Abstract
In this paper, we study an initial value problem for a class of impulsive implicit-type fractional differential equations (FDEs) with proportional delay terms. Schaefer’s fixed point theorem and Banach’s contraction principle are the key tools in obtaining the required results. We apply our results to a numerical problem for demonstration purpose.
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1 Introduction
Differential equations have been proved to be powerful tools to describe many phenomena in real-world problems. There has been a significant advancement in studying various classes of differential equations. We refer to some recent work [1–5]. In the recent years the trend is changed from classical integer-order derivatives to fractional-order derivatives. This is because of the fact that the study of many mathematical models of real-world problems with fractional-order derivatives produces significant results. The main advantage of noninteger-order derivatives is that they are global operators and produce accurate and stable results, whereas integer-order derivatives are local operators. Due to these dominant advantages, various classes of differential equations are reformulated and constructed in terms of fractional-order derivatives. We refer to some problems such as world population growth models, blood alcohol level problems, video tape models, and so on, where fractional-order derivatives are applied as powerful tools. Similarly, fractional-order derivatives have applications in other fields like electrodynamics, fluid dynamics, fluid mechanics, and so on; see [6–15].
The class of implicit differential equations is one of the major classes of differential equations. These equations have applications in managerial and economic sciences. In economical problems the differential equations in equilibrium state we dealing with are mostly of implicit type. Similarly, with the help of implicit functions, we can investigate noteworthy features of most real-life isolines or surface geometry. We refer to some recent work on implicit differential equations [16–18].
On other hand, differential equations with impulsive conditions play an important role almost in every subject of science. Dynamical systems with impulsive phenomena have applications in physics, biology, economics, engineering, and so forth. Differential equations with impulsive conditions are used to model certain processes with discontinuous jumps and abrupt changes that cannot be modeled by classical differential equations; see [19–26]. Considerable attention has been given to impulsive differential equations, but it is worth noticing that many aspects of these equations yet need to be studied and explored.
There are several types of delay differential equations. One type is proportional delay differential equations, also known as pantograph differential equations. The importance of these equations is due to their ability to model several problems in economics, chemistry, medicine, biology, infectious diseases, physiological and pharmaceutical kinetics, chemical kinetics, absorption of light by the interstellar matter, physics, population studies, number theory, the navigational control of ships and aircraft, electronic systems, electrodynamics, quantum mechanics, and so on [27–30].
Implicit differential equations with proportional delay occur in many applied physical applications. In economics the sudden rise and fall in stock exchange or in the status at time t as a function of that time with some delay is inevitable in decision making problems. This is a practical significance of pantograph implicit impulsive differential equations. For some interesting applications of proportional delay differential equations, we refer to [31] and references therein.
Motivated by the aforementioned applications of implicit impulsive pantograph differential equations, in this paper, we study two important aspects; the existence of solutions and the Hyers–Ulam stability of the following initial value problem of implicit impulsive differential equations with proportional delay term:
where \({J}=[0,T], T>0\), \({}_{0}^{c}D_{x_{n}}^{\alpha }\) denotes the Caputo derivative at points other than \(x_{n}\) in J, and \({f}:{J}\times {R}^{3}\rightarrow {R}\) and \(\Phi:C({J},{R})\rightarrow {R}\) are given continuous functions. Further, \(I_{n}:{R}\rightarrow {R}\) are nonlinear impulsive mappings, and \(\Delta z(x_{n})=z(x^{+}_{{n}})-z(x^{-}_{{n}})\), where \(z(x^{+}_{{n}})\) and \(z(x^{-}_{{n}})\) are the right and left limits of z, respectively, at \(x_{n}\), \({n}=1,2,\dots,k\).
It is important to note that in this paper, we use the notations HU for Hyers–Ulam, GHU for generalized Hyers–Ulam, HUR for Hyers–Ulam–Rassias, and GHUR for generalized Hyers–Ulam–Rassis, respectively.
