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New results on Caputo fractional-order neutral differential inclusions without compactness

Abstract

This article deals with existence results of Caputo fractional neutral inclusions without compactness in Banach space using weak topology. In fact, for weakly sequentially closed maps we apply fixed point theorems to obtain the existence of the solution. Furthermore, the results are manifested for fractional neutral system held by nonlocal conditions. To justify the application of the reported results an illustration is presented.

Introduction

The dynamical behavior of real life phenomenon are summarized by essential tools such as fractional differential equations (FDEs) in a precise manner. This aspect is the main convenience of derivatives with fractional-order versus integer-order models. FDEs and inclusions have obtained many interest for their applications in different fields, such as engineering, physics, mechanics, and mathematical modelling, because they are more practical and realistic to describe many natural phenomena. Compared with ordinary and partial differential systems, fractional differential systems have the strong prospect to modulate the real time issues with high efficiency. The goal of analyzing fractional differential systems for the above, major analysis [24, 6, 7, 12, 23, 2528, 3032, 38, 40, 42, 47, 48, 5658] had been carried out. El-Sayed and Ibrahim in [24]. were the first who considered fractional differential inclusions.

Furthermore, differential inclusions are used to model many realistic problems, arising from optimal control, economics, and so on. Recently, by using various techniques, the mild solutions together with other issues for different types of nonlinear fractional evolution inclusions have been studied in [1, 5, 1317, 21, 22, 33, 4345, 4952, 54, 55, 59].

We recall that several techniques and noncompact measures are used to achieve the outcome of the differential systems. Most of these papers, assumed the compactness of the semi group or alternatively a compactness condition on the nonlinear part (generally a measure of noncompactness). In [46], Ravichandran et al. analyzed the controllability of impulsive fractional integro-differential systems utilizing a contraction principle. Li [33] studied the controllability results for neutral impulsive inclusion systems using the Dhage fixed point theorem. The controllability for evolution inclusions without compactness was studied by Benedetti et al. [15]. To the best of our knowledge, the existence of mild solutions of Caputo fractional neutral differential inclusions without compactness has not been studied and this is the main motivation of this work, is to prove the existence of Caputo fractional neutral differential inclusions with weak topology, and without compactness. We investigate the following Caputo fractional neutral inclusion in Banach space:

$$\begin{aligned} &{}^{\mathrm{C}}D^{q}_{u} \bigl[ z(u)-h \bigl(u,z(u) \bigr) \bigr]\in \mathscr {A} z(u)+\mathscr {H} \bigl(u,z(u) \bigr), \quad \mbox{a.e. }u\in [0,b], 0< q\leq 1, \end{aligned}$$
(1.1)
$$\begin{aligned} & z(0)=z_{0} , \end{aligned}$$
(1.2)

where b is positive in nature, \(\mathscr {A}\) represents the infinitesimal generator of a \(C_{0}\)-strongly continuous semigroup \({T(u), u\geq 0,}\) defined from \(\mathscr {A}:D(\mathscr {A})\rightarrow \mathbb{Y}\). Besides, \(z(\cdot )\) assumes values in the Banach space \(\mathbb{Y}\), \(z_{0} \) is for an element of \(\mathbb{Y}\), \(\mathscr {H}:[0,b]\times \mathbb{Y}\multimap \mathbb{Y}\) denotes a multivalued map, \(h:[0,b]\times \mathbb{Y}\rightarrow \mathbb{Y}\) is equicontinuous and bounded.

The aim of this paper is to derive some sufficient conditions for the existence of neutral differential inclusions in Banach space. Furthermore, we expand the result to get the conditions for neutral differential inclusions with nonlocal conditions. In this paper another procedure is considered, it utilize the weak topology of the state space.

Considering the importance of modeling of crisis phenomena, one may extend the analysis to the existence of solutions for a three step crisis integro-differential equation. Also in order to rise the applicability of the fractional calculus, many researchers assumed a new type of fractional derivatives with different kernels. By exploiting it, one can examine the existence of solutions for high-order fractional differential equations using the Caputo–Fabrizio derivative [811, 36, 37].

The layout of this artical is as follows: The preliminaries and notations are listed in Sect. 2. The existence results are discussed in Sect. 3, and in Sect. 4 we investigate the same fractional system supported by nonlocal conditions. An illustration is offered to enhance the abstract technique.

Basic tools

Here, we present a few fundamental facts on fractional theory and theorems in order to use them in the manuscript.

Let \(\mathbb{Y}\) possessing \(\|\cdot \|\). For some constant \(\mathscr {M}_{1} > 0 \) provided \(\sup_{u\in [0,b]}\|T(u)\| \leq \mathscr {M}_{1} \). Let \((\mathbb{Y}, \Vert \cdot \Vert ) \) be a reflexive Banach space and \(\mathbb{Y}_{w} \) denotes \(\mathbb{Y}\) equipped with the weak topology. \(\mathfrak {C}([0,b];\mathbb{Y})\) refers the Banach space of all continuous functions from \([0,b]\) into \(\mathbb{Y}\). For \(\mathscr {D}\subset \mathbb{Y}\), \(\overline{\mathscr {D}}^{w} \) signifies the weak closure of \(\mathscr {D} \). Moreover, the bounded subset \(\mathscr {D} \) of a reflexive Banach space \(\mathbb{Y}\) is weakly relatively compact. Let us express by \(\| \cdot \| _{p} \) both the \(L^{p}([0,b]; \mathbb{Y})\)-norm and the \(L^{p}([0,b];\mathbb {R} )\)-norm and by \(\| \cdot \|_{0} \) the \(\mathfrak {C}([0,b];\mathbb{Y})\)-norm. We evoke (see [18], Theorem 4.3) that the sequence \(\lbrace \chi _{n} \rbrace \subset \mathfrak {C}([0,b];\mathbb{Y}) \) converges weakly to \(\chi \in \mathfrak {C}([0,b];\mathbb{Y}) \) if and only if

  1. (i)

    there exists \(M > 0\) such that, for every \(m \in \mathbb {N} \) and \(u \in [0,b]\), \(\| \chi _{m} (u) \| \leq M \);

  2. (ii)

    \(\chi _{m}(u) \rightharpoonup \chi (u)\), for every \(u \in [0,b]\).

