
In this manuscript, under the matrix measure and some sufficient conditions, we will overcame all difficulties and challenges related to the fundamental matrix for a generalized nonlinear neutral functional differential equations in matrix form with multiple delays. The periodicity of solutions, as well as the uniqueness under the considered conditions has been proved employing the fixed point theory. Our approach expanded and generalized certain previously published findings for example, we studied the uniqueness of a solution that was absent in some literature. Moreover, an example was given to confirm the main results.
Citation: Mouataz Billah Mesmouli, Amir Abdel Menaem, Taher S. Hassan. Effectiveness of matrix measure in finding periodic solutions for nonlinear systems of differential and integro-differential equations with delays[J]. AIMS Mathematics, 2024, 9(6): 14274-14287. doi: 10.3934/math.2024693
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In this manuscript, under the matrix measure and some sufficient conditions, we will overcame all difficulties and challenges related to the fundamental matrix for a generalized nonlinear neutral functional differential equations in matrix form with multiple delays. The periodicity of solutions, as well as the uniqueness under the considered conditions has been proved employing the fixed point theory. Our approach expanded and generalized certain previously published findings for example, we studied the uniqueness of a solution that was absent in some literature. Moreover, an example was given to confirm the main results.
Fractional order differential equations are the generalizations of the classical integer order differential equations. The idea about the fractional order derivative was introduced at the end of the sixteenth century (1695) when Leibniz used the notation dndσn for nth order derivative. By writing a letter to him, L'Hospital asked the question: what would be the result if n=12? Leibniz answered in such words, "An apparent Paradox, from which one day useful consequences will be drawn", and this question became the foundation of fractional calculus. Fractional calculus has become a speedily developing area and its applications can be found in diverse fields ranging from physical sciences, porous media, electrochemistry, economics, electromagnetics, medicine and engineering to biological sciences. Progressively, fractional differential equations play a very important role in fields such as thermodynamics, statistical physics, nonlinear oscillation of earthquakes, viscoelasticity, defence, optics, control, signal processing, electrical circuits, astronomy etc. There are some outstanding articles which provide the main theoretical tools for the qualitative analysis of this research field, and at the same time, shows the interconnection as well as the distinction between integral models, classical and fractional differential equations, see [14,16,18,19,22,25,26,28,30]
Impulsive fractional differential equations are used to describe both physical, social sciences and many dynamical systems such as evolution processes pharmacotherapy. There are two types of impulsive fractional differential equations the first one is instantaneous impulsive fractional differential equations while the other one is non-instantaneous impulsive fractional differential equations. In last few decades, the theory of impulsive fractional differential equations are well utilized in medicine, mechanical engineering, ecology, biology and astronomy etc. There are some remarkable monographs [3,6,8,15,20,23,33,34], considering fractional differential equations with impulses.
The most preferable research area in the field of fractional differential equations (FDE′s), which received great attention from the researchers is the theory regarding the existence of solutions. Many researchers developed some interesting results about the existence of solutions of different boundary value problems (BVPs) using different fixed point theorems. For details we refer the reader to [2,7,9,10,11,13,27]. Most of the time, it is quite intricate to find the exact solutions of nonlinear differential equations, in such a situation different approximation techniques are introduced. The difference between exact and approximate solutions is nowadays dealt with using Hyers-Ulam (HU) type stabilities, which were first introduced in 1940 by Ulam [29] and then answered by Hyers in the following year in the context of Banach spaces. Many researchers investigated HU type stabilities for different problems with different approaches [12,17,31,35,36,37,39,40].
Zada and Dayyan [38], investigated the existence, uniqueness and Ulam's type stability for the implicit fractional differential equation with instantaneous impulses and Riemann-Liouville fractional integral boundary conditions having the following form
{cDα0,σu(σ)−ϕ1(σ,u(σ),cDαu(σ))=0,σ≠σj∈I,0<α≤1,Δu(σj)−Ej(u(σj))=0,j=1,2,…,q−1,η1u(σ)|σ=0+ξ1Iαu(σ)|σ=0=ν1,η2u(σ)|σ=T+ξ2Iαu(σ)|σ=T=ν2, |
where I=[0,T], and cDα0,σ is a generalization of classical Caputo derivative of order α with lower bound at 0, ϕ1:I×R×R→R is a continuous function. Furthermore, u(σ+j) and u(σ+j) represent the right-sided and left-sided limits respectively at σ=σj for j=1,2,…,q−1.
Ali et al. [4], studied a coupled system for the existence and uniqueness of solution using Riemann-Liouville derivative
{Dαu(σ)=ϕ1(σ,v(σ),Dαu(σ)),Dβv(σ)=ϕ2(σ,u(σ),Dβv(σ)),σ∈J,Dα−1u(0+)=β1Dα−1u(T−),Dα−1u(0+)=γ1Dα−1u(T−),Dβ−1v(0+)=β2Dβ−1v(T−),Dβ−1v(0+)=γ2Dβ−1v(T−), |
where σ∈J=[0,T], T>0, α,β∈(1,2], and β1,β2,γ1,γ2≠1. Dα, Dβ are the Riemann-Liouville fractional derivatives and ϕ1,ϕ2:[0,1]×R×R→R are continuous functions.
Wang et al. [32], presented stability of the following coupled system of implicit fractional integro-differential equations having anti-periodic boundary conditions:
{cDαu(σ)−ϕ1(σ,v(σ),cDαu(σ))−1Γ(γ1)∫σ0(σ−s)γ1−1f(s,v(s),cDαu(s))ds=0,∀σ∈J,cDβv(σ)−ϕ2(σ,u(σ),cDβv(σ))−1Γ(γ2)∫σ0(σ−s)γ2−1g(s,u(s),cDβv(s))ds=0,∀σ∈J,u(σ)|σ=0=−u(σ)|σ=T=0,cDr1u(σ)|σ=0=−cDr1u(σ)|σ=T,v(σ)|σ=0=−v(σ)|σ=T=0,cDr2v(σ)|σ=0=−cDr2v(σ)|σ=T, |
where 1<α,β≤2, 0≤r1,r2≤2, γ1,γ2>0, and J=[0,T], T>0. ϕ1,ϕ2,f,g:J×R×R→R are continuous functions.
Motivated by the above work, we focus our attention on the following coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives of the form:
{{Dαu(σ)−ϕ1(σ,Iαu(σ),Iβv(σ))=0, σ∈ω, σ≠σj, j=1,2,…,p,Δu(σj)−Ej(u(σj))=0,Δu′(σj)−E∗j(u(σj))=0, j=1,2,…,p,ν1Dα−2u(σ)|σ=0=u1,μ1u(σ)|σ=T+ν2Iα−1u(σ)|σ=T=u2,{Dβv(σ)−ϕ2(σ,Iαu(σ),Iβv(σ))=0, σ∈ω, σ≠σk, k=1,2,…,q,Δv(σk)−Ek(v(σk))=0,Δv′(σk)−E∗k(v(σk))=0, k=1,2,…,q,ν3Dβ−2v(σ)|σ=0=v1,μ2v(σ)|σ=T+ν4Iβ−1v(σ)|σ=T=v2, | (1.1) |
where 1<α,β≤2, ϕ1,ϕ2:[0,T]×R×R→R being continuous functions and
Δu(σj)=u(σ+j)−u(σ−j),Δu′(σj)=u′(σ+j)−u′(σ−j) |
Δv(σk)=v(σ+k)−v(σ−k),Δv′(σk)=v′(σ+k)−v′(σ−k), |
where u(σ+j),v(σ+k) and u(σ−j),v(σ−k) are the right limits and left limits respectively, Ej,E∗j,Ek,E∗k:R→R are continuous functions, and Dα,Iα are the α-order Riemann-Liouville fractional derivative and integral operators respectively.
The remaining article is arranged as follows: In Section 2, we present some basic definitions, theorems, and lemmas that will be used in our main results. In Section 3, we use suitable cases for the existence and uniqueness of solution for the proposed system (1.1) using Kransnoselskii's type fixed point theorem. In Section 4, we discuss different kinds of stabilities in the sense of Ulam under certain conditions. In Section 5, an example is given to support the main results.
In this section, we present some basics notations, definitions, and results that are used in the whole article.
Let T>0, ω=[0,T]. The Banach space of all continuous functions from ω into R is denoted by C(ω,R) with the norm
‖u‖=sup{|u(σ)|:σ∈ω} |
and the product of these spaces is also a Banach space with the norm
‖(u,v)‖=‖u‖+‖v‖. |
The piecewise continuous functions with 1<α,β≤2 are denoted as follows:
ϑ1=PC2−α(ω,R+)={u:ω→R+,u(σ+j),u(σ−j) and Δu′(σ+j),u′(σ−j) exist for j=1,2,…,p}, |
ϑ2=PC2−β(ω,R+)={v:ω→R+,v(σ+k),v(σ−k) and Δv′(σ+k),v′(σ−k) exist for k=1,2,…,q}, |
with the norms
‖u‖ϑ1=sup{|σ2−αu(σ)|:σ∈ω}, |
‖v‖ϑ2=sup{|σ2−βv(σ)|:σ∈ω}, |
respectively. Their product ϑ=ϑ1×ϑ2 is also a Banach space with the norm ‖(u,v)‖ϑ=‖u‖ϑ1+‖v‖ϑ2.
Definition 2.1. [1] The Riemann-Liouville fractional integral of order α>0 for a function u:R+→R is defined as
Iαu(σ)=1Γ(α)∫σ0(σ−π)α−1u(π)dπ, |
where Γ(⋅) is the Euler gamma function defined by Γ(α)=∫∞0e−σσα−1dσ,α>0.
Definition 2.2. For a function u:R+→R, the Riemann-Liouville derivative of fractional order α>0, p=[α]+1, is defined as
Dαu(σ)=1Γ(p−α)(ddσ)p∫σ0(σ−π)p−α−1u(π)dπ, |
provided that integral on the right side exists. [α] denotes the integer part of the real number α. For more properties, the reader may refer to [1].
Lemma 2.1. [1] Let u be any function, and let α>0, then the Riemann-Liouville fractional derivative for the Homogeneous differential equation
Dαu(σ)=0,α>0, |
has a solution
u(σ)=c1σα−1+c2σα−2+⋯+cp−1σα−p−1+cpσα−p, |
and for non-homogeneous differential equation
Dαu(σ)=ϕ1(σ),α>0, |
has a solution
IαDαu(σ)=Iαϕ1(σ)+c1σα−1+c2σα−2+⋯+cp−1σα−p−1+cpσα−p, |
where p=[α]+1 and ci,i=1,2,…,p, are real constants.
Theorem 2.1. (Altman [5]) Let Λ≠0 be a convex and closed subset of Banach space ϑ. Consider two operators ℑ1,ℑ2 such that
(1) ℑ1(u,v)+ℑ2(u,v)∈Λ;
(2) ℑ1 is a contractive operator;
(3) ℑ2 is a compact and continuous operator.
Then there exists (u,v)∈Λ such that ℑ1(u,v)+ℑ2(u,v)=(u,v)∈ϑ.
The following definitions and remarks are taken from [21,24].
Definition 2.3. The given system (1.1) is HU stable if there exists Nα,β=max{Nα,Nβ}>0 such that, for κ=max{κα,,κβ}>0 and for every solution (ξ,ζ)∈ϑ of the inequality
{{|Dαξ(σ)−ϕ1(σ,Iαξ(σ),Iβζ(σ))|≤κα, σ∈ω,|Δξ(σj)−Ej(ξ(σj))|≤κα, j=1,2,…,p,|Δξ′(σj)−E∗j(ξ(σj))|≤κα, j=1,2,…,p,{|Dβζ(σ)−ϕ2(σ,Iαξ(σ),Iβζ(σ))|≤κβ, σ∈ω,|Δζ(σk)−Ek(ζ(σk))|≤κβ, k=1,2,…,q,|Δζ′(σk)−E∗k(ζ(σk))|≤κβ, k=1,2,…,q, | (2.1) |
there exists a solution (u,v)∈ϑ with
‖(u,v)−(ξ,ζ)‖ϑ≤Nα,βκ,σ∈ω. |
Definition 2.4. The given system (1.1) is generalized HU stable if there exists N′∈C(R+,R+) with N′(0)=0 such that, for any approximate solution (ξ,ζ)∈ϑ of inequality (2.1), there exists a solution (u,v)∈ϑ of (1.1) satisfying
‖(u,v)−(ξ,ζ)‖ϑ≤N′(κ),σ∈ω. |
Definition 2.5. The given system (1.1) is HUR stable with respect to ψα,β=max{ψα,ψβ} with ψα,β∈C(ω,R) if there exists a constant Nψα,ψβ=max{Nψα,Nψβ}>0 such that, for any κ=max{κα,,κβ}>0 and for any approximate solution (ξ,ζ)∈ϑ of the inequality
{{|Dαξ(σ)−ϕ1(σ,Iαξ(σ),Iβζ(σ))|≤ψα(σ)κα, σ∈ω,|Δξ(σj)−Ej(ξ(σj))|≤ψα(σ)κα, j=1,2,…,p,|Δξ′(σj)−E∗j(ξ(σj))|≤ψα(σ)κα, j=1,2,…,p,{|Dβζ(σ)−ϕ2(σ,Iαξ(σ),Iβζ(σ))|≤ψβ(σ)κβ, σ∈ω,|Δζ(σk)−Ek(ζ(σk))|≤ψβ(σ)κβ, k=1,2,…,q,|Δζ′(σk)−E∗k(ζ(σk))|≤ψβ(σ)κβ, k=1,2,…,q, | (2.2) |
there exists a solution (u,v)∈ϑ with
‖(u,v)−(ξ,ζ)‖ϑ≤Nψα,ψβψα,β(σ)κ,σ∈ω. |
Definition 2.6. The given system (1.1) is generalized HUR stable with respect to ψα,β=max{ψα,ψβ} with ψα,β∈C(ω,R) if there exists a constant Nψα,ψβ=max{Nψα,Nψβ}>0 such that, for any approximate solution (ξ,ζ)∈ϑ of inequality (2.2), there exists a solution (u,v)∈ϑ of (1.1) satisfying
‖(u,v)−(ξ,ζ)‖ϑ≤Nψα,ψβψα,β(σ),σ∈ω. |
Remark 2.1. Let (ξ,ζ)∈ϑ be a solution of inequalities (2.1) if there exist functions Kϕ1,Lϕ2∈C(ω,R) depending on ξ,ζ respectively such that
(1) |Kϕ1(σ)|≤κα,|Lϕ2(σ)|≤κβ,σ∈ω;
(2)
{{Dαξ(σ)=ϕ1(σ,Iαξ(σ),Iβζ(σ))+Kϕ1(σ),Δξ(σj)=Ej(ξ(σj))+Kϕ1j, j=1,2,…,p,Δξ′(σj)=E∗j(ξ(σj))+Kϕ1j, j=1,2,…,p,{Dβζ(σ)=ϕ2(t,Iαξ(σ),Iβζ(σ))+Lϕ2(σ),Δζ(σk)=Ek(ζ(σk))+Lϕ2k, k=1,2,…,q,Δζ′(σk)=E∗k(ζ(σk))+Lϕ2k, k=1,2,…,q. | (2.3) |
In this section, we discuss the existence and uniqueness of solution of the proposed system (1.1).
