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Research article

Inventory model with nonlinear price-dependent demand for non-instantaneous decaying items via advance payment and installment facility

  • Received: 27 April 2022 Revised: 04 August 2022 Accepted: 10 August 2022 Published: 07 September 2022
  • MSC : 90B05

  • Determining the joint pricing and ordering policy is a challenging task for policy-makers dealing with perishable items. This research deals with the inventory coordination for a decaying commodity under a non-linear price-sensitive demand structure where the policy-maker completes the payment partially in advance, exploiting the multiple installments facility to control supply disruptions. Moreover, an inventory-out situation is incorporated to make the model more representative; shortages are backlogged partially through a variable rate in exponential form, depending on the customer waiting times. Though the formulated inventory coordination creates a highly complex optimization problem, the existence of the joint optimal pricing and ordering policy is explored by developing several theoretical outcomes. Three numerical illustrations are adopted to ensure the effectiveness of the model in providing the joint optimal pricing and ordering policy for the decision manager. Furthermore, to visualize the concavity of the average profit of the policy manager, as well as to demonstrate the adequacy of the optimum condition, MATLAB software was utilized. Finally, sensitivity studies for known key parameters are done using graphic presentation and a few supportive guidelines for the manager are also shown. The inventory manager should motivate the supplier to allow a higher installment frequency to implement the prepayment regulation, thus reducing the capital cost against the prepayment amount.

    Citation: Avijit Duary, Md. Al-Amin Khan, Sayan Pani, Ali Akbar Shaikh, Ibrahim M. Hezam, Adel Fahad Alrasheedi, Jeonghwan Gwak. Inventory model with nonlinear price-dependent demand for non-instantaneous decaying items via advance payment and installment facility[J]. AIMS Mathematics, 2022, 7(11): 19794-19821. doi: 10.3934/math.20221085

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  • Determining the joint pricing and ordering policy is a challenging task for policy-makers dealing with perishable items. This research deals with the inventory coordination for a decaying commodity under a non-linear price-sensitive demand structure where the policy-maker completes the payment partially in advance, exploiting the multiple installments facility to control supply disruptions. Moreover, an inventory-out situation is incorporated to make the model more representative; shortages are backlogged partially through a variable rate in exponential form, depending on the customer waiting times. Though the formulated inventory coordination creates a highly complex optimization problem, the existence of the joint optimal pricing and ordering policy is explored by developing several theoretical outcomes. Three numerical illustrations are adopted to ensure the effectiveness of the model in providing the joint optimal pricing and ordering policy for the decision manager. Furthermore, to visualize the concavity of the average profit of the policy manager, as well as to demonstrate the adequacy of the optimum condition, MATLAB software was utilized. Finally, sensitivity studies for known key parameters are done using graphic presentation and a few supportive guidelines for the manager are also shown. The inventory manager should motivate the supplier to allow a higher installment frequency to implement the prepayment regulation, thus reducing the capital cost against the prepayment amount.



    As we know today, the role of fractional operators or, rather, fractional calculus, in the study and investigation of natural phenomena is undeniable, if not irreplaceable. Certainly, the most important reason for the stunning growth of fractional calculus in the last decade can be seen in its ability and application in modeling biological [1,2,3,4] and physical [5,6,7,8,9] phenomena. As one of the most prominent features of fractional operators, we can mention their non-locality. Based on the available results and evidence, modeling by ordinary calculus is not capable of describing the real behavior of phenomena and is often associated with the error of estimating the phenomenon [10]. Researchers in the fields of science and engineering have different approaches to the non-local character of fractional calculus. Physicists' approach to this issue has led to interesting modeling schemes for physical phenomena such as heat flow, hereditary polarization in dielectrics, viscoelasticity and so on [11]. Such phenomena have been modeled with equations which are influenced by the past values of one or more variables, and they are called equations with memory in the literature. Mathematicians have also provided the basis for extending the existing models for different fields by generalizing the fractional operators. The most basic fractional operator is the derivative given by Riemann and Liouville (namely, the Riemann-Liouville derivative). With the publication of a book in 1999, Podlubny may have contributed the most to the systematic presentation of the theory of fractional operators [12]. Taking ideas from his works, we have witnessed the introduction of various operators, such as the Caputo, Hilfer, Atangana-Baleanu, Hadamard, fractal fractional, Caputo-Fabrizio, fractional q-derivative and (p,q)-derivative operators, etc., in the last two decades. For get more information about these contributions, one can refer to [13,14,15,16,17,18,19,20,21]. There is a certain type of kernel dependency included in all of those definitions. That is, we can consider a general operator from which fractional integral and derivative operators can be extracted by selecting specific kernels [22,23,24].

