The dynamic model of mobile wheelchair technology requires developing and implementing an intelligent control system to improve protection, increasing performance efficiency, and creating precise maneuvering in indoor and outdoor spaces. This work aims to design a robust tracking control algorithm based on a reference model for operating the kinematic model of powered wheelchairs under the variation of system parameters and unknown disturbance signals. The control algorithm was implemented using the pole placement method in combination with the sliding mode control (PP-SMC) approach. The design also adopted a neural network approach to eliminate system uncertainties from perturbations. The designed method utilized the sinewave signal as an essential input signal to the reference model. The stability of a closed-loop control system was achieved by adopting the Goa reaching law. The performance of the proposed tracking control system was evaluated in three scenarios under different conditions. These included assessing the tracking under normal operation conditions, considering the tracking performance by changing the dynamic system's parameters and evaluating the control system in the presence of uncertainties and external disturbances. The findings demonstrated that the proposed control method efficiently tracked the reference signal within a small error based on mean absolute error (MAE) measurements, where the range of MAE was between 0.08 and 0.12 in the presence of uncertainties or perturbations.
Citation: Mohsen Bakouri, Abdullah Alqarni, Sultan Alanazi, Ahmad Alassaf, Ibrahim AlMohimeed, Mohamed Abdelkader Aboamer, Tareq Alqahtani. Robust dynamic control algorithm for uncertain powered wheelchairs based on sliding neural network approach[J]. AIMS Mathematics, 2023, 8(11): 26821-26839. doi: 10.3934/math.20231373
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The dynamic model of mobile wheelchair technology requires developing and implementing an intelligent control system to improve protection, increasing performance efficiency, and creating precise maneuvering in indoor and outdoor spaces. This work aims to design a robust tracking control algorithm based on a reference model for operating the kinematic model of powered wheelchairs under the variation of system parameters and unknown disturbance signals. The control algorithm was implemented using the pole placement method in combination with the sliding mode control (PP-SMC) approach. The design also adopted a neural network approach to eliminate system uncertainties from perturbations. The designed method utilized the sinewave signal as an essential input signal to the reference model. The stability of a closed-loop control system was achieved by adopting the Goa reaching law. The performance of the proposed tracking control system was evaluated in three scenarios under different conditions. These included assessing the tracking under normal operation conditions, considering the tracking performance by changing the dynamic system's parameters and evaluating the control system in the presence of uncertainties and external disturbances. The findings demonstrated that the proposed control method efficiently tracked the reference signal within a small error based on mean absolute error (MAE) measurements, where the range of MAE was between 0.08 and 0.12 in the presence of uncertainties or perturbations.
The classical derivatives are local in nature, i.e., using classical derivatives we can describe changes in the neighborhood of a point, but using fractional derivatives we can describe changes in an interval. Namely, a fractional derivative is nonlocal in nature. This property makes these derivatives suitable to simulate more physical phenomena such as earthquake vibrations, polymers, etc.
On the other hand, in the case of the heat conduction equation, the fractional order parameter α means the level of thermal conductivity. If α=1, the medium's thermal conductivity is normal; if α<1, the medium has weak conductivity; and if α≥1, the medium has strong conductivity.
Further, in modeling various memory phenomena, it is observed that a memory process usually consists of two stages. One is short with permanent retention, and the other is governed by a simple model of fractional derivative. With the numerical least squares method, the fractional model perfectly fits the test data of memory phenomena in different disciplines, not only in mechanics but also in biology and psychology. Based on this model, it is found that the physical meaning of the fractional order is an index of memory. For more details, see [1,2].
Fractional calculus and its applications have acquired a lot of interest in several disciplines of engineering and science such as biology, chemistry, physics, economics, control theory, signal and image processing, etc, see [3,4,5] and the references therein. Variant definitions for the fractional derivative have emerged over the years. The most famous ones are the Riemann-Liouville and Caputo fractional derivatives. In recent years, many nonlinear phenomena in numerous fields have been modeled by fractional differential equations. Due to the evolution of fractional calculus, these equations have emerged as a new branch of applied mathematics. Several works on the existence and multiplicity of solutions to fractional boundary value problems (FBVPs) have appeared in view of the qualitative properties of fractional differential equations.
Among the used methods to solve a FBVP, there are the variational methods used by Fix and Roop in [6] and Erwin and Roop in [7]. Also, some fixed point techniques have been applied successfully to ensure the existence of solutions of some FBVPs. Here, we may cite the works of Agarwal et al. [8], Benchohra et al. [9], Zhang [10], Ahmad and Nieto [11], etc. Going in the same direction, the critical point theory has been used to investigate the solutions for some FBVPs. For instance, see the works Jiao and Zhou [12] and Tang and Wu [13]. On the other hand, stability analysis of fractional differential equations with different types of initial and boundary conditions have attracted many researchers who discussed the analysis of stability in the setting of Ulam-Hyers (UH) and generalized UH theory. For more details, see [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30].
In 2016, the boundary value problem (BVP) with 4-order Riemann-Liouville fractional (RLF) derivatives is studied by Niyom et al. [31]:
{υDρ(z(τ))+(1−υ)Dθ(z(τ))=Ξ(τ,z(τ)), τ∈[0,G], ρ∈[1,2),z(0)=0, ϱ1Dη1z(G)+(1−ϱ1)Dη2z(G)=ξ1, | (1.1) |
under appropriate conditions. Also, Niyom et al. [32], modified the above problem under multiple orders of fractional integrals and derivatives as follows:
{υDρ(z(τ))+(1−υ)Dθ(z(τ))=Ξ(τ,z(τ)), τ∈[0,G], ρ∈[1,2),z(0)=0, ϱ2Is1z(G)+(1−ϱ2)Is2z(G)=ξ2. | (1.2) |
In 2018, Xu et al. [33] examined the existence of solutions and UH stability for the FDEs
{υDρ(z(τ))+Dθ(z(τ))=Ξ(τ,z(τ)), τ∈[0,G], ρ∈[1,2),z(0)=0, ϱ1Dη1z(G)+Is2z(G)=ξ2. | (1.3) |
They focused on the RLF derivative and integral issues of the two-term class of three-point BVPs, where the notions and parameters in (1.1) and (1.2) are defined below the system (1.4).
Now, utilizing the concepts from the works described above and combining them, we investigate a new category of coupled boundary value problems (CBVPs) that includes a multi-order RLF equation plus various linear integro-derivative boundary stipulations as follows:
{υDρ(z(τ))+(1−υ)Dθ(z(τ))=Ξ(τ,z(τ),r(τ)), τ∈[0,G], ρ∈[2,3), υ∗Dρ∗(r(τ))+(1−υ∗)Dθ∗(r(τ))=Ξ∗(τ,r(τ),z(τ)), τ∈[0,G], ρ∗∈[2,3),z(0)=0, ϱ1Dη1z(G)+(1−ϱ1)Dη2z(G)=ξ1, r(0)=0, ϱ∗1Dη∗1r(G)+(1−ϱ∗1)Dη∗2r(G)=ξ∗1, ϱ2Is1z(G)+(1−ϱ2)Is2z(G)=ξ2, ϱ∗2Is∗1r(G)+(1−ϱ∗2)Is∗2r(G)=ξ∗2, | (1.4) |
where 2<θ<ρ, 2<θ∗<ρ∗, υ,υ∗,ϱ1,ϱ2,ϱ∗1,ϱ∗2∈(0,1], 0≤η1,η2<ρ−θ, 0≤η∗1,η∗2<ρ∗−θ∗, s1,s2,s∗1,s∗2∈R+, Dq is the RLF derivative of order q∈{ρ,θ,ρ∗,θ∗,η1,η2,η∗1,η∗2}, Im is the RLF integral of order m∈{s1,s2,s∗1,s∗2} and Ξ,Ξ∗:[0,G]×R2→R are continuous functions.
As many scholars are interested in exploring the idea of stability for various CBVPs, this can serve as inspiration for us to research the stability of complicated systems with added broad boundary stipulations. Consequently, to be more precise, the main objective of the current manuscript is to find some existing criteria for the solutions to a new general CBVP that includes a two-term fractional differential equation (FDE) (1.4) and multi-order RLF derivatives and integrals. The well-known standard fixed point (FP) theorems are employed in order to achieve this goal. Furthermore, in the follow-up, we examine the HU stability of the suggested problem (1.4) in the unique scenario when ϱ1=ϱ2=1 and ϱ∗1=ϱ∗2=1. Ultimately, to demonstrate the applicability of our theoretical results, two examples are provided. We think that the BVP that has been proposed is a generic one that incorporates a lot of fractional dynamical systems as special examples in the fields of physics and other applied disciplines.
Let G>0 and U=[0,G]. Assume that the piecewise continuous function space PC(U,R+) equipped with the norms ‖z‖=max{|z(υ)|:υ∈U} and ‖r‖=max{|r(υ)|:υ∈U} is a Banach space (BS), then the products of these norms are also a BS under the norm ‖(z,r)‖=‖z‖+‖r‖.
Assume also ℑ1 and ℑ2 represent the piecewise continuous function spaces described as
ℑ1=PC1(U,R)={z:U→R} and ℑ2=PC2(U,R)={r:U→R}, |
with norms
‖z‖ℑ1=sup{|z(υ)|, υ∈U} and ‖r‖ℑ2=sup{|r(υ)|, υ∈U}, |
respectively. Clearly, the product ℑ=ℑ1×ℑ2 is a BS endowed with ‖(z,r)‖ℑ=‖z‖ℑ1+‖r‖ℑ2.
Definition 2.1. [34] For a real valued function z:(0,∞)→R, the RLF integral operator of order ρ is described as
Iρz(τ)=1Γ(ρ)τ∫0(τ−ℏ)ρ−1z(ℏ)dℏ, |
where Γ(.) is the Euler gamma function.
Definition 2.2. [34] The RLF derivative of order ρ of a function z:(0,∞)→R takes the form
Dρz(τ)=1Γ(n−ρ)(ddτ)nτ∫0(τ−ℏ)n−ρ−1z(ℏ)dℏ, n=[ρ]+1. |
where [ρ] refers to the integer part of real number ρ.
Lemma 2.1. [34,35] Assume that ρ>0 and z∈C(0,1). Then the FDE Dρz(τ)=0 owns a general solution z(τ)=n∑j=1Ojτρ−j, where j−1<ρ≤j and the constants O1, O2,...,On∈R.
Lemma 2.2. [34] Assume that ρ>0 and z∈C(0,1). Then, we have
IρDρz(τ)=z(τ)+n∑j=1Ojτρ−j, |
where j−1<ρ≤j and the constants O1, O2,...Oj∈R.
