Keywords

AMS Subject Classification

1 Introduction

There are many physical models which are subject to sudden changes in its states, such rapid changes are known as impulsive response. In the current hypothesis, there are two types of impulsive system, one is instantaneous and another one is known as non-instantaneous impulsive system. In the instantaneous impulsive system, the duration of these abrupt changes is very little correlation to the duration of the whole process, for example pulses, stuns and cataclysmic events [7, 16], while in the non-instantaneous impulses, the duration of these changes continues over a finite time interval. For the initial studies related with the existence, uniqueness, and controllability of non-instantaneous impulsive systems of integer and fractional order, we refer to [10, 15, 18, 21] and the references cited therein. Further, stability analysis of dynamical systems becomes an important research area and various form of stabilities have been developed including Lyapunov stability, Mittag-Leffler function and exponential for dynamical equations. Moreover, an interesting type of stability was introduced by Ulam and Hyers is known as Ulam-Hyers stability which is highly useful in numerical analysis and optimization for dynamical equations. The Ulam-Hyper’s stability for many dynamical equations of integer and fractional order has been studied in lots of articles [4, 5, 25, 26].

In 1988, Hilger presented the time scales calculus. The investigation of analytics on time scales incorporates the continuous and discrete analysis, therefore the investigation of dynamical system on time scales has picked up an awesome consideration and numerous scientists have discovered the uses of time scales in heat transfer system [19], population dynamics [28] and economics [11, 12]. For more details about time scales one can refer the book [8, 9] and papers [2, 3, 17]. Further over the most recent couple of years, many authors talked about the existence, uniqueness and stability of dynamical system on time scales [1, 6, 13, 14, 20, 22,23,24, 27]. Particularly, Geng [13], presented the concepts of lower and upper solutions for a PBVP on time scales.

According as far as anyone is concerned, there is no manuscript which examined the existence, uniqueness and stability investigation of integro differential equations with non-instantaneous impulses on time scales. Spurred by the above actualities, we take the differential equations with periodic boundary condition and non-instantaneous impulses on time scale of the form:

$$\begin{aligned} v^\Delta (\theta )&=\mathcal {C} \left( \theta ,v(\theta ),\int \limits _0^\theta h(\theta ,\tau ,v(\tau ))\Delta \tau \right) , \quad \theta \in \cup _{k=0}^l (\lambda _k,\theta _{k+1}]_ \mathbb {T}, \nonumber \\ v(\theta )&=\dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1} g_k(\zeta ,v(\theta _k^-)) \Delta \zeta , \quad \theta \in (\theta _k,\lambda _k]_ \mathbb {T},\ k=1,2,\ldots ,l, \ q \in (0,1) \\ v(0)&=v(T) \nonumber \end{aligned}$$
(1.1)

where \(\mathbb {T}\) is a time scale with \(\theta _k,\lambda _k \in \mathbb {T}\) are right dense points with \(0=\lambda _0=\theta _0<\theta _1<\lambda _1<\theta _2< \cdots \lambda _l<\theta _{l+1}=T\), \(v(\theta _k^-)=\lim _{h\rightarrow 0^+}v(\theta _k-h), v(\theta _k^+)=\lim _{h\rightarrow 0^+}v(\theta _k+h)\), represent the left and right limits of \(v(\theta )\) at \(\theta =\theta _k\). The functions \(g_k(\theta ,v(\theta _k^-)) \in C(I,\mathbb {R})\) represent non-instantaneous impulses during the intervals \((\theta _k, \lambda _k]_{\mathbb {T}}, \ k=1,2,\ldots ,l\), so impulses at \(\theta _k\) have some duration, namely on intervals \((\theta _k, \lambda _k]_{\mathbb {T}}\). \(\mathcal {C} : I=[0,T]_\mathbb {T} \times \mathbb {R} \rightarrow \mathbb {R}\) and \(h:\mathcal {Q} \times \mathbb {R} \rightarrow \mathbb {R}\) are given functions, where \(\mathcal {Q}=\{ (\theta ,\tau ) \in I \times I : 0 \le \tau \le \theta \le T \}\).

Throughout the manuscript, we impose

$$\mathcal {M}(v(\theta ))=\int \limits _0^\theta h(\theta ,\tau ,v(\tau ))\Delta \tau .$$

The structure of the manuscript is as: In second section, we give preliminaries, fundamental definitions, useful lemmas and some important results. In the subsequent sections, the main results of the manuscript are discussed. Finally, an example is given to outline the utilization of these outcomes.

2 Preliminaries

Below, we give basic notations, fundamental definitions and useful lemmas. Let \((X,\Vert .\Vert )\) be a Banach space. \(C(I,\mathbb {R})\) be the set of all continuous functions. In order to define the solution of the Eq. (1.1), we define the space \(PC(I,\mathbb {R})\) of piecewise continuous functions defined as \( PC(I,\mathbb {R})=\{v:I\rightarrow \mathbb {R}:v\in C(\theta _k,\theta _{k+1}]_{\mathbb {T}},\mathbb {R}),k=0,1,\ldots ,l\) and there exists \( v(\theta _k^-)\) and \( v(\theta _k^+), k=1,2,\ldots ,l\) with \(v(\theta _k^-)=v(\theta _k) \}\). It can be seen easily that \(PC(I,\mathbb {R})\) is a Banach space with the TZ-norm

$$\Vert v\Vert _{\Omega }=\sup _{\theta \in [a,b]} \dfrac{\Vert v(\theta )\Vert }{e_{\Omega }(\theta ,a)}, \text {for some}~\Omega \in \mathcal {R}^+.$$

A closed non-empty subset of real number is called time scales \(\mathbb {T}\). A time scale interval is defined as \([i,m ]_\mathbb {T}=\{\theta \in \mathbb {T}:i\le \theta \le m\}\), accordingly, we define \((i,m)_{\mathbb {T}},[i,m)_{\mathbb {T}}\) and so on. Now onwards, we used a time scale interval [i, m] instead of \([i,m]_{\mathbb {T}}\). Also, now onward if \(\max \mathbb {T}\) exists, then we take \(\mathbb {T}^k= \mathbb {T} \backslash \{ \max \mathbb {T}\}\), otherwise \(\mathbb {T}^k=\mathbb {T}\). The forward jump operator \(\sigma : \mathbb {T}^k \rightarrow \mathbb {T}\) is defined by \(\sigma (\theta ) := \inf \{{r\in \mathbb {T} : r>\theta }\}\) with the substitution \( \inf \{\phi \}=\sup \mathbb {T}\) and the graininess function \(\mu : \mathbb {T}^k \rightarrow [0,\infty )\) is define as \(\mu (\theta ):=\sigma (\theta )-\theta , \forall \theta \in \mathbb {T}^k\).

