On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives

1.   Introduction

The choice of certain fractional operator in modeling real world problems is always of interest to many authors working the filed of fractional calculus and its applications. The authors in ^[1], have shown how suitable fractional formulations are really extensions of the integer order definitions currently used in signal processing. The recent works ^{[2,3,4,5,6,7,8,9]} reflect several theoretical aspects and applications about types of fractional operators and their possible applying. For example, the reference [5] deals with the $\Psi$-Hilfer case and the articles ^[7,8] deal with the $\Psi$-Caputo and Caputo operators, respectively.

The investigation of positive solutions of differential equations is vital due to their presence naturally when we model some real applications appearing in economics, physics, engineering and biology. As a generalization to the ordinary derivative calculus, the theory of fractional calculus appeared and started to develop since $1695$ (see ^[10,11,12]). Since that date, many authors have authored several works to investigate qualitatively the positive attributes of FDE solutions ^[13,14]. Part of the studies have been conducted exclusively on investigating the existence of solutions of problems using Caputo and generalized Caputo fractional derivatives (CFD). Indeed, $\Psi$-Caputo derivatives ($\Psi$-CFD) have been considered. Of special interest is the logarithmic case kernel which is called Hadamard (see ^{[15,16,20,25,33]}).

A pantograph is a mechanical linkage system consisting of four bars of equal length hinged at their ends. The equations that describe the motion of a pantograph are based on the principle of similar triangles. These equations determine the scaling factor or the relationship between the lengths of the bars and the size of the image produced. The equations are usually derived using trigonometry and vector algebra and they take into account the angles formed between the bars and the lengths of the individual bars. The goal of the equations is to determine the path of the stylus (or the end of the fourth bar) given the movement of the original object being traced, see for instance ^{[17,18,19,21,22,23]} and the references therein.

Exclusively, the authors in ^[24], for $1 < \alpha\leq2$, and $^{C}D^{\alpha}$ is the usual CFD, examine the uniqueness and existence of the PS of the following FDE

where $f\in C\left(\left[ 0, 1\right] \times\left[ 0, \infty\right), \left[ 0, \infty\right) \right)$. By utilizing ULS technique and FPTs, the authors in ^[25] got positive results. The novelty in this work is to generalize the results in ^[25] by utilizing the so-called $\Psi$-CFD. The Caputo Hadamard fractional derivatives fall within this class of operators by taking $\Psi(t) = \ln t$. Therefore, the idea of the pantograph is to be considered more generally. In fact, our real concern in this paper is to deal with the problems of PS to pantograph FDEs. It is worth to mentioning that the above works have been motivated and inspired by the papers ^{[26,27,28,29,30,31,32,33,34,35,36,37]}.

Our main concern in this work is to deal with the PS of the below pantograph FDE:

where $\vartheta\in\left(0, \frac{\mathcal{T}-\ell}{\mathcal{T}}\right),$ $\phi\left(\ell\left(1+\vartheta\right) \right) = \phi_{0} > 0,$ $\mathcal{D} _{\ell}^{\alpha; \Psi}$ is $\Psi$-CFD of order $1 < \alpha\leq2$, $\mathcal{G}, \mathcal{F}:\left[ \ell, \mathcal{T}\right] \times\left[ 0, \infty\right) \times\left[ 0, \infty\right) \rightarrow \left[ 0, \infty\right) \mathcal{\ }$ are continuous functions (CFs), $\mathcal{G}$ is non-decreasing on $\phi$ and $\mu_{2} > \mathcal{G}(1, \phi_{0})$. By applying the Riemann-Liouville fractional integral with respect to the function $\Psi$ to (1.1), we transform it to an equivalent integral equation on which we utilize ULS and SFPT, BFPT to prove the existence and uniqueness of the PS.

Our manuscript is to be divided as follows. Section 2 includes some key concepts, definitions, lemmas and theories that will be used in proving the main results. Section 3 will be devoted to our theoretical main results. Our main results generale those obtained in ^[24,25]. In Section 4, we shall give an illustrative example. Section 5 includes our conclusions.

2.   Essential preliminaries

The basic tools to be presented in this section can be recalled from ^{[10,12,13,14,30,31,32,33,38,39]}, where more details can be found.

