(ω,c)-Periodic Solution to Semilinear Integro-Differential Equations with Hadamard Derivatives

Abstract

This study aims to prove the existence and uniqueness of the $(\omega ,c)$-periodic solution as a specific solution to Hadamard impulsive boundary value integro-differential equations with fixed lower limits. The results are proven using the Banach contraction, Schaefer’s fixed point theorem, and the Arzelà–Ascoli theorem. Furthermore, we establish the necessary conditions for a set of solutions to the explored boundary values with impulsive fractional differentials. Finally, we present two examples as applications for our results.

1. Introduction

The invention and continuous progress of modern technologies have heightened interest in systems with impulsive automated trajectories, which are examples of discontinuous trajectory control and impulsive computer systems. These systems have gained prominence and are currently used to tackle a wide variety of technological issues in a range of fields, including medicine, biology, threshold phenomena, hypotheses of bursting rhythms, economics, pharmacokinetics, and optimal control models for frequency-modulated systems [1]. Short-term disruptions with short durations in relation to the entire process times are introduced into these processes (see [2,3,4]). As a result of the significant interest in understanding their behaviors and characteristics, there is a compelling urge to study the qualitative aspects of these impulsive system solutions. A significant number of mathematicians are currently engaged in active research on periodic function theory. In [5], the existence and uniqueness of $(\omega ,c)$-periodic solutions were investigated for the similar evolution equation $y^{\prime }=\lambda y+f(t,y)$ in complex Banach spaces, where $\lambda$ is a bounded linear operator. Wang et al. [6] presented novel linear noninstantaneous impulsive differential equations and obtained solution representations and asymptotic stability for nonlinear and linear problems.

Alvarez et al. [7,8] defined $(\omega ,c)$-periodic functions, including periodic, Bloch periodic functions, and antiperiodic functions, among others. This notion was driven by the Mathieu’s equation $x^{″}\left(t\right)+[a+2qcos2t]x\left(t\right)=0$. One class of $(\omega ,c)$-periodic time-varying impulsive differential equations exists and is unique. A continuous function $g:\mathbb{R}\to X$, where X is a complex Banach space, is $(\omega ,c)$-periodic if $g(x+\omega )=cg\left(x\right)$ holds for all $x\in \mathbb{R}$, where $\omega >0$ and $c\in \mathbb{C}\setminus \left\{0\right\}$. Ren and Wang [9] established a required and sufficient criterion for $(\omega ,c)$-periodic solutions to impulsive fractional differential equations and investigated the existence and uniqueness of solutions via Caputo derivatives to impulsive fractional differential equations in Banach spaces. For further information on the existence and uniqueness of solutions to impulsive and regular fractional differential equations, see [10,11,12,13,14]. We also examined fractional oscillators, fractional dynamical systems, and the periodic solution of fractional oscillation equations in [15,16,17]. To our knowledge, there have not been any investigations into the existence of $(\omega ,c)$- periodic solutions for impulsive Hadamard fractional differential equations. This study expands on previous research on $(\omega ,c)$-periodic solutions for linear and semilinear problems with ordinary and fractional order derivatives. We concentrate on impulsive Hadamard fractional differential equations with boundary value constraints and set lower limits.

2. Preliminary

Consider impulsive fra. integro-diff. eqs. with lower limits, as illustrated below.

$$\begin{array}{ccc}\hfill D^{\vartheta }p\left(t\right)& =& \rho \left(t,p\left(t\right)\right)+{\int }_{t_0}^t\kappa (t-s)\sigma (s,p\left(s\right))ds,\vartheta \in (0,1),t\ne t_n,t\in [t_0,\infty ],\hfill \\ \hfill p\left(t_n^{+}\right)& =& p\left(t_n^{-}\right)+{\Delta }_n,n\in \mathbb{N},\hfill \end{array}$$

(1)

where $D^{\vartheta }p\left(t\right)$ is the Hadamard fra. derivative (where $\vartheta \in (0,1)$ via the lower limit at $t_0$) and for $n\in \mathbb{N}$, $t_n<t_{n+1}$, $li{m}_{n\to \infty }{t}_n=\infty$.

Next, we present some definitions and results, which are used in this study. To begin, consider the following definitions:

 Definition 1 

([18]). The Hadamard fra. integral of the $\vartheta >0$ of a function $\rho \left(t\right)\in \left(C\right[x,y\left]\right)$, $1\le x\le y<\infty$, is defined as

$$I^{\vartheta }\rho \left(t\right)={\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_x^t{\left(log{\displaystyle \frac{t}{s}}\right)}^{\vartheta -1}{\displaystyle \frac{\rho \left(s\right)}{s}}ds$$

for $t\ge x$, assuming the integral exists.

 Definition 2 

([18]). Let $1\le x\le y<\infty$, $\Delta =t\frac{d}{dt}$, and $A{C}_{\Delta }^n[x,y]=\{\rho :[x,y]\to \mathbb{R}:{\Delta }^{n-1}\left[\rho \left(t\right)\right]\in AC[x,y]\}$. The Hadamard fra. derivative of $\vartheta >0$ for a function $\rho \in A{C}_{\Delta }^n[x,y]$ is defined by

$$D^{\vartheta }\rho \left(t\right)={\displaystyle \frac{1}{\Gamma (n-\vartheta )}}{\left(t{\displaystyle \frac{d}{dt}}\right)}^{\left(n\right)}{\int }_x^t{\left(log{\displaystyle \frac{t}{s}}\right)}^{n-\vartheta -1}{\displaystyle \frac{\rho \left(s\right)}{s}}ds$$

for $t\ge x$ and $\vartheta \in (n-1,n)$, where $n=\left[\vartheta \right]+1$, such that $\left[\vartheta \right]$ signifies the integer component of a real number ϑ and $log(.)={log}_e(.)$.

 Lemma 1. 

Let $\rho :\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ and $\sigma :\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$be continuous functions. A solution $p\in PC(\mathbb{R},{\mathbb{R}}^n)$ of the impulsive frac. integro-diff. eq. is shown below with a fixed lower limit.

$$\begin{array}{cc}\hfill D^{\vartheta }p\left(t\right)=& \rho \left(t,p\left(t\right)\right)+{\int }_{t_0}^t\kappa (t-s)\sigma (s,p\left(s\right))ds,\vartheta \in (0,1),t\ne t_n,t\in [t_0,\infty ],\hfill \\ \hfill p\left(t_n^{+}\right)=& p\left(t_n^{-}\right)+{\Delta }_n,n\in \mathbb{N},\hfill \\ \hfill p\left(t_0\right)=& p_{t_0}.\hfill \end{array}$$

(2)

 is given by 

$$\begin{array}{ccc}\hfill p\left(t\right)& =& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_{t_0}^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_{t_0}^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill & & +p\left(t_0\right)+{\displaystyle \sum _{t_0<t_i<t}}{\Delta }_i,\mathrm{for}\mathrm{all}t\ge t_0.\hfill \end{array}$$

(3)

 Proof. 

