Existence and stability results for impulsive $(k, \psi)$-Hilfer fractional double integro-differential equation with mixed nonlocal conditions

1.   Introduction

Fractional calculus (FC) has gained considerable importance in many fields of applied sciences and engineering for solving various differential equations and investigating behaviors of mathematical models simulating real-world problems. Its amazing presence is evident in the modeling of several natural phenomena such as Hamiltonian chaos and fractional dynamics ^[1], bio-engineering ^[2], viscoelasticity ^[3], vibrations and diffusion ^[4], physics ^[5,6], financial economics ^[7] and references cited therein. For more detailed theoretical aspect of FC, see ^{[8,9,10,11,12]} and references therein. One of the significant factors that make FC advantageous to ordinary calculus is that fractional-order (non-integer order) derivatives and integral operators (FDO/FIOs) are more effective for describing real-life problems than integer-order ones. Many researchers have attempted to propose various fractional operators that deal with derivatives and integrals of non-integer orders with successful applications to solve many problems. Different definitions of FDO and FIOs have been employed in research papers, mainly focusing on Riemann-Liouville (RL) ^[13], Caputo ^[13], Hadamard ^[13], Katugampola ^[13], Erdélyi-Kober ^[13], Hilfer ^[6], $\psi$-RL ^[13], $k$-RL ^[14], $\psi$-Hilfer ^[15], $(k, \psi)$-RL ^[14], $(k, \psi)$-Hilfer ^[16], and so on.

The study of nonlocal boundary value problems (BVP) is expanding quickly. In addition to the theoretical interest, this type of problems can be used to represent several phenomena in engineering, physics, and biological sciences. The nonlocal conditions have been used to describe some properties that occur at various points inside the domain instead of handling initial or boundary conditions. For historical backgrounds, see, e.g., ^[17,18,19]. At present, research on fractional differential equations (FDE) under various FDO/FIOs has developed rapidly in numerous directions. Fractional integro-differential equations (FIDEs) are the popular subjects that attract many researchers, some of which studied the existence of the solution for FIDEs. We recommend a series of recent works as in $2017$, Jalilian and Ghasmi ^[20] established the existence and uniqueness of solutions for FIDEs with pantograph type by applying lipchitz condition. In $2020$, the authors ^[21] studied the existence and stability of a class of nonlocal BVPs for integro-differential Langevin equation under the generalized Caputo proportional FDO by means of Banach's, Krasnoselskii's, Schaefer's fixed point theorems, and Ulam's stability approach. In $2021$, Sudsutad et al. ^[22] used fixed point theories of Banach's, Krasnoselskii's, and Leray-Schauder nonlinear alternative types to discuss the existence, uniqueness, and stability of BVP for $\psi$-Hilfer FIDEs with mixed nonlocal boundary conditions, which include multi-point, fractional derivative multi-order, and fractional integral multi-order conditions:

where the description of parameters can be found in ^[22]. Later, in $2022$, Thaiprayoon et al. ^[23] studied a class of $\psi$-Hilfer implicit fractional integro-differential equations with mixed nonlocal conditions:

where the description of parameters can be found in ^[23]. The existence and uniqueness of a solution for their problem were verified employing Banach's, Schaefer's, and Krasnosellskii's fixed point theorems, and the analysis of the problem was established via various kinds of Ulam stability results. In $2023$, Sitho et al. ^[24] utilized the Banach contraction principle to show the uniqueness of the solution and the Leray-Schauder nonlinear alternative to prove the existence of solutions for a new class of BVP consisting of a mixed-type $\psi_1$-Hilfer and $\psi_2$-Caputo fractional order differential equation with integro-differential nonlocal boundary conditions as follows

where the description of parameters can be found in ^[24]. For more interesting works on existence, uniqueness, and Ulam's stability results of these topics, we refer to ^{[25,26,27,28,29,30]} and reference cited therein. In parallel with FIDEs, impulsive differential equations play an important role in dynamical systems of evolutionary procedures by exhibiting instantaneous state changes at specific moments. It is thus regarded as an effective instrument for comprehending numerous real-world situations in applied sciences and engineering; see ^[31,32,33]. Many researchers have attempted FDEs and FIDEs with impulse conditions to develop an excellent qualitative theory. Over the years, they have produced crucial and fascinating insights that greatly aided the mathematical understanding of FDEs with impulse effects. We refer to ^{[34,35,36,37,38,39]} for further fascinating works on the subject.

