Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives

1  Introduction

Drugs are often inoculated in human blood during the treatment of a disease. One interesting fact is that the concentration of the drug decreases over time once it is inoculated. The initial concentration of a drug and many other parameters are used by medical practitioners to determine the frequency at which a specific drug should be administrated to reach the goal of curing a disease.

The clinical development of a new drug involves many phases. During phase 1 of drug development, the experimenter routinely evaluates time elapse until a new regimen reaches its steady state. The steady state here refers to the state at which the concentration of a drug in the blood does not change consistently anymore over time. In practice, patients have to take several shots of the drug in order to recover. However, each time a dose of the drug is administrated, part of it spreads all over the body and exists through excretions. The remaining part will accumulate to form sediment. The steady-state is reached when the concentration of the new dose coincides with the concentration of sediment. To identify the time from the first inoculation to the steady-state, the patient’s blood is sampled at regular time span intervals and the drug concentration is measured from the sample [1–5].

Many research works were produced in line with the evaluation of the steady-state of a drug in blood plasma. An overview of the selected method was proposed by Maganti et al. [6] and included approaches such as a Bayesian approach based on Markov Chain Monte Carlo (MCMC); No-statistical-significance-of-trend (NOSTASOT; spline regression among others. In the same line, a non-Newtonian-based important factors approach was introduced by Johnston et al. [7].

Besides the few but consistent methods of evaluating the steady-state during phase I study are many other approaches that we believe can be used successfully. We think of Fractional Calculus (FC) in this regard. Indeed, over the past decades, many researchers have successfully applied FC to solve real-life problems. One can name a few of these successes just to mention. For instance, Reference [8] was a review of recent trends in the successful application of FC in Biology and medical sciences. Margin has shown in [9] that FC could be successfully used to describe the dynamic of the events that occur in human blood tissues. Toledo-Hernandez et al. [10] have produced a piece on fermentation process modeling using FC. Fatma et al. [11] have studied the spray of omicron SARS-CoV-2 variant of COVID-19. Naik et al. [12] have investigated the effect of prostitution on HIV transmission dynamics. Qureshi et al. [13] have studied the concentration of ethanol in blood using a fractional derivative. They are many such examples in which FC has been successfully applied in modeling real-life phenomena [14,15]. The list is non-exhaustive however, we would recommend to a reader with an interest in the application of fractional calculus in Biology and Medicine to read the following selected papers (see [16–25]).

Almost eighty years ago, more precisely in the year 1940, the mathematician Ulam asked about the stability of functional equations [26], and a year later the mathematician Heyer responded to the question. By setting the necessary conditions for the stability of functional equations [27]. This result was generalized to differential equations, and it was believed that Obloza was the first to make this generalization [28], after which Alsina and Ger published a scientific paper containing what is known today as the Hyers-Ulam stability [29], Rus investigated the Hyers-Ulam stability of differential and integral equations using the Gronwall lemma [30]. Recently, the Hyers-Ulam stability problems of several differential equations were investigated by using the method of integral operators, see [31–34].

In this work, we kept a similar approach to earlier discussed examples. Indeed, we introduced modeling of the drug’s concentration blood using two fractional derivatives, namely the Caputo and the Caputo-Fabrizio fractional derivatives. Moreover, some qualitative analyses of the aforementioned models were obtained.

The remaining work is organized as follows. A review of FC definitions is provided in Section 2. The Section 3 focuses on a review of drug concentration modeling in blood. Section 4 establishes the existence and uniqueness of FC based approach proposed in this work. Section 5 establishes the Ulam-Hyers stability of the proposed fractional model. Section 6 is dedicated to applications on real-life data and the final section is the conclusion and discussion.

2  Some Useful Fractional Calculus Elements

FC has a strong foundation built upon definitions. Nowadays, they exit many fractional derivatives, each of them having its specific advantages and down points. This section focuses on some FC concepts required for the rest of the paper.

