The prevention and control of the spread of Cystic Echinococcosis is an important public health issue. Health education has been supported by many governments because it can increase public awareness of echinococcosis, promote the development of personal hygiene habits, and subsequently reduce the transmission of echinococcosis. In this paper, a dynamic model of echinococcosis is used to integrate all aspects of health education. Theoretical analysis and numerical model fitting were used to quantitatively analysed by the impact of health education on the spread of echinococcosis. Theoretical findings indicate that the basic reproduction number is crucial in determining the prevalence of echinococcosis within a given geographical area. The parameters of the model were estimated and fitted by using data from the Ningxia Hui Autonomous Region in China, and the sensitivity of the basic reproduction number was analysed by using the partial rank correlation coefficient method. These findings illustrate that all aspects of health education demonstrate a negative correlation with the basic reproduction number, suggesting the effectiveness of health education in reducing the basic reproduction number and mitigating the transmission of echinococcosis, which is consistent with reality. Particularly, the basic reproduction number showed a strong negative correlation with the burial rate of infected livestock (b) and the incidence of infected livestock viscera that is not fed to dogs (q). This paper further analyzes the implementation plan for canine deworming rates and sheep immunity rates, as well as the transmission of infected hosts over time under different parameters b and q. According to the findings, emphasizing the management of infected livestock in health education has the potential to significantly reduce the risk of echinococcosis transmission. This study will provide scientific support for the creation of higher quality health education initiatives.
Citation: Qianqian Cui, Qiang Zhang, Zengyun Hu. Modeling and analysis of Cystic Echinococcosis epidemic model with health education[J]. AIMS Mathematics, 2024, 9(2): 3592-3612. doi: 10.3934/math.2024176
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The prevention and control of the spread of Cystic Echinococcosis is an important public health issue. Health education has been supported by many governments because it can increase public awareness of echinococcosis, promote the development of personal hygiene habits, and subsequently reduce the transmission of echinococcosis. In this paper, a dynamic model of echinococcosis is used to integrate all aspects of health education. Theoretical analysis and numerical model fitting were used to quantitatively analysed by the impact of health education on the spread of echinococcosis. Theoretical findings indicate that the basic reproduction number is crucial in determining the prevalence of echinococcosis within a given geographical area. The parameters of the model were estimated and fitted by using data from the Ningxia Hui Autonomous Region in China, and the sensitivity of the basic reproduction number was analysed by using the partial rank correlation coefficient method. These findings illustrate that all aspects of health education demonstrate a negative correlation with the basic reproduction number, suggesting the effectiveness of health education in reducing the basic reproduction number and mitigating the transmission of echinococcosis, which is consistent with reality. Particularly, the basic reproduction number showed a strong negative correlation with the burial rate of infected livestock (b) and the incidence of infected livestock viscera that is not fed to dogs (q). This paper further analyzes the implementation plan for canine deworming rates and sheep immunity rates, as well as the transmission of infected hosts over time under different parameters b and q. According to the findings, emphasizing the management of infected livestock in health education has the potential to significantly reduce the risk of echinococcosis transmission. This study will provide scientific support for the creation of higher quality health education initiatives.
Numerous phenomena are mathematically described using fractional order differential and integral operators. The fundamental advantage of these operators is that they are nonlocal. This makes it possible to describe the components and procedures used throughout the phenomenon's history. As a result, fractional-order models are more precise and useful than their equivalents in integer order. Because fractional calculus (FC) techniques are frequently used in a number of real-world applications, numerous scholars established this significant branch of mathematical analysis (see e.g., [1,2,3,4,5,6,7,8,9]).
Recent studies on fractional differential equations (FDEs) with various boundary conditions (BCs) have been carried out by several researchers. Nonlocal nonlinear fractional-order boundary value issues, especially, have received a lot of attention. In the work of Bitsadze and Samarski (see [10]), when nonlocal conditions were first presented, they were used to describe physical occurrences that occurred within a specific domain's bounds. Due to a blood vessel's shifting form throughout the vessel, it is difficult to defend the assumption of a circular cross section in computational fluid dynamics analyses of blood flow problems. To overcome this issue, integral BCs have been introduced. Additionally, ill-posed parabolic backward problems are resolved using integral BCs. The mathematical models of bacterial self-regularization also depend heavily on integral BCs. In mathematical models of bacterial self-regularization, integral BCs are also essential.
In the mathematical modelling of a number of practical issues, coupled systems of FDEs represent the main tools. Examples include fractional dynamics, chaos, financial economics, ecology, and bio-engineering, etc (see e.g., [1,2,3,4,5,6,7,8,9]), also see the recent interesting results in e.g., [11,12,13,14]. The study of fractional differential systems has been a well-liked and significant field of science, supplemented by many types of BCs. The advancement of this topic has been aided by several researchers who have published countless outputs. Modern functional analysis techniques greatly aid in obtaining existence (Exs.) and uniqueness (Unq.) findings for these issues. We recommend the reader study a number of papers for some recent research on fractional or sequential FDEs with nonlocal integral BCs (e.g., [15,16]).
In [17], by using fixed point theorems (FPTs), the authors looked into the possibility of solving an initial value problem (IVP) involving a sequential FDE. In [18], using the method of upper and lower solutions and the monotone iterative technique, the Exs. and Unq. results for a periodic boundary value issue of nonlinear sequential FDEs were discovered. Since they contain multipoint and integral BCs as special examples, Riemann-Stieltjes BCs are highly general (see [19]). The astronomer T. J. Stieltjes generalization of the Riemann integral, the Riemann-Stieltjes integral, has potential uses in probability theory (see e.g., [20,21]).
Banach and Schaefer FPTs have been employed in [22] (see also e.g., [23,24,25]) to study the Exs. and Unq. of solutions for a coupled system of nonlinear fractional integro-differential equations (Int-DifEqn.) involving Riemann-Liouville integrals with several continuous functions.
{DαU(ρ)=f1(ρ,U(ρ),V(ρ))+∑mi=1∫ρ0(ρ−λ)αi−1Γ(αi)φi(λ)gi(λ,U(λ),V(λ))dλ,DβV(ρ)=f2(ρ,U(ρ),V(ρ))+∑mi=1∫ρ0(ρ−λ)βiΓ(βi)ϕi(λ)hi(λ,U(λ),V(λ))dλ,U(0)=a>0,V(0)=b>0,ρ∈[0,1], |
where Dα,Dβ denote the Caputo fractional derivatives (CFD), 0<α,β<1;αi;βi are nonnegative real numbers, φi and ϕi are some continuous functions. It should be remarked that the authors in [26] considered the short-memory which can be considered in some work.
The authors in [27] investigated a boundary value problem of coupled systems of nonlinear Riemann-Liouvillle fractional Int-DifEqn. supplemented with nonlocal Riemann-Liouvillle fractional Int-Dif. BCs. The results obtained by using some standard FPTs (with ρ∈[0,T], 1<α,β≤2)
{DαU(ρ)=A(ρ,U(ρ),V(ρ),(ϕ1U)(ρ),(ψ1V)(ρ)),DβV(ρ)=B(ρ,U(ρ),V(ρ),(ϕ2U)(ρ),(ψ2V)(ρ)), |
with the following coupled Riemann-Liouville Int-Dif. BCs (with 0<η<T, 0<σ<T)
{Dα−2u(0+)=0,Dα−1u(0+)=νIα−1υ(η),Dβ−2u(0+)=0,Dβ−1υ(0+)=μIβ−1u(σ), |
where the Riemann-Liouville derivatives is denoted by D(.), and I(.) denotes the Riemann-Liouville integral of fractional order (.), and f,g:[0,T]×R4→R are given continuous functions, ν,μ are real constants, and ϕi,ψi,i=1,2 are given operators.
For a nonlinear coupled system of Liouville-Caputo type fractional Int-DifEqn. with non-local discrete and integral BCs, the Exs. and Unq. of solutions have been studied in [28]. The Exs. results are obtained by usng Leray-Schauder FPT, while the Unq. results by the concept of Banach FPT.
{CDqx(r1)=A(r1,x(r1),y(r1))),CDpy(r1)=B(r1,x(r1),y(r1))),x′(0)=α∫ξ0x′(r2)dr2,x(1)=β∫10g(x′(r2))dr2,y′(0)=α1∫θ0y′(r2)dr2,y(1)=β1∫10g(y′(r2))dr2,r1∈[0,1],1<q,p≤2,0≤ξ,θ≤1, |
where CDq,CDp denote the Caputo fractional derivatives (CFDs) of order q,p,A,B:[0,1]×R×R→R are given continuous functions, and α,β,α1,β1 are real constants.
In [29], the authors discussed the FDEs with integral and ordinary-fractional flux BCs
{CDp1x(κ)=F(s,x(κ),y(κ))),CDp2y(κ)=G(s,x(κ),y(κ))),x(0)+x(1)=a∫10x(r2)dr2,x′(0)=bCDq1x(1),y(0)+y(1)=z∫10y(r2)dr2,y′(0)=bC1De1y(1),s∈[0,1],1<p1,p2≤2,0≤q,e1≤1, |
where CDp1,CDp2,CDq1,CDe1 denote the CFDs of order p1,p2,F,G:[0,1]×R×R→R are given continuous functions, and a,z,b,b1 are real constants. The Exs. results have been analyzed in [30] for coupled system of FDEs (with μ∈(0,1),1<α,β<2,0<η<1)
{DαU(μ)=A(μ,V(μ),DpV(μ))),DβV(μ)=B(μ,U(μ),DqU(μ))),U(0)=0,U(1)=γU(η),V(0)=0,V(1)=γV(η), |
where D⋅ denotes the Riemann-Liouville FDs of order (⋅), A,B:[0,1]×R2→R, are given continuous functions, and γ is a real constant.
Exs. of solutions for nonlinear coupled Caputo fractional Int-DifEqn has been investigated in [16],
{CDαu(ρ)=f(ρ,u(ρ),v(ρ),CDζ1υ(ρ),Iξυ(ρ)),ρ∈[0,T]:=U,CDβυ(ρ)=g(ρ,u(ρ),u(ρ),CDι1υ(ρ),Iςυ(ρ)),ρ∈[0,T]:=U, |
with nonlocal integral and multi-point BCs
{U(0)=ψ1(V),U′(0)=ε1∫ν10V′(θ)dθ,U′′(0)=0,⋯,Un−2(0)=0,U(T)=λ1∫δ10V(θ)dθ+μ1∑k−2j=1wjV(θj),V(0)=ψ2(V),V′(0)=ε2∫ν20U′(θ)dθ,V′′(0)=0,⋯,Vn−2(0)=0,V(T)=λ2∫δ20U(θ)dθ+μ2∑k−2j=1wjU(φj), |
where CDα,CDβ,CDζ1,CDι1 are the Caputo FDs of order n−1<α,β<n, 0<ζ1,ι1<1, Iξ,Iς are the Riemann-Liouville fractional integrals (FI) of order ξ,ς>0.
