1 Introduction

In many real world problems, the fractional order models are found to be more suitable than integer order ones. More specifically, we can find the applications of fractional order derivatives and integrals in electrodynamics of complex medium, aerodynamics, polymer rheology, physics, chemistry, and so forth. For basic study and applications, we recommend the books and papers provided in [1–10]. Due to these applications, fractional order derivatives and integrals are gaining much importance and consideration from researchers nowadays. We refer to some recent work [11–18].

Problems with integral boundary conditions naturally arise in applied fields of science like thermal conduction problems, semiconductor problems, chemical engineering, blood flow problems, underground water flow problems, hydrodynamic problems, population dynamics, and so forth. For detailed study of integral boundary value problems, we recommend the papers [19–22].

Similarly, the dynamical systems with impulsive phenomena have been an object of great interest in many subjects such as physics, biology, economics, and engineering. The differential equations with impulsive conditions are used to model certain processes with discontinuous jumps and abrupt changes. Such processes cannot be modeled with classical differential equations (see [23–26]).

On the other hand, stability analysis has got greater interest in the last few years for FODEs. Because during numerical and optimal analysis such tools are greatly needed, see for instance [27]. Recently, for FODEs, the Ulam stability results have been considered very well. This concept of stability was introduced for the first time by Ulam and Hyers [28, 29] during 1941. It is nowadays known as Hyers–Ulam stability. This approach has stimulated a number of people to investigate stability of various mathematical problems. So, a large number of papers on Hyers–Ulam stability have been reported in the literature (see [30–34]). Motivated by the applications of impulsive and integral boundary value problems, in this paper we study the following nonlinear problem of implicit FODEs with impulsive and integral boundary conditions:

(1)

where \({}_{0}^{C}D_{t}^{\varsigma }\) represents the Caputo derivative, \(\mathrm{g}:[0, 1]\times \mathbb {R}^{3}\rightarrow \mathbb {R}\) and are continuous functions, \(p>0\), \(q\geq 0\) are real numbers. Moreover, ; , are the right- and left-hand limits, respectively, at for .

Our considered problem (1) involves proportional delay term which includes a famous class of differential equations called pantograph. The pantograph differential equations form an important class of differential equations which has a wide range of applications in various applied fields of science, engineering and in economics. In economics the sudden rise and fall in stock exchange or in its status at time t as a function of that time with some delay which is inevitable in decision making problems is a practical significance of impulsive delay differential equations. Further, second order boundary value problems can be used to describe a large number of physical, mechanical, biological, and chemical phenomena. For some physical significance, we refer to some valuable work in [35–38].

2 Preliminaries

The space \(\mathscr{W}=C([0, 1],\mathbb {R})=\{\upsilon (t):\upsilon \in C([0, 1]) \}\) is a Banach space with respect to the norm defined by

$$ \|\upsilon \|_{\mathscr{W}}=\max_{t\in [0, 1]}\bigl\{ \bigl\vert \upsilon (t) \bigr\vert : t \in [0, 1]\bigr\} .$$
(2)

Definition 2.1

([1])

The noninteger order integral of a function \(\mathrm{g}\in L^{1}([a,b],\mathbb {R}^{+})\) of order \(\varsigma \in \mathbb {R}^{+}\) is defined by

$$ \mathcal{I}^{\varsigma }_{a} \mathrm{g}(t)= \int _{a}^{t} \frac{(t-z)^{\varsigma -1}}{\Gamma (\varsigma )} \mathrm{g}(z)\,dz, $$
(3)

where Γ is the gamma function.

Definition 2.2

([1])

For a function g given on the interval \([a,b]\), the Caputo fractional order derivative of υ is defined by

$$ {}^{C}D^{\varsigma }\mathrm{g}(t)=\frac{1}{\Gamma (\ell -\varsigma )} \int _{a}^{t} (t-z)^{\ell -\varsigma -1} \mathrm{g}^{(\ell )}(z)\,dz, $$
(4)

where \(\ell =[\varsigma ]+1\).

Lemma 2.3

([39])

For \(\varsigma >0\), the given result holds:

$$ \mathcal{I}^{\varsigma } \bigl({}^{C}D^{\varsigma } \mathrm{g}(t) \bigr)= \mathrm{g}(t)-\sum_{\jmath =0}^{\ell -1} \frac{\mathrm{g}^{(\jmath )}(0)}{\jmath !}t^{\jmath },\quad \textit{where } \ell =[\varsigma ]+1.$$

We construct the following three inequalities for studying Hyers–Ulam stability of problem (1). Let \(x\in C([0, 1], \mathbb {R}_{+})\) be a nondecreasing function, \(\xi \geq 0\), \(\psi \in \mathscr{W}\), such that for , , the following sets of inequalities are satisfied:

(5)
(6)
(7)

where ς and m are the same as defined in problem (1).

Definition 2.4

([40])

If for \(\epsilon >0\) there exists a constant \(C_{\mathrm{g}}>0\) such that, for any solution \(\psi \in \mathscr{W}\) of inequality (5), there is a unique solution \(\upsilon \in \mathscr{W}\) of system (1) that satisfies

$$ \bigl\vert \psi (t)-\upsilon (t) \bigr\vert \leq C_{\mathrm{g}}\epsilon ,\quad t\in I, $$

then problem (1) is Hyers–Ulam stable.

Definition 2.5

If for \(\epsilon >0\) and a set of positive real numbers \(\mathbb {R}^{+}\) there exists \(x\in C(\mathbb {R}^{+},\mathbb {R}^{+})\) with \(x(0)=0\) such that, for any solution \(\psi \in \mathscr{W}\) of inequality (6), there exist \(\epsilon >0\) and a unique solution \(\upsilon \in \mathscr{W}\) of problem (1) that satisfies

$$ \bigl\vert \psi (t)-\upsilon (t) \bigr\vert \leq x(\epsilon ),\quad t\in I, $$

then problem (1) is generalized Hyers–Ulam stable.

