On Nonlinear Analysis for Multi-term Delay Fractional Differential Equations Under Hilfer Derivative

Abstract

In this manuscript, a class of multi-term delay fractional differential equations (FDEs) under the Hilfer derivative is considered. Some newly updated results are established under boundary conditions. For the required results, we utilize the fixed point theory and tools of the nonlinear functional analysis. Further kee** in mind the importance of stability results, we develop some adequate results about the said aspect. The Hyers-Ulam (H-U)-type concept is used to derive the required stability for the solution of the considered problem. Finally, by appropriate test problems, we justify our findings.

1 Introduction

The area of non-integer-order derivatives and integrals has emerged as important tools in the modeling of many biological phenomena and dynamical systems. The said type of operator is more reliable, and accurate, having a greater degree of freedom and non-locality in nature. Kee** the importance of such a type of differential operator in mind, the model formulated via the aforementioned operators has been investigated from different perspectives. In the previous few decades, the aforesaid area has become very popular, and large numbers of articles and books have been written in this area. Some interesting applications of fractional calculus have been used to study different phenomena and infectious diseases (see [6, 7]). Further, some qualitative results related to fractional differential equations (FDEs) have been derived in [21]. Researchers have studied various results for the existence of solutions to different problems of FDEs, we refer to some papers as [17, 23, 25, 29, 34]).

The analysis of the well-posedness of the model formulated via the fractional differential operator is another key part of the investigation of FDEs. In this regard, monotone iterative techniques [14,15,16,], the topological degree theory [22], the method of successive approximation [32], the fixed point index theory [24], and tools of the fixed point theory [28] have been utilized to obtain the conditions for the existence and uniqueness of the solution. For more information, see the recent article cited as [5, 8, 12].

It is evident that many real-world problems modeled via FDEs are nonlinear. Therefore, obtaining the exact solution to such equations is impossible or very difficult. In this respect, various approximate techniques have been established. A good numerical procedure stability is an important aspect. So in such cases, the stability analysis plays a significant role in the investigation. In this connection, many stability concepts such as the Lyapunov stability, exponential stability, the asymptotic stability, and the Mittag-Leffler stability have been discussed [9]. In addition to the above, an interesting type of stability known as the H-U has got considerable attention from researchers. The aforementioned stability had been initiated by Ulam [30]. Later on, it was modified to generalized H-U by Hyers [16]. The said stability has been further expanded to the H-U-Rassias (H-U-R) stability by Rassias [27]. Moreover, results related to such types of stability have been derived for ordinary differential equations as well as for FDEs (for more detail see [4, 9, 11, 20, 25]).

It is remarkable that the fractional differential operator is not a unique definition. The first definition given by Riemann-Liouville has been very well used. After that, Hadamard, Hilfer, and Caputo defined the said derivative in their own way. It is remarkable that concepts of long and short memory terms are explained well using fractional derivatives. Due to this importance, researchers have worked very well in the said area from different aspects. The reader should see details in [10]. Furati et al. [13] studied a class of Hilfer FDEs with the initial condition for the existence theory using the fixed point theory. The said problem is described by

$$\left\{ \begin{aligned} \begin{aligned}&D_{a}^{\alpha ,\beta }{\mathbb {V}}_1(t)=f_1(t,{\mathbb {V}}_1(t)),\\ {}&I_{a}^{1-\gamma }{\mathbb {V}}_1(a^+) ={\mathbb {V}}_a. \end{aligned} \end{aligned}\right.$$

(1)

Here, ${\mathbb {V}}_a$ and $a<b\in {\mathbf{R}}$, $t\in (a,b],$ and the function $f_1{:\,}(a,b]\,\times {\bf{R}} \mapsto {\bf{R}}$ is weighted continuously. The positive real numbers $\alpha ,\beta ,\gamma$ should be chosen as $0<\alpha <1$, $\beta \in [0,1]$, and $\gamma =\alpha +\beta -\alpha \beta$. Furthermore, $D_{a}^{\alpha ,\beta }$ and $I_{a}^{1-\gamma }$ represent the Hilfer fractional derivative and the Riemann-Liouville fractional integral of order $\alpha$, type $\beta$ and $1-\gamma$, respectively. Via the fixed point theory, the aforesaid problem has been investigated.

Wang and Zhang [33] have analyzed the FDEs with the initial condition on the variables as

$$\left\{\begin{aligned} \begin{aligned}&D_{a}^{\alpha ,\beta }{\mathbb {V}}_1(t)=f_1(t,{\mathbb {V}}_1(t)),\\ {}&I_{a}^{1-\gamma }{\mathbb {V}}_1(a^+) =\sum _{i=1}^{m}b_i{\mathbb {V}}_1(\sigma _i). \end{aligned}\end{aligned}\right.$$

(2)

Here, $a,b,b_i\,(i=1,2, \cdots ,m\in N)$, $t\in (a,b],$ and the domain of the unknown functions is (a, b], while the nonlinear function $f_1{:}\,(a,b]\,\times (-\infty ,\infty ) \times \mapsto (-\infty ,\infty )$ falls in $C_{1-\gamma }[a,b]$, which is the weighted space of the continuous function on the interval [a, b]. The aforementioned FDEs utilized Hilfer fractional derivatives of order $0<\alpha <1$ and type $\beta \in [0,1]$ symbolized as $D_{a}^{\alpha ,\beta }$. The parameter $\gamma =\alpha +\beta -\alpha \beta \in [\alpha ,1).$ Furthermore, the arbitrary constant should be fixed in such a way that $\Gamma (\gamma )-\sum _{i+1}^{m}b_i(\sigma _i-a)^{\gamma -1}\ne 0$. In this regard, the authors gave some assumptions on the given function and then converted the FDE containing Hilfer derivative into an integral equation. Furthermore, they have also shown that the obtained integral equation is equivalent to the underlying FDE. The qualitative aspects have been investigated using the fixed point theory as a strong tool. Afterward, conditions for the existence of a solution have been developed with the help of three well-known fixed point theorems of Krasnoselskii, Schauder, and Schaefer. Further, the required theory has been developed in the weighted space $C_{1-\gamma }^{\alpha ,\beta }.$

In the same fashion, Kamocki and Obczynski [18] in 2016 have investigated some linear and nonlinear FDEs under the Hilfer derivative with homogenous and inhomogeneous initial conditions as

$$\begin{aligned}&D_{a}^{\alpha ,\beta }{\mathbb {V}}_1(t)=f_1(t,{\mathbb {V}}_1(t)), \end{aligned}$$

(3)

$$\begin{aligned}&I_{a}^{1-\gamma }{\mathbb {V}}_1(a^+) =0, \end{aligned}$$

(4)

$$\begin{aligned}&I_{a}^{1-\gamma }{\mathbb {V}}_1(a^+) =b_a, \end{aligned}$$

(5)

and

$$\begin{aligned}&D_{a}^{\alpha ,\beta }{\mathbb {V}}_1(t)=A{\mathbb {V}}_1(t)+g(t), \end{aligned}$$

(6)

$$\begin{aligned}&I_{a}^{1-\gamma }{\mathbb {V}}_1(a^+) =0, \end{aligned}$$

(7)

$$\begin{aligned}&I_{a}^{1-\gamma }{\mathbb {V}}_1(a^+) =b_a. \end{aligned}$$

(8)

Here $t\in [a,b]$, where $a<b\in {\bf{R}},$ the function $f_1{:}\,[a,b]\times {\bf{R}}^n\mapsto {\bf{R}}^n,$ $A\in {\bf{R}}^{n\times n},$ and $g(t)\in I_{a}^{\beta (1-\alpha )(L^1)}$. The arbitrary constants $b_a\ne 0 \in {\bf{R}}^n$. Authors have analyzed the nonlinear problem (3) with the homogenous initial condition (4) and also with the inhomogeneous initial condition (5). Furthermore, the criteria for the uniqueness of the solution for the linear problem (6) have been obtained under the homogenous initial condition (7) and the inhomogeneous condition (8). Further, the required analysis has been developed with the help of the Banach contraction principle under the Bielecki norm in the space $I_{a}^{\alpha }(L^1([a,b]), {\bf{R}}^n)$.

Subsequently, the qualitative aspect of boundary value problems (BVPs) formulated via the Hifer fractional derivative has been investigated recently. For instance, Abdo et al. [

$$\left\{\begin{aligned} \begin{aligned}&D_{a}^{\alpha ,\beta }{\mathbb {V}}_1(t)=f_1(t,{\mathbb {V}}_1(t)),\\ {}&I_{a}^{1-\gamma }[g_1{\mathbb {V}}_1(a^+)+g_2{\mathbb {V}}_1(b^-)] =g_3. \end{aligned} \end{aligned}\right.$$

(9)

Here, $g_i\,(i=1,2,3)\in {\bf{R}}$, $t\in (a,b],$ and the nonlinear function $f_1{:}\,(a,b]\,\times {\bf{R}} \mapsto {\bf{R}}\in$ $C_{1-\gamma }[a,b]$, $0<\alpha <1,$ and type $\beta \in [0,1]$ represents as $D_{a}^{\alpha ,\beta }$. The parameter $\gamma _1=\alpha +\beta -\alpha \beta \in [\alpha ,1).$

Motivated by the established work discussed above, in this manuscript, we present some novel results for multi-term Hilfer FDEs under some general boundary conditions based on fractional integral operators and nonlocal points. To the best of our knowledge, no contribution to the formulation of a coupled system of FDEs with multi-term fractional differential operators exists. As a result, the first innovative component of our study is the construction of FDE with multi-term fractional differential operators. The second unique element of the newly formulated model is the investigation of H-U-type stabilities. Furthermore, the auxiliary conditions imposed on the considered model are in the form of a fractional integral. Also, a proportional type delay term in the newly constructed coupled system of n-fractional differentiation operators is also involved. Our considered problem is given as

$$\left\{ \begin{aligned} \begin{aligned}&\sigma _1D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)+\sum _{i=2}^{n}\sigma _iI_{a}^ {\psi _i}D_{a}^{\alpha _i,1-\beta _1}{\mathbb {V}}_1(t)=f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t)),\\ {}&a_1I_{a}^{q_1}{\mathbb {V}}_1(a^+) =a_2+a_3\sum _{j=1}^{m_1}\eta _j{\mathbb {V}}_1(\tau _j) +a_4\sum _{l=1}^{m_2}\kappa _lI_{a}^{p_l}{\mathbb {V}}_1(\zeta _l). \end{aligned} \end{aligned}\right.$$

(10)

In the considered problem (10), arbitrary constants $a_1, a_2,a_3,a_4,\sigma _i,\eta _j,\kappa _l \in {\bf{R}},\, \tau _j,\zeta _l\in (1,\tau ]$ for $i=1,2,\cdots ,n,\, j=1,2,\cdots ,m_1,\,$ and $l=1,2,\cdots ,m_2,\,$ together with $q_1,p_l \,(l=1,2,\cdots ,m_2)$ are taken from the set of positive real numbers and $\sigma _1\ne 0$. Arbitrary constants should be chosen as

$$\begin{aligned} a_1\mathfrak {A}(q_1)-a_3\sum _{j=1}^{m_1}\eta _j(\tau _j-a)^{\gamma _1-1}\ne \frac{a_4\Gamma (\gamma _1)}{\Gamma (\gamma _1+p_l)}\sum _{l=1}^{m_2}\kappa _l(\zeta _l-a)^{\gamma _1+p_l-1}, \end{aligned}$$

where $\gamma _1=\alpha _1+\beta _1-\alpha _1\beta _1\geqslant 1-q_1$, $\psi _i=2\beta _1+\alpha _i-1-\beta _1(\alpha _1+\alpha _i)>0,$ and $\phi _i=\alpha _1+2\beta _1-\beta _1(\alpha _1+\alpha _i)-1>0$. In addition, $D_{a}^{\alpha _i,\beta _1}$ represent Hilfer fractional derivatives of order $0<\alpha _i<1$, and type $0\leqslant \beta _1\leqslant 1 \,(i=1,2, \cdots,n).$ Furthermore, the nonlinear function $f_1{:}\,(a,\tau ]\times {\mathbf{R}} \times {\mathbf{R}} \mapsto {\bf{R}}$ is weighted continuously. Via fixed point approach the required results will be investigated. For this purpose, some growth conditions are imposed on nonlinear functions known as data dependence results. Also using the powerful tools of the nonlinear functional analysis, sufficient results are established for different kinds of H-U stabilities. A different form of stabilities is generalized H-U (g-H-U), H-U-Rassia (H-U-R), and generalized (g-H-U-R).