In this paper, we investigate the Exs. and Unq. of solutions for the following nonlinear coupled system of FDEs involving Riemann-Liouville and Caputo derivatives with coupled Riemann-Stieltjes integro-multipoint boundary conditions
{RLDf1[(cDh1+α1)x(t)+β1Is1H(t,x(t),y(t))]=ϕ(t,x(t),y(t)),1<h1,f1≤2,t∈[q,p],RLDf2[(cDh2+α2)y(t)+β2Is2U(t,x(t),y(t))]=ψ(t,x(t),y(t)),1<h2,f2≤2,t∈[q,p], | (1.1) |
with coupled non-conjugate Riemann-Stieltjes integro-multipoint BCs:
{x(q)=τ−2∑i=1ηiy(ξi)+∫pqy(κ)dΛ(κ),x′(q)=0,x(p)=0,x′(p)=0,y(q)=τ−2∑i=1ηix(ξi)+∫pqx(κ)dΛ(κ),y′(q)=0,y(p)=0,y′(p)=0, | (1.2) |
where cDa denotes the Caputo fractional differential operator of order a with (a=h1,h2), RlDb denotes the Riemann-Liouville fractional differential operator of order b with (b=f1,f2), with h1+f1>3,h2+f2>3, Is1,Is2 are Riemann-Liouville FI of order s1,s2>1, αi,βi∈R,i=1,2, H,ϕ,U,ψ:[q,p]×R2→R are given continuous functions, Λ is a function of bounded variation, q<ξ1<ξ2<⋯<ξn−2<p, ηj∈R, j=1,2,⋯n−2. It should be remarked that some fundamental assumptions for orders of fractional derivatives are postulated in our study and potential relaxation of this limitations can be considered in some further study. The main contribution of this article can be seen as follows:
(1) A generalization of the results obtained in [16].
(2) A generalization of the results obtained in [29].
(3) A generalization of the results obtained in [30].
Here we emphasize that the present work is motivated by a recent work [31]. Next section recalls some basic definitions of FC and present an auxiliary lemma. In section 3, we discuss the existence of solutions for the given problem while the uniqueness results is presented in section 4, section 5 shows examples that illustrate our results, and section 6 concludes our work.
Now, we recall some basic definitions of fractional calculus.
Definition 2.1. [8] For β>0, the Riemann-Liouville FI of order β for ϑ∈L1[q,p], existing almost everywhere on [q,p], (with −∞<q<p<∞) is defined by
Iβϑ(t)=∫tq(t−s)β−1Γ(β)ϑ(s)ds, |
where Γ denotes the Euler gamma function.
Definition 2.2. [8] For, β∈(n−1,n],n∈N, and g∈ACn[q,p], the Riemann-Liouville and CFDs of order β are respectively defined by
RLDβϑ(t)=dndtn∫tq(t−s)n−β−1Γ(n−β)ϑ(s)dsandcDβϑ(r)=∫rq(r−s)n−β−1Γ(n−β)ϑ(n)(s)ds. |
Lemma 2.1. For m−1<β≤m,t∈[q,p], the general solution of the FDE cDβx(b)=0, is
x(b)=r0+r1(b−q)+r2(b−q)2+...+rm−1(b−q)m−1, |
ri∈R,i=0,1,...,m−1. Moreover,
(IβcDβx)(b)=x(b)+m−1∑i=0ri(b−q)i. |
Lemma 2.2. [8] For β>0 and x∈C(q,p)∩L(q,p), the general solution of (RLDβx)(b)=0 is
x(b)=σ1(b−q)β−1+σ2(b−q)β−2+⋯+σm−1(b−q)β−m−1+σm(b−q)β−m, |
where σj∈R,j=1,2,⋯,m, and
(IβRLDβx)(b)=x(b)+σ1(b−q)β−1+σ2(b−q)β−2+⋯+σm−1(b−q)β−m−1+σm(b−q)β−m=x(b)+m∑j=1σi(b−q)β−j. |
On the other hand, (RLDβIβx)(b)=x(b).
See also Lemma A.1 in Appendix. A for more details.
Denote by X∗={x(t)|x(t)∈C([q,p],R)} as the Banach space (BSp.) of all functions (continuous) from [q,p] into R equipped with the norm ‖x‖=supt∈[q,p]|x(t)|. Obviously (X∗,‖.‖) is a BSp. and as a result, the product space (X∗×X∗,‖.‖) is a BSp. with the norm ‖(r,s)‖=‖r‖+‖s‖ for (r,s)∈X∗×X∗.
By Lemma A.1, we define an operator A:X∗×X∗→X∗×X∗ as
A(x,y)(t):=(A1(x,y)(t),A2(x,y)(t)), | (3.1) |
where
A1(x,y)(t)=−α1∫tq(t−κ)h1−1Γ(h1)x(κ)dκ−β1∫tq(t−κ)s1+h1−1Γ(s1+h1)H(κ,x(κ),y(κ))dκ+∫tq(t−κ)h1+f1−1Γ(h1+f1)ϕ(κ,x(κ),y(κ))dκ+V1(t)[α1∫pq(p−κ)h1−1Γ(h1)x(κ)dκ+β1∫pq(p−κ)s1+h1−1Γ(s1+h1)H(κ,x(κ),y(κ))dκ−∫pq(p−κ)h1+f1−1Γ(h1+f1)ϕ(κ,x(κ),y(κ))dκ]+V2(t)[α1∫pq(p−κ)h1−2Γ(h1−1)x(κ)dκ+β1∫pq(p−κ)s1+h1−2Γ(s1+h1−1)H(κ,x(κ),y(κ))dκ−∫pq(p−κ)h1+f1−2Γ(h1+f1−1)ϕ(κ,x(κ),y(κ))dκ]+V3(t)[−α1n−2∑i=1ηi∫ξiq(ξi−κ)h1−1Γ(h1)x(κ)dκ−β1τ−2∑i=1ηi∫ξiq(ξi−κ)s1+h1−1Γ(s1+h1)H(κ,x(κ),y(κ))dκ+τ−2∑i=1ηi∫ξiq(ξi−κ)h1+f1−1Γ(h1+f1)ϕ(κ,x(κ),y(κ))dκ+∫pq(−α1∫κq(κ−u)h1−1Γ(h1)x(u)du−β1∫κq(κ−u)s1+h1−1Γ(s1+h1)H(u,x(u),y(u))du+∫κq(κ−u)h1+f1−1Γ(h1+f1)ϕ(u,x(u),y(u))du)dΛ(κ)]+V4(t)[α2∫pq(p−κ)h2−1Γ(h2)y(κ)dκ+β2∫pq(p−κ)s2+h2−1Γ(s2+h2)U(κ,x(κ),y(κ))dκ−∫pq(p−κ)h2+f2−1Γ(h2+f2)ψ(κ,x(κ),y(κ))dκ]+V5(t)[α2∫pq(p−κ)h2−2Γ(h2−1)y(κ)dκ+β2∫pq(p−κ)s2+h2−2Γ(s2+h2−1)U(κ,x(κ),y(κ))dκ−∫pq(p−κ)h2+f2−2Γ(h2+f2−1)ψ(κ,x(κ),y(κ))dκ]+V6(t)[−α2τ−2∑i=1ηi∫ξiq(ξi−κ)h2−1Γ(h2)y(κ)dκ−β2τ−2∑i=1ηi∫ξiq(ξi−κ)s2+h2−1Γ(s2+h2)U(κ,x(κ),y(κ))dκ+τ−2∑i=1ηi∫ξiq(ξi−κ)h2+f2−1Γ(h2+f2)ψ(κ,x(κ),y(κ))dκ+∫pq(−α2∫κq(κ−u)h2−1Γ(h2)y(u)du−β2∫κq(κ−u)s2+h2−1Γ(s2+h2)U(u,x(u),y(u))du+∫κq(κ−u)h2+f2−1Γ(h2+f2)ψ(u,x(u),y(u))du)dΛ(κ)]], | (3.2) |
A2(x,y)(t)=−α2∫tq(t−κ)h2−1Γ(h2)y(κ)dκ−β2∫tq(t−κ)s2+h2−1Γ(s2+h2)U(κ,x(κ),y(κ))dκ+∫tq(t−κ)h2+f2−1Γ(h2+f2)ψ(κ,x(κ),y(κ))dκ+W1(t)[α1∫pq(p−κ)h1−1Γ(h1)x(κ)dκ+β1∫pq(p−κ)s1+h1−1Γ(s1+h1)H(κ,x(κ),y(κ))dκ−∫pq(p−κ)h1+f1−1Γ(h1+f1)ϕ(κ,x(κ),y(κ))dκ]+W2(t)[α1∫pq(p−κ)h1−2Γ(h1−1)x(κ)dκ+β1∫pq(p−κ)s1+h1−2Γ(s1+h1−1)H(κ,x(κ),y(κ))dκ−∫pq(p−κ)h1+f1−2Γ(h1+f1−1)ϕ(κ,x(κ),y(κ))dκ]+W3(t)[−α1τ−2∑i=1ηi∫ξiq(ξi−κ)h1−1Γ(h1)x(κ)dκ−β1τ−2∑i=1ηi∫ξiq(ξi−κ)s1+h1−1Γ(s1+h1)H(κ,x(κ),y(κ))dκ+τ−2∑i=1ηi∫ξiq(ξi−κ)h1+f1−1Γ(h1+f1)ϕ(κ,x(κ),y(κ))dκ+∫pq(−α1∫κq(κ−u)h1−1Γ(h1)x(u)du−β1∫κq(κ−u)s1+h1−1Γ(s1+h1)H(u,x(u),y(u))du+∫κq(κ−u)h1+f1−1Γ(h1+f1)ϕ(u,x(u),y(u))du)dΛ(κ)]+W4(t)[α2∫pq(p−κ)h2−1Γ(h2)y(κ)dκ+β2∫pq(p−κ)s2+h2−1Γ(s2+h2)U(κ,x(κ),y(κ))dκ−∫pq(p−κ)h2+f2−1Γ(h2+f2)ψ(κ,x(κ),y(κ))dκ]+W5(t)[α2∫pq(p−κ)h2−2Γ(h2−1)y(κ)dκ+β2∫pq(p−κ)s2+h2−2Γ(s2+h2−1)U(κ,x(κ),y(κ))dκ−∫pq(p−κ)h2+f2−2Γ(h2+f2−1)ψ(κ,x(κ),y(κ))dκ]+W6(t)[−α2τ−2∑i=1ηi∫ξiq(ξi−κ)h2−1Γ(h2)y(κ)dκ−β2τ−2∑i=1ηi∫ξiq(ξi−κ)s2+h2−1Γ(s2+h2)U(κ,x(κ),y(κ))dκ+τ−2∑i=1ηi∫ξiq(ξi−κ)h2+f2−1Γ(h2+f2)ψ(κ,x(κ),y(κ))dκ+∫pq(−α2∫κq(κ−u)h2−1Γ(h2)y(u)du−β2∫κq(κ−u)s2+h2−1Γ(s2+h2)U(u,x(u),y(u))du+∫κq(κ−u)h2+f2−1Γ(h2+f2)ψ(u,x(u),y(u))du)dΛ(κ)], | (3.3) |