Let $\Psi :[\ell _{1}, \ell _{2}]\rightarrow \mathbb{R}$ be an increasing with $\Psi ^{\prime }(\eta) > 0,$ $\forall \eta.$ The symbol $Y = C\left([\ell, \mathcal{T}], \mathbb{R}\right)$ represents the Banach space of CFs $\phi :[\ell, \mathcal{T}]\rightarrow \mathbb{R}$ by norm $\Vert \phi \Vert = \sup \{|\phi (\eta)|:\eta \in \lbrack \ell, \mathcal{T}]\}.$

We define $\mathcal{A} = \left\{ \phi \in Y:\phi \left(\eta \right) \geq 0, \ \eta \in \lbrack \ell, \mathcal{T}]\right\}$ subset of $Y$ consisting of all positive functions in $Y$. Suppose $\hslash _{1}, \hslash _{2}\in \mathbb{ R}^{+}$ with $\hslash _{2} > \hslash _{1}$. For any $\phi, \chi \in \lbrack \hslash _{1}, \hslash _{2}]$, we associate the lower-control function

and the upper-control function

The function $\mathcal{F}$ was defined above in Section 1. On the arguments $\phi, \chi,$ $L$ and $U$ are monotonous non-decreasing and

Definition 2.1. ^[10,12,39] For a function $\phi:\left[ 0, +\infty\right) \rightarrow \mathbb{R}, \ $the Riemann-Liouville fractional integral (RLFI) of order $\alpha > 0$ is defined as

where the Euler gamma function $\Gamma$ is given by

Definition 2.2. ^[10,12,39] The $\Psi$-RLFI of order $\alpha > 0$ for a CF $\phi:\left[ \ell, \mathcal{T}\right] \rightarrow \mathbb{R}$ is defined as

Definition 2.3. ^[10,12,39] The CFD of order $\alpha > 0$ for a $\phi:\left[ 0, +\infty\right) \rightarrow \mathbb{R}$ is intended by

Definition 2.4. ^[10,12,39] The $\Psi$-CFD of order $\alpha > 0$ for a $\phi :\left[ \ell, \mathcal{T} \right] \rightarrow \mathbb{R}$ is defined by

where $\partial_{\Psi}^{n} = \left(\frac{1}{\Psi^{\prime}(\eta)}\frac{d} {d\eta}\right) ^{n}, n\in\mathbb{N}$.

Lemma 2.1. ^[10] Suppose $q, \ell > 0$ and $\phi\in C([\ell, \hslash], \mathbb{R})$. Then, $\forall\eta\in\lbrack\ell, \hslash]$ and by assuming ${ F}_{\ell}(\eta) = \Psi(\eta)-\Psi(\ell)$, we have

● $\mathcal{D}_{\ell}^{q; \Psi}\mathcal{I}_{\ell}^{q; \Psi}\phi(t) = \phi(t),$

● $\mathcal{I}_{\ell}^{q; \Psi}(F_{\ell}(\eta))^{\ell-1} = \frac{\Gamma(\ell)}{\Gamma(\ell+q)}(F_{\ell}(\eta))^{\ell+q-1},$

● $\mathcal{D}_{\ell}^{q; \Psi}(F_{\ell}(\eta))^{\ell-1} = \frac{\Gamma(\ell)}{\Gamma(\ell-q)}(F_{\ell}(\eta))^{\ell-q-1},$

● $\mathcal{D}_{\ell}^{q; \Psi}(F_{\ell}(\eta))^{k} = 0, \ k\in\{0, \ldots, n-1\}, \ n\in\mathbb{N}, \ q\in(n-1, n]$.

Lemma 2.2. ^[10,32] Let $n-1 < \alpha_{1}\leq n, \alpha _{2} > 0, \ \ell > 0, \ \phi\in\mathbb{L}(\ell, \mathcal{T}), \ \mathcal{D}_{\ell _{1}}^{\alpha_{1};\Psi}\phi\in\mathbb{L}(\ell, \mathcal{T})$. Then, the differential equation

has the unique solution

and

with $w_{\ell}\in\mathbb{R}, \ \ell = 0, 1, \ldots, n-1$. Furthermore,

and

Lemma 2.3. Let $\phi\in C^{1}\left(\left[ \ell, \mathcal{T}\right] \right)$, $\phi^{(2)}$ and $\frac{\partial\mathcal{G}}{\partial\eta}$ exist. Then, $\phi$ is a solution of (1.1) if and only if

Proof. Let $\phi$ be a solution of (1.1). Then, we have

From Lemma 2.2, we got

The converse can be proven straightforward as well. □

In what follows, we recall the FPTs that will be used to prove the uniqueness and existence of PS for Eq (1.1).

