1 Introduction

In recent years, fractional calculus and fractional differential equations are emerging as a useful tool in modeling the dynamics of many physical systems and electrical phenomena, which has been demonstrated by many researchers in the fields of mathematics, science, and engineering; see [3, 4, 18, 19, 22, 23, 30, 31, 35–40]. Recently, considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations with Hilfer fractional derivative [15, 16, 18, 20, 32, 34] and other problems with Hilfer–Hadamard fractional derivative [28, 29].

The measure of weak noncompactness was introduced by De Blasi [14]. The strong measure of noncompactness was developed first by Banaś and Goebel [8] and subsequently developed and used in many papers; see, for example, Akhmerov et al. [6], Alvárez [7], Benchohra et al. [12], Guo et al. [17], and the references therein. In [12, 26], the authors considered some existence results by the technique of measure of noncompactness. Recently, several researchers obtained other results by the technique of measure of weak noncompactness; see [2, 4, 10, 11] and the references therein.

Consider the following coupled system of implicit Hilfer–Hadamard fractional differential equations:

$$ \textstyle\begin{cases} ({}^{H}D_{1}^{\alpha,\beta }u_{1})(t)=f_{1}(t,u_{1}(t),u_{2}(t),({}^{H}D_{1}^{\alpha,\beta }u_{1})(t),({}^{H}D_{1}^{\alpha,\beta}u_{2})(t)), &\\ ({}^{H}D_{1}^{\alpha,\beta }u_{2})(t)=f_{2}(t,u_{1}(t),u_{2}(t),({}^{H}D_{1}^{\alpha,\beta }u_{1})(t),({}^{H}D_{1}^{\alpha,\beta}u_{2})(t)), \end{cases}\displaystyle t\in I, $$
(1)

with the initial conditions

$$ \textstyle\begin{cases} ({}^{H}I_{1}^{1-\gamma}u_{i})(t)|_{t=1}=\phi_{1}, &\\ ({}^{H}I_{1}^{1-\gamma}u_{2})(t)|_{t=1}=\phi_{2}, \end{cases} $$
(2)

where \(I:=[1,T], T>1, \alpha\in(0,1), \beta\in[0,1], \gamma =\alpha+\beta-\alpha\beta, \phi_{i}\in E\), \(f_{i}:I\times E^{4}\to E, i=1,2\), are given continuous functions, E is a real (or complex) Banach space with norm \(\|\cdot\|_{E}\) and dual \(E^{*}\), such that E is the dual of a weakly compactly generated Banach space X, \({}^{H}I_{1}^{1-\gamma}\) is the left-sided mixed Hadamard integral of order \(1-\gamma\), and \({}^{H}D_{1}^{\alpha,\beta}\) is the Hilfer–Hadamard fractional derivative of order α and type β. In this paper, we prove the existence of weak solutions for a coupled system of implicit fractional differential equations of Hilfer–Hadamard type.

2 Preliminaries

Let C be the Banach space of all continuous functions v from I into E with the supremum (uniform) norm

$$\Vert v \Vert _{\infty}:= \sup _{t\in I} \bigl\Vert v(t) \bigr\Vert _{E}. $$

As usual, \(\mathrm{AC}(I)\) denotes the space of absolutely continuous functions from I into E. We define the space

$$\mathrm{AC}^{1}(I):=\bigl\{ w:I\to E:w'\in\mathrm{AC}(I) \bigr\} , $$

where \(w'(t)=\frac{\mathrm{d}}{\mathrm{d}t}w(t), t\in I\). Let

$$\delta=t\frac{\mathrm{d}}{\mathrm{d}t},\qquad n=[q]+1, $$

where \([q]\) is the integer part of \(q>0\). Define the space

$$\mathrm{AC}_{\delta}^{n}:=\bigl\{ u:[1,T]\to E: \delta^{n-1}(u)\in\mathrm {AC}(I)\bigr\} . $$

Let \(\gamma\in(0,1]\). By \(\mathrm{C}_{\gamma}(I), \mathrm{C}^{1}_{\gamma}(I)\), and \(\mathrm{C}_{\gamma,\ln}(I)\) we denote the weighted spaces of continuous functions defined by

$$\mathrm{C}_{\gamma}(I)=\bigl\{ w:(1,T]\to E: \in\bar{w}\in\mathrm{C}\bigr\} , $$

where \(\bar{w}(t)= t^{1-\gamma}w(t), t\in(1,T]\), with the norm

$$\begin{aligned} &\Vert w \Vert _{\mathrm{C}_{\gamma}}:= \sup _{t\in I} \bigl\Vert \bar{w}(t) \bigr\Vert _{E}, \\ &\mathrm{C}^{1}_{\gamma}(I)=\bigl\{ w\in\mathrm{C}: w'\in\mathrm{C}_{\gamma}\bigr\} \end{aligned}$$

with the norm

$$\Vert w \Vert _{\mathrm{C}^{1}_{\gamma}}:= \Vert w \Vert _{\infty}+ \bigl\Vert w' \bigr\Vert _{\mathrm{C}_{\gamma}}, $$

and

$$\mathrm{C}_{\gamma,\ln}(I)=\{w:I\to E:\widetilde{w}\in\mathrm{C}\}, $$

where \(\widetilde{w}(t)= (\ln t)^{1-\gamma}w(t), t\in I\), with the norm

$$\Vert w \Vert _{\mathrm{C}_{\gamma,\ln}}:= \sup _{t\in I} \bigl\Vert \widetilde{w}(t) \bigr\Vert _{E}. $$

We further denote \(\|w\|_{\mathrm{C}_{\gamma,\ln}}\) by \(\|w\|_{C}\).

Define the weighted product space \({\mathcal {C}}:=C_{\gamma,\ln}(I)\times C_{\gamma,\ln}(I)\) with the norm

$$\bigl\Vert (w_{1},w_{2}) \bigr\Vert _{\mathcal{C}}:= \Vert w_{1} \Vert _{C}+ \Vert w_{2} \Vert _{C}. $$

In the same way, we can define the the weighted product space \({\overline{C}}:=(C_{\gamma,\ln}(I))^{n}\) with the norm

$$\bigl\Vert (w_{1},w_{2},\ldots,w_{n}) \bigr\Vert _{\overline{C}}:= \sum _{k=1}^{n} \Vert w_{k} \Vert _{C}. $$

Let \((E,w)=(E,\sigma(E,E^{*}))\) be the Banach space E with weak topology.

Definition 2.1

A Banach space X is said to be weakly compactly generated (WCG) if it contains a weakly compact set whose linear span is dense in X.

Definition 2.2

A function \(h:E\rightarrow E\) is said to be weakly sequentially continuous if h takes each weakly convergent sequence in E to a weakly convergent sequence in E (i.e., for any \((u_{n})\) in E with \(u_{n}\rightarrow u\) in \((E,w)\), we have \(h(u_{n})\rightarrow h(u)\) in \((E,w)\)).