2 Preliminaries

We have gathered some fundamental facts, definitions, and lemmas which will be utilized for the investigation of the main work. So in this connection, we need the following spaces. The weighted space of the continuous functions on the finite interval $[a, \tau ]$ is denoted by $C_{\gamma }[a,\tau ]$ and $0 \leqslant \gamma <1$, precisely the space is given as

$$\begin{aligned} C_{\gamma }[a,\tau ]=\{{{\mathcal {V}}_1(t)}{:}\, (0,T] \mapsto {\bf{R}}{:}\,(t-a)^{\gamma }{\mathcal {V}}_1(t)\in C[a,\tau ]\}. \end{aligned}$$

The aforementioned space is Banach space with norm defined as

$$\begin{aligned} ||{{\mathcal {V}}_1}||_{C_{\gamma }}=||(t-a)^\gamma {{\mathcal {V}}_1(t)}||_{\infty } =\max \{|(t-a)^r{\mathcal {V}}_1(t)|{:}\,t\in [a,\tau ]\}. \end{aligned}$$

Meanwhile, $C_{\gamma }^n[a,\tau ]=\{{{\mathcal {V}}_1}{:}\, C^{n-1}C[a,\tau ]{:} \,{{\mathcal {V}}_1}^{(n)}\in C_{\gamma }[a,\tau ]\}$ is the Banach space with norm defined as $||{{\mathcal {V}}_1}||_{C_{\gamma }^n}=\sum _{k=0}^{n-1}|| {\mathcal {V}}_1^{(k)}||_C+||{\mathcal {V}}_1^{(n)}||_{C_{\gamma }}, n\in N.$ Moreover, $C_{\gamma }^0[a,\tau ]=C_{\gamma }[a,\tau ]$. Also

$$\begin{aligned} C_{1-\gamma }^{\alpha ,\beta }[a,\tau ]=\left\{ {\mathcal {V}}_1\in C_{1-\gamma }[a,\tau ]{:}\, D_{0^+}^{\alpha ,\beta }\,{\mathcal {V}}_1\in C_{1-\gamma }[a,\tau ]\right\}, \end{aligned}$$

(11)

and

$$\begin{aligned} C_{1-\gamma }^{\gamma }[a,\tau ]=\left\{ {\mathcal {V}}_1\in C_{1-\gamma }[a,\tau ]{:}\, D_{0^+}^{\gamma }{\mathcal {V}}_1\in C_{1-\gamma }[a,\tau ]\right\} . \end{aligned}$$

(12)

It is obvious that

$$\begin{aligned} C_{1-\gamma }^{\gamma }[a,\tau ]\subset C_{1-\gamma }^{\alpha ,\beta }[a,\tau ]. \end{aligned}$$

Definition 1

[23] For a function ${\mathbb {V}}(t)$, the Riemann-Liouville fractional integral of order $\alpha$ is defined as

$$\begin{aligned} {\mathbb {I}}^{\alpha }{\mathbb {V}}(t)=\frac{1}{\Gamma (\alpha )}\int _0^t (t-\chi )^{\alpha -1}{\mathbb {V}}(\chi ){\text{d}}\chi . \end{aligned}$$

Definition 2

[23] The fractional Riemman-Liouivlle derivative for a function ${\mathbb {V}}(t)$ on the interval $[a,\tau ]$ of order $\alpha \in (n-1,n)$ is defined as

$$\begin{aligned} {D}^{\alpha } {\mathbb {V}}(t)=\frac{1}{\Gamma ( n- \alpha )}\left( \frac{\text{d}}{\text{d}t}\right) ^n\int _0^t \frac{{\mathbb {V}}(\chi )}{(t-\chi )^{\alpha +1-n}} {\text{d}}\chi , \end{aligned}$$

where $n=[\alpha ]+1$ and $[\alpha ]$ denotes the integer part of the real number $\alpha$.

Definition 3

[25] The fractional-order Caputo derivative for a function ${\mathbb {V}}(t)$ of order $\alpha \in (n-1,n)$ is defined as

$$\begin{aligned} ^{\text{c}}{D}^{\alpha } {\mathbb {V}}(t)=\frac{1}{\Gamma ( n- \alpha )}\int _0^t \frac{{\mathbb {V}}^{n}(\chi )}{(t-\chi )^{\alpha +1-n}} {\text{d}}\chi , \end{aligned}$$

where $n=[\alpha ]+1$ and $[\alpha ]$ denotes the integer part of the real number $\alpha$.

Definition 4

[1] Let $\alpha \in (0,1),\, \beta \in [0,1],$ ${\mathcal {V}}_1\in L^1(I),$ and $I_{0}^{(1-\alpha )(1-\beta )}{\mathcal {V}}_1\in AC^1(I)$. The Hilfer fractional derivative of the function ${\mathcal {V}}_1$ of order $\alpha$ and type $\beta$ is defined as

$$\begin{aligned} \big(D_{0}^{\alpha ,\beta }{\mathcal {V}}_1\big )(t) =\big (I_{0}^{\beta (1-\alpha )}D\big (I_{0^+}^{(1-\beta )(1-\alpha )}{\mathcal {V}}_1\big )\big )(t) \,\, \text {for a.e.}\, t\in I. \end{aligned}$$

(13)

Remark 1

[1] Let $\alpha \in (0,1),\, \beta \in [0,1],\, \gamma =\alpha +\beta -\alpha \beta ,$ and ${{\mathcal {V}}_1}\in L^1(I).$

• (i) The operator $D_{0}^{\alpha ,\beta }$ can be written as

  $$\begin{aligned} \big (D_{0}^{\alpha ,\beta }{{\mathcal {V}}_1\big )(t)}= & {} \big (I_{0}^{\beta (1-\alpha )}D\big (I_{0^+}^{(1-\beta )(1-\alpha )}{\mathcal {V}}_1\big )\big )(t)\\= & {} \big (I_{0}^{\beta (1-\alpha )}D_{0}^{\gamma }{\mathcal {V}}_1\big )(t)\,\, {\text{ for a.e.}} \, t \in I. \end{aligned}$$

  Moreover, the parameter $\gamma$ satisfies

  $$\begin{aligned}\gamma \in (0,1], \gamma \geqslant \alpha , \, \gamma >\beta ,\, 1-\gamma <1-\beta (1-\alpha ). \end{aligned}$$

• (ii) The special case of (13) with $\beta =0$, coincides with the Riemann-Liouville derivative, and with $\beta = 1$, it coincides with the Caputo derivative. In addition

  $$\begin{aligned} D_{0}^{\alpha ,0}=D_{0}^{\alpha },\,\,D_{0}^{\alpha ,1}=\,^{\text{c}}D_{0}^{\alpha },\end{aligned}$$

• (iii) If $D_0^{\gamma }{\mathcal {V}}_1$ exist and belong to $L^1(I)$, then

  $$\begin{aligned} I_{0}^{\alpha }D_{0}^{\alpha ,\beta }{\mathcal {V}}_1= & \,\, I_{0}^{\gamma }D_{0}^{\gamma }{\mathcal {V}}_1\\ = & \,\, {\mathcal {V}}_1(t)-\frac{I_0^{1-\gamma }{\mathcal {V}}_1(0^+)}{\Gamma (\gamma )}t^{\gamma -1}\,\, \text {for a.e} \,\,t\in I. \end{aligned}$$

Lemma 1

[19] Let $\alpha >0$, $0\leqslant \gamma <1$, and ${{\mathcal {V}}_1}_{n-\alpha }(t)=(I_{a^+}^{n-\alpha }{{\mathcal {V}}_1)(t)}.$ If ${\mathbb {V}}_1(t)\in C_{\gamma [a,b]}$ and ${{\mathcal {V}}_1}_{n-\alpha }(t)\in C_{\gamma }^n[a,\tau ]$, then

$$\begin{aligned} I_{a^+}^\alpha D_{a^+}^\alpha {\mathcal {V}}_1(t)={\mathcal {V}}_1(t)-\frac{\sum _{j=1}^{n}{{\mathcal {V}}_1} _{n-\alpha }^{n-j}(a^+)}{\Gamma (\alpha -j+1)}(t-a)^{\alpha -j},\,\,\, t\in [a,\tau ], \end{aligned}$$

where ${{\mathcal {V}}_1}_{n-\alpha }^{n-j}(a^+)=\lim _{t\mapsto a^+}{{\mathcal {V}}_1}_{n-\alpha }^{n-j}(t)$.

Lemma 2

[33] If $\alpha ,\beta \in {\bf{R}}^+$, then

$$\begin{aligned}{}[I_{a^+}^{\alpha } (s-a)^{\beta -1}](t)=\frac{\Gamma (\beta )}{\Gamma (\beta +\alpha )} (t-a)^{\beta +\alpha -1} \end{aligned}$$

and

$$\begin{aligned}{}[D_{a^+}^{\alpha } (s-a)^{\alpha -1}](t)=0,\quad 0<\alpha <1. \end{aligned}$$

Lemma 3

[31] Let $0<\alpha <1$, $0\leqslant \gamma <1$. If $f\in C_{\gamma }[J, {\bf{R}}]$ and $(I_{0^+}^{1-\alpha }f)(0)\in C_{\gamma }^1[J, {\bf{R}}]$, then

$$\begin{aligned} I_{0^+}^{\alpha }D_{0^+}^{\alpha }f(t)=f(t)-\frac{(I_{0^+}^{1-\alpha }f)(0)}{\Gamma (\alpha )}t^{\alpha -1}, \quad \,t\in J. \end{aligned}$$

(14)

Lemma 4

[33] Let $0\leqslant \gamma <1$ and $f\in C_{\gamma }[J, {\bf{R}}]$. Then

$$\begin{aligned} (I_{a^+}^{\alpha }f)(a)=\lim _{t\mapsto 0^+}I_{a^+}^{\alpha }f(t)=0, \quad 0\leqslant \gamma <\alpha . \end{aligned}$$

(15)

Theorem 1

[15] The Riemann-Liouville fractional integral $I^{\alpha }, \, \alpha \geqslant 0$, is a linear map of the space $C_{\gamma }, \gamma \geqslant -1$, into itself, that is $I^{\alpha }{:}\, C_{\gamma }\mapsto C_{\alpha +\gamma }\subset C_{\gamma }.$

Theorem 2

[2] Let C be a nonempty convex subset of the Banach space X, U be a nonempty open subset of C with $0\in U,$ and ${\mathcal {F}}\!:\overline{U}\mapsto C$ be a continuous and compact operator. Then, one of the given holds as

• (i) ${\mathcal {F}}$ has fixed points,  or

• (ii) there exist $x\in \partial U$ and $\lambda _*\in (0,1)$ with $x=\lambda _*{\mathcal {F}}(x).$

3 Main Result

This section is devoted to the exhibition of our main results associated with our newly formulated system.

Theorem 3

Let $f_1{:}\,(a,\tau ] \times {\mathbf{R}} \times {\mathbf{R}} \mapsto {\mathbf{R}}$ be a function such that $f(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda t))\in C_{1-\gamma _1} [a,\tau ]$ for any ${\mathbb {V}}_1\in C_{1-\gamma _1} [a,\tau ]$. A function ${\mathbb {V}}_1\in C_{1-\gamma _1}^{\gamma _1} [a,\tau ]$ is a solution of the following MFDDE:

$$\begin{aligned} \left\{ \begin{aligned}&\sigma _1D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)+\sum _{i=2}^{n}\sigma _iI_{a}^ {\psi _i}D_{a}^{\alpha _i,1-\beta _1}{\mathbb {V}}_1(t))=f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t)),\, t\in [a,\tau ],\, 0\leqslant a <\tau \in {\mathbf{R}}^+,\\ {}&a_1I_{a}^{q_1}{\mathbb {V}}_1(a^+) =a_2+a_3\sum _{j=1}^{m_1}\eta _j{\mathbb {V}}_1(\tau _j) +a_4\sum _{l=1}^{m_2}\kappa _lI_{a}^{p_l}{\mathbb {V}}_1(\zeta _l), \lambda _1\in [0,1], \end{aligned}\right. \end{aligned}$$

(16)

if and only if ${\mathbb {V}}_1$ satisfies the following integral equation:

$$\begin{aligned} \begin{aligned} {\mathbb {V}}_1(t)=\;&(t-a)^{\gamma _1-1}\Bigg [ \frac{a_2}{d_1}-\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\sum _{i=2}^{n} \eta _j\sigma _i\int _a^{\tau _j}\frac{(\tau _j-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\phi _i)}{\mathbb {V}}_1({\mathbb {U}}){\text{d}}{\mathbb {U}} \\ {}&+\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\eta _j\int _a^{\tau _j} \frac{(\tau _j-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_ 1(\lambda _1 {\mathbb {U}})){\text{d}}{\mathbb {U}} \\ {}&-\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\sum _{i=2}^{n}\kappa _l \sigma _i\int _a^{\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\phi _i+p_l-1}}{\phi _i+p_l}{\mathbb {V}}_1({\mathbb {U}}){\text{d}}{\mathbb {U}} \\ {}&+\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\kappa _l\int _a^{\zeta _l} \frac{(\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)} f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1 (\lambda _1 {\mathbb {U}})){\text{d}}{\mathbb {U}}\Bigg ] \\ {}&+\sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1}\int _a^{t}\frac{(t-{\mathbb {U}})^ {\phi _i-1}}{\Gamma (\phi _i)}{\mathbb {V}}_1({\mathbb {U}}){\text{d}}{\mathbb {U}} \\ {}&+\frac{1}{\sigma _1}\int _a^{t}\frac{(t-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1 (\lambda _1 {\mathbb {U}})){\text{d}}{\mathbb {U}}, \end{aligned}\end{aligned}$$

(17)

where $d_1=a_1\mathfrak {A}(q_1)-a_3\sum _{j=1}^{m_1}\eta _j(\tau _j-a)^ {\gamma _1-1}-\sum _{l=1}^{m_2}\frac{a_4\Gamma (\gamma _1)}{\Gamma (\gamma _1+p_l)}\kappa _l(\zeta _l-a)^{\gamma _1+p_l-1}\ne 0$,  $\gamma _1=\alpha _1+\beta _1-\alpha _1\beta _1$, $\psi _i=2\beta _1+\alpha _i-1-\beta _1(\alpha _1+\alpha _i),$ and $\phi _i=\alpha _1+2\beta _1-\beta _1(\alpha _1+\alpha _i)-1$,

$$\begin{aligned} \mathfrak {A}(q_1)=\left\{ \begin{aligned}&\quad 0,\quad \quad \,\quad \quad {if}\quad q_1>1-\gamma _1,\\ {}&\,\frac{\Gamma (\gamma _1)}{\Gamma (\gamma _1+q_1)},\quad {if}\quad q_1=1-\gamma _1. \end{aligned}\right. \end{aligned}$$