Definition 2.6

([40])

If for \(\epsilon >0\) there exists a real number \(C_{\mathrm{g}}>0\) such that, for any solution \(\psi \in \mathscr{W}\) of inequality (7), there is a unique solution \(\upsilon \in \mathscr{W}\) of problem (1) that satisfies

$$ \bigl\vert \psi (t)-\upsilon (t) \bigr\vert \leq C_{\mathrm{g}}\epsilon \bigl(\xi +x(t)\bigr), \quad t \in I, $$

then problem (1) is Hyers–Ulam–Rassias stable with respect to \((\xi ,x)\).

Definition 2.7

([40])

If there exists a constant \(C_{\mathrm{g}}>0\) such that, for any solution \(\psi \in \mathscr{W}\) of inequality (6), there is a unique solution \(\upsilon \in \mathscr{W}\) of problem (1) that satisfies

$$ \bigl\vert \psi (t)-\upsilon (t) \bigr\vert \leq C_{\mathrm{g}}\bigl(\xi +x(t)\bigr),\quad t\in I, $$

then problem (1) is generalized Hyers–Ulam–Rassias stable with respect to \((\xi ,x)\).

Remark 1

The function \(\psi \in \mathscr{W}\) is a solution of inequality (5) if there exist a function \(y\in \mathscr{W}\) and a sequence , , which depends on ψ such that

  1. (i)

    \(|y(t)|\leq \epsilon \), , \(t\in I\),

  2. (ii)

    \({}_{0}^{C}D_{t}^{\varsigma }\psi (t)=\mathrm{g}(t, \psi (t), \psi (mt), {}_{0}^{C}D_{t}^{\varsigma }{\psi (t)})+y(t)\),

  3. (iii)

    ,

  4. (iv)

    .

Remark 2

The function \(\psi \in \mathscr{W}\) is a solution of inequality (6) if there exist a function \(y\in \mathscr{W}\) and a sequence , , which depends on ψ such that

  1. (i)

    \(|y(t)|\leq x(t)\), , \(t\in I\),

  2. (ii)

    \({}_{0}^{C}D_{t}^{\varsigma }\psi (t)=\mathrm{g}(t, \psi (t), \psi (mt), {}_{0}^{C}D_{t}^{\varsigma }{\psi (t)})+y(t)\),

  3. (iii)

    ,

  4. (iv)

    .

Remark 3

The function \(\psi \in \mathscr{W}\) is a solution of inequality (6) if there exist a function \(y\in \mathscr{W}\) and a sequence , , which depends on ψ such that

  1. (i)

    \(|y(t)|\leq \epsilon x(t)\), , \(t\in I\),

  2. (ii)

    \({}_{0}^{C}D_{t}^{\varsigma }\psi (t)=\mathrm{g}(t, \psi (t), \psi (mt), {}_{0}^{C}D_{t}^{\varsigma }{\psi (t)})+y(t)\),

  3. (iii)

    ,

  4. (iv)

    .

We give the proof of the following lemma, which provides a base for obtaining a solution to problem (1).

Lemma 2.8

Let \(\varsigma \in (1, 2]\), \(\alpha :[0, 1]\rightarrow \mathbb {R}\) be a continuous function, then the function \(\upsilon \in \mathscr{W}\) is the solution to the following problem:

(8)

if and only if υ satisfies the following integral equation:

(9)

where

Proof

Assume that, for \(t\in [0, t_{1}]\), υ is a solution of (8). Then, by Lemma 2.3, there exist \(a_{1},a_{2}\in \mathbb {R}\) such that

$$ \upsilon (t) = {}_{0}I_{t}^{\varsigma } \alpha (t)-a_{1}-a_{2}t= \frac{1}{\Gamma (\varsigma )} \int _{0}^{t}(t-z)^{\varsigma -1}\alpha (z)\,dz-a_{1}-a_{2}t,$$
(10)

which also yields

$$ \upsilon '(t) = \frac{1}{\Gamma (\varsigma -1)} \int _{0}^{t}(t-z)^{ \varsigma -2}\alpha (z)\,dz-a_{2}. $$
(11)

Let, for \(t\in (t_{1}, t_{2}]\), us have \(d_{1},d_{2}\in \mathbb {R}\) with

$$\begin{aligned}& \upsilon (t) = \frac{1}{\Gamma (\varsigma )} \int _{t_{1}}^{t}(t-z)^{ \varsigma -1}\alpha (z)\,dz-d_{1}-d_{2}(t-t_{1}), \\& \upsilon '(t) = \frac{1}{\Gamma (\varsigma -1)} \int _{t_{1}}^{t}(t-z)^{ \varsigma -2}\alpha (z)\,dz-d_{2}. \end{aligned}$$
(12)

This leads us to

$$\begin{aligned}& \upsilon \bigl(t_{1}^{-}\bigr) = \frac{1}{\Gamma (\varsigma )} \int _{t_{0}}^{t_{1}}(t_{1}-z)^{ \varsigma -1} \alpha (z)\,dz-a_{1}-a_{2}t_{1},\qquad \upsilon \bigl(t_{1}^{+}\bigr)=-d_{1}, \\& \upsilon '\bigl(t_{1}^{-}\bigr) = \frac{1}{\Gamma (\varsigma -1)} \int _{0}^{t_{1}}(t_{1}-z)^{ \varsigma -2} \alpha (z)\,dz-a_{2},\qquad \upsilon '\bigl(t_{1}^{+} \bigr)=-d_{2}. \end{aligned}$$

Corresponding to impulsive conditions, we have

$$ \Delta \upsilon (t_{1})=\upsilon \bigl(t_{1}^{+} \bigr)-\upsilon \bigl(t_{1}^{-}\bigr)= \mathcal{F}_{1}\bigl(\upsilon (t_{1})\bigr) \quad \text{and}\quad \Delta \upsilon '(t_{1})= \upsilon '\bigl(t_{1}^{+}\bigr)-\upsilon '\bigl(t_{1}^{-}\bigr)=\bar{ \mathcal{F}}_{1}\bigl( \upsilon (t_{1})\bigr), $$

one has

$$\begin{aligned}& -d_{1} = \frac{1}{\Gamma (\varsigma )} \int _{t_{0}}^{t_{1}}(t_{1}-z)^{ \varsigma -1} \alpha (z)\,dz-a_{1}-a_{2}t_{1}+ \mathcal{F}_{1}\bigl(\upsilon (t_{1})\bigr), \\& -d_{2} = \frac{1}{\Gamma (\varsigma -1)} \int _{0}^{t_{1}}(t_{1}-z)^{ \varsigma -2} \alpha (z)\,dz-a_{2}+\bar{\mathcal{F}}_{1}\bigl(\upsilon (t_{1})\bigr). \end{aligned}$$

