Aflatoxins are secondary carcinogenic fungal metabolites derived from the toxic various Aspergillus species. These fungi can easily infect nuts and grains. A global systematic review was done to extract data on the concentration of aflatoxins in different nuts. Besides, risk assessment was conducted on data. The scientific databases were searched systematically from 2000 to 2020. Based on the results, aflatoxin B1 (AFB1) had the most frequency in nut samples. The mean concentration of aflatoxin total (AFT) and AFB1 in nuts were as follows: peanut (37.85, 32.82 μg/kg), pistachio (31.42, 39.44 μg/kg), almond (3.54, 3.93 μg/kg), walnut (42.27, 22.23 μg/kg), hazelnut (17.33, 10.54 μg/kg), Brazil nut (4.61, 3.35 μg/kg), and other nuts (26.22, 7.38 μg/kg). According to country the margin of exposure (MOE) value for adult was as Argentina (21) > Congo (67) > India (117) > Bangladesh (175) > Cameroon (238) > Iran (357) > Bahrain (438) > Brazil (447) > Ghana (606) > South Africa (1017) > Egypt (1176) > USA (1505) > China (1526) > Cyprus (1588). The MOE of the consumers in some countries was considerably below the safety margin of 10,000. To conclude, nuts are highly consumed by different consumers, so it is necessary to emphasize strict control measures to prevent contamination of these foods with aflatoxins.
Citation: Arezoo Ebrahimi, Alireza Emadi, Majid Arabameri, Ahmad Jayedi, Anna Abdolshahi, Behdad Shokrolahi Yancheshmeh, Nabi Shariatifar. The prevalence of aflatoxins in different nut samples: A global systematic review and probabilistic risk assessment[J]. AIMS Agriculture and Food, 2022, 7(1): 130-148. doi: 10.3934/agrfood.2022009
[1] | Ridha Dida, Hamid Boulares, Bahaaeldin Abdalla, Manar A. Alqudah, Thabet Abdeljawad . On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives. AIMS Mathematics, 2023, 8(10): 23032-23045. doi: 10.3934/math.20231172 |
[2] | Hamid Boulares, Manar A. Alqudah, Thabet Abdeljawad . Existence of solutions for a semipositone fractional boundary value pantograph problem. AIMS Mathematics, 2022, 7(10): 19510-19519. doi: 10.3934/math.20221070 |
[3] | Reny George, Fahad Al-shammari, Mehran Ghaderi, Shahram Rezapour . On the boundedness of the solution set for the ψ-Caputo fractional pantograph equation with a measure of non-compactness via simulation analysis. AIMS Mathematics, 2023, 8(9): 20125-20142. doi: 10.3934/math.20231025 |
[4] | Saeed M. Ali, Mohammed S. Abdo, Bhausaheb Sontakke, Kamal Shah, Thabet Abdeljawad . New results on a coupled system for second-order pantograph equations with ABC fractional derivatives. AIMS Mathematics, 2022, 7(10): 19520-19538. doi: 10.3934/math.20221071 |
[5] | Weerawat Sudsutad, Chatthai Thaiprayoon, Aphirak Aphithana, Jutarat Kongson, Weerapan Sae-dan . Qualitative results and numerical approximations of the (k,ψ)-Caputo proportional fractional differential equations and applications to blood alcohol levels model. AIMS Mathematics, 2024, 9(12): 34013-34041. doi: 10.3934/math.20241622 |
[6] | Choukri Derbazi, Zidane Baitiche, Mohammed S. Abdo, Thabet Abdeljawad . Qualitative analysis of fractional relaxation equation and coupled system with Ψ-Caputo fractional derivative in Banach spaces. AIMS Mathematics, 2021, 6(3): 2486-2509. doi: 10.3934/math.2021151 |
[7] | Abdelkader Moumen, Ramsha Shafqat, Zakia Hammouch, Azmat Ullah Khan Niazi, Mdi Begum Jeelani . Stability results for fractional integral pantograph differential equations involving two Caputo operators. AIMS Mathematics, 2023, 8(3): 6009-6025. doi: 10.3934/math.2023303 |
[8] | Iyad Suwan, Mohammed S. Abdo, Thabet Abdeljawad, Mohammed M. Matar, Abdellatif Boutiara, Mohammed A. Almalahi . Existence theorems for Ψ-fractional hybrid systems with periodic boundary conditions. AIMS Mathematics, 2022, 7(1): 171-186. doi: 10.3934/math.2022010 |
[9] | Mohamed Houas, Kirti Kaushik, Anoop Kumar, Aziz Khan, Thabet Abdeljawad . Existence and stability results of pantograph equation with three sequential fractional derivatives. AIMS Mathematics, 2023, 8(3): 5216-5232. doi: 10.3934/math.2023262 |
[10] | Karim Guida, Lahcen Ibnelazyz, Khalid Hilal, Said Melliani . Existence and uniqueness results for sequential ψ-Hilfer fractional pantograph differential equations with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(8): 8239-8255. doi: 10.3934/math.2021477 |
Aflatoxins are secondary carcinogenic fungal metabolites derived from the toxic various Aspergillus species. These fungi can easily infect nuts and grains. A global systematic review was done to extract data on the concentration of aflatoxins in different nuts. Besides, risk assessment was conducted on data. The scientific databases were searched systematically from 2000 to 2020. Based on the results, aflatoxin B1 (AFB1) had the most frequency in nut samples. The mean concentration of aflatoxin total (AFT) and AFB1 in nuts were as follows: peanut (37.85, 32.82 μg/kg), pistachio (31.42, 39.44 μg/kg), almond (3.54, 3.93 μg/kg), walnut (42.27, 22.23 μg/kg), hazelnut (17.33, 10.54 μg/kg), Brazil nut (4.61, 3.35 μg/kg), and other nuts (26.22, 7.38 μg/kg). According to country the margin of exposure (MOE) value for adult was as Argentina (21) > Congo (67) > India (117) > Bangladesh (175) > Cameroon (238) > Iran (357) > Bahrain (438) > Brazil (447) > Ghana (606) > South Africa (1017) > Egypt (1176) > USA (1505) > China (1526) > Cyprus (1588). The MOE of the consumers in some countries was considerably below the safety margin of 10,000. To conclude, nuts are highly consumed by different consumers, so it is necessary to emphasize strict control measures to prevent contamination of these foods with aflatoxins.
