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Research article

A characterization for totally real submanifolds using self-adjoint differential operator

  • Received: 29 August 2021 Accepted: 26 September 2021 Published: 30 September 2021
  • MSC : 53C05, 53C20, 53C40

  • In this article, we study totally real submanifolds in Kaehler product manifold with constant scalar curvature using self-adjoint differential operator . Under this setup, we obtain a characterization result. Moreover, we discuss δinvariant properties of such submanifolds and get an obstruction result as an application of the inequality derived. The results in the article are supported by non-trivial examples.

    Citation: Mohd. Aquib, Amira A. Ishan, Meraj Ali Khan, Mohammad Hasan Shahid. A characterization for totally real submanifolds using self-adjoint differential operator[J]. AIMS Mathematics, 2022, 7(1): 104-120. doi: 10.3934/math.2022006

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  • In this article, we study totally real submanifolds in Kaehler product manifold with constant scalar curvature using self-adjoint differential operator . Under this setup, we obtain a characterization result. Moreover, we discuss δinvariant properties of such submanifolds and get an obstruction result as an application of the inequality derived. The results in the article are supported by non-trivial examples.



    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 α1, 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))+(1υ)Dθ(z(τ))=Ξ(τ,z(τ)), τ[0,G], ρ[1,2),z(0)=0, ϱ1Dη1z(G)+(1ϱ1)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))+(1υ)Dθ(z(τ))=Ξ(τ,z(τ)), τ[0,G], ρ[1,2),z(0)=0, ϱ2Is1z(G)+(1ϱ2)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))+(1υ)Dθ(z(τ))=Ξ(τ,z(τ),r(τ)), τ[0,G], ρ[2,3),       υDρ(r(τ))+(1υ)Dθ(r(τ))=Ξ(τ,r(τ),z(τ)), τ[0,G], ρ[2,3),z(0)=0, ϱ1Dη1z(G)+(1ϱ1)Dη2z(G)=ξ1,                                          r(0)=0, ϱ1Dη1r(G)+(1ϱ1)Dη2r(G)=ξ1,                                      ϱ2Is1z(G)+(1ϱ2)Is2z(G)=ξ2, ϱ2Is1r(G)+(1ϱ2)Is2r(G)=ξ2,          (1.4)

    where 2<θ<ρ, 2<θ<ρ, υ,υ,ϱ1,ϱ2,ϱ1,ϱ2(0,1], 0η1,η2<ρθ, 0η1,η2<ρθ, s1,s2,s1,s2R+, Dq is the RLF derivative of order q{ρ,θ,ρ,θ,η1,η2,η1,η2}, Im is the RLF integral of order m{s1,s2,s1,s2} and Ξ,Ξ:[0,G]×R2R 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 ϱ1=ϱ2=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)|:υU} and r=max{|r(υ)|:υU} is a Banach space (BS), then the products of these norms are also a BS under the norm (z,r)=z+r.

    Assume also 1 and 2 represent the piecewise continuous function spaces described as

    1=PC1(U,R)={z:UR} and 2=PC2(U,R)={r:UR},

    with norms

    z1=sup{|z(υ)|, υU} and r2=sup{|r(υ)|, υU},

    respectively. Clearly, the product =1×2 is a BS endowed with (z,r)=z1+r2.

    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)τ0(τ)ρ1z()d,

    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(nρ)(ddτ)nτ0(τ)nρ1z()d, n=[ρ]+1.

    where [ρ] refers to the integer part of real number ρ.

    Lemma 2.1. [34,35] Assume that ρ>0 and zC(0,1). Then the FDE Dρz(τ)=0 owns a general solution z(τ)=nj=1Ojτρj, where j1<ρj and the constants O1, O2,...,OnR.

    Lemma 2.2. [34] Assume that ρ>0 and zC(0,1). Then, we have

    IρDρz(τ)=z(τ)+nj=1Ojτρj,

    where j1<ρj and the constants O1, O2,...OjR.

    Lemma 2.3. [4] Assume that ρ,θ>0 with ρ>θ, then Iρ0+Dθ0+=Iρθ0+.

    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 . Let Ω,Ω:SS be operators such that

    (1) Ω(z)+Ω(r)S, where z,rS;

    (2) Ω is a contraction mapping;

    (3) Ω is completely continuous.

    Then there exists zS 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)=υ1υΓ(ρθ)τ0(τ)ρθ1z()d+1υΓ(ρ)τ0(τ)ρ1Ξ(,z(),r())d+τρ1Φ(ϱ14(υ1)υIρθη1z(G)ϱ22(υ1)υIρθs1z(G)+4(1ϱ1)(υ1)υIρθη2z(G)2(1ϱ2)(υ1)υIρθs2z(G)+ϱ14υIρη1Ξ(G,z(G),r(G))ϱ22υIρ+s1Ξ(G,z(G),r(G))+2ξ24ξ1+(1ϱ1)4υIρη2Ξ(G,z(G),r(G))(1ϱ2)2υIρ+s2Ξ(G,z(G),r(G)))τρ2Φ(ϱ13(υ1)υIρθη1z(G)ϱ21(υ1)υIρθs1z(G)+3(1ϱ1)(υ1)υIρθη2z(G)1(1ϱ2)(υ1)υIρθs2z(G)+ϱ13υIρη1Ξ(G,z(G),r(G))ϱ21υIρ+s1Ξ(G,z(G),r(G))+1ξ23ξ1+(1ϱ1)3υIρη2Ξ(G,z(G),r(G))(1ϱ2)1υIρ+s2Ξ(G,z(G),r(G))), (3.1)

    and

    r(τ)=υ1υΓ(ρθ)τ0(τ)ρθ1r()d+1υΓ(ρ)τ0(τ)ρ1Ξ(,r(),z())d+τρ1Φ(ϱ14(υ1)υIρθη1r(G)ϱ22(υ1)υIρθ+s1r(G)+4(1ϱ1)(υ1)υIρθη2r(G)2(1ϱ2)(υ1)υIρθ+s2r(G)+ϱ14υIρη1Ξ(G,r(G),z(G))ϱ22υIρ+s1Ξ(G,r(G),z(G))+2ξ24ξ1+(1ϱ1)4υIρη2Ξ(G,r(G),z(G))(1ϱ2)2υIρ+s2Ξ(G,r(G),z(G)))τρ2Φ(ϱ13(υ1)υIρθη1r(G)ϱ21(υ1)υIρθ+s1r(G)+3(1ϱ1)(υ1)υIρθη2r(G)1(1ϱ2)(υ1)υIρθ+s2r(G)ϱ13υIρη1Ξ(G,r(G),z(G))ϱ21υIρ+s1Ξ(G,r(G),z(G))+1ξ23ξ1+(1ϱ1)3υIρη2Ξ(G,r(G),z(G))(1ϱ2)1υIρ+s2Ξ(G,r(G),z(G))), (3.2)