Thus (12) implies

$$\begin{aligned} \upsilon (t) =&\frac{1}{\Gamma (\varsigma )} \int _{t_{1}}^{t}(t-z)^{ \varsigma -1}\alpha (z)\,dz+ \frac{1}{\Gamma (\varsigma )} \int _{0}^{t_{1}}(t_{1}-z)^{ \varsigma -1} \alpha (z)\,dz \\ &{}+\frac{t-t_{1}}{\Gamma (\varsigma -1)} \int _{0}^{t_{1}}(t_{1}-z)^{ \varsigma -2} \alpha (z)\,dz+\mathcal{F}_{\jmath }\bigl(\upsilon (t_{1}) \bigr)+(t-t_{1}) \bar{\mathcal{F}}_{1}\bigl(\upsilon (t_{1})\bigr) \\ &{}-a_{1}-a_{2}t, \quad t\in (t_{1}, t_{2}]. \end{aligned}$$

Similarly, for , one has

(13)

which by differentiation gives the result

(14)

Using the given boundary conditions in (10), (11), we obtain

$$ -pa_{1}-qa_{2}= \int _{0}^{1}h_{1}\bigl(\upsilon (z) \bigr)\,dz $$
(15)

and

Thus, in view of \(p\upsilon (1)+q\upsilon '(1)=\int _{0}^{1}h_{2}(\upsilon (z))\,dz\) and the result (15), we get the following values for \(-a_{1}\) and \(-a_{2}\):

Putting these values for \(-a_{1}\) and \(-a_{2}\) in (10) and (13), we get (9). □

3 Main results

Corollary 3.1

As a result of Lemma 2.8, problem (1) has the following solution:

(16)

where

For the purpose of simplicity, we take \(\Phi _{\upsilon }(t)=\mathrm{g}(t, \upsilon (t), \upsilon (mt), {}_{0}^{C}D_{t}^{\varsigma }{\upsilon (t)})\). To transform problem (1) to a fixed point problem, here we define the operator \(\mathscr{Z}:\mathscr{W}\rightarrow \mathscr{W}\) by

The given assumptions are necessary for obtaining our main results.

\((H_{1})\):

Let \(\mathrm{g}:[0, 1]\times \mathbb {R}\times \mathbb {R}\times \mathbb {R} \rightarrow [0,\infty )\) be a jointly continuous function;

\((H_{2})\):

For every \(\upsilon , \bar{\upsilon }\in C([0, 1],\mathbb {R})\) and \(L_{\mathrm{g}}>0\), \(0< N_{\mathrm{g}}<1\), let the following inequality

$$\begin{aligned}& \bigl\vert \mathrm{g}\bigl(t, \upsilon (t), \upsilon (mt), \Phi _{\upsilon }(t)\bigr)- \mathrm{g}\bigl(t, \bar{\upsilon }(t), \bar{\upsilon }(mt), \Phi _{ \bar{\upsilon }}(t)\bigr) \bigr\vert \\& \quad \leq L_{\mathrm{g}}\bigl( \bigl\vert \upsilon (t)-\bar{\upsilon }(t) \bigr\vert + \bigl\vert \upsilon (mt)-\bar{\upsilon }(mt) \bigr\vert \bigr)+N_{\mathrm{g}}\bigl|\Phi _{\upsilon }(t)- \Phi _{\bar{\upsilon }}(t)\bigr| \end{aligned}$$

hold;

\((H_{3})\):

There exist \(C_{1}, C_{2}>0\) such that the following relations hold true:

\((H_{4})\):

There exist constants \(C_{3}\), \(C_{4}>0\) such that, for all \(\upsilon \in \mathbb {R}\), the following inequalities hold true:

$$\begin{aligned}& \bigl\vert h_{1}\bigl(\upsilon (t)\bigr)-h_{1}\bigl( \bar{\upsilon }(t)\bigr) \bigr\vert \leq C_{3} \bigl\vert \upsilon (t)- \bar{\upsilon }(t) \bigr\vert , \\& \bigl\vert h_{2}\bigl(\upsilon (t)\bigr)-h_{2}\bigl( \bar{\upsilon }(t)\bigr) \bigr\vert \leq C_{4} \bigl\vert \upsilon (t)- \bar{\upsilon }(t) \bigr\vert ; \end{aligned}$$
\((H_{5})\):

There exist constants \(C_{5}\), \(C_{6}>0\) such that, for all \(\upsilon \in \mathbb {R}\),

$$\begin{aligned}& \bigl\vert h_{1}\bigl(\upsilon (t)\bigr) \bigr\vert \leq C_{5}, \\& \bigl\vert h_{2}\bigl(\upsilon (t)\bigr) \bigr\vert \leq C_{6}; \end{aligned}$$
\((H_{6})\):

There exist functions \(\theta _{1}, \theta _{2}, \theta _{3} \in C([0, 1],\mathbb {R}^{+})\) with

$$\begin{aligned}& \bigl\vert \mathrm{g}\bigl(t, \upsilon (t), \upsilon (m t), {}_{0}^{C}D_{t_{\jmath }}^{\varsigma }\upsilon (t) \bigr) \bigr\vert \\& \quad \leq \theta _{1}(t)+\theta _{2}(t) \bigl( \vert \upsilon \vert + \bigl\vert \upsilon (m t) \bigr\vert \bigr)+ \theta _{3}(t) \bigl\vert {}_{0}^{C}D_{t_{\jmath }}^{\varsigma } \upsilon (t) \bigr\vert \quad \text{for } t \in [0, 1], \upsilon \in \mathscr{W}, \end{aligned}$$

such that \(\theta _{3}^{*}=\max_{t\in I}|\theta _{3}(t)|<1\);

\((H_{7})\):

If g, , are continuous functions and there exist constants B, \(B^{*}\), M, \(M^{*}>0\) such that, for all \(\upsilon \in \mathscr{W}\), the following inequalities are satisfied:

Theorem 3.2

If assumptions \((H_{1})\)–\((H_{6})\) are satisfied, then problem (1) has at least one solution.