(18)

Proof

Applying the fractional-order integral ${I_{a}^{\alpha _1}}$ in (16), we get

$$\begin{aligned} \begin{aligned} I_{a}^{\alpha _1+\beta _1-\alpha _1\beta _1}D_{a}^{\alpha _1 +\beta _1-\alpha _1\beta _1}{\mathbb {V}}_1(t) = &-\sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1}I_{a}^{\alpha _1 +\psi _i+1-\beta _1-\alpha _i(1-\beta _1)}D_{a}^{\alpha _i+1 -\beta _1-\alpha _i(1-\beta _1)}{\mathbb {V}}_1(t)\\ & +\frac{1}{\sigma _1}I_{a}^{\alpha _1}f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t)). \end{aligned} \end{aligned}$$

(19)

In view of Lemma 1, we have

$$\begin{aligned} {\mathbb {V}}_1(t)=C_1(t-a)^{\gamma _1-1} -\sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1}I_{a}^{\phi _i}\,{\mathbb {V}}_1(t) +\frac{1}{\sigma _1}I_{a}^{\alpha _1}f_1(t, {\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t)). \end{aligned}$$

(20)

Now, computing the fractional integral of order p at $t=a$ of (20) and using (18), we obtain the following equation:

$$\begin{aligned} I_{a}^{q_1}(a)=\mathfrak {A}(q_1)C_1. \end{aligned}$$

(21)

One can obtain the following equation using (20):

$$\begin{aligned} \begin{aligned} a_3\sum _{j=1}^{m_1}\eta _j{\mathbb {V}}_1(\tau _j) & = a_3C_1 \sum _{j=1}^{m_1}\eta _j(\tau _j-a)^{\gamma _1-1}-a_3\sum _{j=1}^{m_1} \eta _j\sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1}I_{a}^ {\phi _i}{\mathbb {V}}_1(\tau _j)\\ & \quad +a_3\sum _{j=1}^{m_1}\eta _j\frac{1}{\sigma _1}I_{a}^ {\alpha _1}f_1(\tau _j,{\mathbb {V}}_1(\tau _j),{\mathbb {V}}_1(\lambda _1 \tau _j)),\\ a_4\sum _{l=1}^{m_2}\kappa _lI_{a}^{p_l}{\mathbb {V}}_1(\zeta _l) & = \frac{a_4C_1\Gamma (\gamma _1)}{\Gamma (\gamma _1+p_l)} \sum _{l=1}^{m_2}\kappa _l(\zeta _l-a)^{\gamma _1+p_l-1} -a_4\sum _{l=1}^{m_2} \kappa _l\sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1}I_{a}^{\phi _i+p_l}{\mathbb {V}}_1(\zeta _l)\\ & \quad +a_4\sum _{l=1}^{m_2}\kappa _l\frac{1}{\sigma _1}I_{a}^ {\alpha _1+p_l}f_1(\zeta _l,{\mathbb {V}}_1(\zeta _l),{\mathbb {V}}_1(\lambda _1 \zeta _l)). \end{aligned} \end{aligned}$$

(22)

Using (21), (22), and the subsidiary condition involved in (10), we obtain the expression given as

$$\begin{aligned} \begin{aligned}a_1\mathfrak {A}(q_1)C_1 = \,\, & a_2+a_3C_1\sum _{j=1}^{m_1}\eta _j(\tau _j-a)^{\gamma _1-1}-a_3\sum _{j=1}^{m_1} \eta _j\sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1}I_{a}^{\phi _i}{\mathbb {V}}_1(\tau _j)\\ {}&+a_3\sum _{j=1}^{m_1}\eta _j\frac{1}{\sigma _1}I_{a}^{\alpha _1}f_1 (\tau _j,{\mathbb {V}}_1(\tau _j),{\mathbb {V}}_1(\lambda _1 \tau _j))\\ {}&+\frac{a_4C_1\Gamma (\gamma _1)}{\Gamma (\gamma _1+p_l)}\sum _{l=1}^ {m_2}\kappa _l(\zeta _l-a)^{\gamma _1+p_l-1} -a_4\sum _{l=1}^{m_2} \kappa _l\sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1} I_{a}^{\phi _i+p_l}{\mathbb {V}}_1(\zeta _l)\\ {}&+a_4\sum _{l=1}^{m_2}\kappa _l\frac{1}{\sigma _1}I_{a}^{\alpha _1+p_l}f_1 (\zeta _l,{\mathbb {V}}_1(\zeta _l),{\mathbb {V}}_1(\lambda _1 \zeta _l)). \end{aligned}\end{aligned}$$

(23)

By rearranging, one has

$$\begin{aligned} \begin{aligned}&C_1 \bigg [a_1\mathfrak {A}(q_1)-a_3\sum _{j=1}^{m_1}\eta _j(\tau _j-a)^ {\gamma _1-1}-\frac{a_4\Gamma (\gamma _1)}{\Gamma (\gamma _1+p_l)} \sum _{l=1}^{m_2}\kappa _l(\zeta _l-a)^{\gamma _1+p_l-1}\bigg ] \\ = & a_2 -a_3\sum _{j=1}^{m_1}\sum _{i=2}^{n}\frac{\eta _j\sigma _i}{\sigma _1} I_{a}^{\phi _i}{\mathbb {V}}_1(\tau _j) +a_3\sum _{j=1}^{m_1}\frac{\eta _j}{\sigma _1}I_{a}^{\alpha _1}f_1 (\tau _j,{\mathbb {V}}_1(\tau _j),{\mathbb {V}}_1(\lambda _1 \tau _j)) \\ & - a_4\sum _{l=1}^{m_2}\sum _{i=2}^{n}\frac{\kappa _l\sigma _i}{\sigma _1 }I_{a}^{\phi _i+p_l}{\mathbb {V}}_1(\zeta _l) +a_4\sum _{l=1}^{m_2}\frac{\kappa _l}{\sigma _1}I_{a}^{\alpha _1+p_l}f_1 (\zeta _l,{\mathbb {V}}_1(\zeta _l),{\mathbb {V}}_1(\lambda _1 \zeta _l)). \end{aligned} \end{aligned}$$

(24)

For simplicity, we use notation as

$$\begin{aligned} d_1=a_1\mathfrak {A}(q_1)-a_3\sum _{j=1}^{m_1} \eta _j(\tau _j-a)^{\gamma _1-1}- \frac{a_4\Gamma (\gamma _1)}{\Gamma (\gamma _1+p_l)}a_4 \sum _{l=1}^{m_2}\kappa _l(\zeta _l-a)^{\gamma _1+p_l-1}\ne 0, \end{aligned}$$

one can obtain the desired solution (17) using (24) and (20).

Conversely, assuming the solution of (17) is ${\mathbb {V}}_1$, we have to prove that ${\mathbb {V}}_1$ also satisfies the FDE (16). To achieve the desired result, one can obtain the following relations with the help of the given integral equation (17):

$$\begin{aligned} \begin{aligned} &a_1I_{a}^{q_1}{\mathbb {V}}_1(a^+)\\ =\,\, &a_1\mathfrak {A}(q_1)\Bigg [ \frac{a_2}{d_1} -\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\sum _{i=2}^{n}\eta _j \sigma _iI_{a}^{\phi _i}{\mathbb {V}}_1(\tau _j) +\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\eta _jI_{a}^{\alpha _1}f_1 (\tau _j,{\mathbb {V}}_1(\tau _j),{\mathbb {V}}_1(\lambda _1 \tau _j)) \\ {}&-\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\sum _{i=2}^{n}\kappa _l \sigma _iI_{a}^{\phi _i+p_l}{\mathbb {V}}_1(\zeta _l) +\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\kappa _lI_{a}^{\alpha _1 +p_l}f_1(\zeta _l,{\mathbb {V}}_1(\zeta _l),{\mathbb {V}}_1(\lambda _1 \zeta _l))\Bigg ],\\ & {\begin{aligned} a_3\sum _{j=1}^{m_1}\eta _j{\mathbb {V}}_1(\tau _j) & =a_3\sum _{j=1}^{m_1}\eta _j(\tau _j-a)^{\gamma _1-1}\Bigg [ \frac{a_2}{d_1} -\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\sum _{i=2}^{n}\eta _j \sigma _iI_{a}^{\phi _i}{\mathbb {V}}_1(\tau _j) \\ & \quad +\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\eta _jI_{a}^{\alpha _1} f_1(\tau _j,{\mathbb {V}}_1(\tau _j),{\mathbb {V}}_1(\lambda _1 \tau _j)) \\ & \quad -\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\sum _{i=2}^{n}\kappa _l \sigma _iI_{a}^{\phi _i+p_l}{\mathbb {V}}_1(\zeta _l) +\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\kappa _lI_{a}^{\alpha _1 +p_l}f_1(\zeta _l,{\mathbb {V}}_1(\zeta _l),{\mathbb {V}}_1(\lambda _1 \zeta _l))\Bigg ] \\ & \quad -a_3 \sum _{j=1}^{m_1}\eta _j \sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1}I_{a}^{\phi _i}{\mathbb {V}}_1(\tau _j) +a_3\sum _{j=1}^{m_1}\eta _l\frac{1}{\sigma _1}I_{a}^{\alpha _1} f_1(\tau _j,{\mathbb {V}}_1(\tau _j),{\mathbb {V}}_1(\lambda _1 \tau _j)), \end{aligned}} \\ & {\begin{aligned} a_4\sum _{l=1}^{m_2}\kappa _lI_{a}^{p_l}{\mathbb {V}}_1(\zeta _l) & = \frac{a_4\Gamma (\gamma _1)}{\Gamma (\gamma _1+p_l)}\sum _{l=1}^ {m_2}\kappa _l(\zeta _l-a)^{\gamma _1+p_l-1}\Bigg [ \frac{a_2}{d_1} -\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\sum _{i=2}^{n}\eta _j \sigma _iI_{a}^{\phi _i}{\mathbb {V}}_1(\tau _j) \\ &\quad +\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\eta _jI_{a}^{\alpha _1} f_1(\tau _j,{\mathbb {V}}_1(\tau _j),{\mathbb {V}}_1(\lambda _1 \tau _j)) -\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\sum _{i=2}^{n}\kappa _l \sigma _iI_{a}^{\phi _i+p_l}{\mathbb {V}}_1(\zeta _l) \\ &\quad +\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\kappa _lI_{a}^{\alpha _1+p_l} f_1(\zeta _l,{\mathbb {V}}_1(\zeta _l),{\mathbb {V}}_1(\lambda _1 \zeta _l))\Bigg ] -a_4 \sum _{l=1}^{m_2}\kappa _l \sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1} I_{a}^{\phi _i+p_l}{\mathbb {V}}_1(\zeta _l) \\ &\quad +a_4\sum _{l=1}^{m_2}\kappa _l\frac{1}{\sigma _1}I_{a}^{\alpha _1+p_l}f_1 (\zeta _l,{\mathbb {V}}_1(\zeta _l),{\mathbb {V}}_1(\lambda _1 \zeta _l)). \end{aligned}}\end{aligned} \end{aligned}$$

(25)

Hence from (25), we get

$$\begin{aligned} \begin{aligned} a_1I_{a}^{q_1}{\mathbb {V}}_1(a^+) =a_2+a_3\sum _{j=1}^{m_1}\eta _j{\mathbb {V}}_1(\tau _j) +a_4\sum _{l=1}^{m_2}\kappa _lI_{a}^{p_l}{\mathbb {V}}_1(\zeta _l). \end{aligned}\end{aligned}$$

(26)

Now, we have to prove ${\mathbb {V}}_1(t)$ defined by (17) also satisfies MFDDE (16). For this, by applying the fractional derivative $D_{a}^{\alpha _1+\beta _1-\alpha _1\beta _1}$ to both sides of (17), we obtain

$$\begin{aligned} \begin{aligned} D_{a}^{\alpha _1+\beta _1-\alpha _1\beta _1}{\mathbb {V}}_1(t)&=- \sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1} D_{a}^{\alpha _i\beta _1+1-\beta _1} {\mathbb {V}}_1(t) \\&\quad +\frac{1}{\sigma _1}D_{a}^{\beta _1-\alpha _1\beta _1}f_1 (t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t)), \end{aligned}\end{aligned}$$

(27)

$$\begin{aligned} \begin{aligned} D_{a}^{\alpha _1+\beta _1-\alpha _1\beta _1}{\mathbb {V}}_1(t)&=- \sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1} D_{a}^{\alpha _i+(1-\beta _1)-\alpha _i(1-\beta _1)} {\mathbb {V}}_1(t) \\&\quad +\frac{1}{\sigma _1}D_{a}^{\beta _1-\alpha _1\beta _1} f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t)). \end{aligned} \end{aligned}$$

(28)

Now by applying the fractional-order integral $I_{a}^{\beta _1-\alpha _1\beta _1}$ to (28), we get

$$\begin{aligned}{} & {} \begin{aligned} I_{a}^{\beta _1-\alpha _1\beta _1}D_{a}^{\alpha _1+\beta _1-\alpha _1\beta _1}{\mathbb {V}}_1(t)&=- \sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1} I_{a}^{\beta _1-\alpha _1\beta _1}D_{a}^{\alpha _i+(1-\beta )-\alpha _i(1-\beta _1)} {\mathbb {V}}_1(t) \\&\quad +\frac{1}{\sigma _1}I_{a}^{\beta _1-\alpha _1\beta _1}D_{a}^{\beta _1 -\alpha _1\beta _1}f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t)), \end{aligned} \end{aligned}$$

(29)

$$\begin{aligned} \begin{aligned}&D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)\\ = & - \sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1} I_{a}^{\psi _i}D_{a}^{\alpha _i,1-\beta _1} {\mathbb {V}}_1(t) +f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t))-\frac{ (I_0^{1-\beta _1+\alpha _1\beta _1}f_1)(a)}{\Gamma (\beta _1-\alpha _1\beta _1)}t^{\beta _1-\alpha _1\beta _1-1}, \end{aligned} \end{aligned}$$

(30)

where $(I_0^{1-\beta _1+\alpha _1\beta _1}f_1)(0)=0$ is implied by Lemma 4. Hence, we get

$$\begin{aligned} \begin{aligned} D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)+ \sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1} I_{a}^{\psi _i}D_{a}^{\alpha _i,1-\beta _1} {\mathbb {V}}_1(t)=f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t)). \end{aligned}\end{aligned}$$

(31)

Hence the proof is completed.

