Katugampola-type fractional differential equations with delay and impulses

doi:10.15406/oajmtp.2018.01.00012

Open Access Journal of

eISSN: 2641-9335

Mathematical and Theoretical Physics

Research Article Volume 1 Issue 3

Katugampola-type fractional differential equations with delay and impulses

M Janaki,¹ K Kanagarajan,¹ EM Elsayed^2,3

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¹Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, India
²Department of Mathematics,King Abdulaziz University, Saudi Arabia
³Department of Mathematics, Mansoura University, Egypt

Correspondence: Elsayed EM, Department of Mathematics,Faculty of Science, Mansoura University, Mansoura 35516,Egypt

Received: February 03, 2018 | Published: May 1, 2018

Citation: Janaki M, Elsayed EM, Kanagarajan K. Katugampola-type fractional differential equations with delay and impulses. Open Acc J Math Theor Phy. 2018;1(3):73-77. DOI: 10.15406/oajmtp.2018.01.00012

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Abstract

Our aim in this note is to study the existence of solutions of a Katugampola-type fractional impulsive differential equation with delay. We use successive approximation method to show the existence of solutions. In the end, an example is given to verify the hypothetical results.

Keywords: katugampola fractional derivative, impulsive equations, time delay

Introduction

Because of its wide applicability in biology, medicine and in more and more fields, the theory of fractional differential equations(FDEs) has recently been attracting increasing interest, see for instance^1‒8 and references therein. Impulsive differential equations have played an important role in modelling phenomena, especially in describing dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases, and so forth, some authors have used impulsive differential systems to describe the model since the last century. For the basic theory on impulsive differential equations, the reader can refer to the books^9‒12 and the papers.^1,13‒16 In addition, some modelling is done via impulsive functional differential equations when these processes involve hereditary phenomena such as biological and social macrosystems. For fractional functional differential equations, the initial value problem, for a class of nonlinear fractional functional differential equations is discussed. For more details, see.^17‒24 Motivated by the papers,^25,26 the aim of this note is to discuss the existence and uniqueness of solutions of Katugampola-type FDEs with delay and impulses.

Consider the Katugampola-type FDEs with delay and impulse of the form,

$(\begin{matrix} _{c}^{ρ} D_{0^{+}}^{ω} ℨ (t)= ℌ (t, ℨ_{t}), t \neq t_{k}; t \in ℑ :=[0, T]; \\ Δ ℨ (t_{k})= I_{k} (ℨ (t_{k})), k =1,2,..., m; \\ ℨ (t)= ψ (t), t \in [- μ,0], \end{matrix}$ (1)

where is the generalized fractional derivative in Caputo sense, and are given functions satisfying some assumptions that will be specified later. and represent the right and left limits of at respectively, and they satisfy that . If , then for any , define by for , here represents the history of the state from time to present time and .

The rest of this paper is organized as follows. In Section 2, we give some notations and recall some concepts and preliminary results. In Section 3, the existence and uniqueness of the problem(1) are obtained by successive approximation method. In Section 4, an example is given to demonstrate the effectiveness of the main results.

Preliminaries

In this section, we recollect several definitions of fractional derivatives and integals from the papers^27‒30

Definition The fractional (arbitrary) order integral of the function of order is defined by

$I_{a}^{ω} ℌ (t) = \int_{a}^{t} \frac{{(t - s)}^{ω - 1}}{Γ (ω)} ℌ (s) d s,$ (2)

where $Γ$ is the gamma function.

Definition For a function given on the interval , the Caputo fractional order derivative of , is defined by

$(^{c} D_{a^{+}}^{ω} ℌ)(t)= \frac{1}{Γ (n - ω)} \int_{a}^{t} {(t - s)}^{n - ω - 1} ℌ^{(n)} (s) d s,$ (3)

where

$n =[ω] + 1$ . Definition 4.3 The generalized left-sided fractional integral of order is defined by

$(^{ρ} I_{a^{+}}^{ω} ℌ)(t)= \frac{ρ^{1 - ω}}{Γ (ω)} \int_{a}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s,$ (4)

for

$t > a$ , if the integral exists.

