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Review

Human–computer interaction in healthcare: Comprehensive review

  • Technological advancements have fundamentally transformed healthcare systems and significantly altered the interactions between medical professionals and information interfaces. This study provides a comprehensive review of the role of human–computer interaction (HCI) in healthcare, emphasizing the importance of user-centered design and the integration of emerging technologies. The paper reviews the evolution of healthcare interfaces, exploring key assumptions in foundational HCI principles and theoretical frameworks that guide the design processes. The analysis delves into the application of HCI principles, particularly user-centered approaches, to enhance feedback mechanisms, ensure consistency, and improve visibility within medical settings, all of which contribute to creating practical, usable, and memorable interfaces. The review provides an overview of successful and unsuccessful cases, demonstrating what determines efficacy in healthcare interface design. The discussion extends to cover the role of interactive interfaces in streamlining clinical workflows, facilitating communication and collaboration, and supporting informed decision-making among healthcare providers. This paper focuses on historical views and milestones of interface design, emphasizing the significance of interactive interfaces. The discussion extends to cover the role of interactive interfaces in streamlining clinical workflows, facilitating communication and collaboration, and supporting informed decision-making among healthcare providers. Further, the review concludes with an examination of future trends in HCI for healthcare, particularly focusing on the rapid integration of emerging technologies and their implications for the ongoing evolution of interface design in this critical field.

    Citation: Meher Langote, Saniya Saratkar, Praveen Kumar, Prateek Verma, Chetan Puri, Swapnil Gundewar, Palash Gourshettiwar. Human–computer interaction in healthcare: Comprehensive review[J]. AIMS Bioengineering, 2024, 11(3): 343-390. doi: 10.3934/bioeng.2024018

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  • Technological advancements have fundamentally transformed healthcare systems and significantly altered the interactions between medical professionals and information interfaces. This study provides a comprehensive review of the role of human–computer interaction (HCI) in healthcare, emphasizing the importance of user-centered design and the integration of emerging technologies. The paper reviews the evolution of healthcare interfaces, exploring key assumptions in foundational HCI principles and theoretical frameworks that guide the design processes. The analysis delves into the application of HCI principles, particularly user-centered approaches, to enhance feedback mechanisms, ensure consistency, and improve visibility within medical settings, all of which contribute to creating practical, usable, and memorable interfaces. The review provides an overview of successful and unsuccessful cases, demonstrating what determines efficacy in healthcare interface design. The discussion extends to cover the role of interactive interfaces in streamlining clinical workflows, facilitating communication and collaboration, and supporting informed decision-making among healthcare providers. This paper focuses on historical views and milestones of interface design, emphasizing the significance of interactive interfaces. The discussion extends to cover the role of interactive interfaces in streamlining clinical workflows, facilitating communication and collaboration, and supporting informed decision-making among healthcare providers. Further, the review concludes with an examination of future trends in HCI for healthcare, particularly focusing on the rapid integration of emerging technologies and their implications for the ongoing evolution of interface design in this critical field.



    Fractional differential equations have attracted much attention and been widely used in engineering, physics, chemistry, biology, and other fields. For more details, see [1,2,3]. The theory is a beautiful mixture of pure and applied analysis. Over the years, the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena.

    In particular, fixed-point techniques have been applied in many areas of mathematics, sciences, and engineering. Various fixed-point theorems have been utilized to establish sufficient conditions for the existence and uniqueness of solutions for different types of fractional differential problems; see, for example, [4,5,6,7,8,9,10].

    The Caputo-Hadamard fractional differential equations (CHFDEs), with their non-integer order derivatives, offer a distinctive perspective on modeling complex phenomena. Incorporating boundary conditions adds depth to the study by constraining solutions, and this is essential for practical applications and system analysis. There have been investigations into the existence, uniqueness, and properties of solutions under specific constraints by using diverse techniques like Laplace transformation and numerical methods. Unveiling new insights into fractional dynamics under constraints has broad applications in physics, engineering, biology, and finance. The development of tailored analytical and numerical tools for fractional contexts presents hurdles and opportunities for solving real-world problems effectively. Ongoing research promises advancements in mathematical methods, algorithms, and theoretical frameworks, potentially refining existing models and solving complex problems. Moreover, researchers have made great efforts in the study of the properties of Caputo-Hadamard fractional derivatives, and they have established the existence of the (CHFDEs) by applying some fixed-point theorems; see [11,12,13,14,15,16,17].

    Furthermore, the Lp-integrable solutions for fractional differential equations have been intensively studied by many mathematicians. For example, the authors of [18] discussed the existence of fractional boundary-value problems in Lp-spaces. Agrwal et al. [19] derived the existence of Lp-solutions for differential equations with fractional derivatives under compactness conditions. The investigations of Lp-integrable solutions can be see in [20,21,22,23,24]. Nowadays, the Ulam-Hyers stability is a crucial topic in nonlinear differential equation research, as it studies the effective flexibility of solutions along small perturbations and focuses on how the differential equations behave when the initial state or parameters are slightly changed. Several articles have been published related to this subject; see [25,26,27,28,29,30,31].

    Dhaigude and Bhairat [32] discussed the existence and Ulam-type stability of solutions for the following fractional differential equation:

    DW1S(ζ)=M(ζ,S(ζ),DW1S(ζ)),ζ[1,β],β>1,S(ˆk)(1)=SˆkRn,ˆk=0,1,..p1,

    where p1<Wp and DW denotes the Caputo-Hadamard derivative of order W.

    In [33], utilizing the O'Regan fixed-point theorem and Burton-Kirk fixed-point theorem, Derbazi and Hammouche presented a new result on the existence and stability for the boundary-value problem of nonlinear fractional differential equations, as follows:

    CDW0+S(ζ)=M(ζ,S(ζ),Dβ0+S(ζ)),ζ[0,1],2<W3,S(0)=g1(S),S(0)=m1Iδ10+S(σ1),0<σ1<1,CDβ10+S(1)=m2Iδ20+S(σ2),0<σ2<1,

    where DW,Dβ, and Dβ1 are the Caputo fractional derivatives such that 0<β,β11 and Iδ10+,Iδ20+ are the Riemann-Liouville fractional integral and m1,m2,δ1,δ2 are real constants. In [34], Hu and Wang investigated the existence of solutions of the following nonlinear fractional differential equation:

    DWS(ζ)=M(ζ,S(ζ),DHS(ζ)),1<W2,0<H<1,

    with the following integral boundary conditions:

    S(0)=0,S(1)=10g(s)S(s)ds,

    where Dαt is the Riemann-Liouville fractional derivative, M:[0,1]×R×RR, and gL1[0,1].