Theorem 3.1. Let α,β∈(1,2] and ϕ1 be any linear and continuous function. The fractional impulsive differential equation
{Dαu(σ)=ϕ1(σ,Iαu(σ),Iβv(σ)),σ∈ω,σ≠σj,j=1,2,…,p,Δu(σj)=Ej(u(σj)),Δu′(σj)=E∗j(u(σj)),j=1,2,…,p,ν1Dα−2u(σ)|σ=0=u1,μ1u(σ)|σ=T+ν2Iα−1u(σ)|σ=T=u2, | (3.1) |
has a solution
u(σ)={{σα−1u2μ1Tα−1−σα−1u1Tν1Γ(α−1)+σα−2u1ν1Γ(α−1)+1Γ(α)∫σ0(σ−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ−σα−1T1−αΓ(α)∫Tσ1(T−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ−ν2σα−1T1−αμ1Γ(α−1)∫T0(T−π)α−2u(π)dπ−σα−1T[((α−1)−(α−2)Tσ−11)σ2−α1E1(u(σ1))+(T−σ1)σ2−α1E∗1(u(σ1))+(T−σ1)σ2−α1Γ(α−1)∫σ10(σ1−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+((α−1)−(α−2)Tσ−11)σ2−α1Γ(α)∫σ10(σ1−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ],σ∈[0,σ1],{σα−1u2μ1Tα−1−σα−1u1Tν1Γ(α−1)+σα−2u1ν1Γ(α−1)+1Γ(α)∫σσz(σ−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ−σα−1T1−αΓ(α)∫Tσz(T−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ−ν2σα−1T1−αμ1Γ(α−1)∫T0(T−π)α−2u(π)dπ−σα−1Tz∑j=1[((α−1)−(α−2)Tσ−1j)σ2−αjEj(u(σj))+(T−σj)σ2−αjE∗j(u(σj))+(T−σj)σ2−αjΓ(α−1)∫σjσj−1(σj−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+((α−1)−(α−2)Tσ−1j)σ2−αjΓ(α)∫σjσj−1(σj−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ]+z∑j=1[((α−1)−(α−2)σσ−1j)σα−2σ2−αjEj(u(σj))+(σ−σj)σα−2σ2−αjE∗j(u(σj))+(σ−σj)σα−2σ2−αjΓ(α−1)∫σjσj−1(σj−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+((α−1)−(α−2)σσ−1j)σα−2σ2−αjΓ(α)∫σjσj−1(σj−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ],σ∈(σj,σj+1];z=1,2,…,p. | (3.2) |
Proof. Consider
Dαu(σ)=ϕ1(σ,Iαu(σ),Iβv(σ)),σ∈ω,α∈(1,2]. | (3.3) |
For σ∈[0,σ1], Lemma 2.1 gives
{u(σ)=1Γ(α)∫σ0(σ−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ+a1σα−1+a2σα−2,u′(σ)=1Γ(α−1)∫σ0(σ−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+a1(α−1)σα−2+a2(α−2)σα−3. | (3.4) |
Again, for σ∈(σ1,σ2], Lemma 2.1 gives
{u(σ)=1Γ(α)∫σσ1(σ−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ+b1σα−1+b2σα−2,u′(σ)=1Γ(α−1)∫σσ1(σ−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+b1(α−1)σα−2+b2(α−2)σα−3. | (3.5) |
Hence it follows that
{u(σ−1)=1Γ(α)∫σ10(σ1−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ+a1σα−11+a2σα−21,u(σ+1)=b1σα−11+b2σα−21,u′(σ−1)=1Γ(α−1)∫σ10(σ1−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+a1(α−1)σα−21+a2(α−2)σα−31,u′(σ+1)=b1(α−1)σα−21+b2(α−2)σα−31. |
Using
{Δu(σ1)=u(σ+1)−u(σ−1)=E1(u(σ1)),Δu′(σ1)=u′(σ+1)−u′(σ−1)=E∗1(u(σ1)), |
we obtain
{b1=a1−(α−2)σ1−α1E1(u(σ1))+σ2−α1E∗1(u(σ1))+σ2−α1Γ(α−1)∫σ10(σ1−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ−(α−2)σ1−α1Γ(α)∫σ10(σ1−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ,b2=a2+(α−1)σ2−α1E1(u(σ1))−σ3−α1E∗1(u(σ1))−σ3−α1Γ(α−1)∫σ10(σ1−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+(α−1)σ2−α1Γ(α)∫σ10(σ1−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ. |
Substituting the values of b1, b2 in (3.5), we get
{u(σ)=a1σα−1+a2σα−2+((α−1)−(α−2)σσ−11)σα−2σ2−α1E1(u(σ1))+(σ−σ1)σα−2σ2−α1E∗1(u(σ1))+1Γ(α)∫σσ1(σ−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ+((α−1)−(α−2)σσ−11)σα−2σ2−α1Γ(α)∫σ10(σ1−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ+(σ−σ1)σα−2σ2−α1Γ(α−1)∫σ10(σ1−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ,u′(σ)=a1(α−1)σα−2+a2(α−2)σα−3+(α−1)(α−2)(σ−1−σ−11)σα−2σ2−α1E1(u(σ1))+((α−1)−(α−2)σ−1σ1)σα−2σ2−α1E∗1(u(σ1))+1Γ(α−1)∫σσ1(σ−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+((α−1)−(α−2)σ−1σ1)σα−2σ2−α1Γ(α−1)∫σ10(σ1−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+(α−1)(α−2)(σ−1−σ−11)σα−2σ2−α1Γ(α)∫σ10(σ1−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ. |
Similarly, for σ∈(σj,σj+1],
u(σ)=a1σα−1+a2σα−2+z∑j=1((α−1)−(α−2)σσ−1j)σα−2σ2−αjEj(u(σj))+z∑j=1(σ−σj)σα−2σ2−αjE∗j(u(σj))+1Γ(α)∫σσz(σ−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ+z∑j=1(σ−σj)σα−2σ2−αjΓ(α−1)∫σjσj−1(σj−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+z∑j=1((α−1)−(α−2)σσ−1j)σα−2σ2−αjΓ(α)∫σjσj−1(σj−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ. | (3.6) |
Finally, after applying conditions ν1Dα−2u(σ)|σ=0=u1, and μ1u(σ)|σ=T+ν2Iα−1u(σ)|σ=T=u2 to (3.6) and finding the values of a1 and a2, we obtain Eq (2.2).
Corollary 1. In view of Theorem 3.1, our coupled system (1.1) has the following solution:
u(σ)={{σα−1u2μ1Tα−1−σα−1u1Tν1Γ(α−1)+σα−2u1ν1Γ(α−1)+1Γ(α)∫σ0(σ−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ−σα−1T1−αΓ(α)∫Tσ1(T−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ−ν2σα−1T1−αμ1Γ(α−1)∫T0(T−π)α−2u(π)dπ−σα−1T[((α−1)−(α−2)Tσ−11)σ2−α1E1(u(σ1))+(T−σ1)σ2−α1E∗1(u(σ1))+(T−σ1)σ2−α1Γ(α−1)∫σ10(σ1−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+((α−1)−(α−2)Tσ−11)σ2−α1Γ(α)∫σ10(σ1−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ],σ∈[0,σ1],{σα−1u2μ1Tα−1−σα−1u1Tν1Γ(α−1)+σα−2u1ν1Γ(α−1)+1Γ(α)∫σσz(σ−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ−σα−1T1−αΓ(α)∫Tσz(T−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ−ν2σα−1T1−αμ1Γ(α−1)∫T0(T−π)α−2u(π)dπ−σα−1Tz∑j=1[((α−1)−(α−2)Tσ−1j)σ2−αjEj(u(σj))+(T−σj)σ2−αjE∗j(u(σj))+(T−σj)σ2−αjΓ(α−1)∫σjσj−1(σj−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+((α−1)−(α−2)Tσ−1j)σ2−αjΓ(α)∫σjσj−1(σj−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ]+z∑j=1[((α−1)−(α−2)σσ−1j)σα−2σ2−αjEj(u(σj))+(σ−σj)σα−2σ2−αjE∗j(u(σj))+(σ−σj)σα−2σ2−αjΓ(α−1)∫σjσj−1(σj−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+((α−1)−(α−2)σσ−1j)σα−2σ2−αjΓ(α)∫σjσj−1(σj−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ],σ∈(σj,σj+1];z=1,2,…,p. | (3.7) |
and
v(σ)={{σβ−1v2μ2Tβ−1−σβ−1v1Tν3Γ(β−1)+σβ−2v1ν3Γ(β−1)+1Γ(β)∫σ0(σ−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ−σβ−1T1−βΓ(β)∫Tσ1(T−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ−ν4σβ−1T1−βμ2Γ(β−1)∫T0(T−π)β−2v(π)dπ−σβ−1T[((β−1)−(β−2)Tσ−11)σ2−β1E1(v(σ1))+(T−σ1)σ2−β1E∗1(v(σ1))+(T−σ1)σ2−β1Γ(β−1)∫σ10(σ1−π)β−2ϕ2(π,Iαu(π),Iβv(π))dπ+((β−1)−(β−2)Tσ−11)σ2−β1Γ(β)∫σ10(σ1−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ],σ∈[0,σ1],{σβ−1v2μ2Tβ−1−σβ−1v1Tν3Γ(β−1)+σβ−2v1ν3Γ(β−1)+1Γ(β)∫σσz(σ−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ−σβ−1T1−βΓ(β)∫Tσz(T−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ−ν4σβ−1T1−βμ2Γ(β−1)∫T0(T−π)β−2v(π)dπ−σβ−1Tz∑k=1[((β−1)−(β−2)Tσ−1k)σ2−βkEk(v(σk))+(T−σk)σ2−βkE∗k(v(σk))+(T−σk)σ2−βkΓ(β−1)∫σkσk−1(σk−π)β−2ϕ2(π,Iαu(π),Iβv(π))dπ+((β−1)−(β−2)Tσ−1k)σ2−βkΓ(β)∫σkσk−1(σk−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ]+z∑k=1[((β−1)−(β−2)σσ−1k)σβ−2σ2−βkEk(v(σk))+(σ−σk)σβ−2σ2−βkE∗k(v(σk))+(σ−σk)σβ−2σ2−βkΓ(β−1)∫σkσk−1(σk−π)β−2ϕ2(π,Iαu(π),Iβv(π))dπ+((β−1)−(β−2)σσ−1k)σβ−2σ2−βkΓ(β)∫σkσk−1(σk−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ],σ∈(σk,σk+1];z=1,2,…,q. | (3.8) |
Now, for transformation of the given system (1.1) into a fixed point problem, let the operators ℑ1,ℑ2:ϑ→ϑ be define as follows:
ℑ1(u,v)(σ)=(ℑ∗1(u(σ)),ℑ∗∗1(v(σ))),ℑ2(u,v)(σ)=(ℑ∗2(u,v)(σ),ℑ∗∗2(u,v)(σ)), |
ℑ1(u,v)(σ)={ℑ∗1(u(σ))={σα−1u2μ1Tα−1−σα−1u1Tν1Γ(α−1)+σα−2u1ν1Γ(α−1)−ν2σα−1T1−αμ1Γ(α−1)∫T0(T−π)α−2u(π)dπ−σα−1Tz∑j=1[((α−1)−(α−2)Tσ−1j)σ2−αjEj(u(σj))+(T−σj)σ2−αjE∗j(u(σj))]+z∑j=1[((α−1)−(α−2)σσ−1j)σα−2σ2−αjEj(u(σj))+(σ−σj)σα−2σ2−αjE∗j(u(σj))],σ∈(σj,σj+1];z=1,2,…,p,ℑ∗∗1(v(σ))={σβ−1v2μ2Tβ−1−σβ−1v1Tν3Γ(β−1)+σβ−2v1ν3Γ(β−1)−ν4σβ−1T1−βμ2Γ(β−1)∫T0(T−π)β−2v(π)dπ−σβ−1Tz∑k=1[((β−1)−(β−2)Tσ−1k)σ2−βkEk(v(σk))+(T−σk)σ2−βkE∗k(v(σk))]+z∑k=1[((β−1)−(β−2)σσ−1k)σβ−2σ2−βkEk(v(σk))+(σ−σk)σβ−2σ2−βkE∗k(v(σk))],σ∈(σk,σk+1];z=1,2,…,q, | (3.9) |
and
ℑ2(u,v)(σ)={ℑ∗2(u,v)(σ)={1Γ(α)∫σσz(σ−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ−σα−1T1−αΓ(α)∫Tσz(T−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ−σα−1Tz∑j=1[(T−σj)σ2−αjΓ(α−1)∫σjσj−1(σj−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+((α−1)−(α−2)Tσ−1j)σ2−αjΓ(α)∫σjσj−1(σj−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ]+z∑j=1[(σ−σj)σα−2σ2−αjΓ(α−1)∫σjσj−1(σj−π)α−2ϕ1(π,Iαu(π),Iβv(π))dπ+((α−1)−(α−2)σσ−1j)σα−2σ2−αjΓ(α)∫σjσj−1(σj−π)α−1ϕ1(π,Iαu(π),Iβv(π))dπ],σ∈(σj,σj+1];z=1,2,…,p,ℑ∗∗2(u,v)(σ)={1Γ(β)∫σσz(σ−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ−σβ−1T1−βΓ(β)∫Tσz(T−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ−σβ−1Tz∑k=1[(T−σk)σ2−βkΓ(β−1)∫σkσk−1(σk−π)β−2ϕ2(π,Iαu(π),Iβv(π))dπ+((β−1)−(β−2)Tσ−1k)σ2−βkΓ(β)∫σkσk−1(σk−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ]+z∑k=1[(σ−σk)σβ−2σ2−βkΓ(β−1)∫σkσk−1(σk−π)β−2ϕ2(π,Iαu(π),Iβv(π))dπ+((β−1)−(β−2)σσ−1k)σβ−2σ2−βkΓ(β)∫σkσk−1(σk−π)β−1ϕ2(π,Iαu(π),Iβv(π))dπ],σ∈(σk,σk+1];z=1,2,…,q. | (3.10) |
For additional analysis, the following hypothesis needs to hold:
(H1) ● For σ∈ω there exist bounded functions o,τ,υ∈ϑ such that
|ϕ1(σ,x1(σ),x2(σ))|≤o(σ)+τ(σ)|x1(σ)|+υ(σ)|x2(σ)| |
with o1=supσ∈ωo(σ), τ1=supσ∈ωτ(σ), and υ1=supσ∈ωυ(σ)<1.
● Similarly, for σ∈ω there exist bounded functions o∗,τ∗,υ∗∈ϑ such that
|ϕ2(σ,x1(σ),x2(σ))|≤o∗(σ)+τ∗(σ)|x1(σ)|+υ∗(σ)|x2(σ)| |
with o2=supσ∈ωo∗(σ), τ2=supσ∈ωτ∗(σ), and υ2=supσ∈ωυ∗(σ)<1.
(H2) Ej,E∗j:R→R are continuous and there exist constants GE,GE∗,G′E,G′E∗,ˆGE,ˆGE∗,ˆG′E,ˆG′E∗>0 such that, for any (u,v)∈ϑ,
|Ez(u)|≤GE|u|+G′E,|Ez(v)|≤ˆGE|v|+ˆG′E,|E∗z(u)|≤GE∗|u|+G′E∗,|E∗z(v)|≤ˆGE∗|v|+ˆG′E∗, |
where z=1,2,…,p.