From [19] in Lemma 3.2, a solution p of Equation (2) is provided by

$$\begin{array}{ccc}\hfill p\left(t\right)& =& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_{t_0}^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_{t_0}^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill & & +p\left(t_0\right)+{\displaystyle \sum _{i=1}^n}{\Delta }_i,\mathrm{for}\mathrm{every}t\in (t_n,t_{n+1}].\hfill \end{array}$$

(4)

Using

$$\sum _{i=1}^n{\Delta }_i=\sum _{t_0<t_i<t}{\Delta }_i,\mathrm{for}\mathrm{every}t\in (t_n,t_{n+1}],$$

we obtain that the Equation (4) is equivalent to

$$\begin{array}{ccc}\hfill p\left(t\right)& =& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_{t_0}^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_{t_0}^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill & & +p\left(t_0\right)+{\displaystyle \sum _{t_0<t_i<t}}{\Delta }_i,\hfill \end{array}$$

(5)

for $t\in (t_n,t_{n+1}]$. Using the arbitrariness of n, we find that Equation (5) holds for ${\cup }_{n=1}^{\infty }(t_n,t_{n+1}]$. Since Equation (5) is independent of n, we have that Equation (3) holds for $[t_0,\infty )$. □

 Definition 3 

([7]). Let $c\in \mathbb{C}\setminus \left\{0\right\}$, where $\omega >0$ and X represents a complex Banach space with a norm $\parallel ·\parallel$. A function $\gamma :\mathbb{R}\to X$ is called $(\omega ,c)$-periodic if $\gamma (t+\omega )=c\gamma \left(t\right)$∀$t\in \mathbb{R}$, where γ is continuous.

 Lemma 2 

([5]). Let ${\Phi }_{\omega ,c}:=\{p:p\in PC(\mathbb{R},{\mathbb{R}}^n)\}$, and $p(·+\omega )=cp(·)$. Then, $p\in {\Phi }_{\omega ,c}$ if $p\left(\omega \right)=cp\left(t_0\right)$.

3. The $(\mathbf{\omega },\mathbf{c})$-Periodic Solution to Semilinear Integro Differential Equations

For $t_0=1$, we investigate the $(\omega ,c)$-periodic solution of impulsive fra. integro-diff. eqs. with determined lower limits.

$$\begin{array}{cc}\hfill D^{\vartheta }p\left(t\right)=& \rho \left(t,p\left(t\right)\right)+{\int }_1^t\kappa (t-s)\sigma (s,p\left(s\right))ds,\vartheta \in (0,1),t\ne t_n,t\in [1,\infty ],\hfill \\ \hfill p\left(t_n^{+}\right)=& p\left(t_n^{-}\right)+{\Delta }_n,n\in \mathbb{N},\hfill \\ \hfill p\left(1\right)=& p_0.\hfill \end{array}$$

(6)

We present the following assumptions:

(I):

There are continuous functions $\rho :\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ and $\sigma :\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, such that

$$\rho (t+\omega ,cp)=c\rho (t,p),\forall t\in \mathbb{R},\mathrm{and}p\in {\mathbb{R}}^n$$

$$\sigma (t+\omega ,cp)=c\sigma (t,p),\forall t\in \mathbb{R},\mathrm{and}p\in {\mathbb{R}}^n$$

(II):

There are constants $A_1,A_2>0$ and $B_1,B_2>0,$ such that

$$\parallel \rho (t,p)\parallel \le A_1\parallel p\parallel +B_1,\forall t\in \mathbb{R},\mathrm{and}p\in {\mathbb{R}}^n$$

$$\parallel \sigma (t,p)\parallel \le A_2\parallel p\parallel +B_2,\forall t\in \mathbb{R},\mathrm{and}p\in {\mathbb{R}}^n$$

(III):

There is a constant ${ϵ}_1,{ϵ}_2>0,$ such that

$$\parallel \rho (t,p)-\rho (t,q)\parallel \le {ϵ}_1\parallel p-q\parallel ,\forall t\in \mathbb{R},\mathrm{and}p,q\in {\mathbb{R}}^n$$

$$\parallel \sigma (t,p)-\sigma (t,q)\parallel \le {ϵ}_2\parallel p-q\parallel ,\forall t\in \mathbb{R},\mathrm{and}p,q\in {\mathbb{R}}^n$$

(IV):

${\Delta }_n\in {\mathbb{R}}^n$ and $N\in \mathbb{N}$, such that $\omega =t_N$, $t_{n+N}=t_n+\omega$ and ${\Delta }_{n+N}={\Delta }_n$ for any $n\in \mathbb{N}$

(V):

The kernel map $\kappa :\mathbb{R}\to \mathbb{R}$ is locally integrable on $[1,\infty )$ and ${\int }_1^{\omega }\parallel k\left(s\right)\parallel ds=a_T>0$.

 Lemma 3. 

Assume that$c\ne 1$ and conditions (I) and (IV) are satisfied. Then, the solution  $p\in \Psi :=PC([1,\omega ],{\mathbb{R}}^n)$ of Equation (6) satisfying Lemma 2 is provided by 

$$\begin{array}{cc}\hfill p\left(t\right)=& {\displaystyle \frac{1}{(c-1)\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left[\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right]\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill +& {(c-1)}^{-1}\sum _{i=1}^n{\Delta }_i+{\displaystyle \sum _{1<t_i<t}}{\Delta }_i,t\in [1,\omega ].\hfill \end{array}$$

(7)

 Proof. 

The solution $p\in \Psi :=PC([1,\omega ],{\mathbb{R}}^n)$ is provided by Equation (3), such that

$$\begin{array}{ccc}\hfill p\left(t\right)& =& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left[log{\displaystyle \frac{t}{\xi }}\right]}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill & & +p\left(1\right)+{\displaystyle \sum _{t_0<t_i<t}}{\Delta }_i,\mathrm{for}\mathrm{all}t\in [1,\omega ].\hfill \end{array}$$

(8)

So, we have

$$\begin{array}{ccc}\hfill p\left(\omega \right)& =& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill & & +p\left(1\right)+{\displaystyle \sum _{t_0<t_i<\omega }}{\Delta }_i=cp\left(1\right),\hfill \end{array}$$

(9)

which is the same as

$$\begin{array}{ccc}\hfill p\left(1\right)& =& {\displaystyle \frac{1}{(c-1)\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill & & +{(c-1)}^{-1}{\displaystyle \sum _{t_0<t_i<\omega }}{\Delta }_i.\hfill \end{array}$$

(10)

From Equations (8) and (10), we obtain

$$\begin{array}{cc}\hfill p\left(t\right)=& {\displaystyle \frac{1}{|c-1|\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left[]log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\left[\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right]\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left[]log{\displaystyle \frac{t}{\xi }}\right]}^{\vartheta -1}\left[]\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right]\frac{d\xi }{\xi }\hfill \\ \hfill +& {(c-1)}^{-1}{\displaystyle \sum _{i=1}^n}{\Delta }_i+{\displaystyle \sum _{1<t_i<t}}{\Delta }_i.\hfill \end{array}$$

(11)

The proof is complete. □

 Theorem 1. 

