Review Special Issues

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

  • Received: 29 November 2015 Accepted: 23 December 2015 Published: 30 December 2015
  • 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.



    Fractional differential equations (FDEs) appeared as an excellent mathematical tool for, modeling of many physical phenomena appearing in various branches of science and engineering, such as viscoelasticity, statistical mechanics, dynamics of particles, etc. Fractional calculus is a recently developing work in mathematics which studies derivatives and integrals of functions of fractional order [26].

    The most used fractional derivatives are the Riemann-Liouville (RL) and Caputo derivatives. These derivatives contain a non-singular derivatives but still conserves the most important peculiarity of the fractional operators [1,2,10,11,23,24]. Atangana and Baleanu described a derivative with a generalized Mittag-leffler (ML) function. This derivative is often called the Atangana-Baleanu (AB) fractional derivative. The AB-derivative in the senses of Riemman-Liouville and Caputo are denoted by ABR-derivative and ABC-derivative, respectively.

    The AB fractional derivative is a nonlocal fractional derivative with nonsingular kernel which is connected with various applications [3,5,6,8,9,13,14,15,16]. Using the advantage of the non-singular ML kernal present in the AB fractional derivatives, operators, many authors from various branches of applied mathematics have developed and studied mathematical models involving AB fractional derivatives [18,22,29,30,31,32,35,36,37].

    Mohamed et al. [25] considered a system of multi-derivatives for Caputo FDEs with an initial value problem, examined the existence and uniqueness results and obtained numerical results. Sutar et al. [32,33] considered multi-derivative FDEs involving the ABR derivative and examined existence, uniqueness and dependence results. Kucche et al. [12,19,20,21,34] enlarged the work of multi-derivative fractional differential equations involving the Caputo fractional derivative and studied the existence, uniqueness and continuous dependence of the solution.

    Inspired by the preceding work, we perceive the multi-derivative nonlinear neutral fractional integro-differential equation with AB fractional derivative of the Riemann-Liouville sense of the problem:

    $ dVdȷ+0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=φ(ȷ,V(ȷ),ȷ0K(ȷ,θ,V(θ))dθ,T0χ(ȷ,θ,V(θ))dθ),ȷI
    $
    (1.1)
    $ \mathcal{V}\left(0\right) = \mathcal{V}_{0} \in \mathscr{R}, $ (1.2)

    where $ ^{\star}_{0} D^{\delta}_{\jmath} $ denotes the ABR fractional derivative of order $ \delta\in(0, 1) $, and $ \varphi \in \mathscr{C}(\mathscr{I} \times \mathscr{R} \times \mathscr{R} \times \mathscr{R}, \mathscr{R}) $ is a non-linear function. Let $ \mathcal{P}_{1}\mathcal{V}(\jmath) = \int^{\jmath}_{0}\mathcal{K}(\jmath, \theta, \mathcal{V}(\theta))d\theta $ and $ \mathcal{P}_{2}\mathcal{V}(\jmath) = \int^{T}_{0}\chi(\jmath, \theta, \mathcal{V}(\theta))d\theta $. Now, (1.1) becomes,

    $ dVdȷ+0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)),ȷI,
    $
    (1.3)
    $ V(0)=V0R.
    $
    (1.4)

    In this work, we derive a few supplemental results using the characteristics of the fractional integral operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $. The existence results are obtained by Krasnoselskii's fixed point theorem and the uniqueness and data dependence results are obtained by the Gronwall-Bellman inequality.

    Definition 2.1. [14] The Sobolev space $ H^{\mathfrak{q}}(X) $ is defined as $ H^{\mathfrak{q}}\left(X\right) = \left\{\varphi\in L^{2}\left(X\right):D^{\beta}\varphi\in L^{2}(X), \forall \left|\beta\right|\leq \mathfrak{q}\right\}. $ Let $ \mathfrak{q}\in[1, \infty) $ and $ X $ be open, $ X\subset\mathbb{R} $.

    Definition 2.2. [11,17] The generalized ML function $ E^{\alpha}_{\delta, \beta}\left(u\right) $ for complex $ \delta, \beta, \alpha $ with Re$ (\delta) > 0 $ is defined by

    $ Eαδ,β(u)=t=0(α)tα(δt+β)utt!,
    $

    and the Pochhammer symbol is $ (\alpha)_{t} $, where $ (\alpha)_{0} = 1, (\alpha)_{t} = \alpha(\alpha+1)...(\alpha+t-1), $ $ t = 1, 2...., $ and $ E^{1}_{\delta, \beta}\left(u\right) = E_{\delta, \beta}\left(u\right), E^{1}_{\delta, 1}\left(u\right) = E_{\delta}\left(u\right). $

    Definition 2.3. [4] The ABR fractional derivative of $ \mathcal{V} $ of order $ \delta $ is

    $ 0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=B(δ)1δddȷȷ0Eδ[δ1δ(ȷθ)δ]V(θ)dθ,
    $

    where $ \mathcal{V} \in H^{1}(0, 1) $, $ \delta \in(0, 1) $, $ B(\delta) > 0 $. Here, $ E_{\delta} $ is a one parameter ML function, which shows $ B(0) = B(1) = 1 $.

    Definition 2.4. [4] The ABC fractional derivative of $ \mathcal{V} $ of order $ \delta $ is

    $ 0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ]V(θ)dθ,
    $

    where $ \mathcal{V} \in H^{1}(0, 1) $, $ \delta \in(0, 1) $, and $ B(\delta) > 0 $. Here, $ E_{\delta} $ is a one parameter ML function, which shows $ B(0) = B(1) = 1 $.

