2 Preliminary results

Let $I:=[a,b]$ . Throughout this work, $C(I,\mathbb {E})$ denotes all $\mathbb {E}$ -valued continuous functions on I with the sup norm

$$ \begin{align*}\|z\|_{\infty}=\sup_{t\in I}\|z(t)\|. \end{align*} $$

We endow the space $L^{p}_{\varphi }(I,{\mathbb {E}})$ , $1\leq p<\infty $ , of Bochner integrable functions z on I for which $\|z\|_{L^p_{\varphi }}< \infty $ , with the norm $\|z\|_{L^p_{\varphi }} =\left (\displaystyle \int _a^b \varphi '(s)\| z(s)\|^p d s \right )^{\frac {1}{p}}$ . If $\varphi (t)=t$ , the space $L^{p}_{\varphi }(I,\mathbb {E})$ coincides with the usual $L^{p}(I,\mathbb {E})$ space.

If $p=\infty $ , $L^{\infty }(I,\mathbb {E})$ is the Banach space of all equivalence classes of essentially bounded measurable functions on I equipped with the norm

$$ \begin{align*}\|z\|_{L^\infty}=\displaystyle \operatorname*{\mbox{ess sup}}_{t\in I}\|z(t)\|=\inf\{M>0;\,\|z(t)\|\leq M\,\,\text{for almost every }\,t\in I \}. \end{align*} $$

We also define

$$ \begin{align*}\mathbb{S}_{a^+}^{1,+}(I,\mathbb{R}) =\{\varphi ~:~ \varphi\in C^{1}(I,\mathbb{R})~~\text{and}~~ \varphi'(t)> 0~~\text{for all}~~ t\in I\}. \end{align*} $$

For $ \varphi \in \mathbb {S}_{a^+}^{1,+}(I, \mathbb {R}) $ and $t, s \in I$ , $(t> s)$ , we pose

$$ \begin{align*}(\varphi(t) -\varphi(s))^\alpha=\varphi(t,s)^\alpha,\quad \text{for }~\alpha\in \mathbb{R}. \end{align*} $$

Definition 2.1 [Reference Diaz and Teruel15]

For $ \alpha , \beta \in (0,+\infty ) $ , the gamma and beta functions are given by

$$ \begin{align*}\Gamma(\alpha)=\int^{\infty}_{0} t^{\alpha -1}e^{-t}dt, \qquad B(\alpha,\beta)=\int^1_0 t^{\alpha-1} (1-t)^{\beta -1}dt. \end{align*} $$

Lemma 2.2 [Reference Nguyen26]

Let $\psi \in \mathbb {S}_{a^+}^{1,+}(I, \mathbb {R})$ , $0< \alpha < 1$ , $0\leqslant \beta < \alpha $ , $a \leq \tau \leq \zeta \leq t \leq b$ , and let

$$ \begin{align*}\Theta_{\alpha,\beta}(\tau,\zeta,t,\varphi) = \int_{\tau}^{\zeta} \varphi(t,s)^{\alpha -1} \varphi(s,a)^{-\beta}\varphi'(s)ds. \end{align*} $$

Then, for all $t \in I $ , we have

$$ \begin{align*}\Theta_{\alpha,\beta}(a,t,t,\varphi) = \varphi(t,a)^{\alpha -\beta} B(\alpha,1-\beta) \end{align*} $$

and

$$ \begin{align*}0 \leq \Theta_{\alpha,\beta}(\tau,\zeta,t,\varphi) \leq \left( \frac{2^{\beta}}{\alpha} + \frac{2^{1-\alpha}}{1-\beta} \right) \max\{1, \varphi(b,a)^{\alpha -\beta}\}\varphi(\zeta,\tau)^{\min\{\alpha,1-\beta,\alpha -\beta\} }. \end{align*} $$

Remark 2.4 By Lemma 2.2, we get

$$ \begin{align*}\psi(t,\cdot)^{\alpha_i -1}\psi(\cdot,a)^{\gamma_j -1} \psi'(\cdot) \in L^{1}(I,\mathbb{R}), \quad i,j=1,2. \end{align*} $$

So it is possible to choose $\aleph $ such that

(2.1) $$ \begin{align} \mathcal{L}_{i,j}(\aleph) &:= \sup_{t \in I} \frac{2 \Vert \eta_{i,j} \Vert_{L^{\infty}}\varphi(b,a)^{1- \gamma_i }}{\Gamma(\alpha_i)} \int_{a}^{t} \varphi'(s) \varphi(t,s)^{\alpha_i-1} \varphi(s,a)^{\gamma_j-1} e^{-\aleph (t-s)} ds \\[0.2cm] & <1/4.\nonumber \end{align} $$

Definition 2.5 The left-sided $\varphi $ -fractional integral of a function f of order $\alpha>0$ is defined as

$$ \begin{align*}\left(\! \mathcal{I}^{\alpha,\varphi}_{a^+} f\right)(t)=\frac{1}{\Gamma(\alpha)} \int_{a}^{t}\varphi(t,s)^{\alpha -1} \varphi^{\prime}(s) f(s) d s, \quad t>a, \end{align*} $$

with $\varphi \in \mathbb {S}_{a^+}^{1,+}(I,\mathbb {R})$ .

Definition 2.6 Let $ n-1 <\alpha \leq n $ with $ n\in \mathbb {N}$ , $ \varphi \in \mathbb {S}_{a^+}^{1,+}(I,\mathbb {R})$ . The left-sided $\varphi $ -Hilfer FD of a function f of order $\alpha $ and type $ 0\leq \beta \leq 1$ is defined as

$$ \begin{align*} \left({^{H}} {\mathcal{D}}_{a^+}^{\alpha, \beta;\varphi} f\right)(t) =\left( \mathcal{I}_{a^+}^{\beta(n-\alpha),\varphi}\left(\frac{1}{\varphi^{\prime}(t)} \frac{d}{d t}\right)^n \left( \mathcal{I}_{a^+}^{(1-\beta)(n-\alpha ),\varphi} f \right)\right)(t). \end{align*} $$

To define a solution of the system (1.1), for each $i=1,2$ , we consider the Banach space

$$ \begin{align*} {\mathcal{C}}_{1-\gamma_i}^{\varphi} (I,\mathbb{E})=\left\{z\in C(I',\mathbb{E}):\, \varphi(\cdot,a)^{1-\gamma_i} z(\cdot)\in C(I,\mathbb{E})\right\}, \end{align*} $$

normed by

(2.2) $$ \begin{align} \|z\|_{\mathcal{C}_{1-\gamma_i}^{\varphi}}=\sup_{t\in I} \varphi(t,a)^{1-\gamma_i}\|z(t)\|. \end{align} $$

Also, by $\mathcal {C}_{1-\gamma _1}^{\varphi } (I,\mathbb {E}) \times \mathcal {C}_{1-\gamma _2}^{\varphi } (I,\mathbb {E})$ we denote the product weighted space with the norm

$$ \begin{align*}\|(z_1,z_2)\|_{\mathcal{C}_{1-\gamma_1}^{\varphi}\times \mathcal{C}_{1-\gamma_2}^{\varphi} } = \|z_1\|_{\mathcal{C}_{1-\gamma_1}^{\varphi} } + \|z_2\|_{\mathcal{C}_{1-\gamma_2}^{\varphi}}. \end{align*} $$

Henceforth, for a subset $\mathbb {U}$ of the space $\mathcal {C}_{1-\gamma _i}^{\varphi }(I,\mathbb {E})$ , define $\mathbb {U}_{\gamma _i}$ by

$$ \begin{align*}\mathbb{U}_{\gamma_i}=\{z_{\gamma_i}~:~ z\in \mathbb{U}\}, \end{align*} $$

where

(2.3) $$ \begin{align} z_{\gamma_i}(t) =\left\{\hspace{-4pt}\begin{array}{l} \varphi(t,a)^{1-\gamma_i} z(t),\quad t\in I',\\[0.3cm] \lim_{t\to a^{+}} \varphi(t,a)^{1-\gamma_i} z(t),\quad t=a. \end{array} \right. \end{align} $$

It is clear that $z_{\gamma _i}\in C(I,\mathbb {E})$ .