    In [35], Murad and Hadid, by means of the Schauder fixed-point theorem and the Banach contraction principle, considered the boundary-value problem of the fractional differential equation:

    DWS(ζ)=M(ζ,S(ζ),DHS(ζ)),ζ(0,1),S(0)=0,S(1)=I10US(ζ),

    where DW,DH are Riemann-Liouville fractional derivatives, 1<W2, 0<H1, and 0<U1.

    The focal point of the originality of this work is that we deal with the existence of Lp-integrable solutions for CHFDEs by applying the rarely used Burton-Kirk fixed-point theorem under sufficient conditions with the help of the Kolmogorov compactness criterion and Hölder inequality. The Burton-Kirk fixed-point theorem, a pivotal result in the field of functional analysis and nonlinear analysis, is a tool for addressing existence problems in systems of differential equations; see [36,37,38]. This theorem combines Krasnoselskii's fixed-point theorem on the sum of two operators with Schaefer's fixed-point theorem. Schaefer's theorem eliminates a difficult hypothesis in Krasnoselskii's theorem, but it requires an a priori bound on solutions. Motivated by the above works, we extended the previous results obtained in [34,35] to study the existence, uniqueness, and Ulam stability of solutions for fractional differential equations of the Caputo-Hadamard type with integral boundary conditions of the following form:

    CHDWS(ζ)=M(ζ,S(ζ),CHDHS(ζ)),ζI=[ˆa,T], (1.1)
    S(ˆa)=0,S(ˆT)=1Γ(U)Tˆa(lnTθ)U1S(θ)θdθ, (1.2)

    where CHDW,CHDH are Caputo-Hadamard fractional derivatives of order W(1,2], H(0,1], ITUˆa is the Caputo-Hadamard fractional integral, U(0,1], and M:I×R×RR.

    To the best of our knowledge, up to now, no work has been reported to drive the (CHFDEs) with the rarely used Bourten-Kirk fixed-point in Lebesgue space (Lp). The main contribution is summarized as follows:

    1) (CHFDEs) with integral boundary conditions are formulated.

    2) Initially, we establish the uniqueness result by applying the Banach fixed-point theorem together with the Hölder inequality.

    3) The arguments are based on the Bourtin-Kirk fixed-point theorem, in combination with the technique of measures of noncompactness, to prove the existence of Lp-integrable solutions for Eq (1.1). A necessary and sufficient condition for a subset of Lebesgue space to be compact is given in what is often called the Kolmogorov compactness theorem.

    4) Ulam-Hyers stability is also investigated by applying the Hölder inequality for the Lp-integrable solutions.

    5) Appropriate examples with figures and tables are also provided to demonstrate the applicability of our results.

    The paper is organized as follows. In Section 2, we recall some definitions and results required for this study. Section 3 deals with the existence and uniqueness of Lp-integrable solutions for CHFDEs. In Section 4, we show the stability of this solution by using the Ulam-Hyers with Ulam-Hyers-Rassias stability. Examples are given to illustrate our main results in Section 5.

    Definition 2.1. [2] Let S:[ˆa,T]R be a continuous function. Then, the Hadamard fractional integral is defined by

    IWˆaS(ζ)=1Γ(W)ζˆa(ln(ζ))W1S()d,

    provided that the integral exists.

    Definition 2.2. [2] Let S be a continuous function. Then, the Hadamard fractional derivative is defined by

    DWˆaS(ζ)=1Γ(νW)(ζddζ)νζˆa(ln(ζ))νW1S()d,

    where ν=[W]+1, [W] denotes the integer part of the real number W, and Γ is the gamma function.

    Definition 2.3. [2] Let S be a continuous function. Then, the Caputo-Hadamard derivative of order W is defined as follows

    DWˆaS(ζ)=1Γ(νW)ζˆa(ln(ζ))νW1ΔνS()d,

    where ν=[W]+1, Δ=(ζddζ), and [W] denotes the integer part of the real number W.

    Lemma 2.4. [2] Let WR+ and ν=[W]+1. If SACνW([ˆa,T],R), then the Caputo-Hadamard differential equation CHDWaS(ζ)=0 has a solution

    S(ζ)=ν1p=0gp(lnζˆa)p,

    and the next formula hold:

    IWˆaCHDWˆaS(ζ)=S(ζ)+ν1p=0gp(lnζˆa)p,

    where gpR,p=0,1,2,...,ν1.

    Definition 2.5. [39] If there exists a real number cf>0 such that ˆε>0, for each solution ˆΨLp([ˆa,T],R) of the inequality

    |CHDWˆΨ(ζ)M(ζ,ˆΨ(ζ),CHDHˆΨ(ζ))|ˆε,ζ[ˆa,T], (2.1)

    there exists a solution SLp([ˆa,T],R) of Eq (1.1) with

    |ˆΨ(ζ)S(ζ)|cfˆε,ζ[ˆa,T].

    Then, Eq (1.1) is Ulam-Hyers-stable

    Definition 2.6. [39] If there exists a real number cfˆΦ>0 such that ˆε>0, for each solution ˆΨLp([ˆa,T],R) of the inequality

    |CHDWˆΨ(ζ)M(ζ,ˆΨ(ζ),CHDHˆΨ(ζ))|ˆεˆΦ(ζ),ζ[ˆa,T], (2.2)

    there exists a solution SLp([ˆa,T],R) of Eq (1.1) with

    |ˆΨ(ζ)S(ζ)|cfˆΦˆε,ζ[ˆa,T].

    Then, Eq (1.1) is Ulam-Hyers-Rassias-stable with respect to ˆΦ.

    Theorem 2.7. [40] (Kolmogorov compactness criterion)

    Let ˆνLp[ˆa,T], 1p<. If

    (ⅰ) ˆν is bounded in Lp[ˆa,T] and

    (ⅱ) ˆν is compact (relatively) in Lp[ˆa,T] then ˆψhˆψ as h0 uniformly with respect to ˆψˆν, where

    ˆψh(ζ)=1hζ+hζˆψ(θ)dθ.

