Abstract
The aim of this paper is to study the existence and uniqueness of periodic solutions for a certain type of nonlinear fractional pantograph differential equation with a \(\psi \)-Caputo derivative. The proofs are based on the coincidence degree theory of Mawhin. To show the efficiency of the results, some illustrative examples are included.
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1 Introduction
In last few decades, nonlinear fractional differential equations (NFDEs) have been the focus of many studies due to the intensive development of the theory of fractional calculus and to their frequent applications in many areas such as mechanics, physics, chemistry, engineering, and many other scientific disciplines [15, 16].
Recently, many definitions and results about fractional derivatives and integrals operators have been generalized [1, 2, 4, 17, 22]. Almeida [5, 6] introduced a new generalized fractional derivative, the \(\psi \)-Caputo fractional derivative; some recent work on the subject of existence and uniqueness for NFDEs with \(\psi \)-Caputo fractional derivative can be found in [7, 13].
In the current paper, we study the nonlinear pantograph fractional equation with \(\psi \)-Caputo fractional derivative
where \(^{c}{\mathfrak {D}}_{0^{+}}^{\alpha ;\psi }\) denotes the \(\psi \)-Caputo fractional derivative of order \(0<\alpha <1\), \(\varepsilon \in (0,1)\), and \(h: J\times {{\mathbb {R}}}\times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) is a continuous function.
Pantograph equations have been widely used in the fields of quantum mechanics and dynamical system [20, 21]. Several researchers have investigated some new existence and uniqueness results for NFDE pantograph models and others by applying fixed point theorems, the nonlinear alternative on cones, or coincidence degree theory [3, 8,9,10,11,12].
In [23], Shah et al. studied a class of \(\psi \)-Caputo fractional pantograph equations with nonlocal boundary conditions
where n, m, and \({\mathfrak {c}}\) are real constants with \(n+m\ne 0\), and obtained some existence and uniqueness results by using the Banach contraction theorem and Schaefer’s fixed point theorem. However, if \(n+m=0\), which includes the periodic case, the problem cannot be studied this way.
In this work, we construct a suitable operator and use the coincidence degree theory of Mawhin [14] to study the existence of solutions for NFDEs (1) with periodic boundary conditions (2).
The present article is organized as follows: In Sect. 2, some basic definitions and lemmas related to fractional calculus are recalled. In Sect. 3, the existence and uniqueness of periodic solutions for the NFDEs (1)–(2) are obtained. Finally, in Sect. 4, we give two examples to illustrate our main findings.
2 Basic concepts
We consider \(C(J,{{\mathbb {R}}})\) and \(C^{m}(J,{{\mathbb {R}}})\), the spaces of continuous and m times continuously differentiable functions on J, respectively, with the supremum norm \(\Vert \cdot \Vert _{\infty }\). We begin this section with the concept of a fractional integral or fractional derivative with respect to another function.
Definition 2.1
([5]) Let \(J=[0, {\mathfrak {b}}]\), \(0< {\mathfrak {b}} < \infty \), be a finite or infinite interval, \(\alpha > 0\), u be an integrable function defined on J, and \(\psi \in C^{1}(J,{{\mathbb {R}}})\) be a positive increasing function, such that \(\psi ^{\prime }(t)\ne 0\) for all \(t \in J\). Fractional integrals and fractional derivatives of a function u with respect to another function \(\psi \) are defined as
and
respectively, where \(n=[\alpha ]+1\).
In particular, if \(0<\alpha <1,\) we have
We will need several lemmas, some of which have appeared in the literature, to prove our main results.
