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Review

Ferroptosis-molecular mechanisms and newer insights into some diseases

  • Ferroptosis is a recently discovered iron dependent form of programmed cell death, characterized by accumulation of lipid reactive oxygen species (ROS). It shows a strikingly different set of morphological characteristics from other forms of cell death, like reduced mitochondrial volume, increased bi-layer membrane density, and reduction of mitochondrial cristae with absence of any nuclear changes. Ferroptosis is mainly regulated by two core biochemical processes, namely iron accumulation and lipid peroxidation. Lipid peroxides exert their toxic effects by disturbing the integrity, structure and composition of bi-lipid cell membranes. However, being highly reactive compounds, they further propagate the generation of ROS, leading to cross-linking of DNA and proteins. Key regulators of ferroptosis include various genes involved in the above pathways, inhibition of the antioxidant system and upregulation of the oxidant system. Recent studies have shown the ferroptotic pathway to be involved in the patho-physiology of many diseases, including cancer. Understanding the biochemical mechanisms and key substances upregulating/inhibiting this pathway, may have an implication towards development of targeted therapies for various cancers, and, hence, has become a hotspot for biomedical research. This review article summarizes the core biochemical processes involved in ferroptosis, with a brief summary of its role in various diseases and possible therapeutic targets.

    Citation: Naba Hasan, Waleem Ahmad, Feroz Alam, Mahboob Hasan. Ferroptosis-molecular mechanisms and newer insights into some diseases[J]. AIMS Molecular Science, 2023, 10(1): 22-36. doi: 10.3934/molsci.2023003

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  • Ferroptosis is a recently discovered iron dependent form of programmed cell death, characterized by accumulation of lipid reactive oxygen species (ROS). It shows a strikingly different set of morphological characteristics from other forms of cell death, like reduced mitochondrial volume, increased bi-layer membrane density, and reduction of mitochondrial cristae with absence of any nuclear changes. Ferroptosis is mainly regulated by two core biochemical processes, namely iron accumulation and lipid peroxidation. Lipid peroxides exert their toxic effects by disturbing the integrity, structure and composition of bi-lipid cell membranes. However, being highly reactive compounds, they further propagate the generation of ROS, leading to cross-linking of DNA and proteins. Key regulators of ferroptosis include various genes involved in the above pathways, inhibition of the antioxidant system and upregulation of the oxidant system. Recent studies have shown the ferroptotic pathway to be involved in the patho-physiology of many diseases, including cancer. Understanding the biochemical mechanisms and key substances upregulating/inhibiting this pathway, may have an implication towards development of targeted therapies for various cancers, and, hence, has become a hotspot for biomedical research. This review article summarizes the core biochemical processes involved in ferroptosis, with a brief summary of its role in various diseases and possible therapeutic targets.



    Fractional calculus (FC), also known as non-integer calculus, has been widely studied in recent over the past decades (beginning 1695) in fields of applied sciences and engineering. FC deals with fractional-order integral and differential operators, which establishes phenomenon model as an increasingly realistic tool for real-world problems. In addition, it has been properly specified the term "memory" particularly in mathematics, physics, chemistry, biology, mechanics, electricity, finance, economics and control theory, we recommend these books to readers who require to learn more about the core ideas of fractional operators [1,2,3,4,5,6]. However, recently, several types of fractional operators have been employed in research education that mostly focus on the Riemann-Liouville (RL) [3], Caputo [3], Hadamard [3], Katugampola [7], conformable [8] and generalized conformable [9]. In 2017, Jarad et al. [10] introduced generalized RL and Caputo proportional fractional derivatives including exponential functions in their kernels. After that, in 2021, new fractional operators combining proportional and classical differintegrals have been introduced in [11]. Moreover, Akgül and Baleanu [12] studied the stability analysis and experiments of the proportional Caputo derivative.

    Recently, one type of fractional operator that is popular with researchers now is proportional fractional derivative and integral operators (PFDOs/PFIOs) with respect to another function; for more details see [13,14]. For α>0, ρ(0,1], ψC1([a,b]), ψ>0, the PFIO of order α of hL1([a,b]) with respect to ψ is given as:

    ρIα,ψa[h(t)]=1ραΓ(α)taρHα1ψ(t,s)[h(s)]ψ(s)ds, (1.1)

    where Γ(α)=0sα1esds, s>0, and

    ρHα1ψ(t,s)=eρ1ρ(ψ(t)ψ(s))(ψ(t)ψ(s))α1. (1.2)

    The Riemann-Liouville proportional fractional derivative (RL-PFD) of order α of hCn([a,b]) with respect to ψ is given by

    ρDα,ψa[h(t)]=ρDn,ψρaInα,ψ[h(t)]=ρDn,ψtρnαΓ(nα)taρHnα1ψ(t,s)[h(s)]ψ(s)ds, (1.3)

    with n=[α]+1, where [α] denotes the integer part of order α, ρDn,ψ=ρDψρDψρDψntimes, and ρDψ[h(t)]=(1ρ)h(t)+ρh(t))/ψ(t). The PFD in Caputo type is given as in

    CρDα,ψa[h(t)]=ρInα,ψa[ρDn,ψ[h(t)]]=ραnΓ(nα)taρHnα1ψ(t,s)ρDn,ψ[h(s)]ψ(s)ds. (1.4)

    The relation of PFI and Caputo-PFD which will be used in this manuscript as

    ρIα,ψa[CρDα,ψa[h(t)]]=h(t)n1k=0ρDk,ψ[h(a)]ρkk!ρHk+1ψ(t,a). (1.5)

    Moreover, for α, β>0 and ρ(0,1], we have the following properties

    ρIα,ψa[ρHβ1ψ(t,a)]=Γ(β)ραΓ(β+α)ρHβ+α1ψ(t,a), (1.6)
    CρDα,ψa+[ρHβ1ψ(t,a)]=ραΓ(β)Γ(βα)ρHβα1ψ(t,a). (1.7)

    Notice that if we set ρ=1 in (1.1), (1.3) and (1.4), then we have the RL-fractional operators [3] with ψ(t)=t, the Hadamard fractional operators [3] with ψ(t)=logt, the Katugampola fractional operators [7] with ψ(t)=tμ/μ, μ>0, the conformable fractional operators [8] with ψ(t)=(ta)μ/μ, μ>0 and the generalized conformable fractional operators [9] with ψ(t)=tμ+ϕ/(μ+ϕ), respectively. Recent interesting results on PFOs with respect to another function could be mention in [15,16,17,18,19,20,21,22,23,24,25].

    Exclusive investigations in concepts of qualitative property in fractional-order differential equations (FDEs) have recently gotten a lot of interest from researchers as existence property (EP) and Ulam's stability (US). The EP of solutions for FDEs with initial or boundary value conditions has been investigated applying classical/modern fixed point theorems (FPTs). As we know, US is four types like Hyers-Ulam stability (HU), generalized Hyers-Ulam stability (GHU), Hyers-Ulam-Rassias stability (HUR) and generalized Hyers-Ulam-Rassias stability (GHUR). Because obtaining accurate solutions to fractional differential equations problems is extremely challenging, it is beneficial in various of optimization applications and numerical analysis. As a result, it is requisite to develop concepts of US for these issues, since studying the properties of US does not need us to have accurate solutions to the proposed problems. This qualitative theory encourages us to obtain an efficient and reliable technique for solving fractional differential equations because there exists a close exact solution when the purpose problem is Ulam stable. We suggest some interesting papers about qualitative results of fractional initial/boundary value problems (IVPs/BVPs) involving many types of non-integer order, see [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43] and references therein.

