On solutions of a hybrid generalized Caputo-type problem via the noncompactness measure in the generalized version of Darbo’s criterion

Abstract

In this manuscript, we study the existence and uniqueness of solutions for a new neutral hybrid nonlinear differential equation in the context of a fractional generalized operator in the sense of ψ-Caputo. To emphasize the novelty of the manuscript, a pure technique of the noncompactness measures is applied to a hybrid system based on the notion of the modulus of continuity in Darbo’s criterion that covers the existing results of other works published before. The Ulam–Hyers and generalized Ulam–Hyers stabilities are explored for the given neutral nonhybrid nonlinear problem. An application is prepared in the framework of an example to ensure the validity of theorems for different cases.

1 Introduction

Fractional differential equations under different numerical algorithms show their ability and applicability in modeling and analyzing numerous fields of science, such as engineering, material science, chemistry, models based on the bloodstream, image processing, etc. [1, 2]. In the meantime, we have witnessed various studies in recent years, in all of which fractional differential equations have shown their constructive role well. All these results and achievements are due to the definition of fractional-order operators in the context of fractional calculus, which proved their accuracy in describing phenomena and modeling processes in the universe.

In this regard, researchers first studied the properties of these new operators and then analyzed the issues of existence theory in relation to the solutions of these fractional systems on various abstract fractional boundary value problems. The important and decisive method in these studies was a combination of the existing concepts of fixed-point theory and fractional calculus theory, which led to finding useful existence results; for example, [3–13]. With the development of new theorems and techniques related to fixed-point theorems, various models of boundary value problems could be studied and investigated, including the hybrid fractional boundary value problems of the first and second type.

In fact, these equations belong to the class of quadratic perturbations of fractional differential equation. Dhage et al. introduced the first version of hybrid equations of the first (integer) order. These two authors turned to studying inequalities, their extremal solutions, and some results in relation to the comparison notion [14]. Then, various versions of hybrid boundary value problems were studied in which the order of the existing equations was equal to the arbitrary fractional (noninteger) values. In 2020, Salem et al. introduced and analyzed the properties of solutions to the following fractional boundary value problem in the context of the nonlinear hybrid Langevin fractional differential equation:

$$ \textstyle\begin{cases} {{}^{c}\mathcal{D}}^{\alpha _{1}} [{{}^{c}\mathcal{D}}^{\alpha _{2}} [ \frac {\varkappa (z)}{ \mathcal{F}(z, \varkappa (\vartheta (z)))} ]- \lambda \varkappa (z) ]= H (z, \varkappa (\mu (z)), \mathcal{I}^{\alpha _{3}}\varkappa (\mu (z)) ),& \\ \varkappa (0)=0, & \\ {{}^{c}\mathcal{D}}^{\alpha _{2}} [ \frac{\varkappa (z)}{ \mathcal{F}(z, \varkappa (\vartheta (z)))} ]_{z=0} = 0, & \\ \vartheta (1) = \eta \vartheta (a), \quad \eta \in \mathbb{R}, a \in (0,1),& \end{cases} $$

for $z\in [0, 1]$, where ${{}^{c}\mathcal{D}}^{\alpha _{1}}$ and ${{}^{c}\mathcal{D}}^{\alpha _{2}}$ are Caputo derivatives of orders $\alpha _{i} \in (0, 1]$ and $\alpha _{2} \in (1, 2]$, respectively, $0 \neq \lambda \in \mathbb{R}$, $\mathcal{I}^{\alpha _{3}}$ is the Riemann–Liouville fractional integral of order $\alpha _{3} \in (0,1)$, [15]

$$\eta = \frac{\mathcal{F}( 1, \varkappa (\vartheta (1)))}{\mathcal{F}(a, \varkappa (\vartheta (a)))}, \quad \mathcal{F} \in C\bigl([0,1] \times \mathbb{R}, \mathbb{R} \setminus \{0\} \bigr). $$

In the same year, Yokus used the expansion method to produce several new cases with the exact solutions for some nonlinear partial differential equations to obtain the new wave solution of mathematical equations used in physics, engineering, and many applied sciences and they analyzed the nonlinear Klein–Gordon equation models associated with the Schrödinger equation as follows

$$\varkappa _{tt} - \omega ^{2} \varkappa _{xx} + \alpha \varkappa - \beta \varkappa ^{3} + \gamma \varkappa ^{5} =0, $$

here ω, α, β, and γ are arbitrary constants [16]. Also, Jena et al. focused on the unsteady one-dimensional motion of the viscous fluid through a tube for which the equations of motion of a viscous fluid were the Navier–Stokes equations

$$\frac{\partial}{\partial t} \varkappa (x, t) = - \frac{\partial p}{\rho \partial z} + v \biggl( \frac{\partial ^{2}}{\partial x^{2}} \varkappa (x, t) + \frac{1}{x} \frac{\partial}{\partial r} \varkappa (x, t) \biggr), $$

here t, ϰ, ρ, p, and v are the time, velocity, density, pressure, and kinematic viscosity of the fluid, respectively [17]. Etemad et al. in [18], investigated the existence notion to the inclusion type of the hybrid multiterm Caputo integrofractional differential equation

$${{}^{c}\mathcal{D}_{0}}^{\alpha _{1}} \biggl( \frac{ \varkappa (z)}{ \mathcal{F}(z, \varkappa (z), \vartheta _{1}(\varkappa (z)), \dots , \vartheta _{n}(\varkappa (z)) ) } \biggr) \in \mathfrak{G}\bigl(z, \varkappa (z), \vartheta _{1}\bigl(\varkappa (z)\bigr), \dots , \vartheta _{n}\bigl( \varkappa (z)\bigr) \bigr), $$

under integrohybrid conditions in three points

$$\textstyle\begin{cases} \hslash _{1} ( \frac{ \varkappa (z)}{ \mathcal{F}(z, \varkappa (z), \vartheta _{1}(\varkappa (z)), \dots , \vartheta _{n}(\varkappa (z)) ) } )|_{z=0} \\ \quad {} +\hslash _{2} ( \frac{ \varkappa (z)}{ \mathcal{F}(z, \varkappa (z), \vartheta _{1}(\varkappa (z)), \dots , \vartheta _{n}(\varkappa (z)) ) } )|_{z=1}= c_{1}, \\ \hslash _{3} {{}^{c}\mathcal{D}_{0}}^{\beta _{1}} ( \frac{ \varkappa (z)}{ \mathcal{F}(z, \varkappa (z), \vartheta _{1}(\varkappa (z)), \dots , \vartheta _{n}(\varkappa (z)) ) } )|_{z=\eta} \\ \quad {} +\hslash _{4} \int _{0}^{1} {{}^{c}\mathcal{D}_{0}}^{ \beta _{1}} ( \frac{ \varkappa (\xi )}{ \mathcal{F}(\xi , \varkappa (\xi ), \vartheta _{1}(\varkappa (\xi )), \dots , \vartheta _{n}(\varkappa (\xi )) ) } ) \,\mathrm{d}\xi = c_{2}, \end{cases} $$

where $z \in [0,1]$, $\alpha _{1} \in (1,2]$, $\beta _{1} \in (0,1]$, $\hslash _{i} \in \mathbb{R}^{\geq 0}$ ($i=1,2,3,4$), $\eta \in (0,1)$, $c_{j} \in \mathbb{R}^{\geq 0}$, ($j=1,2$), ${{}^{c}\mathcal{D}_{0}}^{\sigma}$ shows the Caputo derivative of order $\sigma \in \{ \alpha _{1}, \beta _{1}\}$, $\mathcal{F}:[0,1] \times \mathbb{R}^{m_{1}+1} \to \mathbb{R} \setminus \{0\}$ is continuous, and $\mathfrak{G}:[0,1] \times \mathbb{R}^{m_{2}+1} \to \mathcal{P}( \mathbb{R})$, is a multifunction. In 2021, Boutiara et al. implemented an analysis on the generalized Ulam–Hyers and Ulam–Hyers stability and the existence property regarding the system of nonlinear hybrid integrofractional differential equations in ψ-fractional calculus:

$$\textstyle\begin{cases} {{}^{c}\mathcal{D}_{a^{+}}}^{\alpha _{1};\psi} [ \frac{ \varkappa (z) - \sum_{j=1}^{m_{1}} {\mathcal{I}_{a^{+}}}^{\beta _{j};\psi} \mathfrak{G}_{j} (z, \varkappa (z), \vartheta (z))}{ \mathfrak{M}_{1} (z, \varkappa (z), \vartheta (z))} ]=\mathfrak{N}_{1} (z, \varkappa (z), \vartheta (z)), \\ {{}^{c}\mathcal{D}_{a^{+}}}^{\alpha _{2};\psi} [ \frac{ \vartheta (z) - \sum_{j=1}^{m_{2}} {\mathcal{I}_{a^{+}}}^{\gamma _{j};\psi} \tilde{\mathfrak{G}}_{j} (z, \varkappa (z), \vartheta (z))}{ \mathfrak{M}_{2} (z, \varkappa (z), \vartheta (z))} ] = \mathfrak{N}_{2} (z, \varkappa (z), \vartheta (z)), \end{cases} $$

for $z \in [a, b]$, with the initial conditions $\varkappa (a)=\vartheta (a)=0$, where ${{}^{c}\mathcal{D}_{a^{+}}}^{\sigma ;\psi}$ is the ψ-Caputo derivative of order $\sigma \in \{\alpha _{1}, \alpha _{2} \} \subseteq (0,1)$, ${\mathcal{I}_{a^{+}}}^{\sigma ;\psi}$ is the ψ-fractional Riemann–Liouville integral for $0<\sigma \in \{\beta _{j}, \gamma _{j'} \}$, the nonlinear functions $\mathfrak{M}_{1}, \mathfrak{M}_{2}: [0,1]\times \mathbb{R}^{2} \to \mathbb{R}\setminus \{0\}$ along with

$$\mathfrak{G}_{j}, \tilde{\mathfrak{G}}_{j'}, \mathfrak{N}_{1}, \mathfrak{N}_{2}: [0,1]\times \mathbb{R}^{2} \to \mathbb{R}, $$

are continuous [19]. The authors in [20] investigated the following fractional neutral functional differential equations with infinite delay

$$\mathcal{D}^{\alpha} \bigl[ \varkappa (z) - \mathcal{F}_{1}(z, \varkappa _{z}) \bigr] = \mathcal{F}_{2}(z, \varkappa _{z}), \quad \forall z \in [0, b], $$

