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Research article Special Issues

Study of modified prism networks via fractional metric dimension

  • Received: 07 November 2022 Revised: 08 February 2023 Accepted: 09 February 2023 Published: 06 March 2023
  • MSC : 05C12, 05C76, 05C90

  • For a connected network Γ, the distance between any two vertices is the length of the shortest path between them. A vertex c in a connected network is said to resolve an edge e if the distances of c from its endpoints are unequal. The collection of all the vertices which resolve an edge is called the local resolving neighborhood set of this edge. A local resolving function is a real-valued function is defend as η:V(Γ)[0,1] such that η(Rx(e))1 for each edge eE(Γ), where Rx(e) represents the local resolving neighborhood set of a connected network. Thus the local fractional metric dimension is defined as dimLF(Γ)=min{|η|:ηistheminimallocalresolvingfunctionofΓ}, where |η|=aRx(e)η(a). In this manuscript, we have established sharp bounds of the local fractional metric dimension of different types of modified prism networks and it is also proved that local fractional metric dimension remains bounded when the order of these networks approaches to infinity.

    Citation: Ahmed Alamer, Hassan Zafar, Muhammad Javaid. Study of modified prism networks via fractional metric dimension[J]. AIMS Mathematics, 2023, 8(5): 10864-10886. doi: 10.3934/math.2023551

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  • For a connected network Γ, the distance between any two vertices is the length of the shortest path between them. A vertex c in a connected network is said to resolve an edge e if the distances of c from its endpoints are unequal. The collection of all the vertices which resolve an edge is called the local resolving neighborhood set of this edge. A local resolving function is a real-valued function is defend as η:V(Γ)[0,1] such that η(Rx(e))1 for each edge eE(Γ), where Rx(e) represents the local resolving neighborhood set of a connected network. Thus the local fractional metric dimension is defined as dimLF(Γ)=min{|η|:ηistheminimallocalresolvingfunctionofΓ}, where |η|=aRx(e)η(a). In this manuscript, we have established sharp bounds of the local fractional metric dimension of different types of modified prism networks and it is also proved that local fractional metric dimension remains bounded when the order of these networks approaches to infinity.



    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:

    AwC|w0|+Nχwα+MχforeverywC. (3.5)

    Proof. At first, we shall show that the operator A is Lipschitz with the constant Lχ. To do this, let w,sC; then, we have

    |Aw(κ)As(κ)||χ(w)χ(s)|;

    the hypothesis (H1) yields that

    |Aw(κ)As(κ)|Lχws,

    and taking the supremum over κ implies that

    AwAsLχws;

    hence, A is Lipschitz with Lχ. In view of Lemma 2.1, it follows that A is ρ-Lipschitz with the same constant Lχ. Now, to prove (3.5), let wC; then, we have

    |Aw(κ)|=|w0χ(w)||w0|+|χ(w)|;

    by using the assumption (H2), we get

    Aw|w0|+Nχwα+Mχ.

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

    Tw1Γ(η+1)(Nχwβ+Mχ)(ψ(K)ψ(0))η,wC. (3.6)

    Proof. For T to be continuous, assume that wnw in C; hence, δ>0, given that wnδ and wδ. Now, let κK; we can write

    |Twn(κ)Tw(κ)|1Γ(η)κ0ψ(p)(ψ(κ)ψ(p))η1|h(p,wn(p),wn(εp))h(p,w(p),w(εp))|dp;

    since h is continuous, then

    limnh(p,wn(p),wn(εp))=h(p,w(p),w(εp)).