The rest of the paper is organized as follows. In Sect. 2, we give some definitions and preliminary results necessary in this study. In Sect. 3, we give results concerning the existence of solutions. In Sect. 4, we study the Hyers–Ulam stability and Hyers–Ulam–Rassias stability of problem (1). In Sect. 5, we illustrate the applications of our main results by providing a self-illustrative example. In Sect. 6, we conclude our work.
2 Preliminaries
This section is concerned with introduction to some basic results and definitions.
Let \({J}=[0,T]\) with \(0=x_{0}< x_{1}< x_{2}<\cdots <x_{n}=T\). We denote by E the space \(PC({J},{R})\) of all piecewise continuous functions on J. More precisely, let \({J}= {J}_{0} \cup {J}_{1} \cup {J}_{2} \cup \cdots \cup {J}_{n}\), where \({J}_{0}=[x_{0},x_{1}],{J}_{1}=(x_{1},x_{2}],{J}_{2}=(x_{2},x_{3}], \dots,{J}_{n}=(x_{n},x_{{n}+1}] \), \({n}=1,2,3,\dots,k\), and \({J}'={J}\setminus \{x_{1},x_{2},\dots,x_{n}\}\). Then
and \(\Delta z(x_{n})=z(x^{+}_{{n}})-z(x^{-}_{{n}}) \text{ for } {n}=1,2, \dots,k\). The space \(({E},\|\cdot \|_{E})\) is a Banach space with respect to the norm \(\|z\|_{{E}}=\max \{|z(x)|: x\in {J}\}\).
Definition 1
([32])
The Riemann fractional integral of a function \(z\in L^{1}([0,T],{R}_{+})\) is defined as
where \(\alpha \in (0, \infty )\) is the order of integration, and the integral on the right-hand side is pointwise defined on \((0, \infty )\).
Definition 2
([6])
The Caputo fractional-order derivative of a function \(z:(0,\infty )\rightarrow {R}\) is defined as
where n=\([\alpha ]+1\), and \([\alpha ]\) is the integral part of a real number α.
Lemma 1
([33])
For \(\alpha >0\), the solution of FDE
is given by
where \(n=[\alpha ]+1\).
Let \(\varphi \in C({J},{R}_{+})\) be a nondecreasing function. We assume that for \(s\in {E}\), \(\epsilon >0\), and \(\omega \geq 0\), the following inequalities hold:
and
Definition 3
([34])
Problem (1) is HU stable if there exists a real number \(C_{f}>0\) such that for any solution \(s\in {E}\) of inequality (2), there are a unique solution \(z\in {E}\) of (1) and \(\epsilon >0\) such that
Definition 4
([34])
Problem (1) is GHU stable if there exists a real function \(\vartheta \in C({R}_{+},{R}_{+})\) with \(\vartheta (0)=0\) such that for any solution \(s\in {E}\) of inequality (2), there exist a unique solution \(z\in {E}\) of (1) and \(\epsilon >0\) such that
Definition 5
([34])
Problem (1) is HUR stable with respect to \((\omega,\varphi )\) if there exists a real number \(C_{f}>0\) such that for any solution \(s\in {E}\) of inequality (4), there is a unique solution \(z\in {E}\) of (1) such that
Definition 6
([34])
Problem (1) is GHUR stable with respect to \((\omega,\varphi )\) if there exists \(C_{f}\in {R}>0\) such that for any solution \(s\in {E}\) of inequality (3), there is a unique solution \(z\in {E}\) of (1) such that
Remark 1
A function \(s\in {E}\) is a solution of (2) if there are a function \(\theta \in {E}\) and a sequence \(\theta _{n}\) (which depends on s) such that
-
(i)
\(|\theta (x)|\leq \epsilon \), \(|\theta _{n}|\leq \epsilon \), \(x\in {J}_{n}, {n}=1,2,\dots,k\);
-
(ii)
\({_{0}^{c}D_{x_{n}}^{\alpha }} s(x)={f}(x,s(x),s({m} x),{_{0}^{c}D_{x_{n}}^{ \alpha }} s(x))+\theta (x)\), \(x\in {J}_{n}\), \({n}=1,2,\dots,k\);
-
(iii)
\(\Delta s(x_{n})=I_{n}(s(x_{n}))+\theta _{n}\), \(x\in {J}_{n}\), \({n}=1,2,\dots,k\).