Assume that $c\ne 1$ and (I), (III), and (IV) of the conditions are true if

$$0<{\displaystyle \frac{\left[{ϵ}_1+{ϵ}_2{a}_T\right]{\left(log\omega \right)}^{\vartheta }\left({1+|c-1|}^{-1}\right)}{\Gamma (\vartheta +1)}}<1.$$

Then, Equation (6) has one unique $(\omega ,c)$-periodic solution $p\in {\Phi }_{\omega ,c}$. In addition, we obtain

$$\begin{array}{cc}\hfill {\parallel p\parallel }_{\infty }\le & {\displaystyle \frac{{(log\omega )}^{\vartheta }\left({\delta }_1+{\delta }_2{a}_T\right)\left({|c-1|}^{-1}+1\right)+\Gamma (\vartheta +1)\left({|c-1|}^{-1}+1\right){\sum }_{i=1}^n\parallel {\Delta }_i\parallel }{\Gamma (\vartheta +1)-{(log\omega )}^{\vartheta }\left({ϵ}_1+{ϵ}_2{a}_T\right)\left({|c-1|}^{-1}+1\right)}}.\hfill \end{array}$$

 where ${\delta }_1=su{p}_{t\in [1,\omega ]}\parallel \rho (t,0)\parallel$ and ${\delta }_2=su{p}_{t\in [1,\omega ]}\parallel \sigma (t,0)\parallel$.

 Proof. 

We can deduce from (I) that for every $p\in {\Phi }_{\omega ,c}$, the following are true:

$$\rho (t+\omega ,p(t+\omega \left)\right)=\rho (t+\omega ,cp(t\left)\right)=c\rho (t,p(t\left)\right),\mathrm{for}\mathrm{all}t\in \mathbb{R},$$

and

$$\sigma (t+\omega ,p(t+\omega \left)\right)=\sigma (t+\omega ,cp(t\left)\right)=c\sigma (t,p(t\left)\right),\mathrm{for}\mathrm{all}t\in \mathbb{R},$$

which imply that $\rho (·,p(·\left)\right)$ and $\sigma (·,p(·))\in {\Phi }_{\omega ,c}$.

Now, we determine the operator $\Omega :\Psi \to \Psi$ as

$$\begin{array}{cc}\hfill (\Omega p)\left(t\right)=& {\displaystyle \frac{1}{|c-1|\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill +& {(c-1)}^{-1}\sum _{i=1}^n{\Delta }_i+\sum _{1<t_i<t}{\Delta }_i.\hfill \end{array}$$

(12)

Lemmas 2 and 3 show that the fixed points of $\Omega$ define the $(\omega ,c)$-periodic solution of Equation (6). It is straightforward to find $\Omega (\Psi )\subseteq \Psi$. For all $p_1,p_2\in \Psi$, we obtain

$$\begin{array}{cc}\hfill \parallel (\Omega & p_1)\left(t\right)-(\Omega p_2)\left(t\right)\parallel \hfill \\ \hfill =& ∥{\displaystyle \frac{1}{(c-1)\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\left[\rho \left(\xi ,p_1\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_1\left(s\right))ds\right]\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p_1\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_1\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill -& {(c-1)}^{-1}{\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\left[\rho \left(\xi ,p_2\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_2\left(s\right))ds\right]\frac{d\xi }{\xi }\hfill \\ \hfill -& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p_2\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_2\left(s\right))ds\right)\frac{d\xi }{\xi }∥\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {(c-1)}^{-1}{\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\left(\parallel \rho (\xi ,p_1\left(\xi \right))-\rho (\xi ,p_2\left(\xi \right))\parallel \right.\hfill \\ \hfill +& \left.{\int }_1^{\xi }\parallel \kappa (\xi -s)\parallel \parallel \sigma (s,p_1\left(s\right))-\sigma (s,p_2\left(s\right))\parallel ds\frac{d\xi }{\xi }\right)\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left({\displaystyle \frac{}{}}\parallel \rho (\xi ,p_1\left(\xi \right))-\rho (\xi ,p_2\left(\xi \right))\parallel \right.\hfill \\ \hfill +& \left.{\int }_1^{\xi }\parallel \kappa (\xi -s)\parallel \parallel \sigma (s,p_1\left(s\right))-\sigma (s,p_2\left(s\right))\parallel ds\frac{d\xi }{\xi }\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {|c-1|}^{-1}{\displaystyle \frac{{ϵ}_1+{ϵ}_2{a}_T}{\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\parallel p_1\left(\xi \right)-p_2\left(\xi \right)\parallel \frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{{ϵ}_1+{ϵ}_2{a}_T}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\parallel p_1\left(\xi \right)-p_2\left(\xi \right)\parallel \frac{d\xi }{\xi }\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{{ϵ}_1+{ϵ}_2{a}_T}{\Gamma \left(\vartheta \right)}}{\parallel p_1-p_2\parallel }_{\infty }\left({|c-1|}^{-1}{\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }+{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{\left[{ϵ}_1+{ϵ}_2{a}_T\right]{\left(log\omega \right)}^{\vartheta }\left({|c-1|}^{-1}+1\right)}{\Gamma (\vartheta +1)}}{\parallel p_1-p_2\parallel }_{\infty },\hfill \end{array}$$

and hence, we have

$$\parallel (\Omega p_1)\left(t\right)-(\Omega p_2)\left(t\right){\parallel }_{\infty }\le {\displaystyle \frac{\left[{ϵ}_1+{ϵ}_2{a}_T\right]{\left(log\omega \right)}^{\vartheta }\left({1+|c-1|}^{-1}\right)}{\Gamma (\vartheta +1)}}{\parallel p_1-p_2\parallel }_{\infty }.$$

From the assumption

$$0<{\displaystyle \frac{\left[{ϵ}_1+{ϵ}_2{a}_T\right]{\left(log\omega \right)}^{\vartheta }\left({1+|c-1|}^{-1}\right)}{\Gamma (\vartheta +1)}}<1.$$