Proof

To prove this result, here we apply Schaefer’s fixed point theorem.

Step 1: We will show that \(\mathscr{Z}\) is continuous. We take a sequence \(\upsilon _{n}\in \mathscr{W}\) with \(\upsilon _{n}\rightarrow \upsilon \in \mathscr{W}\). We consider

(17)

where \(\Phi _{\upsilon _{n}}(t), \Phi _{\upsilon }(t)\in \mathscr{W}\) satisfy the following functional equations:

$$\begin{aligned}& \Phi _{\upsilon _{n}}(t) = \mathrm{g}\bigl(t, \upsilon _{n}(t), \upsilon _{n}(mt), \Phi _{\upsilon _{n}}(t)\bigr), \\& \Phi _{\upsilon }(t) = \mathrm{g}\bigl(t, \upsilon (t), \upsilon (mt), \Phi _{\upsilon }(t)\bigr). \end{aligned}$$

By the application of assumption \((H_{2})\), we have

$$\begin{aligned} \bigl\vert \Phi _{\upsilon _{n}}(t)-\Phi _{\upsilon }(t) \bigr\vert =& \bigl\vert \mathrm{g}\bigl(t, \upsilon _{n}(t), \upsilon _{n}(mt), \Phi _{\upsilon _{n}}(t)\bigr)- \mathrm{g}\bigl(t, \upsilon (t), \upsilon (mt), \Phi _{\upsilon }(t)\bigr) \bigr\vert \\ \leq &L_{\mathrm{g}}\bigl( \bigl\vert \upsilon _{n}(t)- \upsilon (t) \bigr\vert + \bigl\vert \upsilon _{n}(mt)- \upsilon (mt) \bigr\vert \bigr)+N_{\mathrm{g}}\bigl|\Phi _{\upsilon _{n}}(t)-\Phi _{ \upsilon }(t)\bigr|. \end{aligned}$$

Then

$$ \Vert \Phi _{\upsilon _{n}}-\Phi _{\upsilon } \Vert _{PC} \leq \frac{2L_{\mathrm{g}}}{1-N_{\mathrm{g}}} \Vert \upsilon _{n}-\upsilon \Vert _{PC}. $$

Now we see as \(n\rightarrow \infty \), \(\upsilon _{n}\rightarrow \upsilon \), which implies that \(\Phi _{\upsilon ,n}\rightarrow \Phi _{\upsilon }\). Let there exist \(\mathbf{b}>0\) such that, for each t, \(|\Phi _{\upsilon ,n}(t)| \leq \mathbf{b}\) and \(|\Phi _{\upsilon }(t)|\leq \mathbf{b}\). Then

$$\begin{aligned}& \begin{aligned} (t-z)^{\varsigma -1} \bigl\vert \Phi _{\upsilon ,n}(z)-\Phi _{\upsilon }(z) \bigr\vert & \leq (t-z)^{\varsigma -1} \bigl( \bigl\vert \Phi _{\upsilon ,n}(z) \bigr\vert + \bigl\vert \Phi _{ \upsilon }(z) \bigr\vert \bigr) \\ &\leq 2\mathbf{b}(t-z)^{\varsigma -1},\end{aligned} \\& \begin{aligned} (t_{\jmath }-z)^{\varsigma -1} \bigl\vert \Phi _{\upsilon ,n}(z)- \Phi _{\upsilon }(z) \bigr\vert & \leq (t_{\jmath }-z)^{\varsigma -1} \bigl( \bigl\vert \Phi _{\upsilon ,n}(z) \bigr\vert + \bigl\vert \Phi _{\upsilon }(z) \bigr\vert \bigr) \\ &\leq 2\mathbf{b}(t_{\jmath }-z)^{\varsigma -1},\end{aligned} \\& \begin{aligned} (t_{\jmath }-z)^{\varsigma -2} \bigl\vert \Phi _{\upsilon ,n}(z)- \Phi _{\upsilon }(z) \bigr\vert & \leq (t_{\jmath }-z)^{\varsigma -2} \bigl( \bigl\vert \Phi _{\upsilon ,n}(z) \bigr\vert + \bigl\vert \Phi _{\upsilon }(z) \bigr\vert \bigr) \\ &\leq 2\mathbf{b}(t_{\jmath }-z)^{\varsigma -2},\end{aligned} \\& \begin{aligned} (1-z)^{\varsigma -1} \bigl\vert \Phi _{\upsilon ,n}(z)-\Phi _{\upsilon }(z) \bigr\vert & \leq (1-z)^{\varsigma -1} \bigl( \bigl\vert \Phi _{\upsilon ,n}(z) \bigr\vert + \bigl\vert \Phi _{ \upsilon }(z) \bigr\vert \bigr) \\ &\leq 2\mathbf{b}(1-z)^{\varsigma -1},\end{aligned} \\& \begin{aligned} (1-z)^{\varsigma -2} \bigl\vert \Phi _{\upsilon ,n}(z)-\Phi _{\upsilon }(z) \bigr\vert & \leq (1-z)^{\varsigma -2} \bigl( \bigl\vert \Phi _{\upsilon ,n}(z) \bigr\vert + \bigl\vert \Phi _{ \upsilon }(z) \bigr\vert \bigr) \\ &\leq 2\mathbf{b}(1-z)^{\varsigma -2}.\end{aligned} \end{aligned}$$

For each \(t\in [0, t]\), the functions \(z\rightarrow 2\mathbf{b}(t-z)^{\varsigma -1}\), \(z\rightarrow 2\mathbf{b}(t_{\jmath }-z)^{\varsigma -1}\), \(z\rightarrow 2\mathbf{b}(t_{\jmath }-z)^{\varsigma -2}\), \(z\rightarrow 2\mathbf{b}(1-z)^{\varsigma -1}\), \(z\rightarrow 2\mathbf{b}(1-z)^{\varsigma -2}\) are integrable. Hence, applying Lebesgue dominated convergent theorem, we have \(|\mathscr{Z}\psi _{n}(t)-\mathscr{Z}\psi (t)|\rightarrow 0\) as \(t\rightarrow \infty \). This implies \(\|\mathscr{Z}\psi _{n}-\mathscr{Z}\psi \|\rightarrow 0\) as \(t\rightarrow \infty \). Therefore, \(\mathscr{Z}\) is continuous.