Remark 2
A function \(s\in {E}\) is a solution of (3) if there are a function \(\theta \in {E}\) and a sequence \(\theta _{n}\) (which depends on s) such that
-
(i)
\(|\theta (x)|\leq \varphi (x)\), \(|\theta _{n}|\leq \omega \), \(x\in {J}_{n}, {n}=1,2,\dots,k\);
-
(ii)
\({_{0}^{c}D_{x_{n}}^{\alpha }} s(x)={f}(x,s(x),s({m} x),{_{0}^{c}D_{x_{n}}^{ \alpha }} s(x))+\theta (x)\), \(x\in {J}_{n}\), \({n}=1,2,\dots,k\);
-
(iii)
\(\Delta s(x_{n})=I_{n}(s(x_{n}))+\theta _{n}\), \(x\in {J}_{n}\), \({n}=1,2,\dots,k\).
Remark 3
A function \(s\in {E}\) is a solution of (4) if there are a function \(\theta \in {E}\) and sequence \(\theta _{n}\) (which depends on s) such that
-
(i)
\(|\theta (x)|\leq \epsilon \varphi (x)\), \(|\theta _{n}|\leq \epsilon \omega \), \(x\in {J}_{n}, {n}=1,2,\dots,k\);
-
(ii)
\({_{0}^{c}D_{x_{n}}^{\alpha }} s(x)={f}(x,s(x),s({m} x),{_{0}^{c}D_{x_{n}}^{ \alpha }} s(x))+\theta (x)\), \(x\in {J}_{n}\), \({n}=1,2,\dots,k\);
-
(iii)
\(\Delta s(x_{n})=I_{n}(s(x_{n}))+\theta _{n}\), \(x\in {J}_{n}\), \({n}=1,2,\dots,k\).
Theorem 1
([35])
Let \(\mathscr{W}:{E}\rightarrow {E}\) be a completely continuous operator, where E is a Banach space, and \(\Omega =\{z\in {E}: z=\delta \mathscr{W}z, 0<\delta <1\}\) is a bounded set. Then \(\mathscr{W}\) has at least one fixed point in E.
3 Existence of solution: main results
In this section, we derive conditions for the existence and uniqueness of a solution for problem (1).
Lemma 2
Let \(\psi:{J}\rightarrow {R}\) be a continuous function, and let \(0<\alpha \leq 1\). Then a function \(z\in {E}\) is a solution of the impulsive problem
if and only if z satisfies the integral equation
Proof
Let z be a solution of (5). Then applying Lemma 1, for each \(x\in {J}_{0}\), we have
which implies
Using the initial condition \(z(0)=z_{0}\), from (7) we have
Similarly, for \(x\in {J}_{1}\), we have
and
From
we get
Putting for \(z(x_{1})\), (9) implies
Generalizing in this way, for \(x\in {J}_{n}\), we have
Thus from (8) and (10) we get (6).
Now, conversely, let z be a solution of (6). Then taking the αth-order derivative of (6) gives the differential equation in (5). This completes the proof. □
Corollary 1
From Lemma 2we get the following solution for our problem (1):
We use the notation \({u}_{z}(x)={f}(x,z(x),z({m} x),{_{0}^{c}D_{x_{n}}^{\alpha }}{z(x))}={f}(x,z(x),z({m} x),{u}_{z}(x))\).
The following assumptions are necessary in obtaining the main results.
