The COVID-19 pandemic still gains the attention of many researchers worldwide. Over the past few months, China faced a new wave of this pandemic which increases the risk of its spread to the rest of the world. Therefore, there has become an urgent demand to know the expected behavior of this pandemic in the coming period. In this regard, there are many mathematical models from which we may obtain accurate predictions about the behavior of this pandemic. Such a target may be achieved via updating the mathematical models taking into account the memory effect in the fractional calculus. This paper generalizes the power-law growth model of the COVID-19. The generalized model is investigated using two different definitions in the fractional calculus, mainly, the Caputo fractional derivative and the conformable derivative. The solution of the first-model is determined in a closed series form and the convergence is addressed. At a specific condition, the series transforms to an exact form. In addition, the solution of the second-model is evaluated exactly. The results are applied on eight European countries to predict the behavior/variation of the infected cases. Moreover, some remarks are given about the validity of the results reported in the literature.
Citation: Weam G. Alharbi, Abdullah F. Shater, Abdelhalim Ebaid, Carlo Cattani, Mounirah Areshi, Mohammed M. Jalal, Mohammed K. Alharbi. Communicable disease model in view of fractional calculus[J]. AIMS Mathematics, 2023, 8(5): 10033-10048. doi: 10.3934/math.2023508
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The COVID-19 pandemic still gains the attention of many researchers worldwide. Over the past few months, China faced a new wave of this pandemic which increases the risk of its spread to the rest of the world. Therefore, there has become an urgent demand to know the expected behavior of this pandemic in the coming period. In this regard, there are many mathematical models from which we may obtain accurate predictions about the behavior of this pandemic. Such a target may be achieved via updating the mathematical models taking into account the memory effect in the fractional calculus. This paper generalizes the power-law growth model of the COVID-19. The generalized model is investigated using two different definitions in the fractional calculus, mainly, the Caputo fractional derivative and the conformable derivative. The solution of the first-model is determined in a closed series form and the convergence is addressed. At a specific condition, the series transforms to an exact form. In addition, the solution of the second-model is evaluated exactly. The results are applied on eight European countries to predict the behavior/variation of the infected cases. Moreover, some remarks are given about the validity of the results reported in the literature.
This paper considers a fractional coupled system on an infinite interval involving the Erdélyi-Kober derivative:
{Dγ,δ1βu(x)+F(x,u(x),v(x))=0,x∈(0,+∞),Dγ,δ2βv(x)+G(x,u(x),v(x))=0,x∈(0,+∞),limx→0xβ(2+γ)Iδ1+γ,2−δ1u(x)=0,limx→∞xβ(1+γ)Iδ1+γ,2−δ1u(x)=0,limx→0xβ(2+γ)Iδ2+γ,2−δ2v(x)=0,limx→∞xβ(1+γ)Iδ2+γ,2−δ2v(x)=0, | (1.1) |
where δ1,δ2∈(1,2], γ∈(−2,−1), and β>0. Dγ,δ1β, Dγ,δ2β are Erdélyi-Kober fractional derivatives (EKFDs for short), and Iδ1+γ,2−δ1,Iδ2+γ,2−δ2 are the Erdélyi-Kober fractional integrals. F,G are continuous functions. We discuss the existence of positive solutions for (1.1).
During the past several decades, fractional equations have been studied widely; see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] for instance. From the literature, we can see that there are many fractional derivatives used in differential equations. Among these various definitions, the widely used ones are the Riemann-Liouville and Caputo fractional derivatives, in many works. To generalize the Riemann-Liouville fractional derivative, Erdélyi-Kober defined a new fractional derivative, and we call it the Erdélyi-Kober fractional derivative. Moreover, the Erdélyi-Kober operator is very useful; we can refer to [6,9,14,15,16,17] and the references therein. The Erdélyi-Kober operator is a fractional integration operation which was given by Arthur Erdélyi and Hermann Kober in 1940 [23]. Some of these definitions and results were given in Samko et al. [3], Kiryakova [19], and McBride [20].
Nowadays, the theory of fractional operators in the Erdélyi-Kober frame has attracted much interest from researchers. The study of fractional systems is also very important, as these systems appear in various applications, especially in biological sciences. Recently, some problems of Erdélyi-Kober type fractional differential equations on infinite intervals received widespread attention from many scholars; see [8,21,22] for example.
Recently, in [8], the authors investigated the following equation:
{(Dϑ,σθu)(x)+F(u(x))=0,0≤x<∞,limt→0xθ(2−σ)Iσ+ϑ,2−σu(x)=0,limt→+∞xθ(2−σ)Iσ+ϑ,2−σu(x)=0, |
where σ∈(1,2), ϑ∈(1,2), θ>0, and F is a given continuous function, Dϑ,σθ denotes the EKFD, and Iσ+ϑ,2−σ denotes the Erdélyi-Kober fractional integral. The authors studied the existence and nonexistence of positive solutions for this problem by utilizing a fixed point result which uses the strongly positive-like operators and eigenvalue criteria.
In [9], the authors studied a fractional coupled system:
{cDϱu(τ)=F(τ,u(τ),z(τ),cDς1z(τ),Iξz(τ)),τ∈[0,T]:=K,2<ϱ≤3,1<ς1<2,cDςz(τ)=G(τ,u(τ),cDϱ1u(τ),Iζu(τ),z(τ),τ∈[0,T]:=K,2<ς≤3,1<ϱ1<2,u(0)=ϕ1(z),u′(0)=ε1z′(k1),u(T)=γρϑ−ρ(ϖ+v)Γ(ϖ)∫ϑ0σρv+ρ−1z(σ)(ϑρ−σρ)1−ϖdσ:=γJv,ϖρv(ϑ),z(0)=ϕ2(u),z′(0)=ε2z′(k2),z(T)=δvφ−v(θ+ω)Γ(θ)∫φ0σvω+υ−1u(σ)(φv−σv)1−θdσ:=δJω,θvu(φ), |
where cDϱ,cDς1,cDς,cDϱ1 are the Liouville-Caputo fractional derivatives of order 2<ϱ,ς≤3, 1<ς1,ϱ1<2. Iξ,Iζ are the Riemann-Liouville fractional integrals of order 1<ξ,ζ<2. Jυ,ϖρ,Jω,θv are the Erdélyi-Kober fractional integrals of order ϖ,θ>0, with v,ω>0, ρ, ϑ∈(−∞,+∞). F,G:K×(−∞,+∞)4→(−∞,+∞) and ϕ1,ϕ2:C(K,(−∞,+∞))→(−∞,+∞) are continuous functions. γ,δ,ε1,ε2 are positive real constants. The existence result was given by the Leray-Schauder alternative, and the uniqueness result was obtained due to Banach's fixed-point theorem. By the same methods, Arioua and Titraoui [18] studied system (1.1). Moreover, In [10], Arioua and Titraoui also investigated a new fractional problem involving the Erdélyi-Kober derivative. Inspired by the above articles, we use different methods to consider the fractional coupled system involving Erdélyi-Kober derivative (1.1). We employ the Guo-Krasnosel'skii fixed point theorem to discuss (1.1) in a special Banach space, and we also use the monotone iterative technique to study this system. Some existence results of positive solutions for system (1.1) are obtained, including the existence results of at least two positive solutions.
Definition 2.1. (see [2]) Let α∈(−∞,+∞). Cnα, n∈N, denotes a set of all functions f(t),t>0, with f(t)=tpf1(t) with p>α and f1∈Cn[0,∞).
Definition 2.2. (see [1,2]) For a function u∈Cα, the σ-order right-hand Erdélyi-Kober fractional integral is
(Iγ,σβu)(t)=βt−β(γ+σ)Γ(σ)∫t0sβ(γ+1)−1u(s)(tβ−sβ)1−σds,σ,β>0,γ∈(−∞,+∞), |
in which, Γ is the Euler gamma function.
Definition 2.3. (see [2]) Let n−1<δ≤n,n∈N, and for u∈Cα, the σ-order right-hand Erdélyi-Kober fractional derivative is
(Dγ,σβu)(t)=n∏j=1(γ+j+tβddt)(Iγ+σ,n−σβu)(t), |
where
n∏j=1(γ+j+tβddt)(Iγ+σ,n−σβu)=(γ+1+tβddt)⋯(γ+n+tβddt)(Iγ+σ,n−σβu). |
Lemma 2.1. (see [10]) Let 1<σ≤2, −2<γ<−1, β>0, and h∈C2α, with ∫∞0sβ(γ+m)−1h(τ)dτ<∞, m=1,2. The fractional problem
{Dγ,σβu(x)+h(x)=0,x>0,limx→0xβ(2+γ)Iσ+γ,2−δu(x)=0,limx→∞xβ(1+γ)Iσ+γ,2−σu(x)=0, |
has a unique solution given by u(x)=∫∞0Gσ(x,s)sβ(γ+1)−1h(s)ds, where
Gσ(x,s)={βΓ(σ)[x−β(γ+1)−x−β(δ+γ)(xβ−sβ)σ−1],0<s≤x<∞,βΓ(σ)x−β(γ+1),0<x≤s<∞. | (2.1) |
Lemma 2.2. (see [10]) For 1<σ≤2, −2<γ<−1, and β>0, the function Gσ, defined in (2.1), has the following properties:
(i) Gσ(x,s)1+x−β(1+γ)>0, for x,s>0;
(ii) Gσ(x,s)1+x−β(1+γ)≤βΓ(σ), for x,s>0;
(iii) for 0<τλ≤x≤τ and s>τλ2, where λ>1,τ>0, we have
Gσ(x,s)1+x−β(1+γ)≥β(σ−1)τ−β(1+γ)Γ(σ)λ−β(1−γ)(1+τ−β(1+γ))=βp(τ)Γ(σ), |
where p(τ)=(σ−1)τ−β(1+γ)λβ(1+γ)(1+τ−β(1+γ)).
Lemma 2.3. (see [18]) Let 0<σ1,σ2≤1 and F,G∈C2α with
∫∞0sβ(γ+m)−1F(s,u(s),v(s))ds<∞,m=1,2, |
∫∞0sβ(γ+m)−1G(s,u(s),v(s))ds<∞,m=1,2. |
Then, (1.1) has a unique solution given by
u(x)=∫∞0Gσ1(x,s)sβ(γ+1)−1F(s,u(s),v(s))ds, |
v(x)=∫∞0Gσ2(x,s)sβ(γ+1)−1G(s,u(s),v(s))ds, |
where
Gσ1(x,s)={βΓ(σ1)[x−β(γ+1)−x−β(σ1+γ)(xβ−sβ)σ1−1],0<s≤x<∞,βΓ(σ1)x−β(γ+1),0<x≤s<∞, | (2.2) |
Gσ2(x,s)={βΓ(σ2)[x−β(γ+1)−x−β(σ2+γ)(xβ−sβ)σ2−1],0<s≤x<∞,βΓ(σ2)x−β(γ+1),0<x≤s<∞. | (2.3) |
The following result is our main tool.