The inspiration for this paper is based on the previous works mentioned above. The novelty and differences from the others are considered in our work. In this paper, we produce qualitative results of the solutions for the following nonlinear impulsive $(\rho_{k}, \psi_{k})$-Hilfer FIDEs supplemented with mixed nonlocal boundary conditions:

where ${_{\rho_{k}}^{H}}\mathfrak{D}_{t_{k}^+}^{\alpha_{k}, \beta_{K}; \psi_{k}}$ denotes the $(\rho_{k}, \psi_{k})$-Hilfer-FDO of order $\alpha_{k} \in (1, 2]$ and type $\beta_{k} \in [0, 1]$, $\rho_{k} \in {R}^{+}$, ${J}_{k} : = (t_{k}, t_{k+1}] \subset (a, b]$ for $k = 0, 1, 2, \ldots, m$, with ${J}_{0}: = [a, t_{1}]$, ${J} : = [a, b]$, $0 \leq a = t_{0} < t_{1} < \cdots < t_{m} < t_{m+1} = b \leq T$, ${_{\rho_{k}}^{}}{I}_{t_{k}^{+}}^{q; \psi_{k}}$ is the $(\rho_{k}, \psi_{k})$-RL-FIO with order $q \in \{\rho_{k}(2-\gamma_{k}), \rho_{k-1}(2-\gamma_{k-1}), \nu_{k}, \sigma_{k}, \theta_{l} \}$, $q > 0$, $k = 1, 2, \ldots, m$, $l = 0, 1, \ldots, n$, ${_{\rho_{k}}^{{RL}}}\mathfrak{D}_{t_{k}^{+}}^{p; \psi_{k}}$, is the $(\rho_{k}, \psi_{k})$-RL fractional derivative with order $p \in \{\rho_{k}(\gamma_{k}-1), \rho_{k-1}(\gamma_{k-1}-1) \}$ with $p \in (1, 2)$, $k = 1, 2, \ldots, m$, ${_{\rho_{k}}^{}}{I}_{t_{k}^{+}}^{\rho_{k}(2-\gamma_{k}); \psi_{k}} u(t_{k}^{+}) = \lim_{t \to 0^+} {_{\rho_{k}}^{}}{I}_{t_{k}^{+}}^{\rho_{k}(2-\gamma_{k}); \psi_{k}} u(t_{k} + h)$, ${_{\rho_{k}}^{{RL}}}\mathfrak{D}_{t_{k}^{+}}^{\rho_{k}(\gamma_{k}-1); \psi_{k}} u(t_{k}^{+}) = \lim_{h \to 0^+} {_{\rho_{k}}^{{RL}}}\mathfrak{D}_{t_{k}^{+}}^{\rho_{k}(\gamma_{k}-1); \psi_{k}} u(t_{k} + h)$, $\phi_{k}$, $\phi_{k}^{*} \in {C}({R}, {R})$, $k = 1, 2, \ldots, m$, $f \in {C}({J} \times {R}^{3}, {R})$, ${A}$, $\mu_i$, $\lambda_l \in {R}$, $\eta_{i}, \in (t_{i}, t_{i+1}]$, $\xi_{l} \in (t_{l}, t_{l+1}]$, $i = 1, 2, \ldots, m$, and $l = 0, 1, \ldots, n$.

The remaining sections of this work are structured as follows: Section 2 presents the prerequisite and relevant facts for the concepts of the ($\rho, \psi$)-Hilfer fractional operators, as well as some necessary lemmas that examine the solution of the linear variant of the proposed problem in terms of an integral equation. Section 3 proves the existence of the solution using O'Regan's fixed point theorem, while the uniqueness of the solution is investigated by utilizing Banach's fixed point theorem. Later, various Ulam's stability results, such as Ulam-Hyers (UH), generalized Ulam-Hyers (GUH), Ulam-Hyers-Rassias (UHR), and generalized Ulam-Hyers-Rassias (GUHR), are established to ensure the existence results in Section 4. Finally, some illustrative examples are provided to support the main theoretical results in the last section.

2.   Preliminaries

Definition 2.1. ^[40] Let $f \in L^{1}({J}, b)$ and an increasing function $\psi(t) : {J} \to {R}$ with $\psi^{\prime}(t) \neq 0$ for $t \in [a, b]$. The $(\rho, \psi)$-RL-FIO of a function $f$ of order $\alpha > 0$ is defined by

where $\Gamma_{\rho}(\cdot)$ is the $\rho$-Gamma function introduced by Diaz and Pariguan ^[41],

Some other useful properties of (2.1) are well known: $\Gamma_{\rho}(z + \rho) = z \Gamma_{\rho}(z)$, $\Gamma_{\rho}(\rho) = 1$, $\Gamma_{\rho}(z) = (\rho)^{\frac{z}{\rho}-1}\Gamma\left({z}/{\rho}\right)$, $\Gamma(z) = \lim_{\rho \to 1} \Gamma_{\rho}(z)$.

Definition 2.2. ^[16] Let $f \in {C}^{n}({J}, {R})$ and a function $\psi(t) \in {C}^{n}({J}, {R})$ with $\psi^{\prime}(t) \neq 0$ for $t \in {J}$. Then, the $(\rho, \psi)$-RL-FDO of a function $f$ of order $\alpha$, $\rho \in {R}^{+}$, is defined by

Definition 2.3. ^[16] Let $f \in {C}^{n}({J}, {R})$, $\psi \in {C}^{n}({J}, {R})$, $\psi^{\prime}(t) \neq 0$, for $t \in {J}$, $\alpha$, $\rho \in {R}^{+}$, and $\beta \in [0, 1]$. The $(\rho, \psi)$-Hilfer FDO of a function $f$ of order $\alpha$ and type $\beta$ is given by

where $(1-\beta)(n\rho-\alpha) = n\rho - \gamma_{\rho}$, $\delta_{\psi}^{n} = \left(\frac{\rho}{\psi^{\prime}(t)} \frac{d}{dt} \right)^{n}$ and $n = \left\lceil \alpha/\rho \right\rceil$.

