Existence and stability of fractional integro differential equation with non-instantaneous integrable impulses and periodic boundary condition on time scales

1 Introduction

In this paper, we consider the following fractional integro-differential equation with non-instantaneous impulses and periodic boundary condition on time scale:

A lot of certifiable issues are seen with sudden changes in their states, for example, cataclysmic events, stuns, and heartbeats. Such sudden changes are called impulses. Some of the times, these abrupt changes stay over a period of time and that impulses are known as non-instantaneous impulses. For the comprehensive studies of non-instantaneous impulsive systems, one can see (Abbas et al., 2017; Feĉkan and Wang, 2015; Gautam and Dabas, 2016; Hernández and O’Regan, 2013; Kumar et al., 2016; Pandey et al., 2014; Muslim et al., 2018) and the references therein. Further, the theory of fractional calculus is an extended version of the theory of integer order. Since fractional differential equations define the fundamental properties of the system more accurately, therefore fractional calculus plays a significant role in the qualitative theory of differential equations. In addition, the stability analysis is an important feature of the research area of fractional calculus. Moreover, an interesting type of stability was introduced by Hyers and Ulam which is known as Hyers-Ulam stability. The Hyers-Ulam stability for several dynamical equations of the integer as well as the fractional order has been reported in Agarwal et al. (2017) and Wang and Li (2016). However to the best of author’s knowledge, there is no work related to existence and stability analysis of integro fractional differential equations with non-instantaneous impulses on time scales. Motivated by the above facts, in this paper we obtain existence and Ulam type stability results for the Eq. (1).

The paper is organized in the following manner, in Section 2, we give some preliminaries, fundamental definitions, useful lemmas and some important results. In Sections 3 and 4, the main results of the manuscript are discussed. In the last, an example is given to illustrate the implementation of the obtained results.

2 Preliminaries

Below, we briefly described 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 and $\mathit{PC}(I,\mathbb{R})$ be the space of piecewise continuous functions. We define the space of piecewise continuous functions as $\mathit{PC}(I,\mathbb{R})=\{u:I\to \mathbb{R}:u\in C(({\theta }_k,{\theta }_{k+1}],\mathbb{R}),k=0,1,\dots ,p$ and there exists $u({\theta }_k^{-})$ and $u({\theta }_k^{+}),k=1,2,\dots ,p,$ with $u({\theta }_k^{-})=u({\theta }_k)\}$ . Moreover, $\mathit{PC}(I,\mathbb{R})$ forms a Banach space endowed with the norm $\Vert u{\Vert }_0={\sup }_{\theta \in [a,b]}|u(\theta )|$ . Further, we define ${\mathit{PC}}^1(I,\mathbb{R})=\{u\in \mathit{PC}(I,\mathbb{R}):u^{\mathrm{\Delta }}\in \mathit{PC}(I,\mathbb{R})\}$ . ${\mathit{PC}}^1(I,\mathbb{R})$ form a Banach space endowed with the norm $\Vert u{\Vert }_1=\max \{\Vert u{\Vert }_0,\Vert u^{\mathrm{\Delta }}{\Vert }_0\}$ .

An arbitrary non-empty closed subset of the real numbers is called a Time scales. As usual, we denote a time scales by $\mathbb{T}$ . The examples of time scales are $\mathbb{R},\mathbb{N},h\mathbb{Z}$ , where $h>0$ . A time scale interval such that $[a,b]=\{\theta \in \mathbb{T}:a⩽\theta ⩽b\}$ , accordingly, we define $(a,b),[a,b),(a,b]$ and so on. Also, ${\mathbb{T}}^{\kappa }=\mathbb{T}⧹\{\max \mathbb{T}\}$ if $\max \mathbb{T}$ exists, otherwise ${\mathbb{T}}^{\kappa }=\mathbb{T}$ .

The forward jump operator $\sigma :{\mathbb{T}}^{\kappa }\to \mathbb{T}$ is defined by $\sigma (\theta )≔\text{inf}\{s\in \mathbb{T}:s>\theta \}$ with the substitution $\text{inf}\{\varphi \}=\sup \mathbb{T}$ and the graininess function $\mu :{\mathbb{T}}^{\kappa }\to [0\text{,}\infty )$ by $\mu (\theta ):=\sigma (\theta )-\theta \text{,}\forall \theta \in {\mathbb{T}}^{\kappa }$ .