In addition to that, we state some results that will be utilized further as a part of this manuscript.

Theorem 2.1

(Donal O’Regan fixed point theorem [39])

LetFbe a metrizable locally convex linear topological space andUbe a weakly compact, convex subset ofFand \(\mathbb {C}(U)\)the family of nonempty closed, convex subsets ofU. If \(G:U \rightarrow \mathbb {C}(U)\)possesses weakly sequentially closed graph thenGadmits a fixed point.

Theorem 2.2

([29])

LetΨbe a subset of the Banach space \(\mathbb{Y}\)then the subsequent affirmations are equivalent, namely:

  1. (i)

    Ψis relatively weakly compact;

  2. (ii)

    Ψis relatively weakly sequentially compact.

Remark 2.1

([29])

Let Ψ be a subset of the Banach space \(\mathbb{Y}\), then the subsequent affirmations are equivalent:

  1. (a)

    Ψ is weakly compact;

  2. (b)

    Ψ is weakly sequentially compact.

Theorem 2.3

(Krein–Smulian theorem [23, p. 434])

The convex hull of a weakly compact set in a Banach space \(\mathbb{Y}\)is weakly compact.

Theorem 2.4

(Pettis measurability theorem [41])

Let \((E, \varSigma ) \)be a measurable space, \(\mathbb{Y}\)be a separable Banach space. Then \(f:E\longrightarrow \mathbb{Y}\)is measurable if and only if, for every \(e\in E^{\prime }\), the function \(e\circ f: E\longrightarrow \mathbb{R}\)is measurable with respect toand the Borelσ-algebra in \(\mathbb{R}\).

Now we summarize the subsequent interpretations [7, 31, 32, 38, 42].

Definition 2.1

The form of the fractional integral for f is

$$ I^{\alpha } g(p)= \frac{1}{\varGamma (\alpha )} \int _{0}^{p}\frac{g(w)}{(p-w)^{1- \alpha }}\,dw, \quad p>0, \alpha >0, $$

as the right hand-side is point-wise on \([0,\infty )\) and \(\varGamma ( \alpha )=\int _{0}^{\infty }p^{\alpha -1}e^{-p}\,dp\).

Definition 2.2

([31])

The form of the R-L fractional derivative for \(g:[0, \infty )\rightarrow \mathbb{R}\) is characterized by

$$ {}^{(\text{R--L})}D_{0+}^{\alpha }g(p)=\frac{1}{\varGamma (m-\alpha )} \biggl(\frac{d}{dp} \biggr) ^{m} \int _{0}^{p} (p-w)^{m-\alpha -1}g(w)\,dw, \quad p>0, m-1< \alpha < m, $$

such that the function \(g(p)\) posses absolutely continuous derivative up to \((m-1)\).

Definition 2.3

([31])

The expression of the Caputo derivative for \(g:[0,\infty )\rightarrow \mathbb{R}\) is

$$ {}^{\mathrm{C}}D^{\alpha }g(p)= {}^{(\text{R--L})}D^{\alpha } \Biggl(g(p)-\sum_{k=0}^{m-1} \frac{p ^{k}}{k!}g^{(k)}(0) \Biggr),\quad p>0, m-1 < \alpha < m. $$

Remark 2.2

(i) If \(g(p)\in C^{m}[0,\infty )\), then

$$ {}^{\mathrm{C}}D^{\alpha }g(p)= \frac{1}{\varGamma (m-\alpha )} \int _{0}^{p} \frac{g ^{(m)}(w)}{(p-w)^{\alpha +1-m}}\,dw = I^{m-\alpha }g^{(m)}(p),\quad p>0, m-1< \alpha < m. $$

Existence results

Below, we demonstrate some sufficient conditions for the existence of (1.1)–(1.2) coupled with weak topology.

\((\mathbf{H_{1}})\):

For \(\{T(u)\}_{u\geq 0}\) in \(\mathbb{Y}\), there is a constant \(\mathscr {M}_{1} \geq 1 \) fulfilling \(\sup_{u\in [0,b]}\|T(u)\|\leq \mathscr {M}_{1}\).

Additionally, we require that the multivalued nonlinearity function \(\mathscr {H}:[0,b] \times \mathbb{Y}\multimap \mathbb{Y}\) possess nonempty convex and weakly compact values.

\((\mathbf{H_{2}})\):

For all \(z \in \mathbb{Y}\), the multivalued function \(\mathscr {H}(\cdot , z):[0,b]\multimap \mathbb{Y}\) has a measurable selection.

\((\mathbf{H_{3}})\):

\(\mathscr {H}(u ,\cdot ):\mathbb{Y}\multimap \mathbb{Y}\) is weakly sequentially closed for almost everywhere u in \([0,b]\).

\((\mathbf{H_{4}})\):

For a real valued function \(h:[0,b] \times \mathbb{Y}\rightarrow \mathbb{Y}\), for all \(u>0\) and some constant \(\mathfrak{M}_{h} >0 \) we have \(\|h(u, \cdot ) \| \leq \mathfrak{M}_{h}\).

\((\mathbf{H_{5}})\):

For \(\kappa _{1} \in (0,q) \), for every \(r >0 \) and a function \(\delta _{r} \in L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb {R}_{+}) \) as for each \(d \in \mathbb{Y}\), \(\|d\| \leq r\):

$$ \bigl\Vert h(u,d) \bigr\Vert = \sup \bigl\{ \Vert z \Vert :z \in h(u,d) \bigr\} \leq \delta _{r} {(u)}, $$

for almost everywhere \(u \in [0,b] \).

\((\mathbf{H_{6}})\):

For \(\kappa _{1} \in (0,q) \), for every \(r >0 \) and a function \(\mu _{r} \in L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb {R}_{+}) \) as for each \(d \in \mathbb{Y}\), \(\|d\| \leq r\):

$$ \bigl\Vert \mathscr {H}(u,d) \bigr\Vert = \sup \bigl\{ \Vert z \Vert :z \in \mathscr {H}(u,d) \bigr\} \leq \mu _{r} {(u)}, $$

for almost everywhere \(u \in [0,b] \).

In connection with the above consideration, we determine the solution of (1.1)–(1.2).

