Ascaris lumbricoides is a soil-transmitted parasite responsible for ascariasis. It ranks as the most prevalent parasitic helminth infection globally and causes significant health problems in humans, including lung diseases and intestinal obstructions [1]. The impact of this infection is particularly severe in impoverished populations that do not have access to clean water, sanitation, and hygiene facilities, especially in tropical and subtropical regions, due to the parasite’s life cycle relying on fecal soil contamination [2].

Beyond its substantial health implications, the migration of Ascaris larvae through the lungs can contribute to various pulmonary diseases, including asthma. In addition, chronic intestinal ascariasis can result in growth stunting, malnutrition, and severe abdominal pain [3]. The immune response to this parasite plays a role in the pathogenesis of allergic diseases, studies indicating that antibody responses to its proteins are linked to asthma symptoms [4]. According to [5], recent advances focus on understanding tissue-specific Type 2 immune responses to helminths, including the discovery of immune cells and cytokine pathways that contribute to immunity, tissue repair, and tolerance to parasites, as well as comparisons with immune-related diseases such as asthma and allergies. Upon exposure to allergens, crosslinking of IgE triggers the release of histamine, leukotrienes, and prostaglandins, leading to bronchoconstriction [6].

Pulmonary function tests are commonly used to assess lung function by measuring lung volumes and capacities. The connection between ascariasis and asthma remains unclear and is currently being investigated in regions that transition from high to low prevalence of helminthiasis [7]. Previous studies in Iraq have examined various aspects of ascariasis [8–10], while global research has been conducted in Brazil, Colombia, and Europe [11–13].

Recently, mathematical models and fractional calculus have been widely used to analyze the spread and control of infectious diseases, providing insight into epidemic dynamics, as demonstrated in studies on dengue infection [14] and breast cancer [15]. In [16], the authors analyze the dynamics of monkeypox in the UK and evaluate the impact of vaccination using a fractional mathematical model based on real data to inform effective disease control strategies. The stability of the Bcl-2/Bax ratio over time in reproductive cancer has been studied using Atangana-Baleanu fractional derivative operators, along with an investigation of the effect of the ABT-737 inhibitor on mitochondrial apoptosis through mathematical modeling and numerical simulations [17]. A fractal-fractional cancer model has been developed to examine the interactions between stem cells, effector cells, and tumor cells, both with and without chemotherapy, as well as to assess the role of chemotherapy in cancer treatment [18]. In addition, the study in [19] explored the effectiveness of seasonally timed treatment programs to control Ascaris lumbricoides infections using mathematical modeling for the four different stages of the life cycle of A. lumbricoides, intending to educate public health strategies by optimizing treatment schedules and maximizing the impact of intervention.

Over the years, fixed-point theory has become a crucial and effective tool for studying nonlinear phenomena. For example, the authors of [20] discussed that the Banach theorem has been utilized to establish the uniqueness, stability, and existence of stable solutions for the fractional-order mathematical model for cervical cancer in the sense of the Caputo-Fabrizio operator. Various fixed-point theorems and stability analyzes have been used to determine the conditions under which solutions exist and remain unique for different types of fractional differential problems; see [21–24]. Recently, stability has been an important topic in differential equations and guaranteeing that there is a close and exact solution. Several articles have been published related to Ulam-Hyers and Ulam-Hyers-Rassias stability; see [25–28]. Some researchers have begun to focus on the existence and stability of p-integrable solutions under specific conditions, with the aid of Hölder’s inequality, which helps prove the continuity or boundedness of operators in the space. In particular, the existence and uniqueness results of fractional differential equations with boundary conditions when (0<α≤1) in Łp space have been studied in [29,30]; for additional details, one can refer to [31–33].