Definition 2.3

([27])

The function \(u:I\rightarrow E\) is said to be Pettis integrable on I if and only if there is an element \(u_{J}\in E\) corresponding to each \(J\subset I\) such that \(\phi(u_{J})=\int_{J} \phi(u(s))\,\mathrm{d}s\) for all \(\phi\in E^{\ast}\), where the integral on the right-hand side is assumed to exist in the Lebesgue sense (by definition \(u_{J}=\int_{J}u(s)\,\mathrm{d}s)\).

Let \(\mathrm{P}(I,E)\) be the space of all E-valued Pettis-integrable functions on I, and let \(L^{1}(I,E)\) be the Banach space of Bochner-integrable measurable functions \(u:I\to E\). Define the class

$$\mathrm{P}_{1}(I,E)=\bigl\{ u\in P(I,E): \varphi(u)\in L^{1}(I,{\mathbb {R}}) \text{ for every } \varphi\in E^{*}\bigr\} . $$

The space \(\mathrm{P}_{1}(I,E) \) is normed by

$$\Vert u \Vert _{\mathrm{P}_{1}}= \sup _{\varphi\in E^{*}, \Vert \varphi \Vert \leq1} \int _{1}^{T} \bigl\vert \varphi\bigl(u(x)\bigr) \bigr\vert \,\mathrm{d}\lambda x, $$

where λ is the Lebesgue measure on I.

The following result is due to Pettis [27, Thm. 3.4 and Cor. 3.41].

Proposition 2.4

([27])

If \(u\in\mathrm{P}_{1}(I,E)\) and h is a measurable and essentially bounded E-valued function, then \(uh\in\mathrm{P}_{1}(I,E)\).

In what follows, the symbol “∫” denotes the Pettis integral.

Now, we give some results and properties of fractional calculus.

Definition 2.5

([3, 22, 30])

The left-sided mixed Riemann–Liouville integral of order \(r>0\) of a function \(w\in L^{1}(I)\) is defined by

$$\bigl( I_{1}^{r}w\bigr) (t) =\frac{1}{\Gamma(r)} \int_{1}^{t}( t-s) ^{r-1}w(s)\,\mathrm{d}s\quad \text{for a.e. } t\in I, $$

where Γ is the (Euler) gamma function defined by

$$\Gamma(\xi)= \int_{0}^{\infty}t^{\xi-1}e^{-t}\,{ \mathrm {d}}t,\quad \xi>0. $$

Notice that, for all \(r,r_{1},r_{2}>0\) and \(w\in\mathrm{C}\), we have \(I_{0}^{r}w\in\mathrm{C}\) and

$$\bigl(I_{1}^{r_{1}}I_{1}^{r_{2}}w\bigr) (t)= \bigl(I_{1}^{r_{1}+r_{2}}w\bigr) (t);\quad \mbox{for a.e. } t\in I. $$

Definition 2.6

([3, 22, 30])

The Riemann–Liouville fractional derivative of order \(r>0\) of a function \(w\in L^{1}(I)\) is defined by

$$\begin{aligned} \bigl(D^{r}_{1} w\bigr) (t)&=\biggl(\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}I_{1}^{n-r}w \biggr) (t) \\ &=\frac{1}{\Gamma(n-r)}\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}} \int _{1}^{t}(t-s)^{n-r-1}w(s)\,\mathrm{d}s\quad \text{for a.e. } t\in I, \end{aligned}$$

where \(n=[r]+1\), and \([r]\) is the integer part of r.

In particular, if \(r\in(0,1]\), then

$$\begin{aligned} \bigl(D^{r}_{1} w\bigr) (t)&= \biggl(\frac{\mathrm{d}}{\mathrm{d}t}I_{1}^{1-r}w \biggr) (t) \\ &=\frac{1}{\Gamma(1-r)}\frac{\mathrm{d}}{\mathrm{d}t} \int _{1}^{t}(t-s)^{-r}w(s)\,\mathrm{d}s\quad \text{for a.e. } t\in I. \end{aligned}$$

Let \(r\in(0,1], \gamma\in[0,1)\), and \(w\in\mathrm{C}_{1-\gamma }(I)\). Then the following expression leads to the left inverse operator:

$$\bigl(D_{1}^{r}I_{1}^{r}w\bigr) (t)=w(t) \quad\text{for all } t\in(1,T]. $$

Moreover, if \(I_{1}^{1-r}w\in C^{1}_{1-\gamma}(I)\), then the following composition is proved in [30]:

$$\bigl(I_{1}^{r}D_{1}^{r}w\bigr) (t)=w(t)-\frac{(I_{1}^{1-r}w)(1^{+})}{\Gamma (r)}t^{r-1} \quad\text{for all } t\in(1,T]. $$

Definition 2.7

([3, 22, 30])

The Caputo fractional derivative of order \(r>0\) of a function \(w\in L^{1}(I)\) is defined by

$$\begin{aligned} \bigl({}^{c}D^{r}_{1}w\bigr) (t)&= \biggl(I_{1}^{n-r}\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}w \biggr) (t) \\ &=\frac{1}{\Gamma(n-r)} \int_{1}^{t}(t-s)^{n-r-1}\frac {\mathrm{d}^{n}}{\mathrm{d}s^{n}}w(s)\,{ \mathrm {d}}s \quad\text{for a.e. } t\in I. \end{aligned}$$

In particular, if \(r\in(0,1]\), then

$$\begin{aligned} \bigl({}^{c}D^{r}_{1}w\bigr) (t)&= \biggl(I_{1}^{1-r}\frac{\mathrm{d}}{\mathrm{d}t}w \biggr) (t) \\ &=\frac{1}{\Gamma(1-r)} \int_{1}^{t}(t-s)^{-r}\frac{d}{\mathrm {d}s}w(s)\,{ \mathrm {d}}s \quad\text{for a.e. } t\in I. \end{aligned}$$

Let us recall some definitions and properties of Hadamard fractional integration and differentiation. We refer to [22] for more details.

Definition 2.8

([22])

The Hadamard fractional integral of order \(q>0\) for a function \(g\in L^{1}(I,E)\) is defined as

$$\bigl({}^{H}I_{1}^{q}g\bigr) (x)= \frac{1}{\Gamma(q)} \int_{1}^{x} \biggl(\ln\frac {x}{s} \biggr)^{q-1}\frac{g(s)}{s}\,\mathrm{d}s, $$

provided that the integral exists.

Example 2.9

Let \(0< q<1\). Then

$$^{H}I_{1}^{q} \ln t=\frac{1}{\Gamma(2+q)}(\ln t)^{1+q} \quad\text{for a.e. } t\in[0,e]. $$

Remark 2.10

Let \(g\in\mathrm{P}_{1}(I, E)\). For every \(\varphi\in E^{*}\), we have

$$\varphi\bigl({}^{H}I_{1}^{q}g\bigr) (t)= \bigl({}^{H}I_{1}^{q}\varphi g\bigr) (t) \quad\text{for a.e. } t\in I. $$

Similarly to the Riemann–Liouville fractional calculus, the Hadamard fractional derivative is defined in terms of the Hadamard fractional integral as follows.