with $\varkappa (z) = \varphi (z)$, for $z \in (-\infty , 0]$, where $0 < \alpha < 1$, $\mathcal{D}^{\alpha}$ is the standard Riemann–Liouville fractional derivative, $\varphi (0)=0$, $\mathcal{F}_{i} [0, b] \times B \to \mathbb{R}$, $(i=1,2)$, where B is called a phase space, are given functions satisfying some assumptions with $\mathcal{F}_{1}(0, \varphi ) =0$. Ali et al. discussed the existence theorems of boundary value problems for Hilfer-type fractional differential equations with respect to ψ and boundary conditions

$$\textstyle\begin{cases} \mathcal{D}_{a^{+}}^{\alpha _{1},\alpha _{2}, \psi} \varkappa (z) = \mathcal{F}(z, \varkappa (\vartheta (z))),\quad z\in (a, b], \\c_{1} \mathcal{I}_{a^{+}}^{1-\alpha , \psi} \varkappa (z) |_{z=a} + c_{2} \vartheta (z) |_{z=b}= d,& \end{cases} $$

where $0 < \alpha _{1} <1$ and $0 \leq \alpha _{2} \leq 1$, $\alpha = \alpha _{1} + \alpha _{2}(1- \alpha _{1})$, $c_{1}, c_{2},d \in \mathbb{R}$, $\mathcal{D}_{a^{+}}^{\alpha _{1},\alpha _{2}, \psi}$ and $\mathcal{I}_{a^{+}}^{1-\alpha , \psi}$ are the Hilfer fractional derivative and Riemann–Liouville fractional derivative with respect to ψ, respectively and $\mathcal{F} \in C([a, b]\times \mathbb{R})$ [21]. Also, they considered the following ψ-Caputo-type fractional integrodifferential equation

$${}^{C}\mathcal{D}_{a^{+}}^{\alpha , \psi} \varkappa (z) + c_{1} {}^{C} \mathcal{D}_{a^{+}}^{\alpha _{1}, \psi} \varkappa (z) + c_{2} \mathcal{I}_{a^{+}}^{\alpha _{2}, \psi} \varkappa (z)= \int _{a}^{z} \mathcal{F}\bigl(\eta , \varkappa ^{\prime}(\eta ) \bigr) {\mathrm {d}}\eta , $$

with $\varkappa (a) =0$, where $0 < \alpha _{1}< \alpha < 1$, $\alpha _{2} >0$, $c_{1}, c_{2} \in \mathbb{C}$, ${}^{C}\mathcal{D}_{a^{+}}^{\sigma , \psi}$ denotes the generalized Caputo fractional derivative of order $\sigma \in \{\alpha , \alpha _{1}\}$, $\mathcal{I}_{a^{+}}^{\alpha _{2}, \psi}$ means the generalized Riemann–Liouville fractional integral of order $\alpha _{2}$, $\mathcal{F} \in C([a, b]\times \mathbb{R})$ and $\varkappa \in AC_{0}[a, b]$ such that $\mathcal{I}_{a^{+}}^{\alpha _{2}, \psi}$ and ${}^{C}\mathcal{D}_{a^{+}}^{\alpha , \psi}$ exist and are both continuous in $[a, b]$ [22].

The technique of noncompactness measure is a theoretical method for the solvability of various types of fractional boundary value problems and integral equations; see [23, 24]. Furthermore, this technique has been frequently utilized in the existing branches of nonlinear analysis. The first noncompactness measure was defined by Kuratowski [25]. With regard to this notion, Darbo presented a fixed-point criterion with respect to it and further, some generalizations of Darbo’s criterion were introduced by other mathematicians [26, 27].

In recent years, authors have conducted some researches via a combination of two of the above notions to establish the existence–uniqueness of solutions for given fractional boundary value problems in Banach spaces and also for q-fractional fractional differential equations. For example, Benchohra et al. [28] used this criterion for a special category of differential equations and Samei [13] implemented this technique for a singular q-fractional boundary value problem via numerical results.

In this paper, the properties of the existence, uniqueness, Ulam–Hyers stability, and generalized Ulam–Hyers stability of solutions are studied for fractional neutral hybrid nonlinear differential equations with a ψ-Caputo derivative of the form

$$ \textstyle\begin{cases} { {}^{c}\mathcal{D}}_{a^{+}}^{\rho ;\psi } [ \frac{\varkappa (z)}{ \mathfrak{F}(z, \varkappa (\vartheta (z)))} ] =\mathcal{F}(z, \varkappa (\mu (z))),\quad z\in \mathrm {J}:=[a, b], \\ \alpha [ \frac{\varkappa (z)}{\mathfrak{F}(z, \varkappa ( \vartheta (z)))} ]_{z=a} +\beta [ \frac{ \varkappa (z)}{ \mathfrak{F}(z, \varkappa (\vartheta (z)))} ]_{z=b} \\\quad = \lambda \mathcal{I}^{ \sigma ;\psi}_{a^{+}} [ \frac{\varkappa (z)}{\mathfrak{F}(z, \varkappa (\vartheta (z)))} ]_{z=\eta}+\delta , \end{cases} $$

(1)

where μ and ϑ are two mappings from J into itself, $\mathfrak{F}\in C(\mathrm {J}\times \mathbb{R},\mathbb{R}\setminus \{0 \})$ and $\mathcal{F}\in C(\mathrm {J}\times \mathbb{R},\mathbb{R})$. Also, $\eta \in (a, b)$ and α, β, λ, δ are real constants, ${ {}^{c}\mathcal{D}}_{a^{+}}^{\rho ;\psi }$ is the ψ-Caputo derivative of order $\rho \in (0, 1]$, and $\mathcal{I}_{a^{+}}^{\sigma ;\psi }$ is the ψ-Riemann–Liouville fractional integral of order $\sigma >0$.

The main novelty of this work is that we establish our results with the help of the technique of noncompactness measure based on the modulus of continuity for a generalized hybrid neutral fractional boundary value problem furnished with generalized operators that leads to some general theoretical findings involving the following special cases:

• Our results for the neutral hybrid nonlinear fractional boundary value problem (1) are valid for the following cases:

  • Caputo-type fractional boundary value problem: $\psi (z)=z$.

  • Caputo–Hadamard-type fractional boundary value problem: $\psi (z)=\log z$.

• The obtained results for the neutral hybrid nonlinear fractional boundary value problem (1) includes the results of Boutiara et al. [29], and Baitiche et al. [30]; in particular:

  • for $\psi (z)=\log z$ and $\mathfrak{F}(z, \varkappa (\vartheta (z)))=1$, the results obtained in the current study incorporate that of Boutiara et al. [29] for Caputo–Hadamard-type fractional boundary value problems;

  • for $\psi (z)=z$ and $\lambda =0$, the results obtained in the current study incorporate that of Baitiche et al. [30] for hybrid Caputo-type fractional boundary value problem.

We here present the organization of the paper briefly. Section 2 collects the basic definitions and preliminary lemmas that we will require. The corresponding integral equation of the given neutral hybrid nonlinear fractional boundary value problem (1) is derived in Section 3, and then we prove the desired result by means of the noncompactness measure. Two types of stability notion are investigated for the case of the mentioned neutral hybrid nonlinear fractional boundary value problem (1) in Section 4. An example is given in Section 5. The conclusion is indicated in Section 6.

2 Preliminary notions

Let $\psi \in \mathcal{C}^{1} = C^{1}(J,\mathbb{R})$ be increasing and differentiable with $\psi ^{\prime }(z)\neq 0$, for all $z \in \mathrm{J.}$ Now, we follow by defining ψ-operators first.

Definition 2.1

([1])

The ρth-ψ–Riemann–Liouville fractional integral for $\varkappa \colon \mathrm{J} \rightarrow \mathbb{R}$ is formulated by

$$ \mathcal{I}_{a^{+}}^{\rho ;\psi }\varkappa (z)= \frac{1}{\Gamma (\rho )}\int _{a}^{z}\psi ^{\prime }(s) \bigl(\psi (z)-\psi (s)\bigr)^{\rho -1} \varkappa (s) \,\mathrm{d}s, \quad \rho >0, $$

(2)

if it is finite valued.

Definition 2.2

([1])

Let $n-1<\rho <n$, $\varkappa : J\rightarrow \mathbb{R}$ be integrable, and $\psi \in C^{n}(\mathrm{J},\mathbb{R})$. The ρth-ψ–Riemann–Liouville fractional derivative of ϰ is given by

$$\mathcal{D}_{a^{+}}^{\rho ;\psi }\varkappa (z) = \biggl( \frac{D_{z}}{\psi ^{\prime }(z)} \biggr) ^{n}\mathcal{I}_{a^{+}}^{n- \rho ;\psi } \varkappa (z), $$

where $n=[\rho ]+1$ and $D_{z} =\frac{\mathrm{d}}{\mathrm{d}z}$.

Definition 2.3

([31])

For $n-1<\rho <n$ and $\varkappa ,\psi \in C^{n}(\mathrm{J}, \mathbb{R})$, the ρth-ψ–Caputo derivative of ϰ is formulated as

$${}^{c}\mathcal{D}_{a^{+}}^{\rho ;\psi }\varkappa (z) = \mathcal{I}_{a^{+}}^{ n- \rho ;\psi }\varkappa _{\psi }^{[n]}(z), $$

where $n=[\rho ]+1$, and for $n=\rho \in \mathbb{N}$,

$$\varkappa _{\psi }^{[n]}(z)= \biggl( \frac{D_{z}}{\psi ^{\prime }(z)} \biggr)^{n}\varkappa (z). $$

By the above notation, we have

$$ {}^{c}\mathcal{D}_{a^{+}}^{\rho ;\psi }\varkappa (z)= \textstyle\begin{cases} \int _{a}^{z} \frac{\psi ^{\prime }(s)(\psi (z)-\psi (s))^{n-\rho -1} }{\Gamma (n-\rho )}\varkappa _{\psi }^{[n]}(s) \,\mathrm{d}s,& \text{if } \rho \notin \mathbb{N}, \\\varkappa _{\psi }^{[n]}(z), & \text{if } \rho \in \mathbb{N}.\end{cases} $$

(3)

In addition, the following identity is valid for this derivative:

$${}^{c}\mathcal{D}_{a^{+}}^{\rho ;\psi }\varkappa (z )= \mathcal{D}_{a^{+}}^{ \rho ;\psi } \Biggl[ \varkappa (z )-\sum _{\jmath =0}^{n-1} \frac{\varkappa _{\psi }^{[\jmath ]}(a)}{\jmath !}\bigl(\psi (z)-\psi (a) \bigr)^{ \jmath} \Biggr]. $$

Lemma 2.4

([1])

For $\rho ,\sigma >0$, we have

$$\mathcal{I}_{a^{+}}^{\rho ;\psi }\mathcal{I}_{a^{+}}^{\sigma ;\psi } \varkappa (z ) = \mathcal{I}_{a^{+}}^{\rho +\sigma ;\psi }\varkappa (z ), \quad \textit{a.e. } z \in \mathrm{J}, \varkappa \in C(\mathrm{J}, \mathbb{R}). $$

Lemma 2.5

([32])

Let $\rho >0$. Then, for $z \in \mathrm{J}$,

$${}^{c}\mathcal{D}_{a^{+}}^{\rho ;\psi }\mathcal{I}_{a^{+}}^{ \rho ;\psi } \varkappa (z)=\varkappa (z ), \quad \textit{if } \varkappa \in C(\mathrm{J}, \mathbb{R}) $$

and

$$\mathcal{I}_{a^{+}}^{\rho ;\psi } {}^{c}\mathcal{D}_{a^{+}}^{ \rho ;\psi } \varkappa (z)=\varkappa (z )-\sum_{\jmath =0}^{n-1} \frac{\varkappa _{\psi}^{[\jmath ]}(a)}{\jmath !} \bigl[ \psi (z)- \psi (a) \bigr]^{\jmath}, $$

if $\varkappa \in C^{n-1}(\mathrm{J},\mathbb{R})$.