    One of the most recent generalizations of fractional operators is related to the work of the Portuguese mathematician Ricardo Almeida. In 2017, he presented a new definition of the Caputo derivative, namely, ψ-Caputo, with respect to another non-decreasing function, such as ψ [25]. In his new model, the Riemann-Liouville and Hadamard fractional operators are obtained by choosing ψ(κ)=κ and ψ(κ)=lnκ. A year later, he and colleagues investigated the existence and uniqueness of the solution for an initial value problem with his new fractional operator, and, using it, he presented a model for the growth of the world population and the gross domestic product growth rate in the USA [26]. In 2019, Abdo et al. studied the existence and uniqueness of the solution for initial and boundary value problems with the ψ-Caputo derivative [27,28]. In 2020, Wahash et al. investigated fractional differential equations with singularities by applying the ψ-Caputo operator using the Picard iteration method [29]. Voyiadjis and Sumelka used this new type of fractional operator to provide an important model of brain damage in the framework of anisotropic hyperelasticity [30]. Also, Ahmed et al, presented a model for thermostats by using this fractional operator [31]. For more contributions that include this new fractional operator, the reader can see [32,33,34,35]. Here, we are going to present a model for the pantograph equation using the ψ-Caputo fractional operator; however, we will continue this section with a discussion about the pantograph.

    Considering the issues raised and the potential in fractional operators, it is not far from expected that we also want to present a model for one of the most important and widely used equations in applied sciences, that is, the pantograph equation. Usually, the pantograph reminds us of a device that is installed on the top of the roof of electric buses. The pantograph problem was first raised by Mr. E. A. Cardwell of the British Railways Technical Center, at the Conference on Applications of Differential Equations in 1969. Ockendon and Tayler, in 1971, presented a mathematical model for Pantograph motion [36]. Today, the pantograph in electric trains is a tool that converts the electric current from direct to alternating current [37]. The pantograph differential equation is widely used in various fields, including number theory, quantum mechanics, statistics and electrodynamics [38,39]. Many researchers have investigated the pantograph equation from different aspects. For example, numerical solutions via Chebyshev polynomials are presented in [40], the existence of mild solutions to pantograph equations are investigated in [41], the stability of solutions was studied in [42] and a coupled system of pantograph problems using sequential fractional derivatives was examined in [43]. To access more information, see [44,45,46,47,48].

    The standard mode of the pantograph equation is formulated as follows:

    {w(κ)=c1w(κ)+c2w(εκ),0κK,w(0)=w0,

    such that 0<ε<1 [49]. The fractional case of the pantograph equation investigated by Balachandran et al. involving the Caputo operator is as follows:

    {CDηw(κ)=h(κ,w(κ),w(εκ)),0κK,w(0)=w0,

    where 0<ȷ and ε<1 [49].

    To the best of our knowledge, topological degree theory for condensing maps has not been applied to nonlinear pantograph differential equations with ψ-Caputo fractional derivatives under nonlocal boundary conditions. Therefore, inspired by the history mentioned above and previous works, in this paper, we investigate the existence of solutions for the following nonlinear fractional pantograph differential equation:

    {CDη,ψw(κ)=h(κ,w(κ),w(εκ)),κK=[0,K],w(0)=0,w(0)+χ(w)=w0, (1.1)

    where CDη,ψ is the ψ-Caputo fractional derivative of order η(1,2), ε(0,1), K>0, hC(K×R2,R), w0R and χ is the nonlocal term that satisfies some given conditions. The importance of the nonlocal condition, which is better than the classical initial condition, is explained in [50]. Furthermore, in recent research in 2021, Sabatier and Farges showed, by designing some problems and numerical analysis, that the use of fractional derivatives is problematic in some fractional models [51]. According to the above topics, in this work, we also considered the boundary conditions as nonlocal, although the authors, as mentioned have other works with fractional initial conditions applied in this matter [43].

    The rest of the paper is structured as follows. In Section 2, we state what we need from fractional calculus and topological degree theory as preliminaries to prove our main results. In Section 3, we first introduce three hypotheses, prove four auxiliary lemmas and prove the existence of solutions for the pantograph equation given by (1.1). In Section 4, we present examples with numerical and graphical simulations to validate our results. The last section concludes this paper.

    This section deals with some preliminaries and notations which are used throughout this paper. For more details, we refer the reader to [25].

    Definition 2.1. [26] Assume that w is an integrable function on K=[0,K], and that ψCn(K,R), where κK, ψ(κ)>0. Then, the ψ-Riemann-Liouville (ψ-RL) integral and derivative of w of fractional order ȷ is expressed as follows:

    Iȷ,ψw(κ)=1Γ(ȷ)κ0ψ(p)(ψ(κ)ψ(p))ȷ1w(p)dp,

    and

    Dȷ,ψw(κ)=1Γ(ȷn)(1ψ(κ)ddκ)nκ0ψ(p)(ψ(κ)ψ(p))nȷ1w(p)dp=(1ψ(κ)ddκ)nInȷ,ψw(κ),

    where n=[ȷ]+1.

    Definition 2.2. [26] Suppose that wCn1(K,R) and ψCn(K,R) such that κK, ψ(κ)>0. Then, the Ψ-Caputo operator of fractional order ȷ is formulated as follows:

    CDȷ,ψw(κ)=1Γ(nȷ)κ0ψ(p)(ψ(κ)ψ(p))nȷ1w[n]ψ(p)dp, (2.1)

    where

    w[n]ψ(p)=(1ψ(p)ddp)nw(p),n=[ȷ]+1.

    Remark 2.1. It is obvious that, with correct placement, namely, ψ(κ)=κ and ψ(κ)=ln(κ), in (2.1), the Caputo and Caputo-Hadamard derivatives can be reached.