    where

    1=ϱ1Γ(ρ)Γ(ρη1)Gρη11+(1ϱ1)Γ(ρ)Γ(ρη2)Gρη21,    1=ϱ1Γ(ρ)Γ(ρη1)Gρη11(1ϱ1)Γ(ρ)Γ(ρη2)Gρη21,    2=ϱ1Γ(ρ1)Γ(ρη11)Gρη12+(1ϱ1)Γ(ρ1)Γ(ρη21)Gρη22,2=ϱ1Γ(ρ1)Γ(ρη11)Gρη12(1ϱ1)Γ(ρ1)Γ(ρη21)Gρη22,3=ϱ2Γ(ρ)Γ(ρ+s1)Gρ+s11+(1ϱ2)Γ(ρ)Γ(ρ+s2)Gρ+s21,     3=ϱ2Γ(ρ)Γ(ρ+s1)Gρ+s11(1ϱ2)Γ(ρ)Γ(ρ+s2)Gρ+s21,     4=ϱ2Γ(ρ1)Γ(ρ+s11)Gρ+s12+(1ϱ2)Γ(ρ1)Γ(ρ+s21)Gρ+s22,4=ϱ2Γ(ρ1)Γ(ρ+s11)Gρ+s12(1ϱ2)Γ(ρ1)Γ(ρ+s21)Gρ+s22,Φ=3214,                                 Φ=3214.                                      (3.3)

    Proof. Let (z0,r0) be a solution for the Eq (1.4), then, we get

    {Dρz0(τ)=(υ1)υDθz0(τ)+1υΞ(τ,z0(τ),r0(τ)),Dρr0(τ)=(υ1)υDθz0(τ)+1υΞ(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)=υ1υΓ(ρθ)τ0(τ)ρθ1z0()d+1υΓ(ρ)τ0(τ)ρ1Ξ(τ,z0(τ),r0(τ))d+O1τρ1+O2τρ2+O3τρ3,

    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)=υ1υIρθz0(τ)+1υIρΞ(τ,z0(τ),r0(τ))+O1τρ1+O2τρ2. (3.5)

    Using the RLF integral and derivative of order η and s, respectively with η{η1,η2}, s{s1,s2}, 0<η<ρθ and 2<θ<ρ, we obtain

    Dρz0(τ)=υ1υΓ(ρθη)τ0(τ)ρθη1z0()d+O1Γ(ρ)Γ(ρη)τρη1+1υΓ(ρη)τ0(τ)ρη1Ξ(,z0(),r0())d+O2Γ(ρ1)Γ(ρη1)τρη2.

    and

    Iρz0(τ)=υ1υΓ(ρθ+s)τ0(τ)ρθ+s1z0()d+O1Γ(ρ)Γ(ρ+s)τρ+s1+1υΓ(ρ+s)τ0(τ)ρ+s1Ξ(,z0(),r0())d+O2Γ(ρ1)Γ(ρ+s1)τρ+s2.

    Replacing η=η1, η=η2, s=s1, s=s2 and using the boundary stipulations ϱ1Dη1z(G)+(1ϱ1)Dη2z(G)=ξ1 and ϱ2Is1z(G)+(1ϱ2)Is2z(G)=ξ2, we can write

    ξ1=ϱ1(υ1)υΓ(ρθη1)G0(G)ρθη11z0()d+(1ϱ1)(υ1)υΓ(ρθη2)G0(G)ρθη21z0()d+ϱ1υΓ(ρη1)G0(G)ρη11Ξ(,z0(),r0())d+O11+(1ϱ1)υΓ(ρη2)G0(G)ρη21Ξ(,z0(),r0())d+O22,

    and

    ξ2=ϱ2(υ1)υΓ(ρθ+s1)G0(G)ρθ+s11z0()d+(1ϱs)(υ1)υΓ(ρθ+s2)G0(G)ρθ+s21z0()d+ϱ2υΓ(ρ+s1)G0(G)ρ+s11Ξ(,z0(),r0())d+O13+(1ϱ2)υΓ(ρ+s2)G0(G)ρ+s21Ξ(,z0(),r0())d+O24,

    which yields that

    O1=ϱ14(υ1)υIρθη1z0(G)ϱ22(υ1)υIρθ+s1z0(G)+4(1ϱ1)(υ1)υIρθη2z0(G)2(1ϱ2)(υ1)υIρθ+s2z0(G)+ϱ14υIρη1Ξ(G,z0(G),r0(G))ϱ22υIρ+s1Ξ(G,z0(G),r0(G))+2ξ24ξ1+(1ϱ1)3υIρη2Ξ(G,z0(G),r0(G))(1ϱ2)1υIρ+s2Ξ(G,z0(G),r0(G)).

    and

    O2=ϱ13(υ1)υIρθη1z0(G)ϱ21(υ1)υIρθ+s1z0(G)+3(1ϱ1)(υ1)υIρθη2z0(G)1(1ϱ2)(υ1)υIρθ+s2z0(G)+ϱ13υIρη1Ξ(G,z0(G),r0(G))ϱ21υIρ+s1Ξ(G,z0(G),r0(G))+1ξ23ξ1+(1ϱ1)3υIρη2Ξ(G,z0(G),r0(G))(1ϱ2)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: by

    Ω(z,r)=(Ω1(z,r),Ω2(z,r)),

    where

    Ω1(z,r)=υ1υΓ(ρθ)τ0(τ)ρθ1z()d+1υΓ(ρ)τ0(τ)ρ1Ξ(,z(),r())d+τρ1Φ(ϱ14(υ1)υIρθη1z(G)ϱ22(υ1)υIρθ+s1z(G)+4(1ϱ1)(υ1)υIρθη2z(G)2(1ϱ2)(υ1)υIρθ+s2z(G)+ϱ14υIρη1Ξ(G,z(G),r(G))ϱ22υIρ+s1Ξ(G,z(G),r(G))+2ξ24ξ1+(1ϱ1)4υIρη2Ξ(G,z(G),r(G))(1ϱ2)2υIρ+s2Ξ(G,z(G),r(G)))τρ2Φ(ϱ13(υ1)υIρθη1z(G)ϱ21(υ1)υIρθ+s1z(G)+3(1ϱ1)(υ1)υIρθη2z(G)1(1ϱ2)(υ1)υIρθ+s2z(G)+ϱ13υIρη1Ξ(G,z(G),r(G))ϱ21υIρ+s1Ξ(G,z(G),r(G))+1ξ23ξ1+(1ϱ1)3υIρη2Ξ(G,z(G),r(G))(1ϱ2)1υIρ+s2Ξ(G,z(G),r(G))), (3.6)