Motivated by the above work, the objective of this study is to assess the relationship between ascariasis and asthma in humans. This research aims to increase public awareness about the health complications associated with this parasite, highlighting the importance of risk reduction and early treatment. Furthermore, by using fresh mathematical techniques, this study generalizes the model presented in [19] using the fractal-fractional Caputo-Fabrizio (FFCF) derivative of order α as follows: (1)Dα,cFFCFJ(η)=BL(η)−(1τ1+M1)J(η),Dα,cFFCFM(η)=1τ1J(η)−M2M(η),Dα,cFFCFE(η)=sNλM(η)−(1τ2+γ1)E(η),Dα,cFFCFL(η)=1τ2E(η)−(BN+γ2)L(η),with the initial conditions J(0)=J0,M(0)=M0,E(0)=E0,L(0)=L0.

Here, we present an analysis of the existence and uniqueness of a fractal-fractional Ascaris lumbricoides mathematical model by applying Banach’s contraction mapping principle and Krasnosel’skii’s fixed point theorem, along with the Hölder inequality in the Łp space. Furthermore, the stability of the model was investigated under sufficient conditions through the Ulam-Hyers and Ulam-Hyers-Rassias stability approaches. To the best of our knowledge, this is the first attempt to investigate the existence and stability of a fractal-fractional Ascaris lumbricoides model in Łp space.

The structure of the manuscript is as follows: Section 1 presents the introduction and motivation. Section 2 outlines the materials and methods, while Section 3 displays the results. The definitions and fundamental concepts relevant to this study are introduced in Section 4. The fractal-fractional extension of the mathematical model for Ascaris infection is formulated in Section 5. The existence and uniqueness theorems for the fractal-fractional model are established in Section 6. The Ulam-Hyers and Ulam-Hyers-Rassias stability is analyzed in Section 7. Finally, the discussion and conclusion are provided in Sections 8 & 9, respectively.

This section outlines the study design, including participant grouping, laboratory procedures to detect ascariasis, serological tests for anti-Ascaris lumbricoides antibodies, data collection, and statistical analysis methods.

Patients: This study included 270 individuals with asthma and 130 controls who visited general hospitals in Duhok city, Iraq. A specialist physician confirmed the asthma diagnosis prior to the enrollment of patients in the study. The participants ranged in age from 15 to 80 years. A questionnaire was developed to collect information from each participant.

Pulmonary function tests: Pulmonary function tests (PFT) were conducted on all subjects using a MicroMedical Spirometer (MIR SpirolabIII Diagnostic Spirometer, Ltd., England), which is effective for accurate diagnosis of respiratory conditions such as asthma and pulmonary diseases. All participants underwent spirometry and received instructions to forcefully and continuously exhale into the instrument’s mouthpiece. This test was performed as a confirmatory diagnosis for asthma.

Blood Sample Collection: A volume of five milliliters of venous blood was collected from participants using a sterile syringe. One milliliter of this blood was placed in a tube containing anticoagulants for blood parameter analysis using the Coulter Count machine (Swelab, Germany). In [34], the remaining four milliliters was transferred to a second tube without anticoagulants, allowed to clot for 20 min, and then centrifuged for 10 min at 3000 rpm to obtain the serum. The collected serum was stored in sterile Eppendorf tubes at −20°C until it was needed for analysis in [35].

Measurement of blood parameters by Coulter Count machine: A volume of one milliliter of the blood sample was placed into a tube containing anticoagulant and analyzed using the Coulter Count machine to assess blood parameters [36].

Parasite examination: This study assessed A. lumbricoides using anti-A. lumbricoides IgM (AFG Bioscience, USA) and anti-E. granulosus IgG antibodies (AFG Bioscience, USA).

Serum total IgE measurement: The measurement of total IgE levels in serum was conducted using an ELISA kit (AFG Bioscience, USA), which allows for a quantitative assessment of human IgE in vitro. The procedure was carried out in accordance with the manufacturer’s guidelines.

Measurement of IL-4 using ELISA: In this study, interleukin-4 was measured using a kit (AFG Bioscience, USA), following the manufacturer’s guidelines. The optical density was recorded at 450 nm with a BioTek ELISA plate reader (USA). The procedure was carried out following the manufacturer’s guidelines.