Definition 2.1

Let \(z :\mathbb {T}\rightarrow \mathbb {R}\) and \( \theta \in \mathbb {T}^k\). The delta derivative \(z^\Delta (\theta )\) is the number (when it exists) such that given any \(\epsilon >0\), there is a neighbourhood U of \(\theta \) such that

$$ |[z(\sigma (\theta ))-\ z (\tau )]- z^\Delta (\theta )[\sigma (\theta )-\tau ] |\le \epsilon |\sigma (\theta )-\tau |, \quad \forall \ \tau \in U.$$

Definition 2.2

Function Z is said to be antiderivative of \(z : \mathbb {T}\rightarrow \mathbb {R}\) provided \(Z^\Delta (\theta ) = z (\theta )\) for each \(\theta \in \mathbb {T}^k\), then the delta integral is defined by

$$\int \limits _{\theta _0}^{\theta } z(\zeta )\Delta \zeta =Z(\theta )-Z(\theta _0).$$

A function \( z :\mathbb {T} \rightarrow \mathbb {R}\) is called rd-continuous on \(\mathbb {T}\), if z has finite left-sided limits at points \(\theta \in \mathbb {T}\) with \(\sup \{ r\in \mathbb {T}:r<\theta \}=\theta \) and z is continuous at points \(\theta \in \mathbb {T}\) with \(\sigma (\theta )=\theta \). The collection of all rd-continuous functions \(z :\mathbb {T} \rightarrow \mathbb {R}\) will be denoted by \(C_{rd}(\mathbb {T},\mathbb {R})\).

Definition 2.3

A function \({p} : \mathbb {T} \rightarrow \mathbb {R}\) is said to be regressive (positive regressive) if \(1+\mu (\theta )p(\theta )\ne 0( >0)\), \(\forall \theta \in \mathbb {T}\) and the set of all regressive (positive regressive) functions are denoted by \(\mathcal {R}( \mathcal {R^+})\).

Definition 2.4

The generalized exponential function is defined as

$$e_p(\theta ,r)=\exp \left( \int \limits _r^\theta \xi _{\mu (\zeta )}(p(\zeta ))\Delta \zeta \right) , \quad \theta , r \in \mathbb {T}, \ p \in \mathcal {R},$$

where \(\xi _{\mu (\beta )}(p(\beta ))\) is given by

$$ \xi _{\mu (\beta )}(\varkappa )= {\left\{ \begin{array}{ll} \dfrac{1}{\mu (\beta )}Log(1+\mu (\beta )\varkappa ),&{} \text {if } \mu (\beta ) \ne 0.\\ \varkappa , &{} \text {if } \mu (\beta )=0. \end{array}\right. } $$

Lemma 2.5

([17]) Let \(\theta _1, \theta _2 \in \mathbb {T}\), such that \(\theta _1 \le \theta _2\) and \(z : \mathbb {R} \rightarrow \mathbb {R}\) be a non-decreasing continuous function. Then,

$$\begin{aligned} \int \limits _{\theta _1}^{\theta _2} z(\zeta ) \Delta \zeta \le \int \limits _{\theta _1}^{\theta _2} z(\zeta )d \zeta . \end{aligned}$$
(2.1)

Lemma 2.6

Let \(g : I \rightarrow \mathbb {R}\) be a right dense continuous function. Then, for any \(k=1,2,\ldots ,l\), the solution of the following problem

$$\begin{aligned} v^\Delta (\theta )&=g(\theta ), \quad \theta \in \cup _{k=0}^l (\lambda _k,\theta _{k+1}],\\ v(\theta )&=\dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1} g_k(\zeta ,v(\theta _k^-)) \Delta \zeta , \quad \theta \in (\theta _k,\lambda _k],\ k=1,2,\ldots ,l,\\ v(0)&=v(T), \end{aligned}$$

is given by the following integral equation

$$\begin{aligned} v(\theta )&=\dfrac{1}{\Gamma (q)} \int \limits _{\theta _l}^{\lambda _l} (\lambda _l-\zeta )^{q-1} g_l(\zeta ,v(\theta _l^-)) \Delta \zeta + \int \limits _{\lambda _l}^T g(\zeta ) \Delta \zeta + \int \limits _0^\theta g(\zeta ) \Delta \zeta , \quad \forall \ \theta \in [0,\theta _1], \\ v(\theta )&= \dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1} g_k(\zeta ,v(\theta _k^-)) \Delta \zeta , \quad \forall \ \theta \in (\theta _k,\lambda _k], \ k=1,2,\ldots ,l,\\ v(\theta )&=\dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1} g_k(\zeta ,v(\theta _k^-)) \Delta \zeta + \int \limits _{\lambda _k}^\theta g(\zeta ) \Delta \zeta , \quad \forall \ \theta \in (\lambda _k,\theta _{k+1}], \ k=1,2,\ldots ,l. \end{aligned}$$

 

(H1)::

The non-linear function \(\mathcal {C}:J_1 \times \mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}, \ J_1= \cup _{k=0}^l[\lambda _k,\theta _{k+1}]\) is continuous and \(\exists \) positive constants \(L_{\mathcal {C}_1}, \ L_{\mathcal {C}_2}\) such that

$$\begin{aligned} |\mathcal {C}(\theta ,v_1,v_2)-\mathcal {C}(\theta ,w_1,w_2)|&\le L_{\mathcal {C}_1} |v_1-w_1|+ L_{\mathcal {C}_2} |v_2-w_2|, \\&\qquad \forall \ \theta \in I, \ v_j, w_j \in \mathbb {R}, \ j=1,2. \end{aligned}$$

Also, \(\exists \) positive constants \(C_{\mathcal {C}},\ M_{\mathcal {C}}\) and \(N_{\mathcal {C}}\) such that

$$\begin{aligned} |\mathcal {C}(\theta ,v,w)|\le C_{\mathcal {C}} + M_{\mathcal {C}}|v| + N_{\mathcal {C}} |w|, \quad \forall \ \theta \in I, \ v,w \in \mathbb {R}. \end{aligned}$$
(H2)::

\(h: \mathcal {Q} \times \mathbb {R} \rightarrow \mathbb {R}\) is continuous and \(\exists \) positive constant \(L_h\) such that

$$\begin{aligned} |h(\theta ,\tau ,v)-h(\theta ,\tau ,w)| \le L_{h} |v-w|, \quad \forall \ \theta ,\tau \in \mathcal {Q}, \ v, w\in \mathbb {R}. \end{aligned}$$

Also, \(\exists \) positive constants \(C_h,\ M_h\) such that

$$\begin{aligned} |h(\theta ,\tau ,v)|\le C_h + M_h|v|, \quad \forall \ \theta , \tau \in \mathcal {Q}, \ v \in \mathbb {R}. \end{aligned}$$
(H3)::

The functions \(g_k:I_k \times \mathbb {R} \rightarrow \mathbb {R}, \ I_k= [\theta _k,\lambda _k], \ k=1,2,\ldots ,l\) are continuous and \(\exists \) a positive constant \(L_{g}\) such that

$$|g_k(\theta ,v)-g_k(\theta ,w)|\le L_{g}|v-w|, \quad \forall \ v,w \in \mathbb {R} , \theta \in I_k, \ k=1,2,\ldots ,l.$$

Also, \(\exists \) a positive constant \(M_g\) such that \(|g_k(\theta ,v)|\le M_g, \quad \forall \ \theta \in I_k \ \text {and} \ v \in \mathbb {R}\).

