Research article Topical Sections

Stress and serum cortisol levels in major depressive disorder: a cross-sectional study

  • Major depressive disorder (MDD) is one of the disorders that most causes disability and affects about 265 million people worldwide, according to the World Health Organization (WHO). Chronic stress is one of the most prevalent factors that trigger MDD. Among the most relevant biological mechanisms that mediate stress and MDD are changes in the hypothalamic-pituitary-adrenal (HPA) axis function. Hypercortisolism is one of the relevant mechanisms involved in response to stress and is present in many people with MDD and in animals subjected to stress in the laboratory. This study aimed to investigate the levels of stress and cortisol in individuals diagnosed with MDD from the Basic Health Unit (BHU) in a small city in the western region of Santa Catarina, Brazil. Depression scores were assessed using Beck's inventory. For the investigation of stress, an adaptation with twenty-four questions of the Checklist-90-R manual was performed. The analysis of the cortisol levels in the individuals' serum was by the chemiluminescence method. Depression and stress scores were significantly higher in individuals with MDD than in control subjects (p < 0.001). Cortisol levels were also significantly higher in individuals with MDD (p < 0.05). Besides, depression scores were positively correlated with stress scores in individuals with MDD (Pearson's “r” = 0.70). Conclusion: Individuals with MDD had higher stress levels and cortisol than control subjects. The positive correlation between the levels of stress and depression in MDD individuals suggests that these conditions are related to a dysregulation of the HPA axis function.

    Citation: Amanda G Bertollo, Roberta E Grolli, Marcos E Plissari, Vanessa A Gasparin, João Quevedo, Gislaine Z Réus, Margarete D Bagatini, Zuleide M Ignácio. Stress and serum cortisol levels in major depressive disorder: a cross-sectional study[J]. AIMS Neuroscience, 2020, 7(4): 459-469. doi: 10.3934/Neuroscience.2020028

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  • Major depressive disorder (MDD) is one of the disorders that most causes disability and affects about 265 million people worldwide, according to the World Health Organization (WHO). Chronic stress is one of the most prevalent factors that trigger MDD. Among the most relevant biological mechanisms that mediate stress and MDD are changes in the hypothalamic-pituitary-adrenal (HPA) axis function. Hypercortisolism is one of the relevant mechanisms involved in response to stress and is present in many people with MDD and in animals subjected to stress in the laboratory. This study aimed to investigate the levels of stress and cortisol in individuals diagnosed with MDD from the Basic Health Unit (BHU) in a small city in the western region of Santa Catarina, Brazil. Depression scores were assessed using Beck's inventory. For the investigation of stress, an adaptation with twenty-four questions of the Checklist-90-R manual was performed. The analysis of the cortisol levels in the individuals' serum was by the chemiluminescence method. Depression and stress scores were significantly higher in individuals with MDD than in control subjects (p < 0.001). Cortisol levels were also significantly higher in individuals with MDD (p < 0.05). Besides, depression scores were positively correlated with stress scores in individuals with MDD (Pearson's “r” = 0.70). Conclusion: Individuals with MDD had higher stress levels and cortisol than control subjects. The positive correlation between the levels of stress and depression in MDD individuals suggests that these conditions are related to a dysregulation of the HPA axis function.


    Recently, fractional calculus has attained assimilated bounteous flow and significant importance due to its rife utility in the areas of technology and applied analysis. Fractional derivative operators have given a new rise to mathematical models such as thermodynamics, fluid flow, mathematical biology, and virology, see [1,2,3]. Previously, several researchers have explored different concepts related to fractional derivatives, such as Riemann-Liouville, Caputo, Riesz, Antagana-Baleanu, Caputo-Fabrizio, etc. As a result, this investigation has been directed at various assemblies of arbitrary order differential equations framed by numerous analysts, (see [4,5,6,7,8,9,10]). It has been perceived that the supreme proficient technique for deliberating such an assortment of diverse operators that attracted incredible presentation in research-oriented fields, for example, quantum mechanics, chaos, thermal conductivity, and image processing, is to manage widespread configurations of fractional operators that include many other operators, see the monograph and research papers [11,12,13,14,15,16,17,18,19,20,21,22].

    In [23], the author proposed a novel idea of fractional operators, which is called GPF operator, that recaptures the Riemann-Liouville fractional operators into a solitary structure. In [24], the authors analyzed the existence of the FDEs as well as demonstrated the uniqueness of the GPF derivative by utilizing Kransnoselskii's fixed point hypothesis and also dealt with the equivalency of the mixed type Volterra integral equation.

    Fractional calculus can be applied to a wide range of engineering and applied science problems. Physical models of true marvels frequently have some vulnerabilities which can be reflected as originating from various sources. Additionally, fuzzy sets, fuzzy real-valued functions, and fuzzy differential equations seem like a suitable mechanism to display the vulnerabilities marked out by elusiveness and dubiousness in numerous scientific or computer graphics of some deterministic certifiable marvels. Here we broaden it to several research areas where the vulnerability lies in information, for example, ecological, clinical, practical, social, and physical sciences [25,26,27].

    In 1965, Zadeh [28] proposed fuzziness in set theory to examine these issues. The fuzzy structure has been used in different pure and applied mathematical analyses, such as fixed-point theory, control theory, topology, and is also helpful for fuzzy automata and so forth. In [29], authors also broadened the idea of a fuzzy set and presented fuzzy functions. This concept has been additionally evolved and the bulk of the utilization of this hypothesis has been deliberated in [30,31,32,33,34,35] and the references therein. The concept of HD has been correlated with fuzzy Riemann-Liouville differentiability by employing the Hausdorff measure of non-compactness in [36,37].