Inclusion criteria: Subjects who agreed to participate in the current study were included.

Exclusion criteria: Individuals who declined to participate in this study, along with those suffering from infectious, non-infectious, chronic, and autoimmune diseases, were excluded.

Statistical analysis: All data were analyzed using the statistical program R and a chi-square test. Descriptive statistics included means, standard deviations, and ranges for numerical variables, along with frequencies (n) and percentages (%) for categorical data. The results are presented in tables and histogram charts. A p-value of less than 0.05 was considered statistically significant.

In this section, the results of the study are presented, including the distribution of ascariasis infection among different groups, the serological findings, and statistical analysis of the data.

A total of 400 subjects aged 15 to 80 years participated in this study, with 170 (42.5%) found to be infected with ascariasis. Among the infected cases, 45 individuals (26.5%) tested positive for anti-A. lumbricoides IgM antibodies, while 125 (73.5%) had IgG antibodies. The asthmatic group included 270 subjects and 130 controls. Within the asthmatic patients, 140 (51.9%) tested positive for ascariasis, while only 30 infections (23.1%) had been detected in the control group, as explained in Table 1.

Table 2 displays the distribution of various demographic characteristics between asthmatic and control subjects. Concerning asthmatic patients, the majority were male, with 170 cases (63.0%), while females made up 100 cases (37.5%). The age group with the highest occurrence was 48 to 58 years, accounting for (37.0%) of the cases, whereas only 8 cases (3.0%) observed in the 70 to 80-year age range. The data indicated that 160 cases (59.3%) from urban areas, in contrast to 110 cases (40.7%) from rural locations. A notable proportion of asthmatic patients had completed secondary education, totaling 190 cases (70.4%), compared to 16 cases (5.9%) among individuals with lower educational qualifications, within the asthmatic cohort, 160 cases (59.2%) ported having a low income, while 35 patients (13.0%) reported having a high income. In addition, 150 cases (55.6%) reported did not wash their hands before meals, while (44.4%) practiced hand washing. Finally, 140 asthmatic patients (51.9%) had exposure to soil, while 130 cases (48.1%) did not.

The analysis of ascariasis among the examined subjects is illustrated in Fig. 1. Out of 270 asthmatic patients, 140 (51.9%) tested seropositive for Ascaris, while 130 (48.1%) showed no infection.

Fig. 2 shows the sociodemographic features among individuals. Regarding asthmatic patients, males comprised the majority of cases of 170 (63.0%), followed by 100 females (37.0%). The highest cases belonged to age group 48–58 years represented 100 (37.0%) and the lowest found to be 8 (3.0%) as 70–80 years old. The rates of asthmatic patients from urban and rural areas accounted for (59.3%) and (40.7%), respectively. Secondary qualification reported 190 (70.4%) and illiterates 16 (5.9%). In terms of monthly income, 160 (59.2%) cases recorded a monthly income below 150.000 Dinars, and 35 subjects (13.0%) had more than 300.000 Dinars. About, 140 (51.9%) cases had a soil contact and 130 (48.1%) did not have it.

Fig. 3 shows the infected and non-infected cases among asthmatic patients. About, the infection rate among males and females are 98 (70.0%) and 42 (30.0%), respectively. The age group 48–58 years old assessed 45 (32.1%) and 6 cases (4.3%) among 70–80 years old. Infected rural patients reported 72 (51.4%) and urban ones 68 (48.6%).