A function $\varphi :\mathbb{T}\to \mathbb{R}$ is said to be rd-continuous on $\mathbb{T}$ , if $\varphi$ is continuous at points $\theta \in \mathbb{T}$ with $\sigma (\theta )=\theta$ and has finite left-sided limits at points $\theta \in \mathbb{T}$ with $$\sup \{r\in \mathbb{T}:r<\theta \}=\theta$$ and the set of all rd-continuous functions $\varphi :\mathbb{T}\to \mathbb{R}$ will be denoted by $C_{\mathit{rd}}(\mathbb{T},\mathbb{R})$ .

For $∊>0,\psi ⩾0$ , and nondecresing $\phi \in \mathit{PC}(I,{\mathbb{R}}^{+})$ , consider the following inequalities

1. $|\mathsf{G}(\theta )|⩽∊,\forall \theta \in {\cup }_{k=0}^p({\eta }_k,{\theta }_{k+1}]$ and $|{\mathsf{G}}_k(\theta )|⩽∊,\forall \theta \in ({\theta }_k,{\eta }_k],k=1,2,\dots ,p$ .

2. ${}^c{\mathrm{\Delta }}^{q}v(\theta )=\mathcal{M}(\theta ,v(\theta ),\mathcal{N}(v(\theta )))+\mathsf{G}(\theta ),\theta \in ({\eta }_k,{\theta }_{k+1}],k=0,1,\dots ,p$ .

3. $v(\theta )=\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\theta }_k}^{\theta }{(\theta -z)}^{q-1}{g}_k(z,v({\theta }_k^{-}))\mathrm{\Delta }z+{\mathsf{G}}_k(\theta ),\theta \in ({\theta }_k,{\eta }_k],k=1,2,\dots ,p$ .

We have similar lemma for the inequality (13).

From the Lemma 2.13, we have $$\left\{\begin{array}{c}\hfill {}^c{\mathrm{\Delta }}^{q}v(\theta ))=\mathcal{M}(\theta ,v(\theta ),\mathcal{N}(v(\theta )))+\mathsf{G}(\theta ),\theta \in ({\eta }_k,{\theta }_{k+1}],k=0,1,\dots ,p\text{.}\hfill \\ \hfill v(\theta )=\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\theta }_k}^{\theta }{(\theta -z)}^{q-1}{g}_k(z,v({\theta }_k^{-}))\mathrm{\Delta }z+{\mathsf{G}}_k(\theta ),\theta \in ({\theta }_k,{\eta }_k],k=1,2,\dots ,p\text{.}\hfill \end{array}\right.$$ Also, by Lemma 2.8, one can find that the solution v with $v(0)=v(T)$ of the above equation is given by $$\begin{array}{cc}\hfill v(\theta )& =\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\theta }_p}^{{\eta }_p}{({\eta }_p-z)}^{q-1}{g}_p(z,v({\theta }_p^{-}))\mathrm{\Delta }z+\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\eta }_p}^T{(T-z)}^{q-1}(\mathcal{M}(z,v(z),\mathcal{N}(v(z)))+\mathsf{G}(z))\mathrm{\Delta }z\hfill \\ \hfill & +{\mathsf{G}}_p(\theta )+\frac{1}{\mathrm{\Gamma }(q)}{\int }_0^{\theta }{(\theta -z)}^{q-1}(\mathcal{M}(z,v(z),\mathcal{N}(v(z)))+\mathsf{G}(z))\mathrm{\Delta }z,\forall \theta \in [0,{\theta }_1],\hfill \end{array}$$ $$\begin{array}{cc}\hfill v(\theta )& =\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\theta }_k}^{\theta }{(\theta -z)}^{q-1}{g}_k(z,v({\theta }_k^{-}))\mathrm{\Delta }z+{\mathsf{G}}_k(\theta ),\theta \in ({\theta }_k,{\eta }_k],k=1,2,\dots ,p,\hfill \\ \hfill v(\theta )& =\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\theta }_k}^{{\eta }_k}{({\eta }_k-z)}^{q-1}{g}_k(z,v({\theta }_k^{-}))\mathrm{\Delta }z+\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\eta }_k}^{\theta }{(\theta -z)}^{q-1}(\mathcal{M}(z,v(z),\mathcal{N}(v(z)))+\mathsf{G}(z))\mathrm{\Delta }z\hfill \\ \hfill & +{\mathsf{G}}_k(\theta ),\forall \theta \in ({\eta }_k,{\theta }_{k+1}],k=1,2,\dots ,p\text{.}\hfill \end{array}$$ Therefore, for $\theta \in ({\eta }_k,{\theta }_{k+1}],k=1,2,\dots ,p$ , we have $$\begin{array}{cc}\hfill |v(\theta )-\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\theta }_k}^{{\eta }_k}{({\eta }_k-z)}^{q-1}{g}_k(z,v({\theta }_k^{-}))\mathrm{\Delta }z& -\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\eta }_k}^{\theta }{(\theta -z)}^{q-1}\mathcal{M}(z,v(z),\mathcal{N}(v(z)))\mathrm{\Delta }z|\hfill \\ \hfill & ⩽|{\mathsf{G}}_k(\theta )|+\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\eta }_k}^{\theta }{(\theta -z)}^{q-1}|\mathsf{G}(z)|\mathrm{\Delta }z\hfill \\ \hfill & ⩽∊\left(1+\frac{T^q}{\mathrm{\Gamma }(q+1)}\right)\text{.}\hfill \end{array}$$ Also, for $\theta \in [0,{\theta }_1]$ , $$\begin{array}{cc}\hfill \left|v(\theta )-\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\theta }_p}^{{\eta }_p}{({\eta }_p-z)}^{q-1}\right.& g_p(z,v({\theta }_p^{-}))\mathrm{\Delta }z-\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\eta }_p}^T{(T-z)}^{q-1}\mathcal{M}(z,v(z),\mathcal{N}(v(z))\mathrm{\Delta }z\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(q)}{\int }_0^{\theta }{(\theta -z)}^{q-1}\mathcal{M}(z,v(z),\mathcal{N}(v(z)))\mathrm{\Delta }z\right|\hfill \\ \hfill & ⩽|{\mathsf{G}}_p(\theta )|+\frac{1}{\mathrm{\Gamma }(q)}{\int }_{{\eta }_p}^T{(T-z)}^{q-1}\mathsf{G}(z)\mathrm{\Delta }z+\frac{1}{\mathrm{\Gamma }(q)}{\int }_0^{\theta }{(\theta -z)}^{q-1}\mathsf{G}(z)\mathrm{\Delta }z\hfill \\ \hfill & ⩽∊\left(1+\frac{2T^q}{\mathrm{\Gamma }(q+1)}\right)\text{.}\hfill \end{array}$$ Similarly, when $\theta \in ({\theta }_k,{\eta }_k],k=1,2,\dots ,p$ ,