    and

    Ω2(z,r)=υ1υΓ(ρθ)τ0(τ)ρθ1r()d+1υΓ(ρ)τ0(τ)ρ1Ξ(,r(),z())d+τρ1Φ(ϱ14(υ1)υIρθη1r(G)ϱ22(υ1)υIρθs1r(G)+4(1ϱ1)(υ1)υIρθη2r(G)2(1ϱ2)(υ1)υIρθs2r(G)+ϱ14υIρη1Ξ(G,r(G),z(G))ϱ22υIρ+s1Ξ(G,r(G),z(G))+2ξ24ξ1+(1ϱ1)4υIρη2Ξ(G,r(G),z(G))(1ϱ2)2υIρ+s2Ξ(G,r(G),z(G)))τρ2Φ(ϱ13(υ1)υIρθη1r(G)ϱ21(υ1)υIρθs1r(G)+3(1ϱ1)(υ1)υIρθη2r(G)1(1ϱ2)(υ1)υIρθs2r(G)ϱ13υIρη1Ξ(G,r(G),z(G))ϱ21υIρ+s1Ξ(G,r(G),z(G))+1ξ23ξ1+(1ϱ1)3υIρη2Ξ(G,r(G),z(G))(1ϱ2)1υIρ+s2Ξ(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=|υ1|(4+3G1)|Φ|(ϱ1G2ρθη11υΓ(ρθη1+1)+(1ϱ1)G2ρθη21υΓ(ρθη2+1))+|υ1|(2+1G1)|Φ|(ϱ2G2ρθ+s11υΓ(ρθ+s1+1)+(1ϱ2)G2ρθ+s21υΓ(ρθ+s2+1))+|υ1|GρθυΓ(ρθ+1). (3.8)
    Λ1=|υ1|(4+3G1)|Φ|(ϱ1G2ρθη11υΓ(ρθη1+1)+(1ϱ1)G2ρθη21υΓ(ρθη2+1))+|υ1|(4+3G1)|Φ|(ϱ2G2ρθ+s11υΓ(ρθ+s1+1)+(1ϱ2)G2ρθ+s21υΓ(ρθ+s2+1))+|υ1|GρθυΓ(ρθ+1). (3.9)
    Λ2=GρυΓ(ρ+1)+4+3G1|Φ|(ϱ1G2ρη11υΓ(ρη1+1)+(1ϱ1)G2ρη21υΓ(ρη2+1))+2+1G1|Φ|(ϱ2G2ρ+s11υΓ(ρ+s1+1)+(1ϱ2)G2ρ+s21υΓ(ρ+s2+1)). (3.10)
    Λ2=GρυΓ(ρ+1)+(4+3G1)|Φ|(ϱ1G2ρη11υΓ(ρη1+1)+(1ϱ1)G2ρη21υΓ(ρη2+1))+4+3G1|Φ|(ϱ2G2ρ+s11υΓ(ρ+s1+1)+(1ϱ2)G2ρ+s21υΓ(ρ+s2+1)). (3.11)

    Now, our main theorem is as follows:

    Theorem 3.1. Assume that the mappings Ξ,Ξ:U×R2R 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 τU and z1,z2,˜z1,˜z2,r1,r2,˜r1,˜r2R. 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+Λ1=Λ3 and Λ2+Λ2=Λ4 and Λ1, Λ1, Λ2, Λ2 are described as (3.8)–(3.11), respectively.

    Proof. Set supυUΞ(τ,0,0)=N<, supυUΞ(τ,0,0)=N< and choose

    yˆNΛ41ˆTΛ4Λ3+(|Φ|+|Φ|)Gρ1(|2ξ1|+|4ξ2|+|2ξ1|+|4ξ2|)|Φ||Φ|(1ˆTΛ4Λ3)+Gρ2(|2ξ1|+|4ξ2|+|2ξ1|+|4ξ2|)|Φ||Φ|(1ˆTΛ4Λ3), (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 ΩQyQy, where Qy={(z,r):(z,r)y}. For any (z,r)Qy, we have

    Ω(z,r)=Ω1(z,r)1+Ω2(z,r)2 (3.13)

    From (3.6) and (3.7), we get

    |Ω1(z,r)||υ1|υΓ(ρθ)τ0(τ)ρθ1|z()|d+1υΓ(ρ)τ0(τ)ρ1(|Ξ(,z(),r())Ξ(,0,0)|+|Ξ(,0,0)|)d+Gρ1|Φ|(ϱ14|υ1|υIρθη1|z(G)|ϱ22|υ1|υIρθ+s1|z(G)|+4|1ϱ1|(|υ1|)υIρθη2|z(G)|2|1ϱ2||υ1|υIρθ+s2|z(G)|+ϱ14υIρη1[|Ξ(G,z(G),r(G))Ξ(G,0,0)|+|Ξ(G,0,0)|]ϱ22υIρ+s1[|Ξ(G,z(G),r(G))Ξ(G,0,0)|+|Ξ(G,0,0)|]+|2ξ2|+|4ξ1|+(1ϱ1)4υIρη2[|Ξ(G,z(G),r(G))Ξ(G,0,0)|+|Ξ(G,0,0)|](1ϱ2)2υIρ+s2[|Ξ(G,z(G),r(G))Ξ(G,0,0)|+|Ξ(G,0,0)|])Gρ2Φ(ϱ13|υ1|υIρθη1|z(G)|ϱ21|υ1|υIρθ+s1|z(G)|+3|1ϱ1|(|υ1|)υIρθη2|z(G)|1|1ϱ2||υ1|υIρθ+s2|z(G)|+ϱ13υIρη1[|Ξ(G,z(G),r(G))Ξ(G,0,0)|+|Ξ(G,0,0)|]ϱ22υIρ+s1[|Ξ(G,z(G),r(G))Ξ(G,0,0)|+|Ξ(G,0,0)|]+|1ξ2|+|3ξ1|+(1ϱ1)3υIρη2[|Ξ(G,z(G),r(G))Ξ(G,0,0)|+|Ξ(G,0,0)|](1ϱ2)1υIρ+s2[|Ξ(G,z(G),r(G))Ξ(G,0,0)|+|Ξ(G,0,0)|]),

    which implies that

    Ω1(z,r)1(T(z,r)+N)Λ2+(z,r)Λ1+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]=(TΛ2+Λ1)(z,r)+NΛ2+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]. (3.14)

    In the same scenario, we can write

    Ω2(z,r)2(TΛ2+Λ1)(z,r)+NΛ2+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]. (3.15)

    Applying (3.14) and (3.15) in (3.13) and using (3.12), we have

    Ω(z,r)=Ω1(z,r)1+Ω2(z,r)2=(TΛ2+Λ1+TΛ2+Λ1)(z,r)+NΛ2+NΛ2+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)](TΛ2+Λ1+TΛ2+Λ1)y+NΛ2+NΛ2+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)](ˆTΛ4+Λ3)y+ˆNΛ4+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]y+

