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Some inequalities on multi-functions for applying in the fractional Caputo–Hadamard jerk inclusion system

Abstract

Results reported in this paper establish the existence of solutions for a class of generalized fractional inclusions based on the Caputo–Hadamard jerk system. Under some inequalities between multi-functions and with the help of special contractions and admissible maps, we investigate the existence criteria. Fixed points and end points are key roles in this manuscript, and the approximate property for end points helps us to derive the desired result for existence theory. An example is prepared to demonstrate the consistency and correctness of analytical findings.

1 Introduction

With the presentation of new analytical results in recent years, the power of fractional calculus in describing processes and modeling physical events and engineering tools has become clear to everyone. In most published papers we are able to observe different generalized fractional modelings of standard equations in which the Caputo or Riemann–Liouville derivatives or their extensions have been utilized in fractional differential equations (FDEs) and fractional differential inclusions (FDIs) such as pantograph inclusion [1], hybrid thermostat inclusion [2], q-differential inclusion on time scale [3], Langevin inclusion [4], and higher order fractional differential inequalities [5]. One can find many published works on various applications of fractional calculus in different fields of science (see, for example, [616]).

In 2016, the authors considered the following mixed initial value problem involving Hadamard derivative and Riemann–Liouville fractional integrals given by

$$ \textstyle\begin{cases} {}^{H }\mathbb{D}^{q} (y(t) - \sum_{i=1}^{m} {}^{RC } \mathbb{I}_{r}^{\sigma _{i}} w_{i} (t, y(t) ) \in \mathfrak{H}( t, y(t)), \quad t \in [1, M], \\ y(1) =0, \end{cases} $$

where \({}^{H }\mathbb{D}^{q}\) denotes the Hadamard fractional derivative of order \(0 < q\leq 1\), \({}^{RC }\mathbb{I}_{r}^{\sigma}\) is the Riemann–Liouville fractional integral of order \(\sigma >0\), \(\sigma \in \{\sigma _{1}, \sigma _{2}, \dots , \sigma _{m}\}\), \(\mathfrak{H}: [1, M ] \times \mathbb{R} \to \mathcal{P}(\mathbb{R})\), \(w_{i} \in C([1, M] \times \mathbb{R}, \mathbb{R})\) with \(w_{i}(1,0)=0\), \(i=1,2,\dots , m\) [17]. In 2017, Ahmad et al. considered the existence and uniqueness of solutions to the initial value problem of Caputo–Hadamard sequential fractional order neutral functional differential equations as follows:

$$ \textstyle\begin{cases} {}^{C }\mathbb{D}^{\sigma _{1}} [ {}^{C }\mathbb{D}^{\sigma _{2}} y(t) - f_{1}(t, y_{t} ) ]= f_{2}(t, y_{t}),& t \in [1, M], \\ y(t) = \phi (t), & t \in [1-r, 1], \\ {}^{C }\mathbb{D}^{\sigma _{2}} y(1) = \eta \in \mathbb{R}, \end{cases} $$

where \({}^{C }\mathbb{D}^{\sigma _{1}}\), \({}^{C }\mathbb{D}^{\sigma _{2}}\) are the Caputo–Hadamard fractional derivatives, \(0 < \alpha , \beta <1\), \(f_{i}: [1, M] \times C([-r, 0], \mathbb{R}) \to \mathbb{R}\) is a given function, \(i=1,2\), and \(\phi \in C([1-r, 1], \mathbb{R})\) [18]. The authors in [19] introduced a new class of boundary value problems consisting of Caputo–Hadamard type fractional differential equations and Hadamard type fractional integral boundary conditions:

$$ \textstyle\begin{cases} ( {}^{C }\mathbb{D}^{\sigma _{1}} + \lambda {}^{C }\mathbb{D}^{ \sigma _{1}-1} ) y(t) = w_{1} ( t, y(t), z(t),{}^{C } \mathbb{D}^{\beta} z(t) ),& 1 < \sigma _{2} \leq 2, \\ ( {}^{C }\mathbb{D}^{\sigma _{2}} + \lambda {}^{C }\mathbb{D}^{ \sigma _{2} -1} ) z(t) = w_{2} ( t, y(t), {}^{C } \mathbb{D}^{\beta} y(t), z(t) ),& 1 < \sigma _{2} \leq 2, \\ y(1) =0, a_{11} \mathbb{I}^{\gamma _{1}} v(\eta _{1}) + a_{12} u(\delta )= K_{1}, & \gamma _{1} >0, 1 < \eta _{1} < \delta , \\ z(1) =0, a_{21} \mathbb{I}^{\gamma _{2}} u(\eta _{2}) + a_{22} v(\delta )= K_{2}, & \gamma _{2} >0, 1 < \eta _{2} < \delta , \end{cases} $$

where \(0 <\beta <1\), \({}^{C }\mathbb{D}^{(.)}\), \(\mathbb{I}^{(.)}\) respectively denote the Caputo–Hadamard fractional derivative and Hadamard fractional integral (to be defined later), \(w_{i}: [0, \delta ] \times \mathbb{R}^{3} \to \mathbb{R}\) is a given appropriate function and \(a_{ij}\), \(K_{i}\) are real constants, here \(i,j=1,2\) [19].

More precisely, in [1], Thabet et al. formulated a version of FDI taken from the pantograph BVP in the sense of Caputo-conformable equipped with three-point RL-conformable integral conditions:

$$ \textstyle\begin{cases} {}^{CC }\mathbb{D}_{r}^{q,\sigma _{1}} y(t) \in \mathfrak{H}( t, \varkappa (t), \varkappa (\lambda ^{\ast}t) ), \quad t \in [r, M], \\\varkappa (r) =0, \\ p_{1} \varkappa (M) + p_{2} {}^{RC }\mathbb{I}_{r}^{q,\sigma _{2}} \varkappa (\zeta ) = \varkappa ^{\ast}. \end{cases} $$

Here, \({}^{CC }\mathbb{D}_{r}^{q,\sigma _{1}}\) indicates the derivative of the Caputo-conformable type of order \(1 < \sigma _{1} < 2\) along with \(0 < q <1\), \({}^{RC }\mathbb{I}_{r}^{q,\sigma _{2}}\) is the integral of the RL-conformable type of order \(\sigma _{2}>0\), \(\zeta \in (r, M)\), \(p_{1}\), \(p_{2}\), \(y^{\ast}\in \mathbb{R}\), \(0 < \lambda <1\), and \(\mathfrak{H}: [r, M] \times \mathbb{R}^{2} \to \mathcal{P} ( \mathbb{R})\) is a multifunction. Also, Baleanu et al. in [2] investigated the hybrid problem caused by the thermostat model

$$ \textstyle\begin{cases} {}^{C}\mathbb{D}_{0}^{q} ( \frac {y(t) }{\varkappa (t, y(t))} ) + w (t, y(t)) =0, \\ \mathbb{D} ( \frac {y(t) }{\varkappa (t, y(t))} ) |_{t=0} =0, \\ \eta {}^{C}\mathbb{D}_{0}^{q-1} ( \frac {y(t) }{\varkappa (t, y(t))} ) |_{t=1} + ( \frac {y(t) }{\varkappa (t, y(t))} ) |_{t= a} =0, \end{cases} $$

so that \({}^{C }\mathbb{D}_{0}^{q}\) is the Caputo derivation of fractional order \(1 < q\leq 2\), \(\mathbb{D} = \frac{\mathrm{d}}{\mathrm{d}t}\) the function \(w : [0,1] \times \mathbb{R} \to \mathbb{R} \) is continuous, \(\varkappa \in C([0,1]\times \mathbb{R}, \mathbb{R} \setminus \{0\} )\), η is a positive real parameter and \(0 \leq a \leq 1\). Furthermore, Samei et al. in [3] discussed the fractional q-differential inclusion

$$ \textstyle\begin{cases} {}^{C }\mathbb{D}_{q}^{\sigma} y(t) \in \mathfrak{H} (t, y(t) , y'(t), {}^{C }\mathbb{D}_{q}^{p_{1}} y(t), \dots , {}^{C } \mathbb{D}_{q}^{p_{m}} y(t) ), \quad t \in [0, 1], \\y(0) + c y''(0) =0, \qquad a_{1}y(\tau )+ a_{2}\varrho (1)=0, \qquad c y'(1) + a_{3} \varrho (1) = 0. \end{cases} $$

Here, \({}^{C }\mathbb{D}_{q}^{\sigma}\) denotes the Caputo fractional quantum derivative of order \(2 < \sigma \leq 3\), \(1< p_{i}\leq 2\), (\(i=1,2,\dots , m\)), \(0 < \tau <1\), \(c = \sum_{j=1}^{m} c_{i}\), \(c_{j} \in \mathbb{R}\), \(\varrho : [0, \infty ) \to [0, \infty )\) defined by

$$ \varrho _{i}(z) = \int _{0}^{z} \varphi \bigl(y(\xi )\bigr) \, \mathrm{d}\xi ,$$

\(\varphi : [0, \infty ) \to [0, \infty )\), \(\mathfrak{H}: [0,1]\times \mathbb{R}^{m+2} \to \mathcal{P}( \mathbb{R})\) is a compact-valued multifunction and \(a_{1}, a_{2}, a_{3} \in \mathbb{R}\). Recently, Rezapour et al. introduced and investigated a new BVP consisting of a generalized fractional integro-Langevin equation with constant coefficient and nonlocal fractional boundary conditions (BCs) given by

$$ \textstyle\begin{cases} {}^{C }\mathbb{D}_{0^{+}}^{q_{1}} [ {}^{C }\mathbb{D}_{0^{+}}^{q_{2}} y(t) - \beta y(t) ] = {}^{RL}\mathbb{I}_{0^{+}}^{p} h( t, y(t)), & t \in [0, 1], \\y(0) =0, \qquad {}^{C }\mathbb{D}_{0^{+}}^{q_{2}} y(0) = 0,& \\ {}^{C }\mathbb{D}_{0^{+}}^{1} y(1) {}^{C }\mathbb{D}_{0^{+}}^{1} y( \tau ) =0,& \tau \in (0,1), \end{cases} $$

where \(0 < q_{1} < 1\), \(1 < q_{2} < 2\), \(p>0\), \(\beta \in \mathbb{R}^{+}\), \({}^{C }\mathbb{D}_{0^{+}}^{\eta}\), (\(\eta \in \{ q_{1}, q_{2}\}\)) and \({}^{RL }\mathbb{I}_{0^{+}}^{p}\) denote the Caputo fractional derivative operators and the Riemann–Liouville fractional integral of orders p and η, respectively, and the function \(h: [0,1] \times \mathbb{R} \to \mathbb{R}\) is continuous [4].

The authors in [5] showed how fractional differential inequalities can be useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. The nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative has been obtained [5]. On the other hand in [20] the authors analyzed the properties of fractional operators with fixed memory length in the context of Laplace transform of the Riemann–Liouville fractional integral and derivative with fixed memory length [20] on the fractional differential equation

$$ {}_{a }\mathbb{D}_{t^{+}}^{\eta} y(t) \sim {}_{t-L }\mathbb{D}_{t}^{ \eta} y(t), \quad t > a+L.$$

These facts could be used to better explain the motivation behind the present study [20]. Jleli et al. studied the wave inequality with a Hardy potential

$$ \partial _{tt} y - \Delta y + \frac{\lambda}{ \vert x \vert ^{2}} y \geq \vert y \vert ^{p} \quad \text{in } (0, \infty )\times \Omega ,$$

where Ω is the exterior of the unit ball in \(\mathbb{R}^{N}\), (\(N\geq 2\)), \(p>1\), and

$$ \lambda \geq - \biggl(\frac{N-2}{2} \biggr)^{2},$$

under the inhomogeneous boundary condition \(\alpha \frac{\partial y}{\partial x} (t, x) + \beta y(t, x) \geq w(x)\) on \((0, \infty )\times \partial \Omega \), where \(\alpha , \beta \geq 0\) and \((\alpha , \beta ) \neq (0,0)\) [21]. The Caputo–Hadamard derivation operator [22] is another extension of the above operators that many researchers got help from it in their modelings. For instance, we can find the applications of this generalized operator in modeling of the Sturm–Liouville–Langevin problem [23], investigation of the combination synchronization of a Caputo–Hadamard system [24], description of an uncertain BVP‘ [25], studying the proportional Langevin BVP [26], etc.