    Remark 2.2. If 0<ȷ<1, then we have

    CDȷ,ψw(κ)=1Γ(1ȷ)(1ψ(p)ddp)1κ0(ψ(κ)Ψ(p))ȷw(p)dp.

    Theorem 2.1. [26] If ȷ>0 and wCn1(K,R), then the following assertions hold:

    1) CDȷ,ψIȷ,ψw(κ)=w(κ).

    2) Iȷ,ψCDȷ,ψw(κ)=w(κ)n1i=0w[i]ψ(0)i!(ψ(κ)ψ(0))i.

    Theorem 2.2. [26] Let μ>ν>0 and κK; then, we have the following:

    1) Iμ,ψ(ψ(κ)ψ(0))ν1=Γ(ν)Γ(μ+ν)(ψ(t)ψ(0))μ+ν1.

    2) Dμ,ψ(ψ(t)ψ(0))ν1=Γ(ν)Γ(νμ)(ψ(κ)ψ(0))νμ1.

    3) Dμ,ψ(ψ(κ)ψ(0))p=0, p<nN.

    Definition 2.3. [52] Suppose that X is a Banach space and BX={YX:Y,Yis bounded}. The function ρ : BX[0,+) is called the Kuratowski measure of non-compactness and defined as follows:

    ρ(Y)=inf{r>0:Yadmits a finite cover by sets of diameter r}.

    Theorem 2.3. [52] The measure ρ defined above, namely, Definition 2.3, applies to the following properties.

    (1) ρ(Y)=0 iff Y is relativity compact.

    (2) ρ(aY)=|a|ρ(Y),aR.

    (3) ρ(Y1+Y2)ρ(Y1)+ρ(Y2).

    (4) If Y1Y2, then ρ(Y1)ρ(Y2).

    (5) ρ(Y1Y2)=max{ρ(Y1), ρ(Y2)}.

    (6) ρ(Y)=ρ(¯Y)=ρ(convY), where ¯Y and convY denote the closure and convex hull of Y, respectively.

    Definition 2.4. [52] Assume that the function Θ:YXX is a continuous and bounded map. The function Θ is called ρ-Lipschitz if 0, given that

    ρ(Θ(Y))ρ(Y),YY.

    Definition 2.5. [52] The function Θ, which is defined in Definition 2.4 is called ρ-condensing if, for every bounded subset Y of Y, the following inequality holds:

    ρ(Θ(Y))<ρ(Y),

    such that ρ(Y)>0. Indeed,

    ρ(Θ(Y))ρ(Y)ρ(Y)=0.

    Moreover, we denote the class of all ρ-condensing maps Θ:YX by Cρ(Y).

    Definition 2.6. [52] The function w:YX is called Lipschitz if >0, given that

    w(y1)w(y2)∥≤y1y2,y1,y2Y.

    Lemma 2.1. [52] Suppose that w is a Lipschitz function with a constant ; then, w is ρLipschitz with the same constant.

    Lemma 2.2. [52] Consider the ρ-Lipschitz functions Θ,Δ:YX with constants 1,2, respectively. Then, the following statements are true:

    Θ+Δ:YX, is ρ-Lipschitz because of the 1+2 constant.

    If Θ is compact, then 1=0.

    Theorem 2.4. [53] Let Θ:YX such that YX is open and bounded; also suppose that

    T={(IΘ,Y,x):ΘCρ(ˉY),xX(IΘ)(Y)}

    is a family of the admissible triplets. Then, there exists one degree function, deg:TZ, such that the following properties are satisfied:

    deg(I,Y,y)=1 for every yY.

    For every disjoint, open set Y1,Y2Y, and every x(IΘ)(ˉY(Y1Y2)), we have

    deg(IΘ,Y,x)=deg(IΘ,Y1,x)+deg(IΘ,Y2,x).

    deg(If(t,.),Y,x(t)) is independent of t[0,1] for every continuous, bounded map f:[0,1]×ˉYX, which satisfies

    ρ(f([0,1]×Ω))<ρ(Ω),ΩˉY,withρ(Ω)>0,

    and every continuous function x:[0,1]X, which satisfies

    x(t)zf(t,z),t[0,1],zY.

    deg(IΘ,Y,x)0 implies that x(IΘ)(Y).

    deg(IΘ,Y,x)=deg(IΘ,Y1,x) for every open set Y1Y, and every x(IΘ)(ˉYY1).

    Theorem 2.5. [53] Assume that the map Θ:XX is ρ-condensing, τ[0,1] and EτX such that

    Eτ={xX:x=τΘxfor someτ}.

    Now, if Eτ is a bounded subset of X, then q>0, given that EτBq(0), and we have

    deg(IδΘ,Bq(0),0)=1,δ[0,1].

    As a result, Θ has at least one fixed point and the set of the fixed points of Θ lies in Bq.

    Here, to continue the work, we first introduce the necessary notations and three hypotheses which play a fundamental role in providing a suitable space for using the results of fixed-point theory and its contractions in the sequel. It is worth noting that, as a reminder, we are referring to a closed ball centered at 0 with radius q>0 by using Bq. Also, our Banach space C:=C(K,R) is equipped with the supreme norm, namely, w∥=supκKw(κ).

    (H1) Lχ>0, where

    |χ(w)χ(s)|Lχwsfor eachw,sC.