Lemma 2.2
([5]) Let \(\alpha > 0\) and \(\beta >0.\) Then, we have
Lemma 2.3
([17]) Let \(\alpha > 0\), \(\rho > 0\), and \(t \in J\). If \(u(t)=\left( \psi (t)-\psi (0)\right) ^{\rho -1},\) then
Definition 2.4
([5]) Let \(n-1<\alpha < n\) with \(n\in {{\mathbb {N}}}\) and \(u,\psi \in C^{n}(J,{{\mathbb {R}}})\) be two functions, such that \(\psi \) is increasing and positive with \(\psi ^{\prime }(t)\ne 0\) for any \(t\in J.\) The left \(\psi \)-Caputo fractional derivative of u of order \(\alpha \) is given by
In particular, if \(0<\alpha <1,\) we have
Lemma 2.5
([5]) If \(u\in C^{n}(J,{{\mathbb {R}}})\) and \(n-1<\alpha <n,\) then
In particular, when \(0<\alpha <1,\) we have
Lemma 2.6
([5]) If \(u\in C^{1}(J,{{\mathbb {R}}})\) and \(\alpha >0\), we have
Lemma 2.7
([5]) Let \(u,{\mathfrak {\upsilon }}\in C^{1}(J,{{\mathbb {R}}})\) and \(0<\alpha <1.\) Then
Remark 2.8
If \({\mathfrak {\upsilon }}\equiv 0,\) then
Next, we present definitions and concepts from coincidence degree theory that are essential in the proofs of our results (see [14, 18]).
Definition 2.9
Let \({\mathcal {X}}\) and \({\mathcal {Y}}\) be normed spaces. A Fredholm operator of index zero is a linear operator \({\mathfrak {L}}:{\text {Dom}} ({\mathfrak {L}})\subset {\mathcal {X}}\rightarrow {\mathcal {Y}}\), such that
-
(a)
\(\dim \ker {\mathfrak {L}}= {\text {codim}}\; {\text {Img}} {\mathfrak {L}} < +\infty \).
-
(b)
\({\text {Img}}\,{\mathfrak {L}}\) is a closed subset of \({\mathcal {Y}}\).
By Definition 2.9, there exist continuous projectors \(Q:{\mathcal {Y}}\rightarrow {\mathcal {Y}}\) and \({\mathcal {P}}:{\mathcal {X}}\rightarrow {\mathcal {X}}\) satisfying
Thus, the restriction of \({\mathfrak {L}}\) to \({\text {Dom}}\,{\mathfrak {L}}\cap \ker {\mathcal {P}}\), denoted by \({\mathfrak {L}}_{{\mathcal {P}}}\), is an isomorphism onto its image.
Definition 2.10
Let \(\Omega \subseteq {\mathcal {X}}\) be a bounded subset and \({\mathfrak {L}}\) be a Fredholm operator of index zero with \({\text {Dom}}\, {\mathfrak {L}}\cap \Omega \ne \emptyset \). Then, the operator \({{\mathcal {N}}}:{{\overline{\Omega }}}\rightarrow {\mathcal {Y}}\) is said to be \({\mathfrak {L}}-\)compact in \({{\overline{\Omega }}}\) if
-
(a)
the mapping \({Q{\mathcal {N}}}:{{\overline{\Omega }}}\rightarrow {\mathcal {Y}}\) is continuous and \({Q{\mathcal {N}}}\left( {{\overline{\Omega }}}\right) \subseteq {\mathcal {Y}} \) is bounded.
-
(b)
the mapping \({\left( {\mathfrak {L}}_{{\mathcal {P}}}\right) ^{-1}(id-Q){\mathcal {N}}}:{{\overline{\Omega }}}\rightarrow {\mathcal {X}}\) is completely continuous.
Lemma 2.11
([19]) Let \({\mathcal {X}}\) and \({\mathcal {Y}}\) be Banach spaces and \(\Omega \subset {\mathcal {X}}\) be a bounded open set that is symmetric with \(0\in \Omega .\) Suppose that \({\mathfrak {L}}: {\text {Dom}}\, {\mathfrak {L}}\subset {\mathcal {X}}\rightarrow {\mathcal {Y}}\) is a Fredholm operator of index zero with \( {\text {Dom}}\, {\mathfrak {L}} \cap {{\overline{\Omega }}} \ne \emptyset \), and \({\mathcal {N}}:{\mathcal {X}}\rightarrow {\mathcal {Y}}\) is a \({\mathfrak {L}}-\)compact operator on \({{\overline{\Omega }}}\). Assume, moreover, that
for any \(x\in {\text {Dom}}\, {\mathfrak {L}}\cap \partial \Omega \) and any \(\zeta \in (0,1],\) where \(\partial \Omega \) is the boundary of \(\Omega \) with respect to \({\mathcal {X}}\). Then, there exists at least one solution of the equation \({\mathfrak {L}}x={{\mathcal {N}}}x\) on \({\text {Dom}}\, {\mathfrak {L}}\cap {{\overline{\Omega }}}\).