Definition 2.7 [Reference Banas and Goebel10]

The Hausdorff MNC is the map $\Lambda : \mathcal {P}(\mathbb {E}) \to [0,\infty )$ defined by

$$ \begin{align*}\Lambda (\mathbb{U} ) =\inf \left\{\epsilon>0 : \mathbb{U} \text{ has a finite } \epsilon-\text{net in } \mathbb{E} \right\}, \end{align*} $$

where $\mathcal {P}(\mathbb {E})$ denotes the family of all bounded subsets of $\mathbb {E}$ .

Lemma 2.9 [Reference Banas and Goebel11]

Let $\mathbb {B}\subseteq C(I, \mathbb {E})$ be a bounded set. Then, for all $t\in I$ ,

$$ \begin{align*}\Lambda (\mathbb{B}(t))\leq \Lambda_c(\mathbb{B}),\end{align*} $$

where $\mathbb {B}(t)=\{u(t)\,:\, u\in \mathbb {B}\}$ . Furthermore, if $\mathbb {B}$ is equicontinuous on I, then $\Lambda (\mathbb {B}(\cdot ))$ is continuous on I and

(2.4) $$ \begin{align} \Lambda_c(\mathbb{B})=\displaystyle\sup_{t\in I}\Lambda(\mathbb{B}(t)), \end{align} $$

where $\Lambda _c$ is the Hausdorff MNC in $C(I, \mathbb {E})$ .

Next, we extend the result of Lemma 2.9 to the space $\mathcal {C}_{1-\gamma }^{\varphi } (I,\mathbb {E})$ . Let us confirm that, in general, the expression (2.4) may not be well-defined, since bounded sets in $\mathcal {C}_{1-\gamma }^{\varphi } (I,\mathbb {E})$ are not necessarily bounded in $C(I,\mathbb {E})$ . Consider, for instance, the set

$$ \begin{align*}\widetilde{\mathcal{Q}}(t)=\left\{\varphi(t,a)^{\gamma -1} z(t) ,\,\,z \in \mathcal{Q}\right\} , \end{align*} $$

where $\mathcal {Q}$ is bounded in $C(I,\mathbb {E})$ . Obviously, $\widetilde {\mathcal {Q}}$ is unbounded in $C(I,\mathbb {E})$ , this indicates that the map $t\mapsto \Lambda (\widetilde {\mathcal {Q}}(t))$ is not well-defined, therefore it is wrong to consider the expression (2.4). However, clearly the set $\widetilde {\mathcal {Q}}$ is bounded with respect to the norm (2.2) (i.e., $\widetilde {\mathcal {Q}}\subset \mathcal {C}_{1-\gamma }^{\varphi }(I,\mathbb {E})$ ).

Lemma 2.10 Let $\mathbb {B} \subseteq \mathcal {C}_{1-\gamma }^{\varphi }(I,\mathbb {E})$ be a bounded set. Then, for all $t\in I$ , we have

$$ \begin{align*}\Lambda(\mathbb{B}_\gamma(t))\leq \Lambda_{\mathcal{C}_{1-\gamma}^{\varphi}}(\mathbb{B}). \end{align*} $$

Additionally, assume that $\mathbb {B}$ is equicontinuous on I, then $\Lambda (\mathbb {B}_\gamma (\cdot ))$ is continuous on I and

$$ \begin{align*}\Lambda_{\mathcal{C}_{1-\gamma}^{\varphi}}(\mathbb{B})=\sup_{t\in I}\Lambda(\mathbb{B}_\gamma(t)). \end{align*} $$

Proof For every $\epsilon>0$ , there exists $\mathbb {B}_i \subseteq \mathcal {C}_{1-\gamma }^{\varphi }(I,\mathbb {E})$ , ( $i=1,\dots ,n$ ) such that ${\mathbb {B}=\displaystyle\bigcup\limits _{i=1}^{n}\mathbb {B}_i}$ and

(2.5) $$ \begin{align} \delta(\mathbb{B}_i)\leq 2 \Lambda(\mathbb{B}(t))+2\epsilon,\quad i=1,\dots,n, \end{align} $$

where $\delta (\cdot )$ denotes the diameter of a bounded set in $\mathcal {C}_{1-\gamma }^{\varphi }(I,\mathbb {E})$ . So, we have

$$ \begin{align*}\mathbb{B}(t)=\displaystyle\bigcup_{i=1}^{n}\mathbb{B}_i(t) ~~\text{for each}~~ t\in I\end{align*} $$

and

(2.6) $$ \begin{align} \|u_\gamma(t)-v_\gamma(t)\| \leq \|u-v\|_\gamma \leq \delta(\mathbb{B}_i),\quad \text{for}\,u,v\in \mathbb{B}_i,\,i=1,\dots,n. \end{align} $$

From (2.5) and (2.6), it follows that

$$ \begin{align*}2\Lambda(\mathbb{B}_\gamma(t))\leq \delta(\mathbb{B}_i(t))\leq \delta(\mathbb{B}_i)\leq 2 \Lambda(\mathbb{B}(t))+2\epsilon. \end{align*} $$

Since $\epsilon $ is arbitrary, one has

$$ \begin{align*}\Lambda(\mathbb{B}_\gamma(t))\leq \Lambda(\mathbb{B}(t)),\quad\text{for every }\, t\in I. \end{align*} $$

Consequently, we have

$$ \begin{align*}\displaystyle\sup_{t\in I}\Lambda(\mathbb{B}_\gamma(t))\leq \Lambda_{\mathcal{C}_{1-\gamma}^{\varphi}}(\mathbb{B}). \end{align*} $$

Now, let us prove the converse inequality. Assume that $\mathbb {B}$ is a bounded subset in $\mathcal {C}_{1-\gamma }^{\varphi }(I,\mathbb {E})$ and equicontinuous on I. Obviously, $\mathbb {B}_\gamma $ is a bounded subset in $C(I,\mathbb {E})$ and equicontinuous on I. From Lemma 2.9, we obtain that

$$ \begin{align*}\Lambda(\mathbb{B}_\gamma)\leq \sup_{t\in I}\Lambda( \mathbb{B}_\gamma(t)). \end{align*} $$

Consider the isometric map $ \Upsilon : \mathcal {C}_{1-\gamma }^{\varphi }(I,\mathbb {E}) \to C(I,\mathbb {E})$ defined by $z \mapsto z_\gamma $ . Then, we get

$$ \begin{align*}\Lambda_{\mathcal{C}_{1-\gamma}^{\varphi}}(\mathbb{B})=\sup_{t\in I}\Lambda(\mathbb{B}_\gamma(t)), \end{align*} $$

and the result is reached.

Lemma 2.11 [Reference Kamenskii, Obukhovskii and Zecca18]

Let $\{x_n\}_{n=1}^{+\infty }$ belongs to $L^{1}(I,\mathbb {E})$ such that $\|x_n(t)\|\leq \varsigma (t)$ almost everywhere on $I (n=1,2,\dots )$ for some $\varsigma \in L^{1}(I,\mathbb {R}_+)$ . Then, the map $t\mapsto \Lambda (\{x_n(t)\}_{n=1}^{+\infty })$ is integrable on $\mathbb {R}_+$ and

(2.7) $$ \begin{align} \Lambda\left(\left\{\int_0^t x_n(s)ds\ \right\}_{n=1}^{+\infty} \right) \leq 2\int_0^t \Lambda(\{x_n(s) \}_{n=1}^{+\infty})ds. \end{align} $$

Lemma 2.12 [Reference Aghajani and Sabzali3]

Let $\mathbb {B}\in \mathcal {P}(\mathbb {E})$ . Then for each $\epsilon> 0$ , there exists a sequence $\{x_n\}_{n=1}^{+\infty }\subseteq \mathbb {B}$ , satisfies

(2.8) $$ \begin{align} \Lambda_{c}\left( \mathbb{B} \right) \leq 2 \Lambda_{c}(\{x_n\}_{n=1}^{+\infty}) + \epsilon. \end{align} $$

Theorem 2.13 (Darbo[Reference Darbo12])

Let $\mathbb {E}$ be a Banach space, let $\mathbb {V}\subset \mathbb {E}$ be a nonempty, bounded, closed, convex, and let $N: \mathbb {V} \rightarrow \mathbb {V}$ be a continuous mapping. Assume that there exists $k \in [0,1)$ such that

(2.9) $$ \begin{align} \Lambda (N(\mathbb{V})) \leq k \Lambda (\mathbb{V}). \end{align} $$

Then N admits a fixed point in $\mathbb {V}$ .