    (H2) Nχ>0, Mχ0 and 0α1, where

    |χ(w)|Nχwα+Mχfor eachwC.

    (H3) Nh,Mh>0 and 0β1, where

    |h(κ,w(κ),w(εκ))|Nhwβ+Mhfor eachwC.

    Lemma 3.1. The solution to problem (1.1) is equivalent to the following integral equation:

    w(κ)=w0χ(w)+1Γ(η)κ0ψ(p)(ψ(κ)Ψ(p))η1h(t,w(p),w(εp))dp. (3.1)

    Proof. Suppose that w is a solution of (1.1); then, by applying operator Iη,ψ on (1.1), we obtain

    Iη,ψCDη,ψw(κ)=Iη,ψh(κ,w(κ),w(εκ)),

    and by employing Proposition 2.1, we get

    w(κ)=c0+(ψ(κ)ψ(0))c1+Iη,ψh(κ,w(κ),w(εκ)),

    where c0,c1R. Hence,

    w(κ)=c1ψ(κ)+1Γ(η)κ0(ψ(p)(ψ(κ)ψ(p))η1h(p,w(p),w(εp)))dp;

    since w(0)+χ(w)=w0 and w(0)=0, then c0=w0χ(w) and c1=0. Hence, (3.1) holds.

    To show that (3.1) has at least one solution wC, we define two operators A,T : CC as follows:

    Aw(κ)=w0χ(w),κK, (3.2)

    and

    Tw(κ)=1Γ(η)κ0ψ(p)(ψ(κ)Ψ(p))η1h(p,w(p),w(εp))dp,κK. (3.3)

    Thus, (3.1) can be formulated as follows:

    Fw(κ)=Aw(κ)+Tw(κ),κK. (3.4)

    Lemma 3.2. The operator A is ρ- Lipschitz with the constant Lχ. Moreover, A satisfies the following inequality:

    (3.5)

    Proof. At first, we shall show that the operator \mathcal{A} is Lipschitz with the constant L\chi . To do this, let \boldsymbol{w}, \boldsymbol{s}\in \mathcal{C} ; then, we have

    \begin{equation*} \left\vert \mathcal{A} \boldsymbol{w}(\kappa)-\mathcal{A} \boldsymbol{s}(\kappa) \right\vert \leq |\chi(\boldsymbol{w})-\chi(\boldsymbol{s})|; \end{equation*}

    the hypothesis (H_1) yields that

    \begin{equation*} \left\vert \mathcal{A} \boldsymbol{w}(\kappa)-\mathcal{A} \boldsymbol{s}(\kappa) \right\vert\leq L_{\chi}\Vert \boldsymbol{w}-\boldsymbol{s}\Vert, \end{equation*}

    and taking the supremum over \kappa implies that

    \begin{equation*} \left\Vert \mathcal{A} \boldsymbol{w}-\mathcal{A} \boldsymbol{s} \right\Vert\leq L_{\chi}\Vert \boldsymbol{w}-\boldsymbol{s}\Vert; \end{equation*}

    hence, \mathcal{A} is Lipschitz with L_\chi . In view of Lemma 2.1, it follows that \mathcal{A} is \rho -Lipschitz with the same constant L_\chi . Now, to prove (3.5), let \boldsymbol{w}\in \mathcal{C} ; then, we have

    \begin{equation*} \left\vert \mathcal{A}\boldsymbol{w}(\kappa)\right\vert = \vert \boldsymbol{w}_0-\chi(\boldsymbol{w}) \vert\leq\vert \boldsymbol{w}_0\vert+ \vert\chi(\boldsymbol{w}) \vert; \end{equation*}

    by using the assumption (H_2) , we get

    \begin{equation*} \left\Vert \mathcal{A} \boldsymbol{w}\right\Vert \leq \vert \boldsymbol{w}_0\vert + N_\chi \Vert \boldsymbol{w}\Vert^{\alpha}+M_\chi. \end{equation*}

    Lemma 3.3. The operator \mathcal{T} , which is formulated in (3.3), is continuous and satisfies the following inequality:

    \begin{equation} \Vert \mathcal{T}\boldsymbol{w}\Vert\leq \frac{ 1}{\Gamma(\eta+1)}(N_\chi \Vert \boldsymbol{w} \Vert^{\beta}+M_\chi)(\psi(K)-\psi(0))^{\eta}, \; \; \forall \boldsymbol{w}\in \mathcal{C}. \end{equation} (3.6)

    Proof. For \mathcal{T} to be continuous, assume that \boldsymbol{w}_n\to \boldsymbol{w} in \mathcal{C} ; hence, \exists\delta > 0 , given that \Vert \boldsymbol{w}_n \Vert\leq \delta and \Vert \boldsymbol{w}\Vert\leq \delta . Now, let \kappa\in \mathcal{K} ; we can write

    \begin{align*} &\left\vert \mathcal{T} \boldsymbol{w}_n(\kappa)-\mathcal{T} \boldsymbol{w}(\kappa) \right\vert\\ & \leq \frac{1}{\Gamma(\eta)} \int_{0}^\kappa \psi'(p)(\psi(\kappa)-\psi(p))^{\eta-1} \left\vert h(p, \boldsymbol{w}_n(p), \boldsymbol{w}_n(\varepsilon p))-h(p, \boldsymbol{w}(p), \boldsymbol{w}(\varepsilon p)) \right\vert dp; \end{align*}

    since h is continuous, then

    \begin{equation*} \lim\limits_{n\rightarrow \infty}h(p, \boldsymbol{w}_n(p), \boldsymbol{w}_n(\varepsilon p)) = h(p, \boldsymbol{w}(p), \boldsymbol{w}(\varepsilon p)). \end{equation*}