3.1 Assumptions and Existence Results

In this subsection of the research work, we represent the desired solution for MFDDEs (10), in the form of an operator equation and provide some assumptions for the investigation of existence results for the proposed problem (10). Let us define an operator ${\mathbb {F}}_1{:}\,C_{1-\gamma _1}[a,\tau ]\mapsto C_{1-\gamma _1}[a,\tau ]$ given by

$$\begin{aligned}\begin{aligned} {\mathbb {F}}_1{\mathbb {V}}_1(t)&=(t-a)^{\gamma _1-1}\Bigg [ \frac{a_2}{d_1} -\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\sum _{i=2}^{n}\eta _j\sigma _i \int _a^{\tau _j}\frac{(\tau _j-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\phi _i)} {\mathbb {V}}_1(\tau _j){\text{d}}{\mathbb {U}} \\ {}&\quad +\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\eta _j\int _a^{\tau _j}\frac{ (\tau _j-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1 (\lambda _1 {\mathbb {U}})){\text{d}}{\mathbb {U}} \\ {}&\quad -\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\sum _{i=2}^{n}\kappa _l \sigma _i\int _a^{\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\phi _i+p_l-1}}{\Gamma (\phi _i+p_l)}{\mathbb {V}}_1({\mathbb {U}}){\text{d}}{\mathbb {U}} \\ {}&\quad +\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\kappa _l\int _a^{\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)} f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}})){\text{d}}{\mathbb {U}}\Bigg ] \\ {}&\quad +\sum _{i=2}^{n}\frac{\sigma _i}{\sigma _1}\int _a^{t}\frac{(t-{\mathbb {U}}) ^{\phi _i-1}}{\Gamma (\phi _i)}{\mathbb {V}}_1({\mathbb {U}}){\text{d}}{\mathbb {U}} \\ {}&\quad +\frac{1}{\sigma _1}\int _a^{t}\frac{(t-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}})){\text{d}}{\mathbb {U}},\quad \quad \forall \,{\mathbb {V}}_1(t)\in C_{1-\gamma _1}[a,\tau ]. \end{aligned} \end{aligned}$$

These assumptions hold.

$(\text{H}_1)$:

Let $f_1{:}\,(a,\tau ] \times {\bf{R}} \times {\bf{R}} \mapsto {\bf{R}}$ be a function such that $f_1\in C_{1-\gamma _1}^{\beta _1-\beta _1\alpha _1}$ for any ${\mathbb {V}}_1,{\mathbb {V}}_1^*\in C_{1-\gamma _1}[a,\tau ]$.

$(\text{H}_2)$:

For any ${\mathbb {V}}_1,{\mathbb {V}}_1^* \in C_{1-\gamma _1}[a,\tau ]$, there exist ${\mathbb {L}}_{f_1}, {\mathbb {L}}_{f_1^{\lambda }}\geqslant 0$, such that

$$\begin{aligned} & |f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t))-f_1(t,{\mathbb {V}}_1^*(t), {\mathbb {V}}_1^* (\lambda _1 t))|\\ \leqslant \, & {\mathbb {L}}_{f_1}|{\mathbb {V}}_1-{\mathbb {V}}_1^*|+{\mathbb {L}}_{f_1^{\lambda }}|{\mathbb {V}}_1(\lambda t)-{\mathbb {V}}_1^*(\lambda _1 t)|. \end{aligned}$$

$(\text{H}_3)$:

For any ${\mathbb {V}}_1\in C_{1-\gamma _1}[a,\tau ]$, there exist ${\mathbb {H}}_{f_1^a},{\mathbb {H}}_{f_1^b},{\mathbb {H}}_{f_1^c}>0$, such that

$$\begin{aligned}&|f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t)),{\mathbb {V}}_2(t)|\leqslant {\mathbb {H}}_{f_1^a}(t)+{\mathbb {H}}_{f_1^b}(t)|{\mathbb {V}}_1(t)|+{\mathbb {H}}_{f_1^c}(t)|{\mathbb {V}}_2(\lambda t)|. \end{aligned}$$

$(\text{H}_4)$:

For any non-decreasing positive function $\chi \in C_{1-\gamma _1}[a,\tau ],$ there exist positive constants ${\mathbb {M}}_{\alpha _1},{\mathbb {M}}_{\tau _j},{\mathbb {M}}_{\zeta _l}$ for $j=1,2,\cdots ,m_1$ and $l=1,2,\cdots ,m_2,$ such that

$$\begin{aligned}I_{a}^{\alpha _1} \chi (t)\leqslant {\mathbb {M}}_{\alpha _1}\chi (t),\ I_{a}^{\alpha _1} \chi (\tau _j)\leqslant {\mathbb {M}}_{\tau _j}\chi (t),\ I_{a}^{\alpha _1+p_l} \chi (\zeta _l)\leqslant {\mathbb {M}}_{\zeta _l}\chi (t),\quad \forall t\in \, [a,\tau ]. \end{aligned}$$

Remark 2

For computational convenience, we introduce the following notations:

$$\begin{aligned} \begin{aligned}&{\mathbb {L}}= \frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}\sum _{i=2}^{n}|\eta _j||\sigma _i| \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}(\tau _j-a)^{\phi _i+\gamma _1-1} \\ {}& \qquad +\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j| \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}(\tau _j-a)^{\alpha _1 +\gamma _1-1}({\mathbb {L}}_{f_1}+{\mathbb {L}}_{f_1^{\lambda }}) \\ {}& \qquad +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}|\kappa _l||\sigma _i| \frac{\beta (\gamma _1, \phi _i+p_l)}{\Gamma (\phi _i+p_l)}(\zeta _l-a)^{\phi _i+p_l+\gamma _1-1} \\ {}& \qquad +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l| \frac{\beta (\gamma _1, \alpha _1+p_l)}{\Gamma (\alpha _1+p_l)}(\zeta _l-a)^{\alpha _ 1+p_l+\gamma _1-1}({\mathbb {L}}_{f_1}+{\mathbb {L}}_{f_1^{\lambda }}) \\ {}& \qquad +\sum _{i=2}^{n}\frac{(\tau -a)^{\phi _i}|\sigma _i|}{|\sigma _1|} \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)} +\frac{(\tau -a)^{\alpha _1}}{|\sigma _1|} \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}({\mathbb {L}}_{f_1}+{\mathbb {L}}_{f_1^{\lambda }}). \end{aligned} \end{aligned}$$

(32)

Also, for the set of parameters used in the newly formulated model (10), we assume ${\mathbb {C}}_1,{\mathbb {C}}_2\in {\mathbf{R}}^+,$ such that

$$\begin{aligned} \begin{aligned} {\mathbb {C}}_1> & \frac{|a_2|}{|d_1|} +\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j| \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)} (\tau _j-a)^{\alpha _1+\gamma _1-1}{\mathbb {H}}_{f_1^a} \\ {}&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l| \frac{\beta (\gamma _1, \alpha _1+p_l)}{\Gamma (\alpha _1+p_l)} (\zeta _l-a)^{\alpha _1+p_l+\gamma _1-1}{\mathbb {H}}_{f_1^a} +\frac{(\tau -a)^{\alpha _1}}{|\sigma _1|} \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}{\mathbb {H}}_{f_1^a}, \end{aligned} \end{aligned}$$

(33)

and

$$\begin{aligned} \begin{aligned}{\mathbb {C}}_2> & \frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}\sum _{i=2}^{n}|\eta _j||\sigma _i| \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}(\tau _j-a)^{\phi _i+\gamma _1-1} \\ {}&+\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j| \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}(\tau _j-a)^{\alpha _1 +\gamma _1-1}({\mathbb {H}}_{f_1^b}+{\mathbb {H}}_{f_1^c}) \\ {}&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}|\kappa _l| |\sigma _i|\frac{\beta (\gamma _1, \phi _i+p_l)}{\Gamma (\phi _i+p_l)}(\zeta _l-a)^{\phi _i+p_l+\gamma _1-1} \\ {}&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l| \frac{\beta (\gamma _1, \alpha _1+p_l)}{\Gamma (\alpha _1+p_l)}(\zeta _l-a)^ {\alpha _1+p_l+\gamma _1-1}({\mathbb {H}}_{f_1^b}+{\mathbb {H}}_{f_1^c}) \\ {}&+\sum _{i=2}^{n}\frac{(\tau -a)^{\phi _i}|\sigma _i|}{|\sigma _1|} \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)} +\frac{(\tau -a)^{\alpha _1}}{|\sigma _1|} \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}({\mathbb {H}}_{f_1^b}+{\mathbb {H}}_{f_1^c}). \end{aligned} \end{aligned}$$

(34)

Theorem 4

Assume that $(\text{H}_1)$ and $(\text{H}_2)$ are satisfied with ${\mathbb {L}}<1,$ where ${\mathbb {L}}$ is defined by (32). Then, the system (10) of BVPs has a unique solution.

Proof

Let ${\mathbb {V}}_1(t),{\mathbb {V}}_1^*(t)\in C_{1-\gamma _1}[a,\tau ]$. We have

$$\begin{aligned} & \Big | \big ({\mathbb {F}}_1{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1^*(t) \big )(t-a)^{1-\gamma _1}\Big | \\ & \leqslant \frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}\sum _{i=2}^{n}|\eta _j||\sigma _ i|\int _a^{\tau _j}\frac{(\tau _j-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\phi _i)}\big | {\mathbb {V}}_1({\mathbb {U}})-{\mathbb {V}}_1^*({\mathbb {U}})\big |{\text{d}}{\mathbb {U}} \\ & +\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j|\int _a^{\tau _j}\frac{ (\tau _j-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \big |f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}}))-f_1({\mathbb {U}},{\mathbb {V}}_1^*({\mathbb {U}}),{\mathbb {V}}_ 1^*(\lambda _1 {\mathbb {U}}))\big |{\text{d}}{\mathbb {U}} \\ & +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}|\kappa _l||\sigma _ i|\int _a^{\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\phi _i+p_l-1}}{\Gamma (\phi _i+p_l)} \big |{\mathbb {V}}_1({\mathbb {U}})-{\mathbb {V}}_1^*({\mathbb {U}})\big |{\text{d}}{\mathbb {U}} \\ & +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l|\int _a^{\zeta _l} \frac{(\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)} \big |f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}}))-f_1({\mathbb {U}},{\mathbb {V}}_1^*({\mathbb {U}}),{\mathbb {V}} _1^*(\lambda _1 {\mathbb {U}}))\big |{\text{d}}{\mathbb {U}} \\ & +\sum _{i=2}^{n}\frac{(t-a)^{1-\gamma _1}|\sigma _i|}{|\sigma _1|}\int _a^{t}\frac{(t-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\phi _i)}\big |{\mathbb {V}}_1({\mathbb {U}}) -{\mathbb {V}}_1^*({\mathbb {U}})\big |{\text{d}}{\mathbb {U}} \\ & +\frac{(t-a)^{1-\gamma _1}}{|\sigma _1|}\int _a^{t}\frac{(t-{\mathbb {U}})^ {\alpha _1-1}}{\Gamma (\alpha _1)} \big |f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}}))-f_1({\mathbb {U}},{\mathbb {V}}_1^*({\mathbb {U}}),{\mathbb {V}}_ 1^*(\lambda _1 {\mathbb {U}}))\big |{\text{d}}{\mathbb {U}}. \end{aligned}$$

(35)