- \((A_{1})\):
-
\({f}:{J}\times {R}^{3}\rightarrow {R}\) is continuous;
- \((A_{2})\):
-
There exist constants \(M_{f}>0\) and \(0< N_{f}<1\) such that for all \(x\in {J}\) and \(z,\bar{z}\in {R}\), we have the following relation:
$$\begin{aligned} & \bigl\vert {f} \bigl(x,z(x),z({m} x),{u}_{z}(x) \bigr)-{f} \bigl(x, \bar{z}(x), \bar{z}({m} x),{u}_{\bar{z}}(x) \bigr) \bigr\vert \\ &\quad\leq M_{f} \bigl( \bigl\vert z(x)-\bar{z}(x) \bigr\vert + \bigl\vert z({m} x)- \bar{z}({m} x) \bigr\vert \bigr) \\ &\qquad{} +N_{f} \bigl\vert {u}_{z}(x)-{u}_{\bar{z}}(x) \bigr\vert . \end{aligned}$$ - \((A_{3})\):
-
For any \(z,\bar{z}\in {E}\), there exists a constant \(A^{*}_{I}>0\) such that
$$ \bigl\vert I_{i} \bigl(z(x_{i}) \bigr)-I_{i} \bigl(\bar{z}(x_{i}) \bigr) \bigr\vert \leq A^{*}_{I} \bigl\vert z(x_{i})- \bar{z}(x_{i}) \bigr\vert ; $$ - \((A_{4})\):
-
there exist functions \(a, b, c\in C({J},{R}^{+})\) such that
$$ \bigl\vert {f} \bigl(x,z(x),z({m} x),{u}_{z}(x) \bigr) \bigr\vert \leq a(x)+b(x) \bigl( \bigl\vert z(x) \bigr\vert + \bigl\vert z({m} x) \bigr\vert \bigr)+c(x) \vert {u}_{z} \vert $$with \(c^{*}=\sup_{x\in {J}}c(x)<1\);
- \((A_{5})\):
-
Let for each \(z\in {R}\), \(i=1,2,\dots,{n}\), there exist constants \(K, L>0\), such that \(|I_{i}(z)|\leq K|z|+L\);
- \((A_{6})\):
-
Let for a nondecreasing function \(\varphi \in PC({J},{R}_{+})\) and constant \(\varrho _{\varphi }>0\), the following inequality hold:
$$ I^{\alpha }{\varphi (x)}\leq \varrho _{\varphi }\varphi (x),\quad x \in {J}. $$
We convert our problem to a fixed-point problem by defining the operator \(\mathscr{W}:C({J},{R})\rightarrow C({J},{R})\) as
Theorem 2
Suppose conditions \((A_{1})\)–\((A_{5})\) are satisfied. Then problem (1) has at least one solution in the given interval.
Proof
The proof is divided into four steps.
Step 1: \(\mathscr{W}\) is continuous. Let \(\{z_{n}\}\in C({J},{R})\) be a sequence such that \(z_{n}\rightarrow z\in C({J},{R})\). For each \(x\in {J}_{n}\), we have
where \({u}_{z_{n}},{u}_{z}\in C({J},{R})\) satisfy
By \((A_{2})\) we get
Now \(z_{n}\rightarrow z\) as \(n\rightarrow \infty \) implies \({u}_{z_{n}}(x)\rightarrow {u}_{z}(x)\) for each \(x\in {J}_{n}\). We know that every convergent sequence is bounded. So let \(\aleph >0\) be such that for each \(x\in {J}\), we have \(|{u}_{z_{n}}(x)|\leq \aleph \) and \(|{u}_{z}(x)|\leq \aleph \). Then
For each \(x\in {J}_{n}\), the functions \(\tau \rightarrow 2\aleph (x-\tau )^{\alpha -1}\) and \(\tau \rightarrow 2\aleph (x_{i}-\tau )^{\alpha -1}\) are integrable. Also, f and I are continuous. Thus applying the Lebesgue dominated convergence theorem, we have \(|(\mathscr{W}z_{n})(x)-(\mathscr{W}z)(x)|\rightarrow 0\) as \(n\rightarrow \infty \). Hence, in particular, \(\max_{x\in {J}}|\mathscr{W}z_{n}(x)-\mathscr{W}z(x)|\rightarrow 0\) as \(n\rightarrow \infty \), which implies that \(\|\mathscr{W}z_{n}-\mathscr{W}z\|_{E}\rightarrow 0 \text{ as } n \rightarrow \infty \). Similarly, for each \(x\in {J}_{0}\), we can show that \(\|\mathscr{W}z_{n}-\mathscr{W}z\|_{E}\rightarrow 0\) as \(n\rightarrow \infty \). Therefore \(\mathscr{W}\) is continuous.