Definition 2.11

([22])

The Hadamard fractional derivative of order \(q>0\) applied to a function \(w\in\mathrm{AC}_{\delta}^{n}\) is defined as

$$\bigl({}^{H}D_{1}^{q}w\bigr) (x)= \delta^{n} \bigl({}^{H}I_{1}^{n-q}w\bigr) (x). $$

In particular, if \(q\in(0,1]\), then

$$\bigl({}^{H}D_{1}^{q}w\bigr) (x)=\delta \bigl({}^{H}I_{1}^{1-q}w\bigr) (x). $$

Example 2.12

Let \(0< q<1\). Then

$$^{H}D_{1}^{q} \ln t=\frac{1}{\Gamma(2-q)}(\ln t)^{1-q} \quad\text{for a.e. } t\in[0,e]. $$

It has been proved (see, e.g., Kilbas [21, Thm. 4.8]) that, in the space \(L^{1}(I,E)\), the Hadamard fractional derivative is the left-inverse operator to the Hadamard fractional integral, that is,

$$\bigl({}^{H}D_{1}^{q}\bigr) \bigl({}^{H}I_{1}^{q}w \bigr) (x)=w(x). $$

From [22, Thm. 2.3] we have

$$\bigl({}^{H}I_{1}^{q}\bigr) \bigl({}^{H}D_{1}^{q}w \bigr) (x)=w(x)-\frac {({}^{H}I_{1}^{1-q}w)(1)}{\Gamma(q)}(\ln x)^{q-1}. $$

Similarly to the Hadamard fractional calculus, the Caputo–Hadamard fractional derivative is defined as follows.

Definition 2.13

The Caputo–Hadamard fractional derivative of order \(q>0\) applied to a function \(w\in\mathrm{AC}_{\delta}^{n}\) is defined as

$$\bigl({}^{Hc}D_{1}^{q}w\bigr) (x)= \bigl({}^{H}I_{1}^{n-q}\delta^{n}w\bigr) (x). $$

In particular, if \(q\in(0,1]\), then

$$\bigl({}^{Hc}D_{1}^{q}w\bigr) (x)= \bigl({}^{H}I_{1}^{1-q}\delta w\bigr) (x). $$

Hilfer [18] studied applications of the generalized fractional operator having the Riemann–Liouville and the Caputo derivatives as particular cases (see also [20, 32]).

Definition 2.14

Let \(\alpha\in(0,1), \beta\in[0,1], w\in L^{1}(I)\) and \(I_{1}^{(1-\alpha)(1-\beta)}w\in\mathrm{AC}^{1}(I)\). The Hilfer fractional derivative of order α and type β of w is defined as

$$ \bigl(D_{1}^{\alpha,\beta}w\bigr) (t)= \biggl(I_{1}^{\beta(1-\alpha)}\frac{d}{\mathrm {d}t} I_{1}^{(1-\alpha)(1-\beta)}w \biggr) (t) \quad\text{for a.e. } t\in I. $$
(3)

Properties

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

  1. 1.

    The operator \((D_{1}^{\alpha,\beta}w)(t)\) can be written as

    $$\bigl(D_{1}^{\alpha,\beta}w\bigr) (t)= \biggl(I_{1}^{\beta(1-\alpha)} \frac{d}{\mathrm {d}t} I_{1}^{1-\gamma}w \biggr) (t)= \bigl(I_{1}^{\beta(1-\alpha)} D_{1}^{\gamma}w \bigr) (t)\quad \text{for a.e. } t\in I. $$

    Moreover, the parameter γ satisfies

    $$\gamma\in(0,1],\qquad \gamma\geq\alpha,\qquad \gamma>\beta,\qquad 1-\gamma < 1-\beta(1-\alpha). $$
  2. 2.

    For \(\beta=0\), generalization (3) coincides with the Riemann–Liouville derivative and for \(\beta=1\), with the Caputo derivative:

    $$D_{1}^{\alpha,0}=D_{1}^{\alpha}\quad \mbox{and}\quad D_{1}^{\alpha,1}= ^{c}D_{1}^{\alpha}. $$
  3. 3.

    If \(D_{1}^{\beta(1-\alpha)}w\) exists and is in \(L^{1}(I)\), then

    $$\bigl(D_{1}^{\alpha,\beta}I_{1}^{\alpha}w\bigr) (t)= \bigl(I_{1}^{\beta(1-\alpha )}D_{1}^{\beta(1-\alpha)}w\bigr) (t)\quad \text{for a.e. } t\in I. $$

    Furthermore, if \(w\in C_{\gamma}(I)\) and \(I_{1}^{1-\beta(1-\alpha )}w\in C^{1}_{\gamma}(I)\), then

    $$\bigl(D_{1}^{\alpha,\beta}I_{1}^{\alpha}w\bigr) (t)=w(t) \quad\text{for a.e. } t\in I. $$
  4. 4.

    If \(D_{1}^{\gamma}w\) exists and is in \(L^{1}(I)\), then

    $$\bigl(I_{1}^{\alpha}D_{1}^{\alpha,\beta}w\bigr) (t)= \bigl(I_{1}^{\gamma}D_{1}^{\gamma}w\bigr) (t) =w(t)-\frac{I_{1}^{1-\gamma}(1^{+})}{\Gamma(\gamma)}t^{\gamma-1} \quad\text{for a.e. } t\in I. $$

Based on the Hadamard fractional integral, the Hilfer–Hadamard fractional derivative (introduced for the first time in [28]) is defined as follows.

Definition 2.15

Let \(\alpha\in(0,1), \beta\in[0,1]\), \(\gamma=\alpha+\beta -\alpha\beta, w\in L^{1}(I)\), and \({}^{H}I_{1}^{(1-\alpha)(1-\beta)}w\in\mathrm{AC}^{1}(I)\). The Hilfer–Hadamard fractional derivative of order α and type β applied to a function w is defined as

$$\begin{aligned} \bigl({}^{H}D_{1}^{\alpha,\beta}w \bigr) (t)&= \bigl({}^{H}I_{1}^{\beta(1-\alpha )} \bigl({}^{H}D_{1}^{\gamma}w\bigr) \bigr) (t) \\ &= \bigl({}^{H}I_{1}^{\beta(1-\alpha)}\delta\bigl({}^{H}I_{1}^{1-\gamma }w \bigr) \bigr) (t)\quad \text{for a.e. } t\in I. \end{aligned}$$
(4)

This new fractional derivative (4) may be viewed as interpolation of the Hadamard and Caputo–Hadamard fractional derivatives. Indeed, for \(\beta=0\), this derivative reduces to the Hadamard fractional derivative, and, for \(\beta=1\), we recover the Caputo–Hadamard fractional derivative:

$$^{H}D_{1}^{\alpha,0}= ^{H}D_{1}^{\alpha}\quad \mbox{and}\quad ^{H}D_{1}^{\alpha,1}= ^{Hc}D_{1}^{\alpha}. $$

From [29, Thm. 21] we have the following lemma.