    Numerous researchers paid attention to illustrating the actual verification of certain fuzzy integral equations by employing the appropriate compactness type assumptions. Different methodologies and strategies, in light of HD or generalized HD (see [38]) have been deliberated in several credentials in the literature (see for instance [39,40,41,42,43,44,45,46,47,48,49]) and we presently sum up quickly a portion of these outcomes. In [50], the authors proved the existence of solutions to fuzzy FDEs considering Hukuhara fractional Riemann-Liouville differentiability as well as the uniqueness of the aforesaid problem. In [51,52], the authors investigated the generalized Hukuhara fractional Riemann-Liouville and Caputo differentiability of fuzzy-valued functions. Bede and Stefanini [39] investigated and discovered novel ideas for fuzzy-valued mappings that correlate with generalized differentiability. In [43], Hoa introduced the subsequent fuzzy FDE with order ϑ(0,1):

    {(cDϑσ+1Φ)(ζ)=F(ζ,Φ(ζ)),Φ(σ1)=Φ0E, (1.1)

    where a fuzzy function is F:[σ1,σ2]×EE with a nontrivial fuzzy constant Φ0E. The article addressed certain consequences on clarification of the fractional fuzzy differential equations and showed that the aforesaid equations in both cases (differential/integral) are not comparable in general. A suitable assumption was provided so that this correspondence would be effective. Hoa et al. [53] proposed the Caputo-Katugampola FDEs fuzzy set having the initial condition:

    {(cDϑ,ρσ+1Φ)(ζ)=F(ζ,Φ(ζ)),Φ(σ1)=Φ0, (1.2)

    where 0<σ1<ζσ2, cDϑ,ρσ+1 denotes the fuzzy Caputo-Katugampola fractional generalized Hukuhara derivative and a fuzzy function is F:[σ1,σ2]×EE. An approach of continual estimates depending on generalized Lipschitz conditions was employed to discuss the actual as well as the uniqueness of the solution. Owing to the aforementioned phenomena, in this article, we consider a novel fractional derivative (merely identified as Hilfer GPF-derivative). Consequently, in the framework of the proposed derivative, we establish the basic mathematical tools for the investigation of GPF-FFHD which associates with a fractional order fuzzy derivative. We investigated the actuality and uniqueness consequences of the clarification to a fuzzy fractional IVP by employing GPF generalized HD by considering an approach of continual estimates via generalized Lipschitz condition. Moreover, we derived the FVFIE using a generalized fuzzy GPF derivative is presented. Finally, we demonstrate the problems of actual and uniqueness of the clarification of this group of equations. The Hilfer-GPF differential equation is presented as follows:

    {Dϑ,q,βσ+1Φ(ζ)=F(ζ,Φ(ζ)),ζ[σ1,T],0σ1<TI1γ,βσ1Φ(σ1)=mj=1RjΦ(νj),ϑγ=ϑ+qϑq,νj(σ1,T], (1.3)

    where Dϑ,q,βσ+1(.) is the Hilfer GPF-derivative of order ϑ(0,1),I1γ,βσ1(.) is the GPF integral of order 1γ>0,RjR, and a continuous function F:[σ1,T]×RR with νj[σ1,T] fulfilling σ<ν1<...<νm<T for j=1,...,m. To the furthest extent that we might actually know, nobody has examined the existence and uniqueness of solution (1.3) regarding FVFIEs under generalized fuzzy Hilfer-GPF-HD with fuzzy initial conditions. An illustrative example of fractional-order in the complex domain is proposed and provides the exact solution in terms of the Fox-Wright function.

    The following is the paper's summary. Notations, hypotheses, auxiliary functions, and lemmas are presented in Section 2. In Section 3, we establish the main findings of our research concerning the existence and uniqueness of solutions to Problem 1.3 by means of the successive approximation approach. We developed the fuzzy GPF Volterra-Fredholm integrodifferential equation in Section 4. Section 5 consists of concluding remarks.

    Throughout this investigation, E represents the space of all fuzzy numbers on R. Assume the space of all Lebsegue measureable functions with complex values F on a finite interval [σ1,σ2] is identified by χrc(σ1,σ2) such that

    Fχrc<,cR,1r.

    Then, the norm

    Fχrc=(σ2σ1|ζcF(ζ)|rdζζ)1/r.

    Definition 2.1. ([53]) A fuzzy number is a fuzzy set Φ:R[0,1] which fulfills the subsequent assumptions:

    (1) Φ is normal, i.e., there exists ζ0R such that Φ(ζ0)=1;

    (2) Φ is fuzzy convex in R, i.e, for δ[0,1],

    Φ(δζ1+(1δ)ζ2)min{Φ(ζ1),Φ(ζ2)}foranyζ1,ζ2R;

    (3) Φ is upper semicontinuous on R;

    (4) [z]0=cl{z1R|Φ(z1)>0} is compact.

    C([σ1,σ2],E) indicates the set of all continuous functions and set of all absolutely continuous fuzzy functions signifys by AC([σ1,σ2],E) on the interval [σ1,σ2] having values in E.

    Let γ(0,1), we represent the space of continuous mappings by

    Cγ[σ1,σ2]={F:(σ1,σ2]E:eβ1β(ζσ1)(ζσ1)1γF(ζ)C[σ1,σ2]}.

    Assume that a fuzzy set Φ:R[0,1] and all fuzzy mappings Φ:[σ1,σ2]E defined on L([σ1,σ2],E) such that the mappings ζˉD0[Φ(ζ),ˆ0] lies in L1[σ1,σ2].

    There is a fuzzy number Φ on R, we write [Φ]ˇq={z1R|Φ(z1)ˇq} the ˇq-level of Φ, having ˇq(0,1].

    From assertions (1) to (4); it is observed that the ˇq-level set of ΦE, [Φ]ˇq is a nonempty compact interval for any ˇq(0,1]. The ˇq-level of a fuzzy number Φ is denoted by [Φ_(ˇq),ˉΦ(ˇq)].

    For any δR and Φ1,Φ2E, then the sum Φ1+Φ2 and the product δΦ1 are demarcated as: [Φ1+Φ2]ˇq=[Φ1]ˇq+[Φ2]ˇq and [δ.Φ1]ˇq=δ[Φ1]ˇq, for all ˇq[0,1], where [Φ1]ˇq+[Φ2]ˇq is the usual sum of two intervals of R and δ[Φ1]ˇq is the scalar multiplication between δ and the real interval.

    For any ΦE, the diameter of the ˇq-level set of Φ is stated as diam[μ]ˇq=ˉμ(ˇq)μ_(ˇq).

    Now we demonstrate the notion of Hukuhara difference of two fuzzy numbers which is mainly due to [54].

    Definition 2.2. ([54]) Suppose Φ1,Φ2E. If there exists Φ3E such that Φ1=Φ2+Φ3, then Φ3 is known to be the Hukuhara difference of Φ1 and Φ2 and it is indicated by Φ1Φ2. Observe that Φ1Φ2Φ1+()Φ2.