Step 2: Here we show that \(\mathscr{W}\) is bounded. Let for \(\gamma >0\), there exist a positive real number \(\Theta >0\) such that for each \(z\in \Omega _{\gamma }=\{z\in {E}: \|z\|_{E}\leq \gamma \}\), we have \(\|\mathscr{W}(z)\|_{E}\leq \Theta \).
For \(x\in {J}_{n}\), consider
By \((A_{4})\) for \(x\in {J}_{n}\), we have
where \(a^{*}=\sup_{x\in {J}_{n}}a(x), b^{*}=\sup_{x\in {J}_{n}}b(x)\), and \(c^{*}=\sup_{x\in {J}_{n}}c(x)<1\).
Then
By application of assumptions \((A_{4})\)–\((A_{5})\) we obtain
Thus
Similarly, for \(x\in {J}_{0}\), we can show that
Step 3: We show \(\mathscr{W}\) maps bounded sets into equicontinuous sets of E.
Let \(x_{1}, x_{2}\in {J}_{n}\) be such that \(x_{1}< x_{2}\). As in Step 2, take a bounded set \(\Omega _{\gamma ^{*}}\). For \(z\in \Omega _{\gamma ^{*}}\), we have
We see that the right-hand side of (13) approaches 0 as \(x_{1}\rightarrow x_{2}\). Therefore \(|(\mathscr{W}z)(x_{2})-(\mathscr{W}z)(x_{1})|\rightarrow 0\) as \(x_{1}\rightarrow x_{2}\). Similarly, in a subinterval \({J}_{0}\), we can show that \(|(\mathscr{W}z)(x_{2})-(\mathscr{W}z)(x_{1})|\rightarrow 0\) as \(x_{1}\rightarrow x_{2}\). Therefore, as a result of Steps 1–3 and the Ascoli–Arzelà theorem, we conclude that \(\mathscr{W}: {E}\rightarrow {E}\) is completely continuous.
Step 4: In this final step, we define the set \(\mho =\{z\in {E}: z=\delta \mathscr{W}z \text{ for some } 0<\delta <1 \}\). We need to show that ℧ is bounded. Let \(z\in \mho \). Then \(z=\delta \mathscr{W}z\) for some \(0<\delta <1\). Hence for each \(x\in {J}_{n}\), we have
Now since \(0<\delta <1\), from the above we have
from which we have
A similar result can be achieved for \(x\in {J}_{0}\). Therefore ℧ is a bounded set. Hence, by applying Schaefer’s fixed point theorem, \(\mathscr{W}\) has at leat one confirmed fixed point. □
Theorem 3
Assume that conditions \((A_{1})\)–\((A_{3})\) together with the inequality
are satisfied. Then (1) has a unique solution.
Proof
For \(z,\bar{z}\in {E}\) and \(x\in {J}_{n}\), we have
where \({u}_{z},{w}_{z}\in C({J},{R})\) are given by
By \(({A_{2}})\) we have
Then
Thus by assumptions \((A_{2})\)–\((A_{3})\) inequality (15) implies
Taking the maximum norm, we get
Similarly, for \(z,\bar{z}\in {E}\) and \(x\in {J}_{0}\), we get
Since
\(\mathscr{W}\) is contraction operator, and therefore by Banach’s contraction principle \(\mathscr{W}\) has a unique fixed point in the given interval J. Thus problem (1) has a unique solution. □
4 Hyers–Ulam-type stability analysis
Here we study the Hyers–Ulam stability of problem (1).