Lemma 2.16

Let \(f_{i}:I\times E^{4}\rightarrow E, i=1,2\), be such that \(f_{i}(\cdot ,u,v,\bar{u},\bar{v})\in\mathrm{C}_{\gamma,\ln}(I)\) for any \(u,v,\bar{u},\bar{v}\in\mathrm{C}_{\gamma,\ln}(I)\). Then system (1)–(2) is equivalent to the problem of obtaining the solution of the coupled system

$$\textstyle\begin{cases} g_{1}(t)=f_{1} (t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+({}^{H}I_{1}^{\alpha}g_{1})(t), \frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+({}^{H}I_{1}^{\alpha }g_{2})(t),g_{1}(t),g_{2}(t) ),\\ g_{2}(t)=f_{2} (t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+({}^{H}I_{1}^{\alpha}g_{1})(t), \frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+({}^{H}I_{1}^{\alpha }g_{2})(t),g_{1}(t),g_{2}(t) ), \end{cases} $$

and if \(g_{i}(\cdot)\in\mathrm{C}_{\gamma,\ln}\) are the solutions of this system, then

$$\textstyle\begin{cases} u_{1}(t)=\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+({}^{H}I_{1}^{\alpha}g_{1})(t),\\ u_{2}(t)=\frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+({}^{H}I_{1}^{\alpha}g_{2})(t). \end{cases} $$

Definition 2.17

([14])

Let E be a Banach space, let \(\Omega_{E}\) be the set of bounded subsets of E, and let \(B_{1}\) be the unit ball of E. The De Blasi measure of weak noncompactness is the map \(\mu:\Omega_{E}\rightarrow[0, \infty)\) defined by

$$\mu(X)=\inf\{\varepsilon>0: \text{there exists a weakly compact set } \Omega \subset E \text{ such that } X\subset\varepsilon B_{1}+ \Omega\}. $$

The De Blasi measure of weak noncompactness satisfies the following properties:

  1. (a)

    \(A\subset B\Rightarrow\mu(A)\leq\mu(B)\),

  2. (b)

    \(\mu(A)= 0 \Leftrightarrow A \) is weakly relatively compact,

  3. (c)

    \(\mu(A\cup B)=\max\{\mu(A), \mu(B)\}\),

  4. (d)

    \(\mu(\overline{A}^{\omega})=\mu(A)\), where \(\overline{A}^{\omega}\) denotes the weak closure of A,

  5. (e)

    \(\mu(A+B)\leq\mu(A)+\mu(B)\),

  6. (f)

    \(\mu(\lambda A)=|\lambda| \mu(A)\),

  7. (g)

    \(\mu(\operatorname{conv}(A))=\mu(A)\),

  8. (h)

    \(\mu(\bigcup_{|\lambda|\leq h} \lambda A)= h \mu(A)\).

The next result follows directly from the Hahn–Banach theorem.

Proposition 2.18

If E is a normed space and \(x_{0}\in E-\{0\}\), then there exists \(\varphi\in E^{\ast}\) with \(\|\varphi\|=1\) and \(\varphi(x_{0})=\|x_{0}\|\).

For a given set V of functions \(v: I\to E\), let us denote

$$V(t)=\bigl\{ v(t): v\in V\bigr\} ; \quad t\in I \quad\textit{and}\quad V(I)=\bigl\{ v(t):v\in V, t\in I\bigr\} . $$

Lemma 2.19

([17] )

Let \(H\subset C\) be a bounded equicontinuous subset. Then the function \(t\to\mu(H(t))\) is continuous on I,

$$\mu_{C}(H)= \max _{t\in I}\mu\bigl(H(t)\bigr), $$

and

$$\mu \biggl( \int_{I}u(s)\,\mathrm{d}s \biggr)\leq \int_{I}\mu\bigl(H(s)\bigr)\,\mathrm{d}s, $$

where \(H(t)=\{u(t):u\in H\}, t\in I\), and \(\mu_{C}\) is the De Blasi measure of weak noncompactness defined on the bounded sets of C.

For our purpose, we will need the following fixed point theorem.

Theorem 2.20

([25])

Let Q be a nonempty, closed, convex, and equicontinuous subset of a metrizable locally convex vector space \(C(I,E)\) such that \(0\in{Q}\). Suppose \(T:Q\rightarrow Q\) is weakly sequentially continuous. If the implication

$$ \overline{V}=\overline{\operatorname{conv}}\bigl(\{0\}\cup T(V)\bigr) \Rightarrow V \quad\textit{is relatively weakly compact} $$
(5)

holds for every subset \(V\subset Q\), then the operator T has a fixed point.

3 Existence of weak solutions

Let us start by the definition of a weak solution of problem (1).

Definition 3.1

By a weak solution of the coupled system (1)–(2) we mean a coupled measurable functions \((u_{1},u_{2})\in{\mathcal{C}}\) such that \(({}^{H}I_{1}^{1-\gamma }u_{i})(1^{+})=\phi_{i}, i=1,2\), and the equations \(({}^{H}D_{1}^{\alpha,\beta }u_{i})(t)=f_{i}(t,u_{1}(t),u_{2}(t),({}^{H}D_{1}^{\alpha,\beta }u_{1})(t),({}^{H}D_{1}^{\alpha,\beta}u_{2})(t))\) are satisfied on I.

We further will use the following hypotheses.

\((H_{1})\) :

The functions \(v\to f_{i}(t,v,w,\bar{v},\bar{w}), w\to f_{i}(t,v,w,\bar{v},\bar{w}), \bar{v}\to f_{i}(t,v,w,\bar{v},\bar{w})\), and \(\bar{w}\to f_{i}(t,v,w,\bar{v},\bar{w}), i=1,2\), are weakly sequentially continuous for a.e. \(t\in I\),

\((H_{2})\) :

For all \(v,w,\bar{v},\bar{w}\in E\), the functions \(t\to f_{i}(t,v,w,\bar{v},\bar{w}), i=1,2\), are Pettis integrable a.e. on I,

\((H_{3})\) :

There exist \(p_{i},q_{i}\in C(I,[0,\infty))\) such that, for all \(\varphi\in E^{*}\),

$$\bigl\vert \varphi\bigl(f_{i}(t,u,v,\bar{u},\bar{v})\bigr) \bigr\vert \leq\frac{p_{i}(t) \Vert u \Vert _{E}+q_{i}(t) \Vert v \Vert _{E}}{1+ \Vert \varphi \Vert + \Vert u \Vert _{E}+ \Vert v \Vert _{E}+ \Vert \bar{u} \Vert _{E}+ \Vert \bar{v} \Vert _{E}} $$

\(\text{ for a.e. } t\in I \text{ and all } u,v,\bar{u},\bar{v}\in E\),

\((H_{4})\) :

For all bounded measurable sets \(B_{i}\subset E, i=1,2\), and all \(t\in I\), we have

$$\mu\bigl(f_{1}\bigl(t,B_{1},B_{2},^{H}D_{1}^{\alpha,\beta}B_{1},^{H}D_{1}^{\alpha,\beta }B_{2} \bigr),0\bigr)\leq p_{1}(t)\mu(B_{1})+q_{1}(t) \mu(B_{2}) $$

and

$$\mu\bigl(0,f_{2}\bigl(t,B_{1},B_{2},^{H}D_{1}^{\alpha,\beta}B_{1},^{H}D_{1}^{\alpha,\beta }B_{2} \bigr)\bigr)\leq p_{2}(t)\mu(B_{1})+q_{2}(t) \mu(B_{2}), $$

where \(^{H}D_{1}^{\alpha,\beta}B_{i}=\{^{H}D_{1}^{\alpha,\beta}w:w\in B_{i}\}, i=1,2\).