Lemma 2.1. ^[16] Let $\alpha$, $\rho \in {R}^{+}$ and $\beta \in {R}$, such that $\beta/\rho > -1$. Then we have

$(i)$ $_\rho I_{a^+}^{\alpha; \psi} \Big[\left(\psi(t) - \psi(a) \right)^{\frac{\beta}{\rho}} \Big] = \frac{\Gamma_{\rho}(\beta+\rho)}{\Gamma_{\rho}(\beta + \rho + \alpha)} \left(\psi(t) - \psi(a) \right)^{\frac{\beta + \alpha}{\rho}}$.

$(ii)$ ${_{\rho}^{{RL}}}\mathfrak{D}_{a^+}^{\alpha; \psi} \Big[\left(\psi(t) - \psi(a) \right)^{\frac{\beta}{\rho}} \Big] = \frac{\Gamma_{\rho}(\beta + \rho)}{\Gamma_{\rho}(\beta + \rho - \alpha)} \left(\psi(t) - \psi(a) \right)^{\frac{\beta - \alpha}{\rho}}$.

$(iii)$ $_{\rho }I{{_{{{a}^{+}}}^{\alpha ;\phi }}_{\rho }}I_{{{a}^{+}}}^{\beta ;\psi }f(t){{=}_{\rho }}I_{{{a}^{+}}}^{\alpha +\beta ;\psi }f(t){{=}_{\rho }}I{{_{{{a}^{+}}}^{\beta ;\phi }}_{\rho }}I_{{{a}^{+}}}^{\alpha ;\psi }f(t)$.

Lemma 2.2. ^[35] If $f \in {C}^{n}({J}, {R})$, $\rho$, $\alpha \in {R}^{+}$, $\beta \in[0, 1]$ and $n \in {N}$, then

where $\gamma = \frac{1}{\rho}\left(\beta\left(\rho n - \alpha \right) + \alpha \right)$ and $n = \left\lceil \alpha/\rho \right\rceil$.

For convenience's sake, we set the notation as follows:

Next, we establish the following auxiliary result:

Lemma 2.3. Let $\nu \in (m-1, m)$, $\rho$, $\alpha \in {R}^{+}$, $m \in {N}$. If $h \in {C}^{n}([a, b], {R})$, then

Proof. By applying Definition 2.2 and (ⅲ) of Lemma 2.1, we have

By using Definition 2.1, for $n = 1$, we obtain

In the same way, for $n = 2$, we have

Repeating the above method, we obtain

The proof is completed.

Denote the weighted space

where ${C}_{\psi}^{2-\gamma} = {C}_{\psi}^{2-\gamma}({J}, {R})$. The weighted space of piece-wise continuous functions is defined by

Observe that ${PC} = {PC}_{\psi_{k}}^{2-\gamma_{k}}({J}, {R})$ is a Banach space equipped with

Lemma 2.4. Let $\alpha_{k} \in (1, 2)$, $\beta_{k} \in [0, 1]$, $\rho_{k} > 0$, $\mu_{k} > 0$, $\nu_{k} > 0$, $\gamma_{k} = (1/\rho_{k})(\beta_{k}(2\rho_{k}-\alpha_{k})+\alpha_{k})$, $\psi_{k} \in {C}({J}, {R})$ with $\psi^{\prime}_{k} > 0$, $k = 0, 1, 2, \ldots, m$, $\mathfrak{h} \in {C}_{\psi_{k}}^{2-\gamma_{k}}$. Then the following linear variant impulsive $(\rho_{k}, \psi_{k})$-Hilfer fractional boundary value problem

satisfies the following integral equation, $u \in {PC}$, as

where

Proof. Suppose $u\in {PC}$ is a solution of the impulsive $(\rho_{k}, \psi_{k})$-Hilfer problem (2.5).