    Definition 2.3. ([54]) We say that ¯D0[Φ1,Φ2] is the distance between two fuzzy numbers if

    ¯D0[Φ1,Φ2]=supˇq[0,1]H([Φ1]ˇq,[Φ2]ˇq),Φ1,Φ2E,

    where the Hausdroff distance between [Φ1]ˇq and [Φ2]ˇq is defined as

    H([Φ1]ˇq,[Φ2]ˇq)=max{|Φ_(ˇq)ˉΦ(ˇq)|,|ˉΦ(ˇq)Φ_(ˇq)|}.

    Fuzzy sets in E is also refereed as triangular fuzzy numbers that are identified by an ordered triple Φ=(σ1,σ2,σ3)R3 with σ1σ2σ3 such that [Φ]ˇq=[Φ_(ˇq),ˉΦ(ˇq)] are the endpoints of ˇq-level sets for all ˇq[0,1], where Φ_(ˇq)=σ1+(σ2σ1)ˇq and ˉΦ(ˇq)=σ3(σ3σ2)ˇq.

    Generally, the parametric form of a fuzzy number Φ is a pair [Φ]ˇq=[Φ_(ˇq),ˉΦ(ˇq)] of functions Φ_(ˇq),ˉΦ(ˇq),ˇq[0,1], which hold the following assumptions:

    (1) μ_(ˇq) is a monotonically increasing left-continuous function;

    (2) ˉμ(ˇq) is a monotonically decreasing left-continuous function;

    (3) μ_(ˇq)ˉμ(ˇq),ˇq[0,1].

    Now we mention the generalized Hukuhara difference of two fuzzy numbers which is proposed by [38].

    Definition 2.4. ([38]) The generalized Hukuhara difference of two fuzzy numbers Φ1,Φ2E (gH-difference in short) is stated as follows

    Φ1gHΦ2=Φ3Φ1=Φ2+Φ3orΦ2=Φ1+(1)Φ3.

    A function Φ:[σ1,σ2]E is said to be d-increasing (d-decreasing) on [σ1,σ2] if for every ˇq[0,1]. The function ζdiam[Φ(ζ)]ˇq is nondecreasing (nonincreasing) on [σ1,σ2]. If Φ is d-increasing or d-decreasing on [σ1,σ2], then we say that Φ is d-monotone on [σ1,σ2].

    Definition 2.5. ([39])The generalized Hukuhara derivative of a fuzzy-valued function F:(σ1,σ2)E at ζ0 is defined as

    FgH(ζ0)=limh0F(ζ0+h)gHF(ζ0)h,

    if (F)gH(ζ0)E, we say that F is generalized Hukuhara differentiable (gH-differentiable) at ζ0.

    Moreover, we say that F is [(i)gH]-differentiable at ζ0 if

    [FgH(ζ0)]ˇq=[[limh0F_(ζ0+h)gHF_(ζ0)h]ˇq,[limh0ˉF(ζ0+h)gHˉF(ζ0)h]ˇq]=[(F_)(ˇq,ζ0),(ˉF)(ˇq,ζ0)], (2.1)

    and that F is [(ii)gH]-differentiable at ζ0 if

    [FgH(ζ0)]ˇq=[(ˉF)(ˇq,ζ0),(F_)(ˇq,ζ0)]. (2.2)

    Definition 2.6. ([49]) We state that a point ζ0(σ1,σ2), is a switching point for the differentiability of F, if in any neighborhood U of ζ0 there exist points ζ1<ζ0<ζ2 such that

    Type Ⅰ. at ζ1 (2.1) holds while (2.2) does not hold and at ζ2 (2.2) holds and (2.1) does not hold, or

    Type Ⅱ. at ζ1 (2.2) holds while (2.1) does not hold and at ζ2 (2.1) holds and (2.2) does not hold.

    Definition 2.7. ([23]) For β(0,1] and let the left-sided GPF-integral operator of order ϑ of F is defined as follows

    Iϑ,βσ+1F(ζ)=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1F(ν)dν,ζ>σ1, (2.3)

    where β(0,1], ϑC, Re(ϑ)>0 and Γ(.) is the Gamma function.

    Definition 2.8. ([23]) For β(0,1] and let the left-sided GPF-derivative operator of order ϑ of F is defined as follows

    Dϑ,βσ+1F(ζ)=Dn,ββnϑΓ(nϑ)ζσ1eβ1β(ζν)(ζν)nϑ1F(ν)dν, (2.4)

    where β(0,1], ϑC,Re(ϑ)>0,n=[ϑ]+1 and Dn,β represents the nth-derivative with respect to proportionality index β.

    Definition 2.9. ([23]) For β(0,1] and let the left-sided GPF-derivative in the sense of Caputo of order ϑ of F is defined as follows

    cDϑ,βσ+1F(ζ)=1βnϑΓ(nϑ)ζσ1eβ1β(ζν)(ζν)nϑ1(Dn,βF)(ν)dν, (2.5)

    where β(0,1], ϑC,Re(ϑ)>0 and n=[ϑ]+1.

    Let ΦL([σ1,σ2],E), then the GPF integral of order ϑ of the fuzzy function Φ is stated as:

    Φβϑ(ζ)=(Iϑ,βσ+1Φ)(ζ)=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1Φ(ν)dν,ζ>σ1. (2.6)

    Since [Φ(ζ)]ˇq=[Φ_(ˇq,ζ),ˉΦ(ˇq,ζ)] and 0<ϑ<1, we can write the fuzzy GPF-integral of the fuzzy mapping Φ depend on lower and upper mappingss, that is,

    [(Iϑ,βσ+1Φ)(ζ)]ˇq=[(Iϑ,βσ+1Φ_)(ˇq,ζ),(Iϑ,βσ+1ˉΦ)(ˇq,ζ)],ζσ1, (2.7)

    where

    (Iϑ,βσ+1Φ_)(ˇq,ζ)=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1Φ_(ˇq,ν)dν, (2.8)

    and

    (Iϑ,βσ+1ˉΦ)(ˇq,ζ)=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1ˉΦ(ˇq,ν)dν. (2.9)

    Definition 2.10. For nN, order ϑ and type q hold n1<ϑn with 0q1. The left-sided fuzzy Hilfer-proportional gH-fractional derivative, with respect to ζ having β(0,1] of a function ζCβ1γ[σ1,σ2], is stated as

    (Dϑ,q,βσ+1Φ)(ζ)=(Iq(1ϑ),βσ+1Dβ(I(1q)(1ϑ),βσ+1Φ))(ζ),

    where DβΦ(ν)=(1β)Φ(ν)+βΦ(ν) and if the gH-derivative Φ(1ϑ),β(ζ) exists for ζ[σ1,σ2], where

    Φβ(1ϑ)(ζ):=(I(1ϑ),βσ+1Φ)(ζ)=1β1ϑΓ(1ϑ)ζσ1eβ1β(ζν)(ζν)ϑΦ(ν)dν,ζσ1.