    Lemma 2.5. [4] If $ L\left\{g(\jmath); b\right\} = \bar{G}(b) $, then $ L\left\{^{\star}_{0}D^{\delta}_{\jmath}g(\jmath); b\right\} = \frac{B(\delta)}{1-\delta}\frac{b^{\delta}\bar{G}(b)}{b^{\delta}+\frac{\delta}{1-\delta}}. $

    Lemma 2.6. [26] $ L\left[\jmath^{m\delta+\beta-1}E^{(m)}_{\delta, \beta}\left(\pm a\jmath^{\delta}\right); b\right] = \frac{m!b^{\delta-\beta}}{\left(b^{\delta}\pm a\right)^{m+1}}, E^{m}(\jmath) = \frac{d^{m}}{d\jmath^{m}}E(\jmath). $

    Definition 2.7. [17,27] The operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $ on class $ L(m, n) $ is

    $ (εαδ,η,V;c+)[V(ȷ)x(ȷ,y(ȷ))]=t0(ȷθ)α1Eαδ,η[V(ȷθ)δ]Θ(θ)dθ,ȷ[c,d],
    $

    where $ \delta, \eta, \mathcal{V}, \alpha\in \mathbb{C}\left(Re(\delta), Re(\eta) > 0\right) $, and $ n > m $.

    Lemma 2.8. [17,27] The operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $ is bounded on $ C[m, n] $, such that $ \left\|\left(\varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+}\right)[\mathcal{V}(\jmath)-x(\jmath, y(\jmath))]\right\| \leq \mathcal{P} \left\|\Theta\right\|, $ where

    $ P=(nm)Re(η)t=0|(α)t||α(δt+η)|[Re(δ)t+Re(η)]|V(nm)Re(δ)|tt!.
    $

    Here, $ \delta, \eta, \mathcal{V}, \alpha\in \mathbb{C}\left(Re(\delta), Re(\eta) > 0\right) $, and $ n > m $.

    Lemma 2.9. [17,27] The operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $ is invertible in the space $\textrm{L}(m, n) $ and $ \varphi \in \textrm{L}(m, n) $ its left inversion is given by

    $ ([εαδ,η,V;c+]1)[V(ȷ)x(ȷ,y(ȷ))]=(Dη+ςc+εαδ,η,V;c+)[V(ȷ)x(ȷ,y(ȷ))],ȷ(m,n],
    $

    where $ \delta, \eta, \mathcal{V}, \alpha\in \mathbb{C}\left(Re(\delta), Re(\eta) > 0\right) $, and $ n > m $.

    Lemma 2.10. [17,27] Let $ \delta, \eta, \mathcal{V}, \alpha\in \mathbb{C}\left(Re(\delta), Re(\eta) > 0\right), n > m $ and suppose that the integral equation is

    $ ȷ0(ȷθ)α1Eαδ,η[V(ȷθ)δ]Θ(θ)dθ=φ(ȷ),ȷ(m,n],
    $

    is solvable in the space $ \textrm{L}(m, n) $.Then, its unique solution $ \Theta(\jmath) $ is given by

    $ Θ(ȷ)=(Dη+ςc+εαδ,η,V;c+)[V(ȷ)x(ȷ,y(ȷ))],ȷ(m,n].
    $

    Lemma 2.11. [7] (Krasnoselskii's fixed point theorem) Let $ A $ be a Banach space and $ X $ be bounded, closed, convex subset of $ A $. Let $ \mathscr{F}_{1}, \mathscr{F}_{2} $ be maps of S into $ A $ such that $ \mathscr{F}_{1}\mathcal{V}+\mathscr{F}_{2}\varphi \in X $ $ \forall $ $ \mathcal{V}, \varphi \in U $. The equation $ \mathscr{F}_{1}\mathcal{V}+\mathscr{F}_{2}\mathcal{V} = \mathcal{V} $ has a solution on S, and $ \mathscr{F}_{1} $, $ \mathscr{F}_{2} $ is a contraction and completely continuous.

    Lemma 2.12. [28] (Gronwall-Bellman inequality) Let $ \mathcal{V} $ and $ \varphi $ be continuous and non-negative functions defined on $ \mathscr{I} $. Let $ \mathcal{V}(\jmath)\leq \mathcal{A}+\int^{\jmath}_{a}\varphi(\theta)\mathcal{V}(\theta)d\theta, \jmath \in \mathscr{I} $; here, $ \mathcal{A} $ is a non-negative constant.

    $ V(ȷ)Aexp(ȷaφ(θ)dθ),ȷI.
    $

    In this part, we need some fixed-point-techniques-based hypotheses for the results:

    $ ({\rm{H1}}) $ Let $ \mathcal{V} \in C\left[0, T\right] $, function $ \varphi \in \left(C[0, T]\times \mathscr{R} \times \mathscr{R} \times \mathscr{R}, \mathscr{R} \right) $ is a continuous function, and there exist $ +^{ve} $ constants $ \zeta _{1}, \zeta_{2} $ and $ \zeta $. $ \left\|\varphi(\jmath, \mathcal{V}_{1}, \mathcal{V}_{2}, \mathcal{V}_{3})-\varphi(\jmath, \varphi_{1}, \varphi_{2}, \varphi_{3})\right\|\leq \zeta_{1}\left(\left\|\mathcal{V}_{1}-\varphi_{1}\right\|+\left\|\mathcal{V}_{2}-\varphi_{2}\right\|+\left\|\mathcal{V}_{3}-\varphi_{3}\right\|\right) $ for all $ \mathcal{V}_{1}, \mathcal{V}_{2}, \mathcal{V}_{3}, \varphi_{1}, \varphi_{2}, \varphi_{3} $ in $ Y $, $ \zeta _{2} = max_{\mathcal{V} \in \mathscr{R}}\left\|f(\jmath, 0, 0, 0)\right\| $, and $ \zeta = max\left\{ \zeta _{1}, \zeta _{2}\right\} $.