For $t \in [t_{0}, t_{1}]$, we have

where $c_{1} = {_{\rho_{0}}^{{RL}}}\mathfrak{D}_{t_{0}}^{\rho_{0}(\gamma_{0}-1); \psi_{0}} u(t_{0}^{+})$ and $c_{2} = {_{\rho_{0}}^{}}{I}_{t_{0}}^{\rho_{0}(2-\gamma_{0}); \psi_{0}} u(t_{0}^{+})$. By using Lemma 2.1 and Lemma 2.3, we obtain

Putting $t = t_{1}$ into (2.8) and (2.9), we have

For $t \in (t_{1}, t_{2}]$, we obtain

From the impulsive conditions, that is ${_{\rho_{1}}^{{RL}}}\mathfrak{D}_{t_{1}^{+}}^{\rho_{1}(\gamma_{1}-1); \psi_{1}} u(t_{1}^{+}) = {_{\rho_{0}}^{{RL}}}\mathfrak{D}_{t_{0}^{+}}^{\rho_{0}(\gamma_{0}-1); \psi_{0}} u(t_{1}) + \phi_{1}(u(t_{1}))$ and ${_{\rho_{1}}^{}}{I}_{t_{1}^{+}}^{\rho_{1}(2-\gamma_{1}); \psi_{1}} u(t_{1}^{+}) = {_{\rho_{0}}^{}}{I}_{t_{0}^{+}}^{\rho_{0}(2-\gamma_{0}); \psi_{0}} u(t_{1}) + \phi_{1}^{*}(u(t_{1}))$, it implies that

By applying Lemmas 2.1 and 2.3, we get

In particular for $t = t_{2}$, we have

Under the impulsive conditions, ${_{\rho_{2}}^{{RL}}}\mathfrak{D}_{t_{2}^{+}}^{\rho_{2}(\gamma_{2}-1); \psi_{2}} u(t_{2}^{+}) = {_{\rho_{1}}^{{RL}}}\mathfrak{D}_{t_{1}^{+}}^{\rho_{1}(\gamma_{1}-1); \psi_{1}} u(t_{2}) + \phi_{2}(u(t_{2}))$ and ${_{\rho_{2}}^{}}{I}_{t_{2}^{+}}^{\rho_{2}(2-\gamma_{2}); \psi_{2}} u(t_{2}^{+}) = {_{\rho_{1}}^{}}{I}_{t_{1}^{+}}^{\rho_{1}(2-\gamma_{1}); \psi_{1}} u(t_{2}) + \phi_{2}^{*}(u(t_{2}))$, for $t \in (t_{2}, t_{3}]$, we get

Then for $t \in (t_{3}, t_{4}]$, we have

Repeating the above process, for any $t \in (t_{k}, t_{k+1}]$, $k = 0, 1, \ldots, m$, one has

By applying the first boundary condition, $u(0) = 0$, we get $c_{2} = 0$. From the second boundary condition, $\sum_{i = 0}^{m+1}\mu_{i} u(\eta_{i}) + \sum_{l = 0}^{n}\lambda_{l} {_{\rho_{l}}^{}}{I}_{t_{l}}^{\theta_{l}; \psi_{l}} u(\xi_{l}) = {A}$, we have

where $\Lambda$ is given by (2.7). Taking the values $c_{1}$ and $c_{2}$ in (2.12), we obtain the solution (2.6).

3.   Existence results

By applying Lemma 2.4 and ${F}_{u}(t) = f(t, u(t), {_{\rho_{k}}^{}}{I}_{t_{k}}^{\sigma_{k}; \psi_{k}} u(t), {_{\rho_{k}}^{}}{I}_{t_{k}}^{\nu_{k}; \psi_{k}} u(t))$, we set an operator ${Q}: {PC} \to {PC}$ by

Note that, the considered problem (1.4) has a solution if and only if ${Q}$ has fixed points.

We assign notation for constants that will be used throughout this work

3.1.   Uniqueness result under Banach's fixed point theorem

Lemma 3.1. (Banach's fixed point theorem ^[42]) Let ${D}$ be a non-empty closed subset of a Banach space $\mathfrak{E}$. Then any contraction mapping ${Q}$ from ${D}$ into itself has a unique fixed-point.

Theorem 3.1. Assume $\psi_{k} \in {C}^{2}({J})$ where $\psi_{k}^{\prime}(t) > 0$, $k = 0, 1, 2, \ldots, m$, $t \in {J}$ and $f \in {C}({J}\times{R}^{3}, {R})$, $\phi_{k}$, $\phi_{k}^{*} \in {C}({R}, {R})$, $k = 1, 2, \ldots, m$, corresponding to the following conditions:

$({H}_{1})$ There are real constants ${L}_{i} > 0$, $i = 1, 2, 3$, so that, for any $t \in {J}$ and $u_{i}$, $v_{i} \in {R}$, $i = 1, 2, 3$,

$({H}_{2})$ There are real constants ${I}_{i} > 0$, $i = 1, 2$, so that, for any $t \in {J}$ and $u$, $v \in {R}$, $k = 1, 2, \ldots, m$,

Then, the considered problem (1.4) has a unique solution on ${J}$, if

where

Proof. Clearly, the considered problem (1.4) is corresponding to fixed-point problem $u = {Q}u$. Then we will show that ${Q}$ has a fixed-point by the Banach's fixed-point theorem.

