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Impact of Physical Activity on Frailty Status and How to Start a Semiological Approach to Muscular System

  • Introduction: The world population is aging, and this demographic fact is associated with an increased prevalence of sedentary lifestyles, sarcopenia and frailty; all of them with impact on health status. Biologic reserve determination in the elderly with comorbidity poses a challenge for medical activities. Frailty is an increasingly used concept in the geriatric medicine literature, which refers to an impairment in biologic reserve. There is a close and multidirectional relationship between physical activity, the muscular system function, and a fit status; decline in this dimensions is associated with poor outcomes. The aim of this article is to make a narrative review on the relationship between physical activity, sarcopenia and frailty syndrome. Results: The low level of physical activity, sarcopenia and frailty, are important predictors for development of disability, poor quality of life, falls, hospitalizations and all causes mortality. For clinical practice we propose a semiological approach based on measurement of muscle performance, mass and also level of physical activity, as a feasible way to determine the biologic reserve. This evidence shows us that the evaluation of muscle mass and performance, provides important prognostic information because the deterioration of these variables is associated with poor clinical outcomes in older adults followed up in multiple cohorts. Conclusions: Low activity is a mechanism and at the same time part of the frailty syndrome. The determination of biologic reserve is important because it allows the prognostic stratification of the patient and constitutes an opportunity for intervention. The clinician should be aware of the clinical tools that evaluate muscular system and level of physical activity, because they place us closer to the knowledge of health status.

    Citation: Maximiliano Smietniansky, Bruno R. Boietti, Mariela A. Cal, María E. Riggi, Giselle P.Fuccile, Luis A. Camera, Gabriel D. Waisman. Impact of Physical Activity on Frailty Status and How to Start a Semiological Approach to Muscular System[J]. AIMS Medical Science, 2016, 3(1): 52-60. doi: 10.3934/medsci.2016.1.52

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  • Introduction: The world population is aging, and this demographic fact is associated with an increased prevalence of sedentary lifestyles, sarcopenia and frailty; all of them with impact on health status. Biologic reserve determination in the elderly with comorbidity poses a challenge for medical activities. Frailty is an increasingly used concept in the geriatric medicine literature, which refers to an impairment in biologic reserve. There is a close and multidirectional relationship between physical activity, the muscular system function, and a fit status; decline in this dimensions is associated with poor outcomes. The aim of this article is to make a narrative review on the relationship between physical activity, sarcopenia and frailty syndrome. Results: The low level of physical activity, sarcopenia and frailty, are important predictors for development of disability, poor quality of life, falls, hospitalizations and all causes mortality. For clinical practice we propose a semiological approach based on measurement of muscle performance, mass and also level of physical activity, as a feasible way to determine the biologic reserve. This evidence shows us that the evaluation of muscle mass and performance, provides important prognostic information because the deterioration of these variables is associated with poor clinical outcomes in older adults followed up in multiple cohorts. Conclusions: Low activity is a mechanism and at the same time part of the frailty syndrome. The determination of biologic reserve is important because it allows the prognostic stratification of the patient and constitutes an opportunity for intervention. The clinician should be aware of the clinical tools that evaluate muscular system and level of physical activity, because they place us closer to the knowledge of health status.



    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 $ \mathcal{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 $ \mathcal{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 $ \mathcal{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 $ \mathcal{HD} $ or generalized $ \mathcal{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 $ \vartheta\in(0, 1): $

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

    where a fuzzy function is $ \mathcal{F}:[\sigma_{1}, \sigma_{2}]\times\mathfrak{E}\rightarrow \mathfrak{E} $ with a nontrivial fuzzy constant $ \Phi_{0}\in\mathfrak{E} $. 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 < \sigma_{1} < \zeta\leq\sigma_{2}, $ $ \, _{c}\mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \rho} $ denotes the fuzzy Caputo-Katugampola fractional generalized Hukuhara derivative and a fuzzy function is $ \mathcal{F}:[\sigma_{1}, \sigma_{2}]\times\mathfrak{E}\rightarrow \mathfrak{E}. $ 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 $ \mathcal{GPF} $-derivative). Consequently, in the framework of the proposed derivative, we establish the basic mathematical tools for the investigation of $ \mathcal{GPF} $-$ \mathcal{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 $ \mathcal{GPF} $ generalized $ \mathcal{HD} $ by considering an approach of continual estimates via generalized Lipschitz condition. Moreover, we derived the $ \mathcal{FVFIE} $ using a generalized fuzzy $ \mathcal{GPF} $ derivative is presented. Finally, we demonstrate the problems of actual and uniqueness of the clarification of this group of equations. The Hilfer-$ \mathcal{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 $ \mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \mathfrak{q}, \beta}(.) $ is the Hilfer $ \mathcal{GPF} $-derivative of order $ \vartheta\in(0, 1), \, \mathcal{I}_{\sigma_{1}}^{1-\gamma, \beta}(.) $ is the $ \mathcal{GPF} $ integral of order $ 1-\gamma > 0, \, \mathcal{R}_{j}\in\mathbb{R}, $ and a continuous function $ \mathcal{F}:[\sigma_{1}, \mathcal{T}]\times\mathbb{R}\rightarrow \mathbb{R} $ with $ \nu_{j}\in[\sigma_{1}, \mathcal{T}] $ fulfilling $ \sigma < \nu_{1} < ... < \nu_{m} < \mathcal{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 $ \mathcal{FVFIE}s $ under generalized fuzzy Hilfer-$ \mathcal{GPF} $-$ \mathcal{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 $ \mathcal {GPF } $ Volterra-Fredholm integrodifferential equation in Section 4. Section 5 consists of concluding remarks.

    Throughout this investigation, $ \mathfrak{E} $ represents the space of all fuzzy numbers on $ \mathbb{R}. $ Assume the space of all Lebsegue measureable functions with complex values $ \mathcal{F} $ on a finite interval $ [\sigma_{1}, \sigma_{2}] $ is identified by $ \chi_{c}^{r}(\sigma_{1}, \sigma_{2}) $ 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 $ \Phi:\mathbb{R}\rightarrow [0, 1] $ which fulfills the subsequent assumptions:

    $ (1) $ $ \Phi $ is normal, i.e., there exists $ \zeta_{0}\in\mathbb{R} $ such that $ \Phi(\zeta_{0}) = 1; $

    $ (2) $ $ \Phi $ is fuzzy convex in $ \mathbb{R}, $ i.e, for $ \delta\in[0, 1], $

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

    $ (3) $ $ \Phi $ is upper semicontinuous on $ \mathbb{R}; $

    $ (4) $ $ [z]^{0} = cl\big\{z_{1}\in\mathbb{R}\, \vert\, \Phi(z_{1}) > 0\big\} $ is compact.

    $ \mathcal{C}\big([\sigma_{1}, \sigma_{2}], \mathfrak{E}\big) $ indicates the set of all continuous functions and set of all absolutely continuous fuzzy functions signifys by $ \mathcal{AC}\big([\sigma_{1}, \sigma_{2}], \mathfrak{E}\big) $ on the interval $ [\sigma_{1}, \sigma_{2}] $ having values in $ \mathfrak{E}. $

    Let $ \gamma\in(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 $ \Phi:\mathbb{R}\mapsto[0, 1] $ and all fuzzy mappings $ \Phi:[\sigma_{1}, \sigma_{2}]\rightarrow \mathfrak{E} $ defined on $ L\big([\sigma_{1}, \sigma_{2}], \mathfrak{E}\big) $ such that the mappings $ \zeta\rightarrow \bar{\mathcal{D}}_{0}[\Phi(\zeta), \hat{0}] $ lies in $ L_{1}[\sigma_{1}, \sigma_{2}]. $

    There is a fuzzy number $ \Phi $ on $ \mathbb{R}, $ we write $ [\Phi]^{\check{q}} = \big\{z_{1}\in\mathbb{R}\, \vert\, \Phi(z_{1})\geq \check{q}\big\} $ the $ \check{q} $-level of $ \Phi, $ having $ \check{q}\in(0, 1]. $

    From assertions $ (1) $ to $ (4); $ it is observed that the $ \check{q} $-level set of $ \Phi\in\mathfrak{E}, $ $ [\Phi]^{\check{q}} $ is a nonempty compact interval for any $ \check{q}\in(0, 1]. $ The $ \check{q} $-level of a fuzzy number $ \Phi $ is denoted by $ \big[\underline{\Phi}(\check{q}), \bar{\Phi}(\check{q})\big]. $

    For any $ \delta\in\mathbb{R} $ and $ \Phi_{1}, \Phi_{2}\in\mathfrak{E}, $ then the sum $ \Phi_{1}+\Phi_{2} $ and the product $ \delta\Phi_{1} $ are demarcated as: $ [\Phi_{1}+\Phi_{2}]^{\check{q}} = [\Phi_{1}]^{\check{q}}+[\Phi_{2}]^{\check{q}} $ and $ [\delta.\Phi_{1}]^{\check{q}} = \delta[\Phi_{1}]^{\check{q}}, $ for all $ \check{q}\in[0, 1], $ where $ [\Phi_{1}]^{\check{q}}+[\Phi_{2}]^{\check{q}} $ is the usual sum of two intervals of $ \mathbb{R} $ and $ \delta[\Phi_{1}]^{\check{q}} $ is the scalar multiplication between $ \delta $ and the real interval.