    Definition 2.11. Let ΦL([σ1,σ2],E) and the fractional generalized Hukuhara GPF-derivative of fuzzy-valued function Φ is stated as:

    (gHDϑ,βσ+1Φ)(ζ)=I1ϑ,βσ+1(ΦgH)(ζ)=1β1ϑΓ(1ϑ)ϑσ1eβ1β(ζν)(ζν)ϑΦgH(ν)dν,ν(σ1,ζ). (2.10)

    Furthermore, we say that Φ is GPF[(i)gH]-differentiable at ζ0 if

    [(gHDϑ,βσ+1)]ˇq=[[1β1ϑΓ(1ϑ)ϑσ1eβ1β(ζν)(ζν)ϑΦ_gH(ν)dν]ˇq,[1β1ϑΓ(1ϑ)ϑσ1eβ1β(ζν)(ζν)ϑˉΦgH(ν)dν]ˇq]=[(gHD_ϑ,βσ+1)(ˇq,ζ),(gHˉDϑ,βσ+1)(ˇq,ζ)] (2.11)

    and that Φ is GPF[(i)gH]-differentiable at ζ0 if

    [(gHDϑ,βσ+1)]ˇq=[(gHˉDϑ,βσ+1)(ˇq,ζ),(gHD_ϑ,βσ+1)(ˇq,ζ)]. (2.12)

    Definition 2.12. We say that a point ζ0(σ1,σ2), is a switching point for the differentiability of F, if in any neighborhood U of ζ0 there exist points ζ1<ζ0<ζ2 such that

    Type Ⅰ. at ζ1 (2.11) holds while (2.12) does not hold and at ζ2 (2.12) holds and (2.11) does not hold, or

    Type Ⅱ. at ζ1 (2.12) holds while (2.11) does not hold and at ζ2 (2.11) holds and (2.12) does not hold.

    Proposition 1. ([23]) Let ϑ,ϱC such that Re(ϑ)>0 and Re(ϱ)>0. Then for any β(0,1], we have

    (Iϑ,βσ+1eβ1β(sσ1)ϱ1)(ζ)=Γ(ϱ)βϑΓ(ϱ+ϑ)eβ1β(ζσ1)(ζσ1)ϱ+ϑ1,(Dϑ,βσ+1eβ1β(sσ1)ϱ1)(ζ)=Γ(ϱ)βϑΓ(ϱϑ)eβ1β(ζσ1)(ζσ1)ϱϑ1,(Iϑ,βσ+1eβ1β(σ2s)ϱ1)(ζ)=Γ(ϱ)βϑΓ(ϱ+ϑ)eβ1β(σ2s)(σ2ζ)ϱ+ϑ1,(Dϑ,βσ+1eβ1β(σ2s)ϱ1)(ζ)=Γ(ϱ)βϑΓ(ϱϑ)eβ1β(σ2s)(σ2s)ϱϑ1.

    Lemma 2.13. ([24])For β(0,1], ϑ>0, 0γ<1. If ΦCγ[σ1,σ2] and I1ϑσ+1ΦC1γ[σ1,σ2], then

    (Iϑ,βσ+1Dϑ,βσ+1Φ)(ζ)=Φ(ζ)eβ1β(ζσ1)(ζσ1)ϑ1βϑ1Γ(ϑ)(I1ϑ,βσ+1Φ)(σ1).

    Lemma 2.14. ([24]) Let ΦL1(σ1,σ2). If Dq(1ϑ),βσ+1Φ exists on L1(σ1,σ2), then

    Dϑ,q,βσ+1Iϑ,βσ+1Φ=Iq(1ϑ),βσ+1Dq(1ϑ),βσ+1Φ.

    Lemma 2.15. Suppose there is a d-monotone fuzzy mapping ΦAC([σ1,σ2],E), where [Φ(ζ)]ˇq=[Φ_(ˇq,ζ),ˉΦ(ˇq,ζ)] for 0ˇq1,σ1ζσ2, then for 0<ϑ<1 and β(0,1], we have

    (i)[(Dϑ,q,βσ+1Φ)(ζ)]ˇq=[Dϑ,q,βσ+1Φ_(ˇq,ζ),Dϑ,q,βσ+1ˉΦ(ˇq,ζ)] for ζ[σ1,σ2], if Φ is d-increasing;

    (ii)[(Dϑ,q,βσ+1Φ)(ζ)]ˇq=[Dϑ,q,βσ+1ˉΦ(ˇq,ζ),Dϑ,q,βσ+1Φ_(ˇq,ζ)] for ζ[σ1,σ2], if Φ is d-decreasing.

    Proof. It is to be noted that if Φ is d-increasing, then [Φ(ζ)]ˇq=[ddζΦ_(ˇq,ζ),ddζˉΦ(ˇq,ζ)]. Taking into account Definition 2.10, we have

    [(Dϑ,q,βσ+1Φ)(ζ)]ˇq=[Iq(1ϑ),βσ+1Dβ(I(1q)(1ϑ),βσ+1Φ_)(ˇq,ζ),Iq(1ϑ),βσ+1Dβ(I(1q)(1ϑ),βσ+1ˉΦ)(ˇq,ζ)]=[Dϑ,q,βσ+1Φ_(ˇq,ζ),Dϑ,q,βσ+1ˉΦ(ˇq,ζ)].

    If Φ is d-decreasing, then [Φ(ζ)]ˇq=[ddζˉΦ(ˇq,ζ),ddζΦ_(ˇq,ζ)], we have

    [(Dϑ,q,βσ+1Φ)(ζ)]ˇq=[Iq(1ϑ),βσ+1Dβ(I(1q)(1ϑ),βσ+1ˉΦ)(ˇq,ζ),Iq(1ϑ),βσ+1Dβ(I(1q)(1ϑ),βσ+1Φ_)(ˇq,ζ)]=[Dϑ,q,βσ+1ˉΦ(ˇq,ζ),Dϑ,q,βσ+1Φ_(ˇq,ζ)].