Now by making use of $(\text{H}_2)$ in (35), we get

$$\begin{aligned} \begin{aligned}& \, \Big |\big ({\mathbb {F}}_1{\mathbb {V}}_1(t)-{\mathbb {F}}_ 1{\mathbb {V}}_1^*(t)\big )(t-a)^{1-\gamma _1}\Big | \\ {}\leqslant & \frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}\sum _{i=2}^{n}|\eta _j||\sigma _i| \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}(\tau _j-a)^{\phi _i +\gamma _1-1}\big |\big |{\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}} \\ & \,+\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j| \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}(\tau _j-a)^{\alpha _1 +\gamma _1-1}({\mathbb {L}}_{f_1}+{\mathbb {L}}_{f_1^{\lambda }})\big |\big | {\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}} \\ & +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}|\kappa _l|| \sigma _i|\frac{\beta (\gamma _1, \phi _i+p_l)}{\Gamma (\phi _i+p_l)}(\zeta _l-a) ^{\phi _i+p_l+\gamma _1-1}\big |\big |{\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}} \\ & +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l| \frac{\beta (\gamma _1, \alpha _1+p_l)}{\Gamma (\alpha _1+p_l)}(\zeta _l-a)^ {\alpha _1+p_l+\gamma _1-1}({\mathbb {L}}_{f_1}+{\mathbb {L}}_{f_1^{\lambda }}) \big |\big |{\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}} \\ & +\sum _{i=2}^{n}\frac{(t-a)^{1-\gamma _1}|\sigma _i|}{|\sigma _1|} \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}(t-a)^{\phi _i+\gamma _1-1} \big |\big |{\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}} \\ & +\frac{(t-a)^{1-\gamma _1}}{|\sigma _1|} \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}(t-a)^{\alpha _1+\gamma _1-1} ({\mathbb {L}}_{f_1}+{\mathbb {L}}_{f_1^{\lambda }})\big |\big |{\mathbb {V}}_1 -{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}}. \end{aligned}\end{aligned}$$

(36)

By further simplification (36), we obtain

$$\begin{aligned}\begin{aligned} & \Big |\Big |{\mathbb {F}}_1{\mathbb {V}}_1(t) -{\mathbb {F}}_1 {\mathbb {V}}_1^*(t)\Big |\Big |_{C_{1-\gamma _1}}\\ \leqslant & \frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}\sum _{i=2}^ {n}|\eta _j||\sigma _i| \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}(\tau _j-a)^{\phi _i +\gamma _1-1}\big |\big |{\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}} \\ & +\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j| \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}(\tau _j-a)^ {\alpha _1+\gamma _1-1}({\mathbb {L}}_{f_1} +{\mathbb {L}}_{f_1^{\lambda }})\big |\big |{\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}} \\ & +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}|\kappa _l||\sigma _i| \frac{\beta (\gamma _1, \phi _i+p_l)}{\Gamma (\phi _i+p_l)}(\zeta _l-a)^{\alpha _1+p_l+\gamma _1-1- \alpha _i}\big |\big |{\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}} \\ & +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l| \frac{\beta (\gamma _1, \alpha _1+p_l)}{\Gamma (\alpha _1+p_l)}(\zeta _l-a)^{\alpha _1+p_l+ \gamma _1-1}({\mathbb {L}}_{f_1}+{\mathbb {L}}_{f_1^{\lambda }})\big |\big | {\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}} \\ & +\sum _{i=2}^{n}\frac{(\tau -a)^{\phi _i}|\sigma _i|}{|\sigma _1|} \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}\big |\big |{\mathbb {V}} _1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}} \\ & +\frac{(\tau -a)^{\alpha _1}}{|\sigma _1|}\frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}({\mathbb {L}}_{f_1}+{\mathbb {L}}_{f_1^{\lambda }})\big |\big | {\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |_{C_{1-\gamma _1}}\!, \end{aligned}\end{aligned}$$

or

$$\begin{aligned} \Big |\Big |{\mathbb {F}}_1{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1^*(t) \Big |\Big |_{C_{1-\gamma _1}}\leqslant {\mathbb {L}}\big |\big |{\mathbb {V}}_1-{\mathbb {V}}_1^*\big |\big |. \end{aligned}$$

Hence by the Banach contraction principle, we conclude that (10) has a unique solution.

Theorem 5

Consider $(\text{H}_1) \, \text{and} \, (\text{H}_2)$ hold, then the MFDDE (10) has at least one solution in $C_{1-\gamma _1}^{\gamma _1}\subset C_{1-\gamma _1}^{\alpha _1,\beta _1}$.

Proof

Consider ${\mathbb {H}}=\{{\mathbb {V}}_1\in C_{1-\gamma _1}[a,\tau ]{:}\, ||({\mathbb {V}}_1||_{C_{1-\gamma _1}}\leqslant {\mathbb {R}}\},$ with positive radius ${\mathbb {R}}>\frac{{\mathbb {C}}_1}{1-{\mathbb {C}}_2}$, where  ${\mathbb {C}}_1 {\text {and}} {\mathbb {C}}_2$ are defined by (33) and (34), respectively.

Step 1 We prove that ${\mathbb {F}}\big ({\mathbb {V}}_1(t)\big )\in {\mathbb {H}}\subset C_{1-\gamma _1}[a,\tau ]$ for every ${\mathbb {V}}_1(t)\in {\mathbb {H}}$, for obtaining the desired result, consider

$$\begin{aligned} & \;\; \Big |{\mathbb {F}}_1\big ({\mathbb {V}}_1(t)\big )(t-a)^{1-\gamma _1}\Big | \\& \leqslant \frac{|a_2|}{|d_1|} +\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}\sum _{i=2}^{n}|\eta _j|| \sigma _i|\int _a^{\tau _j}\frac{(\tau _j-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\phi _i)}\big |{\mathbb {V}}_1(\tau _j)\big |{\text{d}}{\mathbb {U}} \\ & +\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j|\int _a^{\tau _j} \frac{(\tau _j-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \big |f_1(\tau _j,{\mathbb {V}}_1(\tau _j),{\mathbb {V}}_1(\lambda _1 \tau _j))\big |{\text{d}}{\mathbb {U}} \\ & +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}|\kappa _l|| \sigma _i|\int _a^{\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\phi _i+p_l-1}}{\Gamma (\phi _i+p_l)}\big |{\mathbb {V}}_1(\zeta _l)\big |{\text{d}}{\mathbb {U}} \\ & +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l|\int _a^{\zeta _l} \frac{(\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)} \big |f_1(\zeta _l,{\mathbb {V}}_1(\zeta _l),{\mathbb {V}}_1(\lambda _1 \zeta _l))\big |{\text{d}}{\mathbb {U}} \\ & +\sum _{i=2}^{n}\frac{(t-a)^{1-\gamma _1}|\sigma _i|}{|\sigma _1|}\int _a^{t} \frac{(t-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\phi _i)}\big |{\mathbb {V}}_1(t)\big |{\text{d}}{\mathbb {U}} \\ & +\frac{(t-a)^{1-\gamma _1}}{|\sigma _1|}\int _a^{t}\frac{(t-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \big |f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_ 1(\lambda _1 {\mathbb {U}}))\big |{\text{d}}{\mathbb {U}}. \end{aligned}$$

(37)

Thanks to $(\text{H}_3)$ in the inequality (37), one has

$$\begin{aligned} \begin{aligned} & \;\; \Big |{\mathbb {F}}_1\big ({\mathbb {V}}_1(t)\big )(t-a)^{1-\gamma _1}\Big | \\ & \leqslant \frac{|a_2|}{|d_1|}+\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^ {m_1}\sum _{i=2}^{n}|\eta _j||\sigma _i| \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}(\tau _j-a)^{\phi _i +\gamma _1-1}{\mathbb {R}}\\ & +\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j|\frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}(\tau _j-a)^{\alpha _1+\gamma _1-1} ({\mathbb {H}}_{f_1^a}+{\mathbb {H}}_{f_1^b}{\mathbb {R}}+{\mathbb {H}}_{f_1^c}{\mathbb {R}})\\ & +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}|\kappa _l|| \sigma _i|\frac{\beta (\gamma _1,\phi _i+p_l)}{\Gamma (\phi _i+p_l)}(\zeta _l-a) ^{\phi _i+p_l+\gamma _1-1}{\mathbb {R}} \\ & +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l|\frac{\beta (\gamma _1,\alpha _1+p_l)}{\Gamma (\alpha _1+p_l)}(\zeta _l-a)^{\alpha _1+p_l +\gamma _1-1}({\mathbb {H}}_{f_1^a}+{\mathbb {H}}_{f_1^b}{\mathbb {R}}+{\mathbb {H}}_{f_1^c}{\mathbb {R}}) \\ & +\sum _{i=2}^{n}\frac{(t-a)^{1-\gamma _1}|\sigma _i|}{|\sigma _1|} \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}(t-a)^{\phi _i+\gamma _1-1}{\mathbb {R}} \\ & +\frac{(t-a)^{1-\gamma _1}}{|\sigma _1|}\frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}(t-a)^{\alpha _1+\gamma _1-1}({\mathbb {H}}_{f_1^a} +{\mathbb {H}}_{f_1^b}{\mathbb {R}}+{\mathbb {H}}_{f_1^c}{\mathbb {R}}). \end{aligned}\end{aligned}$$

(38)

After simplification, (38) can be written as

$$\begin{aligned} \begin{aligned} \Big |\Big |{\mathbb {F}}_1\big ({\mathbb {V}}_1(t)\big )\Big |\Big |_{C_{1-\gamma _1}} \leqslant & \frac{|a_2|}{|d_1|} +\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}\sum _{i=2}^{n}|\eta _j||\sigma _i| \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}(\tau _j-a)^{\phi _i+\gamma _1-1}{\mathbb {R}} \\ {}&+\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j| \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}(\tau _j-a)^{\alpha _1 +\gamma _1-1}({\mathbb {H}}_{f_1^a}+{\mathbb {H}}_{f_1^b}{\mathbb {R}} +{\mathbb {H}}_{f_1^c}{\mathbb {R}}) \\ {}&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}|\kappa _l||\sigma _i| \frac{\beta (\gamma _1, \phi _i+p_l)}{\Gamma (\phi _i+p_l)}(\zeta _l-a)^{\phi _i +p_l+\gamma _1-1}{\mathbb {R}} \\ {}&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l| \frac{\beta (\gamma _1, \alpha _1+p_l)}{\Gamma (\alpha _1+p_l)}(\zeta _l-a)^ {\alpha _1+p_l+\gamma _1-1}({\mathbb {H}}_{f_1^a}+{\mathbb {H}}_{f_1^b} {\mathbb {R}}+{\mathbb {H}}_{f_1^c}{\mathbb {R}}) \\ {}&+\sum _{i=2}^{n}\frac{(\tau -a)^{\phi _i}|\sigma _i|}{|\sigma _1|} \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}{\mathbb {R}} \\ {}&+\frac{(\tau -a)^{\alpha _1}}{|\sigma _1|} \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}({\mathbb {H}}_{f_1^a} +{\mathbb {H}}_{f_1^b}{\mathbb {R}}+{\mathbb {H}}_{f_1^c}{\mathbb {R}}), \end{aligned}\end{aligned}$$

(39)

or

$$\begin{aligned} \Big |\Big |{\mathbb {F}}_1\big ({\mathbb {V}}_1(t)\big )\Big |\Big |_{C_{1-\gamma _1}}< {\mathbb {R}}. \end{aligned}$$

(40)

Hence, from (40), one can infer that ${\mathbb {F}}_1\big ({\mathbb {V}}_1(t)\big )\in H$, for each ${\mathbb {V}}_1(t)\in {\mathbb {H}}\subset {C_{1-\gamma _1}}$.

Step 2 Now we claim that ${\mathbb {F}}_1$ is continuous, let ${\mathbb {V}}_{1_n}\in {\mathbb {H}}$ be a sequence which converges to ${\mathbb {V}}_1$. That is to derive that $<{\mathbb {F}}{\mathbb {V}}_{1_n}>\rightarrow <{\mathbb {F}}{\mathbb {V}}_1>$, as $n \mapsto \infty$. So, consider

$$\begin{aligned} \begin{aligned}&\bigg |\big ({\mathbb {F}}_1{\mathbb {V}}_{1_n}(t)-{\mathbb {F}}_1 {\mathbb {V}}_1(t)\big )(t-a)^{1-\gamma _1}\bigg |\\ {} \leqslant & \frac{|a_2|}{|d_1|} +\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}\sum _{i=2}^{n} |\eta _j||\sigma _i|\int _a^{\tau _j}\frac{(\tau _j-{\mathbb {U}})^ {\phi _i-1}}{\Gamma (\phi _i)}\big |{\mathbb {V}}_{1,n}(\tau _j)- {\mathbb {V}}_1(\tau _j)\big |{\text{d}}{\mathbb {U}} \\ {}&+\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j|\int _a^ {\tau _j}\frac{(\tau _j-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \big |f_1(\tau _j,{\mathbb {V}}_{1,n}(\tau _j),{\mathbb {V}}_{1,n} (\lambda _1 \tau _j))-f_1(\tau _j,{\mathbb {V}}_1(\tau _j), {\mathbb {V}}_1(\lambda _1 \tau _j))\big |{\text{d}}{\mathbb {U}} \\ {}&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}| \kappa _l||\sigma _i|\int _a^{\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^ {\phi _i+p_l-1}}{\Gamma (\phi _i+p_l)}\big |{\mathbb {V}}_{1,n} (\zeta _l)-{\mathbb {V}}_1(\zeta _l)\big |{\text{d}}{\mathbb {U}} \\ {}&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l|\int _a^ {\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)} \big |f_1(\zeta _l,{\mathbb {V}}_{1,n}(\zeta _l),{\mathbb {V}}_{1,n}( \lambda _1 \zeta _l))-f_1(\zeta _l,{\mathbb {V}}_1(\zeta _l),{\mathbb {V}} _1(\lambda _1 \zeta _l))\big |{\text{d}}{\mathbb {U}} \\ {}&+\sum _{i=2}^{n}\frac{(t-a)^{1-\gamma _1}|\sigma _i|}{|\sigma _1|} \int _a^{t}\frac{(t-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\phi _i)}\big | {\mathbb {V}}_{1,n}({\mathbb {U}})-{\mathbb {V}}_1({\mathbb {U}})\big |{\text{d}}{\mathbb {U}} \\ {}&+\frac{(t-a)^{1-\gamma _1}}{|\sigma _1|}\int _a^{t}\frac{(t-{\mathbb {U}})^ {\alpha _1-1}}{\Gamma (\alpha _1)} \big |f_1({\mathbb {U}},{\mathbb {V}}_{1,n}({\mathbb {U}}),{\mathbb {V}}_ {1,n}(\lambda _1 {\mathbb {U}}))-f_1({\mathbb {U}},{\mathbb {V}}_1 ({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}}))\big |{\text{d}}{\mathbb {U}}. \end{aligned} \end{aligned}$$