Table 3 clarifies the prevalence of ascariasis according to various factors. About asthmatic patients, 98 males (70.0%) had been infected, while females constituted 42 cases (30.0%). The highest infection rate, at (32.1%), was observed in the 48–58 age group, whereas only (4.3%) of those aged 70–80 were infected. Infection rates in rural areas stood at (51.4%), compared to (48.6%) in urban settings. A significantly high rate of infection was found among individuals with secondary education, accounting for (75.0%), in contrast to just (7.1%) among those with higher education. Concerning monthly income, the majority of infections were found in the low-income group (64.3%), while only (5.7%) noted among high-income individuals. Also, the rate of infection was (53.6%) among subjects who did not practice hand washing, compared to (46.4%) among hand washing. Finally, individuals with soil contact had an infection rate of (58.6%), while those without such contact had a rate of (41.4%). All the studied risk factors had an association with infection with a p-value <0.05.

Table 4 presents the mean values ± standard deviation (SD) for different blood parameters, IgE and IL-4, for both infected asthmatic individuals and the control group. In the infected asthmatic group, there was a notable reduction in the number of red blood cells and hemoglobin concentration, while their white blood cell count showed a significant increase compared to controls. Furthermore, both IgE and IL-4 levels were significantly higher in the patients. All the studied parameters had an association with infection with p-value <0.05.

This section introduces the fundamental definitions, lemmas, and prerequisite results that are crucial for developing the theoretical framework of the study.

The Caputo-Fabrizio derivative of order 0<α<1 (in the sense of Caputo) is defined as:

Definition 1. [37] Let a∈R, 0<α≤1 and ϝ be a continuous on (a,b), then the fractional derivative of order α in the sense of Caputo-Fabrizio is given by DaαCFϝ(η)=S(α)1−α ∫aηϝ′(θ) e(−α1−α(η−θ))dθ,0<α≤1,where S(0)=S(1)=1.

According to Riemann’s definition, it is formulated as follows: DaαCFRϝ(η)=S(α)1−α ddη∫aηϝ(θ) e(−α1−α(η−θ))dθ,0<α≤1.

Then the corresponding Caputo Fabrizio integral is defined by IaαCFϝ(η)=1−αS(α)ϝ(η)+αS(α)∫aηϝ(θ)dθ,so that IaαCFDaαCFRϝ(η)=DaαCFRIaαCFϝ(η)=ϝ(η),and (2)IaαCFDaαCFϝ(η)=ϝ(η)−ϝ(a).

However, it has been observed that (3)DaαCFIaαCFϝ(η)=ϝ(η)−S(α)ϝ(a)eλα(t−a)1−α,λα=−α1−α.

The fractal of order c>0, initiated at a, for a function ϝ is defined as dϝ(θ)dθc=limη→θϝ(η)−ϝ(θ)(η−a)c−(θ−a)c.

It is clear that ϝ(θ) is differentiable, and by applying L’Hôpital’s rule, the following is obtained dϝ(η)dηc=ϝ′(η)c(η−a)c−1.

Definition 2. [38,39] Suppose that ϝ(η) is a continuous function and fractal differentiable on an open interval (a,b) with order c. Then, a α-order fractal-fractional derivative of ϝ(η) in a Caputo sense with an exponential decay type kernel is given by Daα,cFFCFϝ(η)=S(α)1−α ∫aηdϝ(θ)dθc e(−α1−α(η−θ))dθ,0<α≤1,c>0.where S(0)=S(1)=1.

The fractal-fractional derivative in the Riemann sense with an exponential law is defined as Daα,cFFCFRϝ(η)=S(α)1−α ddηc∫aηϝ(θ) e(−α1−α(η−θ))dθ,0<α≤1,c>0.

Definition 3. [39] Suppose that ϝ is a continuous function on (a,b), the fractal-fractional integral of ϝ(η) with an exponential decaying type kernel is given by Iaα,cϝ(η)=(1−α) cS(α)(η−a)c−1ϝ(η)+α cS(α)∫aη(η−θ)c−1ϝ(θ)dθ,=IaαCF(c(η−a)c−1ϝ(η)),where S(0)=S(1)=1.

Notice that for c>1, it holds that (Iaα,cϝ)(a)=0.