    For any $ \Phi\in\mathfrak{E}, $ the diameter of the $ \check{q} $-level set of $ \Phi $ is stated as $ diam[\mu]^{\check{q}} = \bar{\mu}(\check{q})-\underline{\mu}(\check{q}). $

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

    Definition 2.2. ([54]) Suppose $ \Phi_{1}, \Phi_{2}\in\mathfrak{E}. $ If there exists $ \Phi_{3}\in\mathfrak{E} $ such that $ \Phi_{1} = \Phi_{2}+\Phi_{3}, $ then $ \Phi_{3} $ is known to be the Hukuhara difference of $ \Phi_{1} $ and $ \Phi_{2} $ and it is indicated by $ \Phi_{1}\ominus\Phi_{2}. $ Observe that $ \Phi_{1}\ominus\Phi_{2}\neq\Phi_{1}+(-)\Phi_{2}. $

    Definition 2.3. ([54]) We say that $ {\bar{\mathcal{D}_{0}}}[\Phi_{1}, \Phi_{2}] $ 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 $ [\Phi_{1}]^{\check{q}} $ and $ [\Phi_{2}]^{\check{q}} $ is defined as

    $ \mathcal{H}\Big([\Phi_{1}]^{\check{q}},[\Phi_{2}]^{\check{q}}\Big) = \max\big\{\vert\underline{\Phi}(\check{q})-\bar{\Phi}(\check{q})\vert,\vert\bar{\Phi}(\check{q})-\underline{\Phi}(\check{q})\vert\big\}. $

    Fuzzy sets in $ \mathfrak{E} $ is also refereed as triangular fuzzy numbers that are identified by an ordered triple $ \Phi = (\sigma_{1}, \sigma_{2}, \sigma_{3})\in\mathbb{R}^{3} $ with $ \sigma_{1}\leq \sigma_{2}\leq \sigma_{3} $ such that $ [\Phi]^{\check{q}} = [\underline{\Phi}(\check{q}), \bar{\Phi}(\check{q})] $ are the endpoints of $ \check{q} $-level sets for all $ \check{q}\in[0, 1], $ where $ \underline{\Phi}(\check{q}) = \sigma_{1}+(\sigma_{2}-\sigma_{1})\check{q} $ and $ \bar{\Phi}(\check{q}) = \sigma_{3}-(\sigma_{3}-\sigma_{2})\check{q}. $

    Generally, the parametric form of a fuzzy number $ \Phi $ is a pair $ [\Phi]^{\check{q}} = [\underline{\Phi}(\check{q}), \bar{\Phi}(\check{q})] $ of functions $ \underline{\Phi}(\check{q}), \bar{\Phi}(\check{q}), \check{q}\in[0, 1], $ which hold the following assumptions:

    $ (1) $ $ \underline{\mu}(\check{q}) $ is a monotonically increasing left-continuous function;

    $ (2) $ $ \bar{\mu}(\check{q}) $ is a monotonically decreasing left-continuous function;

    $ (3) $ $ \underline{\mu}(\check{q})\leq\bar{\mu}(\check{q}), \, \check{q}\in[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 $ \Phi_{1}, \Phi_{2}\in\mathfrak{E} $ ($ \mathfrak{g}H $-difference in short) is stated as follows

    $ Φ1gHΦ2=Φ3Φ1=Φ2+Φ3orΦ2=Φ1+(1)Φ3. $

    A function $ \Phi:[\sigma_{1}, \sigma_{2}]\rightarrow \mathfrak{E} $ is said to be $ \mathfrak{d} $-increasing ($ \mathfrak{d} $-decreasing) on $ [\sigma_{1}, \sigma_{2}] $ if for every $ \check{q}\in[0, 1]. $ The function $ \zeta\rightarrow \, \, diam[\Phi(\zeta)]^{\check{q}} $ is nondecreasing (nonincreasing) on $ [\sigma_{1}, \sigma_{2}] $. If $ \Phi $ is $ \mathfrak{d} $-increasing or $ \mathfrak{d} $-decreasing on $ [\sigma_{1}, \sigma_{2}] $, then we say that $ \Phi $ is $ \mathfrak{d} $-monotone on $ [\sigma_{1}, \sigma_{2}]. $

    Definition 2.5. ([39])The generalized Hukuhara derivative of a fuzzy-valued function $ \mathcal{F}:(\sigma_{1}, \sigma_{2})\rightarrow \mathfrak{E} $ at $ \zeta_{0} $ is defined as

    $ FgH(ζ0)=limh0F(ζ0+h)gHF(ζ0)h, $

    if $ (\mathcal{F})_{\mathfrak{g}H}^{\prime}(\zeta_{0})\in\mathfrak{E}, $ we say that $ \mathcal{F} $ is generalized Hukuhara differentiable ($ \mathfrak{g}H $-differentiable) at $ \zeta_{0}. $

    Moreover, we say that $ \mathcal{F} $ is $ [(i)-\mathfrak{g}H] $-differentiable at $ \zeta_{0} $ 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 $ \mathcal{F} $ is $ [(ii)-\mathfrak{g}H] $-differentiable at $ \zeta_{0} $ if

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

    Definition 2.6. ([49]) We state that a point $ \zeta_{0}\in(\sigma_{1}, \sigma_{2}), $ is a switching point for the differentiability of $ \mathcal{F}, $ if in any neighborhood $ U $ of $ \zeta_{0} $ there exist points $ \zeta_{1} < \zeta_{0} < \zeta_{2} $ such that

    Type Ⅰ. at $ \zeta_{1} $ (2.1) holds while (2.2) does not hold and at $ \zeta_{2} $ (2.2) holds and (2.1) does not hold, or

    Type Ⅱ. at $ \zeta_{1} $ (2.2) holds while (2.1) does not hold and at $ \zeta_{2} $ (2.1) holds and (2.2) does not hold.

    Definition 2.7. ([23]) For $ \beta\in(0, 1] $ and let the left-sided $ \mathcal{GPF} $-integral operator of order $ \vartheta $ of $ \mathcal{F} $ is defined as follows

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

    where $ \beta\in(0, 1] $, $ \vartheta\in\mathbb{C}, $ $ Re(\vartheta) > 0 $ and $ \Gamma(.) $ is the Gamma function.

    Definition 2.8. ([23]) For $ \beta\in(0, 1] $ and let the left-sided $ \mathcal{GPF} $-derivative operator of order $ \vartheta $ of $ \mathcal{F} $ is defined as follows

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

    where $ \beta\in(0, 1] $, $ \vartheta\in\mathbb{C}, \, Re(\vartheta) > 0, \; n = [\vartheta]+1 $ and $ \mathcal{D}^{n, \beta} $ represents the $ nth $-derivative with respect to proportionality index $ \beta. $

    Definition 2.9. ([23]) For $ \beta\in(0, 1] $ and let the left-sided $ \mathcal{GPF} $-derivative in the sense of Caputo of order $ \vartheta $ of $ \mathcal{F} $ is defined as follows

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

    where $ \beta\in(0, 1] $, $ \vartheta\in\mathbb{C}, \, Re(\vartheta) > 0 $ and $ n = [\vartheta]+1. $

    Let $ \Phi\in L([\sigma_{1}, \sigma_{2}], \mathfrak{E}), $ then the $ \mathcal{GPF} $ integral of order $ \vartheta $ of the fuzzy function $ \Phi $ is stated as:

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

    Since $ [\Phi(\zeta)]^{\check{q}} = [\underline{\Phi}(\check{q}, \zeta), \bar{\Phi}(\check{q}, \zeta)] $ and $ 0 < \vartheta < 1, $ we can write the fuzzy $ \mathcal{GPF} $-integral of the fuzzy mapping $ \Phi $ 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 $ n\in\mathbb{N}, $ order $ \vartheta $ and type $ \mathfrak{q} $ hold $ n-1 < \vartheta\leq n $ with $ 0\leq\mathfrak{q}\leq1. $ The left-sided fuzzy Hilfer-proportional $ \mathfrak{g}H $-fractional derivative, with respect to $ \zeta $ having $ \beta\in(0, 1] $ of a function $ \zeta\in\mathcal{C}_{1-\gamma}^{\beta}[\sigma_{1}, \sigma_{2}], $ is stated as

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

    where $ \mathcal{D}^{\beta}\Phi(\nu) = (1-\beta)\Phi(\nu)+\beta \Phi^{\prime}(\nu) $ and if the $ \mathfrak{g}H $-derivative $ \Phi_{(1-\vartheta), \beta}^{\prime}(\zeta) $ exists for $ \zeta\in[\sigma_{1}, \sigma_{2}], $ where

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

    Definition 2.11. Let $ \Phi^{\prime}\in L([\sigma_{1}, \sigma_{2}], \mathfrak{E}) $ and the fractional generalized Hukuhara $ \mathcal{GPF} $-derivative of fuzzy-valued function $ \Phi $ 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 $ \Phi $ is $ \, ^{GPF}[(i)-\mathfrak{g}H] $-differentiable at $ \zeta_{0} $ 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 $ \Phi $ is $ \, ^{GPF}[(i)-\mathfrak{g}H] $-differentiable at $ \zeta_{0} $ 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 $ \zeta_{0}\in(\sigma_{1}, \sigma_{2}), $ is a switching point for the differentiability of $ \mathcal{F}, $ if in any neighborhood $ U $ of $ \zeta_{0} $ there exist points $ \zeta_{1} < \zeta_{0} < \zeta_{2} $ such that

    Type Ⅰ. at $ \zeta_{1} $ (2.11) holds while (2.12) does not hold and at $ \zeta_{2} $ (2.12) holds and (2.11) does not hold, or

    Type Ⅱ. at $ \zeta_{1} $ (2.12) holds while (2.11) does not hold and at $ \zeta_{2} $ (2.11) holds and (2.12) does not hold.