(41)

Evaluation of (41) yields

$$\begin{aligned} \begin{aligned}&\bigg |\bigg |{\mathbb {F}}_1{\mathbb {V}}_{1_n}(t)-{\mathbb {F}}_1 {\mathbb {V}}_1(t)\bigg |\bigg |_{C_{1-\gamma }}\\{} \leqslant & \frac{|a_2|}{|d_1|}+\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1} \sum _{i=2}^{n}|\eta _j||\sigma _i|\frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)} (\tau _j-a)^{\phi _i+\gamma _1-1}\big |\big |{\mathbb {V}}_{1,n}(t)-{\mathbb {V}}_1(t)\big |\big |_{C_{1-\gamma _1}} \\&+\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j|\frac{(\tau _j -{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}(\tau _j-a)^{\alpha _1 +\gamma _1-1}\big | \big |f_1({\mathbb {U}},{\mathbb {V}}_{1,n}({\mathbb {U}}), {\mathbb {V}}_{1,n}(\lambda _1 {\mathbb {U}}))\\&-f_1({\mathbb {U}},{\mathbb {V}}_1 ({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}}))\big |\big |_{C_{1-\gamma _1}} \\&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}|\kappa _l||\sigma _i |\frac{\beta (\gamma _1, \phi _i+p_l)}{\Gamma (\phi _i+p_l)}(\zeta _l-a)^{\phi _i+p_l +\gamma _1-1}\big |\big |{\mathbb {V}}_{1,n}(t)-{\mathbb {V}}_1(t)\big |\big |_{C_{1-\gamma _1}} \\&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l| \frac{\beta (\gamma _1, \alpha _1+p_l)}{\Gamma (\alpha _1+p_l)}(\zeta _l-a)^{\alpha _1 +p_l+\gamma _1-1}\big | \big |f_1({\mathbb {U}},{\mathbb {V}}_{1,n}({\mathbb {U}}), {\mathbb {V}}_{1,n}(\lambda _1 {\mathbb {U}}))\\&-f_1({\mathbb {U}},{\mathbb {V}}_1 ({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}}))\big |\big |_{C_{1-\gamma _1}} \\&+\sum _{i=2}^{n}\frac{(\tau -a)^{\phi _i}|\sigma _i|}{|\sigma _1|} \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}\big |\big |{\mathbb {V}}_{1,n}(t) -{\mathbb {V}}_1(t)\big |\big |_{C_{1-\gamma _1}} \\&+\frac{(\tau -a)^{\alpha _1}}{|\sigma _1|} \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}\big | \big |f_1({\mathbb {U}}, {\mathbb {V}}_{1,n}({\mathbb {U}}),{\mathbb {V}}_{1,n}(\lambda _1 {\mathbb {U}}))\\&-f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}}))\big |\big |_{C_{1-\gamma _1}}. \end{aligned} \end{aligned}$$

(42)

Hence, from (42), we infer that ${\mathbb {F}}_1$ is continuous.

Step 3 Need to show that ${\mathbb {F}}_1$ assigns bounded sets into equi-continuous sets, let $t_1,t_2\in [a,\tau ]$, $t_1< t_2,$ and ${\mathbb {V}}_1\in {\mathbb {H}}\subset C_{1-\gamma _1}[a,\tau ]$ as

$$\begin{aligned} \begin{aligned}&\bigg |(t_2-a)^{1-\gamma _1}{\mathbb {F}}_1{\mathbb {V}}_1(t_2)-(t_1-a)^ {1-\gamma _1}{\mathbb {F}}_1{\mathbb {V}}_1(t_1)\bigg |\\ {}\leqslant & \Big ((t_2-a)^{\gamma _1-1}-(t_1-a)^{\gamma _1-1}\Big )\Bigg | \frac{a_2}{d_1} -\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\sum _{i=2}^{n}\eta _j\sigma _i\int _a^ {\tau _j}\frac{(\tau _j-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\phi _i)}{\mathbb {V}}_1(\tau _j){\text{d}}{\mathbb {U}} \\ {}&+\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\eta _j\int _a^{\tau _j}\frac{(\tau _j -{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}})){\text{d}}{\mathbb {U}} \\ {}&-\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\sum _{i=2}^{n}\kappa _l\sigma _i\int _a^ {\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\phi _i+p_l-1}}{\Gamma (\phi _i+p_l)} {\mathbb {V}}_1({\mathbb {U}}){\text{d}}{\mathbb {U}} \\ {}&+\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\kappa _l\int _a^{\zeta _l}\frac{ (\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)} f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}})){\text{d}}{\mathbb {U}}\Bigg | \\ {}&+\sum _{i=2}^{n}\frac{|\sigma _i|}{|\sigma _1|}\int _a^{t_1}\frac{(t_2-a)^ {\gamma _1-1}(t_2-{\mathbb {U}})^{\phi _i-1}-(t_1-a)^{\gamma _1-1}(t_1-{\mathbb {U}})^ {\phi _i-1}}{\Gamma (\phi _i)}|{\mathbb {V}}_1({\mathbb {U}})|{\text{d}}{\mathbb {U}} \\ {}&+\sum _{i=2}^{n}\frac{(t_2-a)^{\gamma _1-1}|\sigma _i|}{|\sigma _1|}\int _{t_1}^{t_2} \frac{(t_2-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\phi _i)}|{\mathbb {V}}_1(t)|{\text{d}}{\mathbb {U}} \\ {}&+\frac{1}{|\sigma _1|}\int _a^{t_1}\frac{(t_2-a)^{\gamma _1-1}(t_2-{\mathbb {U}}) ^{\phi _i-1}-(t_1-a)^{\gamma _1-1}(t_1-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\alpha _1)} \Big |f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}}))\Big |{\text{d}}{\mathbb {U}} \\ {}&+\frac{(t_2-a)^{\gamma _1-1}}{|\sigma _1|}\int _{t_1}^{t_2}\frac{(t_2 -{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \Big |f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}}))\Big |{\text{d}}{\mathbb {U}}. \end{aligned}\end{aligned}$$

(43)

One can obtain the following expression using $(\text{H}_3)$ and evaluating integral involved in (43):

$$\begin{aligned} & \;\; \bigg |(t_2-a)^{1-\gamma _1}{\mathbb {F}}_1{\mathbb {V}}_1(t_2)-(t_1-a)^ {1-\gamma _1}{\mathbb {F}}_1{\mathbb {V}}_1(t_1)\bigg | \\ & \leqslant \Big ((t_2-a)^{\gamma _1-1}-(t_1-a)^{\gamma _1-1}\Big )\Bigg |\frac{a_2}{d_1}\\ &-\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\sum _{i=2}^{n}\eta _j\sigma _i \int _a^{\tau _j}\frac{(\tau _j-{\mathbb {U}})^{\phi _i-1}}{\Gamma (\phi _i)}{\mathbb {V}}_1(\tau _j){\text{d}}{\mathbb {U}}\\ {}&+\frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\eta _j\int _a^{\tau _j}\frac{ (\tau _j-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 {\mathbb {U}})){\text{d}}{\mathbb {U}}\\ {}&-\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\sum _{i=2}^{n}\kappa _l\sigma _i \int _a^{\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\phi _i+p_l-1}}{\Gamma (\phi _i+p_l)}{\mathbb {V}}_1({\mathbb {U}}){\text{d}}{\mathbb {U}}\\ {}&+\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\kappa _l\int _a^{\zeta _l}\frac{ (\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)} f_1({\mathbb {U}},{\mathbb {V}}_1({\mathbb {U}}),{\mathbb {V}}_1(\lambda _1 { \mathbb {U}})){\text{d}}{\mathbb {U}}\Bigg |\\ {}&+\sum _{i=2}^{n}\frac{|\sigma _i|}{|\sigma _1|}\frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}[-(t_2-t_1)^{\phi _i+\gamma _1-1}+(t_2-a)^{\phi _i+\gamma _1-1}-(t_1-a)^{\phi _i+\gamma _1-1}] \Big |\Big |{\mathbb {V}}_1\Big |\Big |_{C_{1-\gamma _1}}\\ {}&+\sum _{i=2}^{n}\frac{(t_2-a)^{\gamma _1-1}|\sigma _i|}{|\sigma _1|} \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}(t_2-t_1)^{\phi _i+\gamma _1-1} \Big |\Big |{\mathbb {V}}_1\Big |\Big |_{C_{1-\gamma _1}}\\ {}&+\frac{1}{|\sigma _1|} \frac{\beta (\gamma _1,\alpha _1)}{\Gamma (\alpha _1)}[-(t_2-t_1)^{\alpha _1+\gamma _1-1}\\&+(t_2-a)^{\alpha _1+\gamma _1-1}-(t_1-a)^{\alpha _1+\gamma _1-1}]\left( {\mathbb {H}}_{f_1^a}+{\mathbb {H}}_{f_1^b}\Big | \Big |{\mathbb {V}}_1\Big |\Big |_{C_{1-\gamma _1}}+{\mathbb {H}}_{f_1^c}\Big | \Big |{\mathbb {V}}_1\Big |\Big |_{C_{1-\gamma _1}}\right) \\ {}&+\frac{(t_2-a)^{\gamma _1-1}}{|\sigma _1|}\frac{\beta (\gamma _1,\alpha _1)}{\Gamma (\alpha _1)}(t_2-t_1)^{\alpha _1+\gamma _1-1}\left( {\mathbb {H}}_ {f_1^a}+{\mathbb {H}}_{f_1^b}\Big |\Big |{\mathbb {V}}_1\Big |\Big |_{C_{1-\gamma _1}} +{\mathbb {H}}_{f_1^c}\Big |\Big |{\mathbb {V}}_1\Big |\Big |_{C_{1-\gamma _1}}\right) . \end{aligned}$$

Clearly as $t_1 \mapsto t_2,$ we have $|(t_2-a)^{1-\gamma _1}{\mathbb {F}}_ 1{\mathbb {V}}_1(t_2)-(t_1-a)^{1-\gamma _1}{\mathbb {F}}_1{\mathbb {V}}_1(t_1),{\mathbb {V}}_2(t_1)|\mapsto 0$.

Therefore, in view of aforesaid steps and the Arzelá-Ascoli Theorem, the operator ${\mathbb {F}}_1{:} \,C_{1-\gamma _1}[a,\tau ]\mapsto _{1-\gamma _1}[a,\tau ]$ is completely continuous.

Step 4 We prove that there exists an open set ${\mathbb {W}}\subseteq C_{1-\gamma _1}[a,\tau ],$ such that ${\mathbb {V}}_1 \ne \lambda _*{\mathbb {F}}({\mathbb {V}}_1(t))$ for $\lambda _* \in (0,1)$ and ${\mathbb {V}}_1\in \partial U$. We assume ${\mathbb {V}}_1\in C_{1-\gamma _1}[a,\tau ]$ such that ${\mathbb {V}}_1=\lambda _*{\mathcal {F}}({\mathbb {V}}_1(t))$ where $\lambda _*\in (0,1)$. So for $t\in [a,\tau ],$ one can infer

$$\begin{aligned} \begin{aligned} \Big |\Big |{\mathbb {V}}_1(t)\Big |\Big |_{C_{1-\gamma _1}}= & \,\, \Big |\Big | \lambda _*{\mathbb {F}}_1{\mathbb {V}}_1(t)\Big |\Big |_{C_{1-\gamma _1}} \\ {} \leqslant & \,\, \frac{|a_2|}{|d_1|} +\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}\sum _{i=2}^{n}|\eta _j||\sigma _i| \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\phi _i)}(\tau _j-a)^{\phi _i +\gamma _1-1}||{\mathbb {V}}_1||_{C_{1-\gamma _1}[a,\tau ]} \\ {}&+\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j|\frac{(\tau _j -{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}(\tau _j-a)^ {\alpha _1+\gamma _1-1}({\mathbb {H}}_{f_1^a}+{\mathbb {H}}_{f_1^b}|| {\mathbb {V}}_1||_{C_{1-\gamma _1}[a,\tau ]}+{\mathbb {H}}_{f_1^c}|| {\mathbb {V}}_1||_{C_{1-\gamma _1}[a,\tau ]}) \\ {}&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}\sum _{i=2}^{n}|\kappa _l||\sigma _i| \frac{\beta (\gamma _1, \phi _i+p_l)}{\Gamma (\phi _i+p_l)}(\zeta _l-a)^ {\phi _i+p_l+\gamma _1-1}||{\mathbb {V}}_1||_{C_{1-\gamma _1}[a,\tau ]} \\ {}&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l| \frac{\beta (\gamma _1, \alpha _1+p_l)}{\Gamma (\alpha _1+p_l)}(\zeta _l-a)^ {\alpha _1+p_l+\gamma _1-1}({\mathbb {H}}_{f_1^a}+{\mathbb {H}}_{f_1^b}|| {\mathbb {V}}_1||_{C_{1-\gamma _1}[a,\tau ]}+{\mathbb {H}}_{f_1^c}|| {\mathbb {V}}_1||_{C_{1-\gamma _1}[a,\tau ]}) \\ {}&+\sum _{i=2}^{n}\frac{(\tau -a)^{\phi _i}|\sigma _i|}{|\sigma _1|} \frac{\beta (\gamma _1,\phi _i)}{\Gamma (\alpha _i\beta _1-\alpha _1\beta _1 +\alpha _1-\alpha _i)}||{\mathbb {V}}_1||_{C_{1-\gamma _1}[a,\tau ]} \\ {}&+\frac{(\tau -a)^{\alpha _1}}{|\sigma _1|} \frac{\beta (\gamma _1, \alpha _1)}{\Gamma (\alpha _1)}({\mathbb {H}}_{f_1^a} +{\mathbb {H}}_{f_1^b}||{\mathbb {V}}_1||_{C_{1-\gamma _1}[a,\tau ]}+{\mathbb {H}}_{f_1^c}| |{\mathbb {V}}_1||_{C_{1-\gamma _1}[a,\tau ]}). \end{aligned}\end{aligned}$$