(H4)::

\(\max _{1\le k \le l}\left( e_{\Omega } (T,\lambda _k) \bigg ( \dfrac{M_{\mathcal {C}}}{\Omega } + \dfrac{N_{\mathcal {C}} M_h}{\Omega ^2} \bigg ) \right) <1\).

 

3 Existence and Uniqueness

Theorem 3.1

Let the assumptions (H1)–(H4) are holds, then Eq. (1.1) has a unique solution provided,

$$e_{\Omega }(T,\lambda _l) \bigg (\dfrac{L_{\mathcal {C}_1}}{\Omega } + \dfrac{L_{\mathcal {C}_2} L_h}{\Omega ^2} \bigg )<1.$$

Proof

Consider a subset \(\mathcal {D} \subseteq PC(I,\mathbb {R})\) such that

$$\mathcal {D}=\{v \in PC(I,\mathbb {R}) : \Vert v\Vert _{\Omega } \le \beta \},$$

where

$$\begin{aligned} \beta = \max _{1\le k \le l} \left( \dfrac{ \dfrac{M_g T^q}{\Gamma (q+1)} + C_{\mathcal {C}}( T +\theta _1) + N_{\mathcal {C}} C_h (T^2+\theta _1^2)}{1- (1+e_{\Omega } (T,\lambda _k)) \bigg ( \dfrac{M_{\mathcal {C}}}{\Omega } + \dfrac{N_{\mathcal {C}} M_h }{\Omega ^2} \bigg ) } \right) . \end{aligned}$$

Now, define an operator \(\Pi :\mathcal {D} \rightarrow \mathcal {D}\) given by

$$\begin{aligned} (\Pi v)(\theta )&= \int \limits _0^\theta \mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta ))) \Delta \zeta +\dfrac{1}{\Gamma (q)} \int \limits _{\theta _l}^{\lambda _l} (\lambda _l-\zeta )^{q-1} g_l(\zeta ,v(\theta _l^-)) \Delta \zeta \\&\quad + \, \int \limits _{\lambda _l}^T \mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta ))) \Delta \zeta , \;\; \forall \ \theta \in [0,\theta _1], \nonumber \\ (\Pi v)(\theta )&=\dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1} g_k(\zeta ,v(\theta _k^-)) \Delta \zeta , \quad \forall \ \theta \in (\theta _k,\lambda _k], \ k=1,2,\ldots ,l,\\ (\Pi v)(\theta )&=\dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1} g_k(\zeta ,v(\theta _k^-)) \Delta \zeta + \int \limits _{\lambda _k}^\theta \mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta ))) \Delta \zeta , \\&\qquad \qquad \qquad \qquad \qquad \qquad \forall \ \theta \in (\lambda _k,\theta _{k+1}], \ k=1,2,\ldots ,l. \end{aligned}$$

The proof of this theorem are divided into two steps.

Step 1: To use the Banach contraction theorem, we have to show that \(\Pi : \mathcal {D} \rightarrow \mathcal {D}\). For this, we are taking three cases as follows:

Case 1: For \(\theta \in (\lambda _k,\theta _{k+1}], k=1,2,\ldots ,l\) and \(v \in \mathcal {D}\), we have:

$$\begin{aligned} |(\Pi v)(\theta )|&\le \dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1} |g_k(\zeta ,v(\theta _k^-))| \Delta \zeta + \int \limits _{\lambda _k}^\theta |\mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta )))| \Delta \zeta \\&\le \dfrac{M_g}{\Gamma (q)} \int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1} \Delta \zeta + \int \limits _{\lambda _k}^\theta (C_{\mathcal {C}} + M_{\mathcal {C}} |v(\zeta )| + N_{\mathcal {C}}|\mathcal {M}(v(\zeta ))|)\Delta \zeta \\&\le \dfrac{M_g (\lambda _k-\theta _k)^q}{\Gamma (q+1)} + (C_{\mathcal {C}} + N_{\mathcal {C}} C_h \theta _{k+1})(\theta _{k+1}-\lambda _k) \\&\quad +\, \left( M_{\mathcal {C}} \beta +\dfrac{N_{\mathcal {C}} M_h \beta }{\Omega } \right) \int \limits _{\lambda _k}^\theta e_{\Omega }(\zeta ,\lambda _k) \Delta \zeta \\&\le \dfrac{M_g T^q}{\Gamma (q+1)} + (C_{\mathcal {C}}+N_{\mathcal {C}} C_hT)T + \dfrac{M_{\mathcal {C}} \beta e_{\Omega }(\theta ,\lambda _k)}{\Omega } + \dfrac{N_{\mathcal {C}} M_h \beta e_{\Omega }(\theta ,\lambda _k)}{\Omega ^2}. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert \Pi v\Vert _{\Omega } \le \dfrac{M_g T^q}{\Gamma (q+1)} + (C_{\mathcal {C}}+N_{\mathcal {C}} C_hT)T + \dfrac{M_{\mathcal {C}} \beta }{\Omega } + \dfrac{N_{\mathcal {C}} M_h \beta }{\Omega ^2}. \end{aligned}$$
(3.1)

Case 2: For \(\theta \in [0,\theta _1]\) and \(v \in \mathcal {D} \), we have:

$$\begin{aligned} |(\Pi v)(\theta )|&\le \int \limits _0^\theta |\mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta )))| \Delta \zeta + \dfrac{1}{\Gamma (q)} \int \limits _{\theta _l}^{\lambda _l} (\lambda _l-\zeta )^{q-1} |g_l(\zeta ,v(\theta _l^-))| \Delta \zeta \\&\quad +\, \int \limits _{\lambda _l}^T |\mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta )))| \Delta \zeta \\&\le \dfrac{M_g (\lambda _l-\theta _l)^q}{\Gamma (q+1)}+ C_{\mathcal {C}} (T-\lambda _l) + M_{\mathcal {C}} \beta \int \limits _{\lambda _l}^T e_{\Omega }(\zeta ,\lambda _l) \Delta \zeta + N_{\mathcal {C}} C_h T(T-\lambda _l) \\&\quad + \,\dfrac{N_{\mathcal {C}} M_h \beta }{\Omega } \int \limits _{\lambda _l}^T e_{\Omega }(\zeta ,\lambda _l) \Delta \zeta + C_{\mathcal {C}} \theta _1 + N_{\mathcal {C}} C_h \theta _1^2 \\&\quad + \, \left( M_{\mathcal {C}} \beta + \dfrac{N_{\mathcal {C}} M_h \beta }{\Omega } \right) \int \limits _{0}^\theta e_{\Omega }(\zeta ,0) \Delta \zeta \\&\le \dfrac{M_g T^q}{\Gamma (q+1)} + C_{\mathcal {C}}( T +\theta _1) + \dfrac{M_{\mathcal {C}} \beta e_{\Omega }(T,\lambda _l)}{\Omega } + N_{\mathcal {C}} C_h (T^2+\theta _1^2) \\&\quad +\, \dfrac{N_{\mathcal {C}} M_h \beta e_{\Omega }(T,\lambda _l)}{\Omega ^2} + \dfrac{M_{\mathcal {C}} \beta e_{\Omega }(\theta ,0)}{\Omega } + \dfrac{N_{\mathcal {C}} M_h \beta e_{\Omega }(\theta ,0)}{\Omega ^2} . \end{aligned}$$