Definition 3.1

([56])

\(z:[0,b]\rightarrow \mathbb{Y}\) is a mild solution of (1.1)–(1.2) if the accompanying recognize \(z(0) = z_{0} \) and there is \(\chi \in L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb{Y})\) provided \(\chi (u) \in \mathscr {H}(u, z(u)) \) on \(u \in [0,b]\) and z fulfills

$$\begin{aligned} z(u) ={}& \mathbb{T}(u) \bigl[z_{0}-h(0,z_{0}) \bigr]+h \bigl(u,z(u) \bigr)+ \int _{0}^{u}(u-w)^{q-1}\mathscr {A} \mathbb{S}(u-w)h \bigl(w,z(w) \bigr)\,dw \\ &{} + \int _{0}^{u}(u-w)^{q-1} \mathbb{S}(u-w)\chi (w)\,dw,\quad u\in [0,b], \end{aligned}$$

where

$$ \mathbb{T}(u) = \int _{0}^{\infty }\xi _{q}(\theta )T \bigl(u^{q}\theta \bigr)\,d\theta , \qquad \mathbb{S}(u)= q \int _{0}^{\infty }\theta \xi _{q}(\theta )T \bigl(u^{q} \theta \bigr)\,d\theta , $$

and, for \(\theta \in (0,\infty )\),

$$\begin{aligned}& \xi _{q}(\theta )=\frac{1}{q} \theta ^{-1-\frac{1}{q}} \overline{w}_{q} \bigl( \theta ^{-\frac{1}{q}} \bigr)\geq 0, \\& \overline{w}_{q}(\theta ) = \frac{1}{\pi }\sum _{n=1}^{\infty }(-1)^{n-1} \theta ^{-nq-1}\frac{\varGamma (nq+1)}{n!}\sin (n\pi q), \\& \int _{0}^{\infty }\xi _{q}(\theta )\,d \theta = 1. \end{aligned}$$

Remark 3.1

Obviously, for \(\nu \in [0,1]\),

$$ \int _{0}^{\infty }\theta ^{\nu }\xi _{q}(\theta )\,d\theta = \int _{0}^{ \infty }\theta ^{-q\nu } \overline{w}_{q}(\theta )\,d\theta = \frac{ \varGamma (1+\nu )}{\varGamma (1+q\nu )}. $$

Lemma 3.1

(See [56])

\(\mathbb{T}\)and \(\mathbb{S}\)obey the subsequent assertions:

  1. (i)

    For a constant \(\mathscr {M}_{2}\geq 1 \), for any \(z\in \mathbb{Y}\), fixed \(u\geq 0\)and for the bounded linear operators \(\mathbb{T}\)and \(\mathbb{S}\)we have

    $$\begin{aligned}& \bigl\Vert \mathbb{T}(u)z \bigr\Vert \leq \mathscr {M}_{1} \Vert z \Vert \quad \textit{and} \quad \bigl\Vert \mathbb{S}(u)z \bigr\Vert \leq \frac{q \mathscr {M}_{1}}{\varGamma (1+q)} \Vert z \Vert , \\& \bigl\Vert \mathscr {A} \mathbb{S}(u)z \bigr\Vert \leq \frac{q \mathscr {M}_{1}\mathscr {M}_{2}}{\varGamma (1+q)} \Vert z \Vert . \end{aligned}$$
  2. (ii)

    The operators \(\{\mathbb{T}(u),u\geq 0\}\)and \(\{\mathbb{S}(u),u\geq 0\}\)are strongly continuous.

Construct the set \(\varUpsilon _{\wp }\), for given \(\wp \in \mathfrak {C}([0,b];\mathbb{Y}) \) as \(\varUpsilon _{\wp }= \{ \chi \in L^{\frac{1}{ \kappa _{1}}}([0,b];\mathbb{Y}):\chi (u) \in \mathscr {H}(u,\wp (u))\mbox{ for almost everywhere } u \in [0,b] \} \). \(\varUpsilon _{\wp }\) is nonempty as the next Proposition 3.1 mentions.

Proposition 3.1

(See [15, 59])

Let us assume that a multivalued map \(\mathscr {H}:[0,b] \times \mathbb{Y}\multimap \mathbb{Y}\)obeys \((\mathbf{H_{2}})\)\(( \mathbf{H_{6}})\), the set \(\varUpsilon _{\wp }\)is nonempty for any \(\wp \in \mathfrak {C}([0,b];\mathbb{Y}) \).

We define the operator \(\varLambda : \mathfrak {C}([0,b];\mathbb{Y})\multimap \mathfrak {C}([0,b];\mathbb{Y})\) by

$$\begin{aligned} \varLambda (\wp ) =& \bigl\lbrace z\in \mathfrak {C} \bigl([0,b];\mathbb{Y}\bigr):z(u) = \mathbb{T}(u) \bigl(z _{0}-h(0,z_{0}) \bigr)+ h \bigl(u, \wp (u) \bigr) \\ &{}+ S_{1} (z) (u) + S_{2} \chi (u), \chi \in \varUpsilon _{\wp } \bigr\rbrace , \end{aligned}$$

where

$$\begin{aligned}& S_{1} :\mathfrak {C} \bigl([0,b];\mathbb{Y}\bigr)\rightarrow \mathfrak {C} \bigl([0,b]; \mathbb{Y}\bigr), \\& S_{1} (z) = \int _{0}^{u}(u-w)^{q-1}\mathscr {A} \mathbb{S}(u-w)h \bigl(w,\wp (w) \bigr)\,dw, \end{aligned}$$

and

$$\begin{aligned}& S_{2} : L^{\frac{1}{\kappa _{1}}} \bigl([0,b];\mathbb{Y}\bigr)\rightarrow \mathfrak {C} \bigl([0,b];\mathbb{Y}\bigr), \\& S_{2}(\chi ) = \int _{0}^{u}(u-w)^{q-1} \mathbb{S}(u-w)\chi (w)\,dw. \end{aligned}$$

At first, we show that \(S_{1} \) and \(S_{2} \) are continuous.