    We are going to present some of the researches that inspired this manuscript. In recent years, pantograph equation (PE) is a type of proportional delay differential equation emerging in deterministic situations which first studied by Ockendon and Taylor [44]:

    {u(t)=μu(t)+κu(λt),a<t<T,u(0)=u0=A,0<λ<1,μ,κR. (1.8)

    The problem (1.8) has been a broad area of applications in applied branchs such as science, medicine, engineering and economics that use the sake of PEs to model some phenomena of the problem at present which depend on the previous states. For more evidences of PEs, see [45,46,47,48,49,50,51]. There are many researches of literature on nonlinear fractional differential equations involving a specific function with initial, boundary, or nonlocal conditions, for examples in 2013, Balachandran et al. [52] discussed the initial nonlinear PEs as follows:

    {CDα[u(t)]=h(t,u(t),u(σt)),α(0,1],0<t<1,u(0)=u0,0<σ<1, (1.9)

    where u0R, CDα denotes the Caputo fractional derivative of order α and hC([0,1]×R2,R). FC and FPTs were applied to discuss the existence properties of the solutions in their work. In 2018, Harikrishman and co-workers [53] examined the existence properties of ψ-Hilfer fractional derivative for nonlocal problem for PEs:

    {HDα,β;ψa+[u(t)]=h(t,u(t),u(σt)),t(a,b),σ(0,1),α(0,1)I1γ;ψa+[u(a)]=ki=1ciu(τi),τi(a,b],αγ=α+βαβ, (1.10)

    where HDα,β;ψa+ represents the ψ-Hilfer fractional derivative of order α and type β[0,1], I1γ;ψa+ is RL-fractional integral of order 1γ with respect to ψ so that ψ>0 and hC([a,b]×R2,R). Asawasamrit and co-workers [54] used Schaefer's and Banach's FPTs to establish the existence properties of FDEs with mixed nonlocal conditions (MNCs) in 2019. In 2021, Boucenna et al. [55] investigated the existence and uniqueness theorem of solutions for a generalized proportional Caputo fractional Cauchy problem. They solved the proposed problem based on the decomposition formula. Amongst important fractional equations, one of the most interesting equations is the fractional integro-differential equations, which provide massive freedom to explain processes involving memory and hereditary properties, see [56,57].

    Recognizing the importance of all parts that we mentioned above, motivated us to generate this paper which deals with the qualitative results to the Caputo proportional fractional integro-differential equation (PFIDE) with MNCs:

    {CρDα,ψa+[u(t)]=f(t,u(t),u(λt),ρIω,ψa+[u(λt)],CρDα,ψa+[u(λt)]),t(a,T),mi=1γiu(ηi)+nj=1κjCρDβj,ψa+[u(ξj)]+kr=1σrρIδr,ψa+[u(θr)]=A, (1.11)

    where CρDq,ψa+ is the Caputo-PFDO with respect to another increasing differentiable function ψ of order q={α,βj} via 0<βj<α1, j=1,2,,n, 0<ρ1, 0<λ<1, ρIp,ψa+ is the PFIO with respect to another increasing differentiable function ψ of order p={ω,δr}>0 for r=1,2,,k, 0<ρ1, γi,κj,σr,AR, 0aηi,ξj,θrT, i=1,2,,m, fC(J×R4,R), J=[a,T]. We use the help of the famous FPTs like Banach's, Leray-Schauder's nonlinear alternative and Krasnoselskii's to discuss the existence properties of the solutions for (1.11). Moreover, we employ the context of different kinds of US to discuss the stability analysis. The results are well demonstrated by numerical examples at last section.

    The advantage of defining MNCs of the problem (1.11) is it covers many cases as follows:

    ● If we set κj=σr=0, then (1.11) is deducted to the proportional multi-point problem.

    ● If we set γi=σr=0, then (1.11) is deducted to the PFD multi-point problem.

    ● If we set γi=κj=0, then (1.11) is deducted to the PFI multi-point problem.

    ● If we set α=ρ=1, (1.11) emerges in nonlocal problems [58].

    This work is collected as follows. Section 2 provides preliminary definitions. The existence results of solutions for (1.11) is studied in Section 3. In Section 4, stability analysis of solution for (1.11) in frame of HU, GHU, HUR and GHUR are given is established. Section 5 contains the example to illustrate the theoretical results. In addition, the summarize is provided in the last part.

    Before proving, assume that E=C(J,R) is the Banach space of all continuous functions from J into R provided with u=suptJ{|u(t)|}. The symbol ρIq,ψa+[Fu(s)(c)] means that

    ρIq,ρ,ψa+[Fu(c)]=1ρqΓ(q)caρHq1ψ(c,s)[Fu(s)]ψ(s)ds,

    where q={α,αβj,α+δr}, c={t,ηi,ξj,θr}, and

    Fu(t):=f(t,u(t),u(λt),ρIω,ψa+[u(λt)],Fu(λt)). (2.1)

    In order to convert the considered problem into a fixed point problem, (1.11) must be transformed to corresponding an integral equation. We discuss the following key lemma.

    Lemma 2.1. Let 0<βj<α1, j=1,2,,n, ρ>0, δr, >0, r=1,2,,k and Ω0. Then, the Caputo-PFIDE with MNCs:

    {CρDα,ψa+[u(t)]=Fu(t),t(a,T),mi=1γiu(ηi)+nj=1κjCρDβj,ψa+[u(ξj)]+kr=1σrρIδr,ψa+[u(θr)]=A, (2.2)

    is equivalent to the integral equation

    u(t)=ρIα,ψa+[Fu(t)]+eρ1ρ(ψ(t)ψ(a))Ω(Ami=1γiρIα,ψa+[Fu(ηi)]nj=1κjρIαβj,ψa+[Fu(ξj)]kr=1σrρIα+δr,ψa+[Fu(θr)]), (2.3)

    where

    Ω=mi=1γieρ1ρ(ψ(ηi)ψ(a))+kr=1σrρHδr+1ψ(θr,a)ρδrΓ(1+δr). (2.4)

    Proof. Suppose u is the solution of (2.2). By using (1.5), the integral equation can be rewritten as

    u(t)=ρIα,ψa+[Fu(t)]+c1eρ1ρ(ψ(t)ψ(a)), (2.5)

    where c1R. Taking CρDβj,ψa+ and ρIδr,ψa+ into (2.5) with (1.6) and (1.7), we obtain

    CρDβj,ψa+[u(t)]=ρIαβj,ψa+[Fu(t)],ρIδr,ψa+[u(t)]=ρIα+δr,ψa+[Fu(t)]+c1ρHδr+1ψ(t,a)ρδrΓ(1+δr).

    Applying the nonlocal conditions in (2.2), we have

    A=mi=1γiρIα,ψa+[Fu(ηi)]+nj=1κjρIαβj,ψa+[Fu(ξj)]+kr=1σrρIα+δr,ψa+[Fu(θr)]+c1(mi=1γieρ1ρ(ψ(ηi)ψ(a))+kr=1σrρHδr+1ψ(θr,a)ρδrΓ(1+δr)).