Define constants ${M}_{i}$, $i = 1, 2, 3$, by ${M}_{1} : = \sup_{t \in {J}}\left\vert f(t, 0, 0, 0) \right\vert$, ${M}_{2} : = \max\left\{ \phi_{k}(0) : k = 1, 2, \ldots, m \right\}$ and ${M}_{3} : = \max\left\{ \phi_{k}^{*}(0) : k = 1, 2, \ldots, m \right\}$. Let $\mathfrak{B}_{\mathfrak{R}_{1}} : = \left\{ u \in \mathfrak{E} : \left\Vert u \right\Vert \leq \mathfrak{R}_{1} \right\}$ where

The remaining proof is divided into two steps:

Step Ⅰ. We will prove that ${Q}\mathfrak{B}_{\mathfrak{R}_{1}} \subset \mathfrak{B}_{\mathfrak{R}_{1}}$.

Suppose that $u \in \mathfrak{B}_{\mathfrak{R}_{1}}$ and $t \in {J}$, we obtain

By using the property $(i)$ in Lemma 2.1, we get

From the conditions $({H}_{1})$, $({H}_{2})$ and (3.13), we can find that

Inserting (3.14)–(3.16) into (3.12), we see that

From the property $(i)$ in Lemma 2.1, it implies that

Hence, $\left\Vert {Q}u \right\Vert_{{PC}} \leq \mathfrak{R}_{1}$, which yields that ${Q}\mathfrak{B}_{\mathfrak{R}_{1}} \subset \mathfrak{B}_{\mathfrak{R}_{1}}$.

Step Ⅱ. We will prove that ${Q}$ is a contraction.

Suppose that $u$, $v \in \mathfrak{B}_{\mathfrak{R}_{1}}$ and $t \in {J}$, we have

By applying the property $(i)$ in Lemma 2.1, we have

Using the conditions $({H}_{1})$, $({H}_{2})$ and (3.18), we can find that

Inserting (3.19)–(3.21) into (3.17), which yields that

It follows that $\left\Vert {Q}u - {Q}v \right\Vert_{{PC}} \leq [\Delta_{1} + \Delta_{2}]\Vert u - v \Vert_{{PC}}$. Condition (3.8) stated that $\Delta_{1} + \Delta_{2} < 1$. Thus ${Q}$ is a contraction. By Lemma 3.1, problem (1.4) has a unique solution on ${J}$.

3.2.   Existence result under O'Regan's fixed point theorem

Lemma 3.2. (O'Regan's fixed point theorem ^[43]) Let ${O}$ be an open subset of a closed, convex set ${D}$ in a Banach space $\mathfrak{E}$ such that $0 \in {O}$. Moreover, let ${Q}:\overline{{O}} \to {D}$ be such that ${Q}(\overline{{O}})$ is bounded and that ${Q} = {Q}_{1} + {Q}_{2}$, where ${Q}_{1}:\overline{{O}} \to {D}$ is continuous and completely continuous and ${Q}_{2}:\overline{{O}} \to {D}$ is a nonlinear contraction, i.e., there exists a nonnegative nondecreasing function $\Theta: [0, \infty) \to [0, \infty)$, such that $\Theta(w) < w$ for any $w \in {R}^{+}$ and $\left\Vert {Q}_{2} u - {Q}_{2}v \right\Vert \leq \Theta(\left\Vert u - v \right\Vert)$ for all $u$, $v \in \overline{{O}}$. Then either $(a_{1}).$ ${Q}$ has a fixed point $u \in \overline{{O}}$ or $(a_{2}).$ there exist a point $u \in \partial{K}$ and $\theta \in (0, 1)$, such that $u = \theta{Q}u$. Here, $\overline{{O}}$ and $\partial{O}$ represent the closure and the boundary of ${O}$, respectively.

Theorem 3.2. Assume $\psi_{k} \in {C}^{2}({J})$ where $\psi_{k}^{\prime}(t) > 0$, $k = 0, 1, 2, \ldots, m$, $t \in {J}$, $f \in {C}({J}\times{R}^{3}, {R})$, $\phi_{k}$, $\phi_{k}^{*} \in {C}({R}, {R})$, $k = 1, 2, \ldots, m$ satisfying the following conditions:

$({H}_{3})$ There exist positive real numbers ${M}_{1}$, ${M}_{2}$ such that

$({H}_{4})$ There exist a continuous nondecreasing function $\Theta:[0, \infty) \to [0, \infty)$ and $g_{i} \in {C}({J}, {R}^{+})$, $i = 1, 2, 3$, such that

for any $u$, $v$, $w \in {R}$, $t \in {J}$, $k = 1, 2, \ldots, m$.

$({H}_{5})$ There exist continuous nondecreasing functions ${K}_{i}: [0, \infty) \to [0, \infty)$, and $\Xi_{i}$, $i = 1, 2$, such that

for any $u$, $v \in {R}$, $k = 1, 2, \ldots, m$, satisfying $[(\Omega_{5} + \Omega_{1} \Omega_{4}) \Xi_{1} + (m \Psi_{*}^{\gamma_{m}} + \Omega_{1} \Omega_{6}) \Xi_{2}] < 1$ where $\Omega_{1}$, $\Omega_{4}$, $\Omega_{5}$, $\Omega_{6}$ are given by (3.2) and (3.5)–(3.7), respectively.

