A coupled non-separated system of Hadamard-type fractional differential equations

Available online at www.atnaa.org Research Article

A coupled non-separated system of Hadamard-type fractional dierential equations

Saleh S. Redhwan^a, Suad Y. Al-Mayyahi^b, Sadikali L. Shaikh^c, Mohammed S. Abdo^d

aDepartment of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004 (M.S.), India.

bDepartment of Mathematics, Dept. of mathematics, college of education of pure science, University of Wasit, Iraq.

cDepartment of Mathematics, Maulana Azad College of arts, Science and Commerce, RozaBagh, Aurangabad 431004 (M.S.), India.

dDepartment of Mathematics, Department of Mathematics, Hodeidah University, Al-Hodeidah, Yemen.

Abstract

In this article, we discuss the existence and uniqueness of solutions of a coupled non-separated system for fractional dierential equations involving a Hadamard fractional derivative. The existence and uniqueness results obtained in the present study are not only new but also cover some results corresponding to special values of the parameters involved in the Caputo problems. These developed results are obtained by apply- ing Banach's xed point theorem and Leray-Schauder's nonlinear alternative. An example is presented to illustrate our main results.

Keywords: Hadamard-type fractional dierential equations xed point theorems boundary conditions.

2010 MSC: 26A33 34A12 30E25

1. Introduction

Fractional calculus (FC) is one of the sections of mathematics which is a popularization of classical calculus that include integrals and derivatives of non-integer order. FC has a substantial role in numerous

elds of science, engineering, and economics. FC tools have been found to support the development of mathematical methods which is more pragmatic to applied problems an expression of fractional dierential equations (FDEs). Recently, many versions of the fractional derivatives, such as Reimann-Liouville, Caputo,

Email addresses: saleh.redhwan909@gmail.com (Saleh S. Redhwan), Suadluck7@gmail.com (Suad Y. Al-Mayyahi), sad.math@gmail.com (Sadikali L. Shaikh), msabdo1977@gmail.com (Mohammed S. Abdo)

Received April 26, 2021, Accepted October 09, 2021, Online October 12, 2021

Hadamard, Katugampola, Hilfer, ψ-Caputo, ψ-Hilfer, Atangana-Baleanu were presented, Here we refer to [2, 4, 15, 16, 19, 20, 21, 23, 24, 26, 27, 30].

Initial and boundary value problems for FDEs have won considerable signicance because of their many employment in applied sciences and engineering. Many authors have shown great interest in this topic and obtaining a variety of results for FDEs involving dierent kinds of the conditions and the fractional operators, see [1, 9, 28, 29, 34] and the references therein. Some numerical methods such as the Adomian decomposition method for solving the nonlinear FDEs and system of nonlinear FDEs were studied by Jafari and Daftardar-Gejji in [12, 17, 18].

Several authors studied the existence and uniqueness theory for FDEs involving Hadamard-type operator, see [2, 3, 5, 11, 14, 22, 25, 31] and references therein. Coupled systems of FDEs are of great value and an important eld. This type of system arises in business mathematics, management sciences, and other managerial sciences and so forth. To model such problems and some theoretical works on the coupled systems of FDEs, we refer the reader to some studied works [6, 7, 8, 10, 32, 33]. Motivated by the above discussion, in this article, we consider a coupled system of Hadamard-type FDEs:

 D^ρ1ω(ϑ) = }1(ϑ, ω(ϑ), $(ϑ)), ϑ ∈ [1, e], 1 < ρ ≤ 2

D^σ₁$(ϑ) = }2(ϑ, ω(ϑ), $(ϑ)), ϑ ∈ [1, e], 1 < σ ≤ 2 , (1) with the following non-separated coupled boundary conditions:

 ω(1) = ξ1$(e)

$(1) = `1ω(e)

 ω⁰(1) = ξ2$⁰(e)

$⁰(1) = `2ω⁰(e) , (2)

where ξ1, ξ2,`1, `2 are real constants, D^ρ₁, D^σ1 are the Hadamard fractional derivatives of order ρ and σ respectively, and }1, }2 :[1, e] × R × R → R are appropriate functions.

The existence and uniqueness of solutions for the system (1)-(2) are mainly investigated. To the best of our knowledge, a coupled system of Hadamard-type FDEs (1) with non-separated coupled boundary conditions (2) have were not widely studied. In consequence, the coupled system of FDEs with non-separated coupled boundary conditions will be studied by the Hadamard fractional derivative. Moreover, the main results are obtained by applying Banach xed point theorem and Leray-Schauder xed point theorem.

The outline of the paper is the following. In Section 2, we present some basic denitions and known results related to fractional calculus. Section 3 is devoted to proving the existence and uniqueness of the Hadamard coupled system (1)-(2). In the end, we present an illustrative example to justify our main results.

2. Preliminaries:

First of all, we present some denitions and properties from fractional calculus used throughout this article.

Denition 2.1. [20] For a continuous function ω : [1, +∞) → R, the Hadamard fractional integral of order ρ > 0 is dened by

I^ρ₁ω(ϑ) = 1 Γ(ρ)

Z ϑ 1

(logϑ

s)^ρ−1ω(s)ds s , provided the right-hand side is point-wise dened on [1, +∞).

Denition 2.2. [16] Let n−1 < ρ < n, and ω(ϑ) has an absolutely continuous derivative up to order (n−1).