Hence,

$$\begin{aligned} \Vert \Pi v\Vert _{\Omega }&\le \dfrac{M_g T^q}{\Gamma (q+1)} + C_{\mathcal {C}}( T +\theta _1) + \dfrac{M_{\mathcal {C}} \beta e_{\Omega }(T,\lambda _l)}{\Omega }+ N_{\mathcal {C}} C_h (T^2+\theta _1^2) \nonumber \\&\quad + \, \dfrac{N_{\mathcal {C}} M_h \beta e_{\Omega }(T,\lambda _l)}{\Omega ^2} + \dfrac{M_{\mathcal {C}} \beta }{\Omega } + \dfrac{N_{\mathcal {C}} M_h \beta }{\Omega ^2}. \end{aligned}$$
(3.2)

Case 3: For \(\theta \in (\theta _k, \lambda _k], \ k=1,2,\ldots ,l\) and \( v \in \mathcal {D}\), we can easily get:

$$\begin{aligned} \Vert \Pi v\Vert _{\Omega }= \dfrac{M_g T^q}{\Gamma (q+1)}. \end{aligned}$$
(3.3)

After summarizing the above inequalities (3.1)–(3.3), we get:

$$\Vert \Pi v\Vert _{\Omega }\le \beta .$$

Therefore, \(\Pi : \mathcal {D} \rightarrow \mathcal {D}\).

Step 2: In this step, we will show that the operator \(\Pi \) is a contracting operator. Here also, we are taking three cases as follows:

Case 1: For any \(v, w \in \mathcal {D}, \ \theta \in (\lambda _k, \theta _{k+1}], k=1,2,\ldots ,l\), we have:

$$\begin{aligned} |(\Pi v)(\theta )-(\Pi w)(\theta )|&\le \dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1} |g_k(\zeta ,v(\theta _k^-))-g_k(\zeta ,w(\theta _k^-))| \Delta \zeta \\&\quad +\, \int \limits _{\lambda _k}^\theta |\mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta )))-\mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(w(\zeta )))| \Delta \zeta \\&\le \dfrac{ L_{g} }{\Gamma (q)} \int \limits _{\theta _k}^{\lambda _k} \dfrac{(\lambda _k-\zeta )^{q-1} |v(\theta _k^-) - w(\theta _k^-)|e_{\Omega }(\theta _k^-,\theta _k)}{e_{\Omega }(\theta _k^-,\theta _k)} \Delta \zeta \\&\quad + \, L_{\mathcal {C}_1} \int \limits _{\lambda _k}^\theta \dfrac{|v(\zeta )-w(\zeta ) | e_{\Omega }(\zeta ,\lambda _k)}{e_{\Omega }(\zeta ,\lambda _k)} \Delta \zeta \\&\quad + \, L_{\mathcal {C}_2} \int \limits _{\lambda _k}^\theta |\mathcal {M}(v(\zeta ))-\mathcal {M}(w(\zeta ))|\Delta \zeta \\&\le \dfrac{\Vert v - w\Vert _{\Omega } L_{g} e_\Omega (\theta _k^-,\theta _k)(\lambda _k-\theta _k)^{q} }{\Gamma (q+1)} \\&\quad +\, L_{\mathcal {C}_1} \Vert v-w \Vert _{\Omega } \int \limits _{\lambda _k}^\theta e_{\Omega }(\zeta ,\lambda _k)\Delta \zeta \\&\quad + \, \dfrac{L_{\mathcal {C}_2} L_h \Vert v-w \Vert _{\Omega }}{\Omega } \int \limits _{\lambda _k}^\theta e_{\Omega }(\zeta ,\lambda _k) \Delta \zeta \\&\le \dfrac{L_{g} e_\Omega (\theta _k^-,\theta _k)(\lambda _k-\theta _k)^{q} \Vert v - w\Vert _{\Omega }}{\Gamma (q+1)} + \dfrac{L_{\mathcal {C}_1}e_{\Omega }(\theta ,\lambda _k) \Vert v-w \Vert _{\Omega }}{\Omega } \\&\quad + \, \dfrac{L_{\mathcal {C}_2} L_he_{\Omega }(\theta ,\lambda _k) \Vert v-w \Vert _{\Omega }}{\Omega ^2} . \end{aligned}$$

Thus, we have:

$$\begin{aligned} \Vert \Pi v-\Pi w\Vert _{\Omega } \le \bigg [\dfrac{L_{g} e_\Omega (\theta _k^-,\theta _k)T^{q} }{\Gamma (q+1)} + \dfrac{L_{\mathcal {C}_1}}{\Omega } + \dfrac{L_{\mathcal {C}_2} L_h}{\Omega ^2} \bigg ] \Vert v - w\Vert _{\Omega }. \end{aligned}$$
(3.4)

Case 2: For any \(v, w \in \mathcal {D}, \theta \in [0,\theta _1]\), we have:

$$\begin{aligned} |(\Pi v)(\theta )-(\Pi w)(\theta )|&\le \int \limits _0^\theta |\mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta )))-\mathcal {C}(\zeta ,w(\zeta ),\mathcal {M}(w(\zeta )))| \Delta \zeta \\&\quad + \, \dfrac{1}{\Gamma (q)} \int \limits _{\theta _l}^{\lambda _l} (\lambda _l-\zeta )^{q-1} |g_l(\zeta ,v(\theta _l^-))-g_l(\zeta ,w(\theta _l^-))| \Delta \zeta \\&\quad + \, \int \limits _{\lambda _l}^T |\mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta )))-\mathcal {C}(\zeta ,w(\zeta ),\mathcal {M}(w(\zeta )))| \Delta \zeta \\&\le \dfrac{ L_{g} }{\Gamma (q)} \int \limits _{\theta _l}^{\lambda _l} \dfrac{(\lambda _l-\zeta )^{q-1} |v(\theta _l^-) - w(\theta _l^-)|e_{\Omega }(\theta _l^-,\theta _l)}{e_{\Omega }(\theta _l^-,\theta _l)}\Delta \zeta \\&\quad +\, L_{\mathcal {C}_1} \int \limits _{\lambda _l}^T \dfrac{ |v(\zeta )-w(\zeta ) | e_{\Omega }(\zeta ,\lambda _l)}{e_{\Omega }(\zeta ,\lambda _l)}\Delta \zeta \\&\quad +\, L_{\mathcal {C}_2} \int \limits _{\lambda _l}^T |\mathcal {M}(v(\zeta ))-\mathcal {M}(w(\zeta ))|\Delta \zeta \\&\quad +\, L_{\mathcal {C}_1} \int \limits _{0}^\theta \dfrac{ |v(\zeta )-w(\zeta ) | e_{\Omega }(\zeta ,0)}{e_{\Omega }(\zeta ,0)}\Delta \zeta \\&\quad +\, L_{\mathcal {C}_2} \int \limits _{0}^\theta |\mathcal {M}(v(\zeta ))-\mathcal {M}(w(\zeta ))|\Delta \zeta \\&\le \dfrac{L_{g} e_\Omega (\theta _l^-,\theta _l)(\lambda _l-\theta _l)^{q} \Vert v - w\Vert _{\Omega }}{\Gamma (q+1)} + L_{\mathcal {C}_1} \Vert v-w \Vert _{\Omega } \int \limits _{\lambda _l}^T e_{\Omega }(\zeta ,\lambda _l)\Delta \zeta \\&\quad + \, \dfrac{L_{\mathcal {C}_2} L_h \Vert v-w \Vert _{\Omega }}{\Omega } \int \limits _{\lambda _l}^T e_{\Omega }(\zeta ,\lambda _l) \Delta \zeta + L_{\mathcal {C}_1} \Vert v-w \Vert _{\Omega } \int \limits _{0}^\theta e_{\Omega }(\zeta ,0)\Delta \zeta \\&\quad + \, \dfrac{L_{\mathcal {C}_2} L_h \Vert v-w \Vert _{\Omega }}{\Omega } \int \limits _{0}^\theta e_{\Omega }(\zeta ,0) \Delta \zeta \\&\le \dfrac{L_{\mathcal {C}_2} L_he_{\Omega }(T,\lambda _l) \Vert v-w \Vert _{\Omega }}{\Omega ^2} + \dfrac{L_{g} e_\Omega (\theta _l^-,\theta _l)(\lambda _l-\theta _l)^{q} \Vert v - w\Vert _{\Omega }}{\Gamma (q+1)} \\&\quad +\, \dfrac{ \Vert v-w \Vert _{\Omega }L_{\mathcal {C}_1}e_{\Omega }(T,\lambda _l)}{\Omega } + \dfrac{L_{\mathcal {C}_1}e_{\Omega }(\theta ,0) \Vert v-w \Vert _{\Omega }}{\Omega } \\&\quad + \, \dfrac{L_{\mathcal {C}_2} L_he_{\Omega }(\theta ,0) \Vert v-w \Vert _{\Omega }}{\Omega ^2}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert \Pi v-\Pi w\Vert _{\Omega } \le \bigg [\dfrac{L_{g} e_\Omega (\theta _l^-,\theta _l)T^{q} }{\Gamma (q+1)} +(1+e_{\Omega }(T,\lambda _l)) \bigg (\dfrac{L_{\mathcal {C}_1}}{\Omega } + \dfrac{L_{\mathcal {C}_2} L_h}{\Omega ^2} \bigg )\bigg ] \Vert v - w\Vert _{\Omega }. \end{aligned}$$
(3.5)

Case 3: Similarly, for \(\theta \in (\theta _k,\lambda _k], \ k=1,2,\ldots ,l\), we get:

$$\begin{aligned} |(\Pi v)(\theta )-(\Pi w)(\theta )|&\le \dfrac{L_{g} e_\Omega (\theta _k^-,\theta _k)T^{q} }{\Gamma (q+1)} \Vert v-w\Vert _{\Omega }. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert \Pi v-\Pi w\Vert _{\Omega } \le \dfrac{L_{g} T^q}{e_{\Omega }(\theta _k,\theta _{k}^-) \Gamma (q+1)} \Vert v - w\Vert _{\Omega }. \end{aligned}$$
(3.6)

After summarizing the inequalities (3.4)–(3.6), we get:

$$ \Vert \Pi v-\Pi w\Vert _{\Omega } \le L_\Pi \Vert v-w\Vert _{\Omega },$$

where

$$\begin{aligned} L_\Pi = \max _{1\le k \le l} \bigg [\dfrac{L_{g} T^{q} e_\Omega (\theta _k^-,\theta _k) }{\Gamma (q+1)} +(1+e_{\Omega }(T,\lambda _l) ) \bigg (\dfrac{L_{\mathcal {C}_1}}{\Omega } + \dfrac{L_{\mathcal {C}_2} L_h}{\Omega ^2} \bigg ) \bigg ]. \end{aligned}$$

Hence, for sufficiently large \(\Omega \), \(\Pi \) is a strict contraction mapping. Therefore, \(\Pi \) has a unique fixed point and that fixed point is the solution of the taken Eq. (1.1). \(\square \)

Let us consider a special case when \(\mathcal {C}\left( \theta ,v(\theta ),\int _0^\theta h(\theta ,\tau ,v(\tau ))\Delta \tau \right) =\mathcal {P}(\theta ,v)+ \int _0^\theta h(\theta ,\tau ,v(\tau ))\Delta \tau \) then (1.1) becomes:

$$\begin{aligned} v^\Delta (\theta )&=\mathcal {P}(\theta ,v)+ \int \limits _0^\theta h(\theta ,\tau ,v(\tau ))\Delta \tau , \quad \theta \in \cup _{k=0}^l (\lambda _k,\theta _{k+1}], \nonumber \\ v(\theta )&=\dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1} g_k(\zeta ,v(\theta _k^-)) \Delta \zeta , \quad \theta \in (\theta _k,\lambda _k],\ k=1,2,\ldots ,l,\\ v(0)&=v(T). \nonumber \end{aligned}$$
(3.7)

 

(H5): :

\(\mathcal {P}: J_1 \times \mathbb {R} \rightarrow \mathbb {R}\) is a non-linear continuous function and \(\exists \) a positive constant \(L_{\mathcal {P}}\) such that

$$\begin{aligned} |\mathcal {P}(\theta ,v)-\mathcal {P}(\theta ,w)| \le L_{\mathcal {P}} |v-w|, \quad \forall \ \theta \in I, \ v, w \in \mathbb {R}. \end{aligned}$$

Also, \(\exists \) positive constants \(C_\mathcal {P}\) and \(M_\mathcal {P}\) such that

$$\begin{aligned} |\mathcal {P}(\theta ,v)|\le C_\mathcal {P} + M_\mathcal {P}|v|, \quad \forall \ \theta \in I, \ v \in \mathbb {R}. \end{aligned}$$
(H6): :

\(\max _{1\le k \le l}\left( e_{\Omega } (T,\lambda _k) \bigg ( \dfrac{M_\mathcal {P}}{\Omega } + \dfrac{M_h }{\Omega ^2} \bigg ) \right) <1\).

 

Corollary 3.2

If the assumptions (H2)–(H3) and (H5)–(H6) are holds, then the Eq. (3.7) has a unique solution, provided

$$e_{\Omega } (T,\lambda _l) \bigg ( \dfrac{L_\mathcal {P}}{\Omega } + \dfrac{L_h }{\Omega ^2} \bigg )<1. $$

4 Hyer-Ulam’s Stability

For \(\epsilon >0, \psi \ge 0\), and nondecreasing \(\varphi \in PC(I,\mathbb {R}^+)\), consider the below inequalities

$$\begin{aligned} {\left\{ \begin{array}{ll} |w^\Delta (\theta )-\mathcal {C}(\theta ,w(\theta ),\mathcal {M}(w(\theta )))| \le \epsilon , \quad \theta \in \cup _{k=0}^l (\lambda _k,\theta _{k+1}].\\ \bigg |w(\theta )-\dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1} g_k(\zeta ,w(\theta _k^-)) \Delta \zeta \bigg | \le \epsilon , \quad \theta \in (\theta _k,\lambda _k],\ k=1,2,\ldots ,l. \end{array}\right. } \end{aligned}$$
(4.1)
$$\begin{aligned} {\left\{ \begin{array}{ll} |w^\Delta (\theta )-\mathcal {C}(\theta ,w(\theta ),\mathcal {M}(w(\theta ))| \le \epsilon \varphi (\theta ), \quad \theta \in \cup _{k=0}^l (\lambda _k,\theta _{k+1}]. \\ \bigg | w(\theta )-\dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1} g_k(\zeta ,w(\theta _k^-)) \Delta \zeta \bigg | \le \epsilon \psi , \quad \theta \in (\theta _k,\lambda _k],\ k=1,2,\ldots ,l. \end{array}\right. } \end{aligned}$$
(4.2)

