γ+Nφ.
From Equation (25) and Equation (26), we deduce that ϒχ,ϑ≤γ.
Next, forχ1,ϑ1,χ2,ϑ2∈C×C,∀τ∈0,1, we have
(27)ϒ1χ1,ϑ1τ−ϒ1χ2,ϑ2τ≤λℏsup0≤τ≤11ΓrΓq∫0ττ−zr−1∫z1u−zq−1ψz,χ1z,ϑ1z−ψz,χ2z,ϑ2zdudz+τr+11−δζr+1ΓrΓq×δ∫0ζζ−zr−1∫z1u−zq−1ψz,χ1z,ϑ1z−ψz,χ2z,ϑ2zdudz−∫011−zr−1∫z1u−zq−1ψz,χ1z,ϑ1z−ψz,χ2z,ϑ2zdudz≤λℏυ1χ1−χ2+υ2ϑ1−ϑ2sup0≤τ≤11ΓrΓq∫0ττ−zr−1∫z1u−zq−1dudz+τr+11−δζr+1ΓrΓq×δ∫0ζζ−zr−1∫z1u−zq−1dudz−∫011−zr−1∫z1u−zq−1dudz≤λℏΛ1υ1χ1−χ2+υ2ϑ1−ϑ2≤λℏΛ1υ1+υ2χ1−χ2+ϑ1−ϑ2.
Similarly, we can find(28)ϒ2χ1,ϑ1−ϒ2χ2,ϑ2≤λƛΛ2ℓ1+ℓ2χ1−χ2+ϑ1−ϑ2.
Combining Equation (27) and Equation (28) yields
(29)ϒχ1,ϑ1−ϒχ2,ϑ2≤λℏΛ1υ1+υ2+λƛΛ2ℓ1+ℓ2×χ1−χ2+ϑ1−ϑ2.
Equation (29) becomes ϒχ1,ϑ1−ϒχ2,ϑ2≤χ1−χ2+ϑ1−ϑ2. That is, ϒ is a contraction; consequently, Banach fixed point theorem applies; thus, the uniqueness of solutions for Equation (6) holds on 0,1.Theorem 8.
If (C1), (C3), and (C4) are satisfied, and if λℏΛ1ω1+λƛΛ2θ1<1 and λℏΛ1ω2+λƛΛ2θ2<1, then Equation (6) has at least one solution.Proof.
In the first step, we verify that the operatorϒ:C×C⟶C×C is completely continuous; obviously, the operator is continuous as a result that ℏ, ƛ, ψ, and φ are all assumed to be continuous.
With the aid of (C4), ∀χ,ϑ∈S, we have
(30)ϒ1χ,ϑτ≤λℏsup0≤τ≤11ΓrΓq∫0ττ−zr−1∫z1u−zq−1ψz,χz,ϑzdudz+τr+11−δζr+1ΓrΓq×δ∫0ζζ−zr−1∫z1u−zq−1ψz,χz,ϑzdudz−∫011−zr−1∫z1u−zq−1ψz,χz,ϑzdudz≤λℏΛ1κ1.Similarly,
(31)ϒ2χ,ϑ≤λƛΛ2κ2.Combining the inequalities (30) and (31) yields ϒχ,ϑ≤λℏΛ1κ1+λƛΛ2κ2, implying that ϒ is uniformly bounded.
Next, to verify the equicontinuity for the operatorϒ, we let τ1,τ2∈0,1,τ1<τ2 then
(32)ϒ1χ,ϑτ2−ϒ1χ,ϑτ1≤λℏsup0≤τ≤11ΓrΓq∫0τ1τ1−zr−1−τ2−zr−1∫z1u−zq−1ψz,χz,ϑzdudz−1ΓrΓq∫τ1τ2τ2−zr−1∫z1u−zq−1ψz,χz,ϑzdudz+τ2r+1−τ1r+11−δζr+1ΓrΓq×δ∫0ζζ−zr−1∫z1u−zq−1ψz,χz,ϑzdudz−∫011−zr−1∫z1u−zq−1ψz,χz,ϑzdudz,(33)ϒ1χ,ϑτ2−ϒ1χ,ϑτ1≤λℏκ1×1ΓrΓq∫0τ1τ1−zr−1−τ2−zr−1∫z1u−zq−1dudz−1ΓrΓq∫τ1τ2τ2−zr−1∫z1u−zq−1dudz+τ2r+1−τ1r+11−δζr+1ΓrΓq×δ∫0ζζ−zr−1∫z1u−zq−1dudz−∫011−zr−1∫z1u−zq−1dudz,(34)ϒ2χ,ϑτ2−ϒ2χ,ϑτ1≤λƛκ2×1ΓpΓq∫0τ1τ1−zp−1−τ2−zp−1∫s1u−sq−1duds−1ΓpΓq∫t1t2t2−sp−1∫s1u−sq−1duds+τ2p+1−τ1p+11−εξp+1ΓpΓq×ε∫0ξξ−zp−1∫z1u−zq−1dudz−∫011−zp−1∫z1u−zq−1dudz.
The R.H.S for both inequalities (33) and (34) tend to zero as τ1⟶τ2, and they are both independent on χ,ϑ. So, operator ϒχ,ϑ is equicontinuous and yields; ϒχ,ϑ is completely continuous.
Finally, we establish the bounded set given byΩ=x,y∈C×Cx,y=βIx,y,β∈0,1; then, ∀τ∈0,1; the equation χ,ϑ=βϒχ,ϑ gives
(35)χτ=βϒ1χ,ϑτ,ϑτ=βϒ2χ,ϑτ.
Using the hypothesis (C3), we get
(36)χ≤λℏΛ1ω0+ω1χ+ω2ϑ,ϑ≤λƛΛ2θ0+θ1χ+θ2ϑ.
Consequently, we have(37)χ+ϑ≤λℏΛ1ω0+λƛΛ2θ0+λℏΛ1ω1+λƛΛ2θ1χ+λℏΛ1ω2+λƛΛ2θ2ϑ.
Inequality (37) can be written as follows:
(38)χ,ϑ≤λℏΛ1ω0+λƛΛ2θ0Λ0,
whereΛ0=min1−λℏΛ1ω1+λƛΛ2θ1,1−λℏΛ1ω2+λƛΛ2θ2.
Inequality (38) shows that Ω is bounded. Hence, Leray-Schauder alternative applies, implying the existence of the solution for Equation (6).