and Vi(t)(i=1,⋯,6) and Wj(t),j=1,...,6 are given by (A.4) and (A.5) respectively. From now on, we impose that H,ϕ,U,ψ:[q,p]×R2→R are continuous functions satisfying the following condition:
(H1) For all t∈[q,p],x,y∈R, ∃ real constants ϖi,εi,ni,mi≥0(i=1,2),ϖ0,ε0,n0,m0>0:
|H(t,x,y)|≤ϖ0+ϖ1|x|+ϖ2|y|, |
|ϕ(t,x,y)|≤ε0+ε1|x|+ε2|y|, |
|U(t,x,y)|≤n0+n1|x|+n2|y|, |
|ψ(t,x,y)|≤m0+m1|x|+m2|y|. |
For simplicity, we use the following notations:
F0=|α1|{(p−q)h1Γ(h1+1)+˜V1(p−q)h1Γ(h1+1)+˜V2(p−q)h1−1Γ(h1)+˜V3(τ−2∑i=1|ηi|(ξi−q)h1Γ(h1+1)+∫pq(κ−q)h1Γ(h1+1)dΛ(κ))},F1=|β1|{(p−q)s1+h1Γ(s1+h1+1)+˜V1(p−q)s1+h1Γ(s1+h1+1)+˜V2(p−q)s1+h1−1Γ(s1+h1)+˜V3(τ−2∑i=1|ηi|(ξi−q)s1+h1Γ(s1+h1+1)+∫pq(κ−q)s1+h1Γ(s1+h1+1)dΛ(κ))},F2={(p−q)h1+f1Γ(h1+f1+1)+˜V1(p−q)h1+f1Γ(h1+f1+1)+˜V2(p−q)h1+f1−1Γ(h1+f1)+˜V3(τ−2∑i=1|ηi|(ξi−q)h1+f1Γ(h1+f1+1)+∫pq(κ−q)h1+f1Γ(h1+f1+1)dΛ(κ))},F3=|α2|{˜V4(p−q)h2Γ(h2+1)+˜V5(p−q)h2−1Γ(h2)+˜V6(τ−2∑i=1|ηi|(ξi−q)h2Γ(h2+1)+∫pq(κ−q)h2Γ(h2+1)dΛ(κ))},F4=|β2|{˜V4(p−q)s2+h2Γ(s2+h2+1)+˜V5(p−q)s2+h2−1Γ(s2+h2)+˜V6(τ−2∑i=1|ηi|(ξi−q)s2+h2Γ(s2+h2+1)+∫pq(κ−q)s2+h2Γ(s2+h2+1)dΛ(κ))},F5={˜V4(p−q)h2+f2Γ(h2+f2+1)+˜V5(p−q)h2+f2−1Γ(h2+f2)+˜V6(τ−2∑i=1|ηi|(ξi−q)h2+f2Γ(h2+f2+1)+∫pq(κ−q)h2+f2Γ(h2+f2+1)dΛ(κ))}, | (3.4) |
G0=|α1|{˜W1(p−q)h1Γ(h1+1)+˜W2(p−q)h1−1Γ(h1)+˜W3(τ−2∑i=1|ηi|(ξi−q)h1Γ(h1+1)+∫pq(κ−q)h1Γ(h1+1)dΛ(κ))},G1=|β1|{˜W1(p−q)s1+h1Γ(s1+h1+1)+˜W2(p−q)s1+h1−1Γ(s1+h1)+˜W3(τ−2∑i=1|ηi|(ξi−q)s1+h1Γ(s1+h1+1)+∫pq(κ−q)s1+h1Γ(s1+h1+1)dΛ(κ))},G2={˜W1(p−q)h1+f1Γ(h1+f1+1)+˜W2(p−q)h1+f1−1Γ(h1+f1)+˜W3(τ−2∑i=1|ηi|(ξi−q)h1+f1Γ(h1+f1+1)+∫pq(κ−q)h1+f1Γ(h1+f1+1)dΛ(κ))},G3=|α2|{(p−q)h2Γ(h2+1)+˜W4(p−q)h2Γ(h2+1)+˜W5(p−q)h2−1Γ(h2)+˜W6(τ−2∑i=1|ηi|(ξi−q)h2Γ(h2+1)+∫pq(κ−q)h2Γ(h2+1)dΛ(κ))},G4=|β2|{(p−q)s2+h2Γ(s2+h2+1)+˜W4(p−q)s2+h2Γ(s2+h2+1)+˜W5(p−q)s2+h2−1Γ(s2+h2)+˜W6(τ−2∑i=1|ηi|(ξi−q)s2+h2Γ(s2+h2+1)+∫pq(κ−q)s2+h2Γ(s2+h2+1)dΛ(κ))},G5={(p−q)h2+f2Γ(h2+f2+1)+˜W4(p−q)h2+f2Γ(h2+f2+1)+˜W5(p−q)h2+f2−1Γ(h2+f2)+˜W6(τ−2∑i=1|ηi|(ξi−q)h2+f2Γ(h2+f2+1)+∫pq(κ−q)h2+f2Γ(h2+f2+1)dΛ(κ))}, | (3.5) |
where ˜Vi=supt∈[q,p]|Vi(t)|,i=1,...,6 and ˜Wj=supt∈[q,p]|Wj(t)|,j=1,...,6,
O0=(F1+G1)ϖ0+(F2+G2)ε0+(F4+G4)n0+(F5+G5)m0, | (3.6) |
O1=(F0+G0)+(F1+G1)ϖ1+(F2+G2)ε1+(F4+G4)|n1+(F5+G5)m1, | (3.7) |
O2=(F1+G1)ϖ2+(F2+G2)ε2+(F3+G3)+(F4+G4)n2+(F5+G5)m2, | (3.8) |
O=max{O1,O2}. | (3.9) |
Now we introduce our Exs. results. In the first method we use Leray-Schauder alternative to show the Exs. of solution for the systems (1.1) and (1.2).
Lemma 3.1. (Leray-Schauder alternative [32]): Let a completely continuous operator S:J⟶J. Assume that E(S)={y∈J:y=λS(y), 0<λ<1}. Then:
(1) the set E(S) is unbounded, or
(2) S has at lest one FP.
Theorem 3.1. If continuous functions H,ϕ,U,ψ:[q,p]×R2→R satisfying (H1). Then the systems (1.1) and (1.2) has at least one solution on [q,p] if O<1, where O is given by (3.9).
Proof. We start by proving that the operator A:X∗×X∗→X∗×X∗ is completely continuous. Since the functions H,ϕ,U and ψ, are continuous, then the operator A is continuous.
Let P⊂X∗×X∗ be bounded. Then ∃ constants ζi>0(i=1,...,4): |H(t,x(t),y(t))|≤ζ1,|ϕ(t,x(t),y(t))|≤ζ2,|U(t,x(t),y(t))|≤ζ3,|ψ(t,x(t),y(t))|≤ζ4,∀(x,y)∈P. Then, for any (x,y)∈P, we have
|A1(x,y)(t)|≤|α1|∫tq(t−κ)h1−1Γ(h1)|x(κ)|dκ+|β1|∫tq(t−κ)s1+h1−1Γ(s1+h1)ζ1dκ+∫tq(t−κ)h1+f1−1Γ(h1+f1)ζ2dκ+|V1(t)|[|α1|∫pq(p−κ)h1−1Γ(h1)|x(κ)|dκ+|β1|∫pq(p−κ)s1+h1−1Γ(s1+h1)ζ1dκ+∫pq(p−κ)h1+f1−1Γ(h1+f1)ζ2dκ]+|V2(t)|[|α1|∫pq(p−κ)h1−2Γ(h1−1)|x(κ)|dκ+|β1|∫pq(p−κ)s1+h1−2Γ(s1+h1−1)ζ1dκ+∫pq(p−κ)h1+f1−2Γ(h1+f1−1)ζ2dκ]+|V3(t)|[|α1|τ−2∑i=1|ηi|∫ξiq(ξi−κ)h1−1Γ(h1)|x(κ)|dκ+|β1|τ−2∑i=1|ηi|∫ξiq(ξi−κ)s1+h1−1Γ(s1+h1)ζ1dκ+τ−2∑i=1|ηi|∫ξiq(ξi−κ)h1+f1−1Γ(h1+f1)ζ2dκ+∫pq(|α1|∫κq(κ−u)h1−1Γ(h1)|x(u)|du+|β1|∫κq(κ−u)s1+h1−1Γ(s1+h1)ζ1du+∫κq(κ−u)h1+f1−1Γ(h1+f1)ζ2du)dΛ(κ)]+|V4(t)|[|α2|∫pq(p−κ)h2−1Γ(h2)|y(κ)|dκ+|β2|∫pq(p−κ)s2+h2−1Γ(s2+h2)ζ3dκ+∫pq(p−κ)h2+f2−1Γ(h2+f2)ζ4dκ]+|V5(t)|[|α2|∫pq(p−κ)h2−2Γ(h2−1)|y(κ)|dκ+|β2|∫pq(p−κ)s2+h2−2Γ(s2+h2−1)ζ3dκ+∫pq(p−κ)h2+f2−2Γ(h2+f2−1)ζ4dκ]+|V6(t)|[|α2|τ−2∑i=1|ηi|∫ξiq(ξi−κ)h2−1Γ(h2)|y(κ)|dκ+|β2|τ−2∑i=1|ηi|∫ξiq(ξi−κ)s2+h2−1Γ(s2+h2)ζ3dκ+τ−2∑i=1|ηi|∫ξiq(ξi−κ)h2+f2−1Γ(h2+f2)ζ4dκ+∫pq(|α2|∫κq(κ−u)h2−1Γ(h2)|y(u)|du+|β2|∫κq(κ−u)s2+h2−1Γ(s2+h2)ζ3du+∫κq(κ−u)h2+f2−1Γ(h2+f2)ζ4du)dΛ(κ)]≤F0|x(t)|+F1ζ1+F2ζ2+F3|y(t)|+F4ζ3+F5ζ4, |
which implies that,
‖A1(x,y)‖≤F0‖x‖+F1ζ1+F2ζ2+F3‖y‖+F4ζ3+F5ζ4. |
Similarly, we can get
‖A2(x,y)‖≤G0‖x‖+G1ζ1+G2ζ2+G3‖y‖+G4ζ3+G5ζ4. |
Hence, the operator A is uniformly bounded, since ‖A(x,y)‖≤(F0+G0)‖x‖+(F1+G1)ζ1+(F2+G2)ζ2+(F3+G3)‖y‖+(F4+G4)ζ3+(F5+G5)ζ4.