Definition 4.1

([25]) Equation (1.1) is called Hyer’s-Ulam stable if there exists a positive constant \(H_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g)}\) such that for \(\epsilon >0\) and for each solution w of inequality (4.1), there exist a unique solution v of Eq. (1.1) satisfies the following inequality

$$|w(\theta )-v(\theta )| \le H_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g)} \epsilon , \quad \forall \ \theta \in I. $$

Definition 4.2

([25]) Equation (1.1) is said to be generalized Hyer’s-Ulam stable if there exists \( \mathcal {H}_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g)} \) \( \in C(\mathbb {R}^+, \mathbb {R}^+)\), \(\mathcal {H}_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g)}(0)=0\) such that for each solution w of inequalities (4.1), there exists a unique solution v of Eq. (1.1) satisfies the following inequality

$$|w(\theta )-v(\theta )| \le \mathcal {H}_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g)} (\epsilon ), \quad \forall \ \theta \in I. $$

Remark 4.3

Definition (4.1) \(\implies \) Definition (4.2).

Definition 4.4

([25]) Equation (1.1) is said to be Hyers-Ulam-Rassias stable w.r.t \((\varphi ,\psi )\), if there exists \(H_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g,\varphi )}\) such that for \(\epsilon >0\) and for each solution w of inequality (4.2), there exist a unique solution v of Eq. (1.1) satisfies the following inequality

$$|w(\theta )-v(\theta )| \le H_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g,\varphi )} \epsilon (\varphi (\theta ),\psi ), \quad \forall \ \theta \in I. $$

Remark 4.5

A function \(w \in PC(I,\mathbb {R})\) is a solution of inequality (4.1) if and only if there is \( \mathsf {G}\in PC(I,\mathbb {R})\) and a sequence \(\mathsf {G}_k, \ k=1,2,\ldots ,l\), such that

  1. (a)

    \(|\mathsf {G}(\theta )| \le \epsilon , \forall \ \theta \in \cup _{k=0}^l (\lambda _k,\theta _{k+1}]\) and \(|\mathsf {G}_k| \le \epsilon , \ \forall \ \theta \in (\theta _k,\lambda _k], k=1,2,\ldots ,l\).

  2. (b)

    \(w^\Delta (\theta )=\mathcal {C}(\theta ,w(\theta ),\mathcal {M}(w(\theta )))+ \mathsf {G}(\theta ) , \ \theta \in (\lambda _k,\theta _{k+1}],\ k=0,1,\ldots ,l\).

  3. (c)

    \(w(\theta )=\dfrac{1}{\Gamma (q)}\int _{\theta _k}^\theta (\theta -\zeta )^{q-1}g_k(\zeta ,w(\theta _k^-))\Delta \zeta +\mathsf {G}_k, \theta \in (\theta _k,\lambda _k],\ k=1,2,\ldots ,l\).

Now, by the above Remark 4.5, we have:

$$\begin{aligned} {\left\{ \begin{array}{ll} w^\Delta (\theta )= \mathcal {C}(\theta ,w(\theta ),\mathcal {M}(w(\theta )))+ \mathsf {G}(\theta ), \ \theta \in (\lambda _k,\theta _{k+1}],\ k=0,1,\ldots ,l, \\ w(\theta )=\dfrac{1}{\Gamma (q)}\int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1}g_k(\zeta ,w(\theta _k^-))\Delta \zeta +\mathsf {G}_k, \quad \theta \in (\theta _k,\lambda _k],\ k=1,2,\ldots ,l. \end{array}\right. } \end{aligned}$$

From Lemma 2.6, one can find that the solution w with \(w(0)=w(T) \) of the above equation is given by

$$\begin{aligned} w(\theta )&=\int \limits _0^\theta (\mathcal {C}(\zeta ,w(\zeta ),\mathcal {M}(w(\zeta )))+ \mathsf {G}(\zeta ) ) \Delta \zeta +\dfrac{1}{\Gamma (q)} \int \limits _{\theta _l}^{\lambda _l} (\lambda _l-\zeta )^{q-1} g_l(\zeta ,w(\theta _l^-)) \Delta \zeta + \mathsf {G}_l \\&\quad + \, \int \limits _{\lambda _l}^T (\mathcal {C}(\zeta ,w(\zeta ),\mathcal {M}(w(\zeta )))+ \mathsf {G}(\zeta ) )\Delta \zeta , \quad \forall \ \theta \in [0,\theta _1],\\ w(\theta )&=\dfrac{1}{\Gamma (q)}\int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1}g_k(\zeta ,w(\theta _k^-))\Delta \zeta +\mathsf {G}_k, \quad \forall \ \theta \in (\theta _k,\lambda _k],\ k=1,2,\ldots ,l, \\ w(\theta )&=\dfrac{1}{\Gamma (q)}\int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1}g_k(\zeta ,w(\theta _k^-))\Delta \zeta +\mathsf {G}_k+ \int \limits _{\lambda _k}^\theta (\mathcal {C}(\zeta ,w(\zeta ),\mathcal {M}(w(\zeta )))+ \mathsf {G}(\zeta ) ) \Delta \zeta ,\\&\qquad \qquad \qquad \qquad \forall \ \theta \in (\lambda _k,\theta _{k+1}], \ k=1,2,\ldots ,l. \end{aligned}$$

Therefore, for \(\theta \in (\lambda _k,\theta _{k+1}], \ k=1,2,\ldots ,l\), we have:

$$\begin{aligned}&\bigg |w(\theta )- \dfrac{1}{\Gamma (q)}\int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1}g_k(\zeta ,w(\theta _k^-))\Delta \zeta - \int \limits _{\lambda _k}^\theta \mathcal {C}(\zeta ,w(\zeta ), \mathcal {M}(w(\zeta ))) \Delta \zeta \bigg | \\&\quad \le |\mathsf {G}_k| + \int \limits _{\lambda _k}^\theta |\mathsf {G}(\zeta )| \Delta \zeta \le \epsilon (1+T). \end{aligned}$$

Also, for \(\theta \in [0,\theta _1]\), we have:

$$\begin{aligned}&\bigg |w(\theta ) - \dfrac{1}{\Gamma (q)} \int \limits _{\theta _l}^{\lambda _l} (\lambda _l-\zeta )^{q-1} g_l(\zeta ,w(\theta _l^-)) \Delta \zeta - \int \limits _{\lambda _l}^T \mathcal {C}(\zeta ,w(\zeta ),\mathcal {M}(w(\zeta ))) \Delta \zeta \\&\qquad -\, \int \limits _0^\theta \mathcal {C}(\zeta ,w(\zeta ),\mathcal {M}(w(\zeta ))) \Delta \zeta \bigg | \le |\mathsf {G}_l| + \int \limits _{\lambda _l}^T |\mathsf {G}(\zeta )| \Delta \zeta + \int \limits _0^\theta |\mathsf {G}(\zeta )| \Delta \zeta \\&\quad \le \epsilon (1+2T). \end{aligned}$$

Similarly, for \(\theta \in (\theta _k,\lambda _k], \ k=1,2,\ldots ,l\), we have:

$$\begin{aligned} \bigg |w(\theta )- \dfrac{1}{\Gamma (q)}\int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1}g_k(\zeta ,w(\theta _k^-))\Delta \zeta \bigg | \le \epsilon . \end{aligned}$$

We have similar remark for the inequality (4.2).