Then the Caputo-Hadamard fractional derivative of order ρ is dened as

D^ρ₁ω(ϑ) = 1 Γ(n − ρ)

Z ϑ 1

(logϑ

s)^n−ρ−1δⁿ(ω)(s)ds s , where δⁿ= (ϑ_dϑ^d )ⁿ, and n = [ρ] + 1.

Lemma 2.3. [16] Let ρ > 0 and ω ∈ Cⁿ[1, +∞) such that δ⁽ⁿ⁾(ω) exists a.e. on any bounded interval of [1, +∞). Then we have

I^ρ₁[D^ρ₁ω(ϑ)] = ω(ϑ) −

n−1

X

k=0

δ^(k)ω(1)

Γ(k + 1)(log ϑ)^k. In particular, if 0 < ρ < 1, then we have I^ρ₁[D^ρ₁ω(ϑ)] = ω(ϑ) − ω(1).

Lemma 2.4. [20] For all ` > 0 and ν > −1, then we have 1

Γ(`) Z ϑ

1

(logϑ

s)^`−1(log s)^νds

s = Γ(ν)

Γ(` + ν)(log ϑ)<sup>ν+`−1</sup>.

Theorem 2.5. [20] (Banach xed point theorem) Let (W, d) be a nonempty complete metric space with Π : W → W is a contraction mapping. Then map Π has a xed point.

Theorem 2.6. [13] (Leray-Schauder Nonlinear Alternative). Let W be a Banach space and S ⊆ W closed and convex. Assume that K is a relatively open subset of S with 0 ∈ K and Π : K −→ S is a compact and continuous mapping. Then ethier

1. Π has a xed point in K, or

2. there exists ω ∈ ∂K such that ω = λΠω for some λ ∈ (0, 1), where ∂K is boundary of K.

Now we present an auxiliary lemma which plays a key role in the sequel.

Lemma 2.7. Let u, v ∈ C([1, e], R). Then the solution of the linear fractional dierential system











D^ρ₁ω(ϑ) = u(ϑ), ϑ ∈ [1, e], 1 < ρ ≤ 2 D^σ1$(ϑ) = v(ϑ), ϑ ∈ [1, e], 1 < σ ≤ 2 ω(1) = ξ₁$(e), ω⁰(1) = ξ₂$⁰(e)

$(1) = `1ω(e), $<sup>0</sup>(1) = `2ω⁰(e)

, (3)

is equivalent to the system of integral equations ω(ϑ) = ξ1

η₂ 1 Γ(ρ)

Z e 1

(loge

s)^ρ−1u(s)ds

s +`1ξ1

η₂ 1 Γ(σ)

Z e 1

(loge

s)^σ−1v(s)ds s + `1ξ1ξ2

η₁η₂ +`2ξ1ξ2

eη₁η₂ +`2ξ2

eη₁ log ϑ

 1

Γ(ρ − 1) Z e

1

(loge

s)^ρ−2u(s)ds s + `<sub>1</sub>`₂ξ₁ξ₂

eη₁η₂ + `₂ξ₁ η₁η₂ + ξ₂

η₁ log ϑ

 1

Γ(σ − 1) Z e

1

(loge

s)^σ−2v(s)ds s + 1

Γ(ρ) Z ϑ

1

(logϑ

s)^ρ−1u(s)ds

s, (4)

and

$(ϑ) = `1

η₂ 1 Γ(σ)

Z _e

1

(loge

s)^σ−1v(s)ds

s +`1ξ1

η₂ 1 Γ(ρ)

Z _e

1

(loge

s)^ρ−1u(s)ds s + `1`2ξ1ξ2

eη₁η₂ + `1ξ2

η₁η₂ + `1

η₁log ϑ

 1

Γ(ρ − 1) Z _e

1

(loge

s)^ρ−2u(s)ds s + `<sub>1</sub>`₂ξ₁

η1η2

+`<sub>1</sub>`₂ξ₂ eη1η2

+`₂ξ₂ eη1

log ϑ

 1

Γ(σ − 1) Z e

1

(loge

s)^σ−2v(s)ds s + 1

Γ(σ) Z ϑ

1

(logϑ

s)^σ−1v(s)ds

s , (5)

where η1= 1 − <sup>`2</sup><sub>e</sub><sup>ξ</sup>2<sup>2</sup> 6= 0, η<sub>2</sub> = 1 − `₁ξ₁ 6= 0.

Proof. We know that the general solution of Hadamard coupled system in (3) can be written as

ω(ϑ) = c0+ c1(log ϑ) + 1 Γ(ρ)

Z ϑ 1

(logϑ

s)^ρ−1u(s)ds

s , (6)

$(ϑ) = d₀+ d₁(log ϑ) + 1 Γ(σ)

Z ϑ 1

(logϑ

s)^σ−1v(s)ds

s , (7)

where ci, d_i (i = 0, 1) are arbitrary real constants.