Set

$$p_{i}^{*}= \sup _{t\in I}p_{i}(t) \quad\mbox{and}\quad q_{i}^{*}= \sup _{t\in I}q_{i}(t),\quad i=1,2. $$

Theorem 3.2

Assume that the hypotheses \((H_{1})\)–\((H_{4}) \) hold. If

$$ L:=\frac{(p_{1}^{*}+p_{2}^{*}+q_{1}^{*}+q_{2}^{*})(\ln T)^{\alpha}}{\Gamma (1+\alpha)}< 1, $$
(6)

then the coupled system (1)–(2) has at least one weak solution defined on I.

Proof

Consider the operators \(N_{i}:C_{\gamma,\ln }\rightarrow C_{\gamma,\ln}, i=1,2\), defined by

$$(N_{i}u_{i}) (t)=\frac{\phi_{i}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+ \bigl({}^{H}I_{1}^{\alpha}g_{i}\bigr) (t), $$

where \(g_{i}\in C_{\gamma,\ln}, i=1,2\), are defined as

$$g_{i}(t)=f_{i} \biggl(t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+\bigl({}^{H}I_{1}^{\alpha}g_{1} \bigr) (t), \frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+\bigl({}^{H}I_{1}^{\alpha }g_{2} \bigr) (t),g_{1}(t),g_{2}(t) \biggr). $$

Consider the operator \(N:{\mathcal{C}}\to{\mathcal{C}}\) such that, for any \((u_{1},u_{2})\in{\mathcal{C}}\),

$$ \bigl(N(u_{1},u_{2})\bigr) (t)= \bigl((N_{1}u_{1}) (t),(N_{2}u_{2}) (t) \bigr). $$
(7)

First, notice that the hypotheses imply that, for each \(g_{i}\in C_{\gamma,\ln}, i=1,2\), the function

$$t\mapsto \biggl(\ln\frac{t}{s} \biggr)^{\alpha-1}g_{i}(s) $$

is Pettis integrable over I, and

$$t\mapsto f_{i} \biggl(t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+ \bigl({}^{H}I_{1}^{\alpha}g_{1}\bigr) (t), \frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+\bigl({}^{H}I_{1}^{\alpha }g_{2} \bigr) (t),g_{1}(t),g_{2}(t) \biggr) $$

\(\text{for a.e. } t\in I\) is Pettis integrable. Thus, the operator N is well defined. Let \(R>0\) be such that \(R>L_{1}+L_{2}\), where

$$L_{i}:=\frac{(p_{i}^{*}+q_{i}^{*})(\ln T)^{1-\gamma+\alpha}}{\Gamma (1+\alpha)},\quad i=1,2, $$

and consider the set

$$\begin{aligned} Q={}& \biggl\{ (u_{1},u_{2})\in{ \mathcal{C}}: \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{C}}\leq R \text{ and } \bigl\Vert (\ln t_{2})^{1-\gamma}u_{i}(t_{2})-( \ln t_{1})^{1-\gamma}u_{i}(t_{1}) \bigr\Vert _{E} \\ \leq{}& L_{i} \biggl(\ln\frac{t_{2}}{t_{1}} \biggr)^{\alpha} \\ &{} +\frac{p_{i}^{*}+q_{i}^{*}}{\Gamma(\alpha)} \int_{1}^{t_{1}} \biggl\vert (\ln t_{2})^{1-\gamma} \biggl(\ln\frac{t_{2}}{s} \biggr)^{\alpha-1} -(\ln t_{1})^{1-\gamma} \biggl(\ln \frac{t_{1}}{s} \biggr)^{\alpha -1} \biggr\vert \,\mathrm{d}s, i=1,2 \biggr\} . \end{aligned}$$

Clearly, the subset Q is closed, convex, and equicontinuous. We will show that the operator N satisfies all the assumptions of Theorem 2.20. The proof will be given in several steps.

Step 1. N maps Q into itself. Let \((u_{1},u_{2})\in Q, t\in I\), and assume that \((N(u_{1},u_{2}))(t)\neq(0.0)\). Then there exists \(\varphi\in E^{*}\) such that \(\|(\ln t)^{1-\gamma}(N_{i}u_{i})(t)\|_{E}=|\varphi((\ln t)^{1-\gamma }(N_{i}u_{i})(t))|, i=1,2\). Thus, for any \(i\in\{1,2\}\), we have

$$\bigl\Vert (\ln t)^{1-\gamma}(N_{i}u_{i}) (t) \bigr\Vert _{E}=\varphi \biggl(\frac{\phi _{i}}{\Gamma(\gamma)}+\frac{(\ln t)^{1-\gamma}}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac{t}{s} \biggr)^{\alpha-1}g_{i}(s)\frac {\mathrm{d}s}{s} \biggr), $$

where \(g_{i}\in C_{\gamma,\ln}\) are defined as

$$g_{i}(t)=f_{i} \biggl(t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+\bigl({}^{H}I_{1}^{\alpha}g_{1} \bigr) (t), \frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+\bigl({}^{H}I_{1}^{\alpha }g_{2} \bigr) (t),g_{1}(t),g_{2}(t) \biggr). $$

Then from \((H_{3})\) we get

$$\bigl\vert \varphi\bigl(g_{i}(t)\bigr) \bigr\vert \leq p_{i}^{*}+q_{i}^{*}. $$

Thus

$$\begin{aligned} &\bigl\Vert (\ln t)^{1-\gamma}(N_{i}u_{i}) (t) \bigr\Vert _{E}\\ &\quad\leq\frac{(\ln t)^{1-\gamma }}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac{t}{s} \biggr)^{\alpha-1} \bigl\vert \varphi \bigl(g_{i}(s)\bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad\leq \frac{(p_{i}^{*}+q_{i}^{*})(\ln T)^{1-\gamma}}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac{t}{s} \biggr)^{\alpha-1}\frac{\mathrm {d}s}{s} \\ &\quad\leq \frac{(p_{i}^{*}+q_{i}^{*})(\ln T)^{1-\gamma+\alpha}}{\Gamma (1+\alpha)} \\ &\quad= L_{i}. \end{aligned}$$