Recently, fractional calculus methods became of great interest, because it is a powerful tool for calculating the derivation of multiples systems. These methods study real world phenomena in many areas of natural sciences including biomedical, radiography, biology, chemistry, and physics [1,2,3,4,5,6,7]. Abundant publications focus on the Caputo fractional derivative (CFD) and the Caputo-Hadamard derivative. Additionally, other generalization of the previous derivatives, such as Ψ-Caputo, study the existence of solutions to some FDEs (see [8,9,10,11,12,13,14]).
In general, an m-point fractional boundary problem involves a fractional differential equation with fractional boundary conditions that are specified at m different points on the boundary of a domain. The fractional derivative is defined using the Riemann-Liouville fractional derivative or the Caputo fractional derivative. Solving these types of problems can be challenging due to the non-local nature of fractional derivatives. However, there are various numerical and analytical methods available for solving such problems, including the spectral method, the finite difference method, the finite element method, and the homotopy analysis method. The applications of m-point fractional boundary problems can be found in various fields, including physics, engineering, finance, and biology. These problems are useful in modeling and analyzing phenomena that exhibit non-local behavior or involve memory effects (see [15,16,17,18]).
Pantograph equations are a set of differential equations that describe the motion of a pantograph, which is a mechanism used for copying and scaling drawings or diagrams. The equations are based on the assumption that the pantograph arms are rigid and do not deform during operation, we can simply say that see [19]. One important application of the pantograph equations is in the field of drafting and technical drawing. Before the advent of computer-aided design (CAD) software, pantographs were commonly used to produce scaled copies of drawings and diagrams. By adjusting the lengths of the arms and the position of the stylus, a pantograph can produce copies that are larger or smaller than the original [20], electrodynamics [21] and electrical pantograph of locomotive [22].
Many authors studied a huge number of positive solutions for nonlinear fractional BVP using fixed point theorems (FPTs) such as SFPT, Leggett-Williams and Guo-Krasnosel'skii (see [23,24]). Some studies addressed the sign-changing of solution of BVPs [25,26,27,28,29].
In this work, we use Schauder's fixed point theorem (SFPT) to solve the semipostone multipoint Ψ-Caputo fractional pantograph problem
Dν;ψrϰ(ς)+F(ς,ϰ(ς),ϰ(r+λς))=0, ς in (r,ℑ) | (1.1) |
ϰ(r)=ϑ1, ϰ(ℑ)=m−2∑i=1ζiϰ(ηi)+ϑ2, ϑi∈R, i∈{1,2}, | (1.2) |
where λ∈(0,ℑ−rℑ),Dν;ψr is Ψ-Caputo fractional derivative (Ψ-CFD) of order ν, 1<ν≤2, ζi∈R+(1≤i≤m−2) such that 0<Σm−2i=1ζi<1, ηi∈(r,ℑ), and F:[r,ℑ]×R×R→R.
The most important aspect of this research is to prove the existence of a positive solution of the above m-point FBVP. Note that in [30], the author considered a two-point BVP using Liouville-Caputo derivative.
The article is organized as follows. In the next section, we provide some basic definitions and arguments pertinent to fractional calculus (FC). Section 3 is devoted to proving the the main result and an illustrative example is given in Section 4.
In the sequel, Ψ denotes an increasing map Ψ:[r1,r2]→R via Ψ′(ς)≠0, ∀ ς, and [α] indicates the integer part of the real number α.
Definition 2.1. [4,5] Suppose the continuous function ϰ:(0,∞)→R. We define (RLFD) the Riemann-Liouville fractional derivative of order α>0,n=[α]+1 by
RLDα0+ϰ(ς)=1Γ(n−α)(ddς)n∫ς0(ς−τ)n−α−1ϰ(τ)dτ, |
where n−1<α<n.
Definition 2.2. [4,5] The Ψ-Riemann-Liouville fractional integral (Ψ-RLFI) of order α>0 of a continuous function ϰ:[r,ℑ]→R is defined by
Iα;Ψrϰ(ς)=∫ςr(Ψ(ς)−Ψ(τ))α−1Γ(α)Ψ′(τ)ϰ(τ)dτ. |
Definition 2.3. [4,5] The CFD of order α>0 of a function ϰ:[0,+∞)→R is defined by
Dαϰ(ς)=1Γ(n−α)∫ς0(ς−τ)n−α−1ϰ(n)(τ)dτ, α∈(n−1,n),n∈N. |
Definition 2.4. [4,5] We define the Ψ-CFD of order α>0 of a continuous function ϰ:[r,ℑ]→R by
Dα;Ψrϰ(ς)=∫ςr(Ψ(ς)−Ψ(τ))n−α−1Γ(n−α)Ψ′(τ)∂nΨϰ(τ)dτ, ς>r, α∈(n−1,n), |
where ∂nΨ=(1Ψ′(ς)ddς)n,n∈N.