    $ ({\rm{H2}}) $ $ \mathcal{P}_{1} $ is a continuous function, and there exist $ +^{ve} $ constants $ \mathscr{C}_{1}, \mathscr{C}_{2} $ and $ \mathscr{C} $. $ \left\|\mathcal{P}_{1}(\jmath, \theta, \mathcal{V}_{1})-\mathcal{P}_{1}(\jmath, \theta, \varphi_{1})\right\| \leq \mathscr{C}_{1} \left(\left\|\mathcal{V}_{1}-\varphi_{1}\right\|\right) \forall \, \mathcal{V}_{1}, \varphi_{1} $ in $ Y $, $ \mathscr{C}_{2} = max_{(\jmath, \theta) \in D}\left\|\mathcal{P}_{1}(\jmath, \theta, 0)\right\| $, and $ \mathscr{C} = max\left\{ \mathscr{C} _{1}, \mathscr{C} _{2} \right\} $.

    $ ({\rm{H3}}) $ $ \mathcal{P}_{2} $ is a continuous function and there are $ +^{ve} $ constants $ \mathcal{D}_{1}, \mathcal{D}_{2} $ and $ \mathcal{D} $. $ \left\|\mathcal{P}_{2}(\jmath, \theta, \mathcal{V}_{1})-\mathcal{P}_{2}(\jmath, \theta, \varphi_{1})\right\| \leq \mathcal{D}_{1} \left(\left\|\mathcal{V}_{1}-\varphi_{1}\right\|\right) $ for all $ \mathcal{V}_{1}, \varphi_{1} $ in $ Y $, $ \mathcal{D}_{2} = max_{(\jmath, \theta) \in D}\left\|\mathcal{P}_{2}(\jmath, \theta, 0)\right\| $ and $ \mathcal{D} = max\left\{ \mathcal{D} _{1}, \mathcal{D} _{2} \right\} $.

    $ ({\rm{H4}}) $ Let $ x \in c[0, I] $, function $ u \in (c[0, I] \times \mathscr{R}, \mathscr{R}) $ is a continuous function, and there is a $ +^{ve} $ constant $ k > 0 $, such that $ \left\|u(\jmath, x)-u(\jmath, y)\right\|\leq k \left\| x-y \right\| $. Let $ Y = C[\mathscr{R}, X] $ be the set of continuous functions on $ \mathscr{R} $ with values in the Banach space $ X $.

    Lemma 2.13. If $ {\bf{(H_2)}} $ and $ {\bf{(H_3)}} $ are satisfied the following estimates, $ \left\|\mathcal{P}_{1}\mathcal{V}(\jmath)\right\|\leq \jmath(\mathscr{C}_{1}\left\|\mathcal{V}\right\|+\mathscr{C} _{2}), \left\|\mathcal{P}_{1}\mathcal{V}(\jmath)-\mathcal{P}_{1}\varphi(\jmath)\right\|\leq \mathscr{C}\jmath\left\|\mathcal{V}-\varphi\right\| $, and $ \left\|\mathcal{P}_{2}\mathcal{V}(\jmath)\right\|\leq \jmath(\mathcal{D}_{1}\left\|\mathcal{V}\right\|+\mathcal{D} _{2}), \left\|\mathcal{P}_{2}\mathcal{V}(\jmath)-\mathcal{P}_{2}\varphi(\jmath)\right\|\leq \mathcal{D}\jmath\left\|\mathcal{V}-\varphi\right\| $.

    Theorem 3.1. The function $ \varphi \in \mathscr{C}\left(\mathscr{I} \times \mathscr{R} \times \mathscr{R} \times \mathscr{R}, \mathscr{R}\right) $ and $ \mathcal{V}\in\mathscr{C}(\mathscr{I}) $ is a solution for the problem of Eqs (1.3) and (1.4), iff $ \mathcal{V} $ is a solution of the fractional equation

    $ V(ȷ)=V0B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ,ȷI.
    $
    (3.1)

    Proof. (1) By using Definition 2.3 and Eq (1.3), we get

    $ ddȷ(V(ȷ)+B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ)=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)).
    $

    Integrating both sides of the above equation with limits $ 0 $ to $ \jmath $, we get

    $ V(ȷ)+B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθV(0)=ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ,ȷI.
    $

    Conversely, with differentiation on both sides of Eq (3.1) with respect to $ \jmath $, we get

    $ dVdȷ+B(δ)1δddȷȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)),ȷI.
    $

    Using Definition 2.3, we get Eq (1.3) and substitute $ \jmath = 0 $ in Eq (3.1), we get Eq (1.4).