Our main novelty in this work is to use the Caputo–Hadamard operator for generalizing the standard jerk problem in the form of a fractional inclusion problem. In fact, a jerk system is a simple form of a nonlinear ODE of third order depicted by

$$ \frac {\mathrm{d}^{3}y}{\mathrm{d}t^{3}} = \mathcal{F} \biggl(y, \frac {\mathrm{d}y}{\mathrm{d}t}, \frac {\mathrm{d}^{2}y}{\mathrm{d}t^{2}} \biggr),$$

where, in mechanics, the nonlinear mapping \(\mathcal{F}(\cdot , \cdot , \cdot )\) is equivalent to the 1st-derivative of acceleration. For this reason, it is introduced as a jerk [27, 28].

The mathematical analysis of this generalized system is our main purpose in this work. To do this, we decided to utilize a new family of multi-functions belonging to ϕ-admissibles and ϕ-ψ-contractions for proving theorems based on fixed point methods. Also, those multi-functions that have approximate property for their end points play a fundamental role in our analysis. These items present the novelty and contribution of our work in this regard, because most researchers get help from standard fixed point techniques in their papers. For example, the Leray–Schauder, the Banach principle, Krasnoselskii, degree principle, Schaefer are the most famous of them, and they are applied in more papers including the generalized proportional equation by Das et al. in [29], impulsive implicit problem by Ali et al. in [30], nonlinear ϕ-Hilfer problem on compact domain by Mottaghi et al. in [31], multi-term multi-strip coupled system by Ahmad et al. in [32], ψ-Hilfer system of coupled Langevin equations by Sudsutad et al. in [33], sequential RL-Hadamard–Caputo problem by Ntouyas et al. in [34], sequential post-quantum integro-difference problem by Soontharanon et al. in [35] and Samei in [3638], Neumann symmetric Hahn problem by Dumrongpokaphan et al. in [39], etc.

By virtue of the idea of a standard jerk equation and extending it to the generalized fractional Caputo–Hadamard settings, we here introduce and study new existence methods based on some special multi-functions to guarantee the existence of solution for the extended fractional jerk inclusion problem illustrated as

$$ \textstyle\begin{cases} ({}^{CH }\mathbb{D}_{1^{+}}^{\iota _{1}} ( {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}}({}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y) ) )(t) \\\quad \in \mathbb{G} ( t,y(t), {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} ( {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y(t) ) ), \\y(1) + y(e) = 0,\qquad {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} y ( \eta ) = 0,\qquad {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} ( {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} y (e) ) = 0, \end{cases} $$
(1)

in which \(\iota _{1},\iota _{2},\iota _{3} \in (0,1]\) and \({}^{CH }\mathbb{D}_{1^{+}}^{p}\) displays the derivative operator in the sense of Caputo–Hadamard subject to \(p \in \{ \iota _{1}, \iota _{2}, \iota _{3} \}\) and also \(t\in I := [1,e]\) and \(\eta \in (1,e)\). In addition to these, we have considered the operator \(\mathbb{G}: I \times \mathbb{R}^{3} \rightarrow \mathcal{P}( \mathbb{R})\) as a multi-function in which \(\mathcal{P}(\mathbb{R})\) illustrates all nonempty subsets of \(\mathbb{R}\).

This research is conducted as follows. Section 2 is fundamental and necessary in its nature since it collects definitions and required results. Section 3 is divided into two parts: one is in relation to the existence criterion via fixed points and the second is in relation to the existence criterion via end points. In fact, in Sect. 3.1, some inequalities between multi-functions and contractions and admissible functions play the role to prove the desired results via fixed point notion. Accordingly, Sect. 3.2 is devoted to proving similar results via end points and approximate property for end points. Section 4 discusses an example for simulating and analyzing the results numerically. Section 5 completes our research via conclusions.

2 Preliminaries

Here, we shall review some primitive and fundamental concepts in the direction of used approaches and techniques in the present study. As you will observe, these notions and properties are utilized throughout the paper. The readers can find more details in [22, 40, 41].

Definition 2.1

([40, 41])

Let \(q \geq 0\). Then the Hadamard fractional \(q^{\mathrm{th}}\)-integral of a continuous function \(y:(a,\infty )\to \mathbb{R}\) of order q is formulated by \({}^{H}\mathbb{I}_{a^{+}}^{0} y(t)=y(t) \) and

$$ {}^{H}\mathbb{I}_{a^{+}}^{q} y(t)= \frac{1}{\Gamma (q )} \int _{a}^{t} \biggl( \ln \frac{t}{r} \biggr)^{q-1}y(r) \frac{\mathrm{dr}}{r},\quad q>0.$$

Definition 2.2

([22])

The Caputo–Hadamard fractional \(q^{\mathrm{th}}\)-derivative for \(y \in AC_{\delta}^{n}([a, b], \mathbb{R})\) is illustrated as

$$ {}^{CH }\mathbb{D}_{a^{+}}^{q} y(t)= {}^{H}\mathbb{I}_{a^{+}}^{n-q} \delta ^{n} y(t) = \frac{1}{\Gamma ( n - q )} \int _{a}^{t} \biggl( \ln \frac{t}{r} \biggr)^{n-q-1} \delta ^{n} y(r) \frac{\mathrm{dr}}{r},$$

in which \(n-1 < q < n\) and \(\delta = t \frac {\mathrm{d}}{\mathrm{dt}}\). Note that, for \(q=n\in \mathbb{N}\), we have

$$ {}^{CH }\mathbb{D}_{a^{+}}^{n} y(t) = \delta ^{n} y(t) = \biggl(t \frac {\mathrm{d}}{\mathrm{dt}} \biggr)^{n} y(t), \qquad {}^{CH } \mathbb{D}_{a^{+}}^{0} y(t) = y(t).$$

From here onwards, we denote the abbreviations HF-integral and CHF-derivative for the above fractional operators. To find other information on the CHF-operators, we direct the interested readers to [22].

Lemma 2.3

([22, 40, 41])

Let \(q,p \in \mathbb{R}^{+}\). Then:

  1. (1)

    \({}^{H}\mathbb{I}_{a^{+}}^{q} {}^{H}\mathbb{I}_{a^{+}}^{p} y(t)= {}^{H} \mathbb{I}_{a^{+}}^{ q + p} y(t)\), (Semi-group property for HF-integrals);

  2. (2)

    For \(n-1< q<n\), \(m-1< p< m\) and \(y(t) \in C_{\delta}^{m+n} [a,b]\), we have

    $$ {}^{CH }\mathbb{D}_{a^{+}}^{q} {}^{CH } \mathbb{D}_{a^{+}}^{p} y(t)= {}^{CH } \mathbb{D}_{a^{+}}^{ q + p} y(t),$$

    (Semi-group property for CHF-derivatives);

  3. (3)

    For \(q>p\),

    $$ {}^{CH }\mathbb{D}_{a^{+}}^{p} {}^{H} \mathbb{I}_{a^{+}}^{q} y(t)= {}^{H} \mathbb{I}_{a^{+}}^{ q - p} y(t),$$

    (Composition property for HF-CHF-operators).

Example 2.4

([40, 41])

Let \(q , \iota \in \mathbb{R}^{+}\). For \(y(t) = ( \ln \frac {t}{a} )^{\iota}\), we have

$$ {}^{H}\mathbb{I}_{a^{+}}^{q} y(t) = {}^{H}\mathbb{I}_{a^{+}}^{q} \biggl( \ln \frac {t}{a} \biggr)^{\iota}= \frac{\Gamma (\iota +1)}{\Gamma (q+\iota +1)} \biggl( \ln \frac {t}{a} \biggr)^{q+\iota},\quad \forall t>a. $$

Further, if \(y(t) \equiv c \in \mathbb{R}\), then

$$ {}^{H}\mathbb{I}_{a^{+}}^{q} y(t) = {}^{H}\mathbb{I}_{a^{+}}^{q}c= \frac{c}{\Gamma (q+1)} \biggl( \ln \frac {t}{a} \biggr)^{q},\quad \forall t>a. $$

Lemma 2.5

([22])

Let \(y \in AC_{\delta}^{n}([a, b], \mathbb{R})\) and \(n -1 < q < n\).

$$ {}^{H}\mathbb{I}_{a^{+}}^{q} \bigl({}^{CH } \mathbb{D}_{a^{+}}^{q} y \bigr) (t)= y (t)-\sum _{i=0}^{n-1} \frac{\delta ^{i} y(a)}{i!} \biggl( \ln \frac{t}{a} \biggr)^{i},\quad \forall t>a.$$

For the homogeneous CHF-differential equation \({}^{CH }\mathbb{D}_{a^{+}}^{q} y(t)=0\), its general solution, by virtue of Lemma 2.5, is obtained by

$$ y(t) = s_{0} + s_{1} \biggl( \ln \frac {t}{a} \biggr) + s_{2} \biggl( \ln \frac {t}{a} \biggr)^{2} + \cdots + s_{n-1} \biggl( \ln \frac {t}{a} \biggr)^{n-1},$$

subject to \(s_{i} \in \mathbb{R}\) and \(n = [ q ] + 1\) [22]. Hence

$$\begin{aligned} {}^{H}\mathbb{I}_{a^{+}}^{q} \bigl( {}^{CH }\mathbb{D}_{a^{+}}^{q} y \bigr) (t) & = y(t) + s_{0} + s_{1} \biggl( \ln \frac {t}{a} \biggr) + s_{2} \biggl( \ln \frac {t}{a} \biggr)^{2} \\ & \quad {}+ \cdots + s_{n-1} \biggl( \ln \frac {t}{a} \biggr)^{n-1} \end{aligned}$$

for \(t>a\) [22].

In what follows we give a brief introduction to some special function spaces and multi-valued operators. We assume \((A,\|\cdot \|)\) as a normed space. We mean by \({\mathcal {P}}_{CL}( A )\), \({\mathcal {P}}_{BN}( A )\), \({\mathcal {P}}_{CP}( A )\), and \({\mathcal {P}}_{CV}( A )\) the category of all closed, bounded, compact, and convex sets, respectively, belonging to A.

Definition 2.6

([42])

The (Pompeiu–Hausdorff) metric, displayed by

$$ H_{\rho}: \bigl( \mathcal{P}(A) \bigr)^{2} \to \mathbb{R} \cup \{\infty \},$$

is introduced as

$$ H_{\rho}(W_{1}, W_{2}) = \max \Bigl\{ \sup _{\nu _{1} \in W_{1}}\rho ( \nu _{1},\nu _{2}), \sup _{\nu _{2} \in W_{2}}\rho (W_{1},\nu _{2}) \Bigr\} ,$$

in which ρ is a metric of A and

$$ \rho (W_{1},\nu _{2}) = \inf_{\nu _{1} \in W_{1}} \rho (\nu _{1}, \nu _{2}), \qquad \rho (\nu _{1},W_{2}) = \inf_{\nu _{2} \in W_{2}} \rho (\nu _{1},\nu _{2}).$$

Definition 2.7

([42])

For \(\mathbb{G} : A \to {\mathcal {P}}_{CL} (A)\) and \(y_{1}, y_{2} \in A\), let

$$ H_{\rho}\bigl(\mathbb{G}(y_{1}),\mathbb{G}(y_{2}) \bigr) \leq L \rho (y_{1},y_{2}).$$

Then \(\mathbb{G}\) is called: (1) Lipschitz if \(L > 0\); (2) a contraction if \(L \in (0,1)\).