Hence we get

$$\bigl\Vert N(u_{1},u_{1}) \bigr\Vert _{\mathcal{C}}\leq L_{1}+L_{2}< R. $$

Next, let \(t_{1},t_{2}\in I\) be such that \(t_{1}< t_{2}\), and let \(u\in Q\) be such that

$$(\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1})\neq0. $$

Then there exists \(\varphi\in E^{*}\) such that

$$\begin{aligned} &\bigl\Vert (\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1}) \bigr\Vert _{E}\\ &\quad= \bigl\vert \varphi\bigl((\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1})\bigr) \bigr\vert \end{aligned}$$

and \(\|\varphi\|=1\). Then, for any \(i\in\{1,2\}\), we have

$$\begin{aligned} & \bigl\Vert (\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1}) \bigr\Vert _{E} \\ &\quad = \bigl\vert \varphi\bigl((\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1})\bigr) \bigr\vert \\ &\quad\leq\varphi \biggl((\ln t_{2})^{1-\gamma} \int_{1}^{t_{2}} \biggl(\ln \frac{t_{2}}{s} \biggr)^{\alpha-1}\frac{g_{i}(s)}{s\Gamma(\alpha)}\,\mathrm{d}s- (\ln t_{1})^{1-\gamma} \int_{1}^{t_{1}} \biggl(\ln\frac{t_{1}}{s} \biggr)^{\alpha-1}\frac{g_{i}(s)}{s\Gamma(\alpha)}\,\mathrm{d}s \biggr), \end{aligned}$$

where \(g_{i}\in\mathrm{C}_{\gamma,\ln}\) are defined as

$$g_{i}(t)=f_{i} \biggl(t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+\bigl({}^{H}I_{1}^{\alpha}g_{1} \bigr) (t),\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+\bigl({}^{H}I_{1}^{\alpha}g_{2} \bigr) (t),g_{1}(t),g_{2}(t) \biggr). $$

Then

$$\begin{aligned} & \bigl\Vert (\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1}) \bigr\Vert _{E} \\ &\quad\leq(\ln t_{2})^{1-\gamma} \int_{t_{1}}^{t_{2}} \biggl(\ln\frac {t_{2}}{s} \biggr)^{\alpha-1}\frac{ \vert \varphi(g_{i}(s)) \vert }{s\Gamma(\alpha )}\,\mathrm{d}s \\ &\qquad{} + \int_{1}^{t_{1}} \biggl\vert (\ln t_{2})^{1-\gamma} \biggl(\ln\frac {t_{2}}{s} \biggr)^{\alpha-1} -(\ln t_{1})^{1-\gamma} \biggl(\ln \frac{t_{1}}{s} \biggr)^{\alpha -1} \biggr\vert \frac{ \vert \varphi(g_{i}(s)) \vert }{s\Gamma(\alpha)}\, \mathrm{d}s \\ &\quad\leq(\ln t_{2})^{1-\gamma} \int_{t_{1}}^{t_{2}} \biggl(\ln\frac {t_{2}}{s} \biggr)^{\alpha-1}\frac{p_{i}(s)+q_{i}(s)}{s\Gamma(\alpha)}\,\mathrm {d}s \\ & \qquad{}+ \int_{1}^{t_{1}} \biggl\vert (\ln t_{2})^{1-\gamma} \biggl(\ln\frac {t_{2}}{s} \biggr)^{\alpha-1} -(\ln t_{1})^{1-\gamma} \biggl(\ln \frac{t_{1}}{s} \biggr)^{\alpha -1} \biggr\vert \frac{p_{i}(s)+q_{i}(s)}{s\Gamma(\alpha)}\, \mathrm{d}s. \end{aligned}$$

Thus, we get

$$\begin{aligned} & \bigl\Vert (\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1}) \bigr\Vert _{E} \\ &\quad\leq L_{i} \biggl(\ln \frac{t_{2}}{t_{1}} \biggr)^{\alpha} \\ & \qquad{}+\frac{p_{i}^{*}+q_{i}^{*}}{\Gamma(\alpha)} \int_{1}^{t_{1}} \biggl\vert (\ln t_{2})^{1-\gamma} \biggl(\ln\frac{t_{2}}{s} \biggr)^{\alpha-1} -(\ln t_{1})^{1-\gamma} \biggl(\ln \frac{t_{1}}{s} \biggr)^{\alpha -1} \biggr\vert \,\mathrm{d}s. \end{aligned}$$

Hence \(N(Q)\subset Q\).

Step 2. N is weakly sequentially continuous. Let \(\{(u_{n},v_{n})\}_{n}\) be a sequence in Q, and let \((u_{n}(t),v_{n}(t)\to (u(t),v(t)) \) in \((E,\omega)\times(E,\omega)\) for each \(t\in I\). Fix \(t\in I\). Since for any \(i\in{1,2}\), the function \(f_{i}\) satisfies assumption \((H_{1})\), we have that \(f_{i}(t,u_{n}(t),v_{n}(t),({}^{H}D_{1}^{\alpha,\beta }u_{n})(t), ({}^{H}D_{1}^{\alpha,\beta}v_{n})(t))\) converges weakly uniformly to \(f_{i}(t,u(t),v(t),(D_{0}^{\alpha,\beta}u)(t),(D_{0}^{\alpha ,\beta}v)(t))\). Hence the Lebesgue dominated convergence theorem for Pettis integral implies that \((N(u_{n},v_{n}))(t)\) converges weakly uniformly to \((N(u,v))(t)\) in \((E,\omega)\) for each \(t\in I\). Thus \(N(u_{n},v_{n})\to N(u,v)\). Hence \(N:Q\to Q\) is weakly sequentially continuous.

Step 3. Implication ( 5 ) holds. Let V be a subset of Q such that \(\overline{V}=\overline{ {\operatorname{conv}}}(N(V)\cup\{(0,0)\})\). Obviously,

$$V(t)\subset\overline{\operatorname{conv}}(NV) (t))\cup\bigl\{ (0,0)\bigr\} ),\quad t\in I. $$