Theorem 4.6

If the assumptions of Theorem 3.1 are holds, then the Eq. (1.1) is Hyer-Ulam stable.

Proof

Let \(w \in PC(I,\mathbb {R})\) be the solution of inequality (4.1) and \(v \in PC(I,\mathbb {R})\) be a unique solution of the Eq. (1.1). Therefore, for \(\theta \in (\lambda _k,\theta _{k+1}], k=1,2,\ldots ,l\), we have:

$$\begin{aligned} |w(\theta )-v(\theta )|&\le \bigg | w(\theta )- \int \limits _{\lambda _k}^\theta \mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta ))) \Delta \zeta \bigg | - \dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1} g_k(\zeta ,v(\theta _k^-)) \Delta \zeta \nonumber \\&\le \bigg | w(\theta ) - \dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1} g_k(\zeta ,w(\theta _k^-)) \Delta \zeta \\&\quad - \, \int \limits _{\lambda _k}^\theta \mathcal {C}(\zeta ,w(\zeta ),\mathcal {M}(w(\zeta ))) \Delta \zeta \bigg | \nonumber \\&\quad + \, \bigg |\dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1} (g_k(\zeta ,w(\theta _k^-))-g_k(\zeta ,v(\theta _k^-))) \Delta \zeta \bigg | \nonumber \\&\quad +\, \bigg | \int \limits _{\lambda _k}^\theta (\mathcal {C}(\zeta ,w(\zeta ),\mathcal {M}(w(\zeta )))-\mathcal {C}(\zeta ,w(\zeta ),\mathcal {M}(v(\zeta )))) \Delta \zeta \bigg | \nonumber \\&\le \epsilon (1+T) + \dfrac{ L_{g} }{\Gamma (q)} \int \limits _{\theta _k}^{\lambda _k} (\lambda _k-\zeta )^{q-1}|w(\theta _k^-) - v(\theta _k^-)| \Delta \zeta \nonumber \\&\quad + \, L_{\mathcal {C}_1} \int \limits _{\lambda _k}^\theta |w(\zeta )-v(\zeta ) | \Delta \zeta + L_{\mathcal {C}_2} \int \limits _{\lambda _k}^\theta |\mathcal {M}(w(\zeta ))-\mathcal {M}(v(\zeta ))|\Delta \zeta \\&\le \epsilon (1+T)+\dfrac{L_{g} e_\Omega (\theta _k^-,\theta _k)(\lambda _k-\theta _k)^{q} \Vert v - w\Vert _{\Omega }}{\Gamma (q+1)} \\&\quad + \,\dfrac{L_{\mathcal {C}_1}e_{\Omega }(\theta ,\lambda _k) \Vert v-w \Vert _{\Omega }}{\Omega } + \dfrac{L_{\mathcal {C}_2} L_he_{\Omega }(\theta ,\lambda _k) \Vert v-w \Vert _{\Omega }}{\Omega ^2} . \end{aligned}$$

Hence,

$$\begin{aligned} \Vert w-v\Vert _{\Omega } \le \epsilon (1+T) +\bigg [\dfrac{L_{g} e_\Omega (\theta _k^-,\theta _k)T^{q} }{\Gamma (q+1)} + \dfrac{L_{\mathcal {C}_1}}{\Omega } + \dfrac{L_{\mathcal {C}_2} L_h}{\Omega ^2} \bigg ] \Vert v - w\Vert _{\Omega }. \end{aligned}$$
(4.3)

Also, for \(\theta \in [0,\theta _{1}]\), we have:

$$\begin{aligned} |w(\theta )-v(\theta )|&\le \bigg | w(\theta ) - \int \limits _0^\theta \mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta ))) \Delta \zeta \bigg | \nonumber \\&\quad -\, \dfrac{1}{\Gamma (q)} \int \limits _{\theta _l}^{\lambda _l} (\lambda _l-\zeta )^{q-1} g_l(\zeta ,v(\theta _l^-)) \Delta \zeta -\int \limits _{\lambda _l}^T \mathcal {C}(\zeta ,v(\zeta ),\mathcal {M}(v(\zeta ))) \Delta \zeta \nonumber \\&\le \epsilon (1+2T) +\dfrac{ L_{g} }{\Gamma (q)} \int \limits _{\theta _l}^{\lambda _l} (\lambda _l-\zeta )^{q-1}|v(\theta _l^-) - w(\theta _l^-)| \Delta \zeta \\&\quad +\, L_{\mathcal {C}_1} \int \limits _{\lambda _l}^T |v(\zeta )-w(\zeta ) | \Delta \zeta + L_{\mathcal {C}_2} \int \limits _{\lambda _l}^T |\mathcal {M}(v(\zeta ))-\mathcal {M}(w(\zeta ))|\Delta \zeta \\&\quad +\, L_{\mathcal {C}_1} \int \limits _{0}^\theta |v(\zeta )-w(\zeta ) | \Delta \zeta + L_{\mathcal {C}_2} \int \limits _{0}^\theta |\mathcal {M}(v(\zeta ))-\mathcal {M}(w(\zeta ))|\Delta \zeta \\&\le \epsilon (1+2T) + \dfrac{L_{g} e_\Omega (\theta _l^-,\theta _l)(\lambda _l-\theta _l)^{q} \Vert v - w\Vert _{\Omega }}{\Gamma (q+1)} + \dfrac{L_{\mathcal {C}_1}e_{\Omega }(T,\lambda _l) \Vert v-w \Vert _{\Omega }}{\Omega } \\&\quad +\, \dfrac{L_{\mathcal {C}_2} L_he_{\Omega }(T,\lambda _l) \Vert v-w \Vert _{\Omega }}{\Omega ^2} + \dfrac{L_{\mathcal {C}_1}e_{\Omega }(\theta ,0) \Vert v-w \Vert _{\Omega }}{\Omega } \\&\quad +\, \dfrac{L_{\mathcal {C}_2} L_he_{\Omega }(\theta ,0) \Vert v-w \Vert _{\Omega }}{\Omega ^2}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert w-v\Vert _{\Omega } \le \epsilon (1+2T) + \bigg [\dfrac{L_{g} e_\Omega (\theta _l^-,\theta _l)T^{q} }{\Gamma (q+1)} +(1+e_{\Omega }(T,\lambda _l)) \bigg (\dfrac{L_{\mathcal {C}_1}}{\Omega } + \dfrac{L_{\mathcal {C}_2} L_h}{\Omega ^2} \bigg )\bigg ] \Vert v - w\Vert _{\Omega }. \end{aligned}$$
(4.4)