    Theorem 2.8. [41] (Burton-Kirk fixed-point theorem)

    Assume that H is a Banach space and that there are two operators F1,F2:HH such that F1 is a contraction and F2 is completely continuous. Then, either

    - Ξ={SH:ˆγF2(Sˆγ)+ˆγF1(S)=Sis unbounded forˆγ(0,1)}, or

    - the operator equation S=F1(S)+F2(S) has a solution.

    Then, zH exists such that z=F1z+F2z.

    Lemma 2.9. [42] (Bochner integrability)

    If ||ˆV|| is Lebesgue integrable, then a measurable function ˆV:[ˆa,T]×RR is Bochner integrable.

    Lemma 2.10. [43] (Hölder's inequality)

    Assume that ˆQ is a measurable space and that a and b satisfy the condition that 1a+1b=1.1a<,1b< and (ej) belongs to L(ˆQ), which is satisfied if eLa(ˆQ) and jLb(ˆQ).

    ˆQ|ej|dζ(ˆQ|e|adζ)1a(ˆQ|j|bdζ)1b.

    Lemma 2.11. [44] If 0<W<1, then

    ζ1(lnζθ)a(W1)1θadθ(lnζ)a(W1)+1a(W1)+1,

    where 1<a<1/(1W).

    Lemma 2.12. A function SLp(I,R) is a unique solution of the boundary-value problem given by Eqs (1.1) and (1.2) if and only if S satisfies the integral equation

    S(ζ)=1Γ(W)ζˆa(lnζθ)W1M(θ,S(θ),DHS(θ))dθθ+ˆ(lnζˆa)Γ(W)Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ.

    Proof. Equation (1.1) can be reduced to the corresponding integral equation by using Lemma 2.4:

    S(ζ)=1Γ(W)ζˆa(lnζθ)W1M(θ,S(θ),DHS(θ))dθθ+g1(lnζˆa)+g0, (2.3)

    for g0,g1R and S(ˆa)=0; we can obtain g0=0. Then, we can write Eq (2.3) as

    S(ζ)=1Γ(W)ζˆa(lnζθ)W1M(θ,S(θ),DHS(θ))dθθ+g1(lnζˆa),

    and it follows from the condition S(ˆT)=ITUˆaS that

    g1=ˆΓ(W)Γ(U)Tˆaθˆa(lnTθ)U1(lnθϖ)W1M(ϖ,S(ϖ),DHS(ϖ))dϖϖdθθˆΓ(W)Tˆa(lnTθ)W1M(θ,S(θ),DHS(θ))dθθ,g1=ˆΓ(W)Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ,

    where ˆ=Γ(U)(lnTˆa)(Γ(U)(lnTˆa)Uβ(U,2)).

    Hence, the solution of the problem defined by Eqs (1.1) and (1.2) is given by

    S(ζ)=1Γ(W)ζˆa(lnζθ)W1M(θ,S(θ),DHS(θ))dθθ+ˆ(lnζˆa)Γ(W)Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ. (2.4)

    In this section, we study the existence of a solution for the boundary-value problem given by Eqs (1.1) and (1.2) under certain conditions and assumptions. For measurable functions denoted by M:I×R×RR, define the space ={ζ:SLp(I,R),CHDHSLp(I,R)},

    equipped with the norm

    Spp=Spp+||DHS||pp=Tˆa|S(ζ)|pdζ+Tˆa|DHS(ζ)|pdζ,(1p<),

    where Lp(I,R) represents the Banach space containing all Lebesgue measurable functions.

    Our results are based on the following assumptions:

    (O1) a constant Z>0 such that |M(ζ,S1,S2)|Z(|S1|+|S2|),

    for each ζI and for all S1,S2R.

    (O2) M is continuous and a constant Q1>0 such that

    |M(ζ,S1,S2)M(ζ,¯S1,¯S2)|Q1(|S1¯S1|+|S2¯S2|),

    for each S1,S2,¯S1,¯S2R.

    To make things easier, we set the notation as follows:

    V1=(2p(Γ(W))p(p1pW1)p1(lnTˆa)pWpW+22pˆp(Γ(W))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1]T(lnTˆa)p),
    V2=(2p(Γ(WH))p(p1(WH)p1)p1(lnTˆa)(WH)p(WH)p+22pˆp(Γ(W))p(Γ(2H))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1]T(lnTˆa)p(1H)),V3=1(Γ(W))p(p1pW1)p1(lnTˆa)pWpW,V4=1(Γ(WH))p(p1(WH)p1)p1(lnTˆa)(WH)p(WH)p,Δ1=22pˆp(Γ(W))p(β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1)T(lnTˆa)p,Δ2=22pˆp(Γ(W))p(Γ(2H))p(β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆapW1(pW1p1)p1)T(lnTˆa)p(1H).JJ=((lnTˆa)pW(Γ(W+1))p+(lnTˆa)(WH)p(Γ(WH+1))p)T.ω=2(V1+V2)1pQ1.

    The first theorem is based on Banach contraction mapping.

    Theorem 3.1. Let M:[ˆa,T]×R×RR be a continuous function that satisfies the conditions (O1) and (O2). If ω<1, then the problem defined by Eqs (1.1) and (1.2) has only one solution.

    Proof. First, define the operator F by

    (FS)(ζ)=1Γ(W)ζˆa(lnζθ)W1M(θ,S(θ),DHS(θ))dθθ++ˆ(lnζˆa)Γ(W)Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ.

    It is necessary to derive the fixed-point of the operator F on the following set:

    YA={SLp(I,R):||S||ppAp,A>0}. For SYA, we have

    |(FS)(ζ)|p2p(Γ(W))p(ζˆa(lnζθ)W1|M(θ,S(θ),DHS(θ))|dθθ)p+22pˆp(lnζˆa)p(Γ(W))p(Γ(U))p(TˆaTθ(lnTϖ)U1(lnϖθ)W1|M(θ,S(θ),DHS(θ))|dϖϖdθθ)p+22pˆp(lnζˆa)p(Γ(W))p(Tˆa(lnTθ)W1|M(θ,S(θ),DHS(θ))|dθθ)p. (3.1)

    From Hölder's inequality and Lemma 2.11, the first term of Eq (3.1) can be simplified as follows:

    (ζˆa(lnζθ)W1|M(θ,S(θ),DHS(θ))|dθθ)p(ln(ζˆa)pW1(pW1p1)p1ζˆa|M(θ,S(θ),DHS(θ))|pdθ. (3.2)