    Proposition 1. ([23]) Let $ \vartheta, \varrho\in\mathbb{C} $ such that $ Re(\vartheta) > 0 $ and $ Re(\varrho) > 0. $ Then for any $ \beta\in(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 $ \beta\in(0, 1], $ $ \vartheta > 0, $ $ 0\leq\gamma < 1. $ If $ \Phi\in\mathcal{C}_{\gamma}[\sigma_{1}, \sigma_{2}] $ and $ \mathcal{I}_{\sigma_{1}^{+}}^{1-\vartheta}\Phi\in\mathcal{C}_{\gamma}^{1}[\sigma_{1}, \sigma_{2}], $ then

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

    Lemma 2.14. ([24]) Let $ \Phi\in L_{1}(\sigma_{1}, \sigma_{2}). $ If $ \mathcal{D}_{\sigma_{1}^{+}}^{\mathfrak{q}(1-\vartheta), \beta}\Phi $ exists on $ L_{1}(\sigma_{1}, \sigma_{2}), $ then

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

    Lemma 2.15. Suppose there is a $ \mathfrak{d} $-monotone fuzzy mapping $ \Phi\in\mathcal{AC}\big([\sigma_{1}, \sigma_{2}], \mathfrak{E}\big), $ where $ \big[\Phi(\zeta)\big]^{\check{q}} = \big[\underline{\Phi}(\check{q}, \zeta), \bar{\Phi}(\check{q}, \zeta)\big] $ for $ 0\leq\check{q}\leq1, \, \sigma_{1}\leq\zeta\leq \sigma_{2}, $ then for $ 0 < \vartheta < 1 $ and $ \beta\in(0, 1], $ we have

    $ (i) \quad\big[\big(\mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \mathfrak{q}, \beta}\Phi\big)(\zeta)\big]^{\check{q}} = \big[\mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \mathfrak{q}, \beta}\underline{\Phi}(\check{q}, \zeta), \mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \mathfrak{q}, \beta}\bar{\Phi}(\check{q}, \zeta)\big] $ for $ \zeta\in[\sigma_{1}, \sigma_{2}], $ if $ \Phi $ is $ \mathfrak{d} $-increasing;

    $ (ii)\quad \big[\big(\mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \mathfrak{q}, \beta}\Phi\big)(\zeta)\big]^{\check{q}} = \big[\mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \mathfrak{q}, \beta}\bar{\Phi}(\check{q}, \zeta), \mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \mathfrak{q}, \beta}\underline{\Phi}(\check{q}, \zeta)\big] $ for $ \zeta\in[\sigma_{1}, \sigma_{2}], $ if $ \Phi $ is $ \mathfrak{d} $-decreasing.

    Proof. It is to be noted that if $ \Phi $ is $ \mathfrak{d} $-increasing, then $ \big[\Phi^{\prime}(\zeta)\big]^{\check{q}} = \big[\frac{d}{d\zeta}\underline{\Phi}(\check{q}, \zeta), \frac{d}{d\zeta}\bar{\Phi}(\check{q}, \zeta)\big]. $ 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 $ \Phi $ is $ \mathfrak{d} $-decreasing, then $ \big[\Phi^{\prime}(\zeta)\big]^{\check{q}} = \big[\frac{d}{d\zeta}\bar{\Phi}(\check{q}, \zeta), \frac{d}{d\zeta}\underline{\Phi}(\check{q}, \zeta)\big], $ 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 $ \beta\in(0, 1], \, \vartheta\in(0, 1). $ If $ \Phi\in\mathcal{AC}([\sigma_{1}, \sigma_{2}], \mathfrak{E}) $ is a $ \mathfrak{d} $-monotone fuzzy function. We take

    $ z_{1}(\zeta): = \Big(\mathcal{I}_{\sigma_{1}^{+}}^{\vartheta,\beta}\Phi\Big)(\zeta) = \frac{1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\sigma_{1}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\Phi(\nu)d\nu, $

    and

    $ z_{1}^{(1-\vartheta),\beta}: = \Big(\mathcal{I}_{\sigma_{1}^{+}}^{(1-\vartheta),\beta}\Phi\Big)(\zeta) = \frac{1}{\beta^{1-\vartheta}\Gamma(1-\vartheta)}\int\limits_{\sigma_{1}}^{\vartheta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta}\Phi_{\mathfrak{g}H}^{\prime}(\nu)d\nu, $

    is $ \mathfrak{d} $-increasing on $ (\sigma_{1}, \sigma_{2}], $ 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 $ z_{1}(\zeta) $ is $ \mathfrak{d} $-increasing on $ [\sigma_{1}, \sigma_{2}] $ or $ z_{1}(\zeta) $ is $ \mathfrak{d} $-decreasing on $ [\sigma_{1}, \sigma_{2}] $ and $ z_{1}^{(1-\vartheta), \beta}(\zeta) $ is $ \mathfrak{d} $-increasing on $ (\sigma_{1}, \sigma_{2}]. $

    Utilizing the Definitions 2.6, 2.10 and Lemma 2.13 with the initial condition $ (\mathcal{I}_{\sigma_{1}^{+}}^{1-\gamma, \beta}\Phi)(\sigma_{1}) = 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 $ \Phi\in\mathcal{AC}([\sigma_{1}, \sigma_{2}], \mathfrak{E}) $, there exists a constant $ \mathcal{K} $ such that $ \mathcal{K} = \sup\limits_{\zeta\in[\sigma_{1}, \sigma_{2}]}\bar{\mathcal{D}_{0}}[\Phi(\zeta), \hat{0}]. $

    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 $ \big\vert e^{\frac{\beta-1}{\beta}\zeta}\big\vert < 1 $ and $ \mathcal{I}_{\sigma_{1}^{+}}^{\vartheta, \beta}\Phi(\zeta) = 0 $ and $ \zeta = \sigma_{1}. $

    This completes the proof.

    Lemma 2.17. Let there be a continuous mapping $ \Phi:[\sigma_{1}, \sigma_{2}]\rightarrow \mathbb{R}^{+} $ on $ [\sigma_{1}, \sigma_{2}] $ and hold $ \mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \mathfrak{q}, \beta}\Phi(\zeta)\leq\mathcal{F}(\xi, \Phi(\xi)), \xi\geq\sigma_{1}, $ where $ \mathcal{F}\in\mathcal{C}([\sigma_{1}, \sigma_{1}]\times\mathbb{R}^{+}, \mathbb{R}^{+}). $ Assume that $ m(\zeta) = m(\zeta, \sigma_{1}, \xi_{0}) $ is the maximal solution of the IVP

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

    on $ [\sigma_{1}, \sigma_{2}]. $ Then, if $ \Phi(\sigma_{1})\leq\xi_{0}, $ we have $ \Phi(\zeta)\leq m(\zeta), \zeta\in[\sigma_{1}, \sigma_{2}]. $

    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 $ \alpha > 0 $ be a given constant and $ \mathfrak{B}(\Phi_{0}, \alpha) = \big\{\Phi\in\mathbb{R}:\vert\Phi-\Phi_{0}\vert\leq\alpha\big\}. $ Assume that the real-valued functions $ \mathcal{F}:[\sigma_{1}, \sigma_{2}]\times[0, \alpha]\rightarrow \mathbb{R}^{+} $ satisfies the following assumptions:

    $ (i) $ $ \mathcal{F}\in\mathcal{C}\big([\sigma_{1}, \sigma_{2}]\times[0, \alpha], \mathbb{R}^{+}\big), \, \mathcal{F}(\zeta, 0)\equiv0, \, 0\leq\mathcal{F}(\zeta, \Phi)\leq\mathcal{M}_{\mathcal{F}} $ for all $ (\zeta, \Phi)\in[\sigma_{1}, \sigma_{2}]\times[0, \alpha]; $

    $ (ii) $ $ \mathcal{F}(\zeta, \Phi) $ is nondecreasing in $ \Phi $ for every $ \zeta\in[\sigma_{1}, \sigma_{2}]. $ Then the problem (2.15) has at least one solution defined on $ [\sigma_{1}, \sigma_{2}] $ and $ \Phi(\zeta)\in\mathfrak{B}(\Phi_{0}, \alpha). $

    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 $ \gamma = \vartheta+\mathfrak{q}(1-\vartheta), \, \, \vartheta\in(0, 1), \mathfrak{q}\in[0, 1] $ with $ \beta\in(0, 1], $ and let there is a fuzzy function $ \mathcal{F}:(\sigma_{1}, \sigma_{2}]\times\mathfrak{E}\rightarrow \mathfrak{E} $ such that $ \zeta\rightarrow \mathcal{F}(\zeta, \Phi) $ belongs to $ \mathcal{C}_{\gamma}^{\beta}([\sigma_{1}, \sigma_{2}], \mathfrak{E}) $ for any $ \Phi\in\mathfrak{E}. $ Then a $ \mathfrak{d} $-monotone fuzzy function $ \Phi\in\mathcal{C}([\sigma_{1}, \sigma_{2}], \mathfrak{E}) $ is a solution of IVP (1.3) if and only if $ \Phi $ 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 $ \zeta\rightarrow \mathcal{I}_{\sigma_{1}^{+}}^{1-\gamma}\mathcal{F}(\zeta, \Phi) $ is $ \mathfrak{d} $-increasing on $ (\sigma_{1}, \sigma_{2}]. $