Lemma 2.6

([1, 31])

For $z >a$, $\rho \geq 0$, $\sigma >0$, and $\mathcal{F} (z)=\psi (z )-\psi (a)$,

• 1 $\mathcal{I}_{a^{+}}^{\rho ;\psi }(\mathcal{F}(z ))^{\sigma -1}= \frac {\Gamma (\sigma )}{\Gamma (\sigma +\rho )}(\mathcal{F} (z ))^{\sigma + \rho -1}$;

• 2 ${}^{c}\mathcal{D}_{a^{+}}^{\rho ;\psi }(\mathcal{F} (z ))^{ \sigma -1}=\frac {\Gamma (\sigma )}{\Gamma (\sigma -\rho )}( \mathcal{F} (z ))^{\sigma -\rho -1}$;

• 3 ${}^{c}\mathcal{D}_{a^{+}}^{\rho ;\psi }(\mathcal{F} (z ))^{ \jmath}=0$, $\forall \jmath = 0,\dots ,n-1$.

A closed ball centered at ϰ with radius θ is specified by $\mathfrak{P}(\varkappa , \theta )$ in the Banach space $\mathbb{E}$; for $\varkappa =0$, we use the notation $\mathfrak{P}_{\theta}$ instead of $\mathfrak{P}(0, \theta )$. For $\mathbb{X} \neq \emptyset$ and $\mathbb{X} \subseteq \mathbb{E}$, the notations $\bar{\mathbb{X}}$ and $\mathrm{Conv} \mathbb{X}$ stand for the closure and a convex closure of $\mathbb{X}$. Throughout the manuscript, the category of the nonempty bounded subsets of $\mathbb{E}$ is represented by $\mathfrak{M}_{\mathbb{E}}$, and the subcategory of all relatively compact subsets of $\mathfrak{M}_{\mathbb{E}}$ is represented by $\mathfrak{N}_{\mathbb{E}}$.

Definition 2.7

([33])

A mapping $\chi : \mathfrak{M}_{\mathbb{E}} \rightarrow [0, \infty )$ is termed a noncompactness measure in $\mathbb{E}$ if:

• 1 $\operatorname{ker} \chi = \{\mathbb{X} \in \mathfrak{M}_{ \mathbb{E}}: \chi (\mathbb{X})=0 \}$ is nonempty and $\operatorname{ker} \chi \in \mathfrak{N}_{\mathbb{E}}$;

• 2 $\mathbb{A} \subset \mathbb{X} \Rightarrow \chi (\mathbb{A}) \leq \chi (\mathbb{X})$;

• 3 $\chi (\mathbb{A})=\chi (\bar{\mathbb{A}})=\chi (\operatorname{Conv} \mathbb{A})$;

• 4 $\chi (\lambda _{1} \mathbb{A}+\lambda _{2} \mathbb{X} ) \leq \lambda _{1} \chi (\mathbb{A})+\lambda _{2} \chi (\mathbb{X})$, $\lambda _{1}+\lambda _{2}=1$;

• 5 If $(\mathbb{Y}_{n} )$ is a decreasing sequence of closed sets in $\mathfrak{M}_{\mathbb{E}}$ and

  $$\lim_{n \rightarrow \infty} \chi ( \mathbb{Y}_{n} )=0, $$

  then $\bigcap_{n=1}^{\infty} \mathbb{Y}_{n} \neq \emptyset$.

Definition 2.8

([33])

Let $\mathbb{Y}$ be a bounded nonempty set in the Banach space $\mathfrak{C}(\mathrm{J})$, where $\mathfrak{C}(\mathrm{J})$ consists of all continuous maps on J with real values, and it is a Banach algebra with usual multiplication between real numbers. The modulus of continuity of the function $m \in \mathcal{M}\subset \mathbb{Y}$, denoted by $\varrho (\mathcal{M}, \epsilon )$, is formulated by the following if $\forall \epsilon >0$,

$$\varrho (m, \epsilon )=\sup \bigl\{ \bigl\vert m(z) - m(\varsigma ) \bigr\vert : z, \varsigma \in \mathrm{J}, \vert z-\varsigma \vert \leq \epsilon \bigr\} . $$

Also,

$$\varrho (\mathcal{M}, \epsilon ) = \sup \bigl\{ \varrho (m, \epsilon ): m \in \mathcal{M} \bigr\} $$

and

$$\varrho _{0}(\mathbb{Y})=\lim_{\epsilon \rightarrow 0} \varrho ( \mathbb{Y}, \epsilon ). $$

Definition 2.9

([34])

Let $\mathfrak{C}(\mathrm{J})$ be a Banach algebra. A noncompactness measure χ in $\mathfrak{C}(\mathrm{J})$ fulfills the condition $(m)$ if

$$ \chi (\mathcal{M} \mathcal{N}) \leq \Vert \mathcal{M} \Vert \chi ( \mathcal{N}) + \Vert \mathcal{N} \Vert \chi (\mathcal{M}),\quad \forall \mathcal{M}, \mathcal{N} \in \mathfrak{M}_{\mathfrak{C}(\mathrm{J})}. $$

Lemma 2.10

([30])

The condition $(\mathrm{m})$ is valid for the noncompactness measure $\varrho _{0}$ on $\mathfrak{C}(\mathrm{J})$; in other words for each $\mathcal{M},\mathcal{N} \in \mathfrak{M}_{\mathfrak{C}(\mathrm{J})}$,

$$\varrho _{0}(\mathcal{M}\mathcal{N}) \leq \Vert \mathcal{M} \Vert \varrho _{0}(\mathcal{N}) + \Vert \mathcal{N} \Vert \varrho _{0}( \mathcal{M}). $$

Consider the category $\mathcal{S}$ of all mappings $\phi :(0, \infty ) \rightarrow (1, \infty )$ such that

$$\forall \{ t_{n} \} \subset (0, \infty ), \quad \lim _{n \rightarrow \infty} \phi (t_{n} )=1 \quad \Longleftrightarrow \quad \lim _{n \rightarrow \infty} t_{n}=0. $$

We recall Darbo’s fixed-point theorem, which one can utilize to establish the existence of a fixed point based on the noncompactness measure.

Theorem 2.11

([26, 27])

Regard the Banach space $\mathbb{E}$ involving a bounded, convex, and closed nonempty subset Σ. Let $\mathcal{K}: \Sigma \rightarrow \Sigma$ be continuous and $\exists \theta \in [0,1)$ with χ as a noncompactness measure in $\mathbb{E}$ fulfilling

$$\chi (\mathcal{K} \mathbb{A}) \leq \theta \chi (\mathbb{A}), \quad \emptyset \neq \mathbb{A} \subseteq \Sigma . $$

Then, $\mathcal{K}$ possesses a fixed point in Σ.

The extension of the above theorem is applied here for further results.

Theorem 2.12

([27])

Regard the Banach space $\mathbb{E}$ involving a bounded, convex, and closed nonempty subset Σ. Let $\mathcal{K}: \Sigma \rightarrow \Sigma$ be continuous and $\exists \phi \in \mathcal{S}$ and $0\leq \theta <1$ such that $\forall \emptyset \neq \mathbb{Y} \subseteq \Sigma$ with $\chi (\mathbf{\mathcal{K}} \mathbb{Y})>0$,

$$\phi \bigl(\chi (\mathbf{\mathcal{K}} \mathbb{Y})\bigr) \leq \bigl(\phi \bigl( \chi ( \mathbb{Y})\bigr)\bigr)^{\theta}. $$

Then, $\mathcal{K}$ possesses a fixed point in Σ.

3 Exitsence results

Before establishing the desired theorems, we start by the next lemma that gives an equivalent integral equation.