    Solving the above equation, we get the value

    c1=1Ω(Ami=1γiρIα,ψa+[Fu(ηi)]nj=1κjρIαβj,ψa+[Fu(ξj)]kr=1σrρIα+δr,ψa+[Fu(θr)]),

    where Ω is given as in (2.4). Taking c1 in (2.5), we obtain (2.3).

    On the other hand, it is easy to show by direct computing that u(t) is provided as in (2.3) verifies (2.2) via the given MNCs. The proof is done.

    By using Lemma 2.1, we will set the operator K:EE

    (Ku)(t)=ρIα,ψa+[Fu(t)]+eρ1ρ(ψ(t)ψ(a))Ω(Ami=1γiρIα,ψa+[Fu(ηi)]nj=1κjρIαβj,ψa+[Fu(ξj)]kr=1σrρIα+δr,ψa+[Fu(θr)]), (3.1)

    where K1, K2:EE defined by

    (K1u)(t)=ρIα,ψa+Fu(t), (3.2)
    (K2u)(t)=eρ1ρ(ψ(t)ψ(a))Ω(Ami=1γiρIα,ψa+[Fu(ηi)]nj=1κjρIαβj,ψa+[Fu(ξj)]kr=1σrρIα+δr,ψa+[Fu(θr)]). (3.3)

    Notice that Ku=K1u+K2u. It should be noted that (1.11) has solutions if and only if K has fixed points. Next, we are going to examine the existence properties of solutions for (1.11), which is discussed by employing Banach's FPT, Leray-Schauder's nonlinear alternative and Krasnoselskii's FPT. For the benefit of calculation in this work, we will provide the constants:

    Θ(χ,σ)=(ψ(χ)ψ(a))σρσΓ(σ+1), (3.4)
    Λ=Θ(T,α)+1|Ω|(mi=1|γi|Θ(ηi,α)+nj=1|κj|Θ(ξj,αβj)+kr=1|σr|Θ(θr,α+δr)). (3.5)

    Firstly, the uniqueness result for (1.11) will be stidied by applying Banach's FPT.

    Lemma 3.1. (Banach contraction principle [59]) Assume that B is a non-empty closed subset of a Banach space E. Then any contraction mapping K from B into itself has a unique fixed point.

    Theorem 3.2. Let fC(J×R4,R)so that:

    (H1) there exist constants L1>0, L2>0, 0<L3<1 so that

    |f(t,u1,v1,w1,z1)f(t,u2,v2,w2,z2)|L1(|u1u2|+|v1v2|)+L2|w1w2|+L3|z1z2|,

    ui, vi, wi, ziR, i=1,2, tJ.

    If

    (2L1+L2Θ(T,ω)1L3)Λ<1, (3.6)

    then the Caputo-PFIDE with MNCs (1.11) has a unique solution on J, where (3.4) and (3.5) refers to Θ(T,ω) and Λ.

    Proof. First, we will convert (1.11) into u=Ku, where K is given as in (3.1). Clearly, the fixed points of K are solutions to (1.11). By using the Banach's FPT, we are going to prove that K has a FP which is a unique solution of (1.11).

    Define suptJ|f(t,0,0,0,0)|:=M1< and setting Br1:={uE:ur1} with

    r1M1Λ1L3+|A||Ω|1(2L1+L2Θ(T,ω)1L3)Λ, (3.7)

    where Ω, Θ(T,ω) and Λ are given as in (2.4), (3.4) and (3.5). Clearly, Br1 is a bounded, closed and convex subset of E. We have divided the method of the proof into two steps:

    Step I. We prove that KBr1Br1.

    For each uBr1, we obtain

    |(Ku)(t)|ρIα,ψa+|Fu(t)|+eρ1ρ(ψ(t)ψ(a))|Ω|(|A|+mi=1|γi|ρIα,ψa+|Fu(ηi)|+nj=1|κj|ρIαβj,ψa+|Fu(ξj)|+kr=1|σr|ρIα+δr,ψa+|Fu(θr)|).

    From the assumption (H1), it follows that

    |Fu(t)||f(t,u(t),u(λt),ρIω,ψa+[u(λt)],Fu(λt))f(t,0,0,0,0)|+|f(t,0,0,0,0)|L1(|u(t)|+|u(λt)|)+L2|ρIω,ψa+[u(λt)]|+L3|Fu(λt)|+M12L1u+L2uρIω,ψa+[1](t)+L3Fu()+M1=(2L1+L2(ψ(T)ψ(a))ωρωΓ(ω+1))u+L3Fu()+M1.

    Then

    Fu()(2L1+L2Θ(T,ω))u+M11L3.

    This implies that

    |(Ku)(t)|ρIα,ψa+((2L1+L2Θ(T,ω))u+M11L3)(t)+eρ1ρ(ψ(t)ψ(a))|Ω|[|A|+mi=1|γi|ρIα,ψa+((2L1+L2Θ(T,ω))u+M11L3)(ηi)+nj=1|κj|ρIαβj,ψa+((2L1+L2Θ(T,ω))u+M11L3)(ξj)+kr=1|σr|ρIα+δr,ψa+((2L1+L2Θ(T,ω))u+M11L3)(θr)].

    By using the fact of 0<eρ1ρ(ψ(t)ψ(s))1, as<tT, it follows that

    |(Ku)(t)|((2L1+L2Θ(T,ω))u+M11L3)((ψ(T)ψ(a))αραΓ(α+1)+1|Ω|[mi=1|γi|(ψ(ηi)ψ(a))αραΓ(α+1)+nj=1|κj|(ψ(ξj)ψ(a))αβjραβjΓ(αβj+1)+kr=1|σr|(ψ(θr)ψ(a))α+δrρα+δrΓ(α+δr+1)])+|A||Ω|=((2L1+L2Θ(T,ω))u+M11L3)(Θ(T,α)+1|Ω|[mi=1|γi|Θ(ηi,α)+nj=1|κj|Θ(ξj,αβj)+kr=1|σr|Θ(θr,α+δr)])+|A||Ω|=(2L1+L2Θ(T,ω)1L3)Λr1+M1Λ1L3+|A||Ω|r1,

    which implies that Kur1. Thus, KBr1Br1.

    Step II. We prove that K:EE is contraction.

    For each u, vE, tJ, we obtain

    |(Ku)(t)(Kv)(t)|ρIα,ψa+|FuFv|(T)+eρ1ρ(ψ(T)ψ(a))|Ω|(mi=1|γi|ρIα,ψa+|FuFv|(ηi)+nj=1|κj|ρIαβj,ψa+|FuFv|(ξj)+kr=1|σr|ρIα+δr,ψa+|FuFv|(θr)). (3.8)

    From (H1) again, we can compute that

    |Fu(t)Fv(t)||f(t,u(t),u(λt),ρIω,ψa+[u(λt)],Fu(λt))f(t,v(t),v(λt),ρIω,ψa+[v(λt)],Fv(λt))|L1(|u(t)v(t)|+|u(λt)v(λt)|)+L2ρIω,ψa+|u(λt)v(λt)|+L3|Fu(λt)Fv(λt)|2L1uv+L2(ψ(T)ψ(a))ωρωΓ(ω+1)uv+L3Fu()Fv().