Definition 2.1 [35,36]. Given τ>0, a positive real number, the one-parameter Mittag-Leffer function is defined as follows:

Eτ(x)=∑r=0∞xrΓ(τr+1),

where x∈R and Γ stands for the commonly known gamma function, which is defined as follows:

Γ(x)=∫0∞e−μμx−1dμ,∀x>0

This function was introduced by the author of the same name in 1903 (see [37]). With the advent and spread of research in the field of FC, the function quickly became an essential tool for FC and particularly for fractional differential equations (FDE). In practice, most solutions of FDE are expressed in terms of the Mittag-Leffer function.

Another important derivative that is among the basis of FC is the Caputo fractional derivative. It is an integral form-based derivative defined as follows.

Definition 2.2 [35]. The Reimann-Liouville (RL) fractional integrals is defined by

(RLI0+δϑ)(ϖ):=1Γ(δ)∫0ϖ(ϖ−t)δ−1ϑ(t)dt,ϖ>0,Re(δ)>0.

Definition 2.3 [35]. The Caputo fractional derivative of order ν of a function ϑ:R+→R is given by

(CD0+νϑ)(τ)=∫0τ(τ−z)p−ν−1ϑ(p)(z)Γ(p−ν)dz,p−1<ν<p,p=[ν]+1.

where [ν] stands for the integer part of the real number ν.

Definition 2.4 [35]. Given f, a function defined such that, φ∈H1(c,d),c>d and γ∈[0,1], then it is possible to define another form of the Caputo Fractional derivative. This form is often called the new Caputo fractional derivative, and it expressed as follows:

(CFD0γφ)(τ)=M(γ)(1−γ)∫cτφ′(r)exp⁡[−γ1−γ(τ−r)]dr.

It is even possible to derive another useful fractional derivative that is used in this work. Indeed, if the function f introduced in Definition 2.3 is such that φ∉H1[c,d] then, the new fractional derivative called Caputo-Fabrizio fractional derivative is obtained as follows:

(CFD0qφ)(τ)=M(γ)(1−γ)∫0τφ′(r)exp⁡[−γ1−γ(τ−r)]dr.

The Caputo-Fabrizio fractional integral exists and is also useful in solving FDE. The said integral is defined as follows:

Definition 2.5 [35]. Given γ>0, a positive and real number denoting the order (fractional) of integral. The Caputo-Fabrizio fractional integral of a continuous function f:[0,+∞)→R is defined by

(CFI0+γf)(x)=2(1−γ)2M(γ)−γM(γ)f(x)+2γ2M(γ)−γM(γ)∫0xf(u)du,x≥0,

where M is a function with the property M(0)=M(1)=1. It is called the normalization function.

We end the shortlist of fractional derivatives and integrals of this section with Riemann-Liouville definitions.

Definition 2.6 [35]. Reimann-Liouville (RL) fractional integral is given by the following formula:

(RLI0+δϑ)(ϖ):=1Γ(δ)∫0ϖ(ϖ−t)δ−1ϑ(t)dt,ϖ>0,Re(δ)>0.

So far in this section, three fractional derivatives and two fractional integrals have been introduced. To end this section, we will state without proving some important Lemmas. These lemmas are used in sequel in the process of proving the existence of a solution to the problem and the uniqueness of the existing solution to the core problem which is also defined in the coming sections.

Lemma 2.3 [38] (Krasnoselskii’s Theorem).

Let us consider a Banach space B, and a convex set Q such that Q ⊂ B. Moreover, let q ∈ Q, and let us assume that H can be written as H=H1+H2. If in addition, H(Q¯) is a bounded set in B satisfying the following two conditions:

i)   H1:Q¯→B is continuous and completely continuous;

ii)   H2:Q¯→B is a contraction, a continuous non-decreasing function ω:[0,∞]→[0,∞], with ω(q1)>q1,q1>0, such that |H2(q1)−H2(q2)|≤ω(||q1−q2||), for any q1,q2∈Q¯.

Then it follows that:

1)   H(q0)=q0,q0∈Q¯, or

2)   ∃ q ∈ ∂Q and λ∈(0,1) with q=λH(q)+(1−λ)q.