Lemma 3.1

For given $\mathfrak{K} \in \mathfrak {C}(\mathrm{J}, \mathbb{R})$ and $\rho \in (0, 1]$, the hybrid fractional differential equation,

$$ \textstyle\begin{cases} { {}^{c}\mathcal{D}}_{a^{+}}^{\rho ;\psi } [ \frac {\varkappa (z)}{ \mathfrak{F}(z, \varkappa ( \vartheta (z)))} ] =\mathfrak{K}(z), \quad z\in \mathrm {J}:=[a, b], \\ \alpha [ \frac {\varkappa ( z)}{\mathfrak{F}(z, \varkappa ( \vartheta (z)))} ]_{z=a} +\beta [ \frac {\varkappa (z)}{\mathfrak{F}(z, \varkappa ( \vartheta (z)))} ]_{z=b} \\\quad = \lambda \mathcal{I}_{ a^{+}}^{\sigma ;\psi} [ \frac { \varkappa (z)}{ \mathfrak{F}(z, \varkappa (\vartheta (z)))} ]_{z=\eta}+\delta ,& \end{cases} $$

(4)

has a solution formulated of the form

$$\begin{aligned} \varkappa (z)&=\mathfrak{F}\bigl(z, \varkappa \bigl(\vartheta (z) \bigr)\bigr) \biggl[ \mathcal{I}_{a^{+}}^{\rho ;\psi }\mathfrak{K}(z)+ \frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{\rho +\sigma ;\psi } \mathfrak{K}(\eta )-\beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \mathfrak{K}(b)+\delta \bigr\} \biggr] \\ &= \mathfrak{F}\bigl(z, \varkappa \bigl(\vartheta (z)\bigr)\bigr) \biggl[ \int _{a}^{b} G_{ \psi }(z,s) \mathfrak{K}(s) \,\mathrm{d}s + \frac{\delta}{\Lambda} \biggr], \end{aligned}$$

(5)

where $z\in [a, b]$, and

$$\begin{aligned} \Lambda = \biggl[\beta + \alpha -\lambda \frac{(\psi (\eta ) - \psi (a))^{\sigma}}{ \Gamma (\sigma +1)} \biggr] \neq 0. \end{aligned}$$

(6)

Proof

We apply the conclusion of Lemma 2.5, and operating the ρth-ψ-Riemann–Liouville integral on both sides of (4), we obtain

$$\frac{\varkappa (z)}{\mathfrak{F}(z,\varkappa (\vartheta (z)))}= \mathcal{I}_{a^{+}}^{\rho ;\psi} \mathfrak{K}(z)+c_{0}. $$

Consequently,

$$ \varkappa (z)=\mathfrak{F}\bigl(z,\varkappa \bigl(\vartheta (z) \bigr)\bigr) \bigl[ \mathcal{I}_{a^{+}}^{\rho ;\psi}\mathfrak{K}(z)+c_{0} \bigr],\quad c_{0} \in \mathbb{R}. $$

(7)

Substituting the values of the given boundary conditions (4) into (7), we have

$$\alpha c_{0}+\beta \bigl(\mathcal{I}_{a^{+}}^{\rho ;\psi} \mathfrak{K}(b)+c_{0} \bigr)=\lambda \mathcal{I}_{a^{+}}^{\rho + \sigma ;\psi} \mathfrak{K}\bigl(\eta ,\varkappa \bigl(\mu (\eta )\bigr)\bigr)+\lambda \frac{(\psi (\eta )-\psi (a))^{\sigma}c_{0}}{\Gamma (\sigma +1)}+ \delta . $$

Therefore, we have

$$c_{0} \biggl(\beta + \alpha -\lambda \frac{(\psi (\eta )-\psi (a))^{\sigma}}{\Gamma (\sigma +1)} \biggr)= \lambda \mathcal{I}_{a^{+}}^{\rho +\sigma ;\psi}\mathfrak{K}(\eta )- \beta \mathcal{I}_{a^{+}}^{\rho ;\psi}\mathfrak{K}(b)+\delta . $$

Then,

$$c_{0}=\frac{1}{\Lambda} \bigl[\lambda \mathcal{I}_{a^{+}}^{\rho + \sigma ;\psi} \mathfrak{K}(\eta )-\beta \mathcal{I}_{a^{+}}^{\rho ; \psi} \mathfrak{K}(b)+\delta \bigr], $$

where,

$$\Lambda = \biggl[\beta + \alpha -\lambda \frac{(\psi (\eta )-\psi (a))^{\sigma}}{\Gamma (\sigma +1)} \biggr] \neq 0. $$

The obtained value of $c_{0}$ is substituted into (7), and equation (5) is derived as

$$\varkappa (z)=\mathfrak{F}\bigl(z, \varkappa \bigl(\vartheta (z)\bigr)\bigr) \biggl[ \mathcal{I}_{a^{+}}^{\rho ;\psi }\mathfrak{K}(z)+ \frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{\rho +\sigma ;\psi } \mathfrak{K}(\eta )-\beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \mathfrak{K}(b)+\delta \bigr\} \biggr]. $$

 □

Now, regarding the existence of a solution to the neutral hybrid nonlinear fractional boundary value problem (1), we collect some hypotheses in the next remark:

Remark 3.1

Assume that:

$(C_{1})$:

The functions $\mu ,\vartheta :\mathrm{J}\rightarrow \mathrm{J}$ are continuous.

$(C_{2})$:

$\mathcal{F}\in C(\mathrm {J}\times \mathbb{R}, \mathbb{R})$, $\mathfrak{F}\in C(J\times \mathbb{R},\mathbb{R}\setminus \{0\})$.

$(C_{3})$:

$\exists \ell \in (0,1)$ s.t.

$$ \bigl\vert \mathfrak{F}( z,\varkappa _{1})-\mathfrak{F}(z,\varkappa _{2}) \bigr\vert \leq \bigl( \vert \varkappa _{1}- \varkappa _{2} \vert +1\bigr)^{\ell}-1,\quad z\in \mathrm{J}, \varkappa _{1},\varkappa _{2}\in \mathbb{R}. $$

$(C_{4})$:

There exists a continuous nondecreasing $\phi : \mathbb{R}_{+}\rightarrow (0,+\infty )$ with

$$\bigl\vert \mathcal{F}(z,\varkappa ) \bigr\vert \leq \phi \bigl( \vert \varkappa \vert \bigr), \quad z\in \mathrm {J}, \varkappa \in \mathbb{R}_{+}. $$

$(C_{5})$:

$\exists \theta > 0$ s.t.

$$\begin{aligned} \theta \geq \Delta \bigl[(\theta +1)^{\ell}-1+N \bigr], \end{aligned}$$

(8)

where $\Delta \leq 1$, by assuming

$$\begin{aligned} \Delta & = \biggl\{ \frac{\phi (\theta )+\frac{\beta}{\Lambda}\phi (\theta )}{\Gamma (\rho +1)}\bigl( \psi (b)-\psi (a) \bigr)^{\rho} \\ & \quad {}+ \frac{\lambda \phi (\theta )}{\Lambda \Gamma (\rho +\sigma +1)}\bigl(\psi ( \eta )-\psi (a) \bigr)^{\rho +\sigma}+\frac{\delta}{\Lambda} \biggr\} \end{aligned}$$

(9)

and $N = \sup \{ |\mathfrak{F}(z,0)|:z\in \mathrm{J} \}$.

Theorem 3.2

Assume that the hypotheses $(C_{1})$–$(C_{5})$ are satisfied. Then, there exists a solution to the neutral hybrid nonlinear fractional boundary value problem (1) in the Banach algebra $\mathfrak{C}(\mathrm{J})$.

Proof

In order to use Theorem 2.12 to prove our main result, we define a subset $\mathfrak{P}_{\theta}$ of $\mathfrak {C}(\mathrm{J})$ by

$$ \mathfrak{P}_{\theta}= \bigl\{ \varkappa \in \mathfrak {C}( \mathrm{J}): \Vert \varkappa \Vert _{\mathfrak {C}} \leq \theta \bigr\} . $$

(10)

Note that $\mathfrak{P}_{\theta}$ is closed in $\mathfrak {C}(\mathrm{J})$ and also it admits the convexity and boundedness properties. We formulate the operator $\mathcal{H}$ on $\mathfrak{C}(\mathrm{J})$ as

$$\begin{aligned} \mathcal{H} \varkappa (z) & = \mathfrak{F}\bigl(z, \varkappa \bigl(\vartheta (z) \bigr) \bigr) \biggl[ \mathcal{I}_{a^{+}}^{\rho ;\psi }\mathcal{F}\bigl(z, \varkappa \bigl( \mu (z)\bigr)\bigr) \\ & \quad {}+ \frac{1}{ \Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{ \rho +\sigma ;\psi } \mathcal{F}\bigl(\eta , \varkappa \bigl(\mu (\eta )\bigr)\bigr)- \beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \mathcal{F}\bigl(b, \varkappa \bigl( \mu (b)\bigr)\bigr) +\delta \bigr\} \biggr], \end{aligned}$$

for $z \in \mathrm{J}$. In the light of Lemma 3.1, a fixed point of $\mathcal{H}$ provides us with the desired results. Based on the above operator, define the operators $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ on the Banach algebra $\mathfrak{C}(\mathrm{J})$ by

$$ (\mathcal{H}_{1}\varkappa ) (z)=\mathfrak{F}\bigl(z, \varkappa \bigl(\vartheta (z)\bigr)\bigr), \quad z\in \mathrm{J} $$

(11)

and

$$\begin{aligned} (\mathcal{H}_{2}\varkappa ) (z) & = \biggl[ \mathcal{I}_{a^{+}}^{\rho ; \psi } \mathcal{F}\bigl( z, \varkappa \bigl(\mu (z)\bigr)\bigr) + \frac{1}{\Lambda} \\ & \quad {}\times \bigl\{ \lambda \mathcal{I}_{a^{+}}^{\rho +\sigma ; \psi } \mathcal{F}\bigl(\eta , \varkappa \bigl(\mu (\eta )\bigr)\bigr)-\beta \mathcal{I}_{a^{+}}^{ \rho ;\psi } \mathcal{F}\bigl(b,\varkappa \bigl(\mu (b)\bigr)\bigr)+\delta \bigr\} \biggr]. \end{aligned}$$

(12)

Then,

$$\mathcal{H}\varkappa = (\mathcal{H}_{1} \varkappa )\cdot ( \mathcal{H}_{2} \varkappa ), $$

for any $\varkappa \in \mathfrak{C}(\mathrm{J})$. To satisfy the hypotheses of Theorem 2.12, we divide the argument into several steps.