Using the non-separated coupled boundary conditions in (6) and (7), we have

ω(1) = ξ1$(e) ⇒ c₀= ξ1



d0+ d1+ 1 Γ(σ)

Z e 1

(loge

s)^σ−1v(s)ds s



, (8)

$(1) = `<sub>1</sub>ω(e) ⇒ h<sub>o</sub> = `₁



c₀+ c₁+ 1 Γ(ρ)

Z e 1

(loge

s)^ρ−1u(s)ds s



. (9)

Similarly, by using the non-separated coupled boundary conditions in (6) and (7), we have

ω⁰(1) = ξ₂$⁰(e) ⇒ c₁= ξ₂ d₁

e + 1

Γ(σ − 1) Z e

1

(loge

s)^σ−2v(s)ds s



, (10)

$⁰(1) = `<sub>2</sub>ω<sup>0</sup>(e) ⇒ d<sub>1</sub> = `₂[c₁

e + 1

Γ(ρ − 1) Z e

1

(loge

s)^ρ−2u(s)ds

s ]. (11)

From the relations (10) and (11), we get c₁ = `₂ξ₂

eη1

1 Γ(ρ − 1)

Z e 1

(loge

s)^ρ−2u(s)ds s +ξ₂

η1

1 Γ(σ − 1)

Z e 1

(loge

s)^σ−2v(s)ds

s , (12)

d1 = `2ξ2

eη₁ 1 Γ(σ − 1)

Z e 1

(loge

s)^σ−2v(s)ds s +`2

η₁ 1 Γ(ρ − 1)

Z e 1

(loge

s)^ρ−2u(s)ds

s. (13)

Substituting c1 and d1 into (8) and (9), we get c₀ = ξ₁

η₂ 1 Γ(ρ)

Z e 1

(loge

s)^ρ−1u(s)ds

s +`₁ξ₁ η₂

1 Γ(σ)

Z e 1

(loge

s)^σ−1v(s)ds s + `₁ξ₁ξ₂

η₁η₂ +`₂ξ₁ξ₂ eη₁η₂

 1

Γ(ρ − 1) Z e

1

(loge

s)^ρ−2u(s)ds s + `<sub>1</sub>`₂ξ₁ξ₂

eη1η2

+ `₂ξ₁ η1η2

 1

Γ(σ − 1) Z e

1

(loge

s)^σ−2v(s)ds s , and

d0 = `1

η₂ 1 Γ(σ)

Z e 1

(loge

s)^σ−1v(s)ds

s +`1ξ1

η₂ 1 Γ(ρ)

Z e 1

(loge

s)^ρ−1u(s)ds s + `<sub>1</sub>`2ξ1ξ2

eη₁η₂ + `1ξ2

η₁η₂

 1

Γ(ρ − 1) Z e

1

(loge

s)^ρ−2u(s)ds s + `1`2ξ1

η₁η₂ +`1`2ξ2

eη₁η₂

 1

Γ(σ − 1) Z _e

1

(loge

s)^σ−2v(s)ds s.

Substituting the values of ci, d_i(i = 0, 1)in (6) and (7), we get solutions (4) and (5). The converse follows by direct computation. This completes the proof.

3. Main results:

Let us introduce the space

W = {ω(ϑ)|ω(ϑ) ∈ C([1, e], R)}, endowed with the norm

kωk = sup{|ω(ϑ)| , ϑ ∈ [1, e]}.

Clearly, (W, k·k) is a Banach space. Then the product space W × G = {ω × $ : kω, $k = sup

ϑ∈[1,e]

|(ω, $)(ϑ)|}, is also a Banach space equipped with the norm

k(ω, $)k = kωk + k$k .

In view of Lemma (2.7), we dene the operator Π : W × G → W × G by Π(ω, $)(ϑ) =Π₁(ω, $)(ϑ)

Π₂(ω, $)(ϑ)

 , where

Π1(ω, $)(ϑ) = ξ1

η₂ 1 Γ(ρ)

Z e 1

(loge

s)^ρ−1}1(s, ω(s), $(s))ds s +`1ξ1

η₂ 1 Γ(σ)

Z e 1

(loge

s)^σ−1}2(s, ω(s), $(s))ds s + `₁ξ1ξ2

η₁η₂ +`2ξ1ξ2

eη₁η₂ +`2ξ2

eη₁ log ϑ



×

 1

Γ(ρ − 1) Z _e

1

(loge

s)^ρ−2}1(s, ω(s), $(s))ds s



+ `<sub>1</sub>`₂ξ₁ξ₂ eη1η2

+ `₂ξ₁ η1η2

+ξ₂ η1

log ϑ



×

 1

Γ(σ − 1) Z e

1

(loge

s)^σ−2}2(s, ω(s), $(s))ds s



+ 1 Γ(ρ)

Z ϑ 1

(logϑ

s)^ρ−1}1(s, ω(s), $(s))ds s , and

Π₂(ω, $)(ϑ) = `₁ η₂

1 Γ(σ)

Z e 1

(loge

s)^σ−1}2(s, ω(s), $(s))ds s +`₁ξ₁

η₂ 1 Γ(ρ)

Z _e

1

(loge

s)^ρ−1}1(s, ω(s), $(s))ds s + `<sub>1</sub>`₂ξ₁ξ₂

eη₁η₂ + `<sub>1</sub>ξ<sub>2</sub> η<sub>1</sub>η<sub>2</sub> +`₁

η₁log ϑ



×

 1

Γ(ρ − 1) Z e

1

(loge

s)^ρ−2}1(s, ω(s), $(s))ds s



+ `<sub>1</sub>`2ξ1

η1η2

+`1`2ξ2

eη1η2

+`2ξ2

eη1

log ϑ



×

 1

Γ(σ − 1) Z e

1

(loge

s)^σ−2}2(s, ω(s), $(s))ds s



+ 1 Γ(σ)

Z ϑ 1

(logϑ

s)^σ−1}2(s, ω(s), $(s))ds s .

