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Effect of fractional temporal variation on the vibration of waves on elastic substrates with spatial non-homogeneity

  • Received: 06 March 2022 Revised: 02 May 2022 Accepted: 04 May 2022 Published: 23 May 2022
  • MSC : 73D05, 26A33

  • The current manuscript examines the effect of the fractional temporal variation on the vibration of waves on non-homogeneous elastic substrates by applying the Laplace integral transform and the asymptotic approach. Four different non-homogeneities, including linear and exponential forms, are considered and scrutinized. In the end, it is reported that the fractional temporal variation significantly affects the respective vibrational fields greatly as the vibrations increase with a decrease in the fractional-order μ. Besides, the two approaches employed for the cylindrical substrates are also shown to be in good agreement for very small non-homogeneity parameter α. More so, the present study is set to play a vital role in the fields of material science, and non-homogenization processes to state a few.

    Citation: Ahmed SM Alzaidi, Ali M Mubaraki, Rahmatullah Ibrahim Nuruddeen. Effect of fractional temporal variation on the vibration of waves on elastic substrates with spatial non-homogeneity[J]. AIMS Mathematics, 2022, 7(8): 13746-13762. doi: 10.3934/math.2022757

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  • The current manuscript examines the effect of the fractional temporal variation on the vibration of waves on non-homogeneous elastic substrates by applying the Laplace integral transform and the asymptotic approach. Four different non-homogeneities, including linear and exponential forms, are considered and scrutinized. In the end, it is reported that the fractional temporal variation significantly affects the respective vibrational fields greatly as the vibrations increase with a decrease in the fractional-order μ. Besides, the two approaches employed for the cylindrical substrates are also shown to be in good agreement for very small non-homogeneity parameter α. More so, the present study is set to play a vital role in the fields of material science, and non-homogenization processes to state a few.



    FDEs, which provide a very important class of DEs for describing many processes in the real world, differ from ODEs. FDEs can be found in a variety of areas, including control theory, physics, et cetera. In the literature, many authors focused on R-L and Caputo type derivatives in investigating fractional differential equations. A generalization of derivatives of both R-L and Caputo was given by Hilfer in [1], the known as the Hilfer fractional derivative of order α and a type β[0,1], which interpolates between the R-L and Caputo derivative, respectively. This justify the utilization of the Hilfer fractional operator and their generalization in integro-differential equations. In recent years, many researchers have studied the existence, uniqueness and stability of different boundary value problems via Hilfer operators and their generalization.

    Asawasamrit et al. [2] studied the ψ-Caputo (or, more appropriately, ψ-Liouville-Caputo) fractional derivative and non-instantaneous impulsive BVPs. Abdo et al. [3] discussed the ψ-Hilfer fractional derivative involving boundary conditions. Ali et al. in [4] found solution of fractional Volterra-Fredholm integro-differential equations under mixed boundary conditions by using the HOBW method. Anguraj et al. in [5] established new existence results for FIDEs with impulsive and integral conditions. Agarwal et al. in [6] investigated non-instantaneous impulses in Caputo FDEs. Abdo et al. in [7] considered fractional BVP with ψ-Caputo fractional derivative. Kailasavalli et al. in [8] derived existence of solutions for fractional BVPs involving integro-differential equations in Banach spaces. Karthikeyan et al. in [9] investigated existence results for fractional impulsive integro differential equations with integral conditions of Katugampola type. Nuchpong et al. in [10] considered BVPs of Hilfer-type FIDEs and inclusions with nonlocal integro-multipoint boundary conditions. Kilbas et al. in [11] give some basic theory and applications of FDEs. Podlubny in [12] investigated some FDEs. Srivastava in [13] overview recent developments of fractional-order derivatives and integrals. Srivastava in [14] considered some parametric and argument variations of the operators of fractional calculus and related special functions, and integral transformations. Srivastava in [15] give an introductory overview of fractional-calculus operators based upon the Fox-Wright and related higher transcendental functions.