Lemma 2.3. [4] Assume that ρ,θ>0 with ρ>θ, then Iρ0+Dθ0+=Iρ−θ0+.
The auxiliary theorems that follows is also required.
Theorem 2.1. (Krasnoselskii's FP theorem [36]) Assume that S is a non-empty, closed, bounded and convex subset of a BS ℑ. Let Ω,Ω∗:S→S be operators such that
(1) Ω(z)+Ω∗(r)∈S, where z,r∈S;
(2) Ω∗ is a contraction mapping;
(3) Ω is completely continuous.
Then there exists z∈S so that z=Ω(z)+Ω∗(z).
Theorem 2.2. (Banach FP theorem [37]) Every contraction self-mapping defined on a complete metric space admits a unique FP.
We begin this section with the lemma below.
Lemma 3.1. The mappings z0,r0 are a solution for CBVP (1.4) if z0,r0 are solutions to the following integral equations:
z(τ)=υ−1υΓ(ρ−θ)∫τ0(τ−ℏ)ρ−θ−1z(ℏ)dℏ+1υΓ(ρ)∫τ0(τ−ℏ)ρ−1Ξ(ℏ,z(ℏ),r(ℏ))dℏ+τρ−1Φ(ϱ1∇4(υ−1)υIρ−θ−η1z(G)−ϱ2∇2(υ−1)υIρ−θ−s1z(G)+∇4(1−ϱ1)(υ−1)υIρ−θ−η2z(G)−∇2(1−ϱ2)(υ−1)υIρ−θ−s2z(G)+ϱ1∇4υIρ−η1Ξ(G,z(G),r(G))−ϱ2∇2υIρ+s1Ξ(G,z(G),r(G))+∇2ξ2−∇4ξ1+(1−ϱ1)∇4υIρ−η2Ξ(G,z(G),r(G))−(1−ϱ2)∇2υIρ+s2Ξ(G,z(G),r(G)))−τρ−2Φ(ϱ1∇3(υ−1)υIρ−θ−η1z(G)−ϱ2∇1(υ−1)υIρ−θ−s1z(G)+∇3(1−ϱ1)(υ−1)υIρ−θ−η2z(G)−∇1(1−ϱ2)(υ−1)υIρ−θ−s2z(G)+ϱ1∇3υIρ−η1Ξ(G,z(G),r(G))−ϱ2∇1υIρ+s1Ξ(G,z(G),r(G))+∇1ξ2−∇3ξ1+(1−ϱ1)∇3υIρ−η2Ξ(G,z(G),r(G))−(1−ϱ2)∇1υIρ+s2Ξ(G,z(G),r(G))), | (3.1) |
and
r(τ)=υ∗−1υ∗Γ(ρ∗−θ∗)∫τ0(τ−ℏ)ρ∗−θ∗−1r(ℏ)dℏ+1υ∗Γ(ρ∗)∫τ0(τ−ℏ)ρ∗−1Ξ∗(ℏ,r(ℏ),z(ℏ))dℏ+τρ∗−1Φ∗(ϱ∗1∇∗4(υ∗−1)υ∗Iρ∗−θ∗−η∗1r(G)−ϱ∗2∇∗2(υ∗−1)υ∗Iρ∗−θ∗+s∗1r(G)+∇∗4(1−ϱ∗1)(υ∗−1)υ∗Iρ∗−θ∗−η∗2r(G)−∇∗2(1−ϱ∗2)(υ∗−1)υ∗Iρ∗−θ∗+s∗2r(G)+ϱ∗1∇∗4υ∗Iρ∗−η∗1Ξ∗(G,r(G),z(G))−ϱ∗2∇∗2υ∗Iρ∗+s∗1Ξ∗(G,r(G),z(G))+∇∗2ξ∗2−∇∗4ξ∗1+(1−ϱ∗1)∇∗4υ∗Iρ∗−η∗2Ξ∗(G,r(G),z(G))−(1−ϱ∗2)∇∗2υ∗Iρ∗+s∗2Ξ∗(G,r(G),z(G)))−τρ∗−2Φ∗(ϱ∗1∇∗3(υ∗−1)υ∗Iρ∗−θ∗−η∗1r(G)−ϱ∗2∇∗1(υ∗−1)υ∗Iρ∗−θ∗+s∗1r(G)+∇∗3(1−ϱ∗1)(υ∗−1)υ∗Iρ∗−θ∗−η∗2r(G)−∇∗1(1−ϱ∗2)(υ∗−1)υ∗Iρ∗−θ∗+s∗2r(G)ϱ∗1∇∗3υ∗Iρ∗−η∗1Ξ∗(G,r(G),z(G))−ϱ∗2∇∗1υ∗Iρ∗+s∗1Ξ∗(G,r(G),z(G))+∇∗1ξ∗2−∇∗3ξ∗1+(1−ϱ∗1)∇∗3υ∗Iρ∗−η∗2Ξ∗(G,r(G),z(G))−(1−ϱ∗2)∇∗1υ∗Iρ∗+s∗2Ξ∗(G,r(G),z(G))), | (3.2) |
where
∇1=ϱ1Γ(ρ)Γ(ρ−η1)Gρ−η1−1+(1−ϱ1)Γ(ρ)Γ(ρ−η2)Gρ−η2−1, ∇∗1=ϱ∗1Γ(ρ∗)Γ(ρ∗−η∗1)Gρ∗−η∗1−1−(1−ϱ∗1)Γ(ρ∗)Γ(ρ∗−η∗2)Gρ∗−η∗2−1, ∇2=ϱ1Γ(ρ−1)Γ(ρ−η1−1)Gρ−η1−2+(1−ϱ1)Γ(ρ−1)Γ(ρ−η2−1)Gρ−η2−2,∇∗2=ϱ∗1Γ(ρ∗−1)Γ(ρ∗−η∗1−1)Gρ∗−η∗1−2−(1−ϱ∗1)Γ(ρ∗−1)Γ(ρ∗−η∗2−1)Gρ∗−η∗2−2,∇3=ϱ2Γ(ρ)Γ(ρ+s1)Gρ+s1−1+(1−ϱ2)Γ(ρ)Γ(ρ+s2)Gρ+s2−1, ∇∗3=ϱ∗2Γ(ρ∗)Γ(ρ∗+s∗1)Gρ∗+s∗1−1−(1−ϱ∗2)Γ(ρ∗)Γ(ρ∗+s∗2)Gρ∗+s∗2−1, ∇4=ϱ2Γ(ρ−1)Γ(ρ+s1−1)Gρ+s1−2+(1−ϱ2)Γ(ρ−1)Γ(ρ+s2−1)Gρ+s2−2,∇∗4=ϱ∗2Γ(ρ∗−1)Γ(ρ∗+s∗1−1)Gρ∗+s∗1−2−(1−ϱ∗2)Γ(ρ∗−1)Γ(ρ∗+s∗2−1)Gρ∗+s∗2−2,Φ=∇3∇2−∇1∇4, Φ∗=∇∗3∇∗2−∇∗1∇∗4. | (3.3) |
Proof. Let (z0,r0) be a solution for the Eq (1.4), then, we get
{Dρz0(τ)=(υ−1)υDθz0(τ)+1υΞ(τ,z0(τ),r0(τ)),Dρ∗r0(τ)=(υ∗−1)υ∗Dθ∗z0(τ)+1υ∗Ξ(τ,r0(τ),z0(τ)). | (3.4) |
Taking the RLF integration of order ρ from both sides of the first equation in (3.4), we have
z0(τ)=υ−1υΓ(ρ−θ)∫τ0(τ−ℏ)ρ−θ−1z0(ℏ)dℏ+1υΓ(ρ)∫τ0(τ−ℏ)ρ−1Ξ(τ,z0(τ),r0(τ))dℏ+O1τρ−1+O2τρ−2+O3τρ−3, |
where O1, O2 and O3 are real constants. From the first boundary stipulation of (1.4), for ρ∈(2,3), we have O3=0. By Lemma 2.3, we can write
z0(τ)=υ−1υIρ−θz0(τ)+1υIρΞ(τ,z0(τ),r0(τ))+O1τρ−1+O2τρ−2. | (3.5) |
Using the RLF integral and derivative of order η and s, respectively with η∈{η1,η2}, s∈{s1,s2}, 0<η<ρ−θ and 2<θ<ρ, we obtain
Dρz0(τ)=υ−1υΓ(ρ−θ−η)∫τ0(τ−ℏ)ρ−θ−η−1z0(ℏ)dℏ+O1Γ(ρ)Γ(ρ−η)τρ−η−1+1υΓ(ρ−η)∫τ0(τ−ℏ)ρ−η−1Ξ(ℏ,z0(ℏ),r0(ℏ))dℏ+O2Γ(ρ−1)Γ(ρ−η−1)τρ−η−2. |
and
Iρz0(τ)=υ−1υΓ(ρ−θ+s)∫τ0(τ−ℏ)ρ−θ+s−1z0(ℏ)dℏ+O1Γ(ρ)Γ(ρ+s)τρ+s−1+1υΓ(ρ+s)∫τ0(τ−ℏ)ρ+s−1Ξ(ℏ,z0(ℏ),r0(ℏ))dℏ+O2Γ(ρ−1)Γ(ρ+s−1)τρ+s−2. |
Replacing η=η1, η=η2, s=s1, s=s2 and using the boundary stipulations ϱ1Dη1z(G)+(1−ϱ1)Dη2z(G)=ξ1 and ϱ2Is1z(G)+(1−ϱ2)Is2z(G)=ξ2, we can write
ξ1=ϱ1(υ−1)υΓ(ρ−θ−η1)∫G0(G−ℏ)ρ−θ−η1−1z0(ℏ)dℏ+(1−ϱ1)(υ−1)υΓ(ρ−θ−η2)∫G0(G−ℏ)ρ−θ−η2−1z0(ℏ)dℏ+ϱ1υΓ(ρ−η1)∫G0(G−ℏ)ρ−η1−1Ξ(ℏ,z0(ℏ),r0(ℏ))dℏ+O1∇1+(1−ϱ1)υΓ(ρ−η2)∫G0(G−ℏ)ρ−η2−1Ξ(ℏ,z0(ℏ),r0(ℏ))dℏ+O2∇2, |
and
ξ2=ϱ2(υ−1)υΓ(ρ−θ+s1)∫G0(G−ℏ)ρ−θ+s1−1z0(ℏ)dℏ+(1−ϱs)(υ−1)υΓ(ρ−θ+s2)∫G0(G−ℏ)ρ−θ+s2−1z0(ℏ)dℏ+ϱ2υΓ(ρ+s1)∫G0(G−ℏ)ρ+s1−1Ξ(ℏ,z0(ℏ),r0(ℏ))dℏ+O1∇3+(1−ϱ2)υΓ(ρ+s2)∫G0(G−ℏ)ρ+s2−1Ξ(ℏ,z0(ℏ),r0(ℏ))dℏ+O2∇4, |
which yields that
O1=ϱ1∇4(υ−1)υIρ−θ−η1z0(G)−ϱ2∇2(υ−1)υIρ−θ+s1z0(G)+∇4(1−ϱ1)(υ−1)υIρ−θ−η2z0(G)−∇2(1−ϱ2)(υ−1)υIρ−θ+s2z0(G)+ϱ1∇4υIρ−η1Ξ(G,z0(G),r0(G))−ϱ2∇2υIρ+s1Ξ(G,z0(G),r0(G))+∇2ξ2−∇4ξ1+(1−ϱ1)∇3υIρ−η2Ξ(G,z0(G),r0(G))−(1−ϱ2)∇1υIρ+s2Ξ(G,z0(G),r0(G)). |
and
O2=ϱ1∇3(υ−1)υIρ−θ−η1z0(G)−ϱ2∇1(υ−1)υIρ−θ+s1z0(G)+∇3(1−ϱ1)(υ−1)υIρ−θ−η2z0(G)−∇1(1−ϱ2)(υ−1)υIρ−θ+s2z0(G)+ϱ1∇3υIρ−η1Ξ(G,z0(G),r0(G))−ϱ2∇1υIρ+s1Ξ(G,z0(G),r0(G))+∇1ξ2−∇3ξ1+(1−ϱ1)∇3υIρ−η2Ξ(G,z0(G),r0(G))−(1−ϱ2)∇1υIρ+s2Ξ(G,z0(G),r0(G))), |
Substituting O1 and O2 in (1.4), we have the first part of the solution (3.1). With the same scenario followed above, the second part of the solution (3.2) can easily be obtained.