Step 1: $\mathcal{H} (\mathfrak{C}(\mathrm{J})) \subseteq \mathfrak{C}( \mathrm{J})$: We know that the product of two continuous functions is continuous, so it is enough to derive

$$\mathcal{H}_{1}\varkappa ,\mathcal{H}_{2}\varkappa \in \mathfrak{C}( \mathrm{J}), \quad \forall \varkappa \in \mathfrak{C}(\mathrm{J}). $$

From assumptions $(C_{1})$ and $(C_{2})$, this gives that if $\varkappa \in \mathfrak{C}(\mathrm{J})$ then immediately $\mathcal{H}_{1}\varkappa \in \mathfrak{C}(\mathrm{J})$. Further, we investigate this implication that if $\varkappa \in \mathfrak{C}(\mathrm{J})$, then $\mathcal{H}_{2}\varkappa \in \mathfrak{C}(\mathrm{J})$. To see this, fix $\epsilon >0$, and take $\varkappa \in \mathfrak{C}(\mathrm{J})$ and $z_{1} \leq z_{2}\in \mathrm{J}$ along with $z_{2}-z_{1}<\epsilon$. Then, in view of assumption $(C_{4})$, we obtain

$$\begin{aligned} \bigl\vert \mathcal{H}_{2} \varkappa ( z_{2}) - \mathcal{H}_{2} \varkappa (z_{1}) \bigr\vert & \leq \frac{1}{\Gamma (\rho )} \int ^{z_{1}}_{a} \psi '(s) \bigl[ \bigl( \psi (z_{2})-\psi (s)\bigr)^{\rho -1} \\ & \quad {}- \bigl(\psi (z_{1})-\psi (s)\bigr)^{\rho -1} \bigr] \bigl\vert \mathcal{F}\bigl(s, \varkappa \bigl(\mu (s)\bigr)\bigr) \bigr\vert \mathrm{d}s \\ &\quad {}+ \int ^{z_{2}}_{z_{1}} \psi '(s) \bigl[\bigl( \psi (z_{2}) - \psi (s)\bigr)^{ \rho -1} \bigr] \bigl\vert \mathcal{F}\bigl(s,\varkappa \bigl(\mu (s)\bigr)\bigr) \bigr\vert \mathrm{d}s \\ &\leq \frac{\phi ( \Vert \varkappa \Vert )}{ \Gamma (\rho +1)} \bigl[\bigl( \psi (z_{2})- \psi (a) \bigr)^{\rho}-\bigl(\psi (z_{1})-\psi (a)\bigr)^{\rho} \\ & \quad {}+ 2 \bigl(\psi (z_{2})-\psi (z_{1}) \bigr)^{\rho} \bigr]. \end{aligned}$$

From the above inequality, we conclude that $|\mathcal{H}_{2}\varkappa (z_{2})-\mathcal{H}_{2}\varkappa (z_{1})| \rightarrow 0$ when $\epsilon \rightarrow 0$. Then, $\mathcal{H}_{2}\varkappa \in \mathfrak{C}(\mathrm{J})$. These yield the desired implication that if $\varkappa \in \mathfrak{C}(\mathrm{J})$, then $\mathcal{H}\in \mathfrak{C}(\mathrm{J})$.

Step 2: An estimate of $\|\mathcal{H}\varkappa \|$, $\forall \varkappa \in \mathfrak{C}(\mathrm{J})$: Fix $\varkappa \in \mathfrak{C}(\mathrm{J})$. Using our condition for $z\in \mathrm{J}$, we obtain

$$\begin{aligned} \bigl\vert (\mathcal{H}\varkappa ) (z) \bigr\vert &= \biggl\vert \mathfrak{F}\bigl(z, \varkappa \bigl( \vartheta (z)\bigr)\bigr) \biggl[ \mathcal{I}_{a^{+}}^{\rho ;\psi }\mathcal{F}\bigl(z, \varkappa \bigl(\mu (z)\bigr)\bigr) \\ &\quad {}+\frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{\rho + \sigma ;\psi } \mathcal{F}\bigl(\eta ,\varkappa \bigl(\mu (\eta )\bigr)\bigr) -\beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \mathcal{F}\bigl(b,\varkappa \bigl(\mu (b)\bigr)\bigr)+ \delta \bigr\} \biggr] \biggr\vert \\ & \leq \bigl( \bigl\vert \mathfrak{F}\bigl(z, \varkappa \bigl( \vartheta (z)\bigr)\bigr) - \mathfrak{F}(z,0) \bigr\vert + \bigl\vert \mathfrak{F}(z,0) \bigr\vert \bigr) \biggl[ \mathcal{I}_{a^{+}}^{\rho ;\psi } \bigl\vert \mathcal{F}\bigl(z, \varkappa \bigl(\mu (z)\bigr)\bigr) \bigr\vert \\ &\quad {}+\frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{\rho + \sigma ;\psi } \bigl\vert \mathcal{F}\bigl(\eta ,\varkappa \bigl(\mu (\eta )\bigr)\bigr) \bigr\vert +\beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \bigl\vert \mathcal{F} \bigl(b,\varkappa \bigl(\mu (b)\bigr)\bigr) \bigr\vert + \delta \bigr\} \biggr] \\ &\leq \bigl[\bigl( \bigl\vert \varkappa \bigl(\vartheta (z)\bigr) \bigr\vert +1\bigr)^{\ell}-1+N \bigr] \biggl\{ \frac{\phi ( \Vert \varkappa \Vert )}{\Gamma (\rho +1)}\bigl( \psi (z)-\psi (a)\bigr)^{ \rho} \\ &\quad {}+\frac{1}{ \Lambda} \biggl\{ \frac{\lambda \phi ( \Vert \varkappa \Vert )}{\Gamma (\rho +\sigma +1)}\bigl(\psi ( \eta )-\psi (a)\bigr)^{\rho +\sigma} \\ & \quad {}+ \frac{ \beta \phi ( \Vert \varkappa \Vert )}{ \Gamma (\rho +1)}\bigl( \psi (b) - \psi (a) \bigr)^{\rho}+\delta \biggr\} \biggr\} \\ &\leq \bigl[ \bigl( \Vert \varkappa \Vert +1\bigr)^{\ell}-1+N \bigr] \biggl\{ \frac{ \phi ( \Vert \varkappa \Vert )+\frac{\beta}{ \Lambda}\phi ( \Vert \varkappa \Vert )}{ \Gamma (\rho +1)} \bigl( \psi (b)-\psi (a)\bigr)^{\rho} \\ & \quad {}+ \frac{\lambda \phi ( \Vert \varkappa \Vert )}{\Lambda \Gamma (\rho +\sigma +1)}\bigl( \psi (\eta )-\psi (a) \bigr)^{\rho +\sigma}+\frac{\delta}{\Lambda} \biggr\} . \end{aligned}$$

Therefore, by Assumption $(C_{5})$,

$$\Vert \mathcal{H}\varkappa \Vert \leq \Delta \bigl[(\theta +1)^{\ell}-1+N \bigr] \leq \theta , $$

where Δ is given by (9). We deduce that $\mathcal{H}$ maps $\mathfrak{P}_{\theta}$ into itself. Moreover, let us observe, from the last estimate, that

$$ \textstyle\begin{cases} \Vert \mathcal{H}_{1}\mathfrak{P}_{\theta} \Vert \leq [(\theta +1)^{ \ell}-1+N ], \\ \Vert \mathcal{H}_{2}\mathfrak{P}_{\theta} \Vert \leq \{ \frac {\phi (\theta )+\frac{\beta}{\Lambda}\phi (\theta )}{\Gamma (\rho +1)}( \psi (b)-\psi (a))^{\rho} \\\hphantom{ \Vert \mathcal{H}_{2}\mathfrak{P}_{\theta} \Vert \leq}{} + \frac {\lambda \phi (\theta )}{ \Lambda \Gamma (\sigma + \rho +1)}( \psi (\eta )-\psi (a))^{\rho +\sigma}+ \frac {\delta}{\Lambda} \}. \end{cases} $$

(13)

Step 3: Both $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are continuous on $\mathfrak{P}_{\theta}$: First, we check this regarding the operator $\mathcal{H}_{1}$. Fix $\epsilon > 0$ and take $\varkappa ,w\in \mathfrak{P}_{\theta}$ arbitrarily so that $\|\varkappa -w\|\leq \epsilon$. Then, $\forall z\in \mathrm{J}$, and we have

$$\begin{aligned} \bigl\vert (\mathcal{H}_{1}\varkappa ) (z)-(\mathcal{H}_{1}w) (z) \bigr\vert &= \bigl\vert \mathfrak{F}\bigl(z, \varkappa \bigl(\vartheta (z)\bigr)\bigr)-\mathfrak{F}\bigl(z,w\bigl(\vartheta (z)\bigr)\bigr) \bigr\vert \\ &\leq \bigl( \bigl\vert \varkappa \bigl(\vartheta (z)\bigr)-w\bigl( \vartheta (z)\bigr) \bigr\vert +1 \bigr)^{ \ell}-1 \\ &\leq \bigl( \Vert \varkappa -w \Vert +1 \bigr)^{\ell}-1 \\ &\leq (\epsilon +1)^{\ell}-1 \end{aligned}$$

and, because of $(\epsilon +1)^{\ell}-1\rightarrow 0$ by assuming $\epsilon \rightarrow 0$, $\mathcal{H}_{1}$ is continuous on $\mathfrak{P}_{\theta}$. Next, to prove the same result in relation to $\mathcal{H}_{2}$, consider the sequence $\{\varkappa _{n}\}\subset \mathfrak{P}_{\theta}$ and $\varkappa \in \mathfrak{P}_{\theta}$ such that $\|\varkappa _{n}-\varkappa \|\rightarrow 0$ when $n\rightarrow \infty$. We show that

$$\Vert \mathcal{H}_{2} \varkappa _{n}- \mathcal{H}_{2}\varkappa \Vert \rightarrow 0,\quad n\rightarrow \infty . $$

Since $\mathcal{F}(z,\varkappa )$ has uniform continuity on the compact interval $\mathrm{J}\times [-\theta ,\theta ]$, we may define

$$\mathcal{A}=\sup \bigl\{ \bigl\vert \mathcal{F}(z,\varkappa ) \bigr\vert : z \in \mathrm{J}, \varkappa \in [-\theta , \theta ] \bigr\} . $$

Since $\mu :\mathrm{J}\rightarrow \mathrm{J}$ is continuous, then $\forall n \in \mathbb{N}$ and $\forall z\in \mathrm{J}$, we obtain

$$\bigl\vert \varkappa _{n}\bigl( \mu (z)\bigr) \bigr\vert \leq \theta . $$

Then,

$$ \bigl\vert \mathcal{F}(z,\varkappa _{n}\bigl(\vartheta (z) \bigr) \bigr\vert \leq \mathcal{A}, \quad z\in \mathrm{J}. $$

(14)

By applying the Lebesgue dominated convergence theorem,

$$\begin{aligned} \lim_{n \to +\infty}( \mathcal{H}_{2}\varkappa _{n}) (z)&=\lim_{n \to +\infty}\frac{1}{\Gamma (\rho )} \int ^{z}_{a}\psi '(s) \bigl(\psi (z)- \psi (s)\bigr)^{\rho -1}\mathcal{F}\bigl(s,\varkappa _{n}\bigl( \vartheta (s)\bigr)\bigr) \,\mathrm{d}s \\ & \quad {}+\lim_{n \to +\infty} \frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{\rho +\sigma ;\psi} \mathcal{F}\bigl(\eta ,\varkappa _{n}\bigl( \vartheta (\eta )\bigr)\bigr) \\ & \quad {}-\beta \mathcal{I}_{a^{+}}^{ \sigma ;\psi} \mathcal{F} \bigl(b, \varkappa _{n}\bigl(\vartheta (b)\bigr)\bigr)+\delta \bigr\} \\ &= \frac{1}{\Gamma (\rho )} \int ^{z}_{a}\psi '(s) \bigl(\psi (z)- \psi (s)\bigr)^{ \rho -1}\mathcal{F}\bigl(s, \varkappa \bigl( \vartheta (s) \bigr)\bigr) \,\mathrm{d}s \\ &\quad {}+\frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{ \rho + \sigma ;\psi} \mathcal{F}\bigl(\eta ,\varkappa \bigl(\vartheta (\eta )\bigr)\bigr) - \beta \mathcal{I}_{a^{+}}^{ \sigma ;\psi}\mathcal{F}\bigl(b, \varkappa \bigl( \vartheta (b)\bigr)\bigr)+\delta \bigr\} \\ &=(\mathcal{H}_{2}\varkappa ) (z). \end{aligned}$$

This equality confirms the continuity of $\mathcal{H}_{2}$ on $\mathfrak{P}_{\theta}$. Hence, $\mathcal{H}$ is continuous on $\mathfrak{P}_{\theta}$.