Step 2: In this step we need to show that \(\mathscr{Z}\) is bounded. Consequently, for each \(\upsilon \in \mathscr{E}=\{\upsilon \in \mathscr{W} : \|\upsilon \|_{PC} \leq \mathbf{r}^{*}\}\), we have to show that \(\|\mathscr{Z}\upsilon \|_{\mathscr{W}}\leq \eta \), where η is a positive real number. Then, for , we have

(18)

Apply assumption \((H_{6})\) to get

$$\begin{aligned} \bigl\vert \Phi _{\upsilon }(\mathrm{t}) \bigr\vert =& \bigl\vert \mathrm{g}\bigl(t, \upsilon (t), \upsilon (\varsigma t), \Phi _{\upsilon }( \mathrm{t})\bigr) \bigr\vert \\ \leq &\theta _{1}(t)+\theta _{2}(t) \bigl( \vert \upsilon \vert + \bigl\vert \upsilon ( \varsigma t) \bigr\vert \bigr)+\theta _{3}(t) \bigl\vert \Phi _{\upsilon }(\mathrm{t}) \bigr\vert . \end{aligned}$$

Taking maximum of both sides, we have

$$ \bigl\vert \Phi _{\upsilon }(\mathrm{t}) \bigr\vert \leq \theta _{1}^{*}+2\theta _{2}^{*} \mathbf{r}^{*}+\theta _{3}^{*} \bigl\vert \Phi _{\upsilon }(\mathrm{t}) \bigr\vert , $$

which further implies

$$ \max_{t\in I} \bigl\vert \Phi _{\upsilon }(t) \bigr\vert \leq \frac{\theta _{1}^{*}+2\theta _{2}^{*} \mathbf{r}^{*}}{1-\theta _{3}^{*}}:= \zeta , $$
(19)

where \(\theta _{1}^{*}=\max_{t\in I}|\theta _{1}(t)|\), \(\theta _{2}^{*}=\max_{t\in I}|\theta _{2}(t)|\), and \(\theta _{3}^{*}=\max_{t\in I}|\theta _{3}(t)|<1\).

Hence from (18) we get

$$\begin{aligned} \Vert \mathscr{Z}\upsilon \Vert \leq &\zeta \biggl( \frac{(\aleph +1)(2p+q)}{p\Gamma (\varsigma +1)}+ \frac{2\aleph p^{2}+pq(2\aleph +1)+q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr) \\ &{} +\frac{1}{p} \biggl(2+\frac{q}{p} \biggr)C_{5}+ \frac{1}{p} \biggl(1+ \frac{q}{p} \biggr)C_{6}+ \aleph (1+p+q ) \bigl(B\mathbf{r}^{*}+B^{*}\bigr) \\ &{}+ \aleph \biggl(1+q+\frac{q}{p}+\frac{q^{2}}{p^{2}} \biggr) \bigl(M \mathbf{r}^{*}+M^{*}\bigr):= \eta . \end{aligned}$$

This shows that \(\mathscr{Z}\) is bounded.

Step 3: \(\mathscr{Z}\) maps bounded sets into equicontinuous sets of \(\mathscr{W}\). Let such that \(\tau _{1}<\tau _{2}\). And let \(\mathscr{E}\) be a bounded set of \(\mathscr{W}\) as in Step 2, let \(\upsilon \in \mathscr{E}\). Then

Taking into account the assumptions, we obtain

(20)

We see as \(\tau _{1}\) tends to \(\tau _{2}\), the right-hand side of (20) tends to 0. Thus by “Arzelà–Ascoli theorem” \(\mathscr{Z}\) is completely continuous.

Step 4: Here the set defined by \(\mathscr{E}_{\sigma }=\{\upsilon \in \mathscr{W} :\upsilon = \sigma \mathscr{Z}\upsilon \text{ for } 0<\sigma <1\}\) is bounded. Let \(\upsilon \in \mathscr{E}_{\varsigma }\). Then by definition \(\upsilon = \varsigma \mathscr{Z}\upsilon \). From Step 2, we get

$$\begin{aligned} \Vert \mathscr{Z}\upsilon \Vert \leq& \sigma \biggl[\zeta \biggl( \frac{(\aleph +1)(2p+q)}{p\Gamma (\varsigma +1)} + \frac{2\aleph p^{2}+pq(2\aleph +1)+q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr) \\ &{} +\frac{1}{p} \biggl(2+\frac{q}{p} \biggr)C_{5}+ \frac{1}{p} \biggl(1+ \frac{q}{p} \biggr)C_{6}+ \aleph (1+p+q ) \bigl(B\mathbf{r}^{*}+B^{*}\bigr) \\ &{}+ \aleph \biggl(1+q+\frac{q}{p}+\frac{q^{2}}{p^{2}} \biggr) \bigl(M \mathbf{r}^{*}+M^{*}\bigr) \biggr]=\sigma \eta \leq \eta, \end{aligned}$$

which shows that \(\mathscr{E}_{\sigma }\) is bounded. Therefore, by Schaefer’s fixed point theorem, \(\mathscr{Z}\) has at least one fixed point and hence problem (1) has at least one solution. □

Theorem 3.3

If assumptions \((H_{1})\)–\((H_{4})\) and the inequality

$$ \begin{aligned} K={}&\frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{(\aleph +1)(2p+q)}{p\Gamma (\varsigma +1)}+ \frac{2\aleph p^{2}+pq(2\aleph +1) +q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr) \\ &{} +\aleph C_{1}(p+q+1)+\aleph C_{2} \biggl(\frac{p^{2}(p+q+1)+q(p+q)}{p^{2}} \biggr) \\ &{}+C_{3} \biggl(\frac{2p+q}{p^{2}} \biggr)+C_{4} \biggl( \frac{p+q}{p^{2}} \biggr)< 1 \end{aligned} $$
(21)

are satisfied, then the problem has a unique solution in the interval \([0, 1]\).