Now, we convert the problem to the FP problem. Based on Lemma 3.1, define an operator Ω:ℑ→ℑ by
Ω(z,r)=(Ω1(z,r),Ω2(z,r)), |
where
Ω1(z,r)=υ−1υΓ(ρ−θ)∫τ0(τ−ℏ)ρ−θ−1z(ℏ)dℏ+1υΓ(ρ)∫τ0(τ−ℏ)ρ−1Ξ(ℏ,z(ℏ),r(ℏ))dℏ+τρ−1Φ(ϱ1∇4(υ−1)υIρ−θ−η1z(G)−ϱ2∇2(υ−1)υIρ−θ+s1z(G)+∇4(1−ϱ1)(υ−1)υIρ−θ−η2z(G)−∇2(1−ϱ2)(υ−1)υIρ−θ+s2z(G)+ϱ1∇4υIρ−η1Ξ(G,z(G),r(G))−ϱ2∇2υIρ+s1Ξ(G,z(G),r(G))+∇2ξ2−∇4ξ1+(1−ϱ1)∇4υIρ−η2Ξ(G,z(G),r(G))−(1−ϱ2)∇2υIρ+s2Ξ(G,z(G),r(G)))−τρ−2Φ(ϱ1∇3(υ−1)υIρ−θ−η1z(G)−ϱ2∇1(υ−1)υIρ−θ+s1z(G)+∇3(1−ϱ1)(υ−1)υIρ−θ−η2z(G)−∇1(1−ϱ2)(υ−1)υIρ−θ+s2z(G)+ϱ1∇3υIρ−η1Ξ(G,z(G),r(G))−ϱ2∇1υIρ+s1Ξ(G,z(G),r(G))+∇1ξ2−∇3ξ1+(1−ϱ1)∇3υIρ−η2Ξ(G,z(G),r(G))−(1−ϱ2)∇1υIρ+s2Ξ(G,z(G),r(G))), | (3.6) |
and
Ω2(z,r)=υ∗−1υ∗Γ(ρ∗−θ∗)∫τ0(τ−ℏ)ρ∗−θ∗−1r(ℏ)dℏ+1υ∗Γ(ρ∗)∫τ0(τ−ℏ)ρ∗−1Ξ∗(ℏ,r(ℏ),z(ℏ))dℏ+τρ∗−1Φ∗(ϱ∗1∇∗4(υ∗−1)υ∗Iρ∗−θ∗−η∗1r(G)−ϱ∗2∇∗2(υ∗−1)υ∗Iρ∗−θ∗−s∗1r(G)+∇∗4(1−ϱ∗1)(υ∗−1)υ∗Iρ∗−θ∗−η∗2r(G)−∇∗2(1−ϱ∗2)(υ∗−1)υ∗Iρ∗−θ∗−s∗2r(G)+ϱ∗1∇∗4υ∗Iρ∗−η∗1Ξ∗(G,r(G),z(G))−ϱ∗2∇∗2υ∗Iρ∗+s∗1Ξ∗(G,r(G),z(G))+∇∗2ξ∗2−∇∗4ξ∗1+(1−ϱ∗1)∇∗4υ∗Iρ∗−η∗2Ξ∗(G,r(G),z(G))−(1−ϱ∗2)∇∗2υ∗Iρ∗+s∗2Ξ∗(G,r(G),z(G)))−τρ∗−2Φ∗(ϱ∗1∇∗3(υ∗−1)υ∗Iρ∗−θ∗−η∗1r(G)−ϱ∗2∇∗1(υ∗−1)υ∗Iρ∗−θ∗−s∗1r(G)+∇∗3(1−ϱ∗1)(υ∗−1)υ∗Iρ∗−θ∗−η∗2r(G)−∇∗1(1−ϱ∗2)(υ∗−1)υ∗Iρ∗−θ∗−s∗2r(G)ϱ∗1∇∗3υ∗Iρ∗−η∗1Ξ∗(G,r(G),z(G))−ϱ∗2∇∗1υ∗Iρ∗+s∗1Ξ∗(G,r(G),z(G))+∇∗1ξ∗2−∇∗3ξ∗1+(1−ϱ∗1)∇∗3υ∗Iρ∗−η∗2Ξ∗(G,r(G),z(G))−(1−ϱ∗2)∇∗1υ∗Iρ∗+s∗2Ξ∗(G,r(G),z(G))). | (3.7) |
Remember that the solution to CBVP (1.4) is (z0,r0) iff (z0,r0) is a FP of Ω. We employ the following notation to streamline calculations:
Λ1=|υ−1|(∇4+∇3G−1)|Φ|(ϱ1G2ρ−θ−η1−1υΓ(ρ−θ−η1+1)+(1−ϱ1)G2ρ−θ−η2−1υΓ(ρ−θ−η2+1))+|υ−1|(∇2+∇1G−1)|Φ|(ϱ2G2ρ−θ+s1−1υΓ(ρ−θ+s1+1)+(1−ϱ2)G2ρ−θ+s2−1υΓ(ρ−θ+s2+1))+|υ−1|Gρ−θυΓ(ρ−θ+1). | (3.8) |
Λ∗1=|υ∗−1|(∇∗4+∇∗3G−1)|Φ∗|(ϱ∗1G2ρ∗−θ∗−η∗1−1υ∗Γ(ρ∗−θ∗−η∗1+1)+(1−ϱ∗1)G2ρ∗−θ∗−η∗2−1υ∗Γ(ρ∗−θ∗−η∗2+1))+|υ∗−1|(∇∗4+∇∗3G−1)|Φ∗|(ϱ∗2G2ρ∗−θ∗+s∗1−1υ∗Γ(ρ∗−θ∗+s∗1+1)+(1−ϱ∗2)G2ρ∗−θ∗+s∗2−1υ∗Γ(ρ∗−θ∗+s∗2+1))+|υ∗−1|Gρ∗−θ∗υ∗Γ(ρ∗−θ∗+1). | (3.9) |
Λ2=GρυΓ(ρ+1)+∇4+∇3G−1|Φ|(ϱ1G2ρ−η1−1υΓ(ρ−η1+1)+(1−ϱ1)G2ρ−η2−1υΓ(ρ−η2+1))+∇2+∇1G−1|Φ|(ϱ2G2ρ+s1−1υΓ(ρ+s1+1)+(1−ϱ2)G2ρ+s2−1υΓ(ρ+s2+1)). | (3.10) |
Λ∗2=Gρ∗υ∗Γ(ρ∗+1)+(∇∗4+∇∗3G−1)|Φ∗|(ϱ∗1G2ρ∗−η∗1−1υ∗Γ(ρ∗−η∗1+1)+(1−ϱ∗1)G2ρ∗−η∗2−1υ∗Γ(ρ∗−η∗2+1))+∇∗4+∇∗3G−1|Φ∗|(ϱ∗2G2ρ∗+s∗1−1υ∗Γ(ρ∗+s∗1+1)+(1−ϱ∗2)G2ρ∗+s∗2−1υ∗Γ(ρ∗+s∗2+1)). | (3.11) |
Now, our main theorem is as follows:
Theorem 3.1. Assume that the mappings Ξ,Ξ∗:U×R2→R are continuous and there are constants TΞ,˜TΞ,TΞ∗,˜TΞ∗>0 so that
|Ξ(τ,z1(τ),z2(τ))−Ξ(τ,˜z1(τ),˜z2(τ))|≤TΞ|z1−˜z1|+˜TΞ|z2−˜z2|, |
and
|Ξ∗(τ,r1(τ),r2(τ))−Ξ∗(τ,˜r1(τ),˜r2(τ))|≤TΞ∗|r1−˜r1|+˜TΞ∗|r2−˜r2|, |
for all τ∈U and z1,z2,˜z1,˜z2,r1,r2,˜r1,˜r2∈R. If ˆTΛ4+Λ3<1, then the considered problem (1.4) has a unique solution (US), where ˆT=max{T,T∗}, T=max{TΞ,˜TΞ}, T∗=max{TΞ∗,˜TΞ∗}, Λ1+Λ∗1=Λ3 and Λ2+Λ∗2=Λ4 and Λ1, Λ∗1, Λ2, Λ∗2 are described as (3.8)–(3.11), respectively.