    Recent theories regarding IDEs arise in many fields like, biology, physics, engineering and medicine, where objects change their state rapidly at certain points, see [16,17,18,19,20]. Hernandez et al. in [21] introduced non-instantaneous IDE. Practical problems in the area of psychology related to non-instantaneous impulses can be found in [22,23,24,25,26,27]. Asawasamrit et al. [28] considered the nonlocal BVPs for Hilfer FDEs. Mahmudov et al. [29] investigated the fractional-order BVPs with the so-called Katugampola (or, equivalently, the Erdélyi-Kober type) fractional integral conditions. da Costa Sousa et al. [30] studied a Gronwall inequality via the ψ-Hilfer operator. Phuangthong et al. [31] investigated the nonlocal sequential BVPs for Hilfer type FIDEs and inclusions. Sitho et al. in [32], studied the BVPs regarding ψ-Hilfer type sequential FDEs. Sudsutad et al. in [33], investigated the existence and stability results for ψ-Hilfer FIDE. Subashini et al. [34] obtained some results of fractional order regarding Hilfer integro-differential equations. Wang et al. [35] studied the existence results for FDEs with integral and multipoint boundary conditions. Yu [36] investigated β-Ulam-Hyers stability for a special class of FDEs. Zhang et al. [37] studied the FDEs with not instantaneous impulses. ψ-Hilfer FDEs with impulsive conditions were studied in [38,39].

    Abbas [16] studied the following proportional fractional derivatives:

    a1Dp,q,g(ȷ)=Y(ȷ,(ȷ),a1Ir,q,g(ȷ)),ȷ(rk,ȷk+1],(ȷ)=ψk(ȷ,(t+k)),ȷ(ȷk,rk],k=1,,ϱ,I1p,q,g(a1)=0R,

    where a1Dp,q,g and a1Ir,q,g denote the proportional fractional derivative and the proportional fractional integral and the function Y is continuous.

    Nuchpong et al. [10] discussed the Hilfer fractional derivative with non-local boundary conditions of the form given by

    HDp,q(ȷ)=Y(ȷ,(ȷ),Iδ(ȷ)),  ȷ[a1,a2],(a1)=0,  +a2a1(ı)dı=ϱ2k=1ςk(ϑk),

    where we have used the HDp,q-Hilfer fractional derivative and the Iδ-R-L, and the function Y is continuous.

    Salim et al. [23] studied the BVP for implicit fractional-order generalized Hilfer-type fractional derivative with non-instantaneous impulses of the form:

    (αDp,qτ+)(ȷ)=Y(ȷ,(ȷ),(αDp,q)(ȷ)),  ȷJk,(ȷ)=Hk(ȷ,(ȷ)),  ȷ(ȷk,rk],  k=1,,ϱ,φ1(αI1ϵa+1)(a1)+φ2(αI1ϵτ+)(a2)=φ3,

    where αDp,qτ+ and αI1ϵa+1 are the generalized Hilfer-type fractional derivative and fractional integral and the function Y is continuous.

    Inspired by the above works, we study here new important class of FIDEs namely ψ-Hilfer FIDEs with non-instantaneous impulsive multi-point boundary conditions of the form given by

    HDp,q;ψ(ȷ)=Y(ȷ,(ȷ),ψ(ȷ)),ȷ(rk,ȷk+1], (1.1)
    (ȷ)=Hk(ȷ,(ȷ)),  ȷ(ȷk,rk],  k=1,,ϱ, (1.2)
    (0)=0,  (T)=ϱk=1νkIςk(υk),  νkR,  υk[0,T], (1.3)

    where the order p(1,2) and with the parameters q[0,1], νkR, υk[0,T], and Iςk-is ψ-R-L of order ςk>0, and 0=r0<ȷ1ȷ2<<ȷϱrϱrϱ+1=T, which is pre-fixed, Y:[0,T]×R×RR with Hk:[ȷk,rk]×RR that are continuous. Moreover, ψ(ȷ)=ȷ0k(ȷ,ı)(ı)dı and kC(D,R+) with domain D:={(ȷ,r)R2:0rȷT}.

    Motivated from above results, we introduce ψ-Hilfer FIDEs class with multi-point boundary conditions via the ψ-Hilfer fractional derivative. Moreover, we investigate via Krasnoselskii's and Banach's fixed point theorems, the existence and uniqueness of solutions of the problem given by the Eqs (1.1)–(1.3). Also, we extend the results studied in [28] by including the ψ-Hilfer fractional derivative, nonlinear integral terms and non-instantaneous impulsive conditions.

    This paper is organized as follows: In Section 2, we recall several known results. In Section 3, we use the suitable conditions for existence and uniqueness of solution for the problem given by the Eqs (1.1)–(1.3). Moreover, we prove its boundedness of the method. In Section 4, we consider an application to explain the consistency of our theoretical results.