    Then

    Fu()Fv()(2L1+L2Θ(T,ω)1L3)uv. (3.9)

    By inserting (3.9) into (3.8), one has

    |(Ku)(t)(Kv)(t)|ρIα,ψa+((2L1+L2Θ(T,ω)1L3)uv)(T)+eρ1ρ(ψ(T)ψ(a))|Ω|[mi=1|γi|ρIα,ψa+((2L1+L2Θ(T,ω)1L3)uv)(ηi)+nj=1|κj|ρIαβj,ψa+((2L1+L2Θ(T,ω)1L3)uv)(ξj)+kr=1|σr|ρIα+δr,ψa+((2L1+L2Θ(T,ω)1L3)uv)(θr)](2L1+L2Θ(T,ω)1L3)[(ψ(T)ψ(a))αραΓ(α+1)+1|Ω|(mi=1|γi|(ψ(ηi)ψ(a))αραΓ(α+1)+nj=1|κj|(ψ(ξj)ψ(a))αβjραβjΓ(αβj+1)+kr=1|σr|(ψ(θr)ψ(a))α+δrρα+δrΓ(α+δr+1))]uv=(2L1+L2Θ(T,ω)1L3)Λuv,

    also, KuKv(2L1+L2Θ(T,ω))/(1L3)Λuv. It follows from [(2L1+L2Θ(T,ω))/(1L3)]Λ<1, that K is contraction. Then, (from Lemma 3.1), we conclude that K has the unique fixed point, that is the unique solution to (1.11) in E.

    Next, Leray-Schauder's nonlinear alternative is employed to analyze in the second property.

    Lemma 3.3. (Leray-Schauder's nonlinear alternative [59]) Assume that E is a Banach space, C is a closed and convex subset of M, X is an open subset of C and 0X. Assume that F:¯XC is continuous, compact (that is, F(¯X) is a relatively compact subset of C) map. Then either (i) F has a fixed point in ¯X, or (ii) there is xX (the boundary of X in C) and ϱ(0,1) with z=ϱF(z).

    Theorem 3.4. Assume that fC(J×R4,R) so that:

    (H2) there exists ΨC(R+,R+) and Ψ is non-decreasing, p, fC(J,R+), qC(J,R+{0}) so that

    |f(t,u,v,w,z)|p(t)Ψ(|u|+|v|)+f(t)|w|+q(t)|z|,tJ,u,v,w,zR,

    where p0=suptJ{p(t)}, f0=suptJ{f(t)}, q0=suptJ{q(t)}<1.

    (H3) there exists a constant N>0 so that

    N|A||Ω|+(f0Θ(T,ω)N+2p0Ψ(N)1q0)Λ>1,

    where Θ(T,ω) and Λ are given as in (3.4) and (3.5).

    Then the Caputo-PFIDE with MNCs (1.11) has at least one solution.

    Proof. Assume that K is given as in (3.1). Next, we are going to prove that K maps bounded sets (balls) into bounded sets in E. For any r2>0, assume that Br2:={uE:ur2}E, we have, for each tJ,

    |(Ku)(t)|ρIα,ψa+|Fu(T)|+eρ1ρ(ψ(T)ψ(a))|Ω|(|A|+mi=1|γi|ρIα,ψa+|Fu(ηi)|+nj=1|κj|ρIαβj,ψa+|Fu(ξj)|+kr=1|σr|ρIα+δr,ψa+|Fu(θr)|).

    It follows from (H2) that

    |CρDα,ψa+[u(t)]|p(t)Ψ(|u(t)|+|u(λt)|)+f(t)|ρIω,ψa+[u(λt)]|+q(t)|CρDα,ψa+[u(λt)]|p(t)Ψ(2u)+f(t)(ψ(T)ψ(a))ωρωΓ(ω+1)u+q(t)|CρDα,ψa+[u(t)]|.

    Then, we have

    |CρDα,ψa+u(t)|p(t)Ψ(2u)+f(t)Θ(T,ω)u1q(t).

    For any as<tT, we have 0<eρ1ρ(ψ(t)ψ(s))1, then

    |(Ku)(t)|[(ψ(T)ψ(a))αραΓ(α+1)+1|Ω|(mi=1|γi|(ψ(ηi)ψ(a))αραΓ(α+1)+nj=1|κj|(ψ(ξj)ψ(a))αβjραβjΓ(αβj+1)+kr=1|σr|(ψ(θr)ψ(a))α+δrρα+δrΓ(α+δr+1))](p0Ψ(2u)+f0Θ(T,ω)u1q0)+|A||Ω|=(2p0Ψ(u)+f0Θ(T,ω)u1q0)Λ+|A||Ω|,

    which leads to

    Ku(2p0Ψ(u)+f0Θ(T,ω)u1q0)Λ+|A||Ω|:=N.

    Now, we will prove that K maps bounded sets into equicontinuous sets of E.

    Given τ1<τ2 where τ1, τ2J, and for each uBr2. Then, we obtain

    |(Ku)(τ2)(Ku)(τ1)||ρIα,ψa+[Fu(τ2)]ρIα,ψa+[Fu(τ1)]|+|eρ1ρ(ψ(τ2)ψ(a))eρ1ρ(ψ(τ1)ψ(a))||Ω|(|A|+mi=1|γi|ρIα,ψa+|Fu(ηi)|+nj=1|κj|ρIαβj,ψa+|Fu(ξj)|+kr=1|σr|ρIα+δr,ψa+|Fu(θr)|)(2p0Ψ(u)+f0Θ(T,ω)u1q0)(1ραΓ(α)τ2τ1ρHα1ψ(τ2,s)ψ(s)ds+1ραΓ(α)τ1a|ρHα1ψ(τ2,s)ρHα1ψ(τ1,s)|ψ(s)ds)+|eρ1ρ(ψ(τ2)ψ(a))eρ1ρ(ψ(τ1)ψ(a))||Ω|(2p0Ψ(u)+f0Θ(T,ω)u1q0)×(|A|+mi=1|γi|(ψ(ηi)ψ(a))αραΓ(α+1)+nj=1|κj|(ψ(ξj)ψ(a))αβjραβjΓ(αβj+1)+kr=1|σr|(ψ(θr)ψ(a))α+δrραδrΓ(α+δr+1))[1ραΓ(α+1)(|(ψ(τ2)ψ(a))α(ψ(τ1)ψ(a))α|+2(ψ(τ2)ψ(τ1))α)+1|Ω|(|A|+mi=1|γi|Θ(ηi,α)+nj=1|κj|Θ(ξj,αβj)+kr=1|σr|Θ(θr,α+δr))×|eρ1ρ(ψ(τ2)ψ(a))eρ1ρ(ψ(τ1)ψ(a))|][2p0Ψ(u)+f0Θ(T,ω)u1q0].

    Clearly, which independent of uBr2 the above inequality, |(Ku)(τ2)(Ku)(τ1)|0 as τ2τ1. Hence, by the Arzelá-Ascoli property, K:EE is completely continuous.