    Proof. Let $ \Phi\in\mathcal{C}([\sigma_{1}, \sigma_{2}], \mathfrak{E}) $ be a $ \mathfrak{d} $-monotone solution of (1.3), and considering $ z_{1}(\zeta): = \Phi(\zeta)\ominus_{\mathfrak{g}H}\big(\mathcal{I}_{\sigma_{1}^{+}}^{1-\gamma, \beta}\Phi\big)(\sigma_{1}), \zeta\in(\sigma_{1}, \sigma_{2}]. $ Since $ \Phi $ is $ \mathfrak{d} $-monotone on $ [\sigma_{1}, \sigma_{2}], $ it follows that $ \zeta\rightarrow z_{1}(\zeta) $ is $ \mathfrak{d} $-increasing on $ [\sigma_{1}, \sigma_{2}] $ (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 $ \mathcal{F}(\zeta, \Phi)\in\mathcal{C}_{\gamma}([\sigma_{1}, \sigma_{2}], \mathfrak{E}) $ for any $ \Phi\in\mathfrak{E}, $ 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 $ z_{1}(\zeta) $ is $ \mathfrak{d} $-increasing on $ (\sigma_{1}, \sigma_{2}]. $ Also, we observe that $ \zeta\rightarrow \mathcal{F}^{\vartheta, \beta}(\zeta, \Phi) $ is also $ \mathfrak{d} $-increasing on $ (\sigma_{1}, \sigma_{2}]. $

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

    Further, suppose $ \Phi\in\mathcal{C}([\sigma_{1}, \sigma_{2}], \mathfrak{E}) $ be a $ \mathfrak{d} $-monotone fuzzy function fulfills (3.1) and such that $ \zeta\rightarrow \mathcal{F}^{\vartheta, \beta}(\zeta, \Phi) $ is $ \mathfrak{d} $-increasing on $ (\sigma_{1}, \sigma_{2}]. $ By the continuity of the fuzzy mapping $ \mathcal{F}, $ the fuzzy mapping $ \zeta\rightarrow \mathcal{F}^{\vartheta, \beta}(\zeta, \Phi) $ is continuous on $ (\sigma_{1}, \sigma_{2}] $ with $ \mathcal{F}^{\vartheta, \beta}(\sigma_{1}, \Phi(\sigma_{1})) = \lim\limits_{\zeta\rightarrow \sigma_{1}^{+}}\mathcal{F}^{\vartheta, \beta}(\zeta, \Phi) = 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 $ \zeta\rightarrow \mathcal{F}^{\vartheta, \beta}(\zeta, \Phi) $ is $ \mathfrak{d} $-increasing on $ (\sigma_{1}, \sigma_{2}]. $ Applying, the operator $ \mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \mathfrak{q}, \beta} $ 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 $ \hbar > 0 $, and let $ \mathfrak{B}(\Phi_{0}, \hbar) = \big\{\Phi\in\mathfrak{E}:\bar{\mathcal{D}_{0}}[\Phi, \Phi_{0}]\leq\hbar\big\}. $

    Theorem 3.2. Let $ \mathcal{F}\in\mathcal{C}\big([\sigma_{1}, \sigma_{2}]\times\mathfrak{B}(\Phi_{0}, \hbar), \mathfrak{E}\big) $ and suppose that the subsequent assumptions hold:

    $ (i) $ there exists a positive constant $ \mathcal{M}_{\mathcal{F}} $ such that $ \bar{\mathcal{D}_{0}}[\mathcal{F}(\zeta, z_{1}), \hat{0}]\leq\mathcal{M}_{\mathcal{F}}, $ for all $ (\zeta, z_{1})\in[\sigma_{1}, \sigma_{2}]\times\mathfrak{B}(\Phi_{0}, \hbar) $;

    $ (ii) $ for every $ \zeta\in[\sigma_{1}, \sigma_{2}] $ and every $ z_{1}, \omega\in\mathfrak{B}(\Phi_{0}, \hbar), $

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

    where $ \mathfrak{g}(\zeta, .)\in\mathcal{C}\big([\sigma_{1}, \sigma_{2}]\times[0, \beta], \mathbb{R}^{+}\big) $ satisfies the assumption in Lemma 2.18 given that problem (2.15) has only the solution $ \phi(\zeta)\equiv0 $ on $ [\sigma_{1}, \sigma_{2}]. $ Then the subsequent successive approximations given by $ \Phi^{0}(\zeta) = \Phi_{0} $ 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 $ [\sigma_{1}, \mathcal{T}] $ for some $ \mathcal{T}\in(\sigma_{1}, \sigma_{2}] $ given that the mapping $ \zeta\rightarrow \mathcal{I}_{\sigma_{1}^{+}}^{\vartheta, \beta}\mathcal{F}(\zeta, \Phi^{n}(\zeta)) $ is $ \mathfrak{d} $-increasing on $ [\sigma_{1}, \mathcal{T}]. $

    Proof. Take $ \sigma_{1} < \zeta^{*} $ such that $ \zeta^{*}\leq\big[\frac{\beta^{\vartheta}\hbar.\Gamma(1+\vartheta)}{\mathcal{M}}+\sigma_{1}\big]^{\frac{1}{\vartheta}}, $ where $ \mathcal{M} = \max\big\{\mathcal{M}_{\mathfrak{g}}, \mathcal{M}_{\mathcal{F}}\big\} $ and put $ \mathcal{T}: = \min\{\zeta^{*}, \sigma_{2}\}. $ Let $ \mathbb{S} $ be a set of continuous fuzzy functions $ \Phi $ such that $ \omega(\sigma_{1}) = \Phi_{0} $ and $ \omega(\zeta)\in\mathfrak{B}(\Phi_{0}, \hbar) $ for all $ \zeta\in[\sigma_{1}, \mathcal{T}]. $ Further, we suppose the sequence of continuous fuzzy function $ \{\Phi^{n}\}_{n = 0}^{\infty} $ given by $ \Phi^{0}(\zeta) = \Phi_{0}, \, \forall \zeta\in[\sigma_{1}, \mathcal{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 $ \Phi^{n}(\zeta)\in\mathcal{C}([\sigma_{1}, \mathcal{T}], \mathfrak{B}(\Phi_{0}, \hbar)) $. For $ n\geq1 $ and for any $ \zeta_{1}, \zeta_{2}\in[\sigma_{1}, \mathcal{T}] $ with $ \zeta_{1} < \zeta_{2}, $ 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 $ \vert e^{\frac{\beta-1}{\beta}\zeta}\vert < 1, $ then, on the right-hand side from the last inequality, the subsequent integral becomes $ \frac{1}{\beta^{\vartheta}\Gamma(1+\vartheta)}(\zeta_{2}-\zeta_{1})^{\vartheta}. $ Therefore, with the similar assumption as we did above, the first integral reduces to $ \frac{1}{\beta^{\vartheta}\Gamma(1+\vartheta)}\big[(\zeta_{1}-\sigma_{1})^{\vartheta}-(\zeta_{2}-\sigma_{1})^{\vartheta}+(\zeta_{2}-\zeta_{1})^{\vartheta}\big]. $ 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 $ \zeta_{1}\rightarrow \zeta_{2}, $ then the last expression of the above inequality tends to $ 0, $ which shows $ \Phi^{n} $ is a continuous function on $ [\sigma_{1}, \mathcal{T}] $ for all $ n\geq1. $

    Moreover, it follows that $ \Phi^{n}\in\mathfrak{B}(\Phi_{0}, \hbar) $ for all $ n\geq0, \, \zeta\in[\sigma_{1}, \mathcal{T}] $ if and only if $ \Phi^{n}(\zeta)\ominus_{\mathfrak{g}H}\frac{\sum\limits_{j = 1}^{m}R_{j}\Phi(\zeta_{j})}{\beta^{\gamma}\Gamma(\gamma)}e^{\frac{\beta-1}{\beta}(\zeta-\sigma_{1})}(\zeta-\sigma_{1})^{\gamma-1}\in\mathfrak{B}(0, \hbar) $ for all $ \zeta\in[\sigma_{1}, \mathcal{T}] $ and for all $ n\geq0. $

    Also, if we assume that $ \Phi^{n-1}(\zeta)\in\mathbb{S} $ for all $ \zeta\in[\sigma_{1}, \mathcal{T}], \, n\geq2, $ 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 $ \Phi^{n}(\zeta)\in\mathbb{S}, \, \forall\in[\sigma_{1}, \mathcal{T}]. $

    Henceforth, by mathematical induction, we have $ \Phi^{n}(\zeta)\in\mathbb{S}, \, \, \forall \zeta\in[\sigma_{1}, \mathcal{T}] $ and $ \forall\, n\geq1. $

    Further, we show that the sequence $ \Phi^{n}(\zeta) $ converges uniformly to a continuous function $ \Phi\in\mathcal{C}([\sigma_{1}, \mathcal{T}], \mathfrak{B}(\Phi_{0}, \hbar)). $ By assertion $ (ii) $ and mathematical induction, we have for $ \zeta\in[\sigma_{1}, \mathcal{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 $ \phi^{n}(\zeta) $ 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 $ \vert e^{\frac{\beta-1}{\beta}\zeta}\vert < 1 $ and $ \phi^{0}(\zeta) = \frac{\mathcal{M}(\zeta-\sigma_{1})^{\vartheta}}{\beta^{{\vartheta}}\Gamma({\vartheta}+1)}. $ Thus, we have, for $ \zeta\in[\sigma_{1}, \mathcal{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 $ n\leq m $ and $ \zeta\in[\sigma_{1}, \mathcal{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 $ \phi(\zeta) = 0 $ is a unique solution of problem (2.15) and $ \mathfrak{g}(., \phi^{n-1}):[\sigma_{1}, \mathcal{T}]\rightarrow [0, \mathcal{M}_{\mathfrak{g}}] $ uniformly converges to $ 0 $, for every $ \epsilon > 0, $ there exists a natural number $ n_{0} $ such that