To simplify, we put

N₁= 1 Γ(ρ)

 |`₁| |ξ₁| |ξ₂|

|η₁| |η₂| + |`₂| |ξ₁| |ξ₂|

e |η₁| |η₂| +|`₂| |ξ₂| e |η₁|



+ 1

Γ(ρ + 1)

 |ξ₁|

|η₂|+ 1



, (14)

N₂ = 1 Γ(σ)

 |`<sub>1</sub>| |`₂| |ξ₁| |ξ₂|

e |η1| |η₂| + |`₂| |ξ₁|

|η₁| |η₂|+ |ξ₂|

|η₁|



+ 1

Γ(σ + 1)

 |`₁| |ξ₁|

|η₂|



, (15)

N3= 1 Γ(ρ)

 |`<sub>1</sub>| |`₂| |ξ₁| |ξ₂|

e |η₁| |η₂| + |`₁| |ξ₂|

|η₁| |η₂|+ |`₁|

|η₁|



+ 1

Γ(ρ + 1)

 |`₁| |ξ₁|

|η₂|



, (16)

N₄= 1 Γ(σ)

 |`1| |`₂| |ξ₁|

|η₁| |η₂| + |`<sub>1</sub>| |`₂| |ξ₂|

e |η₁| |η₂| +|`₂| |ξ₂| e |η₁|



+ 1

Γ(σ + 1)

 |`1|

|η₂|+ 1



. (17)

The rst result is based on Banach's xed point theorem. To this end, we need the following hypotheses:

(H1) }1, }2 : [1, e] × R × R → R are continuous and there exist two positive constants κ1 and κ2 such that

|}1(ϑ, ω1, ω2) − }1(ϑ, $1, $2)| ≤ κ₁(|ω1− $₁| + |ω₂− $₂|) ,

|}2(ϑ, ω1, ω2) − }2(ϑ, $1, $2)| ≤ κ₂(|ω1− $₁| + |ω₂− $₂|) , for all ϑ ∈ [1, e] and ωi, $_i ∈ R i = 1, 2.

(H2) There exist real constants δi, βi ≥ 0, i = 0, 1, 2,such that ∀ωj ∈ R(j = 1, 2), we have

|}1(ϑ, ω₁, ω₂)| ≤ δ₀+ δ₁|ω₁| + δ₂|ω₂| ,

|}2(ϑ, ω₁, ω₂)| ≤ β₀+ β₁|ω₁| + β₂|ω₂| . Theorem 3.1. Assume that (H1) holds. If

φ₁ := [(N₁+ N₃) κ₁+ (N₂+ N₄) κ₂] < 1, (18) then system (1)-(2) has a unique solution, where Ni, i = 1, 2, 3, 4 are given by (14)-(17).

Proof. Let supϑ∈[1,e]}1(ϑ, 0, 0) = 1 < ∞and supϑ∈[1,e]}2(ϑ, 0, 0) = 2 < ∞and r > 0, we show that ΠBr⊂ Br, where

Br = {(ω, $) ∈ W × G : k(ω, $)k < r}, with r ≥ _1−φ^φ2₁,where φ1< 1 and

φ₂:= (N₁+ N₃) ₁+ (N₂+ N₄) ₂. By hypotheses (H1) and for (ω, $) ∈ Br, ϑ ∈ [1, e], we have

|}1(ϑ, ω(ϑ), $(ϑ))| ≤ |}1(ϑ, ω(ϑ), $(ϑ)) − }1(ϑ, 0, 0)| + |}1(ϑ, 0, 0)|

≤ κ₁(|ω(ϑ)| + |$(ϑ)|) + ₁

≤ κ₁r + 1. As same way, we have

|}2(ϑ, ω(ϑ), $(ϑ))| ≤ κ₂r + ₂,

which give to

|Π₁(ω, $)(ϑ)| ≤  |ξ₁|

|η₂|

 1 Γ(ρ)

Z e 1

(loge

s)^ρ−1|}1(s, ω(s), $(s))|ds s + |`1| |ξ₁|

|η₂|

 1 Γ(σ)

Z e 1

(loge

s)^σ−1|}2(s, ω(s), $(s))|ds s + |`₁| |ξ₁| |ξ₂|

|η₁| |η₂| + |`₂| |ξ₁| |ξ₂|

e |η₁| |η₂| +|`₂| |ξ₂| e |η₁| |log ϑ|



×

 1

Γ(ρ − 1) Z e

1

(loge

s)^ρ−2|}1(s, ω(s), $(s))|ds s



+ |`<sub>1</sub>| |`₂| |ξ₁| |ξ₂|

e |η1| |η₂| + |`₂| |ξ₁|

|η₁| |η₂|+ |ξ₂|

|η₁||log ϑ|



×

 1

Γ(σ − 1) Z e

1

(loge

s)^σ−2|}2(s, ω(s), $(s))|ds s



≤  |ξ₁|

|η₂|+ 1 (κ₁r + ₁)

Γ(ρ + 1) + |`₁| |ξ₁|

|η₂|

 (κ₂r + ₂) Γ(σ + 1) + |`₁| |ξ₁| |ξ₂|

|η₁| |η₂| + |`₂| |ξ₁| |ξ₂|

e |η1| |η₂| +|`₂| |ξ₂| e |η1|

 (κ₁r + ₁) Γ(ρ) + |`<sub>1</sub>| |`₂| |ξ₁| |ξ₂|

e |η1| |η₂| + |`₂| |ξ₁|

|η₁| |η₂|+ |ξ₂|

|η₁|

 (κ₂r + 2) Γ(σ)

≤ (κ₁N₁+ κ₂N₂)r + (₁N₁+ ₂N₂).

