
Several mathematical models of two competing viruses (or viral strains) that have been published in the literature assume that the infection rate is determined by bilinear incidence. These models do not show co-existence equilibrium; moreover, they might not be applicable in situations where the virus concentration is high. In this paper, we developed a mathematical model for the co-dynamics of two competing viruses with saturated incidence. The model included the latently infected cells and three types of time delays: discrete (or distributed): (ⅰ) The formation time of latently infected cells; (ⅱ) The activation time of latently infected cells; (ⅲ) The maturation time of newly released virions. We established the mathematical well-posedness and biological acceptability of the model by examining the boundedness and nonnegativity of the solutions. Four equilibrium points were identified, and their stability was examined. Through the application of Lyapunov's approach and LaSalle's invariance principle, we demonstrated the global stability of equilibria. The impact of saturation incidence, latently infected cells, and time delay on the viral co-dynamics was examined. We demonstrated that the saturation could result in persistent viral coinfections. We established conditions under which these types of viruses could coexist. The coexistence conditions were formulated in terms of saturation constants. These findings offered new perspectives on the circumstances under which coexisting viruses (or strains) could live in stable viral populations. It was shown that adding the class of latently infected cells and time delay to the coinfection model reduced the basic reproduction number for each virus type. Therefore, fewer treatment efficacies would be needed to keep the system at the infection-free equilibrium and remove the viral coinfection from the body when utilizing a model with latently infected cells and time delay. To demonstrate the associated mathematical outcomes, numerical simulations were conducted for the model with discrete delays.
Citation: Ahmed M. Elaiw, Ghadeer S. Alsaadi, Aatef D. Hobiny. Global co-dynamics of viral infections with saturated incidence[J]. AIMS Mathematics, 2024, 9(6): 13770-13818. doi: 10.3934/math.2024671
[1] | Sunisa Theswan, Sotiris K. Ntouyas, Jessada Tariboon . Coupled systems of ψ-Hilfer generalized proportional fractional nonlocal mixed boundary value problems. AIMS Mathematics, 2023, 8(9): 22009-22036. doi: 10.3934/math.20231122 |
[2] | Saima Rashid, Fahd Jarad, Khadijah M. Abualnaja . On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative. AIMS Mathematics, 2021, 6(10): 10920-10946. doi: 10.3934/math.2021635 |
[3] | Sotiris K. Ntouyas, Bashir Ahmad, Jessada Tariboon . Nonlocal integro-multistrip-multipoint boundary value problems for ¯ψ∗-Hilfer proportional fractional differential equations and inclusions. AIMS Mathematics, 2023, 8(6): 14086-14110. doi: 10.3934/math.2023720 |
[4] | Weerawat Sudsutad, Wicharn Lewkeeratiyutkul, Chatthai Thaiprayoon, Jutarat Kongson . Existence and stability results for impulsive (k,ψ)-Hilfer fractional double integro-differential equation with mixed nonlocal conditions. AIMS Mathematics, 2023, 8(9): 20437-20476. doi: 10.3934/math.20231042 |
[5] | Donny Passary, Sotiris K. Ntouyas, Jessada Tariboon . Hilfer fractional quantum system with Riemann-Liouville fractional derivatives and integrals in boundary conditions. AIMS Mathematics, 2024, 9(1): 218-239. doi: 10.3934/math.2024013 |
[6] | Ayub Samadi, Sotiris K. Ntouyas, Jessada Tariboon . Coupled systems of nonlinear sequential proportional Hilfer-type fractional differential equations with multi-point boundary conditions. AIMS Mathematics, 2024, 9(5): 12982-13005. doi: 10.3934/math.2024633 |
[7] | Adel Lachouri, Mohammed S. Abdo, Abdelouaheb Ardjouni, Bahaaeldin Abdalla, Thabet Abdeljawad . On a class of differential inclusions in the frame of generalized Hilfer fractional derivative. AIMS Mathematics, 2022, 7(3): 3477-3493. doi: 10.3934/math.2022193 |
[8] | Sunisa Theswan, Ayub Samadi, Sotiris K. Ntouyas, Jessada Tariboon . Hilfer iterated-integro-differential equations and boundary conditions. AIMS Mathematics, 2022, 7(8): 13945-13962. doi: 10.3934/math.2022770 |
[9] | Ugyen Samdrup Tshering, Ekkarath Thailert, Sotiris K. Ntouyas . Existence and stability results for a coupled system of Hilfer-Hadamard sequential fractional differential equations with multi-point fractional integral boundary conditions. AIMS Mathematics, 2024, 9(9): 25849-25878. doi: 10.3934/math.20241263 |
[10] | Abdelkrim Salim, Sabri T. M. Thabet, Imed Kedim, Miguel Vivas-Cortez . On the nonlocal hybrid (k,φ)-Hilfer inverse problem with delay and anticipation. AIMS Mathematics, 2024, 9(8): 22859-22882. doi: 10.3934/math.20241112 |
Several mathematical models of two competing viruses (or viral strains) that have been published in the literature assume that the infection rate is determined by bilinear incidence. These models do not show co-existence equilibrium; moreover, they might not be applicable in situations where the virus concentration is high. In this paper, we developed a mathematical model for the co-dynamics of two competing viruses with saturated incidence. The model included the latently infected cells and three types of time delays: discrete (or distributed): (ⅰ) The formation time of latently infected cells; (ⅱ) The activation time of latently infected cells; (ⅲ) The maturation time of newly released virions. We established the mathematical well-posedness and biological acceptability of the model by examining the boundedness and nonnegativity of the solutions. Four equilibrium points were identified, and their stability was examined. Through the application of Lyapunov's approach and LaSalle's invariance principle, we demonstrated the global stability of equilibria. The impact of saturation incidence, latently infected cells, and time delay on the viral co-dynamics was examined. We demonstrated that the saturation could result in persistent viral coinfections. We established conditions under which these types of viruses could coexist. The coexistence conditions were formulated in terms of saturation constants. These findings offered new perspectives on the circumstances under which coexisting viruses (or strains) could live in stable viral populations. It was shown that adding the class of latently infected cells and time delay to the coinfection model reduced the basic reproduction number for each virus type. Therefore, fewer treatment efficacies would be needed to keep the system at the infection-free equilibrium and remove the viral coinfection from the body when utilizing a model with latently infected cells and time delay. To demonstrate the associated mathematical outcomes, numerical simulations were conducted for the model with discrete delays.
Fractional calculus has attempted to be accessed as a promising technique in fluid mechanics [1], nano-material [2], thermal energy [3], epidemics [4] and other scientific disciplines over recent decades. For example, by provoking interest in both cutting-edge and conventional pure and applied analytical techniques, it has reinforced creative collaboration between different disciplines, existence, and relevant applications in real-world manifestations, see [5,6,7]. In 1965, the possibility of fractional calculus depended on the conversation between L'hospital and Leibnitz as letters in [8]. After that, many researchers started experimenting in this field, and a large portion of them concentrated on describing novel fractional formulations [9,10,11]. Various classifications were raised in this process and were expressed by the advancement of research.
Recently, extensive investigation has been proposed for the qualitative characterization of verification for various fractional differential equations (FDEs) with initial and boundary value problems. Several significant approaches regarding the existence, uniqueness, multiplicity and stability have been reported by proposing certain fixed point theorems. Although many of the important problems have been tackled by the classical fractional derivatives (RL and Caputo) [12,13,14], it has a few limitations when used to design physical issues as a result of the necessary assumptions are themselves fractional and may be unsuitable for physical problems. The Caputo derivative has the opportunity of being appropriate for physical problems because it only necessitates classical initial conditions [15,16].
Khalil et al. [17] invented an interesting definition of fractional derivative, which is known to be a conformable derivative. In actuality, this so-called derivative is not a fractional derivative, however, it is it is essentially a first derivative duplicated by an extra straightforward factor. Consequently, this novel concept appears to be a regular extension of the classical derivative. More characterizations and the extended form of this derivative have been expounded in [18]. Then authors [19] explored an extension of the conformable derivative by considering proportional derivative. This fact leads to the modified conformable (proportional) differential operator of order λ. The researchers investigated numerous integral inequalities using classical [20], conformable and generalized conformable fractional integrals [21,22]. Qurashi et al. [23] proposed new fractional derivatives and integrals that have nonsingular kernels. By using the generalized proportional Hadamard fractional integral operator, Zhou et al. [24] investigated some general inequalities and their variant forms.
This recently characterized local derivative approaches to the original function as λ↦0. In this manner, they had the option to improve the conformable derivative. Jarad et al. [25] presented another kind of fractional operators created from the extended conformable derivatives. The exponential function appeared as a kernel in their examination with outstanding performance [26,27,28]. For the interest of readers, we draw in their thoughtfulness regarding some new papers [29,30].
In parallel with the concentrated exploration of the fractional derivative, the existence-uniqueness of verification fits to the intense prominent qualitative characterizations of FDEs, see [31,32,33,34].
Inspired by the work, we utilize a novel fractional derivative which is known as Hilfer-GPF derivative for finding the existence-uniqueness of solutions for a new class of nonlinear FDEs having non-local boundary conditions. For this we consider the subsequent BVP for a class of Hilfer-FDEs:
{Dλ,ζ;ϑϖ+1y(℘)=G(℘,y(℘)),λ∈(0,1),ζ∈[0,1],℘∈(ϖ1,ϖ2],I1−δ,;ϑϖ+1[m1y(ϖ+1)+m2y(ϖ−2)]=eȷ,δ=λ+ζ(1−λ),eȷ∈R, | (1.1) |
where G:(ϖ1,ϖ2]×R↦R be a continuous mapping, Dλ,ζ;ϑϖ+1(.) is the Hilfer-GPF derivative of order λ∈(0,1) and I1−δ;ϑϖ+1(.) is the GPF integral of order 1−δ>0. We find the existence consequences by the fixed point techniques of Schauder, Schaefer and Kransnoselskii. Additionally, the investigation of nonlinear FDEs as far as their information sources (fractional orders, related boundaries, and suitable function) has fascinated the interest of mathematicians because of its importance in Orlicz space (see [35]). Rely upon this, the subject of coherence of verification of the Hilfer-FDEs regarding inputs is significant and worth assuming.