    Proof. (2) In Equation (1.3), taking the Laplace Transform on both sides, we get

    $ L[V(ȷ);b]+L[0Dδȷ;b][V(ȷ)x(ȷ,y(ȷ))]=L[φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ));b].
    $

    Now, using the Laplace Transform formula for the AB fractional derivative of the RL sense, as given in Lemma 2.5, we get

    $ bˉX(b)[V(ȷ)x(ȷ,y(ȷ))]V(0)+B(δ)1δbδˉX(b)bδ+δ1δ=ˉG(b),
    $

    $ \bar{X}(b) = \left[\mathcal{V}(\jmath); b\right] $ and $ \bar{G}(b) = L\left[\varphi\left(\jmath, \mathcal{V}\left(\jmath\right), P_{1}\mathcal{V}(\jmath), P_{2}\mathcal{V}(\jmath)\right); b\right]. $ Using Eq (1.4), we get

    $ ˉX(b)=V01bB(δ)1δbδ1ˉX(b)bδ+δ1δ[V(ȷ)x(ȷ,y(ȷ))]+1bˉG(b).
    $
    (3.2)

    In Eq (3.2) applying the inverse Laplace Transform on both sides using Lemma 2.6 and the convolution theorem, we get

    $ L1[ˉX(b);ȷ]=V0L1[1b;ȷ]B(δ)1δ(L1[bδ1bδ+δ1δ][V(ȷ)x(ȷ,y(ȷ))]L1[ˉX(b);ȷ])+L1[ˉG(b);ȷ]L1[1b;ȷ]=V0B(δ)1δ(Eδ[δ1δȷδ][V(ȷ)x(ȷ,y(ȷ))])+φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ))=V0B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ.V(ȷ)=V0B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ.
    $
    (3.3)

    Theorem 3.2. Let $ \delta\in(0, 1) $. Define the operator $ \mathscr{F} $ on $ \mathscr{C}(\mathscr{I}) $:

    $ (FV)(ȷ)=V0B(δ)1δ(ε1δ,1,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],VC(I).
    $
    (3.4)

    $(A)$ $ \mathscr{F} $ is a bounded linear operator on $ \mathscr{C}(\mathscr{I}) $.

    $(B)$ $ \mathscr{F} $ satisfying the hypotheses.

    $(C)$ $ \mathscr{F}(X) $ is equicontinuous, and $ X $ is a bounded subset of $ \mathscr{C}(\mathscr{I}) $.

    $(D)$ $ \mathscr{F} $ is invertible, function $ \varphi\in \mathscr{C}(\mathscr{I}) $, and the operator equation $ \mathscr{F}\mathcal{V} = \varphi $ has a unique solution in $ \mathscr{C}(\mathscr{I}) $.

    Proof. (A) From Definition 2.7 and Lemma 2.8, the fractional integral operator $ \varepsilon^{1}_{\delta, 1, \frac{-\delta}{1-\delta}; 0^{+}} $ is a bounded linear operator on $ \mathscr{C}(\mathscr{I}) $, such that

    $ ε1δ,1,δ1δ;0+[V(ȷ)x(ȷ,y(ȷ))]PV,ȷI,where
    $
    $ P=Tn=0(1)nα(δn+1)(δn+1)|δ1δTδ|nn!=Tn=0(δ1δ)nTδnα(δn+2)=TEδ,2(δ1δTδ),
    $

    and we have

    $ FV=|B(δ)1δ|ε1δ,1,δ1δ;0+[V(ȷ)x(ȷ,y(ȷ))]PB(δ)1δV,VC(I).
    $
    (3.5)

    Thus, $ \mathscr{F}\mathcal{V} = \varphi $ is a bounded linear operator on $ \mathscr{C}(\mathscr{I} $).

    (B) We consider $ \mathcal{V}, \varphi\in \mathscr{C}(\mathscr{I}) $. By using linear operator $ \mathscr{F} $ and bounded operator $ \varepsilon^{1}_{\delta, 1, \frac{-\delta}{1-\delta}; 0^{+}} $, for any $ \jmath \in \mathscr{I} $,

    $ |(FV)(ȷ)(Fφ)(ȷ)|=|F(Vφ)[V(ȷ)x(ȷ,y(ȷ))]|B(δ)1δ(ε1δ,1,δ1δ;0+Vφ)[V(ȷ)x(ȷ,y(ȷ))]PB(δ)1δVφ.
    $

    Where, $ P = TE_{\delta, 2}\left(\frac{\delta}{1-\delta}T^{\delta}\right) $, then the operator $ \mathscr{F} $ is satisfied the hypotheses with constant $ P\frac{B(\delta)}{1-\delta} $.

    (C) Let $ U = \left\{\mathcal{V}\in \mathscr{C}(\mathscr{I}) : \left\|\mathcal{V}\right\|\leq R\right\} $ be a bounded and closed subset of $ \mathscr{C}(\mathscr{I}) $, $ \mathcal{V}\in U $, and $ \jmath_{1}, \jmath_{2}\in \mathscr{I} $ with $ \jmath_{1}\leq \jmath_{2} $.