(H3) ● For all x1,x2,x∗1,x∗2∈R and for each σ∈ω, there exist constants Lϕ1>0, 0<L∗ϕ1<1 such that
|ϕ1(σ,x1,x2)−ϕ1(σ,x∗1,x∗2)|≤Lϕ1|x1−x∗1|+L∗ϕ1|x2−x∗2|. |
● Similarly, for all x1,x2,x∗1,x∗2∈R and for each σ∈ω, there exist constants Lϕ2>0, 0<L∗ϕ2<1 such that
|ϕ2(σ,x1,x2)−ϕ2(σ,x∗1,x∗2)|≤Lϕ2|x1−x∗1|+L∗ϕ2|x2−x∗2|. |
(H4) Ez,E∗z:R→R are continuous and there exist constants LE,LE∗,L∗E,L∗E∗ such that, for any (u,v),(u∗,v∗)∈ϑ,
|Ez(u(σ))−Ez(u∗(σ))|≤LE|u−u∗|,|Ez(v(σ))−Ez(v∗(σ))|≤L∗E|v−v∗|,|E∗z(u(σ))−E∗z(u∗(σ))|≤LE∗|u−u∗|,|E∗z(v(σ))−E∗z(v∗(σ))|≤L∗E∗|v−v∗|. |
Here we use Kransnoselskii's fixed point theorem to show that the operator ℑ1+ℑ2 has at least one fixed point. Therefore, we choose a closed ball
ϑr={(u,v)∈ϑ,‖(u,v)‖≤r,‖u‖≤r2and‖v‖≤r2}⊂ϑ, |
where
r≥G1+G∗1+o1G3+o2G∗31−(G2+G∗2+G3G4+G∗3G∗4). |
Theorem 3.2. If hypotheses (H1)–(H4) are hold, then the given system (1.1) has at least one solution.
Proof. 1) For any (u,v)∈ϑr, we have
‖ℑ1(u,v)+ℑ2(u,v)‖ϑ≤‖ℑ∗1(u)‖ϑ1+‖ℑ∗∗1(v)‖ϑ2+‖ℑ∗2(u,v)‖ϑ1+‖ℑ∗∗2(u,v)‖ϑ2. | (3.11) |
From (3.9), we get
|σ2−αℑ∗1(u(σ))|≤|σu2μ1Tα−1|+|σu1Tν1Γ(α−1)|+|u1ν1Γ(α−1)|+ν2|σ||T1−α|μ1Γ(α−1)∫T0|(T−π)α−2||u(π)|dπ+z∑j=1|((α−1)−(α−2)σσ−1j)−σT((α−1)−(α−2)Tσ−1j)||σ2−αj||Ej(u(σj))|+z∑j=1|(σ−σj)−σT(T−σj)||σ2−αj||E∗j(u(σj))|,z=1,2,…,p. | (3.12) |
This implies that
‖ℑ∗1(u)‖ϑ1≤|σu2μ1Tα−1|+|σu1Tν1Γ(α−1)|+|u1ν1Γ(α−1)|+ν2|σ|μ1Γ(α)‖u‖+z(α−1)|σ2−αz||1−σT|(GE‖u‖+G′E)+z|σ3−αz||σT−1|(GE∗‖u‖+G′E∗)≤G1+G2‖u‖. | (3.13) |
Similarly, we can obtain
‖ℑ∗∗1(v)‖ϑ2≤G∗1+G∗2‖v‖, | (3.14) |
where
G1=zG′E(α−1)|σ2−αz||1−σT|+zG′E∗|σ3−αz||σT−1|+|σu2μ1Tα−1|+|σu1Tν1Γ(α−1)|+|u1ν1Γ(α−1)|,G2=zGE(α−1)|σ2−αz||1−σT|+zGE∗|σ3−αz||σT−1|+ν2|σ|μ1Γ(α),forz=1,2,…,p,andG∗1=zˆG′E(β−1)|σ2−βz||1−σT|+zˆG′E∗|σ3−βz||σT−1|+|σv2μ2Tβ−1|+|σv1Tν3Γ(β−1)|+|v1ν3Γ(β−1)|,G∗2=zˆGE(β−1)|σ2−βz||1−σT|+zˆGE∗|σ3−βz||σT−1|+ν4|σ|μ2Γ(β),forz=1,2,…,q. |
Also, we have
|σ2−αℑ∗2(u,v)|≤|σ2−α|Γ(α)∫σσz|(σ−π)α−1||y(π)|dπ+|σ||T1−α|Γ(α)∫Tσz|(T−π)α−1||y(π)|dπ+σTz∑j=1[|(T−σj)||σ2−αj|Γ(α−1)∫σjσj−1|(σj−π)α−2||y(π)|dπ+|((α−1)−(α−2)Tσ−1j)||σ2−αj|Γ(α)∫σjσj−1|(σj−π)α−1||y(π)|dπ]+z∑j=1[|(σ−σj)||σ2−αj|Γ(α−1)∫σjσj−1|(σj−π)α−2||y(π)|dπ+|((α−1)−(α−2)σσ−1j)||σ2−αj|Γ(α)∫σjσj−1|(σj−π)α−1||y(π)|dπ] forz=1,2,…,p. | (3.15) |
Now by (H1)
|y(σ)|=|ϕ1(σ,Iαu(σ),Iβv(σ))|≤o(σ)+τ(σ)|Iαu(σ)|+υ(σ)|Iβv(σ)|≤o(σ)+τ(σ)1Γ(α)∫σ0|(σ−π)α−1||u(π)|dπ+υ(σ)1Γ(β)∫σ0|(σ−π)β−1||v(π)|dπ. |
Now, taking supσ∈ω on both sides, we get
‖y‖≤o1+τ1|σα|‖u‖Γ(α+1)+υ1|σβ|‖v‖Γ(β+1). | (3.16) |
Now taking supσ∈ω of (3.15) and using (3.16) in (3.15), we get
‖ℑ∗2(u,v)‖ϑ1≤(o1+τ1|σα|‖u‖Γ(α+1)+υ1|σβ|‖v‖Γ(β+1))(|σ2−α||(σ−σz)α|Γ(α+1)+|σ||T1−α||(T−σz)α|Γ(α+1)+z|σ||σ2−αz|T[|(T−σz)||(σz−σz−1)α−1|Γ(α)+|((α−1)−(α−2)Tσ−1z)||(σz−σz−1)α|Γ(α+1)]+z|(σ−σz)||σ2−αz||(σz−σz−1)α−1|Γ(α)+z|((α−1)−(α−2)σσ−1z)||σ2−αz||(σz−σz−1)α|Γ(α+1))≤o1G3+τ1|σα|‖u‖G3Γ(α+1)+υ1|σβ|‖v‖G3Γ(β+1)≤o1G3+G3G4‖(u,v)‖. | (3.17) |
Similarly,
‖ℑ∗∗2(u,v)‖ϑ2≤o2G∗3+G∗3G∗4‖(u,v)‖, | (3.18) |
where
G3=|σ2−α||(σ−σz)α|Γ(α+1)+|σ||T1−α||(T−σz)α|Γ(α+1)+z|σ||σ2−αz|T[|(T−σz)||(σz−σz−1)α−1|Γ(α)+|((α−1)−(α−2)Tσ−1z)||(σz−σz−1)α|Γ(α+1)]+z|(σ−σz)||σ2−αz||(σz−σz−1)α−1|Γ(α)+z|((α−1)−(α−2)σσ−1z)||σ2−αz||(σz−σz−1)α|Γ(α+1),z=1,2,…,p,G∗3=|σ2−β||(σ−σz)β|Γ(β+1)+|σ||T1−β||(T−σz)β|Γ(β+1)+z|σ||σ2−βz|T[|(T−σz)||(σz−σz−1)β−1|Γ(β)+|((β−1)−(β−2)Tσ−1z)||(σz−σz−1)β|Γ(β+1)]+z|(σ−σz)||σ2−βz||(σz−σz−1)β−1|Γ(β)+z|((β−1)−(β−2)σσ−1z)||σ2−βz||(σz−σz−1)β|Γ(β+1),z=1,2,…,q,G4=max{τ1|σα|Γ(α+1),υ1|σβ|Γ(β+1)}andG∗4=max{τ2|σα|Γ(α+1),υ2|σβ|Γ(β+1)}. |
Putting (3.13), (3.14), (3.17) and (3.18) in (3.11), we get
‖ℑ1(u,v)+ℑ2(u,v)‖ϑ≤G1+G2‖u‖+G∗1+G∗2‖v‖+o1G3+G3G4‖(u,v)‖+o2G∗3+G∗3G∗4‖(u,v)‖≤G1+G∗1+o1G3+o2G∗3+(G2+G∗2+G3G4+G∗3G∗4)‖(u,v)‖≤r. |
Hence, ‖ℑ1(u,v)+ℑ2(u,v)‖ϑ∈ϑr.
2) Next, for any σ∈ω, (u,v),(ξ,ζ)∈ϑ
‖ℑ1(u,v)−ℑ1(ξ,ξ)‖ϑ≤‖ℑ∗1(u)−ℑ∗1(ξ)‖ϑ1+‖ℑ∗∗1(v)−ℑ∗∗1(ξ)‖ϑ2≤|ν2||σ||T1−α||μ1|Γ(α−1)∫T0|(T−π)α−2||u(π)−ξ(π)|dπ+z∑j=1|((α−1)−(α−2)σσ−1j)−σT((α−1)−(α−2)Tσ−1j)|×|σ2−αj||Ej(u(σj))−Ej(ξ(σj))|+z∑j=1|(σ−σj)−σT(T−σj)||σ2−αj||E∗j(u(σj))−E∗j(ξ(σj))|+|ν4||T1−β||μ2|Γ(β−1)∫T0|(T−π)β−2||v(π)−ζ(π)|dπ+z∑k=1|((β−1)−(β−2)σσ−1k)−σT((β−1)−(β−2)Tσ−1k)|×|σ2−βk||Ek(v(σk))−Ek(ζ(σk))|+z∑k=1|(σ−σk)−σT(T−σk)||σ2−βk||E∗k(v(σk))−E∗k(ζ(σk))|≤(z(α−1)|σ2−αz||1−σT|LE+z|σ3−αz||σT−1|LE∗+|ν2||σ||μ1|Γ(α))‖u−ξ‖+(z(β−1)|σ2−β|z|1−σT|L∗E+z|σ3−βz||σT−1|L∗E∗+|ν4||σ||μ2|Γ(β))‖v−ζ‖≤L(ϱ1+ϱ2)‖(u−ξ,v−ζ)‖. |
Here \mathcal{L} = \max\{\mathcal{L}_{\mathcal{E}}, \mathcal{L}_{\mathcal{E}^*}, \mathcal{L}_{\mathcal{E}}^*, \mathcal{L}_{\mathcal{E}^*}^*\},
\varrho_{1} = z(\alpha-1)|\sigma_{z}^{2-\alpha}|\left|1-\frac{\sigma}{ \mathbb{T}}\right|+z|\sigma_{z}^{3-\alpha}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|+\frac{|\nu_{2}||\sigma|}{|\mu_{1}|\Gamma(\alpha)},\; \; z = 1,2,\dots,p, |
and
\varrho_{2} = z(\beta-1)|\sigma_{z}^{2-\beta|}\left|1-\frac{\sigma}{ \mathbb{T}}\right|+z|\sigma_{z}^{3-\beta}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|+\frac{|\nu_{4}||\sigma|}{|\mu_{2}|\Gamma(\beta)},\; \; z = 1,2,\dots,q. |
Therefore, \Im_{1} is a contractive operator.
3) Now, for the continuity and compactness of \Im_{2} , we make a sequence T_{s} = (\texttt{u}_{s}, \texttt{v}_{s}) in \vartheta_{r} such that (\texttt{u}_{s}, \texttt{v}_{s})\rightarrow(\texttt{u}, \texttt{v}) for s\rightarrow\infty in \vartheta_{r} . Thus, we have
\begin{align*} \|\Im_{2}&(\texttt{u}_{s},\texttt{v}_{s})-\Im_{2}(\texttt{u},\texttt{v})\|_{\vartheta}\\\leq&\|\Im_2^*(\texttt{u}_{s},\texttt{v}_{s})-\Im_2^*(\texttt{u},\texttt{v})\|_{\vartheta_{1}}+\|\Im_2^{**}(\texttt{u}_{s},\texttt{v}_{s})-\Im_2^{**}(\texttt{u},\texttt{v})\|_{\vartheta_{2}}\\ \leq&\Bigg(\frac{\mathcal{L}_{\phi_1}|\sigma^{\alpha}|\|\texttt{u}_{s}-\texttt{u}\|}{\Gamma(\alpha+1)}+\frac{\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|\|\texttt{v}_{s}-\texttt{v}\|}{\Gamma(\beta+1)}\Bigg)\Bigg(\frac{\left|\sigma^{2-\alpha}\right|\left|(\sigma-\sigma_{z})^{\alpha}\right|}{\Gamma(\alpha+1)}+\frac{\left|\sigma\right|\left| \mathbb{T}^{1-\alpha}\right|\left|( \mathbb{T}-\sigma_{z})^{\alpha}\right|}{\Gamma(\alpha+1)}\\ &+\frac{z\left|\sigma\right|\left|\sigma_{z}^{2-\alpha}\right|}{ \mathbb{T}}\Bigg[\frac{\left|\left( \mathbb{T}-\sigma_{z}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha-1}\right|}{\Gamma(\alpha)}+\frac{\left|\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{z}^{-1}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha}\right|}{\Gamma(\alpha+1)}\Bigg]\\ &+\frac{z\left|\left(\sigma-\sigma_{z}\right)\right|\left|\sigma_{z}^{2-\alpha}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha-1}\right|}{\Gamma(\alpha)}+\frac{z\left|\left((\alpha-1)-(\alpha-2)\sigma\sigma_{z}^{-1}\right)\right|\left|\sigma_{z}^{2-\alpha}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha}\right|}{\Gamma(\alpha+1)}\Bigg)\\& +\Bigg(\frac{\mathcal{L}_{\phi_2}|\sigma^{\alpha}|\|\texttt{u}_{s}-\texttt{u}\|}{\Gamma(\alpha+1)}+\frac{\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|\|\texttt{v}_{s}-\texttt{v}\|}{\Gamma(\beta+1)}\Bigg)\Bigg(\frac{\left|\sigma^{2-\beta}\right|\left|(\sigma-\sigma_{z})^{\beta}\right|}{\Gamma(\beta+1)}+\frac{\left|\sigma\right|\left| \mathbb{T}^{1-\beta}\right|\left|( \mathbb{T}-\sigma_{z})^{\beta}\right|}{\Gamma(\beta+1)}\\ &+\frac{z\left|\sigma\right|\left|\sigma_{z}^{2-\beta}\right|}{ \mathbb{T}}\Bigg[\frac{\left|\left( \mathbb{T}-\sigma_{z}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\beta-1}\right|}{\Gamma(\beta)}+\frac{\left|\left((\beta-1)-(\beta-2) \mathbb{T}\sigma_{z}^{-1}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\beta}\right|}{\Gamma(\beta+1)}\Bigg]\\ &+\frac{z\left|\left(\sigma-\sigma_{z}\right)\right|\left|\sigma_{z}^{2-\beta}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\beta-1}\right|}{\Gamma(\beta)}+\frac{z\left|\left((\beta-1)-(\beta-2)\sigma\sigma_{z}^{-1}\right)\right|\left|\sigma_{z}^{2-\beta}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\beta}\right|}{\Gamma(\beta+1)}\Bigg)\\ \leq&\mathcal{G}_{3}\left(\frac{\mathcal{L}_{\phi_1}|\sigma^{\alpha}|\|\texttt{u}_{s}-\texttt{u}\|}{\Gamma(\alpha+1)}+\frac{\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|\|\texttt{v}_{s}-\texttt{v}\|}{\Gamma(\beta+1)}\right)+\mathcal{G}_{3}^*\left(\frac{\mathcal{L}_{\phi_2}|\sigma^{\alpha}|\|\texttt{u}_{s}-\texttt{u}\|}{\Gamma(\alpha+1)}+\frac{\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|\|\texttt{v}_{s}-\texttt{v}\|}{\Gamma(\beta+1)}\right). \end{align*} |
This implies \|\Im_2(\texttt{u}_{s}, \texttt{v}_{s})-\Im_2(\texttt{u}, \texttt{v})\|_{\vartheta}\rightarrow0 as s\rightarrow\infty , therefore \Im_{2} is continuous.