3 Main results
Let the spaces
and
be endowed with the norms
We define the operator \({\mathfrak {L}} : {\text {Dom}} \,{\mathfrak {L}}\subseteq {\mathcal {X}}\rightarrow {\mathcal {Y}}\) by
where
Lemma 3.1
For the operator \({\mathfrak {L}}\) given in (3), we have
and
Proof
By Remark 2.8, we have for all \(u\in {{\mathcal {X}}}\) the equation \({\mathfrak {L}}u=^{c}{\mathfrak {D}}_{0^{+}}^{\alpha ;\psi }u=0 \) in J, has a solution of the form
so
For \({\mathfrak {\upsilon }}\in {\text {Img}}\,{\mathfrak {L}}\), there exists \(u\in {\text {Dom}}\,{\mathfrak {L}} \), such that \({\mathfrak {\upsilon }}= {\mathfrak {L}}u \in {{\mathcal {Y}}}\). From Lemma 2.5, we obtain that for every \(t\in J\)
Since \(u\in {\text {Dom}}\,{\mathfrak {L}}\), we have \(u(0)=u({\mathfrak {b}})\). Thus
Furthermore, if \({\mathfrak {\upsilon }}\in {{\mathcal {Y}}}\) satisfies
then for any \(u(t)={\mathfrak {I}}_{0^{+}}^{\alpha ;\psi }{\mathfrak {\upsilon }}(t)\), using Lemma 2.6, we obtain \({\mathfrak {\upsilon }}(t)= {}^{c}{\mathfrak {D}}_{0^{+}}^{\alpha ;\psi }u(t)\). Therefore
which implies that \(u\in {\text {Dom}}\,{\mathfrak {L}}\), and so, \({\mathfrak {\upsilon }}\in {\text {Img}}\,{\mathfrak {L}}\). Hence
which completes the proof of the lemma. \(\square \)
Lemma 3.2
Let \({\mathfrak {L}}\) be defined by (3). Then, \({\mathfrak {L}} \) is a Fredholm operator of index zero, and the linear continuous projector operators \( {Q} : {{\mathcal {Y}}} \rightarrow {{\mathcal {Y}}}\) and \({{\mathcal {P}}} :{{\mathcal {X}}}\rightarrow {{\mathcal {X}}} \) can be written as
and
Furthermore, the operator \( {\mathfrak {L}}_{{{\mathcal {P}}}}^{-1}: {\text {Img}}\,{\mathfrak {L}}\rightarrow {{\mathcal {X}}}\cap \ker {{\mathcal {P}}}\) becomes
Proof
Clearly, for each \({\mathfrak {\upsilon }}\in {{\mathcal {Y}}}\), \({Q}^{2}{\mathfrak {\upsilon }}={Q}{\mathfrak {\upsilon }}\) and \({\mathfrak {\upsilon }}={Q}({\mathfrak {\upsilon }})+({\mathfrak {\upsilon }}-{Q}({\mathfrak {\upsilon }}))\), where \( ({\mathfrak {\upsilon }}-{Q}({\mathfrak {\upsilon }}))\in \ker {Q}={\text {Img}}\,{\mathfrak {L}}\). Using these facts, we see that \({\text {Img}}\,{Q}\cap {\text {Img}}\,{\mathfrak {L}}={0}\), so
Similarly, \({\text {Img}}\,{{\mathcal {P}}} = \ker {\mathfrak {L}}\) and \({{\mathcal {P}}}^{2}={{\mathcal {P}}}\). It follows that for each \(u\in {{\mathcal {X}}}\), \(u=\left( u-{{\mathcal {P}}}(u)\right) +{{\mathcal {P}}}(u)\), so \({{\mathcal {X}}}=\ker {{\mathcal {P}}}+\ker {\mathfrak {L}}\). Clearly, we have \(\ker {{\mathcal {P}}}\cap \ker {\mathfrak {L}}={0}\). Hence
Therefore
Consequently, \({\mathfrak {L}}\) is a Fredholm operator of index zero.