Further, as V is bounded and equicontinuous, by [13, Lemma 3] the function \(t\to\mu(V(t))\) is continuous on I. From \((H_{3}), (H_{4})\), Lemma 2.19, and the properties of the measure μ, for any \(t\in I\), we have

$$\begin{aligned} &\mu\bigl((\ln t)^{1-\gamma}V(t)\bigr)\\ &\quad\leq\mu\bigl((\ln t)^{1-\gamma}(NV) (t)\cup \bigl\{ (0,0)\bigr\} \bigr) \\ &\quad\leq\mu\bigl((\ln t)^{1-\gamma}(NV) (t)\bigr) \\ &\quad\leq\mu(\bigl\{ \bigl((\ln t)^{1-\gamma}(N_{1}v_{1}) (t),(\ln t)^{1-\gamma }(N_{2}v_{2}) (t):(v_{1},v_{2}) \in V\bigr\} \bigr) \\ &\quad \leq\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac {t}{s} \biggr)^{\alpha-1}\mu\bigl(\bigl\{ (\ln s)^{1-\gamma }\bigl(f_{1} \bigl(s,v_{1}(s),v_{2}(s), \\ &\qquad\bigl({}^{H}D_{1}^{\alpha,\beta}v_{1}\bigr) (t), \bigl({}^{H}D_{1}^{\alpha,\beta }v_{2}\bigr) (t) \bigr),0\bigr):(v_{1},v_{2})\in V\bigr\} \bigr) \frac{\mathrm{d}s}{s} \\ &\qquad{}+\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac {t}{s} \biggr)^{\alpha-1}\mu\bigl(\bigl\{ (\ln s)^{1-\gamma }\bigl(0,f_{2} \bigl(s,v_{1}(s),v_{2}(s), \\ &\qquad\bigl({}^{H}D_{1}^{\alpha,\beta}v_{1}\bigr) (t), \bigl({}^{H}D_{1}^{\alpha,\beta }v_{2}\bigr) (t)\bigr) \bigr):(v_{1},v_{2})\in V\bigr\} \bigr)\frac{\mathrm{d}s}{s} \\ &\quad\leq\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac {t}{s} \biggr)^{\alpha-1}\bigl[p_{1}(s)\mu\bigl(\bigl\{ (\ln s)^{1-\gamma }\bigl(v_{1}(s),0\bigr):(v_{1},0)\in V\bigr\} \bigr) \\ &\qquad{}+q_{1}(s)\mu\bigl(\bigl\{ (\ln s)^{1-\gamma}\bigl(0,v_{2}(s) \bigr):(0,v_{2})\in V\bigr\} \bigr)\bigr]\frac {\mathrm{d}s}{s} \\ &\qquad{}+\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac {t}{s} \biggr)^{\alpha-1}\bigl[p_{2}(s)\mu\bigl(\bigl\{ (\ln s)^{1-\gamma }\bigl(v_{1}(s),0\bigr):(v_{1},0)\in V\bigr\} \bigr) \\ &\qquad{}+q_{2}(s)\mu\bigl(\bigl\{ (\ln s)^{1-\gamma}\bigl(0,v_{2}(s) \bigr):(0,v_{2})\in V\bigr\} \bigr)\bigr]\frac {\mathrm{d}s}{s}. \end{aligned}$$

Thus

$$\begin{aligned} &\mu\bigl((\ln t)^{1-\gamma}V(t)\bigr) \\ &\quad \leq\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac {t}{s} \biggr)^{\alpha-1}\bigl(p_{1}(s)+q_{1}(s)+p_{2}(s)+q_{2}(s) \bigr) \\ &\qquad{}\times\mu\bigl((\ln s)^{1-\gamma}V(s)\bigr)\frac{\mathrm{d}s}{s} \\ &\quad \leq\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac{t}{s} \biggr)^{\alpha -1}\bigl(p_{1}(s)+q_{1}(s)+p_{2}(s)+q_{2}(s) \bigr) \\ &\qquad{}\times \sup _{s\in I}\mu\bigl((\ln s)^{1-\gamma}V(s)\bigr) \frac{\mathrm {d}s}{s} \\ &\quad\leq\frac{(p_{1}^{*}+p_{2}^{*}+q_{1}^{*}+q_{2}^{*})(\ln T)^{\alpha }}{\Gamma(1+\alpha)} \sup _{t\in I}\mu\bigl((\ln t)^{1-\gamma}V(t) \bigr). \end{aligned}$$

Hence

$$\sup _{t\in I}\mu\bigl((\ln t)^{1-\gamma}V(t)\bigr)\leq L \sup _{t\in I}\mu \bigl((\ln t)^{1-\gamma}V(t)\bigr). $$

From (6) we get \(\sup _{t\in I}\mu((\ln t)^{1-\gamma }V(t))=0\), that is, \(\mu(V(t))=0\) for each \(t\in I\). Then by [24, Thm. 2] V is weakly relatively compact in \({\mathcal{C}}\). From Theorem 2.20 we conclude that N has a fixed point, which is a weak solution of the coupled system (1)–(2). □

As a consequence of the theorem, we get the following corollary.

Corollary 3.3

Consider the following system of implicit Hilfer–Hadamard fractional differential equations:

$$\begin{aligned} &\textstyle\begin{cases} ({}^{H}D_{1}^{\alpha,\beta}u_{1})(t)\\ \quad=f_{1}(t,u_{1}(t),u_{2}(t),\ldots,u_{n}(t),\\ \qquad({}^{H}D_{1}^{\alpha,\beta}u_{1})(t),({}^{H}D_{1}^{\alpha,\beta }u_{2})(t),\ldots,({}^{H}D_{1}^{\alpha,\beta}u_{n})(t)), \\ ({}^{H}D_{1}^{\alpha,\beta}u_{2})(t)\\ \quad=f_{2}(t,u_{1}(t),u_{2}(t),\ldots,u_{n}(t),\\ \qquad({}^{H}D_{1}^{\alpha,\beta}u_{1})(t),({}^{H}D_{1}^{\alpha,\beta }u_{2})(t),\ldots,({}^{H}D_{1}^{\alpha,\beta}u_{n})(t)), \\ \vdots\\ ({}^{H}D_{1}^{\alpha,\beta}u_{n})(t)\\ \quad=f_{n}(t,u_{1}(t),u_{2}(t),\ldots,u_{n}(t),\\ \qquad({}^{H}D_{1}^{\alpha,\beta}u_{1})(t),({}^{H}D_{1}^{\alpha,\beta }u_{2})(t),\ldots,({}^{H}D_{1}^{\alpha,\beta}u_{n})(t)), \end{cases}\displaystyle t\in I, \end{aligned}$$
(8)
$$\begin{aligned} & \bigl({}^{H}I_{1}^{1-\gamma}u_{i} \bigr) (t)|_{t=1}=\phi_{i},\quad i=1,2,\dots,n, \end{aligned}$$
(9)

\(I:=[1,T], T>1, \alpha\in(0,1), \beta\in[0,1], \gamma=\alpha +\beta-\alpha\beta, \phi_{i}\in E, f_{i}:I\times E^{2n}\to E, i=1,2,\dots,n\), are given continuous functions, E is a real (or complex) Banach space with norm \(\|\cdot\|_{E}\) and dual \(E^{*}\), such that E is the dual of a weakly compactly generated Banach space X, \({}^{H}I_{1}^{1-\gamma}\) is the left-sided mixed Hadamard integral of order \(1-\gamma\), and \({}^{H}D_{1}^{\alpha,\beta}\) is the Hilfer–Hadamard fractional derivative of order α and type β.