Theorem 4
Assume that conditions \((A_{1})\)–\((A_{5})\) and (14) hold. Then proposed problem (1) is HU stable.
Proof
Let \(s\in {E}\) be any solution of (2), and let z be the unique solution of
By Lemma 2, for \(x\in {J}_{n}\), we have
By Remark 1 we get
The solution of (18) is
where \(\bar{{u}_{z}}\in C({J},{R})\) is given by
Therefore, for each \(x\in {J}_{n}\), we have
By \(({A_{2}})\) we obtain
Hence applying assumptions \(({A_{1}})\)–\(({A_{4}})\) and Remark 1, we obtain
By taking the maximum norm and simplification we get
from which we obtain
Similarly, for \(x\in {J}_{0}\), we have
Using (19) and (20), for \(x\in {J}\), we have
Thus
where
Therefore problem (1) is HU stable. □
Corollary 2
If we set \(\vartheta (\epsilon )=C_{f}(\epsilon )\) with \(\vartheta (0)=0\), then problem (1) becomes GHU stable.
Theorem 5
If assumptions \(({A_{1}})\)–\(({A_{8}})\) and (14) hold, then problem (1) is HUR stable with respect to \((\omega,\varphi )\).
Proof
Let \(s\in {E}\) be any solution of (4), and let z be a unique solution of
Then from the proof of Theorem 4, for each \(x\in {J}_{n}\), we have
By \(({A_{2}})\) we obtain
By assumptions \(({A_{1}})\)–\(({A_{4}})\) and Remark 3 we have
Using \(({A_{8}})\) and taking the maximum norm, we get
which yields
Similarly, for \(x\in {J}_{0}\), we have
Using (21) and (22), for \(x\in {J}\), we have
Thus
where
Therefore problem (1) is HUR stable. □
5 Applications
Example 1
where \(\alpha =\frac{1}{2}\), \({J}_{0}=[0, \frac{1}{3}]\), \({J}_{1}=(\frac{1}{3},1]\).
Set
with \(\alpha =\frac{1}{2}\) and \({m}=\frac{1}{4}\). It is clear that f is a jointly continuous function.
Using \((H_{2})\), for any \(z, \bar{z} \in {R}\), we have
Hence \((H_{2})\) holds with \(M_{{f}}=\frac{1}{19}\) and \(N_{{f}}=\frac{1}{38}\). Set
where \(v \in {R}\). Then for \(v, \bar{v}\in {R}\) and \(k=1\), we have
Hence \(({A_{3}})\) holds with \(A^{*}_{I}=\frac{1}{10}\).
Also, the condition
is satisfied with \(T=1\) and \({n}=1\). Therefore by Theorem 3 problem (23) has a unique solution.
Let \(\varphi (x)=x\) and \(\omega =1\). Then for any \(x\in [0,1]\), we have
We see that condition \(({A_{8}})\) holds with \(\varrho _{\varphi }=\frac{2}{\sqrt{\pi }}\). Applying 5, problem (23) is HUR stable.
6 Conclusion
Pantograph differential equations is a special type of delay differential equations with proportional delay terms. In this research work, we studied important aspects such as the existence theory and stability analysis to an IVP of pantograph implicit fractional differential equations with impulsive conditions. Using Schaefer’s fixed point theorem, we derived a result of at least one solution to system (1), and applying the Banach contraction theorem, we obtained conditions for a unique solution to the mentioned problem. Similarly, using the Hyers–Ulam concept, we studied the stability of the considered problem.
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Ibrahim Mahariq and Bahaa Al-Sheikh would like to acknowledge College of Engineering and Technology at American University of the Middle East. Thabet Abdeljawad would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17.
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Ali, A., Mahariq, I., Shah, K. et al. Stability analysis of initial value problem of pantograph-type implicit fractional differential equations with impulsive conditions. Adv Differ Equ 2021, 55 (2021). https://doi.org/10.1186/s13662-021-03218-x
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DOI: https://doi.org/10.1186/s13662-021-03218-x