    On the other hand, by using (H3), we obtain

    1Γ(η)(ψ(p)(ψ(κ)ψ(p))η1h(p,wn(p),wn(εp))h(p,w(p),w(εp))(Nχδβ+Mχ)×1Γ(η)(ψ(p)(ψ(κ)ψ(p))η1;

    since p1Γ(η)(ψ(p)(ψ(κ)ψ(p))η1 is an integrable function on [0,κ], then Lebesgue's dominated convergence theorem implies that

    limn+1Γ(η)(ψ(p)(ψ(κ)ψ(p))η1h(p,wn(p),wn(εp))h(p,w(p),w(εp))dp=0,

    which yields that

    limn+TwnTw∥=0;

    hence, T is continuous. To show (3.6), let w(κ)C; then, we have

    |Tw(κ)|1Γ(η)t0ψ(p)(ψ(κ)ψ(p))η1|h(p,w(p),w(εp))|dp;

    from (H3), we obtain

    |Tw(κ)|(Nχwβ+Mχ)Γ(η)κ0ψ(p)(ψ(κ)ψ(p))η1dp.

    Finally, we obtain

    Tw∥≤(Nχwβ+Mχ)(ψ(K)ψ(0))ηΓ(η+1).

    Lemma 3.4. The operator T:CC is compact.

    Proof. We shall show that TBq is relatively compact in C. To do this, let wBq; then, from (3.6), we get

    Tw∥≤(Nχqβ+Mχ)(ψ(T)ψ(0))ηΓ(η+1):=ξ.

    It follows that TBqBξ. Hence, TBq is bounded. To prove that TBq is equicontinuous, let wTBq and κ1,κ2K such that κ1<κ2; then, we have

    |Tw(κ2)Tw(κ1)|Nχ|w|p+MχΓ(η)κ2κ1ψ(p)(ψ(κ2)ψ(p))η1dp,
    |Tw(κ2)Tw(κ1)|Nχqβ+MχΓ(η)κ2κ1ψ(p)(ψ(κ2)ψ(p))η1dp,
    |Tw(κ2)Tw(κ1)|Nχqβ+MχΓ(η+1)(ψ(κ2)ψ(κ1))η.

    Since Ψ is a continuous function, then we obatin

    limκ1κ2|Tw(κ1)Tw(κ2)|=0,

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

    Corollary 3.1. T:CC is ρ-Lipschitz with a zero constant.

    Proof. From the compactness of the operator T, and Lemma 2.2, it follows that T is ρ-Lipschitz with a zero constant.

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

    Theorem 3.1. Suppose that the hypotheses (H1)(H3) are true; then, the fractional pantograph differential equation mentioned in (1.1) has at least one solution: wC. Moreover, the set of all solutions for (1.1) is bounded in C(K,R).

    Proof. Let A,T,F: CC be the operators formulated in (3.2)–(3.4), respectively. A,T,F are continuous and bounded. Furthermore, in view of Lemma 3.2 and Corollary 3.1, the operator A is ρ-Lipschitz given Lχ[0,1), and ρ-Lipschitz with a zero constant. By using Lemma 2.2, we deduce that F is a strict ρ-contraction with a constant Lχ. Now, for some τ[0,1], we set

    Eτ={wC:w=τFw}.

    We claim that Eτ is bounded in C. To prove this claim, suppose that wEτ; then,

    w=τFw=τ(Aw+Tw),

    which yields that

    w=τFwτ(Aw+Tw);

    by using Lemmas 3.2 and 3.3, we get

    w(|w0|+Nχwα+Mχ+(Nhwβ+Mh)(ψ(K)ψ(0))ηΓ(η+1)). (3.7)

    The above inequality, namely, (3.7), yields that Eτ is bounded in C given α<1 and β<1.

    Suppose that our claim is not true; in this case, let ξ:=w. Dividing both sides of (3.7) by ξ, and taking ξ, then we obtain

    1limξ(|w0|+Nχξα+Mχ+(Nhξβ+Mh)(ψ(T)ψ(0))ηΓ(η+1))ξ=0,

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

    Remark 3.1. If we set α=β=1 in hypotheses (H2) and (H3), then the result of Theorem 3.1 will be as follows:

    Nχ+Nh(ψ(K)ψ(0))ηΓ(η+1)<1.