    Next, we will prove that there is BE where B is an open set, uϱK(u) for ϱ(0,1) and uB. Assume that uE is a solution of u=ϱKu, ϱ(0,1). Hence, it follows that

    |u(t)|=|ϱ(Ku)(t)||A||Ω|+(2p0Ψ(u)+f0Θ(T,ω)u1q0)×[Θ(T,α)+1|Ω|(mi=1|γi|Θ(ηi,α)+nj=1|κj|Θ(ξj,αβj)+kr=1|σr|Θ(θr,α+δr))]=|A||Ω|+(2p0Ψ(u)+f0Θ(T,ω)u1q0)Λ,

    which yields

    u|A||Ω|+(2p0Ψ(u)+f0Θ(T,ω)u1q0)Λ.

    Consequently,

    u|A||Ω|+(2p0Ψ(u)+f0Θ(T,ω)u1q0)Λ1.

    By (H3), there is N so that uN. Define

    B={uE:u<N}andQ=BBr2.

    Note that K:¯QE is continuous and completely continuous. By the option of Q, there is no uQ so that u=ϱKu, ϱ(0,1). Thus, (by Lemma 3.3), we conclude that K has fixed point u¯Q which verifies that (1.11) has at least one solution.

    By applying Krasnoselskii's FPT, the existence property will be achieved.

    Lemma 3.5. (Krasnoselskii's fixed point theorem [60]) Let M be a closed, bounded, convex and nonempty subset of a Banach space. Let K1, K2 be the operators such that (i) K1x+K2yM whenever x,yM; (ii) K1 is compact and continuous; (iii) K2 is contraction mapping. Then there exists zM such that z=K1z+K2z.

    Theorem 3.6. Suppose that (H1) holds and fC(J×R4,R) so that:

    (H4) gC(J,R+) so that

    f(t,u,v,w,z)|g(t),(t,u,v,w,z)J×R4.

    If

    (2L1+L2Θ(T,ω)1L3)(ΛΘ(T,α))<1, (3.10)

    then the Caputo-PFIDE with MNCs (1.11) has at least one solution.

    Proof. Define suptJ|g(t)|=g and picking

    r3|A||Ω|+gΛ, (3.11)

    we consider Br3={uE:ur3}. Define K1 and K2 on Br3 as (3.2) and (3.3).

    For any u,vBr3, we obtain

    |(K1u)(t)+(K2v)(t)|suptJ{ρIα,ψa+|Fu(t)|+eρ1ρ(ψ(t)ψ(a))|Ω|(|A|+mi=1|γi|ρIα,ψa+|Fv(ηi)|+nj=1|κj|ρIαβj,ψa+|Fv(ξj)|+kr=1|σr|ρIα+δr,ψa+|Fv(θr)|)}|A||Ω|+g{(ψ(T)ψ(a))αραΓ(α+1)+1|Ω|(mi=1|γi|(ψ(ηi)ψ(a))αραΓ(α+1)+nj=1|κj|(ψ(ξj)ψ(a))αβjραβjΓ(αβj+1)+kr=1|σr|(ψ(θr)ψ(a))α+δrρα+δrΓ(α+δr+1))}=|A||Ω|+gΛr3.

    This implies that K1u+K2vBr3, which verifies Lemma 3.5 (i).

    Next, we are going to show that Lemma 3.5 (ii) is verified.

    Assume that un is a sequence so that unuE as n. Hence, we get

    |(K1un)(t)(K1u)(t)|ρIα,ψa+|FunFu|(T)Θ(T,α)FunFu.

    Since f is continuous, verifies that Fu is also continuous. By the Lebesgue dominated convergent theorem, we have

    |(K1un)(t)(K1u)(t)|0asn.

    Therefore,

    K1unK1u0asn.

    Thus, implies that K1u is continuous. Also, the set K1Br3 is uniformly bounded as

    K1uΘ(T,α)g.

    Next step, we will show the compactness of K1.

    Define sup{|f(t,u,v,w,z)|;(t,u,v,w,z)J×R4}=f<, thus, for each τ1,τ2J with τ1τ2, it follows that

    |(K1u)(τ2)(K1u)(τ1)|=|ρIα,ψa+[Fu(τ2)]ρIα,ψa+[Fu(τ1)]|1ραΓ(α+1)(|(ψ(τ2)ψ(a))α(ψ(τ1)ψ(a))α|+2(ψ(τ2)ψ(τ1))α)f.

    Clearly, the right-hand side of the above inequality is independent of u and |(K1u)(τ2)(K1u)(τ1)|0, as τ2τ1. Hence, the set K1Br3 is equicontinuous, also K1 maps bounded subsets into relatively compact subsets, which implies that K1Br3 is relatively compact. By the Arzelá-Ascoli theorem, then K1 is compact on Br3.

    Finally, we are going to show that K2 is contraction.

    For each u, vBr3 and tJ, we get

    |(K2u)(t)(K2v)(t)|1|Ω|(mi=1|γi|ρIα,ψa+|FuFv|(ηi)+nj=1|κj|ρIαβj,ψa+|FuFv|(ξj)+kr=1|σr|ρIα+δr,ψa+|FuFv|(θr))1|Ω|(mi=1|γi|(ψ(ηi)ψ(a))αραΓ(α+1)+nj=1|κj|(ψ(ξj)ψ(a))αβjραβjΓ(αβj+1)+kr=1|σr|(ψ(θr)ψ(a))α+δrρα+δrΓ(α+δr+1))×(2L1+L2Θ(T,ω)1L3)uv=(2L1+L2Θ(T,ω)1L3)(ΛΘ(T,α))uv.

    Since (3.10) holds, implies that K2 is contraction and also Lemma 3.5 (iii) verifies.

    Therefore, the assumptions of Lemma 3.5 are verified. Then, (by Lemma 3.5) which verifies that (1.11) has at least one solution.

    This part is proving different kinds of US like HU stable, GHU stable, HUR stable and GHUR stable of the Caputo-PFIDE with MNCs (1.11).

    Definition 4.1. The Caputo-PFIDE with MNCs (1.11) is called HU stable if there is a constant Δf>0 so that for every ϵ>0 and the solution zE of

    |CρDα,ψa+[z(t)]f(t,z(t),z(λt),ρIω,ψa+[z(λt)],CρDα,ψa+[z(λt)])|ϵ, (4.1)

    there exists the solution uE of (1.11) so that

    |z(t)u(t)|Δfϵ,tJ. (4.2)

    Definition 4.2. The Caputo-PFIDE with MNCs (1.11) is called GHU stable if there is a function ΦC(R+,R+) via Φ(0)=0 so that, for every solution zE of

    |CρDα,ψa+[z(t)]f(t,z(t),z(λt),ρIω,ψa+[z(λt)],CρDα,ψa+[z(λt)])|ϵΦ(t), (4.3)

    there is the solution uE of (1.11) so that

    |z(t)u(t)|Φ(ϵ),tJ. (4.4)

    Definition 4.3. The Caputo-PFIDE with MNCs (1.11) is called HUR stable with respect to ΦC(J,R+) if there is a constant Δf,Φ>0 such that for every ϵ>0 and for any the solution zE of (4.3) there is the solution uE of (1.11) so that

    |z(t)u(t)|Δf,ΦϵΦ(t),tJ. (4.5)

    Definition 4.4. The Caputo-PFIDE with MNCs (1.11) is called GHUR stable with respect to ΦC(J,R+) if there is a constant Δf,Φ>0 so that for any the solution zE of

    |CρDα,ψa+[z(t)]f(t,z(t),z(λt),ρIω,ψa+[z(λt)],CρDα,ψa+[z(λt)])|Φ(t), (4.6)

    there is the solution uE of (1.11) so that

    |z(t)u(t)|Δf,ΦΦ(t),tJ. (4.7)

    Remark 4.5. Cleary, (i) Definition 4.1 Definition 4.2; (ii) Definition 4.3 Definition 4.4; (iii) Definition 4.3 for Φ(t)=1 Definition 4.1.

