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Implicit coupled Hilfer–Hadamard fractional differential systems under weak topologies
Advances in Difference Equations volume 2018, Article number: 328 (2018)
Abstract
In this article, we present some existence of weak solutions for a coupled system of implicit fractional differential equations of Hilfer–Hadamard type. Our approach is based on Mönch’s fixed point theorem associated with the technique of measure of weak noncompactness.
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:
with the initial conditions
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
As usual, \(\mathrm{AC}(I)\) denotes the space of absolutely continuous functions from I into E. We define the space
where \(w'(t)=\frac{\mathrm{d}}{\mathrm{d}t}w(t), t\in I\). Let
where \([q]\) is the integer part of \(q>0\). Define the space
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
where \(\bar{w}(t)= t^{1-\gamma}w(t), t\in(1,T]\), with the norm
with the norm
and
where \(\widetilde{w}(t)= (\ln t)^{1-\gamma}w(t), t\in I\), with the norm
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
In the same way, we can define the the weighted product space \({\overline{C}}:=(C_{\gamma,\ln}(I))^{n}\) with the norm
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
The space \(\mathrm{P}_{1}(I,E) \) is normed by
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
The left-sided mixed Riemann–Liouville integral of order \(r>0\) of a function \(w\in L^{1}(I)\) is defined by
where Γ is the (Euler) gamma function defined by
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
Definition 2.6
The Riemann–Liouville fractional derivative of order \(r>0\) of a function \(w\in L^{1}(I)\) is defined by
where \(n=[r]+1\), and \([r]\) is the integer part of r.
In particular, if \(r\in(0,1]\), then
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:
Moreover, if \(I_{1}^{1-r}w\in C^{1}_{1-\gamma}(I)\), then the following composition is proved in [30]:
Definition 2.7
The Caputo fractional derivative of order \(r>0\) of a function \(w\in L^{1}(I)\) is defined by
In particular, if \(r\in(0,1]\), then
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
provided that the integral exists.
Example 2.9
Let \(0< q<1\). Then
Remark 2.10
Let \(g\in\mathrm{P}_{1}(I, E)\). For every \(\varphi\in E^{*}\), we have
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
In particular, if \(q\in(0,1]\), then
Example 2.12
Let \(0< q<1\). Then
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,
From [22, Thm. 2.3] we have
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
In particular, if \(q\in(0,1]\), then
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
Properties
Let \(\alpha\in(0,1), \beta\in[0,1], \gamma =\alpha+\beta-\alpha\beta\), and \(w\in L^{1}(I)\).
-
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.
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.
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.
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
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:
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
and if \(g_{i}(\cdot)\in\mathrm{C}_{\gamma,\ln}\) are the solutions of this system, then
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
The De Blasi measure of weak noncompactness satisfies the following properties:
-
(a)
\(A\subset B\Rightarrow\mu(A)\leq\mu(B)\),
-
(b)
\(\mu(A)= 0 \Leftrightarrow A \) is weakly relatively compact,
-
(c)
\(\mu(A\cup B)=\max\{\mu(A), \mu(B)\}\),
-
(d)
\(\mu(\overline{A}^{\omega})=\mu(A)\), where \(\overline{A}^{\omega}\) denotes the weak closure of A,
-
(e)
\(\mu(A+B)\leq\mu(A)+\mu(B)\),
-
(f)
\(\mu(\lambda A)=|\lambda| \mu(A)\),
-
(g)
\(\mu(\operatorname{conv}(A))=\mu(A)\),
-
(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
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,
and
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
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
Theorem 3.2
Assume that the hypotheses \((H_{1})\)–\((H_{4}) \) hold. If
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
where \(g_{i}\in C_{\gamma,\ln}, i=1,2\), are defined as
Consider the operator \(N:{\mathcal{C}}\to{\mathcal{C}}\) such that, for any \((u_{1},u_{2})\in{\mathcal{C}}\),
First, notice that the hypotheses imply that, for each \(g_{i}\in C_{\gamma,\ln}, i=1,2\), the function
is Pettis integrable over I, and
\(\text{for a.e. } t\in I\) is Pettis integrable. Thus, the operator N is well defined. Let \(R>0\) be such that \(R>L_{1}+L_{2}\), where
and consider the set
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
where \(g_{i}\in C_{\gamma,\ln}\) are defined as
Then from \((H_{3})\) we get
Thus
Hence we get
Next, let \(t_{1},t_{2}\in I\) be such that \(t_{1}< t_{2}\), and let \(u\in Q\) be such that
Then there exists \(\varphi\in E^{*}\) such that
and \(\|\varphi\|=1\). Then, for any \(i\in\{1,2\}\), we have
where \(g_{i}\in\mathrm{C}_{\gamma,\ln}\) are defined as
Then
Thus, we get
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,
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
Thus
Hence
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:
\(I:=[1,T], T>1, \alpha\in(0,1), \beta\in[0,1], \gamma=\alpha +\beta-\alpha\beta, \phi_{i}\in E, f_{i}:I\times E^{2n}\to E, i=1,2,\dots,n\), 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
where
then the coupled system (8)–(9) has at least one weak solution defined on I.
4 An example
Let
be the Banach space with the norm
As an application of our results, we consider the coupled system of Hilfer–Hadamard fractional differential equations
where
and
with
Set
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
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,
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.
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The work was supported by the National Natural Science Foundation of China (No. 11671339).
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Abbas, S., Benchohra, M., Hamidi, N. et al. Implicit coupled Hilfer–Hadamard fractional differential systems under weak topologies. Adv Differ Equ 2018, 328 (2018). https://doi.org/10.1186/s13662-018-1787-4
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DOI: https://doi.org/10.1186/s13662-018-1787-4
MSC
- 26A33
- 45D05
- 45G05
- 45M10
Keywords
- Coupled fractional differential system
- Left-sided mixed Pettis–Hadamard integral of fractional order
- Hilfer–Hadamard fractional derivative
- Weak solution
- Implicit
- Fixed point