Theorem 2.14 [Reference Banas and Goebel10]

Suppose $\varpi _1,\varpi _2,\dots ,\varpi _n$ are MNCs in the Banach spaces $\mathbb {E}_1,\mathbb {E}_2,\dots ,\mathbb {E}_n$ , respectively. Let $G : [0,\infty )^{n} \to [0,\infty ) $ be a convex function such that $G(x_1,x_2,\dots ,x_n) =0 $ if and only if $x_i = 0$ for $i=1,2,\dots ,n$ . Then

(2.10) $$ \begin{align} \varpi(X)= G(\varpi_1(X_1),\varpi_2(X_2),\dots,\varpi_n(X_n)) \end{align} $$

defines an MNC in $\mathbb {E}_1 \times \mathbb {E}_2 \times \dots \times \mathbb {E}_n$ , where $X_i$ denotes the natural projection of X into $\mathbb {E}_i$ for $i=1,2,\dots ,n$ .

3 Main results

We initiate this section by introducing the following hypotheses which are needed in the sequel:

1. (H1) The functions $t\mapsto g_i(t,u,v); i=1,2$ are measurable on I for each $(u,v)\in C(I, \mathbb {E})\times C(I, \mathbb {E})$ , and the functions $(u,v) \mapsto g_i(t,u,v)$ are continuous for a.e. $t \in I$ .

2. (H2) There exist functions $h_i\in L^{\frac {1}{\mu _i}}_{\varphi }(I,\mathbb {R}_+)$ , $0\leq \mu _i <\alpha _i $ such that

   $$ \begin{align*}\|g_i(t,v_1,v_2)\|\leq h_i(t) \left( 1 + \|v_1\| + \|v_2\| \right),\quad i=1,2, \end{align*} $$
   for $v_1, v_2 \in \mathbb {E}$ , and a.e. $t\in I$ .

3. (H3) There exist functions $\eta _i, \widehat {\eta }_i \in L^\infty (I,\mathbb {R}_+)$ , $i = 1,2$ , such that for any bounded subsets $\mathbb {A}^1\times \mathbb {A}^2\subset \mathcal {C}_{1-\gamma _1}^{\varphi }(I,\mathbb {E})\times \mathcal {C}_{1-\gamma _2}^{\varphi }(I,\mathbb {E})$ , we have

   (3.1) $$ \begin{align} \Lambda(g_i(t,\mathbb{A}^1,\mathbb{A}^2)) \leq \displaystyle\sum_{j=1}^{2} \eta_{i,j}(t) \Lambda(\mathbb{A}^j) ,\quad \text{for all}~~ t\in I. \end{align} $$

4. (H4) The following inequality holds:

   (3.2) $$ \begin{align} 2 \widehat{K}_i\leq K,\quad i=1,2, \end{align} $$
   where
   $$ \begin{align*} K_i &:= \| \xi_i\| + \frac{ \varphi(b,a)^{1 + \alpha_i - \gamma_i -\mu_i }\Vert h_i\Vert_{L^{\frac{1}{\mu_i}}_{\varphi}}}{\Gamma(\alpha_i) \theta_{i}^{1-\mu_i}} \\[4pt] &\qquad \qquad + K \displaystyle\sum_{j=1}^{2}\frac{\varphi(b,a)^{1-\gamma_i} \Vert h_i\Vert_{L^{\frac{1}{\mu_i}}_{\varphi}} }{\Gamma(\alpha_i)} \left( \Theta_{\theta_i, \vartheta_{i,j}} (a,b,b,\varphi) \right)^{1-\mu_i}, \end{align*} $$
   $$ \begin{align*}\theta_i = \frac{\alpha_i -\mu_i}{1-\mu_i},\quad \text{and}\quad \vartheta_{i,j} =\frac{1 -\gamma_{j}}{1-\mu_i},\quad i,j=1,2. \end{align*} $$

Now, we prove our main result for the system (1.1), which is based on Theorem 2.13.

Proof First, let us introduce an operator $\mathcal {H}: \mathcal {C}_{1-\gamma _1}^{\varphi }(I,\mathbb {E})\times \mathcal {C}_{1-\gamma _2}^{\varphi }(I,\mathbb {E}) \to \mathcal {C}_{1-\gamma _1}^{\varphi }(I,\mathbb {E})\times \mathcal {C}_{1-\gamma _2}^{\varphi }(I,\mathbb {E})$ associated with the system (1.1) as

(3.3) $$ \begin{align} (\mathcal{H}(y_1,y_2))(t)=((\mathcal{H}_1(y_1,y_2))(t),(\mathcal{H}_2(y_1,y_2))(t))\,, \end{align} $$

where the operators $\mathcal {H}_i:\mathcal {C}_{1-\gamma _1}^{\varphi }(I,\mathbb {E})\times \mathcal {C}_{1-\gamma _2}^{\varphi }(I,\mathbb {E}) \to \mathcal {C}_{1-\gamma _i}^{\varphi }(I,\mathbb {E})$ , $i=1,2$ are defined by

$$ \begin{align*}(\mathcal{H}_i(y_1,y_2))(t)= \xi_{i} \varphi(t,a)^{\gamma_i - 1}+ \frac{1}{\Gamma(\alpha_i)} \int_{a}^{t} \varphi^{\prime}(s)\varphi(t,s)^{\alpha_i -1} g_i(s, y_1(s), y_2(s)) d s,\, t\in I'. \end{align*} $$

According to [Reference Kucche and Mali22, Lemma 3.1], the solutions of the system (1.1) are fixed points of the operator $\mathcal {H}$ . Consider the following bounded closed convex set:

$$ \begin{align*}\Omega_{K} = \left\lbrace (y_1,y_2)\in \mathcal{C}_{1-\gamma_1}^{\varphi}(I,\mathbb{E})\times \mathcal{C}_{1-\gamma_2}^{\varphi}(I,\mathbb{E})~:~ \Vert y_1\Vert_{\mathcal{C}_{1-\gamma_1}^{\varphi}} \leq K, ~~ \Vert y_2\Vert_{\mathcal{C}_{1-\gamma_2}^{\varphi}} \leq K \right\rbrace. \end{align*} $$

Then we divide the proof into four steps.

Step 1. $\mathcal {H}$ transforms $\Omega _{K}$ into itself. Indeed, for each $(y_1,y_2)\in \Omega _{K}$ and every $t\in I'$ , one has

$$ \begin{align*} & \left\| \varphi(t,a)^{1-\gamma_i} \left( \mathcal{H}_i(y_1,y_2)\right)(t)\right\| \\[5pt] &\leq \| \xi_i\| + \frac{\varphi(t,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{a}^{t} \varphi'(s)\varphi(t,s)^{\alpha_i -1} \left\| g_i(s, y_1(s),y_2(s)) \right\|d s \\[5pt] &\leq \| \xi_i\| + \frac{\varphi(t,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{a}^{t} \varphi'(s)\varphi(t,s)^{\alpha_i -1} h_i(s) \left( 1 + \|y_1(s)\| + \|y_2(s)\| \right) d s \\[5pt] & \leq \| \xi_i\| + \frac{\varphi(t,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{a}^{t} \varphi'(s)\varphi(t,s)^{\alpha_i -1} h_i(s) (1+ \varphi(s,a)^{\gamma_1 -1}\|y_1\|_{\mathcal{C}_{1-\gamma_1}^{\varphi}} \\[5pt] & \qquad + \varphi(s,a)^{\gamma_2 -1}\|y_2\|_{\mathcal{C}_{1-\gamma_2}^{\varphi}} )ds \\[5pt] & \leq \| \xi_i\| + \frac{\varphi(t,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{a}^{t} \varphi(t,s)^{\alpha_i -1} h_i(s) \left(1 +K \displaystyle\sum_{j=1}^{2} \varphi(s,a)^{\gamma_{j} -1} \right) \varphi'(s) ds. \end{align*} $$