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

    Using the fact that $ \bar{\mathcal{D}_{0}}\big[\Phi^{n}(\sigma_{1}), \Phi^{m}(\sigma_{1})\big] = 0 < \epsilon $ and by using Lemma 2.17, we have for $ \zeta\in[\sigma_{1}, \mathcal{T}] $

    $ ¯D0[Φn(ζ),Φm(ζ)]δϵ(ζ),n0nm, $ (3.8)

    where $ \delta_{\epsilon}(\zeta) $ 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 $ [\phi_{\epsilon}(., \omega)] $ converges uniformly to the maximal solution $ \phi(\zeta) \equiv0 $ of (2.15) on $ [\sigma_{1}, \mathcal{T}] $ as $ \epsilon\rightarrow 0. $

    Therefore, in view of (3.8), we can obtain $ n_{0}\in\mathbb{N} $ is large enough such that, for $ n_{0} < 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 $ (\mathfrak{E}, \bar{\mathcal{D}_{0}}) $ is a complete metric space and (3.9) holds, thus $ \big\{\Phi^{n}(\zeta)\} $ converges uniformly to $ \Phi\in\mathcal{C}([\sigma_{1}, \sigma_{2}], \mathfrak{B}(\Phi_{0}, \hbar)). $ 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 $ \Phi(\zeta) $ is the solution to (1.3) on $ [\sigma_{1}, \mathcal{T}]. $

    In order to find the unique solution, assume that $ \Psi:[\sigma_{1}, \mathcal{T}]\rightarrow \mathfrak{E} $ is another solution of problem (1.3) on $ [\sigma_{1}, \mathcal{T}]. $ We denote $ \kappa(\zeta) = \bar{\mathcal{D}_{0}}[\Phi(\zeta), \Psi(\zeta)]. $ Then $ \kappa(\sigma_{1}) = 0 $ and for every $ \zeta\in[\sigma_{1}, \mathcal{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 $ \kappa(\zeta)\leq m(\zeta), $ where $ m $ is a maximal solution of the IVP $ \mathcal{D}_{\sigma_{1}^{+}}^{\vartheta, \mathfrak{q}, \beta}m(\zeta)\leq\mathfrak{g}(\zeta, m(\zeta)), \, \big(\mathcal{I}_{\sigma_{1}^{+}}^{1-\gamma}m\big)(\sigma_{1}) = 0. $ By asseration $ (ii), $ we have $ m(\zeta) = 0 $ and hence $ \Phi(\zeta) = \Psi(\zeta), \, \forall\in[\sigma_{1}, \mathcal{T}]. $

    This completes the proof.

    Corollary 1. For $ \beta\in(0, 1] $ and let $ \mathcal{C}([\sigma_{1}, \sigma_{2}], \mathfrak{E}). $ Assume that there exist positive constants $ \mathcal{L}, \mathcal{M}_{\mathcal{F}} $ such that, for every $ z_{1}, \omega\in\mathfrak{E}, $

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

    Then the subsequent successive approximations given by $ \Phi^{0}(\zeta) = \Phi_{0} $ 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 $ [\sigma_{1}, \mathcal{T}] $ for certain $ \mathcal{T}\in(\sigma_{1}, \sigma_{2}] $ given that the mapping $ \zeta\rightarrow \mathcal{I}_{\sigma_{1}^{+}}^{\vartheta, \beta}\mathcal{F}(\zeta, \Phi^{n}(\zeta)) $ is $ \mathfrak{d} $-increasing on $ [\sigma_{1}, \mathcal{T}]. $

    Example 3.3. For $ \beta\in(0, 1], \, \gamma = \vartheta+\mathfrak{q}(1-\vartheta), \, \vartheta\in(0, 1), \, \mathfrak{q}\in[0, 1] $ and $ \delta\in\mathbb{R}. $ Assume that the linear fuzzy $ \mathcal{GPF} $-$ FDE $ under Hilfer-$ \mathcal{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 $ \eta\in\mathcal{C}((\sigma_{1}, \sigma_{2}], \mathfrak{E}) $ and furthermore, assuming the diameter on the right part of the aforementioned equation is increasing. Observing $ \mathcal{F}(\zeta, \Phi): = \delta\Phi+\eta $ fulfill the suppositions of Corollary 1.

    In order to find the analytical view of (3.12), we utilized the technique of successive approximation. Putting $ \Phi^{0}(\zeta) = \Phi_{0} $ 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, \; \delta > 0, $ assuming there is a $ \mathfrak{d} $-increasing mapping $ \Phi, $ 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 $ \delta < 0 $ and $ \Phi $ is $ \mathfrak{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 $ \delta > 0 $ and there is $ \mathfrak{d} $-increasing mapping $ \Phi $, we have

    $ (1)(mj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1gHΦ2(ζ))=mj=1RjΦ(ζj)[δ(ζσ1)ϑβϑΓ(ϑ+1)+δ2(ζσ1)2ϑβ2ϑΓ(2ϑ+1)]+(Iϑ,βσ+1η)(ζ)+(I2ϑ,βσ+1η)(ζ), $

    and there is $ \delta < 0, \, and\, \mathfrak{d} $-increasing mapping $ \Phi. $ So, continuing inductively and in the limiting case, when $ n\rightarrow \infty, $ we attain the solution

    $ Φ(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=mj=1RjΦ(ζj)l=1δl(ζσ1)lϑβlϑΓ(lϑ+1)+ζσ1l=1δl1(ζσ1)lϑ1βlϑ1Γ(lϑ)η(ν)dν=mj=1RjΦ(ζj)l=1δl(ζσ1)lϑβlϑΓ(lϑ+1)+ζσ1l=0δl(ζσ1)lϑ+(ϑ1)βlϑ+(ϑ1)Γ(lϑ+ϑ)η(ν)dν=mj=1RjΦ(ζj)l=1δl(ζσ1)lϑβlϑΓ(lϑ+1)+1βϑ1ζσ1(ζσ1)ϑ1l=0δl(ζσ1)lϑβlϑΓ(lϑ+ϑ)η(ν)dν, $

    for every $ \delta > 0 $ and $ \Phi $ is $ \mathfrak{d} $-increasing, or $ \delta < 0 $ and $ \Phi $ is $ \mathfrak{d} $-decreasing, accordingly. Therefore, by means of Mittag-Leffler function $ \mathcal{E}_{\vartheta, \mathfrak{q}}(\Phi) = \sum\limits_{l = 1}^{\infty}\frac{\Phi^{\kappa}}{\Gamma(l\vartheta+\mathfrak{q})}, \, \vartheta, \mathfrak{q} > 0, $ the solution of problem (3.12) is expressed by

    $ Φ(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=mj=1RjΦ(ζj)Eϑ,1(δ(ζσ1)ϑ)+1βϑ1ζσ1(ζσ1)ϑ1Eϑ,ϑ(δ(ζσ1)ϑ)η(ν)dν, $

    for every of $ \delta > 0 $ and $ \Phi $ is $ \mathfrak{d} $-increasing. Alternately, if $ \delta < 0 $ and $ \Phi $ is $ \mathfrak{d} $-decreasing, then we get the solution of problem (3.12)

    $ Φ(ζ)gHmj=1RjΦ(ζj)βγΓ(γ)eβ1β(ζσ1)(ζσ1)γ1=mj=1RjΦ(ζj)Eϑ,1(δ(ζσ1)ϑ)(1)1βϑ1ζσ1(ζσ1)ϑ1Eϑ,ϑ(δ(ζσ1)ϑ)η(ν)dν. $

    Consider IVP

    $ \begin{equation} \begin{cases} \big(\,_{\mathfrak{g}H}\mathcal{D}_{\sigma_{1}^{+}}^{\vartheta,\beta}\Phi\big)(\zeta) = \mathcal{F}\big(\zeta,\Phi(\zeta),\mathcal{H}_{1}\Phi(\zeta),\mathcal{H}_{2}\Phi(\zeta)\big),\quad\quad \zeta\in[\zeta_{0},\mathcal{T}]\\\Phi(\zeta_{0}) = \Phi_{0}\in\mathfrak{E}, \end{cases} \end{equation} $ (4.1)

    where $ \beta\in(0, 1] $ and $ \vartheta\in(0, 1) $ is a real number and the operation $ _{\mathfrak{g}H}\mathcal{D}_{\sigma_{1}^{+}}^{\vartheta} $ denote the $ \mathcal{GPF} $ derivative of order $ \vartheta, $ $ \mathcal{F}:[\zeta_{0}, \mathcal{T}]\times\mathfrak{E}\times\mathfrak{E}\times\mathfrak{E}\rightarrow \mathfrak{E} $ is continuous in $ \zeta $ which fulfills certain supposition that will be determined later, and

    $ \begin{eqnarray} \mathcal{H}_{1}\Phi(\zeta) = \int\limits_{\zeta_{0}}^{\zeta}\mathcal{H}_{1}(\zeta,s)\Phi(s)ds,\quad\quad\quad\mathcal{H}_{2}\Phi(\zeta) = \int\limits_{\zeta_{0}}^{\mathcal{T}}\mathcal{H}_{2}(\zeta,s)\Phi(s)ds, \end{eqnarray} $ (4.2)

    with $ \mathcal{H}_{1}, \mathcal{H}_{2}:[\zeta_{0}, \mathcal{T}]\times[\zeta_{0}, \mathcal{T}]\rightarrow \mathbb{R} $ such that