The organization of the paper is as follows. In Section 2, we proceed with some basics concepts and detailed consequences as a review literature. In Section 3, we establish an equivalence criterion of integral equation of BVP (1.1) and then proposed the existence consequences for GPF-derivative by well-noted fixed point theorem. Also, in Section 4, illustrative examples are presented to check the applicability of the findings developed in Section 3. The conclusion with some open problems is presented in Section 5.
In what follows, we demonstrate some preliminaries, initial results and spaces which are essential for proving further consequences. Throughout this investigation, let Lp(ϖ1,ϖ2),p≥1, is the space of Lebesgue integrable mappings on (ϖ1,ϖ2).
Assume that ϖ1,ϖ2∈(−∞,+∞) be a finite and infinite intervals on R.
Furthermore, we elaborate the subsequent weighted spaces with induced norms defined by (see[8]). Suppose that C[ϖ1,ϖ2] is said to be the space of continuous functions defined on [ϖ1,ϖ2] and the norm is defined as follows:
‖G‖C[ϖ1,ϖ2]=max℘∈[ϖ1,ϖ2]|G|, |
and ACn[ϖ1,ϖ2] represents the space of n-times absolutely continuous differentiable mappings defined as follows:
ACn[ϖ1,ϖ2]={G:(ϖ1,ϖ2]↦R:Gn−1∈AC[ϖ1,ϖ2]}, |
Cδ[ϖ1,ϖ2] denotes the weighted space of G on (ϖ1,ϖ2] is defined as
Cδ[ϖ1,ϖ2]={G:(ϖ1,ϖ2]↦R:(℘−ϖ1)δG(℘)∈C[ϖ1,ϖ2]},δ∈[0,1) |
with the norm
‖G‖Cδ[ϖ1,ϖ2]=‖(℘−ϖ1)δG(℘)‖C[ϖ1,ϖ2]=max℘∈[ϖ1,ϖ2]|(℘−ϖ1)δG|. |
Also, the weighted space of a function G on (ϖ1,ϖ2] is denoted by Cnδ(ϖ1,ϖ2] defined as
Cnδ[ϖ1,ϖ2]={G:(ϖ1,ϖ2]↦R:G(℘)∈Cn−1[ϖ1,ϖ2];Gn(℘)∈Cδ[ϖ1,ϖ2]},δ∈[0,1) |
with the norm
‖G‖Cnδ[ϖ1,ϖ2]=n−1∑κ=0‖Gκ‖C[ϖ1,ϖ2]+‖Gn‖Cδ[ϖ1,ϖ2],∀n∈N. |
For n=0, Cnδ[ϖ1,ϖ2] coincides with Cδ[ϖ1,ϖ2].
Definition 2.1. ([8]) Assume that G∈L1([ϖ1,ϖ2],R), then the RL fractional integral operator of G of order λ>0 is stated as
Jλϖ+1G(℘)=1Γ(λ)℘∫ϖ1(℘−ℓ)λ−1G(ℓ)dℓ,℘>ϖ1, | (2.1) |
where Γ(.) represents the classical Gamma function.
Definition 2.2. ([8]) Assume that G∈C([ϖ1,ϖ2]), then the RL fractional derivative operator of G of order λ>0 is stated as
Dλϖ+1G(℘)=1Γ(n−λ)dnd℘n℘∫ϖ1(℘−ℓ)n−λ−1G(ℓ)dℓ,℘>ϖ1,n−1<λ<n,n∈N, | (2.2) |
where Γ(.) represents the Gamma function.
Definition 2.3. ([8]) Assume that G∈Cn([ϖ1,ϖ2]), then the Caputo fractional derivative operator of G of order λ>0 is stated as
cDλϖ+1G(℘)=1Γ(n−λ)℘∫ϖ1(℘−ℓ)n−λ−1Gn(ℓ)dℓ,℘>ϖ1,n−1<λ<n,n∈N, | (2.3) |
where Γ(.) represents the Gamma function.
Definition 2.4. ([25]) For ϑ∈(0,1],λ∈C,ℜ(λ)>0, then the left-sided generalized proportional integral of G of order λ>0 is stated as
Jλ;ϑϖ+1G(℘)=1ϑλΓ(λ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)λ−1G(ℓ)dℓ,℘>ϖ1. | (2.4) |
Definition 2.5. ([25]) For ϑ∈(0,1],λ∈C,ℜ(λ)>0, then the left-sided generalized proportional derivative of G of order λ>0 is stated as
Dλ;ϑϖ+1G(℘)=Dn,ϑϑn−λΓ(n−λ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)n−λ−1G(ℓ)dℓ,℘>ϖ1, | (2.5) |
where n=[λ]+1.
Definition 2.6. ([25]) For ϑ∈(0,1],λ∈C,ℜ(λ)>0, then the left-sided generalized proportional integral of G of order λ>0 is stated as
cDλ;ϑϖ+1G(℘)=1ϑn−λΓ(n−λ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)n−λ−1(Dn,ϑG)(ℓ)dℓ,℘>ϖ1, | (2.6) |
where n=[λ]+1.
Remark 2.1. Specifically, if ϑ=1 Definitions 2.4–2.6 reduces to Definitions 2.1–2.3, respectively.
Definition 2.7. ([25]) For n∈N,λ∈(n−1,n),ϑ∈(0,1],ζ∈[0,1], then the left/right-sided Hilfer-GPF derivative having order λ, type ζ of G is stated as follows:
(Dϖ+1G)(y)=Iζ(n−λ),ϑϖ+1[Dϑ(I(1−ζ)(n−λ),ϑϖ+1G)](y), | (2.7) |
where DϑG(y)=(1−ϑ)G(y)+ϑG′(y) and I assumed to be GPF-integral stated in 2.4.
Specifically, if n=1, then Definition 2.7 reduces to
(Dϖ+1G)(y)=Iζ(1−λ),ϑϖ+1[Dϑ(I(1−ζ)(1−λ),ϑϖ+1G)](y). | (2.8) |
In the present investigation, we discuss the case where n=1,λ∈(0,1),ζ∈[0,1] and δ=λ+ζ−λζ.
Remark 2.2. It is remarkable to mention that:
(a)The Hilfer fractional derivative can be considered as an interpolator between the GPF-derivative and Caputo GPF-derivative, respectively, as
Dλ,ζ;ϑϖ+1G={DϑI(1−λ);ϑϖ+1G,ζ=0(seeDefinition2.5),I1−λ;ϑϖ+1DϑG,ζ=1(seeDefinition2.6), | (2.9) |
(b) The following assumptions holds true:
0<δ≤1δ≥λ,δ>ζ,1−δ<1−ζ(1−λ). |
(c) Particularly, if λ∈(0,1),ζ∈[0,1] and δ=λ+ζ−λζ, then
(Dλ,ζ;ϑϖ+1G)(℘)=(Iζ(1−λ)ϖ+1[Dϑ(I(1−ζ)(1−λ)ϖ+1G)])(y), |
therefore, we have
(Dλ,ζ;ϑϖ+1G)(℘)=(Iζ(1−λ)ϖ+1(Dδ;ϑϖ+1G))(℘), |
where (Dδ;ϑϖ+1g1)(℘)=dd℘(I(1−ζ)(1−λ);ϑϖ+1G)(℘).
Now we define the weighted spaces of continuous mappings on (ϖ1,ϖ2]:
Cλ,ζ;ϑ1−δ[ϖ1,ϖ2]={G∈C1−δ[ϖ1,ϖ2],Dλ,ζ;ϑϖ+1G∈C1−δ[ϖ1,ϖ2]},δ=λ+ζ(1−λ) | (2.10) |
and
Cδ1−δ[ϖ1,ϖ2]={G∈C1−δ[ϖ1,ϖ2],Dδ;ϑϖ+1G∈C1−δ[ϖ1,ϖ2]}. | (2.11) |
Since Dλ,ζ;ϑϖ+1=Iζ(1−λ),ϑϖ+1Dδ;ϑϖ+1, therefore, we have Cδ1−δ[ϖ1,ϖ2]⊂Cλ,ζ1−δ[ϖ1,ϖ2].