Now, we will show that the inverse of \({\mathfrak {L}}|_{{\text {Dom}}{\mathfrak {L}}\cap \ker {{\mathcal {P}}}}\) is \({\mathfrak {L}}_{{{\mathcal {P}}}}^{-1}\). For \({\mathfrak {\upsilon }}\in {\text {Img}}\,{\mathfrak {L}}\), by Lemma 2.6, we have
In addition, for \(u\in {\text {Dom}}\,{\mathfrak {L}}\cap \ker {{\mathcal {P}}}\),
Using the fact that \(u \in {\text {Dom}}\,{\mathfrak {L}}\cap \ker {{\mathcal {P}}}\), we see that
Thus
From (4) and (5), it follows that \({\mathfrak {L}}_{{{\mathcal {P}}}}^{-1}=\left( {\mathfrak {L}}|_{{\text {Dom}}{\mathfrak {L}}\cap \ker {{\mathcal {P}}}}\right) ^{-1}\), which completes the proof. \(\square \)
We will need the following condition in the sequel.
-
(H1) There exist positive constants \(\gamma \) and \(\eta \), such that
$$\begin{aligned} |h(t,u,{\mathfrak {\upsilon }})-h(t,{\bar{u}},\bar{{\mathfrak {\upsilon }}})|\leqslant \gamma |u-{\bar{u}}|+\eta |{\mathfrak {\upsilon }}-\bar{{\mathfrak {\upsilon }}}|, \end{aligned}$$for every \(t \in J\) and u, \({\bar{u}}\), \({\mathfrak {\upsilon }}\), \(\bar{{\mathfrak {\upsilon }}}\in {{\mathbb {R}}}\).
Define \({{\mathcal {N}}_{\varepsilon }}:{{\mathcal {X}}}\rightarrow {{\mathcal {Y}}}\) by
Then, the problem (1)–(2) is equivalent to the problem \({\mathfrak {L}}u(t)={{\mathcal {N}}_{\varepsilon }}u(t), \; t\in J \; \text {and} \; u \in {{\mathcal {X}}}\).
Lemma 3.3
If (H1) holds, then for any bounded open set \(\Omega \subset {{\mathcal {X}}},\) the operator \({{\mathcal {N}}_{\varepsilon }}\) is \({\mathfrak {L}}-\)compact.
Proof
For \({{\mathcal {M}}}>0\), consider the bounded open set \(\Omega =\{u\in {{\mathcal {X}}}: \Vert u\Vert _{{{\mathcal {X}}}}<{{\mathcal {M}}}\}\). We divide the proof into three steps.
Step 1: \({\mathcal {QN_{\varepsilon }}}\) is continuous. Let \(\left( u_n\right) _{\mathfrak {n}\in {{{\mathbb {N}}}}}\) be a sequence, such that \(u_n \rightarrow u\) in \({{\mathcal {Y}}}\); then, for each \(t \in J\), we have
By \((H1 )\), we have
Thus, for each \(t \in J\)
and hence
Therefore, \({\mathcal {QN_{\varepsilon }}}\) is continuous.
Step 2: \({\mathcal {QN_{\varepsilon }}}({{\overline{\Omega }}})\) is bounded. For \(t\in J\) and \(u\in {{\overline{\Omega }}}\), we have
where \(h^*=\left\| h(\cdot ,0,0)\right\| _{\infty }\). Thus
so \({\mathcal {QN_{\varepsilon }}}({{\overline{\Omega }}})\) is a bounded set in \({{\mathcal {Y}}}\).