Lemma 2.1. [4,5] Suppose q,ℓ>0, and ϰinC([r,ℑ],R). Then ∀ς∈[r,ℑ] and by assuming Fr(ς)=Ψ(ς)−Ψ(r), we have
1) Iq;ΨrIℓ;Ψrϰ(ς)=Iq+ℓ;Ψrϰ(ς),
2) Dq;ΨrIq;Ψrϰ(ς)=ϰ(ς),
3) Iq;Ψr(Fr(ς))ℓ−1=Γ(ℓ)Γ(ℓ+q)(Fr(ς))ℓ+q−1,
4) Dq;Ψr(Fr(ς))ℓ−1=Γ(ℓ)Γ(ℓ−q)(Fr(ς))ℓ−q−1,
5) Dq;Ψr(Fr(ς))k=0, k=0,…,n−1, n∈N, qin(n−1,n].
Lemma 2.2. [4,5] Let n−1<α1≤n,α2>0, r>0, ϰ∈L(r,ℑ), Dα1;Ψrϰ∈L(r,ℑ). Then the differential equation
Dα1;Ψrϰ=0 |
has the unique solution
ϰ(ς)=W0+W1(Ψ(ς)−Ψ(r))+W2(Ψ(ς)−Ψ(r))2+⋯+Wn−1(Ψ(ς)−Ψ(r))n−1, |
and
Iα1;ΨrDα1;Ψrϰ(ς)=ϰ(ς)+W0+W1(Ψ(ς)−Ψ(r))+W2(Ψ(ς)−Ψ(r))2+⋯+Wn−1(Ψ(ς)−Ψ(r))n−1, |
with Wℓ∈R, ℓ∈{0,1,…,n−1}.
Furthermore,
Dα1;ΨrIα1;Ψrϰ(ς)=ϰ(ς), |
and
Iα1;ΨrIα2;Ψrϰ(ς)=Iα2;ΨrIα1;Ψrϰ(ς)=Iα1+α2;Ψrϰ(ς). |
Here we will deal with the FDE solution of (1.1) and (1.2), by considering the solution of
−Dν;ψrϰ(ς)=h(ς), | (2.1) |
bounded by the condition (1.2). We set
Δ:=Ψ(ℑ)−Ψ(r)−Σm−2i=1ζi(Ψ(ηi)−Ψ(r)). |
Lemma 2.3. Let ν∈(1,2] and ς∈[r,ℑ]. Then, the FBVP (2.1) and (1.2) have a solution ϰ of the form
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)h(τ)Ψ′(τ)dτ, |
where
ϖ(ς,τ)=1Γ(ν){[(Ψ(ℑ)−Ψ(r))ν−1−Σm−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1]Ψ(ς)−Ψ(r)Δ−(Ψ(ς)−Ψ(τ))ν−1,τ≤ς,ηi−1<τ≤ηi,[(Ψ(ℑ)−Ψ(τ))ν−1−Σm−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1]Ψ(ℑ)−Ψ(r)Δ,ς≤τ,ηi−1<τ≤ηi, | (2.2) |
i=1,2,...,m−2.
Proof. According to the Lemma 2.2 the solution of Dν;ψrϰ(ς)=−h(ς) is given by
ϰ(ς)=−1Γ(ν)∫ςr(Ψ(ς)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+c0+c1(Ψ(ς)−Ψ(r)), | (2.3) |
where c0,c1∈R. Since ϰ(r)=ϑ1 and ϰ(ℑ)=∑m−2i=1ζiϰ(ηi)+ϑ2, we get c0=ϑ1 and
c1=1Δ(−1Γ(ν)m−2∑i=1ζi∫ηjr(Ψ(ηi)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+1Γ(ν)∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+ϑ1[m−2∑i=1ζi−1]+ϑ2). |
By substituting c0,c1 into Eq (2.3) we find,
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2−1Γ(ν)(∫ςr(Ψ(ς)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+(Ψ(ς)−Ψ(r))Δm−2∑i=1ζi∫ηjr(Ψ(ηi)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ−Ψ(ς)−Ψ(r)Δ∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2+∫ℑrϖ(ς,τ)h(τ)Ψ′(τ)dτ, |
where ϖ(ς,τ) is given by (2.2). Hence the required result.
Lemma 2.4. If 0<∑m−2i=1ζi<1, then
i) Δ>0,
ii) (Ψ(ℑ)−Ψ(τ))ν−1−∑m−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1>0.
Proof. i) Since ηi<ℑ, we have
ζi(Ψ(ηi)−Ψ(r))<ζi(Ψ(ℑ)−Ψ(r)), |
−m−2∑i=1ζi(Ψ(ηi)−Ψ(r))>−m−2∑i=1ζi(Ψ(ℑ)−Ψ(r)), |
Ψ(ℑ)−Ψ(r)−m−2∑i=1ζi(Ψ(ηi)−Ψ(r))>Ψ(ℑ)−Ψ(r)−m−2∑i=1ζi(Ψ(ℑ)−Ψ(r))=(Ψ(ℑ)−Ψ(r))[1−m−2∑i=1ζi]. |
If 1−Σm−2i=1ζi>0, then (Ψ(ℑ)−Ψ(r))−Σm−2i=1ζi(Ψ(ηi)−Ψ(r))>0. So we have Δ>0.