Remark 4. Following [39], in Definition (3), the author selects the fractal order c to lie between 0 and 1. However, this choice prevents the corresponding integral operator from vanishing at the initial time η=0. As a result, the solution generated in the form of an integral equation does not satisfy the initial condition unless the right-hand side of the model also vanishes at η=0. This imposes a constraint on the initial population size. Moreover, it becomes impossible to verify the solution in the backward direction. To overcome this issue, we choose the value of c to be greater than one; for instance, 1<c≤2.

Further, it is clear that Iaα,cDaα,cFFCFRϝ(η)=Daα,cFFCFRIaα,cϝ(η)=ϝ(η),but not able to prove that Iaα,cDaα,cFFCFϝ(η)=ϝ(η)−ϝ(a).

Due to this fact and Remark 4, the following new fractal fractional derivative with exponential law is defined in the Caputo sense as

Definition 5. Suppose that ϝ(η) is a continuous function and fractal differentiable on an open interval (a,b) with order c. Then, a α-order fractal-fractional derivative of ϝ(η) in a Caputo sense with an exponential decay type kernel is given by Daα,cFFCFϝ(η)=DaαCFϝ(η)c(η−a)c−1,0<α≤1, c>1.

The following lemma is crucial for advancing the solution representation of the system that describes the investigated model. The solution will satisfy the dynamic equation on both sides and verify the initial data.

Lemma 6. For c>1,0< α<1 and a function ϝ:[a,b]→ℛ whose derivative is integrable, it follows that

(4)Iaα,cDaα,cFFCFϝ(η)=ϝ(η)−ϝ(a).

Proof.   •  From definition and (2), the result is Iaα,cDaα,cFFCFϝ(η)=IaαCF[c(η−a)c−1DaαCFϝ(η)c(η−a)c−1]=ϝ(η)−ϝ(a).

From definition of the fractal fractional operators, (3), and that (η−a)c−1 vanishes at a, one can has Daα,cFFCFIaα,cϝ(η)=DaαCFIaαCF[c(η−a)c−1ϝ(η)]c(η−a)c−1ϝ(η)=ϝ(η).◻

Lemma 7. [40] The measurable function ϝ:[a,b]×ℛ→ℛ is Bochner integrable, if ||ϝ|| is Lebesgue integrable.

Theorem 8. (Kolmogorov) [40]

Suppose ϝ^ is a set of functions in Łp[0,J], 1≤p<∞. To ensure that this set is relatively compact, it is essential and adequate for both of the subsequent conditions to be fulfilled:

(A) The set ϝ^ is bounded in Łp;

(B) limg→0||Hg−H||=0 uniformly with respect H∈ϝ^, where Hg(η)=1g∫ηη+gH(θ)dθ.

Lemma 9. (Hölder’s inequality) [40]

Let B be a measurable space and f1 and f2 satisfy the condition 1f1+1f2=1. 1≤f1<∞, 1≤f2<∞ with (mn)∈Ł(B), which is satisfied if m belong to Łf1(B) and n belong to Łf2(B). ∫B|mn|dη≤(∫B|m|df1η)1f1(∫B|n|df2η)1f2.

Theorem 10. (Krasnosel’skii theorem) [41]. Let M be a closed, bounded, convex, and nonempty subset of a Banach space V. Let A and B be two operators such that (1)

Az1+Bz2∈M whenever z1,z2∈M;

Then there exists z∈M such that z=Az+Bz.