Proof. Set supυ∈UΞ(τ,0,0)=N<∞, supυ∈UΞ∗(τ,0,0)=N∗<∞ and choose
y≥ˆNΛ41−ˆTΛ4−Λ3+(|Φ|+|Φ∗|)Gρ−1(|∇2ξ1|+|∇4ξ2|+|∇∗2ξ∗1|+|∇∗4ξ∗2|)|Φ||Φ∗|(1−ˆTΛ4−Λ3)+Gρ−2(|∇2ξ1|+|∇4ξ2|+|∇∗2ξ∗1|+|∇∗4ξ∗2|)|Φ||Φ∗|(1−ˆTΛ4−Λ3), | (3.12) |
where ∇i and ∇∗i, i∈{1,2,3,4}, Φ and Φ∗ are defined by (3.3) and ˆN=max{N,N∗}. As a first step, we show that ΩQy⊂Qy, where Qy={(z,r)∈ℑ:‖(z,r)‖≤y}. For any (z,r)∈Qy, we have
‖Ω(z,r)‖ℑ=‖Ω1(z,r)‖ℑ1+‖Ω2(z,r)‖ℑ2 | (3.13) |
From (3.6) and (3.7), we get
|Ω1(z,r)|≤|υ−1|υΓ(ρ−θ)∫τ0(τ−ℏ)ρ−θ−1|z(ℏ)|dℏ+1υΓ(ρ)∫τ0(τ−ℏ)ρ−1(|Ξ(ℏ,z(ℏ),r(ℏ))−Ξ(ℏ,0,0)|+|Ξ(ℏ,0,0)|)dℏ+Gρ−1|Φ|(ϱ1∇4|υ−1|υIρ−θ−η1|z(G)|−ϱ2∇2|υ−1|υIρ−θ+s1|z(G)|+∇4|1−ϱ1|(|υ−1|)υIρ−θ−η2|z(G)|−∇2|1−ϱ2||υ−1|υIρ−θ+s2|z(G)|+ϱ1∇4υIρ−η1[|Ξ(G,z(G),r(G))−Ξ(G,0,0)|+|Ξ(G,0,0)|]−ϱ2∇2υIρ+s1[|Ξ(G,z(G),r(G))−Ξ(G,0,0)|+|Ξ(G,0,0)|]+|∇2ξ2|+|∇4ξ1|+(1−ϱ1)∇4υIρ−η2[|Ξ(G,z(G),r(G))−Ξ(G,0,0)|+|Ξ(G,0,0)|]−(1−ϱ2)∇2υIρ+s2[|Ξ(G,z(G),r(G))−Ξ(G,0,0)|+|Ξ(G,0,0)|])−Gρ−2Φ(ϱ1∇3|υ−1|υIρ−θ−η1|z(G)|−ϱ2∇1|υ−1|υIρ−θ+s1|z(G)|+∇3|1−ϱ1|(|υ−1|)υIρ−θ−η2|z(G)|−∇1|1−ϱ2||υ−1|υIρ−θ+s2|z(G)|+ϱ1∇3υIρ−η1[|Ξ(G,z(G),r(G))−Ξ(G,0,0)|+|Ξ(G,0,0)|]−ϱ2∇2υIρ+s1[|Ξ(G,z(G),r(G))−Ξ(G,0,0)|+|Ξ(G,0,0)|]+|∇1ξ2|+|∇3ξ1|+(1−ϱ1)∇3υIρ−η2[|Ξ(G,z(G),r(G))−Ξ(G,0,0)|+|Ξ(G,0,0)|]−(1−ϱ2)∇1υIρ+s2[|Ξ(G,z(G),r(G))−Ξ(G,0,0)|+|Ξ(G,0,0)|]), |
which implies that
‖Ω1(z,r)‖ℑ1≤(T‖(z,r)‖+N)Λ2+‖(z,r)‖Λ1+1|Φ|[Gρ−1(|∇2ξ2|+|∇4ξ1|)+Gρ−2(|∇1ξ2|+|∇3ξ1|)]=(TΛ2+Λ1)‖(z,r)‖+NΛ2+1|Φ|[Gρ−1(|∇2ξ2|+|∇4ξ1|)+Gρ−2(|∇1ξ2|+|∇3ξ1|)]. | (3.14) |
In the same scenario, we can write
‖Ω2(z,r)‖ℑ2≤(T∗Λ∗2+Λ∗1)‖(z,r)‖+N∗Λ∗2+1|Φ∗|[Gρ−1(|∇∗2ξ∗2|+|∇∗4ξ∗1|)+Gρ−2(|∇∗1ξ∗2|+|∇∗3ξ∗1|)]. | (3.15) |
Applying (3.14) and (3.15) in (3.13) and using (3.12), we have
‖Ω(z,r)‖ℑ=‖Ω1(z,r)‖ℑ1+‖Ω2(z,r)‖ℑ2=(TΛ2+Λ1+T∗Λ∗2+Λ∗1)‖(z,r)‖+NΛ2+N∗Λ∗2+1|Φ|[Gρ−1(|∇2ξ2|+|∇4ξ1|)+Gρ−2(|∇1ξ2|+|∇3ξ1|)]+1|Φ∗|[Gρ−1(|∇∗2ξ∗2|+|∇∗4ξ∗1|)+Gρ−2(|∇∗1ξ∗2|+|∇∗3ξ∗1|)]≤(TΛ2+Λ1+T∗Λ∗2+Λ∗1)y+NΛ2+N∗Λ∗2+1|Φ|[Gρ−1(|∇2ξ2|+|∇4ξ1|)+Gρ−2(|∇1ξ2|+|∇3ξ1|)]+1|Φ∗|[Gρ−1(|∇∗2ξ∗2|+|∇∗4ξ∗1|)+Gρ−2(|∇∗1ξ∗2|+|∇∗3ξ∗1|)]≤(ˆTΛ4+Λ3)y+ˆNΛ4+1|Φ|[Gρ−1(|∇2ξ2|+|∇4ξ1|)+Gρ−2(|∇1ξ2|+|∇3ξ1|)]+1|Φ∗|[Gρ−1(|∇∗2ξ∗2|+|∇∗4ξ∗1|)+Gρ−2(|∇∗1ξ∗2|+|∇∗3ξ∗1|)]≤y+ |
Hence, ‖Ω(z,r)‖ℑ≤y and so ΩQy⊂Qy. For each υ∈U and for z,r,˜z,˜r∈ℑ, we get
|Ω1(z,r)(τ)−Ω1(˜z,˜r)(υ)|≤|υ−1|υΓ(ρ−θ)∫τ0(τ−ℏ)ρ−θ−1|z(ℏ)−˜z(ℏ)|dℏ+1υΓ(ρ)∫τ0(τ−ℏ)ρ−1|Ξ(ℏ,z(ℏ),r(ℏ))−Ξ(ℏ,˜z(ℏ),˜r(ℏ))|dℏ+Gρ−1|Φ|(ϱ1∇4|υ−1|υIρ−θ−η1|z(G)−˜z(G)|−ϱ2∇2|υ−1|υIρ−θ+s1|z(G)−˜z(G)|+∇4|1−ϱ1|(|υ−1|)υIρ−θ−η2|z(G)−˜z(G)|−∇2|1−ϱ2||υ−1|υIρ−θ+s2|z(G)−˜z(G)|+ϱ1∇4υIρ−η1|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))| |
−ϱ2∇2υIρ+s1|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|+(1−ϱ1)∇4υIρ−η2|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|−(1−ϱ2)∇2υIρ+s2|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|)−Gρ−2Φ(ϱ1∇3|υ−1|υIρ−θ−η1|z(G)|−ϱ2∇1|υ−1|υIρ−θ+s1|z(G)|+∇3|1−ϱ1|(|υ−1|)υIρ−θ−η2|z(G)−˜z(G)|−∇1|1−ϱ2||υ−1|υIρ−θ+s2|z(G)−˜z(G)|+ϱ1∇3υIρ−η1|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|−ϱ2∇2υIρ+s1|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|+(1−ϱ1)∇3υIρ−η2|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|−(1−ϱ2)∇1υIρ+s2|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|), |
which leads to
‖Ω1(z,r)−Ω1(˜z,˜r)‖ℑ1≤TΛ2(‖z−˜z‖+‖r−˜r‖)+‖z−˜z‖Λ1. |
Similarly, one can obtain
‖Ω2(z,r)−Ω2(˜z,˜r)‖ℑ2≤T∗Λ∗2(‖z−˜z‖+‖r−˜r‖)+‖r−˜r‖Λ∗1. |
Hence,
‖Ω(z,r)−Ω(˜z,˜r)‖ℑ≤‖Ω1(z,r)−Ω1(˜z,˜r)‖ℑ1+‖Ω2(z,r)−Ω2(˜z,˜r)‖ℑ2=TΛ2(‖z−˜z‖+‖r−˜r‖)+‖z−˜z‖Λ1+T∗Λ∗2(‖z−˜z‖+‖r−˜r‖)+‖r−˜r‖Λ∗1=(Λ1+TΛ2+T∗Λ∗2)‖z−˜z‖+(Λ∗1+T∗Λ∗2+TΛ2)‖r−˜r‖≤(Λ1+ˆTΛ4)‖z−˜z‖+(Λ∗1+ˆTΛ4)‖r−˜r‖≤(ˆTΛ4+Λ3)‖(z,r)−(˜z,˜r)‖. |
Since ˆTΛ4+Λ3<1, then Ω is a contraction mapping. Using the contraction principle, Ω has a unique FP, which is the US for the CBVP (1.4).
Now, we present an existence result by applying Krasnoselskii's FP theorem.
Theorem 3.2. Suppose that the mappings Ξ,Ξ∗:U×R2→R are continuous and there are positive constants TΞ,˜TΞ,TΞ∗,˜TΞ∗ so that
|Ξ(τ,z1(τ),z2(τ))−Ξ(τ,˜z1(τ),˜z2(τ))|≤TΞ|z1−˜z1|+˜TΞ|z2−˜z2|, |
and
|Ξ∗(τ,r1(τ),r2(τ))−Ξ∗(τ,˜r1(τ),˜r2(τ))|≤TΞ∗|r1−˜r1|+˜TΞ∗|r2−˜r2|, |
for all τ∈U and z1,z2,˜z1,˜z2,r1,r2,˜r1,˜r2∈R. If there are V(τ),V∗(τ)∈C(U,R+) so that
Ξ(τ,z(τ),r(τ))≤V(τ)andΞ∗(τ,z(τ),r(τ))≤V∗(τ), |
for all (τ,z,r)∈U×R×R and Λ3<1, then, the CBVP (1.4) has at least one solution.