    Remark 4.6. zE is the solution of (4.1) if and only if there is the function wE (which depends on z) so that: (i) |w(t)|ϵ, tJ; (ii) CρDα,ψa+[z(t)]=Fz(t)+w(t), tJ.

    Remark 4.7. zE is the solution of (4.3) if and only if there is the function vE (which depends on z) so that: (i) |v(t)|ϵΦ(t), tJ; (ii) CρDα,ψa+[z(t)]=Fz(t)+v(t), tJ.

    From Remark 4.6, the solution of

    CρDα,ψa+[z(t)]=f(t,z(t),z(λt),ρIω,ψa+[z(λt)],CρDα,ψa+[z(λt)])+w(t),tJ,

    can be rewritten as

    z(t)=ρIα,ψa+[Fz(t)]+eρ1ρ(ψ(t)ψ(a))Ω(Ami=1γiρIα,ψa+[Fz(ηi)]nj=1κjρIαβj,ψa+[Fz(ξj)]kr=1σrρIα+δr,ψa+[Fz(θr)])+ρIα,ψa+[w(t)]eρ1ρ(ψ(t)ψ(a))Ω(mi=1γiρIα,ψa+[w(ηi)]+nj=1κjρIαβj,ψa+[w(ξj)]+kr=1σrρIα+δr,ψa+[w(θr)]). (4.8)

    Firstly, the key lemma that will be applied in the presents of HU stable and GHU stable.

    Lemma 4.8. Assume that 0<ϵ,ρ1. If zE verifies (4.1), hence z is the solution of

    |z(t)(Kz)(t)|Λϵ, (4.9)

    where Λ is given as in (3.5).

    Proof. By Remark 4.6 with (4.8), it follows that

    |z(t)(Kz)(t)|=|ρIα,ψa+[w(t)]eρ1ρ(ψ(t)ψ(a))Ω(mi=1γiρIα,ψa+[w(ηi)]+nj=1κjρIαβj,ψa+[w(ξj)]+kr=1σrρIα+δr,ψa+[w(θr)])|ρIα,ψa+|w(T)|+1|Ω|(mi=1|γi|ρIα,ψa+|w(ηi)|+nj=1|κj|ρIαβj,ψa+|w(ξj)|+kr=1|σr|ρIα+δr,ψa+|w(θr)|)[(ψ(T)ψ(a))αραΓ(α+1)+1|Ω|(mi=1|γi|(ψ(ηi)ψ(a))αραΓ(α+1)+nj=1|κj|(ψ(ξj)ψ(a))αβjραβjΓ(αβj+1)+kr=1|σr|(ψ(θr)ψ(a))α+δrρα+δrΓ(α+δr+1))]ϵ=Λϵ,

    where Λ is given by (3.5), from which (4.9) is achieved.

    Next, we will show the HU and GHU stability results.

    Theorem 4.9. Suppose that fC(J×R4,R). If (H1) is verified with (3.6) trues. Hence the Caputo-PFIDE with MNCs (1.11) is HU stable as well as GHU stable on J.

    Proof. Assume that zE is the solution of (4.1) and assume that u is the unique solution of

    {CρDα,ψa+[u(t)]=f(t,u(t),u(λt),ρIω,ψa+[u(λt)],CρDα,ψa+[u(λt)]),t(a,T),λ(0,1),mi=1γiu(ηi)+nj=1κjCρDβj,ψa+[u(ξj)]+kr=1σrρIδr,ψa+[u(θr)]=A.

    By using |xy||x|+|y| with Lemma 4.8, one has

    |z(t)u(t)|=|z(t)ρIα,ψa+[Fu(t)]eρ1ρ(ψ(t)ψ(a))Ω(Ami=1γiρIα,ψa+[Fu(ηi)]nj=1κjρIαβj,ψa+[Fu(ξj)]kr=1σrρIα+δr,ψa+[Fu(θr)])|=|z(t)(Kz)(t)+(Kz)(t)(Ku)(t)||z(t)(Kz)(t)|+|(Kz)(t)(Ku)(t)|Λϵ+(2L1+L2Θ(T,ω)1L3)Λ|z(t)u(t)|,

    where Λ is given as in (3.5). This offers |z(t)u(t)|Δfϵ, where

    Δf=Λ1(2L1+L2Θ(T,ω)1L3)Λ. (4.10)

    Then, the Caputo-PFIDE with MNCs (1.11) is HU stable. In addition, if we input Φ(ϵ)=Δfϵ via Φ(0)=0, hence (1.11) is GHU stable.

    Thanks of Remark 4.7, the solution

    CρDα,ψa+[z(t)]=f(t,z(t),z(λt),ρIω,ψa+[z(λt)],CρDα,ψa+[z(λt)])+v(t),t(a,T],

    can be rewritten as

    z(t)=ρIα,ψa+[Fz(t)]+eρ1ρ(ψ(t)ψ(a))Ω(Ami=1γiρIα,ψa+[Fz(ηi)]nj=1κjρIαβj,ψa+[Fz(ξj)]kr=1σrρIα+δr,ψa+[Fz(θr)])+ρIα,ψa+[v(t)]eρ1ρ(ψ(t)ψ(a))Ω(mi=1γiρIα,ψa+[v(ηi)]+nj=1κjρIαβj,ψa+[v(ξj)]+kr=1σrρIα+δr,ψa+[v(θr)]). (4.11)

    For the next proving, we state the following assumption:

    (H5) there is an increasing function ΦC(J,R+) and there is a constant nΦ>0, so that, for each tJ,

    ρIα,ψa+[Φ(t)]nΦΦ(t). (4.12)

    Lemma 4.10. Assume that zE is the solution of (4.3). Hence, z verifies

    |z(t)(Kz)(t)|ΛϵnΦΦ(t),0<ϵ1. (4.13)

    where Λ is given as in (3.5).

    Proof. From (4.11), we have

    |z(t)(Kz)(t)|=|ρIα,ψa+v(t)eρ1ρ(ψ(t)ψ(a))Ω(mi=1γiρIα,ψa+[v(ηi)]+nj=1κjρIαβj,ψa+[v(ξj)]+kr=1σrρIα+δr,ψa+[v(θr)])|[ρIα,ψa+[Φ(T)]+1|Ω|(mi=1|γi|ρIα,ψa+[Φ(ηi)]+nj=1|κj|ρIαβj,ψa+[Φ(ξj)]+kr=1|σr|ρIα+δr,ψa+[Φ(θr)])]ϵ[(ψ(T)ψ(a))αραΓ(α+1)+1|Ω|(mi=1|γi|(ψ(ηi)ψ(a))αραΓ(α+1)+nj=1|κj|(ψ(ξj)ψ(a))αβjραβjΓ(αβj+1)+kr=1|σr|(ψ(θr)ψ(a)big)α+δrρα+δrΓ(α+δr+1))]ϵnΦΦ(t)=ΛϵnΦΦ(t),

    where Λ is given by (3.5), which leads to (4.13).