Step 3: \({\mathfrak {L}}_{{{\mathcal {P}}}}^{-1}(id-{Q}){{\mathcal {N}}_{\varepsilon }}:{{\overline{\Omega }}}\rightarrow {{\mathcal {X}}} \) is completely continuous. We will use the Arzelà–Ascoli theorem, so we need to show that \({\mathfrak {L}}_{{{\mathcal {P}}}}^{-1}(id-{Q}){{\mathcal {N}}_{\varepsilon }}({{\overline{\Omega }}})\subset {{\mathcal {X}}} \) is uniformly bounded and equicontinuous. First, for any \(u\in {{\overline{\Omega }}} \) and \(t \in J\)
Therefore
which means that \({\mathfrak {L}}_{{{\mathcal {P}}}}^{-1}(id-{Q}){{\mathcal {N}}_{\varepsilon }}({{\overline{\Omega }}})\) is uniformly bounded on \({{\mathcal {X}}}\).
To show that \({\mathfrak {L}}_{{{\mathcal {P}}}}^{-1}(id-{Q}){{\mathcal {N}}_{\varepsilon }}({{\overline{\Omega }}})\) is equicontinuous, let \(0< t_{1}< t_{2}\leqslant {\mathfrak {b}}\) and \(u\in {\overline{\Omega }}\). Then
where
Since the right-hand side of the above inequality tends to zero as \( t_{1}\rightarrow t_{2}\) and the limit is independent of u, the operator \( {\mathfrak {L}}_{{{\mathcal {P}}}}^{-1}(id-{Q}){{\mathcal {N}}_{\varepsilon }}({{\overline{\Omega }}})\) is equicontinuous. By the Arzelà–Ascoli theorem, \({\mathfrak {L}}_{{{\mathcal {P}}}}^{-1}(id-{Q}){{\mathcal {N}}_{\varepsilon }}({{\overline{\Omega }}})\) is relatively compact in \({{\mathcal {X}}}\). As a consequence of Steps 1–3, we see that \({{\mathcal {N}}_{\varepsilon }} \) is \({\mathfrak {L}}-\)compact in \({{\overline{\Omega }}}\), which completes the proof of the lemma. \(\square \)
Lemma 3.4
In addition to condition (H1), assume that
Then, there exists \({{\mathcal {A}}}>0\), which is independent of \(\zeta \), such that
Proof
Let \(u\in {\mathcal {{\mathcal {X}}}}\) satisfy
then
From the expressions for \({\mathfrak {L}}\) and \({{\mathcal {N}}_{\varepsilon }}\), for any \(t\in J\)
By applying \({\mathfrak {I}}_{0^{+}}^{\alpha ;\psi }\) to both sides of the above equation, we have
Since the operator \({\mathfrak {I}}_{0^{+}}^{\alpha ;\psi }\) is linear
Now, using Lemma 2.5 gives
Thus, for every \(t \in J\)
where \(h^*=\left\| h(\cdot ,0,0)\right\| _{\infty }\) as in Step 2 of the proof of Lemma 3.3. Therefore
and so
This proves the lemma. \(\square \)
Lemma 3.5
If conditions (H1) and (6) hold, then there exist a bounded open set \(\Omega \subset {{\mathcal {X}}} \) with
for any \(u\in \partial \Omega \) and any \(\zeta \in (0,1].\)
Proof
Using Lemma 3.4, there exists a positive constant \({{\mathcal {A}}}\) independent of \(\zeta \), such that, if u satisfies
then \(\Vert u\Vert _{{{\mathcal {X}}}}\leqslant {{\mathcal {A}}}\). Thus, if
with \(\vartheta >{{\mathcal {A}}}\), then
for all \(u\in \partial \Omega =\{u\in {{\mathcal {X}}}; \Vert u\Vert _{{{\mathcal {X}}}}=\vartheta \}\) and \(\zeta \in (0,1]\), which is what we wanted to show.
Our first existence result is given in the following theorem. \(\square \)
Theorem 3.6
If (H1) and (6) hold, then there exist at least one solution to the problem (1)–(2) in \({\text {Dom}}\,{\mathfrak {L}}\cap {{\overline{\Omega }}}\).
Proof
It is clear that the set \(\Omega \) defined in (8) is symmetric, \(0\in \Omega \), and \({{\mathcal {X}}}\cap {{\overline{\Omega }}}={{\overline{\Omega }}} \ne \emptyset \). By Lemma 3.5
for each \(u\in {{\mathcal {X}}}\cap \partial \Omega =\partial \Omega \) and each \(\zeta \in (0,1]. \) By Lemma 2.11, problem (1)–(2) has at least one solution in \({\text {Dom}}\,{\mathfrak {L}}\cap {{\overline{\Omega }}}\).