Proof. Consider supτ∈U|V(τ)|=‖V‖, suptau∈U|V∗(τ)|=‖V∗‖ and the set Qx={(z,r)∈ℑ:‖(z,r)‖≤x}, where
x≥ˆVΛ31−Λ4+(|Φ|+|Φ∗|)Gρ−1(|∇2ξ2|+|∇4ξ1|+|∇∗2ξ∗2|+|∇∗4ξ∗1|)|Φ||Φ∗|(1−Λ4)+(|Φ|+|Φ∗|)Gρ−2(|∇1ξ2|+|∇1ξ1|+|∇∗1ξ∗2|+|∇∗3ξ∗1|)|Φ||Φ∗|(1−Λ4), |
and ∇i, ∇∗i, i∈{1,2,3,4}, Φ and Φ∗ are defined by (3.3), ˆV=max{V,V∗} and Λ3=Λ1+Λ∗1. For any (z,r)∈Qx, define the operators Ω,Ω∗:ℑ→ℑ by
Ω(z,r)=˜Ω1(z,r)+˜Ω2(z,r) and Ω∗(z,r)=ˆΩ1(z,r)+ˆΩ2(z,r), |
where
˜Ω1(z,r)=υ−1υΓ(ρ−θ)∫τ0(τ−ℏ)ρ−θ−1z(ℏ)dℏ+τρ−1Φ×(ϱ1∇4(υ−1)υIρ−θ−η1z(G)−ϱ2∇2(υ−1)υIρ−θ+s1z(G)+∇4(1−ϱ1)(υ−1)υIρ−θ−η2z(G)−∇2(1−ϱ2)(υ−1)υIρ−θ+s2z(G))τρ−2Φ(ϱ1∇3(υ−1)υIρ−θ−η1z(G)−ϱ2∇1(υ−1)υIρ−θ+s1z(G)+∇3(1−ϱ1)(υ−1)υIρ−θ−η2z(G)−∇1(1−ϱ2)(υ−1)υIρ−θ+s2z(G)), | (3.16) |
ˆΩ1(z,r)=1υΓ(ρ)∫τ0(τ−ℏ)ρ−1Ξ(ℏ,z(ℏ),r(ℏ))dℏ+τρ−1Φ×(ϱ1∇4υIρ−η1Ξ(G,z(G),r(G))−ϱ2∇2υIρ+s1Ξ(G,z(G),r(G))+∇2ξ2−∇4ξ1+(1−ϱ1)∇4υIρ−η2Ξ(G,z(G),r(G))−(1−ϱ2)∇2υIρ+s2Ξ(G,z(G),r(G)))−τρ−2Φ(ϱ1∇3υIρ−η1Ξ(G,z(G),r(G))−ϱ2∇1υIρ+s1Ξ(G,z(G),r(G))+(1−ϱ1)∇3υIρ−η2Ξ(G,z(G),r(G))−(1−ϱ2)∇1υIρ+s2Ξ(G,z(G),r(G))+∇1ξ2−∇3ξ1), | (3.17) |
˜Ω2(z,r)=υ∗−1υ∗Γ(ρ∗−θ∗)∫τ0(τ−ℏ)ρ∗−θ∗−1r(ℏ)dℏ+τρ∗−1Φ∗×(ϱ∗1∇∗4(υ∗−1)υ∗Iρ∗−θ∗−η∗1r(G)−ϱ∗2∇∗2(υ∗−1)υ∗Iρ∗−θ∗+s∗1r(G)∇∗4(1−ϱ∗1)(υ∗−1)υ∗Iρ∗−θ∗−η∗2r(G)−∇∗2(1−ϱ∗2)(υ∗−1)υ∗Iρ∗−θ∗+s∗2r(G))−τρ∗−2Φ∗(ϱ∗1∇∗3(υ∗−1)υ∗Iρ∗−θ∗−η∗1r(G)−ϱ∗2∇∗1(υ∗−1)υ∗Iρ∗−θ∗+s∗1r(G)+∇∗3(1−ϱ∗1)(υ∗−1)υ∗Iρ∗−θ∗−η∗2r(G)−∇∗1(1−ϱ∗2)(υ∗−1)υ∗Iρ∗−θ∗+s∗2r(G)), | (3.18) |
and
ˆΩ2(z,r)=1υ∗Γ(ρ∗)∫τ0(τ−ℏ)ρ∗−1Ξ∗(ℏ,r(ℏ),z(ℏ))dℏ+τρ∗−1Φ∗×(+ϱ∗1∇∗4υ∗Iρ∗−η∗1Ξ∗(G,r(G),z(G))−ϱ∗2∇∗2υ∗Iρ∗+s∗1Ξ∗(G,r(G),z(G))+∇∗2ξ∗2−∇∗4ξ∗1+(1−ϱ∗1)∇∗4υ∗Iρ∗−η∗2Ξ∗(G,r(G),z(G))−(1−ϱ∗2)∇∗2υ∗Iρ∗+s∗2Ξ∗(G,r(G),z(G)))−τρ∗−2Φ∗(ϱ∗1∇∗3υ∗Iρ∗−η∗1Ξ∗(G,r(G),z(G))−ϱ∗2∇∗1υ∗Iρ∗+s∗1Ξ∗(G,r(G),z(G))++(1−ϱ∗1)∇∗3υ∗Iρ∗−η∗2Ξ∗(G,r(G),z(G))−(1−ϱ∗2)∇∗1υ∗Iρ∗+s∗2Ξ∗(G,r(G),z(G))+∇∗1ξ∗2−∇∗3ξ∗1). | (3.19) |
We shall show that Ω(z,r)+Ω∗(z,r)∈Qx, for all (z,r)∈Qx. From (3.16) and (3.17), we have
|˜Ω1(z,r)(τ)+ˆΩ1(z,r)(τ)|≤‖V‖[GρυΓ(ρ+1)+∇4+∇3G−1|Φ|(ϱ1G2ρ−η1−1υΓ(ρ−η1+1)+(1−ϱ1)G2ρ−η2−1υΓ(ρ−η2+1))+∇2+∇1G−1|Φ|(ϱ2G2ρ+s1−1υΓ(ρ+s1+1)+(1−ϱ2)G2ρ+s2−1υΓ(ρ+s2+1))]+‖(z,r)‖[|υ−1|Gρ−θυΓ(ρ−θ+1)+|υ−1|(∇4+∇3G−1)|Φ|(ϱ1G2ρ−θ−η1−1υΓ(ρ−θ−η1+1)+(1−ϱ1)G2ρ−θ−η2−1υΓ(ρ−θ−η2+1))+|υ−1|(∇2+∇1G−1)|Φ|(ϱ2G2ρ−θ+s1−1υΓ(ρ−θ+s1+1)+(1−ϱ2)G2ρ−θ+s2−1υΓ(ρ−θ+s2+1))+1|Φ|[Gρ−1(|∇2ξ2|+|∇4ξ1|)+Gρ−2(|∇1ξ2|+|∇3ξ1|)]≤‖V‖Λ1+yΛ2+1|Φ|[Gρ−1(|∇2ξ2|+|∇4ξ1|)+Gρ−2(|∇1ξ2|+|∇3ξ1|)]. | (3.20) |
Analogously, using (3.18) and (3.19), we get
|˜Ω2(z,r)(τ)+ˆΩ2(z,r)(τ)|≤‖V∗‖Λ∗1+yΛ∗2+1|Φ∗|[Gρ−1(|∇∗2ξ∗2|+|∇∗4ξ∗1|)+Gρ−2(|∇∗1ξ∗2|+|∇∗3ξ∗1|)]. | (3.21) |
Combining (3.20) and (3.21), we obtain that
|Ω(z,r)+Ω∗(z,r)|≤|˜Ω1(z,r)(τ)+ˆΩ1(z,r)(τ)|+|˜Ω2(z,r)(τ)+ˆΩ2(z,r)(τ)|=‖V‖Λ1+yΛ2+1|Φ|[Gρ−1(|∇2ξ2|+|∇4ξ1|)+Gρ−2(|∇1ξ2|+|∇3ξ1|)]+‖V∗‖Λ∗1+yΛ∗2+1|Φ∗|[Gρ−1(|∇∗2ξ∗2|+|∇∗4ξ∗1|)+Gρ−2(|∇∗1ξ∗2|+|∇∗3ξ∗1|)]=ˆVΛ3+Λ4y+1|Φ|[Gρ−1(|∇2ξ2|+|∇4ξ1|)+Gρ−2(|∇1ξ2|+|∇3ξ1|)]+1|Φ∗|[Gρ−1(|∇∗2ξ∗2|+|∇∗4ξ∗1|)+Gρ−2(|∇∗1ξ∗2|+|∇∗3ξ∗1|)]≤x. |
Thus, Ω(z,r)+Ω∗(z,r)∈Qx. Hence the condition (1) of Theorem 2.1 is true. Next, we prove that Ω(z,r) is a contraction mapping. Let (z,r),(˜z,˜r)∈Qx, then by (3.16), one has
|˜Ω1(z,r)(τ)−˜Ω1(˜z,˜r)(τ)|≤|υ−1|υΓ(ρ−θ)∫τ0(τ−ℏ)ρ−θ−1|z(ℏ)−˜z(ℏ)|dℏ+Gρ−1Φ×+(ϱ1∇4|υ−1|υIρ−θ−η1|z(G)−˜z(G)|−ϱ2∇2|υ−1|υIρ−θ+s1|z(G)−˜z(G)|+∇4(1−ϱ1)|υ−1|υIρ−θ−η2|z(G)−˜z(G)|−∇2(1−ϱ2)|υ−1|υIρ−θ+s2|z(G)−˜z(G)|)+Gρ−2Φ(ϱ1∇3|υ−1|υIρ−θ−η1|z(G)−˜z(G)|−ϱ2∇1|υ−1|υIρ−θ+s1|z(G)−˜z(G)|+∇3(1−ϱ1)|υ−1|υIρ−θ−η2|z(G)−˜z(G)|−∇1(1−ϱ2)|υ−1|υIρ−θ+s2|z(G)−˜z(G)|)≤Λ1‖z−z‖. |
Similarly, we can write
|˜Ω2(z,r)(τ)−˜Ω2(˜z,˜r)(τ)|≤Λ∗1‖r−r‖. |
It follows that
‖Ω(z,r)−Ω(˜z,˜r)‖ℑ≤‖˜Ω1(z,r)−˜Ω1(˜z,˜r)‖ℑ1+‖˜Ω2(z,r)−˜Ω2(˜z,˜r)‖ℑ2=Λ3‖(z,˜z)−(z,˜z)‖. |
Since Λ3<1, then Ω1 is a contraction mapping. Hence the condition (2) of Theorem 2.1 holds. The continuity of Ξ and Ξ∗ lead to the continuity of Ω∗. If (z,r)∈Qx, then
‖ˆΩ1(z,r)‖ℑ1≤‖V‖[GρυΓ(ρ+1)+∇4+∇3G−1|Φ|(ϱ1G2ρ−η1−1υΓ(ρ−η1+1)+(1−ϱ1)G2ρ−η2−1υΓ(ρ−η2+1))+∇2+∇1G−1|Φ|(ϱ2G2ρ+s1−1υΓ(ρ+s1+1)+(1−ϱ2)G2ρ+s2−1υΓ(ρ+s2+1))=Λ2‖(z,r)‖. |
Similarly, we have
‖ˆΩ2(z,r)‖ℑ2≤Λ∗2‖(z,r)‖. |
Hence,
‖Ω∗(z,r)‖ℑ≤‖ˆΩ1(z,r)‖ℑ1+‖ˆΩ1(z,r)‖ℑ2≤Λ4‖(z,r)‖, where Λ4=Λ2+Λ∗2. |
This means that Ω∗ is a uniformly bounded operator on Qx. Finally, we prove that the operator Ω∗ is completely continuous. Set for (z,r)∈Qx, supτ∈UΞ(τ,z(τ),r(τ))=R, and supτ∈UΞ∗(τ,z(τ),r(τ))=R∗. Then, for each τ1,τ2∈U with τ1<τ2, we get