Lemma 2.4. (Guo-Krasnosel'skii fixed point theorem; see [37]) P is a cone in a Banach space E, and D1 and D2 are bounded open sets in E with θ∈D1, ¯D1⊂D2. A:P∩(¯D2∖D1)→P is a completely continuous operator. Consider the following conditions (ⅰ), (ⅱ):
(i) ‖Aw‖≤‖w‖ for w∈P∩∂D1, ‖Aw‖≥‖w‖ for w∈P∩∂D2;
(ii) ‖Aw‖≥‖w‖ for w∈P∩∂D1, ‖Aw‖≤‖w‖ for w∈P∩∂D2.
If one of the preceding conditions (ⅰ), (ⅱ) holds, then A has at least one fixed point in P∩(¯D2∖D1).
Next, we present some hypotheses that will play an important role in the subsequent discussion:
(H1) F,G:(0,+∞)×(−∞,+∞)×(−∞,+∞)→(0,+∞) are continuous and nondecreasing with respect to the second, third variables on (0,+∞).
(H2) For (x,u,v)∈(0,+∞)×(−∞,+∞)×(−∞,+∞),
F1(x,u,v)=xβ(1+γ)−1F(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v), |
F2(x,u,v)=xβ(1+γ)−1G(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v), |
such that
F1(x,u,v)≤φ1(x)ω1(∣u∣)+ψ1(x)ω2(∣v∣), |
F2(x,u,v)≤φ2(x)~ω1(∣u∣)+ψ2(x)~ω2(∣v∣), |
with ωi,~ωi∈C((0,+∞),(0,+∞)) nondecreasing and φi,ψi∈L1(0,+∞), i=1,2.
(H3) There are positive functions qi,˜qi,i=1,2, with
q∗i=∫∞0(1+x−β(1+γ))qi(x)dx<∞, |
˜q∗i=∫∞0(1+x−β(1+γ))˜qi(x)dx<∞, |
such that
xβ(γ+1)−1∣F(x,u,v)−F(x,˜u,˜v)∣≤q1(x)∣u−˜u∣+˜q1(x)∣v−˜v∣, |
xβ(γ+1)−1∣G(x,u,v)−G(x,˜u,˜v)∣≤q2(x)∣u−˜u∣+˜q2(t)∣v−˜v∣, |
for any u,v,˜u,˜v∈(−∞,+∞) and x∈(0,+∞).
(H4) F,G:(0,+∞)×(0,+∞)×(0,+∞)→(0,+∞) are continuous, such that
xβ(1+γ)−1F(x,u,v)=a1(x)F1(x,u,v), |
xβ(1+γ)−1G(x,u,v)=a2(x)G1(x,u,v), |
where a1,a2∈L1((0,+∞),(0,+∞)), F1,G1∈C((0,+∞)×(0,+∞)×(0,+∞),(0,+∞)), 0<∫ττλa1(x)dx<∞, 0<∫ττλa2(x)dx<∞, with τ>0, λ>1. Moreover, xβ(1+γ)−1F(x,u,v), xβ(1+γ)−1G(x,u,v):[0,+∞)×(0,+∞)×(0,+∞)→[0,+∞) also are continuous.
Remark 2.1. These conditions ensure the continuity and integrability of nonlinear terms in an infinite interval, which play a very important role in the proof of completely continuity for the relevant integral operators.
In this section, we use two Banach spaces defined by
X={u∈C((0,+∞),(−∞,+∞))∣limx→0u(x)1+x−β(1+γ) and limt→+∞u(x)1+x−β(1+γ) exist}, |
with the norm
‖u‖X=supx>0∣u(x)1+x−β(1+γ)∣, |
and
Y={v∈C((0,+∞),(−∞,+∞))∣limx→0v(x)1+x−β(1+γ) and limx→+∞v(x)1+x−β(1+γ) exist}, |
with the norm
‖v‖Y=supx>0∣v(x)1+x−β(1+γ)∣. |
So, (X×Y,‖(u,v)‖X×Y) is a Banach space, with the norm ‖(u,v)‖X×Y=‖u‖X+‖v‖Y.
Lemma 3.1. If F,G are continuous, then (u,v)∈X×Y is a solution of system (1.1)⇔(u,v)∈X×Y is a solution of the following equations:
{u(x)=∫∞0Gσ1(x,s)sβ(γ+1)−1F(s,u(s),v(s))ds,v(x)=∫∞0Gσ2(x,s)sβ(γ+1)−1G(s,u(s),v(s))ds. |
For (u,v)∈X×Y, we define an operator A:X×Y→X×Y as follows:
A(u,v)(x)=(A1(u,v)(x),A2(u,v)(x)), |
where
A1(u,v)(x)=∫∞0Gσ1(x,s)sβ(γ+1)−1F(s,u(s),v(s))ds, |
A2(u,v)(x)=∫∞0Gσ2(x,s)sβ(γ+1)−1G(s,u(s),v(s))ds, |
with Gσi(x,s),i=1,2, given by (2.2) and (2.3).
Remark 3.1. Let σ1,σ2,β,γ,λ,τ∈R, such that 1<σ1,σ2≤2,β>0,−2<γ<−1,λ>1,τ>0. If (H2) and (H4) hold, then for (u,v)∈X×Y with u(x),v(x)>0,
∫∞0sβ(γ+1)−1F(s,u(s),v(s))ds≤η∫∞τλ2sβ(γ+1)−1F(s,u(s),v(s))ds, |
∫∞0sβ(γ+1)−1G(s,u(s),v(s))ds≤η∫∞τλ2sβ(γ+1)−1G(s,u(s),v(s))ds, |
where η=max{η1,η2} with η1=1+ιϱ1(λ2−1),η2=1+ι∗ϱ2(λ2−1)>1, ϱ1,ϱ2,ι,ι∗>0.
Proof. By (H4), for x∈[τλ2,τ], we know that there exist two constants ϱ1,ϱ2>0, such that
xβ(γ+1)−1F(s,u,v)≥ϱ1,xβ(γ+1)−1G(s,u,v)≥ϱ2,u,v∈(0,+∞). |
So, for (u,v)∈X×Y with u(x),v(x)>0,
∫∞τλ2sβ(γ+1)−1F(s,u(s),v(s))ds≥∫ττλ2sβ(γ+1)−1F(s,u(s),v(s))ds≥τ(λ2−1)λ2ϱ1, |
∫∞τλ2sβ(γ+1)−1G(s,u(s),v(s))ds≥∫ττλ2sβ(γ+1)−1G(s,u(s),v(s))ds≥τ(λ2−1)λ2ϱ2, |
and hence,
λ2τ(λ2−1)ϱ1∫∞τλ2sβ(γ+1)−1F(s,u(s),v(s))ds≥1, |
λ2τ(λ2−1)ϱ2∫∞τλ2sβ(γ+1)−1G(s,u(s),v(s))ds≥1. |
By (H4), we know that there exist two constants ι,ι∗>0, such that
xβ(γ+1)−1F(x,u(x),v(x))≤ι,xβ(γ+1)−1G(x,u(x),v(x))≤ι∗,for ∀x∈[0,τλ2]. |
Thus,
∫τλ20sβ(γ+1)−1F(s,u(s),v(s))ds≤ιτλ2, |
∫τλ20sβ(γ+1)−1G(s,u(s),v(s))ds≤ι∗τλ2. |
Therefore, we can obtain
∫∞0sβ(γ+1)−1F(s,u(s),v(s))ds=∫τλ20sβ(γ+1)−1F(s,u(s),v(s))ds+∫∞τλ2sβ(γ+1)−1F(s,u(s),v(s))ds≤ιτλ2+∫∞τλ2sβ(γ+1)−1F(s,u(s),v(s))ds≤(1+ιϱ1(λ2−1))∫∞τλ2sβ(γ+1)−1F(s,u(s),v(s))ds=η1∫∞τλ2sβ(γ+1)−1F(s,u(s),v(s))ds. |
Similarly,
∫∞0sβ(γ+1)−1G(s,u(s),v(s))ds≤(1+ιϱ2(λ2−1))∫∞τλ2sβ(γ+1)−1G(s,u(s),v(s))ds=η2∫∞τλ2sβ(γ+1)−1G(s,u(s),v(s))ds. |
Take η=max{η1,η2}, and thus
∫∞0sβ(γ+1)−1F(s,u(s),v(s))ds≤η∫∞τλ2sβ(γ+1)−1F(s,u(s),v(s))ds, |
∫∞0sβ(γ+1)−1G(s,u(s),v(s))ds≤η∫∞τλ2sβ(γ+1)−1G(s,u(s),v(s))ds, |
hold.
Define two cones
K1={u∈X∣u(x)>0,x>0;minx∈[τλ,τ]u(x)1+x−β(1+γ)≥p(τ)η‖u‖X}, |
K2={v∈Y∣v(x)>0,x>0;minx∈[τλ,τ]v(x)1+x−β(1+γ)≥p(τ)η‖v‖Y}. |
Obviously, K1×K2={(u,v)∈X×Y∣u(x)>0,v(x)>0,∀x>0; minx∈[τλ,τ]u(x)1+x−β(1+γ)≥p(τ)η‖u‖X,minx∈[τλ,τ]v(x)1+x−β(1+γ)≥p(τ)η‖v‖Y} is also a cone. For convenience, we first list the following definitions:
F0=lim(u,v)→(0+,0+)supx>0F1(t,(1+x−β(1+γ))u,(1+x−β(1+γ))v)u+v, |
f∞=lim(u,v)→(+∞,+∞)infx>0F1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)u+v, |
f0=lim(u,v)→(0+,0+)infx>0F1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)u+v, |
F∞=lim(u,v)→(+∞,+∞)supx>0F1(t,(1+x−β(1+γ))u,(1+x−β(1+γ))v)u+v, |
G∗0=lim(u,v)→(0+,0+)supx>0G1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)u+v, |
g∗∞=lim(u,v)→(+∞,+∞)infx>0G1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)u+v, |
g∗0=lim(u,v)→(0+,0+)infx>0G1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)u+v, |
G∗∞=lim(u,v)→(+∞,+∞)supx>0G1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)u+v. |
Lemma 4.1. If assumptions (H1) and (H2) hold, then A:K1×K2→K1×K2 is completely continuous.