## 4. Stability

In this part, we address the issue of stability of solutions to the system of equations defined by Equation (6) via U-H definition.Definition 9.
The system of the coupled sequential fractional differential BVPs Equation (6) is stable in U–H sense if a real number c=maxc1,c2>0 exists so that, for any ε=maxε1,ε2>0 and for any χ¯,ϑ¯∈C×C satisfying
(39)DC1−qDRL0+rχ¯τℏτ,χ¯τ,ϑ¯τ−ψτ,χ¯τ,ϑ¯τ<ε1,τ∈0,1,DC1−qDRL0+pϑ¯τƛτ,χ¯τ,ϑ¯τ−φτ,χ¯τ,ϑ¯τ<ε2,there exists a unique solution χ,ϑ∈C×C of (6) with
(40)χ,ϑ−χ¯,ϑ¯<cε.
It is clear thatχ¯,ϑ¯∈C×C satisfies the inequalities (39) if there exists a function h1,h2∈C×C (which depends on χ¯,ϑ¯), such that
(i)
h1τ<ε1andh2τ<ε2,τ∈0,1(ii)
Forτ∈0,1(41)DC1−qDRL0+rχ¯τℏτ,χ¯τ,ϑ¯τ=ψτ,χ¯τ,ϑ¯τ+h1τ,DC1−qDRL0+pϑ¯τƛτ,χ¯τ,ϑ¯τ=φτ,χ¯τ,ϑ¯τ+h2τ.Theorem 10.
Suppose that (C2) is fulfilled. Moreover
(42)λℏΛ1υ1+υ2<1,λƛΛ2ℓ1+ℓ2<1,Δ=1−λℏΛ1υ1+υ21−λƛΛ2ℓ1+ℓ2−λℏΛ1υ1+υ2λƛΛ2ℓ1+ℓ2>0.
Then, the system of coupled sequential fractional differential BVPs (6) is U–H stable.Proof.
Assume that forε1,ε2>0 a couple χ¯,ϑ¯∈C×C satisfies the inequalities (39). Introduce the following operator
(43)K1τ;h=1Γr∫0ττ−zr−1IRL1−qhzdz,K2τ;h=1Γp∫0ττ−up−1IRL1−qhudu.
Then(44)χ¯τ=ℏτ,χ¯τ,ϑ¯τK1τ;ψ·,χ¯·,ϑ¯·+τr+11−δζr+1δK1ζ;ψ·,χ¯·,ϑ¯·−K11;ψ·,χ¯·,ϑ¯·+ℏτ,χ¯τ,ϑ¯τK1τ;h1+τr+11−δζr+1δK1ζ;h1−K11;h1,(45)ϑ¯τ=ƛτ,χ¯τ,ϑ¯τK2τ;φ·,χ¯·,ϑ¯·+τr+11−δζr+1δK2ζ;φ·,χ¯·,ϑ¯·−K21;φ·,χ¯·,ϑ¯·+ƛτ,χ¯τ,ϑ¯τK2τ;h2+τr+11−δζr+1δK2ζ;h2−K21;h2.
From Equation (44) and Equation (45), we obtain
(46)χ¯τ−ℏτ,χ¯τ,ϑ¯τK1τ;ψ·,χ¯·,ϑ¯·+τr+11−δζr+1δK1ζ;ψ·,χ¯·,ϑ¯·−K11;ψ·,χ¯·,ϑ¯·≤ℏτ,χ¯τ,ϑ¯τK1τ;h1+τr+11−δζr+1δK1ζ;h1−K11;h1,(47)ϑ¯τ−ƛτ,χ¯τ,ϑ¯τK2τ;φ·,χ¯·,ϑ¯·+τr+11−δζr+1δK2ζ;φ·,χ¯·,ϑ¯·−K21;φ·,χ¯·,ϑ¯·≤ƛτ,χ¯τ,ϑ¯τK2τ;h2+τr+11−δζr+1δK2ζ;h2−K21;h2.