For any \(z_{n}, z\in \mathfrak {C}([0,b];\mathbb{Y})\) and \(z_{n}\rightarrow z\) (\(n\rightarrow \infty \)), using \((\mathbf{H_{5}})\), for every \(u\in [0,b] \), we get

$$ (u-w)^{q-1} \bigl\Vert z_{n}(w)-z(w) \bigr\Vert \leq 2(u-w)^{q-1} \mu _{r}(w),\quad \text{almost everywhere } w\in [0,u). $$

Also for any \(\chi _{n}, \chi \in L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb{Y}) \) and \(\chi _{n}\rightarrow \chi\) (\(n\rightarrow \infty \)), using \((\mathbf{H_{6}})\), we can have, for every \(u\in [0,b] \),

$$ (u-w)^{q-1} \bigl\Vert \chi _{n}(w)-\chi (w) \bigr\Vert \leq 2(u-w)^{q-1} \delta _{r}(w),\quad \text{almost everywhere } w\in [0,u). $$

Moreover, the functions

$$ \int _{0}^{u}(u-w)^{q-1} \mu _{r}(w)\,dw = \biggl[ \biggl({\frac{1-\kappa _{1}}{q- \kappa _{1}}} \biggr)b^{\frac{q-\kappa _{1}}{1-\kappa _{1}}} \biggr]^{1-\kappa _{1}} \Vert \mu _{r} \Vert _{\frac{1}{\kappa _{1}}} $$

and

$$ \int _{0}^{u}(u-w)^{q-1} \delta _{r}(w)\,dw = \biggl[ \biggl({\frac{1-\kappa _{1}}{q-\kappa _{1}}} \biggr)b^{\frac{q-\kappa _{1}}{1-\kappa _{1}}} \biggr]^{1- \kappa _{1}} \Vert \delta _{r} \Vert _{\frac{1}{\kappa _{1}}} $$

becomes integrable for \(u\in [0,b]\). Taking into account the Lebesgue theorem, we conclude, as \(n\rightarrow \infty \),

$$ \int _{0}^{u}(u-w)^{q-1} \bigl\Vert z_{n}(w)-z(w) \bigr\Vert \,dw\rightarrow 0\quad \text{and}\quad \int _{0}^{u}(u-w)^{q-1} \bigl\Vert \chi _{n}(w)-\chi (w) \bigr\Vert \,dw\rightarrow 0. $$

Therefore

$$\begin{aligned}& \begin{aligned} \bigl\Vert S_{1}(z_{n})-S_{1}(z) \bigr\Vert &\leq \biggl\Vert \int _{0}^{u}(u-w)^{q-1}\mathscr {A} \mathbb{S}(u-w) \bigl(z_{n}(w)-z(w) \bigr)\,dw \biggr\Vert & \\ &\leq \frac{q\mathscr {M}_{1}\mathscr {M}_{2}}{\varGamma { (1+q)}} \int _{0}^{u}(u-w)^{q-1} \bigl\Vert z_{n}(w)-z(w) \bigr\Vert \,dw\rightarrow 0, \quad \text{as } n\rightarrow \infty , \end{aligned} \\& \begin{aligned} \bigl\Vert S_{2}(\chi _{n})-S_{2}(\chi ) \bigr\Vert &\leq \biggl\Vert \int _{0}^{u}(u-w)^{q-1} \mathbb{S}(u-w) \bigl(\chi _{n}(w)-\chi (w) \bigr)\,dw \biggr\Vert & \\ &\leq \frac{q \mathscr {M}_{1}}{\varGamma { (1+q)}} \int _{0}^{u}(u-w)^{q-1} \bigl\Vert \chi _{n}(w)-\chi (w) \bigr\Vert \,dw \rightarrow 0,\quad \text{as } n\rightarrow \infty . \end{aligned} \end{aligned}$$

It indicates that the operators \(S_{1} \) and \(S_{2} \) are continuous.

For \(n\in \mathbb {N}\), \(\varPhi _{n}\), the closed ball of radius n in \(\mathfrak {C}([0,b];\mathbb{Y}) \) described by \(\varLambda _{n} = \varLambda \mid \varPhi _{n} : \varPhi _{n}\multimap \mathfrak {C}([0,b];\mathbb{Y}) \), the limitation of Λ on \(\varPhi _{n} \). Next we illustrate the qualities of \(\varLambda _{n} \).

Proposition 3.2

\(\varLambda _{n} \)possess a weakly sequentially closed graph.

Proof

Let a sequence \(\lbrace \wp _{m}\rbrace \subset \varPhi _{n} \), \(\lbrace z_{m}\rbrace \subset \mathfrak {C}([0,b];\mathbb{Y}) \) obeying \({z_{m}} \subset \varLambda _{n}(\wp _{m}) \), for all m and \(\wp _{m}\rightharpoonup \wp \), \(z_{m}\rightharpoonup z\) in \(\mathfrak {C}([0,b];\mathbb{Y}) \); we claim \(z\in \varLambda _{n}(\wp ) \).

Since \(\wp _{m}\in \varPhi _{n} \), for each m and \(\wp _{m}(u)\rightharpoonup \wp (u) \), for every \(u\in [0,b] \), we conclude \(\|\wp (u)\| \leq \lim_{\inf _{m}\rightarrow \infty } \| \wp _{m}(u)\|\leq n \), for all u (see [19]). By \(z_{m}\in \varLambda _{n}(p_{m}) \), there is a sequence \(\lbrace \chi _{m}\rbrace \), \(\chi _{m}\in \varUpsilon _{\wp m, }\) provided for all \(u\in [0,b] \), we get

$$\begin{aligned} \begin{aligned} z_{m}(u)={}& \mathbb{T}(u) \bigl(z_{0}-h(0,z_{0}) \bigr)+ h \bigl(u,\wp _{m}(u) \bigr) + \int _{0}^{u}(u-w)^{q-1}\mathscr {A} \mathbb{S}(u-w)h \bigl(w,\wp _{m}(w) \bigr)\,dw \\ &{} + \int _{0}^{u}(u-w)^{q-1} \mathbb{S}(u-w)\chi _{m}(w)\,dw. \end{aligned} \end{aligned}$$

By reference to \((\mathbf{H_{6}})\), \(\|\chi _{m}(u)\|\leq \mu _{n}(u)\), for almost everywhere u and every m. It means that \(\lbrace \chi _{m}\rbrace \) is bounded, uniformly integrable and \(\lbrace \chi _{m}(u)\rbrace \) is bounded in \(\mathbb{Y}\) for almost everywhere \(u\in [0,b]\). By the Dunford-Pettis theorem [13], we can conclude that there exist a subsequence, represented as the sequence, and functions \(g_{1}\), \(g_{2}\) provided \(z_{m}\rightharpoonup g_{1} \) in \(\mathfrak {C}([0,b];\mathbb{Y})\) and \(\chi _{m}\rightharpoonup g_{2} \) in \(L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb{Y})\).