Now, we find that $\Omega$ is a contraction mapping. Therefore, p is a unique fixed point of Equation (12) that satisfies $p\left(\omega \right)=cp\left(1\right)$. Then, we obtain from Lemma 2 that $p\in {\Phi }_{\omega ,c}$. Hence, Equation (6) has a $(\omega ,c)$-periodic $p\in {\Phi }_{\omega ,c}$, which is a unique. In addition, we have

$$\begin{array}{cc}\hfill & \parallel p\left(t\right)\parallel \hfill \\ \hfill \le & {|c-1|}^{-1}{\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left(\parallel \rho (\xi ,p\left(\xi \right))-\rho (\xi ,0)\parallel \right.\hfill \\ \hfill & +\left.{\int }_1^{\xi }\parallel \kappa (\xi -s)\parallel \parallel \sigma (s,p\left(s\right))-\sigma (s,0)\parallel ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{|c-1|\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\left[\parallel \rho (\xi ,0)\parallel +{\int }_1^{\xi }\parallel \kappa (\xi -s)\parallel \parallel \sigma (s,0)\parallel ds\right]\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\parallel \rho (\xi ,p\left(\xi \right))-\rho (\xi ,0)\parallel +{\int }_1^{\xi }\parallel \kappa (\xi -s)\parallel \parallel \sigma (s,p\left(s\right))-\sigma (s,0))\parallel ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\parallel \rho (\xi ,0)\parallel +{\int }_1^{\xi }\parallel \kappa (\xi -s)\parallel \parallel \sigma (s,0)\parallel ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill +& {|c-1|}^{-1}\sum _{i=1}^n\parallel {\Delta }_i\parallel +\sum _{1<t_i<t}\parallel {\Delta }_i\parallel \hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{{ϵ}_1+{ϵ}_2{a}_T}{|c-1|\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\parallel p\left(\xi \right)\parallel \frac{d\xi }{\xi }+{|c-1|}^{-1}{\displaystyle \frac{{\delta }_1+{\delta }_2{a}_T}{\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{{ϵ}_1+{ϵ}_2{a}_T}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\parallel p\left(\xi \right)\parallel \frac{d\xi }{\xi }+{\displaystyle \frac{{\delta }_1+{\delta }_2{a}_T}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }\hfill \\ \hfill +& \left({|c-1|}^{-1}+1\right)\sum _{i=1}^n\parallel {\Delta }_i\parallel \hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel p\left(t\right)\parallel \le & {\displaystyle \frac{{(log\omega )}^{\vartheta }\left({ϵ}_1+{ϵ}_2{a}_T\right)\left({|c-1|}^{-1}+1\right)}{\Gamma (\vartheta +1)}}{\parallel p\parallel }_{\infty }+{\displaystyle \frac{{(log\omega )}^{\vartheta }\left({\delta }_1+{\delta }_2{a}_T\right)\left({|c-1|}^{-1}+1\right)}{\Gamma (\vartheta +1)}}\hfill \\ \hfill +& \left({|c-1|}^{-1}+1x\right)\sum _{i=1}^n\parallel {\Delta }_i\parallel ,\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill {\parallel p\parallel }_{\infty }\le & {\displaystyle \frac{{(log\omega )}^{\vartheta }\left({\delta }_1+{\delta }_2{a}_T\right)\left({|c-1|}^{-1}+1\right)+\Gamma (\vartheta +1)\left({|c-1|}^{-1}+1\right){\sum }_{i=1}^n\parallel {\Delta }_i\parallel }{\Gamma (\vartheta +1)-{(log\omega )}^{\vartheta }\left({ϵ}_1+{ϵ}_2{a}_T\right)\left({|c-1|}^{-1}+1\right)}}.\hfill \end{array}$$

The proof is complete. □

 Theorem 2. 

Assume $c\ne 1$ and the conditions (I), (II), and (IV) are satisfied. If

$$\Gamma (\vartheta +1)>{(log\omega )}^{\vartheta }\left(A_1+A_2{a}_T\right)\left({1+|c-1|}^{-1}\right),$$

 then there is at least one $(\omega ,c)$ -periodic solution to the impulsive fra. integro-diff. Equation (6), such that $p\in {\Phi }_{\omega ,c}$.

 Proof. 

Let ${\mathbb{B}}_{\lambda }=\{p\in \Psi :\parallel p\parallel \le \lambda \}$, where

$$\begin{array}{cc}\hfill \lambda \ge & {\displaystyle \frac{{(log\omega )}^{\vartheta }\left(B_1+B_2{a}_T\right)\left({1+|c-1|}^{-1}\right)+\Gamma (\vartheta +1)\left({1+|c-1|}^{-1}\right){\sum }_{i=1}^n\parallel {\Delta }_i\parallel }{\Gamma (\vartheta +1)-{(log\omega )}^{\vartheta }\left(A_1+A_2{a}_T\right)\left({|c-1|}^{-1}+1\right)}}.\hfill \end{array}$$

We assume the $\Omega$ operator given in Equation (12) for ${\mathbb{B}}_{\lambda }$. In any case where $t\in [1,\omega ]$ and $p\in {\mathbb{B}}_{\lambda }$, we have

$$\begin{array}{cc}\hfill & \parallel (\Omega p)\left(t\right)\parallel \hfill \\ \hfill \le & {\displaystyle \frac{1}{|c-1|\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\left[\parallel \rho (\xi ,p\left(\xi \right))\parallel +{\int }_1^{\xi }\parallel \kappa (\xi -s)\parallel \parallel \sigma (s,p\left(s\right))ds\parallel \right]\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\parallel \rho (\xi ,p\left(\xi \right))\parallel +{\int }_1^{\xi }\parallel \kappa (\xi -s)\parallel \parallel \sigma (s,p\left(s\right))ds\right)\parallel \frac{d\xi }{\xi }\hfill \\ \hfill +& {|c-1|}^{-1}\sum _{i=1}^n\parallel {\Delta }_i\parallel +\sum _{1<t_i<t}\parallel {\Delta }_i\parallel \hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{1}{|c-1|\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left[\left(A_1+A_2{a}_T\right)\parallel p\left(\xi \right)\parallel +\left(B_1+B_2{a}_T\right)\right]\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left[\left(A_1+A_2{a}_T\right)\parallel p\left(\xi \right)\parallel +\left(B_1+B_2{a}_T\right)\right]\parallel \frac{d\xi }{\xi }\hfill \\ \hfill +& {|c-1|}^{-1}\sum _{i=1}^n\parallel {\Delta }_i\parallel +\sum _{1<t_i<t}\parallel {\Delta }_i\parallel \hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{{(log\omega )}^{\vartheta }\left(A_1+A_2{a}_T\right)\left({1+|c-1|}^{-1}\right)}{\Gamma (\vartheta +1)}}{\parallel p\parallel }_{\infty }+{\displaystyle \frac{{(log\omega )}^{\vartheta }\left(B_1+B_2{a}_T\right)\left({1+|c-1|}^{-1}\right)}{\Gamma (\vartheta +1)}}\hfill \\ \hfill +& \left({1+|c-1|}^{-1}\right)\sum _{i=1}^n\parallel {\Delta }_i\parallel .\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill {\parallel \Omega p\parallel }_{\infty }\le & {\displaystyle \frac{{(log\omega )}^{\vartheta }\left(B_1+B_2{a}_T\right)\left({|c-1|}^{-1}+1\right)+\Gamma (\vartheta +1)\left({|c-1|}^{-1}+1\right){\sum }_{i=1}^n\parallel {\Delta }_i\parallel }{\Gamma (\vartheta +1)-{(log\omega )}^{\vartheta }\left(A_1+A_2{a}_T\right)\left({|c-1|}^{-1}+1\right)}}\le \lambda ,\hfill \end{array}$$

which implies that ${\parallel \Omega p\parallel }_{\infty }\le \lambda$. So, $\Omega \left({\mathbb{B}}_{\lambda }\right)\subseteq {\mathbb{B}}_{\lambda }$.