Proof. First, we show A:K1×K2→K1×K2. By (H1) and (H2), for (u,v)∈K1×K2,
‖A1(u,v)‖X=supt>0|A1(u,v)(x)|1+x−β(1+γ)=supx>0∣∫∞0Gσ1(x,s)1+x−β(1+γ)sβ(γ+1)−1F(s,u(s),v(s))ds∣≤βΓ(σ1)∫∞0∣sβ(γ+1)−1F(s,u(s),v(s))∣ds=βΓ(σ1)∫∞0∣sβ(γ+1)−1F(s,(1+s−β(1+γ))u(s)1+s−β(1+γ),(1+s−β(1+γ))v(s)1+s−β(1+γ))∣ds=βΓ(σ1)∫∞0∣F1(s,u(s)1+s−β(1+γ),v(s)1+s−β(1+γ))∣≤βΓ(σ1)[ω1(‖u‖X)∫∞0φ1(s)ds+ω2(‖v‖Y)∫∞0ψ1(s)ds]<+∞. |
Similarly,
‖A2(u,v)‖Y≤βΓ(σ1)[~ω1(‖u‖X)∫∞0φ2(s)ds+~ω2(‖v‖Y)∫∞0ψ2(s)ds]<+∞. |
By (H1) and Lemma 2.2, for (u,v)∈K1×K2, we have A1(u,v)(x)>0,A2(u,v)(x)>0,x>0. From Lemma 2.2 and Remark 3.1, for x∈[τλ,τ],τ>0, and λ>1,
|A1(u,v)(x)|1+x−β(1+γ)=∫∞0Gσ1(x,s)1+x−β(1+γ)sβ(γ+1)−1F(s,u(s),v(s))ds=∫τλ20Gσ1(x,s)1+x−β(1+γ)sβ(γ+1)−1F(s,u(s),v(s))ds+∫0τλ2Gσ1(x,s)1+x−β(1+γ)sβ(γ+1)−1F(s,u(s),v(s))ds≥∫0τλ2Gσ1(t,s)1+t−β(1+γ)sβ(γ+1)−1F(s,u(s),v(s))ds≥βp(τ)Γ(σ1)∫0τλ2sβ(γ+1)−1F(s,u(s),v(s))ds≥βp(τ)ηΓ(σ1)∫∞0sβ(γ+1)−1F(s,u(s),v(s))ds≥p(τ)η‖A1(u,v)‖X. |
So, A1(u,v)(x)1+x−β(1+γ)≥p(τ)η‖A1(u,v)‖X. Similarly, A2(u,v)(x)1+x−β(1+γ)≥p(τ)η‖A2(u,v)‖Y. Therefore,
minx∈[τλ,τ]A1(u,v)(x)1+x−β(1+γ)≥p(τ)η‖A1(u,v)‖X, |
minx∈[τλ,τ]A2(u,v)(x)1+x−β(1+γ)≥p(τ)η‖A2(u,v)‖Y. |
That is, A:K1×K2→K1×K2 is true.
Second, it will give a simply prove that A is continuous. Let D={(u,v)|(u,v)∈K1×K2,‖(u,v)‖X×Y≤K,K>0}, a bounded subset in K1×K2. Let (un,vn)∈D be a sequence that converges to (u,v) in ∈K1×K2. Then ‖(un,vn)‖X×Y≤K. From Lemma 2.2,
‖A1(un,vn)−A1(u,v)‖X=supx>0∣A1(un,vn)(x)−A1(u,v)(x)1+x−β(1+γ)∣≤βΓ(σ1)∣∫∞0sβ(γ+1)−1F(s,un(s),vn(s))ds−∫∞0sβ(γ+1)−1F(s,u(s),v(s))ds∣≤βΓ(σ1)∫∞0∣sβ(γ+1)−1(F(s,un(s),vn(s))−F(s,u(s),v(s)))∣ds. |
By (H2),
∣sβ(γ+1)−1F(s,un(s),vn(s))∣=∣sβ(γ+1)−1F(s,(1+s−β(1+γ))un(s)1+s−β(1+γ),(1+s−β(1+γ))vn(s)1+s−β(1+γ))∣=F1(s,un(s)1+s−β(1+γ),vn(s)1+s−β(1+γ))≤φ1(s)ω1(‖un‖X)+ψ1(s)ω2(‖vn‖Y)∈L1(0,∞). |
By the continuity of sβ(γ+1)−1F(s,u(s),v(s)) and the Lebesgue dominated convergence theorem,
∫∞0sβ(γ+1)−1F(s,un(s),vn(s))ds→∫∞0sβ(γ+1)−1F(s,u(s),v(s))ds,n→∞. |
Therefore, ‖A1(un,vn)−A1(u,v)‖X→0,n→∞. Similarly, ‖A2(un,vn)−A2(u,v)‖Y→0,n→∞.
So, ‖A(un,vn)−A(u,v)‖X×Y→0,n→∞. That is, A is continuous in D. In the end, we know that A(D) is relatively compact on (0,∞) and is equi-convergent at ∞ by [18]. Therefore, A:K1×K2→K1×K2 is completely continuous.
Theorem 4.1. Assume that (H2) and (H4) hold. If F0=0,G∗0=0,f∞=∞,g∗∞=∞, then the system (1.1) has at least one positive solution.
Proof. We divide the proof into several steps.
Step 1. A:K1×K2→K1×K2 is completely continuous. This result easily follows from Lemma 4.1.
Step 2. We show that there exist R1>0 and D1={(u,v)∈X×Y,‖(u,v)‖X×Y<R1} such that ‖A(u,v)‖X×Y≤‖(u,v)‖X×Y, (u,v)∈(K1×K2)∩∂D1.
Because F0=0,G∗0=0, we choose R1>0, such that
F1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)≤ϵ1(u+v), |
G1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)≤ϵ2(u+v), |
for 0<u+v≤R1,x>0, where ϵ1,ϵ2>0 satisfy
ϵ1≤12Γ(σ1)β∫∞0a1(s)ds,ϵ2≤12Γ(σ2)β∫∞0a2(s)ds. |
So, for (u,v)∈K1×K2 and ‖(u,v)‖X×Y=R1, by Lemma 2.2,
A1(u,v)(x)1+x−β(1+γ)=∫∞0Gσ1(x,s)1+x−β(1+γ)sβ(γ+1)−1F(s,u(s),v(s))ds≤βΓ(σ1)∫∞0sβ(γ+1)−1F(s,u(s),v(s))ds, |
A2(u,v)(x)1+x−β(1+γ)=∫∞0Gσ2(x,s)1+x−β(1+γ)sβ(γ+1)−1G(s,u(s),v(s))ds≤βΓ(σ2)∫∞0sβ(γ+1)−1G(s,u(s),v(s))ds. |
By (H4),
A1(u,v)(x)1+x−β(1+γ)≤βΓ(σ1)∫∞0a1(s)F1(s,u(s),v(s))ds=βΓ(σ1)∫∞0a1(s)F1(s,(1+s−β(1+γ))u(s)1+s−β(1+γ),(1+s−β(1+γ))v(s)1+s−β(1+γ))ds≤βΓ(σ1)∫∞0a1(s)ϵ1u(s)+v(s)1+s−β(1+γ)ds≤βΓ(σ1)ϵ1‖(u,v)‖X×Y∫∞0a1(s)ds≤12‖(u,v)‖X×Y. |
Similarly,
A2(u,v)(x)1+x−β(1+γ)≤βΓ(σ2)ϵ2‖(u,v)‖X×Y∫∞0a2(s)ds≤12‖(u,v)‖X×Y. |
Therefore,
‖A(u,v)‖X×Y≤‖(u,v)‖X×Y, for (u,v)∈K1×K2, and ‖(u,v)‖X×Y=R1. |
Let D1={(u,v)∈X×Y,‖(u,v)‖X×Y<R1}. Then,
‖A(u,v)‖X×Y≤‖(u,v)‖X×Y, for (u,v)∈(K1×K2)∩∂D1. |
Step 3. We show that there exist R2>0 and D2={(u,v)∈X×Y,‖(u,v)‖X×Y<R2} such that
‖A(u,v)‖X×Y≥‖(u,v)‖X×Y, for (u,v)∈(K1×K2)∩∂D2. |
Because f∞=∞,g∗∞=∞, there exists R>0, such that
F1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)≥m1(u+v), |
G1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)≥m2(u+v), |
for u+v≥R,x>0, where m1,m2>0 satisfy
m1≥12η1ηΓ(σ1)βp2(τ)∫τλτa1(s)ds,m2≥12η2ηΓ(σ2)βp2(τ)∫τλτa2(s)ds,η=max{η1,η2}. |
Let R2≥max{R1,ηRp(τ)}, and D2={(u,v)∈X×Y,‖(u,v)‖X×Y<R2}. Then, D1⊂D2.
Thus, for (u,v)∈K1×K2, ‖(u,v)‖X×Y=R2, we have
u(x)1+x−β(1+γ)≥minx∈[τλ,τ]u(x)1+x−β(1+γ)≥p(τ)η1‖u‖X, |
v(x)1+x−β(1+γ)≥minx∈[τλ,τ]v(x)1+x−β(1+γ)≥p(τ)η2‖v‖Y. |
So,
u(x)+v(x)1+x−β(1+γ)≥p(τ)η1‖u‖X+p(τ)η2‖v‖Y≥p(τ)η(‖u‖X+‖v‖Y)=p(τ)η‖(u,v)‖X×Y=p(τ)ηR2≥R. |
By (H4), for x∈[τλ,τ], we can obtain
A1(u,v)(x)1+x−β(1+γ)≥βp(τ)η1Γ(σ1)∫∞0sβ(γ+1)−1F(s,u(s),v(s))ds=βp(τ)η1Γ(σ1)∫∞0a1(s)F1(s,u(s),v(s))ds=βp(τ)η1Γ(σ1)∫∞0a1(s)F1(s,(1+s−β(1+γ))u(s)1+s−β(1+γ),(1+s−β(1+γ))v(s)1+s−β(1+γ))ds≥βp(τ)η1Γ(σ1)m1∫∞0a1(s)u(s)+v(s)1+s−β(1+γ)ds≥βp(τ)η1Γ(σ1)m1∫∞0a1(s)dsp(τ)η1‖u‖X+βp(τ)η1Γ(σ1)m1∫∞0a1(s)dsp(τ)η2‖v‖Y≥βp(τ)η1Γ(σ1)m1∫τλτa1(s)dsp(τ)η1‖u‖X+βp(τ)η1Γ(σ1)m1∫τλτa1(s)dsp(τ)η2‖v‖Y=βp2(τ)η1Γ(σ1)m1∫τλτa1(s)ds(1η1‖u‖X+1η2‖v‖Y)≥βp2(τ)η1Γ(σ1)m1∫τλτa1(s)ds1η‖(u,v)‖X×Y≥12‖(u,v)‖X×Y. |
Similarly, A2(u,v)(x)1+x−β(1+γ)≥12‖(u,v)‖X×Y. Therefore,
‖A(u,v)‖X×Y≥‖(u,v)‖X×Y, for (u,v)∈(K1×K2)∩∂D2. |
Finally, by Lemma 2.4, A has a fixed point in (K1×K1)∩∂(¯D2∖D1). So, (1.1) has at least one positive solution.
Theorem 4.2. Assume that (H2) and (H4) hold. If f0=∞,g∗0=∞,F∞=0,G∗∞=0, then (1.1) has at least one positive solution.