    On the other hand, by using (H_3) , we obtain

    \begin{align*} &\frac{1}{\Gamma(\eta)} (\psi'(p)(\psi(\kappa)-\psi(p))^{\eta-1}\left\Vert h(p, \boldsymbol{w}_n(p), \boldsymbol{w}_n(\varepsilon p))-h(p, \boldsymbol{w}(p), \boldsymbol{w}(\varepsilon p)) \right\Vert\\ &\leq \left( N_\chi \delta^{\beta}+M_\chi\right) \times \frac{1}{\Gamma(\eta)} (\psi'(p)(\psi(\kappa)-\psi(p))^{\eta-1}; \end{align*}

    since p\mapsto \frac{1}{\Gamma(\eta)} (\psi'(p)(\psi(\kappa)-\psi(p))^{\eta-1} is an integrable function on [0, \kappa] , then Lebesgue's dominated convergence theorem implies that

    \begin{equation*} \lim\limits_{n\mapsto +\infty}\frac{1}{\Gamma(\eta)} (\psi'(p)(\psi(\kappa)-\psi(p))^{\eta-1}\left\Vert h(p, \boldsymbol{w}_n(p), \boldsymbol{w}_n(\varepsilon p))-h(p, \boldsymbol{w}(p), \boldsymbol{w}(\varepsilon p)) \right\Vert dp = 0, \end{equation*}

    which yields that

    \begin{equation*} \lim\limits_{n\mapsto +\infty} \parallel \mathcal{T} \boldsymbol{w}_n-\mathcal{T} \boldsymbol{w}\parallel = 0; \end{equation*}

    hence, \mathcal{T} is continuous. To show (3.6), let \boldsymbol{w}(\kappa)\in \mathcal{C} ; then, we have

    \begin{equation*} \left\vert \mathcal{T} \boldsymbol{w}(\kappa) \right\vert \leq \frac{1}{\Gamma(\eta)} \int_{0}^t \psi'(p)(\psi(\kappa)-\psi(p))^{\eta-1} \left\vert h(p, \boldsymbol{w}(p), \boldsymbol{w}(\varepsilon p)) \right\vert dp; \end{equation*}

    from (H_3) , we obtain

    \begin{equation*} \left\vert \mathcal{T} \boldsymbol{w}(\kappa) \right\vert \leq \frac{( N_\chi \Vert \boldsymbol{w}\Vert^{\beta}+M_\chi)}{\Gamma(\eta)} \int_{0}^\kappa \psi'(p)(\psi(\kappa)-\psi(p))^{\eta-1} dp. \end{equation*}

    Finally, we obtain

    \begin{equation*} \parallel \mathcal{T}\boldsymbol{w}\parallel \leq \frac{ ( N_\chi \Vert \boldsymbol{w}\Vert^{\beta}+M_\chi)(\psi(K)-\psi(0))^{\eta}}{\Gamma(\eta+1)}. \end{equation*}

    Lemma 3.4. The operator \mathcal{T}:\mathcal{C} \rightarrow \mathcal{C} is compact.

    Proof. We shall show that \mathcal{T}\mathcal{B}_q is relatively compact in \mathcal{C} . To do this, let \boldsymbol{w}\in \mathcal{B}_q ; then, from (3.6), we get

    \begin{equation*} \parallel \mathcal{T}\boldsymbol{w}\parallel \leq \frac{ ( N_\chi q^{\beta}+M_\chi)(\psi(T)-\psi(0))^{\eta}}{\Gamma(\eta+1)}: = \xi. \end{equation*}

    It follows that \mathcal{T}\mathcal{B}_q \subset \mathcal{B}_\xi . Hence, \mathcal{T}\mathcal{B}_q is bounded. To prove that \mathcal{T}\mathcal{B}_q is equicontinuous, let \boldsymbol{w}\in \mathcal{T}\mathcal{B}_q and \kappa_1, \kappa_2\in \mathcal{K} such that \kappa_1 < \kappa_2 ; then, we have

    \begin{equation*} \left| \mathcal{T}\boldsymbol{w}(\kappa_2)-\mathcal{T}\boldsymbol{w}(\kappa_1)\right|\leq\frac{N_\chi \vert \boldsymbol{w}\vert ^{p}+M_\chi}{\Gamma(\eta)} \int_{\kappa_1}^{\kappa_2} \psi'(p)(\psi(\kappa_2)-\psi(p))^{\eta-1}dp, \end{equation*}
    \begin{equation*} \left| \mathcal{T}\boldsymbol{w}(\kappa_2)-\mathcal{T}\boldsymbol{w}(\kappa_1)\right|\leq\frac{N_\chi q^{\beta}+M_\chi}{\Gamma(\eta)} \int_{\kappa_1}^{\kappa_2} \psi'(p)(\psi(\kappa_2)-\psi(p))^{\eta-1}dp, \end{equation*}
    \begin{equation*} \left| \mathcal{T}\boldsymbol{w}(\kappa_2)-\mathcal{T}\boldsymbol{w}(\kappa_1)\right|\leq\frac{N_\chi q^{\beta}+M_\chi}{\Gamma(\eta+1)} (\psi(\kappa_2)-\psi(\kappa_1))^{\eta}. \end{equation*}