Hence

kΠ₁(ω, $)(ϑ)k ≤ (κ₁N₁+ κ₂N₂)r + (₁N₁+ ₂N₂).

Similarly, we obtain

|Π₂(ω, $)(ϑ)| ≤  |`₁|

|η₂|

 (κ₂r + ₂)

Γ(σ + 1) + |`₁| |ξ₁|

|η₂|

 (κ₁r + ₁) Γ(ρ + 1) + |`<sub>1</sub>| |`₂| |ξ₁| |ξ₂|

e |η1| |η₂| + |`₁| |ξ₂|

|η₁| |η₂|+ |`₁|

|η₁|

 (κ₁r + 1) Γ(ρ) + |`1| |`₂| |ξ₁|

|η₁| |η₂| +|`<sub>1</sub>| |`₂| |ξ₂|

e |η₁| |η₂| + |`₂| |ξ₂| e |η₁|

 (κ2r + 2)

Γ(σ) +(κ2r + 2) Γ(σ + 1)

=

 1 Γ(ρ)

 |`<sub>1</sub>| |`₂| |ξ₁| |ξ₂|

e |η₁| |η₂| + |`₁| |ξ₂|

|η₁| |η₂|+ |`₁|

|η₁|



+ 1

Γ(ρ + 1)

 |`₁| |ξ₁|

|η₂|



(κ₁r + ₁) +

 1 Γ(σ)

 |`<sub>1</sub>| |`₂| |ξ₁|

|η₁| |η₂| +|`<sub>1</sub>| |`₂| |ξ₂|

e |η1| |η₂| +|`₂| |ξ₂| e |η1|



+ 1

Γ(σ + 1)

 |`1|

|η₂|+ 1



(κ2r + 2)

= (κ2N3+ κ1N4)r + (2N3+ 1N4).

Hence

kΠ₂(ω, $)(ϑ)k ≤ (κ₁N₃+ κ₂N₄)r + (₁N₃+ ₂N₄).

Consequently,

kΠ(ω, $)(ϑ)k ≤ kΠ₁(ω, $)(ϑ)k + kΠ2(ω, $)(ϑ)k

≤ [(N₁+ N₃)κ₁+ (N₂+ N₄)κ₂] r [N₁+ N₃] ₁+ [N₂+ N₄] ₂

≤ φ₁r + φ2= r.

Now, for (ω1, $₁), (ω₂, $₂) ∈ W × Gand for any ϑ ∈ [1, e], we get

|Π₁(ω2, $2)(ϑ) − Π1(ω1, $1)(ϑ)|

≤

 1 Γ(ρ)

 |`₁| |ξ₁| |ξ₂|

|η₁| |η₂| +|`₂| |ξ₁| |ξ₂|

e |η1| |η₂| +|`₂| |ξ₂| e |η1|



+ 1

Γ(ρ + 1)

 |ξ₁|

|η₂|+ 1



× κ₁(kω2− ω₁k + k$₂− $₁k) +

 1 Γ(σ)

 |`<sub>1</sub>| |`₂| |ξ₁| |ξ₂|

e |η₁| |η₂| + |`₂| |ξ₁|

|η₁| |η₂|+|ξ₂|

|η₁|



+ 1

Γ(σ + 1)

 |`₁| |ξ₁|

|η₂|



× κ₂(kω₂− ω₁k + k$₂− $₁k)

≤ (N₁κ1+ N2κ2) (kω2− ω₁k + k$₂− $₁k) . Hence

kΠ₁(ω2, $2)(ϑ) − Π1(ω1, $1)(ϑ)k ≤ (N1κ1+ N2κ2) (kω2− ω₁k + k$₂− $₁k) . (19) In the same way,

kΠ₂(ω₂, $₂)(ϑ) − Π₂(ω₁, $₁)(ϑ)k ≤ (N₃κ₁+ N₄κ₂) (kω₂− ω₁k + k$₂− $₁k) . (20) It follows from (19) and (20) that

kΠ(ω₂, $₂)(ϑ) − Π(ω₁, $₁)(ϑ)k

≤ [(N₁+ N₃) κ₁+ (N₂+ N₄) κ₂] (kω₂− ω₁k + k$₂− $₁k) .

Since φ1 < 1, the operator Π is a contraction mapping. So, we conclude that the Hadamard coupled system (1)-(2) has a unique solution due to Banach xed point theorem. The proof is completed.

The next result is based on Leray-Schauder's xed point theorem.

Theorem 3.2. Assume that (H1) and (H2) holds. If

[(N₁+ N₃) δ₁+ (N₂+ N₄) β₁] < 1, [(N1+ N3) δ2+ (N2+ N4) β2] < 1, and

[(N₁+ N₃) µ₁+ (N₂+ N₄) µ₂] < 1,

where µ1 > 0, µ2 > 0 and Ni, i = 1, 2, 3, 4, are given by (14)-(17), then the Hadamard coupled system (1)-(2) has at least one solution.