Proof

Assume that \((H_{1})\)–\((H_{3})\) and inequality (21) are satisfied. Then, for \(\upsilon , \bar{\upsilon }\in \mathscr{W}\), we consider

(22)

where

$$\begin{aligned}& \Phi _{\upsilon }(t) = \mathrm{g}\bigl(t, \upsilon (t), \upsilon (mt), \Phi _{\upsilon }(t)\bigr), \\& \Phi _{\bar{\upsilon }}(t) = \mathrm{g}\bigl(t, \bar{\upsilon }(t), \bar{\upsilon }(mt), \Phi _{\bar{\upsilon }}(t)\bigr). \end{aligned}$$

By the application of assumption \((H_{2})\), we have

$$\begin{aligned} \bigl\vert \Phi _{\upsilon }(t)-\Phi _{\bar{\upsilon }}(t) \bigr\vert =& \bigl\vert \mathrm{g}\bigl(t, \upsilon (t), \upsilon (mt), \Phi _{\upsilon }(t)\bigr)-\mathrm{g}\bigl(t, \bar{\upsilon }(t), \bar{\upsilon }(mt), \Phi _{\bar{\upsilon }}(t)\bigr) \bigr\vert \\ \leq &L_{\mathrm{g}}\bigl( \bigl\vert \upsilon (t)-\bar{\upsilon }(t) \bigr\vert + \bigl\vert \upsilon (mt)- \bar{\upsilon }(mt) \bigr\vert \bigr)+N_{\mathrm{g}}\bigl|\Phi _{\upsilon }(t)-\Phi _{ \bar{\upsilon }}(t)\bigr|. \end{aligned}$$

Then

$$ \Vert \Phi _{\upsilon }-\Phi _{\bar{\upsilon }} \Vert _{PC} \leq \frac{2L_{\mathrm{g}}}{1-N_{\mathrm{g}}} \Vert \upsilon -\bar{\upsilon } \Vert _{PC}. $$

Thus from (22) we get the following result in its simplified form:

$$\begin{aligned} \Vert \mathscr{Z}\upsilon - \mathscr{Z}\bar{\upsilon } \Vert _{PC} \leq& \biggl[ \frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{(\aleph +1)(2p+q)}{p\Gamma (\varsigma +1)}+ \frac{2\aleph p^{2}+pq(2\aleph +1)+q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr) \\ &{} +\aleph C_{1}(p+q+1)+\aleph C_{2} \biggl( \frac{p^{2}(p+q+1)+q(p+q)}{p^{2}} \biggr) \\ &{}+C_{3} \biggl(\frac{2p+q}{p^{2}} \biggr)+C_{4} \biggl(\frac{p+q}{p^{2}} \biggr) \biggr] \Vert \upsilon - \bar{\upsilon } \Vert _{PC}, \end{aligned}$$

where

$$\begin{aligned} K =&\frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{(\aleph +1)(2p+q)}{p\Gamma (\varsigma +1)}+ \frac{2\aleph p^{2}+pq(2\aleph +1) +q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr)+\aleph C_{1}(p+q+1) \\ &{} +\aleph C_{2} \biggl(\frac{p^{2}(p+q+1)+q(p+q)}{p^{2}} \biggr)+C_{3} \biggl(\frac{2p+q}{p^{2}} \biggr)+C_{4} \biggl( \frac{p+q}{p^{2}} \biggr)< 1. \end{aligned}$$

Therefore, by the Banach contraction principle, operator \(\mathscr{Z}\) has a unique fixed point. □

4 Stability analysis of problem (1)

Here we derive results about “Hyers–Ulam and Hyers–Ulam–Rassias” stability for problem (1).

Theorem 4.1

If assumptions \((H_{1})\)–\((H_{4})\) and inequality (21) are satisfied, then problem (1) is Hyers–Ulam stable.

Proof

Let \(\psi \in \mathscr{W}\) be any solution of inequality (5) and υ be a unique solution of (1). Then, using Remark 1, for \(t\in [0,1]\), (), we have

(23)

By Corollary 3.1, the solution of (23) is given by

(24)

where

Then, for , we have

(25)

Using assumptions \((H_{1})\)–\((H_{4})\) and (i) of Remark 1, we get the result

$$\begin{aligned} \Vert \psi -\upsilon \Vert _{PC} \leq& \biggl[ \frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{(\aleph +1)(2p+q)}{p\Gamma (\varsigma +1)}+ \frac{2\aleph p^{2}+pq(2\aleph +1)+q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr) \\ &{} +\aleph C_{1}(p+q+1)+\aleph C_{2} \biggl( \frac{p^{2}(p+q+1)+q(p+q)}{p^{2}} \biggr) \\ &{}+C_{3} \biggl(\frac{2p+q}{p^{2}} \biggr)+C_{4} \biggl(\frac{p+q}{p^{2}} \biggr) \biggr] \Vert \psi -\upsilon \Vert _{PC} \\ &{} +\frac{1}{\Gamma (\varsigma +1)} \biggl[ \frac{\aleph +p+(\aleph +1)(p+q)}{p} \biggr]\epsilon \\ &{}+ \frac{1}{\Gamma (\varsigma )} \biggl[ \frac{2\aleph p(p+q)+pq+q^{2}(\aleph +1)}{p^{2}} \biggr]\epsilon+ \aleph \biggl[\frac{p(p+1)(2p+q)+q^{2}}{p^{2}} \biggr]\epsilon , \end{aligned}$$

where

$$\begin{aligned} K =&\frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{(\aleph +1)(2p+q)}{p\Gamma (\varsigma +1)}+ \frac{2\aleph p^{2}+pq(2\aleph +1)+q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr)+\aleph C_{1}(p+q+1) \\ &{} +\aleph C_{2} \biggl(\frac{p^{2}(p+q+1)+q(p+q)}{p^{2}} \biggr)+C_{3} \biggl(\frac{2p+q}{p^{2}} \biggr)+C_{4} \biggl( \frac{p+q}{p^{2}} \biggr)< 1. \end{aligned}$$