    Now, by the same technique, the second term can be found as follows

    (TˆaTθ(lnTϖ)U1(lnϖθ)W1|M(θ,S(θ),DHS(θ))|dϖϖdθθ)pβ(W,U)(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)Tˆa|M(θ,S(θ),DHS(θ))|pdθ, (3.3)

    where β(W,U) and βp1p(p(W+U)1p1,1) are beta functions. Now, the last term of Eq (3.1) needs to be found, as follows:

    (Tˆa(lnTθ)W1|M(θ,S(θ),DHS(θ))|dθθ)p(lnTˆa)pW1(pW1p1)p1Tˆa|M(θ,S(θ),DHS(θ))|pdθ. (3.4)

    Thus, (Tˆa(lnTθ)W1|M(θ,S(θ),DHS(θ))|dθθ)p,(ζˆa(lnζθ)W1|M(θ,S(θ),DHS(θ))|dθθ)p and (TˆaTθ(lnTϖ)U1(lnϖθ)W1|M(θ,S(θ),DHS(θ))|dϖϖdθθ)p are Lebesgue-integrable; by Lemma 2.9, we conclude that (lnTθ)W1M(θ,S(θ),DHS(θ)), Tθ(lnTϖ)U1(lnϖθ)W1dϖϖM(θ,S(θ),DHS(θ)) and (lnζθ)W1M(θ,S(θ),DHS(θ)) are Bochner-integrable with respect to θ[ˆa,ζ] for all ζI. From Eqs (3.2)–(3.4), Eq (3.1) gives

    Tˆa|(FS)(ζ)|pdζ2p(Γ(W))pTˆa(lnζˆa)pW1(pW1p1)p1ζˆa|M(θ,S(θ),DHS(θ))|pdθdζ+22pˆp(Γ(W))p(β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1)Tˆa(lnζˆa)pTˆa|M(θ,S(θ),DHS(θ))|pdθdζ,Tˆa|(FS)(ζ)|pdζ2pZp(Γ(W))pTˆa(lnζˆa)pW1(pW1p1)p1ζˆa|S(θ)+DHS(θ)|pdθdζ+22pZpˆp(Γ(W))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1]Tˆa(lnζˆa)pTˆa|S(θ)+DHS(θ)|pdθdζ. (3.5)

    By using integration by parts, Eq (3.5) becomes

    2p(2pZp(Γ(W))p(p1pW1)p1(lnTˆa)pWpW+22pZpˆp(Γ(W))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1]T(lnTˆa)p)(Tˆa|S(θ)|pdθ+Tˆa|DHS(θ)|pdθ)dζ,2p(2p(Γ(W))p(p1pW1)p1(lnTˆa)pWpW+22pˆp(Γ(W))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1]T(lnTˆa)p)Zp||S||pp.||FS||pp2pV1ZpAp, (3.6)

    and

    Tθ|DH(FS)(ζ)|pdζ2p(Γ(WH))pTˆa(ζˆa(lnζθ)WH1|M(θ,S(θ),DHS(θ))|dθθ)pdζ+2p(ˆΓ(W)Γ(2H))pTˆa(lnζˆa)p(1H)(Tˆa[Tθ(lnTϖ)U1(Γ(U))(lnϖθ)W1dϖϖ+(lnTθ)W1]|M(θ,S(θ),DHS(θ))|dθθ)pdζ.

    By (O1) and the Hölder inequality, we can find that

    ||DHFS||pp2p(2p(Γ(WH))p(p1(WH)p1)p1(lnTˆa)(WH)p(WH)p+22pˆp(Γ(W))p(Γ(2H))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1]T(lnTˆa)p(1H))Zp||S||pp,||DHFS||pp2pV2ZpAp. (3.7)

    Combining Eq (3.6) with Eq (3.7), we get

    ||FS||pp=||FS||pp+||DHFS||pp,||FS||p2(V1+V2)1pZA,

    which implies that FYAYA. Hence, F(S)(ζ) is Lebesgue-integrable and F maps YA into itself.

    Now, to show that F is a contraction mapping, considering that S1,S2Lp(I,R), we obtain

    Tˆa|(FS1)(ζ)(FS2)(ζ)|pdζ2p(Γ(W))pTˆa(ζˆa(lnζθ)W1|M(θ,S1(θ),DHS1(θ))M(θ,S2(θ),DHS2(θ))|dθθ)pdζ+Tˆa2pˆp(lnζˆa)p(Γ(W))p(Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ+(lnTθ)W1]|M(θ,S1(θ),DHS1(θ))M(θ,S2(θ),DHS2(θ))|dθθ)pdζ.

    Some computations give

    2pQp1(Γ(W))p(p1pW1)p1Tˆa(lnζˆa)pW1ζˆa(|S1(θ)S2(θ)|+|DHS1(θ)DHS2(θ)|)pdθdζ+22pQp1ˆp(Γ(W))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(p1pW1)p1]Tˆa(lnζˆa)pTˆa(|S1(θ)S2(θ)|+|DHS1(θ)DHS2(θ)|)pdθdζ,
    Qp1(2p(Γ(W))p(p1pW1)p1(lnTˆa)pWpW+22pˆp(Γ(W))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1]T(lnTˆa)p)Tˆa(|S1(θ)S2(θ)|+|DHS1(θ)DHS2(θ)|)pdθ,
    ||FS1FS2||pp2p(2p(Γ(W))p(p1pW1)p1(lnTˆa)pWpW+22pˆp(Γ(W))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1]T(lnTˆa)p)Qp1||S1S2||pp,
    ||FS1FS2||pp2pV1Qp1||S1S2||pp. (3.8)

    Using similar techniques, we obtain

    Tˆa|DH(FS1)(ζ)DH(FS2)(ζ)|pdζ2p(Γ(WH))pTˆa(ζˆa(lnζθ)WH1|M(θ,S1(θ),DHS1(θ))M(θ,S2(θ),DHS2(θ))|dθθ)pdζ+22pˆp(Γ(W))p(Γ(2H))pTˆa(lnζˆa)p(1H)(Tˆa[Tθ(lnTϖ)U1Γ(U)(lnϖθ)W1dϖϖ+(lnTθ)W1]|M(θ,S1(θ),DHS1(θ))M(θ,S2(θ),DHS2(θ))|dθθ)pdζ.