|ˆΩ1(z,r)(τ2)−ˆΩ1(z,r)(τ1)||1υΓ(ρ)∫τ20(τ2−ℏ)ρ−1Ξ(ℏ,z(ℏ),r(ℏ))dℏ−1υΓ(ρ)∫τ10(τ1−ℏ)ρ−1Ξ(ℏ,z(ℏ),r(ℏ))dℏτρ−12−τρ−11Φ(ϱ1∇4υIρ−η1Ξ(G,z(G),r(G))−ϱ2∇2υIρ+s1Ξ(G,z(G),r(G))+∇2ξ2−∇4ξ1+(1−ϱ1)∇4υIρ−η2Ξ(G,z(G),r(G))−(1−ϱ2)∇2υIρ+s2Ξ(G,z(G),r(G)))−τρ−12−τρ−11Φ(ϱ1∇3υIρ−η1Ξ(G,z(G),r(G))−ϱ2∇1υIρ+s1Ξ(G,z(G),r(G))(1−ϱ1)∇3υIρ−η2Ξ(G,z(G),r(G))−(1−ϱ2)∇1υIρ+s2Ξ(G,z(G),r(G))+∇1ξ2−∇3ξ1)|≤R[2(τ2−τ1)ρ+|τρ2−τρ1|]υΓ(ρ+1)+τρ−12−τρ−11|Φ|[Rϱ1∇4Iρ−η1υΓ(ρ−η1+1)+Rϱ2∇2Iρ+s1υΓ(ρ+s1+1)+|∇2ξ2|+|∇4ξ1|+R(1−ϱ1)∇4Iρ−η2υΓ(ρ−η2+1)+R(1−ϱ2)∇2Iρ+s2υΓ(ρ+s2+1)]+τρ−22−τρ−21|Φ|[Rϱ1∇3Iρ−η1υΓ(ρ−η1+1)+Rϱ2∇1Iρ+s1υΓ(ρ+s1+1)+R(1−ϱ1)∇3Iρ−η2υΓ(ρ−η2+1)+R(1−ϱ2)∇1Iρ+s2υΓ(ρ+s2+1)+|∇1ξ2|+|∇3ξ1|], |
which implies that
|ˆΩ1(z,r)(τ2)−ˆΩ1(z,r)(τ1)|→0 as τ1→τ2. |
Similarly
|ˆΩ2(z,r)(τ2)−ˆΩ2(z,r)(τ1)|→0 as τ1→τ2. |
Hence
|Ω∗(z,r)(τ2)−Ω∗(z,r)(τ1)|≤|ˆΩ1(z,r)(τ2)−ˆΩ1(z,r)(τ1)|+|ˆΩ2(z,r)(τ2)−ˆΩ2(z,r)(τ1)|→0 as τ1→τ2, |
which yields that Ω∗ is equicontinuous, and so Ω∗ is relatively compact on Qx. Since every compact operator is completely continuous, then by the Arzela-Ascoli theorem, Ω∗ is completely continuous. Thus, condition (3) of Theorem 2.1 is satisfied. Hence, all conditions of Theorem 2.1 are satisfied. Consequently, the CBVP (1.4) has at least one solution.
In this part, we discuss the Hyers–Ulam stability of the CBVP
{υDρ(z(τ))+(1−υ)Dθ(z(τ))=Ξ(τ,z(τ),r(τ)), υ∗Dρ∗(r(τ))+(1−υ∗)Dθ∗(r(τ))=Ξ∗(τ,r(τ),z(τ)),z(0)=0, r(0)=0, Dη1z(G)=ξ1, Dη∗1r(G)=ξ∗1, Is1z(G)=ξ2, Is∗1r(G)=ξ∗2, | (4.1) |
for each τ∈[0,G] and ρ∈[2,3). The CBVP (4.1) is a special case of (1.4) when we take ϱ1=ϱ2=1 and ϱ∗1=ϱ∗2=1.
Definition 4.1. The CBVP (4.1) is called HU stable if there is a positive constant ˆΔ>0 so that, for each ϵ,ϵ∗>0 and (z,r)∈ℑ as a solution to the inequalities
{|υDρ(z(τ))+(υ−1)Dθ(z(τ))−Ξ(τ,z(τ),r(τ))|≤ϵ, |υ∗Dρ∗(r(τ))+(υ∗−1)Dθ∗(r(τ))−Ξ∗(τ,r(τ),z(τ))|≤ϵ∗, |
there is a US (˜z,˜r)∈ℑ with
‖(z,r)−(˜z,˜r)‖ℑ≤ˆΔˆϵ, for all υ∈U, |
where ˆϵ=max{ϵ,ϵ∗}.
Theorem 4.1. Assume that Ξ,Ξ∗:U×R2→R are continuous maps and there are constants TΞ,˜TΞ,TΞ∗,˜TΞ∗>0 so that
|Ξ(τ,z1(τ),z2(τ))−Ξ(τ,˜z1(τ),˜z2(τ))|≤TΞ|z1−˜z1|+˜TΞ|z2−˜z2|, |
and
|Ξ∗(τ,r1(τ),r2(τ))−Ξ∗(τ,˜r1(τ),˜r2(τ))|≤TΞ∗|r1−˜r1|+˜TΞ∗|r2−˜r2|, |
for all τ∈U and z1,z2,˜z1,˜z2,r1,r2,˜r1,˜r2∈R. Then, the CBVP (4.1) is HU stable provided that ℶ=1−⅁⅁∗(1−℘)(1−℘∗)>0.
Proof. Let ϵ,ϵ∗>0 and (z,r)∈ℑ be so that
{|υDρ(z(τ))+(υ−1)Dθ(z(τ))−Ξ(τ,z(τ),r(τ))|≤ϵ, |υ∗Dρ∗(r(τ))+(υ∗−1)Dθ∗(r(τ))−Ξ∗(τ,r(τ),z(τ))|≤ϵ∗. |
Choose the functions ζ and ζ∗ satisfying
{υDρ(z(τ))+(υ−1)Dθ(z(τ))=Ξ(τ,z(τ),r(τ))+ζ(τ), υ∗Dρ∗(r(τ))+(υ∗−1)Dθ∗(r(τ))=Ξ∗(τ,r(τ),z(τ))+ζ∗(τ), |
such that |ζ(τ)|≤ϵ and |ζ∗(τ)|≤ϵ∗ for all τ∈U. Then, we get
z(τ)=υ−1υIρ−θz(τ)+1υIρΞ(τ,z(τ),r(τ))+1υIρζ(τ)+τρ−1Φ(∇4(υ−1)υIρ−θ−η1z(G)−∇2(υ−1)υIρ−θ+s1z(G)+∇4υIρ−η1Ξ(G,z(G),r(G))−∇2υIρ+s1Ξ(G,z(G),r(G))+∇4υIρ−η1ζ(G)−∇2υIρ+s1ζ(G)+∇2ξ2−∇4ξ1)+τρ−2Φ(∇3(υ−1)υIρ−θ−η1z(G)−∇1(υ−1)υIρ−θ+s1z(G)+∇3υIρ−η1Ξ(G,z(G),r(G))−∇1υIρ+s1Ξ(G,z(G),r(G))+∇3υIρ−η1ζ(G)−∇1υIρ+s1ζ(G)+∇1ξ2−∇3ξ1), |
and
r(τ)=υ∗−1υ∗Iρ∗−θ∗r(τ)+1υ∗Iρ∗Ξ∗(τ,r(τ),z(τ))+1υIρ∗ζ∗(τ)+τρ∗−1Φ∗(∇∗4(υ∗−1)υ∗Iρ∗−θ∗−η∗1r(G)−∇∗2(υ∗−1)υ∗Iρ∗−θ∗+s∗1r(G)+∇∗4υ∗Iρ∗−η∗1Ξ∗(G,r(G),z(G))−∇∗2υ∗Iρ∗+s∗1Ξ∗(G,r(G),z(G))+∇∗4υ∗Iρ∗−η∗1ζ∗(G)−∇∗2υ∗Iρ∗+s∗1ζ∗(G)+∇∗2ξ∗2−∇∗4ξ∗1)−τρ∗−2Φ∗(∇∗3(υ∗−1)υ∗Iρ∗−θ∗−η∗1r(G)−∇∗1(υ∗−1)υ∗Iρ∗−θ∗+s∗1r(G)+∇∗3υ∗Iρ∗−η∗1Ξ∗(G,r(G),z(G))−∇∗1υ∗Iρ∗+s∗1Ξ∗(G,r(G),z(G))+∇∗3υ∗Iρ∗−η∗1ζ∗(G)−∇∗1υ∗Iρ∗+s∗1ζ∗(G)+∇∗1ξ∗2−∇∗3ξ∗1). |
Let (˜z,˜r) be a US of the CBVP (4.1), then ˜z(τ) and ˜r(τ) are given by
˜z(τ)=υ−1υIρ−θ˜z(τ)+1υIρΞ(τ,˜z(τ),˜r(τ))+1υIρζ(τ)+τρ−1Φ(∇4(υ−1)υIρ−θ−η1˜z(G)−∇2(υ−1)υIρ−θ+s1˜z(G)+∇4υIρ−η1Ξ(τ,˜z(τ),˜r(τ))−∇2υIρ+s1Ξ(τ,˜z(τ),˜r(τ))+∇4υIρ−η1ζ(G)−∇2υIρ+s1ζ(G)+∇2ξ2−∇4ξ1)+τρ−2Φ(∇3(υ−1)υIρ−θ−η1˜z(G)−∇1(υ−1)υIρ−θ+s1˜z(G)+∇3υIρ−η1Ξ(τ,˜z(τ),˜r(τ))−∇1υIρ+s1Ξ(τ,˜z(τ),˜r(τ))+∇3υIρ−η1ζ(G)−∇1υIρ+s1ζ(G)+∇1ξ2−∇3ξ1), |
and
˜r(τ)=υ∗−1υ∗Iρ∗−θ∗˜r(τ)+1υ∗Iρ∗Ξ∗(τ,˜r(τ),˜z(τ))+1υIρ∗ζ∗(τ)+τρ∗−1Φ∗(∇∗4(υ∗−1)υ∗Iρ∗−θ∗−η∗1˜r(G)−∇∗2(υ∗−1)υ∗Iρ∗−θ∗+s∗1˜r(G)+∇∗4υ∗Iρ∗−η∗1Ξ∗(τ,˜r(τ),˜z(τ))−∇∗2υ∗Iρ∗+s∗1Ξ∗(τ,˜r(τ),˜z(τ))+∇∗4υ∗Iρ∗−η∗1ζ∗(G)−∇∗2υ∗Iρ∗+s∗1ζ∗(G)+∇∗2ξ∗2−∇∗4ξ∗1)−τρ∗−2Φ∗(∇∗3(υ∗−1)υ∗Iρ∗−θ∗−η∗1˜r(G)−∇∗1(υ∗−1)υ∗Iρ∗−θ∗+s∗1˜r(G)+∇∗3υ∗Iρ∗−η∗1Ξ∗(τ,˜r(τ),˜z(τ))−∇∗1υ∗Iρ∗+s∗1Ξ∗(τ,˜r(τ),˜z(τ))+∇∗3υ∗Iρ∗−η∗1ζ∗(G)−∇∗1υ∗Iρ∗+s∗1ζ∗(G)+∇∗1ξ∗2−∇∗3ξ∗1). |
Hence,
|z(τ)−˜z(τ)|≤|υ−1|υIρ−θ|z(τ)−˜z(τ)|+1υIρ|Ξ(τ,z(τ),r(τ))−Ξ(τ,˜z(τ),˜r(τ))|+Gρ−1|Φ|(∇4(υ−1)υIρ−θ−η1|z(G)−˜z(G)|+∇2(υ−1)υIρ−θ+s1|z(G)−˜z(G)|+∇4υIρ−η1|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|+∇2υIρ+s1|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|)+Gρ−2|Φ|(∇3(υ−1)υIρ−θ−η1|z(G)−˜z(G)|+∇1(υ−1)υIρ−θ+s1|z(G)−˜z(G)|+∇3υIρ−η1|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|+∇1υIρ+s1|Ξ(G,z(G),r(G))−Ξ(G,˜z(G),˜r(G))|)+1υIρ|ζ(τ)|+Gρ−1|Φ|(∇4υIρ−η1|ζ(G)|+∇2υIρ+s1|ζ(G)|)+Gρ−2|Φ|(∇3υIρ−η1|ζ(G)|+∇1υIρ+s1|ζ(G)|), |
which implies that