    $ |(FV)(ȷ1)(FV)(ȷ2)|=|B(δ)1δ(ε1δ,1,δ1δ;0+)[V(ȷ1)u(l1,x(l))]B(δ)1δ(ε1δ,1,δ1δ;0+)[V(ȷ2)u(l2,x(l))]|B(δ)1δ|ȷ10{Eδ[δ1δ(ȷ1θ)δ]Eδ[δ1δ(ȷ2θ)δ]}[V(ȷ)x(ȷ,y(ȷ))]dθ|+B(δ)1δ|ȷ2ȷ1Eδ[δ1δ(ȷ2θ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ|B(δ)1δn=0|(δ1δ)n|1α(nδ+1)ȷ10|(ȷ1θ)nδ(ȷ2θ)nδ||[V(ȷ)x(ȷ,y(ȷ))]|dθ+B(δ)1δn=0|(δ1δ)n|1α(nδ+1)ȷ2ȷ1|(ȷ2θ)nδ||[V(ȷ)x(ȷ,y(ȷ))]|dθLB(δ)1δn=0(δ1δ)n1α(nδ+1)ȷ10(ȷ2θ)nδ(ȷ1θ)nδdθ+LB(δ)1δn=0(δ1δ)n1α(nδ+1)ȷ2ȷ1(ȷ2θ)nδdθRB(δ)1δn=0(δ1δ)n1α(nδ+1){(ȷ2ȷ1)nδ+1+ȷnδ+12ȷnδ+11+(ȷ2ȷ1)nδ+1}RB(δ)1δn=0(δ1δ)n1α(nδ+2){ȷnδ+12ȷnδ+11}|(FV)(ȷ1)(FV)(ȷ2)|RB(δ)1δn=0(δ1δ)n1α(nδ+2){ȷnδ+12ȷnδ+11}.
    $
    (3.6)

    Hence, if $ \left|\jmath_{1}-\jmath_{2}\right|\rightarrow 0 $ then $ \left|(\mathscr{F}\mathcal{V})(\jmath_{1})-(\mathscr{F}\mathcal{V})(\jmath_{2})\right|\rightarrow 0. $

    $ \therefore $ $ (\mathscr{F}\mathcal{V}) $ is equicontinuous on $ \mathscr{I}. $

    (D) By Lemmas 2.9 and 2.10, $ \varphi\in \mathscr{C}(\mathscr{I}) $, and we get

    $ (ε1δ,1,δ1δ;0+)1[V(ȷ)x(ȷ,y(ȷ))]=(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],ȷ(m,n).
    $
    (3.7)

    By Eqs (3.4) and (3.5), we have

    $ (F1)[V(ȷ)x(ȷ,y(ȷ))]=(B(δ)1δε1δ,1,δ1δ;0+)1[V(ȷ)x(ȷ,y(ȷ))]=1δB(δ)(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],ȷ(m,n),
    $

    where $ \beta \in \mathbb{C} $ with $ Re(\beta) > 0 $. This shows $ \mathscr{F} $ is invertible on $ \mathscr{C}(\mathscr{I}) $ and

    $ (FV)[V(ȷ)x(ȷ,y(ȷ))]=[V(ȷ)x(ȷ,y(ȷ))],ȷI,
    $

    has the unique solution,

    $ V(ȷ)=(F1[V(ȷ)x(ȷ,y(ȷ))])=1δB(δ)(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ)),y(ȷ))],ȷ(m,n).
    $
    (3.8)

    Theorem 4.1. Let $ \varphi \in \mathscr{C}\left(\mathscr{I}\times \mathscr{R} \times \mathscr{R} \times \mathscr{R}, \mathscr{R} \right) $. Then, the ABR derivative $ ^{\star}_{0}D^{\delta}_{\jmath}[\mathcal{V}(\jmath)-x(\jmath, y(\jmath))] = \varphi\left(\jmath, \mathcal{V}\left(\jmath\right), \mathcal{P}_{1}\mathcal{V}(\jmath), \mathcal{P}_{2}\mathcal{V}(\jmath)\right), \jmath \in \mathscr{I} $, is solvable in $ \mathscr{C}(\mathscr{I} $), and the solution in $ \mathscr{C}(\mathscr{I}) $ is

    $ V(ȷ)=1δB(δ)(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],ȷI,
    $
    (4.1)

    where $ \beta\in \mathbb{C}, Re(\beta) > 0 $, and $ \hat{\varphi}(\jmath) = \int^{\jmath}_{0}\varphi\left(\theta, \mathcal{V}\left(\theta\right), \mathcal{P}_{1}\mathcal{V}(\theta), \mathcal{P}_{2}\mathcal{V}(\theta)\right)d\theta, \jmath \in \mathscr{I} $.

    Proof. The corresponding fractional equation of the ABR derivative

    $ 0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)),ȷI,
    $

    is given by

    $ B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ=ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ,ȷI.
    $

    Using operator $ \mathscr{F} $ of Eq (3.4), we get

    $ (FV)(s)=ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ=ˆφ(ȷ),ȷI.
    $
    (4.2)

    Equations (3.7) and (4.2) are solvable, and we get

    $ V(ȷ)=1δB(δ)(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],ȷI;βC,Re(β)>0.
    $
    (4.3)

    Theorem 4.2. Let $ \varphi \in \mathscr{C}\left(\mathscr{I}\times {R \times R \times R}, \mathscr{R}\right) $ satisfy $ {\bf{(H_1)}} $–$ {\bf{(H_3)}} $ with $ L = \sup_{\jmath\in \mathscr{I}}\omega(\jmath), $ where $ \omega(\jmath) = \zeta(1+\mathscr{C}\jmath+\mathcal{D}T) $, if $ L = \min\left\{1, \frac{1}{2T}\right\} $. Then problem of (1.3) and (1.4) has a solution in $ \mathscr{C}(\mathscr{I}) $ provided