Next, we show that A is equicontinuous. For t1,t2∈[q,p] with t1<t2, we obtain
|A1(x,y)(t2)−A1(x,y)(t1)|≤|α1|[|∫t1q[(t2−κ)h1−1−(t1−κ)h1−1]Γ(h1)x(κ)dκ|+|∫t2t1(t2−κ)h1−1Γ(h1)x(κ)dκ|]+|β1|[|∫t1q[(t2−κ)s1+h1−1−(t1−κ)s1+h1−1]Γ(s1+h1)H(κ,x(κ),y(κ))dκ|+|∫t2t1(t2−κ)s1+h1−1Γ(s1+h1)H(κ,x(κ),y(κ))dκ|]+|∫t1q[(t2−κ)h1+f1−1−(t1−κ)h1+f1−1]Γ(h1+f1)ϕ(κ,x(κ),y(κ))dκ|+|∫t2t1(t2−κ)h1+f1−1Γ(h1+f1)ϕ(κ,x(κ),y(κ))dκ|+|V1(t2)−V1(t1)|[|α1|∫pq(p−κ)h1−1Γ(h1)|x(κ)|dκ+|β1|∫pq(p−κ)s1+h1−1Γ(s1+h1)|H(κ,x(κ),y(κ))|dκ+∫pq(p−κ)h1+f1−1Γ(h1+f1)|ϕ(κ,x(κ),y(κ))|dκ]+|V2(t2)−V2(t1)|[|α1|∫pq(p−κ)h1−2Γ(h1−1)|x(κ)|dκ+|β1|∫pq(p−κ)s1+h1−2Γ(s1+h1−1)|H(κ,x(κ),y(κ))|dκ+∫pq(p−κ)h1+f1−2Γ(h1+f1−1)|ϕ(κ,x(κ),y(κ))|dκ]+|V3(t2)−V3(t1)|[|α1|τ−2∑i=1|ηi|∫ξiq(ξi−κ)h1−1Γ(h1)|x(κ)|dκ+|β1|τ−2∑i=1|ηi|∫ξiq(ξi−κ)s1+h1−1Γ(s1+h1)|H(κ,x(κ),y(κ))|dκ+τ−2∑i=1|ηi|∫ξiq(ξi−κ)h1+f1−1Γ(h1+f1)|ϕ(κ,x(κ),y(κ))|dκ+∫pq(|α1|∫κq(κ−u)h1−1Γ(h1)|x(u)|du+|β1|∫κq(κ−u)s1+h1−1Γ(s1+h1)|H(u,x(u),y(u))|du+∫κq(κ−u)h1+f1−1Γ(h1+f1)|ϕ(u,x(u),y(u))|du)dΛ(κ)] |
+|V4(t2)−V4(t1)|[|α2|∫pq(p−κ)h2−1Γ(h2)|y(κ)|dκ+|β2|∫pq(p−κ)s2+h2−1Γ(s2+h2)|U(κ,x(κ),y(κ))|dκ+∫pq(p−κ)h2+f2−1Γ(h2+f2)|ψ(κ,x(κ),y(κ))|dκ]+|V5(t2)−V5(t1)|[|α2|∫pq(p−κ)h2−2Γ(h2−1)|y(κ)|dκ+|β2|∫pq(p−κ)s2+h2−2Γ(s2+h2−1)|U(κ,x(κ),y(κ))|dκ+∫pq(p−κ)h2+f2−2Γ(h2+f2−1)|ψ(κ,x(κ),y(κ))|dκ]+|V6(t2)−V6(t1)|[|α2|τ−2∑i=1|ηi|∫ξiq(ξi−κ)h2−1Γ(h2)|y(κ)|dκ+|β2|τ−2∑i=1|ηi|∫ξiq(ξi−κ)s2+h2−1Γ(s2+h2)|U(κ,x(κ),y(κ))|dκ+τ−2∑i=1|ηi|∫ξiq(ξi−κ)h2+f2−1Γ(h2+f2)|ψ(κ,x(κ),y(κ))|dκ+∫pq(|α2|∫κq(κ−u)h2−1Γ(h2)|y(u)|du+|β2|∫κq(κ−u)s2+h2−1Γ(s2+h2)|U(u,x(u),y(u))|du+∫κq(κ−u)h2+f2−1Γ(h2+f2)|ψ(u,x(u),y(u))|du)dΛ(κ)]≤|α1|‖x‖Γ(h1+1)(|(t2−q)h1−(t1−q)h1|+2(t2−t1)h1)+|β1|‖ζ1Γ(s1+h1+1)(|(t2−q)s1+h1−(t1−q)s1+h1|+2(t2−t1)s1+h1) |
+ζ2Γ(h1+f1+1)(|(t2−q)h1+f1−(t1−q)h1+f1|+2(t2−t1)h1+f1)+|V1(t2)−V1(t1)|[|α1|∫pq(p−κ)h1−1Γ(h1)|x(κ)|dκ+|β1|∫pq(p−κ)s1+h1−1Γ(s1+h1)ζ1dκ+∫pq(p−κ)h1+f1−1Γ(h1+f1)ζ2dκ]+|V2(t2)−V2(t1)|[|α1|∫pq(p−κ)h1−2Γ(h1−1)|x(κ)|dκ+|β1|∫pq(p−κ)s1+h1−2Γ(s1+h1−1)ζ1dκ+∫pq(p−κ)h1+f1−2Γ(h1+f1−1)ζ2dκ]+|V3(t2)−V3(t1)|[|α1|τ−2∑i=1|ηi|∫ξiq(ξi−κ)h1−1Γ(h1)|x(κ)|dκ+|β1|τ−2∑i=1|ηi|∫ξiq(ξi−κ)s1+h1−1Γ(s1+h1)ζ1dκ+τ−2∑i=1|ηi|∫ξiq(ξi−κ)h1+f1−1Γ(h1+f1)ζ2dκ+∫pq(|α1|∫κq(κ−u)h1−1Γ(h1)|x(u)|du+|β1|∫κq(κ−u)s1+h1−1Γ(s1+h1)ζ2du+∫κq(κ−u)h1+f1−1Γ(h1+f1)|ζ3du)dΛ(κ)]+|V4(t2)−V4(t1)|[|α2|∫pq(p−κ)h2−1Γ(h2)|y(κ)|dκ+|β2|∫pq(p−κ)s2+h2−1Γ(s2+h2)ζ3dκ+∫pq(p−κ)h2+f2−1Γ(h2+f2)ζ4dκ]+|V5(t2)−V5(t1)|[|α2|∫pq(p−κ)h2−2Γ(h2−1)|y(κ)|dκ+|β2|∫pq(p−κ)s2+h2−2Γ(s2+h2−1)ζ3dκ+∫pq(p−κ)h2+f2−2Γ(h2+f2−1)ζ4dκ]+|V6(t2)−V6(t1)|[|α2|τ−2∑i=1|ηi|∫ξiq(ξi−κ)h2−1Γ(h2)|y(κ)|dκ+|β2|τ−2∑i=1|ηi|∫ξiq(ξi−κ)s2+h2−1Γ(s2+h2)ζ3dκ+τ−2∑i=1|ηi|∫ξiq(ξi−κ)h2+f2−1Γ(h2+f2)ζ4dκ+∫pq(|α2|∫κq(κ−u)h2−1Γ(h2)|y(u)|du+|β2|∫κq(κ−u)s2+h2−1Γ(s2+h2)ζ3du+∫κq(κ−u)h2+f2−1Γ(h2+f2)ζ4du)dΛ(κ)]. |
Similarly, we can find that ‖A2(x,y)−A2(x,y)‖→0 independent of x and y as t2→t1. Therefore, the operator A(x,y) is equicontinuous. As a consequence of our steps together with the Arzela-Ascoli theorem, the operator A is completely continuous. Next, we prove that the set E={(x,y)∈X∗×X∗|(x,y)=σA(x,y),0≤σ≤1} is bounded. Take (x,y)∈E, then (x,y)=σA(x,y) and ∀ t∈[q,p], we have
x(t)=σA1(x,y)(t),y(t)=σA2(x,y)(t). |
In consequence, we have
|x(t)|≤F0|x|+F1(ϖ0+ϖ1|x|+ϖ2|y|)+F2(ε0+ε1|x|+ε2|y|)+F3|y|+F4(n0+n1|x|+n2|y|)+F5(m0+m1|x|+m2|y|), |
which yields
‖x‖≤F0‖x‖+F1(ϖ0+ϖ1‖x‖+ϖ2‖y‖)+F2(ε0+ε1‖x‖+ε2‖y‖)+F3‖y‖+F4(n0+n1‖x‖+n2‖y‖)+F5(m0+m1‖x‖+m2‖y‖). | (3.10) |
In a similar manner, we can find that
‖y‖≤G0‖x‖+G1(ϖ0+ϖ1‖x‖+ϖ2‖y‖)+G2(ε0+ε1‖x‖+ε2‖y‖)+G3‖y‖+G4(n0+n1‖x‖+n2‖y‖)+G5(m0+m1‖x‖+m2‖y‖). | (3.11) |
From (3.10) and (3.11) together with notations (3.6)–(3.9) lead to
‖x‖+‖y‖≤[(F1+G1)ϖ0+(F2+G2)ε0+(F4+G4)n0+(F5+G5)m0]+[(F0+G0)+(F1+G1)|ϖ1+(F2+G2)ε1+(F4+G4)n1+(F5+G5)m1]‖x‖+[(F1+G1)ϖ2+(F2+G2)ε2+(F3+G3)+(F4+G4)n2+(F5+G5)m2]‖y‖. |
Which implies,
‖(x,y)‖≤O0+max{O1+O2}‖(x,y)‖≤O0+O‖(x,y)‖, |
consequently,
‖(x,y)‖≤O01−O. |
This prove that the set E is bounded. Thus, by Lemma 3.1, the operator A has at least one FP. Therefore, the systems (1.1) and (1.2) has at least one solution on [q,p].