Now, we show that $\Omega$ is continuous for ${\mathbb{B}}_{\lambda }$.

Let ${\left\{p_i\right\}}_{i\ge 1}\subseteq {\mathbb{B}}_{\lambda }$ and $p_i\to p$ for ${\mathbb{B}}_{\lambda }$ as $i\to \infty$. By the continuity of $\rho$ and $\sigma$, we obtain $\rho (\xi ,p_i\left(\xi \right))\to \rho (\xi ,p\left(\xi \right))$ and $\sigma (\xi ,p_i\left(\xi \right))\to \sigma (\xi ,p\left(\xi \right))$ as $i\to \infty$. As a result, we have

$$\begin{array}{ccc}\hfill {\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\rho \left(\xi ,p_i\left(\xi \right)\right)& \to {\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\rho \left(\xi ,p\left(\xi \right)\right)\hfill & \\ \hfill \mathrm{as}i\to \infty .& \\ \hfill {\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_i\left(s\right))ds\right)& \to {\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\hfill & \\ \hfill \mathrm{as}i\to \infty .& \\ \hfill {\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\rho \left(\xi ,p_i\left(\xi \right)\right)& \to {\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\rho \left(\xi ,p\left(\xi \right)\right)\hfill & \\ \hfill \mathrm{as}i\to \infty .& \\ \hfill {\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_i\left(s\right))ds\right)& \to {\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\hfill & \\ \hfill \mathrm{as}i\to \infty .\end{array}$$

Using assumption (II), we find that for any $1\le i\le t\le \omega$,

(i)

$$\begin{array}{cc}\hfill {\int }_1^{\omega }& ∥{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\rho \left(\xi ,p_i\left(\xi \right)\right)-{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\rho \left(\xi ,p\left(\xi \right)\right)∥\frac{d\xi }{\xi }\hfill \\ \hfill \le & 2(A_1{\parallel p\parallel }_{\infty }+B_1){\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }=2(\lambda A_1+B_1){\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }\hfill \\ \hfill \le & 2(\lambda A_1+B_1){\displaystyle \frac{{(log\omega )}^{\vartheta }}{\vartheta }}<\infty ,\hfill \end{array}$$

(ii)

$$\begin{array}{cc}\hfill {\int }_1^{\omega }& ∥{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\left[{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_i\left(s\right))ds\right]-{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\left[{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right]∥\frac{d\xi }{\xi }\hfill \\ \hfill \le & 2a_T(A_2{\parallel p\parallel }_{\infty }+B_2){\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }=2a_T(\lambda A_2+B_2){\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }\hfill \\ \hfill \le & 2a_T(\lambda A_2+B_2){\displaystyle \frac{{(log\omega )}^{\vartheta }}{\vartheta }}<\infty ,\hfill \end{array}$$

(iii)

$$\begin{array}{cc}\hfill {\int }_1^t& ∥{\left[log{\displaystyle \frac{t}{\xi }}\right]}^{\vartheta -1}\rho \left(\xi ,p_i\left(\xi \right)\right)-{\left[log{\displaystyle \frac{t}{\xi }}\right]}^{\vartheta -1}\rho \left(\xi ,p\left(\xi \right)\right)∥\frac{d\xi }{\xi }\hfill \\ \hfill \le & 2(A_1{\parallel p\parallel }_{\infty }+B_1){\int }_1^t{\left[log{\displaystyle \frac{t}{\xi }}\right]}^{\vartheta -1}\frac{d\xi }{\xi }=2(\lambda A_1+B_1){\int }_1^t{\left[log{\displaystyle \frac{t}{\xi }}\right]}^{\vartheta -1}\frac{d\xi }{\xi }\hfill \\ \hfill \le & 2(\lambda A_1+B_1){\displaystyle \frac{{(log\omega )}^{\vartheta }}{\vartheta }}<\infty ,\hfill \end{array}$$

(iv)

$$\begin{array}{cc}\hfill {\int }_1^t& ∥{\left[log{\displaystyle \frac{t}{\xi }}\right]}^{\vartheta -1}\left[{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_i\left(s\right))ds\right]-{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)∥\frac{d\xi }{\xi }\hfill \\ \hfill \le & 2a_T(A_2{\parallel p\parallel }_{\infty }+B_2){\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }=2a_T(\lambda A_2+B_2){\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }\hfill \\ \hfill \le & 2a_T(\lambda A_2+B_2){\displaystyle \frac{{(log\omega )}^{\vartheta }}{\vartheta }}<\infty .\hfill \end{array}$$

Then, by applying the theorem of Lebesgue dominated convergence, we obtain

(i)

$${\int }_1^{\omega }∥{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\rho \left(\xi ,p_i\left(\xi \right)\right)-{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\rho \left(\xi ,p\left(\xi \right)\right)∥\frac{d\xi }{\xi }\to 0\mathrm{as}i\to \infty ,$$

(ii)

$$\begin{array}{c}\hfill {\int }_1^{\omega }∥{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_i\left(s\right))ds\right)\\ \hfill -{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)∥\frac{d\xi }{\xi }\to 0\mathrm{as}i\to \infty \end{array}$$

(iii)

$${\int }_1^t∥{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\rho \left(\xi ,p_i\left(\xi \right)\right)-{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\rho \left(\xi ,p\left(\xi \right)\right)∥\frac{d\xi }{\xi }\to 0\mathrm{as}i\to \infty ,$$

(iv)

$$\begin{array}{c}\hfill {\int }_1^t∥{\left[log{\displaystyle \frac{t}{\xi }}\right]}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_i\left(s\right))ds\right)\\ \hfill -{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)∥\frac{d\xi }{\xi }\to 0\mathrm{as}i\to \infty .\end{array}$$