Proof. We divide the proof into several steps.
Step 1. A:K1×K2→K1×K2 is completely continuous. This result easily follows from Lemma 4.1.
Step 2. We show that there exist r1>0 and D1={(u,v)∈X×Y,‖(u,v)‖X×Y<r1} such that
‖A(u,v)‖X×Y≥‖(u,v)‖X×Y, for (u,v)∈(K1×K2)∩∂D1. |
Because f0=∞,g∗0=∞, there exists r1>0 such that
F1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)≥M1(u+v), |
G1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)≥M2(u+v), |
for 0<u+v≤r1,x>0, where M1,M2>0, satisfy
M1≥12η1ηΓ(σ1)βp2(τ)∫τλτa1(s)ds,M2≥12η2ηΓ(σ2)βp2(τ)∫τλτa2(s)ds,η=max{η1,η2}. |
Let D1={(u,v)∈X×Y,‖(u,v)‖X×Y<r1}. So, for (u,v)∈K1×K2 with ‖(u,v)‖X×Y=r1, and x∈[τλ,τ], then by (H4),
A1(u,v)(x)1+x−β(1+γ)≥βp(τ)η1Γ(σ1)∫∞0sβ(γ+1)−1F(s,u(s),v(s))ds=βp(τ)η1Γ(σ1)∫∞0a1(s)F1(s,u(s),v(s))ds=βp(τ)η1Γ(σ1)∫∞0a1(s)F1(s,(1+s−β(1+γ))u(s)1+s−β(1+γ),(1+s−β(1+γ))v(s)1+s−β(1+γ))ds≥βp(τ)η1Γ(σ1)M1∫∞0a1(s)u(s)+v(s)1+s−β(1+γ)ds≥βp(τ)η1Γ(σ1)M1∫∞0a1(s)dsp(τ)η1‖u‖X+βp(τ)η1Γ(σ1)M1∫∞0a1(s)dsp(τ)η2‖v‖Y≥βp(τ)η1Γ(σ1)M1∫τλτa1(s)dsp(τ)η1‖u‖X+βp(τ)η1Γ(σ1)M1∫τλτa1(s)dsp(τ)η2‖v‖Y=βp2(τ)η1Γ(σ1)M1∫τλτa1(s)ds(1η1‖u‖X+1η2‖v‖Y)≥βp2(τ)η1Γ(σ1)M1∫τλτa1(s)ds1η‖(u,v)‖X×Y≥12‖(u,v)‖X×Y. |
Similarly, A2(u,v)(x)1+x−β(1+γ)≥12‖(u,v)‖X×Y. Thus,
‖A(u,v)‖X×Y≥‖(u,v)‖X×Y, for (u,v)∈(K1×K2)∩∂D1. |
Step 3. We show that there exist r2>0 and D2={(u,v)∈X×Y,‖(u,v)‖X×Y<r2} such that
‖A(u,v)‖X×Y≤‖(u,v)‖X×Y for (u,v)∈(K1×K2)∩∂D2. |
Because F∞=0,G∗∞=0, there exists r>0, such that
F1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)≤ϵ1(u+v), |
G1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)≤ϵ2(u+v), |
for u+v>r,x>0, where ϵ1,ϵ2>0 satisfy
ϵ1≤12Γ(σ1)β∫∞0a1(s)ds,ϵ2≤12Γ(σ2)β∫∞0a2(s)ds. |
Let D2={(u,v)∈X×Y,‖(u,v)‖X×Y<r2}, where r2>max{r1,r}. Then D1⊂D1. We define two functions U1,U2 as follows:
U1:(−∞,+∞)→(−∞,+∞),U1(a)=sup0<u+v≤asupx>0F1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v), |
U2:(−∞,+∞)→(−∞,+∞),U2(a)=sup0<u+v≤asupx>0G1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v). |
For (u,v)∈K1×K2 and ‖(u,v)‖X×Y=r2,
U1(r2)=sup0<u+v≤r2supx>0F1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)≤ϵ1sup0<u+v≤r2(u+v)=ϵ1r2=ϵ1‖(u,v)‖X×Y, |
U2(r2)=sup0<u+v≤r2supx>0G1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)≤ϵ2sup0<u+v≤r2(u+v)=ϵ2r2=ϵ2‖(u,v)‖X×Y. |
By Lemma 2.2 and (H4),
A1(u,v)(x)1+x−β(1+γ)≤βΓ(σ1)∫∞0sβ(γ+1)−1F(s,u(s),v(s))ds=βΓ(σ1)∫∞0a1(s)F1(s,u(s),v(s))ds=βΓ(σ1)∫∞0a1(s)F1(s,(1+s−β(1+γ))u(s)1+s−β(1+γ),(1+s−β(1+γ))v(s)1+s−β(1+γ))ds≤βΓ(σ1)∫∞0a1(s)sup0<u+v≤r2supx>0F1(x,(1+x−β(1+γ))u,(1+x−β(1+γ))v)ds=βΓ(σ1)∫∞0a1(s)U1(r2)ds≤βΓ(σ1)∫∞0a1(s)dsϵ1‖(u,v)‖X×Y≤12‖(u,v)‖X×Y. |
Similarly, A2(u,v)(x)1+x−β(1+γ)≤12‖(u,v)‖X×Y. Therefore, ‖A(u,v)‖X×Y≤‖(u,v)‖X×Y, for (u,v)∈(K1×K2)∩∂D2. Finally, by Lemma 2.4, A has a fixed point in (K1×K1)∩∂(¯D2∖D1). So, the system (1.1) has at least one positive solution.
In the section, we obtain the multiplicity of positive solution of (1.1) by using the monotone iterative technique.
Theorem 5.1. If (H1) and (H2) hold, then (1.1) has two positive solutions (u∗,v∗) and (w∗,z∗) satisfying 0≤‖(u∗,v∗)‖X×Y≤Υ and 0≤‖(w∗,z∗)‖X×Y≤Υ, where Υ is a positive preset constant. Moreover, limn→∞(un,vn)=(u∗,v∗) and limn→∞(wn,zn)=(w∗,z∗), where (un,vn) and (wn,zn) are given by
(un(x),vn(x))=(A1(un−1,vn−1)(x),A2(un−1,vn−1)(x)),n=1,2,…, | (5.1) |
with
(u0(x),v0(x))=(Υ1[1+x−β(γ+1)],Υ2[1+x−β(γ+1)]),Υ1,Υ2>0,Υ1+Υ2≤Υ, |
and
(wn(x),zn(x))=(A1(wn−1,zn−1)(x),A2(wn−1,zn−1)(x)),n=1,2,…, | (5.2) |
with (w0(x),z0(x))=(0,0). In addition,
(w0(x),z0(x))≤(w1(x),z1(x))≤⋯≤(wn(x),zn(x))≤⋯≤(w∗,z∗)≤(u∗,v∗)≤⋯≤(un(x),vn(x))≤⋯≤(u1(x),v1(x))≤(u0(x),v0(x)). | (5.3) |
Proof. First, from Lemma 4.1, A(K_{1}\times K_{2})\subset K_{1}\times K_{2} for (u, v)\in K_{1}\times K_{2} . Let
\Upsilon_{1} = \frac{\beta}{\Gamma(\sigma_{1})}[\omega_{1}(\Upsilon)\int_0^\infty \varphi_{1}(s)ds+\omega_{2}(\Upsilon)\int_0^\infty \psi_{1}(s)ds] < \infty, |
\Upsilon_{2} = \frac{\beta}{\Gamma(\sigma_{2})}[\widetilde{\omega_{1}}(\Upsilon)\int_0^\infty \varphi_{2}(s)ds+\widetilde{\omega_{2}}(\Upsilon)\int_0^\infty \psi_{2}(s)ds] < \infty, |
and \Upsilon\geq\Upsilon_{1}+\Upsilon_{2} with D_{\Upsilon} = \{(u, v)\in K_{1}\times K_{2}:\|(u, v)\|_{X\times Y}\leq\Upsilon\} . For any (u, v)\in D_{\Upsilon} , from (H_{2}) and Lemma 2.2,
\begin{eqnarray*} \|A_{1}(u, v)\|_{X}& = &\sup\limits_{x > 0}\frac{|A_{1}(u, v)(x)|}{1+x^{-\beta(1+\gamma)}}\\ & = &\sup\limits_{x > 0}\mid\int_0^\infty \frac{G_{\sigma_{1}}(x, s)}{1+t^{-\beta(1+\gamma)}}s^{\beta(\gamma+1)-1}F(s, u(s), v(s))ds\mid \\ &\leq&\frac{\beta}{\Gamma(\sigma_{1})}\int_0^\infty \mid s^{\beta(\gamma+1)-1}F(s, u(s), v(s))ds\mid \\ &\leq&\frac{\beta}{\Gamma(\sigma_{1})}[\omega_{1}(\frac{\mid u(s)\mid}{1+s^{-\beta(1+\gamma)}})\int_0^\infty \varphi_{1}(s)ds+\omega_{2}(\frac{\mid v(s)\mid}{1+s^{-\beta(1+\gamma)}})\int_0^\infty \psi_{1}(s)ds]\\ &\leq&\frac{\beta}{\Gamma(\sigma_{1})}[\omega_{1}(\|u\|_{X})\int_0^\infty \varphi_{1}(s)ds+\omega_{2}(\|v\|_{Y})\int_0^\infty \psi_{1}(s)ds]\\ &\leq&\frac{\beta}{\Gamma(\sigma_{1})}[\omega_{1}(\Upsilon)\int_0^\infty \varphi_{1}(s)ds+\omega_{2}(\Upsilon)\int_0^\infty \psi_{1}(s)ds] = \Upsilon_{1}. \end{eqnarray*} |
Similarly, \|A_{2}(u, v)\|_{Y}\leq\Upsilon_{2} for (u, v)\in D_{\Upsilon} . Thus,
\|A(u, v)\|_{X\times Y} = \|A_{1}(u, v)\|_{X}+\|A_{2}(u, v)\|_{Y}\leq\Upsilon_{1}+\Upsilon_{2}\leq\Upsilon. |
That is, A(D_{\Upsilon})\subset D_{\Upsilon} . We construct two sequences as follows:
(u_{n}, v_{n}) = A(u_{n-1}, v_{n-1}), (w_{n}, z_{n}) = A(w_{n-1}, z_{n-1}), \ \ n = 1, 2, 3, \ldots. |
Obviously, (u_{0}(x), v_{0}(x)), (w_{0}(x), z_{0}(x))\in D_{\Upsilon} . Because A(D_{\Upsilon})\subset D_{\Upsilon} , (u_{n}, v_{n}), (w_{n}, z_{n})\in D_{\Upsilon}, n = 1, 2, \ldots . We need to show that there exist (u^{\ast}, v^{\ast}) and (w^{\ast}, z^{\ast}) satisfying \lim\limits_{n\rightarrow \infty}(u_{n}, v_{n}) = (u^{\ast}, v^{\ast}) and \lim\limits_{n\rightarrow \infty}(w_{n}, z_{n}) = (w^{\ast}, z^{\ast}) which are two monotone sequences for approximating positive solutions of the system (1.1).