In the next step, we recall a specific family of multi-functions introduced by Amini-Harandi [42] in 2010 which we utilize in our proofs.

Definition 2.8

([42])

Let A be a metric space and \(\mathbb{G}\) be a multi-valued operator on it. Then

  1. (1)

    \(y \in A\) is an end point of \(\mathbb{G} :A\to \mathcal{P}(A)\) if \(\mathbb{G}y=\{ y\}\).

  2. (2)

    \(\mathbb{G}\) admits the AEP-property (approximate end point property) whenever

    $$ \inf_{v\in A}\sup_{y\in \mathbb{G}v}\rho (v,y)=0.$$

Later, in 2013, Mohammadi, Rezapour, and Shahzad [43] provided another family of multi-functions based on two operators ψ and ϕ which is a generalized structure of a similar notion pertinent to single-valued operators given by Samet et al. [44] in 2012.

Definition 2.9

([43])

Let Ψ be a family of all increasing mappings \(\psi : \mathbb{R}^{\geq 0}\to \mathbb{R}^{\geq 0}\) s.t. \(\forall t>0\), \(\sum_{i=1}^{\infty}\psi ^{i}(t)<\infty \) and \(\psi (t)< t\). Let \(\mathbb{G}: A\to \mathcal{P}(A)\) and \(\phi : A\times A \to \mathbb{R}^{\geq 0}\). In this case:

  1. (1)

    \(\mathbb{G}: A \to \mathcal{P}_{CL,BN}(A)\) is ϕ-ψ-contraction if \(\forall y_{1}, y_{2}\in A\),

    $$ \phi (y_{1},y_{2}) H_{\rho}(\mathbb{G} y_{1}, \mathbb{G} y_{2})\leq \psi \bigl(\rho (y_{1}, y_{2})\bigr). $$
  2. (2)

    \(\mathbb{G}\) is ϕ-admissible if \(\forall y_{1}\in A\) and \(\forall y_{2}\in \mathbb{G}y_{1}\),

    $$ \phi (y_{1},y_{2})\geq 1 \quad \Longrightarrow\quad \phi ( y_{2},y_{3})\geq 1,\quad \forall y_{3}\in \mathbb{G}y_{2}.$$
  3. (3)

    A admits the property \((C_{\phi})\) if for each \(\{ y_{n} \}_{n \geq 1} \subset A\) with \(y_{n}\to y\) and \(\phi (y_{n} , y_{n+1}) \geq 1\),

    $$ \exists \{ y_{n_{i}} \} \subset \{ y_{n} \}, \quad \text{s.t.}\quad \phi (y_{n_{i}}, y)\geq 1, \quad \forall i\in \mathbb{N}.$$

To follow the required arguments on the existence of a solution for the Caputo–Hadamard fractional jerk problem (CHF-jerk problem) (1), we begin this section by introducing a Banach space as follows:

$$ A = \bigl\{ y(t): y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y(t)\bigr) \in C(I, \mathbb{R}) \bigr\} ,$$

equipped with

$$ \Vert y \Vert _{A}=\sup_{t\in I} \bigl\vert y(t) \bigr\vert + \sup_{t\in I} \bigl\vert {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr\vert + \sup _{t\in I} \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl( {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}y(t) \bigr) \bigr\vert $$

for all \(y \in A\).

3 Existence results via fixed-points and end points

Now, in the next proposition, the solution’s structure for the supposed CHF-jerk problem (1) is exhibited in the format of an integral equation.

Proposition 3.1

Let \(\iota _{1},\iota _{2},\iota _{3} \in (0,1]\), \(\eta \in (1,e)\) and \(T \in C(I,\mathbb{R})\). Then the solution of the linear CHF-jerk problem

$$ \textstyle\begin{cases} ( {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{1}} ( {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} ({}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y) ) )(t) = T(t),&t\in I, \\ y(1) + y(e) = 0,\qquad {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} y (\eta ) = 0,& \\ {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} ( {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}} y (e) ) = 0,& \end{cases} $$
(2)

is obtained as

$$\begin{aligned} y(t)&= \frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} T(r) \frac{\mathrm{dr}}{r} \\ &\quad {}- \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} T(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{1}(t)}{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} T(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{2}(t)}{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \\ & \quad {}\times \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} T(r) \frac{\mathrm{dr}}{r}, \end{aligned}$$
(3)

where

$$ \textstyle\begin{cases} F_{1}(t) = 1- 2 ( \ln t )^{\iota _{3}}, \\F_{2}(t) = \Gamma (1+\iota _{2}) \Gamma (1+ \iota _{3}) [ 1- 2 ( \ln t )^{\iota _{2} + \iota _{3}} ] \\\hphantom{F_{2}(t) = } {}- \Gamma (1 + \iota _{2} + \iota _{3}) ( \ln \eta )^{\iota _{2}} [ 1- 2 ( \ln t )^{\iota _{3}} ]. \end{cases} $$
(4)

Proof

Let y satisfy the linear CHF-jerk problem (2). In view of the semi-group property for HF-integrals given in Lemma 2.3, since \(\iota _{1} \in (0,1]\), so by utilizing the HF-integral of order \(\iota _{1}\), we get

$$ {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl( {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}} y \bigr) (t) = \frac{1}{\Gamma (\iota _{1})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1} -1} T(r) \frac{\mathrm{dr}}{r} + c_{0}, $$
(5)

where \(c_{0} \in \mathbb{R}\). Again, utilizing the HF-integral of order \(\iota _{2} \in (0,1]\) to both sides of (5), we get

$$ {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} y(t) = \frac{1}{\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1} + \iota _{2} -1} T(r) \frac{\mathrm{dr}}{r} + c_{0} \frac{ ( \ln t )^{\iota _{2}}}{\Gamma (1+ \iota _{2})} + c_{1}, $$
(6)

where \(c_{1} \in \mathbb{R}\). At last, utilizing the HF-integral of order \(\iota _{3} \in (0,1]\) to both sides of (6), the general series solution of (2) can be derived by

$$\begin{aligned} y(t) & = \frac{1}{\Gamma (\iota _{1} + \iota _{2} + \iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1} + \iota _{2} + \iota _{3} -1} T(r) \frac{\mathrm{dr}}{r} \\ & \quad {}+ c_{0} \frac{ ( \ln t )^{\iota _{2} + \iota _{3}}}{\Gamma (1+ \iota _{2} + \iota _{3})} + c_{1} \frac{ ( \ln t )^{\iota _{3}}}{\Gamma (1+ \iota _{3})} + c_{2}, \end{aligned}$$
(7)

where \(c_{2} \in \mathbb{R}\). To obtain the values \(c_{i}\) (\(i=0,1,2\)), we first consider the third boundary condition and (5), and so the coefficient \(c_{0}\) is obtained as

$$ c_{0} = - \frac{1}{\Gamma (\iota _{1} )} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} T(r) \frac{\mathrm{dr}}{r}. $$
(8)

In the sequel, the second boundary condition and the obtained value for \(c_{0}\) in (8) yield

$$ \begin{aligned} c_{1} & = - \frac{1}{\Gamma (\iota _{1} + \iota _{2} )} \int _{1}^{\eta}\biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} T(r) \frac{\mathrm{dr}}{r} \\ & \quad {}+ \frac{ ( \ln \eta )^{\iota _{2}} }{\Gamma (1 + \iota _{2})\Gamma (\iota _{1})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} T(r) \frac{\mathrm{dr}}{r}. \end{aligned} $$
(9)

Finally, (8) and (9) and the first boundary condition give

$$\begin{aligned} c_{2} &= - \frac{1}{2\Gamma (\iota _{1} + \iota _{2} + \iota _{3} )} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} + \iota _{2} + \iota _{3} -1} T(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{1}{2\Gamma (1+ \iota _{3})\Gamma (\iota _{1} + \iota _{2} )} \int _{1}^{\eta}\biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} T(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{\Gamma (1+ \iota _{2}) \Gamma (1+ \iota _{3}) - \Gamma (1 + \iota _{2} + \iota _{3}) ( \ln \eta )^{\iota _{2}} }{2\Gamma ( 1 + \iota _{2} + \iota _{3})\Gamma (1 + \iota _{2})\Gamma (1+\iota _{3})\Gamma (\iota _{1})} \\ & \quad {}\times \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} T(r) \frac{\mathrm{dr}}{r}. \end{aligned}$$
(10)

At this moment, we insert the value of the coefficients \(c_{i}\), by (8)–(10), into (7) and obtain

$$\begin{aligned} y(t)&= \frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} T(r) \frac{\mathrm{dr}}{r} \\ &\quad{}- \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} T(r) \frac{\mathrm{dr}}{r} \\ &\quad{}+ \frac{F_{1}(t)}{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} T(r) \frac{\mathrm{dr}}{r} \\ &\quad{}+ \frac{F_{2}(t)}{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} T(r) \frac{\mathrm{dr}}{r}, \end{aligned}$$

showing that y satisfies (3) and \(F_{1}(t)\), \(F_{2}(t)\) are continuous functions represented in (4). This ends the proof. □

3.1 Fixed-point and jerk model (1)

In this part, we define the solution to the CHF-jerk problem (1).

Definition 3.2

The function \(y \in C(I, A)\) is named the solution to the supposed CHF-jerk problem (1) whenever it fulfills the given BCs and \(\exists g \in L^{1}(I)\) s.t.

$$ g(t)\in \mathbb{G} \bigl( t,y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl( {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}y(t) \bigr) \bigr)$$

for almost all \(t\in I\) and

$$\begin{aligned} y(t)&= \frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}- \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{1}(t)}{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{2}(t)}{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} g(r) \frac{\mathrm{dr}}{r}. \end{aligned}$$

\(\forall t\in I\). For each \(y \in A\), we specify selections of \(\mathbb{G}\) as

$$\begin{aligned} \mathit{S}_{\mathbb{G}, y} &= \bigl\{ g \in L^{1}(I) : g(t) \in \mathbb{G} \bigl( t, y(t), {}^{ CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y(t), \\ & \quad {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) \bigr) \text{ (a.e.) } t\in I \bigr\} . \end{aligned}$$

In the sequel, define the multi-function \(K:A\to \mathcal{P}(A)\) by

$$ K(y)= \bigl\{ z\in A: \text{there exists } g\in \mathit{S}_{\mathbb{G}, y} \text{ such that } z(t)= \pi (t) \ \forall t\in I \bigr\} , $$
(11)

for which

$$\begin{aligned} \pi (t)&=\frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}- \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{1}(t)}{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{2}(t)}{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} g(r) \frac{\mathrm{dr}}{r}. \end{aligned}$$
(12)

By making use of the following theorem relying on some inequalities between special multi-functions such as ϕ-ψ-contractions and ϕ-admissible, we establish the first criterion guaranteeing the existence of solution for the CHF-jerk problem (1).

Theorem 3.3

([43])

Regard the complete metric space \((A,\rho )\), \(\psi \in \Psi \), \(\phi : A \times A \to \mathbb{R}^{\geq 0}\) and \(\mathbb{G}: A \to \mathcal{P}_{CL,BN}(A)\). Assume that:

  1. (1)

    \(\mathbb{G}\) is ϕ-admissible and ϕ-ψ-contraction;

  2. (2)

    \(\phi (y_{0}, y_{1} )\geq 1\) for some \(y_{0} \in A\) and \(y_{1} \in \mathbb{G}y_{0}\);

  3. (3)

    A involves the \((C_{\phi})\)-property.