Lemma 2.4 [38]. (Banach’s Contraction mapping principle).

Let X is a Banach space and let the operator Q: X → X be a contraction. Then, Q admits a unique fixed point x ∈ X with Qx = x.

3  A Review of Literature on Modeling of Drug Concentration in Blood

This section provides an overview of pharmacokinetics theories and the classical approach to building the dynamic model. A common approach to overcoming disease is to provide the sick person appropriate drug that will target harmful germs. It is therefore very substantial to inoculate the appropriate dose of the drug at suitable time intervals. Phase I of the clinical study of any drug focus on determining these parameters. Pharmacokinetics is defined as the branch of medicine that studies the dynamic (kinetics) of the drug when it is inoculated in a living body. In phase I of a clinical study, blood samples are taken at a regular time interval to evaluate the drug concentration. This is a purely experimental approach. There also exists a theoretical approach that consists of using a mathematical model and some historical data to develop a predictive model. A model that would predict drug concentration levels in the blood over time. In general, interest among researchers in computational biology has increased significantly over the past decades. Juinn et al. have dedicated research [39] to the study of Pharmacokinetics and Metabolism during a drug development process. Absorption, distribution, metabolism, and elimination (ADME), Reference [40] was the set of processes through which a drug is utilized by an individual leading to a concentration decrease over time.

It was mentioned in an earlier section that drug concentration decreases over time. Hence, a suitable mathematical model to describe such a process can be an exponential decay model. The following differential equation describes the decrease observed over time.

dC(t)dt=−μC(t),(1)

where μ is a constant coefficient that is experimentally estimated. It is called the exponential decay constant. This constant varies from one drug to another. Let us consider that at the initial state, at t = 0 a patient has inoculated a dose C0, of a drug, then one can estimate the drug concentration level, C in the patient’s blood at any time t by solving the differential equation, Eq. (1) (see [41]), and its solution is written as

C(t)=C0e−μt(2)

This study aims to investigate the performance of fractional differential equations in modeling while comparing them with classical differential equations used for solving the same problem. In the next sections, Caputo’s derivative and Caputo-Fabrizio’s derivative are used to build a fractional counterpart of the differential equation Eq. (1). The said models are applied to real-life data sets and the error rates they produce are used for a comparative study. The corresponding counterpart of Eq. (1) based on Caputo’s derivative and Caputo-Fabrizio’s derivative are respectively defined as follows:

CDαC(t)=−μCC(t),α∈(0,1)∪(1,2),CC(0)=CC0,(3)

CFDαC(t)=−μCFC(t),α∈(0,1)∪(1,2),CFC(0)=CFC0,(4)

In order the simplify notation and for conformity purpose, let a kernel function f:[a,T]×R→R be chosen as a continuous function and satisfying f(0,CFC(0))=0, then Eqs. (3) and (4) become respectively Eqs. (5) and (6) shown below:

CDαCC(t)=f(t,CC(t)),α∈(0,1)∪(1,2),CC(0)=CC0,(5)

CFDαCFC(t)=f(t,CFC(t)),α∈(0,1)∪(1,2),CFC(0)=CFC0,(6)

The following Lemmas introduce more generalized formulae of the solutions to the fractional drug concentration models with the initial condition, as defined by Eqs. (5) and (6).

Lemma 3.1: (see [35]). Given γ>0, and let n be the integer part of the real number γ. Let us consider the following initial value problem:

CDγψ(t)=βψ(t)+ϕ(t),(7)

where ψ(i)(0)=ψ0(i), i = 0, 1, …, n − 1, and the function ϕ∈C[0,T] is defined to be continuous.

There exists a solution to Eq. (7) and, the said solution can be written as follows:

ψ(t)=∑i=0n−1ψ0(i)wi(t)+ψ~(t),(8)

withψ~(t)={J0γφ(t)ifβ=01β∫0tϕ(t−s)w0′(s)dsifβ≠0,(9)

where wi(t)=J0ieγ(t), i = 0, 1, …, n − 1, eγ(t)=Eγ(βtγ).