    Hence, Ω(z,r)y and so ΩQyQy. For each υU and for z,r,˜z,˜r, we get

    |Ω1(z,r)(τ)Ω1(˜z,˜r)(υ)||υ1|υΓ(ρθ)τ0(τ)ρθ1|z()˜z()|d+1υΓ(ρ)τ0(τ)ρ1|Ξ(,z(),r())Ξ(,˜z(),˜r())|d+Gρ1|Φ|(ϱ14|υ1|υIρθη1|z(G)˜z(G)|ϱ22|υ1|υIρθ+s1|z(G)˜z(G)|+4|1ϱ1|(|υ1|)υIρθη2|z(G)˜z(G)|2|1ϱ2||υ1|υIρθ+s2|z(G)˜z(G)|+ϱ14υIρη1|Ξ(G,z(G),r(G))Ξ(G,˜z(G),˜r(G))|
    ϱ22υIρ+s1|Ξ(G,z(G),r(G))Ξ(G,˜z(G),˜r(G))|+(1ϱ1)4υIρη2|Ξ(G,z(G),r(G))Ξ(G,˜z(G),˜r(G))|(1ϱ2)2υIρ+s2|Ξ(G,z(G),r(G))Ξ(G,˜z(G),˜r(G))|)Gρ2Φ(ϱ13|υ1|υIρθη1|z(G)|ϱ21|υ1|υIρθ+s1|z(G)|+3|1ϱ1|(|υ1|)υIρθη2|z(G)˜z(G)|1|1ϱ2||υ1|υIρθ+s2|z(G)˜z(G)|+ϱ13υIρη1|Ξ(G,z(G),r(G))Ξ(G,˜z(G),˜r(G))|ϱ22υIρ+s1|Ξ(G,z(G),r(G))Ξ(G,˜z(G),˜r(G))|+(1ϱ1)3υIρη2|Ξ(G,z(G),r(G))Ξ(G,˜z(G),˜r(G))|(1ϱ2)1υIρ+s2|Ξ(G,z(G),r(G))Ξ(G,˜z(G),˜r(G))|),

    which leads to

    Ω1(z,r)Ω1(˜z,˜r)1TΛ2(z˜z+r˜r)+z˜zΛ1.

    Similarly, one can obtain

    Ω2(z,r)Ω2(˜z,˜r)2TΛ2(z˜z+r˜r)+r˜rΛ1.

    Hence,

    Ω(z,r)Ω(˜z,˜r)Ω1(z,r)Ω1(˜z,˜r)1+Ω2(z,r)Ω2(˜z,˜r)2=TΛ2(z˜z+r˜r)+z˜zΛ1+TΛ2(z˜z+r˜r)+r˜rΛ1=(Λ1+TΛ2+TΛ2)z˜z+(Λ1+TΛ2+TΛ2)r˜r(Λ1+ˆTΛ4)z˜z+(Λ1+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×R2R 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 τU and z1,z2,˜z1,˜z2,r1,r2,˜r1,˜r2R. 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 supτU|V(τ)|=V, suptauU|V(τ)|=V and the set Qx={(z,r):(z,r)x}, where

    xˆVΛ31Λ4+(|Φ|+|Φ|)Gρ1(|2ξ2|+|4ξ1|+|2ξ2|+|4ξ1|)|Φ||Φ|(1Λ4)+(|Φ|+|Φ|)Gρ2(|1ξ2|+|1ξ1|+|1ξ2|+|3ξ1|)|Φ||Φ|(1Λ4),

    and i, i, i{1,2,3,4}, Φ and Φ are defined by (3.3), ˆV=max{V,V} and Λ3=Λ1+Λ1. For any (z,r)Qx, define the operators Ω,Ω: by

    Ω(z,r)=˜Ω1(z,r)+˜Ω2(z,r) and Ω(z,r)=ˆΩ1(z,r)+ˆΩ2(z,r),

    where

    ˜Ω1(z,r)=υ1υΓ(ρθ)τ0(τ)ρθ1z()d+τρ1Φ×(ϱ14(υ1)υIρθη1z(G)ϱ22(υ1)υIρθ+s1z(G)+4(1ϱ1)(υ1)υIρθη2z(G)2(1ϱ2)(υ1)υIρθ+s2z(G))τρ2Φ(ϱ13(υ1)υIρθη1z(G)ϱ21(υ1)υIρθ+s1z(G)+3(1ϱ1)(υ1)υIρθη2z(G)1(1ϱ2)(υ1)υIρθ+s2z(G)), (3.16)
    ˆΩ1(z,r)=1υΓ(ρ)τ0(τ)ρ1Ξ(,z(),r())d+τρ1Φ×(ϱ14υIρη1Ξ(G,z(G),r(G))ϱ22υIρ+s1Ξ(G,z(G),r(G))+2ξ24ξ1+(1ϱ1)4υIρη2Ξ(G,z(G),r(G))(1ϱ2)2υIρ+s2Ξ(G,z(G),r(G)))τρ2Φ(ϱ13υIρη1Ξ(G,z(G),r(G))ϱ21υIρ+s1Ξ(G,z(G),r(G))+(1ϱ1)3υIρη2Ξ(G,z(G),r(G))(1ϱ2)1υIρ+s2Ξ(G,z(G),r(G))+1ξ23ξ1), (3.17)
    ˜Ω2(z,r)=υ1υΓ(ρθ)τ0(τ)ρθ1r()d+τρ1Φ×(ϱ14(υ1)υIρθη1r(G)ϱ22(υ1)υIρθ+s1r(G)4(1ϱ1)(υ1)υIρθη2r(G)2(1ϱ2)(υ1)υIρθ+s2r(G))τρ2Φ(ϱ13(υ1)υIρθη1r(G)ϱ21(υ1)υIρθ+s1r(G)+3(1ϱ1)(υ1)υIρθη2r(G)1(1ϱ2)(υ1)υIρθ+s2r(G)), (3.18)

    and

    ˆΩ2(z,r)=1υΓ(ρ)τ0(τ)ρ1Ξ(,r(),z())d+τρ1Φ×(+ϱ14υIρη1Ξ(G,r(G),z(G))ϱ22υIρ+s1Ξ(G,r(G),z(G))+2ξ24ξ1+(1ϱ1)4υIρη2Ξ(G,r(G),z(G))(1ϱ2)2υIρ+s2Ξ(G,r(G),z(G)))τρ2Φ(ϱ13υIρη1Ξ(G,r(G),z(G))ϱ21υIρ+s1Ξ(G,r(G),z(G))++(1ϱ1)3υIρη2Ξ(G,r(G),z(G))(1ϱ2)1υIρ+s2Ξ(G,r(G),z(G))+1ξ23ξ1). (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+3G1|Φ|(ϱ1G2ρη11υΓ(ρη1+1)+(1ϱ1)G2ρη21υΓ(ρη2+1))+2+1G1|Φ|(ϱ2G2ρ+s11υΓ(ρ+s1+1)+(1ϱ2)G2ρ+s21υΓ(ρ+s2+1))]+(z,r)[|υ1|GρθυΓ(ρθ+1)+|υ1|(4+3G1)|Φ|(ϱ1G2ρθη11υΓ(ρθη1+1)+(1ϱ1)G2ρθη21υΓ(ρθη2+1))+|υ1|(2+1G1)|Φ|(ϱ2G2ρθ+s11υΓ(ρθ+s1+1)+(1ϱ2)G2ρθ+s21υΓ(ρθ+s2+1))+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]VΛ1+yΛ2+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]. (3.20)