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

    Example 4.1. Consider the following problem:

    {CD32,eκw(κ)=κ277(w(κ)+sin2(w(κ))+17πcos(w(κ2)),κK=[0,1]w(0)=0,w(0)=20j=1θj|w(κj)|,θj>0,0<κj<1,j=1,2,..,20. (4.1)

    Here, ε=12, η=32, K=1 and ψ(κ)=eκ, and, in this case, we let χ(w)=20j=1θj|w(κj)| with 20j=1θj<1. Clearly, (H1) and (H2) hold with Nχ=Lχ=20i=jθj, Mχ=0 and q=1.

    Indeed, we can write

    |χ(w(κ))|=|20j=1θj|w(κj)|;

    hence,

    |χ(w)|20j=1θjw;

    thus, Nχ=20j=1θj, Mχ=0 and α=1. Alternatively, we have

    |χ(w(κ))χ(s(κ))|=|20j=1θj|w(κj)20j=1θj|s(κj)|;

    hence,

    |ω(w)ω(s)|20j=1θj|ws|;

    thus, Lχ=20j=1θj.

    To check the fulfillment of (H3), let κK and wR; then, we have

    |h(κ,w(κ),w(εκ))|=|κ277(w(κ)+sin2(w(κ))+17πcos(w(κ2))|,

    which implies that

    |h(κ,w(κ),w(εκ))|177|w|+0.1946.

    Thus, (H3) holds with Nh=177, Mh=0.1946 and β=1. Consequently, Theorem 3.1 implies that problem (4.1) has at least one solution. Moreover, from the inequality (3.7), we get

    wξ:0.1946(e1)(η)Γ(η+1)177(e1)(η)=0.1946(e1)(3/2)Γ(5/2)177(e1)(3/2)=0.4086. (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 ψ(κ).

    Figure 1.  The graph of h(κ,w(κ)) for Example 4.1.
    Figure 2.  The graph of h(κ,w(κ),w(εκ)) for Example 4.1.
    Table 1.  Numerical results for ξ based on ψ(κ) selection in Example 4.1.
    ψ(κ) ξ
    κ 0.1601<1
    eκ 0.4086<1
    2κ 0.1601<1
    3κ 0.5467<1
    4κ 1.3721>1
    5κ 3.7306>1

     | Show Table
    DownLoad: CSV

    Example 4.2. Consider the following problem:

    {CD95,κw(κ)=eκ11+κ2(w(κ)+sin(κ)+cos(w(κ2))π+κ2),κK=[0,1]w(0)=0,w(0)=20j=1θj|w(κj)|,θj>0,0<κj<1,j=1,2,..,20. (4.3)

    In this case, ε=12, η=95, K=1, ψ(κ)=κ and χ(w)=20j=1θj|w(κj)| with 20j=1θj<1. Similar to the previous example, hypotheses (H1) and (H2) are valid with Nχ=Lχ=20i=jθj, Mχ=0, q=1 and α=1. To check the fulfillment of (H3), we can write

    |h(κ,w(κ),w(εκ))|=|eκ11+κ2(w(κ)+sin(κ)+cos(w(κ2))π+κ2)|,

    which implies that

    |h(κ,w(κ),w(εκ))|111|w|+0.4716.

    Hence, (H3) holds with Nh=111, Mh=0.4716 and β=1. Consequently, Theorem 3.1 implies that problem (4.3) has at least one solution. Moreover, from the inequality (3.7), we get

    wξ:0.4716Γ(η+1)111=0.4716Γ(14/5)111=0.3430. (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 ψ(κ).

    Figure 3.  The graph of h(κ,w(κ)) for Example 4.2.
    Figure 4.  The graph of h(κ,w(κ),w(εκ)) for Example 4.2.
    Table 2.  Numerical results for ξ based on ψ(κ) selection in Example 4.2.
    ψ(κ) ξ
    κ 0.3430<1
    eκ 1.4238>1
    2κ 0.3430<1
    3κ 2.6209>1
    4κ 6.7896<0
    5κ 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 ψ-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 ψ(κ) 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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