Next, we show that \Im_{2} is uniformly bounded on \vartheta_{r} . From (3.17) and (3.18), we have
\begin{align*} \|\Im_{2}(\texttt{u},\texttt{v})\|_{\vartheta}&\leq\|\Im_{2}^*(\texttt{u},\texttt{v})\|_{\vartheta_{1}}+\|\Im_{2}^{**}(\texttt{u},\texttt{v})\|_{\vartheta_{2}}\\ &\leq o_{1}\mathcal{G}_{3}+ o_{2}\mathcal{G}_{3}^*+(\mathcal{G}_{3}\mathcal{G}_{4}+\mathcal{G}_{3}^*\mathcal{G}_{4}^*)\|(\texttt{u},\texttt{v})\|\\&\leq r. \end{align*} |
Thus, \Im_{2} is uniformly bounded on \vartheta_{r} .
For equicontinuity, suppose \eta_{1}, \eta_{2}\in\omega with \eta_{1} < \eta_{2} , and for any (\texttt{u}, \texttt{v})\in\vartheta_{r}\subset\vartheta where \vartheta_{r} is clearly bounded, we have
\begin{align*} \|\Im_{2}^{*}&(\texttt{u},\texttt{v})(\eta_{1})-\Im_{2}^*(\texttt{u},\texttt{v})(\eta_{2})\|_{\vartheta_{1}}\\ = &\max|\sigma^{2-\alpha}(\Im_{2}^*(\texttt{u},\texttt{v})(\eta_{1})-\Im_{2}^*(\texttt{u},\texttt{v})(\eta_{2}))|\\ \leq&\Bigg(o_{1}+\tau_{1}\frac{|\sigma^{\alpha}|\|\texttt{u}\|}{\Gamma(\alpha+1)}+\upsilon_{1}\frac{|\sigma^{\beta}|\|\texttt{v}\|}{\Gamma(\beta+1)}\Bigg)\Bigg(\frac{\left|\sigma^{2-\alpha}\right|\left|((\eta_{1}-\sigma_{z})^{\alpha}-(\eta_{2}-\sigma_{z})^{\alpha})\right|}{\Gamma(\alpha+1)}\\&+\frac{\left|\sigma^{2-\alpha}\right|\left|\eta_{1}^{\alpha-1}-\eta_{2}^{\alpha-1}\right|\left| \mathbb{T}^{1-\alpha}\right|\left|( \mathbb{T}-\sigma_{z})^{\alpha}\right|}{\Gamma(\alpha+1)}+\Bigg[\left|\left(\eta_{1}^{\alpha-2}-\eta_{2}^{\alpha-2}\right)\right|+\frac{\left|\left(\eta_{1}^{\alpha-1}-\eta_{2}^{\alpha-1}\right)\right|}{ \mathbb{T}}\Bigg]\\ &\times\Bigg[\frac{z\left|\sigma^{2-\alpha}\right|\left|\sigma_{z}^{3-\alpha}\right|\left|(\sigma_{z}-\sigma_{z-1})^{\alpha-1}\right|}{\Gamma(\alpha)}+\frac{z(\alpha-1)\left|\sigma^{2-\alpha}\right|\left|\sigma_{z}^{2-\alpha}\right|\left|(\sigma_{z}-\sigma_{z-1})^{\alpha}\right|}{\Gamma(\alpha+1)}\Bigg]\Bigg). \end{align*} |
This implies that
\|\Im_{2}^*(\texttt{u},\texttt{v})(\eta_{1})-\Im_{2}^*(\texttt{u},\texttt{v})(\eta_{2})\|_{\vartheta_{1}}\rightarrow0 \; \; as \; \; \eta_{1}\rightarrow\eta_{2}. |
In the same way, we have
\|\Im_{2}^{**}(\texttt{u},\texttt{v})(\eta_{1})-\Im_{2}^{**}(\texttt{u},\texttt{v})(\eta_{2})\|_{\vartheta_{2}}\rightarrow0 \; \; as \; \; \eta_{1}\rightarrow\eta_{2}. |
Hence
\|\Im_{2}(\texttt{u},\texttt{v})(\eta_{1})-\Im_{2}(\texttt{u},\texttt{v})(\eta_{2})\|_{\vartheta}\rightarrow0 \; \; as \; \; \eta_{1}\rightarrow\eta_{2}. |
Thus, \Im_{2} is equicontinuous. So \Im_{2} is relatively compact on \vartheta_{r} . Hence, by the Arzel \grave{a} –Ascoli Theorem, \Im_{2} is compact on \vartheta_{r}. Thus all the condition of Theorem 2.1 are satisfied. So the given system (1.1) has at least one solution.
Theorem 3.3. Let hypotheses (\boldsymbol{H}_{3}) , (\boldsymbol{H}_{4}) be satisfied with
\begin{equation} \Delta_{1}+\Delta_{3}+\frac{(\Delta_{2}\mathcal{L}_{\phi_1}+\Delta_{4}\mathcal{L}_{\phi_2})|\sigma^{\alpha}|}{\Gamma(\alpha+1)}+\frac{(\Delta_{2}\mathcal{L}_{\phi_1}^{*}+\Delta_{4}\mathcal{L}_{\phi_2}^{*})|\sigma^{\beta}|}{\Gamma(\beta+1)} \lt 1, \end{equation} | (3.19) |
then the given system (1.1) has unique solution.
Proof. First we define an operator \varphi = (\varphi_{1}, \varphi_{2}):\vartheta\rightarrow\vartheta , i.e., \varphi(\texttt{u}, \texttt{v})(\sigma) = (\varphi_{1}(\texttt{u}, \texttt{v}), \varphi_{2}(\texttt{u}, \texttt{v}))(\sigma) , where
\begin{align*} \varphi_{1}(\texttt{u},\texttt{v})(\sigma) = &\frac{\sigma^{\alpha-1}\texttt{u}_{2}}{\mu_{1} \mathbb{T}^{\alpha-1}}-\frac{\sigma^{\alpha-1}\texttt{u}_{1}}{ \mathbb{T}\nu_{1}\Gamma(\alpha-1)}+\frac{\sigma^{\alpha-2}\texttt{u}_{1}}{\nu_{1}\Gamma(\alpha-1)}+\frac{1}{\Gamma(\alpha)}\int_{\sigma_{z}}^{\sigma}(\sigma-\pi)^{\alpha-1}\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi\\ &-\frac{\sigma^{\alpha-1} \mathbb{T}^{1-\alpha}}{\Gamma(\alpha)}\int_{\sigma_{z}}^{ \mathbb{T}}( \mathbb{T}-\pi)^{\alpha-1}\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi-\frac{\nu_{2}\sigma^{\alpha-1} \mathbb{T}^{1-\alpha}}{\mu_{1}\Gamma(\alpha-1)}\int_{0}^{ \mathbb{T}}( \mathbb{T}-\pi)^{\alpha-2}\texttt{u}(\pi)d\pi\\ &-\frac{\sigma^{\alpha-1}}{ \mathbb{T}}\sum\limits_{j = 1}^{z}\Bigg[\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{j}^{-1}\right)\sigma_{j}^{2-\alpha}\mathcal{E}_{j}(\texttt{u}(\sigma_{j}))+\left( \mathbb{T}-\sigma_{j}\right)\sigma_{j}^{2-\alpha}\mathcal{E}_{j}^{*}(\texttt{u}(\sigma_{j}))\\ &+\frac{\left( \mathbb{T}-\sigma_{j}\right)\sigma_{j}^{2-\alpha}}{\Gamma(\alpha-1)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left(\sigma_{j}-\pi\right)^{\alpha-2}\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi\\ &+\frac{\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{j}^{-1}\right)\sigma_{j}^{2-\alpha}}{\Gamma(\alpha)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left(\sigma_{j}-\pi\right)^{\alpha-1}\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi\Bigg]\\ &+\sum\limits_{j = 1}^{z}\Bigg[\left((\alpha-1)-(\alpha-2)\sigma\sigma_{j}^{-1}\right)\sigma^{\alpha-2}\sigma_{j}^{2-\alpha}\mathcal{E}_{j}(\texttt{u}(\sigma_{j}))+\left(\sigma-\sigma_{j}\right)\sigma^{\alpha-2}\sigma_{j}^{2-\alpha}\mathcal{E}_{j}^{*}(\texttt{u}(\sigma_{j}))\\ &+\frac{\left(\sigma-\sigma_{j}\right)\sigma^{\alpha-2}\sigma_{j}^{2-\alpha}}{\Gamma(\alpha-1)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left(\sigma_{j}-\pi\right)^{\alpha-2}\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi\\ &+\frac{\left((\alpha-1)-(\alpha-2)\sigma\sigma_{j}^{-1}\right)\sigma^{\alpha-2}\sigma_{j}^{2-\alpha}}{\Gamma(\alpha)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left(\sigma_{j}-\pi\right)^{\alpha-1}\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi\Bigg],\\ &\quad for \ \ z = 1,2,\dots,p, \end{align*} |
and
\begin{align*} \varphi_{2}(\texttt{u},\texttt{v})(\sigma) = &\frac{\sigma^{\beta-1}\texttt{v}_{2}}{\mu_{2} \mathbb{T}^{\beta-1}}-\frac{\sigma^{\beta-1}\texttt{v}_{1}}{ \mathbb{T}\nu_{3}\Gamma(\beta-1)}+\frac{\sigma^{\beta-2}\texttt{v}_{1}}{\nu_{3}\Gamma(\beta-1)}+\frac{1}{\Gamma(\beta)}\int_{\sigma_{z}}^{\sigma}(\sigma-\pi)^{\beta-1}\phi_{2}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi\\ &-\frac{\sigma^{\beta-1} \mathbb{T}^{1-\beta}}{\Gamma(\beta)}\int_{\sigma_{z}}^{ \mathbb{T}}( \mathbb{T}-\pi)^{\beta-1}\phi_{2}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi-\frac{\nu_{4}\sigma^{\beta-1} \mathbb{T}^{1-\beta}}{\mu_{2}\Gamma(\beta-1)}\int_{0}^{ \mathbb{T}}( \mathbb{T}-\pi)^{\beta-2}\texttt{v}(\pi)d\pi\\ &-\frac{\sigma^{\beta-1}}{ \mathbb{T}}\sum\limits_{k = 1}^{z}\Bigg[\left((\beta-1)-(\beta-2) \mathbb{T}\sigma_{k}^{-1}\right)\sigma_{k}^{2-\beta}\mathcal{E}_{k}(\texttt{v}(\sigma_{k}))+\left( \mathbb{T}-\sigma_{k}\right)\sigma_{k}^{2-\beta}\mathcal{E}_{k}^{*}(\texttt{v}(\sigma_{k}))\\ &+\frac{\left( \mathbb{T}-\sigma_{k}\right)\sigma_{k}^{2-\beta}}{\Gamma(\beta-1)}\int_{\sigma_{k-1}}^{\sigma_{k}}\left(\sigma_{k}-\pi\right)^{\beta-2}\phi_{2}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi\\ &+\frac{\left((\beta-1)-(\beta-2) \mathbb{T}\sigma_{k}^{-1}\right)\sigma_{k}^{2-\beta}}{\Gamma(\beta)}\int_{\sigma_{k-1}}^{\sigma_{k}}\left(\sigma_{k}-\pi\right)^{\beta-1}\phi_{2}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi\Bigg]\\ &+\sum\limits_{k = 1}^{z}\Bigg[\left((\beta-1)-(\beta-2)\sigma\sigma_{k}^{-1}\right)\sigma^{\beta-2}\sigma_{k}^{2-\beta}\mathcal{E}_{k}(\texttt{v}(\sigma_{k}))+\left(\sigma-\sigma_{k}\right)\sigma^{\beta-2}\sigma_{k}^{2-\beta}\mathcal{E}_{k}^{*}(\texttt{v}(\sigma_{k}))\\ &+\frac{\left(\sigma-\sigma_{k}\right)\sigma^{\beta-2}\sigma_{k}^{2-\beta}}{\Gamma(\beta-1)}\int_{\sigma_{k-1}}^{\sigma_{k}}\left(\sigma_{k}-\pi\right)^{\beta-2}\phi_{2}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi\\ &+\frac{\left((\beta-1)-(\beta-2)\sigma\sigma_{k}^{-1}\right)\sigma^{\beta-2}\sigma_{k}^{2-\beta}}{\Gamma(\beta)}\int_{\sigma_{k-1}}^{\sigma_{k}}\left(\sigma_{k}-\pi\right)^{\beta-1}\phi_{2}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))d\pi\Bigg],\\ &\quad for \ \ z = 1,2,\dots,q. \end{align*} |
In view of Theorem 3.2, we have
\begin{align*} \nonumber |\sigma^{2-\alpha}&(\varphi_{1}(\texttt{u},\texttt{v})-\varphi_{1}(\xi,\zeta))|\\ \leq&\Bigg(\frac{\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\Bigg)\Bigg(\frac{\left|\sigma^{2-\alpha}\right|\left|(\sigma-\sigma_{z})^{\alpha}\right|}{\Gamma(\alpha+1)}+\frac{\left|\sigma\right|\left| \mathbb{T}^{1-\alpha}\right|\left|( \mathbb{T}-\sigma_{z})^{\alpha}\right|}{\Gamma(\alpha+1)}\\ &+\frac{z\left|\sigma\right|\left|\sigma_{z}^{2-\alpha}\right|}{ \mathbb{T}}\Bigg[\frac{\left|\left( \mathbb{T}-\sigma_{z}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha-1}\right|}{\Gamma(\alpha)}+\frac{\left|\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{z}^{-1}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha}\right|}{\Gamma(\alpha+1)}\Bigg]\\ &+\frac{z\left|\left(\sigma-\sigma_{z}\right)\right|\left|\sigma_{z}^{2-\alpha}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha-1}\right|}{\Gamma(\alpha)}+\frac{z\left|\left((\alpha-1)-(\alpha-2)\sigma\sigma_{z}^{-1}\right)\right|\left|\sigma_{z}^{2-\alpha}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha}\right|}{\Gamma(\alpha+1)}\Bigg)|\texttt{v}-\zeta|\\& +\Bigg[\left(\frac{\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\right)\bigg(\frac{\left|\sigma^{2-\alpha}\right|\left|(\sigma-\sigma_{z})^{\alpha}\right|}{\Gamma(\alpha+1)}+\frac{\left|\sigma\right|\left| \mathbb{T}^{1-\alpha}\right|\left|( \mathbb{T}-\sigma_{z})^{\alpha}\right|}{\Gamma(\alpha+1)}\\ &+\frac{z\left|\sigma\right|\left|\sigma_{z}^{2-\alpha}\right|}{ \mathbb{T}}\Bigg[\frac{\left|\left( \mathbb{T}-\sigma_{z}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha-1}\right|}{\Gamma(\alpha)}+\frac{\left|\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{z}^{-1}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha}\right|}{\Gamma(\alpha+1)}\Bigg]\\ &+\frac{z\left|\left(\sigma-\sigma_{z}\right)\right|\left|\sigma_{z}^{2-\alpha}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha-1}\right|}{\Gamma(\alpha)}+\frac{z\left|\left((\alpha-1)-(\alpha-2)\sigma\sigma_{z}^{-1}\right)\right|\left|\sigma_{z}^{2-\alpha}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha}\right|}{\Gamma(\alpha+1)}\bigg)\\ &+\left(z(\alpha-1)|\sigma_{z}^{2-\alpha}|\left|1-\frac{\sigma}{ \mathbb{T}}\right|\mathcal{L}_{\mathcal{E}}+z|\sigma_{z}^{3-\alpha}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|\mathcal{L}_{\mathcal{E}^{*}}+\frac{|\nu_{2}||\sigma|}{|\mu_{1}|\Gamma(\alpha)}\right)\Bigg]|\texttt{u}-\xi|. \end{align*} |