Definition 2.5. Let $(Y, \left\Vert.\right\Vert)$ be a Banach space. Then, a mapping $\Theta :Y\rightarrow Y$ is called contraction. If there is a $l\in(0, 1)$ such that for every $\phi, \chi\in Y$, $\Theta$\, we have

Theorem 2.1. (BFPT ^[38]) Let\ $\Omega\mathcal{\neq\emptyset}$ be a closed convex subset of a Banach space $Y$ and $\Theta :\Omega\rightarrow \Omega$ be a contraction mapping. Then, there is a unique $\phi\in\Omega$ with $\Theta \phi = \phi$.

Theorem 2.2. (BFPT ^[38]) Let\ $\Omega\mathcal{\neq\emptyset}$\ be a closed convex subset of a Banach space $Y$\ and $\Theta :\Omega \rightarrow \Omega$ be a continuous compact operator. So, $\Theta$ has a fixed point in $\Omega$.

3.   Main results

In this section, we present the results of the existence of FDE (1.1). We also provide the necessary hypotheses for the uniqueness of (1.1).

We set the operator $\Theta:\mathcal{A}\longrightarrow Y$ by inversion Eq (2.1) and then apply SFPT

where the fixed point is needed to fulfill the identity operator equation $\Theta\phi = \phi$.

For the next step of our main results, the following forms are adopted.

$(\Sigma1)$ Let $\phi_{\ast}, \phi^{\ast}\in\mathcal{A}$, as well as $\hslash_{1}\leq\phi_{\ast}(\eta)\leq\phi^{\ast}(\eta)\leq\hslash_{2},$

for any $\eta\in\lbrack\ell, \mathcal{T}]$.

$(\Sigma2)$ For $\eta\in\lbrack\ell, \mathcal{T}]\ $and $\phi_{1}, \phi_{2}, \chi_{1}, \chi_{2}\in Y$, there exist $\beta_{1}, \beta_{2}, \beta_{3} > 0$ such that

For (1.1), the functions $\phi^{\ast}$ and $\phi_{\ast}$ are known as the ULS.

Theorem 3.1. If $(\Sigma1)$ is satisfied, FDE (1.1) posses at least one solution $\phi\in Y$ and fulfills $\phi_{\ast}(\eta)\leq\phi(\eta)\leq \phi^{\ast}(\eta), \eta\in\lbrack\ell, \mathcal{T}].$

Proof. Set $\Omega = \left\{ \phi\in\mathcal{A}:\phi_{\ast}(\eta)\leq\phi (\eta)\leq\phi^{\ast}(\eta), \ \eta\in\lbrack\ell, \mathcal{T}]\right\}$. If we use the norm $\left\Vert \phi\right\Vert = \max_{\eta\in\lbrack\ell, \mathcal{T}]}\left\vert \phi(\eta)\right\vert$, we see that $\left\Vert \phi\right\Vert \leq\hslash_{2}$. So, we deduce that $\Omega$ is a convex and closed, bounded subset of $Y$. Moreover, the functions $\mathcal{G}$ and $\mathcal{F}$ being CF implies that $\Theta$, marked by (3.1), is a CF on $\Omega$. If $\phi\in \Omega$, there exist $c_{\mathcal{F}}, c_{\mathcal{G}} > 0$ constants as well as

and

Then,

where $\left\vert \mathcal{G}\left(\ell, \phi_{0}\right) \right\vert = c_{0}.$ Thus,

From which it follows that $\Theta (\Omega)$ is uniformly bounded. The equicontinuity of $\Theta (\Omega)$ is then can be handled. Let $\phi\in \Omega$ and $\ell\leq\eta_{1} < \eta_{2}\leq\mathcal{T}$. Then,