    Then,

    ||DHFS1DHFS2||ppQp1[2p(Γ(WH))p(p1(WH)p1)p1(lnTˆa)(WH)p(WH)p+22pˆp(Γ(W))p(Γ(2H))p(β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1)T(lnTˆa)p(1H)]Tˆa(|S1(θ)S2(θ)|+|DHS1(θ)DHS2(θ)|)pdθ,||DHFS1DHFS2||pp2pQp1V2||S1S2||pp. (3.9)

    Combining Eqs (3.8) and (3.9), we get

    ||FS1FS2||p2Q1(V1+V2)1p||S1S2||p,
    ||FS1FS2||pω||S1S2||p.

    If ω<1, then the Banach theorem guarantees that there is only one fixed-point which is a solution of the problem defined by Eqs (1.1) and (1.2).

    The following outcome is the Burton-Kirk theorem.

    Theorem 3.2. Suppose that (O1) and (O2) hold. Then, the problem defined by Eqs (1.1) and (1.2) has at least one solution.

    Proof. Let F:; we define the operators as follows

    (F1S)(ζ)=ζˆa(lnζθ)W1Γ(W)M(θ,S(θ),DHS(θ))dθθ
    (F2S)(ζ)=ˆ(lnζˆa)Γ(W)Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ.

    Step 1: The operator F1 is continuous.

    Tˆa|(F1Sj)(ζ)(F1S)(ζ)|pdζ1(Γ(W))pTˆa(ζˆa(lnζθ)W1|M(θ,Sj(θ),DHSj(θ))M(θ,S(θ),DHS(θ))|dθθ)pdζ. (3.10)

    It follows from the Hölder inequality and the integration by parts that Eq (3.10) becomes

    1(Γ(W))p(p1pW1)p1(lnTˆa)pWpW||M(θ,Sj(θ),DHSj(θ))M(θ,S(θ),DHS(θ))||pp,||(F1Sj)(F1S)||ppV3||M(θ,Sj(θ),DHSj(θ))M(θ,S(θ),DHS(θ))||pp,

    for all ζI. In a similar manner, we obtain

    Tˆa|DH(F1Sj)(ζ)DH(F1S)(ζ)|pdζ1(Γ(WH))pTˆa(ζˆa(lnζθ)WH1|M(θ,Sj(θ),DHSj(θ))M(θ,S(θ),DHS(θ))|dθθ)pdζ,
    ||DH(F1Sj)DH(F1S)||ppV4||M(θ,Sj(θ),DHSj(θ))M(θ,S(θ),DHS(θ))||pp.

    Then,

    ||(F1Sj)(F1S)||p(V3+V4)1p||M(θ,Sj(θ),DHSj(θ))M(θ,S(θ),DHS(θ))||p.

    According to the Lebesgue dominated convergence theorem, since M is of Caratheodory type, we have that ||(F1Sj)(F1S)||p0asj.

    Step 2: Consider the set Υ={SLp(I,R):||S||ppΥp,Υ>0}.

    For SΥ and ζI, we will prove that F1(Υ) is bounded and equicontinuous, and that

    ||(F1S)||pp2p(Γ(W))p(p1pW1)p1(lnTˆa)pWpWZpAp,||(F1S)||pp2pV3ZpAp.

    In a like manner,

    ||DH(F1S)||pp2pV4ZpAp.

    Then,

    ||(F1S)||p2(V3+V4)1pZA.

    Hence, F1(Υ) is bounded.

    Now, Theorem 2.7-(Kolmogorov compactness criterion) will be applied to prove that F1 is completely continuous. Assume that a bounded subset of Υ is ˆδ. Hence, F1(ˆδ) is bounded in Lp(I,R) and condition (i) of Theorem 2.7 is satisfied. Next, we will demonstrate that, uniformly with regard to ζˆδ, (F1S)h(F1S) in Lp(I,R) as h0. We estimate the following:

    ||(F1S)h(F1S)||pp=Tˆa|(F1S)h(ζ)(F1S)(ζ)|pdζ,Tˆa|1hζ+hζ(F1S)(θ)dθ(F1S)(ζ)|pdζ,Tˆa1hζ+hζ|IWM(θ,S(θ),DHS(θ))IWM(ζ,S(ζ),DHS(ζ))|pdθdζ.

    Similar, the following is obtained:

    ||DH(F1S)hDH(F1S)||pp=Tˆa|DH(F1S)h(ζ)DH(F1S)(ζ)|pdζ,Tˆa1hζ+hζ|IWHM(θ,S(θ),DHS(θ))IWHM(ζ,S(ζ),DHS(ζ))|pdθdζ.

    Since MLp(I,R), we get that IWM,IWHMLp(I,R), as well as that

    1hζ+hζ|IWM(θ,S(θ),DHS(θ))IWM(ζ,S(ζ),DHS(ζ))|pdθ0,

    and

    1hζ+hζ|IWHM(θ,S(θ),DHS(θ))IWHM(ζ,S(ζ),DHS(ζ))|pdθ0.

    Hence, ||(F1S)h(F1S)||p0.

    (F1S)h(F1S),uniformlyash0.

    Then, we conclude that F1(ˆδ) is relatively compact, i.e., F1 is a compact, by using Theorem 2.7.

    Step 3: F2 is contractive. For all S,ScLp(I,R), we have

    Tˆa|(F2S)(ζ)(F2Sc)(ζ)|pdζˆp(Γ(W))pTˆa(lnζˆa)p(Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ+(lnTθ)W1]|M(θ,S(θ),DHS(θ))M(θ,Sc(θ),DHSc(θ))|dθθ)pdζ,||F2SF2Sc||pp22pˆp(Γ(W))p(β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1)T(lnTˆa)pQp1||SSc||pp.||F2SF2Sc||ppΔ1Qp1||SSc||pp, (3.11)

    and

    Tˆa|DH(F2S)(ζ)DH(F2Sc)(ζ)|pdζˆp(Γ(W))p(Γ(2H))pTˆa(lnζˆa)p(1H)(Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ+(lnTθ)W1]|M(θ,S(θ),DHS(θ))M(θ,Sc(θ),DHSc(θ))|dθθ)pdζ,||DHF2SDHF2Sc||pp22pˆp(Γ(W))p(Γ(2H))p(β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1)T(lnTˆa)p(1H)||SSc||pp,||DHF2SDHF2Sc||ppΔ2Qp1||SSc||pp. (3.12)

    Combining Eqs (3.11) and (3.12), the following is obtained:

    ||F2SF2Sc||p(Δ1+Δ2)1pQ1||SSc||p.