    $ \begin{eqnarray*} \mathcal{H}_{1}^{*} = \sup\limits_{\zeta\in[\zeta_{0},\mathcal{T}]}\int\limits_{\zeta_{0}}^{\zeta}\vert\mathcal{H}_{1}(\zeta,s)\vert ds,\quad\quad\quad\mathcal{H}_{2}^{*} = \sup\limits_{\zeta\in[\zeta_{0},\mathcal{T}]}\int\limits_{\zeta_{0}}^{\mathcal{T}}\vert\mathcal{H}_{2}(\zeta,s)\vert ds. \end{eqnarray*} $

    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 $ \mathcal{F}:[\sigma_{1}, \sigma_{2}]\rightarrow \mathfrak{E} $ be a fuzzy-valued function on $ [\sigma_{1}, \sigma_{2}]. $ Then

    $ (i) $ $ \mathcal{F} $ is $ [(i)-\mathfrak{g}H] $-differentiable at $ c\in[\sigma_{1}, \sigma_{2}] $ iff $ \mathcal{F} $ is $ \, ^{GPF}[(i)-\mathfrak{g}H] $-differentiable at $ c. $

    $ (ii) $ $ \mathcal{F} $ is $ [(ii)-\mathfrak{g}H] $-differentiable at $ c\in[\sigma_{1}, \sigma_{2}] $ iff $ \mathcal{F} $ is $ \, ^{GPF}[(ii)-\mathfrak{g}H] $-differentiable at $ c. $

    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 $ \mathcal{F}:[\zeta_{0}, \mathcal{T}]\rightarrow \mathfrak{E} $ such that $ \mathcal{F}^{\prime}_{\mathfrak{g}H}\in\mathfrak{E}\cap\chi_{c}^{r}(\sigma_{1}, \sigma_{2}), $ then

    $ \begin{eqnarray} \mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\big(\,_{\mathfrak{g}H}\mathcal{D}_{\sigma_{1}^{+}}^{\vartheta,\beta}\mathcal{F}\big)(\zeta) = \mathcal{F}(\zeta)\ominus_{\mathfrak{g}H}\mathcal{F}(\zeta_{0}). \end{eqnarray} $ (4.3)

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

    $ \begin{eqnarray} \Phi(\zeta) = \Phi_{0}+\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu, \end{eqnarray} $ (4.4)

    if $ \Phi(\zeta) $ be $ \, ^{GPF}[(i)-\mathfrak{g}H] $-differentiable,

    $ \begin{eqnarray} \Phi(\zeta) = \Phi_{0}\ominus\frac{-1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu, \end{eqnarray} $ (4.5)

    if $ \Phi(\zeta) $ be $ \, ^{GPF}[(ii)-\mathfrak{g}H] $-differentiable, and

    $ \begin{eqnarray} \Phi(\zeta) = \begin{cases} \Phi_{0}+\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu,\quad\zeta\in[\sigma_{1},\sigma_{3}],\\ \Phi_{0}\ominus\frac{-1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu,\quad\zeta\in[\sigma_{3},\sigma_{2}], \end{cases} \end{eqnarray} $ (4.6)

    if there exists a point $ \sigma_{3}\in(\sigma_{1}, \sigma_{2}) $ such that $ \Phi(\zeta) $ is $ \, ^{GPF}[(i)-\mathfrak{g}H] $-differentiable on $ [\sigma_{1}, \sigma_{3}] $ and $ \, ^{GPF}[(ii)-\mathfrak{g}H] $-differentiable on $ [\sigma_{3}, \sigma_{2}] $ and $ \mathcal{F}(\sigma_{3}, \Phi(\sigma_{3}, \Phi(\sigma_{3}), \mathcal{H}_{1}\Phi(\sigma_{3}))\in\mathbb{R}. $

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

    $ \begin{eqnarray} \mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\big(\,_{\mathfrak{g}H}\mathcal{D}_{\sigma_{1}^{+}}^{\vartheta,\beta}\Phi(\zeta)\big) = \mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\big(\mathcal{F}(\zeta,\Phi(\zeta),\mathcal{H}_{1}\Phi(\zeta),\mathcal{H}_{2}\Phi(\zeta)\big). \end{eqnarray} $ (4.7)

    Utilizing Lemma 4.2 and Definition 2.6, we gat

    $ \begin{eqnarray} \Phi(\zeta)\ominus_{\mathfrak{g}H}\Phi_{0} = \frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu. \end{eqnarray} $ (4.8)

    In view of Defnition 2.17 and Theorem 4.1, if $ \Phi(\zeta) $ be $ \, ^{GPF}[(i)-\mathfrak{g}H] $-differentiable,

    $ \begin{eqnarray} \Phi(\zeta) = \Phi_{0}+\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu \end{eqnarray} $ (4.9)

    and if $ \Phi(\zeta) $ be $ \, ^{GPF}[(ii)-\mathfrak{g}H] $-differentiable

    $ \begin{eqnarray} \Phi(\zeta) = \Phi_{0}\ominus\frac{-1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu. \end{eqnarray} $ (4.10)

    In addition, when we have a switchpoint $ \sigma_{3}\in(\sigma_{1}, \sigma_{2}) $ of type $ (I) $ the $ \, ^{GPF}[\mathfrak{g}H] $-differentiability changes from type $ (I) $ to type $ (II) $ at $ \zeta = \sigma_{3}. $ Then by (4.9) and (4.10) and Definition 2.12, The proof is easy to comprehend.

    Also, we proceed with the following assumptions:

    $ ({\mathbb{A}_{1}}). $ $ \mathcal{F}:[\zeta_{0}, \mathcal{T}]\times\mathfrak{E}\times\mathfrak{E}\times\mathfrak{E}\rightarrow \mathfrak{E} $ is continuous and there exist positive real functions $ \mathcal{L}_{1}, \mathcal{L}_{2}, \mathcal{L}_{3} $ such that

    $ \begin{eqnarray*} &&\bar{\mathcal{D}_{0}}\Big(\mathcal{F}(\zeta,\Phi(\zeta),\mathcal{H}_{1}\Phi(\zeta),\mathcal{H}_{2}\Phi(\zeta)),\mathcal{F}(\zeta,\Psi(\zeta),\mathcal{H}_{1}\Psi(\zeta),\mathcal{H}_{2}\Psi(\zeta))\Big)\nonumber\\&&\leq\mathcal{L}_{1}(\zeta)\bar{\mathcal{D}_{0}}(\Phi,\Psi)+\mathcal{L}_{2}(\zeta)\bar{\mathcal{D}_{0}}(\mathcal{H}_{1}\Phi,\mathcal{H}_{1}\Psi)+\mathcal{L}_{3}(\zeta)\bar{\mathcal{D}_{0}}(\mathcal{H}_{2}\Phi,\mathcal{H}_{2}\Psi). \end{eqnarray*} $

    $ ({\mathbb{A}_{2}}). $ There exist a number $ \epsilon $ such that $ \delta\leq\epsilon < 1, \, \zeta\in[\zeta_{0}, \mathcal{T}] $

    $ \delta = \mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{P}(1+\mathcal{H}_{1}^{*}+\mathcal{H}_{2}^{*}) $

    and

    $ \mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{P} = \sup\limits_{\zeta\in[0,\mathcal{T}]}\big\{\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{L}_{1},\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{L}_{2},\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{L}_{3}\big\}. $

    Theorem 4.4. Let $ \mathcal{F}:[\zeta_{0}, \mathcal{T}]\times\mathfrak{E}\times\mathfrak{E}\times\mathfrak{E}\rightarrow \mathfrak{E} $ be a bounded continuous functions and holds $ (\mathbb{A}_{1}). $ Then the IVP (4.1) has a unique solution which is $ \, ^{GPF}[(i)-\mathfrak{g}H] $-differentiable on $ [\zeta_{0}, \mathcal{T}], $ given that $ \delta < 1, $ where $ \delta $ is given in $ (\mathbb{A}_{2}). $

    Proof. Assuming $ \Phi(\zeta) $ is $ \, ^{GPF}[(i)-\mathfrak{g}H] $-differentiability and $ \Phi_{0}\in\mathfrak{E} $ be fixed. Propose a mapping $ \mathfrak{F}:\mathcal{C}([\zeta_{0}, \mathcal{T}], \mathfrak{E})\rightarrow \mathcal{C}([\zeta_{0}, \mathcal{T}], \mathfrak{E}) $ by

    $ \begin{eqnarray} \big(\mathfrak{F}\Phi\big)(\zeta) = \Phi_{0}+\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu,\quad for\,all\,\zeta\in[\zeta_{0},\mathcal{T}]. \end{eqnarray} $ (4.11)

    Next we prove that $ \mathfrak{F} $ is contraction. For $ \Phi, \Psi\in\mathcal{C}([\zeta_{0}, \mathcal{T}], \mathfrak{E}) $ by considering of $ (\mathbb{A}_{1}) $ and by distance properties (2.3), one has