ii) Since 0<ν−1≤1, we have (Ψ(ηi)−Ψ(τ))ν−1<(Ψ(ℑ)−Ψ(τ))ν−1. Then we obtain
m−2∑j=iζj(Ψ(ηj)−Ψ(τ))ν−1<m−2∑j=iζj(Ψ(ℑ)−Ψ(τ))ν−1≤(Ψ(ℑ)−Ψ(τ))ν−1m−2∑i=1ζi<(Ψ(ℑ)−Ψ(τ))ν−1, |
and so
(Ψ(ℑ)−Ψ(τ))ν−1−m−2∑j=iζj(Ψ(ηj)−Ψ(τ))ν−1>0. |
Remark 2.1. Note that ∫ℑrϖ(ς,τ)Ψ′(τ)dτ is bounded ∀ς∈[r,ℑ]. Indeed
∫ℑr|ϖ(ς,τ)|Ψ′(τ)dτ≤1Γ(ν)∫ςr(Ψ(ς)−Ψ(τ))ν−1Ψ′(τ)dτ+Ψ(ς)−Ψ(r)Γ(ν)Δm−2∑i=1ζi∫ηir(Ψ(ηj)−Ψ(τ))ν−1Ψ′(τ)dτ+Ψ(ς)−Ψ(r)ΔΓ(ν)∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1Ψ′(τ)dτ=(Ψ(ς)−Ψ(r))νΓ(ν+1)+Ψ(ς)−Ψ(r)ΔΓ(ν+1)m−2∑i=1ζi(Ψ(ηi)−Ψ(r))ν+Ψ(ς)−Ψ(r)ΔΓ(ν+1)(Ψ(ℑ)−Ψ(r))ν≤(Ψ(ℑ)−Ψ(r))νΓ(ν+1)+Ψ(ℑ)−Ψ(r)ΔΓ(ν+1)m−2∑i=1ζi(Ψ(ηi)−Ψ(r))ν+(Ψ(ℑ)−Ψ(r))ν+1ΔΓ(ν+1)=M. | (2.4) |
Remark 2.2. Suppose Υ(ς)∈L1[r,ℑ], and w(ς) verify
{Dν;ψrw(ς)+Υ(ς)=0,w(r)=0, w(ℑ)=Σm−2i=1ζiw(ηi), | (2.5) |
then w(ς)=∫ℑrϖ(ς,τ)Υ(τ)Ψ′(τ)dτ.
Next we recall the Schauder fixed point theorem.
Theorem 2.1. [23] [SFPT] Consider the Banach space Ω. Assume ℵ bounded, convex, closed subset in Ω. If ϝ:ℵ→ℵ is compact, then it has a fixed point in ℵ.
We start this section by listing two conditions which will be used in the sequel.
● (Σ1) There exists a nonnegative function Υ∈L1[r,ℑ] such that ∫ℑrΥ(ς)dς>0 and F(ς,ϰ,v)≥−Υ(ς) for all (ς,ϰ,v)∈[r,ℑ]×R×R.
● (Σ2) G(ς,ϰ,v)≠0, for (ς,ϰ,v)∈[r,ℑ]×R×R.
Let ℵ=C([r,ℑ],R) the Banach space of CFs (continuous functions) with the following norm
‖ϰ‖=sup{|ϰ(ς)|:ς∈[r,ℑ]}. |
First of all, it seems that the FDE below is valid
Dν;ψrϰ(ς)+G(ς,ϰ∗(ς),ϰ∗(r+λς))=0, ς∈[r,ℑ]. | (3.1) |
Here the existence of solution satisfying the condition (1.2), such that G:[r,ℑ]×R×R→R
G(ς,z1,z2)={F(ς,z1,z2)+Υ(ς), z1,z2≥0,F(ς,0,0)+Υ(ς), z1≤0 or z2≤0, | (3.2) |
and ϰ∗(ς)=max{(ϰ−w)(ς),0}, hence the problem (2.5) has w as unique solution. The mapping Q:ℵ→ℵ accompanied with the (3.1) and (1.2) defined as
(Qϰ)(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)G(ς,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ, | (3.3) |
where the relation (2.2) define ϖ(ς,τ). The existence of solution of the problems (3.1) and (1.2) give the existence of a fixed point for Q.
Theorem 3.1. Suppose the conditions (Σ1) and (Σ2) hold. If there exists ρ>0 such that
[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≤ρ, |
where L≥max{|G(ς,ϰ,v)|:ς∈[r,ℑ], |ϰ|,|v|≤ρ} and M is defined in (2.4), then, the problems (3.1) and (3.2) have a solution ϰ(ς).
Proof. Since P:={ϰ∈ℵ:‖ϰ‖≤ρ} is a convex, closed and bounded subset of B described in the Eq (3.3), the SFPT is applicable to P. Define Q:P→ℵ by (3.3). Clearly Q is continuous mapping. We claim that range of Q is subset of P. Suppose ϰ∈P and let ϰ∗(ς)≤ϰ(ς)≤ρ, ∀ς∈[r,ℑ]. So
|Qϰ(ς)|=|[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)G(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ|≤[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≤ρ, |
for all ς∈[r,ℑ]. This indicates that ‖Qϰ‖≤ρ, which proves our claim. Thus, by using the Arzela-Ascoli theorem, Q:ℵ→ℵ is compact. As a result of SFPT, Q has a fixed point ϰ in P. Hence, the problems (3.1) and (1.2) has ϰ as solution.
Lemma 3.1. ϰ∗(ς) is a solution of the FBVP (1.1), (1.2) and ϰ(ς)>w(ς) for every ς∈[r,ℑ] iff the positive solution of FBVP (3.1) and (1.2) is ϰ=ϰ∗+w.