Remark: If the real number γ, used in Lemma 3.1 is defined in a way that 0<γ<1, then it follows that solution to Eq. (7) can fully be defined by the mean of Eqs. (8)–(9). In which case the said solution is denoted by

ψ(t)=ψ0(0)Eγ(βtγ)+γ∫0tϕ(t−s)sγ−1Eγ′(βtγ)ds,(10)

Using Lemma 3.1 the solution to Eq. (3), when the derivative is defined in Caputo sense is expressed as follow

CC(t)−CC(0)=1Γ(α)∫0t(t−s)α−1ψ(s,CC(s))ds.(11)

Lemma 3.2 (see [35]). Let 0<γ<1, CF0C0∈R and σ∈C1[a,b] with σ(0)=0, then the following initial value problem

{CFD0+CFγC(t)=σ(t),t≥0, CFC(0)=CFC0(12)

has a unique solution which is moreover expressed by the following expression:

CFC(t)=CFC0+aγσ(t)+bγI1σ(t),t≥0(13)

with I1σ(t) representing the primitive function of σ. Moreover, explicit form of the coefficients found in Eq. (13) are provided here and defined as

aγ=2(1−γ)(2−γ)M(γ),bγ=2γ(2−γ)M(γ), i.e.,

CFC(t)=CFC0+2(1−γ)2M(γ)−γM(γ)σ(t)+2γ2M(γ)−γM(γ)∫0tσ(s)ds.(14)

4  Existence Result for the Drug Concentration

Considering the initial value problem given by Eq. (6), and denoting by H=C([0,T],R) the Banach space of all functions f, which are also selected or built such that f is continuous from the closed interval [0,T] to R, moreover let the space be endowed with the norm defined as

||C,CFC||=sup0≤t≤T|C,CFC(t)|.

Let an operator E: H → H be defined as follows:

(ECFC)(t)=CFC(0)+2(1−γ)2M(γ)−γM(γ)f(t,CFC(t))+2γ2M(γ)−γM(γ)(15)

×∫0tf(u,CFC(u))du.(16)

Eq. (15) is used to prove the existence and uniqueness of the solution. Besides, Lemmas 2.3 and 2.4 are needed in the process of proving the existence of a solution to the model proposed in Eq. (6). It is also used to prove the uniqueness of the solution found.

Theorem 4.1: Let us consider the function f:[0,T]×R→R. Moreover, if f is a continuous function defined in a way that the following two assumptions hold:

i) ∃k1>0 such that

|f(t,CFC1)−f(t,CFC2)|≤k1|CFC1−CFC2|,∀t∈[0,T],

for all CFC1,CFC2∈R.

ii)|f(t,CFC)|≤y1(t),∀(t,CFC)∈[0,T]×R,

where, y1∈C([0,T],R+)withsup0≤t≤T|y1(t)|=||y1||.

Moreover, assume that the following inequality holds for the defined constant:

k12γ2M(γ)−γM(γ)T<1.

Based on above setting, one is sure of the existence of at least one solution to the problem Eq. (6), subject to its initial conditions.

Proof. Let us consider the close set Br={CFC∈H,||CFC||≤r} where the radius, r is such that

r≥CFC0+(|2(1−γ)2M(γ)−γM(γ)|+|2γ2M(γ)−γM(γ)|T)||y1||

Let us introduce two operators, E1 and E2 which are built within the domain of Br and explicitly defined as

(E1CFC)(t)=CFC(0)+2(1−γ)2M(γ)−γM(γ)f(t,CFC(t)),(17)

(E2CFC)(t)=2γ2M(γ)−γM(γ)∫0tf(s,CFC(s))ds.(18)

Given two elements CFC1,CFC2∈Br, then it follows:

||E1CFC1+E2CFC2||≤(|2(1−γ)2M(γ)−γM(γ)|+|2γ2M(γ)−γM(γ)|T)||y1||≤r,

hence E1CFC1+E2CFC2∈Br.