Proof. Firstly, we will prove that the operator Π : W × G → W × G is completely continuous. Since }1 and }2 are continuous functions on [1, e], it is obvious that Π is continuous too. Let S ⊂ W × G be bounded.

Then there exist two positive constants µ1 > 0 and µ2 > 0 such that

|}1(ϑ, ω(ϑ), $(ϑ))| < µ₁, for (ϑ, ω, $) ∈ [1, e] × S,

and

|}2(ϑ, ω(ϑ), $(ϑ))| < µ2, for (ϑ, ω, $) ∈ [1, e] × S.

Then, for any (ω, $) ∈ S, we have

|Π₁(ω, $)(ϑ)| ≤

 1 Γ(ρ)

 |`1| |ξ₁| |ξ₂|

|η₁| |η₂| +|`₂| |ξ₁| |ξ₂|

e |η₁| |η₂| + |`₂| |ξ₂| e |η₁|



+ 1

Γ(ρ + 1)

 |ξ₁|

|η₂|+ 1



µ₁ +

 1 Γ(σ)

 |`<sub>1</sub>| |`₂| |ξ₁| |ξ₂|

e |η1| |η₂| + |`₂| |ξ₁|

|η₁| |η₂|+ |ξ₂|

|η₁|



+ 1

Γ(σ + 1)

 |`₁| |ξ₁|

|η₂|



µ2

≤ N₁µ₁+ N₂µ₂,

which gives kΠ1(ω, $)k ≤ N1µ1+ N2µ2.Analogously, we have kΠ2(ω, $)k ≤ N3µ1+ N4µ2. Thus, it follows from the above inequalities that the operator Π is uniformly bounded, since

kΠ(ω, $)k ≤ [(N₁+ N3) µ1+ (N2+ N4) µ2] < 1.

Next, we show that Π is equicontinuous, i.e. we prove that a bounded set S is mapped into an equicontinuous set of W × G by Π. Let ϑ1, ϑ2 ∈ [1, e]with ϑ1 < ϑ2. Then we have

|Π₁(ω(ϑ₂), $(ϑ₂)) − Π₁(ω(ϑ₁), $(ϑ₁)|

≤ 1

Γ(ρ) Z _ϑ2

1

(logϑ2

s )^ρ−1|}1(s, ω(s), $(s))|ds s + 1

Γ(ρ) Z ϑ1

1

(logϑ₁

s )^ρ−1|}1(s, ω(s), $(s))|ds s

≤ µ₁

 1 Γ(ρ)

Z ϑ1

1

 (logϑ₂

s )^ρ−1− (logϑ₁

s )^ρ−1 ds s + 1

Γ(ρ) Z ϑ2

ϑ1

(logϑ2

s )^ρ−1ds s



≤ µ₁

Γ(ρ + 1){2 (log ϑ₂− log ϑ₁)^ρ+ (log ϑ₂)^ρ− (log ϑ₁)^ρ} . (21) In a similar way, we can easily get

|Π₂(ω(ϑ2), $(ϑ2)) − Π2(ω(ϑ1), $(ϑ1)|

≤ µ2

Γ(σ + 1){2 (log ϑ₂− log ϑ₁)^σ+ (log ϑ2)^σ− (log ϑ₁)^σ} . (22) Since log(ϑ) is uniformly continuous on [1, e], the right-hand sides of the inequalities (21) and (22) tend to zero as ϑ2 → ϑ₁. Therefore, the operator Π(ω, $) is equicontinuous. The Arzela-Ascoli theorem along with the above steps shows that Π is completely continuous mapping.

Finally, we shall verify that the set

S = {(ω, $) ∈ W × G : (ω, $) = λΠ(ω, $), 0 ≤ λ ≤ 1}, is bounded. Indeed, Let (ω, $) ∈ S with (ω, $) = λΠ(ω, $). For any ϑ ∈ [1, e], we have

ω(ϑ) = λΠ₁(ω, $)(ϑ), $(ϑ) = λΠ₂(ω, $)(ϑ).

Then

|ω(ϑ)| < |Π₁(ω, $)(ϑ)|

≤  |ξ₁|

|η₂|

 1 Γ(ρ)

Z e 1

(loge

s)^ρ−1|}1(s, ω(s), $(s))|ds s + |`₁| |ξ₁|

|η₂|

 1 Γ(σ)

Z e 1

(loge

s)^σ−1|}2(s, ω(s), $(s))|ds s + |`₁| |ξ₁| |ξ₂|

|η₁| |η₂| +|`₂| |ξ₁| |ξ₂|

e |η₁| |η₂| +|`₂| |ξ₂| e |η₁| |log ϑ|



×

 1

Γ(ρ − 1) Z e

1

(loge

s)^ρ−2|}1(s, ω(s), $(s))|ds s



+ |`<sub>1</sub>| |`₂| |ξ₁| |ξ₂|

e |η1| |η₂| + |`₂| |ξ₁|

|η₁| |η₂|+|ξ₂|

|η₁||log ϑ|



×

 1

Γ(σ − 1) Z e

1

(loge

s)^σ−2|}2(s, ω(s), $(s))|ds s



+ 1 Γ(ρ)

Z ϑ 1

(logϑ

s)^ρ−1|}1(s, ω(s), $(s))|ds s . Assumption (H2) gives

|ω(ϑ)| ≤ N₁(δ₀+ δ₁|ω| + δ₂|$|) + N₂(β₀+ β₁|ω| + β₂|$|)