Step 4: Estimation of $\varrho _{0}(\mathcal{H}_{1}\mathcal{M})$ and $\varrho _{0}(\mathcal{H}_{2}\mathcal{M})$ for $\emptyset = \mathcal{M}\subset \mathfrak{P}_{\theta}$:

First, we estimate $\varrho _{0}(\mathcal{H}_{1}\mathcal{M})$. Fix $\epsilon > 0$. Due to the uniform continuity of $\vartheta : \mathrm{J}\rightarrow \mathrm{J}$, we can find $\kappa > 0$ (by assuming $\kappa <\epsilon$) so that for $|z_{1} -z_{2}| < \kappa$, we have

$$\bigl\vert \vartheta (z_{1})-\vartheta (z_{2}) \bigr\vert < \epsilon . $$

Let $\varkappa \in \mathcal{M}$ and $z_{1}, z_{2} \in \mathrm{J}$ with $|z_{1} -z_{2}| < \kappa <\epsilon$. Then, in view of Assumption $(C_{3})$, we have

$$\begin{aligned} \bigl\vert (\mathcal{H}_{1}\varkappa ) (z_{1}) -( \mathcal{H}_{1}\varkappa ) (z_{2}) \bigr\vert & = \bigl\vert \mathfrak{F}\bigl(z_{1},\varkappa \bigl(\vartheta (z_{1})\bigr)\bigr)-\mathfrak{F}\bigl(z_{2}, \varkappa \bigl(\vartheta (z_{2})\bigr)\bigr) \bigr\vert \\ & \leq \bigl\vert \mathfrak{F}\bigl(z_{1},\varkappa \bigl(\vartheta (z_{1})\bigr)\bigr)- \mathfrak{F}\bigl(z_{1}, \varkappa \bigl(\vartheta (z_{2})\bigr)\bigr) \bigr\vert + \bigl\vert \mathfrak{F}\bigl(z_{1}, \varkappa \bigl(\vartheta (z_{2})\bigr)\bigr) \\ & \quad {}-\mathfrak{F}\bigl(z_{2},\varkappa \bigl(\vartheta (z_{2})\bigr)\bigr) \bigr\vert \\ & \leq \bigl[ \bigl( \bigl\vert \varkappa \bigl(\vartheta (z_{1})\bigr)-\varkappa \bigl(\vartheta (z_{2})\bigr) \bigr\vert +1\bigr)^{\ell}-1 \bigr] + \varrho (\mathfrak{F},\epsilon )\\ & \leq \bigl[\bigl(\varrho (\mathcal{M},\epsilon )+1 \bigr)^{\ell}-1 \bigr]+ \varrho (\mathfrak{F},\epsilon ), \end{aligned}$$

where

$$\varrho (\mathfrak{F},\epsilon )=\sup \bigl\{ \bigl\vert \mathfrak{F}(z_{1}, \varkappa )-\mathfrak{F}(z_{2},\varkappa ) \bigr\vert : z_{1},z_{2}\in \mathrm{J}, \vert z_{1}-z_{2} \vert \leq \epsilon , \varkappa \in [- \theta ,\theta ] \bigr\} . $$

Thus,

$$\varrho (\mathcal{H}_{1}\mathcal{M},\epsilon )\leq \bigl[\bigl( \varrho ( \mathcal{M},\epsilon )+1\bigr)^{\ell}-1 \bigr]+\varrho ( \mathfrak{F}, \epsilon ). $$

$\mathfrak{F}(z,\varkappa )$ is uniformly continuous on $\mathrm{J}\times [ -\theta , \theta ]$. As a result, we obtain $\varrho ( \mathfrak{F},\epsilon )\rightarrow 0$ when $\epsilon \rightarrow 0$. Thus,

$$ \varrho _{0}(\mathcal{H}_{1}\mathcal{M}) \leq \bigl(\varrho _{0}( \mathcal{M})+1\bigr)^{\ell}-1. $$

(15)

Next, we estimate $\varrho _{0}(\mathcal{H}_{2}\mathcal{M})$. Fix $\epsilon > 0$. Due to the uniform continuity of $\mathcal{H}_{2}$ on J, $\varpi >0$ exists (by assuming $\varpi <\epsilon$) so that $\forall z_{1}< z_{2}\in \mathrm{J}$ with $|z_{2}-z_{1}|\leq \varpi$, we write

$$\int _{a}^{b} \bigl\vert G_{\psi }(z_{2} , s) - G_{\psi }(z_{1},s) \bigr\vert \,\mathrm{d}s \leq \frac{\epsilon }{\mathcal{A}}. $$

We arbitrarily choose $\varkappa \in \mathcal{M}$, thus by (14), we obtain

$$\begin{aligned} \bigl\vert \mathcal{H}_{2}\varkappa (z_{2})- \mathcal{H}_{2}\varkappa (z_{1}) \bigr\vert & \leq \int _{a}^{b} \bigl\vert G_{\psi }(z_{2} , s) - G_{\psi }(z_{1},s) \bigr\vert \bigl\vert \mathcal{F} \bigl(s, \varkappa \bigl(\mu (s)\bigr)\bigr) \bigr\vert \,\mathrm{d}s \\ &\leq \mathcal{A} \int _{a}^{b} \bigl\vert G_{\psi }(z_{2} , s) - G_{\psi }(z_{1},s) \bigr\vert \,\mathrm{d}s \leq \epsilon , \end{aligned}$$

which gives $\varrho (\mathcal{H}_{2} \varkappa , \epsilon ) \leq \epsilon$. By letting $\epsilon \rightarrow 0$, we obtain

$$ \varrho _{0}(\mathcal{H} \mathcal{M})=0. $$

(16)

Step 5: We estimate $\varrho _{0}(\mathcal{H}\mathcal{M})$ for $\emptyset \neq \mathcal{M}\in \mathfrak{P}_{\theta}$: From Lemma 2.10 and the estimates in Equation (13), (15), and (16), we have

$$\begin{aligned} \varrho _{0}(\mathcal{H}\mathcal{M})& =\varrho _{0}( \mathcal{H}_{1} \mathcal{M} \cdot \mathcal{H}_{2}\mathcal{M}) \\ & \leq \Vert \mathcal{H}_{1}\mathcal{M} \Vert \varrho _{0}(\mathcal{H}_{2} \mathcal{M})+ \Vert \mathcal{H}_{2}\mathcal{M} \Vert \varrho _{0}( \mathcal{H}_{1} \mathcal{M}) \\ &\leq \Vert \mathcal{H}_{1}\mathfrak{P}_{\theta} \Vert \varrho _{0}( \mathcal{H}_{2}\mathcal{M})+ \Vert \mathcal{H}_{2}\mathfrak{P}_{\theta} \Vert \varrho _{0}(\mathcal{H}_{1}\mathcal{M}) \\ &\leq \Delta \bigl[\bigl(\varrho _{0}(\mathcal{M})+1 \bigr)^{\ell}-1 \bigr], \end{aligned}$$

in which Δ is illustrated in (9). By Assumption $(C_{5})$, we know that $\Delta \leq 1$. Hence,

$$ \varrho _{0}(\mathcal{H} \mathcal{M})+1\leq \bigl(\varrho _{0}(\mathcal{M})+1\bigr)^{ \ell}. $$

It is seen that the contractive condition indicated in Theorem 2.11 is established with $\phi (z)=z + 1$, in which $\phi \in \mathcal{S}$. By Theorem 2.12, we determine that $\mathcal{H}$ admits at least a fixed point in the given ball $\mathfrak{P}_{\theta}$. As a result, the neutral hybrid nonlinear fractional boundary value problem defined in (1) possesses a solution in $\mathfrak{P}_{\theta}$. □

4 Stability results

In the recent section, we are interested to study Ulam–Hyers and generalized Ulam–Hyers stability [35, 36] to a special case of System (1) by assuming

$$\mathfrak{F}\bigl(z, \varkappa ^{*}\bigl(\vartheta (z)\bigr)\bigr) = 1, $$

given by:

$$ \textstyle\begin{cases} { {}^{c}\mathcal{D}}_{a^{+}}^{\rho ;\psi }\varkappa (z)= \mathcal{F}(z, \varkappa (\mu (z))),\quad z\in \mathrm {J}:=[a, b], \\ \alpha \varkappa (z)\vert _{z=a}+\beta \varkappa (z) \vert _{z=b}=\lambda \mathcal{I}^{\sigma ;\psi}_{a^{+}}\varkappa (z) \vert _{z=\eta}+\delta ,& \end{cases} $$

(17)

where the equivalent solution (by Lemma 3.1) in the integral equation settings is given as

$$\begin{aligned} \varkappa (z ) & = \mathcal{I}_{a^{+}}^{\rho ;\psi } \mathcal{F}\bigl(z, \varkappa \bigl(\mu (z)\bigr)\bigr) + \frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{ \rho +\sigma ;\psi } \mathcal{F}\bigl(\eta , \varkappa \bigl(\mu (\eta )\bigr)\bigr) \\ & \quad {}-\beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \mathcal{F} \bigl(b, \varkappa \bigl(\mu (b)\bigr)\bigr)+\delta \bigr\} . \end{aligned}$$

(18)