    Finally, we are going to show HUR and GHUR stability results.

    Theorem 4.11. Suppose fC(J×R4,R). If (H1) is satisfied with (3.6) trues. Hence, the Caputo-PFIDE with MNCs (1.11) is HUR stable as well as GHUR stable on J.

    Proof. Assume that ϵ>0. Suppose that zE is the solution of (4.6) and u is the unique solution of (1.11). By using the triangle inequality, Lemma 4.8 and (4.11), we estamate that

    |z(t)u(t)|=|z(t)ρIα,ψa+[Fu(t)]eρ1ρ(ψ(t)ψ(a))Ω(Ami=1γiρIα,ψa+[Fu(ηi)]nj=1κjρIαβj,ψa+[Fu(ξj)]kr=1σrρIα+δr,ψa+[Fu(θr)])|=|z(t)(Kz)(t)+(Kz)(t)(Ku)(t)||z(t)(Kz)(t)|+|(Kz)(t)(Kx)(t)|ΛϵnΦΦ(t)+(2L1+L2Θ(T,ω)1L3)Λ|z(t)u(t)|,

    where Λ is given as in (3.5), verifies that |z(t)u(t)|Δf,ΦϵΦ(t), where

    Δf,Φ:=ΛnΦ1(2L1+L2Θ(T,ω)1L3)Λ.

    Then, the Caputo-PFIDE with MNCs (1.11) is HUR stable. In addition, if we input Φ(t)=ϵΦ(t) with Φ(0)=0, then (1.11) is GHUR stable.

    This part shows numerical instances that demonstrate the exactness and applicability of our main results.

    Example 5.1. Discussion the following nonlinear Caputo-PFIDE with MNCs of the form:

    {C23D12,t0+[u(t)]=f(t,u(t),u(t3),23I34,t0+[u(t3)],C23D12,t[u(t3)]0+),t(0,1),2i=1(i+12)u(2i+15)+3j=1(2j15)C23D2j+120,t0+[u(j4)]+2r=1(r3)23Irr+1,t0+[u(r+14)]=1. (5.1)

    Here, α=1/2, ρ=2/3, ψ(t)=t, λ=1/3, ω=3/4, a=0, T=1, m=2, n=3, k=2, γi=(i+1)/2, ηi=(2i+1)/5, i=1,2, κj=(2j1)/5, βj=(2j+1)/20, ξj=j/4, j=1,2,3, σr=r/3, δr=r/(r+1), θr=(r+1)/4, r=1,2. By using Python, we obtain that Ω2.43090 and Λ4.042711.

    (I) If we set the nonlinear function

    f(t,u,v,w)=13t+3(2+cos2t)(|u|1+|u|+|v|1+|v|+|w|1+|w|+|z|1+|z|).

    For any ui, vi, wi, ziR, i=1,2 and t[0,1], one has

    |f(t,u1,v1,w1,z1)f(t,u2,v2,w2,z2)|13t+3(|u1u2|+|v1v2|+|w1w2|+|z1z2|).

    The assumption (H1) is satisfied with L1=L2=L3=127. Hence

    (2L1+L2Θ(T,ω)1L3)Λ0.540287<1.

    All conditions of Theorem 3.2 are verified. Hence, the nonlinear Caputo-PFIDE with MNCs (5.1) has a unique solution on [0,1]. Moreover, we obtain

    Δf=Λ1(2L1+L2Θ(T,ω)1L3)Λ8.793988>0.

    Hence, from Theorem 4.9, the nonlinear Caputo-PFIDE with MNCs (5.1) is HU stable and also GHU stable on [0,1]. In addition, by taking Φ(t)=eρ1ρψ(t)(ψ(t)ψ(0)), we have

    23I12,ψ0+[Φ(t)]=223πe0.5ψ(t)(ψ(t)ψ(0))32223π(ψ(t)ψ(0))12Φ(t).

    Thus, (4.12) is satisfied with nΦ=223π>0. Then, we have

    Δf,Φ:=ΛnΦ1(2L1+L2Θ(T,ω)1L3)Λ8.102058>0.

    Hence, from Theorem 4.11, the nonlinear Caputo-PFIDE with MNCs (5.1) is HUR stable and also GHUR stable.

    (II) If we take the nonlinear function

    f(t,u,v,w)=14t+2(|u|+|v|1+|u|+|v|+12)+12t+3(|w|+|z|1+|w|+|z|+12),

    we have

    |f(t,u,v,w,z)|14t+2(|u|+|v|+12)+12t+3(|w|+|z|+12).

    From the above inequality with (H2)(H3), we get that p(t)=1/4t+2, Ψ(|u|+|v|)=|u|+|v|+1/2 and h(t)=q(t)=1/2t+3. So, we have p0=1/16 and h0=q0=1/8. From all the datas, we can compute that the constant N>0.432489. All assumptions of Theorem 3.4 are verified. Hence, the nonlinear Caputo-PFIDE with MNCs (5.1) has at least one solution on [0,1]. Moreover,

    Δf:=Λ1(2L1+L2Θ(T,ω)1L3)Λ63.662781>0,

    Hence, from Theorem 4.9, the nonlinear Caputo-PFIDE with MNCs 5.1 is HU stable and also GHU stable on [0,1]. In addition, by taking Φ(t)=eρ1ρψ(t)(ψ(t)ψ(0))12, we have

    23I12,ψ0+[Φ(t)]=3π22e0.5ψ(t)(ψ(t)ψ(0))3π22(ψ(t)ψ(0))12Φ(t).

    Thus, (4.12) is satisfied with nΦ=3π22>0. Then, we have

    Δf,Φ:=ΛnΦ1(2L1+L2Θ(T,ω)1L3)Λ69.0997>0,

    Hence, from Theorem 4.11, the nonlinear Caputo-PFIDE with MNCs 5.1 is HUR stable and also GHUR stable on [0,1].

    (III) If we set the nonlinear function

    f(t,u,v,w)=14t+2(|u|+|v|1+|u|+|v|)+12t+3(|w|+|z|1+|w|+|z|).

    For ui, vi, wi, ziR, i=1,2 and t[0,1], we have

    |f(t,u1,v1,w1,z1)f(t,u2,v2,w2,z2)|14t+2(|u1u2|+|v1v2|)+12t+3(|w1w2|+|z1z2|).

    The assumption (H4) is satisfied with L1=L2=116, L3=18 and

    |f(t,u,v,w,z)|14t+2+12t+3,

    which yields that

    g(t)=14t+2+12t+3.

    Then

    (2L1+L2Θ(T,ω)1L3)(ΛΘ(T,α))0.660387<1.

    All assumptions of Theorem 3.6 are verified. Hence the nonlinear Caputo-PFIDE with MNCs (5.1) has at least one solution on [0,1].

    Furthermore, we get

    Δf=Λ1(2L1+L2Θ(T,ω)1L3)Λ43.1392148>0.