    This completes the proof.

    Lemma 2.16. For β(0,1],ϑ(0,1). If ΦAC([σ1,σ2],E) is a d-monotone fuzzy function. We take

    z1(ζ):=(Iϑ,βσ+1Φ)(ζ)=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1Φ(ν)dν,

    and

    z(1ϑ),β1:=(I(1ϑ),βσ+1Φ)(ζ)=1β1ϑΓ(1ϑ)ϑσ1eβ1β(ζν)(ζν)ϑΦgH(ν)dν,

    is d-increasing on (σ1,σ2], then

    (Iϑ,βσ+1Dϑ,q,βσ+1Φ)(ζ)=Φ(ζ)mj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1,

    and

    (Dϑ,q,βσ+1Iϑ,βσ+1Φ)(ζ)=Φ(ζ).

    Proof. If z1(ζ) is d-increasing on [σ1,σ2] or z1(ζ) is d-decreasing on [σ1,σ2] and z(1ϑ),β1(ζ) is d-increasing on (σ1,σ2].

    Utilizing the Definitions 2.6, 2.10 and Lemma 2.13 with the initial condition (I1γ,βσ+1Φ)(σ1)=0, we have

    (Iϑ,βσ+1Dϑ,q,βσ+1Φ)(ζ)=(Iϑ,βσ+1Iq(1ϑ),βσ+1DβI(1q)(1ϑ),βσ+1Φ)(ζ)=(Iγ,βσ+1DβI1γ,βσ+1Φ)(ζ)=(Iγ,βσ+1Dγ,βσ+1Φ)(ζ)=Φ(ζ)I1γ,βσ+1Φβγ1Γ(γ)eβ1β(ζσ1)(ζσ1)γ1. (2.13)

    Now considering Proposition 1, Lemma 2.13 and Lemma 2.14, we obtain

    (Dϑ,q,βσ+1Iϑ,βσ+1Φ)(ζ)=(Iq(1ϑ),βσ+1Dq(1ϑ),βσ+1Φ)(ζ)=Φ(ζ)(I1q(1ϑ),βσ+1Φ)(σ1)eβ1β(ζσ1)βq(1ϑ)Γ(q(1ϑ))(ζσ1)q(1ϑ)1=Φ(ζ).

    On contrast, since ΦAC([σ1,σ2],E), there exists a constant K such that K=supζ[σ1,σ2]¯D0[Φ(ζ),ˆ0].

    Then

    ¯D0[Iϑ,βσ+1Φ(ζ),ˆ0]K1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1dνK1βϑΓ(ϑ)ζσ1|eβ1β(ζν)|(ζν)ϑ1dν=KβϑΓ(ϑ+1)(ζσ1)ϑ,

    where we have used the fact |eβ1βζ|<1 and Iϑ,βσ+1Φ(ζ)=0 and ζ=σ1.

    This completes the proof.

    Lemma 2.17. Let there be a continuous mapping Φ:[σ1,σ2]R+ on [σ1,σ2] and hold Dϑ,q,βσ+1Φ(ζ)F(ξ,Φ(ξ)),ξσ1, where FC([σ1,σ1]×R+,R+). Assume that m(ζ)=m(ζ,σ1,ξ0) is the maximal solution of the IVP

    Dϑ,q,βσ+1ξ(ζ)=F(ζ,ξ),(I1γ,βσ+1ξ)(σ1)=ξ00, (2.14)

    on [σ1,σ2]. Then, if Φ(σ1)ξ0, we have Φ(ζ)m(ζ),ζ[σ1,σ2].

    Proof. The proof is simple and can be derived as parallel to Theorem 2.2 in [53].

    Lemma 2.18. Assume the IVP described as:

    Dϑ,q,βσ+1Φ(ζ)=F(ζ,Φ(ζ)),(I1γ,βσ+1Φ)(σ1)=Φ0=0,ζ[σ1,σ2]. (2.15)

    Let α>0 be a given constant and B(Φ0,α)={ΦR:|ΦΦ0|α}. Assume that the real-valued functions F:[σ1,σ2]×[0,α]R+ satisfies the following assumptions:

    (i) FC([σ1,σ2]×[0,α],R+),F(ζ,0)0,0F(ζ,Φ)MF for all (ζ,Φ)[σ1,σ2]×[0,α];

    (ii) F(ζ,Φ) is nondecreasing in Φ for every ζ[σ1,σ2]. Then the problem (2.15) has at least one solution defined on [σ1,σ2] and Φ(ζ)B(Φ0,α).

    Proof. The proof is simple and can be derived as parallel to Theorem 2.3 in [53].

    In this investigation, we find the existence and uniqueness of solution to problem 1.3 by utilizing the successive approximation technique by considering the generalized Lipschitz condition of the right-hand side.

    Lemma 3.1. For γ=ϑ+q(1ϑ),ϑ(0,1),q[0,1] with β(0,1], and let there is a fuzzy function F:(σ1,σ2]×EE such that ζF(ζ,Φ) belongs to Cβγ([σ1,σ2],E) for any ΦE. Then a d-monotone fuzzy function ΦC([σ1,σ2],E) is a solution of IVP (1.3) if and only if Φ satisfies the integral equation

    Φ(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1F(ν,Φ(ν))dν,ζ[σ1,σ2],j=1,2,...,m. (3.1)

    and the fuzzy function ζI1γσ+1F(ζ,Φ) is d-increasing on (σ1,σ2].

    Proof. Let ΦC([σ1,σ2],E) be a d-monotone solution of (1.3), and considering z1(ζ):=Φ(ζ)gH(I1γ,βσ+1Φ)(σ1),ζ(σ1,σ2]. Since Φ is d-monotone on [σ1,σ2], it follows that ζz1(ζ) is d-increasing on [σ1,σ2] (see [43]).

    From (1.3) and Lemma 2.16, we have

    (Iϑ,βσ+1Dϑ,q,βσ+1Φ)(ζ)=Φ(ζ)mj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1,ζ[σ1,σ2]. (3.2)

    Since F(ζ,Φ)Cγ([σ1,σ2],E) for any ΦE, and from (1.3), observes that

    (Iϑ,βσ+1Dϑ,q,βσ+1Φ)(ζ)=Iϑ,βσ+1F(ζ,Φ(ζ))=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1F(ν,Φ(ν))dν,ζ[σ1,σ2]. (3.3)

    Additionally, since z1(ζ) is d-increasing on (σ1,σ2]. Also, we observe that ζFϑ,β(ζ,Φ) is also d-increasing on (σ1,σ2].