$({H}_{6})$

with $[g_{2}^{*} \Psi_{*}^{\sigma_{m}} + g_{3}^{*} \Psi_{*}^{\nu_{m}}](\Omega_{3} + \Omega_{1} \Omega_{2}) < 1$, $\Omega_{i}$ are given by (3.2)–(3.4), respectively, and $g_{i}^{*} = \sup_{t \in {J}}\vert g_{i}(t) \vert$, $i = 1, 2, 3$.

Then the considered problem (1.4) has at least one solution on ${J}$.

Proof. We will divide the operator ${Q} : {PC} \to {PC}$ defined by (3.1) into two operators, that is ${Q}_{1}$ and ${Q}_{2}$, where $({Q}u)(t) = ({Q}_{1}u)(t) + ({Q}_{2}u)(t)$, for any $t \in {J}$. The operators ${Q}_{1}$ and ${Q}_{2}$ are defined by

Next, assume that $\mathfrak{B}_{\mathfrak{R}_{2}} = \{ u \in \mathfrak{E} : \Vert u \Vert_{{PC}} \leq \mathfrak{R}_{2} \}$ such that

Thanks to Theorem 3.1, we see that ${Q}_{1}$ is continuous. For any $t \in {J}$, we have

From condition $({H}_{4})$, we obtain

Substituting (3.29) into (3.28) and using the property $(i)$ in Lemma 2.1, we have

This yields that ${Q}_{1}(\mathfrak{B}_{\mathfrak{R}_{2}}) \leq (g_{1}^{*}\Theta(\mathfrak{R}_{2}) + [g_{2}^{*} \Psi_{*}^{\sigma_{m}} + g_{3}^{*} \Psi_{*}^{\nu_{m}}] \mathfrak{R}_{2}) (\Omega_{3} + \Omega_{1} \Omega_{2})$.

Now, we will prove that ${Q}_{1}$ maps bounded set $\mathfrak{B}_{\mathfrak{R}_{2}}$ into equicontinuous set of $\mathfrak{E}$. Suppose that $\tau_{1}$, $\tau_{2} \in {J}_{k}$, $k = 0, 1, \ldots, m$, under $\tau_{1} < \tau_{2}$ and for any $u\in\mathfrak{B}_{\mathfrak{R}_{2}}$, we obtain that

Observe that the above result is independent of $u \in \mathfrak{B}_{\mathfrak{R}_{2}}$. This implies that

Since ${Q}_{1}$ maps bounded set $\mathfrak{B}_{\mathfrak{R}_{2}}$ into an equicontinuous set of $\mathfrak{E}$, by the Arzelá-Ascoli theorem, we obtain that ${Q}_{1}$ is completely continuous.

Next, we will prove that ${Q}_{2}$ is a nonlinear contraction. Let $\Theta:{R}^{+}\to{R}^{+}$ be a continuous nondecreasing function given by $\Theta(\epsilon) = [(\Omega_{5} + \Omega_{1} \Omega_{4}) \Xi_{1} + (m \Psi_{*}^{\gamma_{m}} + \Omega_{1} \Omega_{6}) \Xi_{2}] \epsilon$, for all $\epsilon \geq 0$. It is easy to see that $\Theta(0) = 0$. Since $[(\Omega_{5} + \Omega_{1} \Omega_{4}) \Xi_{1} + (m \Psi_{*}^{\gamma_{m}} + \Omega_{1} \Omega_{6}) \Xi_{2}] < 1$, we have $\Theta(\epsilon) < \epsilon$ for all $\epsilon > 0$. For any $u$, $v \in \mathfrak{B}_{\mathfrak{R}_{2}}$, we obtain

By taking $\Theta(\epsilon) = [(\Omega_{5} + \Omega_{1} \Omega_{4}) \Xi_{1} + (m \Psi_{*}^{\gamma_{m}} + \Omega_{1} \Omega_{6}) \Xi_{2}] \epsilon$, we have $\Theta(0) = 0$ and $\Theta(\epsilon) < \epsilon$ for all $\epsilon > 0$. Then

This yields that ${Q}_{2}$ is a nonlinear contraction.

Next, we will prove that ${Q}_{2}(\mathfrak{B}_{\mathfrak{R}_{2}})$ is bounded. By $({H}_{3})$, for any $u \in \mathfrak{B}_{\mathfrak{R}_{2}}$, it follows that

Then, ${Q}_{2}(\mathfrak{B}_{\mathfrak{R}_{2}})$ is bounded with the boundedness of the set ${Q}_{1}(\mathfrak{B}_{\mathfrak{R}_{2}})$.

Lastly, we will prove that the assumption $(a_{2})$ in Lemma 3.2 is not true. Suppose that $(a_{2})$ is true. Then there exists a constant $\theta \in (0, 1)$ such that $u = \theta {Q} u$ for any $u \in \mathfrak{B}_{\mathfrak{R}_{2}}$. We obtain that $\Vert u \Vert_{{PC}} \leq \mathfrak{R}_{2}$ and

which implies

Hence,

where

this contradicts the condition $({H}_{6})$. Therefore, ${Q}_{1}$ and ${Q}_{2}$ satisfy all conditions of Lemma 3.2. Hence, the considered problem (1.4) has a solution on ${J}$.