This implies that

$$ \Vert \psi -\upsilon \Vert _{PC}\leq C_{g}\epsilon . $$

We see that

$$ C_{g}= \biggl[ \frac{\frac{1}{\Gamma (\varsigma +1)} (\frac{\aleph +p+(\aleph +1)(p+q)}{p} ) +\frac{1}{\Gamma (\varsigma )} (\frac{2\aleph p(p+q)+pq+q^{2}(\aleph +1)}{p^{2}} ) +\aleph (\frac{p(p+1)(2p+q)+q^{2}}{p^{2}} )}{1-K} \biggr]>0. $$

Therefore, problem (1) is Hyers–Ulam stable. □

Corollary 4.2

In Theorem 4.1, if we set \(x(\epsilon )=C_{g}(\epsilon )\) such that \(x(0)=0\), then problem (1) becomes generalized Hyers–Ulam stable.

To prove the next result, we further need another assumption as follows:

\((H_{8})\):

Let, for a nondecreasing function \(x \in {C(I, \mathbb {R})}\), there exist constants \(\mu _{x}>0\), \(\epsilon >0\) such that

$$ {{}_{0}I_{t}}^{\varsigma }x(t)\leq \mu _{x}x(t). $$

Theorem 4.3

If assumptions \((H_{1})\)–\((H_{4})\), \((H_{8})\) and inequality (21) are satisfied, then problem (1) is Hyers–Ulam–Rassias stable with respect to \((\xi , x)\).

Proof

Let \(\psi \in \mathscr{W}\) be any solution of inequality (6) and υ be a unique solution of problem (1). Then, from the above proof of Theorem 4.1, we obtain the following result for :

(26)

Using assumptions \((H_{1})\)–\((H_{4})\), \((H_{8})\), and Remark 3, we get the result

$$\begin{aligned} \Vert \psi -\upsilon \Vert _{PC} \leq &\biggl[ \frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{(\aleph +1)(2p+q)}{p\Gamma (\varsigma +1)}+ \frac{2\aleph p^{2}+pq(2\aleph +1)+q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr) \\ &{} +\aleph C_{1}(p+q+1)+\aleph C_{2} \biggl( \frac{p^{2}(p+q+1)+q(p+q)}{p^{2}} \biggr) \\ &{}+C_{3} \biggl(\frac{2p+q}{p^{2}} \biggr)+C_{4} \biggl(\frac{p+q}{p^{2}} \biggr) \biggr] \Vert \psi -\upsilon \Vert _{PC} \\ &{}+\epsilon \mu _{x}x(t) \biggl(2(2\aleph +1)+\frac{q}{p}(3\aleph +2)+ \frac{q^{2}}{p^{2}}( \aleph +1) \biggr) \\ &{}+\aleph \epsilon \xi \biggl(2(p+q+1)+ \frac{q}{p} \biggl(\frac{q}{p}+1\biggr) \biggr) \\ \leq& \biggl[\frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{(\aleph +1) (2p+q)}{p\Gamma (\varsigma +1)}+ \frac{2\aleph p^{2}+pq(2\aleph +1)+q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr) \\ &{} +\aleph C_{1}(p+q+1)+\aleph C_{2} \biggl( \frac{p^{2}(p+q+1)+q(p+q)}{p^{2}} \biggr) \\ &{}+C_{3} \biggl(\frac{2p+q}{p^{2}} \biggr)+C_{4} \biggl(\frac{p+q}{p^{2}} \biggr) \biggr] \Vert \psi -\upsilon \Vert _{PC} \\ &{} +\bigl(x(t)+\xi \bigr) \biggl[\mu _{x} \biggl(2(2\aleph +1)+ \frac{q}{p}(3\aleph +2)+ \frac{q^{2}}{p^{2}}(\aleph +1) \biggr) \\ &{}+ \aleph \biggl(2(p+q+1)+ \frac{q}{p}\biggl(\frac{q}{p}+1\biggr) \biggr) \biggr]\epsilon . \end{aligned}$$
(27)

Simplifying further, we have

$$\begin{aligned}& \Vert \psi -\upsilon \Vert _{PC} \\& \quad \leq \frac{(x(t)+\xi ) [\mu _{x} (2(2\aleph +1)+\frac{q}{p}(3\aleph +2)+\frac{q^{2}}{p^{2}}(\aleph +1) )+ \aleph (2(p+q+1)+\frac{q}{p}(\frac{q}{p}+1) ) ]\epsilon }{1-K}, \end{aligned}$$

where

$$\begin{aligned} K =&\frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{(\aleph +1)(2p+q)}{p\Gamma (\varsigma +1)}+ \frac{2\aleph p^{2}+pq(2\aleph +1)+q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr)+\aleph C_{1}(p+q+1) \\ &{} +\aleph C_{2} \biggl(\frac{p^{2}(p+q+1)+q(p+q)}{p^{2}} \biggr)+C_{3} \biggl(\frac{2p+q}{p^{2}} \biggr)+C_{4} \biggl( \frac{p+q}{p^{2}} \biggr)< 1. \end{aligned}$$

Therefore, problem (1) is Hyers–Ulam–Rassias stable. □

5 Applications

Example 1

(28)

where e is an exponential function, \(p=q=1\).

Here

$$ \mathrm{g}\bigl(t, \upsilon (t), \upsilon (mt), {}_{0}^{C}D_{t}^{\varsigma }{ \upsilon (t)}\bigr) = \frac{e^{-\pi t}}{15}+\frac{e^{-t}}{38+t^{2}} \biggl( \sin \bigl( \bigl\vert \upsilon (t) \bigr\vert \bigr)+ \upsilon \biggl( \frac{1}{4}t \biggr)+\sin \bigl( \bigl\vert {{}_{0}^{C}D_{t}}^{ \frac{3}{2}} \upsilon (t) \bigr\vert \bigr) \biggr)$$

with \(\varsigma =\frac{3}{2}\), \(m=\frac{1}{4}\). The continuity of g is obvious.