    Analogously, using (3.18) and (3.19), we get

    |˜Ω2(z,r)(τ)+ˆΩ2(z,r)(τ)|VΛ1+yΛ2+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]. (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)|=VΛ1+yΛ2+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]+VΛ1+yΛ2+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]=ˆVΛ3+Λ4y+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]+1|Φ|[Gρ1(|2ξ2|+|4ξ1|)+Gρ2(|1ξ2|+|3ξ1|)]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)||υ1|υΓ(ρθ)τ0(τ)ρθ1|z()˜z()|d+Gρ1Φ×+(ϱ14|υ1|υIρθη1|z(G)˜z(G)|ϱ22|υ1|υIρθ+s1|z(G)˜z(G)|+4(1ϱ1)|υ1|υIρθη2|z(G)˜z(G)|2(1ϱ2)|υ1|υIρθ+s2|z(G)˜z(G)|)+Gρ2Φ(ϱ13|υ1|υIρθη1|z(G)˜z(G)|ϱ21|υ1|υIρθ+s1|z(G)˜z(G)|+3(1ϱ1)|υ1|υIρθη2|z(G)˜z(G)|1(1ϱ2)|υ1|υIρθ+s2|z(G)˜z(G)|)Λ1zz.

    Similarly, we can write

    |˜Ω2(z,r)(τ)˜Ω2(˜z,˜r)(τ)|Λ1rr.

    It follows that

    Ω(z,r)Ω(˜z,˜r)˜Ω1(z,r)˜Ω1(˜z,˜r)1+˜Ω2(z,r)˜Ω2(˜z,˜r)2=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)1V[GρυΓ(ρ+1)+4+3G1|Φ|(ϱ1G2ρη11υΓ(ρη1+1)+(1ϱ1)G2ρη21υΓ(ρη2+1))+2+1G1|Φ|(ϱ2G2ρ+s11υΓ(ρ+s1+1)+(1ϱ2)G2ρ+s21υΓ(ρ+s2+1))=Λ2(z,r).

    Similarly, we have

    ˆΩ2(z,r)2Λ2(z,r).

    Hence,

    Ω(z,r)ˆΩ1(z,r)1+ˆΩ1(z,r)2Λ4(z,r), where Λ4=Λ2+Λ2.

    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, supτUΞ(τ,z(τ),r(τ))=R, and supτUΞ(τ,z(τ),r(τ))=R. Then, for each τ1,τ2U with τ1<τ2, we get

    |ˆΩ1(z,r)(τ2)ˆΩ1(z,r)(τ1)||1υΓ(ρ)τ20(τ2)ρ1Ξ(,z(),r())d1υΓ(ρ)τ10(τ1)ρ1Ξ(,z(),r())dτρ12τρ11Φ(ϱ14υIρη1Ξ(G,z(G),r(G))ϱ22υIρ+s1Ξ(G,z(G),r(G))+2ξ24ξ1+(1ϱ1)4υIρη2Ξ(G,z(G),r(G))(1ϱ2)2υIρ+s2Ξ(G,z(G),r(G)))τρ12τρ11Φ(ϱ13υIρη1Ξ(G,z(G),r(G))ϱ21υIρ+s1Ξ(G,z(G),r(G))(1ϱ1)3υIρη2Ξ(G,z(G),r(G))(1ϱ2)1υIρ+s2Ξ(G,z(G),r(G))+1ξ23ξ1)|R[2(τ2τ1)ρ+|τρ2τρ1|]υΓ(ρ+1)+τρ12τρ11|Φ|[Rϱ14Iρη1υΓ(ρη1+1)+Rϱ22Iρ+s1υΓ(ρ+s1+1)+|2ξ2|+|4ξ1|+R(1ϱ1)4Iρη2υΓ(ρη2+1)+R(1ϱ2)2Iρ+s2υΓ(ρ+s2+1)]+τρ22τρ21|Φ|[Rϱ13Iρη1υΓ(ρη1+1)+Rϱ21Iρ+s1υΓ(ρ+s1+1)+R(1ϱ1)3Iρη2υΓ(ρη2+1)+R(1ϱ2)1Iρ+s2υΓ(ρ+s2+1)+|1ξ2|+|3ξ1|],

    which implies that

    |ˆΩ1(z,r)(τ2)ˆΩ1(z,r)(τ1)|0 as τ1τ2.

    Similarly

    |ˆΩ2(z,r)(τ2)ˆΩ2(z,r)(τ1)|0 as τ1τ2.

    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 τ1τ2,

    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))+(1υ)Dθ(z(τ))=Ξ(τ,z(τ),r(τ)),       υDρ(r(τ))+(1υ)Dθ(r(τ))=Ξ(τ,r(τ),z(τ)),z(0)=0, r(0)=0, Dη1z(G)=ξ1, Dη1r(G)=ξ1,  Is1z(G)=ξ2, Is1r(G)=ξ2,                                 (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 ϱ1=ϱ2=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) as a solution to the inequalities

    {|υDρ(z(τ))+(υ1)Dθ(z(τ))Ξ(τ,z(τ),r(τ))|ϵ,          |υDρ(r(τ))+(υ1)Dθ(r(τ))Ξ(τ,r(τ),z(τ))|ϵ,

    there is a US (˜z,˜r) with

    (z,r)(˜z,˜r)ˆΔˆϵ, for all υU,

    where ˆϵ=max{ϵ,ϵ}.

    Theorem 4.1. Assume that Ξ,Ξ:U×R2R 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 τU and z1,z2,˜z1,˜z2,r1,r2,˜r1,˜r2R. Then, the CBVP (4.1) is HU stable provided that =1(1)(1)>0.

    Proof. Let ϵ,ϵ>0 and (z,r) be so that

    {|υDρ(z(τ))+(υ1)Dθ(z(τ))Ξ(τ,z(τ),r(τ))|ϵ,          |υDρ(r(τ))+(υ1)Dθ(r(τ))Ξ(τ,r(τ),z(τ))|ϵ.