Since $\mu _i + (1- \mu _i)=1$ , we can write $ \varphi '(s)$ as the product

$$ \begin{align*}\varphi'(s) =\varphi'(s)^{\mu_i} \varphi'(s)^{1-\mu_i}. \end{align*} $$

Then, applying the Hölder inequality, we get

$$ \begin{align*} &\hspace{-0.3cm} \left\| \varphi(t,a)^{1-\gamma_i} \left( \mathcal{H}_i(y_1,y_2)\right)(t)\right\|\\[5pt] & \leq \| \xi_i\| + \frac{\varphi(t,a)^{1-\gamma_i} \Vert h_i\Vert_{ L^{\frac{1}{\mu_i}}_{\varphi} } }{\Gamma(\alpha_i)}\left( \int_{a}^{t} \varphi(t,s)^{\frac{\alpha_i -1}{1-\mu_i}} d\varphi(s) \right)^{1-\mu_i} \\[5pt] & \quad + K \displaystyle\sum_{j=1}^{2} \frac{\varphi(t,a)^{1-\gamma_i} \Vert h_i\Vert_{ L^{\frac{1}{\mu_i}}_{\varphi} } }{\Gamma(\alpha_i)} \left( \int_{a}^{t} \varphi(t,s)^{\frac{\alpha_i -1}{1-\mu_i}} \varphi(s,a)^{\frac{\gamma_{j} -1}{1-\mu_i}} d\varphi(s) \right)^{1-\mu_i}. \end{align*} $$

Hence, by Lemma 2.2 and Remark 2.3, we obtain

$$ \begin{align*} \left\|\varphi(t,a)^{1-\gamma_i} \left( \mathcal{H}_i(y_1,y_2)\right)(t)\right\| &\leq \| \xi_i\| + \frac{ \Vert h_i\Vert_{ L^{\frac{1}{\mu_i}}_{\varphi} } }{\Gamma(\alpha_i)} \left(\frac{1-\mu_i}{\alpha_i -\mu_i}\right)^{1-\mu_i}\varphi(t,a)^{1 + \alpha_i - \gamma_i -\mu_i } \\ &\quad + K \displaystyle\sum_{j=1}^{2}\frac{\varphi(t,a)^{1-\gamma_i} \Vert h_i\Vert_{ L^{\frac{1}{\mu_i}}_{\varphi} } }{\Gamma(\alpha_i)} \left( \Theta_{\frac{\alpha_i -\mu_i}{1-\mu_i}, \frac{1-\gamma_{j} }{1-\mu_i}} (a,t,t,\varphi) \right)^{1-\mu_i}\\ &\leq \widehat{K}_i,\quad i=1,2. \end{align*} $$

Thus,

$$ \begin{align*}\|\mathcal{H}_i(y_1,y_2)\|_{\mathcal{C}_{1-\gamma_i}^{\varphi}} \leq \widehat{K}_i,\quad i=1,2. \end{align*} $$

Consequently,

$$ \begin{align*}\|\mathcal{H}(y_1,y_2)\|_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}} \leq \widehat{K}_1 + \widehat{K}_2. \end{align*} $$

This shows that $\mathcal {H}$ transforms $\Omega _{K}$ into itself.

Step 2. The continuity of $\mathcal {H}(\cdot ,\cdot )$ .

Let $\{y_{1,n},y_{2,n}\} \in \Omega _q$ such that $(y_{1,n},y_{2,n})\to (y_{1},y_{2})$ as $ n \longrightarrow \infty $ . Making use of the Carathéodory condition of $g_i$ , $i=1,2$ , we easily have

$$ \begin{align*}\Vert g_i(s, y_{1,n}(s), y_{2,n}(s)) - g_i(s,y_{1}(s), y_{2}(s))\Vert \to 0, \quad \text{as}~~ n \longrightarrow \infty. \end{align*} $$

Next, by (H2), one gets

$$ \begin{align*} &\hspace{-0.9cm} \left\|g_i(s, y_{1,n}(s), y_{2,n}(s)) - g_i(s,y_{1}(s), y_{2}(s))\right\|\\[0.2cm] &\leq \left\|g_i(s, y_{1,n}(s), y_{2,n}(s))\Vert + \Vert g_i(s,y_{1}(s), y_{2}(s))\right\|\\[0.1cm] &\leq 2 h_i(s) ( 1 + \Vert y_1(t)\Vert + \Vert y_2(t)\Vert ) \\ &\leq 2 h_i(s)\Big(1+ K \displaystyle\sum_{j=1}^{2} \varphi(s,a)^{\gamma_{j} -1}\Big). \end{align*} $$

Since, the function $s \mapsto h_i(s) \Big (1+ K \displaystyle\sum\limits _{j=1}^{2} \varphi (s,a)^{\gamma _{j} -1}\Big ) $ is Lebesgue integrable over $[a, t]$ , so is the function $s \mapsto h_i(s) \varphi (t,s)^{\alpha _i -1} \Big (1+ K \displaystyle \displaystyle\sum\limits _{j=1}^{2} \varphi (s,a)^{\gamma _{j} -1}\Big ) $ . Then it follows from the Lebesgue dominated convergence theorem that

$$ \begin{align*} &\hspace{-0.2cm} \left\| \varphi(t,a)^{1-\gamma_i}\left[ \mathcal{H}_i(y_{1,n}, y_{2,n})(t)- \mathcal{H}_i(y_{1}, y_{2})(t)\right]\right\| \\[0.3cm] &\hspace{-0.2cm}\leq \frac{\varphi(t,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{a}^{t} \varphi(t,s)^{\alpha_i -1} \varphi'(s) \left\|g_i(s, y_{1,n}(s), y_{2,n}(s)) - g_i(s,y_{1}(s), y_{2}(s))\right\|d s \to 0 \end{align*} $$

as $n \longrightarrow \infty $ for all $t \in I$ , which leads to

$$ \begin{align*}\left\| \left( \mathcal{H}_i(y_{1,n}, y_{2,n})\right)- \left(\mathcal{H}_i(y_{1}, y_{2})\right)\right\|_{\mathcal{C}_{1-\gamma_i}^{\varphi}} \xrightarrow[n \rightarrow \infty]{} 0 \end{align*} $$

for any $t \in I$ . Therefore,

$$ \begin{align*}\Vert \mathcal{H}(y_{1,n}, y_{2,n})- \mathcal{H}(y_{1}, y_{2}) \Vert_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}} \xrightarrow[n \rightarrow \infty]{} 0. \end{align*} $$

Accordingly, the operator $\mathcal {H}(\cdot ,\cdot )$ is continuous.

Step 3. $\mathcal {H}(\Omega _{q})$ is equicontinuous.

Let $(y_1,y_2)\in \Omega _{q}$ and $a<t_1<t_2\leq b$ . Then for $i=1,2$ , we have

$$ \begin{align*}\begin{array}{ll} &\hspace{-0.5cm} \left\| \varphi(t_2,a)^{1-\gamma_i} (\mathcal{H}_i(y_1,y_2))(t_2)-\varphi(t_1,a)^{1-\gamma_i}(\mathcal{H}_i(y_1,y_2))(t_1)\right\| \\[0.3cm] &\leq \displaystyle \Bigg\|\frac{\varphi(t_2,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{a}^{t_{2}}\varphi'(s) \varphi(t_2,s)^{\alpha_i -1} g_i(s, y_1(s) ,y_2(s)) d s \\[0.3cm] &\hspace{2cm} - \displaystyle\frac{\varphi(t_1,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{a}^{t_{1}}\varphi'(s) \varphi(t_1,s)^{\alpha_i -1} g_i(s, y_1(s) ,y_2(s)) d s \Bigg\|\\[0.3cm] &\leq J_0 +J_1, \end{array} \end{align*} $$

where

$$ \begin{align*}J_0 = \frac{\varphi(t_2,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{t_1}^{t_{2}}\varphi'(s) \varphi(t_2,s)^{\alpha_i -1} \|g_i(s, y_1(s) ,y_2(s))\| d s, \end{align*} $$

and

$$ \begin{align*}J_1 &= \Bigg\|\displaystyle \frac{\varphi(t_2,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{a}^{t_{1}}\varphi'(s) \varphi(t_2,s)^{\alpha_i -1} g_i(s, y_1(s) ,y_2(s)) d s \\ &\hspace{2cm} - \displaystyle\frac{\varphi(t_1,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{a}^{t_{1}}\varphi'(s) \varphi(t_1,s)^{\alpha_i -1} g_i(s, y_1(s) ,y_2(s)) d s \Bigg\|. \end{align*} $$