From Equation (27) and Equation (28), we obtain
(48)χ¯τ−ℏτ,χ¯τ,ϑ¯τK1τ;ψ·,χ¯·,ϑ¯·+τr+11−δζr+1δK1ζ;ψ·,χ¯·,ϑ¯·−K11;ψ·,χ¯·,ϑ¯·≤λℏΛ1h1≤λℏΛ1ε1,(49)ϑ¯τ−ƛτ,χ¯τ,ϑ¯τK2τ;φ·,χ¯·,ϑ¯·+τr+11−δζr+1δK2ζ;φ·,χ¯·,ϑ¯·−K21;φ·,χ¯·,ϑ¯·≤λƛΛ2h2≤λƛΛ2ε2.
Letχ,ϑ∈C×C be a solution of Equation (6). Thanks to Lemma 6, it is equivalent to the following integral equations:
(50)χτ=ℏτ,χτ,ϑτK1τ;ψ·,χ·,ϑ·+τr+11−δζr+1δK1ζ;ψ·,χ·,ϑ·−K11;ψ·,χ·,ϑ·,ϑt=ƛτ,χτ,ϑτK2t;φ·,χ·,ϑ·+τr+11−δζr+1δK2ζ;φ·,χ·,ϑ·−K21;φ·,χ·,ϑ·.
By the same arguments in Theorem7, we get
(51)χτ−χ¯τ=ϒ1χ,ϑτ−ϒ1χ¯,ϑ¯τ−ℏτ,χ¯τ,ϑ¯τK1τ;h1+τr+11−δζr+1δK1ζ;h1−K11;h1≤ϒ1χ,ϑτ−ϒ1χ¯,ϑ¯τ+ℏτ,χ¯τ,ϑ¯τK1τ;h1+τr+11−δζr+1δK1ζ;h1−K11;h1≤λℏΛ1υ1+υ2χ−χ¯+ϑ−ϑ¯+λℏΛ1ε1,(52)ϑτ−ϑ¯τ=ϒ2χ,ϑτ−ϒ2χ¯,ϑ¯τ−ƛτ,χ¯τ,ϑ¯τK2τ;h2+τr+11−δζr+1δK2ζ;h2−K21;h2≤ϒ2χ,ϑτ−ϒ2χ¯,ϑ¯τ+ƛτ,χ¯τ,ϑ¯τK2τ;h2+τr+11−δζr+1δK2ζ;h2−K21;h2≤λƛΛ2ℓ1+ℓ2χ−χ¯+ϑ−ϑ¯+λƛΛ2ε2.
It follows that(53)χ−χ¯−λℏΛ1υ1+υ2χ−χ¯+ϑ−ϑ¯≤λℏΛ1ε1,ϑ−ϑ¯−λgΛ2ℓ1+ℓ2χ−χ¯+ϑ−ϑ¯≤λƛΛ2ε2.
Representing these inequalities as matrices, we get(54)1−λℏΛ1υ1+υ2−λℏΛ1υ1+υ21−λƛΛ2ℓ1+ℓ2−λƛΛ2ℓ1+ℓ2χ−χ¯ϑ−ϑ¯≤λℏΛ1ε1λƛΛ2ε2.
Solving the above inequality, we get(55)χ−χ¯≤1−λℏΛ1υ1+υ2ΔλℏΛ1ε1+λℏΛ1υ1+υ2ΔλƛΛ2ε2,ϑ−ϑ¯≤λƛΛ2ℓ1+ℓ2ΔλℏΛ1ε1+1−λƛΛ2ℓ1+ℓ2ΔλƛΛ2ε2,where Δ=1−λℏΛ1υ1+υ21−λƛΛ2ℓ1+ℓ2−λλℏΛ1υ1+υ2λƛΛ2ℓ1+ℓ2≠0.
Thus(56)χ−χ¯+ϑ−ϑ¯≤1−λℏΛ1υ1+υ2Δ+λƛΛ2ℓ1+ℓ2ΔλℏΛ1ε1+1−λƛΛ2ℓ1+ℓ2Δ+λℏΛ1υ1+υ2ΔλƛΛ2ε2.
Forε=maxε1,ε2 and
(57)c=1−λℏΛ1υ1+υ2+λƛΛ2ℓ1+ℓ2λℏΛ11+1−λƛΛ2ℓ1+ℓ2+λℏΛ1υ1+υ2λƛΛ2Δ,we get
(58)χ,χ¯−ϑ,ϑ¯≤χ−χ¯+ϑ−ϑ¯≤cε.
Therefore, with the aid of Definition9, the solution of the problem Equation (6) is U–H stable.