    Choose the functions ζ and ζ satisfying

    {υDρ(z(τ))+(υ1)Dθ(z(τ))=Ξ(τ,z(τ),r(τ))+ζ(τ),         υDρ(r(τ))+(υ1)Dθ(r(τ))=Ξ(τ,r(τ),z(τ))+ζ(τ),

    such that |ζ(τ)|ϵ and |ζ(τ)|ϵ for all τU. Then, we get

    z(τ)=υ1υIρθz(τ)+1υIρΞ(τ,z(τ),r(τ))+1υIρζ(τ)+τρ1Φ(4(υ1)υIρθη1z(G)2(υ1)υIρθ+s1z(G)+4υIρη1Ξ(G,z(G),r(G))2υIρ+s1Ξ(G,z(G),r(G))+4υIρη1ζ(G)2υIρ+s1ζ(G)+2ξ24ξ1)+τρ2Φ(3(υ1)υIρθη1z(G)1(υ1)υIρθ+s1z(G)+3υIρη1Ξ(G,z(G),r(G))1υIρ+s1Ξ(G,z(G),r(G))+3υIρη1ζ(G)1υIρ+s1ζ(G)+1ξ23ξ1),

    and

    r(τ)=υ1υIρθr(τ)+1υIρΞ(τ,r(τ),z(τ))+1υIρζ(τ)+τρ1Φ(4(υ1)υIρθη1r(G)2(υ1)υIρθ+s1r(G)+4υIρη1Ξ(G,r(G),z(G))2υIρ+s1Ξ(G,r(G),z(G))+4υIρη1ζ(G)2υIρ+s1ζ(G)+2ξ24ξ1)τρ2Φ(3(υ1)υIρθη1r(G)1(υ1)υIρθ+s1r(G)+3υIρη1Ξ(G,r(G),z(G))1υIρ+s1Ξ(G,r(G),z(G))+3υIρη1ζ(G)1υIρ+s1ζ(G)+1ξ23ξ1).

    Let (˜z,˜r) be a US of the CBVP (4.1), then ˜z(τ) and ˜r(τ) are given by

    ˜z(τ)=υ1υIρθ˜z(τ)+1υIρΞ(τ,˜z(τ),˜r(τ))+1υIρζ(τ)+τρ1Φ(4(υ1)υIρθη1˜z(G)2(υ1)υIρθ+s1˜z(G)+4υIρη1Ξ(τ,˜z(τ),˜r(τ))2υIρ+s1Ξ(τ,˜z(τ),˜r(τ))+4υIρη1ζ(G)2υIρ+s1ζ(G)+2ξ24ξ1)+τρ2Φ(3(υ1)υIρθη1˜z(G)1(υ1)υIρθ+s1˜z(G)+3υIρη1Ξ(τ,˜z(τ),˜r(τ))1υIρ+s1Ξ(τ,˜z(τ),˜r(τ))+3υIρη1ζ(G)1υIρ+s1ζ(G)+1ξ23ξ1),

    and

    \begin{eqnarray*} &&\widetilde{r}(\tau ) \\ & = &\frac{\upsilon ^{\ast }-1}{\upsilon ^{\ast }}I^{\rho ^{\ast }-\theta ^{\ast }}\widetilde{r}(\tau )+\frac{1}{\upsilon ^{\ast }}I^{\rho ^{\ast }}\Xi ^{\ast }(\tau ,\widetilde{r}(\tau ),\widetilde{z}(\tau ))+\frac{1}{ \upsilon }I^{\rho ^{\ast }}\zeta ^{\ast }(\tau )+\frac{\tau ^{\rho ^{\ast }-1}}{\Phi ^{\ast }}\left( \frac{\nabla _{4}^{\ast }(\upsilon ^{\ast }-1)}{ \upsilon ^{\ast }}I^{\rho ^{\ast }-\theta ^{\ast }-\eta _{1}^{\ast }} \widetilde{r}(G)\right. \\ &&-\frac{\nabla _{2}^{\ast }(\upsilon ^{\ast }-1)}{\upsilon ^{\ast }}I^{\rho ^{\ast }-\theta ^{\ast }+s_{1}^{\ast }}\widetilde{r}(G)+\frac{\nabla _{4}^{\ast }}{\upsilon ^{\ast }}I^{\rho ^{\ast }-\eta _{1}^{\ast }}\Xi ^{\ast }(\tau ,\widetilde{r}(\tau ),\widetilde{z}(\tau ))-\frac{\nabla _{2}^{\ast }}{\upsilon ^{\ast }}I^{\rho ^{\ast }+s_{1}^{\ast }}\Xi ^{\ast }(\tau ,\widetilde{r}(\tau ),\widetilde{z}(\tau )) \\ &&\left. +\frac{\nabla _{4}^{\ast }}{\upsilon ^{\ast }}I^{\rho ^{\ast }-\eta _{1}^{\ast }}\zeta ^{\ast }(G)-\frac{\nabla _{2}^{\ast }}{\upsilon ^{\ast }} I^{\rho ^{\ast }+s_{1}^{\ast }}\zeta ^{\ast }(G)+\nabla _{2}^{\ast }\xi _{2}^{\ast }-\nabla _{4}^{\ast }\xi _{1}^{\ast }\right) -\frac{\tau ^{\rho ^{\ast }-2}}{\Phi ^{\ast }}\left( \frac{\nabla _{3}^{\ast }(\upsilon ^{\ast }-1)}{\upsilon ^{\ast }}I^{\rho ^{\ast }-\theta ^{\ast }-\eta _{1}^{\ast }} \widetilde{r}(G)\right. \\ &&-\frac{\nabla _{1}^{\ast }(\upsilon ^{\ast }-1)}{\upsilon ^{\ast }}I^{\rho ^{\ast }-\theta ^{\ast }+s_{1}^{\ast }}\widetilde{r}(G)+\frac{\nabla _{3}^{\ast }}{\upsilon ^{\ast }}I^{\rho ^{\ast }-\eta _{1}^{\ast }}\Xi ^{\ast }(\tau ,\widetilde{r}(\tau ),\widetilde{z}(\tau ))-\frac{\nabla _{1}^{\ast }}{\upsilon ^{\ast }}I^{\rho ^{\ast }+s_{1}^{\ast }}\Xi ^{\ast }(\tau ,\widetilde{r}(\tau ),\widetilde{z}(\tau )) \\ &&\left. +\frac{\nabla _{3}^{\ast }}{\upsilon ^{\ast }}I^{\rho ^{\ast }-\eta _{1}^{\ast }}\zeta ^{\ast }(G)-\frac{\nabla _{1}^{\ast }}{\upsilon ^{\ast }} I^{\rho ^{\ast }+s_{1}^{\ast }}\zeta ^{\ast }(G)+\nabla _{1}^{\ast }\xi _{2}^{\ast }-\nabla _{3}^{\ast }\xi _{1}^{\ast }\right) . \end{eqnarray*}