Next, applying the Hölder inequality, we conclude that

$$ \begin{align*}J_0 &\leq \displaystyle\frac{\varphi(t_2,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{t_1}^{t_{2}}\varphi'(s) \varphi(t_2,s)^{\alpha_i -1} h_i(s)( 1+ \Vert y_1(s) \Vert +\Vert y_2(s) \Vert ) d s\\[0.3cm] &\leq \displaystyle\frac{ \varphi(t_2,a)^{1-\gamma_i} \Vert h_i\Vert_{L^{\frac{1}{\mu_i}}_{\varphi}} }{\Gamma(\alpha_i)} \left( \int_{t_1}^{t_{2}} \varphi(t_2,s)^{\frac{\alpha_i -1}{1-\mu_i}}d\varphi(s) \right)^{1-\mu_i}\\[0.4cm] & \quad + \displaystyle K \displaystyle\sum_{j=1}^{2}\frac{ \varphi(t_2,a)^{1-\gamma_i} \Vert h_i\Vert_{L^{\frac{1}{\mu_i}}_{\varphi}} }{\Gamma(\alpha_i)} \left( \int_{t_1}^{t_{2}} \varphi(t_2,s)^{\frac{\alpha_i -1}{1-\mu_i}} \varphi(s,a)^{\frac{\gamma_{j} -1}{1-\mu_i}} d\varphi(s) \right)^{1-\mu_i}, \end{align*} $$

and hence,

$$ \begin{align*}\begin{array}{l} J_0 \leq \displaystyle \frac{ \varphi(t_2,a)^{1-\gamma_i} \Vert h_i\Vert_{L^{\frac{1}{\mu_i}}_{\varphi} } }{\Gamma(\alpha_i)} \Bigg[\frac{\varphi(t_2,t_1)^{\alpha_i -1} }{\theta_{i}^{1-\mu_i} } + K \displaystyle\sum_{j=1}^{2} \left( \Theta_{\theta_i, \vartheta_{i,j}} (t_1,t_2,t_2,\varphi) \right)^{1-\mu_i}\Bigg]. \end{array} \end{align*} $$

Now, using Remark 2.3, we obtain

$$ \begin{align*} J_0 \longrightarrow 0\quad \text{as}\quad t_1 \longrightarrow t_2, \end{align*} $$

and

$$ \begin{align*}J_1 \leq J_ 2 + J_ 3, \end{align*} $$

where

$$ \begin{align*}J_2 &= \displaystyle\frac{|\varphi(t_2,a)^{1-\gamma_i} - \varphi(t_1,a)^{1-\gamma_i}|}{\Gamma(\alpha_i)} \int_{a}^{t_{1}}\varphi'(s) \varphi(t_2,s)^{\alpha_i -1} \Vert g_i(s, y_1(s) ,y_2(s))\Vert d s, \\[0.6cm] J_3 &= \displaystyle\frac{\varphi(t_1,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \int_{a}^{t_{1}}\varphi'(s)|\varphi(t_2,s)^{\alpha_i -1} - \varphi(t_1,s)^{\alpha_i -1}| \Vert g_i(s, y_1(s) ,y_2(s))\Vert d s. \end{align*} $$

Then,

$$ \begin{align*}J_2 \xrightarrow[n \rightarrow \infty]{} 0, \end{align*} $$

on the other hand, $\varphi (t_2,s)^{\alpha _i -1} \leq \varphi (t_1,s)^{\alpha _i -1}$ . Therefore

$$ \begin{align*}J_3 &\leq \displaystyle \frac{\varphi(t_1,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} \left( \int_{a}^{t_{1}}h_i(s)\varphi'(s)\varphi(t_1,s)^{\alpha_i -1} d s -\int_{a}^{t_{1}}h_i(s)\varphi'(s) \varphi(t_2,s)^{\alpha_i -1} d s\right)\\[0.3cm] & \quad + \displaystyle \frac{\varphi(t_1,a)^{1-\gamma_i}}{\Gamma(\alpha_i)} K \displaystyle\sum_{j=1}^{2} \int_{a}^{t_{1}}h_i(s)\varphi'(s)\Big(\varphi(t_1,s)^{\alpha_i -1} - \varphi(t_2,s)^{\alpha_i -1}\Big) \varphi(s,a)^{\gamma_{j} -1} ds. \end{align*} $$

Then,

$$ \begin{align*} J_3 &\leq \displaystyle \frac{\varphi(t_1,a)^{1-\gamma_i}\Vert h_i\Vert_{L^{\frac{1}{\mu_i}}_{\varphi}}}{\Gamma(\alpha_i)} \left[\left( \Theta_{\theta_i, 0} (a,t_1,t_1,\varphi) \right)^{1-\mu_i} - \left( \Theta_{\theta_i, 0} (a,t_1,t_2,\varphi) \right)^{1-\mu_i} \right]\\[0.5cm] & \quad + \displaystyle \frac{\varphi(t_1,a)^{1-\gamma_i}\Vert h_i\Vert_{L^{\frac{1}{\mu_i}}_{\varphi}}}{\Gamma(\alpha_i)} K \displaystyle\sum_{j=1}^{2} \Bigg[\left( \Theta_{\theta_i, \vartheta_{i,j}} (a,t_1,t_1,\varphi) \right)^{1-\mu_i} \\[0.5cm] & \qquad -\left( \Theta_{\theta_i, \vartheta_{i,j}} (a,t_1,t_2,\varphi) \right)^{1-\mu_i} \Bigg]. \end{align*} $$

So,

$$ \begin{align*} J_3 \longrightarrow 0\quad \text{as}\quad t_1 \longrightarrow t_2. \end{align*} $$

Hence,

$$ \begin{align*}\left\| \varphi(t_2,a)^{1-\gamma_i} (\mathcal{H}_i(y_1,y_2))(t_2)-\varphi(t_1,a)^{1-\gamma_i}(\mathcal{H}_i(y_1,y_2))(t_1)\right\| \longrightarrow 0 \quad \text{as}\quad t_1 \longrightarrow t_2. \end{align*} $$

Thus, we conclude that $\mathcal {H}(\Omega _{q})$ is equicontinuous.

Step 4. Condition (2.9) holds. For every bounded subset $\mathbb {J}^1\times \mathbb {J}^2\subset \mathcal {C}_{1-\gamma _1}^{\varphi }(I,\mathbb {E})\times \mathcal {C}_{1-\gamma _2}^{\varphi }(I,\mathbb {E})$ , we define the MNC as

(3.4) $$ \begin{align} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}(\mathbb{J}^1\times \mathbb{J}^2) = \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi}}(\mathbb{J}^1) + \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_2}^{\varphi}}(\mathbb{J}^2), \end{align} $$

where

(3.5) $$ \begin{align} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_i}^{\varphi}}(\mathbb{J}^i) =\sup_{t\in I} e^{-\aleph t}\Lambda (\mathbb{J}^i_{\gamma_i}(t));\,\quad i=1,2, ~~ \aleph>0. \end{align} $$

By using Lemma 2.10 and Example 2.15, $\widehat {\Lambda }_{\mathcal {C}_{1-\gamma _1}^{\varphi } \times \mathcal {C}_{1-\gamma _2}^{\varphi }}$ satisfies all properties of the Hausdorff MNC mentioned in Lemma 2.8.