By hypothesis \((H_{2})\), for any \(\upsilon , \bar{\upsilon } \in \mathbb {R}\), we have

$$\begin{aligned}& \bigl\vert \mathrm{g}\bigl(t, \upsilon (t), \upsilon (mt), {}_{0}^{C}D_{t}^{\varsigma }{ \upsilon (t)} \bigr)-\mathrm{g}\bigl(t,\bar{\upsilon }(t), \bar{\upsilon }(mt),{{}_{0}^{C}D_{t}^{\varsigma }} \bar{\upsilon }(t)\bigr) \bigr\vert \\& \quad \leq \frac{1}{38} \bigl[2 \bigl\vert \upsilon (t)- \bar{\upsilon }(t) \bigr\vert + \bigl\vert {{}_{0}^{C}D_{t}}^{\frac{3}{2}} \upsilon (t)-{{}_{0}^{C}D_{t}}^{ \frac{3}{2}} \bar{\upsilon }(t) \bigr\vert \bigr]. \end{aligned}$$

Hence g satisfies hypothesis \((H_{2})\) with \(L_{\mathrm{g}}=N_{\mathrm{g}}=\frac{1}{38}\). Also hypothesis \((H_{4})\) holds with \(\theta _{0}(t)=\frac{e(-\pi t)}{15}\), \(\theta _{1}(t)=\theta _{2}(t)=\frac{e(-t)}{38+t}\), where \(\theta _{0}^{*}(t)=\frac{1}{15}\), \(\theta _{1}^{*}(t)=\theta _{2}^{*}(t)=\frac{1}{38}\).

At \(t_{1}=\frac{1}{3}\) the impulsive conditions are given as follows:

$$\begin{aligned}& \mathcal{F}_{1}\upsilon \biggl(\frac{1}{3} \biggr) = \frac{ \vert \upsilon (\frac{1}{3}) \vert }{60+ \vert \upsilon (\frac{1}{3}) \vert }, \\& \bar{\mathcal{F}}_{1}\upsilon '\biggl( \frac{1}{3}\biggr) = \frac{ \vert \upsilon (\frac{1}{3}) \vert }{80+ \vert \upsilon (\frac{1}{3}) \vert }. \end{aligned}$$

For any \(\upsilon , \bar{\upsilon }\in \mathrm{E}\), we have

$$\begin{aligned}& \biggl\vert \mathcal{F}_{1}\biggl(\upsilon \biggl( \frac{1}{3}\biggr)\biggr)-\mathcal{F}_{1}\biggl( \bar{ \upsilon }\biggl(\frac{1}{3}\biggr)\biggr) \biggr\vert = \biggl\vert \frac{ \vert \upsilon (\frac{1}{3}) \vert }{60+ \vert \upsilon (\frac{1}{3}) \vert }- \frac{ \vert \upsilon (\frac{1}{3}) \vert }{60+ \vert \bar{\upsilon }(\frac{1}{3}) \vert } \biggr\vert \leq \frac{1}{60} \biggl\vert \upsilon \biggl(\frac{1}{3}\biggr)- \bar{\upsilon }\biggl( \frac{1}{3}\biggr) \biggr\vert ,\\& \biggl\vert \bar{\mathcal{F}}_{1}\biggl(\upsilon \biggl( \frac{1}{3}\biggr)\biggr)-\bar{\mathcal{F}}_{1}\biggl( \bar{ \upsilon }\biggl(\frac{1}{3}\biggr)\biggr) \biggr\vert = \biggl\vert \frac{ \vert \upsilon (\frac{1}{3}) \vert }{80+ \vert \upsilon (\frac{1}{3}) \vert }- \frac{ \vert \upsilon (\frac{1}{3}) \vert }{80+ \vert \bar{\upsilon }(\frac{1}{3}) \vert } \biggr\vert \leq \frac{1}{80} \biggl\vert \upsilon \biggl(\frac{1}{3}\biggr)- \bar{\upsilon }\biggl( \frac{1}{3}\biggr) \biggr\vert , \end{aligned}$$

which satisfy \((H_{3})\) with real constants \(C_{1}=\frac{1}{60}\), \(C_{2}=\frac{1}{80}\). And

$$\begin{aligned}& \bigl\Vert h_{1}(\upsilon )-h_{1}(\bar{\upsilon }) \bigr\Vert \leq \frac{1}{16} \Vert \upsilon -\bar{\upsilon } \Vert , \\& \bigl\Vert h_{2}(\upsilon )-h_{2}(\bar{\upsilon }) \bigr\Vert \leq \frac{1}{20} \Vert \upsilon -\bar{\upsilon } \Vert \end{aligned}$$

satisfy \((H_{4})\) with real constants \(C_{3}=\frac{1}{16}\), \(C_{4}=\frac{1}{20}\). So we have

$$\begin{aligned} K =&\frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{(\aleph +1)(2p+q)}{p\Gamma (\varsigma +1)}+ \frac{2\aleph p^{2}+pq(2\aleph +1)+q^{2}(\aleph +1)}{p^{2}\Gamma (\varsigma )} \biggr)+\aleph C_{1}(p+q+1) \\ &{} +\aleph C_{2} \biggl(\frac{p^{2}(p+q+1)+q(p+q)}{p^{2}} \biggr)+C_{3} \biggl(\frac{2p+q}{p^{2}} \biggr)+C_{4} \biggl( \frac{p+q}{p^{2}} \biggr)=0.78< 1. \end{aligned}$$

Therefore, by Theorem 3.3, problem (28) has a unique solution. And by result Theorem 4.1, problem (28) is Hyers–Ulam stable. Similarly, by setting \(x(t)=t\), taking \(\xi =1\), and applying the obtained result Theorem 4.3, it is obvious that, for any \(t\in [0,1]\), the numerical problem (28) is Hyers–Ulam–Rassias stable with respect to \((\xi ,x)\).

6 Conclusion

By using classical fixed point results, we have established some useful results about the existence and stability of Ulam type for an impulsive problem of FODEs under integral boundary conditions. The concerned results have been testified by an example. Hence fixed point approach is a powerful technique to investigate various nonlinear problems of impulsive FODEs which have many applications in dynamics and fluid mechanics.