    Let the space PC([0,T],R):={:[0,T]R:C(ȷk,ȷk+1],R} be continuous. Suppose that there exists (ȷk) and (ȷ+k), where (ȷk)=(ȷ+k) is equipped with the norm given by PC:=sup{|(ȷ)|:0ȷT}. Set

    PC([0,T],R):={PC([0,T],R):PC([0,T],R)}

    with norm PC:=max{PC,PC}. Clearly, PC([0,T],R) equipped with .PC.

    Definition 2.1. [11,12,13,14,15] The R-L derivative of Y with order p>0 is defined by

    Dp0+Y(ȷ)=1Γ(σp)(ddȷ)σȷ0(ȷı)σp1Y(ı)dı,σ1<p<σ.

    Definition 2.2. [11,12,13,14,15] The R-L integral of Y with order p>0 is given as follows:

    IpY(ȷ)=1Γ(p)ȷ0(ȷı)p1Y(ı)dı,

    with Γ(p)=0exp(ı)ıp1dı.

    Definition 2.3. [30] The R-L integrals and derivatives of Y with regard to another function ψ are defined by

    Ip;ψY(ȷ)=1Γ(p)ȷ0ψ(ı)(ψ(ȷ)ψ(ı))p1Y(ı)dı

    and

    Dp;ψY(ȷ)=(1ψ(ȷ)ddȷ)σIσp;ψY(ȷ)=1Γ(σp)(1ψ(ȷ)ddȷ)σȷ0ψ(ı)(ψ(ȷ)ψ(ı))σp1Y(ı)dı,

    respectively.

    Definition 2.4. [3] Let σ1<p<σ, where σN and Y,ψPC([a1,a2],R) such that ψ is increasing, and ψ(ȷ)0 for all ȷ[a1,a2]. The ψ-Hilfer fractional derivative HDp,q;ψ(.) of function Y with order p and parameter 0q1 is given by

    HDp,q;ψY(ȷ)=Iq(σp);ψ(1ψ(ȷ)ddȷ)σI(1q)(σp);ψY(ȷ),

    where σ=[p]+1,[p] represents the integer part of the real number p.

    Lemma 2.1. [3] Let p,ι>0 and δ>0. Then

    (1) Ip;ψIι;ψ(ȷ)=Ip+ι;ψ(ȷ), (semigroup property);

    (2) Ip;ψ(ψ(ȷ)ψ(0))δ1=Γ(δ)Γ(p+δ)(ψ(ȷ)ψ(0))p+δ1.

    Note: HDp,q;ψ(ψ(ȷ)ψ(0))θ1=0.

    Lemma 2.2. [3] Let YL(a1,a2),σ1<pσ,σN with θ=p+σqpq, and I(σp)(1q)YACk[a1,a2]. Then

    (Ip;ψ;ψHDp,q;ψY)(ȷ)=Y(ȷ)σk=1(ψ(ȷ)ψ(0)Γ(θk+1)Y[σk]ψlimȷa1+(I(σp)(1q);ψY)(ȷ),

    where Y[σk]ψ=(1ψ(ȷ)ddȷ)σkY(ȷ).

    Assume that ϵ>0 be a real number. Let σ1<p<σ, where σN and Y,ψPC([a1,a2],R) such that ψ is increasing, and ψ(ȷ)0 for all ȷ[a1,a2], where the parameter 0q1.

    We consider the following inequality:

    |HDp,q;ψ(ȷ)Y(ȷ,(ȷ),ψ(ȷ))|ϵ. (2.1)

    Definition 2.5. [33,40] The problem given by the Eqs (1.1)(1.3) is said to be Ulam-Hyers stable (see [41]), if there exists a real number MY>0 such that for every ϵ>0 and for each solution PC([a1,a2],R) of the inequality (2.1), there exists a solution 1PC([a1,a2],R) of the problem given by the Eqs (1.1)(1.3) with

    |(ȷ)1(ȷ)|MYϵ,ȷ[a1,a2]. (2.2)

    Fixed point theorems play a major role in establishing the existence theory for the problem given by the Eqs (1.1)–(1.3). The following two well-known fixed point theorems will be used in the sequel.

    Theorem 2.1. (Banach's Fixed Point Theorem [42]) Let C([0,T],R) be a Banach space and let N:RR be a contraction mapping. If C is a nonempty closed subset of C([0,T],R), then N has a unique fixed point.