Now, let $\mathscr {A} = (\mathbb {A}^1,\mathbb {A}^2)$ be a bounded set belongs to $ \overline {conv} \mathcal {H}(\Omega _{K})$ , using Lemma 2.12, it follows that for a given $\epsilon _i> 0$ ( $i=1,2$ ), there exists $\{(y^{1,n} ,y^{2,n}) \}_{n=1}^{+\infty }\subseteq \mathscr {A}$ such that, for all $t \in I$ ,

(3.6) $$ \begin{align} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_i}^{\varphi}}\Big(\mathcal{H}_i(\mathscr{A})(t) \Big) &= \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_i}^{\varphi}}\Big(\big\{(\mathcal{H}_i(y^{1},y^{2}))(t) ~:~ (y^{1},y^{2}) \in \mathscr{A} \big\} \Big) \\ & \leq 2 \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_i}^{\varphi}}\Big(\big\{(\mathcal{H}_i(y^{1,n},y^{2,n}))(t)\big\}_{n=1}^{+\infty} \Big) + \varepsilon_i,\quad i=1,2.\nonumber \end{align} $$

From

(3.7) $$ \begin{align} (\mathcal{H}_i(y^{1,n},y^{2,n}))(t)= \varphi(t,a)^{\gamma_i - 1}\xi_{i} + \mathcal{I}_{a^+}^{\alpha_i,\varphi}g_{i}(t,y^{1,n}(t),y^{2,n}(t)),\quad i=1,2, \end{align} $$

wet get

(3.8) $$ \begin{align} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_i}^{\varphi}}\Big(\big\{(\mathcal{H}_i(y^{1,n},y^{2,n}))(\cdot)\big\}_{n=1}^{+\infty} \Big) = \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_i}^{\varphi}}\Big(\big\{ \mathcal{I}_{a^+}^{\alpha_i,\varphi}g_{i}(\cdot,y^{1,n}(\cdot),y^{2,n}(\cdot))\big\}_{n=1}^{+\infty} \Big),\, i=1,2. \end{align} $$

Now, we estimate the quantity $\widehat {\Lambda }_{\mathcal {C}_{1-\gamma _i}^{\varphi }}\Big (\big \{(\mathcal {H}_i(y^{1,n},y^{2,n}))(\cdot )\big \}_{n=1}^{+\infty } \Big )$ . Using (H3), for all $s \in [a,t]$ , one has for $i=1,2$

$$ \begin{align*}\begin{array}{ll} &\hspace{-0.5cm} \displaystyle\Lambda\left(\Big\{ \varphi'(s) \varphi(t,s)^{\alpha_i-1}g_{i}(s,y^{1,n}(s),y^{2,n}(s))\Big\} _{n=1}^{+\infty}\right)\\[0.1cm] &\leq \displaystyle \displaystyle\sum_{j=1}^{2} \varphi'(s) \varphi(t,s)^{\alpha_i-1} \eta_{i,j}(s) \Lambda(\{y^{j,n}(s)\}_{n=1}^{+\infty}) \\[0.3cm] &\leq \displaystyle \displaystyle\sum_{j=1}^{2} \eta_{i,j}(s) \varphi'(s) \varphi(t,s)^{\alpha_i-1} \varphi(s,a)^{\gamma_j-1} \Lambda(\{y_{\gamma_j}^{j,n}(s)\}_{n=1}^{+\infty})\\[0.3cm] &\leq \displaystyle \displaystyle\sum_{j=1}^{2} \eta_{i,j}(s) \varphi'(s) \varphi(t,s)^{\alpha_i-1} \varphi(s,a)^{\gamma_j-1} e^{\aleph s} \sup_{a\leq s\leq t} e^{-\aleph s} \Lambda(\{y_{\gamma_j}^{j,n}(s)\}_{n=1}^{+\infty}) \\[0.5cm] &\leq \displaystyle \displaystyle\sum_{j=1}^{2} \eta_{i,j}(s) \varphi'(s) \varphi(t,s)^{\alpha_i-1} \varphi(s,a)^{\gamma_j-1} e^{\aleph s} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_j}^{\varphi}}(\{y^{j,n}\}_{n=1}^{+\infty}). \end{array} \end{align*} $$

Thus, Lemma 2.11 entails that, for all $t \in I$ and $s \leq t$ ,

$$ \begin{align*}\begin{array}{ll} &\hspace{-0.3cm} \displaystyle\Lambda \Big(\big\{ \mathcal{I}_{a^+}^{\alpha_i,\varphi}g_{i}(t,y^{1,n}(t),y^{2,n}(t))\big\}_{n=1}^{+\infty} \Big) \\[0.3cm] &\leq \displaystyle \displaystyle\sum_{j=1}^{2} \frac{2 \Vert \eta_{i,j} \Vert_{L^{\infty}}}{\Gamma(\alpha_i)} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_j}^{\varphi}}(\{y^{j,n}\}_{n=1}^{+\infty}) \int_{a}^{t} \varphi'(s) \varphi(t,s)^{\alpha_i-1} \varphi(s,a)^{\gamma_j-1} e^{\aleph s} ds. \end{array} \end{align*} $$

Hence,

$$ \begin{align*}\begin{array}{ll} &\hspace{-0.2cm} \displaystyle\Lambda\left( \left\lbrace \varphi(t,a)^{1- \gamma_i } \mathcal{H}_i(y^{1,n}(t),y^{2,n}(t))\right\rbrace _{n=1}^{+\infty}\right) \\[0.3cm] &\hspace{-0.2cm}\leq \displaystyle \displaystyle\sum_{j=1}^{2} \frac{2 \Vert \eta_{i,j} \Vert_{L^{\infty}}}{\Gamma(\alpha_i)} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_j}^{\varphi}}(\{y^{j,n}\}_{n=1}^{+\infty}) \varphi(b,a)^{1- \gamma_i } \int_{a}^{t} \varphi'(s) \varphi(t,s)^{\alpha_i-1} \varphi(s,a)^{\gamma_j-1} e^{\aleph s} ds. \end{array} \end{align*} $$

Multiplying both sides by $e^{-\aleph t}$ , we obtain

$$ \begin{align*}\begin{array}{ll} & \hspace{-0.3cm} \displaystyle \sup_{t \in J} e^{-\aleph t}\Lambda\left( \left\lbrace \varphi(t,a)^{1- \gamma_i } \mathcal{H}_i(y^{1,n}(t),y^{2,n}(t))\right\rbrace _{n=1}^{+\infty}\right) \\[0.3cm] &\leq \displaystyle \displaystyle\sum_{j=1}^{2} \frac{2 \Vert \eta_{i,j} \Vert_{L^{\infty}}}{\Gamma(\alpha_i)} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_j}^{\varphi}}(\{y^{j,n}\}_{n=1}^{+\infty}) \varphi(b,a)^{1- \gamma_i } \times\\[0.3cm] &\qquad \displaystyle \sup_{t \in I} \int_{a}^{t} \varphi'(s) \varphi(t,s)^{\alpha_i-1} \varphi(s,a)^{\gamma_j-1} e^{-\aleph(t-s)} ds. \end{array} \end{align*} $$

So,

$$ \begin{align*}\widehat{\Lambda}_{\mathcal{C}_{1-\gamma_i}^{\varphi}}\Big(\big\{\mathcal{H}_i(y^{1,n},y^{2,n})\big\}_{n=1}^{+\infty} \Big) \leq \displaystyle\sum_{j=1}^{2} \mathcal{L}_{i,j}(\aleph) \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_j}^{\varphi}}(\{y^{j,n}\}_{n=1}^{+\infty}), \end{align*} $$

where $\mathcal {L}_{i,j}(\aleph ), i=1,2, j=1,2$ are defined in (2.1). Hence,

$$ \begin{align*} &\hspace{-0.7cm} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_i}^{\varphi}}\Big(\big\{\mathcal{H}_i(y^{1,n},y^{2,n})\big\}_{n=1}^{+\infty} \Big)\\ &\leq \max_{1\leq j\leq 2}\{\mathcal{L}_{i,j}(\aleph)\} \Big( \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi}}(\{y^{1,n}\}_{n=1}^{+\infty}) + \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_2}^{\varphi}}(\{y^{2,n}\}_{n=1}^{+\infty})\Big)\\ &= \max_{1\leq j\leq 2}\{\mathcal{L}_{i,j}(\aleph)\} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}(\{(y^{1,n},y^{2,n})\}_{n=1}^{+\infty}). \end{align*} $$