    $ \begin{eqnarray} &&\bar{\mathcal{D}_{0}}\big(\mathfrak{F}\Phi(\zeta),\mathfrak{F}\Psi(\zeta)\big)\\&&\leq\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert \bar{\mathcal{D}_{0}}\big(\mathcal{F}(\zeta,\Phi(\zeta),\mathcal{H}_{1}\Phi(\zeta),\mathcal{H}_{2}\Phi(\zeta)),\mathcal{F}(\zeta,\Psi(\zeta),\mathcal{H}_{1}\Psi(\zeta),\mathcal{H}_{2}\Psi(\zeta))\Big)d\nu\\&&\leq\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\big[\mathcal{L}_{1}\bar{\mathcal{D}_{0}}(\Phi,\Psi)+\mathcal{L}_{2}\bar{\mathcal{D}_{0}}(\mathcal{H}_{1}\Phi,\mathcal{H}_{1}\Psi)+\mathcal{L}_{3}\bar{\mathcal{D}_{0}}(\mathcal{H}_{2}\Phi,\mathcal{H}_{2}\Psi) \big]d\nu\\&&\leq\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{1}\bar{\mathcal{D}_{0}}(\Phi,\Psi)d\nu+\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{2}\bar{\mathcal{D}_{0}}(\mathcal{H}_{1}\Phi,\mathcal{H}_{1}\Psi)d\nu\\&&\quad+\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{3}\bar{\mathcal{D}_{0}}(\mathcal{H}_{2}\Phi,\mathcal{H}_{2}\Psi)d\nu. \end{eqnarray} $ (4.12)

    Now, we find that

    $ \begin{eqnarray} &&\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{2}\bar{\mathcal{D}_{0}}(\mathcal{H}_{1}\Phi,\mathcal{H}_{1}\Psi)d\nu\\&&\leq\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}\Big(\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{2}\bar{\mathcal{D}_{0}}(\Phi,\Psi)\int\limits_{\zeta_{0}}^{\nu}\vert\mathcal{H}_{1}(\nu,x)\vert dx \Big)d\nu\\&&\leq\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{L}_{2}\mathcal{H}_{1}^{*}.\bar{\mathcal{D}_{0}}(\Phi,\Psi). \end{eqnarray} $ (4.13)

    Analogously,

    $ \begin{eqnarray} &&\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{3}\bar{\mathcal{D}_{0}}(\mathcal{H}_{2}\Phi,\mathcal{H}_{2}\Psi)d\nu\leq\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{L}_{3}\mathcal{H}_{1}^{*}.\bar{\mathcal{D}_{0}}(\Phi,\Psi),\\ &&\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{1}\bar{\mathcal{D}_{0}}(\Phi,\Psi)d\nu = \mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{L}_{1}\bar{\mathcal{D}_{0}}(\Phi,\Psi). \end{eqnarray} $ (4.14)

    Then we have

    $ \begin{eqnarray} \bar{\mathcal{D}_{0}}\big(\mathfrak{F}\Phi,\mathfrak{F}\Psi\big)&&\leq\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{L}_{1}\bar{\mathcal{D}_{0}}(\Phi,\Psi)+\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{L}_{2}\mathcal{H}_{1}^{*}.\bar{\mathcal{D}_{0}}(\Phi,\Psi)+\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{L}_{3}\mathcal{H}_{2}^{*}.\bar{\mathcal{D}_{0}}(\Phi,\Psi)\\&&\leq\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{P}(1+\mathcal{H}_{1}^{*}+\mathcal{H}_{2}^{*})\bar{\mathcal{D}_{0}}(\Phi,\Psi)\\&& < \bar{\mathcal{D}_{0}}(\Phi,\Psi). \end{eqnarray} $ (4.15)

    Consequently, $ \mathfrak{F} $ is a contraction mapping on $ \mathcal{C}([\zeta_{0}, \mathcal{T}], \mathfrak{E}) $ having a fixed point $ \mathfrak{F}\Phi(\zeta) = \Phi(\zeta). $ Henceforth, the IVP (4.1) has unique solution.

    Theorem 4.5. For $ \beta\in(0, 1] $ and let $ \mathcal{F}:[\zeta_{0}, \mathcal{T}]\times\mathfrak{E}\times\mathfrak{E}\times\mathfrak{E}\rightarrow \mathfrak{E} $ be a bounded continuous functions and satisfies $ (\mathbb{A}_{1}). $ Let the sequence $ \Phi_{n}:[\zeta_{0}, \mathcal{T}]\rightarrow \mathfrak{E} $ is given by

    $ \begin{eqnarray} \Phi_{n+1}(\zeta)&& = \Phi_{0}\ominus\frac{-1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\zeta_{0}}^{\zeta}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi_{n}(\nu),\mathcal{H}_{1}\Phi_{n}(\nu),\mathcal{H}_{2}\Phi_{n}(\nu)\big)d\nu,\quad\\ \Phi_{0}(\zeta)&& = \Phi_{0}, \end{eqnarray} $ (4.16)

    is described for any $ n\in\mathbb{N}. $ Then the sequence $ \{\Phi_{n}\} $ converges to fixed point of problem (4.1) which is $ \, ^{GPF}[(ii)-\mathfrak{g}H] $-differentiable on $ [\zeta_{0}, \mathcal{T}], $ given that $ \delta < 1, $ where $ \delta $ is defined in $ (\mathbb{A}_{2}). $

    Proof. We now prove that the sequence $ \{\Phi_{n}\} $, given in (4.16), is a Cauchy sequence in $ \mathcal{C}([\zeta_{0}, \mathcal{T}], \mathfrak{E}). $ To do just that, we'll require

    $ \begin{eqnarray} \bar{\mathcal{D}_{0}}(\Phi_{1},\Phi_{0})&& = \bar{\mathcal{D}_{0}}\bigg(\Phi_{0}\ominus\frac{-1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi_{0}(\nu),\mathcal{H}_{1}\Phi_{0}(\nu),\mathcal{H}_{2}\Phi_{0}(\nu)\big)d\nu,\Phi_{0}\bigg)\\&&\leq\frac{1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\bar{\mathcal{D}_{0}}\Big(\mathcal{F}\big(\nu,\Phi_{0}(\nu),\mathcal{H}_{1}\Phi_{0}(\nu),\mathcal{H}_{2}\Phi_{0}(\nu)\big), \hat{0}\Big)d\nu\\&&\leq\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{M}, \end{eqnarray} $ (4.17)

    where $ \mathcal{M} = \sup_{\zeta\in[\zeta_{0}, \mathcal{T}]}\bar{\mathcal{D}_{0}}\big(\mathcal{F}(\zeta, \Phi, \mathcal{H}_{1}\Phi, \mathcal{H}_{2}\Phi), \hat{0}\big). $

    Since $ \mathcal{F} $ is Lipschitz continuous, In view of Definition (2.3), we show that

    $ \begin{eqnarray} &&\bar{\mathcal{D}_{0}}(\Phi_{n+1},\Phi_{n})\\&&\leq\frac{1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert \bar{\mathcal{D}_{0}}\big(\mathcal{F}\big(\nu,\Phi_{n}(\nu),\mathcal{H}_{1}\Phi_{n}(\nu),\mathcal{H}_{2}\Phi_{n}(\nu)\big),\mathcal{F}\big(\nu,\Phi_{n-1}(\nu),\mathcal{H}_{1}\Phi_{n-1}(\nu),\mathcal{H}_{2}\Phi_{n-1}(\nu)\big)\Big)d\nu\\&&\leq\frac{1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{1}.\bar{\mathcal{D}_{0}}\big(\Phi_{n},\Phi_{n-1}\big)d\nu\\&&\quad+\frac{1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{2}.\bar{\mathcal{D}_{0}}\big(\mathcal{H}_{1}\Phi_{n},\mathcal{H}_{1}\Phi_{n-1}\big)d\nu\\&&\quad+\frac{1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{3}.\bar{\mathcal{D}_{0}}\big(\mathcal{H}_{2}\Phi_{n},\mathcal{H}_{2}\Phi_{n-1}\big)d\nu\\&&\leq\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{P}(1+\mathcal{H}_{1}^{*}+\mathcal{H}_{2}^{*})\bar{\mathcal{D}_{0}}(\Phi_{n},\Phi_{n-1})\leq\delta \bar{\mathcal{D}_{0}}(\Phi_{n},\Phi_{n-1})\leq\delta^{n}\bar{\mathcal{D}_{0}}(\Phi_{1},\Phi_{0})\leq\delta^{n}\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{M}. \end{eqnarray} $ (4.18)

    Since $ \delta < 1 $ promises that the sequence $ \{\Phi_{n}\} $ is a Cauchy sequence in $ \mathcal{C}([\zeta_{0}, \mathcal{T}], \mathfrak{E}). $ Consequently, there exist $ \Phi\in\mathcal{C}([\zeta_{0}, \mathcal{T}], \mathfrak{E}) $ such that $ \{\Phi_{n}\} $ converges to $ \Phi. $ Thus, we need to illustrate that $ \Phi $ is a solution of the problem (4.1).