Proof. Let ϰ(ς) be a solution of FBVP (3.1) and (1.2). Then
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)G(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)(F(τ,ϰ∗(τ),ϰ∗(r+λτ))+p(τ))Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ+1Γ(ν)∫ℑrϖ(ς,τ)p(τ)Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)G(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ+w(ς). |
So,
ϰ(ς)−w(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ. |
Then we get the existence of the solution with the condition
ϰ∗(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ. |
For the converse, if ϰ∗ is a solution of the FBVP (1.1) and (1.2), we get
Dν;ψr(ϰ∗(ς)+w(ς))=Dν;ψrϰ∗(ς)+Dν;ψrw(ς)=−F(ς,ϰ∗(ς),ϰ∗(r+λς))−p(ς)=−[F(ς,ϰ∗(ς),ϰ∗(r+λς))+p(ς)]=−G(ς,ϰ∗(ς),ϰ∗(r+λς)), |
which leads to
Dν;ψrϰ(ς)=−G(ς,ϰ∗(ς),ϰ∗(r+λς)). |
We easily see that
ϰ∗(r)=ϰ(r)−w(r)=ϰ(r)−0=ϑ1, |
i.e., ϰ(r)=ϑ1 and
ϰ∗(ℑ)=m−2∑i=1ζiϰ∗(ηi)+ϑ2, |
ϰ(ℑ)−w(ℑ)=m−2∑i=1ζiϰ(ηi)−m−2∑i=1ζjw(ηi)+ϑ2=m−2∑i=1ζi(ϰ(ηi)−w(ηi))+ϑ2. |
So,
ϰ(ℑ)=m−2∑i=1ζiϰ(ηi)+ϑ2. |
Thus ϰ(ς) is solution of the problem FBVP (3.1) and (3.2).
We propose the given FBVP as follows
D75ϰ(ς)+F(ς,ϰ(ς),ϰ(1+0.5ς))=0, ς∈(1,e), | (4.1) |
ϰ(1)=1, ϰ(e)=17ϰ(52)+15ϰ(74)+19ϰ(115)−1. | (4.2) |
Let Ψ(ς)=logς, where F(ς,ϰ(ς),ϰ(1+12ς))=ς1+ςarctan(ϰ(ς)+ϰ(1+12ς)).
Taking Υ(ς)=ς we get ∫e1ςdς=e2−12>0, then the hypotheses (Σ1) and (Σ2) hold. Evaluate Δ≅0.366, M≅3.25 we also get |G(ς,ϰ,v)|<π+e=L such that |ϰ|≤ρ, ρ=17, we could just confirm that
[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≅16.35≤17. | (4.3) |
By applying the Theorem 3.1 there exit a solution ϰ(ς) of the problem (4.1) and (4.2).
In this paper, we have provided the proof of BVP solutions to a nonlinear Ψ-Caputo fractional pantograph problem or for a semi-positone multi-point of (1.1) and(1.2). What's new here is that even using the generalized Ψ-Caputo fractional derivative, we were able to explicitly prove that there is one solution to this problem, and that in our findings, we utilize the SFPT. The results obtained in our work are significantly generalized and the exclusive result concern the semi-positone multi-point Ψ-Caputo fractional differential pantograph problem (1.1) and (1.2).
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Small Groups (RGP.1/350/43).
The authors declare no conflict of interest.
[1] |
Witkowska AM, Waśkiewicz A, Zujko ME, et al. (2019) The consumption of nuts is associated with better dietary and lifestyle patterns in polish adults: results of WOBASZ and WOBASZ Ⅱ surveys. Nutrients 11: 1410. https://doi.org/10.3390/nu11061410 doi: 10.3390/nu11061410
![]() |
[2] |
Micha R, Khatibzadeh S, Shi P, et al. (2015) Global, regional and national consumption of major food groups in 1990 and 2010: a systematic analysis including 266 country-specific nutrition surveys worldwide. BMJ Open 5: e008705. https://doi.org/10.1136/bmjopen-2015-008705 doi: 10.1136/bmjopen-2015-008705
![]() |
[3] |
Neale EP, Tran G, Brown RC (2020) Barriers and facilitators to nut consumption: A narrative review. Int J Environ Res Public Health 17: 9127. https://doi.org/10.3390/ijerph17239127 doi: 10.3390/ijerph17239127
![]() |
[4] |
Blomhoff R, Carlsen MH, Andersen LF, et al. (2006) Health benefits of nuts: potential role of antioxidants. Br J Nutr 96: S52-S60. https://doi.org/10.1017/BJN20061864 doi: 10.1017/BJN20061864
![]() |
[5] |
Sheridan MJ, Cooper JN, Erario M, et al. (2007) Pistachio nut consumption and serum lipid levels. J Am Coll Nutr 26: 141-148. https://doi.org/10.1080/07315724.2007.10719595 doi: 10.1080/07315724.2007.10719595
![]() |
[6] |
Varga J, Frisvad JC, Samson R (2011) Two new aflatoxin producing species, and an overview of Aspergillus section Flavi. Stud Mycol 69: 57-80. https://doi.org/10.3114/sim.2011.69.05 doi: 10.3114/sim.2011.69.05
![]() |
[7] |
Pildain MB, Frisvad JC, Vaamonde G, et al. (2008) Two novel aflatoxin-producing Aspergillus species from Argentinean peanuts. Int J Syst Evol Microbiol 58: 725-735. https://doi.org/10.1099/ijs.0.65123-0 doi: 10.1099/ijs.0.65123-0
![]() |
[8] |
Mutegi C, Ngugi H, Hendriks S, et al. (2009) Prevalence and factors associated with aflatoxin contamination of peanuts from Western Kenya. Int J Food Microbiol 130: 27-34. https://doi.org/10.1016/j.ijfoodmicro.2008.12.030 doi: 10.1016/j.ijfoodmicro.2008.12.030
![]() |
[9] |
Ammida N, Micheli L, Piermarini S, et al. (2006) Detection of aflatoxin B1 in barley: Comparative study of immunosensor and HPLC. Anal Lett 39: 1559-1572. https://doi.org/10.1080/00032710600713248 doi: 10.1080/00032710600713248
![]() |
[10] |