    $ 2B(δ)TEδ,2(δ1δ)Tδ1δ1.
    $
    (4.4)

    Proof. Define

    $ R=V0+NφT1LTB(δ)TEδ,2(δ1δ)Tδ1δ,
    $

    where $ N_{\varphi} = \sup_{\jmath\in \mathscr{I}}\left\|\varphi(\jmath, 0, 0, 0)\right\|. $ Let $ U = \left\{\mathcal{V}\in \mathscr{C}(\mathscr{I}):\left\|\mathcal{V}\right\|\leq R\right\} $. Consider $ \mathscr{F}_{1}:X\rightarrow A $ and $ \mathscr{F}_{2}:X\rightarrow A $ given as

    $ (F1V)(ȷ)=V0+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ,ȷI,(F2V)(ȷ)=(F)[V(ȷ)x(ȷ,y(ȷ))],ȷI.
    $

    Let $ \mathcal{V} = \mathscr{F}_{1}\mathcal{V}+\mathscr{F}_{2}\mathcal{V}, \mathcal{V}\in \mathscr{C}(\mathscr{I}) $ is the fractional Eq (3.1) to the problems (1.3) and (1.4).

    Hence, the operators $ \mathscr{F}_{1} $ and $ \mathscr{F}_{2} $ satisfy the Krasnoselskii's fixed point theorem.

    Step (ⅰ) $ \mathscr{F}_{1} $ is a contraction.

    By $ {\bf{(H_1)}} $–$ {\bf{(H_3)}} $ on $ \varphi $, $ \forall $ $ \mathcal{V}, \varphi\in \mathscr{C}(\mathscr{I}) $ and $ \jmath\in \mathscr{I} $,

    $ |F1V(ȷ)F2φ(ȷ)|ω(ȷ)|V(ȷ)φ(ȷ)|RVφ.
    $
    (4.5)

    This gives, $ \left\|\mathscr{F}_{1}\mathcal{V}-\mathscr{F}_{2}\varphi\right\|\leq RT\left\|\mathcal{V}-\varphi\right\|, \mathcal{V}, \varphi\in \mathscr{C}(\mathscr{I}). $

    Step (ⅱ) $ \mathscr{F}_{2} $ is completely continuous. By using Theorem 3.3 and Ascoli-Arzela theorem, $ \mathscr{F}_{2} = -\mathscr{F} $ is completely continuous.

    Step (ⅲ) $ \mathscr{F}_{1}\mathcal{V}+\mathscr{F}_{2}\varphi\in U $, for any $ \mathcal{V}, \varphi\in U $, using Theorem 3.3, we obtain

    $ (F1V+F2φ)(ȷ)(F1V)(ȷ)+(F2φ)(ȷ)V0+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ+ε1δ,1,δ1δ;0+φV0+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ+B(δ)1δTEδ,2(δ1δTδ)φV0+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))φ(θ,0,0,0)dθ+ȷ0φ(θ,0,0,0)dθ+B(δ)1δTEδ,2(δ1δTδ)LV0+ȷ0ζ(V+CȷV+DTV)dθ+Nφȷ0dθ+B(δ)1δTEδ,2(δ1δTδ)LV0+ζ(1+Cȷ+DT)ȷ0Vdθ+Nφȷ0dθ+B(δ)1δTEδ,2(δ1δTδ)LV0+ω(ȷ)Rȷ0dθ+Nφȷ0dθ+B(δ)1δTEδ,2(δ1δTδ)LV0+LRT+NφT+B(δ)1δTEδ,2(δ1δTδ)L.
    $
    (4.6)

    By definition of $ R $, we get

    $ V0+NφT=L(1RT+B(δ)TEδ,2(δ1δTδ)1δ).
    $
    (4.7)

    Using the Eq (4.5) in (4.7), we get condition of Eq (4.4).

    $ (F1V+F2φ)(ȷ)L(2B(δ)TEδ,2(δ1δ)Tδ1δ),ȷI.
    $
    (4.8)

    $ \therefore\; \left\|(\mathscr{F}_{1}\mathcal{V}+\mathscr{F}_{2}\varphi)(\jmath)\right\|\leq L, \jmath\in \mathscr{I}. $ This gives, $ \mathscr{F}_{1}\mathcal{V}+\mathscr{F}_{2}\varphi\in U $, $ \forall\; \mathcal{V}, \varphi\in X. $

    From Steps (ⅰ)–(ⅲ), all the conditions of Lemma 2.11 follow.

    Theorem 4.3. By Theorem 4.2, the Eqs (1.3) and (1.4) have a unique solution in $ \mathscr{C}(\mathscr{I}). $

    Proof. (1) The problems (1.3) and (1.4) have an operator equation form as:

    $ (ε1δ,1,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))]=ˆφ(ȷ),ȷI,
    $
    (4.9)

    where,

    $ ˆφ(ȷ)=1δB(δ)(V0V(ȷ)+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ),ȷI.
    $

    By Theorem 4.2, Eq (4.7) is solvable in $ \mathscr{C}(\mathscr{I}) $, by Lemma 2.10 we get a unique solution of Eqs (1.3) and (1.4),

    $ V(ȷ)=(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],VC(I).
    $

    Proof. (2) Let $ \mathcal{V}, \varphi $ be solutions of Eqs (1.3) and (1.4). By fractional integral operators and $ {\bf{(H_1)}} $–$ {\bf{(H_3)}}, $ we find, for any $ \jmath\in \mathscr{I} $,