Then \(\mathbb{G}\) admits a fixed point.

Remark 3.4

For the sake of simplicity, we define

$$\begin{aligned}& \begin{aligned}[t] \check{\Lambda}_{1} &= \frac{ 3 }{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3}+1)} + \frac{ F_{1}^{*} (\ln \eta )^{\iota _{1}+\iota _{2}} }{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2}+1)} \\ & \quad {}+ \frac{ F_{2}^{*} }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1}+1)}, \end{aligned} \\& \begin{aligned}[t] \check{\Lambda}_{2} &= \frac{1}{ \Gamma (\iota _{1}+\iota _{2}+1)} + \frac{ F_{1}^{**}(\ln \eta )^{\iota _{1}+\iota _{2}} }{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2}+1)} \\ & \quad {}+ \frac{ F_{2}^{**} }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1}+1)}, \end{aligned} \\& \check{\Lambda}_{3} = \frac{1}{ \Gamma (\iota _{1}+1) } + \frac{ F_{2}^{***} }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1}+1)}, \end{aligned}$$
(13)

where for \(t\in I=[1,e]\),

$$ \textstyle\begin{cases} \sup_{t\in I} \vert F_{1}(t) \vert \leq \sup_{t\in I} ( 1 + 2 ( \ln t )^{\iota _{3}} ) = 3 :=F_{1}^{*}>0, \\\sup_{t\in I} \vert F_{2}(t) \vert \leq \sup_{t\in I} ( \Gamma (1+\iota _{2}) \Gamma (1+ \iota _{3}) [ 1 + 2 ( \ln t )^{\iota _{2} + \iota _{3}} ] \\\hphantom{\sup_{t\in I} \vert F_{2}(t) \vert \leq} {}+ \Gamma (1 + \iota _{2} + \iota _{3}) ( \ln \eta )^{ \iota _{2}} [ 1+ 2 ( \ln t )^{\iota _{3}} ] ) \\\hphantom{\sup_{t\in I} \vert F_{2}(t) \vert }\leq 3\Gamma (1+\iota _{2}) \Gamma (1+ \iota _{3}) + 3\Gamma (1 + \iota _{2} + \iota _{3}) ( \ln \eta )^{\iota _{2}} \\\hphantom{\sup_{t\in I} \vert F_{2}(t) \vert }:=F_{2}^{*} >0, \end{cases} $$
(14)

and

$$ \textstyle\begin{cases} \sup_{t\in I} \vert ({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}} F_{1})(t) \vert \leq \sup_{t\in I} ( 2 \Gamma (\iota _{3}+1) ) \\\hphantom{\sup_{t\in I} \vert ({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}} F_{1})(t) \vert } = 2 \Gamma (\iota _{3}+1) :=F_{1}^{**}>0, \\\sup_{t\in I} \vert ({}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}F_{2})(t) \vert \\\quad \leq \sup_{t\in I} ( 2 \Gamma (1+ \iota _{3}) \Gamma (1+ \iota _{2} + \iota _{3}) ( \ln t )^{\iota _{2} } \\ \qquad {}+ 2\Gamma (1 + \iota _{2} + \iota _{3}) \Gamma (1+\iota _{3}) ( \ln \eta )^{\iota _{2}} ) \\\quad \leq 2\Gamma (1 + \iota _{2} + \iota _{3}) \Gamma (1+\iota _{3}) (1 + ( \ln \eta )^{\iota _{2}} ) :=F_{2}^{**} >0, \end{cases} $$
(15)

and

$$ \textstyle\begin{cases} \sup_{t\in I} \vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}}({}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}F_{1})(t) \vert \leq \sup_{t\in I} ( 0 ) = 0 :=F_{1}^{***}, \\\sup_{t\in I} \vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}}({}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}F_{2})(t) \vert \\\quad \leq \sup_{t\in I} ( 2 \Gamma (1+ \iota _{3}) \Gamma (1+ \iota _{2} + \iota _{3}) \Gamma (1+\iota _{2}) ) \\\quad = 2 \Gamma (1+ \iota _{3}) \Gamma (1+ \iota _{2} + \iota _{3}) \Gamma (1+\iota _{2}) :=F_{2}^{***} >0. \end{cases} $$
(16)

Theorem 3.5

Let \(\mathbb{G}:I \times A^{3} \to \mathcal{P}_{CP}(A)\) be a multifunction and assume the following scenario:

\((\mathcal{H}_{1})\):

The multifunction \(\mathbb{G}\) is bounded and integrable with \(\mathbb{G}(\cdot , y_{1}, y_{2}, y_{3}): I \to \mathcal{P}_{CP}(A)\) is measurable for all \(y_{m} \in A\) (\(m=1,2,3\));

\((\mathcal{H}_{2})\):

There exist \(\kappa \in C(I, [0, \infty ))\) and \(\psi \in \Psi \) s.t.

$$ {H}_{\rho} (\mathbb{G}(t, y_{1}, y_{2}, y_{3}), \mathbb{G}(t, \bar{y}_{1}, \bar{y}_{2}, \bar{y}_{3} ) \leq \kappa (t) \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \Biggl(\sum_{m=1}^{3} \vert y_{m} - \bar{y}_{m} \vert \Biggr) $$
(17)

for all \(t \in I\) and \(y_{m}, \bar{y}_{m} \in A \) (\(m= 1,2,3\)), where \(\sup_{t\in I} \vert \kappa (t) \vert = \Vert \kappa \Vert \),

$$ \vartheta ^{\star}= \frac {1}{ \check{\Lambda}_{1} + \check{\Lambda}_{2}+ \check{\Lambda}_{3} },$$

and \(\check{\Lambda}_{m}\) (\(m=1,2,3\)) are given by (13);

\((\mathcal{H}_{3})\):

A function \(\Omega _{\star}:\mathbb{R}^{3} \times \mathbb{R}^{3} \to \mathbb{R}\) exists such that, for all \(y_{m}, \bar{y}_{m} \in A \) (\(m= 1,2,3\)), we have

$$ \Omega _{\star}\bigl((y_{1}, y_{2}, y_{3}), (\bar{y}_{1},\bar{y}_{2}, \bar{y}_{3} )\bigr) \geq 0;$$
\((\mathcal{H}_{4})\):

If \(\{ y_{\jmath}\}_{\jmath \geq 1} \subset A\) s.t. \(y_{\jmath}\to y\) and

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl(y_{\jmath}(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y_{\jmath}(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl( {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y_{\jmath}(t) \bigr) \bigr), \\ &\quad \bigl(y_{\jmath +1}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y_{ \jmath +1}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y_{\jmath +1}(t) \bigr) \bigr) \bigr) \geq 0, \end{aligned}$$

then \(\exists \{ y_{\jmath _{s}} \}_{s \geq 1} \subset \{y_{\jmath}\}\) exists such that, for all \(t \in I\) and \(s \geq 1\), we have

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl(y_{\jmath _{s}}(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y_{\jmath _{s}}(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}}\bigl( {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y_{\jmath _{s}}(t) \bigr) \bigr), \\ &\quad \bigl(y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) \bigr) \bigr) \geq 0; \end{aligned}$$
\((\mathcal{H}_{5})\):

There exist a member \(y^{0} \in A\) and \(\mu \in K(y^{0})\) such that, for any \(t \in I\),

$$\begin{aligned} &\Omega \bigl( \bigl(y^{0}(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y^{0}(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl( {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y^{0}(t) \bigr) \bigr), \\ & \quad \bigl(\mu (t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \mu (t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}\mu (t)\bigr) \bigr) \bigr) \geq 0, \end{aligned}$$

where the multifunction \(K: A \to \mathcal{P}(A)\) is specified by (11);

\((\mathcal{H}_{6})\):

For every \(y \in A\) and \(\mu \in K(y)\) with

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl( y(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) \bigr), \\ & \quad \bigl(\mu (t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \mu (t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}\mu (t) \bigr) \bigr) \bigr) \geq 0, \end{aligned}$$

there exists a member \(\nu \in K(y)\) such that the inequality

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl( \mu (t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}\mu (t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\mu (t) \bigr) \bigr), \\ & \quad \bigl(\nu (t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \nu (t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}\nu (t) \bigr) \bigr) \bigr) \geq 0 \end{aligned}$$

holds for all \(t\in I\).

Then, the CHF-jerk problem (1) owns a solution.

Proof

Definitely, the fixed point of the mapping \(K: A \to \mathcal{P}(A)\) is a solution of the CHF-jerk problem (1). Note that \(S_{\mathbb{G}, y}\) is nonempty. Indeed, the multifunction

$$ t \mapsto \mathbb{G} \bigl(t, y(t), {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t)\bigr) \bigr) $$

is both measurable and closed-valued for any \(y \in A\), so \(S_{\mathbb{G}, y} \neq \emptyset \). Firstly, we will claim that \(K(y)\subseteq A\) is closed \(\forall y\in A\). As for, take a sequence \(\{ y_{n} \}_{n \geq 1}\) in \(K(y)\) such that \(y_{n} \to y\) as \(n\to \infty \). For each \(n \geq 1\), there is \(g_{n}\in S_{\mathbb{G}, y}\) such that

$$\begin{aligned} y_{n}(t)&= \frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g_{n}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}- \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g_{n}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{1}(t)}{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} g_{n}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{2}(t)}{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} g_{n}(r) \frac{\mathrm{dr}}{r} \end{aligned}$$

for all \(t\in I\). Since the multifunction \(\mathbb{G}\) has compact values, there is indeed a subsequence of \(\{g_{n} \}_{n \geq 1}\) (following the same notation) that converges to some \(g\in {L}^{1} (I)\). Subsequently, \(g\in S_{\mathbb{G}, y}\) and

$$\begin{aligned} {y}_{n}(t) &\to y(t) \\ &= \frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}- \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{1}(t)}{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{2}(t)}{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \\ & \quad {}\times \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} g(r) \frac{\mathrm{dr}}{r} \end{aligned}$$

for all \(t\in I\). As a result, we can deduce that \(y\in K(y)\) and K is closed-valued. The boundedness of \(K(y)\) is obvious from the compactness of multifunction \(\mathbb{G}\). Next, we prove that K is a ϕ-ψ-contraction. To do this, we regard \(\phi :A^{2}\mapsto {\mathbb{R}}_{\geq 0}\) by \(\phi (y, \bar{y})= 1\) whenever

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl(y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}} y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y(t)\bigr) \bigr), \\ & \quad \bigl(\bar{y}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \bar{y}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y}(t)\bigr) \bigr) \bigr) \geq 0, \end{aligned}$$

and \(\phi (y, \bar{y})=0\) otherwise, where \(y, \bar{y} \in A\). Consider \(y, \bar{y} \in A\) and \(\ell _{1} \in K(\bar{y})\) and choose \(g_{1}\in S_{\mathbb{G}, \bar{y}}\) such that

$$\begin{aligned} \ell _{1}(t) &= \frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+ \iota _{3} -1} g_{1}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}- \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g_{1}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{1}(t)}{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} g_{1}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{2}(t)}{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \\ & \quad{}\times \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} g_{1}(r) \frac{\mathrm{dr}}{r} \end{aligned}$$

for all \(t\in I\). By making use of (17), we get

$$\begin{aligned} \mathbb{H}_{\rho} & \bigl(\mathbb{G} \bigl( t, y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) \bigr), \\ & \quad \mathbb{G} \bigl( \bar{y}(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}\bar{y}(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl( {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y}(t)\bigr) \bigr) \bigr) \\ & \leq \kappa (t) \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \vert y - \bar{y} \vert + \bigl\vert {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}} y - {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}} \bar{y} \bigr\vert \\ & \quad {}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y \bigr) - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y} \bigr) \bigr\vert \bigr) \end{aligned}$$

with

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl( y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y(t) \bigr) \bigr), \\ & \quad \bigl( \bar{y}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \bar{y}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y}(t) \bigr) \bigr) \bigr) \geq 0. \end{aligned}$$