Assume that the following hypotheses hold:

\((H_{01})\) :

The functions \(v_{j}\to f_{i}(t,v_{1},v_{2},\dots,v_{j},\dots ,v_{2n}), i=1,\dots,n, j=1,\dots,2n\), are weakly sequentially continuous for a.e. \(t\in I\),

\((H_{02})\) :

For each \(v_{j}\in E, j=1,\dots,2n\), the functions \(t\to f_{i}(t,v_{1},v_{2},\dots,v_{j},\dots,v_{2n}), i=1,2\), are Pettis integrable a.e. on I,

\((H_{03})\) :

There exist \(p_{ij}\in C(I,[0,\infty))\) such that, for all \(\varphi\in E^{*}\), we have

$$\bigl\vert \varphi\bigl(f_{i}(t,v_{1},v_{2}, \dots,v_{2n})\bigr) \bigr\vert \leq\frac{ \sum _{i=1}^{n} \sum _{j=1}^{n}p_{ij}(t) \Vert v_{j} \Vert _{E}}{1+ \Vert \varphi \Vert + \sum _{j=1}^{n} \Vert v_{i} \Vert _{E}} $$

\(\textit{for a.e. }t\in I\textit{ and each }v_{i}\in E, i=1,2,\dots,n\),

\((H_{04})\) :

For all bounded measurable sets \(B_{i}\subset E, i=1,\dots,n\), and for each \(t\in I\), we have

$$\begin{aligned} &\mu\bigl(0,\dots,f_{j}\bigl(t,B_{1},B_{2}, \dots,B_{n},^{H}D_{1}^{\alpha,\beta }B_{1},^{H}D_{1}^{\alpha,\beta}B_{2}, \dots, ^{H}D_{1}^{\alpha,\beta}B_{n}\bigr),\dots,0 \bigr) \\ &\quad\leq \sum _{i=1}^{n}p_{ij}(t)\mu(B_{i}),\quad j=1,\dots,n, \end{aligned}$$

where \(^{H}D_{1}^{\alpha,\beta}B_{i}=\{^{H}D_{1}^{\alpha,\beta}w:w\in B_{i}\}, i=1,\dots,n\).

If

$$L^{*}:=\frac{ \sum _{i=1}^{n} \sum _{j=1}^{n}p_{ij}^{*}(\ln T)^{\alpha }}{\Gamma(1+\alpha)}< 1, $$

where

$$p_{ij}^{*}= \sup _{t\in I}p_{ij}(t),\quad i,j=1,\dots,n, $$

then the coupled system (8)–(9) has at least one weak solution defined on I.

4 An example

Let

$$E=l^{1}= \bigl\{ u =(u_{1}, u_{2},\ldots, u_{n},\ldots), \sum ^{\infty }_{n=1} \vert u_{n} \vert < \infty \bigr\} $$

be the Banach space with the norm

$$\Vert u \Vert _{E}= \sum _{n=1}^{\infty} \vert u_{n} \vert . $$

As an application of our results, we consider the coupled system of Hilfer–Hadamard fractional differential equations

$$\begin{aligned} &\textstyle\begin{cases} ({}^{H}D_{1}^{\frac{1}{2},\frac {1}{2}}u_{n})(t)=f_{n}(t,u(t),v(t),({}^{H}D_{1}^{\frac{1}{2},\frac{1}{2}}u_{n})(t), ({}^{H}D_{1}^{\frac{1}{2},\frac{1}{2}}v_{n})(t)),\\ ({}^{H}D_{1}^{\frac{1}{2},\frac {1}{2}}v_{n})(t)=g_{n}(t,u(t),v(t),({}^{H}D_{1}^{\frac{1}{2},\frac{1}{2}}u_{n})(t), ({}^{H}D_{1}^{\frac{1}{2},\frac{1}{2}}v_{n})(t)), \end{cases}\displaystyle t\in[1,e], \end{aligned}$$
(10)
$$\begin{aligned} & \bigl({}^{H}I_{1}^{\frac{1}{4}}u\bigr) (t)|_{t=1}=\bigl({}^{H}I_{1}^{\frac {1}{4}}v\bigr) (t)|_{t=1}=(0,0,\ldots,0,\ldots), \end{aligned}$$
(11)

where

$$f_{n}\bigl(t,u(t),v(t)\bigr)=\frac{ct^{2}}{1+ \Vert u(t) \Vert _{E}+ \Vert v(t) \Vert _{E}+ \Vert \bar{u}(t) \Vert _{E}+ \Vert \bar{v}(t) \Vert _{E}}\frac{u_{n}(t)}{e^{t+4}},\quad t \in[1,e], $$

and

$$g_{n}\bigl(t,u(t),v(t)\bigr)=\frac{ct^{2}}{1+ \Vert v(t) \Vert _{E}+ \Vert v(t) \Vert _{E}+ \Vert \bar{u}(t) \Vert _{E}+ \Vert \bar{v}(t) \Vert _{E}}\frac{u_{n}(t)}{e^{t+4}},\quad t \in[1,e], $$

with

$$u=(u_{1},u_{2},\ldots,u_{n},\ldots),\qquad v=(v_{1},v_{2},\ldots,v_{n},\ldots) \quad\mbox{and}\quad c:=\frac{e^{3}}{16}\sqrt{\pi}. $$

Set

$$f=(f_{1},f_{2},\ldots,f_{n},\ldots) \quad\mbox{and}\quad g=(g_{1},g_{2},\ldots ,g_{n},\ldots). $$

Clearly, the functions f and g are continuous.

For all \(u,v,\bar{u},\bar{v}\in E\) and \(t\in[1,e]\), we have

$$\bigl\Vert f\bigl(t,u(t),v(t),\bar{u}(t),\bar{v}(t)\bigr) \bigr\Vert _{E}\leq ct^{2}\frac{1}{e^{t+4}} \quad\mbox{and}\quad \bigl\Vert g \bigl(t,u(t),v(t),\bar{u}(t),\bar{v}(t)\bigr) \bigr\Vert _{E}\leq ct^{2}\frac {1}{e^{t+4}}. $$

Hence, hypothesis \((H_{3})\) is satisfied with \(p_{i}^{*}=ce^{-3}\) and \(q_{i}^{*}=0, i=1,2\). We will show that condition (6) holds with \(T=e\). Indeed,

$$\frac{(p_{1}^{*}+q_{1}^{*}+p_{2}^{*}+q_{2}^{*})(\ln T)^{\alpha}}{\Gamma (1+\alpha)} =\frac{4ce^{-3}}{\sqrt{\pi}}=\frac{1}{4}< 1. $$

Simple computations show that all conditions of Theorem 3.2 are satisfied. It follows that the coupled system (10)–(11) has at least one weak solution defined on \([1,e]\).

5 Conclusion

In the recent years, implicit functional differential equations have been considered by many authors [1, 5, 9, 33]. In this work, we give some existence results for coupled implicit Hilfer–Hadamard fractional differential systems. This paper initiates the application of the measure of weak noncompactness to such a class of problems.