    Since \Psi is a continuous function, then we obatin

    \begin{equation*} \lim\limits_{\kappa_1\rightarrow \kappa_2}\left| \mathcal{T}\boldsymbol{w}(\kappa_1)-\mathcal{T}\boldsymbol{w}(\kappa_2)\right| = 0, \end{equation*}

    which shows that \mathcal{T}\mathcal{B}_q is equicontinuous. Hence, \mathcal{T}\mathcal{B}_q is uniformly bounded and equicontinuous. The Arzelà-Ascoli theorem [54] permits us to conclude that \mathcal{T}\mathcal{B}_q is relatively compact; thus, \mathcal{T} is compact.

    Corollary 3.1. \mathcal{T}: \mathcal{C} \rightarrow \mathcal{C} is \rho -Lipschitz with a zero constant.

    Proof. From the compactness of the operator \mathcal{T} , and Lemma 2.2, it follows that \mathcal{T} is \rho -Lipschitz with a zero constant.

    Now, we have all of the tools to establish our main result.

    Theorem 3.1. Suppose that the hypotheses (H_1) (H_3) are true; then, the fractional pantograph differential equation mentioned in (1.1) has at least one solution: \boldsymbol{w}\in \mathcal{C} . Moreover, the set of all solutions for (1.1) is bounded in C(\mathcal{K}, \mathbb{R}) .

    Proof. Let \mathcal{A}, \mathcal{T}, \mathcal{F} : \mathcal{C}\rightarrow \mathcal{C} be the operators formulated in (3.2)–(3.4), respectively. \mathcal{A}, \mathcal{T}, \mathcal{F} are continuous and bounded. Furthermore, in view of Lemma 3.2 and Corollary 3.1, the operator \mathcal{A} is \rho -Lipschitz given L_{\chi}\in[0, 1) , and \rho -Lipschitz with a zero constant. By using Lemma 2.2, we deduce that \mathcal{F} is a strict \rho -contraction with a constant L_{\chi} . Now, for some \tau\in [0, 1] , we set

    \begin{equation*} \mathcal{E}_\tau = \{\boldsymbol{w}\in \mathcal{C}: \boldsymbol{w} = \tau \mathcal{F}\boldsymbol{w}\}. \end{equation*}

    We claim that \mathcal{E}_\tau is bounded in \mathcal{C} . To prove this claim, suppose that \boldsymbol{w}\in \mathcal{E}_\tau ; then,

    \begin{equation*} \boldsymbol{w} = \tau \mathcal{F}\boldsymbol{w} = \tau (\mathcal{A}\boldsymbol{w}+\mathcal{T}\boldsymbol{w}), \end{equation*}

    which yields that

    \begin{equation*} \Vert \boldsymbol{w}\Vert = \tau\Vert \mathcal{F}\boldsymbol{w}\Vert\leq\tau(\Vert \mathcal{A} \boldsymbol{w}\Vert+\Vert \mathcal{T} \boldsymbol{w}\Vert); \end{equation*}

    by using Lemmas 3.2 and 3.3, we get

    \begin{equation} \Vert \boldsymbol{w}\Vert \leq\left(\vert \boldsymbol{w}_0 \vert + N_\chi \Vert \boldsymbol{w}\Vert^{\alpha}+M_\chi+\frac{ ( N_h \Vert \boldsymbol{w}\Vert^{\beta}+M_h)(\psi(K)-\psi(0))^{\eta}}{\Gamma(\eta+1)}\right). \end{equation} (3.7)

    The above inequality, namely, (3.7), yields that \mathcal{E}_\tau is bounded in \mathcal{C} given \alpha < 1 and \beta < 1 .

    Suppose that our claim is not true; in this case, let \xi: = \Vert \boldsymbol{w}\Vert \longrightarrow\infty . Dividing both sides of (3.7) by \xi , and taking \xi\rightarrow \infty , then we obtain

    \begin{equation*} 1\leq\lim\limits_{\xi\rightarrow \infty}\frac{\left(\vert \boldsymbol{w}_0\vert + N_\chi \xi^{\alpha}+M_\chi+ \frac{ ( N_h \xi^{\beta}+M_h)(\psi(T)-\psi(0))^{\eta}}{\Gamma(\eta+1)}\right)}{\xi} = 0, \end{equation*}

    which is a contradiction. By using Theorem 2.5, we conclude that \mathcal{F} has at least one fixed point which is the solution of (1.1) and the set of the fixed points of \mathcal{F} is bounded in \mathcal{C} .

    Remark 3.1. If we set \alpha = \beta = 1 in hypotheses (H_2) and (H_3) , then the result of Theorem 3.1 will be as follows:

    \begin{equation*} N_\chi+\frac{ N_h(\psi(K)-\psi(0))^{\eta}}{\Gamma(\eta+1)} < 1. \end{equation*}

    In this section, we give two examples to illustrate the usefulness of our main result.