    Reluctantly, merging (3.2) and (3.3), we get the immediate consequence.

    Further, suppose ΦC([σ1,σ2],E) be a d-monotone fuzzy function fulfills (3.1) and such that ζFϑ,β(ζ,Φ) is d-increasing on (σ1,σ2]. By the continuity of the fuzzy mapping F, the fuzzy mapping ζFϑ,β(ζ,Φ) is continuous on (σ1,σ2] with Fϑ,β(σ1,Φ(σ1))=limζσ+1Fϑ,β(ζ,Φ)=0. Then

    Φ(ζ)=mj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1+(Iϑ,βσ+1F(ζ,ζ))(ζ),I1γ,βσ+1Φ(ζ)=mj=1RjΦ(ζj)+(I1q(1ϑ)σ+1F(ζ,Φ(ζ)))(ζ),

    and

    I1γ,βσ+1Φ(0)=mj=1RjΦ(ζj).

    Moreover, since ζFϑ,β(ζ,Φ) is d-increasing on (σ1,σ2]. Applying, the operator Dϑ,q,βσ+1 on (3.1), yields

    Dϑ,q,βσ+1(Φ(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1)=Dϑ,q,βσ+1(1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1F(ν,Φ(ν))dν)=F(ζ,Φ(ζ)).

    This completes the proof.

    In our next result, we use the following assumption. For a given constant >0, and let B(Φ0,)={ΦE:¯D0[Φ,Φ0]}.

    Theorem 3.2. Let FC([σ1,σ2]×B(Φ0,),E) and suppose that the subsequent assumptions hold:

    (i) there exists a positive constant MF such that ¯D0[F(ζ,z1),ˆ0]MF, for all (ζ,z1)[σ1,σ2]×B(Φ0,);

    (ii) for every ζ[σ1,σ2] and every z1,ωB(Φ0,),

    ¯D0[F(ζ,z1),F(ζ,ω)]g(ζ,¯D0[z1,ω]), (3.4)

    where g(ζ,.)C([σ1,σ2]×[0,β],R+) satisfies the assumption in Lemma 2.18 given that problem (2.15) has only the solution ϕ(ζ)0 on [σ1,σ2]. Then the subsequent successive approximations given by Φ0(ζ)=Φ0 and for n=1,2,...,

    Φn(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1F(ν,Φn1(ν))dν,

    converges consistently to a fixed point of problem (1.3) on certain interval [σ1,T] for some T(σ1,σ2] given that the mapping ζIϑ,βσ+1F(ζ,Φn(ζ)) is d-increasing on [σ1,T].

    Proof. Take σ1<ζ such that ζ[βϑ.Γ(1+ϑ)M+σ1]1ϑ, where M=max{Mg,MF} and put T:=min{ζ,σ2}. Let S be a set of continuous fuzzy functions Φ such that ω(σ1)=Φ0 and ω(ζ)B(Φ0,) for all ζ[σ1,T]. Further, we suppose the sequence of continuous fuzzy function {Φn}n=0 given by Φ0(ζ)=Φ0,ζ[σ1,T] and for n=1,2,..,

    Φn(ζ)gHmj=1RjΦn1(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1F(ν,Φn1(ν))dν. (3.5)

    Firstly, we show that Φn(ζ)C([σ1,T],B(Φ0,)). For n1 and for any ζ1,ζ2[σ1,T] with ζ1<ζ2, we have

    ¯D0(Φn(ζ1)gHmj=1RjΦn1(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1,Φn(ζ2)gHmj=1RjΦn1(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1)1βϑΓ(ϑ)ζ1σ1[eβ1β(ζ1ν)(ζ1ν)ϑ1eβ1β(ζ2ν)(ζ2ν)ϑ1]¯D0[F(ν,Φn1(ν)),ˆ0]dν+1βϑΓ(ϑ)ζ2ζ1eβ1β(ζ2ν)(ζ2ν)ϑ1¯D0[F(ν,Φn1(ν)),ˆ0]dν.

    Using the fact that |eβ1βζ|<1, then, on the right-hand side from the last inequality, the subsequent integral becomes 1βϑΓ(1+ϑ)(ζ2ζ1)ϑ. Therefore, with the similar assumption as we did above, the first integral reduces to 1βϑΓ(1+ϑ)[(ζ1σ1)ϑ(ζ2σ1)ϑ+(ζ2ζ1)ϑ]. Thus, we conclude

    ¯D0[Φn((ζ1),Φn(ζ2))]MFβϑΓ(1+ϑ)[(ζ1σ1)ϑ(ζ2σ1)ϑ+2(ζ2ζ1)ϑ]2MFβϑΓ(1+ϑ)(ζ2ζ1)ϑ.

    In the limiting case as ζ1ζ2, then the last expression of the above inequality tends to 0, which shows Φn is a continuous function on [σ1,T] for all n1.

    Moreover, it follows that ΦnB(Φ0,) for all n0,ζ[σ1,T] if and only if Φn(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1B(0,) for all ζ[σ1,T] and for all n0.

    Also, if we assume that Φn1(ζ)S for all ζ[σ1,T],n2, then

    ¯D0[Φn(ζ)gHmj=1RjΦn1(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1,ˆ0]1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1¯D0[F(ν,Φn1(ν)),ˆ0]dν=MF(ζσ1)ϑβϑΓ(1+ϑ).

    It follows that Φn(ζ)S,[σ1,T].

    Henceforth, by mathematical induction, we have Φn(ζ)S,ζ[σ1,T] and n1.

    Further, we show that the sequence Φn(ζ) converges uniformly to a continuous function ΦC([σ1,T],B(Φ0,)). By assertion (ii) and mathematical induction, we have for ζ[σ1,T]

    ¯D0[Φn+1(ζ)gHmj=1RjΦn(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1,Φn(ζ)gHmj=1RjΦn1(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1]ϕn(ζ),n=0,1,2,..., (3.6)

    where ϕn(ζ) is defined as follows:

    ϕn(ζ)=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1g(ν,ϕn1(ν))dν, (3.7)

    where we have used the fact that |eβ1βζ|<1 and ϕ0(ζ)=M(ζσ1)ϑβϑΓ(ϑ+1). Thus, we have, for ζ[σ1,T] and for n=0,1,2,...,

    ¯D0[Dϑ,qσ+1Φn+1(ζ),Dϑ,qσ+1Φn(ζ)]¯D0[F(ζ,Φn(ζ)),F(ζ,Φn1(ζ))]g(ζ,¯D0[Φn(ζ),Φn1(ζ)])g(ζ,ϕn1(ζ)).