Next, we have a uniqueness result. \(\square \)
Theorem 3.7
In addition to (H1), assume that
-
(H2) There exist constants \({\overline{\gamma }}>0\) and \({\overline{\eta }}\geqslant 0\), such that
$$\begin{aligned} |h(t,u,{\mathfrak {\upsilon }})-h(t,{\bar{u}},\bar{{\mathfrak {\upsilon }}})|\geqslant {\overline{\gamma }}|u-{\bar{u}}|-{\overline{\eta }}|{\mathfrak {\upsilon }}-\bar{{\mathfrak {\upsilon }}}|, \end{aligned}$$for every \(t \in J\) and u, \({\bar{u}}\), \({\mathfrak {\upsilon }}\), \(\bar{{\mathfrak {\upsilon }}}\in {{\mathbb {R}}}\).
If
then the problem (1)–(2) has a unique solution in \({\text {Dom}}\,{\mathfrak {L}}\cap {{\overline{\Omega }}}\).
Proof
Since condition (6) must hold, by Theorem 3.6, the problem (1)–(2) has at least one solution in \({\text {Dom}}\,{\mathfrak {L}} \cap {{\overline{\Omega }}}\).
To prove uniqueness of the solution, suppose that problem (1)–(2) has two different solutions \(u_{1}\), \(u_{2} \in {\text {Dom}}\,{\mathfrak {L}} \cap {{\overline{\Omega }}}\). Then, for each \(t \in J\)
Let \(u(t)=u_{1}(t)-u_{2}(t), \; \text {for all} \; t \in J\). Then
Using the fact that \({\text {Img}}\,{\mathfrak {L}}=\ker {Q}\), we have
Since h is continuous, there exists \(t_{0}\in [0,{\mathfrak {b}}]\), such that
In view of (H2), we have
so
On the other hand, by Lemma 2.5, we have
which implies that
and therefore
Using (11), for every \(t \in J\)
From (10) and (H1), we see that
and so
Substituting (13) into the right-hand side of (12), we obtain
for every \(t \in J\). Therefore
which by (9) implies
That is, for any \(t \in J\), \(u(t)=0\) or \(u_{1}(t)=u_{2}(t)\). This completes the proof.
4 Examples
Example 1
Consider the problem
where
Here \(\alpha =\frac{1}{3}\), \(\psi (t)=2^{t}\), and \(\varepsilon =\displaystyle \frac{1}{\sqrt{2}}\). Clearly, \(h \in C([0,1]\times {{\mathbb {R}}}\times {{\mathbb {R}}},{{\mathbb {R}}})\). Let u, \({\overline{u}}\), \({\mathfrak {\upsilon }}\), \(\overline{{\mathfrak {\upsilon }}} \in {{\mathbb {R}}}\), and \(t \in J\); then
so (H1) is satisfied with \(\gamma =\displaystyle \frac{1}{5}\) and \(\eta =\displaystyle \frac{1}{3\sqrt{\pi }}\). By a simple calculation, we see that
Therefore, by Theorem 3.6, this problem has at least one solution.
Example 2
Consider the problem
where
Here, \( \alpha =\frac{1}{2}\), \(\psi (t)=e^{t}\), and \(\varepsilon =\displaystyle \frac{1}{\sqrt{\pi }}\). It is easy to see that \(h \in C([0,1]\times {{\mathbb {R}}}\times {{\mathbb {R}}},{{\mathbb {R}}})\). For u, \({\overline{u}}\), \({\mathfrak {\upsilon }}\), \(\overline{{\mathfrak {\upsilon }}} \in {{\mathbb {R}}}\) and \(t \in J\)
and
Hence, (H1) and (H2) are satisfied with
A calculation shows that
so by Theorem 3.7, the problem has a unique solution.