Next results are based on Krasnoselskii FPTs. We assume continuous functions H,ϕ,U,ψ:[q,p]×R2→R satisfying the condition:
(H2) For all t∈[q,p] and xj,yj∈R(j=1,2), ∃ Li,i=1,...,4:
|H(t,x1,y1)−H(t,x2,y2)|≤L1(|x1−x2|+|y1−y2|), |
|ϕ(t,x1,y1)−ϕ(t,x2,y2)|≤L2(|x1−x2|+|y1−y2|), |
|U(t,x1,y1)−U((t,x2,y2)|≤L3(|x1−x2|+|y1−y2|), |
|ψ(t,x1,y1)−ψ(t,x2,y2)|≤L4(|x1−x2|+|y1−y2|); |
For simplicity, we introduce the following notations:
N=Δ1+Δ2, | (3.12) |
¯N=¯Δ1+¯Δ2, | (3.13) |
Δ1=F0+L1F1+L2F2, | (3.14) |
Δ2=F3+L3F4+L4F5, | (3.15) |
Δ3=Q0+L1Q1+L2Q2, | (3.16) |
¯Δ1=G0+L1G1+L2G2, | (3.17) |
¯Δ2=G3+L3G4+L4G5, | (3.18) |
¯Δ3=Q3+L3Q4+L4Q5, | (3.19) |
where
Q0=F0−|α1|(p−q)h1Γ(h1+1),Q1=F1−|β1|(p−q)s1+h1Γ(s1+h1+1),Q2=F2−(p−q)h1+f1Γ(h1+f1+1),Q3=G3−|α2|(p−q)h2Γ(h2+1),Q4=G4−|β2|(p−q)s2+h2Γ(s2+h2+1),Q5=G5−(p−q)h2+f2Γ(h2+f2+1), | (3.20) |
and Fi,Gi(i=0,...,5) are given by (3.4) and (3.5).
Lemma 3.2. (Krasnoselskii) Let B≠∅ be a closed, bounded, convex subset of a BSp. K. Let operators M1,M2:B→K:
(a) M1z1+M2z2∈B where z1,z2∈B;
(b) M1 is compact and continuous;
(c) M2 is a contraction mapping.
Then ∃ z∈B: z=M1z+M2z.
Here we prove the Unq. result of solution for the systems (1.1) and (1.2) by applying Banach's FPTs.
For simplicity we use the following notations:
B=B1+B2,B1=Z1F1+Z2F2,B2=Z3F4+Z4F5, | (4.1) |
¯B=¯B1+¯B2,¯B1=Z1G1+Z2G2,¯B2=Z3G4+Z4G5, | (4.2) |
Z1=supt∈[q,p]|H(t,0,0)|<∞,Z2=supt∈[q,p]|ϕ(t,0,0,)|<∞,Z3=supt∈[q,p]|U(t,0,0,)|<∞,Z4=supt∈[q,p]|ψ(t,0,0,)|<∞. | (4.3) |
Theorem 4.1. Let the condition (H2) holds. Then (1.1) and (1.2) has a unique solution on [q,p] if
N+¯N<1, | (4.4) |
where N and ¯N are given by (3.12) and (3.13) respectively.
Proof. Setting S>B+¯B1−N−¯N, where N,¯N,B and ¯B are given by (3.12), (3.13), (4.1) and (4.2) respectively. We show that ASS⊂SS, where SS={(x,y)∈X∗×X∗:‖(x,y)‖≤S}, and the operator A is given by (3.1).
By assumption (H2) together with (4.3), for (x,y)∈SS,e∈[q,p], we have
|H(e,x(e),y(e))|≤|H(e,x(e),y(e))−H(e,0,0)|+|H(e,0,0)|≤L1(‖x‖+‖y‖)+Z1≤L1S+Z1, |
|ϕ(e,x(e),y(e))|≤|ϕ(e,x(e),y(e))−ϕ(e,0,0)|+|ϕ(e,0,0)|≤L2(‖x‖+‖y‖)+Z2≤L2S+Z2. |
|U(e,x(e),y(e))|≤|U(e,x(e),y(e))−U(e,0,0)|+|U(e,0,0)|≤L3(‖x‖+‖y‖)+Z3≤L3S+Z3, |
|ψ(e,x(e),y(e))|≤|ψ(e,x(e),y(e))−ψ(e,0,0)|+|ψ(e,0,0)|≤L4(‖x‖+‖y‖)+Z4≤L4S+Z4. |
By using (3.12) and (4.1), we obtain
|A1(x,y)(e)|≤‖x‖F0+(L1S+Z1)F1+(L2S+Z2)F2+‖y‖F3+(L3S+Z3)F4+(L4S+Z4)F5≤(F0+L1F1+L2F2+|F3+L3F4+L4F5)S+(Z1F1+Z2F2+Z3F4+Z4F5)=(Δ1+Δ2)S+(B1+B2)=NS+B, |
hence,
‖A1(x,y)‖≤NS+B. | (4.5) |
In the same way, by using (3.13) and (4.2), we obtain
|A2(x,y)(e)|≤(G0+L1G1+L2G2+G3+L3G4+L4G5)S+(Z1|G1+Z2G2+|Z3G4+Z4G5)=(¯Δ1+¯Δ2)S+(¯B1+¯B2)=¯NS+¯B, |
which lead to
‖A2(x,y)‖≤¯NS+¯B. | (4.6) |
Consequently, from (4.5) and (4.6) we get
‖A(x,y)‖≤(NS+B)+(¯NS+¯B)≤(N+¯N)S+(B+¯B)≤S. |
Therefore, ASS⊂SS. Now, for any (x1,y1),(x2,y2)∈X∗×X∗,e∈[q,p] and by using conditions (H2), (3.12) and (3.13), we get
‖A1(x1,y1)−A1(x2,y2)‖=supt∈[q,p]|A1(x1,y1)(e)−A1(x2,y2)(e)|≤supe∈[q,p]{|α1|∫eq(e−κ)h1−1Γ(h1)|x1(κ)−x2(κ)|dκ+|β1|∫eq(e−κ)s1+h1−1Γ(s1+h1)|H(κ,x1(κ),y1(κ))−H(κ,x2(κ),y2(κ))|dκ+∫eq(e−κ)h1+f1−1Γ(h1+f1)|ϕ(κ,x1(κ),y1(κ))−ϕ(κ,x2(κ),y2(κ))|dκ+|V1(e)|[|α1|∫pq(p−κ)h1−1Γ(h1)|x1(κ)−x2(κ)|dκ+|β1|∫pq(p−κ)s1+h1−1Γ(s1+h1)|H(κ,x1(κ),y1(κ))−H(κ,x2(κ),y2(κ))|dκ+∫pq(p−κ)h1+f1−1Γ(h1+f1)|ϕ(κ,x1(κ),y1(κ))−ϕ(κ,x2(κ),y2(κ))|dκ]+|V2(e)|[|α1|∫pq(p−κ)h1−2Γ(h1−1)|x1(κ)−x2(κ)|dκ+|β1|∫pq(p−κ)s1+h1−2Γ(s1+h1−1)|H(κ,x1(κ),y1(κ))−H(κ,x2(κ),y2(κ))|dκ+∫pq(p−κ)h1+f1−2Γ(h1+f1−1)|ϕ(κ,x1(κ),y1(κ))−ϕ(κ,x2(κ),y2(κ))|dκ]+|V3(e)|[|α1|τ−2∑i=1|ηi|∫ξiq(ξi−κ)h1−1Γ(h1)|x1(κ)−x2(κ)|dκ+|β1|τ−2∑i=1|ηi|∫ξiq(ξi−κ)s1+h1−1Γ(s1+h1)|H(κ,x1(κ),y1(κ))−H(κ,x2(κ),y2(κ))|dκ+τ−2∑i=1|ηi|∫ξiq(ξi−κ)h1+f1−1Γ(h1+f1)|ϕ(κ,x1(κ),y1(κ))−ϕ(κ,x2(κ),y2(κ))|dκ |
+∫pq(|α1|∫κq(κ−u)h1−1Γ(h1)|x1(u)−x2(u)|du+|β1|∫κq(κ−u)s1+h1−1Γ(s1+h1)|H(u,x1(u),y1(u))−H(u,x2(u),y2(u))|du+∫κq(κ−u)h1+f1−1Γ(h1+f1)|ϕ(u,x1(u),y1(u))−ϕ(u,x2(u),y2(u))|du)dΛ(κ)]+|V4(e)|[|α2|∫pq(p−κ)h2−1Γ(h2)|y1(κ)−y2(κ)|dκ+|β2|∫pq(p−κ)s2+h2−1Γ(s2+h2)|U(κ,x1(κ),y1(κ))−U(κ,x2(κ),y2(κ))|dκ+∫pq(p−κ)h2+f2−1Γ(h2+f2)|ψ(κ,x1(κ),y1(κ))−ψ(κ,x2(κ),y2(κ))|dκ]+|V5(e)|[|α2|∫pq(p−κ)h2−2Γ(h2−1)|y1(κ)−y2(κ)|dκ+|β2|∫pq(p−κ)s2+h2−2Γ(s2+h2−1)|U(κ,x1(κ),y1(κ))−U(κ,x2(κ),y2(κ))|dκ+∫pq(p−κ)h2+f2−2Γ(h2+f2−1)|ψ(κ,x1(κ),y1(κ))−ψ(κ,x2(κ),y2(κ))|dκ]+|V6(e)|[|α2|τ−2∑i=1|ηi|∫ξiq(ξi−κ)h2−1Γ(h2)|y1(κ)−y2(κ)|dκ+|β2|τ−2∑i=1|ηi|∫ξiq(ξi−κ)s2+h2−1Γ(s2+h2)|U(κ,x1(κ),y1(κ))−U(κ,x2(κ),y2(κ))|dκ+τ−2∑i=1|ηi|∫ξiq(ξi−κ)h2+f2−1Γ(h2+f2)|ψ(κ,x1(κ),y1(κ))−ψ(κ,x2(κ),y2(κ))|dκ+∫pq(|α2|∫κq(κ−u)h2−1Γ(h2)|y1(u)−y2(u)|du+|β2|∫κq(κ−u)s2+h2−1Γ(s2+h2)|U(u,x1(u),y1(u))−U(u,x1(u),y1(u))|du+∫κq(κ−u)h2+f2−1Γ(h2+f2)|ψ(u,x1(u),y1(u))−ψ(u,x2(u),y2(u))|du)dΛ(κ)]}≤{F0‖x1−x2‖+L1F1(‖x1−x2‖+‖y1−y2‖)+L2F2(‖x1−x2‖+‖y1−y2‖)+F3‖y1−y2‖+L3F4(‖x1−x2‖+‖y1−y2‖)+L4F5(‖x1−x2‖+‖y1−y2‖)}≤(F0+L1F1+L2F2+F3+L3F4+L4F5)(‖x1−x2‖+‖y1−y2‖)=(Δ1+Δ2)(‖x1−x2‖+‖y1−y2‖)=N(‖x1−x2‖+‖y1−y2‖). |
Similarly
‖A2(x1,y1)−A2(x2,y2)‖=supe∈[q,p]|A2(x1,y1)(e)−A2(x2,y2)(e)|≤(¯Δ1+¯Δ2)(‖x1−x2‖+‖y1−y2‖)=¯N(‖x1−x2‖+‖y1−y2‖). |
Consequently, we obtain
‖A(x1,y1)−A(x2,y2))‖≤(N+¯N)(‖x1−x2‖+‖y1−y2‖), |
which implies that A is a contraction operator by the assumption (4.4). Hence, by Banach's FPT, the operator A has a unique FP, which is the unique solution of systems (1.1) and (1.2) on [q,p].