    Hence, from Theorem 4.9, the nonlinear Caputo-PFIDE with MNCs (5.1) is HU stable and also GHU stable on [0,1]. In addition, by taking Φ(t)=eρ1ρψ(t)(ψ(t)ψ(0))2, we have

    23I12,ψ0+[Φ(t)]=8615πe0.5ψ(t)(ψ(t)ψ(0))528615π(ψ(t)ψ(0))12Φ(t).

    Thus, (4.12) is satisfied with nΦ=8615π>0. Then, we have

    Δf,Φ:=ΛnΦ1(2L1+L2Θ(T,ω)1L3)Λ31.79594>0.

    Hence, from Theorem 4.11, the nonlinear Caputo-PFIDE with MNCs (5.1) is HUR stable and also GHUR stable on [0,1].

    Example 5.2. Discussion the following linear Caputo-PFIDE with MNCs of the form:

    {C23Dα,ψ(t)0+[u(t)]=e12ψ(t)(ψ(t)ψ(0))12,t(0,1),2i=1(i+12)u(2i+15)+3j=1(2j15)C23D2j+120,t0+[u(j4)]+2r=1(r3)23Irr+1,t0+[u(r+14)]=1. (5.2)

    By Lemma 2.1, the implicit solution of the problem (5.2)

    u(t)=Γ(32)(23)αΓ(32+α)e12ψ(t)(ψ(t)ψ(0))12+α+e12(ψ(t)ψ(0))Ω(12i=1(i+12)e12ψ(2i+15)(ψ(2i+15)ψ(0))123j=1(2j15)Γ(32)(23)α2j+120Γ(32+α2j+120)e12ψ(j4)(ψ(j4)ψ(0))12+α2j+1202r=1(r3)Γ(32)(23)α+rr+1Γ(32+α+rr+1)e12ψ(r+14)(ψ(r+14)ψ(0))12+α+rr+1),

    where

    Ω=2i=1(i+12)e12(ψ(t)ψ(0))+2r=1(r3)(ψ(r+14)ψ(0))rr+1e12(ψ(r+14)ψ(0))(23)rr+1Γ(1+rr+1).

    We consider several cases of the following function ψ(t):

    (I) If ψ(t)=tα then the solution of linear Caputo-PFIDE with MNCs (5.2) is given as in

    u(t)=Γ(32)(23)αΓ(32+α)etα2(tα)1/2+α+etα2Ω(12i=1(i+12)e12(2i+15)α(2i+15)α23j=1(2j15)Γ(32)(23)α2j+120Γ(32+α2j+120)e12(j4)α(j4)α(12+α2j+120)2r=1(r3)Γ(32)(23)α+rr+1Γ(32+α+rr+1)e12(r+14)α(r+14)α(12+α+rr+1)),

    where

    Ω=2i=1(i+12)etα2+2r=1(r3)(r+14)αrr+1e12(r+14)α(23)rr+1Γ(1+rr+1).

    (II) If ψ(t)=sintα then the solution of linear Caputo-PFIDE with MNCs (5.2) is given as in

    u(t)=Γ(32)(23)αΓ(32+α)esint2α(sintα)12+α+esint2αΩ(12i=1(i+12)esin(2i+15)2α(sin(2i+15)α)123j=1(2j15)Γ(32)(23)α2j+120Γ(32+α2j+120)esin(j4)2α(sin(j4)α)12+α2j+1202r=1(r3)Γ(32)(23)α+rr+1Γ(32+α+rr+1)esin(r+14)2α(sin(r+14)α)12+α+rr+1),

    where

    Ω=2i=1(i+12)esint2α+2r=1(r3)(sin(r+14)α)rr+1esin(r+14)2α(23)rr+1Γ(1+rr+1).

    (III) If ψ(t)=eαt then the solution of linear Caputo-PFIDE with MNCs (5.2) is given as in

    u(t)=Γ(32)(23)αΓ(32+α)eeαt2(eαt1)12+α+eeαt12Ω(12i=1(i+12)eeα(2i+15)2(eα(2i+15)1)123j=1(2j15)Γ(32)(23)α2j+120Γ(32+α2j+120)eeαj42(eαj41)12+α2j+1202r=1(r3)Γ(32)(23)α+rr+1Γ(32+α+rr+1)eeα(r+14)2(eα(r+14)1)12+α+rr+1),

    where

    Ω=2i=1(i+12)e12(eαt1)+2r=1(r3)(eα(r+14)1)rr+1e12(eα(r+14)1)(23)rr+1Γ(1+rr+1).

    (IV) If ψ(t)=ln(1+t)α then the solution of linear Caputo-PFIDE with MNCs (5.2) is given as in

    u(t)=Γ(32)(23)αΓ(32+α)e12αln(1+t)(ln(1+t)α)1/2+α+e12ln(1+t)Ω(12i=1(i+12α)e12αln(1+2i+15)(1αln(1+2i+15)123j=1(2j15)Γ(32)(23)α2j+120Γ(32+α2j+120)e12αln(1+j4)(1αln(1+j4)12+α2j+1202r=1(r3)Γ(32)(23)α+rr+1Γ(32+α+rr+1)e12αln(1+r+14)(1αln(1+r+14))12+α+rr+1),

    where

    Ω=2i=1(i+12)e12αln(1+t)+2r=1(r3)(1αln(1+r+14))rr+1e12αln(1+r+14)(23)rr+1Γ(1+rr+1).

    Graph representing the solution of the problem (5.2) with various values α via many the functions ψ(t)=tα, ψ(t)=sintα, ψ(t)=eαt and ψ(t)=ln(1+t)α is shown as in Figures 14 by using Python.

    Figure 1.  The graphical of u(t) under ψ(t)=tα.
    Figure 2.  The graphical of u(t) under ψ(t)=sintα.
    Figure 3.  The graphical of u(t) under ψ(t)=eαt.
    Figure 4.  The graphical of u(t) under ψ(t)=ln(1+t)α.

    The qualitative analysis is accomplished in this work. The authors proved the existence, uniqueness and stability of solutions for Caputo-PFIDE with MNCs which consist of multi-point and fractional multi-order boundary conditions. Some famous theorems are employed to obtain the main results such as the Banach's FPT is the important theorem to prove the uniqueness of the solution, while Leray-Schauder's nonlinear alternative and Krasnoselskii's FPT are used to investigate the existence results. Furthermore, we established the various kinds of Ulam's stability like HU, GHU, HUR and GHUR stables. Finally, by using Python, numerical instances allowed to guarantee the accuracy of the theoretical results.

    This research would be a great work to enrich the qualitative theory literature on the problem of nonlinear fractional mixed nonlocal conditions involving a particular function. For the future works, we shall focus on studying the different types of existence results and stability analysis for impulsive fractional boundary value problems.

    B. Khaminsou was supported by the International Science Programme PhD. Research Scholarship from National University of Laos (NUOL). W. Sudsutad was partially supported by Ramkhamhaeng University. C. Thaiprayoon and J. Kongson would like to thank for funding this work through the Center of Excellence in Mathematics (CEM), CHE, Sri Ayutthaya Road, Bangkok, 10400, Thailand and Burapha University.

    On behalf of all authors, the corresponding author states that there is no conflict of interest.


    Acknowledgments



    We are thankful to Dr. Sakib Anwer and Dr. Somya Ranjan.

    Conflicts of Interest



    We declare no conflicts of interest in this paper.

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