## 5. Example

In this part, we present an applied example to support the theoretical results we reached in the previous part, consider the following system:(59)DC1−7/4DRL0+1/4χτ1/2sinχτ+7/5=3eτ+110τ3+15χ1+χ+140tan−1ϑ,τ∈0,1,DC1−7/4DRL0+3/4ϑτ1/3cosϑτ+1=2e−3τsinτ+120tan−1ϑ+tan−1χ,χ0=χ′0=0,χ1=χ12,ϑ0=ϑ′0=0,ϑ1=ϑ13.Here,(60)1=δ=ε,p=74,q=54,ℏτ,χ,ϑ=12sinχτ+75,λτ,χ,ϑ=13cosϑτ+1,ψτ,χτ,ϑτ=3eτ+110τ3+15χ1+χ+140tan−1ϑ,φτ,χτ,ϑτ=2e−3τsinτ+120tan−1ϑ+tan−1χ.Observe that(61)ψτ,χ1,ϑ1−ψτ,χ2,ϑ2≤140χ2−χ1+140ϑ2−ϑ1,φτ,χ1,ϑ1−φτ,χ2,ϑ2≤120χ2−χ1+120ϑ2−ϑ1,λℏΛ1υ1+υ2+λƛΛ2ℓ1+ℓ2≤0.330291<1.Thus, the boundary value problem Equation (59) satisfies all the conditions of Theorem 7; consequently, the uniqueness of solution of Equation (59) is satisfied on 0,1.In order to explain Theorem7, it is clear that (C1) is satisfied as follows:
(62)ℏτ,χ,ϑ≤12sinχτ+75≤2=λℏ,ƛτ,χ,ϑ≤13cosϑτ+1≤32=λƛ,Also, one can easily show that (C3) holds, taking into account that τ∈0,1, then
(63)ψτ,χ,ϑ=3eτ+110τ3+15χ1+χ+140tan−1ϑ≤3e+140χ+140ϑ,φτ,χ,ϑ=2e−3τsinτ+120tan−1ϑ+tan−1χ≤2+120χ+120ϑ.Also, (C4) satisfied with
(64)ψτ,χ,ϑ≤3e+1+2π40,φτ,χ,ϑ≤2+π5.Finally, easy calculations with the data above giveλℏΛ1ω1+λƛΛ2θ1=0.203275<1 and λℏΛ1ω2+λƛΛ2θ2=0.203<1; all conditions of Theorem 8 hold; that is, the problem (59) has at least one solution in 0,1.


## 6. Conclusion

We have studied a coupled hybrid FDEs consisting of mixed fractional derivatives such as Caputo and Riemann-Liouville fractional derivatives and nonlocal boundary conditions. Existence/uniqueness results are established via a nonlinear alternative of the Leray-Schauder and Banach fixed point theorem. We also studied the Ulam-Hyers stability of these couple of hybrid FDEs. The obtained result is well illustrated by a numerical example. The result obtained in this paper is new and significantly contributes to the existing literature on the topic.One possible direction in which to extend the results of this paper is toward different kinds of mixed fractional differential and mixed conformable fractional differential systems of higher order. Another challenge is to find out if similar results can be derived in the case of constant/variable delays in linear/nonlinear terms.



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