Theorem 2.1. ([25]) For ℘≥ϖ1,ϑ∈(0,1],ℜ(λ),ℜ(ζ)>0. If G∈C([ϖ1,ϖ2],R), then
Iλ;ϑϖ+1(Iζ;ϑϖ+1G)(℘)=Iζ;ϑϖ+1(Iλ;ϑϖ+1G)(℘)=(Iλ+ζϖ+1G)(℘). |
Theorem 2.2. ([25]) For ℘≥ϖ1,ϑ∈(0,1] and ℜ(λ)>0 and let G∈L1([ϖ1,ϖ2]), then
Dλ;ϑϖ+1Iλ;ϑϖ+1G(℘)=G(℘),n=[ℜ(λ)]+1. |
Lemma 2.1. ([25]) For λ,ς∈C such that ℜ(λ)≥0 and ℜ(ς)>0. Then for any ϑ∈(0,1] we have
(a)(Iλ;ϑϖ+1eϑ−1ϑℓ(ℓ−ϖ1)ς−1)(℘)=Γ(ς)ϑλΓ(ς+λ)eϑ−1ϑ℘(℘−ϖ1)ς+λ−1, |
(b)(Dλ;ϑϖ+1eϑ−1ϑℓ(ℓ−ϖ1)ς−1)(℘)=ϑλΓ(ς)Γ(ς−λ)eϑ−1ϑ℘(℘−ϖ1)ς−λ−1. |
Lemma 2.2. ([30]) For λ∈(0,1),ϑ∈(0,1],ζ∈(0,1) and δ=λ+ζ−λζ. If G∈Cδ1−δ[ϖ1,ϖ2], then
Iδ;ϑϖ+1Dδ;ϑϖ+1G=Iδ;ϑϖ+1Dλ,ζ;ϑϖ+1G |
and
Dδ;ϑϖ+1Iλ;ϑϖ+1G=Dζ(1−λ);ϑϖ+1G. |
Lemma 2.3. ([30])For y∈(ϖ1,ϖ2],λ∈(0,1),ϑ∈(0,1),ζ∈[0,1] and δ∈(0,1). If G∈C1−δ[ϖ1,ϖ2] and I1−δ,ϑϖ+1G, then
Iλ;ϑϖ+1Dλ,ζ;ϑϖ+1G(y)=G(y)−eϑ−1ϑ(y−ϖ1)(y−ϖ1)δ−1ϑδ−1Γ(δ)(I1−δ;ϑϖ1)(ϖ+1). |
Lemma 2.4. ([30]) For δ∈[0,1),ϑ∈(0,1] and g1∈Cδ. If G∈Cδ[ϖ1,ϖ2], then
Iλ;ϑϖ+1G(ϖ1)=limy↦ϖ+1Iλ;ϑϖ+1G(y)=0,δ∈[0,λ). |
Lemma 2.5. ([30]) For λ∈(0,1),ζ∈[0,1] and δ=λ+ζ−λζ and let G:(ϖ1,ϖ2]×R↦R be a mapping such that G∈C1−δ[ϖ1,ϖ2] for any y∈Cδ1−δ[ϖ1,ϖ2], then y satisfies problem (1.1) if and only if y satisfies the Volterra integral equation
y(℘)=(℘−ϖ1)δ−1eϑ−1ϑ(℘−ϖ1)I1−δ;ϑϖ+1y(ϖ+1)ϑδ−1Γ(δ)+1ϑλΓ(λ)℘∫ϖ1eϑ−1ϑ(℘−s1)(℘−ℓ)λ−1G(ℓ,y(ℓ))dℓ. | (2.12) |
This section consists of the existence of solution to BVP (1.1) in Cλ,ζ;ϑ1−δ[ϖ1,ϖ2].
Lemma 3.1. For λ∈(0,1),ζ∈[0,1], where δ=λ+ζ−λζ and suppose there be a function G:(ϖ1,ϖ2]×R→R such that G∈C1−δ[ϖ1,ϖ2] for any y∈C1−δ[ϖ1,ϖ2]. If y∈Cδ1−δ[ϖ1,ϖ2], then y fulfills BVP (1.1) if and only if y holds the following identity
y(℘)=eȷm1+m2eϑ−1ϑ(℘−ϖ1)(℘−ϖ1)δ−1ϑδΓ(δ)−m2m1+m2eϑ−1ϑ(℘−ϖ1)(℘−ϖ1)δ−1ϑδΓ(δ)×1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)λ−δG(ℓ,y(ℓ))dℓ+1ϑλΓ(λ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)λ−1G(ℓ,y(ℓ))dℓ. | (3.1) |
Proof. By means of Lemma 2.5 and utilizing the solution of (1.1) can be expressed as
y(℘)=J1−δ;ϑϖ+1y(ϖ+1)ϑδ−1Γ(δ)eϑ−1ϑ(℘−ϖ1)(℘−ϖ1)δ−1+1ϑδΓ(δ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)λ−1G(ℓ,y(ℓ))dℓ,℘>ϖ1. | (3.2) |
Employing J1−δ;ϑϖ+1 on (3.2) and applying the limit ℘→ϖ−12, we find
J1−δ;ϑϖ+1y(ϖ−2)=J1−δ;ϑϖ+1y(ϖ+1)+1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)λ−δG(ℓ,y(ℓ))dℓ. | (3.3) |
Again, employing J1−δ;ϑϖ+1 on (3.3), we have
J1−δ;ϑϖ+1y(ϖ−2)=J1−δ;ϑϖ+1y(ϖ+1)+1ϑ1−δ+λΓ(1−δ+λ)℘∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(℘−ℓ)λ−δG(ℓ,y(ℓ))dℓ=J1−δ;ϑϖ+1y(ϖ+1)+J1−ζ(1−λ);ϑϖ+1G(℘,y(℘)). | (3.4) |
Applying limit ℘↦ϖ+1 and utilizing Lemma 2.4 having 1−δ<1−ζ(1−λ), yields
J1−δ;ϑϖ+1y(ϖ+1)=J1−δ;ϑϖ+1y(ϖ+1), | (3.5) |
thus
J1−δ;ϑϖ+1y(ϖ−2)=J1−δ;ϑϖ+1y(ϖ+1)+1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)λ−δG(ℓ,y(ℓ))dℓ. | (3.6) |
From boundary condition (1.1), we have
J1−δ;ϑϖ+1y(ϖ−2)=eȷm2−m1m2J1−δ;ϑϖ+1y(ϖ+1). | (3.7) |
From (3.5) and (3.6), and utilizing (3.4), we have
J1−δ;ϑϖ+1y(ϖ+1)=m2m1+m2(eȷm2−1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)λ−δG(ℓ,y(ℓ))dℓ). | (3.8) |
Setting (3.2) in (3.8), one can find
y(℘)=eȷm1+m2eϑ−1ϑ(℘−ϖ1)(℘−ϖ1)δ−1ϑδ−1Γ(δ)−m2m1+m2eϑ−1ϑ(℘−ϖ1)(℘−ϖ1)δ−1ϑδ−1Γ(δ)×1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)λ−δG(ℓ,y(ℓ))dℓ+1ϑλΓ(λ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)λ−1G(ℓ,y(ℓ))dℓ. | (3.9) |
Conversely, employing J1−δ;ϑϖ+1 on (3.1), utilizing Lemmas 2.1 and 2.2, and simple computations yields
J1−δ;ϑϖ+1m1y(ϖ+1)+J1−δ;ϑϖ+1m2y(ϖ−2)=m1m2m1+m2(eȷm2−1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)λ−δG(ℓ,y(ℓ))dℓ)+m22m1+m2(eȷm2−1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)λ−δG(ℓ,y(ℓ))dℓ)+m2ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)λ−δG(ℓ,y(ℓ))dℓ=eȷ, | (3.10) |
This shows that y(℘) satisfies boundary condition (1.1).
Furthermore, employing Dδ;ϑϖ+1 on (3.1) and applying Lemmas 2.1 and 2.2, we have
Dδ;ϑϖ+1y(℘)=Dζ(1−λ);ϑϖ+1G(℘,y(℘)). | (3.11) |
Since y∈Cδ;ϑ1−δ[ϖ1,ϖ2] and in view of definition of Cδ;ϑ1−δ[ϖ1,ϖ2], we have Dδϖ+1y∈Cϑn−δ[ϖ1,ϖ2], thus, Dζ(1−λ);ϑϖ+1G=DI1−ζ(1−λ)ϖ+1G∈C1−δ[ϖ1,ϖ2].
For G∈C1−δ[ϖ1,ϖ2], it is noting that I1−ζ(1−λ)ϖ+1G∈C1−δ[ϖ1,ϖ2]. So, G and I1−ζ(1−λ);ϑϖ+1G holds the assumptions of Lemma 2.3. Now, employing Iζ(1−λ);ϑϖ+1 on (3.11), we have
Iζ(1−λ);ϑϖ+1Dδ;ϑϖ+1y(℘)=Iζ(1−λ);ϑϖ+1Dζ(1−λ);ϑϖ+1G(℘,y(℘)). | (3.12) |
Considering (2.8), (3.11) and Lemma 2.3, we have
Iδ;ϑϖ+1Dδ;ϑϖ+1y(℘)=G(℘,y(℘))−I1−ζ(1−λ);ϑϖ+1G(ϖ1,y(ϖ1))ϑζ(1−λ)Γ(ζ(1−λ))eϑ−1ϑ(℘−ϖ1)(℘−ϖ1)ζ(1−λ)−1,∀℘∈(ϖ1,ϖ2]. | (3.13) |
By Lemma 2.4, we have I1−ζ(n−λ)ϖ+1G(ϖ1,y(ϖ1))=0. Thus, we have Dλ,ζ;ϑϖ+1y(℘)=G(℘,y(℘)). Hence, this completes the proof.
Let us evoke some essential assumptions which are required to prove the existence of solutions for the problem mentioned.
(A1) Let a function G:(ϖ1,ϖ2]×R↦R with G(.,y(.))∈Cζ(1−λ);ϑ1−δ[ϖ1,ϖ2]. For any y∈Cϑ1−δ[ϖ1,ϖ2] and there exist two constants M1,m such that
|G(℘1,y)|≤M1(1+m‖y‖Cϑ1−δ). | (3.14) |
(A2) The inequality
G:=mM1Γ(δ)ϑλΓ(λ+1)[(ϖ2−ϖ1)λ+(ϖ2−ϖ1)λ+1−δ]<1 | (3.15) |
holds.