    Hence,

    \begin{eqnarray*} &&\left\vert z(\tau )-\widetilde{z}(\tau )\right\vert \\ &\leq &\frac{\left\vert \upsilon -1\right\vert }{\upsilon }I^{\rho -\theta }\left\vert z(\tau )-\widetilde{z}(\tau )\right\vert +\frac{1}{\upsilon } I^{\rho }\left\vert \Xi (\tau ,z(\tau ),r(\tau ))-\Xi (\tau ,\widetilde{z} (\tau ),\widetilde{r}(\tau ))\right\vert \\ &&+\frac{G^{\rho -1}}{\left\vert \Phi \right\vert }\left( \frac{\nabla _{4}(\upsilon -1)}{\upsilon }I^{\rho -\theta -\eta _{1}}\left\vert z(G)- \widetilde{z}(G)\right\vert +\frac{\nabla _{2}(\upsilon -1)}{\upsilon } I^{\rho -\theta +s_{1}}\left\vert z(G)-\widetilde{z}(G)\right\vert \right. \\ &&+\left. \frac{\nabla _{4}}{\upsilon }I^{\rho -\eta _{1}}\left\vert \Xi (G,z(G),r(G))-\Xi (G,\widetilde{z}(G),\widetilde{r}(G))\right\vert +\frac{ \nabla _{2}}{\upsilon }I^{\rho +s_{1}}\left\vert \Xi (G,z(G),r(G))-\Xi (G, \widetilde{z}(G),\widetilde{r}(G))\right\vert \right) \\ &&+\frac{G^{\rho -2}}{\left\vert \Phi \right\vert }\left( \frac{\nabla _{3}(\upsilon -1)}{\upsilon }I^{\rho -\theta -\eta _{1}}\left\vert z(G)- \widetilde{z}(G)\right\vert +\frac{\nabla _{1}(\upsilon -1)}{\upsilon } I^{\rho -\theta +s_{1}}\left\vert z(G)-\widetilde{z}(G)\right\vert \right. \\ &&+\frac{\nabla _{3}}{\upsilon }I^{\rho -\eta _{1}}\left\vert \Xi (G,z(G),r(G))-\Xi (G,\widetilde{z}(G),\widetilde{r}(G))\right\vert \\ &&\left. +\frac{\nabla _{1}}{\upsilon }I^{\rho +s_{1}}\left\vert \Xi (G,z(G),r(G))-\Xi (G,\widetilde{z}(G),\widetilde{r}(G))\right\vert \right) + \frac{1}{\upsilon }I^{\rho }\left\vert \zeta (\tau )\right\vert \\ &&+\frac{G^{\rho -1}}{\left\vert \Phi \right\vert }\left( \frac{\nabla _{4}}{ \upsilon }I^{\rho -\eta _{1}}\left\vert \zeta (G)\right\vert +\frac{\nabla _{2}}{\upsilon }I^{\rho +s_{1}}\left\vert \zeta (G)\right\vert \right) + \frac{G^{\rho -2}}{\left\vert \Phi \right\vert }\left( \frac{\nabla _{3}}{ \upsilon }I^{\rho -\eta _{1}}\left\vert \zeta (G)\right\vert +\frac{\nabla _{1}}{\upsilon }I^{\rho +s_{1}}\left\vert \zeta (G)\right\vert \right) , \end{eqnarray*}

    which implies that

    \begin{eqnarray*} \left\Vert z(\tau )-\widetilde{z}(\tau )\right\Vert &\leq &\left( \frac{ G^{\rho -\theta }\left\vert \upsilon -1\right\vert }{\upsilon \Gamma (\rho -\theta +1)}+\frac{TG^{\rho -1}}{\upsilon \Gamma (\rho )}+\frac{T}{\upsilon \Gamma (\rho +1)}+\frac{\nabla _{4}(\upsilon -1)G^{2\rho -\theta -\eta _{1}-1}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta -\eta _{1}+1)}\right. \\ &&+\frac{\nabla _{2}(\upsilon -1)G^{2\rho -\theta +s_{1}-1}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta +s_{1}+1)}+\frac{T\nabla _{4}G^{2\rho -\eta _{1}-1}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\eta _{1}+1)}+\frac{T\nabla _{2}G^{2\rho +s_{1}-1}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho +s_{1}+1)} \\ &&+\frac{\nabla _{3}(\upsilon -1)G^{2\rho -\theta -\eta _{1}-2}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta -\eta _{1}+1)}+\frac{\nabla _{1}(\upsilon -1)G^{2\rho -\theta +s_{1}-2}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta +s_{1}+1)}+\frac{\nabla _{3}G^{2\rho -\theta -\eta _{1}-2}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta -\eta _{1}+1)} \\ &&\left. +\frac{\nabla _{1}G^{2\rho -\theta +s_{1}-2}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta +s_{1}+1)}\right) \left\Vert z(\tau )- \widetilde{z}(\tau )\right\Vert \\ &&+\left( \frac{TG^{\rho -1}}{\upsilon \Gamma (\rho )}+\frac{T\nabla _{4}G^{2\rho -\eta _{1}-1}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\eta _{1}+1)}+\frac{T\nabla _{2}G^{2\rho +s_{1}-1}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho +s_{1}+1)}+\frac{\nabla _{3}G^{2\rho -\theta -\eta _{1}-2}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta -\eta _{1}+1)}\right. \\ &&\left. \frac{\nabla _{1}G^{2\rho -\theta +s_{1}-2}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta +s_{1}+1)}\right) \left\Vert r(\tau )- \widetilde{r}(\tau )\right\Vert + \\ &&\frac{\varepsilon G^{\rho }}{\Gamma (\rho +1)}+\frac{\varepsilon G^{\rho -1}}{\left\vert \Phi \right\vert }\left( \frac{\nabla _{4}G^{\rho -\eta _{1}} }{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\eta _{1}+1)}+\frac{ \nabla _{2}G^{\rho +s_{1}}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho +s_{1}+1)}\right) \\ &&+\frac{\varepsilon G^{\rho -2}}{\left\vert \Phi \right\vert }\left( \frac{ \nabla _{3}G^{\rho -\eta _{1}}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\eta _{1}+1)}+\frac{\nabla _{1}G^{\rho +s_{1}}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho +s_{1}+1)}\right) , \end{eqnarray*}

    where T = \max \{T_{\Xi }, \widetilde{T}_{\Xi }\} . For simplicity, we consider

    \begin{eqnarray*} \wp & = &\frac{G^{\rho -\theta }\left\vert \upsilon -1\right\vert }{\upsilon \Gamma (\rho -\theta +1)}+\frac{T}{\upsilon \Gamma (\rho +1)}+\frac{\nabla _{4}(\upsilon -1)G^{2\rho -\theta -\eta _{1}-1}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta -\eta _{1}+1)}+\frac{\nabla _{2}(\upsilon -1)G^{2\rho -\theta +s_{1}-1}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta +s_{1}+1)} \\ &&+\frac{\nabla _{3}(\upsilon -1)G^{2\rho -\theta -\eta _{1}-2}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta -\eta _{1}+1)}+\frac{\nabla _{1}(\upsilon -1)G^{2\rho -\theta +s_{1}-2}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta +s_{1}+1)}+\Game , \end{eqnarray*}
    \begin{eqnarray*} \Game & = &\frac{TG^{\rho -1}}{\upsilon \Gamma (\rho )}+\frac{T\nabla _{4}G^{2\rho -\eta _{1}-1}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\eta _{1}+1)}+\frac{T\nabla _{2}G^{2\rho +s_{1}-1}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho +s_{1}+1)} \\ &&+\frac{\nabla _{3}G^{2\rho -\theta -\eta _{1}-2}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta -\eta _{1}+1)}+\frac{\nabla _{1}G^{2\rho -\theta +s_{1}-2}}{\upsilon \left\vert \Phi \right\vert \Gamma (\rho -\theta +s_{1}+1)}, \end{eqnarray*}

    and

    \begin{eqnarray*} S & = &\frac{G^{\rho }}{\Gamma (\rho +1)}+\frac{G^{\rho -1}}{\upsilon \left\vert \Phi \right\vert }\left( \frac{\nabla _{4}G^{\rho -\eta _{1}}}{ \Gamma (\rho -\eta _{1}+1)}+\frac{\nabla _{2}G^{\rho +s_{1}}}{\Gamma (\rho +s_{1}+1)}\right) \\ &&+\frac{G^{\rho -2}}{\upsilon \left\vert \Phi \right\vert }\left( \frac{ \nabla _{3}G^{\rho -\eta _{1}}}{\Gamma (\rho -\eta _{1}+1)}+\frac{\nabla _{1}G^{\rho +s_{1}}}{\Gamma (\rho +s_{1}+1)}\right) . \end{eqnarray*}