Taking \sup_{\sigma\in\omega} , we get
\begin{align*} \|\varphi_{1}(\texttt{u},\texttt{v})-\varphi_{1}(\xi,\zeta)\|_{\vartheta_{1}}\leq&\left( \Delta_{1}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\right)\|(\texttt{u},\texttt{v})-(\xi,\zeta)\|\\& \ \ \ for \; \; z = 1,2,\dots,p, \end{align*} |
where
\begin{align*} \Delta_{1} = &z(\alpha-1)|\sigma_{z}^{2-\alpha}|\left|1-\frac{\sigma}{ \mathbb{T}}\right|\mathcal{L}_{\mathcal{E}}+z|\sigma_{z}^{3-\alpha}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|\mathcal{L}_{\mathcal{E}^{*}}+\frac{|\nu_{2}||\sigma|}{|\mu_{1}|\Gamma(\alpha)},\\ \Delta_{2} = &\frac{\left|\sigma^{2-\alpha}\right|\left|(\sigma-\sigma_{z})^{\alpha}\right|}{\Gamma(\alpha+1)}+\frac{\left|\sigma\right|\left| \mathbb{T}^{1-\alpha}\right|\left|( \mathbb{T}-\sigma_{z})^{\alpha}\right|}{\Gamma(\alpha+1)}\\ &+\frac{z\left|\sigma\right|\left|\sigma_{z}^{2-\alpha}\right|}{ \mathbb{T}}\Bigg[\frac{\left|\left( \mathbb{T}-\sigma_{z}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha-1}\right|}{\Gamma(\alpha)}+\frac{\left|\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{z}^{-1}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha}\right|}{\Gamma(\alpha+1)}\Bigg]\\ &+\frac{z\left|\left(\sigma-\sigma_{z}\right)\right|\left|\sigma_{z}^{2-\alpha}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha-1}\right|}{\Gamma(\alpha)}+\frac{z\left|\left((\alpha-1)-(\alpha-2)\sigma\sigma_{z}^{-1}\right)\right|\left|\sigma_{z}^{2-\alpha}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\alpha}\right|}{\Gamma(\alpha+1)},\\ & \ \ \ for \; \; z = 1,2,\dots,p. \end{align*} |
Similarly,
\begin{align*} \|\varphi_{2}(\texttt{u},\texttt{v})-\varphi_{2}(\xi,\zeta)\|_{\vartheta_{2}}\leq&\left(\Delta_{3}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\right)\|(\texttt{u},\texttt{v})-(\xi,\zeta)\| \\& \ \ \ for \; \; z = 1,2,\dots,q, \end{align*} |
where
\begin{align*} \Delta_{3} = &z(\beta-1)|\sigma_{z}^{2-\beta}|\left|1-\frac{\sigma}{ \mathbb{T}}\right|\mathcal{L}_{\mathcal{E}}^{*}+z|\sigma_{z}^{3-\beta}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|\mathcal{L}_{\mathcal{E}^{*}}^{*}+\frac{|\nu_{4}||\sigma|}{|\mu_{2}|\Gamma(\beta)},\\ \Delta_{4} = &\frac{\left|\sigma^{2-\beta}\right|\left|(\sigma-\sigma_{z})^{\beta}\right|}{\Gamma(\beta+1)}+\frac{\left|\sigma\right|\left| \mathbb{T}^{1-\beta}\right|\left|( \mathbb{T}-\sigma_{z})^{\beta}\right|}{\Gamma(\beta+1)}\\ &+\frac{z\left|\sigma\right|\left|\sigma_{z}^{2-\beta}\right|}{ \mathbb{T}}\Bigg[\frac{\left|\left( \mathbb{T}-\sigma_{z}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\beta-1}\right|}{\Gamma(\beta)}+\frac{\left|\left((\beta-1)-(\beta-2) \mathbb{T}\sigma_{z}^{-1}\right)\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\beta}\right|}{\Gamma(\beta+1)}\Bigg]\\ &+\frac{z\left|\left(\sigma-\sigma_{z}\right)\right|\left|\sigma_{z}^{2-\beta}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\beta-1}\right|}{\Gamma(\beta)}+\frac{z\left|\left((\beta-1)-(\beta-2)\sigma\sigma_{z}^{-1}\right)\right|\left|\sigma_{z}^{2-\beta}\right|\left|\left(\sigma_{z}-\sigma_{z-1}\right)^{\beta}\right|}{\Gamma(\beta+1)},\\ & \ \ \ for\; \; z = 1,2,\dots,q. \end{align*} |
Hence
\begin{align*} \|\varphi(\texttt{u},\texttt{v})-\varphi(\xi,\zeta)\|_{\vartheta}\leq\left(\Delta_{1}+\Delta_{3}+\frac{(\Delta_{2}\mathcal{L}_{\phi_1}+\Delta_{4}\mathcal{L}_{\phi_2})|\sigma^{\alpha}|}{\Gamma(\alpha+1)}+\frac{(\Delta_{2}\mathcal{L}_{\phi_1}^{*}+\Delta_{4}\mathcal{L}_{\phi_2}^{*})|\sigma^{\beta}|}{\Gamma(\beta+1)}\right)\|(\texttt{u},\texttt{v})-(\xi,\zeta)\|. \end{align*} |
This implies that the operator \varphi is a contraction. Therefore, (1.1) has a unique solution.
In this section, we study different kinds of stabilities, like \mathcal{HU} , generalized \mathcal{HU} , \mathcal{HUR} , and generalized \mathcal{HUR} stability of the proposed system.
Theorem 4.1. If assumptions (\boldsymbol{H}_{3}) , (\boldsymbol{H}_{4}) and inequality (3.19) are satisfied and
\begin{equation*} \mathcal{F} = 1-\frac{\bigg(\frac{\Delta_{2}\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg) \bigg(\frac{\Delta_{4}\mathcal{L}_{\phi_2}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)}{\bigg[ 1-\bigg(\Delta_{1}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)\bigg]\bigg[ 1-\bigg(\Delta_{3}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)\bigg] } \gt 0, \end{equation*} |
then the unique solution of the coupled system (1.1) is \mathcal{HU} stable and consequently generalized \mathcal{HU} stable.
Proof. Let (\xi, \zeta)\in\vartheta is a solution of inequality (2.1), and let (\texttt{u}, \texttt{v})\in\vartheta be the unique solution of the coupled system given by
\begin{eqnarray} \left\{\begin{split} &\left\{\begin{split} &\mathcal{D}^\alpha \texttt{u}(\sigma)-\phi_1(\sigma,\mathcal{I}^\alpha \texttt{u}(\sigma),\mathcal{I}^\beta \texttt{v}(\sigma)) = 0,~~\sigma\in\omega,~~\sigma\neq \sigma_{j},~~j = 1,2,\dots,p,\\ &\Delta \texttt{u}(\sigma_{j})-\mathcal{E}_{j}(\texttt{u}(\sigma_{j})) = 0,\qquad\Delta \texttt{u}'(\sigma_{j})-\mathcal{E}_{j}^*(\texttt{u}(\sigma_{j})) = 0,~~j = 1,2,\dots,p,\\ &\nu_{1}\mathcal{D}^{\alpha-2}\texttt{u}(\sigma)|_{\sigma = 0} = \texttt{u}_{1},\qquad\mu_{1}\texttt{u}(\sigma)|_{\sigma = \mathbb{T} }+\nu_{2}\mathcal{I}^{\alpha-1}\texttt{u}(\sigma)|_{\sigma = \mathbb{T} } = \texttt{u}_{2}, \end{split}\right.\\ &\left\{\begin{split} &\mathcal{D}^\beta \texttt{v}(\sigma)-\phi_2(\sigma,\mathcal{I}^\alpha \texttt{u}(\sigma),\mathcal{I}^\beta \texttt{v}(\sigma)) = 0,~~\sigma\in\omega,~~\sigma\neq \sigma_{k},~~k = 1,2,\dots,q,\\ &\Delta \texttt{v}(\sigma_{k})-\mathcal{E}_{k}(\texttt{v}(\sigma_{k})) = 0,\qquad\Delta \texttt{v}'(\sigma_{k})-\mathcal{E}_{k}^*(\texttt{v}(\sigma_{k})) = 0,~~k = 1,2,\dots,q,\\ &\nu_{3}\mathcal{D}^{\beta-2}\texttt{v}(\sigma)|_{\sigma = 0} = \texttt{v}_{1},\qquad\mu_{2}\texttt{v}(\sigma)|_{\sigma = \mathbb{T} }+\nu_{4}\mathcal{I}^{\beta-1}\texttt{v}(\sigma)|_{\sigma = \mathbb{T} } = \texttt{v}_{2}. \end{split}\right. \end{split}\right. \end{eqnarray} | (4.1) |
By Remark 2.1 we have
\begin{eqnarray}\label{eq4.2} \left\{\begin{split} &\left\{\begin{split} &\mathcal{D}^\alpha \xi(\sigma) = \phi_{1}(\sigma,\mathcal{I}^{\alpha}\xi(\sigma),\mathcal{I}^{\beta}\zeta(\sigma))+\mathfrak{K}_{\phi_{1}}(\sigma),\\ &\Delta\xi(\sigma_{j}) = \mathcal{E}_{j}(\xi(\sigma_{j}))+\mathfrak{K}_{\phi_{1j}},~~j = 1,2,\dots,p,\\ &\Delta\xi'(\sigma_{j}) = \mathcal{E}_{j}^*(\xi(\sigma_{j}))+\mathfrak{K}_{\phi_{1j}},~~j = 1,2,\dots,p, \end{split}\right.\\ &\left\{\begin{split} &\mathcal{D}^\beta \zeta(\sigma) = \phi_2(\sigma,\mathcal{I}^\alpha \xi(\sigma),\mathcal{I}^\beta \zeta(\sigma))+\mathfrak{L}_{\phi_2}(\sigma),\\ &\Delta\zeta(\sigma_{k}) = \mathcal{E}_{k}(\zeta(\sigma_{k}))+\mathfrak{L}_{\phi_{2k}},~~k = 1,2,\dots,q,\\ &\Delta\zeta'(\sigma_{k}) = \mathcal{E}_{k}^*(\zeta(\sigma_{k}))+\mathfrak{L}_{\phi_{2k}},~~k = 1,2,\dots,q. \end{split}\right. \end{split}\right. \end{eqnarray} | (4.2) |
By Corollary 1, the solution of problem (4.2) is
\begin{align} \xi(\sigma) = &\frac{\sigma^{\alpha-1}\texttt{u}_{2}}{\mu_{1} \mathbb{T}^{\alpha-1}}-\frac{\sigma^{\alpha-1}\texttt{u}_{1}}{ \mathbb{T}\nu_{1}\Gamma(\alpha-1)}+\frac{\sigma^{\alpha-2}\texttt{u}_{1}}{\nu_{1}\Gamma(\alpha-1)}-\frac{\nu_{2}\sigma^{\alpha-1} \mathbb{T}^{1-\alpha}}{\mu_{1}\Gamma(\alpha-1)}\int_{0}^{ \mathbb{T}}( \mathbb{T}-\pi)^{\alpha-2}\xi(\pi)d\pi\\ &+\frac{1}{\Gamma(\alpha)}\int_{\sigma_{z}}^{\sigma}(\sigma-\pi)^{\alpha-1}(\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{K}_{\phi_{1}}(\pi))d\pi\\ &-\frac{\sigma^{\alpha-1} \mathbb{T}^{1-\alpha}}{\Gamma(\alpha)}\int_{\sigma_{z}}^{ \mathbb{T}}( \mathbb{T}-\pi)^{\alpha-1}(\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{K}_{\phi_{1}}(\pi))d\pi\\ &-\frac{\sigma^{\alpha-1}}{ \mathbb{T}}\sum\limits_{j = 1}^{z}\Bigg[\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{j}^{-1}\right)\sigma_{j}^{2-\alpha}(\mathcal{E}_{j}(\xi(\sigma_{j}))+\mathfrak{K}_{\phi_{1j}})+\left( \mathbb{T}-\sigma_{j}\right)\sigma_{j}^{2-\alpha}(\mathcal{E}_{j}^{*}(\xi(\sigma_{j}))+\mathfrak{K}_{\phi_{1j}})\\ &+\frac{\left( \mathbb{T}-\sigma_{j}\right)\sigma_{j}^{2-\alpha}}{\Gamma(\alpha-1)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left(\sigma_{j}-\pi\right)^{\alpha-2}(\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{K}_{\phi_{1}}(\pi))d\pi\\ &+\frac{\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{j}^{-1}\right)\sigma_{j}^{2-\alpha}}{\Gamma(\alpha)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left(\sigma_{j}-\pi\right)^{\alpha-1}(\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{K}_{\phi_{1}}(\pi))d\pi\Bigg]\\ &+\sum\limits_{j = 1}^{z}\Bigg[\left((\alpha-1)-(\alpha-2)\sigma\sigma_{j}^{-1}\right)\sigma^{\alpha-2}\sigma_{j}^{2-\alpha}(\mathcal{E}_{j}(\xi(\sigma_{j}))+\mathfrak{K}_{\phi_{1j}})+\left(\sigma-\sigma_{j}\right)\sigma^{\alpha-2}\sigma_{j}^{2-\alpha}(\mathcal{E}_{j}^{*}(\xi(\sigma_{j}))+\mathfrak{K}_{\phi_{1j}})\\ &+\frac{\left(\sigma-\sigma_{j}\right)\sigma^{\alpha-2}\sigma_{j}^{2-\alpha}}{\Gamma(\alpha-1)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left(\sigma_{j}-\pi\right)^{\alpha-2}(\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{K}_{\phi_{1}}(\pi))d\pi\\ &+\frac{\left((\alpha-1)-(\alpha-2)\sigma\sigma_{j}^{-1}\right)\sigma^{\alpha-2}\sigma_{j}^{2-\alpha}}{\Gamma(\alpha)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left(\sigma_{j}-\pi\right)^{\alpha-1}(\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{K}_{\phi_{1}}(\pi))d\pi\Bigg],\\ &\quad z = 1,2,\dots,p, \end{align} | (4.3) |
and