Definition Thegeneralized fractional derivative, corresponding to the generalized fractional integral (4), is defined for $0 \leq a < t$ , by

$(^{ρ} D_{a^{+}}^{ω} ℌ)(t)= \frac{ρ^{ω - n + 1}}{Γ (n - 1)} {(t^{1 - ρ} \frac{d}{d t})}^{n} \int_{a}^{t} {(t^{ρ} - s^{ρ})}^{n - ω - 1} s^{ρ - 1} ℌ (s) d s,$ (5)

if the integral exists.

Definition The Caputo-type generalized fractional derivative, $^{ρ} D_{a^{+}}^{ω} ℌ (t)$ is defined via the above generalized fractional derivative (5) as follows

$(_{c}^{ρ} D_{a^{+}}^{ω} ℌ)(t)= (^{ρ} D_{a^{+}}^{ω} [ℌ (s) - \sum_{k =0}^{n - 1} \frac{ℌ^{k} (a)}{k!} {(μ - a)}^{k}]) (t),$ (6)

where

$n = ⌈ R e (ω) ⌉$ . Definition The generalized fractional derivative in Caputo sense, corresponding to the generalized fractional integral in Caputo sense (6), is defined for

$0 \leq a < t$ , by

$(_{c}^{ρ} D_{a^{+}}^{ω} ℌ)(t)= {(t^{1 - ρ} \frac{d}{d t})}^{n} (_{c}^{ρ} I_{a^{+}}^{ω} ℌ)(t)$

$= \frac{ρ^{ω - n + 1}}{Γ (n - ω)} {(t^{1 - ρ} \frac{d}{d t})}^{n} \int_{a}^{t} {(t^{ρ} - s^{ρ})}^{n - ω - 1} s^{ρ - 1} ℌ (s) d s .$ (7)

Remark In Caputo sense, the Katugampola fractional derivative operator $_{c}^{ρ} D_{t}^{ω}$ is a left inverse of the integral operator $_{c}^{ρ} I_{t}^{ω}$ but in general is not a right inverse,

$_{c}^{ρ} D_{t}^{ω} (I_{t}^{ω} ℨ (t)) = ℨ (t)$

and the following holds

$_{c}^{ρ} I_{t}^{ω} (I_{t}^{ω} ℨ (t)) = ℨ (t) - \sum_{k =0}^{n - 1} \frac{{(t^{ρ} - a)}^{k}}{k!} ℨ^{(k)} (a), t \in [a, b].$ (8)

For readers’ understanding, we introduce the following notations for the following lemma and theorem. Let

We denote exist and . Obviously,is a Banach space with the norm

Lemma 4.8 Assume that A function is a solution of the initial value problem

$(\begin{matrix} _{c}^{ρ} D_{0^{+}}^{ω} ℨ (t)= ℌ (t), t \neq t_{k}, t \in J :=[0, T]; \\ Δ ℨ (t_{k})= I_{k} (ℨ (t_{k})), k =1,2,..., m; \\ ℨ (t)= ψ (t), t \in [- μ,0] \end{matrix}$ (9)

if and only if $ℨ$ satisfies the following integral equation

$ℨ (t)= (\begin{matrix} ψ (t), t \in [- μ,0]; \\ \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s + \sum_{j =1}^{k} I_{j} (ℨ (t_{j})) \\ + \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s, t \in (t_{k}, t_{k + 1}], k =0,1,2,..., m . \end{matrix}$ (10)

Proof. Assume that

$ℨ$ satisfies (9). One can see, from Remark 2.7 and

$ψ (0)=0,$ that

$ℨ (t)= \frac{ρ^{1 - ω}}{Γ (ω)} \int_{0}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s, f o r t \in J_{0} =[t_{0}, t_{1}].$

In view of

$ℨ (t_{1}^{+}) - ℨ (t_{1}^{-}) = I_{1} (ℨ (t_{1})),$ we get that

$ℨ (t_{1}^{+}) = I_{1} (ℨ (t_{1})) + \frac{ρ^{1 - ω}}{Γ (ω)} \int_{0}^{t_{1}} {(t_{1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s .$

It follows that, for

$t \in (t_{1}, t_{2}],$

$ℨ (t)= ℨ (t_{1}^{+}) + \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{1}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s$

$= \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{1}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s + \frac{ρ^{1 - ω}}{Γ (ω)} \int_{0}^{t_{1}} {(t_{1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s + I_{1} (ℨ (t_{1})) .$