5 Conclusions
Using Mawhin’s [14] coincidence degree theory, we obtained the existence and uniqueness of solutions to the nonlinear fractional pantograph differential equations involving the \(\psi \)-Caputo derivative. The results are illustrated with examples. It is worth noting that not only do we present a new class of fractional differential equations involving the \(\psi \)-Caputo fractional derivative, but by choosing \(\psi (t) = t\) or \(\psi (t) = ln t\), we have existence results for problems involving the Caputo or the Caputo-Hadamard type fractional derivatives, respectively.
References
Abbas, S.; Benchohra, M.; N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
Abbas, S.; Benchohra, M.; N’Guérékata, G.M.: Advanced Fractional Differential and Integral Equations. Nova Science Publishers, New York (2014)
Abdo, M.S.; Abdeljawad, T.; Kucche, K.D.; Alqudah, M.A.; Ali, S.M.; Jeelani, M.B.: On nonlinear pantograph fractional differential equations with Atangana-Baleanu-Caputo derivative. Adv. Differ. Equ. 2021(65), 1–17 (2021)
Agrawal, O.P.: Some generalized fractional calculus operators and their applications in integral equations. Frac. Cal. Appl. Anal. 15, 700–711 (2012)
Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)
Almeida, R.: Functional differential equations involving the \(\psi \)-Caputo fractional derivative. Fract. Fract. 4(29), 1–8 (2020)
Almeida, R.; Malinowska, A.B.; Odzijewicz, T.: On systems of fractional differential equations with the \(\psi \)-Caputo derivative and their applications. Math. Meth. Appl. Sci. 44, 1–16 (2019)
Balachandran, K.; Kiruthika, S.; Trujillo, J.J.: Existence of solutions of nonlinear fractional pantograph equations. Acta. Math. Sci. Sci. Ser. B (Engl. Ed.) 33, 712–720 (2013)
Benchohra, M.; Bouriah, S.; Henderson, J.: Existence and stability results for nonlinear implicit neutral fractional differential equations with finite delay and impulses. Commun. Appl. Nonlinear Anal. 22, 46–67 (2015)
Benchohra, M.; Bouriah, S.: Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order, Moroccan J. Pure. Appl. Anal. 1, 22–36 (2015)
Benchohra, M.; Bouriah, S.; Nieto, J.J.: Existence of periodic solutions for nonlinear implicit Hadamard’s fractional differential equations. Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat. RACSAM 112, 25–35 (2018)
Benchohra, M.; Bouriah, S.; Graef, J.R.: Nonlinear implicit differential equation of fractional order at resonance. Electron. J. Differ. Equ. 2016(324), 1–10 (2016)
Derbazi, C.; Baitiche, Z.: Coupled systems of \(\psi \)-Caputo differential equations with initial conditions in Banach spaces. Mediter. J. Math. 17(169), 13 (2020)
Gaines, R.E.; Mawhin, J.: Coincidence Degree and Nonlinear Differential Equations, vol. 568. Lecture Notes in Math. Springer, Berlin (1977)
Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204. Elsevier Science B.V, Amsterdam (2006)
Mawhin, J.: Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, 40. American Mathematical Society, Providence, R.I. (1979)
O’Regan, D.; Chao, Y.J.; Chen, Y.Q.: Topological Degree Theory and Application. Taylor and Francis, Boca Raton (2006)
Rahimkhani, P.; Ordokhani, Y.; Babolian, E.: Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. J. Comput. Appl. Math. 309, 493–510 (2017)
Saeed, U.; Rehman, M.: Hermite wavelet method for fractional delay differential equations. J. Differ. Equ. 2014, 8 (2014) Article ID 359093
Samko, S.G.; Kilbas, A.A.; Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993)
Shah, K.; Vivek, D.; Kanagarajan, K.: Dynamics and stability of \(\psi \)-fractional pantograph equations with boundary conditions. Bol. Soc. Paran. Mat. 39, 43–55 (2021)
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Bouriah, S., Foukrach, D., Benchohra, M. et al. Existence and uniqueness of periodic solutions for some nonlinear fractional pantograph differential equations with \(\psi \)-Caputo derivative. Arab. J. Math. 10, 575–587 (2021). https://doi.org/10.1007/s40065-021-00343-z
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DOI: https://doi.org/10.1007/s40065-021-00343-z