Hence, for every $t\in [1,\omega ]$, we obtain

$$\begin{array}{cc}& \parallel (\Omega p_i)\left(t\right)-(\Omega p)\left(t\right)\parallel \hfill \\ \hfill \le & {\displaystyle \frac{{|c-1|}^{-1}}{\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }∥{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\rho \left(\xi ,p_i\left(\xi \right)\right)-{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\rho \left(\xi ,p\left(\xi \right)\right)∥\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{|c-1|\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }∥{\left[log{\displaystyle \frac{\omega }{\xi }}\right]}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_i\left(s\right))ds\right)\hfill \\ \hfill & -{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)∥\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t∥{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p_i\left(s\right))ds\right)\hfill \\ \hfill & -{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)∥\frac{d\xi }{\xi }\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t∥{\left[log{\displaystyle \frac{t}{\xi }}\right]}^{\vartheta -1}\rho \left(\xi ,p_i\left(\xi \right)\right)-{\left[log{\displaystyle \frac{t}{\xi }}\right]}^{\vartheta -1}\rho \left(\xi ,p\left(\xi \right)\right)∥\frac{d\xi }{\xi }\to 0\mathrm{as}i\to \infty .\hfill \end{array}$$

So, $\Omega$ is continuous for ${\mathbb{B}}_{\lambda }$. Now, we show that $\Omega$ is pre-compact.

For every $t_n\le t\le s\le t_{n+1}$, where $n\in {\mathbb{N}}_0$, we obtain

$$\parallel \sum _{1<t_i<t}{\Delta }_i-\sum _{1<t_i<s}{\Delta }_i\parallel =\parallel \sum _{i=1}^n{\Delta }_i-\sum _{i=1}^n{\Delta }_i\parallel =0$$

which implies that

$$\parallel \sum _{1<t_i<t}{\Delta }_i-\sum _{1<t_i<s}{\Delta }_i\parallel \to 0,\mathrm{as}t\to s.$$

So, for every $1\le s_1<s_2\le \omega$, where $p\in {\mathbb{B}}_{\lambda }$, and the operator $\Omega :\Psi \to \Psi$ is provided by

$$\begin{array}{cc}\hfill (\Omega p)\left(t\right)=& {\displaystyle \frac{1}{|c-1|\Gamma \left(\vartheta \right)}}{\int }_1^{\omega }{\left(log{\displaystyle \frac{\omega }{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{\xi }}\right)}^{\vartheta -1}\left(\rho \left(\xi ,p\left(\xi \right)\right)+{\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\frac{d\xi }{\xi }\hfill \\ \hfill +& {(c-1)}^{-1}\sum _{i=1}^n{\Delta }_i-\sum _{1<t_i<t}{\Delta }_i,\hfill \end{array}$$

the following holds:

$$\begin{array}{cc}\hfill \parallel (\Omega p)\left(s_1\right)-& (\Omega p)\left(s_2\right)\parallel \hfill \\ \hfill \le & {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^{s_1}∥{\left[log{\displaystyle \frac{s_1}{\xi }}\right]}^{\vartheta -1}\rho \left(\xi ,p\left(\xi \right)\right)-{\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^{s_2}{\left[log{\displaystyle \frac{s_2}{\xi }}\right]}^{\vartheta -1}\rho \left(\xi ,p\left(\xi \right)\right)∥\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^{s_1}∥{\left(log{\displaystyle \frac{s_1}{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)\hfill \\ \hfill -& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^{s_2}{\left(log{\displaystyle \frac{s_2}{\xi }}\right)}^{\vartheta -1}\left({\int }_1^{\xi }\kappa (\xi -s)\sigma (s,p\left(s\right))ds\right)∥\frac{d\xi }{\xi }\hfill \\ \hfill +& ∥\sum _{1<t_i<s_1}{\Delta }_i-\sum _{1<t_i<s_2}{\Delta }_i∥\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^{s_1}\left[{\left(log{\displaystyle \frac{s_1}{\xi }}\right)}^{\vartheta -1}-{\left(log{\displaystyle \frac{s_2}{\xi }}\right)}^{\vartheta -1}\right]∥\rho \left(\xi ,p\left(\xi \right)\right)∥\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_1^{s_1}\left[{\left(log{\displaystyle \frac{s_1}{\xi }}\right)}^{\vartheta -1}-{\left(log{\displaystyle \frac{s_2}{\xi }}\right)}^{\vartheta -1}\right]{\int }_1^{\xi }\parallel \kappa (\xi -s)\parallel \parallel \sigma (s,p\left(s\right))\parallel ds\frac{d\xi }{\xi }\hfill \\ \hfill +& ∥\sum _{1<t_i<s_1}{\Delta }_i-\sum _{1<t_i<s_2}{\Delta }_i∥-{\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_{s_1}^{s_2}{\left(log{\displaystyle \frac{s_2}{\xi }}\right)}^{\vartheta -1}∥\rho \left(\xi ,p\left(\xi \right)\right)∥\frac{d\xi }{\xi }\hfill \\ \hfill -& {\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_{s_1}^{s_2}{\left(log{\displaystyle \frac{s_2}{\xi }}\right)}^{\vartheta -1}{\int }_1^{\xi }\parallel \kappa (\xi -s)\parallel \parallel \sigma (s,p\left(s\right))\parallel ds\frac{d\xi }{\xi }\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{\lambda A_1+B_1}{\Gamma \left(\vartheta \right)}}{\int }_1^{s_1}\left[{\left(log{\displaystyle \frac{s_1}{\xi }}\right)}^{\vartheta -1}-{\left(log{\displaystyle \frac{s_2}{\xi }}\right)}^{\vartheta -1}\right]\frac{d\xi }{\xi }\hfill \\ \hfill +& {\displaystyle \frac{a_{T\left(\lambda A_2+B_2\right)}}{\Gamma \left(\vartheta \right)}}{\int }_1^{s_1}\left[{\left(log{\displaystyle \frac{s_1}{\xi }}\right)}^{\vartheta -1}-{\left(log{\displaystyle \frac{s_2}{\xi }}\right)}^{\vartheta -1}\right]\frac{d\xi }{\xi }\hfill \\ \hfill +& ∥\sum _{1<t_i<s_1}{\Delta }_i-\sum _{1<t_i<s_2}{\Delta }_i∥\hfill \\ \hfill -& {\displaystyle \frac{\lambda A_1+B_1}{\Gamma \left(\vartheta \right)}}{\int }_{s_1}^{s_2}{\left(log{\displaystyle \frac{s_2}{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }-{\displaystyle \frac{a_{T\left(\lambda A_2+B_2\right)}}{\Gamma \left(\vartheta \right)}}{\int }_{s_1}^{s_2}{\left(log{\displaystyle \frac{s_2}{\xi }}\right)}^{\vartheta -1}\frac{d\xi }{\xi }\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{\lambda A_1+B_1}{\Gamma (\vartheta +1)}}\left[{(log{s}_1)}^{\vartheta }-{(log{s}_2)}^{\vartheta }+2{\left(log\frac{s_2}{s_1}\right)}^{\vartheta }\right]\hfill \\ \hfill +& {\displaystyle \frac{a_{T\left(\lambda A_2+B_2\right)}}{\Gamma (\vartheta +1)}}\left[{(log{s}_1)}^{\vartheta }-{(log{s}_2)}^{\vartheta }+2{\left(log\frac{s_2}{s_1}\right)}^{\vartheta }\right]\hfill \\ \hfill +& ∥\sum _{1<t_i<s_1}{\Delta }_i-\sum _{1<t_i<s_2}{\Delta }_i∥\hfill \end{array}$$