Mathematical model of the dynamics of Ascaris infection

Mathematical modeling is an essential tool for effectively illustrating and understanding the epidemic models and intricacies of scientific phenomena. In this section, we discuss the mathematical model developed by [19] for the four stages of the A. lumbricoides life cycle, as described below. (6)dJdη=BL(η)−(1τ1+M1)J(η),dMdη=1τ1J(η)−M2M(η)dEdη=sNλM(η)−(1τ2+γ1)E(η)dLdη=1τ2E(η)−(BN+γ2)L(η)with initial conditions J(0)=J0>0,M(0)=M0>0,E(0)=E0>0,L(0)=L0>0,where B is ingestion or uptake rate seasonal; τ1 is the maturation rate from juvenile stage to adult worm; τ2 is maturation rate from eggs to infective larvae; μ is death rate of hosts; M1 is death rate of juvenile worms; M2 is death rate of adult worms; γ1 is death rate of immature eggs; γ2 is death rate of infective larval stages; s is sex ratio in adult worms (proportion female); λ is baseline fecundity per adult female worm; N is host population size; J is juvenile worms (inside the host); M is mature worms (inside the host); E is eggs (developing in the environment); and L is larvae (at infectious stage in the environment). These states are denoted by the letters; J and M are taken to be mean values per host, whereas E and L are total values in the environment.

The work mentioned above serves as our inspiration, as we examine the model (6) in the fractal-fractional sense, in the Caputo-Fabrizio (CF) sense: (7)Dα,cFFCFJ(η)=BL(η)−(1τ1+M1)J(η),Dα,cFFCFM(η)=1τ1J(η)−M2M(η),Dα,cFFCFE(η)=sNλM(η)−(1τ2+γ1)E(η),Dα,cFFCFL(η)=1τ2E(η)−(BN+γ2)L(η).

The functions gi for i=1,2,3,4 are provided as follows: g1(η,J(η))=BL(η)−(1τ1+M1)J(η),g2(η,M(η))=1τ1J(η)−M2M(η),g3(η,E(η))=sNλM(η)−(1τ2+γ1)E(η),g4(η,L(η))=1τ2E(η)−(BN+γ2)L(η). Dα,cFFCFS(η)={Dα,cFFCFJ(η),Dα,cFFCFM(η),Dα,cFFCFE(η),Dα,cFFCFL(η),S(η)={J(η),M(η),E(η),L(η), S0=S(0)={J(0),M(0),E(0),L(0),ϖ(η,S(η))={g1(η,J(η)),g2(η,M(η)),g3(η,E(η)),g4(η,L(η)).

The fractal-fractional mathematical model (7) can be reformulated as the following problem (8)Dα,cFFCFS(η)=ϖ(η,S(η)),0<α≤1S(0)=S0,η∈J=[0,Q].

Lemma 11. Let ℳ∈Łp(K,R), then the initial value problem (8) has a solution (9)S(η)=S0+(1−α)cS(α)ηc−1ϖ(η,S(η))+αcS(α)∫0ηθc−1ϖ(θ,S(θ)) dθ.

Proof. Applying the fractal-fractional integral I0α,c to both sides of Eq. (8), and utilizing (4) from Lemma 6 with a=0, yields the solution representation given by Eq. (9). Conversely, applying the fractal-fractional derivative to the solution representation in Eq. (9), and using (5) in Lemma 6, recovers the right-hand side of Eq. (8). Moreover, the solution given in Eq. (9) satisfies the initial data. The proof is complete. □

The solutions can be expressed as follows: (10)J(η) = J(0)+(1−α) cS(α)ηc−1g1(η,J(η))+α cS(α)∫0ηθc−1g1(θ,J(θ))dθ,M(η) = M(0)+(1−α) cS(α)ηc−1g2(η,M(η))+α cS(α)∫0ηθc−1g2(θ,M(θ))dθ,E(η) = E(0)+(1−α) cS(α)ηc−1g3(η,E(η))+α cS(α)∫0ηθc−1g3(θ,E(θ))dθ,L(η) = L(0)+(1−α) cS(α)ηc−1g4(η,L(η))+α cS(α)∫0ηθc−1g4(θ,L(θ))dθ. DαCFS(η)=cηc−1ϖ(η,S(η)),⟹DαCFS(η)cηc−1=ϖ(η,S(η)),c>1.DαFFCFS(η)=ϖ(η,S(η)).