    Let nm and ζ[σ1,T], then one obtains

    Dϑ,qσ+1¯D0[Φn(ζ),Φm(ζ)]¯D0[Dϑ,qσ+1Φn(ζ),Dϑ,qσ+1Φm(ζ)]¯D0[Dϑ,qσ+1Φn(ζ),Dϑ,qσ+1Φn+1(ζ)]+¯D0[Dϑ,qσ+1Φn+1(ζ),Dϑ,qσ+1Φm+1(ζ)]+¯D0[Dϑ,qσ+1Φm+1(ζ),Dϑ,qσ+1Φm(ζ)]2g(ζ,ϕn1(ζ))+g(ζ,¯D0[Φn(ζ),Φm(ζ)]).

    From (ii), we observe that the solution ϕ(ζ)=0 is a unique solution of problem (2.15) and g(.,ϕn1):[σ1,T][0,Mg] uniformly converges to 0, for every ϵ>0, there exists a natural number n0 such that

    Dϑ,qσ+1¯D0[Φn(ζ),Φm(ζ)]g(ζ,¯D0[Φn(ζ),Φm(ζ)])+ϵ,forn0nm.

    Using the fact that ¯D0[Φn(σ1),Φm(σ1)]=0<ϵ and by using Lemma 2.17, we have for ζ[σ1,T]

    ¯D0[Φn(ζ),Φm(ζ)]δϵ(ζ),n0nm, (3.8)

    where δϵ(ζ) is the maximal solution to the following IVP:

    (Dϑ,qσ+1δϵ)(ζ)=g(ζ,δϵ(ζ))+ϵ,(I1γσ+1δϵ)=ϵ.

    Taking into account Lemma 2.17, we deduce that [ϕϵ(.,ω)] converges uniformly to the maximal solution ϕ(ζ)0 of (2.15) on [σ1,T] as ϵ0.

    Therefore, in view of (3.8), we can obtain n0N is large enough such that, for n0<n,m,

    supζ[σ1,T]¯D0[Φn(ζ)gHmj=1RjΦn1(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1,Φm(ζ)gHmj=1RjΦn1(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1]ϵ. (3.9)

    Since (E,¯D0) is a complete metric space and (3.9) holds, thus {Φn(ζ)} converges uniformly to ΦC([σ1,σ2],B(Φ0,)). Hence

    Φ(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=limn(Φn(ζ)gHmj=1RjΦn1(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1)=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1F(ν,Φn1(ν))dν. (3.10)

    Because of Lemma 3.1, the function Φ(ζ) is the solution to (1.3) on [σ1,T].

    In order to find the unique solution, assume that Ψ:[σ1,T]E is another solution of problem (1.3) on [σ1,T]. We denote κ(ζ)=¯D0[Φ(ζ),Ψ(ζ)]. Then κ(σ1)=0 and for every ζ[σ1,T], we have

    Dϑ,q,βσ+1κ(ζ)¯D0[F(ζ,Φ(ζ)),F(ζ,Ψ(ζ))]g(ζ,κ(ζ)). (3.11)

    Further, using the comaprison Lemma 2.17, we get κ(ζ)m(ζ), where m is a maximal solution of the IVP Dϑ,q,βσ+1m(ζ)g(ζ,m(ζ)),(I1γσ+1m)(σ1)=0. By asseration (ii), we have m(ζ)=0 and hence Φ(ζ)=Ψ(ζ),[σ1,T].

    This completes the proof.

    Corollary 1. For β(0,1] and let C([σ1,σ2],E). Assume that there exist positive constants L,MF such that, for every z1,ωE,

    ¯D0[F(ζ,z1),F(ζ,ω)]L¯D0[z1,ω],¯D0[F(ζ,z1),ˆ0]MF.

    Then the subsequent successive approximations given by Φ0(ζ)=Φ0 and for n=1,2,..

    Φn(ζ)gHΦ0=1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1F(ν,Φn1(ν))dν,

    converges consistently to a fixed point of problem (1.3) on [σ1,T] for certain T(σ1,σ2] given that the mapping ζIϑ,βσ+1F(ζ,Φn(ζ)) is d-increasing on [σ1,T].

    Example 3.3. For β(0,1],γ=ϑ+q(1ϑ),ϑ(0,1),q[0,1] and δR. Assume that the linear fuzzy GPF-FDE under Hilfer-GPF-derivative and moreover, the subsequent assumptions hold:

    {(Dϑ,qσ+1Φ)(ζ)=δΦ(ζ)+η(ζ),ζ(σ1,σ2],(I1γ,βσ+1Φ)(σ1)=Φ0=mj=1RjΦ(ζj),γ=ϑ+q(1ϑ). (3.12)

    Applying Lemma 3.1, we have

    Φ(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=δ1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1Φ(ν)dν+1βϑΓ(ϑ)ζσ1eβ1β(ζν)(ζν)ϑ1η(ν)dν,ζ[σ1,σ2]=δ(Iϑ,βσ+1Φ)(ζ)+(Iϑ,βσ+1η)(ζ),

    where ηC((σ1,σ2],E) and furthermore, assuming the diameter on the right part of the aforementioned equation is increasing. Observing F(ζ,Φ):=δΦ+η fulfill the suppositions of Corollary 1.

    In order to find the analytical view of (3.12), we utilized the technique of successive approximation. Putting Φ0(ζ)=Φ0 and

    Φn(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=δ(Iϑ,βσ+1Φn1)(ζ)+(Iϑ,βσ+1η)(ζ),n=1,2,...

    Letting n=1,δ>0, assuming there is a d-increasing mapping Φ, then we have

    Φ1(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=δmj=1RjΦ(ζj)(ζσ1)ϑβϑΓ(ϑ+1)+(Iϑ,βσ+1η)(ζ).

    In contrast, if we consider δ<0 and Φ is d-decreasing, then we have

    (1)(mj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1gHΦ1(ζ))=δmj=1RjΦ(ζj)(ζσ1)ϑβϑΓ(ϑ+1)+(Iϑ,βσ+1η)(ζ).