Definition 4.1

System (17) is Ulam–Hyers stable if $\exists k>0$ s.t. $\forall \varepsilon \in \mathbb{R}^{+}$ and $\forall \varkappa ^{*} \in \mathfrak{C}(\mathrm{J})$ satisfying

$$ \bigl\vert { {}^{c}\mathcal{D}}_{a^{+}}^{\rho ;\psi } \varkappa ^{*}(z) - \mathcal{F}\bigl(z, \varkappa ^{*} \bigl(\mu (z)\bigr)\bigr) \bigr\vert \leq \varepsilon ,\quad z\in \mathrm {J}:=[a, b], $$

(19)

$\exists \varkappa \in \mathfrak{C}(\mathrm{J})$ as the unique solution of (17) with

$$\bigl\Vert \varkappa ^{*}-\varkappa \bigr\Vert \leq k \varepsilon . $$

Definition 4.2

System (17) is generalized Ulam–Hyers stable if $\exists K \in C (\mathbb{R}^{+}, \mathbb{R}^{+} )$ with $K(0)=0$ s.t. $\forall \varepsilon \in \mathbb{R}^{+}$ and $\forall \varkappa ^{*} \in \mathfrak{C}(\mathrm{J})$ satisfying (19), $\exists \varkappa \in \mathfrak{C}(\mathrm{J})$ as the unique solution of (17) with

$$\bigl\Vert \varkappa ^{*}-\varkappa \bigr\Vert \leq K(\varepsilon ). $$

Remark 4.1

$\varkappa ^{*} \in \mathfrak{C}(\mathrm{J})$ is a solution of (4.1) iff $\exists H \in \mathfrak{C}(\mathrm{J})$ (depends on $\varkappa ^{*}$) s.t.

• a) $|H(z)| \leq \varepsilon$, $\forall z \in \mathrm{J}$;

• b) ${ {}^{c}\mathcal{D}}_{a^{+}}^{\rho ;\psi }\varkappa ^{*}(z)= \mathcal{F}(z, \varkappa ^{*}(\mu (z)))+H(z)$.

Now, the Ulam–Hyers and generalized Ulam–Hyers stability of the solution to the nonhybrid fractional boundary value problem (17) are proved.

Theorem 4.3

Assume the condition:

$(A1^{\prime} )$ There exists $\ell _{\mathcal{F}}>0$ such that

$$\bigl\Vert \mathcal{F}(z, \varkappa )-\mathcal{F}\bigl(z, \varkappa '\bigr) \bigr\Vert \leq \ell _{ \mathcal{F}} \bigl\Vert \varkappa - \varkappa ' \bigr\Vert , $$

for each $z \in \mathrm{J}$, and $\varkappa , \varkappa ' \in \mathfrak{C}(\mathrm{J})$, is fulfilled. If $Q\ell _{\mathcal{F}}<1$, then, the system of (17) is Ulam–Hyers and generalized Ulam–Hyers stable, where

$$Q= \biggl\{ \frac{(\psi (b)-\psi (a))^{\rho}}{\Gamma (\rho +1)}+ \frac{\lambda (\psi (\eta )-\psi (a))^{\rho +\sigma}}{\Lambda \Gamma (\rho +\sigma +1)}+ \frac{\beta (\psi (b)-\psi (a))^{\rho}}{\Lambda \Gamma (\rho +1)} \biggr\} . $$

Proof

Let $\varepsilon >0$ and $\varkappa ^{*} \in \mathfrak{C}(\mathrm{J})$ satisfying (19). Since we have assumed that $\varkappa ^{*}$ is a solution of (4.1), by Remark 4.1, we find $H \in \mathfrak{C}(\mathrm{J})$ with $|H(z)| \leq \varepsilon$, $\forall z \in \mathrm{J}$ so that.

$${ {}^{c}\mathcal{D}}_{a^{+}}^{\rho ;\psi }\varkappa ^{*}(z)= \mathcal{F}\bigl(z, \varkappa ^{*}\bigl(\mu (z) \bigr)\bigr)+H(z). $$

Again by (18), we have

$$\begin{aligned} \varkappa ^{*}(z)&=\mathcal{I}_{a^{+}}^{\rho ;\psi } \mathcal{F}\bigl(z, \varkappa ^{*}\bigl(\mu (z)\bigr)\bigr) + \mathcal{I}_{a^{+}}^{\rho ;\psi }H(z) + \frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{\rho +\sigma ; \psi } \mathcal{F}\bigl(\eta , \varkappa ^{*}\bigl(\mu (\eta )\bigr)\bigr) \\ &\quad {}-\beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \mathcal{F} \bigl(b, \varkappa ^{*}\bigl(\mu (b)\bigr)\bigr) + \lambda \mathcal{I}_{a^{+}}^{\rho +\sigma ; \psi } H(\eta )-\beta \mathcal{I}_{a^{+}}^{\rho ;\psi } H(b) +\delta \bigr\} . \end{aligned}$$

(20)

Also, consider $\varkappa \in \mathfrak{C}(\mathrm{J})$ as the unique solution of (17) formulated as

$$ \varkappa (z ) = \mathcal{I}_{a^{+}}^{\rho ;\psi }\mathcal{F}\bigl(z, \varkappa \bigl(\mu (z)\bigr)\bigr) +\frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{ \rho +\sigma ;\psi } \mathcal{F}\bigl(\eta ,\varkappa \bigl( \mu (\eta )\bigr)\bigr) - \beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \mathcal{F}\bigl(b,\varkappa \bigl(\mu (b)\bigr)\bigr)+ \delta \bigr\} . $$

Further, $\forall z\in \mathrm{J}$, we obtain

$$\begin{aligned} \bigl\vert \varkappa ^{*}(z)-\varkappa (z) \bigr\vert & \leq \biggl\vert \varkappa ^{*}(z) - \mathcal{I}_{a^{+}}^{\rho ;\psi } \mathcal{F}\bigl(z, \varkappa \bigl(\mu (z)\bigr)\bigr) + \frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{ \rho +\sigma ;\psi }\mathcal{F}\bigl(\eta , \varkappa \bigl(\mu (\eta )\bigr)\bigr) \\ & \quad {}-\beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \mathcal{F} \bigl(b, \varkappa \bigl(\mu (b)\bigr)\bigr)+\delta \bigr\} \biggr\vert . \end{aligned}$$

By part (a) of Remark 4.1 along with $(A1^{\prime} )$,

$$\begin{aligned} \bigl\vert \varkappa ^{*}(z)-\varkappa (z) \bigr\vert &\leq \mathcal{I}_{a^{+}}^{\rho ; \psi } \bigl\vert \mathcal{F}\bigl(z, \varkappa ^{*}\bigl(\mu (z)\bigr)\bigr) - \mathcal{F}\bigl(z, \varkappa \bigl(\mu (z)\bigr)\bigr) \bigr\vert + \mathcal{I}_{a^{+}}^{ \rho ;\psi } \bigl\vert H(z) \bigr\vert \\ &\quad {}+\frac{1}{\Lambda} \bigl\{ \lambda \mathcal{I}_{a^{+}}^{\rho + \sigma ;\psi } \bigl\vert \mathcal{F}\bigl(\eta , \varkappa ^{*}\bigl(\mu (\eta ) \bigr)\bigr) - \mathcal{F}\bigl(\eta , \varkappa \bigl(\mu (\eta )\bigr)\bigr) \bigr\vert \\ &\quad {}+\beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \bigl\vert \mathcal{F}\bigl(b, \varkappa ^{*}\bigl(\mu (b)\bigr)\bigr) - \mathcal{F}\bigl(b, \varkappa \bigl(\mu (b)\bigr)\bigr) \bigr\vert \\ & \quad {}+ \lambda \mathcal{I}_{a^{+}}^{\rho +\sigma ;\psi } \bigl\vert H(\eta ) \bigr\vert + \beta \mathcal{I}_{a^{+}}^{\rho ;\psi } \bigl\vert H(b) \bigr\vert \bigr\} \\ &\leq \biggl\{ \frac{(\psi (b)-\psi (a))^{\rho}}{\Gamma (\rho +1)} + \frac{\lambda (\psi (\eta ) - \psi (a))^{\rho +\sigma}}{ \Lambda \Gamma (\rho +\sigma +1)} \\ & \quad {}+ \frac{ \beta ( \psi (b) - \psi (a))^{\rho}}{ \Lambda \Gamma (\rho +1)} \biggr\} \ell _{\mathcal{F}} \bigl\Vert \varkappa ^{*}-\varkappa \bigr\Vert \\ & \quad {}+ \biggl\{ \frac{(\psi (b)-\psi (a))^{\rho}}{\Gamma (\rho +1)}+ \frac{\lambda (\psi (\eta )-\psi (a))^{\rho +\sigma}}{\Lambda \Gamma (\rho +\sigma +1)} \\ & \quad {}+ \frac{\beta (\psi (b)-\psi (a))^{\rho}}{\Lambda \Gamma (\rho +1)} \biggr\} \varepsilon \\ & \leq Q\varepsilon +Q \ell _{ \mathcal{F}} \bigl\Vert \varkappa ^{*}- \varkappa \bigr\Vert . \end{aligned}$$

In consequence, it follows that

$$\bigl\Vert \varkappa ^{*}-\varkappa \bigr\Vert \leq \frac{Q}{1-Q\ell _{\mathcal{F}}} \varepsilon . $$

If we let $k=\frac{Q}{1-Q\ell _{\mathcal{F}}}$, then the Ulam–Hyers stability is established. Further, for

$$K( \varepsilon ) = \frac{Q}{1-Q\ell _{\mathcal{F}}}\varepsilon , $$

we obtain $K(0)=0$, and so the generalized Ulam–Hyers stability is also established regarding the system (17). □

5 Application with numerical results

This section presents an example to show the validity of the conclusions of the theorems. Here, we shall consider the following different definitions for the function ψ. The linear case gives the Caputo derivative when $\psi _{1} (z) = z$. Also, in the cases $\psi _{2} (z) = e^{z}$, $\psi _{3} (z)=\ln (z)$, and $\psi _{4} (z)= 2^{z}$, they yield the exponential, the Caputo–Hadamard, and the power generalized derivative, respectively.