    Theorem 2.2. (Krasnoselskii's Fixed Point Theorem [42]) Let U be a Banach space and E be a closed convex, bounded and nonempty subset of U. Suppose that Q and R are two operators that satisfy the following conditions:

    (1) Qx1+Rx2E,x1,x2E;

    (2) Q is completely continuous operator;

    (3) Q is contraction operator.

    Then there exists at least one fixed point z1E such that z1=Qz1+Rz1.

    Other recently published papers related fixed point results can be found in [43,44,45,46].

    Lemma 2.3 below is our first result.

    Lemma 2.3. A function PC([0,T],R) given by

    (ȷ):={Hk(rϱ)+1Γ(p)ȷa1ψ(ı)(ψ(ȷ)ψ(ı))p1ω(ı)dı+(ψ(ȷ)ψ(0))θ1ΔΓ(p)[ϱk=1νkυk0ψ(ȷ)(ψ(υk)ψ(ı))p1ω(ı)dı],  ȷ[0,ȷ1],Hk(ȷ),ȷ(ȷk,rk],k=1,2,,ϱ,Hk(rk)+1Γ(p)ȷ0ψ(ı)(ψ(ȷ)ψ(ı))p1ω(ı)dı1Γ(p)rk0ψ(ı)(ψ(rk)ψ(r))p1ω(ı)dı,  ȷ(rk,ȷk+1],k=1,2,,ϱ (2.3)

    is a solution of the following system:

    HDp,q;ψ(ȷ)=ω(ȷ),ȷ(rk,ȷk+1][0,T],0<p<1,(ȷ)=Hk(ȷ),ȷ(ȷk,rk],k=1,,ϱ,(0)=0,  (T)=ϱk=1νkIςk(υk), (2.4)

    where

    Δ:=(ψ(ȷ)ψ(0))θ1ϱk=1νk(ψ(υk)ψ(0))θ10.

    Proof. Assume that (ȷ) is satisfies for Eq (2.4). Integrating the first equation of (2.4) for ȷ[0,ȷ1], we have

    (ȷ)=(T)+1Γ(p)ȷ0ψ(ı)(ψ(ȷ)ψ(ı))p1ω(ı)dı. (2.5)

    On other hand, if ȷ(rk,ȷk+1],k=1,2,,ϱ, after integrating again (2.4), we get

    (ȷ)=(ık)+1Γ(p)ȷrkψ(ı)(ψ(ȷ)ψ(ı))p1ω(ı)dı. (2.6)

    Applying impulsive condition, (ȷ)=Hk(ȷ), ȷ(ȷk,rk], we obtain

    (ık)=Hk(rk). (2.7)

    Consequently, from (2.6) and (2.7), we get

    (ȷ)=Hk(rk)+1Γ(p)ȷ0ψ(ı)(ψ(ȷ)ψ(ı))p1ω(ı)dı, (2.8)

    and

    (ȷ)=Hk(rk)+1Γ(p)ȷ0ψ(ı)(ψ(ȷ)ψ(ı))p1ω(ı)dı1Γ(p)rk0(ψ(ı)ψ(rk)ψ(r))p1ω(ı)dı. (2.9)

    Now, we prove that satisfies the boundary conditions (2.4). Obviously, (0)=0.

    ϱk=1νkIφk(υk)=ϱk=1νk(ψ(ȷ)ψ(0))p1ΔΓ(θ)[ϱk=1νkIp+φk;ψω(υk)Iα;ψω(a2)]+ϱk=1νkIα+φkω(υk)=(ψ(ȷ)ψ(0))θ1Δ[ϱk=1νkIp+φk;ψω(υk)]+Ip;ψω(T)=(T). (2.10)

    Now, it's clear that (2.5), (2.9) and (2.10) (2.3), which completes the proof.

    First main result is Theorem 3.1 below.

    Theorem 3.1. Let the assumption below holds true:

    (Al1): There exists L,G,M,Lhk>0, such that

    |Y(ȷ,1,ω1)Y(ȷ,2,ω2)|L|12|+G|ω1ω2|,for  ȷ[0,T], 1,2,ω1,ω2R.|k(ȷ,ı,ϑ)k(ȷ,ı,ν)|M|ϑν|,for ȷ[ȷk,rk],ϑ,νR.|Hk(ȷ,v1)Hk(ȷ,v2)|Lhk|v1v2|,for v1,v2R.