The last inequality together with the fact that

$$ \begin{align*}\widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}(\{(y^{1,n},y^{2,n})\}_{n=1}^{+\infty}) \leq \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}(\mathscr{A}) \end{align*} $$

yields

(3.9) $$ \begin{align} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_i}^{\varphi}}\Big(\big\{\mathcal{H}_i(y^{1,n},y^{2,n})\big\}_{n=1}^{+\infty} \Big) \leq \max_{1\leq j\leq 2}\{\mathcal{L}_{i,j}(\aleph)\} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}(\mathscr{A}). \end{align} $$

From (3.6) and (3.9), one gets

(3.10) $$ \begin{align} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_i}^{\varphi}}\Big(\mathcal{H}_i(\mathscr{A})(t) \Big) &\leq 2 \max_{1\leq j\leq 2}\{\mathcal{L}_{i,j}(\aleph)\} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}(\mathscr{A}) + \varepsilon_i,\quad i=1,2. \end{align} $$

Then,

$$ \begin{align*}&\hspace{-0.7cm} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}\Big(\mathcal{H}(\mathscr{A}) \Big)\\ &= \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi}}\Big(\mathcal{H}_i(\mathscr{A}) \Big) + \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_2}^{\varphi}}\Big(\mathcal{H}_i(\mathscr{A})\Big) \\ &\leq 2 \Big( \max_{1\leq j\leq 2}\{\mathcal{L}_{1,j}(\aleph)\} + \max_{1\leq j\leq 2}\{\mathcal{L}_{2,j}(\aleph)\} \Big) \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}(\mathscr{A}) + \varepsilon_3 \\ &\leq 4 \max_{1\leq i\leq 2} \{ \max_{1\leq j\leq 2}\{\mathcal{L}_{i,j}(\aleph)\} \} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}(\mathscr{A}) + \varepsilon_3, \end{align*} $$

where $\varepsilon _3 = \varepsilon _1 +\varepsilon _2 $ . Since $\varepsilon _3> 0$ is arbitrary, we have

$$ \begin{align*}\widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}\Big(\mathcal{H}(\mathscr{A}) \Big) \leq 4 \max_{1\leq i\leq 2} \{ \max_{1\leq j\leq 2}\{\mathcal{L}_{i,j}(\aleph)\} \} \widehat{\Lambda}_{\mathcal{C}_{1-\gamma_1}^{\varphi} \times \mathcal{C}_{1-\gamma_2}^{\varphi}}(\mathscr{A}). \end{align*} $$

From Remark 2.4, we deduce $4 \max _{1\leq i\leq 2} \{ \max _{1\leq j\leq 2}\{\mathcal {L}_{i,j}(\aleph )\} \} <1 $ .

In view of Steps 1 to 4, we can apply Theorem 2.13 and deduce that $\mathcal {H}$ admits in $\Omega _{K}$ , at least one fixed point which is a solution of system (1.1).

4 An example

Consider the Banach space

$$ \begin{align*}\mathbb{E}=\{ z= (z^1, z^2,\dots, z^n, \dots) ~:~ z^n \to 0 ~~\text{as}~~ n \to \infty \}, \end{align*} $$

equipped with the norm $ \Vert z \Vert _{\mathbb {E}} = \sup\limits _{n \geq 1} |z^n|$ .

We recall that the Hausdorff MNC $\Lambda $ in $\left (\mathbb {E}, \Vert \cdot \Vert _{\mathbb {E}} \right ) $ is defined as follows:

$$ \begin{align*}\Lambda(\mathbb{A})=\lim_{n \to \infty}\sup_{z \in \mathbb{A}} \left\Vert \left( \mathrm{I} - \mathrm{P}_n \right)z \right\Vert{}_{\infty}, \end{align*} $$

where $\mathbb {A}$ is a bounded subset in $\mathbb {E}$ and $\mathrm {P}_n$ is the projection onto the linear span of the first n vectors in the standard basis (see [Reference Akhmerov, Kamenskii, Potapov, Rodkina and Sadovskii5, Reference Mursaleen25]).

Consider the following coupled system:

(4.1) $$ \begin{align} \left\{ \begin{array}{l} { }^{H} \mathcal{D}_{a^+}^{\alpha_1, \beta_1; t} y_1(t)= g_1(t,y_1(t),y_2(t)), \quad t\in I' := (0, b],~~ 0< b< \frac{1}{4e},\\[0.3cm] { }^{H} \mathcal{D}_{a^+}^{\alpha_2, \beta_2; t} y_2(t)= g_2(t,y_1(t),y_2(t)), \hspace{7pt} t \in I' := (0, b],~~ 0< b< \frac{1}{4e}, \\[0.3cm] \left(t^{1-\gamma_1} y_1\right)(0^+) = (0, 0,\dots, 0, \dots),\\[0.3cm] \left(t^{1-\gamma_2} y_2\right)(0^+) = (0, 0,\dots, 0, \dots). \\ \end{array} \right. \end{align} $$

Note that (4.1) is a particular case of (1.1), where:

$$ \begin{align*}a=0, ~~ \varphi(t)=L^{-1/\phi_{\min}} b^{\zeta} t, ~~ L= \max_{\substack{ 1\leq i\leq 2 \\ 1\leq j\leq 2 }} L_{i,j} , ~~ \phi_{i,j} = \theta_{i} +\vartheta_{i,i}-\vartheta_{i,j}, ~~ \mu_{\min} = \min_{1\leq i \leq 2}\mu_i \end{align*} $$

$$ \begin{align*}\zeta =\frac{1}{\phi_{\min}(1- \mu_{\min})} - 1, ~~ L_{i,j} = \Bigg\{ \frac{2^{\vartheta_{i,j}}}{\theta_i} + \frac{2^{1-\theta_i}}{1-\vartheta_{i,j}} \Bigg\},~~ \phi_{\min}=\min_{\substack{ 1\leq i\leq 2 \\ 1\leq j\leq 2 }} \phi_{i,j},~~ \end{align*} $$

and for $t\in I$ , $y_i =\{y^{i,n}\} _n \in \mathbb {E}$ ,

$$ \begin{align*}g_1(t,y_1,y_2)= e^t {\left\lbrace \arctan(|y^{1,n}| + |y^{2,n}|) + \frac{1}{n^2} \right\rbrace}_{n=1}^{\infty}, \end{align*} $$

$$ \begin{align*}g_2(t,y_1,y_2)= e^t {\left\lbrace \ln(|y^{1,n}| +1) +|y^{2,n}| + \frac{1}{n^2} \right\rbrace}_{n=1}^{\infty}. \end{align*} $$

One can easily deduce that the functions $g_i$ , ( $i=1,2$ ) satisfy the Carathéodory type hypotheses, so (H1) holds.

To justify hypothesis (H2), let $t \in I$ and $y_i =\{y^{i,n}\} _n \in \mathbb {E} $ , $(i=1,2)$ . We have

$$ \begin{align*} \Vert g_1(t,y_1,y_2) \Vert_{\mathbb{E}} & = e^t \Big\Vert {\left\lbrace \arctan(|y^{1,n}| + |y^{2,n}|) + \frac{1}{n^2} \right\rbrace}_{n=1}^{\infty} \Big\Vert_{\mathbb{E}} \\ &\leq e^t \left( \sup_{n \geq 1}|y^{1,n}|+ \sup_{n \geq 1}|y^{2,n}| +1 \right) \\ &\leq e^t \left( \Vert y_{1}\Vert_{\mathbb{E}}+ \Vert y_{2}\Vert_{\mathbb{E}} +1 \right), \end{align*} $$

and

$$ \begin{align*} \Vert g_2(t,y_1,y_2)\Vert_{\mathbb{E}} & = e^{t}\Big\Vert {\left\lbrace \ln(|1+y^{1,n}|) +|y^{2,n}| + \frac{1}{n^2} \right\rbrace}_{n=1}^{\infty} \Big\Vert_{\mathbb{E}} \\ &\leq e^{t} \left( \sup_{n \geq 1}|y^{1,n}|+ \sup_{n \geq 1}|y^{2,n}| +1 \right) \\ &\leq e^{t}\left( \Vert y_{1}\Vert_{\mathbb{E}}+ \Vert y_{2}\Vert_{\mathbb{E}} +1 \right). \end{align*} $$