In consequence, we can see, by means of

$ℨ (t_{2}^{+}) = ℨ (t_{2}^{-}) + I_{2} (ℨ (t_{2})),$

that

$ℨ (t_{2}^{+}) = \sum_{i =0}^{1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℨ (s) d s + \sum_{j =1}^{2} I_{j} (ℨ (t_{j})),$

which implies that for

$t \in (t_{2}, t_{3}],$

$ℨ (t)= \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{2}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s$

Repeating the above process, the solution

$ℨ (t)$ for

$t \in (t_{k}, t_{k + 1}]$ can be written as

$ℨ (t)= \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s$

$+ \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s + \sum_{j =1}^{k} I_{j} (ℨ (t_{j})).$

Conversely, if

$ℨ$ is a solution of (10), one can obtain by a direct computation, that

$_{c}^{ρ} D_{0^{+}}^{ω} ℨ (t)= ℌ (t),$

$t \neq t_{k},$

$t \in [0, T],$ and

$Δ ℨ (t_{k})= ℨ (t_{k}^{+}) - ℨ (t_{k}^{-}) = I_{k} (ℨ (t_{k})),$ where

$ℨ (t_{k}^{+})= \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s + \sum_{j =1}^{k} I_{j} (ℨ (t_{j})),$

and

$ℨ (t_{k}^{-})= \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k - 1}}^{t_{k}} {(t_{k}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s + \sum_{i =0}^{k - 2} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s) d s$

$+ \sum_{j =1}^{k - 1} I_{j} (ℨ (t_{j})).$

This completes the proof.

Existence and uniqueness results

Initially, set $C_{0} = {v | v \in ℭ (J, ℝ), v (0)=0} .$ For each, $v \in C_{0}$ ,we denote by $\bar{v} (t)= v (t), 0 \leq t \leq T a n d \bar{v} (t)=0, - μ \leq t \leq 0.$ the function defined by

$\bar{v} (t)= v (t), 0 \leq t \leq T a n d \bar{v} (t)=0, - μ \leq t \leq 0.$ (11)

$ℨ$ is a solution of (1), then

$ℨ (.)$ can be decomposed as

$ℨ (t)= \bar{v} (t) + ϕ (t)$ for

$- μ \leq t \leq T,$ which implies that

$ℨ_{t} = {\bar{v}}_{t} + ϕ_{t}$ for

$0 \leq t \leq T,$ where

$ϕ (t)=0, 0 \leq t \leq T, a n d ϕ (t)= ψ (t), - μ \leq t \leq 0.$ (12)

Therefore, the problem (1) can be transformed into the following fixed point problem of the operator

$N : ℭ_{0} \to ℝ,$

$N v (t)= \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s, {\bar{v}}_{s} + ϕ_{s}) d s$

$+ \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s, {\bar{v}}_{s} + ϕ_{s}) d s$

$+ \sum_{j =1}^{k} I_{j} (\bar{v} (t_{j})), t \in (t_{k}, t_{k + 1}], k =0,1,2,..., m .$ (13)

Now, let us present our main results.

Theorem For the functions $ℌ \in ℭ (J \times ℝ, ℝ)$ and $I_{k} : ℝ \to ℝ$ , assume the following conditions hold • There exists a continuous function $α :[0, T] \to ℝ^{+}$ satisfying $| ℌ (t, p_{t}) - ℌ (t, q_{t}) | \leq α (t) s u p_{s \in [0, t]} | p (s) - q (s) |, p, q \in ℝ, t \in [0, T];$

• There exists a constant $L_{k} >0$ such that $| I_{k} (p) - I_{k} (q) | \leq L_{k} | p - q |, k =1,2,..., m;$

$\sum_{i =1}^{m + 1} \frac{α_{i} T^{ρ ω}}{ρ^{ω} Γ (ω + 1)} + \sum_{j =1}^{m} L_{j} <1,$ where $α_{k} =_{t \in (t_{k}, t_{k + 1})}^{s u p} α (t);$ • There exists a constant $M >0$ such that $| ℌ (t, ϕ_{t}) | \leq M$ , where $ϕ$ is defined in (12).