    If

    Z:=max{maxk=1,2,,ϱLhk+(L+GM)Γ(p+1)(ȷpk+1+rpk),Lhk+(L+GM){(ψ(ȷ)ψ(0))θ1|Δ|Γ(θ)[ϱk=1|νk|(ψ(υk)ψ(0))p+φk;ψΓ(p+φk+1)]+(ψ(ȷ)ψ(0))pΓ(p+1)}}<1, (3.1)

    then the problem given by (1.1) to (1.3) has a unique solution on [0,T].

    Proof. Let expand N:PC([0,T],R)PC([0,T],R) by

    (N)(ȷ):={Hϱ(rϱ,(ıϱ))+1Γ(p)ȷ0ψ(ı)(ψ(ȷ)ψ(ı))p1Y(ı,(ı),B(ı))dı+(ψ(ȷ)ψ(0))θ1Δ[ϱk=1νkυk0ψ(ȷ)(ψ(υk)ψ(ı))p1Y(υk,(υk),B(υk)],  ȷ[0,ȷ1],Hk(ȷ),ȷ(ȷk,rk],k=1,2,,ϱ,Hk(rk)+1Γ(p)ȷ0ψ(ı)(ψ(ȷ)ψ(ı))p1Y(ı,(ı),B(ı))dı1Γ(p)rk0ψ(ı)(ψ(rk)ψ(r))p1Y(ı,(ı),B(ı))dı,  ȷ(rk,ȷk+1],k=1,2,,ϱ.

    It is evident that N is well-defined and NPC([0,T],R). We now prove that N is a contraction.

    Case 1. Taking ,¯PC([0,T],R) and ȷ[0,ȷ1], we have

    |(N)(ȷ)(N¯)(ȷ)|Lhk+(L+GM){(ψ(ȷ)ψ(0))θ1|Δ|Γ(θ)[ϱk=1|νk|(ψ(υk)ψ(0))p+φk;ψΓ(p+φk+1)]+(ψ(ȷ)ψ(0))pΓ(p+1)}¯PC.

    Case 2. Choosing ȷ(ȷk,rk], we get

    |(N)(ȷ)(N¯)(ȷ)||Hk(ȷ,(ȷ))Hk(ȷ,¯(ȷ))|Lhk¯PC.

    Case 3. Letting ȷ(rk,ȷk+1], we obtain

    |(N)(ȷ)(N¯)(ȷ)||Hk(rk,(ık)Hk(ık,¯(rk)|+1Γ(p)ȷ0(ȷı)p1|Y(ı,(ı),B(ı))Y(ı,(ı),B(ı))|dı+1Γ(p)rk0(rkr)p1|Y(ı,(ı),B(ı))Y(ı,(ı),B(ı))|dı[Lhk+(L+GM)Γ(p+1)(ȷpk+1+rpk)]¯PC.

    Therefore, N is a contraction since

    Z=[Lhk+(L+GM)Γ(p+1)(ȷpk+1+rpk)]<1.

    Thus, clearly, the problem given by the Eqs (1.1)–(1.3) has a unique solution for each PC([0,T],R).

    Second main result is Theorem 3.2 below.

    Theorem 3.2. Let (Al1) be satisfied and let the assumption below hold true:

    (Al2): There exists Lgk>0 such that

    |Y(ȷ,W1,ω1)|Lgk(1+|W1|+|ω1|),  ȷ[rk,ȷk+1],W1,ω1R.

    (Al3): A function κk(ȷ),k=1,2,,ϱ exists, with

    |Hk(ȷ,W1,ω1)|κk(ȷ),  ȷ[ȷk,rk],W1,ω1R.

    Assume that Mk:=supȷ[ȷk,rk]κk(ȷ)< and K:=maxLgk<1 for all k=1,2,,ϱ. Then the problem given by (1.1) to (1.3) has at least one solution on [0,T].

    Proof. Let us set

    Bp,r:={PC([0,T],R):PCr}.