For x\in(0, +\infty), (u_{n}, v_{n})\in D_{\Upsilon} , from Lemma 2.2 and (5.1),
\begin{eqnarray*} u_{1}(x)& = &A_{1}(u_{0}, v_{0})(x) = \int_0^\infty G_{\sigma_{1}}(x, s)s^{\beta(\gamma+1)-1}F(s, u_{0}(s), v_{0}(s))ds \\ &\leq&\frac{\beta}{\Gamma(\sigma_{1})}\int_0^\infty (1+t^{-\beta(1+\gamma)})s^{\beta(\gamma+1)-1}F(s, u_{0}(s), v_{0}(s))ds \\ &\leq&\frac{\beta}{\Gamma(\sigma_{1})}(1+x^{-\beta(1+\gamma)})[\omega_{1}(\frac{\mid u_{0}(s)\mid}{1+s^{-\beta(1+\gamma)}})\int_0^\infty \varphi_{1}(s)ds+\omega_{2}(\frac{\mid v_{0}(s)\mid}{1+s^{-\beta(1+\gamma)}})\int_0^\infty \psi_{1}(s)ds] \\ &\leq&\frac{\beta}{\Gamma(\sigma_{1})}(1+x^{-\beta(1+\gamma)})[\omega_{1}(\|u_{0}\|_{X})\int_0^\infty \varphi_{1}(s)ds+\omega_{2}(\|v_{0}\|_{Y})\int_0^\infty \psi_{1}(s)ds] \\ &\leq&\frac{\beta}{\Gamma(\sigma_{1})}(1+x^{-\beta(1+\gamma)})[\omega_{1}(\Upsilon)\int_0^\infty \varphi_{1}(s)ds+\omega_{2}(\Upsilon)\int_0^\infty \psi_{1}(s)ds] \\ & = &(1+x^{-\beta(1+\gamma)})\Upsilon_{1} = u_{0}(x) \end{eqnarray*} |
and
\begin{eqnarray*} v_{1}(x) = A_{2}(u_{0}, v_{0})(x)& = &\int_0^\infty G_{\sigma_{2}}(x, s)s^{\beta(\gamma+1)-1}G(s, u_{0}(s), v_{0}(s))ds \\ &\leq&\frac{\beta}{\Gamma(\sigma_{2})}\int_0^\infty (1+x^{-\beta(1+\gamma)})s^{\beta(\gamma+1)-1}G(s, u_{0}(s), v_{0}(s))ds \\ &\leq&\frac{\beta}{\Gamma(\sigma_{2})}(1+t^{-\beta(1+\gamma)})[\widetilde{\omega_{1}}(\frac{\mid u_{0}(s)\mid}{1+s^{-\beta(1+\gamma)}})\int_0^\infty \varphi_{2}(s)ds+\widetilde{\omega_{2}}(\frac{\mid v_{0}(s)\mid}{1+s^{-\beta(1+\gamma)}})\int_0^\infty \psi_{2}(s)ds] \\ &\leq&\frac{\beta}{\Gamma(\sigma_{2})}(1+t^{-\beta(1+\gamma)})[\widetilde{\omega_{1}}(\|u_{0}\|_{X})\int_0^\infty \varphi_{2}(s)ds+\widetilde{\omega_{2}}(\|v_{0}\|_{Y})\int_0^\infty \psi_{2}(s)ds] \\ &\leq&\frac{\beta}{\Gamma(\sigma_{2})}(1+x^{-\beta(1+\gamma)})[\widetilde{\omega_{1}}(\Upsilon)\int_0^\infty \varphi_{2}(s)ds+\widetilde{\omega_{2}}(\Upsilon)\int_0^\infty \psi_{2}(s)ds] \\ & = &(1+x^{-\beta(1+\gamma)})\Upsilon_{2} = v_{0}(x), \end{eqnarray*} |
that is,
(u_{1}(x), v_{1}(x)) = (A_{1}(u_{0}, v_{0})(x), A_{2}(u_{0}, v_{0})(x))\leq((1+x^{-\beta(1+\gamma)})\Upsilon_{1}, (1+x^{-\beta(1+\gamma)})\Upsilon_{2}) = (u_{0}(x), v_{0}(x)). |
So, by the condition (H_{1}) ,
(u_{2}(x), v_{2}(x)) = (A_{1}(u_{1}, v_{1})(x), A_{2}(u_{1}, v_{1})(x))\leq(A_{1}(u_{0}, v_{0})(x), A_{2}(u_{0}, v_{0})(x)) = (u_{1}(x), v_{1}(x)). |
For x\in(0, +\infty) , the sequences \{(u_{n}, v_{n})\}_{n = 0}^{\infty} satisfy (u_{n+1}(x), v_{n+1}(x))\leq(u_{n}(x), v_{n}(x)) . By the iterative sequences (u_{n+1}, v_{n+1}) = A(u_{n}, v_{n}) and the complete continuity of the operator A , (u_{n}, v_{n})\rightarrow (u^{\ast}, v^{\ast}) , and A(u^{\ast}, v^{\ast}) = (u^{\ast}, v^{\ast}) .
Similarly, for the sequences \{(w_{n}, z_{n})\}_{n = 0}^{\infty} , we have
\begin{eqnarray*} (w_{1}(x), z_{1}(x))& = &(A_{1}(w_{0}, z_{0})(x), A_{2}(w_{0}, z_{0})(x))\\ & = &(\int_0^\infty G_{\sigma_{1}}(x, s)s^{\beta(\gamma+1)-1}F(s, w_{0}(s), z_{0}(s))ds, \int_0^\infty G_{\sigma_{2}}(x, s)s^{\beta(\gamma+1)-1}G(s, w_{0}(s), z_{0}(s))ds)\\ &\geq&(0, 0) = (w_{0}(x), z_{0}(x)). \end{eqnarray*} |
Then, by the condition (H_{1}) ,
(w_{2}(x), z_{2}(x)) = (A_{1}(w_{1}, z_{1})(x), A_{2}(w_{1}, z_{1})(x))\geq(A_{1}(w_{0}, z_{0})(x), A_{2}(w_{0}, z_{0})(x)) = (w_{1}(x), z_{1}(x)). |
Analogously, for x\in(0, +\infty) , we have (w_{n+1}(x), z_{n+1}(x))\geq(w_{n}(x), z_{n}(x)) . By the iterative sequences (w_{n+1}, z_{n+1}) = A(w_{n}, z_{n}) and the complete continuity of the operator A , (w_{n}, z_{n})\rightarrow (w^{\ast}, z^{\ast}) , and A(w^{\ast}, z^{\ast}) = (w^{\ast}, z^{\ast}) .
Finally, we prove that (u^{\ast}, v^{\ast}) and (w^{\ast}, z^{\ast}) are the minimal and maximal positive solutions of (1.1). Assume that (\varsigma(x), \mu(x)) is any positive solution of (1.1). Then, A(\varsigma(x), \mu(x)) = (\varsigma(x), \mu(x)) , and
(w_{0}(x), z_{0}(x)) = (0, 0)\leq(\varsigma(x), \mu(x))\leq((1+x^{-\beta(1+\gamma)})\Upsilon_{1}, (1+x^{-\beta(1+\gamma)})\Upsilon_{2}) = (u_{0}(x), v_{0}(x)). |
Therefore,
(w_{1}(x), z_{1}(x)) = (A_{1}(w_{0}, z_{0})(x), A_{2}(w_{0}, z_{0})(x))\leq(\varsigma(x), \mu(x))\leq(A_{1}(u_{0}, v_{0})(x), A_{2} (u_{0}, v_{0})(x)) = (u_{1}(x), v_{1}(x)). |
That is, (w_{1}(x), z_{1}(x))\leq(\varsigma(x), \mu(x))\leq(u_{n}(x), v_{n}(x)) . So, (5.3) holds. By (H_{1}) , (0, 0) is not a solution of (1.1). From (5.1), (w^{\ast}, z^{\ast}) and (u^{\ast}, v^{\ast}) are two extreme positive solutions of (1.1), which can be constructed via limitS of two monotone iterative sequences in (5.1) and (5.2).
Example 6.1. We consider the following system:
\begin{align} \begin{cases} D_{1}^{-\frac{3}{2}, \frac{5}{3}}u(x)+x^{\frac{3}{2}}(\frac{u}{1+x^{\frac{1}{2}}})^{2}e^{-x}+x^{\frac{3}{2}} (\frac{v}{1+x^{\frac{1}{2}}})^{2}e^{-x} = 0, t\in (0, +\infty), \\ D_{1}^{-\frac{3}{2}, \frac{3}{2}}v(x)+x^{\frac{5}{2}}e^{-2x^{2}}(\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u} {1+x^{\frac{1}{2}}})^{2}) +x^{\frac{5}{2}}e^{-2x^{2}}(\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u}{1+x^{\frac{1}{2}}})^{2}), x\in (0, +\infty), \\ \lim\nolimits_{x\rightarrow 0} x^{\frac{1}{2}}I^{\frac{1}{6}, \frac{1}{3}}u(x) = 0, \lim\nolimits_{x\rightarrow \infty} x^{-\frac{1}{2}}I^{\frac{1}{6}, \frac{1}{3}}u(x) = 0, \\ \lim\nolimits_{x\rightarrow 0} x^{\frac{1}{2}}I^{0, \frac{1}{2}}v(x) = 0, \lim\nolimits_{x\rightarrow \infty} x^{-\frac{1}{2}}I^{0, \frac{1}{2}}v(x) = 0, \end{cases} \end{align} | (6.1) |
where \sigma_{1} = \frac{5}{3}, \sigma_{2} = \frac{3}{2}, \gamma = -\frac{3}{2}, \beta = 1 ,
F(x, u, v) = x^{\frac{3}{2}}e^{-x}[(\frac{u}{1+x^{\frac{1}{2}}})^{2}+(\frac{v}{1+x^{\frac{1}{2}}})^{2}], |
G(x, u, v) = x^{\frac{5}{2}}e^{-2x^{2}}[(\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u}{1+x^{\frac{1}{2}}})^{2}) +(\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u}{1+x^{\frac{1}{2}}})^{2})]. |
First, for F_{1}(x, u, v) = x^{\beta(1+\gamma)-1}F(x, (1+x^{-\beta(1+\gamma)})u, (1+x^{-\beta(1+\gamma)})v) = e^{-x}(u^{2}, v^{2}) , we choose \omega_{1}(u) = u^{2}\in C((0, +\infty), (0, +\infty)) , \omega_{2}(v) = v^{2}\in C((0, +\infty), (0, +\infty)) , and \varphi_{1}(x) = \psi_{1}(x) = e^{-x}\in L^{1}(0, +\infty) . Then,
\mid F_{1}(x, u, v)\mid\leq\varphi_{1}(x)\omega_{1}(\mid u\mid)+\psi_{1}(t)\omega_{2}(\mid v\mid), \ \ (0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty). |
Similarly, for F_{2}(x, u, v) = x^{\beta(1+\gamma)-1}G(x, (1+x^{-\beta(1+\gamma)})u, (1+x^{-\beta(1+\gamma)})v) = xe^{-2x^{2}}[u^{2}\ln(u^{2}+1)+ v^{2}\ln(v^{2}+1)] , we choose \widetilde{\omega_{1}}(u) = u^{2}\ln(u^{2}+1)\in C((0, +\infty), (0, +\infty)) , \widetilde{\omega_{2}}(v) = v^{2}\ln(v^{2}+1)\in C((0, +\infty), (0, +\infty)) , and \varphi_{2}(x) = \psi_{2}(x) = xe^{-2x^{2}}\in L^{1}(0, +\infty) . Then,
\mid F_{2}(x, u, v)\mid\leq\varphi_{2}(x)\widetilde{\omega_{1}}(\mid u\mid)+\psi_{2}(x)\widetilde{\omega_{2}}(\mid v\mid), \ \ (0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty). |
So, the condition (H_{2}) holds. Obviously, F, G:(0, +\infty)\times(0, +\infty)\times(0, +\infty)\rightarrow (0, +\infty) are continuous.