Therefore, we have \(S_{1} z_{m}\rightharpoonup S_{1} g_{1}\) and \(S_{2} \chi _{m}\rightharpoonup S_{2} g_{2}\). In this connection, let the linear continuous operator \(e^{\prime } : \mathbb{Y}\rightarrow \mathbb {R}\). The operators \(S_{1}\) and \(S_{2}\) are linear and continuous, therefore we have

$$ g_{1}\rightarrow e^{\prime }(S_{1} g_{1}) (u),\qquad g_{1}\in \mathfrak {C} \bigl([0,b];\mathbb{Y}\bigr), $$

is linear continuous operator on \(\mathfrak {C}([0,b];\mathbb{Y})\) to \(\mathbb{R}\) for every \(u \in [0,b]\). Also,

$$ g_{2}\rightarrow e^{\prime }(S_{2} g_{2}) (u),\qquad g_{2}\in L^{\frac{1}{\kappa _{1}}} \bigl([0,b];\mathbb{Y}\bigr), $$

is linear continuous operator on \(L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb{Y})\) to \(\mathbb{R}\) for every \(u \in [0,b]\). By weak convergence, we get

$$\begin{aligned}& e^{\prime } \biggl( \int _{0}^{u}(u-w)^{q-1}\mathscr {A} \mathbb{S}(u-w)h \bigl(w,\wp _{m}(w) \bigr)\,dw+ \int _{0}^{u}(u-w)^{q-1} \mathbb{S}(u-w)\chi _{m}(w)\,dw \biggr) \\& \quad \rightarrow e^{\prime } \biggl( \int _{0}^{u}(u-w)^{q-1}\mathscr {A} \mathbb{S}(u-w)g _{1}(w)\,dw+ \int _{0}^{u}(u-w)^{q-1} \mathbb{S}(u-w)g_{2}(w)\,dw \biggr). \end{aligned}$$

Therefore

zm(u)T(u)(z0h(0,z0))+h(u,m(u))+0u(uw)q1AS(uw)g1(w)dw+0u(uw)q1S(uw)g2(w)dw=z0(u),

for every \(u\in [0,b] \). This indicates that \(z_{0}(u) = z(u) \), for all \(u\in [0,b]\). Hence by Proposition 3.1, \(g_{2}(u)\in \mathscr {H}(u,\wp (u)) \), for almost everywhere \(u \in [0,b]\). □

Proposition 3.3

\(\varLambda _{n} \)is weakly compact.

Proof

At first, we show that \(\varLambda _{n}(\varPhi _{n}) \) is relatively weakly sequentially compact.

Let us consider \({\wp _{m}}\in \varPhi _{n} \) and \({z_{m}}\subset \mathfrak {C}([0,b];\mathbb{Y}) \) such that \(z_{m}\in \varLambda _{n}(\wp _{m}) \) for all m. For \(\varLambda _{n}\), there exists a sequence \(\lbrace \chi _{m} \rbrace \), \(\chi _{m}\in \varUpsilon _{\wp m} \), provided that

$$\begin{aligned} z_{m}(u)={}& \mathbb{T}(u) \bigl(z_{0}-h(0,z_{0}) \bigr)+ h \bigl(u,\wp _{m}(u) \bigr) + \int _{0}^{u}(u-w)^{q-1}\mathscr {A} \mathbb{S}(u-w)h \bigl(w,\wp _{m}(w) \bigr)\,dw \\ &{} + \int _{0}^{u}(u-w)^{q-1} \mathbb{S}(u-w)\chi _{m}(w)\,dw, \end{aligned}$$

for every \(u\in [0,b]\). Therefore, by Proposition 3.2, there exist a subsequence, represented by the sequence, and functions \(g_{1}\), \(g _{2}\) provided \(z_{m}\rightharpoonup g_{1} \) in \(\mathfrak {C}([0,b];\mathbb{Y})\) and \(\chi _{m}\rightharpoonup g_{2} \) in \(L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb{Y})\). Hence

zm(u)l(u)=T(z0h(0,z0))+h(u,m(u))+0u(uw)q1AS(uw)g1(w)dw+0u(uw)q1S(uw)g2(w)dw.

Furthermore, by the nature of weak convergence of \({\chi _{m}}\), by \((\mathbf{H_{1}})\), we have

zm(u)M1z0+δr(0)+Mh+M1M2qΓ(1+q)[(1κ1qκ1)bqκ11κ1]1κ1δr1κ1+M1qΓ(1+q)[(1κ1qκ1)bqκ11κ1]1κ1μr1κ1,

for all \(m\in \mathbb {N} \) and \(u\in [0,b] \). By utilizing the Proposition 3.2, we ensure that \(z_{m}\rightharpoonup l \) in \(\mathfrak {C}([0,b];\mathbb{Y}) \). Thus, \(\varLambda _{n}(\varPhi _{n}) \) is relatively weakly compact by Theorem 2.2. □

Proposition 3.4

\(\varLambda _{n} \)has convex and weakly compact values.

Proof

Fixing \(\wp \in \varPhi _{n} \), taking into account that \(\mathscr {H} \) is convex valued and the characteristics of \(\mathbb{T}(u) \) and \(\mathbb{S}(u) \), it implies that \(\varLambda _{n}(\wp ) \) is convex. By reference to Proposition 3.2 and Proposition 3.3, \(\varLambda _{n}( \wp )\) has weakly compact values. □

Next we list out the essential outcomes of this part.