    $ |V(ȷ)φ(ȷ)||B(δ)1δ(ε1δ,1,δ1δ;0+(Vφ))[V(ȷ)x(ȷ,y(ȷ))]|+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))φ(θ,φ(θ),P1φ(θ),P2φ(θ))|dθ|B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ](V(θ)φ(θ))dθ|+ȷ0ζ(|V(θ)φ(θ)|+C|V(θ)φ(θ)|+D|V(θ)φ(θ)|)dθB(δ)1δȷ0Eδ(|δ1δTδ|)|V(θ)φ(θ)|dθ+ȷ0ζ(1+C+D)|V(θ)φ(θ)|dθB(δ)1δȷ0Eδ(δ1δTδ)|V(θ)φ(θ)|dθ+ȷ0[V(ȷ)x(ȷ,y(ȷ))]|V(θ)φ(θ)|dθȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]|V(θ)φ(θ)|dθ|V(ȷ)φ(ȷ)|ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]|V(θ)φ(θ)|dθ.
    $
    (4.10)

    Theorem 5.1. By Theorem 4.2, if $ \mathcal{V}(\jmath) $ is a solution of Eqs (1.3) and (1.4), then

    $ |V(ȷ)|{|V0|+NφT}exp(ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]dθ),ȷI,
    $
    (5.1)

    where, $ N_{\varphi} = \sup_{\jmath\in \mathscr{I}}\left|\varphi(\jmath, 0, 0, 0)\right|. $

    Proof. If $ \mathcal{V}(\jmath) $ is a solution of Eqs (1.3) and (1.4), for all $ \jmath\in \mathscr{I}, $

    $|V(ȷ)||V0|B(δ)1δȷ0Eδ(|δ1δ(ȷθ)δ|)|V(θ)|dθ+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))|dθ
    $
    $ |V0|B(δ)1δȷ0Eδ(δ1δ(ȷθ)δ)|V(θ)|dθ+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))φ(θ,0,0,0)|dθ+ȷ0|φ(θ,0,0,0)|dθ|V0|B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)|dθ+ȷ0ζ(|V(θ)|+C|V(θ)|+D|V(θ)|)dθ+Nφȷ|V0|B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)|dθ+ȷ0ζ(1+C+D)|V(ȷ)|dθ+NφT|V0|B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)|dθ+ȷ0[V(ȷ)x(ȷ,y(ȷ))]|V(θ)|dθ+NφT{|V0|+NφT}+ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]|V(θ)|dθ.
    $

    By Lemma 2.12, we get

    $ |V(ȷ)|{|V0|+NφT}exp(ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]dθ),ȷI.
    $
    (5.2)

    We discuss data dependence results for the problem

    $ dφdȷ+0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=˜φ(ȷ,φ(ȷ),P1φ(ȷ),P2φ(ȷ)),ȷI,
    $
    (6.1)
    $ φ(0)=φ0R.
    $
    (6.2)

    Theorem 6.1. Equation (4.2) holds, and $ \xi_{k} > 0, $ where $ k = 1, 2 $ are real numbers such that,

    $ |V0φ0|ξ1,|φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ))˜φ(ȷ,φ(ȷ),P1φ(ȷ),P2φ(ȷ))|ξ2,ȷI.
    $
    (6.3)

    $ \varphi(\jmath) $ is a solution of ABR fractional derivative Eqs (6.1) and (6.2), and $ \mathcal{V}(\jmath) $ is a solution of Eqs (1.3) and (1.4).

    Proof. Let $ \mathcal{V}, \varphi $ are the solution of Eqs (1.3) and (1.4), (6.1) and (6.2) respectively. We find for any

    $ |V(ȷ)φ(ȷ)||V0φ0|+B(δ)1δȷ0Eδ(|δ1δ(ȷθ)δ|)|V(θ)φ(θ)|dθ+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))˜φ(s,φ(θ),P1φ(θ),P2φ(θ))|dθ|V0φ0|+B(δ)1δȷ0Eδ(|δ1δ(ȷθ)δ|)|V(θ)φ(s)|dθ+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))φ(θ,φ(θ),P1φ(θ),P2φ(θ))|dθ+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))˜φ(θ,φ(θ),P1φ(θ),P2φ(θ))|dθξ1+B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)φ(θ)|dθ+ȷ0ζ(|V(θ)φ(θ)|+C|V(θ)φ(θ)|+D|V(θ)φ(θ)|)dθ+ξ2ȷ0dθξ1+B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)φ(θ)|dθ+ȷ0ζ(1+C+D)|V(θ)φ(θ)|dθ+ξ2ȷξ1+B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)φ(θ)|dθ+ȷ0[V(ȷ)x(ȷ,y(ȷ))]|V(θ)φ(θ)|dθ+ξ2T|V(ȷ)φ(ȷ)|ξ1+ξ2T+ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]|V(ȷ)φ(θ)|dθ.
    $

    By Lemma 2.12, we get

    $ |V(ȷ)φ(ȷ)|(ξ1+ξ2T)exp(ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]dθ),ȷI.
    $
    (6.4)

    Let any $ \lambda, \lambda_{0}\in \mathscr{R} $ and

    $ dVdȷ+0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=Θ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ),λ),ȷI,
    $
    (7.1)
    $ V(0)=V0R.
    $
    (7.2)
    $ dVdȷ+0Dδȷ[V(ȷ)x(ȷ,y(ȷ)]=Θ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ),λ0),ȷI,
    $
    (7.3)
    $ V(0)=V0R.
    $
    (7.4)