x^{-\frac{3}{2}}F(x, u, v) = e^{-x}[(\frac{u}{1+x^{\frac{1}{2}}})^{2}+(\frac{v}{1+x^{\frac{1}{2}}})^{2}] = a_{1}(x)F_{1}(x, u, v), |
x^{-\frac{3}{2}}G(x, u, v) = xe^{-2x^{2}}[(\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u}{1+x^{\frac{1}{2}}})^{2}) +(\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u}{1+x^{\frac{1}{2}}})^{2})] = a_{2}(x)G_{1}(x, u, v), |
where a_{1}(x) = e^{-x}, a_{2}(x) = xe^{-2x^{2}} , F_{1}(x, u, v) = (\frac{u}{1+x^{\frac{1}{2}}})^{2}+(\frac{v}{1+x^{\frac{1}{2}}})^{2} , G_{1}(t, u, v) = (\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u}{1+x^{\frac{1}{2}}})^{2}) +(\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u}{1+x^{\frac{1}{2}}})^{2}) . So, x^{-\frac{3}{2}}f(x, u, v), x^{-\frac{3}{2}}G(x, u, v):[0, +\infty)\times(0, +\infty)\times(0, +\infty)\rightarrow [0, +\infty) are continuous. Hence, the condition (H_{4}) holds. Finally,
F_{0} = \lim\limits_{(u, v)\rightarrow (0^{+}, 0^{+})} \frac{u^{2}+v^{2}}{u+v} = 0, G_{0}^{\ast} = \lim\limits_{(u, v)\rightarrow (0^{+}, 0^{+})} \frac{u^{2}\ln(u^{2}+1)+v^{2}\ln(v^{2}+1)}{u+v} = 0, |
f_{\infty} = \lim\limits_{(u, v)\rightarrow (+\infty, +\infty)} \frac{u^{2}+v^{2}}{u+v} = \infty, g_{\infty}^{\ast} = \lim\limits_{(u, v)\rightarrow (+\infty, +\infty)} \frac{u^{2}\ln(u^{2}+1)+v^{2}\ln(v^{2}+1)}{u+v} = \infty. |
Therefore, from Theorem 4.1, (6.1) has at least one positive solution (u(x), v(x)) . Further,
\begin{cases} u(x) = \frac 3{2\Gamma(\frac 23)}[x^{\frac 12}\int_0^\infty s^{-\frac 32}F(s, u(s), v(s))ds-x^{-\frac 83}\int_x^\infty (x-s)^{\frac 23}s^{-\frac 32}F(s, u(s), v(s))ds], \\ v(x) = \frac 2{\sqrt{\pi}}[x^{\frac 12}\int_0^\infty s^{-\frac 32}G(s, u(s), v(s))ds-\int_x^\infty (x-s)^{\frac 12}s^{-\frac 32}G(s, u(s), v(s))ds]. \end{cases} |
Example 6.2. We consider the following system:
\begin{align} \begin{cases} D_{1}^{-\frac{3}{2}, \frac{3}{2}}u(x)+x^{\frac{5}{2}}e^{-2x^{2}+1}[\arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\frac{1}{\pi}]+ x^{\frac{5}{2}}e^{-2x^{2}+1} [\arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\pi] = 0, x\in (0, +\infty), \\ D_{1}^{-\frac{3}{2}, \frac{7}{6}}v(x)+x^{\frac{3}{2}}e^{-x}[\arctan(\ln((\frac{u}{1+x^{\frac{1}{2}}})^{2}+1))+\frac{3}{2}\pi]+ x^{\frac{3}{2}}e^{-x} [\arctan(\ln((\frac{v}{1+x^{\frac{1}{2}}})^{2}+1))+1], x\in (0, +\infty), \\ \lim\nolimits_{x\rightarrow 0} x^{\frac{1}{2}}I^{0 \frac{1}{2}}u(x) = 0, \lim\nolimits_{x\rightarrow \infty} x^{-\frac{1}{2}}I^{0, \frac{1}{2}}u(x) = 0, \\ \lim\nolimits_{x\rightarrow 0} x^{\frac{1}{2}}I^{-\frac{1}{3}, \frac{5}{6}}v(x) = 0, \lim\nolimits_{x\rightarrow \infty} x^{-\frac{1}{2}}I^{-\frac{1}{3}, \frac{5}{6}}v(x) = 0, \end{cases} \end{align} | (6.2) |
where \sigma_{1} = \frac{3}{2}, \sigma_{2} = \frac{7}{6}, \gamma = -\frac{3}{2}, \beta = 1 ,
F(x, u, v) = x^{\frac{5}{2}}e^{-2x^{2}+1}[\arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\frac{1}{\pi}]+x^{\frac{5}{2}}e^{-2x^{2}+1} [\arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\frac{1}{\pi}], |
G(x, u, v) = x^{\frac{3}{2}}e^{-x}[\arctan(\ln((\frac{u}{1+x^{\frac{1}{2}}})^{2}+1))+\frac{3}{2}\pi]+x^{\frac{3}{2}}e^{-x} [\arctan(\ln((\frac{v}{1+x^{\frac{1}{2}}})^{2}+1))+1]. |
First, for
F_{1}(x, u, v) = x^{\beta(1+\gamma)-1}F(x, (1+x^{-\beta(1+\gamma)})u, (1+x^{-\beta(1+\gamma)})v) = xe^{-2x^{2}+1}[\arctan u^{2}+\frac{1}{\pi}+\arctan v^{2}+\pi], |
we choose \omega_{1}(u) = \arctan u^{2}+\frac{1}{\pi}\in C((0, +\infty), (0, +\infty)), \omega_{2}(v) = \arctan v^{2}+\pi\in C((0, +\infty), (0, +\infty)) , and \varphi_{1}(x) = \psi_{1}(x) = xe^{-2x^{2}+1}\in L^{1}(0, +\infty) . Then,
\mid F_{1}(x, u, v)\mid\leq\varphi_{1}(x)\omega_{1}(\mid u\mid)+\psi_{1}(x)\omega_{2}(\mid v\mid), \ (0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty). |
Similarly, for
F_{2}(x, u, v) = x^{\beta(1+\gamma)-1}g(x, (1+x^{-\beta(1+\gamma)})u, (1+x^{-\beta(1+\gamma)})v) = e^{-x}[\arctan(\ln(u^{2}+1))+ \frac{3}{2}\pi+\arctan(\ln(v^{2}+1))+1], |
we choose \widetilde{\omega_{1}}(u) = \arctan(\ln(u^{2}+1))+\frac{3}{2}\pi\in C((0, +\infty), (0, +\infty)) , \widetilde{\omega_{2}}(v) = \arctan(\ln(v^{2}+1))+1\in C((0, +\infty), (0, +\infty)) , and \varphi_{2}(x) = \psi_{2}(x) = e^{-x}\in L^{1}(0, +\infty) . Then,
\mid F_{2}(x, u, v)\mid\leq\varphi_{2}(x)\widetilde{\omega_{1}}(\mid u\mid)+\psi_{2}(x)\widetilde{\omega_{2}}(\mid v\mid), \ \ (0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty). |
That is, (H_{2}) holds. Second, F, G:(0, +\infty)\times(0, +\infty)\times(0, +\infty)\rightarrow (0, +\infty) are continuous. And
x^{-\frac{3}{2}}F(x, u, v) = xe^{-2x^{2}+1}[\arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\frac{1}{\pi}+\arctan(\frac{v} {1+x^{\frac{1}{2}}})^{2}+\pi] = a_{1}(x)F_{1}(x, u, v), |
x^{-\frac{3}{2}}G(x, u, v) = e^{-x}[\arctan(\ln((\frac{u}{1+x^{\frac{1}{2}}})^{2}+1))+\frac{3}{2}\pi+\arctan(\ln((\frac{v} {1+x^{\frac{1}{2}}})^{2}+1))+1] = a_{2}(x)G_{1}(x, u, v), |
where a_{1}(x) = xe^{-2x^{2}+1}, a_{2}(x) = e^{-x} , F_{1}(x, u, v) = \arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\frac{1}{\pi}+\arctan(\frac{v}{1+x^{\frac{1}{2}}})^{2}+\pi , G_{1}(x, u, v) = \arctan(\ln((\frac{u}{1+x^{\frac{1}{2}}})^{2}+1))+\frac{3}{2}\pi+\arctan(\ln((\frac{v}{1+x^{\frac{1}{2}}})^{2}+1))+1 . So, x^{-\frac{3}{2}}F(x, u, v), x^{-\frac{3}{2}}G(x, u, v):[0, +\infty)\times(0, +\infty)\times(0, +\infty)\rightarrow [0, +\infty) are continuous. That is, (H_{4}) holds. In addition,
f_{0} = \lim\limits_{(u, v)\rightarrow (0^{+}, 0^{+})} \frac{\arctan u^{2}+\frac{1}{\pi}+\arctan v^{2}+\pi}{u+v} = \infty, |
g_{0}^{\ast} = \lim\limits_{(u, v)\rightarrow (0^{+}, 0^{+})} \frac{\arctan(\ln(u^{2}+1))+\frac{3}{2}\pi+\arctan(\ln(v^{2}+1))+1}{u+v} = \infty, |
F_{\infty} = \lim\limits_{(u, v)\rightarrow (+\infty, +\infty)} \frac{\arctan u^{2}+\frac{1}{\pi}+\arctan v^{2}+\pi}{u+v} = 0, |
G_{\infty}^{\ast} = \lim\limits_{(u, v)\rightarrow (+\infty, +\infty)} \frac{\arctan(\ln(u^{2}+1))+\frac{3}{2}\pi+\arctan(\ln(v^{2}+1))+1}{u+v} = 0. |
Therefore, from Theorem 4.2, (6.2) has at least one positive solution (u(x), v(x)) . Further,
\begin{cases} u(x) = \frac 2{\sqrt{\pi}}[x^{\frac 12}\int_0^\infty s^{-\frac 32}F(s, u(s), v(s))ds-\int_x^\infty (x-s)^{\frac 12}s^{-\frac 32}F(s, u(s), v(s))ds], \\ v(x) = \frac 6{\Gamma(\frac 16)}[x^{\frac 12}\int_0^\infty s^{-\frac 32}G(s, u(s), v(s))ds-x^{\frac 13}\int_x^\infty (x-s)^{\frac 16}s^{-\frac 32}G(s, u(s), v(s))ds]. \end{cases} |
Example 6.3. We consider the following system:
\begin{align} \begin{cases} D_{1}^{-\frac{3}{2}, \frac{5}{3}}u(x)+x^{\frac{3}{2}}\frac{e^{-x}}{3}\mid\frac{u}{1+x^{\frac{1}{2}}}\mid+ x^{\frac{5}{2}}\ln(\mid\frac{v}{1+x^{\frac{1}{2}}}\mid+1)\frac{e^{-2x^{2}+1}}{10} = 0, x\in (0, +\infty), \\ D_{1}^{-\frac{3}{2}, \frac{3}{2}}v(x)+x^{\frac{5}{2}}e^{-2x^{2}+1}\arctan(\mid\frac{u}{1+x^{\frac{1}{2}}}\mid+ \frac{1}{\sqrt{\pi}})+x^{\frac{5}{2}}\frac{e^{-2x^{2}+1}}{5} \mid\frac{v}{1+x^{\frac{1}{2}}}\mid = 0, x\in (0, +\infty), \\ \lim\nolimits_{x\rightarrow 0} x^{\frac{1}{2}}I^{\frac{1}{6}, \frac{1}{3}}u(x) = 0, \lim\nolimits_{x\rightarrow \infty} x^{-\frac{1}{2}}I^{\frac{1}{6}, \frac{1}{3}}u(x) = 0, \\ \lim\nolimits_{x\rightarrow 0} x^{\frac{1}{2}}I^{0, \frac{1}{2}}v(x) = 0, \lim\nolimits_{x\rightarrow \infty} x^{-\frac{1}{2}}I^{0, \frac{1}{2}}v(x) = 0, \end{cases} \end{align} | (6.3) |
where \sigma_{1} = \frac{5}{3}, \sigma_{2} = \frac{3}{2}, \gamma = -\frac{3}{2}, \beta = 1 ,