Theorem 3.1

Assuming \((\mathbf{H_{1}})\)\((\mathbf{H_{6}})\)hold. Suppose \((\mathbf{H_{7}})\)for a sequence of functions \(\{ u_{n} \} \in L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb {R}_{+})\)provided

$$ \sup_{ \Vert d \Vert \leq n} \bigl\Vert \mathscr {H}(u,d) \bigr\Vert \leq u_{n}(u), $$

for almost everywhere \(u\in [0,b]\), \(n\in \mathbb {N} \)with

$$ \lim_{n\rightarrow \infty }\inf \frac{ \Vert u_{n} \Vert _{\frac{1}{\kappa _{1}}}}{n} = 0. $$
(3.1)

Then (1.1)(1.2) recognizes at least a mild solution.

Proof

We have to confirm that Λ maps \(\varPhi _{n} \) into itself for \(n\in \mathbb {N} \).

Assume by way of contradiction that there exist \(\lbrace z_{n}\rbrace \), \(\lbrace x_{n}\rbrace \) such that \(z_{n}\in \varPhi _{n}\), \(x_{n}\in \varLambda _{n}(z_{n}) \) and \(x_{n}\notin \varPhi _{n}\), for every \(n\in \mathbb {N}\). Therefore for a sequence \(\lbrace \chi _{n}\rbrace \subset L ^{\frac{1}{\kappa _{1}}}([0,b];\mathbb{Y})\), \(\chi _{n}(w)\in \mathscr {H}(w,z_{n}(w)) \), we can have

xn(u)=T(u)(z0h(0,z0))+h(u,zn(u))+0u(uw)q1AS(uw)h(w,zn(w))dw+0u(uw)q1S(uw)χn(w)dw,

for every \(u\in [0,b] \). By Proposition 3.3, we have

nxn0M1z0+δr(0)+Mh+M1MhM2qΓ(1+q)[(1κ1qκ1)bqκ11κ1]1κ1+M1qΓ(1+q)[(1κ1qκ1)bqκ11κ1]1κ1un1κ1.

Then, for \(n\in \mathbb {N} \),

1xn0n1n[M1z0+δr(0)+Mh+M1MhM2qΓ(1+q)[(1κ1qκ1)bqκ11κ1]1κ1+M1qΓ(1+q)[(1κ1qκ1)bqκ11κ1]1κ1un1κ1],

which leads to a contradiction. Therefore \(x_{n}\in \varPhi _{n} \).

Now, fix \(n\in \mathbb {N}\) such that \(\varLambda _{n}(\varPhi _{n})\subset \varPhi _{n} \). By Proposition 3.3, the set \(\mathscr {V}_{n} =\overline{ \varLambda _{n}(\varPhi _{n})}^{w} \) is weakly compact. Let \(\xi _{n} = \overline{ \operatorname{co}}(\mathscr {V}_{n}) \), be the closed convex hull of \(\mathscr {V}_{n} \). According to Theorem 2.3, \(\xi _{n} \) denotes a weakly compact set. In addition to that \(\varLambda _{n}(\varPhi _{n})\subset \varPhi _{n} \) and \(\varPhi _{n} \) is a closed convex set. Furthermore we have \(\xi _{n}\subset \varPhi _{n} \), and we have

$$ \varLambda _{n}(\xi _{n}) = \varLambda _{n} \bigl(\overline{\operatorname{co}} \bigl( \varLambda _{n}(\varPhi _{n}) \bigr) \bigr)\subseteq \varLambda _{n}(\varPhi _{n})\subseteq \overline{ \varLambda _{n}(\varPhi _{n})}^{w} = \mathscr {V}_{n}\subset \xi _{n}. $$

This shows that \(\varLambda _{n} \) possesses a weakly sequentially closed graph. As a result by utilizing Theorem 2.1, we conclude that the system (1.1)–(1.2) recognizes a solution. □

Remark 3.2

There exist \(\alpha \in L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb {R}_{+}) \) and a nondecreasing function \(\phi :[0,\infty ) \rightarrow [0,\infty )\) such that \(\|\mathscr {H}(u,d)\|\leq \alpha (u) \phi (\|d\|) \), for almost everywhere \(u\in [0,b] \) and every \(d\in \mathbb{Y}\). Then the restriction (3.1) is related to

$$ \lim_{n\rightarrow \infty }\inf \frac{\phi (n)}{n} = 0. $$

Theorem 3.2

Assume that \((\mathbf{H_{1}})\)\((\mathbf{H_{5}}) \)holds.

\((\mathbf{H_{8}})\):

There exists \(\rho \in L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb {R}_{+}) \), for almost everywhere \(u\in [0,b] \), for every \(d\in \mathbb{Y}\)provided

$$ \bigl\Vert \mathscr {H}(u,d) \bigr\Vert \leq \rho (u) \bigl(1+ \Vert d \Vert \bigr) $$

and

$$ \frac{\mathscr {M}_{1}q}{\varGamma { (1+q)}} \biggl[ \biggl({\frac{1-\kappa _{1}}{q- \kappa _{1}}} \biggr) b^{\frac{q-\kappa _{1}}{1-\kappa _{1}}} \biggr]^{1- \kappa _{1}} \bigl( \mathscr {M}_{2} \mathfrak{M}_{h} + \Vert \rho \Vert _{\frac{1}{ \kappa _{1}}} \bigr)< 1, $$
(3.2)

then (1.1)(1.2) possess at least a mild solution.

Proof

By reference to Theorem 3.1, assuming that there exist \(\lbrace z _{n}\rbrace \), \(\lbrace x_{n}\rbrace \) provided \(z_{n}\in \varPhi _{n}\), \(x _{n}\in \varLambda _{n}(z_{n}) \) and \(x_{n}\notin \varPhi _{n}\), for every \(n\in \mathbb {N} \), we get