    It follows that

    \begin{equation} \left\Vert z(\tau )-\widetilde{z}(\tau )\right\Vert _{\Im _{1}}-\frac{\Game }{1-\wp }\left\Vert r(\tau )-\widetilde{r}(\tau )\right\Vert _{\Im _{2}}\leq \frac{S\varepsilon }{1-\wp }. \end{equation} (4.2)

    Similarly, one can obtain under T^{\ast } = \max \{T_{\Xi ^{\ast }}, \widetilde{T}_{\Xi ^{\ast }}\} and \left\vert \zeta ^{\ast }(\tau)\right\vert \leq \epsilon ^{\ast } that

    \begin{equation} \left\Vert r(\tau )-\widetilde{r}(\tau )\right\Vert _{\Im _{2}}-\frac{\Game ^{\ast }}{1-\wp ^{\ast }}\left\Vert z(\tau )-\widetilde{z}(\tau )\right\Vert _{\Im _{1}}\leq \frac{S^{\ast }\varepsilon ^{\ast }}{1-\wp ^{\ast }}, \end{equation} (4.3)

    where

    \begin{eqnarray*} \wp ^{\ast } & = &\frac{G^{\rho ^{\ast }-\theta ^{\ast }}\left\vert \upsilon ^{\ast }-1\right\vert }{\upsilon ^{\ast }\Gamma (\rho ^{\ast }-\theta ^{\ast }+1)}+\frac{T^{\ast }}{\upsilon ^{\ast }\Gamma (\rho ^{\ast }+1)}+\frac{ \nabla _{4}^{\ast }(\upsilon ^{\ast }-1)G^{2\rho ^{\ast }-\theta ^{\ast }-\eta _{1}^{\ast }-1}}{\upsilon ^{\ast }\left\vert \Phi ^{\ast }\right\vert \Gamma (\rho ^{\ast }-\theta ^{\ast }-\eta _{1}^{\ast }+1)} \\ &&+\frac{\nabla _{2}^{\ast }(\upsilon ^{\ast }-1)G^{2\rho ^{\ast }-\theta ^{\ast }+s_{1}^{\ast }-1}}{\upsilon ^{\ast }\left\vert \Phi ^{\ast }\right\vert \Gamma (\rho ^{\ast }-\theta ^{\ast }+s_{1}^{\ast }+1)}+\frac{ \nabla _{3}^{\ast }(\upsilon ^{\ast }-1)G^{2\rho ^{\ast }-\theta ^{\ast }-\eta _{1}^{\ast }-2}}{\upsilon ^{\ast }\left\vert \Phi ^{\ast }\right\vert \Gamma (\rho ^{\ast }-\theta ^{\ast }-\eta _{1}^{\ast }+1)} \\ &&+\frac{\nabla _{1}^{\ast }(\upsilon ^{\ast }-1)G^{2\rho ^{\ast }-\theta ^{\ast }+s_{1}^{\ast }-2}}{\upsilon ^{\ast }\left\vert \Phi ^{\ast }\right\vert \Gamma (\rho ^{\ast }-\theta ^{\ast }+s_{1}^{\ast }+1)}+\Game ^{\ast }, \end{eqnarray*}
    \begin{eqnarray*} \Game ^{\ast } & = &\frac{T^{\ast }G^{\rho ^{\ast }-1}}{\upsilon ^{\ast }\Gamma (\rho ^{\ast })}+\frac{T^{\ast }\nabla _{4}^{\ast }G^{2\rho ^{\ast }-\eta _{1}^{\ast }-1}}{\upsilon ^{\ast }\left\vert \Phi ^{\ast }\right\vert \Gamma (\rho ^{\ast }-\eta _{1}^{\ast }+1)}+\frac{T^{\ast }\nabla _{2}^{\ast }G^{2\rho ^{\ast }+s_{1}^{\ast }-1}}{\upsilon ^{\ast }\left\vert \Phi ^{\ast }\right\vert \Gamma (\rho ^{\ast }+s_{1}^{\ast }+1)} \\ &&+\frac{\nabla _{3}^{\ast }G^{2\rho ^{\ast }-\theta ^{\ast }-\eta _{1}^{\ast }-2}}{\upsilon ^{\ast }\left\vert \Phi ^{\ast }\right\vert \Gamma (\rho ^{\ast }-\theta ^{\ast }-\eta _{1}^{\ast }+1)}+\frac{\nabla _{3}^{\ast }G^{2\rho ^{\ast }-\theta ^{\ast }-\eta _{1}^{\ast }-2}}{\upsilon ^{\ast }\left\vert \Phi ^{\ast }\right\vert \Gamma (\rho ^{\ast }-\theta ^{\ast }-\eta _{1}^{\ast }+1)}, \end{eqnarray*}

    and

    \begin{eqnarray*} S^{\ast } & = &\frac{G^{\rho ^{\ast }}}{\upsilon ^{\ast }\Gamma (\rho ^{\ast }+1)}+\frac{G^{\rho ^{\ast }-1}}{\upsilon ^{\ast }\left\vert \Phi ^{\ast }\right\vert }\left( \frac{\nabla _{4}^{\ast }G^{\rho ^{\ast }-\eta _{^{\ast }1}}}{\Gamma (\rho ^{\ast }-\eta _{1}^{\ast }+1)}+\frac{\nabla _{2}^{\ast }G^{\rho ^{\ast }+s_{1}^{\ast }}}{\Gamma (\rho ^{\ast }+s_{1}^{\ast }+1)} \right) \\ &&+\frac{G^{\rho ^{\ast }-2}}{\upsilon ^{\ast }\left\vert \Phi ^{\ast }\right\vert }\left( \frac{\nabla _{3}^{\ast }G^{\rho ^{\ast }-\eta _{1}^{\ast }}}{\Gamma (\rho ^{\ast }-\eta _{1}^{\ast }+1)}+\frac{\nabla _{1}G^{\rho ^{\ast }+s_{1}^{\ast }}}{\Gamma (\rho ^{\ast }+s_{1}^{\ast }+1)} \right) . \end{eqnarray*}

    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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