    For n=2, we have

    Φ2(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=mj=1RjΦ(ζj)[δ(ζσ1)ϑβϑΓ(ϑ+1)+δ2(ζσ1)2ϑβ2ϑΓ(2ϑ+1)]+(Iϑ,βσ+1η)(ζ)+(I2ϑ,βσ+1η)(ζ),

    if and there is -increasing mapping , we have

    and there is -increasing mapping So, continuing inductively and in the limiting case, when we attain the solution

    for every and is -increasing, or and is -decreasing, accordingly. Therefore, by means of Mittag-Leffler function the solution of problem (3.12) is expressed by

    for every of and is -increasing. Alternately, if and is -decreasing, then we get the solution of problem (3.12)

    Consider IVP

    (4.1)

    where and is a real number and the operation denote the derivative of order is continuous in which fulfills certain supposition that will be determined later, and

    (4.2)

    with such that

    Now, we investigate the existence and uniqueness of the solution of problem (4.1). To establish the main consequences, we require the following necessary results.

    Theorem 4.1. Let be a fuzzy-valued function on Then

    is -differentiable at iff is -differentiable at

    is -differentiable at iff is -differentiable at

    Proof. In view of Definition 2.18 and Definition 2.11, the proof is straightforward.

    Lemma 4.2. ([44]) Let there be a fuzzy valued mapping such that then

    (4.3)

    Lemma 4.3. The IVP (4.1) is analogous to subsequent equation

    (4.4)

    if be -differentiable,

    (4.5)

    if be -differentiable, and

    (4.6)

    if there exists a point such that is -differentiable on and -differentiable on and

    Proof. By means of the integral operator (2.6) on both sides of (4.1), yields

    (4.7)

    Utilizing Lemma 4.2 and Definition 2.6, we gat

    (4.8)

    In view of Defnition 2.17 and Theorem 4.1, if be -differentiable,

    (4.9)

    and if be -differentiable

    (4.10)

    In addition, when we have a switchpoint of type the -differentiability changes from type to type at Then by (4.9) and (4.10) and Definition 2.12, The proof is easy to comprehend.

    Also, we proceed with the following assumptions:

    is continuous and there exist positive real functions such that

    There exist a number such that

    and

    Theorem 4.4. Let be a bounded continuous functions and holds Then the IVP (4.1) has a unique solution which is -differentiable on given that where is given in

    Proof. Assuming is -differentiability and be fixed. Propose a mapping by

    (4.11)

    Next we prove that is contraction. For by considering of and by distance properties (2.3), one has

    (4.12)

    Now, we find that

    (4.13)

    Analogously,

    (4.14)

    Then we have

    (4.15)

    Consequently, is a contraction mapping on having a fixed point Henceforth, the IVP (4.1) has unique solution.

    Theorem 4.5. For and let be a bounded continuous functions and satisfies Let the sequence is given by

    (4.16)

    is described for any Then the sequence converges to fixed point of problem (4.1) which is -differentiable on given that where is defined in

    Proof. We now prove that the sequence , given in (4.16), is a Cauchy sequence in To do just that, we'll require

    (4.17)

    where

    Since is Lipschitz continuous, In view of Definition (2.3), we show that

    (4.18)

    Since promises that the sequence is a Cauchy sequence in Consequently, there exist such that converges to Thus, we need to illustrate that is a solution of the problem (4.1).

    (4.19)

    In the limiting case, when Thus we have

    (4.20)

    By Lemma 4.3, we prove that is a solution of the problem (4.1). In order to prove the uniqness of let be another solution of problem (4.1) on Utilizing Lemma 4.3, gets

    Analogously, by employing the distance properties and Lipschitiz continuity of consequently, we deduce that since we have for all Hence, the proof is completed.

    Example 4.6. Suppose the Cauchy problem by means of differential operator (2.4)

    (4.21)

    where is analytic in and is analytic in the unit disk. Therefore, can be written as

    Consider Then the solution can be formulated as follows:

    (4.22)

    where are constants. Putting (4.22) in (4.21), yields

    Since

    then the simple computations gives the expression

    Consequently, we get

    Therefore, we have the subsequent solution

    or equivalently

    where is assumed to be arbitrary constant, we take

    Therefore, for appropriate we have

    where

    The present investigation deal with an IVP for fuzzy and we employ a new scheme of successive approximations under generalized Lipschitz condition to obtain the existence and uniqueness consequences of the solution to the specified problem. Furthermore, another method to discover exact solutions of fuzzy by utilizing the solutions of integer order differential equations is considered. Additionally, the existence consequences for under - with fuzzy initial conditions are proposed. Also, the uniqueness of the so-called integrodifferential equations is verified. Meanwhile, we derived the equivalent integral forms of the original fuzzy whichis utilized to examine the convergence of these arrangements of conditions. Two examples enlightened the efficacy and preciseness of the fractional-order and the other one presents the exact solution by means of the Fox-Wright function. For forthcoming mechanisms, we will relate the numerical strategies for the estimated solution of nonlinear fuzzy

    The authors would like to express their sincere thanks to the support of Taif University Researchers Supporting Project Number (TURSP-2020/217), Taif University, Taif, Saudi Arabia.

    The authors declare that they have no competing interests.


    Abbreviation BHU: Basic health unit; HPA: Hypothalamic-Pituitary-Adrenal axis; MDD: Major depressive disorder; WHO: World Health Organization;
    Acknowledgments



    The Translational Psychiatry Program (USA) is funded by the Department of Psychiatry and Behavioral Sciences, McGovern Medical School, The University of Texas Health Science Center at Houston (UTHealth). Laboratory of Translational Psychiatry (Brazil) is one of the members of the Center of Excellence in Applied Neurosciences of Santa Catarina (NENASC). Its research is supported by grants from CNPq (JQ, GZR, ZMI, and MDB), FAPESC (JQ, GZR, ZMI, and MDB), Instituto Cérebro e Mente (JQ and GZR), UFFS (MDB and ZMI), and UNESC (JQ and GZR). JQ is a 1A CNPq Research Fellow. ZR assisted in writing and revised the manuscript. All authors read and approved the manuscript.

    Author contributions



    VAG and ZMI applied the questionnaires; MGB made the blood collections. AGB, MEP and MGB prepared the samples and tabulated the data. GZR did the statistical analysis. ZMI and AGB wrote the manuscript. JQ and G.

    Conflict of interest



    The authors declare no conflict of interest.

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