Afsah-Hejri L, Jinap S, Arzandeh S, et al. (2011) Optimization of HPLC conditions for quantitative analysis of aflatoxins in contaminated peanut. Food Control 22: 381-388. https://doi.org/10.1016/j.foodcont.2010.09.007 doi: 10.1016/j.foodcont.2010.09.007
![]() |
[11] |
Misihairabgwi J, Ezekiel C, Sulyok M, et al. (2019) Mycotoxin contamination of foods in Southern Africa: A 10-year review (2007-2016). Crit Rev Food Sci Nutr 59: 43-58. https://doi.org/10.1080/10408398.2017.1357003 doi: 10.1080/10408398.2017.1357003
![]() |
[12] |
Sharma S, Gupta D, Sharma YP (2015) Natural Incidence of Aflatoxins, Ochratoxin A, Patulin and Their Co-Occurrence in Chilgoza Pine Nuts Marketed in Jammu, India. Proc Natl Acad Sci India Sect B Biol Sci 85: 45-50. https://doi.org/10.1007/s40011-014-0326-7 doi: 10.1007/s40011-014-0326-7
![]() |
[13] |
Wang Y, Nie J, Yan Z, et al. (2018) Occurrence and co-occurrence of mycotoxins in nuts and dried fruits from China. Food Control 88: 181-189. https://doi.org/10.1016/j.foodcont.2018.01.013 doi: 10.1016/j.foodcont.2018.01.013
![]() |
[14] | Kazemi A, Mohtadi Nia J, Vahed Jabbari M, et al. (2007) Survey of food stuff to carcinogenic mycotoxins. The 3rd Congress of Medical Plants 24: 386. |
[15] |
Cheraghali A, Yazdanpanah H, Doraki N, et al. (2007) Incidence of aflatoxins in Iran pistachio nuts. Food Chem Toxicol 45: 812-816. https://doi.org/10.1016/j.fct.2006.10.026 doi: 10.1016/j.fct.2006.10.026
![]() |
[16] |
Medina A, Gilbert MK, Mack BM, et al. (2017) Interactions between water activity and temperature on the Aspergillus flavus transcriptome and aflatoxin B1 production. Int J Food Microbiol 256: 36-44. https://doi.org/10.1016/j.ijfoodmicro.2017.05.020 doi: 10.1016/j.ijfoodmicro.2017.05.020
![]() |
[17] |
Petroczi A, Nepusz T, Taylor G, et al. (2011) Network analysis of the RASFF database: a mycotoxin perspective. World Mycotoxin J 4: 329-338. https://doi.org/10.3920/WMJ2010.1271 doi: 10.3920/WMJ2010.1271
![]() |
[18] |
Molyneux RJ, Mahoney N, Kim JH, et al. (2007) Mycotoxins in edible tree nuts. Int J Food Microbiol 119: 72-78. https://doi.org/10.1016/j.ijfoodmicro.2007.07.028 doi: 10.1016/j.ijfoodmicro.2007.07.028
![]() |
[19] | International Agency for Research on Cancer (1993) Some naturally occurring substances: Food items and constituents, heterocyclic aromatic amines and mycotoxins: World Health Organization. |
[20] | IARC (2012) Aflatoxins, IARC Monographs on the Evaluation of Carcinogenic Risks on Humans. |
[21] | Wogan G (1975) Dietary factors and special epidemiological situations of liver cancer in Thailand and Africa. Cancer Res 35: 3499-3502. |
[22] | International Agency for Research on Cancer (2002) Some traditional herbal medicines, some mycotoxins, naphthalene and styrene: World Health Organization. |
[23] |
Villa P, Markaki P (2009) Aflatoxin B1 and ochratoxin A in breakfast cereals from Athens market: occurrence and risk assessment. Food Control 20: 455-461. https://doi.org/10.1016/j.foodcont.2008.07.012 doi: 10.1016/j.foodcont.2008.07.012
![]() |
[24] |
Chiavaro E, Dall'Asta C, Galaverna G, et al. (2001) New reversed-phase liquid chromatographic method to detect aflatoxins in food and feed with cyclodextrins as fluorescence enhancers added to the eluent. J Chromatogr A 937: 31-40. https://doi.org/10.1016/S0021-9673(01)01300-0 doi: 10.1016/S0021-9673(01)01300-0
![]() |
[25] |
Juan C, Ritieni A, Mañes J (2012) Determination of trichothecenes and zearalenones in grain cereal, flour and bread by liquid chromatography tandem mass spectrometry. Food Chem 134: 2389-2397. https://doi.org/10.1016/j.foodchem.2012.04.051 doi: 10.1016/j.foodchem.2012.04.051
![]() |
[26] |
Yu FY, Gribas AV, Vdovenko MM, et al. (2013) Development of ultrasensitive direct chemiluminescent enzyme immunoassay for determination of aflatoxin B1 in food products. Talanta 107: 25-29. https://doi.org/10.1016/j.talanta.2012.12.047 doi: 10.1016/j.talanta.2012.12.047
![]() |
[27] | Ghaednia B, Bayat M, Sohrabi Haghdoost I, et al. (2013) Effects of aflatoxin B1 on growth performance, health indices, phagocytic activity and histopathological alteration in Fenneropenaeus indicus. Iran J Fish Sci 12: 723-737. |
[28] |
Nakai VK, de Oliveira Rocha L, Gonçalez E, et al. (2008) Distribution of fungi and aflatoxins in a stored peanut variety. Food Chem 106: 285-290. https://doi.org/10.1016/j.foodchem.2007.05.087 doi: 10.1016/j.foodchem.2007.05.087
![]() |
[29] |
Marin S, Ramos A, Cano-Sancho G, et al. (2013) Mycotoxins: Occurrence, toxicology, and exposure assessment. Food Chem Toxicol 60: 218-237. https://doi.org/10.1016/j.fct.2013.07.047 doi: 10.1016/j.fct.2013.07.047