    Example 4.1. Consider the following problem:

    \begin{equation} \begin{cases} ^C\mathcal{D}^{\frac{3}{2}, e^\kappa} \boldsymbol{w}(\kappa) = \dfrac{\kappa^{2}}{\sqrt{77}}\big(\boldsymbol{w}(\kappa)+sin^{2}(\boldsymbol{w}(\kappa)\big)+\dfrac{1}{7\sqrt{\pi}}cos(\boldsymbol{w}(\dfrac{\kappa}{\sqrt{2}})), \; \; \kappa\in \mathcal{K} = [0, 1]\\[0.3cm] \boldsymbol{w}'(0) = 0, \; \; \boldsymbol{w}(0) = \sum\limits_{j = 1}^{20}\theta_j \vert \boldsymbol{w}(\kappa_j)\vert, \; \; \theta_j > 0, \; \; 0 < \kappa_j < 1, \; \; j = 1, 2, .., 20. \end{cases} \end{equation} (4.1)

    Here, \varepsilon = \frac{1}{\sqrt{2}} , \eta = \frac{3}{2} , K = 1 and \psi(\kappa) = e^\kappa , and, in this case, we let \chi(\boldsymbol{w}) = \sum\limits_{j = 1}^{20}\theta_j \vert \boldsymbol{w}(\kappa_j)\vert\; \; with \sum\limits_{j = 1}^{20}\theta_j < 1 . Clearly, (H_1) and (H_2) hold with N_\chi = L_\chi = \sum\limits_{i = j}^{20}\theta_j , M_\chi = 0 and q = 1 .

    Indeed, we can write

    \begin{equation*} \vert\chi(\boldsymbol{w}(\kappa))\vert = \left\vert\sum\limits_{j = 1}^{20}\theta_j \vert \boldsymbol{w}(\kappa_j)\right\vert; \end{equation*}

    hence,

    \begin{equation*} \vert\chi(\boldsymbol{w})\vert \leq \sum\limits_{j = 1}^{20}\theta_j\left\Vert \boldsymbol{w}\right\Vert; \end{equation*}

    thus, N_\chi = \sum\limits_{j = 1}^{20}\theta_j , M_\chi = 0 and \alpha = 1 . Alternatively, we have

    \begin{equation*} \vert\chi(\boldsymbol{w}(\kappa)) -\chi(\boldsymbol{s}(\kappa))\vert = \left\vert\sum\limits_{j = 1}^{20}\theta_j \vert \boldsymbol{w}(\kappa_j)-\sum\limits_{j = 1}^{20}\theta_j \vert \boldsymbol{s}(\kappa_j)\right\vert; \end{equation*}

    hence,

    \begin{equation*} \vert\omega(\boldsymbol{w}) -\omega(\boldsymbol{s})\vert \leq \sum\limits_{j = 1}^{20}\theta_j \left\vert \boldsymbol{w}-\boldsymbol{s} \right\vert; \end{equation*}

    thus, L_\chi = \sum\limits_{j = 1}^{20}\theta_j .

    To check the fulfillment of (H_3) , let \kappa \in \mathcal{K} and \boldsymbol{w}\in\mathbb{R} ; then, we have

    \begin{equation*} \left| h(\kappa, \boldsymbol{w}(\kappa), \boldsymbol{w}(\varepsilon \kappa)) \right| = \left| \dfrac{\kappa^{2}}{\sqrt{77}}\big(\boldsymbol{w}(\kappa)+sin^{2}(\boldsymbol{w}(\kappa)\big)+\dfrac{1}{7\sqrt{\pi}}cos(\boldsymbol{w}(\dfrac{\kappa}{\sqrt{2}})) \right|, \end{equation*}

    which implies that

    \begin{equation*} \left|h(\kappa, \boldsymbol{w}(\kappa), \boldsymbol{w}(\varepsilon \kappa)) \right|\leq \frac{1}{\sqrt{77}}\vert\boldsymbol{w}\vert+0.1946. \end{equation*}

    Thus, (H_3) holds with N_h = \dfrac{1}{\sqrt{77}} , M_h = 0.1946 and \beta = 1 . Consequently, Theorem 3.1 implies that problem (4.1) has at least one solution. Moreover, from the inequality (3.7), we get

    \begin{equation} \Vert \boldsymbol{w}\Vert\leq \xi_{*}:\frac{0.1946(e-1)^{(\eta)}}{\Gamma(\eta+1)-\frac{1}{\sqrt{77}}(e-1)^{(\eta)}} = \frac{0.1946(e-1)^{(3/2)}}{\Gamma(5/2)-\frac{1}{\sqrt{77}}(e-1)^{(3/2)}} = 0.4086. \end{equation} (4.2)

    Thus, the set of solutions for (4.1) is bounded. To better understand this example, graphs of some functions are provided in Figures 1 and 2. The data from Table 1 indicate that the boundedness of the solution set for (4.1) depends on the choice of \psi(\kappa) .