= N₁δ₀+ N₂β₀+ (N₁δ₁+ N₂β₁) |ω| + (N₁δ₂+ N₂β₂) |$| , and

|$(ϑ)| ≤ N₃(δ0+ δ1|ω| + δ₂|$|) + N₄(β0+ β1|ω| + β₂|$|)

≤ N₃δ0+ N4β0+ (N3δ1+ N4β1) |ω| + (N3δ2+ N4β2) |$| . Hence

kωk ≤ N₁δ₀+ N₂β₀+ (N₁δ₁+ N₂β₁) kωk + (N₁δ₂+ N₂β₂) k$k , and

k$k ≤ N₃δ₀+ N₄β₀+ (N₃δ₁+ N₄β₁) kωk + (N₃δ₂+ N₄β₂) k$k , which imply that

kωk + k$k ≤ (N₁+ N₃) δ₀+ (N₂+ N₄) β₀+ [(N₁+ N₃)δ₁+ (N₂+ N₄)β₁] kωk + [(N1+ N3)δ2+ (N2+ N4)β2] k$k . Consequently,

kω + $k ≤ (N₁+ N₃) δ₀+ (N₂+ N₄) β₀

N₀ ,

where N0= min{1 − (N₁+ N₃)δ₁+ (N₂+ N₄)β₁, 1 − (N₁+ N₃)δ₂+ (N₂+ N₄)β₂},which shows that the set S is bounded. Thus, as a consequence Theorem 2.6, the Hadamard coupled system (1)-(2) has at least one solution. The proof is complete.

Example 3.3. Let ρ = σ = ⁴₃, ξ1= ³₄, ξ₂ = ¹₂, `<sub>1</sub> = <sup>3</sup><sub>2</sub>, `₂ = ¹₂,

}1(ϑ, ω, $) = 1 16(ϑ + 1)²

|ω(ϑ)|

1 + |ω(ϑ)| + 1 + 1

128sin²($(ϑ)) + 1

√

ϑ²+ 1, and

}2(ϑ, ω, $) = 1

128πsin(2πω(ϑ)) + |$(ϑ)|

64 (1 + |$(ϑ)|)+1 2.

Consider the Hadamard-type coupled system (

D

4 3

1ω(ϑ) = }1(ϑ, ω, $), ϑ ∈ [1, e], D

4 3

1$(ϑ) = }2(ϑ, ω, $), ϑ ∈ [1, e], (23)

with the non-separated coupled boundary conditions

 ω(1) = ³₄$(e)

$(1) = ³₂ω(e) ,

 ω⁰(1) = ¹₂$⁰(e)

$⁰(1) = ¹₂ω⁰(e) . (24)

For ϑ ∈ [1, e] and ω, ω^∗, $, $^∗∈ R⁺, we have

|}1(ϑ, ω, ω^∗) − }1(ϑ, $, $^∗)| ≤ 1

64|ω − $| + 1

64|ω^∗− $^∗| and

|}2(ϑ, ω, ω^∗) − }2(ϑ, $, $^∗)| ≤ 1

64|ω − $| + 1

64|ω^∗− $^∗| .

Hence the condition (H1)holds with κ1 = κ₂= ₆₄¹, η₁= 0.966 17 6= 0, η₂ = −0.125 6= 0, sup_ϑ∈[1,e]}1(ϑ, 0, 0) = 1 +^√1₂ = 1 < ∞ and sup_ϑ∈[1,e]}2(ϑ, 0, 0) = ¹₂ = 2 < ∞. We shall check that condition (18) holds. Indeed, by some simple calculations we nd that N1= 11. 841, N2 = 12. 575, N3 = 17. 211 and N4 = 17. 52 With the given data, we see that φ1= 0.924 17 < 1.

Therefore, by Theorem 3.1, we conclude that problem (23)-(24) has a unique solution.

4. conclusion:

In this article, we have established the existence and uniqueness results of a new type of Hadamard-type fractional dierential equations with coupled non-separated boundary conditions. Our analysis is based on the reduction of convert dierential equations to integral equations and using some xed point theorems which are completely general and eective. We are condent the reported results here will have a favorable impact on the expansion of further applications in applied sciences and engineering.

References

[1] S.Y.A. Al-Mayyahi, M.S. Abdo, S.S. Redhwan, and B.N. Abbood, Boundary value problems for a coupled system of Hadamard-type fractional dierential equations, IAENG Int. J. Appl. Math. 51 (1) (2020) 1-10.

[2] Y. Arioua, N. Benhamidouche, Boundary value problem for Caputo-Hadamard fractional dierential equations, Surv.

Math. Appl. 12 (2017) 103-115.

[3] S. Abbas, M. Benchohra, N. Hamidi and Y. Zhou, Implicit coupled Hilfer-Hadamard fractional dierential systems under weak topologies, Adv. Dierence Equ. 2018 (1) (2018) 1-17.

[4] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Si. Numer.

Simul. 44 (2017) 460-481.

[5] Y. Adjabi, F. Jarad, D. Baleanu and T. Abdeljawad, On Cauchy problems with Caputo Hadamard fractional derivatives, J. Comput. Anal. Appl. 21 (1) (2016) 661-681.

[6] H.H. Alsulami, S.K. Ntouyas, R. P. Agarwal, B. Ahmad, and A. Alsaedi, A study of fractional-order coupled systems with a new concept of coupled non-separated boundary conditions, Bound. Value Probl. 2017 (1) (2017) 68.