In the forthcoming step of the proof, we will show that the operator E2 is a contraction. Let us start by observing that ∀t∈[0,T],∀CFC1,CFC2∈Br, the following relation holds:

|E2CFC1(t)−E2CFC2(t)|=|2γ2M(γ)−γM(γ)∫0t(f(s,CFC1(s))−f(s,CFC2(s)))ds|≤2γ2M(γ)−γM(γ)t|f(s,CFC1(s))−f(s,CFC2(s))|,

then

||E2CFC1−E2CFC2||≤2γ2M(γ)−γM(γ)Tk1||CFC1−CFC2||<||CFC1−CFC2||.(19)

Eq. (19) is an evidence that E2 is a contraction.

Moreover, E1 is a continuous operator due to the continuity of CFC. On the other hand, E1 is bounded uniformly, since

||E1CFC||≤CFC0+2(1−γ)2M(γ)−γM(γ)||y1||.

Beside all the above properties, one can prove compactness of E1 in the following approach:

For t1,t2∈[0,T],suchthat(t1<t2), it follows that:

||(E1CFC)(t2)−(E1CFC)(t1)||≤2(1−γ)2M(γ)−γM(γ)|f(t2,CFC(t2))−f(t1,CFC(t1))|.(20)

The quantity defined on the right-hand side of the inequality Eq. (20) goes to zero as t1→t2. Moreover, it is observed that the quantity ||(E1CFC)(t2)−(E1CFC)(t1)|| does not depend on CFC, this implies that E1 is relatively compact. A Recall of Lemma 2.2 is enough to derive the conclusion that E1 is a compact operator on the closed set Br. Moreover, recalling Lemma 2.3, is enough to conclude on the existence of a solution to Eq. (6). Q.E.D.

Theorem 4.2: Let f:[0,T]×R→R be a continuous and assume moreover that

k1(2(1−γ)2M(γ)−γM(γ)+γ2M(γ)−γM(γ)T)<1,

then the solution to the initial value problem Eq. (6) is unique.

Proof. Consider the close set Br={CFC∈H,||CFC||≤r} with

r≥CFC0+N(2(1−γ)2M(γ)−γM(γ)+2γ2M(γ)−γM(γ)T)1−k1(2(1−γ)2M(γ)−γM(γ)+2γ2M(γ)−γM(γ)T),whereN=sup0≤t≤T|f(t,0)|. The first step of the proof is to show that EBr⊂Br. To do so, observe that ∀CFC∈Br,∀t∈[0,T], it follows that

|(ECFC)(t)|≤CFC0+2(1−γ)2M(γ)−γM(γ)|f(t,CFC(t))|+2γ2M(γ)−γM(γ)∫0t|f(s,CFC(s))|ds.(21)

On the other hand, the following inequality is derived from the quantity or norm of |f(t,C(t))|:

|f(t,CFC(t))|=|f(t,CFC(t))−f(t,0)+f(t,0)|≤|f(t,CFC(t))−f(t,0)|+|f(f,0)|≤k1||CFC||+N≤k1r+N,(22)

Using Eqs. (21)–(22) leads to the following relation:

||ECFC||≤CFC0+(2(1−γ)2M(γ)−γM(γ)+2γ2M(γ)−γM(γ)T)(k1r+N)≤r,,(23)

Eq. (23) implies that ECFC∈Br,∀CFC∈Br which means that in general EBr⊂Br.

The second step of this proof is to show that the operator E is a contraction. To do so, observe that ∀CFC1,CFC2∈H, the following relation holds:

|ECFC1(t)−ECFC2(t)|≤2(1−γ)2M(γ)−γM(γ)|f(t,CFC1(t))−f(t,CFC2(t))|+2γ2M(γ)−γM(γ)∫0t|(f(s,CFC1(s))−f(s,CFC2(s)))|ds.(24)

The relation given by Eq. (24) leads to

||ECFC1−ECFC2||≤(2(1−γ)2M(γ)−γM(γ)+2γ2M(γ)−γM(γ)T)k||CFC1−CFC2||<||CFC1−CFC2||.,(25)

On can see from Eq. (25) that E is a contraction. Therefore, recalling Lemma 2.4 is enough to conclude that Eq. (6) has a unique in the closed interval [0,T]. The said solution is built subject to initial values problem conditions. Q.E.D.