The right-hand side of above inequality approaches to zero as $\eta _{1}\rightarrow \eta_{2}$. As a consequence, $\Theta (\Omega)$ is equicontinuous. Therefore, the compactness of $\Theta :\Omega\longrightarrow Y$ follows by Arzelè-Ascoli theorem. Finally, in order to employ SFPT, we need to prove that $\Theta (\Omega)\subseteq\Omega$. Let $\phi\in\Omega$. Then,

and

Consequently, $\phi_{\ast}(\eta)\leq\left(\Theta \phi\right) (\eta)\leq\phi^{\ast}(\eta),$ $\eta\in\lbrack\ell, \mathcal{T}]$, that is, $\Theta (\Omega)\subseteq\Omega$. Hence, SFPT asserts that the mapping $\Theta$ has at least one fixed point $\phi\in\Omega$. This means that FDE (1.1) admits at least one PS $\phi\in Y$ and $\phi_{\ast}(\eta)\leq\phi(\eta)\leq\phi^{\ast}(\eta), \ \eta\in\lbrack\ell, \mathcal{T}]$. □

Next, we adopt and offer a different set of uses for the above theorem.

Corollary 3.1. Suppose that CFs $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$ and $\varphi_{4}\ $exist. So,

and

Then, the FDE (1.1) must have at least one PS $\phi\in Y$. Also,

Proof. Starting from formula (3.12) and control function, we reach $\varphi_{3}(\eta)\leq L(\eta, \phi, \chi)\leq U(\eta, \phi, \chi)\leq\varphi_{4}(\eta)$, $(\eta, \phi(\eta), \chi(\eta))\in\lbrack\ell, \mathcal{T}]\times\lbrack \hslash_{1}, \hslash_{2}]\times\lbrack\hslash_{1}, \hslash_{2}]$. We consider the equations

Equation (3.14) is evidently equivalent to

So, the first part of (3.14) involves

and the second part of (3.14) suggests

Hence, both equations in (3.14) have ULS. Therefore, the FDE (1.1) has at least one solution $\phi\in Y$ fulfilling (3.13) when Theorem 3.1 is conducted. □

Corollary 3.2. Assume (3.11) is satisfied and $0 < \sigma < \varphi(\eta) = \lim_{\phi, \chi\rightarrow \infty}\mathcal{F}(\eta, \phi, \chi) < \infty$ for $\eta\in\lbrack\ell, \mathcal{T}]$. The FDE (1.1) must posses at least one $PS$ $\phi, \chi\in Y$.

Proof. If $\phi, \chi > \rho > 0$ then $0\leq\left\vert \mathcal{F}(\eta, \phi, \chi)-\varphi(\eta)\right\vert < \sigma$ for any $\eta\in\lbrack \ell, \mathcal{T}]$. Hence, $0 < \varphi(\eta)-\sigma\leq\mathcal{F}(\eta, \phi, \chi)\leq\varphi(\eta)+\sigma$ for$\ \eta\in\lbrack\ell, \mathcal{T}]$ and $\rho < \phi, \chi < +\infty$. If $\max\left\{ \mathcal{F}(\eta, \phi, \chi):\eta\in\lbrack\ell, \mathcal{T}], \ \phi, \chi\leq\rho\right\} \leq\nu$ then $\varphi(\eta)-\sigma\leq\mathcal{F}(\eta, \phi, \chi)\leq\varphi (\eta)+\sigma+\nu$ for $\eta\in\lbrack\ell, \mathcal{T}]$ and $0 < \phi, \chi < +\infty$. By Corollary 3.1, the FDE (1.1) has at least one PS $\phi\in Y$ with

□

Corollary 3.3. If we assume that $0 < \sigma < \mathcal{F}(\eta, \phi(\eta), \phi(\ell+\vartheta\eta))\leq\gamma_{1}\phi(\eta)+\gamma_{2}\phi (\ell+\vartheta\eta)+\eta < \infty$ for $\eta\in\lbrack\ell, \mathcal{T}]$ and $\sigma$, $\ \eta,$ $\gamma_{1}$ and $\gamma_{2} > 0$ constants. Then, FDE (1.1) admits at least one PS $\phi\in C\left([\ell, \delta]\right)$ where $\delta > \ell$.