    Theorem 7.1. Let the function $ \Theta $ satisfy Theorem 4.2. Suppose there exists $ \omega, u \in \mathscr{C}(\mathscr{I}, \mathscr{R}^{+}) $ such that,

    $ |Θ(ȷ,V,P1V,P2V,λ)Θ(ȷ,φ,P1φ,P2φ,λ)|ω(ȷ)|Vφ|,|Θ(ȷ,V,P1V,P2V,λ)Θ(ȷ,V,P1V,P2V,λ0)|u(ȷ)|λλ0|.
    $

    If $ \mathcal{V}_{1}, \mathcal{V}_{2} $ are the solutions of Eqs (7.1) and (7.3), then

    $ |V1(ȷ)V2(ȷ)|PT|λλ0|exp(ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]dθ),ȷI,
    $
    (7.5)

    where $ \mathcal{P} = \sup_{\jmath \in \mathscr{I}}u(\jmath). $

    Proof. Let, for any $ \jmath\in \mathscr{I} $,

    $ |V1(ȷ)V2(ȷ)|B(δ)1δ|ȷ0Eδ[δ1δ(ȷθ)δ](V2(θ)V1(θ)dθ)|+ȷ0|Θ(θ,V1(θ),P1V1(θ),P2V1(θ),λ)Θ(θ,V2(θ),P1V2(θ),P2V2(θ),λ0)|dθB(δ)1δȷ0Eδ(|δ1δ(ȷθ)δ|)|V1(θ)V2(θ)|dθ+ȷ0|Θ(θ,V1(θ),P1V1(θ),P2V1(θ),λ)Θ(θ,V2(θ),P1V2(θ),P2V2(θ),λ)|dθ+ȷ0|Θ(θ,V2(θ),P1V2(θ),P2V2(θ),λ)Θ(θ,V2(θ),P1V2(θ),P2V2(θ),λ0)|dθB(δ)1δȷ0Eδ(δ1δ(ȷθ)δ)|V1(θ)V2(θ)|dθ+ȷ0ζ(|V1(θ)V2(θ)|+C|V1(θ)V2(θ)|+D|V1(θ)V2(θ)|)dθ+ȷ0u(θ)|λλ0|dθB(δ)1δȷ0Eδ(δ1δTδ)|V1(θ)V2(θ)|dθ+ȷ0ζ(1+C+D)|V1(θ)V2(θ)|dθ+Pȷ|λλ0|B(δ)1δȷ0Eδ(δ1δTδ)|V1(θ)V2(θ)|dθ+ȷ0[V(ȷ)x(ȷ,y(ȷ))]|V1(θ)V2(θ)|dθ+PT|λλ0|ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]|V1(θ)V2(θ)|dθ+PT|λλ0|.
    $

    By Lemma 2.12,

    $ |V1(ȷ)V2(ȷ)|PT|λλ0|exp(ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]dθ),ȷI.
    $
    (7.6)

    Consider a nonlinear ABR fractional derivative with neutral integro-differential equations of the form:

    $ dVdȷ+0D12ȷ[V(ȷ)x(ȷ,y(ȷ))]=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)),ȷI=[0,2],
    $
    (8.1)
    $ V(0)=1R.
    $
    (8.2)

    $ \varphi:(\mathscr{I}\times \mathscr{R}\times \mathscr{R}\times \mathscr{R})\rightarrow \mathscr{R} $ is a continuous nonlinear function such that,

    $ φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ))=|V(ȷ)|+13+M(ȷ)+N(ȷ),ȷI,
    $

    and

    $ M(ȷ)=B(12){ȷE12,2(ȷ12)+E12(ȷ12)ȷ1},N(ȷ)=B(12){E12,2(ȷ12)+ȷE12(ȷ12)1}.
    $

    We observe that for all $ \mathcal{V}, \varphi\in \mathscr{R} $ and $ \jmath\in \mathscr{I}, $

    $ |φ(ȷ,V,P1V,P2V)φ(ȷ,φ,P1φ,P2φ)|=|(|V(ȷ)|+13+M(ȷ)+N(ȷ))(|φ(ȷ)|+13+M(ȷ)+N(ȷ))|13|Vφ|.
    $
    (8.3)

    The function $ \varphi $ satisfies $ (H_{1}) $–$ (H_{4}) $ with constant $ \frac{1}{3} $. From Theorem 4.2, we have $ \delta = \frac{1}{2} $ and T = 2 which is substitute in Eq (4.2), and we get

    $ B(12)<18E12,2(212).
    $
    (8.4)

    If the function $ B(\delta) $ satisfies Eq (8.4), then Eqs (8.1) and (8.2) have a unique solution.

    $ V(ȷ)=ȷ3+1,ȷ[0,2].
    $
    (8.5)

    In this research article, we explored multi-derivative nonlinear neutral fractional integro-differential equations involving the ABR fractional derivative. The elementary results of the existence, uniqueness and dependence solution on various data are based on the Prabhakar fractional integral operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $ involving a generalized ML function. The existence results are obtained by Krasnoselskii's fixed point theorem, and the uniqueness and data dependence results are obtained by the Gronwall-Bellman inequality with continuous functions.

    The research on Existence and data dependence results for neutral fractional order integro-differential equations by Khon Kaen University has received funding support from the National Science, Research and Innovation Fund.

    The authors declare no conflict of interest.

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