This section presents examples that illustrate our results.
Example 5.1. Assume the coupled system of FDEs given by
{RLD19/11[(cD39/21+1414)x(t)+4407I8/3H(t,x(t),y(t))]=ϕ(t,x(t),y(t)),RLD29/17[(cD38/23+3880)x(t)+1336I16/5U(t,x(t),y(t))]=ψ(t,x(t),y(t)),t∈[−2,−1], | (5.1) |
with the BCs
{x′(−2)=0,x(−1)=0,x′(−1)=0,x(−2)=3∑i=1ηiy(ξi)+∫−1−2y(κ)dΛ(κ),y′(−2)=0,y(−1)=0,y′(−1)=0,y(−2)=3∑i=1ηix(ξi)+∫−1−2x(κ)dΛ(κ). | (5.2) |
where q=−2,p=−1,f1=1911,h1=3921,f2=2917, h2=3823,s1=83,s2=165,α1=1414,β1=4407,α2=3880, β2=1336,ξ1=−74,ξ2=−32,ξ3=−54,η1=−3,η2=94,η3=52
H(t,x(t),y(t))=1ln(5)+sinx(t)933+y(t)(t2+649),ϕ(t,x(t),y(t))=166+x(t)(t8+22)2+y(t)|x(t)|800(1+|x(t)|),U(t,x(t),y(t))=2y(t)23(1+y(t))+sin(2πx(t))900π+y(t)√t4+2400, |
and
ψ(t,x(t),y(t))=1312+t3+sinx(t)|tan−1y(t)|57π+y(t)12(4√t2+6560). |
Using the given data, we have that F0≃0.016050,F1≃0.002735,F2≃1.04237,F3≃0.0294380,F4≃0.000408,F5≃1.26577,G0≃0.017124, G1≃0.002247,G2≃0.918958,G3≃0.245213,G4≃0.006572,G5≃16.2678. Clearly,
|H(t,x(t),y(t))|≤1ln(5)+1933‖x‖+1650‖y‖,|ϕ(t,x(t),y(t))|≤166+1529‖x‖+1800‖y‖, |
|U(t,x(t),y(t))|≤223+1450‖x‖+149‖y‖,|ψ(t,x(t),y(t))|≤1313+1114‖x‖+1108‖y‖, |
with ϖ0=1ln(5),ϖ1=1933,ϖ2=1650,ε0=166,ε1=1529ε2=1800,n0=223,n1=1450,n2=149,m0=1313,m1=1114, and m2=1108. Using (3.7) and (3.8), we find that O1≃0.190707,O2≃0.439600 and O=max{O1,O2}≃0.439600<1. Therefore, by Theorem 3.1, the problems (5.1) and (5.2) have at least one solution on [−2,−1].
Example 5.2. Consider the system (5.1) with the coupled BCs (5.2) and
H(a,x(a),y(a))=e−2acos2a+170(sinx(a)+y(a)),a∈[−2,−1],ϕ(a,x(a),y(a))=30a5+1510(|x(a)|1+|x(a)|+cosy(a)),a∈[−2,−1],U(a,x(a),y(a))=14√a6+6399(x(a)+tan−1y(a)),a∈[−2,−1],ψ(a,x(a),y(a))=2seca+11800(sin2x(a)+2|y(a)|1+|y(a)|),a∈[−2,−1]. |
Clearly,
|H(a,x1,y1)−H(a,x2,y2)|≤170(‖x1−x2‖+‖y1−y2‖),|ϕ(a,x1,y1)−ϕ(a,x2,y2)|≤1510(‖x1−x2‖+‖y1−y2‖),|U(a,x1,y1)−U((a,x2,y2)|≤1320(‖x1−x2‖+‖y1−y2‖),|ψ(a,x1,y1)−ψ(a,x2,y2)|≤1900(‖x1−x2‖+‖y1−y2‖). |
Using the given data in Example (5.1), we find that N+¯N≃0.331246<1. Thus, in view of Theorem 4.1 the problem (5.1) has a unique solution on [−2,−1].
We managed to employ Leray-Schauder alternative, Banach, and the Krasnoselskii fixed point theory to study the Existence and Uniqueness of solutions for a nonlinear coupled system of fractional differential equations involving Riemann-Liouville and Caputo derivatives with coupled Riemann-Stieltjes integro-multipoint boundary conditions. The system under study is a generalized version of many recent studied system. We used some examples to illustrate the results. Potential future work could be to investigate our results based on other fractional derivates such as, e.g., Abu-Shady-Kaabar fractional derivative, Katugampola derivative, and conformable derivative.
The authors declare that there are no competing interests.
Lemma A.1. Let H,Φ,U,Ψ∈C(q,p)∩L(q,p), the solution of the linear system of FDEs:
{RLDf1[(cDh1+α1)x(t)+β1Is1H∗(t)]=Φ(t),1<h1,f1≤2,t∈(q,p),RLDf2[(cDh2+α2)y(t)+β2Is2U∗(t)]=Ψ(t),1<h2,f2≤2,t∈(q,p), | (A.1) |
with the BCs (1.2) is equivalent to the system:
x(t)=−α1∫tq(t−κ)h1−1Γ(h1)x(κ)dκ−β1∫tq(t−κ)s1+h1−1Γ(s1+h1)H∗(κ)dκ+∫tq(t−κ)h1+f1−1Γ(h1+f1)Φ(κ)dκ+V1(t)[α1∫pq(p−κ)h1−1Γ(h1)x(κ)dκ+β1∫pq(p−κ)s1+h1−1Γ(s1+h1)H∗(κ)dκ−∫pq(p−κ)h1+f1−1Γ(h1+f1)Φ(κ)dκ]+V2(t)[α1∫pq(p−κ)h1−2Γ(h1−1)x(κ)dκ+β1∫pq(p−κ)s1+h1−2Γ(s1+h1−1)H∗(κ)dκ−∫pq(p−κ)h1+f1−2Γ(h1+f1−1)Φ(κ)dκ]+V3(t)[−α1τ−2∑i=1ηi∫ξiq(ξi−κ)h1−1Γ(h1)x(κ)dκ−β1τ−2∑i=1ηi∫ξiq(ξi−κ)s1+h1−1Γ(s1+h1)H∗(κ)dκ+τ−2∑i=1ηi∫ξiq(ξi−κ)h1+f1−1Γ(h1+f1)Φ(κ)dκ+∫pq(−α1∫κq(κ−u)h1−1Γ(h1)x(u)du−β1∫κq(κ−u)s1+h1−1Γ(s1+h1)H∗(u)du+∫κq(κ−u)h1+f1−1Γ(h1+f1)Φ(u)du)dΛ(κ)]+V4(t)[α2∫pq(p−κ)h2−1Γ(h2)y(κ)dκ+β2∫pq(p−κ)s2+h2−1Γ(s2+h2)U∗(κ)dκ−∫pq(p−κ)h2+f2−1Γ(h2+f2)Ψ(κ)dκ]+V5(t)[α2∫pq(p−κ)h2−2Γ(h2−1)y(κ)dκ+β2∫pq(p−κ)s2+h2−2Γ(s2+h2−1)U∗(κ)dκ−∫pq(p−κ)h2+f2−2Γ(h2+f2−1)Ψ(κ)dκ]+V6(t)[−α2τ−2∑i=1ηi∫ξiq(ξi−κ)h2−1Γ(h2)y(κ)dκ−β2τ−2∑i=1ηi∫ξiq(ξi−κ)s2+h2−1Γ(s2+h2)U∗(κ)dκ+τ−2∑i=1ηi∫ξiq(ξi−κ)h2+f2−1Γ(h2+f2)Ψ(κ)dκ+∫pq(−α2∫κq(κ−u)h2−1Γ(h2)y(u)du−β2∫κq(κ−u)s2+h2−1Γ(s2+h2)U∗(u)du+∫κq(κ−u)h2+f2−1Γ(h2+f2)Ψ(u)du)dΛ(κ)], | (A.2) |
y(t)=−α2∫tq(t−κ)h2−1Γ(h2)y(κ)dκ−β2∫tq(t−κ)s2+h2−1Γ(s2+h2)U∗(κ)dκ+∫tq(t−κ)h2+f2−1Γ(h2+f2)Ψ(κ)dκ+W1(t)[α1∫pq(p−κ)h1−1Γ(h1)x(κ)dκ+β1∫pq(p−κ)s1+h1−1Γ(s1+h1)H∗(κ)dκ−∫pq(p−κ)h1+f1−1Γ(h1+f1)Φ(κ)dκ]+W2(t)[α1∫pq(p−κ)h1−2Γ(h1−1)x(κ)dκ+β1∫pq(p−κ)s1+h1−2Γ(s1+h1−1)H∗(κ)dκ−∫pq(p−κ)h1+f1−2Γ(h1+f1−1)Φ(κ)dκ]+W3(t)[−α1τ−2∑i=1ηi∫ξiq(ξi−κ)h1−1Γ(h1)x(κ)dκ−β1τ−2∑i=1ηi∫ξiq(ξi−κ)s1+h1−1Γ(s1+h1)H∗(κ)dκ+τ−2∑i=1ηi∫ξiq(ξi−κ)h1+f1−1Γ(h1+f1)Φ(κ)dκ+∫pq(−α1∫κq(κ−u)h1−1Γ(h1)x(u)du−β1∫κq(κ−u)s1+h1−1Γ(s1+h1)H(u)du+∫κq(κ−u)h1+f1−1Γ(h1+f1)Φ(u)du)dΛ(κ)]+W4(t)[α2∫pq(p−κ)h2−1Γ(h2)y(κ)dκ+β2∫pq(p−κ)s2+h2−1Γ(s2+h2)U∗(κ)dκ−∫pq(p−κ)h2+f2−1Γ(h2+f2)Ψ(κ)dκ]+W5(t)[α2∫pq(p−κ)h2−2Γ(h2−1)y(κ)dκ+β2∫pq(p−κ)s2+h2−2Γ(s2+h2−1)U∗(κ)dκ−∫pq(p−κ)h2+f2−2Γ(h2+f2−1)Ψ(κ)dκ]+W6(t)[−α2τ−2∑i=1ηi∫ξiq(ξi−κ)h2−1Γ(h2)y(κ)dκ−β2τ−2∑i=1ηi∫ξiq(ξi−κ)s2+h2−1Γ(s2+h2)U∗(κ)dκ+τ−2∑i=1ηi∫ξiq(ξi−κ)h2+f2−1Γ(h2+f2)Ψ(κ)dκ+∫pq(−α2∫κq(κ−u)h2−1Γ(h2)y(u)du−β2∫κq(κ−u)s2+h2−1Γ(s2+h2)U∗(u)du+∫κq(κ−u)h2+f2−1Γ(h2+f2)Ψ(u)du)dΛ(κ)], | (A.3) |
where
Vi(t)=B1(t)ρi+Q1(t)ωi+ϵi,i=1,...,6, | (A.4) |
Wj(t)=B2(t)τj+Q2(t)λj+δj,j=1,...,6, | (A.5) |
Bℓ(t)=(t−q)hℓ+fℓ−1Γ(fℓ)Γ(hℓ+fℓ),Qℓ(t)=(t−q)hℓ+fℓ−2Γ(fℓ−1)Γ(hℓ+fℓ−1),ℓ=1,2, | (A.6) |
{ρ1=(A4A27−A4)μ3+A4A7μ1σ,ρ2=(−A2A27+A6A7+A2)μ3+(−A2A7+A6)μ1σ,ρ3=A4A7μ3+A4μ1σ,ρ4=−A4μ1σ,ρ5=A4μ2σ,ρ6=A4μ3σ, | (A.7) |
{ω1=−(A3A27−A3)μ3−A3A7μ1σ,ω2=(A1A27−A5A7−A1)μ3+(A1A7−A5)μ1σ,ω3=−(A3A7μ3+A3μ1)σ,ω4=A3μ1σ,ω5=−A3μ2σ,ω6=−A3μ3σ, | (A.8) |
{ϵ1=ν1A7μ3+ν1μ1σ,ϵ2=−(ν2A7μ3+ν2μ1)σ,ϵ3=−(ν3A7μ3+ν3μ1)σ,ϵ4=ν3μ1σ,ϵ5=−ν3μ2σ,ϵ6=−ν3μ3σ, | (A.9) |
{τ1=−B4ν1σ,τ2=B4ν2σ,τ3=B4ν3σ,τ4=(A27B4−B4)ν3+A7B4ν1σ,τ5=(−A27B2+A7B6+B2)ν3−(A7B2−B6)ν1σ,τ6=A7B4ν3+B4ν1σ, | (A.10) |
\begin{eqnarray} \left\{ \begin{array}{rcl} \lambda_{1}& = & \frac{B_3\nu_1}{\sigma}, \; \lambda_{2} = \frac{-B_3\nu_2}{\sigma}, \; \lambda_{3} = \frac{-B_3\nu_3}{\sigma}, \; \lambda_{4} = \frac{-(A^{2}_{7}B_3-B_3)\nu_3-A_7B_3\nu_1}{\sigma}, \\ \lambda_{5}& = & \frac{(A^{2}_{7}B_1-A_7B_5-B_1)\nu_3+(A_7B_1-B_5)\nu_1}{\sigma}, \; \lambda_{6} = \frac{-A_7B_3\nu_3-B_3\nu_1}{\sigma}, \end{array} \right. \end{eqnarray} | (A.11) |
\begin{eqnarray} \left\{ \begin{array}{rcl} \delta_{1}& = & \frac{\mu_3\nu_1}{\sigma}, \; \delta_{2} = \frac{-\mu_3\nu_2}{\sigma}, \; \delta_{3} = \frac{-\mu_3\nu_3}{\sigma}, \; \delta_{4} = \frac{\mu_1A_7\nu_3+\mu_1\nu_1}{\sigma}, \\ \delta_{5}& = & \frac{-\mu_2A_7\nu_3-\mu_2\nu_1}{\sigma}, \; \delta_{6} = \frac{-\mu_3A_7\nu_3-\mu_3\nu_1}{\sigma}, \end{array} \right. \end{eqnarray} | (A.12) |
\begin{eqnarray} \mu_1& = &B_3B_6-B_4B_5, \; \mu_2 = B_1B_6-B_2B_5, \; \mu_3 = B_1B_4-B_2B_3, \end{eqnarray} | (A.13) |
\begin{eqnarray} \nu_1& = &A_3A_6-A_4A_5, \; \nu_2 = A_1A_6-A_2A_5, \; \nu_3 = A_1A_4-A_2A_3, \end{eqnarray} | (A.14) |