Having proven the existence and the uniqueness of a solution to the problem defined by Eq. (6), we finally summarize the analytical solutions (see [34]) for these fractional models as follows: Based on Lemma 3.1, the analytical solution for the drug concentration model via Caputo sense Eq. (5) is given by

CC(t)=CC0Eα(−μtα).(26)

Based on Lemma 3.2, the analytical solution for the drug concentration model via Caputo-Fabrizio sense Eq. (6) is given by

CFC(t)=CFC0e−2α21μ+2(1−α)t.(27)

All the necessary theoretical work on the proposed approach was done in previous sections. In the next section stability of the model is studied.

5  Ulam-Hyers Stability Results

In this section, we discuss the UH and generalized UH (GUH) stability of the considered problem.

Definition 5.1. Let f:[0,T]×R→R be continuous. Then Eq. (6) is UH stable if there exists κf>0 and ε>0 such that each solution CFN∈H of

|CFD0+CFαN(t)−f(t,CFN(t))|≤ε,∀t∈[0,T],(28)

there exists a solution C ∈ H of Eq. (6) with

|CFN(t)−CFC(t)|≤κfε,∀t∈[0,T].(29)

Definition 5.2. Let f:[0,T]×R→R be continuous. Then Eq. (6) is GUH stable with respect to Λf∈C(R+,R+) with Λf(0)=0, for ε>0 such that each solution CFN∈H of Eq. (28), there exists a solution CFC∈H of Eq. (6) with

|CFN(t)−CFC(t)|≤Λf(ε),∀t∈[0,T].(30)

Remark 5.3. CFN∈H satisfies Eq. (28) if and only if there exists a function η∈H with

|η(t)|≤ε,∀t∈[0,T];

CFD0+CFαN(t)=f(t,CFN(t))+η(t),CFN(0)=N0.(31)

Lemma 5.4 (Gronwall’s Lemma, [42]). Let T > 0, and d > 0. Assume that ω,ϖ:[0,T]→R+ are continuous. If

ω(t)≤d+∫0tϖ(s)ω(s)ds,∀t∈[0,T],

then

ω(t)≤dexp⁡(∫0tϖ(s)ds),∀t∈[0,T].

By Remark 5.3, the solution of Eq. (31) can be formulated by

CFN(t)=N0+aα[f(t,CFN(t))+η(t)]+bα∫0t[f(s,CFN(s))+η(s)]ds,(32)

where aα:=2(1−α)2M(α)−αM(α),bα:=2α2M(α)−αM(α).

Hence, we will make the following remark.

Remark 5.5. Let CFN∈H satisfies Eq. (28). Then CFN is a solution to the next integral inequality:

|CFN(t)−N0−aαf(t,CFN(t))−bα∫0tf(s,CFN(s))ds|=|N0+aα[f(t,CFN(t))+η(t)]+bα∫0t[f(s,CFN(s))+η(s)]ds−N0−aαf(t,CFN(t))−bα∫0tf(s,CFN(s))ds|=|aαη(t)+bα∫0tη(s)ds|≤aα|η(t)|+bα∫0t|η(s)|ds≤(aα+bαT)ε.

Now we are ready to announce the results of the stabilization results for UH and GUH.

Theorem 5.6. Under the hypotheses of Theorem 4.1. If aαk1<1, then the solution of Eq. (6) is HU stable on [0, T] and consequently GUH stable.