4.   Ulam stability results

This section discusses a variety of Ulam-Hyers stability of the considered problem (1.4). Before proving, we will state Ulam-Hyers stability ideas for the considered problem (1.4). Assume that $\chi \in {C}({J}, {R}^{+})$ is a non-decreasing function and $\epsilon > 0$, $\delta \geq 0$, $z \in {E}$, so that for any $t \in {J}_{k}$, $k = 1, 2, \ldots, m$ the following important inequalities are satisfied:

Definition 4.1. The considered problem (1.4) is said to be Ulam–Hyers $({UH})$ stable, if there exists a real constant $\mathfrak{C}_{{F}} > 0$ so that for every $\epsilon > 0$ and for any $z \in {E}$ of (4.1) there exists $u \in {E}$ of (1.4) that satisfies

Definition 4.2. The considered problem (1.4) is said to be generalized Ulam-Hyers $({GUH})$ stable, if there exists $\chi \in {C}({R}^{+}, {R}^{+})$ via $\chi(0) = 0$ so that for every $\epsilon > 0$ and for any $z \in {E}$ of (4.2) there exists $u \in {E}$ of (1.4) that satisfies

Definition 4.3. The considered problem (1.4) is said to be Ulam-Hyers-Rassias $({UHR})$ stable with respect to $(\delta, \chi)$, if there exists a real constant $\mathfrak{C}_{{F}, \chi_{{F}}} > 0$ so that for every $\epsilon > 0$ and for any $z \in {E}$ of (4.3) there exists $u \in {E}$ of (1.4) that satisfies

Definition 4.4. The considered problem (1.4) is said to be generalized Ulam-Hyers-Rassias $({GUHR})$ stable with respect to $(\delta, \chi)$, if there exists a real constant $\mathfrak{C}_{{F}, \chi_{{F}}} > 0$ so that for any $z \in {E}$ of (4.2) there exists $u \in {E}$ of (1.4) that satisfies

Remark 4.1. By Definitions 4.1–4.4, we will find out that: $({R}_{1})$ Definition 4.1 $\Rightarrow$ Definition 4.2; $({R}_{2})$ Definition 4.3 $\Rightarrow$ Definition 4.4; and $({R}_{3})$ Definition 4.3 $\Rightarrow$ Definition 4.1.

Remark 4.2. Assume that $z \in {E}$ is the solution of (4.1). If there exists $g \in {E}$ with a sequence $g_{k}$ for $k = 1, 2, \ldots, m$, depending on a function $z$, such that $({A}_{1})$ $\vert g(t) \vert \leq \epsilon$, $\vert g_{k} \vert \leq \epsilon$, $t \in {J}$; $({A}_{2})$ ${_{\rho_{k}}^{H}}\mathfrak{D}_{t_{k}^+}^{\alpha_{k}, \beta_{k}; \psi_{k}} z(t) = f(t, z(t), {_{\rho_{k}}^{}}{I}_{t_{k}}^{\sigma_{k}; \psi_{k}} z(t), {_{\rho_{k}}^{}}{I}_{t_{k}}^{\nu_{k}; \psi_{k}} z(t)) + g(t)$, $t \in {J}$; $({A}_{3})$ ${_{\rho_{k}}^{}}{I}_{t_{k}^{+}}^{\rho_{k}(2-\gamma_{k}); \psi_{k}} z(t_{k}^{+}) - {_{\rho_{k-1}}^{}}{I}_{t_{k-1}^{+}}^{\rho_{k-1}(2-\gamma_{k-1}); \psi_{k-1}} z(t_{k}^{-}) = \phi_{k}(z(t_{k})) + g_{k}$, $t \in {J}$; and $({A}_{4})$ ${_{\rho_{k}}^{{RL}}}\mathfrak{D}_{t_{k}^{+}}^{\rho_{k}(\gamma_{k}-1); \psi_{k}} z(t_{k}^{+}) - {_{\rho_{k-1}}^{{RL}}}\mathfrak{D}_{t_{k-1}^{+}}^{\rho_{k-1}(\gamma_{k-1}-1); \psi_{k-1}} z(t_{k}^{-}) = \phi_{k}^{*}(z(t_{k})) + g_{k}$, $t \in {J}$.