\begin{eqnarray} \left\{ \begin{array}{rcl} A_{1}& = & \frac{(\mathfrak{p}-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-1}\Gamma(\mathfrak{f}_1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1)}, \ A_{2} = \frac{(\mathfrak{p}-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-2}\Gamma(\mathfrak{f}_1-1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1-1)}, \\ A_{3}& = & \frac{(\mathfrak{p}-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-2}\Gamma(\mathfrak{f}_1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1-1)}, \ A_{4} = \frac{(\mathfrak{p}-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-3}\Gamma(\mathfrak{f}_1-1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1-2)}, \\ A_{5}& = & \sum\limits_{i = 1}^{\tau-2}\eta_{i}\frac{(\xi_{i}-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-1}\Gamma(\mathfrak{f}_1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1)}+\int_{\mathfrak{q}}^{\mathfrak{p}}\frac{(\kappa-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-1}\Gamma(\mathfrak{f}_1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1)}d\Lambda(\kappa), \\ A_{6}& = & \sum\limits_{i = 1}^{\tau-2}\eta_{i}\frac{(\xi_{i}-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-2}\Gamma(\mathfrak{f}_1-1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1-1)}+\int_{\mathfrak{q}}^{\mathfrak{p}}\frac{(\kappa-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-2}\Gamma(\mathfrak{f}_1-1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1-1)}d\Lambda(\kappa), \; \\ A_{7}& = & \sum\limits_{i = 1}^{\tau-2}\eta_{i}+\int_{\mathfrak{q}}^{\mathfrak{p}}d\Lambda(\kappa), \end{array} \right. \end{eqnarray} | (A.15) |
\begin{eqnarray} \left\{ \begin{array}{rcl} B_{1}& = & \frac{(\mathfrak{p}-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-1}\Gamma(\mathfrak{f}_2)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2)}, \ B_{2} = \frac{(\mathfrak{p}-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-2}\Gamma(\mathfrak{f}_2-1)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2-1)}, \\ B_{3}& = & \frac{(\mathfrak{p}-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-2}\Gamma(\mathfrak{f}_2)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2-1)}, \ B_{4} = \frac{(\mathfrak{p}-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-3}\Gamma(\mathfrak{f}_2-1)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2-2)}, \\ B_{5}& = & \sum\limits_{i = 1}^{\tau-2}\eta_{i}\frac{(\xi_{i}-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-1}\Gamma(\mathfrak{f}_2)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2)}+\int_{\mathfrak{q}}^{\mathfrak{p}}\frac{(\kappa-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-1}\Gamma(\mathfrak{f}_2)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2)}d\Lambda(\kappa), \\ B_{6}& = & \sum\limits_{i = 1}^{\tau-2}\eta_{i}\frac{(\xi_{i}-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-2}\Gamma(\mathfrak{f}_2-1)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2-1)}+\int_{\mathfrak{q}}^{\mathfrak{p}}\frac{(\kappa-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-2}\Gamma(\mathfrak{f}_2-1)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2-1)}d\Lambda(\kappa), \end{array} \right. \end{eqnarray} | (A.16) |
and it is assumed that
\begin{eqnarray} \sigma = (\nu_3A^{2}_{7}+\nu_1A_7-\nu_3)\mu_3+(\nu_3A_{7}+\nu_1)\mu_1\neq 0, \end{eqnarray} | (A.17) |
Proof. Solving the FDE (A.1) in a standard manner and using Lemmas 2.1 and 2.2, we get
\begin{eqnarray} x(t)& = &-\alpha_1\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{h}_1-1}}{\Gamma(\mathfrak{h}_1)}x(\kappa)d\kappa-\beta_1\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{s}_1+\mathfrak{h}_1-1}}{\Gamma(\mathfrak{s}_1+\mathfrak{h}_1)}H^*(\kappa)d\kappa+\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{h}_1+\mathfrak{f}_1-1}}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1)}\Phi(\kappa)d\kappa\\ &+&c_{1}\frac{(t-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-1}\Gamma(\mathfrak{f}_1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1)}+c_{2}\frac{(t-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-2}\Gamma(\mathfrak{f}_1-1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1-1)} +c_{3}+c_{4}(t-\mathfrak{q}), \end{eqnarray} | (A.18) |
\begin{eqnarray} x'(t)& = &-\alpha_1\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{h}_1-2}}{\Gamma(\mathfrak{h}_1-1)}x(\kappa)d\kappa-\beta_1\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{s}_1+\mathfrak{h}_1-2}}{\Gamma(\mathfrak{s}_1+\mathfrak{h}_1-1)}H^*(\kappa)d\kappa+\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{h}_1+\mathfrak{f}_1-2}}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1-1)}\Phi(\kappa)d\kappa\\ &+&c_{1}\frac{(t-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-2}\Gamma(\mathfrak{f}_1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1-1)}+c_{2}\frac{(t-\mathfrak{q})^{\mathfrak{h}_1+\mathfrak{f}_1-3}\Gamma(\mathfrak{f}_1-1)}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1-2)} +c_{4}. \end{eqnarray} | (A.19) |
\begin{eqnarray} y(t)& = &-\alpha_2\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{h}_2-1}}{\Gamma(\mathfrak{h}_2)}y(\kappa)d\kappa-\beta_2\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{s}_2+\mathfrak{h}_2-1}}{\Gamma(\mathfrak{s}_2+\mathfrak{h}_2)}\mathfrak{U}^*(\kappa)d\kappa+\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{h}_2+\mathfrak{f}_2-1}}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2)}\Psi(\kappa)d\kappa\\ &+&b_{1}\frac{(t-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-1}\Gamma(\mathfrak{f}_2)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2)}+b_{2}\frac{(t-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-2}\Gamma(\mathfrak{f}_2-1)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2-1)} +b_{3}+b_{4}(t-\mathfrak{q}), \end{eqnarray} | (A.20) |
\begin{eqnarray} y'(t)& = &-\alpha_2\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{h}_2-2}}{\Gamma(\mathfrak{h}_2-1)}y(\kappa)d\kappa-\beta_2\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{s}_2+\mathfrak{h}_2-2}}{\Gamma(\mathfrak{s}_2+\mathfrak{h}_2-1)}\mathfrak{U}^*(\kappa)d\kappa+\int_{\mathfrak{q}}^{t}\frac {(t-\kappa)^{\mathfrak{h}_2+\mathfrak{f}_2-2}}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2-1)}\Psi(\kappa)d\kappa\\ &+&b_{1}\frac{(t-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-2}\Gamma(\mathfrak{f}_2)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2-1)}+b_{2}\frac{(t-\mathfrak{q})^{\mathfrak{h}_2+\mathfrak{f}_2-3}\Gamma(\mathfrak{f}_2-1)}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2-2)} +b_{4}. \end{eqnarray} | (A.21) |
c_i, b_i \in \mathbb{R}, i = 1, \cdots, 4 are some unknown arbitrary constants.