Now we are in a position to show the existence results for the BVP (1.1) by employing Schauder's fixed point theorem (see [36]).
Theorem 3.1. Suppose that the assumptions (A1) and (A2) fulfills. Then by Hilfer-BVP (1.1) has at least one solution in Cδ;ϑ1−δ[ϖ1,ϖ2]⊂Cλ,ζ;ϑ1−δ[ϖ1,ϖ2].
Proof. Defining an operator T:C1−δ[ϖ1,ϖ2]↦C1−δ[ϖ1,ϖ2] by
(Ty)(℘)=eȷm1+m2eϑ−1ϑ(℘−ϖ1)(℘−ϖ1)δ−1ϑδ−1Γ(δ)−m2m1+m2eϑ−1ϑ(℘−ϖ1)(℘−ϖ1)δ−1ϑδ−1Γ(δ)×1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)λ−δG(ℓ,y(ℓ))dℓ+1ϑλΓ(λ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)λ−1G(ℓ,y(ℓ))dℓ. | (3.16) |
Assume that Bϱ={y∈C1−δ[ϖ1,ϖ2]:‖y‖C1−δ≤ϱ} having ϱ≥Ω1−G, for G<1, we have
Ω:=eȷ(m1+m2)ϑδ−1Γ(δ)+|m2m1+m2|1ϑδ−1Γ(δ)×M1ϑ1−δ+λ[(ϖ2−ϖ1)λ+1−δΓ(λ−δ+2)+(ϖ2−ϖ1)2λ−δ+1Γ(λ+1)]. | (3.17) |
The proof will be demonstrated by the accompanying three steps:
Case 1. We will prove that T(Bϱ)⊂Bϱ. Utilizing assumption (A2), we have
|(Ty)(℘)(℘−ϖ1)1−δ|≤|eȷ(m1+m2)ϑδ−1Γ(δ)|+|m2m1+m21ϑδ−1Γ(δ)|×1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1|eϑ−1ϑ(ϖ2−ℓ)|(ϖ2−ℓ)λ−δM1(1+m|y|)dℓ+(℘−ϖ1)1−δϑλΓ(λ)℘∫ϖ1|eϑ−1ϑ(℘−ℓ)|(℘−ℓ)λ−1M1(1+m|y|)dℓ≤eȷ(m1+m2)ϑδ−1Γ(δ)+|m2m1+m2|1ϑδ−1Γ(δ)×1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1|eϑ−1ϑ(ϖ2−ℓ)|(ϖ2−ℓ)λ−δM1(1+m‖y‖C1−δ)dℓ+(℘−ϖ1)1−δϑλΓ(λ)℘∫ϖ1|eϑ−1ϑ(℘−ℓ)|(℘−ℓ)λ−1M1(1+m‖y‖C1−δ)dℓ. | (3.18) |
Since |eϑ−1ϑ℘|<1. Observe that, for any y∈Bϱ, and for every ℘∈(ϖ1,ϖ2], we have
1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1|eϑ−1ϑ(ϖ2−ℓ)|(ϖ2−ℓ)λ−δM1(1+m‖y‖C1−δ)dℓ≤M1(ϖ2−ϖ1)λϑ1−δ+λ[(ϖ2−ϖ1)1−δΓ(λ−δ+2)+mϱΓ(δ)Γ(λ+1)],since|eϑ−1ϑ℘|<1 | (3.19) |
and
|(℘−ϖ1)1−δ|ϑλΓ(λ)℘∫ϖ1|eϑ−1ϑ(℘−ℓ)|(℘−ℓ)λ−1M1(1+m‖y‖C1−δ)dℓ≤M1(℘−ϖ1)λ−δ+1ϑλ[(℘−ϖ1)λΓ(λ+1)+mϱΓ(δ)Γ(λ+1)]. | (3.20) |
Therefore, we get
|(Ty)(℘)(℘−ϖ1)1−δ|≤eȷ(m1+m2)ϑδ−1Γ(δ)+|m2m1+m2|1ϑδ−1Γ(δ)×M1(ϖ2−ϖ1)λϑ1−δ+λ[(ϖ2−ϖ1)1−δΓ(λ−δ+2)+mϱΓ(δ)Γ(λ+1)]+M1(℘−ϖ1)λ−δ+1ϑλ[(℘−ϖ1)λΓ(λ+1)+mϱΓ(δ)Γ(λ+1)], | (3.21) |
which leads to
‖Ty‖C1−δ≤eȷ(m1+m2)ϑδ−1Γ(δ)+|m2m1+m2|1ϑδ−1Γ(δ)×M1ϑ1−δ+λ[(ϖ2−ϖ1)λ+1−δΓ(λ−δ+2)+(ϖ2−ϖ1)2λ−δ+1Γ(λ+1)]+M1mϱΓ(δ)ϑλΓ(λ+1)[(℘−ϖ1)λ+(ϖ2−ϖ1)λ+1−δ]. | (3.22) |
of assumption (A2), we conclude that ‖Ty‖C1−δ≤Gϱ+(1−G)ϱ=ϱ, Therefore, T(Bϱ)⊂Bϱ.
Next we will prove that T is completely continuous.
Case 2. We prove that the operator T is completely continuous.
Assume that {ˉzn} is a sequence such that ˉzn↦ˉz in Bϱ as n↦∞. Then for every ℘∈(ϖ1,ϖ2], we have
|((Tˉzn)(℘)−(Tˉz)(℘))(℘−ϖ1)1−δ|=|m2m1+m2|1ϑδ−1Γ(δ)×1ϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1|eϑ−1ϑ(ϖ2−ℓ)|(ϖ2−ℓ)λ−δ|G(ℓ,ˉzn(ℓ))−G(ℓ,ˉz(ℓ))|dℓ+(℘−ϖ1)1−δϑλΓ(λ)℘∫ϖ1|eϑ−1ϑ(℘−ℓ)|(℘−ℓ)λ−1|G(ℓ,ˉzn(ℓ))−G(ℓ,ˉz(ℓ))|dℓ≤|m2m1+m2|1ϑλΓ(λ+1)(ϖ2−ϖ1)λ‖G(.,ˉzn(.))−G(.,ˉz(.))‖C1−δ+Γ(δ)(℘−ϖ1)1−δ+λϑλ−δΓ(λ−δ)(℘−ℓ)1−δ+λ‖G(.,ˉzn(.))−G(.,ˉz(.))‖C1−δ. | (3.23) |
Since |eϑ−1ϑ℘|<1 and G is continuous on (ϖ1,ϖ2] and ˉzn↦ˉz, then
‖(Tˉzn−Tˉz‖C1−δ→0asn→∞, | (3.24) |
which shows that operator T is continuous on Bϱ.
Case 3. We show that T(Bϱ) is relatively compact. In case 1, we have T(Bϱ)⊂Bϱ. It is observed that T(Bϱ) is uniformly bounded. To show operator T is equi-continuous on Bϱ. In fact, for any ϖ1<℘1<℘2<ϖ2 and ˉz∈Bϱ, we have
|(℘2−ϖ1)1−δ(Ty)(℘2)−(℘1−ϖ1)1−δ(Ty)(℘)|≤|(℘2−ϖ1)n−κ−(℘1−ϖ1)n−κ|ϑδ−1Γ(δ)eȷm1+m2+|m2m1+m2||(℘2−ϖ1)n−κ−(℘1−ϖ1)n−κ|ϑδ−1Γ(δ)×1ϑλ−δ+1Γ(λ−δ+1)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)λ−δ|G(ℓ,y(ℓ))|dℓ+1ϑλΓ(λ)|(℘2−ϖ1)1−δ℘2∫ϖ1eϑ−1ϑ(℘2−ℓ)(℘2−ℓ)λ−1G(ℓ,y(ℓ))dℓ−(℘1−ϖ1)1−δ℘1∫ϖ1eϑ−1ϑ(℘1−ℓ)(℘1−ℓ)λ−1G(ℓ,y(ℓ))dℓ|≤|(℘2−ϖ1)n−κ−(℘1−ϖ1)n−κ|ϑδ−1Γ(δ)[eȷm1+m2+|m2m1+m2‖G‖C1−δϑ1−δ+λΓ(1−δ+λ)ϖ2∫ϖ1(ϖ2−ℓ)λ−δ(ℓ−ϖ1)δ−1dℓ]+‖G‖C1−δϑλΓ(λ)|(℘2−ϖ1)1−δ℘2∫ϖ1(℘2−ℓ)λ−1(ℓ−ϖ1)δ−1dℓ−(℘1−ϖ1)1−δ℘1∫ϖ1(℘1−ℓ)λ−1(ℓ−ϖ1)δ−1dℓ|(since|eϑ−1ϑ℘|<1)≤|(℘2−ϖ1)n−κ−(℘1−ϖ1)n−κ|ϑδ−1Γ(δ)[eȷm1+m2+|m2m1+m2Γ(δ)ϑλΓ(λ+1)(ϖ2−ϖ1)λ‖G‖C1−δ]+‖G‖C1−δϑλΓ(λ)B(δ−n+1,λ)|(℘2−ϖ1)λ−(℘1−ϖ1)λ|, | (3.25) |
which approaches to zero as ℘2→℘1, independent of y∈ϱ, where B(.,.) denotes the Euler Beta function.
Therefore, we deduce that T(Bϱ) is equicontinuous on Bϱ, that leads to the relatively compactness. As a result, we conclude that by Arzela-Ascoli theorem, the defined operator T:Bϱ↦Bϱ is completely continuous operator.
By Schauder's fixed point theorem, there exists at least one fixed point y of T in C1−δ[ϖ1,ϖ2]. This fixed point y is the solution of (1.1) in Cδ;ϑ1−δ, and this completes the proof.