Thus, there exists

$$ \wp \in \mathbb{G} \bigl(t, y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y(t) \bigr) \bigr)$$

such that

$$\begin{aligned} \bigl\vert g_{1}(t) - \wp \bigr\vert & \leq \kappa (t) \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \vert y - \bar{y} \vert + \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y - {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y} \bigr\vert \\ & \quad {}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y \bigr) - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y} \bigr) \bigr\vert \bigr). \end{aligned}$$

Now, consider a mapping \(\mathbb{U}: I \to \mathcal{P}(A)\) defined by

$$\begin{aligned} \mathbb{U}(t) &= \biggl\{ \wp \in A : \bigl\vert g_{1}(t) - \wp \bigr\vert \leq \kappa (t) \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi ( \vert y - \bar{y} \vert + \bigl\vert {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}\bar{y} \bigr\vert \\ &\quad{}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y \bigr) - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y} \bigr) \bigr\vert \biggr\} \end{aligned}$$

for any \(t\in I\). Since \(g_{1}\) and

$$\begin{aligned} \mho & = \kappa (t) \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \vert y - \bar{y} \vert + \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y - {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y} \bigr\vert \\ & \quad {}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y \bigr) - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y} \bigr) \bigr\vert \bigr) \end{aligned}$$

are measurable, so the multivalued function

$$ \mathbb{U} (\cdot )\cap \mathbb{G} (\cdot , y(\cdot ), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(\cdot ), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(\cdot ) \bigr)$$

is also measurable. Now, suppose

$$ g_{2} \in \mathbb{G} (t, y(t), {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr),$$

so that we have

$$\begin{aligned} \bigl\vert g_{1}(t) - g_{2}(t) \bigr\vert &\leq \kappa (t) \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \vert y - \bar{y} \vert + \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y - {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y} \bigr\vert \\ & \quad {}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y \bigr) - {}^{CH }\mathbb{D}_{1^{+}}^{q_{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y} \bigr) \bigr\vert \bigr). \end{aligned}$$

Define \(\ell _{2} \in K(y)\) by

$$\begin{aligned} \ell _{2}(t) &= \frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+ \iota _{3} -1} g_{2}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}- \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g_{2}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{1}(t)}{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} g_{2}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{2}(t)}{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \\ & \quad {}\times \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} g_{2}(r) \frac{\mathrm{dr}}{r} \end{aligned}$$

for any \(t\in I\). Then we get the following inequalities as a result.

$$\begin{aligned} & \bigl\vert \ell _{1}(t) - \ell _{2}(t) \bigr\vert \\ &\quad \leq \frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} \bigl\vert g_{1}(r) - g_{2}(r) \bigr\vert \frac{\mathrm{dr}}{r} \\ &\qquad {}+ \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} \bigl\vert g_{1}(r) - g_{2}(r) \bigr\vert \frac{\mathrm{dr}}{r} \\ &\qquad{} + \frac{ \vert F_{1}(t) \vert }{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} \bigl\vert g_{1}(r) - g_{2}(z) \bigr\vert \frac{\mathrm{dr}}{r} \\ &\qquad {}+ \frac{ \vert F_{2}(t) \vert }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \\ & \qquad{} \times \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} \bigl\vert g_{1}(r) - g_{2}(z) \bigr\vert \frac{\mathrm{dr}}{r} \\ &\quad \leq \frac{ (\ln t)^{\iota _{1}+\iota _{2}+\iota _{3}} }{\Gamma (\iota _{1}+\iota _{2}+\iota _{3}+1)} \Vert \kappa \Vert \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \Vert y - \bar{y} \Vert \bigr) \\ &\qquad {}+ \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3}+1)} \Vert \kappa \Vert \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \Vert y - \bar{y} \Vert \bigr) \\ &\qquad {}+ \frac{ F_{1}^{*} (\ln \eta )^{\iota _{1}+\iota _{2}} }{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2}+1)} \Vert \kappa \Vert \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \Vert y - \bar{y} \Vert \bigr) \\ & \qquad {}+ \frac{ F_{2}^{*} }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1}+1)} \\ & \qquad{} \times \Vert \kappa \Vert \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \Vert y - \bar{y} \Vert \bigr) \\ &\quad = \biggl[\frac{ 3 }{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3}+1)} + \frac{ F_{1}^{*} (\ln \eta )^{\iota _{1}+\iota _{2}} }{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2}+1)} \\ & \qquad{} + \frac{ F_{2}^{*} }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1}+1)} \biggr] \\ & \qquad{} \times \Vert \kappa \Vert \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \Vert y - \bar{y} \Vert \bigr) \\ &\quad = \vartheta ^{\star} \check{\Lambda}_{1} \psi \bigl( \Vert y - \bar{y} \Vert \bigr). \end{aligned}$$

Also, we have

$$\begin{aligned} & \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}\ell _{1}(t) - {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \ell _{2}(t) \bigr\vert \\ &\quad \leq \biggl[ \frac{1}{ \Gamma (\iota _{1}+\iota _{2}+1)} + \frac{ F_{1}^{**}(\ln \eta )^{\iota _{1}+\iota _{2}} }{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2}+1)} \\ &\quad\quad{}+ \frac{ F_{2}^{**} }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1}+1)} \biggr] \\ &\quad \quad {}\times \Vert \kappa \Vert \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \Vert y - \bar{y} \Vert \bigr) \\ &\quad= \vartheta ^{\star} \check{\Lambda}_{2} \psi \bigl( \Vert y - \bar{y} \Vert \bigr) \end{aligned}$$

and

$$\begin{aligned} &\bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\ell _{1}(t) \bigr) - {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \ell _{2}(t)\bigr) \bigr\vert \\ &\quad \leq \biggl[ \frac{1}{ \Gamma (\iota _{1}+1) } \\ &\quad \quad {}+ \frac{ F_{2}^{***} }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1}+1)} \biggr] \\ &\quad \quad {}\times \Vert \kappa \Vert \biggl( \frac{\vartheta ^{\star}}{ \Vert \kappa \Vert } \biggr) \psi \bigl( \Vert y - \bar{y} \Vert \bigr) \\ &\quad= \vartheta ^{\star} \check{\Lambda}_{3} \psi \bigl( \Vert y - \bar{y} \Vert \bigr) \end{aligned}$$

for all \(t\in I\). Consequently,

$$\begin{aligned} \Vert \ell _{1} - \ell _{2} \Vert & = \sup _{t\in I} \bigl\vert \ell _{1}(t) - \ell _{2}(t) \bigr\vert + \sup_{t\in I} \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}\ell _{1}(t) - {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \ell _{2}(t) \bigr\vert \\ &\quad {}+ \sup_{t\in I} \bigl\vert {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\ell _{1}(t) \bigr) -{}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}\ell _{2}(t)\bigr) \bigr\vert \\ &\leq \vartheta ^{\star} (\check{\Lambda}_{1} + \check{\Lambda}_{2} + \check{\Lambda}_{3}) \psi \bigl( \Vert y - \bar{y} \Vert \bigr) = \psi \bigl( \Vert y - \bar{y} \Vert \bigr). \end{aligned}$$

Accordingly, \(\phi (y, \bar{y}) \mathbb{H}_{\rho} ( K (y), K(\bar{y}) ) \leq \psi (\Vert y - \bar{y} \Vert )\) for all \(y, \bar{y} \in A\). This confirms that K is a ϕ-ψ-contraction. Next, suppose that \(y \in A\) and \(\bar{y} \in K (y)\) s.t. \(\phi (y, \bar{y}) \geq 1\) and

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl( y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y(t) \bigr) \bigr), \\ & \quad \bigl( \bar{y}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \bar{y}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y}(t) \bigr) \bigr) \bigr) \geq 0, \end{aligned}$$

so there exists \(\wp \in K(\bar{y})\) such that

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl(\bar{y}(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}\bar{y}(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y}(t)\bigr) \bigr), \\ & \quad \bigl(\wp (t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \wp (t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}\wp (t) \bigr) \bigr) \bigr) \geq 0, \end{aligned}$$

which further implies that \(\phi (\bar{y}, \wp ) \geq 1\) and accordingly K is ϕ-admissible. Finally, let \(y^{0} \in A\) and \(\bar{y} \in K(y^{0})\) so that

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl( y^{0}(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y^{0}(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl( {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y^{0}(t) \bigr) \bigr), \\ & \quad \bigl(\bar{y}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \bar{y}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\bar{y}(t)\bigr) \bigr) \bigr) \geq 0 \end{aligned}$$

for all \(t\in I\). It follows that \(\phi (y^{0}, \bar{y}) \geq 1\). Assume \(\{ y_{\jmath}\}_{\jmath \geq 1} \subset A\) s.t. \(y_{\jmath}\to y\) and \(\phi (y_{\jmath}, y_{\jmath +1}) \geq 1\) for all ȷ. Then we have

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl(y_{\jmath}(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y_{\jmath}(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y_{\jmath}(t) \bigr) \bigr), \\ & \quad \bigl(y_{\jmath +1}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y_{ \jmath +1}(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y_{\jmath +1}(t) \bigr) \bigr) \bigr) \geq 0. \end{aligned}$$

Then hypothesis \((\mathcal{H}_{4})\) confirms the existence of a subsequence \(\{ y_{\jmath _{s}} \}_{s\geq 1}\) of \(\{ y_{\jmath}\}\) satisfying

$$\begin{aligned} &\Omega _{\star}\bigl( \bigl( y_{\jmath _{s}}(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y_{\jmath _{s}}(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y_{\jmath _{s}}(t) \bigr) \bigr), \\ & \quad \bigl( y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y(t) \bigr) \bigr) \bigr) \geq 0 \end{aligned}$$

for all \(t\in I\). Thus, \(\phi (y_{\jmath _{s}}, y) \geq 1\) for all t, and accordingly it possesses the \((C_{\phi})\) condition. Hence, Theorem 3.3 allows that K possesses a fixed point which is a solution for the CHF-jerk inclusion (1). □

3.2 End point and jerk model (1)

Now, in the next place, by utilizing another theorem based on some other special multi-functions containing the AEP-property, we derive the second criterion guaranteeing the existence of solution for the supposed CHF-jerk problem (1).

Theorem 3.6

([42])

Consider \((A,\rho )\) as a complete metric space. Assume:

  1. (1)

    \(\psi \in \Psi \) is u.s.c along with \(\liminf_{t\to \infty}(t-\psi (t))>0\) for \(t>0\);

  2. (2)

    \(\mathbb{G} : A \to \mathcal{P}_{CL,BN}(A)\) admits the property

    $$ H_{\rho}(\mathbb{G} y_{1},\mathbb{G} y_{2})\leq \psi \bigl(\rho (y_{1}, y_{2})\bigr), \quad \forall y_{1}, y_{2} \in A.$$

Then \(\mathbb{G}\) admits one and exactly one end point iff \(\mathbb{G}\) contains the AEP-property.