This section focuses on studying the existence and uniqueness of the fractal-fractional model (8) under certain conditions. For a measurable function denoted by ϖ, equipped with the following norm ‖ϖ‖pp=∫0Q|ϖ(η)|pdη,(1≤p<∞).

In this case, Łp(J,ℛ) denotes the Banach space of all Lebesgue measurable functions.

The assumptions below are fundamental to this analysis:

(ℶ1) ∃ a positive functions ψ(η),ψ1(η),ψ2(η),ψ3(η),ψ4(η) such that

|ϖ(η,S(η))|≤ψ(η), |g1(η,J(η))|≤ψ1(η), |g2(η,E(η))|≤ψ2(η), |g3(η,M(η))|≤ψ3(η),|g4(η,L(η))|≤ψ4(η).

(ℶ2) ∃ a constants ℵ,ℵ1,ℵ2,ℵ3,ℵ4>0 such that ℵ=sup|ϖ(θ,0)|, ℵ1=sup|g1(θ,0)|, ℵ2=sup|g2(θ,0)|, ℵ3=sup|g3(θ,0)|, ℵ4=sup|g4(θ,0)|.

For simplicity, we define the notation as follows: 𝒩1=[((1−α)cS(α))pQp(c−1)+1p(c−1)+1+(αcS(α))p(p−1pc−1)p−1Qpc+1pc+1],𝒩2=[((1−α)cS(α))pQp(c−1)+1p(c−1)+1+(αcS(α))p(p−1pc−1)p−1Qpcpc], 𝒢=(((1−α)cS(α))pQp(c−1)+1p(c−1)+1+(αS(α))pQpc+1pc+1),Ω1=2 K1𝒩21p,Ω2=2 M2𝒩21p,Ω3=2 K2𝒩21p,Ω4=2 K3𝒩21p.Ω=2 Q1 𝒩21p.

Lemma 12. The function ϖ(η,S(η)) defined in Eq. (8) satisfies the Lipschitz condition with respect to ϖ, and the following Lipschitz constant is obtained Q1=max((1τ1+M1),M2,(1τ2+γ1),(BN+γ2))

Proof. |g1(η,J(η))−g1(η,J^(η))|≤K1|J(η)−J^(η)|.

In the same way (11)|g2(η,M(η))−g2(η,M^(η))|≤M2|M(η)−M^(η)|,|g3(η,E(η))−g3(η,E^(η))|≤K2|E(η)−E^(η)|,|g4(η,L(η))−g4(η,L^(η))|≤K3|L(η)−L^(η)|.where K1=(1τ1+M1), K2=(1τ2+γ1), K3=(BN+γ2).

By including all equations, it concludes that (12)|ϖ(η,S(η))−ϖ(η,S^(η))|≤Q1|S(η)−S^(η)|.

□

Theorem 13. Assume that ϖ satisfy the condition (12). If Ω1,2,3,4<1, then the problem (8) has a only one solution.

Proof. First, consider the operator H defined by (13)(HJ)(η)=S(0)+(1−α)cS(α)ηc−1g1(η,J(η))+αcS(α)∫0ηθc−1g1(θ,J(θ))dθ,(HM)(η)=M(0)+(1−α)cS(α)ηc−1g2(η,M(η))+αcS(α)∫0ηθc−1g2(θ,M(θ))dθ,(HE)(η)=E(0)+(1−α)cS(α)ηc−1g3(η,E(η))+αcS(α)∫0ηθc−1g3(θ,E(θ))dθ,(HL)(η)=L(0)+(1−α)cS(α)ηc−1g4(η,L(η))+αcS(α)∫0ηθc−1g4(θ,L(θ))dθ.

To derive the Banach fixed point theorem, defined the set:

ℱO={S,J,M,E,L∈Łp:||S||pp≤Op,||J||pp≤O1p, ||M||pp≤O2p, ||E||pp≤O3p, ||L||pp≤O4p,  O,O1,O2,O3,O4>0}.