    Also let Q and R be two operators on Bp,r defined as follows:

    Q(ȷ):={Hϱ(rϱ,(ıϱ)),ȷ[0,ȷ1],Hk(ȷ,(ȷ)),ȷ(ȷk,rk],k=1,2,,ϱ,Hk(rk,(ık)),ȷ(rk,ȷk+1],k=1,2,,ϱ,

    and

    R(ȷ):={1Γ(p)ȷa1ψ(ı)(ψ(ȷ)ψ(ı))p1Y(ı,(ı),B(ı))dı+(ψ(ȷ)ψ(0))θ1Δ[ϱk=1νkυk0ψ(ȷ)(ψ(υk)ψ(ı))p1Y(υk,(υk),B(υk))],  ȷ[0,ȷ1],0,ȷ(ȷk,rk],k=1,2,,ϱ,1Γ(p)ȷ0ψ(ı)(ψ(ȷ)ψ(ı))p1Y(ı,(ı),B(ı))dı1Γ(p)rk0ψ(ı)(ψ(rk)ψ(ı))p1Y(ı,(ı),B(ı))dı,  ȷ(rk,ȷk+1],k=1,2,,ϱ.

    Step 1. For Bp,r, we have Q+RBp,r.

    Case 1. For ȷ[0,ȷ1], we have

    |Q+R¯||Hϱ(rϱ,(ıϱ))|+1Γ(p)ȷ0(ȷı)p1|Y(ı,(ı),B(ı))|dı+(ψ(ȷ)ψ(0))θ1Δ[ϱk=1νkυk0ψ(ȷ)(ψ(υk)ψ(ı))p1Y(υk,(υk),B(υk))dυk],[Lhk+(L+GM){(ψ(ȷ)ψ(0))pΓ(p+1)+(ψ(ȷ)ψ(0))θ1|Δ|Γ(θ)×[ϱk=1|νk|(ψ(υk)ψ(0))p+φk;ψΓ(p+φk+1)]}](1+r)r.

    Case 2. For each ȷ(ȷk,rk], we have

    |Q+R¯||Hk(ȷ,W1(ȷ))|Mk.

    Case 3. For every ȷ(rk,ȷk+1],

    |Q+R¯(ȷ)||Hk(rk,(ık))|+1Γ(p)ȷ0(ȷı)p1|Y(ı,(ı),B(ı))|dı+1Γ(p)rk0(rkr)p1|Y(ı,(ı),B(ı))|dı,Mk+[Lgk(rpk+ȷpk+1)Γ(p+1)](1+r)r.

    Thus

    Q+RBp,r.

    Step 2. Q is contraction on Bp,r.

    Case 1. Let 1,2Bp,r. Then, by taking ȷ[0,ȷ1], we have

    |Q1(ȷ)Q2(ȷ)|Lgϱ|1(rϱ)2(rϱ)|Lgϱ12PC.

    Case 2. For each ȷ(ȷk,rk],k=1,2,,ϱ, we get

    |Q1(ȷ)Q2(ȷ)|Lgk12PC.

    Case 3. For every ȷ(rk,ȷk+1], we obtain

    |Q1(ȷ)Q2(ȷ)|Lgk12PC.

    Hence, we deduce the following inequality:

    |Q1(ȷ)Q2(ȷ)|K12PC.

    Consequently, Q is a contraction.

    Step 3. Let demonstrate that R be continuous.

    Let σ be a σ¯ sequence in PC([0,T],R).

    Case 1. For each ȷ[0,ȷ1], we have

    |Qσ(ȷ)Q(ȷ)|[(ψ(ȷ)ψ(0))θ1|Δ|Γ(θ)[ϱk=1|νk|(ψ(υk)ψ(0))p+φk;ψΓ(p+φk+1)]+(ψ(ȷ)ψ(0))pΓ(p+1)]×Y(.,σ(.),.,)Y(.,(.),.,)PC.

    Case 2. For every \mathrm{\jmath}\in (\mathrm{\jmath}_{\mathrm{k}}, \mathrm{r}_{\mathrm{k}}] , we get

    \begin{align*} \left|\mathcal{Q}\hbar_{\mathrm{\sigma}}\mathrm{(\jmath)}-\mathcal{Q}\hbar\mathrm{(\jmath)}\right| = 0. \end{align*}

    Case 3. For each \mathrm{\jmath} \in (\mathrm{r}_{\mathrm{k}}, \mathrm{\jmath}_{\mathrm{k}+1}], \; \mathrm{k} = 1, 2, \cdots, \mathrm{\varrho} , we obtain

    \begin{align*} \left|\mathcal{Q}\hbar_{\mathrm{\sigma}}\mathrm{(\jmath)}-\mathcal{Q}\hbar\mathrm{(\jmath)}\right| \leqq \frac{(\mathrm{\jmath}_{\mathrm{k}+1}-\mathrm{r}_{\mathrm{k}})}{\Gamma(\mathrm{p}+1)}\left\|\mathcal{Y}(.,\hbar_{\mathrm{\sigma}}(.),.,)-\mathcal{Y}(.,\hbar(.),.,)\right\|_{\mathrm{PC}}. \end{align*}

    We thus conclude from the above cases that \left\|\mathcal{Q}\hbar_{\mathrm{\sigma}}\mathrm{(\jmath)}-\mathcal{Q}\hbar\mathrm{(\jmath)}\right\|_{\mathrm{PC}}\longrightarrow 0 as \mathrm{\sigma}\rightarrow \infty .