Now we present another existence result via Schaefer fixed point theorem. For this, we need the following assumption.
(A3) Suppose a function G:(ϖ1,ϖ2]×R↦R such that G(.,y(.))∈Cζ(1−λ);ϑ1−δ[ϖ1,ϖ2] for any y∈C1−δ[ϖ1,ϖ2] and there exist a mapping η(℘)∈C1−δ[ϖ1,ϖ2] such that
|G(℘,y)|≤η(℘),∀℘∈(ϖ1,ϖ2],y∈R. | (3.26) |
Theorem 3.2. Suppose that assumption (A3) satisfies. then Hilfer-BVP (1.1) has at least one solution in Cδ1−δ⊂Cλ,ζ;ϑ1−δ[ϖ1,ϖ2].
Proof. For the proof of Theorem 3.2, one can adopt the same technique as we did in Theorem 3.1 and easily prove that the operator T:C1−δ[ϖ1,ϖ2]↦C1−δ[ϖ1,ϖ2] stated in (3.16) is completely continuous. Now we show that
Δ={y∈Cn−δ[ϖ1,ϖ2]:y=σTy,forsomeσ∈(0,1)} | (3.27) |
is bounded set. Assume that y∈Δ and σ∈(0,1) be such that y=σTy. By assumption (A3) and (3.16), then for all ℘∈[ϖ1,ϖ2], we have
|Ty(℘)(℘−ϖ1)n−δ|≤n∑κ=1eϑ−1ϑ(℘−ϖ1)(℘−ϖ1)n−κϑδ−κ+1Γ(δ−κ+1)eȷm1+m2+|m2m1+m2|n∑κ=1eϑ−1ϑ(℘−ϖ1)(℘−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)Γ(n−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)n+λ−δ−1η(ℓ)dℓ+|(℘−ϖ1)n−δ|ϑλΓ(λ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)λ−1η(ℓ)dℓ≤n∑κ=1(℘−ϖ1)n−κϑδ−κ+1Γ(δ−κ+1)eȷm1+m2+|m2m1+m2|n∑κ=1(℘−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)Γ(n−δ+λ)ϖ2∫ϖ1(ϖ2−ℓ)n+λ−δ−1(ℓ−ϖ1)δ−n‖η‖Cn−δdℓ+|(℘−ϖ1)n−δ|ϑλΓ(λ)℘∫ϖ1(℘−ℓ)λ−1(ℓ−ϖ1)δ−n‖η‖Cn−δdℓ. | (3.28) |
Since |eϑ−1ϑ℘|<1, we have
‖Ty‖Cn−δ≤n∑κ=1(ϖ2−ϖ1)n−κϑδ−κ+1Γ(δ−κ+1)eȷm1+m2+[|m2m1+m2|n∑κ=1(ϖ2−ϖ1)−κϑδ−κ+1Γ(λ)B(λ,1)+B(δ−n+1,1)ϑλ(ϖ2−ϖ1)δ](ϖ2−ϖ1)n+λ‖η‖Cn−δ:=τ. | (3.29) |
Since σ∈(0,1), then y<Ty. The last inequality with (3.29) leads us to the conclusion that
‖y‖Cn−δ<‖Ty‖Cn−δ≤τ, | (3.30) |
which proves that Δ is bounded. Utilizing Schaefer fixed point postulate, this completes the proof.
Our last result is the existence result for the problem (1.1) by using the Kransnoselskii's fixed point theorem (see [37]), the following assumption is needed:
(A4) Suppose that G:(ϖ1,ϖ2]×R↦R is a function such that G(.y(.))∈Cζ(n−λ);ϑn−δ[ϖ1,ϖ2] for any y∈Cn−δ[ϖ1,ϖ2] and there exists a constant L>0 such that
|G(℘,y)−G(℘,ω)|≤L|y−ω|,∀℘∈(ϖ1,ϖ2],y,ω∈R. | (3.31) |
Also, we note the following assumption as follows: (A5) the inequality
Q:=[|m2m1+m2|n∑κ=1(ϖ2−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)+B(δ−n,λ+1)ϑδΓ(δ−n)]×Γ(δ−n)(ϖ2−ϖ1)λϑλB(δ−n,1)Γ(λ+1)‖˜G‖Cn−δ+eȷm1+m2n∑κ=1(ϖ2−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)L<1 | (3.32) |
is hold.
Theorem 3.3. Suppose that the assumptions (A4) and (A5) are satisfied. If
|m2m1+m2|n∑κ=1(ϖ2−ϖ1)n−κ+λϑn−δ+λΓ(δ−κ+1)Γ(δ−n+1)ϑλΓ(λ+1)L<1. | (3.33) |
Then the Hilfer-BVP (1.1) has at least one solution in Cδn−δ[ϖ1,ϖ2]⊂Cλ,ζ;ϑn−δ.
Proof. Considering the operator T stated in Theorem 3.1.
First, surmise the operator T into sum of two operators T1+T2 as follows
T1y(℘)=−m2m1+m2n∑κ=1eϑ−1ϑ(℘−ϖ)(℘−ϖ1)δ−κϑn−δ+λΓ(δ−κ+1)1Γ(n−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)n−δ+λ−1G(ℓ,y(ℓ))dℓ | (3.34) |
and
T2y(℘)=eȷm1+m2n∑κ=1eϑ−1ϑ(℘−ϖ)(℘−ϖ1)δ−κϑδ−κ+1Γ(δ−κ+1)+1ϑλΓ(λ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)λ−1G(ℓ,y(ℓ))dℓ. | (3.35) |
Setting ˜G=G(ℓ,0) and suppose the ball Bϵ={y∈Cn−δ;ψ([ϖ1,ϖ2]):‖y‖Cn−δ;ψ≤ϵ} having ϵ≥σ1−Q,Q<1, where
σ=[|m2m1+m2|n∑κ=1(ϖ2−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)+B(δ−n,λ+1)ϑδΓ(δ−n)]×Γ(δ−n)(ϖ2−ϖ1)λϑλB(δ−n,1)Γ(λ+1)L<1 | (3.36) |
The proof will be done in three cases.
Case 1. We show that T1y+T1ω∈Bϱ for every y,ω∈Bϵ.
Utilizing assumption (A4), then for every y∈Bϵ and ℘∈(ϖ1,ϖ2], we have
|(℘−ϖ1)n−δT1(℘)|≤|m2m1+m2|n∑κ=1eϑ−1ϑ(℘−ϖ)(℘−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)1Γ(n−δ+λ)×ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)n−δ+λ−1[|G(ℓ,y(ℓ))−G(ℓ,0)|+|G(ℓ,0)|]dℓ≤|m2m1+m2|n∑κ=1eϑ−1ϑ(℘−ϖ)(℘−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)1Γ(n−δ+λ)×ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)n−δ+λ−1(ℓ−ϖ1)δ−n[L‖y‖Cn−δ+‖˜G‖Cn−δ]dℓ≤|m2m1+m2|n∑κ=1(℘−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)Γ(δ−n+1)ϑλΓ(λ+1)[Lϵ+‖˜G‖Cn−δ]. | (3.37) |
Since |eϑ−1ϑ℘|<1. Therefore, we get
‖T1y‖Cn−δ≤|m2m1+m2|n∑κ=1(ϖ2−ϖ1)n−κ+λϑn−δ+λΓ(δ−κ+1)Γ(δ−n+1)ϑλΓ(λ+1)[Lϵ+‖˜G‖Cn−δ]. | (3.38) |
For operator T2, we have
|(℘−ϖ1)n−δT2ω(℘)|≤|eȷm1+m2|n∑κ=1eϑ−1ϑ(℘−ϖ)(℘−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)×(℘−ϖ1)n−δϑλΓ(λ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)λ−1[|G(ℓ,ω(ℓ))−G(ℓ,0)|+|G(ℓ,0)|]dℓ≤|eȷm1+m2|n∑κ=1(℘−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)×(℘−ϖ1)n−δϑλΓ(λ)℘∫ϖ1eϑ−1ϑ(℘−ℓ)(℘−ℓ)λ−1(ℓ−ϖ1)δ−n[L‖ω‖Cn−δ+‖˜G‖Cn−δ]dℓ. | (3.39) |
For every ω∈Bϵ and ℘∈(ϖ1,ϖ2], this shows
‖T1ω‖Cn−δ≤|eȷm1+m2|n∑κ=1(ϖ2−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)×(ϖ2−ϖ1)λϑλΓ(δ−n+λ+1)[Lϵ+‖˜G‖Cn−δ]. | (3.40) |
From (3.38), (3.40) and utilizing assumption (A5) with (3.36), we find
‖T1y+T2ω‖Cn−δ≤‖T1y‖Cn−δ+‖T1ω‖Cn−δ≤(ϖ2−ϖ1)λΓ(δ−n+1)ϑλΓ(λ+1)[Lϵ+‖˜G‖Cn−δ][|m2m1+m2|n∑κ=1(ϖ2−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)+Γ(λ+1)Γ(δ−n+λ+1)]+eȷm1+m2n∑κ=1(ϖ2−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)≤Qϵ+(1−Q)ϵ=ϵ. | (3.41) |
Case 2. We prove that the operator T is a contraction mapping on Bϱ.