[7] B. Ahmad, SK. Ntouyas, A. Alsaedi, On a coupled system of fractional dierential equations with coupled nonlocal and integral boundary conditions, Chaos Solitons Fractals 83 (2016) 234-241.

[8] B. Ahmad, SK. Ntouyas, Existence results for a coupled system of Caputo type sequential fractional dierential equations with nonlocal integral boundary conditions, Appl. Math. Comput. 266 (2015) 615-622.

[9] M.S. Abdo, S.K. Panchal, and H.A. Wahash, Ulam- Hyers-Mittag-Leer stability for a ψ-Hilfer problem with fractional order and innite delay, Results in Applied Mathematics 7 (2020) 100115.

[10] M.S. Abdo, K. Shah, S.K. Panchal, and H.A. Wahash, Existence and Ulam stability results of a coupled system for terminal value problems involving ψ-Hilfer fractional operator, Adv. Dierence Equ. 2020 (1) (2020) 1-21.

[11] D.B. Dhaigude, and S.P. Bhairat, Local Existence and Uniqueness of Solution for Hilfer-Hadamard fractional dierential problem, Nonlinear Dyn. Syst. Theory 18 (2) (2018) 144-153.

[12] V. Daftardar-Gejji, H. Jafari, Adomian decomposition: a tool for solving a system of fractional dierential equations, J.

Math. Anal. Appl. 301 (2)(2005) 508-518.

[13] A. Granas, J. Dugundji, Fixed Point Theory, Springer, New York (2005).

[14] Y. Gambo, F. Jarad, D. Baleanu, F. Jarad, On Caputo modication of the Hadamard fractional derivatives, Adv. Dierence Equ. 10 (2014) 1-12.

[15] R. Hilfer, Applications of fractional calculus in Physics, World Scientic, Singapore, (2000).

[16] F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modication of the Hadamard fractional derivatives, Adv. Dierence Equ. 142 (2012) 1-8.

[17] H. Jafari, V. Daftardar-Gejji, Positive solutions of nonlinear fractional boundary value problems using Adomian decompo- sition method, Appl. Math. Comput. 180 (2) (2006) 700-706.

[18] H. Jafari, V. Daftardar-Gejji, Solving a system of nonlinear fractional dierential equations using Adomian decomposition, J. Comput. Appl. Math. 196 (2) (2006) 644-651.

[19] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (4) (2014) 1-15.

[20] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Dierential Equations, North-Holland Mathematics Studies, 204. Elsevier, Amsterdam (2006).

[21] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Dierential Equations, Wiley, New York, (1993).

[22] L. Palve, M.S. Abdo, S.K. Panchal, Some existence and stability results of Hilfer-Hadamard fractional implicit dierential fractional equation in a weighted space, preprint: arXiv, arXiv:1910.08369v1, math.GM. (2019).

[23] I. Podlubny, Fractional Dierential Equations, Academic Press, San Diego (1999).

[24] S.S. Redhwan, M.S. Abdo, K. Shah, T. Abdeljawad, S. Dawood, H.A. Abdo and S.L. Shaikh, Mathematical modeling for the outbreak of the coronavirus (COVID-19) under fractional nonlocal operator, Results in Physics, 19 (2020), 103610.

[25] S.S. Redhwan, S.L. Shaikh, M. S. Abdo, Implicit fractional dierential equation with anti-periodic boundary condition involving Caputo-Katugampola type, AIMS Math. 5 (4) (2020) 3714-3730.

[26] S.S. Redhwan, S.L.Shaikh, Analysis of implicit type of a generalized fractional dierential equations with nonlinear integral boundary conditions, J. Math. Anal. Model. 1 (1) (2020) 64-76.

[27] S.S. Redhwan, S.L. Shaikh, M.S. Abdo, Some properties of Sadik transform and its applications of fractional-order dynam- ical systems in control theory, Adv. Theory Nonlinear Anal. Appl. 4 (1) (2020) 51-66.

[28] J.V.D.C. Sousa, F. Jarad, and T. Abdeljawad, Existence of mild solutions to Hilfer fractional evolution equations in Banach space, Ann. Funct. Anal. 12 (1) (2021) 1-16.

[29] J.V.C. Sousa, D.D.S. Oliveira, and E. Capelas de Oliveira, On the existence and stability for noninstantaneous impulsive fractional integrodierential equation, Math. Methods Appl. Sci. 42 (4) (2019) 1249-1261.

[30] J.V.C. Sousa, and E.C. de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simula. 60 (2018) 72-91.

[31] W. Saengthong, E. Thailert, and S.K. Ntouyas, Existence and uniqueness of solutions for system of Hilfer-Hadamard sequential fractional dierential equations with two point boundary conditions, Adv. Dierence Equ. 2019 (1) (2019) 525.

[32] B. Senol, C. Yeroglu, Frequency boundary of fractional order systems with nonlinear uncertainties, J. Franklin Inst. 350 (2013) 1908-1925.

[33] J.R. Wang, Y. Zhang, Analysis of fractional order dierential coupled systems, Math. Methods Appl. Sci. 38 (2015) 3322-3338.

[34] J.R. Wang, Y. Zhou, M. Feckan, On recent developments in the theory of boundary value problems for impulsive fractional dierential equations, Comput. Math. Appl. 64 (2012) 3008-3020.