    Figure 1.  The graph of h(\kappa, \boldsymbol{w}(\kappa)) for Example 4.1.
    Figure 2.  The graph of h(\kappa, \boldsymbol{w}(\kappa), \boldsymbol{w}(\varepsilon \kappa)) for Example 4.1.
    Table 1.  Numerical results for \xi_{*} based on \psi(\kappa) selection in Example 4.1.
    \psi(\kappa) \xi_{*}
    \kappa 0.1601 < 1
    e^{\kappa} 0.4086 < 1
    2^{\kappa} 0.1601 < 1
    3^{\kappa} 0.5467 < 1
    4^{\kappa} 1.3721 > 1
    5^{\kappa} 3.7306 > 1

     | Show Table
    DownLoad: CSV

    Example 4.2. Consider the following problem:

    \begin{equation} \begin{cases} ^C\mathcal{D}^{\frac{9}{5}, \kappa} \boldsymbol{w}(\kappa) = \dfrac{e^{-\kappa}}{\sqrt{11+\kappa^{2}}}\big(\boldsymbol{w}(\kappa)+sin(\kappa)+\dfrac{cos(\boldsymbol{w}(\frac{\kappa}{\sqrt{2}}))}{\sqrt{\pi+\kappa^{2}}}\big), \; \; \kappa\in \mathcal{K} = [0, 1]\\[0.3cm] \boldsymbol{w}'(0) = 0, \; \; \boldsymbol{w}(0) = \sum\limits_{j = 1}^{20}\theta_j \vert \boldsymbol{w}(\kappa_j)\vert, \; \; \theta_j > 0, \; \; 0 < \kappa_j < 1, \; \; j = 1, 2, .., 20. \end{cases} \end{equation} (4.3)

    In this case, \varepsilon = \frac{1}{\sqrt{2}} , \eta = \frac{9}{5} , K = 1 , \psi(\kappa) = \kappa and \chi(\boldsymbol{w}) = \sum\limits_{j = 1}^{20}\theta_j \vert \boldsymbol{w}(\kappa_j)\vert\; \; with \; \; \sum\limits_{j = 1}^{20}\theta_j < 1 . Similar to the previous example, hypotheses (H_1) and (H_2) are valid with N_\chi = L_\chi = \sum\limits_{i = j}^{20}\theta_j , M_\chi = 0 , q = 1 and \alpha = 1 . To check the fulfillment of (H_3) , we can write

    \begin{equation*} \left| h(\kappa, \boldsymbol{w}(\kappa), \boldsymbol{w}(\varepsilon \kappa)) \right| = \left| \dfrac{e^{-\kappa}}{\sqrt{11+\kappa^{2}}}\big(\boldsymbol{w}(\kappa)+sin(\kappa)+\dfrac{cos(\boldsymbol{w}(\frac{\kappa}{\sqrt{2}}))}{\sqrt{\pi+\kappa^{2}}}\big) \right|, \end{equation*}

    which implies that

    \begin{equation*} \left|h(\kappa, \boldsymbol{w}(\kappa), \boldsymbol{w}(\varepsilon \kappa)) \right|\leq \frac{1}{\sqrt{11}}\vert\boldsymbol{w}\vert+0.4716. \end{equation*}

    Hence, (H_3) holds with N_h = \dfrac{1}{\sqrt{11}} , M_h = 0.4716 and \beta = 1 . Consequently, Theorem 3.1 implies that problem (4.3) has at least one solution. Moreover, from the inequality (3.7), we get

    \begin{equation} \Vert \boldsymbol{w}\Vert\leq \xi_{*}:\frac{0.4716}{\Gamma(\eta+1)-\frac{1}{\sqrt{11}}} = \frac{0.4716}{\Gamma(14/5)-\frac{1}{\sqrt{11}}} = 0.3430. \end{equation} (4.4)

    Thus the set of solutions for (4.3) is bounded. To better understand this example, graphs of some functions are provided in Figures 3 and 4. The data from Table 2 indicate that the boundedness of the solution set for (4.3) depends on the choice of \psi(\kappa) .

    Figure 3.  The graph of h(\kappa, \boldsymbol{w}(\kappa)) for Example 4.2.
    Figure 4.  The graph of h(\kappa, \boldsymbol{w}(\kappa), \boldsymbol{w}(\varepsilon \kappa)) for Example 4.2.
    Table 2.  Numerical results for \xi_{*} based on \psi(\kappa) selection in Example 4.2.
    \psi(\kappa) \xi_{*}
    \kappa 0.3430 < 1
    e^{\kappa} 1.4238 > 1
    2^{\kappa} 0.3430 < 1
    3^{\kappa} 2.6209 > 1
    4^{\kappa} -6.7896 < 0
    5^{\kappa} -2.8888 < 0

     | Show Table
    DownLoad: CSV

    Today, we see the presence of fractional calculus in the mathematical modeling of natural phenomena. The non-locality of fractional derivatives gives it the special ability to be used to model and describe physical phenomena. Using this capability, we presented a comprehensive analysis of pantograph modeling by using the fractional derivative of the \psi -Caputo type. We guaranteed the existence of the solution with the help of topological degree theory and the Arzela-Ascoli theorem. Finally, we presented numerical and graphical simulations to validate our results. Our results show that the boundedness of the solution set depends on the type of the \psi(\kappa) function.

    The authors declare that they have not used artificial intelligence tools in the creation of this article.

    The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number (IF2/PSAU/2022/01/22923).

    The authors declare no conflict of interest.



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