    Step 4: Let Ξ={SLp(I):ˆγF2(Sˆγ)+ˆγF1(S)=S,ˆγ(0,1)}. For all SΞ, there exists ˆγ(0,1) such that

    (FS)(ζ)=ˆγ[1Γ(W)ζˆa(lnζθ)W1M(θ,S(θ),DHS(θ))dθθ+ˆ(lnζˆa)Γ(W)Tˆa(1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1)M(θ,Sˆγ(θ),DHSˆγ(θ))dθθ],||FS||pp2p(2p(Γ(W))p(p1pW1)p1(lnTˆa)pWpW+22pˆp(Γ(W))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1]T(lnTˆa)p)Zp||S||pp.

    With the same arguments, we have

    DH(FS)(ζ)=ˆγ[1Γ(WH)ζˆa(lnζθ)WH1M(θ,S(θ),DHS(θ))dθθ+ˆΓ(W)Γ(2H)(lnζˆa)(1H)Tˆa(Tθ(lnTϖ)U1Γ(U)(lnϖθ)W1dϖϖ(lnTθ)W1)M(θ,Sˆγ(θ),DHSˆγ(θ))dθθ],||DHFS||pp2p(2p(Γ(WH))p(p1(WH)p1)p1(lnTˆa)(WH)p(WH)p+22pˆp(Γ(W))p(Γ(2H))p[β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1]T(lnTˆa)p(1H))Zp||S||pp.

    Then,

    ||S||p2(V1+V2)1pZ||S||p.

    Hence, the set Ξ is bounded, and, by Theorem 3.2, the problem defined by Eqs (1.1) and (1.2) has a solution.

    In this section, we establish the Ulam-Hyers and Ulam-Hyers-Rassias stability of the problem defined by Eqs (1.1) and (1.2); we set the following condition.

    (O3) ˆΦLp(I,R) is an increasing function and ˆλˆΦ,ˆΩˆΦ>0 such that, for any ζI, we have

    1Γ(W)ζˆa(lnζθ)W1ˆΦ(θ)dθθˆλˆΦˆΦ(ζ),
    1Γ(WH)ζˆa(lnζθ)WH1ˆΦ(θ)dθθˆΩˆΦˆΦ(ζ).

    Theorem 4.1. Let M be a continuous function and (O2) hold with

    22pQp1(V1+V2)<1.

    Then, the problem defined by Eqs (1.1) and (1.2) is Ulam-Hyers-stable.

    Proof. For ˆε>0, ˆΨ is a solution that satisfies the following inequality:

    |CHDWˆΨ(ζ)M(ζ,ˆΨ(ζ),CHDHˆΨ(ζ))|pˆεp. (4.1)

    There exists a solution SLp(I,R) of the boundary-value problem defined by Eqs (1.1) and (1.2). Then, S(ζ) is given by Eq (2.4); from Eq (4.1), and for each ζI, we have

    |S(ζ)1Γ(W)ζˆa(lnζθ)W1M(θ,S(θ),DHS(θ))dθθˆ(lnζˆa)Γ(W)Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ|p(ˆε(lnζˆa)WΓ(W+1))p, (4.2)

    and

    |DHS(ζ)1Γ(WH)ζˆa(lnζθ)WH1M(θ,S(θ),DHS(θ))dθθˆΓ(W)Γ(2H)(lnζˆa)1HTˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ|p(ˆε(lnζˆa)WHΓ(WH+1))p. (4.3)

    For each ζI, we have

    |ˆΨ(ζ)S(ζ)|p|ˆΨ(ζ)1Γ(W)ζˆa(lnζθ)W1M(θ,S(θ),DHS(θ))dθθˆ(lnζˆa)Γ(W)Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ|p.

    Then, from Eq (4.2), we conclude that

    Tˆa|ˆΨ(ζ)S(ζ)|pdζ2pTˆaˆεp(lnζˆa)pW(Γ(W+1))pdζ+22pTˆa(ζˆa(lnζθ)W1Γ(W)|M(θ,ˆΨ(θ),DHˆΨ(θ))M(θ,S(θ),DHS(θ))|dθθ)pdζ+22p(ˆΓ(W))pTˆa(lnζˆa)p(Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ+(lnTθ)W1]|M(θ,ˆΨ(θ),DHˆΨ(θ))M(θ,S(θ),DHS(θ))|dθθ)pdζ.

    By (O2) and the Hölder inequality, it follows that

    ||ˆΨS||pp2pˆεp(lnTˆa)pWT(Γ(W+1))p+2p[22p(Γ(W))p(p1pW1)p1(lnTˆa)pWpWT+23pˆp(Γ(W))p(β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1)T(lnTˆa)p]Qp1||ˆΨS||pp.

    Hence,

    ||ˆΨS||pp2pˆεp(lnTˆa)pWT(Γ(W+1))p+22pQp1V1||ˆΨS||pp. (4.4)

    Now, from Eq (4.3), we have

    Tˆa|DHˆΨ(ζ)DHS(ζ)|pdζ2pTˆaˆεp(lnζˆa)p(WH)(Γ(WH+1))pdζ+22pTˆa(ζˆa(lnζθ)WH1Γ(WH)|M(θ,ˆΨ(θ),DHˆΨ(θ))M(θ,S(θ),DHS(θ))|dθθ)pdζ+22p(ˆΓ(W)Γ(2H))pTˆa(lnζˆa)(1H)p(Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ+(lnTθ)W1]|M(θ,ˆΨ(θ),DHˆΨ(θ))M(θ,S(θ),DHS(θ))|dθθ)pdζ,

    and

    ||DHˆΨDHS||pp2pˆεp(lnTˆa)(WH)pT(Γ(WH+1))p+2p[22p(Γ(WH))p(p1(WH)p1)p1(lnTˆa)(WH)p(WH)pT+23pˆp(Γ(W))p(Γ(2H))p(β(W,U)(Γ(U))p(lnTˆa)p(W+U)1pβp1p(p(W+U)1p1,1)+(lnTˆa)pW1(pW1p1)p1)T(lnTˆa)p(1H)]Qp1||ˆΨS||pp.||DHˆΨDHS||pp2pˆεp(lnTˆa)(WH)pT(Γ(WH+1))p+22pQp1V2||ˆΨS||pp. (4.5)

    Combining Eq (4.4) with Eq (4.5), we have

    ||ˆΨS||pp2p((lnTˆa)pW(Γ(W+1))p+(lnTˆa)(WH)p(Γ(WH+1))p)Tˆεp+22pQp1(V1+V2)||ˆΨS||pp.