For any y,ω∈Bϱ, and for any ℘∈(ϖ1,ϖ2], then by supposition (A4), we have
|(℘−ϖ1)n−δT1y(℘)−(℘−ϖ1)n−δT1ω(℘)|≤|m2m1+m2|n∑κ=1eϑ−1ϑ(℘−ϖ)(℘−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)1Γ(n−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)n−δ+λ−1[G(ℓ,y(ℓ))−G(ℓ,ω(ℓ))]dℓ≤|m2m1+m2|n∑κ=1eϑ−1ϑ(℘−ϖ)(℘−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)1Γ(n−δ+λ)ϖ2∫ϖ1eϑ−1ϑ(ϖ2−ℓ)(ϖ2−ℓ)n−δ+λ−1L|y(ℓ)−ω(ℓ)|dℓ≤|m2m1+m2|n∑κ=1(℘−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)Γ(δ−n+1)ϑλΓ(λ+1)(ϖ2−ℓ)λL‖y−ω‖Cn−δ. | (3.42) |
Since |eϑ−1ϑ℘|<1, this yields
‖Ty−Tω‖Cn−δ≤|m2m1+m2|n∑κ=1(ϖ2−ϖ1)n−κ+λϑϑn−δ+λΓ(δ−κ+1)Γ(δ−n+1)ϑλΓ(λ+1)L‖y−ω‖Cn−δ. | (3.43) |
Due to assumption (3.33), which shows that the operator T is a contraction mapping.
Case 3: Now we show that the operator T2 is completely continuous on Bϵ.
From the continuity of G, we deduce that the operator T2:Bϵ↦Bϵ is continuous on Bϵ. Furthermore, we prove that for all ϵ>0 there exists some ϵ′>0 such that ‖T2y‖Cn−δ<ϵ′. In view of case 1, for y∈Bϵ, we have that
‖T2y‖Cn−δ≤n∑κ=1(ϖ2−ϖ1)n−κϑδ−κ+1Γ(δ−κ+1)eȷm1+m2+B(δ−n+1,λ)(ϖ2−ϖ1)λϑλΓ(λ)[Lϵ+‖˜G‖Cn−δ], | (3.44) |
which is free of ℘ and y, so there exists
ϵ′=eȷm1+m2n∑κ=1(ϖ2−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)+B(δ−n+1,λ)(ϖ2−ϖ1)λϑλΓ(λ)[Lϵ+‖˜G‖Cn−δ] | (3.45) |
such that ‖T2(y)‖Cn−δ≤ϵ′. Therefore, T2 is uniformly bounded set on Bϵ. Finally, to show that T2 is equicontinuous in Bϵ, for any z∈Bϵ and ℘1,℘2∈(ϖ1,ϖ2] having ℘1<℘2, we have
|(℘2−ϖ1)n−δT2y(℘2)−(℘1−ϖ1)n−δT2y(℘1)|=eȷm1+m2|n∑κ=1(℘2−ϖ1)n−κ−(℘1−ϖ1)n−δϑδ−κΓ(δ−κ+1)+(℘2−ϖ1)n−δϑλΓ(λ)℘2∫ϖ1eϑ−1ϑ(℘2−ℓ)(℘2−ℓ)λ−1G(ℓ,y(ℓ))dℓ−(℘1−ϖ1)n−δϑλΓ(λ)℘1∫ϖ1eϑ−1ϑ(℘1−ℓ)(℘1−ℓ)λ−1G(ℓ,y(ℓ))dℓ|≤eȷm1+m2n∑κ=1|(℘2−ϖ1)n−κ−(℘1−ϖ1)n−δ|ϑδ−κΓ(δ−κ+1)+|(℘2−ϖ1)n−δϑλΓ(λ)℘2∫ϖ1eϑ−1ϑ(℘2−ℓ)(℘2−ℓ)λ−1(ℓ−ϖ1)δ−n‖G‖Cn−δ,ψ[ϖ1,ϖ2]dℓ−(℘1−ϖ1)n−δϑλΓ(λ)℘1∫ϖ1eϑ−1ϑ(℘1−ℓ)(℘1−ℓ)λ−1(ℓ−ϖ1)δ−n‖G‖Cn−δ,ψ[ϖ1,ϖ2]dℓ|=n∑κ=1eȷm1+m2|(℘2−ϖ1)n−κ−(℘1−ϖ1)n−δ|ϑδ−κΓ(δ−κ+1)+B(δ−n+1)ϑλ‖G‖Cn−δ;ψ[ϖ1,ϖ2]|(℘2−ϖ1)λ−(℘1−ϖ1)λ|. | (3.46) |
Since |eϑ−1ϑ℘|<1. It is noting that the right hand side of the aforesaid variant is free of y. So,
|(℘2−ϖ1)n−δT2y(℘2)−(℘1−ϖ1)n−δT2y(℘1)|↦0,as|℘2−℘1|↦0. | (3.47) |
This shows that T2 is equicontinuous on Bϵ. According to Arzela-Ascoli Theorem, observed that (T2Bϵ) is relatively compact. By Kransnoselskii's fixed point theorem, the problem (1.1) has at least one solution.
Consider the fractional differential equation with boundary condition which encompasses the Hilfer-GPF derivative of the form
{Dλ,ζ;ϑϖ+1y(℘)=℘−16+℘5/616siny(℘),℘∈J=[0,2],λ∈(0,1),ζ∈[0,1]I1−δ;ϑϖ+1[13y(0+)+23y(2−)]=25,λ≤δ=λ+ζ−λζ, | (4.1) |
By comparison (1.1) with (2.9), we have λ=12,ζ=13,δ=23,m1=13,m2=23,ϑ=1 and e1=25. It is clear that ℘13G(℘,y(℘))=℘16+℘7/616siny(℘)∈C([0,2]), So G(℘,y(℘))∈C13. Thus, it follows that, for any y∈R+ and ℘∈J,
|G(℘,y(℘))|≤℘16(1+℘2/316|℘1/3y(℘)|)≤(1+116‖y‖C13). | (4.2) |
Hence, the assumption (A1) is fulfilled having M=1 and m=116. It is easy to verify that the assumption (A2) is hold too. In fact, by simple computations, we obtain
G:=mM1Γ(δ)ϑλΓ(λ+1)[(ϖ2−ϖ1)λ+(ϖ2−ϖ1)λ+1−δ]≈−0.03510<1. | (4.3) |
Hence, all suppositions of Theorem 3.1 implies that the problem (1.1) has a unique solution in C2313([0,2]).
Also, assume that G(℘,y(℘))=℘−16+℘5/616siny(℘). Thus |G(℘1,y(℘))≤℘−16+℘5/616|=η(℘)∈C1−δ([0,2]). So, (A3) is satisfied. Therefore, in view of Theorem 3.2, we conclude that problem (1.1) has a solution in C2/31/3([0,2]).
Finally, if G(℘,y(℘))=℘−16+℘5/616siny(℘), then for y,ω∈R+ and ℘∈J, we have
|G(℘,y(℘))−G(℘,ω(℘))|≤116|y−ω|. |
Therefore, the assumption (A4) is fulfilled having L=116. Clearly, assumption (A5) and inequality (3.33) are holds. In fact, simple computations yields
Q:=[|m2m1+m2|n∑κ=1(ϖ2−ϖ1)n−κϑn−δ+λΓ(δ−κ+1)+B(δ−n,λ+1)ϑδΓ(δ−n)]×Γ(δ−n)(ϖ2−ϖ1)λϑλB(δ−n,1)Γ(λ+1)L≈0.1456<1, | (4.4) |
and
|m2m1+m2|n∑κ=1(ϖ2−ϖ1)n−κ+λϑn−δ+λΓ(δ−κ+1)Γ(δ−n+1)ϑλΓ(λ+1)L≈0.0495<1 | (4.5) |
of Theorem 3.3, shows that problem (1.1) has a solution in C2313([0,2]).
In this approach, we have established certain existence consequences for the solution of BVP for Hilfer-FDEs depend on the lessening of FDEs to integral equations. The proposed scheme with the fixed point assertions unifies the existing results in the frame of RL and Caputo GPF sense, respectively. Besides that, the analysis's comprehensive improvements are dependent on various techniques such as Schauder's, Schaefer's and Kransnoselskii's fixed point theorems. Also, the Hilffer GPF-derivative comprise two parameters and a proportionality index ϑ.
∙ If ϑ↦1 and λ=[0,1], then the contemplated problem converted to RL and Caputo fractional derivative [8]. If ϑ∈(0,1) and ζ=0,1 we recaptures the RL and Caputo GPF-derivative [25], respectively (see Figure 1).
∙ Clearly, if ϑ,ζ∈(0,1), then the newly employed derivatives amalgamate the existing ones in the adjustment of Hilfer, RL and GPF-derivative, (see Figure 2).
∙ If ϑ↦1 and ζ,λ∈[0,1], then the formulation for this problem enjoys Hilfer factional derivative [8], (see Figure 3).
Moreover, a stimulative example is presented to show the efficacy of the established outcomes. We hope that the testified outcomes here will have a considerable impact for more parameters on the stability and other qualitative features of differential equations in the areas of interest of applied sciences.
The authors declare that there is no conflict of interests.