F(x, u, v) = x^{\frac{3}{2}}\frac{e^{-x}}{3}\mid\frac{u}{1+x^{\frac{1}{2}}}\mid+x^{\frac{5}{2}}\ln(\mid\frac{v} {1+x^{\frac{1}{2}}}\mid+1)\frac{e^{-2x^{2}+1}}{10}, |
G(x, u, v) = x^{\frac{5}{2}}e^{-2x^{2}+1}\arctan(\mid\frac{u}{1+x^{\frac{1}{2}}}\mid+\frac{1}{\sqrt{\pi}})+ x^{\frac{5}{2}}\frac{e^{-2x^{2}+1}}{5} \mid\frac{v}{1+x^{\frac{1}{2}}}\mid. |
Obviously, F, G:(0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty)\rightarrow (0, +\infty) are continuous and nondecreasing with respect to the second and the third variables on (0, +\infty) . That is, (H_{1}) holds. Next,
F_{1}(x, u, v) = x^{\beta(1+\gamma)-1}F(x, (1+x^{-\beta(1+\gamma)})u, (1+x^{-\beta(1+\gamma)})v) = \frac{e^{-x}}{3}\mid u\mid+x\frac{e^{-2x^{2}+1}}{10}\ln(\mid v\mid+1). |
We choose \omega_{1}(u) = \mid u\mid\in C((0, +\infty), (0, +\infty)) , \omega_{2}(v) = \ln(\mid v\mid+1)\in C((0, +\infty), (0, +\infty)) , and \varphi_{1}(x) = \frac{e^{-x}}{3}, \psi_{1}(x) = \frac{xe^{-2x^{2}+1}}{10}\in L^{1}(0, +\infty) . Then,
\mid F_{1}(x, u, v)\mid\leq\varphi_{1}(x)\omega_{1}(\mid u\mid)+\psi_{1}(x)\omega_{2}(\mid v\mid), \ \ (0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty). |
Similarly, for
F_{2}(x, u, v) = x^{\beta(1+\gamma)-1}G(x, (1+x^{-\beta(1+\gamma)})u, (1+x^{-\beta(1+\gamma)})v) = xe^{-2x^{2}+1}\arctan(\mid u\mid+\frac{1}{\sqrt{\pi}})+x\frac{e^{-2x^{2}+1}}{5}\mid v\mid, |
we choose \widetilde{\omega_{1}}(u) = \arctan(\mid u\mid+\frac{1}{\sqrt{\pi}})\in C((0, +\infty), (0, +\infty)) , \widetilde{\omega_{2}}(v) = \mid v\mid\in C((0, +\infty), (0, +\infty)) , and \varphi_{2}(x) = xe^{-2x^{2}+1}, \psi_{2}(x) = x\frac{e^{-2x^{2}+1}}{5}\in L^{1}(0, +\infty) . Then,
\mid F_{2}(x, u, v)\mid\leq\varphi_{2}(x)\widetilde{\omega_{1}}(\mid u\mid)+\psi_{2}(x)\widetilde{\omega_{2}}(\mid v\mid), \ \ (0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty). |
That is, (H_{2}) holds. Therefore, from Theorem 5.1, (6.3) has two positive solutions (u^{\ast}, v^{\ast}) and (w^{\ast}, z^{\ast}) with (0, 0)\leq (u^{\ast}(x), v^{\ast}(x)), (w^{\ast}(x), z^{\ast}(x))\leq ((1+x^{\frac 12})\Upsilon_{1}, (1+x^{\frac 12})\Upsilon_{2}) , where \Upsilon_{1}+\Upsilon_{2}\leq \Upsilon , and \Upsilon satisfies
\frac{95.58}{191.86}\Upsilon-0.69\arctan (\Upsilon+0.56)\geq \frac 1{36}. |
This paper studies the Erdélyi-Kober fractional coupled system (1.1), where the variable is in an infinite interval. We give some proper conditions and set a special Banach space. We obtain the existence of at least one positive solution for (1.1) by using the Guo-Krasnosel'skii fixed point theorem, and we get the existence of at least two positive solutions for (1.1) by using the monotone iterative technique. Our methods and results are different from ones in [18]. Moreover, we give three examples to show the plausibility of our main results. For future work, we intend to use other fixed point theorems to solve some Erdélyi-Kober fractional differential equations.
The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.
This paper is supported by the Fundamental Research Program of Shanxi Province (202303021221068).
The authors declare that they have no competing interests.
[1] |
J. Li, X. Zou, Modeling spatial spread of infectious diseases with a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048–2079. http://dx.doi.org/10.1007/s11538-009-9457-z doi: 10.1007/s11538-009-9457-z
![]() |
[2] |
C. Siettos, L. Russo, Mathematical modeling of infectious disease dynamics, Virulence, 4 (2013), 295–306. http://dx.doi.org/10.4161/viru.24041 doi: 10.4161/viru.24041
![]() |
[3] |
S. Jenness, S. Goodreau, M. Morris, Epimodel: an R package for mathematical modeling of infectious disease over networks, J. Stat. Softw., 84 (2018), 1–47. http://dx.doi.org/10.18637/jss.v084.i08 doi: 10.18637/jss.v084.i08
![]() |
[4] |
A. Mahdy, N. Sweilam, M. Higazy, Approximate solution for solving nonlinear fractional order smoking model, Alex. Eng. J., 59 (2020), 739–752. http://dx.doi.org/10.1016/j.aej.2020.01.049 doi: 10.1016/j.aej.2020.01.049
![]() |
[5] |
A. Shaikh, I. Shaikh, K. Nisar, A mathematical model of COVID-19 using fractional derivative: outbreak in India with dynamics of transmission and control, Adv. Differ. Equ., 2020 (2020), 373. http://dx.doi.org/10.1186/s13662-020-02834-3 doi: 10.1186/s13662-020-02834-3
![]() |
[6] | J. De Abajo, Simple mathematics on COVID-19 expansion, MedRxiv, in press. http://dx.doi.org/10.1101/2020.03.17.20037663 |
[7] |
K. Gepreel, M. Mohamed, H. Alotaibi, A. Mahdy, Dynamical behaviors of nonlinear Coronavirus (COVID-19) model with numerical studies, CMC-Comput. Mater. Con., 67 (2021), 675–686. http://dx.doi.org/10.32604/cmc.2021.012200 doi: 10.32604/cmc.2021.012200
![]() |
[8] |
D. Xenikos, A. Asimakopoulos, Power-law growth of the COVID-19 fatality incidents in Europe, Infect. Dis. Model., 6 (2021), 743–750. http://dx.doi.org/10.1016/j.idm.2021.05.001 doi: 10.1016/j.idm.2021.05.001
![]() |
[9] |
W. Zhu, S. Shen, An improved SIR model describing the epidemic dynamics of the COVID-19 in China, Results Phys., 25 (2021), 104289. http://dx.doi.org/10.1016/j.rinp.2021.104289 doi: 10.1016/j.rinp.2021.104289
![]() |
[10] |
K. Sarkar, S. Khajanchi, J. Nieto, Modeling and forecasting the COVID-19 pandemic in India, Chaos Soliton. Fract., 139 (2020), 110049. http://dx.doi.org/10.1016/j.chaos.2020.110049 doi: 10.1016/j.chaos.2020.110049
![]() |
[11] |
K. Ghosh, A. Ghosh, Study of COVID-19 epidemiological evolution in India with a multi-wave SIR model, Nonlinear Dyn., 109 (2022), 47–55. http://dx.doi.org/10.1007/s11071-022-07471-x doi: 10.1007/s11071-022-07471-x
![]() |
[12] |
S. Margenov, N. Popivanov, I. Ugrinova, T. Hristov, Mathematical modeling and short-term forecasting of the COVID-19 epidemic in Bulgaria: SEIRS model with vaccination, Mathematics, 10 (2022), 2570. http://dx.doi.org/10.3390/math10152570 doi: 10.3390/math10152570
![]() |
[13] |
G. Martelloni, G. Martelloni, Modelling the downhill of the Sars-Cov-2 in Italy and a universal forecast of the epidemic in the world, Chaos Soliton. Fract., 139 (2020), 110064. http://dx.doi.org/10.1016/j.chaos.2020.110064 doi: 10.1016/j.chaos.2020.110064
![]() |
[14] |
P. Naik, M. Yavuz, S. Qureshi, J. Zu, S. Townley, Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan, Eur. Phys. J. Plus, 135 (2020), 795. http://dx.doi.org/10.1140/epjp/s13360-020-00819-5 doi: 10.1140/epjp/s13360-020-00819-5
![]() |
[15] |
J. Zhou, S. Salahshour, A. Ahmadian, N. Senu, Modeling the dynamics of COVID-19 using fractal-fractional operator with a case study, Results Phys., 33 (2022), 105103. http://dx.doi.org/10.1016/j.rinp.2021.105103 doi: 10.1016/j.rinp.2021.105103
![]() |
[16] |
M. Ala'raj, M. Majdalawieh, N. Nizamuddin, Modeling and forecasting of COVID-19 using a hybrid dynamic model based on SEIRD with ARIMA corrections, Infect. Dis. Model., 6 (2021), 98–111. http://dx.doi.org/10.1016/j.idm.2020.11.007 doi: 10.1016/j.idm.2020.11.007
![]() |
[17] |
A. Comunian, R. Gaburro, M. Giudici, Inversion of a SIR-based model: a critical analysis about the application to COVID-19 epidemic, Physica D, 413 (2020), 132674. http://dx.doi.org/10.1016/j.physd.2020.132674 doi: 10.1016/j.physd.2020.132674
![]() |
[18] |
N. Kudryashov, M. Chmykhov, M. Vigdorowitsch, Analytical features of the SIR model and their applications to COVID-19, Appl. Math. Model., 90 (2021), 466–473. http://dx.doi.org/10.1016/j.apm.2020.08.057 doi: 10.1016/j.apm.2020.08.057