    Hence,

    ||ˆΨS||pcfˆε,

    where

    cf=2JJ1p(122pQp1(V1+V2))1p.

    Then, the problem is Ulam-Hyers-stable.

    Theorem 4.2. Let M be a continuous function and (O2) and (O3) hold. Then, the problem defined by Eqs (1.1) and (1.2) is Ulam-Hyers-Rassias-stable.

    Proof. Let ˆΨLp(I,R) be a solution of Eq (2.2) and there exist a solution SLp(I,R) of Eq (1.1). Then, we have

    S(ζ)=1Γ(W)ζˆa(lnζθ)W1M(θ,S(θ),DHS(θ))dθθ+ˆ(lnζˆa)Γ(W)Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ.

    From Eq (2.2), for each ζI, we get

    |S(ζ)1Γ(W)ζˆa(lnζθ)W1M(θ,S(θ),DHS(θ))dθθˆ(lnζˆa)Γ(W)Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ|p(ˆεΓ(W)ζˆa(lnζθ)W1ˆΦ(θ)dθθ)p(ˆεˆλˆΦˆΦ(ζ))p, (4.6)

    and

    |DHS(ζ)1Γ(WH)ζˆa(lnζθ)WH1M(θ,S(θ),DHS(θ))dθθˆΓ(W)Γ(2H)(lnζˆa)1HTˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ(lnTθ)W1]M(θ,S(θ),DHS(θ))dθθ|p(ˆεΓ(WH)ζˆa(lnζθ)WH1ˆΦ(θ)dθθ)p(ˆεˆΩˆΦˆΦ(ζ))p. (4.7)

    On the other hand, for each ζI, from Eq (4.6), the below is found:

    Tˆa|ˆΨ(ζ)S(ζ)|pdζ2pˆεpTˆa(ˆλˆΦˆΦ(ζ))pdζ+22pTˆa(ζˆa(lnζθ)W1Γ(W)|M(θ,ˆΨ(θ),DHˆΨ(θ))M(θ,S(θ),DHS(θ))|dθθ)pdζ+22p(ˆΓ(W))pTˆa(lnζˆa)p(Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ+(lnTθ)W1]|M(θ,ˆΨ(θ),DHˆΨ(θ))M(θ,S(θ),DHS(θ))|dθθ)pdζ. (4.8)

    Thus, by condition (O2) and the Hölder inequality, Eq (4.8) becomes

    ||ˆΨS||pp2pˆεp(ˆΦ(T))pT||ˆΦ||ppV3+22pQp1V1||ˆΨS||pp. (4.9)

    Now, from Eq (4.7), one has

    Tˆa|DHˆΨ(ζ)DHS(ζ)|pdζ2pˆεpTˆa(ˆΩˆΦˆΦ(ζ))pdζ+22pTˆa(ζˆa(lnζθ)WH1Γ(WH)|M(θ,ˆΨ(θ),DHˆΨ(θ))M(θ,S(θ),DHS(θ))|dθθ)pdζ+22p(ˆΓ(W)Γ(2H))pTˆa(lnζˆa)(1H)p(Tˆa[1Γ(U)Tθ(lnTϖ)U1(lnϖθ)W1dϖϖ+(lnTθ)W1]|M(θ,ˆΨ(θ),DHˆΨ(θ))M(θ,S(θ),DHS(θ))|dθθ)pdζ.

    Similarly, we have

    ||DHˆΨDHS||pp2pˆεp(ˆΦ(T))pT||ˆΦ||ppV4+22pQp1V2||ˆΨS||pp. (4.10)

    Combining Eq (4.9) with Eq (4.10), we have

    ||ˆΨS||pp2p(V3+V4)ˆεp(ˆΦ(T))p||ˆΦ||ppT+22pQp1(V1+V2)||ˆΨS||pp.

    Hence,

    ||ˆΨS||pcfˆεˆΦ(T)||ˆΦ||p,

    where    cf=2(V3+V4)1pT1p(122pQp1(V1+V2))1p.

    Then, the problem defined by Eqs (1.1) and (1.2) is Ulam-Hyers-Rassias-stable.

    In this section, we present two examples to illustrate the utility of our main results.

    Example 5.1. Consider the following Bagley-Torvik equation:

    {CHD2G+θCHDHG=1eζ1,ζ[1,2],G(1)=0,G(2)=21IUG. (5.1)

    Here, ˆa=1,T=2,θ=1/25, W=2,H=0.4 and U=0.7. Also, let p=2, by the condition (O2), we have that Q1=0.04. Then, from Theorem 3.1,

    V1=17.38072643,V2=17.46632688,ω=2Q1(V1+V2)1p=0.4722511421<1.

    This indicates that the solution to the problem defined by Eq (5.1) is unique.

    Example 5.2. Consider the following boundary-value problem:

    DWG(ζ)=ln(ζ)12+eζ15(3+ln(ζ))(G+DHG),ζI,ζ[1,e],G(1)=0,G(e)=e1IUG. (5.2)

    Here, ˆa=1,T=e, W=1.7,H=0.3, and U=0.6. By the Lipschitz condition, we have that Q1=0.00817509. Now, to check the obtained results for the Banach contraction mapping and Ulam-Hyers and Ulam-Hyers-Rassias stability, we examine the following cases:

    Case Ⅰ: Let p=2; by a direct calculation and by Theorem 3.1, one can obtain that

    V1=3.24027038e+02,V2=1.77652891e+02,ω=2Q1(V1+V2)1p=0.36621519<1.

    We get that the problem defined by Eq (2) has a unique solution.

    At this moment, to examine the stability, let S=1; we show that Eq (2.1) hold. Indeed,

    |D1.7G(ζ)ln(ζ)12eζ15(3+ln(ζ))(G+D0.3G)|=0.08443313ˆε.

    From Theorem 4.1, we have

    ||ˆΨS||p2JJ1p(12pQp1(V1+V2))1pˆε=0.42244099,

    which shows that the problem defined by Eq (5.2) is Ulam-Hyers-stable.

    Next, let ˆΦ(ζ)=ζ1.8; by applying Theorem 4.2, we have

    ||ˆΦ||p=(T1.8)p+1(ˆa1.8)p+1p+1,andcuˆΦ(T)||ˆΦ||p=1.43585898.

    Hence, the problem defined by Eq (5.2) is Ulam-Hyers-Rassias-stable with

    ||ˆΨS||pcuˆεˆΦ(T)||ˆΦ||p=0.12123407.