Using the BCs (1.2) in Eqs (A.18)–(A.21), together with notations (A.15) and (A.16), we obtain c_4 = 0, \; b_4 = 0, and a system of equations in c_i, \; b_i(i = 1, 2, 3) given by
\begin{eqnarray} \left\{ \begin{array}{rcl} A_1c_1+A_2c_2+c_3& = &K_1, \\ B_1b_1+B_2b_2+b_3& = &E_1, \\ A_3c_1+A_4c_2& = &K_2, \\ B_3b_1+B_4b_2& = &E_2, \\ c_3-B_5b_1-B_6b_2-A_7b_3& = &E_3, \\ b_3-A_5c_1-A_6c_2-A_7c_3& = &K_3, \end{array} \right. \end{eqnarray} | (A.22) |
where A_{i}\; (i = 1, ..., 7), \; B_{j}\; (j = 1, ..., 6) are given by (A.15) and (A.16) and K_{i}, E_{i}, i = 1, 2, 3, are defined by
\begin{eqnarray} K_{1}& = &\alpha_1\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{h}_1-1}}{\Gamma(\mathfrak{h}_1)}x(\kappa)d\kappa+\beta_1\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{s}_1+\mathfrak{h}_1-1}}{\Gamma(\mathfrak{s}_1+\mathfrak{h}_1)}H^*(\kappa)d\kappa-\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{h}_1+\mathfrak{f}_1-1}}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1)}\Phi(\kappa)d\kappa, \\ K_{2}& = &\alpha_1\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{h}_1-2}}{\Gamma(\mathfrak{h}_1-1)}x(\kappa)d\kappa+\beta_1\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{s}_1+\mathfrak{h}_1-2}}{\Gamma(\mathfrak{s}_1+\mathfrak{h}_1-1)}H^*(\kappa)d\kappa-\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{h}_1+\mathfrak{f}_1-2}}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1-1)}\Phi(\kappa)d\kappa, \\ K_{3}& = &-\alpha_1\sum\limits_{i = 1}^{\tau-2}{\eta_{i}}\int_{\mathfrak{q}}^{\xi_{i}}\frac {(\xi_{i}-\kappa)^{\mathfrak{h}_1-1}}{\Gamma(\mathfrak{h}_1)}x(\kappa)d\kappa-\beta_1\sum\limits_{i = 1}^{\tau-2}{\eta_{i}}\int_{\mathfrak{q}}^{\xi_{i}}\frac {(\xi_{i}-\kappa)^{\mathfrak{s}_1+\mathfrak{h}_1-1}}{\Gamma(\mathfrak{s}_1+\mathfrak{h}_1)}H^*(\kappa)d\kappa\\ &+&\sum\limits_{i = 1}^{\tau-2}{\eta_{i}}\int_{\mathfrak{q}}^{\xi_{i}}\frac {(\xi_{i}-\kappa)^{\mathfrak{h}_1+\mathfrak{f}_1-1}}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1)}\Phi(\kappa)d\kappa +\int_{\mathfrak{q}}^{\mathfrak{p}}\Big(-\alpha_1\int_{\mathfrak{q}}^{\kappa}\frac {(\kappa-u)^{\mathfrak{h}_1-1}}{\Gamma(\mathfrak{h}_1)}x(u)du\\ &-&\beta_1\int_{\mathfrak{q}}^{\kappa}\frac {(\kappa-u)^{\mathfrak{s}_1+\mathfrak{h}_1-1}}{\Gamma(\mathfrak{s}_1+\mathfrak{h}_1)}H^*(u)du +\int_{\mathfrak{q}}^{\kappa}\frac {(\kappa-u)^{\mathfrak{h}_1+\mathfrak{f}_1-1}}{\Gamma(\mathfrak{h}_1+\mathfrak{f}_1)}\Phi(u)du\Big)d\Lambda(\kappa), \\ E_{1}& = &\alpha_2\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{h}_2-1}}{\Gamma(\mathfrak{h}_2)}y(\kappa)d\kappa+\beta_2\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{s}_2+\mathfrak{h}_2-1}}{\Gamma(\mathfrak{s}_2+\mathfrak{h}_2)}\mathfrak{U}^*(\kappa)d\kappa-\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{h}_2+\mathfrak{f}_2-1}}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2)}\Psi(\kappa)d\kappa, \\ E_{2}& = &\alpha_2\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{h}_2-2}}{\Gamma(\mathfrak{h}_2-1)}y(\kappa)d\kappa+\beta_2\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{s}_2+\mathfrak{h}_2-2}}{\Gamma(\mathfrak{s}_2+\mathfrak{h}_2-1)}\mathfrak{U}^*(\kappa)d\kappa-\int_{\mathfrak{q}}^{\mathfrak{p}}\frac {(\mathfrak{p}-\kappa)^{\mathfrak{h}_2+\mathfrak{f}_2-2}}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2-1)}\Psi(\kappa)d\kappa, \\ E_{3}& = &-\alpha_2\sum\limits_{i = 1}^{\tau-2}{\eta_{i}}\int_{\mathfrak{q}}^{\xi_{i}}\frac {(\xi_{i}-\kappa)^{\mathfrak{h}_2-1}}{\Gamma(\mathfrak{h}_2)}y(\kappa)d\kappa-\beta_2\sum\limits_{i = 1}^{\tau-2}{\eta_{i}}\int_{\mathfrak{q}}^{\xi_{i}}\frac {(\xi_{i}-\kappa)^{\mathfrak{s}_2+\mathfrak{h}_2-1}}{\Gamma(\mathfrak{s}_2+\mathfrak{h}_2)}\mathfrak{U}^*(\kappa)d\kappa\\ &+&\sum\limits_{i = 1}^{\tau-2}{\eta_{i}}\int_{\mathfrak{q}}^{\xi_{i}}\frac {(\xi_{i}-\kappa)^{\mathfrak{h}_2+\mathfrak{f}_2-1}}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2)}\Psi(\kappa)d\kappa +\int_{\mathfrak{q}}^{\mathfrak{p}}\Big(-\alpha_2\int_{\mathfrak{q}}^{\kappa}\frac {(\kappa-u)^{\mathfrak{h}_2-1}}{\Gamma(\mathfrak{h}_2)}y(u)du\\ &-&\beta_2\int_{\mathfrak{q}}^{\kappa}\frac {(\kappa-u)^{\mathfrak{s}_2+\mathfrak{h}_2-1}}{\Gamma(\mathfrak{s}_2+\mathfrak{h}_2)}\mathfrak{U}^*(u)du +\int_{\mathfrak{q}}^{\kappa}\frac {(\kappa-u)^{\mathfrak{h}_2+\mathfrak{f}_2-1}}{\Gamma(\mathfrak{h}_2+\mathfrak{f}_2)}\Psi(u)du\Big)d\Lambda(\kappa), \end{eqnarray} | (A.23) |
Solving the system (A.22) for c_i, \; b_i (i = 1, 2, 3) , we find that
\begin{eqnarray} c_1& = &\rho_1K1+\rho_2K_2+\rho_3K_3+\rho_4E_1+\rho_5E_2+\rho_6E_3, \end{eqnarray} | (A.24) |
\begin{eqnarray} c_2& = &\omega_1K1+\omega_2K_2+\omega_3K_3+\omega_4E_1+\omega_5E_2+\omega_6E_3, \end{eqnarray} | (A.25) |
\begin{eqnarray} c_3& = &\epsilon_1K1+\epsilon_2K_2+\epsilon_3K_3+\epsilon_4E_1+\epsilon_5E_2+\epsilon_6E_3, \end{eqnarray} | (A.26) |
\begin{eqnarray} b_1& = &\tau_1K1+\tau_2K_2+\tau_3K_3+\tau_4E_1+\tau_5E_2+\tau_6E_3, \end{eqnarray} | (A.27) |
\begin{eqnarray} b_1& = &\lambda_1K1+\lambda_2K_2+\lambda_3K_3+\lambda_4E_1+\lambda_5E_2+\rho_6E_3, \end{eqnarray} | (A.28) |
\begin{eqnarray} b_1& = &\delta_1K1+\delta_2K_2+\delta_3K_3+\delta_4E_1+\delta_5E_2+\delta_6E_3, \end{eqnarray} | (A.29) |
where \rho_{i}, \; \omega_i, \; \epsilon_i, \; \tau_i, \; \lambda_i and \delta_{i}\; (i = 1, ..., 6) are given by (A.7)–(A.12) respectively.
Inserting the values of c_1, c_2, c_3, c_4, b_1, b_2, b_{3} and b_{4} in (A.18) and (A.20), we get (A.2) and (A.3). The converse follows by direct computation. This completes the proof.
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