\begin{align} \zeta(\sigma) = &\frac{\sigma^{\beta-1}\texttt{v}_{2}}{\mu_{2} \mathbb{T}^{\beta-1}}-\frac{\sigma^{\beta-1}\texttt{v}_{1}}{ \mathbb{T}\nu_{3}\Gamma(\beta-1)}+\frac{\sigma^{\beta-2}\texttt{v}_{1}}{\nu_{3}\Gamma(\beta-1)}-\frac{\nu_{4}\sigma^{\beta-1} \mathbb{T}^{1-\beta}}{\mu_{2}\Gamma(\beta-1)}\int_{0}^{ \mathbb{T}}( \mathbb{T}-\pi)^{\beta-2}\zeta(\pi)d\pi\\ &+\frac{1}{\Gamma(\beta)}\int_{\sigma_{z}}^{\sigma}(\sigma-\pi)^{\beta-1}(\phi_{2}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{L}_{\phi_{2}}(\pi))d\pi\\ &-\frac{\sigma^{\beta-1} \mathbb{T}^{1-\beta}}{\Gamma(\beta)}\int_{\sigma_{z}}^{ \mathbb{T}}( \mathbb{T}-\pi)^{\beta-1}(\phi_{2}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{L}_{\phi_{2}}(\pi))d\pi\\ &-\frac{\sigma^{\beta-1}}{ \mathbb{T}}\sum\limits_{k = 1}^{z}\Bigg[\left((\beta-1)-(\beta-2) \mathbb{T}\sigma_{k}^{-1}\right)\sigma_{k}^{2-\beta}(\mathcal{E}_{k}(\zeta(\sigma_{k}))+\mathfrak{L}_{\phi_{2k}})+\left( \mathbb{T}-\sigma_{k}\right)\sigma_{k}^{2-\beta}(\mathcal{E}_{k}^{*}(\zeta(\sigma_{k}))+\mathfrak{L}_{\phi_{2k}})\\ &+\frac{\left( \mathbb{T}-\sigma_{k}\right)\sigma_{k}^{2-\beta}}{\Gamma(\beta-1)}\int_{\sigma_{k-1}}^{\sigma_{k}}\left(\sigma_{k}-\pi\right)^{\beta-2}(\phi_{2}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{L}_{\phi_{2}}(\pi))d\pi\\ &+\frac{\left((\beta-1)-(\beta-2) \mathbb{T}\sigma_{k}^{-1}\right)\sigma_{k}^{2-\beta}}{\Gamma(\beta)}\int_{\sigma_{k-1}}^{\sigma_{k}}\left(\sigma_{k}-\pi\right)^{\beta-1}(\phi_{2}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{L}_{\phi_{2}}(\pi))d\pi\Bigg]\\ &+\sum\limits_{k = 1}^{z}\Bigg[\left((\beta-1)-(\beta-2)\sigma\sigma_{k}^{-1}\right)\sigma_{k}^{2-\beta}(\mathcal{E}_{k}(\zeta(\sigma_{k}))+\mathfrak{L}_{\phi_{2k}})+\left(\sigma-\sigma_{k}\right)\sigma^{\beta-2}\sigma_{k}^{2-\beta}(\mathcal{E}_{k}^{*}(\zeta(\sigma_{k}))+\mathfrak{L}_{\phi_{2k}})\\ &+\frac{\left(\sigma-\sigma_{k}\right)\sigma^{\beta-2}\sigma_{k}^{2-\beta}}{\Gamma(\beta-1)}\int_{\sigma_{k-1}}^{\sigma_{k}}\left(\sigma_{k}-\pi\right)^{\beta-2}(\phi_{2}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{L}_{\phi_{2}}(\pi))d\pi\\ &+\frac{\left((\beta-1)-(\beta-2)\sigma\sigma_{k}^{-1}\right)\sigma^{\beta-2}\sigma_{k}^{2-\beta}}{\Gamma(\beta)}\int_{\sigma_{k-1}}^{\sigma_{k}}\left(\sigma_{k}-\pi\right)^{\beta-1}(\phi_{2}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))+\mathfrak{L}_{\phi_{2}}(\pi))d\pi\Bigg],\\ &\quad z = 1,2,\dots,q. \end{align} | (4.4) |
We consider
\begin{align*} |\sigma^{2-\alpha}(\texttt{u}(\sigma)-\xi(\sigma))| \leq&\frac{\left|\sigma^{2-\alpha}\right|}{\Gamma(\alpha)}\int_{\sigma_{z}}^{\sigma}\left|(\sigma-\pi)^{\alpha-1}\right||\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))-\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))|d\pi\\ &+\frac{|\sigma|| \mathbb{T}^{1-\alpha}|}{\Gamma(\alpha)}\int_{\sigma_{z}}^{ \mathbb{T}}\left|( \mathbb{T}-\pi)^{\alpha-1}\right||\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))-\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))|d\pi\\ &+\frac{|\nu_{2}||\sigma|| \mathbb{T}^{1-\alpha}|}{|\mu_{1}|\Gamma(\alpha-1)}\int_{0}^{ \mathbb{T}}\left|( \mathbb{T}-\pi)^{\alpha-2}\right||\texttt{u}(\pi)-\xi(\pi)|d\pi\\ &+\sum\limits_{j = 1}^{z}\left|\left((\alpha-1)-(\alpha-2)\sigma\sigma_{j}^{-1}\right)-\frac{\sigma}{ \mathbb{T}}\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{j}^{-1}\right)\right|\\ &\times|\sigma_{j}^{2-\alpha}|\left|\mathcal{E}_{j}(\texttt{u}(\sigma_{j}))-\mathcal{E}_{j}(\xi(\sigma_{j}))\right|\\ &+\sum\limits_{j = 1}^{z}\left|\left(\sigma-\sigma_{j}\right)-\frac{\sigma}{ \mathbb{T}}\left( \mathbb{T}-\sigma_{j}\right)\right||\sigma_{j}^{2-\alpha}|\left|\mathcal{E}_{j}^{*}(\texttt{u}(\sigma_{j}))-\mathcal{E}_{j}^{*}(\xi(\sigma_{j}))\right|\\ &+\frac{|\sigma|}{ \mathbb{T}}\sum\limits_{j = 1}^{z}\Bigg[\frac{|\left( \mathbb{T}-\sigma_{j}\right)||\sigma_{j}^{2-\alpha}|}{\Gamma(\alpha-1)}\\ &\times\int_{\sigma_{j-1}}^{\sigma_{j}}\left|\left(\sigma_{j}-\pi\right)^{\alpha-2}\right||\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))-\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))|d\pi\\ &+\frac{\left|\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{j}^{-1}\right)\right||\sigma_{j}^{2-\alpha}|}{\Gamma(\alpha)}\\ &\times\int_{\sigma_{j-1}}^{\sigma_{j}}\left|\left(\sigma_{j}-\pi\right)^{\alpha-1}\right||\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))-\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))|d\pi\Bigg]\\ &+\sum\limits_{j = 1}^{z}\Bigg[\frac{\left|\left(\sigma-\sigma_{j}\right)\right||\sigma_{j}^{2-\alpha}|}{\Gamma(\alpha-1)}\\ &\times\int_{\sigma_{j-1}}^{\sigma_{j}}\left|\left(\sigma_{j}-\pi\right)^{\alpha-2}\right||\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))-\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))|d\pi\\ &+\frac{\left|\left((\alpha-1)-(\alpha-2)\sigma\sigma_{j}^{-1}\right)\right||\sigma_{j}^{2-\alpha}|}{\Gamma(\alpha)}\\ &\times\int_{\sigma_{j-1}}^{\sigma_{j}}\left|\left(\sigma_{j}-\pi\right)^{\alpha-1}\right||\phi_{1}(\pi,\mathcal{I}^{\alpha}\texttt{u}(\pi),\mathcal{I}^{\beta}\texttt{v}(\pi))-\phi_{1}(\pi,\mathcal{I}^{\alpha}\xi(\pi),\mathcal{I}^{\beta}\zeta(\pi))|d\pi\Bigg] \end{align*} |
\begin{align*} &+\frac{\left|\sigma^{2-\alpha}\right|}{\Gamma(\alpha)}\int_{\sigma_{z}}^{\sigma}\left|(\sigma-\pi)^{\alpha-1}\right|\left|\mathfrak{K}_{\phi_{1}}(\pi)\right|d\pi+\frac{\left|\sigma\right|\left| \mathbb{T}^{1-\alpha}\right|}{\Gamma(\alpha)}\int_{\sigma_{z}}^{ \mathbb{T}}\left|( \mathbb{T}-\pi)^{\alpha-1}\right|\left|\mathfrak{K}_{\phi_{1}}(\pi)\right|d\pi\\ &+\sum\limits_{j = 1}^{z}\left|\left((\alpha-1)-(\alpha-2)\sigma\sigma_{j}^{-1}\right)-\frac{\sigma}{ \mathbb{T}}\left((\alpha-1)-(\alpha-2) \mathbb{T}\sigma_{j}^{-1}\right)\right||\sigma_{j}^{2-\alpha}|\left|\mathfrak{K}_{\phi_{1j}}\right|\\ &+\sum\limits_{j = 1}^{z}\left|\left(\sigma-\sigma_{j}\right)-\frac{\sigma}{ \mathbb{T}}\left( \mathbb{T}-\sigma_{j}\right)\right||\sigma_{j}^{2-\alpha}|\left|\mathfrak{K}_{\phi_{1j}}\right|\\ &+\frac{\sigma}{ \mathbb{T}}\sum\limits_{j = 1}^{z}\Bigg[\frac{\left|\left( \mathbb{T}-\sigma_{j}\right)\right|\left|\sigma_{j}^{2-\alpha}\right|}{\Gamma(\alpha-1)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left|\left(\sigma_{j}-\pi\right)^{\alpha-2}\right|\left|\mathfrak{K}_{\phi_{1}}(\pi)\right|d\pi\\ &+\frac{\left|(\alpha-1)-(\alpha-2) \mathbb{T}\left|\sigma_{j}^{-1}\right|\right|\left|\sigma_{j}^{2-\alpha}\right|}{\Gamma(\alpha)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left|\left(\sigma_{j}-\pi\right)^{\alpha-1}\right|\left|\mathfrak{K}_{\phi_{1}}(\pi)\right|d\pi\Bigg]\\ &+\sum\limits_{j = 1}^{z}\Bigg[\frac{\left|\left(\sigma-\sigma_{j}\right)\right|\left|\sigma_{j}^{2-\alpha}\right|}{\Gamma(\alpha-1)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left|\left(\sigma_{j}-\pi\right)^{\alpha-2}\right|\left|\mathfrak{K}_{\phi_{1}}(\pi)\right|d\pi\\ &+\frac{\left|\left((\alpha-1)-(\alpha-2)\sigma\sigma_{j}^{-1}\right)\right|\left|\sigma_{j}^{2-\alpha}\right|}{\Gamma(\alpha)}\int_{\sigma_{j-1}}^{\sigma_{j}}\left|\left(\sigma_{j}-\pi\right)^{\alpha-1}\right|\left|\mathfrak{K}_{\phi_{1}}(\pi)\right|d\pi\Bigg]. \end{align*} |
As in Theorem 3.3, we get
\begin{align} \|\texttt{u}-\xi\|_{\vartheta_{1}}\leq&\bigg(\Delta_{1}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)\|\texttt{u}-\xi\|_{\vartheta_{1}}+\bigg(\frac{\Delta_{2}\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)\|\texttt{v}-\zeta\|_{\vartheta_{1}}\\& +\left(\Delta_{2}+z(\alpha-1)|\sigma_{z}^{2-\alpha}|\left|1-\frac{\sigma}{ \mathbb{T}}\right|+z|\sigma_{z}^{3-\alpha}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|+\frac{|\nu_{2}||\sigma|}{|\mu_{1}|\Gamma(\alpha)}\right)\kappa_{\alpha},\\&z = 1,2,\dots,p, \end{align} | (4.5) |
and
\begin{align} \|\texttt{v}-\zeta\|_{\vartheta_{2}}\leq&\bigg(\frac{\Delta_{4}\mathcal{L}_{\phi_2}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)\|\texttt{u}-\xi\|_{\vartheta_{2}}+\bigg(\Delta_{3}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)\|\texttt{v}-\zeta\|_{\vartheta_{2}}\\& +\left(\Delta_{4}+z(\beta-1)|\sigma_{z}^{2-\beta}|\left|1-\frac{\sigma}{ \mathbb{T}}\right|+z|\sigma_{z}^{3-\beta}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|+\frac{|\nu_{4}||\sigma|}{|\mu_{2}|\Gamma(\beta)}\right)\kappa_{\beta},\\&z = 1,2,\dots,q. \end{align} | (4.6) |
From (4.5) and (4.6), we have
\begin{align*} \|\texttt{u}-\xi\|_{\vartheta_{1}}&-\frac{\bigg(\frac{\Delta_{2}\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)}{1-\bigg(\Delta_{1}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)}\|\texttt{v}-\zeta\|_{\vartheta_{1}}\\& \leq\frac{\left(\Delta_{2}+z(\alpha-1)|\sigma_{z}^{2-\alpha}|\left|1-\frac{\sigma}{ \mathbb{T}}\right|+z|\sigma_{z}^{3-\alpha}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|+\frac{|\nu_{2}||\sigma|}{|\mu_{1}|\Gamma(\alpha)}\right)}{1-\bigg(\Delta_{1}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)}\kappa_{\alpha} \end{align*} |
and
\begin{align*} \|\texttt{v}-\zeta\|_{\vartheta_{2}}&-\frac{\bigg(\frac{\Delta_{4}\mathcal{L}_{\phi_2}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)}{1-\bigg(\Delta_{3}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)}\|\texttt{u}-\xi\|_{\vartheta_{2}}\\& \leq\frac{\left(\Delta_{4}+z(\beta-1)|\sigma_{z}^{2-\beta}|\left|1-\frac{\sigma}{ \mathbb{T}}\right|+z|\sigma_{z}^{3-\beta}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|+\frac{|\nu_{4}||\sigma|}{|\mu_{2}|\Gamma(\beta)}\right)}{1-\bigg(\Delta_{3}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)}\kappa_{\beta} \end{align*} |
respectively. Let
\begin{align*} \mathcal{P}_{1}& = \frac{\bigg(\frac{\Delta_{2}\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)}{1-\bigg(\Delta_{1}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)},\qquad\quad \mathcal{P}_{2} = \frac{\left(\Delta_{2}+z(\alpha-1)|\sigma_{z}^{2-\alpha}|\left|1-\frac{\sigma}{ \mathbb{T}}\right|+z|\sigma_{z}^{3-\alpha}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|+\frac{|\nu_{2}||\sigma|}{|\mu_{1}|\Gamma(\alpha)}\right)}{1-\bigg(\Delta_{1}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)},\\ \mathcal{P}_{3}& = \frac{\bigg(\frac{\Delta_{4}\mathcal{L}_{\phi_2}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)}{1-\bigg(\Delta_{3}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)},\; \; and\; \; \mathcal{P}_{4} = \frac{\left(\Delta_{4}+z(\beta-1)|\sigma_{z}^{2-\beta}|\left|1-\frac{\sigma}{ \mathbb{T}}\right|+z|\sigma_{z}^{3-\beta}|\left|\frac{\sigma}{ \mathbb{T}}-1\right|+\frac{|\nu_{4}||\sigma|}{|\mu_{2}|\Gamma(\beta)}\right)}{1-\bigg(\Delta_{3}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)}. \end{align*} |