Proof
To complete the proof, we shall use the method of successive approximations. Define a sequence of functions $v_{n} :[0, T] \to ℝ,$ $n =1,2,...$ as follows:

$v_{0} (t)=0, v_{n} (t)= N v_{n - 1} (t).$ (14)

Since

$v_{0} (t)=0,$ it is easy to see from(11) that

${({\bar{v}}_{0})}_{s} =0$ for

$s \in [0, T] .$ Thus we have,

$| v_{1} (t) - v_{0} (t) | \leq \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} | ℌ (s, ϕ (s)) | d s + \sum_{j =1}^{k} I_{j} (0)$

$+ \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} | ℌ (s, ϕ (s)) | d s$

$\leq \frac{M (t^{ρ ω} - t_{k}^{ρ ω})}{ρ^{ω} Γ (ω + 1)} + \sum_{i =1}^{k} \frac{M (t_{i}^{ρ ω} - t_{i - 1}^{ρ ω})}{ρ^{ω} Γ (ω + 1)} + \sum_{j =1}^{k} | I_{j} (0) |$

$\leq \sum_{i =1}^{m + 1} \frac{M (t_{i}^{ρ ω} - t_{i - 1}^{ρ ω})}{ρ^{ω} Γ (ω + 1)} + \sum_{j =1}^{k} | I_{j} (0) | := N_{0}, k =1,2,.., m,$

it follows that

$‖ v_{1} (t) - v_{0} (t) ‖ \leq N_{0} .$ Furthermore,

$| v_{n} (t) - v_{n - 1} (t) | \leq \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} | ℌ (s,({\bar{v}}_{n - 1})_{s} + ϕ_{s}) - ℌ (s,({\bar{v}}_{n - 2})_{s} + ϕ_{s}) | d s$

$+ \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} | ℌ (s,({\bar{v}}_{n - 1})_{s} + ϕ_{s}) - ℌ (s,({\bar{v}}_{n - 2})_{s} + ϕ_{s}) | d s$

$+ \sum_{j =1}^{k} | I_{j} ({\bar{v}}_{n - 1})(t_{j}) - I_{j} ({\bar{v}}_{n - 2})(t_{j}) |$

$\leq \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} α {(s)}_{x \in [0, s]}^{s u p} | {\bar{v}}_{n - 1} (x) - {\bar{v}}_{n - 2} (x) | d s$

$+ \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} α {(s)}_{x \in [0, s]}^{s u p} | {\bar{v}}_{n - 1} (x) - {\bar{v}}_{n - 2} (x) | d s$

$+ \sum_{j =1}^{k} I_{j} | {\bar{v}}_{n - 1} (t_{j}) - {\bar{v}}_{n - 2} (t_{j}) |$

$\leq (α_{k} \frac{(t^{ρ ω} - t_{k}^{ρ ω})}{Γ (ω + 1)} + \sum_{i =1}^{k} α_{i} \frac{(t_{i}^{ρ ω} - t_{i - 1}^{ρ ω})}{Γ (ω + 1)} + \sum_{j =1}^{k} L_{j}) . ‖ v_{n - 1} - v_{n - 2} ‖$

$\leq (\sum_{i =1}^{m + 1} α_{i} \frac{T^{ω ρ}}{ρ^{ω} Γ (ω + 1)} + \sum_{j =1}^{m} L_{j}) . ‖ v_{n - 1} - v_{n - 2} ‖$

$:= N_{1} ‖ v_{n - 1} - v_{n - 2} ‖,$ (15)

which implies that

$‖ v_{n} - v_{n - 1} ‖ \leq N_{1} ‖ v_{n - 1} - v_{n - 2} ‖$ with

$N_{1} <1 .$ Note that for any

$r > n >0,$ we have

$‖ v_{r} - v_{n} ‖ \leq ‖ v_{n + 1} - v_{n} ‖ + ‖ v_{n + 2} - v_{n + 1} ‖ + ... + ‖ v_{r} - v_{r - 1} ‖$

$\leq (N_{1}^{n} + N_{1}^{n + 1} + ... + N_{1}^{r - 1}) ‖ v_{1} - v_{0} ‖$

$‖ v_{r} - v_{n} ‖ \leq \frac{N_{1}^{n}}{1 - N_{1}} ‖ v_{1} - v_{0} ‖ .$ (16)

for sufficiently large numbers

$r, n,$ it follows from the above inequalities with

$N_{1} <1$ that

$‖ v_{r} - v_{n} ‖ \to 0.$ Thus,

${v_{n} (t)}$ is a Cauchy sequence in

$P ℭ (J).$ Since

$P ℭ (J)$ is a complete Banach space, then

$‖ v_{n} - v ‖ \to 0 (n \to \infty),$ for some

$v \in P ℭ (J),$ which means that

$v_{n} (t)$ is uniformly convergent to

$v (t)$ with respect to

$t .$ In what follows, we shall show that

$v (t)$ is a solution of the equation (1). Observe that

$| \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s,({\bar{v}}_{n})_{s} + ϕ_{s}) d s - \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s,({\bar{v}}_{s} + ϕ_{s})) d s |$