Proof. Consider the equation

where $\phi\left(\ell\left(1+\vartheta\right) \right) = \phi_{0} > 0$. Equation (3.18) has a solution of the form:

For $\omega$ a positive constant, and $\varpi\in\left(0, 1\right)$, there exists $\delta > \ell$ such that $0 < \frac{\left(\gamma_{1}+\gamma_{2}\right) \left(\Psi\left(\delta\right) -\Psi\left(\ell\right) \right) ^{\alpha }}{\Gamma(\alpha +1)} < \varpi < 1$ and

Then, if $\ell\leq\eta\leq\delta$, the set $\mathfrak{B}_{\omega} = \left\{ \phi\in Y:\left\vert \phi(\eta)\right\vert \leq\omega, \ \ell\leq\eta\leq \delta\right\}$ is a closed, convex and bounded subset of $C\left([\ell, \delta]\right)$. The mapping $\Theta :\mathfrak{B}_{\omega }\longrightarrow\mathfrak{B}_{\omega}$ reported as

is compact with the same approach as in the proof of Theorem 3.1. With same way,

Now, for $\phi\in\mathfrak{B}_{\omega}$ we have

and hence $\left\Vert \Theta \phi\right\Vert \leq\omega$. We conclude, the SFPT emphasize that $\Theta$ posses at least one fixed point in $\mathfrak{B}_{\omega}$, and so Eq (3.18) has at least one PS $\phi^{\ast}(\eta)$ where $\ell < \eta < \delta$. Hence, if $\eta\in\left[ \ell, \mathcal{T}\right]$ one can claim that

The term control function denotes

hence $\phi^{\ast}$ is an upper PS of FDE (1.1). Secondly, one can take

as a lower PS of (1.1). By Theorem 3.1, the FDE (1.1) has at least one PS $\phi\in C\left([\ell, \delta]\right)$ where $\delta > \ell$ and $\phi_{\ast}(\eta)\leq\phi(\eta)\leq\phi^{\ast}(\eta)$. □

The final result is the uniqueness of PS to (1.1) by adopting the Theorem 2.1.

Theorem 3.2. Under the satisfaction of the investigators $(\Sigma 1)$ and $(\Sigma 2)$ and that

the FDE (1.1) has a unique PS $\phi \in \Omega$.

Proof. The FDE (1.1) posses at least one PS in $\Omega$ by Theorem 3.1. The mapping specified in (3.1) is a contraction on $Y$. Indeed, for any $\phi, \chi\in Y$ we get

On the light of (3.26), the mapping $\Theta \ $is contraction and hence the FDE (1.1) has a unique PS $\phi\in\; \Omega$. □

4.   Example

Let $\Psi(\eta) = \ln \eta$. We explore the pantograph FDE in this case (Caputo Hadamard fractional derivative).

where $\mu _{1} = 1$, $\ell = 1,$ $\phi \left(1+\vartheta \right) = \phi _{0} > 0$, $\mathcal{T} = e$, $\mathcal{G}\left(\eta, \phi \right) = \pi +\arctan \left(\phi \right)$ and $\mathcal{F}(\eta, \phi, \chi) = \frac{1}{1+e+\eta } \left(1+e+\eta \sin \left(\phi +\chi \right) \right)$. As $\mathcal{G}$ is non-decreasing on $\phi$,

and

Therefore, we conclude that Eq (4.1) has PS proportional to all the natural results mentioned above. We have

Therefore, by making use of Theorem 3.2, we infer that Eq (4.1) posses a unique PS.

5.   Conclusions

We have investigated and verified the existence and uniqueness of positive solutions of the fractional differential pantograph Eq (1.1) in $\Psi$-Caputo sense. We have followed the method of upper and lower solutions by imposing some of the necessary conditions to show the existence and uniqueness of our positive solution. Further, we have used and applied SFPT and BFPT to gain a positive solution for (1.1).

Use of AI tools declaration

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

Acknowledgements

Manar A. Alqudah: Princess Nourah bint Abdul- rahman University Researchers Supporting Project number (PNURSP2023R14), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The authors B. Abdalla, and T. Abdeljawad would like to thank Prince Sultan University for the support through the TAS research lab.

Conflict of interest

The authors declare that they have no conflict of interest.