Then the last two inequalities can be written in a matrix form as follows:
\begin{eqnarray*} \begin{bmatrix} 1 & -\mathcal{P}_{1} \\ -\mathcal{P}_{3} & 1 \end{bmatrix} \begin{bmatrix} \|\texttt{u}-\xi\|_{\vartheta_{1}} \\ \|\texttt{v}-\zeta\|_{\vartheta_{2}} \end{bmatrix} \le \begin{bmatrix} \mathcal{P}_{2}\kappa_{\alpha} \\ \mathcal{P}_{4}\kappa_{\beta} \end{bmatrix} \end{eqnarray*} |
\begin{eqnarray} \begin{bmatrix} \|\texttt{u}-\xi\|_{\vartheta_{1}} \\ \|\texttt{v}-\zeta\|_{\vartheta_{2}} \end{bmatrix} \le \begin{bmatrix} \frac{1}{\mathcal{F}} & \frac{\mathcal{P}_{1}}{\mathcal{F}} \\ \\ \frac{\mathcal{P}_{3}}{\mathcal{F}} & \frac{1}{\mathcal{F}} \end{bmatrix} \begin{bmatrix} \mathcal{P}_{2}\kappa_{\alpha} \\ \mathcal{P}_{4}\kappa_{\beta} \end{bmatrix}, \end{eqnarray} | (4.7) |
where
\begin{eqnarray*} \mathcal{F} = 1-\frac{\bigg(\frac{\Delta_{2}\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg) \bigg(\frac{\Delta_{4}\mathcal{L}_{\phi_2}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)}{\bigg[ 1-\bigg(\Delta_{1}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)\bigg]\bigg[ 1-\bigg(\Delta_{3}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)\bigg] } \gt 0. \end{eqnarray*} |
From system (4.7) we have
\begin{align*} &\|\texttt{u}-\xi\|_{\vartheta_{1}}\leq\frac{\mathcal{P}_{2}\kappa_{\alpha}}{\mathcal{F}}+\frac{\mathcal{P}_{1}\mathcal{P}_{4}\kappa_{\beta}}{\mathcal{F}},\\ &\|\texttt{v}-\zeta\|_{\vartheta_{2}}\leq\frac{\mathcal{P}_{2}\mathcal{P}_{3}\kappa_{\alpha}}{\mathcal{F}}+\frac{\mathcal{P}_{4}\kappa_{\beta}}{\mathcal{F}}, \end{align*} |
which implies that
\begin{eqnarray*} \|\texttt{u}-\xi\|_{\vartheta_{1}}+\|\texttt{v}-\zeta\|_{\vartheta_{2}}\leq\frac{\mathcal{P}_{2}\kappa_{\alpha}}{\mathcal{F}}+\frac{\mathcal{P}_{1}\mathcal{P}_{4}\kappa_{\beta}}{\mathcal{F}}+\frac{\mathcal{P}_{2}\mathcal{P}_{3}\kappa_{\alpha}}{\mathcal{F}}+\frac{\mathcal{P}_{4}\kappa_{\beta}}{\mathcal{F}}. \end{eqnarray*} |
If \kappa = \max\{\kappa_{\alpha}, \kappa_{\beta}\} and \mathcal{N}_{\alpha, \beta} = \frac{\mathcal{P}_{2}}{\mathcal{F}}+\frac{\mathcal{P}_{1}\mathcal{P}_{4}}{\mathcal{F}}+\frac{\mathcal{P}_{2}\mathcal{P}_{3}}{\mathcal{F}}+\frac{\mathcal{P}_{4}}{\mathcal{F}}, then
\begin{eqnarray*} \|(\texttt{u},\texttt{v})-(\xi,\zeta)\|_{\vartheta}\le\mathcal{N}_{\alpha,\beta}\kappa. \end{eqnarray*} |
Thus system (1.1) is \mathcal{HU} stable. Also, if
\begin{eqnarray*} \|(\texttt{u},\texttt{v})-(\xi,\zeta)\|_{\vartheta}\leq\mathcal{N}_{\alpha,\beta}\mathcal{N'}(\kappa), \end{eqnarray*} |
with \mathcal{N'}(0) = 0, then the given system (1.1) is generalized \mathcal{HU} stable.
For the next result, we assume the following:
(H5) Let there exists two nondecreasing functions w_{\alpha}, w_{\beta}\in\mathcal{C}(\omega, \mathbb{R^+}) such that
\begin{eqnarray} \mathcal{I}^{\alpha}w_{\alpha}(\sigma)\le\mathcal{L}_{\alpha}w_{\alpha}(\sigma)\; \; \; and\; \; \; \mathcal{I}^{\beta}w_{\beta}(\sigma)\leq\mathcal{L}_{\beta}w_{\beta}(\sigma),\; \; where \; \; \mathcal{L}_{\alpha},\mathcal{L}_{\beta} \gt 0. \end{eqnarray} | (4.8) |
Theorem 4.2. If assumptions (boldsymbol) – (\boldsymbol{H}_{5}) and inequality (3.19) are satisfied and
\begin{equation*} \mathcal{F} = 1-\frac{\bigg(\frac{\Delta_{2}\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg) \bigg(\frac{\Delta_{4}\mathcal{L}_{\phi_2}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)}{\bigg[ 1-\bigg(\Delta_{1}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)\bigg]\bigg[ 1-\bigg(\Delta_{3}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)\bigg] } \gt 0, \end{equation*} |
then the unique solution of the given system (1.1) is \mathcal{HUR} stable and accordingly generalized \mathcal{HUR} stable.
Proof. With the help of Definitions 2.5 and 2.6, we can achieve our result doing the same steps as in Theorem 4.1.
Here we present a specific example, as follows.
Example 5.1. Let
\begin{eqnarray}\label{eq5.1} \left\{\begin{split} &\left\{\begin{split} &\mathcal{D}^\frac{6}{5} {\mathit{\mathtt{u}}}(\sigma)-\frac{2+\mathcal{I}^\frac{6}{5} {\mathit{\mathtt{u}}}(\sigma)+\mathcal{I}^\frac{5}{4} {\mathit{\mathtt{v}}}(\sigma)}{80e^{\sigma+90}(1+\mathcal{I}^\frac{6}{5}{\mathit{\mathtt{u}}}(\sigma)+\mathcal{I}^\frac{5}{4}{\mathit{\mathtt{v}}}(\sigma))} = 0,~~\sigma\neq\frac{3}{2},\\ &\Delta{\mathit{\mathtt{u}}}\left(\frac{3}{2}\right) = \mathcal{E}_{1}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \frac{|{\mathit{\mathtt{u}}}(\frac{3}{2})|}{70+|{\mathit{\mathtt{u}}}(\frac{3}{2})|},\\ &\Delta{\mathit{\mathtt{u}}}'\left(\frac{3}{2}\right) = \mathcal{E}_{1}^*\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \frac{|{\mathit{\mathtt{u}}}(\frac{3}{2})|}{70+|{\mathit{\mathtt{u}}}(\frac{3}{2})|},\\ &\mathcal{D}^{-\frac{4}{5}}{\mathit{\mathtt{u}}}(\sigma)|_{\sigma = 0} = {\mathit{\mathtt{u}}}_{1},\qquad-50{\mathit{\mathtt{u}}}(\sigma)|_{\sigma = e}+\frac{1}{85}\mathcal{I}^{\frac{1}{5}}{\mathit{\mathtt{u}}}(\sigma)|_{\sigma = e} = {\mathit{\mathtt{u}}}_{2}, \end{split}\right.\\ &\left\{\begin{split} &\mathcal{D}^\frac{5}{4}{\mathit{\mathtt{v}}}(\sigma)-\frac{\sigma\cos({\mathit{\mathtt{u}}}(\sigma))-{\mathit{\mathtt{v}}}(\sigma)\sin(\sigma)}{95}-\frac{{\mathit{\mathtt{u}}}(\sigma)}{95+{\mathit{\mathtt{u}}}(\sigma)} = 0,~~\sigma\neq\frac{3}{2},\\ &\Delta{\mathit{\mathtt{v}}}\left(\frac{3}{2}\right) = \mathcal{E}_{1}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \frac{|{\mathit{\mathtt{v}}}(\frac{3}{2})|}{70+|{\mathit{\mathtt{v}}}(\frac{3}{2})|},\\ &\Delta{\mathit{\mathtt{v}}}'\left(\frac{3}{2}\right) = \mathcal{E}_{1}^*\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \frac{|{\mathit{\mathtt{v}}}(\frac{3}{2})|}{70+|{\mathit{\mathtt{v}}}(\frac{3}{2})|},\\ &\mathcal{D}^{-\frac{3}{4}}{\mathit{\mathtt{v}}}(\sigma)|_{\sigma = 0} = {\mathit{\mathtt{v}}}_{1},\qquad-50{\mathit{\mathtt{v}}}(\sigma)|_{\sigma = e}+\frac{1}{85}\mathcal{I}^{\frac{1}{4}}{\mathit{\mathtt{v}}}(\sigma)|_{\sigma = e} = {\mathit{\mathtt{v}}}_{2}. \end{split}\right. \end{split}\right. \end{eqnarray} | (5.1) |
From system (5.1), we see that \alpha = \frac{6}{5} , \beta = \frac{5}{4} , \mu_{1} = \mu_{2} = -50 , \nu_{1} = \nu_{3} = 1 , \nu_{2} = \nu_{4} = \frac{1}{85} , \mathbb{T} = e , \sigma_{1} = \frac{3}{2} , and {\mathit{\mathtt{u}}}_{1}, {\mathit{\mathtt{u}}}_{2}, {\mathit{\mathtt{v}}}_{1}, {\mathit{\mathtt{v}}}_{2}\in\mathbb{R} .
Set
\begin{align*} \phi_{1}(\sigma,{\mathit{\mathtt{u}}},{\mathit{\mathtt{v}}})& = \frac{2+\mathcal{I}^\frac{6}{5} {\mathit{\mathtt{u}}}(\sigma)+\mathcal{I}^\frac{5}{4}{\mathit{\mathtt{v}}}(\sigma)}{80e^{\sigma+90}(1+\mathcal{I}^\frac{6}{5}{\mathit{\mathtt{u}}}(\sigma)+\mathcal{I}^\frac{5}{4}{\mathit{\mathtt{v}}}(\sigma))},\\ \phi_{2}(\sigma,{\mathit{\mathtt{u}}},{\mathit{\mathtt{v}}})& = \frac{\sigma\cos({\mathit{\mathtt{u}}}(\sigma))-{\mathit{\mathtt{v}}}(\sigma)\sin(\sigma)}{95}-\frac{{\mathit{\mathtt{u}}}(\sigma)}{95+{\mathit{\mathtt{u}}}(\sigma)}. \end{align*} |
Now, for all {\mathit{\mathtt{u}}}, {\mathit{\mathtt{u}}}^{*}, {\mathit{\mathtt{v}}}, {\mathit{\mathtt{v}}}^{*}\in\mathbb{R} , and \sigma\in[0, e] , we obtain
\begin{align*} |\phi_{1}(\sigma,{\mathit{\mathtt{u}}},{\mathit{\mathtt{v}}})-\phi_{1}(\sigma,{\mathit{\mathtt{u}}}^{*},{\mathit{\mathtt{v}}}^{*})|& = \frac{1}{80e^{90}}|{\mathit{\mathtt{u}}}-{\mathit{\mathtt{u}}}^{*}|+\frac{1}{80e^{90}}|{\mathit{\mathtt{v}}}-{\mathit{\mathtt{v}}}^{*}| \end{align*} |
and
\begin{align*} |\phi_{2}(\sigma,{\mathit{\mathtt{u}}},{\mathit{\mathtt{v}}})-\phi_{1}(\sigma,{\mathit{\mathtt{u}}}^{*},{\mathit{\mathtt{v}}}^{*})|& = \frac{1}{95}|{\mathit{\mathtt{u}}}-{\mathit{\mathtt{u}}}^{*}|+\frac{1}{95}|{\mathit{\mathtt{v}}}-{\mathit{\mathtt{v}}}^{*}|. \end{align*} |
These satisfy condition (\boldsymbol{H}_{3}) with \mathcal{L}_{\phi_{1}} = \mathcal{L}_{\phi_{1}}^* = \frac{1}{80e^{90}} , \mathcal{L}_{\phi_{2}} = \mathcal{L}_{\phi_{2}}^* = \frac{1}{95}.
Set
\begin{align*} \mathcal{E}_{1}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right)& = \frac{|{\mathit{\mathtt{u}}}(\frac{3}{2})|}{70+|{\mathit{\mathtt{u}}}(\frac{3}{2})|},\qquad\qquad\; \; \quad\mathcal{E}_{1}^*\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \frac{|{\mathit{\mathtt{u}}}(\frac{3}{2})|}{70+|{\mathit{\mathtt{u}}}(\frac{3}{2})|},\\ \mathcal{E}_{1}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right)& = \frac{|{\mathit{\mathtt{v}}}(\frac{3}{2})|}{70+|{\mathit{\mathtt{v}}}(\frac{3}{2})|}\; \qquad and\qquad\mathcal{E}_{1}^*\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \frac{|{\mathit{\mathtt{v}}}(\frac{3}{2})|}{70+|{\mathit{\mathtt{v}}}(\frac{3}{2})|}. \end{align*} |
Then we have
\begin{align*} \left|\mathcal{E}_{1}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right)-\mathcal{E}_{1}\left({\mathit{\mathtt{u}}}^{*}\left(\frac{3}{2}\right)\right)\right|& = \frac{1}{70}|{\mathit{\mathtt{u}}}-{\mathit{\mathtt{u}}}^{*}|,\qquad\qquad\left|\mathcal{E}_{1}^{*}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right)-\mathcal{E}_{1}^{*}\left({\mathit{\mathtt{u}}}^{*}\left(\frac{3}{2}\right)\right)\right| = \frac{1}{70}|{\mathit{\mathtt{u}}}-{\mathit{\mathtt{u}}}^{*}|,\\ \left|\mathcal{E}_{1}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right)-\mathcal{E}_{1}\left({\mathit{\mathtt{v}}}^{*}\left(\frac{3}{2}\right)\right)\right|& = \frac{1}{70}|{\mathit{\mathtt{v}}}-{\mathit{\mathtt{v}}}^{*}|\; \quad and\; \quad\left|\mathcal{E}_{1}^{*}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right)-\mathcal{E}_{1}^{*}\left({\mathit{\mathtt{v}}}^{*}\left(\frac{3}{2}\right)\right)\right| = \frac{1}{70}|{\mathit{\mathtt{v}}}-{\mathit{\mathtt{v}}}^{*}|. \end{align*} |
These satisfy condition (\boldsymbol{H}_{4}) with \mathcal{L}_{\mathcal{E}} = \mathcal{L}_{\mathcal{E}}^* = \mathcal{L}_{\mathcal{E}^*} = \mathcal{L}_{\mathcal{E}^*}^* = \frac{1}{70}.