    Step 4. Finally, let us prove that \mathcal{Q} is compact.

    Firstly, \mathcal{Q} is uniformly bounded on \mathcal{B}_{\mathrm{p}, \mathrm{r}} .

    Since \left\|\mathcal{Q}\hbar\right\|\leqq \frac{\mathcal{L}_{\mathrm{g}_{\mathrm{k}}}\mathcal{(T)}}{\Gamma(1+\mathrm{p})} < \mathrm{r} , therefore, we have \mathcal{Q} is uniformly bounded on \mathcal{B}_{\mathrm{p}, \mathrm{r}} .

    We prove that \mathcal{Q} maps a bounded set to a \mathcal{B}_ {\mathrm{p}, \mathrm{r}} equicontinuous set.

    Case 1. For interval \mathrm{\jmath} \in [0, \mathrm{\jmath}_{1}], \; 0 \leqq \mathcal{E}_{1} \leqq \mathcal{E}_{2} \leqq \mathrm{\jmath}_{1}, \hbar \in \mathcal{B}_{\mathrm{r}} , we have

    \begin{align*} \left|\mathcal{Q}\mathcal{E}_{2}-\mathcal{Q}\mathcal{E}_{1}\right| \leqq \frac{\mathcal{L}_{\mathrm{g}_{\mathrm{k}}}(1+\mathrm{r})}{\Gamma(\mathrm{p}+1)}(\mathcal{E}_{2}-\mathcal{E}_{1}). \end{align*}

    Case 2. For each \mathrm{\jmath} \in (\mathrm{\jmath}_{\mathrm{k}}, \mathrm{r}_{\mathrm{k}}], \; \mathrm{\jmath}_{\mathrm{k}} < \mathcal{E}_{1} < \mathcal{E}_{2} \leqq \mathrm{r}_{\mathrm{k}}, \; \hbar \in \mathcal{B}_{\mathrm{p}, \mathrm{r}} , we get

    \begin{align*} \left|\mathcal{Q}\mathcal{E}_{2}-\mathcal{Q}\mathcal{E}_{1}\right| = 0. \end{align*}

    Case 3. For every \mathrm{\jmath} \in (\mathrm{r}_{\mathrm{k}}, \mathrm{\jmath}_{\mathrm{k}+1}], \; \mathrm{r}_{\mathrm{k}} < \mathcal{E}_{1} < \mathcal{E}_{2} \leqq \mathrm{\jmath}_{\mathrm{k}+1}, \; \hbar \in \mathcal{B}_{\mathrm{p}, \mathrm{r}} , we obtain

    \begin{align*} \left|\mathcal{Q}\mathcal{E}_{2}-\mathcal{Q}\mathcal{E}_{1}\right| \leqq \frac{\mathcal{L}_{\mathrm{g}_{\mathrm{k}}}(1+\mathrm{r})}{\Gamma(\mathrm{p}+1)}(\mathcal{E}_{2}-\mathcal{E}_{1}). \end{align*}

    From the above cases, we deduce that \left|\mathcal{Q}\mathcal{E}_{2}-\mathcal{Q}\mathcal{E}_{1}\right|\longrightarrow 0 as \mathcal{E}_{2}\longrightarrow\mathcal{E}_{1} and \mathcal{Q} is equicontinuous. As a result, \mathcal{Q}(\mathcal{B}_{\mathrm{p}, \mathrm{r}}) is relatively compact and \mathcal{Q} is compact, by using the Ascoli–Arzela theorem. Hence, the problem given by (1.1) to (1.3) has at least one fixed point on [0, \mathrm{T_{\star}}] .