![]() |
[30] |
Chain EPoCitF, Schrenk D, Bignami M, et al. (2020) Risk assessment of aflatoxins in food. EFSA Journal 18: e06040. https://doi.org/10.2903/j.efsa.2020.6040 doi: 10.2903/j.efsa.2020.6040
![]() |
[31] |
Jahanbakhsh M, Afshar A, Momeni Feeli S, et al. (2021) Probabilistic health risk assessment (Monte Carlo simulation method) and prevalence of aflatoxin B1 in wheat flours of Iran. Int J Environ Anal Chem 101: 1074-1085. https://doi.org/10.1080/03067319.2019.1676421 doi: 10.1080/03067319.2019.1676421
![]() |
[32] |
Karimi F, Shariatifar N, Rezaei M, et al. (2021) Quantitative measurement of toxic metals and assessment of health risk in agricultural products food from Markazi Province of Iran. Int J Food Contam 8: 2. https://doi.org/10.1186/s40550-021-00083-0 doi: 10.1186/s40550-021-00083-0
![]() |
[33] |
Kiani A, Ahmadloo M, Moazzen M, et al. (2021) Monitoring of polycyclic aromatic hydrocarbons and probabilistic health risk assessment in yogurt and butter in Iran. Food Sci Nutr 9: 2114-2128. https://doi.org/10.1002/fsn3.2180 doi: 10.1002/fsn3.2180
![]() |
[34] |
Yaminifar S, Aeenehvand S, Ghelichkhani G, et al. (2021) The measurement and health risk assessment of polychlorinated biphenyls in butter samples using the QuEChERS/GC-MS method. Int J Dairy Technol 74:737-46. https://doi.org/10.1111/1471-0307.12805 doi: 10.1111/1471-0307.12805
![]() |
[35] | EPA (2015) United States Environmental Protection Agency, Quantitative Risk Assessment Calculations, National Academy Press, Washington 13: 7-9. |
[36] |
Nabizadeh S, Shariatifar N, Shokoohi E, et al. (2018) Prevalence and probabilistic health risk assessment of aflatoxins B 1, B 2, G 1, and G 2 in Iranian edible oils. Environ Sci Pollut Res 25: 35562-35570. https://doi.org/10.1007/s11356-018-3510-0 doi: 10.1007/s11356-018-3510-0
![]() |
[37] | Akbari-Adergani B, Poorasad M, Esfandiari Z (2018) Sunset yellow, tartrazine and sodium benzoate in orange juice distributed in Iranian market and subsequent exposure assessment. Int Food Res J 25: 975-981. |
[38] |
Rasouli Z, Akbari-Adergani B (2016) Assessment of aspartame exposure due to consumption of some imported chewing gums by microwave digestion and high performance liquid chromatography analysis. Orient J Chem 32: 1649-1658. https://doi.org/10.13005/ojc/320342 doi: 10.13005/ojc/320342
![]() |
[39] | EPA (2010) Application of the Margin of Exposure (MoE) Approach to Substances in Food that are Genotoxic and Carcinogenic Example: Benzo[a]pyrene and polycyclic aromatic hydrocarbons, Application of the Margin of Exposure (MoE) Approach to Substances in Food that are Genotoxic and Carcinogenic Example: Benzo[a]pyrene and polycyclic aromatic hydrocarbons 48: 2-24. https://doi.org/10.1016/j.fct.2009.09.039 |
[40] |
Authority EFS (2007) Opinion of the scientific panel on contaminants in the food chain[CONTAM] related to the potential increase of consumer health risk by a possible increase of the existing maximum levels for aflatoxins in almonds, hazelnuts and pistachios and derived products. EFSA Journal 5: 446. https://doi.org/10.2903/j.efsa.2007.446 doi: 10.2903/j.efsa.2007.446
![]() |
[41] |
Foerster C, Muñoz K, Delgado-Rivera L, et al. (2020) Occurrence of relevant mycotoxins in food commodities consumed in Chile. Mycotoxin Res 36: 63-72. https://doi.org/10.1007/s12550-019-00369-5 doi: 10.1007/s12550-019-00369-5
![]() |
[42] | US Environmental Protection Agency (2001) Risk Assessment Guidance for Superfund (RAGS) Volume Ⅲ. Available from: https://www.epa.gov/sites/default/files/2015-09/documents/rags3adt_complete.pdf |
[43] |
Saghafi M, Shariatifar N, Alizadeh Sani M, et al. (2021) Analysis and probabilistic health risk assessment of some trace elements contamination and sulphur dioxide residual in raisins. Int J Environ Anal Chem 2021: 1986037. https://doi.org/10.1080/03067319.2021.1986037 doi: 10.1080/03067319.2021.1986037
![]() |
[44] |
Moazzen M, Shariatifar N, Arabameri M, et al. (2022) Measurement of Polycyclic Aromatic Hydrocarbons in Baby Food Samples in Tehran, Iran With Magnetic-Solid-Phase-Extraction and Gas-Chromatography/Mass-Spectrometry Method: A Health Risk Assessment. Front Nutr 9: 833158. https://doi.org/10.3389/fnut.2022.833158 doi: 10.3389/fnut.2022.833158
![]() |
[45] |
Shariatifar N, Moazzen M, Arabameri M, et al. (2021) Measurement of polycyclic aromatic hydrocarbons (PAHs) in edible mushrooms (raw, grilled and fried) using MSPE-GC/MS method: a risk assessment study. Appl Biol Chem 64: 61. https://doi.org/10.1186/s13765-021-00634-1 doi: 10.1186/s13765-021-00634-1
![]() |
[46] |