Proof. Let CFN∈H be a solution of Eq. (28). From Lemma (3.2),

{CFD0+αCFC(t))=f(t,CFC(t)),0<α<1,t∈[0,T],CFC(0)=CFN(0)(33)

has the unique solution

CFC(t)=C0+aαf(t,CFC(t))+bα∫0tf(s,CFC(s))ds,(34)

Since CFC(0)=CFN(0), then C0=N0. Hence Eq. (34) becomes

CFC(t)=N0+aαf(t,CFC(t))+bα∫0tf(s,CFC(s))ds.(35)

Thus, by our assumptions, Remark 5.5, along with Eqs. (32) and (35), we obtain

|CFN(t)−CFC(t)|=|CFN(t)−N0−aαf(t,CFC(t))−bα∫0tf(s,CFC(s))ds|≤|CFN(t)−N0−aαf(t,CFN(t))−bα∫0tf(s,CFN(s))ds|+aα|f(t,CFN(t))−f(t,CFC(t))|+bα∫0t|f(s,CFN(s))−f(s,CFC(s))|ds≤(aα+bαT)ε+aαk1|CFN(t)−CFC(t)|+bαk1∫0t|CFN(s)−CFC(s)|ds.

As aαk1<1, then

|CFN(t)−CFC(t)|≤(aα+bαT)1−aαk1ε+bαk11−aαk1∫0t|CFN(s)−CFC(s)|ds.

From Lemma 5.4, we have

|CFN(t)−CFC(t)|≤(aα+bαT)1−aαk1εexp⁡(∫0tbαk11−aαk1ds),∀t∈[0,T],

which implies

|CFN(t)−CFC(t)|≤(aα+bαT)1−aαk1εexp⁡(bαk1t1−aαk1),∀t∈[0,T].

Set κf:=(aα+bαT)1−aαk1exp⁡(bαk1T1−aαk1), one has

|CFN(t)−CFC(t)|≤κfε,∀t∈[0,T].

Hence, Eq. (36) is UH stable in H. Moreover, if we set Λf(ε)=κfε, which satisfies Λf(0)=0, it is also concluded that Eq. (6) is GUH stable.

Let us consider the following CF-problem:

CFD0+14CFC(t)=e−3t|CFC(t)|(3+et)(1+|CFC(t)|),t∈[0,3],CFC(0)=0,(36)

and the CF-inequality

|CFD0+14CFC(t))−e−3t(3+et)|CFC(t)|1+|CFC(t)||≤ε,t∈[0,3],(37)

where α=14,T=3,C0=0 and f(t,CFC(t))=e−3t|CFC(t)|(3+et)(1+|CFC(t)|), for (t,CFC)∈[0,3]×R. Distinctly, f is continuous with f(0,CFC(0))=0. Then, M(14)=87,aα=34,bα=14, and for any t ∈ [0, 3], CFC,CFC∗∈R,

|f(t,CFC)−f(t,CFC∗)|=e−3t3+et||CFC|(1+|CFC|)−|CFC∗|(1+|CFC∗|)|≤e−3t3+et||CFC(t)−CFC∗(t)|(1+|CFC|)(1+|CFC∗|)|≤e−3t3+et|CFC−CFC∗|≤e−3t4|CFC−CFC∗|≤14|CFC−CFC∗|=k1|CFC−CFC∗|.

For any t ∈ [0, 3], CFC∈R,

|f(t,CFC)|=e−3t3+et||CFC|(1+|CFC|)|≤e−3t3+et=y1(t)∈C([0,3],R+).

Therefore, i) and ii) hold. Also, k1aαT=916<1. From Theorem 4.1, Eq. (36) has a unique solution on [0, 3].

Let CFN∈C([0,3],R) be a solution of Eq. (37). Then

|CFD0+14CFN(t))−f(t,CFN(t))|=|CFD0+14CFN(t))−e−3t|CFN(t)|(3+et)(1+|CFN(t)|)|≤ε,t∈[0,3].(38)

From Theorem 4.1, we known (s1) with CFC(0)=C0 has the unique solution

CFC(t)=C0−a14f(0,CF)C(0)+a14f(t,CFC(t))+b14∫0tf(s,CFC(s))ds=C0−14+34(e−3t|CFC(t)|(3+et)(1+|CFC(t)|))+14∫0te−3s|CFC(s)|(3+es)(1+|CFC(s)|)ds.