$$\begin{aligned} n < {}& \Vert x_{n} \Vert _{0} \\ \leq{}& \mathscr {M}_{1} \Vert z_{0} \Vert +\delta _{r}(0) + \mathfrak{M}_{h}+\frac{\mathscr {M}_{1}\mathscr {M}_{2}\mathfrak{M}_{h} q}{ \varGamma { (1+q)}} \biggl[ \biggl({\frac{1-\kappa _{1}}{q-\kappa _{1}}} \biggr)b^{\frac{q-\kappa _{1}}{1-\kappa _{1}}} \biggr]^{1-\kappa _{1}} \\ &{} +\frac{\mathscr {M}_{1}q}{\varGamma { (1+q)}} \biggl[ \biggl({\frac{1- \kappa _{1}}{q-\kappa _{1}}} \biggr)b^{\frac{q-\kappa _{1}}{1-\kappa _{1}}} \biggr]^{1-\kappa _{1}} \int _{0}^{b} { \bigl\vert \rho (\xi ) \bigr\vert } ^{\frac{1}{ \kappa _{1}}} \bigl(1+ \bigl\Vert z_{n}(\xi ) \bigr\Vert ^{\frac{1}{\kappa _{1}}} d \xi \bigr)^{\kappa _{1}} \\ \leq {}&\mathscr {M}_{1} \Vert z_{0} \Vert +\delta _{r}(0)+ \mathfrak{M}_{h} \\ &{}+\frac{\mathscr {M}_{1}q}{\varGamma { (1+q)}} \biggl[ \biggl({\frac{1-\kappa _{1}}{q-\kappa _{1}}} \biggr)b^{\frac{q-\kappa _{1}}{1-\kappa _{1}}} \biggr]^{1-\kappa _{1}} \bigl( \mathscr {M}_{2} \mathfrak{M}_{h}+(1+n) \Vert \rho \Vert _{\frac{1}{\kappa _{1}}} \bigr), \quad n\in \mathbb {N}, \end{aligned}$$

which contradicts (3.2). The conclusion refers to Theorem 2.1, like Theorem 3.1. □

Theorem 3.3

Assuming that \((\mathbf{H_{1}})\)\((\mathbf{H_{5}})\)holds. \((\mathbf{H_{9}})\)there exist \(\beta \in L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb {R}_{+}) \)and a nondecreasing function \(\varrho :[0,\infty ) \rightarrow [0, \infty )\)fulfilling

$$ \bigl\Vert \mathscr {H}(u,d) \bigr\Vert \leq \beta (u) \varrho \bigl( \Vert d \Vert \bigr), $$

for almost everywhere \(u\in [0,b]\), \(d\in \mathbb{Y}\), and \(\mathscr {L}>0 \)provided

$$ \frac{\mathscr {L}}{\mathscr {M}_{1} \Vert z_{0} \Vert + \delta _{r}(0) + \mathfrak{M} _{h}+\frac{\mathscr {M}_{1}q}{\varGamma { (1+q)}} [ ({\frac{1-\kappa _{1}}{q-\kappa _{1}}} )b^{\frac{q-\kappa _{1}}{1-\kappa _{1}}} ]^{1- \kappa _{1}} ( \mathscr {M}_{2} \mathfrak{M}_{h} + \Vert \rho \Vert _{\frac{1}{ \kappa _{1}}} )\varrho (\mathscr {L})}>1, $$
(3.3)

then (1.1)(1.2) possess at least a mild solution.

Proof

We have to ensure that Λ maps the ball \(\varPhi _{\mathscr {L}} \) into itself. For any \(z\in \varPhi _{\mathscr {L}}\), \(x\in \varGamma (z) \), we conclude

xn0M1z0+δr(0)+Mh+M1M2MhqΓ(1+q)[(1κ1qκ1)bqκ11κ1]1κ1+M1qΓ(1+q)[(1κ1qκ1)bqκ11κ1]1κ10b|ρ(ξ)|1κ1(ϱz(ξ)1κ1dξ)κ1M1z0+δr(0)+Mh+M1qΓ(1+q)[(1κ1qκ1)bqκ11κ1]1κ1[M2Mh+β1κ1]ϱ(L)<L.

This indicates that (1.1)–(1.2) possess at least a mild solution. □

Nonlocal conditions

The active desire for analyzing fractional systems with nonlocal problems comes mainly from theoretical physics. The outcomes regarding the existence of Cauchy problems using nonlocal conditions were firstly investigated by Byszewski [20]. Many papers [13, 21, 34, 35, 53] have acknowledged the facts of the existence, controllability and uniqueness for varied nonlinear fractional systems and abstract differential systems. Motivated by the above discussions, this part deals with the existence of (1.1)–(1.2) as

$$\begin{aligned} &{}^{\mathrm{C}}D^{q}_{t} \bigl[ z(u)-h \bigl(u,z(u) \bigr) \bigr]\in \mathscr {A}z(u)+\mathscr {H} \bigl(u,z(u) \bigr),\quad u\in [0,b], \end{aligned}$$
(4.1)
$$\begin{aligned} & z(0)+\varphi (z)=z_{0} , \end{aligned}$$
(4.2)

where \(\varphi : \mathfrak {C}([0,b];\mathbb{Y})\rightarrow \mathbb{Y}\) fulfills the following conditions:

\((\mathbf{H_{10}})\):

For some constant \(\mathscr {N}>0 \) provided \(\|\varphi (z)\|\leq \mathscr {N}\), \(z \in \mathfrak {C}([0,b];\mathbb{Y}) \).

\((\mathbf{H_{11}})\):

There is a constant \(\mathscr {L}>0 \) and

$$\begin{aligned}& \frac{\mathscr {L}}{\mathscr {M}_{1} \Vert z_{0} \Vert +\mathscr {N}+\delta _{r}(0)+ \mathfrak{M}_{h}+\frac{\mathscr {M}_{1}q}{\varGamma { (1+q)}} [ ( {\frac{1-\kappa _{1}}{q-\kappa _{1}}} )b^{\frac{q-\kappa _{1}}{1- \kappa _{1}}} ]^{1-\kappa _{1}} ( \mathscr {M}_{2} \mathfrak{M}_{h}+ \Vert \rho \Vert _{\frac{1}{\kappa _{1}}} )\varrho (\mathscr {L})} \\& \quad >1. \end{aligned}$$
(4.3)
In order to show the high accuracy, we always refer the nonlocal condition in the place of initial condition \(z(0) = z_{0}\). Particularly, \(\varphi (z) \) can be formulated as

$$ \varphi (z)=\sum_{i=1}^{n} K_{i} z(t_{i}), $$

where \(K_{i} \) (\(i= 1,2,3,\ldots,n\)) are constants and \(0< t_{1}< t_{2}<\cdots<t _{n}\leq b \).