‖z(τ)−˜z(τ)‖≤(Gρ−θ|υ−1|υΓ(ρ−θ+1)+TGρ−1υΓ(ρ)+TυΓ(ρ+1)+∇4(υ−1)G2ρ−θ−η1−1υ|Φ|Γ(ρ−θ−η1+1)+∇2(υ−1)G2ρ−θ+s1−1υ|Φ|Γ(ρ−θ+s1+1)+T∇4G2ρ−η1−1υ|Φ|Γ(ρ−η1+1)+T∇2G2ρ+s1−1υ|Φ|Γ(ρ+s1+1)+∇3(υ−1)G2ρ−θ−η1−2υ|Φ|Γ(ρ−θ−η1+1)+∇1(υ−1)G2ρ−θ+s1−2υ|Φ|Γ(ρ−θ+s1+1)+∇3G2ρ−θ−η1−2υ|Φ|Γ(ρ−θ−η1+1)+∇1G2ρ−θ+s1−2υ|Φ|Γ(ρ−θ+s1+1))‖z(τ)−˜z(τ)‖+(TGρ−1υΓ(ρ)+T∇4G2ρ−η1−1υ|Φ|Γ(ρ−η1+1)+T∇2G2ρ+s1−1υ|Φ|Γ(ρ+s1+1)+∇3G2ρ−θ−η1−2υ|Φ|Γ(ρ−θ−η1+1)∇1G2ρ−θ+s1−2υ|Φ|Γ(ρ−θ+s1+1))‖r(τ)−˜r(τ)‖+εGρΓ(ρ+1)+εGρ−1|Φ|(∇4Gρ−η1υ|Φ|Γ(ρ−η1+1)+∇2Gρ+s1υ|Φ|Γ(ρ+s1+1))+εGρ−2|Φ|(∇3Gρ−η1υ|Φ|Γ(ρ−η1+1)+∇1Gρ+s1υ|Φ|Γ(ρ+s1+1)), |
where T=max{TΞ,˜TΞ}. For simplicity, we consider
℘=Gρ−θ|υ−1|υΓ(ρ−θ+1)+TυΓ(ρ+1)+∇4(υ−1)G2ρ−θ−η1−1υ|Φ|Γ(ρ−θ−η1+1)+∇2(υ−1)G2ρ−θ+s1−1υ|Φ|Γ(ρ−θ+s1+1)+∇3(υ−1)G2ρ−θ−η1−2υ|Φ|Γ(ρ−θ−η1+1)+∇1(υ−1)G2ρ−θ+s1−2υ|Φ|Γ(ρ−θ+s1+1)+⅁, |
⅁=TGρ−1υΓ(ρ)+T∇4G2ρ−η1−1υ|Φ|Γ(ρ−η1+1)+T∇2G2ρ+s1−1υ|Φ|Γ(ρ+s1+1)+∇3G2ρ−θ−η1−2υ|Φ|Γ(ρ−θ−η1+1)+∇1G2ρ−θ+s1−2υ|Φ|Γ(ρ−θ+s1+1), |
and
S=GρΓ(ρ+1)+Gρ−1υ|Φ|(∇4Gρ−η1Γ(ρ−η1+1)+∇2Gρ+s1Γ(ρ+s1+1))+Gρ−2υ|Φ|(∇3Gρ−η1Γ(ρ−η1+1)+∇1Gρ+s1Γ(ρ+s1+1)). |
It follows that
‖z(τ)−˜z(τ)‖ℑ1−⅁1−℘‖r(τ)−˜r(τ)‖ℑ2≤Sε1−℘. | (4.2) |
Similarly, one can obtain under T∗=max{TΞ∗,˜TΞ∗} and |ζ∗(τ)|≤ϵ∗ that
‖r(τ)−˜r(τ)‖ℑ2−⅁∗1−℘∗‖z(τ)−˜z(τ)‖ℑ1≤S∗ε∗1−℘∗, | (4.3) |
where
℘∗=Gρ∗−θ∗|υ∗−1|υ∗Γ(ρ∗−θ∗+1)+T∗υ∗Γ(ρ∗+1)+∇∗4(υ∗−1)G2ρ∗−θ∗−η∗1−1υ∗|Φ∗|Γ(ρ∗−θ∗−η∗1+1)+∇∗2(υ∗−1)G2ρ∗−θ∗+s∗1−1υ∗|Φ∗|Γ(ρ∗−θ∗+s∗1+1)+∇∗3(υ∗−1)G2ρ∗−θ∗−η∗1−2υ∗|Φ∗|Γ(ρ∗−θ∗−η∗1+1)+∇∗1(υ∗−1)G2ρ∗−θ∗+s∗1−2υ∗|Φ∗|Γ(ρ∗−θ∗+s∗1+1)+⅁∗, |
⅁∗=T∗Gρ∗−1υ∗Γ(ρ∗)+T∗∇∗4G2ρ∗−η∗1−1υ∗|Φ∗|Γ(ρ∗−η∗1+1)+T∗∇∗2G2ρ∗+s∗1−1υ∗|Φ∗|Γ(ρ∗+s∗1+1)+∇∗3G2ρ∗−θ∗−η∗1−2υ∗|Φ∗|Γ(ρ∗−θ∗−η∗1+1)+∇∗3G2ρ∗−θ∗−η∗1−2υ∗|Φ∗|Γ(ρ∗−θ∗−η∗1+1), |
and
S∗=Gρ∗υ∗Γ(ρ∗+1)+Gρ∗−1υ∗|Φ∗|(∇∗4Gρ∗−η∗1Γ(ρ∗−η∗1+1)+∇∗2Gρ∗+s∗1Γ(ρ∗+s∗1+1))+Gρ∗−2υ∗|Φ∗|(∇∗3Gρ∗−η∗1Γ(ρ∗−η∗1+1)+∇1Gρ∗+s∗1Γ(ρ∗+s∗1+1)). |
Inequalities (4.2) and (4.3) can be written as
\left[ \begin{array}{ccc} 1 & & -\frac{\Game }{1-\wp } \\ & & \\ -\frac{\Game ^{\ast }}{1-\wp ^{\ast }} & & 1 \end{array} \right] \left[ \begin{array}{c} \left\Vert z(\tau )-\widetilde{z}(\tau )\right\Vert _{\Im _{1}} \\ \\ \left\Vert r(\tau )-\widetilde{r}(\tau )\right\Vert _{\Im _{2}} \end{array} \right] \leq \left[ \begin{array}{c} \frac{S\varepsilon }{1-\wp } \\ \\ \frac{S^{\ast }\varepsilon ^{\ast }}{1-\wp ^{\ast }} \end{array} \right] . |
Hence
\begin{equation} \left[ \begin{array}{c} \left\Vert z(\tau )-\widetilde{z}(\tau )\right\Vert _{\Im _{1}} \\ \\ \left\Vert r(\tau )-\widetilde{r}(\tau )\right\Vert _{\Im _{2}} \end{array} \right] \leq \left[ \begin{array}{ccc} \frac{1}{\beth } & & \frac{\Game }{1-\wp }\frac{1}{\beth } \\ & & \\ \frac{\Game ^{\ast }}{1-\wp ^{\ast }}\frac{1}{\beth } & & \frac{1}{\beth } \end{array} \right] \left[ \begin{array}{c} \frac{S\varepsilon }{1-\wp } \\ \\ \frac{S^{\ast }\varepsilon ^{\ast }}{1-\wp ^{\ast }} \end{array} \right] , \end{equation} | (4.4) |
where \beth = 1-\frac{\Game \Game ^{\ast }}{\left(1-\wp \right) \left(1-\wp ^{\ast }\right) } > 0. Based on System (4.4), one can write
\left\Vert z(\tau )-\widetilde{z}(\tau )\right\Vert _{\Im _{1}}\leq \frac{1}{ \beth }\frac{S\varepsilon }{1-\wp }+\frac{\Game S^{\ast }\varepsilon ^{\ast } }{\left( 1-\wp ^{\ast }\right) \left( 1-\wp \right) }\frac{1}{\beth }, |
and
\left\Vert r(\tau )-\widetilde{r}(\tau )\right\Vert _{\Im _{2}}\leq \frac{ \Game ^{\ast }S\varepsilon }{\left( 1-\wp ^{\ast }\right) \left( 1-\wp \right) }\frac{1}{\beth }+\frac{1}{\beth }\frac{S^{\ast }\varepsilon ^{\ast } }{1-\wp ^{\ast }}, |
which implies that
\begin{eqnarray*} \left\Vert z(\tau )-\widetilde{z}(\tau )\right\Vert _{\Im _{1}}+\left\Vert r(\tau )-\widetilde{r}(\tau )\right\Vert _{\Im _{2}} &\leq &\frac{1}{\beth } \frac{S\varepsilon }{1-\wp }+\frac{1}{\beth }\frac{S^{\ast }\varepsilon ^{\ast }}{1-\wp ^{\ast }} \\ &&+\frac{\Game S^{\ast }\varepsilon ^{\ast }}{\left( 1-\wp ^{\ast }\right) \left( 1-\wp \right) }\frac{1}{\beth }+\frac{\Game ^{\ast }S\varepsilon }{ \left( 1-\wp ^{\ast }\right) \left( 1-\wp \right) }\frac{1}{\beth }. \end{eqnarray*} |
Let us consider \widehat{\varepsilon } = \max \{\varepsilon, \varepsilon ^{\ast }\} and
\widehat{\Delta } = \frac{1}{\beth }\frac{S}{1-\wp }+\frac{1}{\beth }\frac{ S^{\ast }}{1-\wp ^{\ast }}+\frac{\Game S^{\ast }}{\left( 1-\wp ^{\ast }\right) \left( 1-\wp \right) }\frac{1}{\beth }+\frac{\Game ^{\ast }S}{ \left( 1-\wp ^{\ast }\right) \left( 1-\wp \right) }\frac{1}{\beth } > 0. |
Then, we have
\left\Vert (z,r)-(\widetilde{z},\widetilde{r})\right\Vert _{\Im }\leq \widehat{\Delta }\widehat{\epsilon },\text{ for all }\upsilon \in U, |
which yields that the CBVP (4.1) is HU stable. This completes the required proof.