    Let as consider the following {\psi} -Caputo (or, more appropriately, {\psi} -Liouville–Caputo) fractional boundary value problem:

    \begin{align} &D^{\mathrm{p,q};{\psi}}\hbar\mathrm{(\jmath)} = \frac{\exp(-\mathrm{\jmath})\left|\hbar\mathrm{(\jmath)}\right|}{9+\exp(-\mathrm{\jmath})(1+\left|\hbar\mathrm{(\jmath)}\right|)}+\frac{1}{3}\int^{\mathrm{\jmath}}_{0}\mathrm{e}^{-(\mathrm{\imath}-\mathrm{\jmath})}\hbar\mathrm{(\imath)}\mathrm{d\imath},\ \ \mathrm{\jmath} \in (0,1], \end{align} (4.1)
    \begin{align} &\hbar\mathrm{(\jmath)} = \frac{\left|\hbar\mathrm{(\jmath)}\right|}{2(1+\left|\hbar\mathrm{(\jmath)}\right|)}, \ \mathrm{\jmath} \in \left(\frac{1}{2},1\right], \end{align} (4.2)
    \begin{align} & \hbar(0) = 0,\ \ \hbar(1) = \frac{1}{2}\mathcal{I}^{\frac{2}{3}}\hbar\left(\frac{7}{5}\right)+\frac{2}{3}\mathcal{I}^{\frac{4}{5}}\hbar\left(\frac{9}{5}\right)+\frac{5}{2}\mathcal{I}^{\frac{3}{4}}\hbar\left(\frac{7}{2}\right), \end{align} (4.3)

    together with \mathcal{L} = \mathcal{G} = \frac{1}{10}, \; \mathcal{M} = \frac{1}{3}, \; \mathrm{p} = \frac{5}{7}, \; \theta = \frac{2}{5}, \; \mathcal{L}_{\texttt{h}_{1}} = \frac{1}{3}, \; \nu_{1} = \frac{1}{2}, \; \nu_{2} = \frac{2}{3}, \; \nu_{3} = \frac{2}{5}, \; \upsilon_{1} = \frac{2}{7}, \; \upsilon_{2} = \frac{5}{9}, \; \upsilon_{3} = \frac{1}{7}, \; \varphi_{1} = \frac{2}{3}, \; \varphi_{2} = \frac{4}{5}, \; \varphi_{3} = \frac{3}{4} . We shall check the condition (3.1), for value \mathrm{p}\in (1, 2) . Indeed, by using Theorem 3.1, we determine that

    \begin{align*} &\mathcal{L}_{\mathrm{h}_{\mathrm{k}}} +\frac{\mathcal{(L+GM)}}{\Gamma(\mathrm{p}+1)}(\mathrm{\jmath}^{\mathrm{p}}_{\mathrm{k}+1}+\mathrm{r}^{\mathrm{p}}_{\mathrm{k}})\approx 0.41 < 1,\\ &\mbox{and}\\ &\mathcal{L}_{\mathrm{h}_{\mathrm{k}}} +\mathcal{(L+GM)}\Big\{\frac{({\psi}(\mathrm{\jmath})-{\psi}(0))^{\mathrm{p}}}{\Gamma(\mathrm{p}+1)}+\frac{({\psi}(\mathrm{\jmath})-{\psi}(0))^{\theta-1}}{\left|\Delta\right|\Gamma(\theta)}\Big[\sum^{\mathrm{\varrho}}_{\mathrm{k} = 1}\left|\nu_{\mathrm{k}}\right|\frac{({\psi}(\upsilon_{\mathrm{k}})-{\psi}(0))^{\mathrm{p}+\varphi_{\mathrm{k}};{\psi}}}{\Gamma(\mathrm{p}+\varphi_{\mathrm{k}}+1)}\Big]\Big\}\approx 0.485 < 1. \end{align*}

    Hence, from Theorem 3.1 the problem given by (4.1) to (4.3) has a unique solution on [0, \mathrm{T_{\star}}] .

    We have discussed in this paper {\psi} -Hilfer FIDEs class with non-instantaneous impulsive conditions and with an R-L integral boundary condition. Furthermore, the existence and uniqueness of the derived solution is investigated via two well-known fixed point theorems. Moreover, we have proved its boundedness of the method in Section 3, and hence we don't need stability analysis. Finally, the consistency of our results was demonstrated with an example. For future work, we will give the numerical algorithm for the R-L integral BVPs via different kinds of fractional derivatives. Also, interested researchers can improve our results by using the resolvents operators as well.

    This research has received funding support from the National Science, Research and Innovation Fund (NSRF), Thailand, and the first author would like to thank Prince Sultan University for the support through the TAS research lab.

    The authors declare that they have no conflicts interests.



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