This shows that hypothesis (H2) holds, with

$$ \begin{align*}h(t)=h_1(t)=h_2(t)= e^{t}, \quad \text{for all } t \in I. \end{align*} $$

To prove (3.1), let $t \in I$ and $z_i =\{z^{i,n}\} _n \in \mathbb {A}^{i}\subseteq \mathbb {E} $ , $i=1,2$ . Fix $n \in \mathbb {N}$ ; then we have

$$ \begin{align*}\arctan(|z^{1,k}| +|z^{2,k}|) \leq |z^{1,k}| +|z^{2,k}| \leq \Vert \left( \mathrm{I} - \mathrm{P}_n \right)(z^{1,k})_k \Vert_{\infty} +\Vert \left( \mathrm{I} - \mathrm{P}_n \right)(z^{2,k})_k\Vert_{\infty}, \end{align*} $$

for all $k>n$ , which implies, by taking the supremum, that

$$ \begin{align*}&\hspace{-1cm} \sup_{(z_1,z_2) \in \mathbb{A}^{1}\times\mathbb{A}^{2}} \Vert \left( \mathrm{I} - \mathrm{P}_n \right) (\arctan(|y^{1,k}| +|y^{2,k}|))_k \Vert_{\infty}\\ &\leq \sup_{z_1 \in \mathbb{A}^{1}}\Vert \left( \mathrm{I} - \mathrm{P}_n \right)(z^{1,k})_k \Vert_{\infty} + \sup_{z_2 \in \mathbb{A}^{2}}\Vert \left( \mathrm{I} - \mathrm{P}_n \right)(z^{2,k})_k \Vert_{\infty}. \end{align*} $$

Then

$$ \begin{align*}&\hspace{-0.1cm} \lim_{n \to \infty}\sup_{(z_1,z_2) \in \mathbb{A}^{1}\times\mathbb{A}^{2}} \Vert \left( \mathrm{I} - \mathrm{P}_n \right) (\arctan(|y^{1,k}|+|y^{2,k}|))_k \Vert_{\infty}\\ &\leq \lim_{n \to \infty}\sup_{z_1 \in \mathbb{A}^{1}}\Vert \left( \mathrm{I} - \mathrm{P}_n \right) (z^{1,k})_k \Vert_{\infty} + \lim_{n \to \infty}\sup_{z_2 \in \mathbb{A}^{2}} \Vert \left( \mathrm{I} - \mathrm{P}_n \right) (z^{2,k})_k \Vert_{\infty}, \end{align*} $$

which yields

$$ \begin{align*} &\hspace{-0.1cm} \Lambda (g_1(t,\mathbb{A}^1,\mathbb{A}^2))\\ &= e^{t} \lim_{n \to \infty}\sup_{(z_1,z_2) \in \mathbb{A}^{1}\times\mathbb{A}^{2}} \left\Vert \left( \mathrm{I} - \mathrm{P}_n \right) \left(\arctan(|z^{1,k}| +|z^{2,k}| )+\frac{1}{k^2}\right)_k \right\Vert{}_{\infty}\\ &\leq e^{t} \Big[ \lim_{n \to \infty}\sup_{z_1 \in \mathbb{A}^{1}}\Vert \left( \mathrm{I} - \mathrm{P}_n \right) (z^{1,k} )_k \Vert_{\infty}\\ &\quad + \lim_{n \to \infty}\sup_{z_2 \in \mathbb{A}^{2}} \Vert \left( \mathrm{I} - \mathrm{P}_n \right) (z^{2,k})_k \Vert_{\infty} \Big]\\ &\leq \eta_{1,1}(t) \Lambda(\mathbb{A}^1) +\eta_{1,2}(t)\Lambda(\mathbb{A}^2), \end{align*} $$

where

$$ \begin{align*}\eta_{1,1} =\eta_{1,2}= h. \end{align*} $$

Similarly, one can obtain

$$ \begin{align*}\Lambda (g_2(t,\mathbb{A}^1,\mathbb{A}^2)) \leq \eta_{2,1}(t) \Lambda(\mathbb{A}^1) +\eta_{2,2}(t)\Lambda(\mathbb{A}^2), \end{align*} $$

where

$$ \begin{align*}\eta_{2,1} =\eta_{2,2}= h. \end{align*} $$

Hence condition (H3) is verified. Now, it remains to show that (3.2) holds. To do this, from $\Gamma (\alpha _i)>1 $ for $ 0<\alpha _i <1$ , we have

$$ \begin{align*}\widehat{K}_i \leq \Vert h\Vert_{L^{\frac{1}{\mu_i}}}\frac{\varphi(b,0)^{1 + \alpha_i - \gamma_i -\mu_i }}{ \theta_{i}^{1-\mu_i}} + K \displaystyle\sum_{j=1}^{2} \Vert h\Vert_{L^{\frac{1}{\mu_i}}} \left( \Theta_{\theta_i, \vartheta_{i,j}} (0,b,b,\varphi) \varphi(b,0)^{\vartheta_{i,i}} \right)^{1-\mu_i}. \end{align*} $$

Then, by Lemma 2.2 and $ 0<\varphi (b,0) <1$ , we get

$$ \begin{align*}\widehat{K}_i &\leq \displaystyle \Vert h\Vert_{L^{\frac{1}{\mu_i}}}\frac{\varphi(b,0)^{1 + \alpha_i - \gamma_i -\mu_i }}{ \theta_{i}^{1-\mu_i}} \\ &\qquad + K \displaystyle\sum_{j=1}^{2} \Vert h\Vert_{L^{\frac{1}{\mu_i}}} \left( \left(\frac{2^{\vartheta_{i,j}}}{\theta_i} +\frac{2^{1-\theta_i}}{1-\vartheta_{i,j}} \right) \varphi(b,0)^{\theta_i -\vartheta_{i,j}} \varphi(b,0)^{\vartheta_{i,i}} \right)^{1-\mu_i} \\ &\leq \displaystyle \Vert h\Vert_{L^{\frac{1}{\mu_i}}}\frac{\varphi(b,0)^{1 + \alpha_i - \gamma_i -\mu_i }}{ \theta_{i}^{1-\mu_i}} + 2 K \Vert h\Vert_{L^{\frac{1}{\mu_i}}} \left( L \left( L^{-1/\phi_{\min}} b^{\zeta +1} \right)^{\phi_{\min}} \right)^{1-\mu_i} \\ &\leq \displaystyle \Vert h\Vert_{L^{\frac{1}{\mu_i}}}\frac{\varphi(b,0)^{1 + \alpha_i - \gamma_i -\mu_i }}{ \theta_{i}^{1-\mu_i}} + 2 K \Vert h\Vert_{L^{\frac{1}{\mu_i}}} \left( b^{ \frac{1}{1-\mu_{\min}}} \right)^{1-\mu_{\min}} \\ &\leq \displaystyle \frac{\varphi(b,0)^{1 + \alpha_i - \gamma_i -\mu_i }}{\theta_{i}^{1-\mu_i}} \Vert h\Vert_{L^{\frac{1}{\mu_i}}} + 2 K b \Vert h\Vert_{L^{\frac{1}{\mu_i}}} \end{align*} $$

and

$$ \begin{align*}\Vert h\Vert_{L^{\frac{1}{\mu_i}}} = \left(\mu_i e^{b/\mu_i} - \mu_i \right)^{\mu_i} \leq \left(\mu_i e^{b/\mu_i} \right)^{\mu_i} = \mu_{i}^{\mu_i} e^{b} \leq \mu_{i}^{\mu_i} e^{1} = e^{1 + \mu_i \ln(\mu_i)} \leq e. \end{align*} $$

Now, choose

$$ \begin{align*} K &\geq \frac{2e\varphi(b,0)^{1 + \alpha_i - \gamma_i -\mu_i } }{\theta_{i}^{1-\mu_i}(1- 4be )} ,\quad i=1,2. \end{align*} $$

Hence,

$$ \begin{align*}2 \widehat{K}_i\leq K,\quad i=1,2. \end{align*} $$

Since, all hypotheses in Theorem 3.1 are verified, system (4.1) has at least one solution.