Similarly, for \(\theta \in (\theta _k,\lambda _k], \ k=1,2,\ldots ,l\), we can easily find that

$$\begin{aligned} |w(\theta )-v(\theta )|&\le \bigg | w(\theta )- \dfrac{1}{\Gamma (q)} \int \limits _{\theta _k}^\theta (\theta -\zeta )^{q-1} g_k(\zeta ,v(\theta _k^-)) \Delta \zeta \bigg | \\&\le \epsilon + \dfrac{L_g (\lambda _k-\theta _k)^{q} e_\Omega (\theta _k^-,\theta _k) \Vert w-v\Vert _{\Omega }}{\Gamma (q+1)} . \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert w-v\Vert _{\Omega } \le \epsilon + \dfrac{L_{g} T^q}{e_{\Omega }(\theta _k,\theta _{k}^-) \Gamma (q+1)} \Vert v - w\Vert _{\Omega }. \end{aligned}$$
(4.5)

After summarizing the above inequalities (4.3)–(4.5), we get:

$$\begin{aligned} \Vert w-v\Vert _{\Omega } \le \epsilon (1+2T)&+ \bigg [\dfrac{L_{g} e_\Omega (\theta _l^-,\theta _l)T^{q} }{\Gamma (q+1)} +(1+e_{\Omega }(T,\lambda _l)) \bigg (\dfrac{L_{\mathcal {C}_1}}{\Omega } + \dfrac{L_{\mathcal {C}_2} L_h}{\Omega ^2} \bigg )\bigg ] \\&\quad \times \, \Vert v - w\Vert _{\Omega } ,\ \forall \ \theta \in I. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert w-v\Vert _\Omega&\le H_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g)} \epsilon , \quad \theta \in I, \end{aligned}$$

where \(H_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g)}=\dfrac{1+2T}{1-L_\Pi }>0\). Thus, the Eq. (1.1) is Ulam-Hyer’s stable. Moreover, if we put \(\mathcal {H}_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g)}(\epsilon )= H_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g)} \epsilon \), \(\mathcal {H}_{(L_{\mathcal {C}_1},L_{\mathcal {C}_2},L_h,L_g)}(0)=0\), then the Eq. (1.1) is generalized Ulam-Hyer’s stable. \(\square \)

 

(H7)::

There exists a \(\delta _\varphi > 0\) such that \(\int _0^\theta \varphi (\zeta ) \Delta \zeta \le \delta _\varphi \varphi (\theta ), \ \forall \ \theta \in I\).

 

The following theorem is the consequence of the Theorem 4.6.

Theorem 4.7

If the conditions of Theorem 3.1 and (H7) are holds, then the Eq. (1.1) is Hyer’s-Ulam-Rassias stable.

5 Example

Consider the following equation with impulses on \(\mathbb {T}, (0,{3}/{5}\), 4/5, \(1 \in \mathbb {T})\)

$$\begin{aligned} v^\Delta (\theta )&=\dfrac{ 5+ |v(\theta )| }{20 e^{\theta +3}(1+ |v(\theta )| )}+ \dfrac{1}{10}\int \limits _0^\theta \dfrac{\theta \tau ^2\sin (v(\tau ))}{ e^{\tau +5}} \Delta \tau , \quad \theta \in I'=[0,1]_\mathbb {T} \setminus (\theta _1,\lambda _1]_{\mathbb {T}}, \nonumber \\ v(\theta )&= \dfrac{1}{\Gamma (q)}\int \limits _{\theta _1}^\theta \dfrac{ (\theta -\zeta )^{q-1} (1+\zeta ^2 \sin ( v(\theta _1^-)))}{15} \Delta \zeta , \quad \theta \in (\theta _1,\lambda _1]_{\mathbb {T}}, \\ v(0)&=v(1) . \nonumber \end{aligned}$$
(5.1)

Set,

$$\mathcal {C}(\theta ,v,w)=\dfrac{ 5+ | v(\theta )| }{20 e^{\theta +3}(1+ |v(\theta )|) } +\dfrac{1}{10} w, \ \theta \in I', \ v,w\in \mathbb {R},$$
$$ h(\theta ,\tau ,v)=\dfrac{\theta \tau ^2\sin (v(\tau ))}{ e^{\tau +5}}, \ \forall \ \theta , \tau \in I', v\in \mathbb {R},$$

and

$$g_1(\theta ,v)=\dfrac{1+\theta ^2 \sin (v(\theta _1^-))}{15}, \ \theta \in (\theta _1,\lambda 1], \ v\in \mathbb {R}.$$

Then, \(\forall \ \theta , \tau \in I=[0,1], \ v,w,x,y \in \mathbb {R}\), we have:

$$\begin{aligned} \ |f(\theta ,v,w) -f(\theta ,x,y)|&\le \dfrac{1}{20 e^3 } |v-x| + \dfrac{1}{10}|w-y|, \\ |f(\theta ,v,w)|&\le \dfrac{ 5 + |v| }{20e^{3}}+\dfrac{1}{10}|w|,\\ |g_1(\theta ,v) -g_1(\theta ,w)|&\le \dfrac{1}{15}|v-w|, \ |h(\theta ,\tau ,v)|\le \dfrac{1}{e^5}+\dfrac{1}{e^5}|v|, \\ |h(\theta ,\tau ,v)-h(\theta ,\tau ,w)|&\le \dfrac{1}{ e^{5}}|v-w|. \end{aligned}$$

Hence, the assumptions (H1)–(H4) are holds with \(L_{\mathcal {C}_1}=\dfrac{1}{20e^3}, \ L_{\mathcal {C}_2}=\dfrac{1}{10}, \ C_{\mathcal {C}}=\dfrac{5}{20e^3}, \ M_{\mathcal {C}}=\dfrac{1}{20e^3}, \ N_{\mathcal {C}}=\dfrac{1}{10}, \ L_h=\dfrac{1}{ e^{5}}, \ C_h=\dfrac{1}{e^{5}}, \ M_h=\dfrac{1}{e^{5}},\ L_{g}=\dfrac{1}{15}, \ M_g =\dfrac{2}{15}\). Also, for \(l=1, \ \theta _1=3/5, \ \lambda _1=4/5,\ T=1,\ \Omega =10\), the condition

$$e_{\Omega }(T,\lambda _1) \bigg (\dfrac{L_{\mathcal {C}_1}}{\Omega } + \dfrac{L_{\mathcal {C}_2} L_h}{\Omega ^2} \bigg )=0.0039 \ ({<}1)$$

holds. Thus, from Theorems 3.1 and 4.6, Eq. (5.1) has a Ulam Hyer’s stable solution which is unique.

6 Conclusion

In this manuscript, we have successfully established the existence of a unique solution for the system (1.1) by using the Banach contraction theorem and nonlinear functional analysis. Also, we established the Ulam-Hyer’s stability of the taken problem (1.1). To illustrate the application of obtained results, we have given an example.