(44)

Using (33) and (34) in (44), we get the inequality, given as

$$\begin{aligned} \Big |\Big |{\mathbb {V}}_1(t)\Big |\Big |_{C_{1-\gamma _1}}\leqslant {\mathbb {C}}_1 +{\mathbb {C}}_2\Big |\Big |{\mathbb {V}}_1(t)\Big |\Big |_{C_{1-\gamma _1}}, \end{aligned}$$

or

$$\begin{aligned} \Big |\Big |{\mathbb {V}}_1(t)\Big |\Big |_{C_{1-\gamma _1}} \leqslant {\mathbb {M}}=\frac{{\mathbb {C}}_1}{1-{\mathbb {C}}_2}. \end{aligned}$$

Let ${\mathbb {W}}=\{\,{\mathbb {V}}_1\in C_{1-\gamma _1}[a,\tau ]{:}\, {\mathbb {M}}+1\geqslant ||{\mathbb {V}}_1||_{C_{1-\gamma _1}}\}$, by selecting ${\mathbb {W}}$, one can not find ${\mathbb {V}}_1\,\in \partial \, {\mathbb {W}}$ for which, ${\mathbb {V}}_1=\lambda _*{\mathbb {F}}_1{\mathbb {V}}_1(t),$ for $\lambda _*\in (0,1)$. So due to the Schauder nonlinear alternative, the considered problem (10) has at least one solution in $\overline{{\mathbb {W}}}$.

4 H-U Stability Results

This part is enriched by deriving the stability results for the proposed problem (10).

Definition 5

Let ${\mathbb {V}}_1^*(t)$ be a solution of (10) and one can find a constant positive real number ${\mathcal {B}}$, at every $\epsilon >0$ and any solution ${\mathbb {V}}_1(t) \in C_{1-\gamma _1}[a,\tau ]$ of the underlying inequality,

$$\begin{aligned} & \quad \left| \sigma _1D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)+\sum _{i=2}^{n} \sigma _iI_{a}^{\psi _i}D_{a}^{\alpha _i,1-\beta _1}{\mathbb {V}}_1(t) -f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t))\right| \nonumber \\ & \leqslant \epsilon ,\, t\in [a,\tau ],\, 0\leqslant a <\tau \in {\bf{R}}^+, \end{aligned}$$

(45)

such that, $||{\mathbb {V}}_1-{\mathbb {V}}_1^*||_{C_{1-\gamma _1}}\leqslant {\mathcal {B}}_1 \epsilon$, one can infer that (10) has the H-U stable solution. Moreover, the solution of (10) is g-H-U stable, if there exists a positive function ${\mathcal {K}}{:}\,(0,\infty )\mapsto (0,\infty )$ with ${\mathcal {K}}(0)=0,$ such that $||{\mathbb {V}}_1-{\mathbb {V}}_1^*||_{C_{1-\gamma _1}}\leqslant {\mathcal {B}}_1 {\mathcal {K}}(\epsilon ).$

Definition 6

The problem (10) has an H-U-R stable solution, w.r.t. function $\chi \in C_{1-\gamma _1}[a,\tau ]$ if we have ${\mathcal {B}}_2> 0$ and $\epsilon > 0$ for each solution ${\mathbb {V}}_1(t) \in C_{1-\gamma _1}[a,\tau ]$ of the underlying differential inequality

$$\begin{aligned} & \quad \left| \sigma _1D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)+\sum _{i=2}^{n}\sigma _iI_{a} ^{\psi _i}D_{a}^{\alpha _i,1-\beta _1}{\mathbb {V}}_1(t)-f_1(t, {\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t))\right| \nonumber \\ & \leqslant \chi (t)\epsilon ,\, t\in [a,\tau ],\, 0\leqslant a <\tau \in {\bf{R}}^+, \end{aligned}$$

(46)

and a unique solution ${\mathbb {V}}_1 \in C_{1-\gamma _1}[a,\tau ]$ of the given problem (10) such that $||{\mathbb {V}}_1-{\mathbb {V}}_1^*||_{C_{1-\gamma _1}}\leqslant {\mathcal {B}}_2\chi (t)\epsilon .$

Definition 7

The problem (10) has a g-H-U-R stable solution, w.r.t. function $\chi \in C_{1-\gamma _1}[a,\tau ]$ if we have ${\mathcal {B}}_2> 0$ and $\epsilon > 0$ for each solution ${\mathbb {V}}_1(t) \in C_{1-\gamma _1}[a,\tau ]$ of the underlying differential inequality,

$$\begin{aligned} & \quad \left| \sigma _1D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)+\sum _{i=2}^{n} \sigma _iI_{a}^{\psi _i}D_{a}^{\alpha _i,1-\beta _1} {\mathbb {V}}_1(t)-f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t))\right| \nonumber \\ & \leqslant \chi (t),\, t\in [a,\tau ],\, 0\leqslant a <\tau \in {\bf{R}}^+, \end{aligned}$$

(47)

and a unique solution ${\mathbb {V}}_1 \in C_{1-\gamma _1}[a,\tau ]$ of the given problem (10), such that $||{\mathbb {V}}_1-{\mathbb {V}}_1^*||_{C_{1-\gamma _1}}\leqslant {\mathcal {B}}_2\chi (t)\epsilon .$

Remark 3

The solution of the inequality (45) is ${\mathbb {V}}_1$, if and only if there exists a function $\xi \in C[a,\tau ]$, which depends on $\mathbb {V_1}$ such that

(i):

$\epsilon \geqslant \xi (t), \,\ t\in J;$

(ii):

$\sigma _1D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)+\sum _{i=2}^{n}\sigma _iI_{a}^ {\psi _i}D_{a}^{\alpha _i,1-\beta _1}{\mathbb {V}}_1(t)-f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t))-\xi (t)=0.$

Remark 4

Let ${\mathbb {V}}_1\in C_{1-\gamma -1}[a,\tau ]$ be the solution of (46), if and only if there exists a function $\xi \in C[a,\tau ]$, which depends on ${\mathbb {V}}_1$, such that

(i):

  $\xi (t) \leqslant \chi (t)\epsilon , \, t\in J,$

(ii):

$\sigma _1D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)+\sum _{i=2}^ {n}\sigma _iI_{a}^{\psi _i}D_{a}^{\alpha _i,1-\beta _1}{\mathbb {V}}_1(t)-f_1(t, {\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t))-\xi (t)=0.$

Remark 5

Let ${\mathbb {V}}_1\in C_{1-\gamma -1}[a,\tau ]$ be the solution of (46). Then there exists a function $\xi \in C[a,\tau ]$, which depends on ${\mathbb {V}}_1$, such that

(i):

$\xi (t) \leqslant \chi (t), \, t\in J$,

(ii):

$\sigma _1D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t) +\sum _{i=2}^{n}\sigma _iI_{a}^{\psi _i}D_{a}^{\alpha _i,1-\beta _1} {\mathbb {V}}_1(t)-f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t))-\xi (t)=0.$

Lemma 5

Consider ${\mathbb {V}}_1(t) \in C_{1-\gamma _1}[a,\tau ]$ is the solution of FDDE

$$\begin{aligned}\left\{ \begin{aligned}&\sigma _1D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)+\sum _{i=2}^{n} \sigma _iI_{a}^{\psi _i}D_{a}^{\alpha _i,1-\beta _1} {\mathbb {V}}_1(t)=\xi (t)+f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t)),\, t\in [a,\tau ],\, 0\leqslant a <\tau \in {\bf{R}}^+,\\ {}&a_1I_{a}^{q_1}{\mathbb {V}}_1(a^+) =a_2+a_3\sum _{j=1}^{m_1}\eta _j{\mathbb {V}}_1(\tau _j) +a_4\sum _{l=1}^{m_2}\kappa _lI_{a}^{p_l}{\mathbb {V}}_1(\zeta _l),\quad \lambda _1\in [0,1]. \end{aligned}\right. \end{aligned}$$

Then, ${\mathbb {V}}_1$ satisfies the following relation:

$$\begin{aligned} \left| {\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t)\right| \leqslant {\mathbb {G}}_1 \epsilon , \end{aligned}$$

where

$$\begin{aligned} {\mathbb {G}}_1=\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1} ^{m_1}|\eta _j|\frac{(\tau _j-a)^{\alpha _1}}{\Gamma (\alpha _1+1)} +\sum _{l=1}^{m_2}|\kappa _l|\frac{|a_4|(\zeta _l-a)^{\alpha _1 +p_l}}{|d_1||\sigma _1|\Gamma (\alpha _1+p_l+1)} +\frac{1}{|\sigma _1|}\frac{(t-a)^{\alpha _1+1-\gamma _1}}{\Gamma (\alpha _1+1)}. \end{aligned}$$

Proof

Suppose the solution of (10) is ${\mathbb {V}}_1 \in C_{1-\gamma _1}[a,\tau ]$, then

$$\begin{aligned} \begin{aligned} {\mathbb {V}}_1(t)=& \,\,{\mathbb {F}}_1{\mathbb {V}}_1(t)+ (t-a)^{\gamma _1-1}\Bigg [ \frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\eta _j\int _a^{\tau _j} \frac{(\tau _j-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \xi (\tau _j){\text{d}}{\mathbb {U}} \\ {}&+\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\kappa _l\int _a^{\zeta _l} \frac{(\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)} \xi (\zeta _l){\text{d}}{\mathbb {U}}\Bigg ] +\frac{1}{\sigma _1}\int _a^{t}\frac{(t-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \xi ({\mathbb {U}}){\text{d}}{\mathbb {U}}. \end{aligned} \end{aligned}$$

(48)

Using Remark 3 in (48), we get

$$\begin{aligned} &\;\; ||\big ({\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t) \big )||_{C_{1-\gamma _1}} \\& \leqslant \epsilon \sup _{t\in J}\Bigg \{ \Bigg [\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j| \int _a^{\tau _j}\frac{(\tau _j-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)}{\text{d}}{\mathbb {U}}\\ {}&+\frac{|a_4|}{|d_1|\sigma _1}\sum _{l=1}^{m_2}|\kappa _l| \int _a^{\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)}{\text{d}}{\mathbb {U}}\Bigg ] +\frac{(t-a)^{1-\gamma _1}}{|\sigma _1|}\int _a^{t} \frac{(t-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)}{\text{d}}{\mathbb {U}} \Bigg \}. \end{aligned}$$

(49)

By integrating (49), we have

$$\begin{aligned} \begin{aligned}&||\big ({\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t) \big )||_{C_{1-\gamma _1}} \\ \leqslant & \epsilon \Bigg [\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1} |\eta _j|\frac{(\tau _j-a)^{\alpha _1}}{\Gamma (\alpha _1+1)} +\sum _{l=1}^{m_2}|\kappa _l|\frac{|a_4|(\zeta _l-a)^{\alpha _1+p_l}}{|d_1||\sigma _1|\Gamma (\alpha _1+p_l+1)} +\frac{1}{|\sigma _1|}\frac{(t-a)^{\alpha _1+1-\gamma _1}}{\Gamma (\alpha _1+1)} \Bigg ]. \end{aligned} \end{aligned}$$

(50)

Using (48) in (50), we obtain the desired result,

$$\begin{aligned} |{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t)|&\leqslant {\mathbb {G}}^*\epsilon . \end{aligned}$$

Theorem 6

The problem under consideration (10) has H-U and g-H-U stable solutions if the assumptions $(\text {H}_1)$ and $(\text {H}_2)$ hold together with ${\mathbb {L}}<1$, where ${\mathbb {L}}$ is defined by (32).