    $ \begin{eqnarray} &&\bar{\mathcal{D}}_{0}\bigg(\Phi(\zeta)+\frac{-1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu, \Phi_{0}\bigg)\\&& = \bar{\mathcal{D}}_{0}\bigg(\Phi(\zeta)+\frac{-1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu, \Phi_{n+1}\\&&\quad+\frac{-1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi_{n}(\nu),\mathcal{H}_{1}\Phi_{n}(\nu),\mathcal{H}_{2}\Phi_{n}(\nu)\big)d\nu\bigg)\\&&\leq \bar{\mathcal{D}_{0}}\big(\Phi(\zeta),\Phi_{n+1}\big)+\frac{1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{1}.\bar{\mathcal{D}_{0}}\big(\Phi(\nu),\Phi_{n}\big)d\nu\\&&\quad+\frac{1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{2}.\bar{\mathcal{D}_{0}}\big(\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{1}\Phi_{n}\big)d\nu\\&&\quad+\frac{1}{\beta^{\vartheta}\Gamma(\vartheta)}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert\mathcal{L}_{3}.\bar{\mathcal{D}_{0}}\big(\mathcal{H}_{2}\Phi(\nu),\mathcal{H}_{2}\Phi_{n}\big)d\nu\\&&\leq \bar{\mathcal{D}_{0}}\big(\Phi(\zeta),\Phi_{n+1}\big)+\mathcal{I}_{\zeta_{0}}^{\vartheta,\beta}\mathcal{P}(1+\mathcal{H}_{1}^{*}+\mathcal{H}_{2}^{*})\bar{\mathcal{D}_{0}}(\Phi(\zeta),\Phi_{n}). \end{eqnarray} $ (4.19)

    In the limiting case, when $ n\rightarrow \infty. $ Thus we have

    $ \begin{eqnarray} \Phi(\zeta)+\frac{-1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}e^{\frac{\beta-1}{\beta}(\zeta-\nu)}(\zeta-\nu)^{\vartheta-1}\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu)\big)d\nu = \Phi_{0}. \end{eqnarray} $ (4.20)

    By Lemma 4.3, we prove that $ \Phi $ is a solution of the problem (4.1). In order to prove the uniqness of $ \Phi(\zeta), $ let $ \Psi(\zeta) $ be another solution of problem (4.1) on $ [\zeta_{0}, \mathcal{T}]. $ Utilizing Lemma 4.3, gets

    $ \begin{eqnarray*} \bar{\mathcal{D}_{0}}(\Phi,\Psi)\leq\frac{1}{\beta^{\vartheta}\Gamma(\mathfrak{q})}\int\limits_{\zeta_{0}}^{\zeta}\big\vert e^{\frac{\beta-1}{\beta}(\zeta-\nu)}\big\vert\big\vert(\zeta-\nu)^{\vartheta-1}\big\vert \bar{\mathcal{D}}_{0}\bigg(\mathcal{F}\big(\nu,\Phi(\nu),\mathcal{H}_{1}\Phi(\nu),\mathcal{H}_{2}\Phi(\nu),\mathcal{F}\big(\nu,\Psi(\nu),\mathcal{H}_{1}\Psi(\nu),\mathcal{H}_{2}\Psi(\nu)\big)\bigg)d\nu. \end{eqnarray*} $

    Analogously, by employing the distance properties $ \bar{\mathcal{D}}_{0} $ and Lipschitiz continuity of $ \mathcal{F}, $ consequently, we deduce that $ (1-\delta)\bar{\mathcal{D}_{0}}(\Phi, \Psi)\leq0, $ since $ \delta < 1, $ we have $ \Phi(\zeta) = \Psi(\zeta) $ for all $ \zeta\in[\zeta_{0}, \mathcal{T}]. $ Hence, the proof is completed.

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

    $ \begin{eqnarray} \mathcal{D}_{z}^{\vartheta,\beta}\Phi(z) = \mathcal{F}(z,\Phi(z)), \end{eqnarray} $ (4.21)

    where $ \mathcal{F}(z, \Phi(z)) $ is analytic in $ \Phi $ and $ \Phi(z) $ is analytic in the unit disk. Therefore, $ \mathcal{F} $ can be written as

    $ \begin{eqnarray*} \mathcal{F}(z,\Phi) = \varphi \Phi(z). \end{eqnarray*} $

    Consider $ \mathcal{Z} = z^{\vartheta}. $ Then the solution can be formulated as follows:

    $ \begin{eqnarray} \Phi(\mathcal{Z}) = \sum\limits_{j = 0}^{\infty}\Phi_{j}\mathcal{Z}^{j}, \end{eqnarray} $ (4.22)

    where $ \Phi_{j} $ are constants. Putting (4.22) in (4.21), yields

    $ \begin{eqnarray*} \frac{\partial}{\partial z}\sum\limits_{j = 0}^{\infty}\Upsilon_{\vartheta,\beta,j}\Phi_{j}\mathcal{Z}^{j}-\varphi\sum\limits_{j = 0}^{\infty}\Phi_{j}\mathcal{Z}^{j} = 0. \end{eqnarray*} $

    Since

    $ \begin{eqnarray*} \Upsilon_{\vartheta,\beta,j} = \frac{\beta^{\vartheta}\Gamma\big(\frac{j\vartheta}{\beta}+1\big)}{j\Gamma\big(\frac{j\vartheta}{\beta}+1-\vartheta\big)}, \end{eqnarray*} $

    then the simple computations gives the expression

    $ \begin{eqnarray*} \frac{\beta^{\vartheta}\Gamma\big(\frac{j\vartheta}{\beta}+1\big)}{\Gamma\big(\frac{j\vartheta}{\beta}+1-\vartheta\big)}\Phi_{j}-\varphi\Phi_{j-1} = 0. \end{eqnarray*} $

    Consequently, we get

    $ \begin{eqnarray*} \Phi_{j} = \Big(\frac{\varphi}{\beta^{\vartheta}}\Big)^{j}\frac{\Gamma\big(\frac{(j-1) {\vartheta}}{\beta}+1-\vartheta\big)\Gamma\big(\frac{j\vartheta}{\beta}+1-\vartheta\big)}{\Gamma\big(\frac{(j-1)\vartheta}{\beta}+1\big)\Gamma\big(\frac{j\vartheta}{\beta}+1\big)}. \end{eqnarray*} $

    Therefore, we have the subsequent solution

    $ \begin{eqnarray*} \Phi(\mathcal{Z}) = \sum\limits_{j = 0}^{\infty}\Big(\frac{\varphi}{\beta^{\vartheta}}\Big)^{j}\frac{\Gamma\big(\frac{(j-1) {\vartheta}}{\beta}+1-\vartheta\big)\Gamma\big(\frac{j\vartheta}{\beta}+1-\vartheta\big)}{\Gamma\big(\frac{(j-1)\vartheta}{\beta}+1\big)\Gamma\big(\frac{j\vartheta}{\beta}+1\big)}\mathcal{Z}^{j}, \end{eqnarray*} $

    or equivalently

    $ \begin{eqnarray*} \Phi(\mathcal{Z}) = \sum\limits_{j = 0}^{\infty}\Big(\frac{\varphi}{\beta^{\vartheta}}\Big)^{j}\frac{\Gamma(j+1)\Gamma\big(\frac{(j-1) {\vartheta}}{\beta}+1-\vartheta\big)\Gamma\big(\frac{j\vartheta}{\beta}+1-\vartheta\big)}{\Gamma\big(\frac{(j-1)\vartheta}{\beta}+1\big)\Gamma\big(\frac{j\vartheta}{\beta}+1\big)}\frac{\mathcal{Z}^{j}}{j!}, \end{eqnarray*} $

    where $ \varphi $ is assumed to be arbitrary constant, we take

    $ \varphi: = \beta^{\vartheta}. $

    Therefore, for appropriate $ \vartheta, $ we have

    $ \begin{eqnarray*} \Phi(\mathcal{Z})&& = \sum\limits_{j = 0}^{\infty}\Big(\frac{\varphi}{\beta^{\vartheta}}\Big)^{j}\frac{\Gamma(j+1)\Gamma\big(\frac{(j-1) {\vartheta}}{\beta}+1-\vartheta\big)\Gamma\big(\frac{j\vartheta}{\beta}+1-\vartheta\big)}{\Gamma\big(\frac{(j-1)\vartheta}{\beta}+1\big)\Gamma\big(\frac{j\vartheta}{\beta}+1\big)}\frac{\mathcal{Z}^{j}}{j!}\nonumber\\&& = \,_{3}\Psi_{2}\begin{bmatrix} (1,1),\Big(1-\vartheta-\frac{\vartheta}{\beta},\frac{\vartheta}{\beta}\Big),\Big(1-\vartheta,\frac{\vartheta}{\beta}\Big);\\\qquad\qquad\qquad\quad\quad\quad\quad\quad\qquad\qquad\qquad\qquad\mathcal{Z}\\\Big(1-\frac{\vartheta}{\beta},\frac{\vartheta}{\beta},\Big),\Big(1,\frac{\vartheta}{\beta}\Big); \end{bmatrix}\nonumber\\&& = \,_{3}\Psi_{2}\begin{bmatrix} (1,1),\Big(1-\vartheta-\frac{\vartheta}{\beta},\frac{\vartheta}{\beta}\Big),\Big(1-\vartheta,\frac{\vartheta}{\beta}\Big);\\\qquad\qquad\qquad\quad\quad\quad\quad\quad\qquad\qquad\qquad\qquad z^{\vartheta\beta}\\\Big(1-\frac{\vartheta}{\beta},\frac{\vartheta}{\beta},\Big),\Big(1,\frac{\vartheta}{\beta}\Big); \end{bmatrix}, \end{eqnarray*} $

    where $ \vert z\vert < 1. $

    The present investigation deal with an IVP for $ \mathcal{GPF} $ fuzzy $ FDEs $ 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 $ \mathcal{GPF} $ fuzzy $ FDEs $ by utilizing the solutions of integer order differential equations is considered. Additionally, the existence consequences for $ \mathcal{FVFIDE}s $ under $ \mathcal{GPF} $-$ \mathcal{HD} $ 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 $ \mathcal{FVFIDE}s $ whichis utilized to examine the convergence of these arrangements of conditions. Two examples enlightened the efficacy and preciseness of the fractional-order $ \mathcal{HD} $ 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 $ FDEs. $

    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.

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