Theorem 3.7

Take \(\mathbb{G}:I \times A^{3} \to \mathcal{P}_{CP}(A)\). Assume that

\((\mathcal{H}_{7})\):

There exists a nondecreasing and upper semi-continuous mapping \(\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}\) which satisfies \(\psi (t) \leq t\), \(\forall t>0\) and \(\liminf_{t \to \infty} (t - \psi (t)) \geq 0\);

\((\mathcal{H}_{8})\):

Multifunction \(\mathbb{G}:I \times A^{3} \to \mathcal{P}_{CP}(A)\) is bounded and integrable such that the map \(\mathbb{G}( \cdot , y_{1}, y_{2}, y_{3}): I \to \mathcal{P}_{CP}(A)\) is measurable for all \(y_{m} \in A \) (\(m=1,2,3\));

\((\mathcal{H}_{9})\):

There is a function \(\kappa \in {C}(I, [0, \infty ))\) s.t.

$$ {H}_{\rho} (\mathbb{G}(t, y_{1}, y_{2}, y_{3}), \mathbb{G}(t, \check{y}_{1}, \check{y}_{2}, \check{y}_{3} ) \leq \kappa (t) \varpi ^{\star} \psi \Biggl(\sum_{m=1}^{3} \vert y_{m} - \check{y}_{m} \vert \Biggr) $$
(18)

for all \(t\in I\) and \(y_{m}, \check{y}_{m} \in A \) (\(m=1,2,3\)), where

$$ \varpi ^{\star}= \frac {1}{\Xi _{1} + \Xi _{2} + \Xi _{3}}, \qquad \Xi _{m} = \Vert \kappa \Vert \check{\Lambda}_{m}\quad (m=1,2,3);$$
\((\mathcal{H}_{10})\):

The AEP-property is valid for the multi-function K.

Then the CHF-jerk problem (1) has a solution.

Proof

We want to establish that the multifunction \(K: A \to \mathcal{P}(A)\) possesses an end point. Initially, we claim that \(K(y)\) is closed \(\forall y\in A\). As the multifunction

$$ t \mapsto \mathbb{G} \bigl( t, y(t), {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) \bigr)$$

is both measurable and closed-valued for any \(y \in A\), so the \(\mathbb{G}\) has a measurable selection and \(S_{\mathbb{G}, y} \neq \emptyset \). By using the same procedure as that given in Theorem 3.5, it can be easily deduced that \(K(y)\) is closed-valued. Also, the compactness of \(\mathbb{G}\) ensures the boundedness of \(K(y)\). Next, assume that \(y, \check{y} \in A\) and \(\ell _{1} \in K(\check{y})\) and choose \(g_{1}\in S_{\mathbb{G}, \check{y}}\) such that

$$\begin{aligned} \ell _{1}(t) &= \frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+ \iota _{3} -1} g_{1}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}- \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g_{1}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{1}(t)}{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} g_{1}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{2}(t)}{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} g_{1}(r) \frac{\mathrm{dr}}{r} \end{aligned}$$

for all \(t\in I\). Also, for all \(y, \check{y} \in A\) and \(t\in I\), we have

$$\begin{aligned} \mathbb{H}_{\rho} & ( \mathbb{G} \bigl( t, y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) \bigr), \\ & \quad \mathbb{G} \bigl(\check{y}(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}\check{y}(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\check{y}(t) \bigr) \bigr) \\ & \leq \kappa (t)\varpi ^{\star}\psi \bigl( \bigl\vert y(t) - \check{y}(t) \bigr\vert + \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y(t) - {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}\check{y}(t) \bigr\vert \\ & \quad {}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\check{y}(t) \bigr) \bigr\vert \bigr). \end{aligned}$$

There exists

$$ \grave{\wp} \in \mathbb{G} \bigl(t, y(t), {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}y(t), {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) \bigr)$$

such that

$$\begin{aligned} \bigl\vert g_{1}(t) - \grave{\wp} \bigr\vert &\leq \kappa (t) \varpi ^{\star}\psi \bigl( \bigl\vert y(t) - \check{y}(t) \bigr\vert + \bigl\vert {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}\check{y}(t) \bigr\vert \\ &\quad{}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\check{y}(t) \bigr) \bigr\vert \bigr). \end{aligned}$$

We give a mapping \(\digamma : I \to \mathcal{P}(A)\) by

$$\begin{aligned} \digamma (t) & = \bigl\{ \grave{\wp} \in A : \bigl\vert g_{1}(t) - \grave{\wp} \bigr\vert \leq \kappa (t) \varpi ^{\star}\psi \bigl( \bigl\vert y(t) - \check{y}(t) \bigr\vert \\ & \quad {}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}y(t) - {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}\check{y}(t) \bigr\vert \\ &\quad {}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl( {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\check{y}(t) \bigr) \bigr\vert \bigr) \bigr\} \end{aligned}$$

for any \(t\in I\). Because \(g_{1}\) and

$$\begin{aligned} \Bbbk & = \kappa (t) \varpi ^{\star}\psi \bigl( \vert y - \check{y} \vert + \bigl\vert {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}\check{y} \bigr\vert \\ & \quad {}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y \bigr) - {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}} \check{y} \bigr) \bigr\vert \bigr) \end{aligned}$$

are measurable, thus

$$ \digamma (\cdot )\cap \mathbb{G} \bigl(\cdot , y(\cdot ), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(\cdot ), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(\cdot ) \bigr) \bigr)$$

is too. Take

$$ g_{2} \in \mathbb{G} \bigl(t, y(t), {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}y(t), {}^{CH } \mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) \bigr),$$

s.t. for all \(t\in I\) we get

$$\begin{aligned} \bigl\vert g_{1}(t) - g_{2}(t) \bigr\vert &\leq \kappa (t) \varpi ^{\star}\psi \bigl( \bigl\vert y(t) - \check{y}(t) \bigr\vert + \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}y(t) - {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}}\check{y}(t) \bigr\vert \\ &\quad {}+ \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}y(t) \bigr) - {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{\iota _{3}}\check{y}(t) \bigr) \bigr\vert \bigr). \end{aligned}$$

Define \(\ell _{2} \in K(y)\) by

$$\begin{aligned} \ell _{2}(t) &= \frac{1}{\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{\iota _{1}+\iota _{2}+ \iota _{3} -1} g_{2}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}- \frac{1}{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1}+\iota _{2}+\iota _{3} -1} g_{2}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{1}(t)}{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2})} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{\iota _{1} + \iota _{2} -1} g_{2}(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{2}(t)}{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1})} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{\iota _{1} -1} g_{2}(r) \frac{\mathrm{dr}}{r} \end{aligned}$$

for any \(t\in I\). Using the same techniques that were employed in the proof of Theorem 3.5, we get that

$$\begin{aligned} \Vert \ell _{1} - \ell _{2} \Vert &= \sup _{t\in I} \bigl\vert \ell _{1}(t) - \ell _{2}(t) \bigr\vert + \sup_{t\in I} \bigl\vert {}^{CH }\mathbb{D}_{1^{+}}^{ \iota _{3}}\ell _{1}(t) - {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{3}} \ell _{2}(t) \bigr\vert \\ &\quad {}+ \sup_{t\in I} \bigl\vert {}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}\ell _{1}(t) \bigr) - {}^{CH }\mathbb{D}_{1^{+}}^{\iota _{2}} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{ \iota _{3}}\ell _{2}(t) \bigr)\bigr\vert \\ &\leq \varpi ^{\star} (\Xi _{1} + \Xi _{2} + \Xi _{3} ) \psi \bigl( \Vert y - \check{y} \Vert \bigr) = \psi \bigl( \Vert y - \check{y} \Vert \bigr). \end{aligned}$$

We get

$$ {H}_{\rho} \bigl( K (y) , K(\check{y}) \bigr) \leq \psi \bigl( \Vert y - \check{y} \Vert \bigr), \quad \forall y, \check{y} \in A.$$

Hypothesis \((\mathcal{H}_{10})\) gives the approximate property for the end points of K. Hence, due to Theorem 3.6, \(\exists y_{*} \in A\) s.t. \(K(y_{*})= \{ y_{*} \}\). As a result, \(y_{*}\) is a solution of the CHF-jerk problem (1). □

4 Example

We give an example for simulating and analyzing the results numerically.

Example 4.1

We model the following CHF-jerk inclusion BVP by assuming the constant values \(\iota _{1}=0.5\), \(\iota _{2} = 0.7\), \(\iota _{3} = 0.99\) as

$$ \textstyle\begin{cases} ( {}^{CH }\mathbb{D}_{1^{+}}^{0.5} ( {}^{CH }\mathbb{D}_{1^{+}}^{0.7} ({}^{CH }\mathbb{D}_{1^{+}}^{0.99}y) ) )(t) \in [ 0, \frac {1}{4} ( t \vert \sin (y (t)) \vert + t \vert {}^{CH } \mathbb{D}_{1^{+}}^{0.99} y(t) \vert \\\quad {}+ 0.5t \vert \tan ^{-1} ( {}^{CH }\mathbb{D}_{1^{+}}^{0.7} ({}^{CH }\mathbb{D}_{1^{+}}^{0.99} y)(t) ) \vert ) + 2\exp{(t)} ], \\y(1) + y(e) = 0, \qquad {}^{CH }\mathbb{D}_{1^{+}}^{0.99} y (2.69) = 0, \qquad {}^{CH }\mathbb{D}_{1^{+}}^{0.7} ( {}^{CH }\mathbb{D}_{1^{+}}^{0.99} y (e) ) = 0, \end{cases} $$
(19)

where \(t \in I :=[1,e]\), and we choose \(\eta =2.69 \in (1,e)\). Now, consider the multi-function \(\mathbb{G}:I \times A^{3} \to \mathcal{P}_{CP}(A)\) defined by

$$\begin{aligned} \mathbb{G} \bigl( t, y_{1}(t), y_{2}(t), y_{3}(t) \bigr) &= \biggl[ 0, \frac{1}{4} \bigl( t \bigl\vert \sin \bigl(y_{1} (t) \bigr) \bigr\vert \\ & \quad {}+ t \bigl\vert y_{2}(t) \bigr\vert + 0.5t \bigl\vert \tan ^{-1} \bigl( y_{3}(t) \bigr) \bigr\vert \bigr) + 2 \exp{(t)} \biggr], \end{aligned}$$

where

$$ A = \bigl\{ y(t): y(t), {}^{CH }\mathbb{D}_{1^{+}}^{0.99}y(t), {}^{CH }\mathbb{D}_{1^{+}}^{0.7} \bigl({}^{CH } \mathbb{D}_{1^{+}}^{0.99}y(t)\bigr) \in C\bigl([1,e], \mathbb{R} \bigr) \bigr\} .$$

Some calculations, by the above data and using (13), give \(F_{1}^{*}=3\),

$$\begin{aligned} F_{2}^{*}& = 3\Gamma (1+\iota _{2}) \Gamma (1+ \iota _{3}) + 3\Gamma (1 + \iota _{2} + \iota _{3}) ( \ln \eta )^{\iota _{2}} \\ & = 3\Gamma (1.7)\Gamma (1.99) + 3\Gamma (2.69) ( \ln 2.69 )^{0.7} \simeq 7.278176, \end{aligned}$$

\(F_{1}^{**}= 2 \Gamma (1.99)= 1.991626\),

$$\begin{aligned} F_{2}^{**} & = 2\Gamma (1 + \iota _{2} + \iota _{3}) \Gamma (1+\iota _{3}) \bigl(1 + ( \ln \eta )^{\iota _{2}} \bigr) \\ & =2\Gamma (2.69) \Gamma (1.99) \bigl(1 + ( \ln 2.69 )^{0.7} \bigr) \simeq 6.0818, \end{aligned}$$