Proof

Suppose ${\mathbb {V}}_1 \in C_{1-\gamma _1}[a,\tau ]$ is any solution and ${\mathbb {V}}_1^*$ is a unique solution of (10), then

$$\begin{aligned}\begin{aligned}& || {\mathbb {V}}_1(t)-{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}=||{\mathbb {V}}_1(t)-{\mathbb {F}}_1 {\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}\\ =&||{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t)+{\mathbb {F}}_1{\mathbb {V}}_1(t) -{\mathbb {F}}_1{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}\\ {}\leqslant & ||{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t)||_{C_{1-\gamma _1}}+||{\mathbb {F}} _1{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}\!. \end{aligned}\end{aligned}$$

In light of Theorem 4 and Lemma 5, we obtain

$$\begin{aligned}\begin{aligned}&||{\mathbb {V}}_1(t)-{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}\leqslant {\mathbb {G}}^* \epsilon + {\mathbb {L}}||{\mathbb {V}}_1-{\mathbb {V}}_1^*||_{C_{1-\gamma _1}}\!,\\ {}&||{\mathbb {V}}_1(t)-{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}-{\mathbb {L}}||{\mathbb {V}} _1-{\mathbb {V}}_1^*||_{C_{1-\gamma _1}}\leqslant {\mathbb {G}}^*\epsilon ,\\ {}&||{\mathbb {V}}_1(t)-{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}\leqslant \frac{{\mathbb {G}}^*}{1-{\mathbb {L}}}\epsilon . \end{aligned} \end{aligned}$$

Let ${\mathcal {B}}_1=\frac{{\mathbb {G}}^*}{1-{\mathbb {L}}}$. Then the problem (10) has an H-U stable solution. In addition, set ${\mathcal {K}}(\epsilon )=\epsilon ,$ then the problem (10) has a g-H-U stable solution.

Lemma 6

Consider ${\mathbb {V}}_1 \in C_{1-\gamma _1}[a,\tau ]$ is the solution of FDDE and assumption ($H_4$) holds, then one has

$$\begin{aligned} \left\{ \begin{aligned}&\sigma _1D_{a}^{\alpha _1,\beta _1}{\mathbb {V}}_1(t)+\sum _{i=2}^{n}\sigma _iI_{a}^ {\psi _i}D_{a}^{\alpha _i,1-\beta _1}{\mathbb {V}}_1(t)=f_1(t,{\mathbb {V}}_1(t), {\mathbb {V}}_1(\lambda _1 t))+\xi (t),\, t\in [a,\tau ],\, 0\leqslant a <\tau \in {\bf{R}}^+,\\ {}&a_1I_{a}^{q_1}{\mathbb {V}}_1(a^+) =a_2+a_3\sum _{j=1}^{m_1}\eta _j{\mathbb {V}}_1(\tau _j) +a_4\sum _{l=1}^{m_2}\kappa _lI_{a}^{p_l}{\mathbb {V}}_1(\zeta _l),\quad \lambda _1\in [0,1], \end{aligned}\right. \end{aligned}$$

(51)

which satisfies the following relation:

$$\begin{aligned} ||{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t) ||_{C_{1-\gamma _1}}&\leqslant \epsilon {\mathbb {G}}_2\chi (t), \end{aligned}$$

(52)

where

$$\begin{aligned} {\mathbb {G}}_2=\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}\eta _j {\mathbb {M}}_{\tau _j} +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l|{\mathbb {M}}_{p_l} +\frac{(\tau -a)^{1-\gamma _1}}{|\sigma _1|}{\mathbb {M}}_{\alpha _1}. \end{aligned}$$

Proof

Suppose ${\mathbb {V}}_1 \in C_{1-\gamma _1}[a,\tau ]$ is the solution of (51), then

$$\begin{aligned} \begin{aligned} {\mathbb {V}}_1(t) = &{\mathbb {F}}_1{\mathbb {V}}_1(t) +(t-a)^{\gamma _1-1}\Bigg [ \frac{a_3}{d_1\sigma _1}\sum _{j=1}^{m_1}\eta _j\int _a^{\tau _j} \frac{(\tau _j-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \xi (\tau _j){\text{d}}{\mathbb {U}}\\ {}&+\frac{a_4}{d_1\sigma _1}\sum _{l=1}^{m_2}\kappa _l\int _a^ {\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)} \xi (\zeta _l){\text{d}}{\mathbb {U}}\Bigg ]+\frac{1}{\sigma _1}\int _a^{t} \frac{(t-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \xi ({\mathbb {U}}){\text{d}}{\mathbb {U}}. \end{aligned}\end{aligned}$$

(53)

Using Remark 4 in (53), we get

$$\begin{aligned} \begin{aligned} &\;\; |{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t)|\\& \leqslant \epsilon \Bigg \{ (t-a)^{\gamma _1-1}\Bigg [\frac{|a_3|}{|d_1||\sigma _1|} \sum _{j=1}^{m_1}|\eta _j|\int _a^{\tau _j}\frac{(\tau _j-{\mathbb {U}})^ {\alpha _1-1}}{\Gamma (\alpha _1)} \chi (\tau _j){\text{d}}{\mathbb {U}}\\ {}&+\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l|\int _a^ {\zeta _l}\frac{(\zeta _l-{\mathbb {U}})^{\alpha _1+p_l-1}}{\Gamma (\alpha _1+p_l)} \chi (\zeta _l){\text{d}}{\mathbb {U}}\Bigg ] +\frac{1}{|\sigma _1|}\int _a^{t}\frac{(t-{\mathbb {U}})^{\alpha _1-1}}{\Gamma (\alpha _1)} \chi ({\mathbb {U}}){\text{d}}{\mathbb {U}}\Bigg \}. \end{aligned}\end{aligned}$$

(54)

By using ($(\text{H}_4)$) in (54), we obtain the relation (52), given as

$$\begin{aligned}\begin{aligned} ||{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t)||_{C_{1-\gamma _1}}&\leqslant \epsilon \Bigg \{\frac{|a_3|}{|d_1||\sigma _1|}\sum _{j=1}^{m_1}|\eta _j| {\mathbb {M}}_{\tau _j} +\frac{|a_4|}{|d_1||\sigma _1|}\sum _{l=1}^{m_2}|\kappa _l|{\mathbb {M}}_{p_l} +\frac{(\tau -a)^{1-\gamma _1}}{|\sigma _1|}{\mathbb {M}}_{\alpha _1} \Bigg \} \chi (t). \end{aligned}\end{aligned}$$

Theorem 7

The problem under consideration (10) has H-U-R and g-H-U-R stable solutions if the assumptions $(\text{H}_1)$, $(\text{H}_2)$, and $(\text{H}_4)$ hold together with ${\mathbb {L}}<1$, where ${\mathbb {L}}$ is defined in (32).

Proof

Suppose ${\mathbb {V}}_1 \in C_{1-\gamma _1}[a,\tau ]$ is any solution and ${\mathbb {V}}_1^*$ is a unique solution of (10), then

$$\begin{aligned}\begin{aligned}& ||{\mathbb {V}}_1(t)-{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}=||{\mathbb {V}}_1(t) -{\mathbb {F}}_1{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}\\ {}=& ||{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t)+{\mathbb {F}}_1{\mathbb {V}} _1(t)-{\mathbb {F}}_1{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}\\ {}\leqslant & ||{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1(t)||_{C_{1-\gamma _1}} +||{\mathbb {F}}_1{\mathbb {V}}_1(t)-{\mathbb {F}}_1{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}\!. \end{aligned}\end{aligned}$$

In light of Theorem 4 as well as Lemma 6, we obtain

$$\begin{aligned}\begin{aligned}&||{\mathbb {V}}_1(t)-{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}\leqslant {\mathbb {G}}_2\chi (t)\epsilon + {\mathbb {L}}||{\mathbb {V}}_1-{\mathbb {V}}_1^*||_{C_{1-\gamma _1}}\!,\\ {}&||{\mathbb {V}}_1(t)-{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}-{\mathbb {L}}||{\mathbb {V}}_1 -{\mathbb {V}}_1^*||_{C_{1-\gamma _1}}\leqslant {\mathbb {G}}_2 \chi (t)\epsilon ,\\ {}&||{\mathbb {V}}_1(t)-{\mathbb {V}}_1^*(t)||_{C_{1-\gamma _1}}\leqslant \frac{1}{1-{\mathbb {L}}}{\mathbb {G}}_2\chi (t)\epsilon . \end{aligned}\end{aligned}$$

Let ${\mathcal {B}}_2=\frac{1}{1-{\mathbb {L}}}{\mathbb {G}}_2$. Then the problem (10) has an H-U-R stable solution. In addition, set $\epsilon =1,$ then the problem (10) has a g-H-U-R stable solution.

5 Example

We use an example in this section to demonstrate the utility of our aforesaid findings.

Example 1

Consider the following coupled system of MFDDEs:

$$\begin{aligned} \left\{ \begin{aligned}&D^{\frac{1}{5},\frac{9}{10}}{\mathbb {V}}_1(t)+\sum _{2=1}^{84}\frac{68}{i^4}I^ {\frac{1}{10i^3+40}+\frac{31}{50}}D^{\frac{1}{i^3+4},\frac{1}{10}}{\mathbb {V}}_1(t)= \frac{t^2}{55}{\mathbb {V}}_1(t)+\frac{\text{e}^{-t^2}}{67}{\mathbb {V}}_1\bigg(\frac{t}{4}\bigg) +\frac{\sinh (t^4)+\tanh {t}}{\sin ^{-1}(t)+t^2},\\ {}&13I^{\frac{3}{2}}{\mathbb {V}}_1(0^+) =5+15\sum _{j=1}^{31}(3j+10){\mathbb {V}}_1\bigg(\frac{1}{j+1}\bigg) +12\sum _{l=1}^{42}4lI^{\frac{1}{l+1}}{\mathbb {V}}_1\bigg(\frac{1}{3l^2} \bigg), \,\,\, t\in [0,1]. \, \end{aligned}\right. \end{aligned}$$

(55)

Here

$n=84$	$\sigma _i=\frac{68}{i^4}$	$\alpha _i=\frac{i}{i^3+4}$	$a=0$	$\tau =1$	$a_1=13$	$a_2=5$	$a_3=15$
$a_4=12$	$\lambda _1=\frac{1}{4}$	$m_1=31$	$m_2=42$	$q_1=\frac{3}{2}$	$p_l=\frac{1}{l+1}$	$\tau _j=\frac{1}{j+1}$	$\zeta _l=\frac{1}{3l^2}$
$\eta _j=3j+10$	$\kappa _l=4l$

and

$$\begin{aligned} f_1(t,{\mathbb {V}}_1(t),{\mathbb {V}}_1(\lambda _1 t))= \frac{t^2}{55}{\mathbb {V}}_1(t)+\frac{\text{e}^{-t^2}}{67}{\mathbb {V}}_1 \bigg(\frac{t}{3}\bigg)+\frac{\sinh (t^4)+\tanh {t}}{\sin ^{-1}(t)+t^2}. \end{aligned}$$

(56)

Now the reader can easily obtain from (56) that ${\mathbb {L}}_{f_1}=\frac{1}{55},\,\, {\mathbb {L}}_{f_1^\lambda }=\frac{1}{67},\,$ consequently ${\mathbb {L}}=0.100\,8 <1$. So by Theorem 4, the problem (55) has a unique solution. Set $\chi _1(t)=5t^2+1$, for every $t\in [0,1]$, we can easily find

${\mathbb {M}}_{\alpha _1}=7.260\,8$	${\mathbb {M}}_{\tau _1}=4.106$	${\mathbb {M}}_{\tau _2}= 2.817\,1$	${\mathbb {M}}_{\tau _3}= 2.201\,1$	${\mathbb {M}}_{\tau _4}= 1.841\,9$	${\mathbb {M}}_{\tau _5}= 1.606\,8$
${\mathbb {M}}_{\tau _6}= 1.440\,9$	${\mathbb {M}}_{\tau _7}= 1.317\,3$	${\mathbb {M}}_{\tau _8}= 1.221\,7$	${\mathbb {M}}_{\tau _9}= 1.145\,3$	${\mathbb {M}}_{\tau _{10}}= 1.082\,8$	${\mathbb {M}}_{\tau _{11}}= 1.030\,7$
${\mathbb {M}}_{\tau _{j}}= 1\, {\text{for}}\, j=12,13,\cdots, 31$		${\mathbb {M}}_{\zeta _1}= 2.275$	${\mathbb {M}}_{\zeta _2}= 1.717\,3$	${\mathbb {M}}_{\zeta _3}= 1.439\,6$	${\mathbb {M}}_{\zeta _4}= 1.268\,7$
${\mathbb {M}}_{\zeta _5}= 1.151\,3$	${\mathbb {M}}_{\zeta _6}= 1.065\,0$	${\mathbb {M}}_{\zeta _l}= 1\, {\text{for}} \, l=11,12, \cdots,42$

such that,

$$\begin{aligned}I^{\alpha _1} \chi (t)\leqslant {\mathbb {M}}_{\alpha _1}\chi (t),\quad I^{\alpha _1} \chi (\tau _j)\leqslant {\mathbb {M}}_{\tau _j}\chi (t),\quad I^{\alpha _1+p_l} \chi (\zeta _l)\leqslant {\mathbb {M}}_{\zeta _l}\chi (t),\quad \forall t\in \, [0,1] \end{aligned}$$

for $j=1,2,\cdots ,m_1$ and $l=1,2,\cdots ,m_2$.

As the condition, $(\text{H}_5)$ is satisfied. So by Theorem 7, the solution of MFDDE (55) is the H-U-R and g-H-U-R stable solution. In a similar way, we can see that the MFDDE (55) is H-U and g-H-U stable using Theorem 6.

6 Conclusion

In this work, we have investigated a generalized system of multi-term Hilfer-type FDEs involving proportional type delay under boundary conditions. The concerned system has significant applications in modeling real-world problems in various fields of dynamics, mechanics, etc. By employing the tool of the fixed point theory some adequate conditions have been established for the existence and uniqueness of the solution. Further, via the nonlinear functional analysis, some interesting results about the H-U stability and its different kinds have been developed. The whole findings of the manuscript have been testified by proper test examples.