    Case Ⅱ: Let p=3,ˆε=0.08443313,andˆΦ(ζ)=ζ1.8; we have that ω=0.24914024<1.

    Then, the boundary-value problem defined by Eq (2) has a unique solution.

    Now, according Theorems 4.1 and 4.2, the Ulam-Hyers and Ulam-Hyers-Rassias stability for the boundary-value problem defined by Eq (2) are respectively given as follows

    ||ˆΨS||p0.22796162,and||ˆΨS||pcuˆεˆΦ(T)||ˆΦ||p=0.01175782.

    Case Ⅲ: Let p=4. From Theorem 3.1, we start by computing the following:

    2Q1(V1+V2)1p=0.209580118<1. Hence, the boundary-value problem defined by Eq (2) has a unique solution. Also, it has Ulam-Hyers and Ulam-Hyers-Rassias stable with

    ||ˆΨS||p0.19198746,and ||ˆΨS||pcuˆεˆΦ(T)||ˆΦ||p=0.02696865.

    To show the efficiency of the Banach contraction principle and that the problem has a unique solution, we will evaluate the value of ω for some different fractional orders, i.e., 1<W2 and 0<H1. Table 1 presents the value of ω when p=2 and ζ[1,e] for some specific orders, such as when W=1.2, H=0.2,0.8, when W=1.5, H=0.2,0.5,0.8, and when W=1.8, H=0.2,0.5. Furthermore, the behavior of ω at some selected points is illustrated in Figure 1.

    Table 1.  Values of ω when p=2 and 1<W<2, 0<H1 for Example 5.2.
    ω<1 ω<1 ω<1 ω<1 ω<1 ω<1 ω<1
    W=1.2, W=1.2, W=1.5, W=1.5, W=1.5, W=1.8, W=1.8,
    ζ H=0.2 H=0.8 H=0.2 H=0.5 H=0.8 H=0.2 H=0.8
    1.0000 0.0003 0.1393 0.0000 0.0001 0.0038 0.0000 0.0001
    1.2455 0.0607 0.0913 0.0401 0.0438 0.0532 0.0258 0.0304
    1.4909 0.1105 0.1314 0.0818 0.0868 0.0925 0.0581 0.0630
    1.7364 0.1664 0.1816 0.1313 0.1376 0.1401 0.0990 0.1037
    1.9819 0.2306 0.2417 0.1900 0.1976 0.1972 0.1493 0.1537
    2.2273 0.3048 0.3129 0.2594 0.2684 0.2652 0.2102 0.2143
    2.4728 0.3909 0.3967 0.3415 0.3520 0.3461 0.2837 0.2874
    2.7183 0.4917 0.4956 0.4386 0.4508 0.4423 0.3719 0.3752

     | Show Table
    DownLoad: CSV
    Figure 1.  Results of ω on ζ=[1,e] with p=2 for Example 2 when a) 1<W<2 and H=0.3; b) W=1.7 and 0<H1.

    Figure 2 shows that the problem has a unique solution at p=3 when 1<W<2, H=0.5, and when W=1.5, 0<H1. In addition, for p=4 and 1<W<2, H=0.8 or W=1.2, 0<H1, ω has been plotted in Figure 3 and is presented in Table 2. To illustrate the sufficiency of our results to find the solution and its uniqueness, we chose p=15 as shown in Figure 4.

    Figure 2.  Results of ω on ζ=[1,e] with p=3 for Example 2 when a) 1<W<2, H=0.5; b) W=1.5, 0<H1.
    Figure 3.  Results of ω with p=4 for Example 2 when a) 1<W<2, H=0.8; b) W=1.2, 0<H1.
    Table 2.  Values of ω when p=3,4 and 1<W<2, 0<H<1 for Example 5.2.
    p=3 p=3 p=3 p=4 p=4 p=4
    ω<1 ω<1 ω<1 ω<1 ω<1 ω<1
    W=1.2, W=1.5, W=1.8, W=1.2, W=1.5, W=1.8,
    ζ H=0.8 H=0.5 H=0.2 H=0.8 H=0.5 H=0.2
    1.0000 0.0203 0.0001 0.0001 0.0078 0.0003 0.0003
    1.2455 0.0656 0.0390 0.0299 0.0575 0.0425 0.0353
    1.4909 0.0964 0.0713 0.0554 0.0853 0.0706 0.0589
    1.7364 0.1315 0.1068 0.0843 0.1149 0.1001 0.0837
    1.9819 0.1716 0.1471 0.1178 0.1475 0.1326 0.1112
    2.2273 0.2177 0.1932 0.1571 0.1842 0.1692 0.1424
    2.4728 0.2708 0.2464 0.2034 0.2260 0.2110 0.1785
    2.7183 0.3323 0.3082 0.2581 0.2741 0.2592 0.2209

     | Show Table
    DownLoad: CSV
    Figure 4.  Results of ω with p=15 for Example 2 when a) W=1.2, 0<H1; b) 1<W<2, H=0.8.

    In this paper, we examined the Lp-integrable solutions of nonlinear CHFDEs with integral boundary conditions. We applied the Burton-Kirk fixed-point theorem and Banach contraction principle with the Kolmogorov compactness criterion and Hölder's inequality technique to demonstrate the main results. In addition, the Ulam-Hyers and Ulam-Hyers-Rassias stability of the problem defined by Eqs (1.1) and (1.2) have been studied. Finally, examples have been provided to demonstrate the validity of our conclusions. In future works, one can extend the given fractional boundary-value problem to more fractional derivatives, such as the Hilfer and Caputo-Fabrizio fractional derivatives.

    Shayma Adil Murad: Conceptualization, Methodology, Formal analysis, Investigation, Writing-original draft, Validation, Writing-review and editing; Ava Shafeeq Rafeeq: Methodology, Formal analysis, Investigation, Writing-original draft, Validation; Thabet Abdeljawad: Investigation, Validation, Supervision, Writing-review and editing. All authors have read and agreed to the published version of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The author T. Abdeljawad would like to thank Prince Sultan University for the support through the TAS research lab.

    The authors declare no conflict of interest.



    Conflict of interest



    The authors declare no conflicts of interest.

    Author contributions



    Meher Langote, Saniya Saratkar, and Praveen Kumar wrote the manuscript and drew figures and tables. Prateek Verma, Chetan Puri, Swapnil Gundewar and Palash Gourshettiwar provided the conceptual idea and revised the manuscript. All authors have read and approved the final manuscript.

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