Example 5.1. Consider the CBVP
\begin{equation} \left\{ \begin{array}{c} \frac{57}{64}D^{2.6}(z(\tau ))+\frac{7}{64}D^{2.1}(z(\tau )) = \tau ^{2}\left[ \sin z(\tau )+\cos r(\tau )\right] ,\text{ }\tau \in \lbrack 0,\frac{1}{5}], \\ \frac{47}{54}D^{2.7}(z(\tau ))+\frac{7}{54}D^{2.2}(z(\tau )) = \tau ^{2}\left[ \sin r(\tau )+\cos z(\tau )\right] ,\text{ }\tau \in \lbrack 0,\frac{1}{5}], \\ z(0) = 0, \ \varrho _{1}D^{\frac{1}{4}}z(\frac{1}{5})+\left( 1-\varrho _{1}\right) D^{\frac{1}{8}}z(\frac{1}{5}) = \frac{1}{18},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ r(0) = 0, \ \varrho _{1}^{\ast }D^{\frac{1}{3}}r(\frac{1}{5} )+(1-\varrho _{^{\ast }1})D^{\frac{1}{6}}r(\frac{1}{5}) = \frac{1}{16}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \varrho _{2}I^{\frac{4}{5}}z(\frac{1}{5})+(1-\varrho _{2})I^{\frac{5}{3}}z( \frac{1}{5}) = \frac{5}{13}, \ \varrho _{2}^{\ast }I^{\frac{3}{4}}r( \frac{1}{5})+(1-\varrho _{2}^{\ast })I^{\frac{7}{3}}r(\frac{1}{5}) = \frac{5}{ 17}.\text{ } \end{array} \right. \end{equation} | (5.1) |
where \rho = 2.6, \theta = 2.1, \rho ^{\ast } = 2.7, \theta ^{\ast } = 2.2, \upsilon = \frac{57}{64}, \upsilon ^{\ast } = \frac{47}{54}, \eta _{1} = \frac{1}{4}, \eta _{1}^{\ast } = \frac{1}{3}, \eta _{2} = \frac{1}{8}, \eta _{2}^{\ast } = \frac{1}{6}, s_{1} = \frac{4}{5}, s_{1}^{\ast } = \frac{3 }{4}, s_{2} = \frac{5}{3}, s_{2}^{\ast } = \frac{7}{3}, \xi _{1} = \frac{1 }{18}, \xi _{1}^{\ast } = \frac{1}{16}, \xi _{2} = \frac{5}{13}, \xi _{1}^{\ast } = \frac{5}{17} and G = \frac{1}{5} . Clearly 2 < \theta < \rho, 2 < \theta ^{\ast } < \rho ^{\ast }, \upsilon, \upsilon ^{\ast }\in (0, 1], 0\leq \eta _{1}, \eta _{2} < \rho -\theta, 0\leq \eta _{1}^{\ast }, \eta _{2}^{\ast } < \rho ^{\ast }-\theta ^{\ast }, and s_{1}, s_{2}, s_{1}^{\ast }, s_{2}^{\ast }\in \mathbb{R} ^{+}. Also, we have
\begin{eqnarray*} \left\vert \Xi (\tau ,z(\tau ),r(\tau ))-\Xi (\tau ,\widetilde{z}(\tau ), \widetilde{r}(\tau ))\right\vert &\leq &\left( \frac{1}{5}\right) ^{2}\left( \left\vert \sin z(\tau )-\sin \widetilde{z}(\tau )\right\vert +\left\vert \cos r(\tau )-\cos \widetilde{r}(\tau )\right\vert \right) , \\ \left\vert \Xi ^{\ast }(\tau ,r(\tau ),z(\tau ))-\Xi (\tau ,\widetilde{r} (\tau ),\widetilde{z}(\tau ))\right\vert &\leq &\left( \frac{1}{5}\right) ^{2}\left( \left\vert \sin r(\tau )-\sin \widetilde{r}(\tau )\right\vert +\left\vert \cos z(\tau )-\cos \widetilde{z}(\tau )\right\vert \right) . \end{eqnarray*} |
It follows that T = T^{\ast } = \widehat{T} = \frac{1}{25} and
\begin{eqnarray*} \left\vert \Xi (\tau ,z(\tau ),r(\tau ))\right\vert & = &\left\vert \tau ^{2} \left[ \sin z(\tau )+\cos r(\tau )\right] \right\vert \leq \tau ^{2}\left( \left\vert \sin z(\tau )\right\vert +\left\vert \cos r(\tau )\right\vert \right) \leq \tau ^{2} = V(\tau ), \\ \left\vert \Xi ^{\ast }(\tau ,r(\tau ),z(\tau ))\right\vert & = &\left\vert \tau ^{2}\left[ \sin r(\tau )+\cos z(\tau )\right] \right\vert \leq \tau ^{2}\left( \left\vert \sin r(\tau )\right\vert +\left\vert \cos z(\tau )\right\vert \right) \leq \tau ^{2} = V^{\ast }(\tau ). \end{eqnarray*} |
If we take \varrho _{1} = \varrho _{1}^{\ast } = \frac{1}{4} and \varrho _{2} = \varrho _{2}^{\ast } = \frac{3}{4}, we have \varrho _{1}, \varrho _{2}, \varrho _{1}^{\ast }, \varrho _{2}^{\ast }\in (0, 1] . We can easily calculate
\begin{array}{cccc} \nabla _{1}\approx 0.110255, & \nabla _{2}\approx 0.494979, & \nabla _{3}\approx 0.007777, & \nabla _{4}\approx 0.058922, \\ \nabla _{1}^{\ast }\approx 0.107356, & \nabla _{2}^{\ast }\approx 0.734601, & \nabla _{3}^{\ast }\approx 0.007162, & \nabla _{4}^{\ast }\approx 0.044779, \\ \Phi \approx 0.002646, & \Phi ^{\ast }\approx 0.000454, & \Lambda _{1} = 0.332710, & \Lambda _{2} = 0.300271, \\ \Lambda _{1}^{\ast } = 0.512841, & \Lambda _{2}^{\ast } = 0.530105, & \Lambda _{3} = 0.845551, & \Lambda _{4} = 0.830376. \end{array} |
Hence, \widehat{T}\Lambda _{4}+\Lambda _{3}\approx 0.878766 < 1. From Theorem 3.1, the CBVP (5.1) has a US.
If we take \varrho _{1} = \varrho _{1}^{\ast } = 1 and \varrho _{2} = \varrho _{2}^{\ast } = 1, we get
\Game \approx 0.007583,\text{ }\Game ^{\ast }\approx 0.058179,\text{ }\wp ^{\ast }\approx 0.036841\text{ and }\wp \approx 0.0782149. |
Since \beth = 1-\frac{0.0004412}{0.8878256}\approx 0.999503 > 0, then by Theorem 4.1, the CBVP (5.1) is HU stable with
\widehat{\Delta } = \frac{1}{\beth }\left( \frac{S}{1-\wp }+\frac{S^{\ast }}{ 1-\wp ^{\ast }}+\frac{\Game S^{\ast }}{\left( 1-\wp ^{\ast }\right) \left( 1-\wp \right) }+\frac{\Game ^{\ast }S}{\left( 1-\wp ^{\ast }\right) \left( 1-\wp \right) }\right) = 0.0258741 > 0. |
Fractional calculus has found numerous miscellaneous applications connected with real-world problems as they appear in many fields of science and engineering, including fluid flow, signal and image processing, fractal theory, control theory, electromagnetic theory, fitting of experimental data, optics, potential theory, biology, chemistry, diffusion, and viscoelasticity. Due to the many applications that have been mentioned, this branch has become of interest to many writers. Therefore, in this paper, the existence of solutions to a system of two-term FDEs with a fractional bi-order involving the Riemann-Liouville derivative has been established. Also, the considered boundaries are mixed Riemann-Liouville integro-derivative conditions with four different orders. Further, HU stability is studied, and an illustrative example has been introduced. Ultimately, we conclude that our results are new and are considered a further development of the qualitative analysis of fractional differential equations.
The authors thank the Basque Government for Grant IT1555-22.
The authors declare that they have no conflict of interests.
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