\(F_{1}^{***}=0\),

$$\begin{aligned} F_{2}^{***}& = 2 \Gamma (1+ \iota _{3}) \Gamma (1+ \iota _{2} + \iota _{3}) \Gamma (1+\iota _{2}) \\ & = 2 \Gamma (1.99) \Gamma (2.69) \Gamma (1.7)\simeq 2.773247, \end{aligned}$$

and

$$\begin{aligned}& \begin{aligned} \check{\Lambda}_{1} &= \frac{ 3 }{2\Gamma (\iota _{1}+\iota _{2}+\iota _{3}+1)} + \frac{ F_{1}^{*} (\ln \eta )^{\iota _{1}+\iota _{2}} }{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2}+1)} \\ & \quad {}+ \frac{ F_{2}^{*} }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1}+1)} \\ &= \frac{ 3 }{2\Gamma (3.19)} + \frac{ 3 (\ln 2.69)^{1.2}}{ 2\Gamma (1.99)\Gamma (2.2)} \\ & \quad {}+ \frac{ 7.278176}{2\Gamma (2.69)\Gamma (1.7)\Gamma (1.99)\Gamma (1.5)} \\ & \simeq 4.936355, \end{aligned} \\& \begin{aligned} \check{\Lambda}_{2} &= \frac{1}{ \Gamma (\iota _{1}+\iota _{2}+1)} + \frac{ F_{1}^{**}(\ln \eta )^{\iota _{1}+\iota _{2}} }{2\Gamma (1 + \iota _{3})\Gamma (\iota _{1} + \iota _{2}+1)} \\ & \quad {}+ \frac{ F_{2}^{**} }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1}+1)} \\ & = \frac{1}{ \Gamma (2.2)} + \frac{ 1.991626(\ln 2.69)^{1.2} }{2\Gamma (1.99)\Gamma (2.2)} \\ & \quad {}+ \frac{ 6.0818 }{2\Gamma (2.69)\Gamma (1.7)\Gamma (1.99)\Gamma (1.5)} \\ & \simeq 4.278391, \end{aligned} \\& \begin{aligned} \check{\Lambda}_{3} &= \frac{1}{ \Gamma (\iota _{1}+1) } + \frac{ F_{2}^{***} }{2\Gamma (1+\iota _{2} + \iota _{3} )\Gamma (1 + \iota _{2})\Gamma (1 + \iota _{3})\Gamma (\iota _{1}+1)} \\ & = \frac{1}{ \Gamma (1.5) } + \frac{ 2.7732476 }{2\Gamma (2.69)\Gamma (1.7)\Gamma (1.99)\Gamma (1.5)} \\ & \simeq 2.256758. \end{aligned} \end{aligned}$$

For each \(y_{m}, \bar{y}_{m} \in \mathbb{R} \) (\(m=1,2,3\)), we have

$$\begin{aligned} &{H}_{\rho} \bigl( \mathbb{G} \bigl(t, y_{1}(t), y_{2}(t), y_{3}(t) \bigr), \mathbb{G} \bigl(t, \bar{y}_{1}(t), \bar{y}_{2}(t), \bar{y}_{3}(t) \bigr) \bigr) \\ &\quad\leq \frac{t}{4} \bigl( \bigl\vert \sin \bigl(y_{1}(t) \bigr)- \sin \bigl(\bar{y}_{1} (t)\bigr) \bigr\vert + \bigl\vert y_{2}(t) - \bar{y}_{2}(t) \bigr\vert \\ &\quad \quad {}+ \bigl\vert \tan ^{-1}\bigl( y_{3}(t)\bigr) - \tan ^{-1}\bigl(\bar{y}_{3} (t) \bigr) \bigr\vert \bigr) \\ &\quad\leq \frac{t}{4} \bigl( \bigl\vert y_{1}(t) - \bar{y}_{1} (t) \bigr\vert + \bigl\vert y_{2}(t) - \bar{y}_{2} (t) \bigr\vert + \bigl\vert y_{3}(t) - \bar{y}_{3} (t) \bigr\vert \bigr) \\ &\quad= \frac{t}{4} \Biggl( \sum_{i=1}^{3} \bigl\vert y_{i}(t) - \bar{y}_{i} (t) \bigr\vert \Biggr) \\ &\quad= \frac{t}{4} \psi \Biggl( \sum_{i=1}^{3} \bigl\vert y_{i}(t) - \bar{y}_{i} (t) \bigr\vert \Biggr) \\ &\quad= \kappa (t)\psi \Biggl( \sum_{i=1}^{3} \bigl\vert y_{i}(t) - \bar{y}_{i} (t) \bigr\vert \Biggr) \\ &\quad\leq \kappa (t)\varpi ^{\star}\psi \Biggl( \sum _{i=1}^{3} \bigl\vert y_{i}(t) - \bar{y}_{i} (t) \bigr\vert \Biggr). \end{aligned}$$

Hence, from the above, it is found a function \(\kappa \in {C}(I, [0, \infty ))\) as \(\kappa (t)=\frac {t}{2}\) for all \(t\in I = [1,e]\). Then

$$ \Vert \kappa \Vert = \sup_{t \in I} \biggl\vert \frac {t}{2} \biggr\vert = \frac {e}{2} \simeq 1.355.$$

Next, define \(\psi : [0, \infty ) \to [0, \infty )\) by \(\psi (t) = \frac {t}{2}\) for (a.e.) \(t >0\). It is simple to verify that

$$ \liminf_{t \to \infty} \bigl(t - \psi (t)\bigr) >0,$$

and \(\psi (t) < t\) for all \(t >0\). Also, we obtain

$$ \varpi ^{\star}= \frac {1}{\Xi _{1} + \Xi _{2} + \Xi _{3}},\quad \Xi _{m} = \Vert \kappa \Vert \check{\Lambda}_{m}\ (m=1,2,3),$$

in which

$$\begin{aligned} \Xi _{1} & \simeq \Vert \kappa \Vert \check{\Lambda}_{1} \simeq 6.688761, \\ \Xi _{2} & \simeq \Vert \kappa \Vert \check{ \Lambda}_{2} \simeq 5.797221, \\ \Xi _{3} & \simeq \Vert \kappa \Vert \check{ \Lambda}_{3} \simeq 3.057907. \end{aligned}$$

Thus

$$ \varpi ^{\star}=\frac {1}{\Xi _{1} + \Xi _{2} + \Xi _{1}} \simeq \frac {1}{15.543889} \simeq 0.064333 $$

for all \(t\in I\). In the sequel, we regard the multi-function \(K: A \to \mathcal{P}(A)\) by

$$ K(y)= \bigl\{ z\in A: \text{there exists } g\in \mathit{S}_{\mathbb{G}, y} \text{ such that } z(t)= \pi (t), \forall t\in I \bigr\} , $$

for which

$$\begin{aligned} \pi (t)&=\frac{1}{\Gamma (2.19)} \int _{1}^{t} \biggl( \ln \frac{t}{r} \biggr)^{1.19} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}- \frac{1}{2\Gamma (2.19)} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{1.19} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{1}(t)}{2\Gamma (1.99)\Gamma (1.2)} \int _{1}^{\eta} \biggl( \ln \frac{\eta}{r} \biggr)^{0.2} g(r) \frac{\mathrm{dr}}{r} \\ &\quad {}+ \frac{F_{2}(t)}{2\Gamma (2.69)\Gamma (1.7)\Gamma (1.99)\Gamma (0.5)} \int _{1}^{e} \biggl( \ln \frac{e}{r} \biggr)^{-0.5} g(r) \frac{\mathrm{dr}}{r}, \end{aligned}$$
(20)

where

$$ \textstyle\begin{cases} F_{1}(t) = 1- 2 ( \ln t )^{\iota _{3}}= 1- 2 ( \ln t )^{0.99}, \\F_{2}(t) = \Gamma (1+\iota _{2}) \Gamma (1+ \iota _{3}) [ 1- 2 ( \ln t )^{\iota _{2} + \iota _{3}} ] \\\hphantom{F_{2}(t) =} {}- \Gamma (1 + \iota _{2} + \iota _{3}) ( \ln \eta )^{\iota _{2}} [ 1- 2 ( \ln t )^{\iota _{3}} ] \\\hphantom{F_{2}(t)} = \Gamma (1.7) \Gamma (1.99) [ 1- 2 ( \ln t )^{1.69} ] \\\hphantom{F_{2}(t) =} {}- \Gamma (2.69) ( \ln 2.69 )^{0.7} [ 1- 2 ( \ln t )^{0.99} ]. \end{cases} $$
(21)

One can see the results of \(F_{1}(t)\), \(F_{2}(t)\) for \(t \in [1, e]\) in Table 1 and can see a graphical representation of them in Fig. 1. As the multi-function K possesses an approximate end point property, hence by using Theorem 3.7, the supposed CHF-jerk problem (19) admits a solution.

Figure 1
figure 1

Graphical representation of \(F_{1}(t)\) and \(F_{2}(t)\) for \(t \in I\)

Table 1 Numerical results of \(F_{1}(t)\) and \(F_{2}(t)\) for \(t \in I\)

5 Conclusion

In this research work, a generalization of the standard jerk equation in the context of the Caputo–Hadamard differential inclusion (1) was provided, in which we used some inequalities and important properties of multi-valued functions in the framework of the special contractions and admissible mappings. We extracted existence properties of solutions of the mentioned inclusion (1) by applying two different notions of fixed points and end points in functional analysis. This type of the Caputo–Hadamard structure for a jerk problem is a newly-defined FBVP, and we tried to establish our results based on some new non-routine techniques of fixed point and end point theories. With the help of an example, we described our method numerically and graphically. Due to the importance of jerk in the modern physics, it is necessary that we continue our study on the extended models of such physical structures and investigate other qualitative properties of them.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Thabet, S.T.M., Etemad, S., Rezapour, S.: On a new structure of the pantograph inclusion problem in the Caputo conformable setting. Bound. Value Probl. 2020, 171 (2020). https://doi.org/10.1186/s13661-020-01468-4

    Article  MathSciNet  MATH  Google Scholar 

  2. Baleanu, D., Etemad, S., Rezapour, S.: A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound. Value Probl. 2020, 64 (2020). https://doi.org/10.1186/s13661-020-01361-0

    Article  MathSciNet  MATH  Google Scholar 

  3. Samei, M.E., Rezapour, S.: On a fractional q-differential inclusion on a time scale via endpoints and numerical calculations. Adv. Differ. Equ. 2020, 460 (2020). https://doi.org/10.1186/s13662-020-02923-3

    Article  MathSciNet  MATH  Google Scholar 

  4. Rezapour, S., Ahmad, B., Etemad, S.: On the new fractional configurations of integro-differential Langevin boundary value problems. Alex. Eng. J. 60(5), 4865–4873 (2021). https://doi.org/10.1016/j.aej.2021.03.070

    Article  Google Scholar 

  5. Jleli, M., Samet, B., Vetro, C.: Nonexistence results for higher order fractional differential inequalities with nonlinearities involving Caputo fractional derivative. Mathematics 9, 1866 (2021). https://doi.org/10.3390/math9161866

    Article  Google Scholar 

  6. Mohammadi, H., Kumar, S., Rezapour, S., Etemad, S.: A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos Solitons Fractals 144, 110668 (2021). https://doi.org/10.1016/j.chaos.2021.110668

    Article  MathSciNet  MATH  Google Scholar 

  7. Rezapour, S., Imran, A., Hussain, A., Martinez, F., Etemad, S., Kaabar, M.K.A.: Condensing functions and approximate endpoint criterion for the existence analysis of quantum integro-difference FBVPs. Symmetry 13(3), 469 (2021). https://doi.org/10.3390/sym13030469

    Article  Google Scholar 

  8. Matar, M.M., Abbas, M.I., Alzabut, J., Kaabar, M.K.A., Etemad, S., Rezapour, S.: Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivatives. Adv. Differ. Equ. 2021, 68 (2021). https://doi.org/10.1186/s13662-021-03228-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Thabet, S.T.M., Etemad, S., Rezapour, S.: On a coupled Caputo conformable system of pantograph problems. Turk. J. Math. 45(1), 496–519 (2021). https://doi.org/10.3906/mat-2010-70