Choose O≥(2p|S(0)|pQ+23pℵp𝒩11−23pQ1p𝒩2)1p,for S∈ℱO, we have (14)∫0Q|(HJ)(η)|pdη≤2p|J(0)|pQ+22p((1−α)cS(α))p∫0Qηp(c−1)|g1(η,J(η))−g1(η,0)+g1(η,0)|pdη+22p(α cS(α))p∫0Q(∫0ηθc−1|g1(θ,J(θ))−g1(θ,0)+g1(θ,0)|dθ)pdη.,∫0Q|(HM)(η)|pdη≤2p|M(0)|pQ+22p((1−α)cS(α))p∫0Qηp(c−1)|g2(η,M(η))−g2(η,0)+g2(η,0)|pdη+22p(αcS(α))p∫0Q(∫0ηθc−1|g2(θ,M(θ))−g2(θ,0)+g2(θ,0)|dθ)pdη,∫0Q|(HE)(η)|pdη≤2p|E(0)|pQ+22p((1−α)cS(α))p∫0Qηp(c−1)|g3(η,E(η))−g3(η,0)+g3(η,0)|pdη+22p(α cS(α))p∫0Q(∫0ηθc−1|g3(θ,E(θ))−g3(θ,0)+g3(θ,0)|dθ)pdη,∫0Q|(HL)(η)|pdη≤2p|L(0)|pQ+22p((1−α)cS(α))p∫0Qηp(c−1)|g4(η,L(η))−g4(η,0)+g4(η,0)|pdη+22p(α cS(α))p∫0Q(∫0ηθc−1|g4(θ,L(θ))−g4(θ,0)+g4(θ,0)|dθ)pdη.

Thus, (∫0ηθc−1|g1(θ,J(θ))−g1(θ,0)+g1(θ,0)|dθ)p, along with all other terms in Eq. (14), is Lebesgue integrable. Therefore, it is Bochner-integrable. To simplify the third term of Eq. (14), Hölder’s inequality can now be applied as follows: ∫0Q|(HJ)(η)|pdη≤2p|J(0)|pQ+23p((1−α)cS(α))p(∫0Qηp(c−1)|g1(η,J(η))−g1(η,0)|pdη+∫0Qηp(c−1)|g1(η,0)|pdη)+23p(αcS(α))p(p−1pc−1)p−1∫0Qηpc−1∫0η(|g1(θ,J(θ))−g1(θ,0)|p+|g1(θ,0)|p)dθdη,∫0Q|(HM)(η)|pdη≤2p|M(0)|pQ+23p((1−α)cS(α))p(∫0Qηp(c−1)|g2(η,M(η))−g2(η,0)|pdη+∫0Qηp(c−1)|g2(η,0)|pdη)+23p(αcS(α))p(p−1pc−1)p−1∫0Qηpc−1∫0η(|g2(θ,M(θ))−g2(θ,0)|p+|g2(θ,0)|p)dθdη, ∫0Q|(HE)(η)|pdη≤2p|E(0)|pQ+23p((1−α)cS(α))p(∫0Qηp(c−1)|g3(η,E(η))−g3(η,0)|pdη+∫0Qηp(c−1)|g3(η,0)|pdη)+23p(αcS(α))p(p−1pc−1)p−1∫0Qηpc−1∫0η(|g3(θ,E(θ))−g3(θ,0)|p+|g3(θ,0)|p)dθdη, ∫0Q|(HL)(η)|pdη≤2p|L(0)|pQ+23p((1−α)cS(α))p(∫0Qηp(c−1)|g4(η,L(η))−g4(η,0)|pdη+∫0Qηp(c−1)|g4(η,0)|pdη)+23p(αcS(α))p(p−1pc−1)p−1∫0Qηpc−1∫0η(|g4(θ,L(θ))−g4(θ,0)|p+|g4(θ,0)|p)dθdη.