Example 5.1

Let us consider the hybrid nonlinear fractional boundary value problem (1) with specific data given as

$$ \textstyle\begin{cases} {{}^{c}\mathcal{D}_{0^{+}}^{\frac {1}{4},\psi}} [ \frac {\varkappa (z)}{\sqrt{\varkappa (\frac{e^{(z-1)}}{2} )+1}} ]=\frac{z}{10} ( \frac{\varkappa (\sqrt{z})}{1+\varkappa (\sqrt{z})} ), \\ {}[ \frac {\varkappa (z)}{\sqrt{\varkappa (\frac{e^{(z-1)}}{2} )+1}} ]_{z=a}+ [ \frac {\varkappa (z)}{\sqrt{\varkappa (\frac{e^{(z-1)}}{2} )+1}} ]_{z=b} = \frac {1}{10} + \frac {1}{3} \mathcal{I}_{0^{+}}^{\frac {1}{2},\psi} [ \frac {\varkappa (z)}{\sqrt{\varkappa (\frac{e^{(z-1)}}{2} )+1}} ]_{z=\eta}, \end{cases} $$

(21)

for $z \in \mathrm{J}=[0,1]$. By (21), we have the following data $\rho = \frac{1}{4} \in (0, 1]$, $\sigma = \frac{1}{2}>0$, $a=0$, $b=\alpha = \beta =1$, $\delta =\frac{1}{10}$, $\lambda = \frac{1}{3}$, and

$$\vartheta (z) = \frac{1}{2} e^{(z-1)}, \qquad \mu (z)=z^{\frac{1}{2}}. $$

Also,

$$\mathfrak{F}( z, \varkappa ) =\bigl( \vert \varkappa \vert +1 \bigr)^{\frac{1}{2}},\qquad \mathcal{F} (z, \varkappa ) = \frac{1}{10} \biggl[ \frac{\varkappa}{1+\varkappa} \biggr] $$

and

$$N=\sup_{z \in [0,1]} \bigl\vert \mathfrak{F}(z, 0) \bigr\vert =1. $$

Both (C1) and (C2) hold. Furthermore, Let

$$p(\varkappa )=\sqrt{ \vert \varkappa \vert +1}-1. $$

Since $p''(\varkappa )\leq 0$, it is concave along with $p(0)=0$. Therefore, by applying the subadditive property of the concave function p, we obtain

$$\begin{aligned} \bigl\vert \mathfrak{F} (z, \varkappa _{2} ) - \mathfrak{F} (z, \varkappa _{1} ) \bigr\vert &= \bigl\vert p ( \varkappa _{2} ) - p (\varkappa _{1} ) \bigr\vert \\ & \leq p ( \varkappa _{2}-\varkappa _{1} ) \\ &=\sqrt{ \vert \varkappa _{2}-\varkappa _{1} \vert +1}-1, \end{aligned}$$

for $\varkappa _{1}, \varkappa _{2} \in \mathbb{R}$. As a result, (C3) is satisfied via $\ell = 1/2$. Moreover, $\forall z \in \mathrm{J}$ and $\forall \varkappa \in \mathbb{R}$, we have

$$\bigl\vert \mathcal{F} (z, \varkappa ) \bigr\vert \leq \frac{1}{10} \vert \varkappa \vert , \quad \forall z \in \mathrm{J}. $$

This means that $\phi (\varkappa ) = \frac{1}{10}\varkappa$. The last item (C5) permits us to specify the range of θ, which is clearly $0 <\theta \leq 0.2$ for

$$\psi _{i}(z)\in \bigl\{ z, e^{z}, \ln (z)\bigr\} , \quad i=1,2,3 $$

and $\theta =0.25$ for $\psi _{4}=2^{z}$. Accordingly, we have

$$\begin{aligned} \Lambda _{i} &= \biggl[ \alpha + \beta - \lambda \frac{(\psi _{i}( \eta ) - \psi _{i}(a))^{ \sigma}}{ \Gamma (\sigma +1)} \biggr] = \biggl[ 2 - \frac{(\psi _{i}( \eta ) - \psi _{i}(0))^{ \frac{1}{2}}}{ 3 \Gamma (\frac{3}{2} )} \biggr] \\ & = 1.6239, 1.4557, 1.6276, 1.4681, \quad i=1,\dots ,4, \end{aligned}$$

for $\psi _{i}(z) \in \{z, e^{z}, \ln (z), w^{z}\}$, respectively, whenever $\eta = b$. Also, by using (9) we obtain

$$\begin{aligned} \Delta _{i} & = \biggl\{ \frac{ \phi ( \theta ) + \frac{ \beta}{ \Lambda} \phi (\theta )}{ \Gamma (\rho +1)}\bigl( \psi _{i}(b) - \psi _{i}(a)\bigr)^{\rho} \\ & \quad {}+ \frac{\lambda \phi (\theta )}{ \Lambda \Gamma (\rho +\sigma +1)}\bigl( \psi _{i}(\eta ) - \psi _{i}(a)\bigr)^{ \rho + \sigma} + \frac{\delta}{ \Lambda _{i}} \biggr\} \\ & = 0.1102, 0.1303, 0.1098, 0.1593,\quad i=1,2,3,4, \end{aligned}$$

for

$$\psi _{i}(z)= \bigl\{ z, e^{z}, \ln (z), 2^{z} \bigr\} , $$

respectively, whenever $\eta = b$. Therefore, Theorem 3.2 guarantees that the hybrid nonlinear fractional boundary value problem (21) admits at least one solution in the Banach algebra $\mathfrak{C}(\mathrm{J})$. Next, one can find the following values by terms of given data in Tables 1, 2, 3, and 4. These results are plotted in Figs. 1, 2, 3, and 4, respectively. Figure 1 shows the Caputo generalized derivative. In this case, $\psi _{1}$ is a linear function and there is a jump at point $\eta = 0.6$ for Δ. Figure 3 shows the Caputo–Hadamard generalized derivative. In this case, $\psi _{3}$ is a logarithmic function and there is a large jump at point $\eta = 1.6$ for Δ.

Figure 1

Graphical representation of Λ and Δ for $\eta \in \mathrm{J} = [0,1]$ and $\rho =\frac{1}{4}$, $\sigma =\frac{1}{2}$, where $\psi _{1}(z)= z$ (Caputo-type derivative) in Example 5.1. In this case, ψ is a linear function and the rate of Δ change is low

Figure 2

Graphical representation of Λ and Δ for $\eta \in \mathrm{J} = [0,1]$ and $\rho =\frac{1}{3}$, $\sigma =\frac{3}{4}$, where $\psi _{2}(z)= e^{z}$, in Example 5.1. In this case, ψ is an exponential function and the rate of Δ change is low. There exists a jump at point $\eta = 0.2$ for Δ

Figure 3

Graphical representation of Λ and Δ for $\eta \in \mathrm{J} = [1,e]$ and $\rho =\frac{2}{3}$, $\sigma =\frac{1}{3}$, where $\psi _{3}(z)= \ln (z)$ (Caputo–Hadamard-type), in Example 5.1. In this case, ψ is a logarithmic function and the rate of Δ change is high. There exists a large jump at point $\eta = 1.6$ for Δ

Figure 4

Graphical representation of Λ and Δ for $\eta \in \mathrm{J} = [1,2]$ and $\rho =\frac{1}{2}$, $\sigma =\frac{1}{2}$, where $\psi _{4}(z)= 2^{z}$, in Example 5.1. In this case, ψ is a power function and the rate of Δ change is low. There exists a jump at point $\eta = 1.9$ for Δ

Table 1 Numerical results of Δ and θ for $\psi _{1}(z)=z$ and $\eta \in \mathrm{J} = [0,1]$ in Example 5.1

Table 2 Numerical results of Δ and θ for $\psi _{2}(z)=e^{z}$ and $\eta \in \mathrm{J} = [0,1]$ in Example 5.1

Table 3 Numerical results of Δ and θ for $\psi _{3}(z)=\ln (z)$ and $\eta \in \mathrm{J} = [1,e]$ in Example 5.1

Table 4 Numerical results of Δ and θ for $\psi _{4}(z)=2^{z}$ and $\eta \in \mathrm{J} = [1,2]$ in Example 5.1

Figures 1a and 1b show graphical representations of Λ and Δ for $\eta \in \mathrm{J} = [0,1]$ and $\rho =\frac{1}{4}$, $\sigma =\frac{1}{2}$, where $\psi _{1}(z)= z$ (Caputo-type derivative) in Example 5.1. In this case, ψ is a linear function and the rate of Δ changes is low. Figures 2a and 2b show graphical representation of Λ and Δ for $\eta \in \mathrm{J} = [0,1]$ and $\rho =\frac{1}{3}$, $\sigma =\frac{3}{4}$, where $\psi _{2}(z)= e^{z}$, in Example 5.1. In this case, ψ is an exponential function and the rate of Δ change is low. There exists a jump at point $\eta = 0.2$ for Δ.

Figures 3a and 3b show graphical representation of Λ and Δ for $\eta \in \mathrm{J} = [1,e]$ and $\rho =\frac{2}{3}$, $\sigma =\frac{1}{3}$, where $\psi _{3}(z)= \ln (z)$ (Caputo–Hadamard-type), in Example 5.1. In this case, ψ is a logaritmic function and the rate of Δ change is high. There exists a large jump at point $\eta = 1.6$ for Δ. Figures 4a and 4b show graphical representation of Λ and Δ for $\eta \in \mathrm{J} = [1,2]$ and $\rho =\frac{1}{2}$, $\sigma =\frac{1}{2}$, where $\psi _{4}(z)= 2^{z}$, in Example 5.1. In this case, ψ is a power function and the rate of Δ change is low. There exists a jump at point $\eta = 1.9$ for Δ.

6 Conclusions

In the current paper, we focused on the behaviors of a new hybrid neutral nonlinear fractional boundary value problem via an integrohybrid conditions in which the operators were assumed in a generalized structure with respect to an increasing function ψ, entitled the ψ-Caputo derivative. We recalled some properties of noncompactness measures and the modulus of continuity and then we followed and established the results regarding the existence of solutions for the given hybrid neutral nonlinear fractional boundary value problem (1) in terms of Darbo’s criterion.

We completed the results of the next section in relation to the stability of the nonhybrid neutral system (17) in a special case. Our results for the neutral hybrid nonlinear fractional boundary value problem (1) are valid for the cases of Caputo-type fractional boundary value problem $(\psi (z)=z )$ and the Caputo–Hadamard-type fractional boundary value problem $(\psi (z)=\log z)$. Ulam–Hyers and generalized Ulam–Hyers stabilities were investigated for this purpose. Further, by defining different cases for the increasing function ψ, we simulated our example with the help of the subadditivity property of concave functions to confirm the correctness of the theorems. This hybrid fractional boundary value problem (1) can be extended in the framework of some real models of thermostat or pantograph systems by considering the ψ-Hilfer operators in the future.