$\leq \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} | ℌ (s,({\bar{v}}_{n})_{s} + ϕ_{s}) - ℌ (s,({\bar{v}}_{s} + ϕ_{s})) | d s$

$\leq \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} α (t)(t^{ρ} - s^{ρ})^{ω - 1} s^{ρ - 1}_{x \in [0, s]}^{s u p} | {\bar{v}}_{n} (x) - \bar{v} (x) | d s$

$= \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} α (t)(t^{ρ} - s^{ρ})^{ω - 1} s^{ρ - 1}_{x \in [0, s]}^{s u p} | v_{n} (x) - v (x) | d s .$

Since

$v_{n} (t) \to v (t)$ as

$n \to + \infty,$ for any

$ε >0,$ there exists a sufficiently large number

$n_{0} >0$ such that for all

$n > n_{0},$ we have

$| v_{n} (x) - v (x) | < m i n {\frac{ρ^{ω} Γ (ω + 1)}{\sum_{i =0}^{m} α_{i} T^{ρ ω}} ε, \frac{1}{\sum_{j =1}^{m} L_{j}} ε} .$

Therefore,

$| \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s,({\bar{v}}_{n})_{s} + ϕ_{s}) d s$

$- \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} � (s,({\bar{v}}_{s} + φ_{s})) d s |$

$\leq \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} | ℌ (s,({\bar{v}}_{n})_{s} + ϕ_{s}) - ℌ (s,({\bar{v}}_{s} + ϕ_{s})) | d s$

$\leq \sum_{i =0}^{k - 1} α (t_{i}) \frac{{(t_{i}^{ρ} - t_{i - 1}^{ρ})}^{ω}}{ρ^{ω}} Γ (ω + 1)_{x \in [0, s]}^{s u p} | v_{n} (x) - v (x) | d s < ε .$ (18)

and

$| \sum_{j =1}^{k} I_{j} ({\bar{v}}_{n} (t_{j})) - \sum_{j =1}^{k} I_{j} (\bar{v} (t_{j})) | \leq \sum_{j =1}^{k} L_{j} | {\bar{v}}_{n} (t_{j}) - \bar{v} (t_{j}) |$

$= \sum_{j =1}^{k} L_{j} | v_{n} (t_{j}) - v (t_{j}) | < ε .$ (19)

In consequence, we can see that for a sufficiently large number

$n > n_{0},$

$| v (t) - N v (t) |$

$\leq | v (t) - v_{n + 1} (t) | + | v_{n + 1} (t) - N v_{n} (t) | + | N v_{n} (t) - N v (t) |$

$\leq | v (t) - v_{n + 1} (t) | + | v_{n + 1} (t) - [\frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s,({\bar{v}}_{n})_{s} + ϕ_{s}) d s$

$+ \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s,({\bar{v}}_{n})_{s} + ϕ_{s}) d s + \sum_{j =1}^{k} I_{j} ({\bar{v}}_{n} (t_{j}))] |$

$+ | \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s,({\bar{v}}_{n})_{s} + ϕ_{s}) d s - \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s, {\bar{v}}_{s} + ϕ_{s}) d s |$

$+ | \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s,({\bar{v}}_{n})_{s} + ϕ_{s}) d s$

$- \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} ℌ (s, {\bar{v}}_{s} + ϕ_{s}) d s | + | \sum_{j =1}^{k} I_{j} ({\bar{v}}_{n} (t_{j})) - \sum_{j =1}^{k} I_{j} (\bar{v} (t_{j})) | .$

Thus, in view of the convergence of the two previous and (17)-??, one obtains that $| v (t) - N v (t) | \to 0$ , which implies that $v$ is a solution of (1). Finally, we prove the uniqueness of the solution. Assume that $v_{1}, v_{2} :[0, T] \to ℝ$ are two solutions of (1). Note that

$| v_{1} (t) - v_{2} (t) | \leq \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{k}}^{t} {(t^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} α {(s)}_{x \in [0, s]}^{s u p} | {\bar{v}}_{1} (x) - {\bar{v}}_{2} (x) | d s$