[1] |
L. Lansbury, B. Lim, V. Baskaran, W. S. Lim, Co-infections in people with COVID-19: a systematic review and meta-analysis, J. Infect., 81 (2020), 266–275. https://doi.org/10.1016/j.jinf.2020.05.046 doi: 10.1016/j.jinf.2020.05.046
![]() |
[2] |
K. Lacombe, J. Rockstroh, HIV and viral hepatitis coinfections: advances and challenges, Gut, 61 (2012), 47–58. https://doi.org/10.1136/gutjnl-2012-302062 doi: 10.1136/gutjnl-2012-302062
![]() |
[3] |
M. G. Mavilia, G. Y. Wu, HBV-HCV coinfection: viral interactions, management, and viral reactivation, J. Clin. Transl. Hepatol., 6 (2018), 296–305. https://doi.org/10.14218/JCTH.2018.00016 doi: 10.14218/JCTH.2018.00016
![]() |
[4] | H. O. Hashim, M. K. Mohammed, M. J. Mousa, H. H. Abdulameer, A. T. Alhassnawi, S. A. Hassan, et al., Infection with different strains of SARS-CoV-2 in patients with COVID-19, Arch. Biol. Sci., 72 (2020), 575–585. |
[5] |
S. Shoraka, S. R. Mohebbi, S. M. Hosseini, A. Ghaemi, M. R. Zali, SARS-CoV-2 and chronic hepatitis B: focusing on the possible consequences of co-infection, J. Clin. Virol. Plus, 3 (2023), 100167. https://doi.org/10.1016/j.jcvp.2023.100167 doi: 10.1016/j.jcvp.2023.100167
![]() |
[6] |
M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74–79. https://doi.org/10.1126/science.272.5258.74 doi: 10.1126/science.272.5258.74
![]() |
[7] |
P. de Leenheer, S. S. Pilyugin, Multistrain virus dynamics with mutations: a global analysis, Math. Med. Biol., 25 (2008), 285–322. https://doi.org/10.1093/imammb/dqn023 doi: 10.1093/imammb/dqn023
![]() |
[8] |
L. Pinky, H. M. Dobrovolny, SARS-CoV-2 coinfections: could influenza and the common cold be beneficial? J. Med. Virol., 92 (2020), 2623–2630. https://doi.org/10.1002/jmv.26098 doi: 10.1002/jmv.26098
![]() |
[9] |
M. D. Nowak, E. M. Sordillo, M. R. Gitman, A. E. P. Mondolfi, Coinfection in SARS-CoV-2 infected patients: where are influenza virus and rhinovirus/enterovirus? J. Med. Virol., 92 (2020), 1699–1700. https://doi.org/10.1002/jmv.25953 doi: 10.1002/jmv.25953
![]() |
[10] |
S. Kalinichenko, D. Komkov, D. Mazurov, HIV-1 and HTLV-1 transmission modes: mechanisms and importance for virus spread, Viruses, 14 (2022), 152. https://doi.org/10.3390/v14010152 doi: 10.3390/v14010152
![]() |
[11] |
J. Schmidt, H. E. Blum, R. Thimme, T-cell responses in hepatitis B and C virus infection: similarities and differences, Emerg. Micro. Infect., 2 (2013), e15. https://doi.org/10.1038/emi.2013.14 doi: 10.1038/emi.2013.14
![]() |
[12] |
M. Ruiz Silva, J. A. A. Briseño, V. Upasani, H. van der Ende-Metselaar, J. M. Smit, I. A. Rodenhuis-Zybert, Suppression of chikungunya virus replication and differential innate responses of human peripheral blood mononuclear cells during co-infection with dengue virus, PLoS Negl. Trop. Dis., 11 (2017), e0005712. https://doi.org/10.1371/journal.pntd.0005712 doi: 10.1371/journal.pntd.0005712
![]() |
[13] |
A. Nurtay, M. G. Hennessy, J. Sardanyés, L. Alsedà, S. F. Elena, Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: a bifurcation analysis, R. Soc. Open Sci., 6 (2019), 181179. https://doi.org/10.1098/rsos.181179 doi: 10.1098/rsos.181179
![]() |
[14] |
P. J. Goulder, B. D. Walker, HIV-1 superinfection: a word of caution, New Engl. J. Med., 347 (2002), 756–758. https://doi.org/10.1056/NEJMe020091 doi: 10.1056/NEJMe020091
![]() |
[15] |
Y. He, W. Ma, S. Dang, L. Chen, R. Zhang, S. Mei, et al., Possible recombination between two variants of concern in a COVID-19 patient, Emerg. Micro. Infect., 11 (2022), 552–555. https://doi.org/10.1080/22221751.2022.2032375 doi: 10.1080/22221751.2022.2032375
![]() |
[16] |
A. M. Elaiw, N. H. AlShamrani, Analysis of a within-host HIV/HTLV-I co-infection model with immunity, Virus Res., 295 (2021), 198204. https://doi.org/10.1016/j.virusres.2020.198204 doi: 10.1016/j.virusres.2020.198204
![]() |
[17] |
A. M. Elaiw, R. S. Alsulami, A. D. Hobiny, Modeling and stability analysis of within-host IAV/SARS-CoV-2 coinfection with antibody immunity, Mathematics, 10 (2022), 4382. https://doi.org/10.3390/math10224382 doi: 10.3390/math10224382
![]() |
[18] |
A. M. Elaiw, A. S. Shflot, A. D. Hobiny, Global stability of delayed SARS-CoV-2 and HTLV-I coinfection models within a host, Mathematics, 10 (2022), 4756. https://doi.org/10.3390/math10244756 doi: 10.3390/math10244756
![]() |
[19] |
A. M. Elaiw, A. D. Al Agha, S. A. Azoz, E. Ramadan, Global analysis of within-host SARS-CoV-2/HIV coinfection model with latency, Eur. Phys. J. Plus, 137 (2022), 174. https://doi.org/10.1140/epjp/s13360-022-02387-2 doi: 10.1140/epjp/s13360-022-02387-2
![]() |
[20] |
H. Nampala, S. Livingstone, L. Luboobi, J. Y. T. Mugisha, C. Obua, M. Jablonska-Sabuka, Modelling hepatotoxicity and antiretroviral therapeutic effect in HIV/HBV coinfection, Math. Biosci., 302 (2018), 67–79. https://doi.org/10.1016/j.mbs.2018.05.012 doi: 10.1016/j.mbs.2018.05.012
![]() |
[21] |
R. Birger, R. Kouyos, J. Dushoff, B. Grenfell, Modeling the effect of HIV coinfection on clearance and sustained virologic response during treatment for hepatitis C virus, Epidemics, 12 (2015), 1–10. https://doi.org/10.1016/j.epidem.2015.04.001 doi: 10.1016/j.epidem.2015.04.001
![]() |
[22] |
L. Rong, Z. Feng, A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. Math. Biol., 69 (2007), 2027–2060. https://doi.org/10.1007/s11538-007-9203-3 doi: 10.1007/s11538-007-9203-3
![]() |
[23] |
P. Wu, H. Zhao, Dynamics of an HIV infection model with two infection routes and evolutionary competition between two viral strains, Appl. Math. Modell., 84 (2020), 240–264. https://doi.org/10.1016/j.apm.2020.03.040 doi: 10.1016/j.apm.2020.03.040
![]() |
[24] |
B. J. Nath, K. Sadri, H. K. Sarmah, K. Hosseini, An optimal combination of antiretroviral treatment and immunotherapy for controlling HIV infection, Math. Comput. Simul., 217 (2024), 226–243. https://doi.org/10.1016/j.matcom.2023.10.012 doi: 10.1016/j.matcom.2023.10.012
![]() |
[25] |
Y. Liu, Y. Wang, D. Jiang, Dynamic behaviors of a stochastic virus infection model with Beddington-DeAngelis incidence function, eclipse-stage and Ornstein-Uhlenbeck process, Math. Biosci., 2024 (2024), 109154. https://doi.org/10.1016/j.mbs.2024.109154 doi: 10.1016/j.mbs.2024.109154
![]() |
[26] |
O. Lambotte, M. L. Chaix, B. Gubler, N. Nasreddine, C. Wallon, C. Goujard, et al., The lymphocyte HIV reservoir in patients on long-term HAART is a memory of virus evolution, AIDS, 18 (2004), 1147–1158. https://doi.org/10.1097/00002030-200405210-00008 doi: 10.1097/00002030-200405210-00008
![]() |
[27] |
W. Chen, Z. Teng, L. Zhang, Global dynamics for a drug-sensitive and drug-resistant mixed strains of HIV infection model with saturated incidence and distributed delays, Appl. Math. Comput., 406 (2021), 126284. https://doi.org/10.1016/j.amc.2021.126284 doi: 10.1016/j.amc.2021.126284
![]() |
[28] |
A. Perelson, A. Neumann, M. Markowitz, J. Leonard, D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582–1586. https://doi.org/10.1126/science.271.5255.1582 doi: 10.1126/science.271.5255.1582
![]() |
[29] |
R. V. Culshaw, S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27–39. https://doi.org/10.1016/s0025-5564(00)00006-7 doi: 10.1016/s0025-5564(00)00006-7
![]() |
[30] |
S. K. Sahani, Yashi, Effects of eclipse phase and delay on the dynamics of HIV infection, J. Biol. Syst., 26 (2018), 421–454. https://doi.org/10.1142/S0218339018500195 doi: 10.1142/S0218339018500195
![]() |
[31] |
R. Xu, Global dynamics of an HIV-1 infection model with distributed intracellular delays, Comput. Math. Appl., 61 (2011), 2799–2805. https://doi.org/10.1016/j.camwa.2011.03.050 doi: 10.1016/j.camwa.2011.03.050
![]() |
[32] |
J. Li, X. Wang, Y. Chen, Analysis of an age-structured HIV infection model with cell-to-cell transmission, Eur. Phys. J. Plus, 139 (2024), 78. https://doi.org/10.1140/epjp/s13360-024-04873-1 doi: 10.1140/epjp/s13360-024-04873-1
![]() |
[33] |
D. Ebert, C. D. Zschokke-Rohringer, H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200–209. https://doi.org/10.1007/PL00008847 doi: 10.1007/PL00008847