Definition 4.1

\(z :[0,b] \rightarrow \mathbb{Y}\) is called the mild solution of the neutral fractional differential model (4.1)–(4.2) if the accompanying recognize \(z(0)+ \varphi (z) = z_{0} \) and there exists \(\chi \in L^{\frac{1}{\kappa _{1}}}([0,b];\mathbb{Y})\) provided \(\chi (u) \in \mathscr {H}(u, z(u)) \) on \(u \in [0,b]\) and z satisfies

$$\begin{aligned} z(u) ={}& \mathbb{T}(u) \bigl[z_{0}-\varphi (z)-h(0,z_{0}) \bigr]+h \bigl(u,z(u) \bigr)+ \int _{0} ^{u}(u-w)^{q-1}\mathscr {A} \mathbb{S}(u-w)h \bigl(w,z(w) \bigr)\,dw \\ &{} + \int _{0}^{u}(u-w)^{q-1} \mathbb{S}(u-w)\chi (w)\,dw,\quad u\in [0,b], \end{aligned}$$

such that \(\mathbb {T}(u)\) and \(\mathbb {S}(u)\) are defined as in Definition 3.1.

Theorem 4.1

If Theorem 3.1, Theorem 3.2and Theorem 3.3hold, and in addition hypotheses \((\mathbf{H_{10}})\)and \((\mathbf{H_{11}})\)hold, then the neutral fractional system with inclusion (4.1)(4.2) has at least a mild solution.

Proof

We introduce the solution operator \(\varLambda :\mathfrak {C}([0,b];\mathbb{Y})\multimap \mathfrak {C}([0,b];\mathbb{Y}) \) as

$$\begin{aligned} \varLambda (\wp ) =& \bigl\lbrace z\in \mathfrak {C} \bigl([0,b];\mathbb{Y}\bigr):z(u) = \mathbb{T} \bigl(z _{0}-\varphi (z)-h(0,z_{0}) \bigr)+ h \bigl(u,\wp (u) \bigr) \\ &{}+ S_{1}(z) (u) + S_{2}( \chi ) (u) \bigr\rbrace . \end{aligned}$$

It should be noted that we recognize that Λ possesses a fixed point by employing the techniques utilized in Theorems corresponding to initial conditions. The proof is similar, therefore we omitted it. □

An example

Let us consider the model:

$$\begin{aligned}& {}^{\mathrm{C}} D_{u}^{q} \biggl[v(u,y) + \int _{0}^{\pi }k(\theta ,y)v(u,\theta )\,d\theta \biggr] \in \frac{\partial ^{2}}{\partial y^{2}}v(u,y)+s \bigl(u,v(u,y) \bigr),\quad u\in I=[0, \pi ], \\& v(u,0)= v(u,\pi )= 0, \end{aligned}$$
(5.1)
$$\begin{aligned}& v(0,y)= 0,\quad 0< y< \pi , \end{aligned}$$
(5.2)

where \(0< q\leq 1\), construct \(\mathbb{Y}=L^{2}(0,\pi )\) and \(\mathscr {A}: D(\mathscr {A})\subseteq \mathbb{Y}\rightarrow \mathbb{Y}\) by \(\mathscr {A}z = z''\), together \(D(\mathscr {A})=\{z\in \mathbb{Y}:y''\in \mathbb{Y}\}\), are absolutely continuous. Obviously \(\mathscr {A}\) is the infinitesimal generator of \(\{T(u), u\ge 0\}\) in \(\mathbb{Y}\) and generates the strongly continuous semi group \(T(u) \). Additionally, \(\mathscr {A}\) has eigenvalues in the form of \(-n^{2}, n\in \mathbb {N}\), and it can be denoted as

$$ \mathscr {A}z=\sum_{n=1}^{\infty }n^{2} \langle z,z_{n}\rangle ,\quad z\in D(\mathscr {A}), $$

and \(z_{n}(x)=\sqrt{\frac{2}{\pi }}\sin (nx)\), \(n=1,2,3,\ldots\) , represents the set of eigenvectors of \(\mathscr {A}\) which are orthonormal. Also for any \(z\in \mathbb{Y}\),

$$ T(u)z=\sum_{n=1}^{\infty }e^{-n^{2}} u\langle z,z_{n}\rangle z_{n}. $$

Clearly, \(T(u)\) fulfills \((\mathbf{H_{1}})\). Define \(h:[0,\pi ] \times \mathbb{Y}\rightarrow \mathbb{Y}\) by

$$ h(v) (z)= \int _{0}^{\pi }k(\theta ,z) v(u,\theta ) \,d \theta , $$

where the continuous function \(k: [0,\pi ]\times [0, \pi ]\longrightarrow \mathbb{R}\) provided \(\Vert k(\cdot , z) \Vert \leq 1\), for each \(z\in [0,\pi ]\) and \(v(u)(z)= v(u,z)\), \(\mathscr {H}(u,z(u)) z =s(u,v(u,z)) \). With a suitable choice of \(\mathscr {A}\), \(\mathscr {H}\), h, the above said system can be equivalent to (1.1)–(1.2), that is,

DtqC[z(u)h(u,z(u))]Az(u)+H(u,z(u)),u[0,b],
(5.3)
$$\begin{aligned} & z(0)+\varphi (z)=z_{0} . \end{aligned}$$
(5.4)

Besides assuming \(\mathscr {H}\), h satisfies the concerned hypotheses. As a result, (5.1)–(5.2) has at least a mild solution on \([0,b]\).

Conclusion

This manuscript addresses the existence of Caputo fractional neutral inclusions without compactness in a Banach space by using weak topology. Further, the results are derived for fractional neutral system where nonlocal conditions hold. Our theorem guarantees the effectiveness of the existence, which is the result of the system concerned.

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Acknowledgements

This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

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Correspondence to Thabet Abdeljawad.

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Alqudah, M.A., Ravichandran, C., Abdeljawad, T. et al. New results on Caputo fractional-order neutral differential inclusions without compactness. Adv Differ Equ 2019, 528 (2019). https://doi.org/10.1186/s13662-019-2455-z

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MSC

  • 34A08
  • 93B05
  • 34G20
  • 34G25
  • 47H10
  • 47H20

Keywords

  • Existence
  • Fractional differential inclusions
  • Neutral equations
  • Semigroup theory
  • Multivalued map