$$\begin{array}{cc}\hfill ∥{\displaystyle \frac{}{}}(\Omega p)\left(s_1\right)-(\Omega p)\left(s_2\right)∥\le & {\displaystyle \frac{[\lambda A_1+B_1]+\left[a_{T\left(\lambda A_2+B_2\right)}\right]}{\Gamma (\vartheta +1)}}\left[{(log{s}_1)}^{\vartheta }-{(log{s}_2)}^{\vartheta }+2{\left(log\frac{s_2}{s_1}\right)}^{\vartheta }\right]\hfill \\ \hfill +& ∥\sum _{1<t_i<s_1}{\Delta }_i-\sum _{1<t_i<s_2}{\Delta }_i∥⟶0\mathrm{as}{s}_1⟶s_2.\hfill \end{array}$$

So, $\Omega \left({\mathbb{B}}_{\lambda }\right)$ is equicontinuous. From Equation (13), we obtain that $\Omega \left({\mathbb{B}}_{\lambda }\right)$ is bounded uniformly.

Using the Arzelà–Ascoli theorem, we can show that $\Omega \left({\mathbb{B}}_{\lambda }\right)$ is pre-compact. The impulsive fra. diff. arises from Schauder’s fixed point theorem. At least one $(\omega ,c)$-periodic solution $p\in {\Phi }_{\omega ,c}$ exists in Equation (6). The proof is complete. □

We end this section with two illustrative examples.

 Example 1. 

We investigate the following impulsive fra. diff. boundary value problem:

$$\begin{array}{cc}\hfill D^{\frac{2}{3}}p\left(t\right)=& \lambda cos2tsinp\left(t\right)+{\int }_1^t{\displaystyle \frac{sin(s-t)}{5+cosp\left(s\right)}}ds,t\ne t_n,t\in [1,\infty ),\hfill \\ \hfill p\left(t_n^{+}\right)=& p\left(t_n^{-}\right)+cosn\pi ,n=1,2,\cdots ,m,\hfill \end{array}$$

where $\lambda \in \mathbb{R}$, $t_n=\frac{n\pi }{2}$, and $n=1,2,\cdots ,m.$ Therefore, ${\Delta }_n=cosn\pi$ and

$$\rho (t,p(t\left)\right)=\lambda cos2tsinp\left(t\right)$$

$$\sigma (t,p\left(t\right))={\displaystyle \frac{sin(s-t)}{5+cosp\left(t\right)}}$$

 Let $c=-1$ and  $\omega =\pi$. It is clear that for any $n\in \mathbb{N}$, $t_{n+2}=t_n+\pi$, ${\Delta }_{n+2}={\Delta }_n$. Hence, we have $N=2$ and $\left(IV\right)$ holds. For all $t\in \mathbb{R}$ and $p\left(t\right)\in \mathbb{R}$, we obtain 

$$\rho (t+\omega ,cp(t\left)\right)=\rho (t+\pi ,-p(t\left)\right)=-\lambda cos2tsinp\left(t\right)=-\rho (t,p(t\left)\right)=c\rho (t,p(t\left)\right),$$

$$\sigma (t+\omega ,cp\left(t\right))=\sigma (t+\pi ,-p\left(t\right))=-{\displaystyle \frac{sin(s-t)}{5+cosp\left(t\right)}}=-\sigma (t,p\left(t\right))=c\sigma (t,p\left(t\right))$$

which implies that (I) holds. Also, $\kappa \left(s\right)=1$ and $a_T={\int }_1^{\pi }\parallel 1\parallel ds\le \pi -1$.

For all $t\in [1,\pi ]$, and $p,q\in \mathbb{R}$, we obtain

$$\begin{array}{cc}\hfill \parallel \rho (t,p)-\rho (t,q)\parallel =& \parallel \lambda cos2tsinp\left(t\right)-\lambda cos2tsinq\left(t\right)\parallel \hfill \\ \hfill \le & \parallel \lambda cos2tsinp\left(t\right)-\lambda cos2tsinq\left(t\right)\parallel \hfill \\ \hfill \le & \left|\lambda \right|\parallel p-q\parallel ={ϵ}_1\parallel p-q\parallel .\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel \sigma (t,p)-\sigma (t,q)\parallel =& \parallel {\displaystyle \frac{sin(s-t)}{5+cosp\left(t\right)}}-{\displaystyle \frac{sin(s-t)}{5+cosq\left(t\right)}}\parallel \hfill \\ \hfill \le & {\displaystyle \frac{1}{25}}\parallel p-q\parallel ={ϵ}_2\parallel p-q\parallel .\hfill \end{array}$$

 So, $\left(III\right)$ is satisfied for ${ϵ}_1=\left|\lambda \right|$ and ${ϵ}_2={\displaystyle \frac{1}{25}}$. Note that

$${\displaystyle \frac{\left[{ϵ}_1+{ϵ}_2{a}_T\right]{\left(log\omega \right)}^{\vartheta }\left({|c-1|}^{-1}+1\right)}{\Gamma (\vartheta +1)}}={\displaystyle \frac{{\displaystyle \frac{3}{2}}\left[\left|\lambda \right|+{\displaystyle \frac{\pi -1}{25}}\right]\sqrt[3]{{(log\pi )}^2}}{\Gamma (3/2)}}.$$

Let $0<\parallel \lambda \parallel <{\displaystyle \frac{2\Gamma (3/2)}{3\sqrt[3]{{(log\pi )}^2}}}-{\displaystyle \frac{a_T}{25}}$ (where $a_T<\pi -1$) and we obtain

$$0<{\displaystyle \frac{\left[{ϵ}_1+{ϵ}_2{a}_T\right]{\left(log\omega \right)}^{\vartheta }\left({1+|c-1|}^{-1}\right)}{\Gamma (\vartheta +1)}}=0.456<1.$$