Rezaei H, Moazzen M, Shariatifar N, et al. (2021) Measurement of phthalate acid esters in non-alcoholic malt beverages by MSPE-GC/MS method in Tehran city: chemometrics. Environ Sci Pollut Res 28: 51897-51907. https://doi.org/10.1007/s11356-021-14290-x doi: 10.1007/s11356-021-14290-x
![]() |
[47] |
Yaminifar S, Aeenehvand S, Ghelichkhani G, et al. (2021) The measurement and health risk assessment of polychlorinated biphenyls in butter samples using the QuEChERS/GC-MS method. Int J Dairy Technol 74: 737-46. https://doi.org/10.1111/1471-0307.12805 doi: 10.1111/1471-0307.12805
![]() |
[48] |
Moradi M, Bolandi M, Arabameri M, et al. (2021) Semi-volume gluten-free bread: effect of guar gum, sodium caseinate and transglutaminase enzyme on the quality parameters. J Food Meas Charact 15: 2344-51. https://doi.org/10.1007/s11694-021-00823-y doi: 10.1007/s11694-021-00823-y
![]() |
[49] |
Heydarieh A, Arabameri M, Ebrahimi A, et al. (2020) Determination of Magnesium, Calcium and Sulphate Ion Impurities in Commercial Edible Salt. J Chem Health Risks 10: 93-102. https://doi.org/ 10.22034/jchr.2020.1883343.1067 doi: 10.22034/jchr.2020.1883343.1067
![]() |
[50] | Diella G, Caggiano G, Ferrieri F, et al. (2018) Aflatoxin contamination in nuts marketed in Italy: Preliminary results. Ann Ig 30: 401-409. |
[51] |
Ding XX, Li PW, Bai YZ, et al. (2012) Aflatoxin B-1 in post-harvest peanuts and dietary risk in China. Food Control 23: 143-148. https://doi.org/10.1016/j.foodcont.2011.06.026 doi: 10.1016/j.foodcont.2011.06.026
![]() |
[52] |
El tawila MM, Neamatallah A, Serdar SA (2013) Incidence of aflatoxins in commercial nuts in the holy city of Mekkah. Food Control 29: 121-124. https://doi.org/10.1016/j.foodcont.2012.06.004 doi: 10.1016/j.foodcont.2012.06.004
![]() |
[53] |
Christofidou M, Kafouris D, Christodoulou M, et al. (2015) Occurrence, surveillance, and control of mycotoxins in food in Cyprus for the years 2004-2013. Food Agric Immunol 26: 880-895. https://doi.org/10.1080/09540105.2015.1039499 doi: 10.1080/09540105.2015.1039499
![]() |
[54] |
Hoeltz M, Einloft TC, Oldoni VP, et al. (2012) The occurrence of aflatoxin B 1 contamination in peanuts and peanut products marketed in southern brazil. Braz Arch Biol Technol 55: 313-317. https://doi.org/10.1590/S1516-89132012000200019 doi: 10.1590/S1516-89132012000200019
![]() |
[55] | Commission CA (1995) Codex general standard for contaminants and toxins in food and feed. Codex Standard 193: 1-14. |
[56] |
Fernández Pinto V, Patriarca A, Locani O, et al. (2001) Natural co-occurrence of aflatoxin and cyclopiazonic acid in peanuts grown in Argentina. Food Addit Contam 18: 1017-1020. https://doi.org/10.1080/02652030110057125 doi: 10.1080/02652030110057125
![]() |
[57] |
Freitas VP, Brigido BM (1998) Occurrence of aflatoxins B1, B2, G1, and G2 in peanuts and their products marketed in the region of Campinas, Brazil in 1995 and 1996. Food Addit Contam 15: 807-811. https://doi.org/10.1080/02652039809374714 doi: 10.1080/02652039809374714
![]() |
[58] |
Liu Y, Wu F (2010) Global burden of aflatoxin-induced hepatocellular carcinoma: a risk assessment. Environ Health Perspect 118: 818-824. https://doi.org/10.1289/ehp.0901388 doi: 10.1289/ehp.0901388
![]() |
[59] |
Sabran MR, Jamaluddin R, Mutalib MSA (2012) Screening of aflatoxin M1, a metabolite of aflatoxin B1 in human urine samples in Malaysia: a preliminary study. Food Control 28: 55-58. https://doi.org/10.1016/j.foodcont.2012.04.048 doi: 10.1016/j.foodcont.2012.04.048
![]() |
[60] |
Zhang W, Liu Y, Liang B, et al. (2020) Probabilistic risk assessment of dietary exposure to aflatoxin B 1 in Guangzhou, China. Sci Rep 10: 7973. https://doi.org/10.1038/s41598-020-64295-8 doi: 10.1038/s41598-020-64295-8
![]() |
[61] |
Taghizadeh SF, Rezaee R, Badibostan H, et al. (2020) Aflatoxin B1 in walnuts: a probabilistic cancer risk assessment for Iranians. Toxicol Environ Chem 2020: 1791868. https://doi.org/10.1080/02772248.2020.1791868 doi: 10.1080/02772248.2020.1791868
![]() |
[62] |
Do TH, Tran SC, Le CD, et al. (2020) Dietary exposure and health risk characterization of aflatoxin B1, ochratoxin A, fumonisin B1, and zearalenone in food from different provinces in Northern Vietnam. Food Control 112: 107108. https://doi.org/10.1016/j.foodcont.2020.107108 doi: 10.1016/j.foodcont.2020.107108
![]() |
[63] |
Cano-Sancho G, Marin S, Ramos A, et al. (2010) Biomonitoring of Fusarium spp. mycotoxins: perspectives for an individual exposure assessment tool. Food Sci Technol Int 16: 266-276. https://doi.org/10.1177/1082013210368884 doi: 10.1177/1082013210368884
![]() |
![]() |
![]() |