![]() |
[34] |
X. Song, A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281–297. https://doi.org/10.1016/j.jmaa.2006.06.064 doi: 10.1016/j.jmaa.2006.06.064
![]() |
[35] |
O. A. Razzaq, N. A. Khan, M. Faizan, A. Ara, S. Ullah, Behavioral response of population on transmissibility and saturation incidence of deadly pandemic through fractional order dynamical system, Results Phys., 26 (2021), 104438. https://doi.org/10.1016/j.rinp.2021.104438 doi: 10.1016/j.rinp.2021.104438
![]() |
[36] |
W. Chen, N. Tuerxun, Z. Teng, The global dynamics in a wild-type and drug-resistant HIV infection model with saturated incidence, Adv. Differ. Equations, 2020 (2020), 25. https://doi.org/10.1186/s13662-020-2497-2 doi: 10.1186/s13662-020-2497-2
![]() |
[37] | W. Chen, L. Zhang, N. Wang, Z. Teng, Bifurcation analysis and chaos for a double-strains HIV coinfection model with intracellular delays, saturated incidence and logistic growth, Saturated Incidence Logist. Growth, 2023. https://doi.org/10.21203/rs.3.rs-3132841/v1 |
[38] |
T. Li, Y. Guo, Modeling and optimal control of mutated COVID-19 (Delta strain) with imperfect vaccination, Chaos Solitons Fract., 156 (2022), 111825. https://doi.org/10.1016/j.chaos.2022.111825 doi: 10.1016/j.chaos.2022.111825
![]() |
[39] |
Y. Guo, T. Li, Modeling the competitive transmission of the Omicron strain and Delta strain of COVID-19, J. Math. Anal. Appl., 526 (2023), 127283. https://doi.org/10.1016/j.jmaa.2023.127283 doi: 10.1016/j.jmaa.2023.127283
![]() |
[40] | J. K. Hale, S. M. V. Lunel, Introduction to functional differential equations, Springer-Verlag, 1993. |
[41] | Y. Kuang, Delay differential equations with applications in population dynamics, Academic Press, 1993. |
[42] |
D. Wodarz, D. C. Krakauer, Defining CTL-induced pathology: implications for HIV, Virology, 274 (2000), 94–104. https://doi.org/10.1006/viro.2000.0399 doi: 10.1006/viro.2000.0399
![]() |
[43] | A. S. Perelson, Modeling the interaction of the immune system with HIV, In: C. Castillo-Chavez, Mathematical and statistical approaches to AIDS epidemiology, Springer Berlin Heidelberg, 1989,350–370. https://doi.org/10.1007/978-3-642-93454-4_17 |
[44] | A. S. Perelson, D. E. Kirschner, R. de Boer, Dynamics of HIV Infection of CD4+ T cells, Math. Biosci., 114 (1993), 81–125. |
[45] |
W. A. Woldegerima, M. I. Teboh-Ewungkem, G. A. Ngwa, The impact of recruitment on the dynamics of an immune-suppressed within-human-host model of the Plasmodium falciparum parasite, Bull. Math. Biol., 81 (2019), 4564–4619. https://doi.org/10.1007/s11538-018-0436-0 doi: 10.1007/s11538-018-0436-0
![]() |
[46] |
P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/s0025-5564(02)00108-6 doi: 10.1016/s0025-5564(02)00108-6
![]() |
[47] |
A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879–883. https://doi.org/10.1016/j.bulm.2004.02.001 doi: 10.1016/j.bulm.2004.02.001
![]() |
[48] | H. K. Khalil, Nonlinear systems, 3 Eds., Prentice Hall, 2002. |
[49] |
L. Pinky, G. González-Parran, H. M. Dobrovolny, Superinfection and cell regeneration can lead to chronic viral coinfections, J. Theor. Biol., 466 (2019), 24–38. https://doi.org/10.1016/j.jtbi.2019.01.011 doi: 10.1016/j.jtbi.2019.01.011
![]() |
[50] |
F. Li, W. Ma, Dynamics analysis of an HTLV-1 infection model with mitotic division of actively infected cells and delayed CTL immune response, Math. Methods Appl. Sci., 41 (2018), 3000–3017. https://doi.org/10.1002/mma.4797 doi: 10.1002/mma.4797
![]() |
[51] |
N. H. Alshamrani, Stability of an HTLV-HIV coinfection model with multiple delays and CTL-mediated immunity, Adv. Differ. Equations, 2021 (2021), 270. https://doi.org/10.1186/s13662-021-03416-7 doi: 10.1186/s13662-021-03416-7
![]() |
[52] |
D. S. Callaway, A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29–64. https://doi.org/10.1006/bulm.2001.0266 doi: 10.1006/bulm.2001.0266
![]() |
[53] |
Y. Wang, J. Liu, L. Liu, Viral dynamics of an HIV model with latent infection incorporating antiretroviral therapy, Adv. Differ. Equations, 2016 (2016), 225. https://doi.org/10.1186/s13662-016-0952-x doi: 10.1186/s13662-016-0952-x
![]() |
[54] |
Y. Wang, J. Liu, J. M. Heffernan, Viral dynamics of an HTLV-I infection model with intracellular delay and CTL immune response delay, J. Math. Anal. Appl., 459 (2018), 506–527. https://doi.org/10.1016/j.jmaa.2017.10.027 doi: 10.1016/j.jmaa.2017.10.027
![]() |
[55] |
B. Asquith, C. R. Bangham, Quantifying HTLV-I dynamics, Immunol. Cell Biol., 85 (2007), 280–286. https://doi.org/10.1038/sj.icb.7100050 doi: 10.1038/sj.icb.7100050
![]() |
[56] |
G. Huang, W. Ma, Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690–1693. https://doi.org/10.1016/j.aml.2009.06.004 doi: 10.1016/j.aml.2009.06.004
![]() |
[57] |
X. Zhou, J. Cui, Global stability of the viral dynamics with Crowley-Martin functional response, Bull. Korean Math. Soc., 48 (2011), 555–574. https://doi.org/10.4134/BKMS.2011.48.3.555 doi: 10.4134/BKMS.2011.48.3.555
![]() |
[58] |
K. Hattaf, N. Yousfi, A class of delayed viral infection models with general incidence rate and adaptive immune response, Int. J. Dyn. Control, 4 (2016), 254–265. https://doi.org/10.1007/s40435-015-0158-1 doi: 10.1007/s40435-015-0158-1
![]() |
[59] |
G. Huang, Y. Takeuchi, W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693–2708. https://doi.org/10.1137/090780821 doi: 10.1137/090780821
![]() |
[60] |
K. Hattaf, A new mixed fractional derivative with applications in computational biology, Computation, 12 (2024), 7. https://doi.org/10.3390/computation12010007 doi: 10.3390/computation12010007
![]() |
[61] |
J. Danane, K. Allali, Z. Hammouch, Mathematical analysis of a fractional differential model of HBV infection with antibody immune response, Chaos Solitons Fract., 136 (2020), 109787. https://doi.org/10.1016/j.chaos.2020.109787 doi: 10.1016/j.chaos.2020.109787
![]() |
[62] |
Y. Guo, T. Li, Fractional-order modeling and optimal control of a new online game addiction model based on real data, Commun. Nonlinear Sci. Numer. Simul., 121 (2023), 107221. https://doi.org/10.1016/j.cnsns.2023.107221 doi: 10.1016/j.cnsns.2023.107221
![]() |
[63] |
W. Adel, H. Günerhan, K. S. Nisar, P. Agarwal, A. El-Mesady, Designing a novel fractional order mathematical model for COVID-19 incorporating lockdown measures, Sci. Rep., 14 (2024), 2926. https://doi.org/10.1038/s41598-023-50889-5 doi: 10.1038/s41598-023-50889-5
![]() |
[64] |
N. Bellomo, D. Burini, N. Outada, Multiscale models of COVID-19 with mutations and variants, Networks Heterog. Media, 17 (2022), 293–310. https://doi.org/10.3934/nhm.2022008 doi: 10.3934/nhm.2022008
![]() |
[65] |
D. Burini, D. Knopoff, Epidemics and society-a multiscale vision from the small world to the globally interconnected world, Math. Models Methods Appl. Sci., 34 (2024), 295. https://doi.org/10.1142/S0218202524500295 doi: 10.1142/S0218202524500295
![]() |
1. | Saowaluck Chasreechai, Muhammad Aamir Ali, Surapol Naowarat, Thanin Sitthiwirattham, Kamsing Nonlaopon, On some Simpson's and Newton's type of inequalities in multiplicative calculus with applications, 2023, 8, 2473-6988, 3885, 10.3934/math.2023193 | |
2. | Wasfi Shatanawi, Taqi A. M. Shatnawi, Some fixed point results based on contractions of new types for extended b-metric spaces, 2023, 8, 2473-6988, 10929, 10.3934/math.2023554 | |
3. | Rajaa T. Matoog, Amr M. S. Mahdy, Mohamed A. Abdou, Doaa Sh. Mohamed, A Computational Method for Solving Nonlinear Fractional Integral Equations, 2024, 8, 2504-3110, 663, 10.3390/fractalfract8110663 | |
4. | Mohammed A. Almalahi, Khaled Aldowah, Faez Alqarni, Manel Hleili, Kamal Shah, Fathea M. O. Birkea, On modified Mittag–Leffler coupled hybrid fractional system constrained by Dhage hybrid fixed point in Banach algebra, 2024, 14, 2045-2322, 10.1038/s41598-024-81568-8 | |
5. | Muath Awadallah, Mohamed Hannabou, Hajer Zaway, Jihan Alahmadi, Applicability of Caputo-Hadmard fractional operator in mathematical modeling of pantograph systems, 2025, 1598-5865, 10.1007/s12190-025-02421-3 |