• (H1):

  Function $\mathcal{M}:J_1\times \mathbb{R}\times \mathbb{R}\to \mathbb{R},J_1={\cup }_{k=0}^p[{\eta }_k,{\theta }_{k+1}]$ is continuous and $\exists$ positive constants $L_1,L_2,C_1,M_1$ and $N_1$ such that

  1. $|\mathcal{M}(\theta ,u_1,u_2)-\mathcal{M}(\theta ,v_1,v_2)|⩽L_1|u_1-v_1|+L_2|u_2-v_2|,\forall \theta \in I,u_j,v_j\in \mathbb{R},j=1,2$ .

  2. $|\mathcal{M}(\theta ,u,v)|⩽C_1+M_1|u|+N_1|v|,\forall \theta \in I,u,v\in \mathbb{R}$ .

• (H2):

  $h:\mathcal{Q}\times \mathbb{R}\to \mathbb{R}$ is continuous and $\exists$ positive constants $L_h,C_2,M_2$ such that

  1. $|h(\theta ,s,u)-h(\theta ,z,v)|⩽L_h|u-v|,\forall \theta ,s\in \mathcal{Q},u,v\in \mathbb{R}$ .

  2. $|h(\theta ,s,u)|⩽C_2+M_2|u|,\forall \theta ,s\in \mathcal{Q},u\in \mathbb{R}$ .

• (H3):

  The functions $g_k:I_k\times \mathbb{R}\to \mathbb{R},I_k=[{\theta }_k,{\eta }_k],k=1,2,\dots ,p,$ are continuous and $\exists$ a positive constants $L_g,M_g$ such that

  1. $|g_k(\theta ,u)-g_k(\theta ,v)|⩽L_g|u-v|,\forall u,v\in \mathbb{R},\theta \in I_k,k=1,2,\dots ,p$ .

  2. $|g_k(\theta ,u)|⩽M_g,\forall \theta \in I_k\mathrm{and}u\in \mathbb{R}$ .

• (H4):

  $\frac{2T^q(M_1+N_1{M}_{2}T)}{\mathrm{\Gamma }(q+1)}<1$ .

3 Existence and uniqueness of solutions

In this section, we establish our main results for the Eq. (1). These results are carried out using the Banach contraction theorem.