From Theorem 3.3, we use the inequality and get
\begin{equation*} \Delta_{1}+\Delta_{3}+\frac{(\Delta_{2}\mathcal{L}_{\phi_1}+\Delta_{4}\mathcal{L}_{\phi_2})|\sigma^{\alpha}|}{\Gamma(\alpha+1)}+\frac{(\Delta_{2}\mathcal{L}_{\phi_1}^{*}+\Delta_{4}\mathcal{L}_{\phi_2}^{*})|\sigma^{\beta}|}{\Gamma(\beta+1)}\approx0.976847 \lt 1, \end{equation*} |
hence (5.1) has a unique solution, so (5.1) has a solution ({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})\in\vartheta . The solution of (5.1) is given by
\begin{eqnarray*} \label{eq3} {\mathit{\mathtt{u}}}(\sigma) = \left\{\begin{split} &\left\{\begin{split} &\frac{\sigma^{\frac{1}{5}}{\mathit{\mathtt{u}}}_{2}}{-50e^{\frac{1}{5}}}-\frac{\sigma^{\frac{1}{5}}{\mathit{\mathtt{u}}}_{1}}{e\Gamma(\frac{1}{5})}+\frac{\sigma^{-\frac{4}{5}}{\mathit{\mathtt{u}}}_{1}}{\Gamma(\frac{1}{5})}+\frac{1}{\Gamma(\frac{6}{5})}\int_{0}^{\sigma}(\sigma-\pi)^{\frac{1}{5}}\phi_{1}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\\ &-\frac{\sigma^{\frac{1}{5}}e^{-\frac{1}{5}}}{\Gamma(\frac{6}{5})}\int_{\frac{3}{2}}^{e}(e-\pi)^{\frac{1}{5}}\phi_{1}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi+\frac{\frac{1}{85}\sigma^{\frac{1}{5}}e^{-\frac{1}{5}}}{50\Gamma(\frac{1}{5})}\int_{0}^{e}(e-\pi)^{-\frac{4}{5}}{\mathit{\mathtt{u}}}(\pi)d\pi\\ &-\frac{\sigma^{\frac{1}{5}}}{e}\Bigg[\left(\left(\frac{1}{5}\right)+e\left(\frac{4}{5}\right)\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}\mathcal{E}_{1}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right)+\left(e-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}\mathcal{E}_{1}^{*}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right)\\ &+\frac{\left(e-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}}{\Gamma(\frac{1}{5})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{-\frac{4}{5}}\phi_{1}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\\ &+\frac{\left((\frac{1}{5})+e(\frac{4}{5})\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}}{\Gamma(\frac{6}{5})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{\frac{1}{5}}\phi_{1}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\Bigg],\\ &\qquad\sigma\in\left[0,\frac{3}{2}\right], \end{split}\right.\\ &\left\{\begin{split} &\frac{\sigma^{\frac{1}{5}}{\mathit{\mathtt{u}}}_{2}}{-50e^{\frac{1}{5}}}-\frac{\sigma^{\frac{1}{5}}{\mathit{\mathtt{u}}}_{1}}{e\Gamma(\frac{1}{5})}+\frac{\sigma^{-\frac{4}{5}}{\mathit{\mathtt{u}}}_{1}}{\Gamma(\frac{1}{5})}+\frac{1}{\Gamma(\frac{6}{5})}\int_{\frac{3}{2}}^{\sigma}(\sigma-\pi)^{\frac{1}{5}}\phi_{1}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\\ &-\frac{\sigma^{\frac{1}{5}}e^{-\frac{1}{5}}}{\Gamma(\frac{6}{5})}\int_{\frac{3}{2}}^{e}(e-\pi)^{\frac{1}{5}}\frac{2+\mathcal{I}^\frac{6}{5}{\mathit{\mathtt{u}}}(\pi)+\mathcal{I}^\frac{5}{4}{\mathit{\mathtt{v}}}(\pi)}{80e^{\pi+90}(1+\mathcal{I}^\frac{6}{5}{\mathit{\mathtt{u}}}(\pi)+\mathcal{I}^\frac{5}{4}{\mathit{\mathtt{v}}}(\pi))}d\pi+\frac{\frac{1}{85}\sigma^{\frac{1}{5}}e^{-\frac{1}{5}}}{50\Gamma(\frac{1}{5})}\int_{0}^{e}(e-\pi)^{-\frac{4}{5}}{\mathit{\mathtt{u}}}(\pi)d\pi\\ &-\frac{\sigma^{\frac{1}{5}}}{e}\Bigg[\left(\left(\frac{1}{5}\right)+e\left(\frac{4}{5}\right)\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}\mathcal{E}_{1}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right)+\left(e-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}\mathcal{E}_{1}^{*}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right)\\ &+\frac{\left(e-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}}{\Gamma(\frac{1}{5})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{-\frac{4}{5}}\phi_{1}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\\ &+\frac{\left((\frac{1}{5})+e(\frac{4}{5})\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}}{\Gamma(\frac{6}{5})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{\frac{1}{5}}\phi_{1}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\Bigg]\\ &+\Bigg[\left(\left(\frac{1}{5}\right)+\sigma\left(\frac{4}{5}\right)\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}\sigma^{-\frac{4}{5}}\mathcal{E}_{1}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right)+\left(\sigma-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}\sigma^{-\frac{4}{5}}\mathcal{E}_{1}^{*}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right)\\ &+\frac{\left(\sigma-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}\sigma^{-\frac{4}{5}}}{\Gamma(\frac{1}{5})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{-\frac{4}{5}}\phi_{1}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\\ &+\frac{\left((\frac{1}{5})+\sigma(\frac{4}{5})\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{4}{5}}\sigma^{-\frac{4}{5}}}{\Gamma(\frac{6}{5})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{\frac{1}{5}}\phi_{1}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\Bigg],\\ &\qquad\sigma\in\bigg(\frac{3}{2},e\bigg] \end{split}\right. \end{split}\right. \end{eqnarray*} |
and
\begin{eqnarray*} {\mathit{\mathtt{v}}}(\sigma) = \left\{\begin{split} &\left\{\begin{split} &\frac{\sigma^{\frac{1}{4}}{\mathit{\mathtt{v}}}_{2}}{-50e^{\frac{1}{4}}}-\frac{\sigma^{\frac{1}{4}}{\mathit{\mathtt{v}}}_{1}}{e\Gamma(\frac{1}{4})}+\frac{\sigma^{-\frac{3}{4}}{\mathit{\mathtt{v}}}_{1}}{\Gamma(\frac{1}{4})}+\frac{1}{\Gamma(\frac{5}{4})}\int_{0}^{\sigma}(\sigma-\pi)^{\frac{1}{4}}\phi_{2}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\\ &-\frac{\sigma^{\frac{1}{4}}e^{-\frac{1}{4}}}{\Gamma(\frac{5}{4})}\int_{\frac{3}{2}}^{e}(e-\pi)^{\frac{1}{4}}\phi_{2}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi+\frac{\frac{1}{85}\sigma^{\frac{1}{4}}e^{-\frac{1}{4}}}{50\Gamma(\frac{1}{4})}\int_{0}^{e}(e-\pi)^{-\frac{3}{4}}{\mathit{\mathtt{v}}}(\pi)d\pi\\ &-\frac{\sigma^{\frac{1}{4}}}{e}\Bigg[\left(\left(\frac{1}{4}\right)+e\left(\frac{3}{4}\right)\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}\mathcal{E}_{1}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right)+\left(e-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}\mathcal{E}_{1}^{*}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right)\\ &+\frac{\left(e-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}}{\Gamma(\frac{1}{4})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{-\frac{3}{4}}\phi_{2}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\\ &+\frac{\left(\left(\frac{1}{4}\right)+e\left(\frac{3}{4}\right)\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}}{\Gamma(\frac{5}{4})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{\frac{1}{4}}\phi_{2}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\Bigg],\\ &\qquad\sigma\in\left[0,\frac{3}{2}\right], \end{split}\right.\\ &\left\{\begin{split} &\frac{\sigma^{\frac{1}{4}}{\mathit{\mathtt{v}}}_{2}}{-50e^{\frac{1}{4}}}-\frac{\sigma^{\frac{1}{4}}{\mathit{\mathtt{v}}}_{1}}{e\Gamma(\frac{1}{4})}+\frac{\sigma^{-\frac{3}{4}}{\mathit{\mathtt{v}}}_{1}}{\Gamma(\frac{1}{4})}+\frac{1}{\Gamma(\frac{5}{4})}\int_{\frac{3}{2}}^{\sigma}(\sigma-\pi)^{\frac{1}{4}}\phi_{2}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\\ &-\frac{\sigma^{\frac{1}{4}}e^{-\frac{1}{4}}}{\Gamma(\frac{5}{4})}\int_{\frac{3}{2}}^{e}(e-\pi)^{\frac{1}{4}}\phi_{2}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi-\frac{\frac{1}{85}\sigma^{\frac{1}{4}}e^{-\frac{1}{4}}}{-50\Gamma(\frac{1}{4})}\int_{0}^{e}(e-\pi)^{-\frac{3}{4}}{\mathit{\mathtt{v}}}(\pi)d\pi\\ &-\frac{\sigma^{\frac{1}{4}}}{e}\Bigg[\left(\left(\frac{1}{4}\right)+e\left(\frac{3}{4}\right)\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}\mathcal{E}_{1}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right)+\left(e-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}\mathcal{E}_{1}^{*}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right)\\ &+\frac{\left(e-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}}{\Gamma(\frac{1}{4})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{-\frac{3}{4}}\phi_{2}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\\ &+\frac{\left(\left(\frac{1}{4}\right)+e\left(\frac{3}{4}\right)\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}}{\Gamma(\frac{5}{4})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{\frac{1}{4}}\phi_{2}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\Bigg]\\ &+\Bigg[\left(\left(\frac{1}{4}\right)+\sigma\left(\frac{3}{4}\right)\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}\sigma^{-\frac{3}{4}}\mathcal{E}_{1}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right)+\left(\sigma-\frac{3}{2}\right)\sigma^{-\frac{3}{4}}\left(\frac{3}{2}\right)^{\frac{3}{4}}\mathcal{E}_{1}^{*}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right)\\ &+\frac{\left(\sigma-\frac{3}{2}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}\sigma^{-\frac{3}{4}}}{\Gamma(\frac{1}{4})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{-\frac{3}{4}}\phi_{2}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\\ &+\frac{\left((\frac{1}{4})+\sigma(\frac{3}{4})\left(\frac{3}{2}\right)^{-1}\right)\left(\frac{3}{2}\right)^{\frac{3}{4}}\sigma^{-\frac{3}{4}}}{\Gamma(\frac{5}{4})}\int_{0}^{\frac{3}{2}}\left(\frac{3}{2}-\pi\right)^{\frac{1}{4}}\phi_{2}(\pi,\mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\pi),\mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\pi))d\pi\Bigg],\\ &\qquad\sigma\in\bigg(\frac{3}{2},e\bigg]. \end{split}\right. \end{split}\right. \end{eqnarray*} |
(i) If we take \phi_{1}(\sigma, \mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\sigma), \mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\sigma)) = \frac{1}{80e^{\sigma+90}}, \phi_{2}(\sigma, \mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\sigma), \mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\sigma)) = \frac{\sigma\cos(\sigma)-\sin(\sigma)}{95}-\frac{1}{95}, \mathcal{E}_{1}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}^{*}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}^{*}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \frac{1}{70}, and {\mathit{\mathtt{u}}}(\sigma) = {\mathit{\mathtt{v}}}(\sigma) = \sigma then with the constant values {\mathit{\mathtt{u}}}_{1} = {\mathit{\mathtt{v}}}_{1} = \frac{1}{15} , {\mathit{\mathtt{u}}}_{2} = {\mathit{\mathtt{v}}}_{2} = 2 , the graph of the solution is shown in Figure 1.
(ii) If we take \phi_{1}(\sigma, \mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\sigma), \mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\sigma)) = \frac{\sigma+1}{80e^{\sigma+90}}, \phi_{2}(\sigma, \mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\sigma), \mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\sigma)) = \frac{\sigma^{2}+1}{95}-\frac{\sigma}{95}, \mathcal{E}_{1}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}^{*}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}^{*}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \frac{1}{70}, and {\mathit{\mathtt{u}}}(\sigma) = {\mathit{\mathtt{v}}}(\sigma) = \sigma then with the constant values {\mathit{\mathtt{u}}}_{1} = {\mathit{\mathtt{v}}}_{1} = -\frac{1}{15} , {\mathit{\mathtt{u}}}_{2} = {\mathit{\mathtt{v}}}_{2} = -2 , the graph of the solution is shown in Figure 2.
From Theorem 4.1, we use the inequality and get
\begin{equation*} \mathcal{F} = 1-\frac{\bigg(\frac{\Delta_{2}\mathcal{L}_{\phi_1}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)\bigg(\frac{\Delta_{4}\mathcal{L}_{\phi_2}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)}{\bigg[ 1-\bigg(\Delta_{1}+\frac{\Delta_{2}\mathcal{L}_{\phi_1}|\sigma^{\alpha}|}{\Gamma(\alpha+1)}\bigg)\bigg] \bigg[ 1-\bigg(\Delta_{3}+\frac{\Delta_{4}\mathcal{L}_{\phi_2}^{*}|\sigma^{\beta}|}{\Gamma(\beta+1)}\bigg)\bigg] }\approx1 \gt 0, \end{equation*} |
thus, the given system (5.1) is \mathcal{HU} stable and also generalized \mathcal{HU} stable. Likewise, we can justify the condition of Theorems 3.2 and 4.2.
In this article, we used the Kransnoselskii's fixed point theorem and acquired the necessary cases for the existence and uniqueness of solution for the given fractional integro-differential Eqs (1.1). Furthermore, under specific assumptions and conditions, we proved different kinds of Ulam's stability of system (1.1). The concept of Ulam's stability is very important because it gives a relationship between approximate and exact solutions, so our results may be very helpful in approximation theory and numerical analysis. The mentioned stability is rarely investigated for impulsive fractional integro-differential equations. Finally, we illustrated the main results by giving a suitable example.
This research was supported by the Natural Science Foundation of Jiangxi Province (Grant Nos. 20192BAB201011, 20192BCBL23030 and 20192ACBL21053) and the National Natural Science Foundation of China (Grant No. 11861053).
The authors declare that they have no conflict of interest.
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