Hence, all assumptions in Theorem 1 hold and the eq. in Example 1 has a unique $(\pi ,-1)$-periodic solution $p\in {\Phi }_{\pi ,-1}$. In addition, we now obtain

$$\begin{array}{cc}\hfill {\parallel p\parallel }_{\infty }\le & {\displaystyle \frac{{(log\omega )}^{\vartheta }\left({\delta }_1+{\delta }_2{a}_T\right)\left({|c-1|}^{-1}+1\right)+\Gamma (\vartheta +1)\left({|c-1|}^{-1}+1\right){\sum }_{i=1}^N\parallel {\Delta }_i\parallel }{\Gamma (\vartheta +1)-{(log\omega )}^{\vartheta }\left({ϵ}_1+{ϵ}_2{a}_T\right)\left({|c-1|}^{-1}+1\right)}},\hfill \end{array}$$

where ${\delta }_1=su{p}_{t\in [1,\omega ]}\parallel \rho (t,0)\parallel =0$ and ${\delta }_2=su{p}_{t\in [1,\omega ]}\parallel \sigma (t,0)\parallel ={\displaystyle \frac{1}{6}}$. Furthermore, we have

$$\begin{array}{cc}\hfill {\parallel p\parallel }_{\infty }\le & {\displaystyle \frac{\sqrt[3]{{(log\pi )}^2}\left({\displaystyle \frac{\pi -1}{6}}\right)\left({\displaystyle \frac{3}{2}}\right)+\Gamma (3/2)\left({\displaystyle \frac{3}{2}}\right)2}{\Gamma (3/2)-\sqrt[3]{{(log\pi )}^2}\left(\parallel \lambda \parallel +{\displaystyle \frac{\pi -1}{25}}\right)\left({\displaystyle \frac{3}{2}}\right)}}.\hfill \end{array}$$

 Example 2. 

We consider the following impulsive fra. diff. boundary value problem:

$$\begin{array}{cc}\hfill D^{\frac{1}{3}}\left(t\right)=& \lambda p\left(t\right)cos\frac{p\left(t\right)}{e^t}+{\int }_1^t{e}^{-s}sin\frac{p\left(s\right)}{e^s}ds,t\ne t_n,t\in [1,\infty ),\hfill \\ \hfill p\left(t_n^{+}\right)=& p\left(t_n^{-}\right)+3,n=1,2,\cdots ,m,\hfill \end{array}$$

where $\lambda \in \mathbb{R}$, $t_n=\frac{n\pi }{2},n=1,2,\cdots ,m,$ ${\Delta }_n=3$, and

$$\rho (t,p\left(t\right))=\lambda p\left(t\right)cos\frac{p\left(t\right)}{e^t}$$

$$\sigma (t,p\left(t\right))=e^{-t}sin\frac{p\left(t\right)}{e^t}$$

Let $c=e^{\pi }$ and $\omega =\pi$. It is clear that for all $n\in \mathbb{N}$, $t_{n+2}=t_n+\pi$ and ${\Delta }_{n+2}={\Delta }_n$. Hence, we have $N=2$ and $\left(IV\right)$ holds. For all $t\in \mathbb{R}$ and $p\left(t\right)\in \mathbb{R}$, we obtain

$$\rho (t+\omega ,cp\left(t\right))=\rho (t+\pi ,e^{\pi }p\left(t\right))=\lambda e^{\pi }p\left(t\right)cos\frac{e^{\pi }p\left(t\right)}{e^{t+\pi }}=\lambda e^{\pi }p\left(t\right)cos\frac{p\left(t\right)}{e^t}=e^{\pi }\rho (t,p\left(t\right))$$

$$\sigma (t+\omega ,cp\left(t\right))=\sigma (t+\pi ,e^{\pi }p\left(t\right))=e^{-t+\pi }sin\frac{e^{\pi }p\left(t\right)}{e^{t+\pi }}=e^{\pi }{e}^{-t}sin\frac{p\left(t\right)}{e^t}=e^{\pi }\sigma (t,p\left(t\right))$$

which implies that (I) holds. Also, $\kappa \left(s\right)=1$ and $a_T={\int }_0^1\parallel 1\parallel ds\le 1$.

For any $t\in [1,\pi ]$,and $p,q\in \mathbb{R}$, we have

$$\parallel \rho (t,p)\parallel =\parallel \lambda p\left(t\right)cos\frac{p\left(t\right)}{e^t}\parallel =\parallel \lambda \parallel \parallel p\left(t\right)\parallel \parallel cos\frac{p\left(t\right)}{e^t}\parallel \le \parallel \lambda \parallel \parallel p\left(t\right)\parallel ,$$

$$\parallel \sigma (t,p)\parallel =\parallel e^{-t}sin\frac{p\left(t\right)}{e^t}\parallel =\parallel e^{-t}\parallel \parallel sin\frac{p\left(t\right)}{e^t}\parallel \le \parallel e^{-t}\parallel \le \parallel e^{-1}\parallel ,$$

which implies that $A_1=\parallel \lambda \parallel$, $B_1>0$, $A_2=e^{-1}$, and $B_2>0$ and (II) holds.

Letting $\parallel \lambda \parallel <{\displaystyle \frac{e^{\pi -1}\Gamma (4/3)}{e^{\pi }\sqrt[3]{log\pi }}}-e^{-1}$, we obtain $\Gamma (4/3)>\sqrt[3]{log\pi }\left(\parallel \lambda \parallel +e^{-1}\right)\left({\displaystyle \frac{e^{\pi }}{e^{\pi }-1}}\right)$ and

$$\Gamma (\vartheta +1)>{(log\omega )}^{\vartheta }\left(A_1+A_2{a}_T\right)\left({1+|c-1|}^{-1}\right).$$

Hence, all assumptions in Theorem 2 hold in Example 2. Hence, the eq. in Example 2 has at least one $(\pi ,e^{\pi })$-periodic solution $p\in {\Phi }_{\pi ,e^{\pi }}$.

4. Conclusions

This study demonstrated that $(\omega ,c)$-periodic solutions exist for impulsive Hadamard fractional differential equations with boundary value restrictions on Banach contractions. The study aimed to establish the existence and uniqueness of the $(\omega ,c)$-periodic solution of Equation (6), which applies the Banach contraction mapping concept. Furthermore, the paper demonstrated the existence of an $(\omega ,c)$-periodic solution to Equation (6) using Schaefer’s fixed point theorem and the Arzelà-Ascoli theorem. We concluded the study with two examples demonstrating how the generated results could be used. In the future, we will conduct further investigations on the existence and uniqueness of fractional derivatives. There are also other possible lines of research on this topic, such as $(\omega ,c)$-periodic solutions for impulsive Hadamard fractional differential equations with varying lower limits. Moreover, mildly $(\omega ,c)$-periodic solutions to abstract semilinear Hadamard integro-differential equations and $(\omega ,c)$-almost periodic-type functions could also be studied and discussed.This work opens the door for other possible contributions to this topic by combining types of fractional derivatives and types of periodic solutions.