![]() |
[19] |
P. Naik, J. Zu, M. Naik, Stability analysis of a fractional-order cancer model with chaotic dynamics, Int. J. Biomath., 14 (2021), 2150046. http://dx.doi.org/10.1142/S1793524521500467 doi: 10.1142/S1793524521500467
![]() |
[20] | P. Naik, M. Ghoreishi, J. Zu, Approximate solution of a nonlinear fractional-order HIV model using homotopy analysis method, Int. J. Numer. Anal. Mod., 19 (2022), 52–84. |
[21] |
A. Ahmad, M. Farman, P. Naik, N, Zafar, A. Akgul, M. Saleem, Modeling and numerical investigation of fractional-order bovine babesiosis disease, Numer. Meth. Part. D. E., 37 (2021), 1946–1964. http://dx.doi.org/10.1002/num.22632 doi: 10.1002/num.22632
![]() |
[22] |
M. Ghori, P. Naik, J. Zu, Z. Eskandari, M. Naik, Global dynamics and bifurcation analysis of a fractional-order SEIR epidemic model with saturation incidence rate, Math. Method. Appl. Sci., 45 (2022), 3665–3688. http://dx.doi.org/10.1002/mma.8010 doi: 10.1002/mma.8010
![]() |
[23] |
K. Hattaf, M. El Karimi, A. Mohsen, Z. Hajhouji, M. El Younoussi, N. Yousfi, Mathematical modeling and analysis of the dynamics of RNA viruses in presence of immunity and treatment: a case study of SARS-CoV-2, Vaccines, 11 (2023), 201. http://dx.doi.org/10.3390/vaccines11020201 doi: 10.3390/vaccines11020201
![]() |
[24] |
A. Ebaid, Analysis of projectile motion in view of the fractional calculus, Appl. Math. Model., 35 (2011), 1231–1239. http://dx.doi.org/10.1016/j.apm.2010.08.010 doi: 10.1016/j.apm.2010.08.010
![]() |
[25] |
A. Ebaid, E. El-Zahar, A. Aljohani, B. Salah, M. Krid, J. Tenreiro Machado, Analysis of the two-dimensional fractional projectile motion in view of the experimental data, Nonlinear Dyn., 97 (2019), 1711–1720. http://dx.doi.org/10.1007/s11071-019-05099-y doi: 10.1007/s11071-019-05099-y
![]() |
[26] |
A. Ebaid, C. Cattani, A. Al Juhani1, E. El-Zahar, A novel exact solution for the fractional Ambartsumian equation, Adv. Differ. Equ., 2021 (2021), 88. http://dx.doi.org/10.1186/s13662-021-03235-w doi: 10.1186/s13662-021-03235-w
![]() |
[27] |
A. Aljohani, A. Ebaid, E. Algehyne, Y. Mahrous, C. Cattani, H. Al-Jeaid, The Mittag-Leffler function for re-evaluating the chlorine transport model: comparative analysis, Fractal Fract., 6 (2022), 125. http://dx.doi.org/10.3390/fractalfract6030125 doi: 10.3390/fractalfract6030125
![]() |
[28] |
K. Hattaf, On the stability and numerical scheme of fractional differential equations with application to biology, Computation, 10 (2022), 97. http://dx.doi.org/10.3390/computation10060097 doi: 10.3390/computation10060097
![]() |
[29] |
K. Hattaf, A new generalized definition of fractional derivative with non-singular kernel, Computation, 8 (2020), 49. http://dx.doi.org/10.3390/computation8020049 doi: 10.3390/computation8020049
![]() |
[30] |
O. Arqub, M. Osman, C. Park, J. Lee, H. Alsulami, M. Alhodaly, Development of the reproducing kernel Hilbert space algorithm for numerical pointwise solution of the time-fractional nonlocal reaction-diffusion equation, Alex. Eng. J., 61 (2022), 10539–10550. http://dx.doi.org/10.1016/j.aej.2022.04.008 doi: 10.1016/j.aej.2022.04.008
![]() |
[31] |
O. Arqub, S. Tayebi, D. Baleanu, M. Osman, W. Mahmoud, H. Alsulami, A numerical combined algorithm in cubic B-spline method and finite difference technique for the time-fractional nonlinear diffusion wave equation with reaction and damping terms, Results Phys., 41 (2022), 105912. http://dx.doi.org/10.1016/j.rinp.2022.105912 doi: 10.1016/j.rinp.2022.105912
![]() |
[32] |
S. Rashid, K. Kubra, S. Sultana, P. Agarwal, M. Osman, An approximate analytical view of physical and biological models in the setting of Caputo operator via Elzaki transform decomposition method, J. Comput. Appl. Math., 413 (2022), 114378. http://dx.doi.org/10.1016/j.cam.2022.114378 doi: 10.1016/j.cam.2022.114378
![]() |
[33] | K. Owolabi, R. Agarwal, E. Pindza, S. Bernstein, M. Osman, Complex Turing patterns in chaotic dynamics of autocatalytic reactions with the Caputo fractional derivative, Neural Comput. Applic., in press. http://dx.doi.org/10.1007/s00521-023-08298-2 |
[34] |
C. Park, R. Nuruddeen, K. Ali, L. Muhammad, M. Osman, D. Baleanu, Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg-de Vries equations, Adv. Differ. Equ., 2020 (2020), 627. http://dx.doi.org/10.1186/s13662-020-03087-w doi: 10.1186/s13662-020-03087-w
![]() |
[35] |
A. Ebaid, B. Masaedeh, E. El-Zahar, A new fractional model for the falling body problem, Chinese Phys. Lett., 34 (2017), 020201. http://dx.doi.org/10.1088/0256-307X/34/2/020201 doi: 10.1088/0256-307X/34/2/020201
![]() |
[36] |
S. Khaled, E. El-Zahar, A. Ebaid, Solution of Ambartsumian delay differential equation with conformable derivative, Mathematics, 7 (2019), 425. http://dx.doi.org/10.3390/math7050425 doi: 10.3390/math7050425
![]() |
[37] |
F. Alharbi, D. Baleanu, A. Ebaid, Physical properties of the projectile motion using the conformable derivative, Chinese J. Phys., 58 (2019), 18–28. http://dx.doi.org/10.1016/j.cjph.2018.12.010 doi: 10.1016/j.cjph.2018.12.010
![]() |
[38] |
E. Algehyne, E. El-Zahar, F. Alharbi, A. Ebaid, Development of analytical solution for a generalized Ambartsumian equation, AIMS Mathematics, 5 (2020), 249–258. http://dx.doi.org/10.3934/math.2020016 doi: 10.3934/math.2020016
![]() |
[39] | G. Adomian, Solving Frontier problems of physics: the decomposition method, Dordrecht: Springer, 1994. http://dx.doi.org/10.1007/978-94-015-8289-6 |
[40] |
A. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput., 166 (2005), 652–663. http://dx.doi.org/10.1016/j.amc.2004.06.059 doi: 10.1016/j.amc.2004.06.059
![]() |
[41] |
H. Bakodah, A. Ebaid, Exact solution of Ambartsumian delay differential equation and comparison with Daftardar-Gejji and Jafari approximate method, Mathematics, 6 (2018), 331. http://dx.doi.org/10.3390/math6120331 doi: 10.3390/math6120331
![]() |
[42] |
J. Duan, R. Rach, A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations, Appl. Math. Comput., 218 (2011), 4090–4118. http://dx.doi.org/10.1016/j.amc.2011.09.037 doi: 10.1016/j.amc.2011.09.037
![]() |
[43] |
J. Diblík, M. Kúdelcíková, Two classes of positive solutions of first order functional differential equations of delayed type, Nonlinear Anal., 75 (2012), 4807–4820. http://dx.doi.org/10.1016/j.na.2012.03.030 doi: 10.1016/j.na.2012.03.030
![]() |
[44] |
S. Bhalekar, J. Patade, An analytical solution of fishers equation using decomposition method, American Journal of Computational and Applied Mathematics, 6 (2016), 123–127. http://dx.doi.org/10.5923/j.ajcam.20160603.01 doi: 10.5923/j.ajcam.20160603.01
![]() |
[45] |
A. Alenazy, A. Ebaid, E. Algehyne, H. Al-Jeaid, Advanced study on the delay differential equation y'(t) = ay(t)+by(ct), Mathematics, 10 (2022), 4302. http://dx.doi.org/10.3390/math10224302 doi: 10.3390/math10224302
![]() |
[46] |
K. Abbaoui, Y. Cherruault, Convergence of Adomian's method applied to nonlinear equations, Math. Comput. Model., 20 (1994), 69–73. http://dx.doi.org/10.1016/0895-7177(94)00163-4 doi: 10.1016/0895-7177(94)00163-4
![]() |
[47] |
Y. Cherruault, G. Adomian, Decomposition methods: a new proof of convergence, Math. Comput. Model., 18 (1993), 103–106. http://dx.doi.org/10.1016/0895-7177(93)90233-O doi: 10.1016/0895-7177(93)90233-O
![]() |