    Article  MathSciNet  Google Scholar 

  10. Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020). https://doi.org/10.1016/j.chaos.2020.109705

    Article  MathSciNet  MATH  Google Scholar 

  11. Baleanu, D., Etemad, S., Rezapour, S.: On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators. Alex. Eng. J. 59(5), 3019–3027 (2020). https://doi.org/10.1016/j.aej.2020.04.053

    Article  Google Scholar 

  12. Baleanu, D., Mohammadi, H., Rezapour, S.: Analysis of the model of HIV-1 infection of \(CD4^{+}\) T-cell with a new approach of fractional derivative. Adv. Differ. Equ. 2020, 71 (2020). https://doi.org/10.1186/s13662-020-02544-w

    Article  MathSciNet  MATH  Google Scholar 

  13. Baleanu, D., Rezapour, S., Saberpour, Z.: On fractional integro-differential inclusions via the extended fractional Caputo-Fabrizio derivation. Bound. Value Probl. 2019, 79 (2019). https://doi.org/10.1186/s13661-019-1194-0

    Article  MathSciNet  Google Scholar 

  14. Aydogan, S.M., Baleanu, D., Mousalou, A., Rezapour, S.: On high order fractional integro-differential equations including the Caputo-Fabrizio derivative. Bound. Value Probl. 2018, 90 (2018). https://doi.org/10.1186/s13661-018-1008-9

    Article  MathSciNet  MATH  Google Scholar 

  15. Haghi, R.H., Rezapour, S.: Fixed points of multifunctions on regular cone metric spaces. Expo. Math. 28(1), 71–77 (2010). https://doi.org/10.1016/j.exmath.2009.04.001

    Article  MathSciNet  MATH  Google Scholar 

  16. Baleanu, D., Mohammadi, H., Rezapour, S.: On a nonlinear fractional differential equation on partially ordered metric spaces. Adv. Differ. Equ. 2013, 83 (2013). https://doi.org/10.1186/1687-1847-2013-83

    Article  MathSciNet  MATH  Google Scholar 

  17. Ahmad, B., Ntouyas, S.K., Tariboon, J.: A study of mixed Hadamard and Riemann-Liouville fractional integro-differential inclusions via endpoint theory. Appl. Math. Lett. 52, 9–14 (2016). https://doi.org/10.1016/j.aml.2015.08.002

    Article  MathSciNet  MATH  Google Scholar 

  18. Ahmad, B., Ntouyas, S.K.: Existence and uniqueness of solutions for Caputo-Hadamard sequential fractional order neutral functional differential equations. Electron. J. Differ. Equ. 2017, 36, 1–11 (2017)

    Article  MathSciNet  Google Scholar 

  19. Aljoudi, S., Ahmad, B., Alsaedi, A.: Existence and uniqueness results for a coupled system of Caputo-Hadamard fractional differential equations with nonlocal Hadamard type integral boundary conditions. Fractal Fract. 4, 13 (2020). https://doi.org/10.3390/fractalfract4020013

    Article  Google Scholar 

  20. Ledesma, C.T., Rodríguez, J.A., da C. Sousa, J.V.: Differential equations with fractional derivatives with fixed memory length. Rend. Circ. Mat. Palermo (2022). https://doi.org/10.1007/s12215-021-00713-8

    Article  Google Scholar 

  21. Jleli, M., Samet, B., Vetro, C.: On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain. Adv. Nonlinear Anal. 10(1), 1267–1283 (2021). https://doi.org/10.1515/anona-2020-0181

    Article  MathSciNet  MATH  Google Scholar 

  22. Jarad, F., Baleanu, D., Abdeljawad, T.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 142 (2012). https://doi.org/10.1186/1687-1847-2012-142

    Article  MathSciNet  MATH  Google Scholar 

  23. Salem, A., Mshary, N., El-Shahed, M., Alzahrani, F.: Compact and noncompact solutions to generalized Sturm-Liouville and Langevin equation with Caputo-Hadamard fractional derivative. Math. Probl. Eng. 2021, Article ID 9995969, 1–15 (2021). https://doi.org/10.1155/2021/9995969

    Article  Google Scholar 

  24. Nagy, A.M., Ben Makhlouf, A., Alsenafi, A., Alazemi, F.: Combination synchronization of fractional systems involving the Caputo-Hadamard derivative. Mathematics 9(21), 2781 (2021). https://doi.org/10.3390/math9212781

    Article  Google Scholar 

  25. Liu, Y., Zhu, Y., Lu, Z.: On Caputo-Hadamard uncertain fractional differential equations. Chaos Solitons Fractals 146, 10894 (2021). https://doi.org/10.1016/j.chaos.2021.110894

    Article  MathSciNet  Google Scholar 

  26. Barakat, M.A., Soliman, A.H., Hyder, A.: Langevin equations with generalized proportional Hadamard-Caputo fractional derivative. Comput. Intell. Neurosci. 2021, Article ID 6316477 (2021). https://doi.org/10.1155/2021/6316477

    Article  Google Scholar 

  27. Kengne, J., Negou, A.N., Tchiotsop, D.: Antimonotonicity, chaos and multiple attractors in a novel autonomous memristor-based Jerk circuit. Nonlinear Dyn. 88, 2589–2608 (2017). https://doi.org/10.1007/s11071-017-3397-1

    Article  Google Scholar 

  28. Kengne, J., Njitacke, Z.T., Fotsin, H.B.: Dynamical analysis of a simple autonomous Jerk system with multiple attractors. Nonlinear Dyn. 83, 751–765 (2015). https://doi.org/10.1007/s11071-015-2364-y

    Article  MathSciNet  Google Scholar 

  29. Das, A., Suwan, T., Deuri, B.C., Abdeljawad, T.: On solution of generalized proportional fractional integral via a new fixed point theorem. Adv. Differ. Equ. 2021, 427 (2021). https://doi.org/10.1186/s13662-021-03589-1

    Article  MathSciNet  Google Scholar 

  30. Ali, A., Gupta, V., Abdeljawad, T., Shah, K., Jarad, F.: Mathematical analysis of nonlocal implicit impulsive problem under Caputo fractional boundary conditions. Math. Probl. Eng. 2020, 1–16 (2020). https://doi.org/10.1155/2020/7681479

    Article  MathSciNet  MATH  Google Scholar 

  31. Mottaghi, F., Li, C., Abdeljawad, T., Saadati, R., Ghaemi, M.B.: Existence and Kummer stability for a system of nonlinear ϕ-Hilfer fractional differential equations with application. Fractal Fract. 5(4), 200 (2021). https://doi.org/10.3390/fractalfract5040200

    Article  Google Scholar 

  32. Ahmad, B., Alghamdi, N., Alsaedi, A., Ntouyas, S.K.: Existence theory for a system of coupled multi-term fractional differential equations with integral multi-strip coupled boundary conditions. Math. Methods Appl. Sci. 44(3), 2325–2342 (2021). https://doi.org/10.1002/mma.5788

    Article  MathSciNet  MATH  Google Scholar 

  33. Sudsutad, W., Ntouyas, S.K., Thaiprayoon, C.: Nonlocal coupled system for ψ-Hilfer fractional order Langevin equations. AIMS Math. 6(9), 9731–9756 (2021). https://doi.org/10.3934/math.2021566

    Article  MathSciNet  MATH  Google Scholar 

  34. Ntouyas, S.K., Sitho, S., Khoployklang, T., Tariboon, J.: Sequential Riemann-Liouville and Hadamard-Caputo fractional differential equation with iterated fractional integrals conditions. Axioms 10(4), 277 (2021). https://doi.org/10.3390/axioms10040277

    Article  Google Scholar 

  35. Soontharanon, J., Sitthiwirattham, T.: On sequential fractional Caputo \((p,q)\)-integrodifference equations via three-point fractional Riemann-Liouville \((p,q)\)-difference boundary condition. AIMS Math. 7(1), 704–722 (2022). https://doi.org/10.3934/math.2022044

    Article  MathSciNet  MATH  Google Scholar 

  36. Rezapour, S., Samei, M.E.: On the existence of solutions for a multi-singular pointwise defined fractional q-integro-differential equation. Bound. Value Probl. 2020, 38 (2020). https://doi.org/10.1186/s13661-020-01342-3

    Article  MathSciNet  MATH  Google Scholar 

  37. Samei, M.E., Rezapour, S.: On a system of fractional q-differential inclusions via sum of two multi-term functions on a time scale. Bound. Value Probl. 2020, 135 (2020). https://doi.org/10.1186/s13661-020-01433-1

    Article  MathSciNet  MATH  Google Scholar 

  38. Abdeljawad, T., Samei, M.E.: Applying quantum calculus for the existence of solution of q-integro-differential equations with three criteria. Discrete Contin. Dyn. Syst., Ser. S 14(10), 3351–3386 (2021). https://doi.org/10.3934/dcdss.2020440

    Article  MathSciNet  MATH  Google Scholar 

  39. Dumrongpokaphan, T., Patanarapeelert, N., Sitthiwirattham, T.: Nonlocal Neumann boundary value problem for fractional symmetric Hahn integrodifference equations. Symmetry 13(12), 2303 (2021). https://doi.org/10.3390/sym13122303

    Article  Google Scholar 

  40. Aljoudi, S., Ahmad, B., Nieto, J.J., Alsaedi, A.: A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions. Chaos Solitons Fractals 91, 39–46 (2016). https://doi.org/10.1016/j.chaos.2016.05.005

    Article  MathSciNet  MATH  Google Scholar 

  41. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  42. Amini-Harandi, A.: Endpoints of set-valued contractions in metric spaces. Nonlinear Anal., Theory Methods Appl. 72(1), 132–134 (2010). https://doi.org/10.1016/j.na.2009.06.074

    Article  MathSciNet  MATH  Google Scholar 

  43. Mohammadi, B., Rezapour, S., Shahzad, N.: Some results on fixed points of α-ψ-Ciric generalized multifunctions. Fixed Point Theory Appl. 2013, 24 (2013). https://doi.org/10.1186/1687-1812-2013-24

    Article  MathSciNet  MATH  Google Scholar 

  44. Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Anal., Theory Methods Appl. 75(4), 2154–2165 (2018). https://doi.org/10.1016/j.na.2011.10.014

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

J. Alzabut is thankful to Prince Sultan University and OSTİM Technical University for their endless support throughout this work. The first and fourth authors were supported by Azarbaijan Shahid Madani University. The third author was supported by Bu-Ali Sina University.

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SE: Actualization, methodology, formal analysis, validation, investigation, and initial draft. II: Actualization, validation, methodology, formal analysis, investigation, and initial draft. MES: Actualization, methodology, formal analysis, validation, investigation, software, simulation, initial draft, and was a major contributor in writing the manuscript. SR: Actualization, methodology, formal analysis, validation, investigation, initial draft, and supervision of the original draft and editing. JA: Actualization, methodology, formal analysis, validation, investigation, and initial draft. WS: Actualization, methodology, formal analysis, validation, investigation, and initial draft. IG: Actualization, methodology, formal analysis, validation, investigation, and initial draft. All authors read and approved the final manuscript.

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Correspondence to Shahram Rezapour or Jehad Alzabut.

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Appendix:  Supplement

Appendix:  Supplement

Algorithm 1
figure a

MATLAB lines for calculation of all variables in Example 4.1

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Etemad, S., Iqbal, I., Samei, M.E. et al. Some inequalities on multi-functions for applying in the fractional Caputo–Hadamard jerk inclusion system. J Inequal Appl 2022, 84 (2022). https://doi.org/10.1186/s13660-022-02819-8

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