$+ \sum_{i =0}^{k - 1} \frac{ρ^{1 - ω}}{Γ (ω)} \int_{t_{i}}^{t_{i + 1}} {(t_{i + 1}^{ρ} - s^{ρ})}^{ω - 1} s^{ρ - 1} α {(s)}_{x \in [0, s]}^{s u p} | {\bar{v}}_{1} (x) - {\bar{v}}_{2} (x) | d s$

$+ \sum_{j =1}^{k} I_{j} | {\bar{v}}_{1} (t_{j}) - {\bar{v}}_{2} (t_{j}) |$

$\leq (\sum_{i =1}^{p + 1} \frac{α_{i} T^{ω ρ}}{ρ^{ω} Γ (ω + 1)} + \sum_{j =1}^{p} L_{j}) ‖ v_{1} - v_{2} ‖ .$

According to the conditions $(A_{3})$ , the uniqueness of the problem (1) follows immediately, which completes the proof.

An illustrative example

Consider the following Katugampola-type fractional impulsive differential equation with delay of the form

$(\begin{matrix} _{c}^{ρ} D_{0^{+}}^{ω} ℨ (t)= \frac{e^{- t} | ℨ_{t} |}{(9 + e^{t})(1 + | ℨ_{t}) |}, t \in [0,1], t \neq \frac{1}{2}, 0< ω <1; \\ Δ ℨ (\frac{1}{2}) = \frac{| ℨ ({\frac{1}{2}}^{-}) |}{3 + | ℨ ({\frac{1}{2}}^{-}) |}; \\ ℨ (t)= ψ (t)= \frac{e^{- t} - 1}{2}, - μ \leq t \leq 0. \end{matrix}$ (20)

Let us take,

$ω = \frac{1}{2},$

$ρ =1,$

$Γ (ω + 1)> \frac{4}{10},$

$μ$ is a non-negative constant.

$ℨ_{t} (θ)= ℨ (t + θ)$ for

$- μ \leq θ \leq 0$ and

$0 \leq t \leq 1 .$ Set

$ℌ (t, ℨ)= \frac{e^{- t} ℨ}{(9 + e^{t})(1 + ℨ)},$

$I (ℨ)= \frac{ℨ}{3 + ℨ},$ for

$(t, ℨ) \in [0,1] \times [0, + \infty) .$ Now, we can see that

$| ℌ (t, p_{t}) - ℌ (t, q_{t}) | = \frac{e^{- t}}{(9 + e^{t})} \frac{| p_{t} - q_{t} |}{(1 + | p_{t} |)(1 + | q_{t} |)}$

$\leq \frac{e^{- t}}{(9 + e^{t})} | p_{t} - q_{t} |$

$\leq α {(t)}_{s \in [0, t]}^{s u p} | p (s) - q (s) |,$ where $α (t)= \frac{e^{- t}}{(9 + e^{t})}$ and $α =_{t \in [0,1]}^{s u p} α (t)= \frac{1}{10},$ so the condition $(A_{1})$ is satisfied. On the other hand, we get that

$| I (p) - I (q) | = \frac{3 | p - q |}{(3 + p)(3 + q)} \leq \frac{1}{3} | p - q |, p, q >0,$

which satisfies the condition

$(A_{1})$ of Theorem 3.1 with

$L = \frac{1}{3} .$ By a direct computation, we obtain that

$\sum_{i =1}^{m + 1} \frac{α_{i} T^{ρ ω}}{ρ^{ω} Γ (ω + 1)} + \sum_{j =1}^{m} L_{j} = \frac{2}{10} \frac{1}{Γ (ω + 1)} + \frac{1}{3} <1,$ and

$| ℌ (t, ℨ_{t}) | = \frac{e^{- t}}{(9 + e^{t})} \frac{| ℨ_{t} |}{(1 + | ℨ_{t} |)} \leq \frac{e^{- t}}{9 + e^{t}} \leq \frac{1}{10}, t \in [0,1].$

As a result, the equations in (20) satisfy all the hypotheses in Theorem 3.1 which guarantees that (20) has a unique solution.

Conclusion

In this note, the existences of solutions of a Katugampola-type fractional impulsive differential equation with delay were investigated. The successive approximation method was employed to show the existence of solutions. The example reflects the applicability of the proposed method.