Remark 4.3. Assume that $z \in {E}$ is the solution of (4.2). If there exists $g \in {E}$ and $g_{k}$ for $k = 1, 2, \ldots, m$, depending on a function $z$, such that $({B}_{1})$ $\vert g(t) \vert \leq \chi(t)$, $\vert g_{k} \vert \leq \delta$, $t \in {J}$; $({B}_{2})$ ${_{\rho_{k}}^{H}}\mathfrak{D}_{t_{k}^+}^{\alpha_{k}, \beta_{k}; \psi_{k}} z(t) = f(t, z(t), {_{\rho_{k}}^{}}{I}_{t_{k}}^{\sigma_{k}; \psi_{k}} z(t), {_{\rho_{k}}^{}}{I}_{t_{k}}^{\nu_{k}; \psi_{k}} z(t)) + g(t)$, $t \in {J}$; $({B}_{3})$ ${_{\rho_{k}}^{}}{I}_{t_{k}^{+}}^{\rho_{k}(2-\gamma_{k}); \psi_{k}} z(t_{k}^{+}) - {_{\rho_{k-1}}^{}}{I}_{t_{k-1}^{+}}^{\rho_{k-1}(2-\gamma_{k-1}); \psi_{k-1}} z(t_{k}^{-}) = \phi_{k}(z(t_{k})) + g_{k}$; and $({B}_{4})$ ${_{\rho_{k}}^{{RL}}}\mathfrak{D}_{t_{k}^{+}}^{\rho_{k}(\gamma_{k}-1); \psi_{k}} z(t_{k}^{+}) - {_{\rho_{k-1}}^{{RL}}}\mathfrak{D}_{t_{k-1}^{+}}^{\rho_{k-1}(\gamma_{k-1}-1); \psi_{k-1}} z(t_{k}^{-}) = \phi_{k}^{*}(z(t_{k})) + g_{k}$, $t \in {J}$.

Remark 4.4. Assume that $z \in {E}$ is the solution of (4.3). If there exists $g \in {E}$ and $g_{k}$ for $k = 1, 2, \ldots, m$, depending on a function $z$, such that $({C}_{1})$ $\vert g(t) \vert \leq \epsilon \chi(t)$, $\vert g_{k} \vert \leq \epsilon \delta$, $t \in {J}$; $({C}_{2})$ ${_{\rho_{k}}^{H}}\mathfrak{D}_{t_{k}^+}^{\alpha_{k}, \beta_{k}; \psi_{k}} z(t) = f(t, z(t), {_{\rho_{k}}^{}}{I}_{t_{k}}^{\sigma_{k}; \psi_{k}} z(t), {_{\rho_{k}}^{}}{I}_{t_{k}}^{\nu_{k}; \psi_{k}} z(t)) + g(t)$, $t \in {J}$; $({C}_{3})$ ${_{\rho_{k}}^{}}{I}_{t_{k}^{+}}^{\rho_{k}(2-\gamma_{k}); \psi_{k}} z(t_{k}^{+}) - {_{\rho_{k-1}}^{}}{I}_{t_{k-1}^{+}}^{\rho_{k-1}(2-\gamma_{k-1}); \psi_{k-1}} z(t_{k}^{-}) = \phi_{k}(z(t_{k})) + g_{k}$; and $({C}_{4})$ ${_{\rho_{k}}^{{RL}}}\mathfrak{D}_{t_{k}^{+}}^{\rho_{k}(\gamma_{k}-1); \psi_{k}} z(t_{k}^{+}) - {_{\rho_{k-1}}^{{RL}}}\mathfrak{D}_{t_{k-1}^{+}}^{\rho_{k-1}(\gamma_{k-1}-1); \psi_{k-1}} z(t_{k}^{-}) = \phi_{k}^{*}(z(t_{k})) + g_{k}$, $t \in {J}$.

4.1.   Ulam-Hyers stability results

Theorem 4.1. Assume that $\alpha_{k} \in (1, 2]$, $\beta_{k} \in [0, 1]$, $\rho_{k} \in {R}^{+}$, $\gamma_{k} = (\beta_{k}(2\rho_{k}-\alpha_{k})+\alpha_{k})/\rho_{k}$, $\psi_{k} \in {C}({J}, {R})$ where $\psi_{k}^{\prime} > 0$, $k = 1, 2, \ldots, m$ and $f \in {C}({J}\times{R}^{3}, {R})$. If the assumptions $({H}_{1})$ and $({H}_{2})$ and the inequality (3.8) hold, then the considered problem (1.4) is UH stable on ${J}$.

Proof. Assume that $z \in {PC}$ is the solution of the problem (4.1). Under the conditions $({A}_{2})$ and $({A}_{3})$ of Remark 4.2 and Lemma 2.4, we have

Then, the solution of (4.8) is given by

By applying $({A}_{1})$ of Remark 4.2 with $({H}_{1})$ and $({H}_{2})$, we obtain

This yields that $\Vert z-u \Vert_{{PC}} \leq \mathfrak{C}_{{F}} \epsilon$, where $\mathfrak{C}_{{F}}$ is given by

Hence, the considered problem (1.4) is UH stable in ${E}$.

Corollary 4.1. By taking $\chi(\epsilon) = \mathfrak{C}_{{F}} \epsilon$ and $\chi(0) = 0$ in Theorem 4.1, we obtain the considered problem (1.4) is GUH stable.

4.2.   Ulam-Hyers-Rassias stability results

To prove UHR and GUHR stability results, we will require the following assumption:

$({U}_{1})$ There exist a non-decreasing function $\chi \in {C}({J}, {R})$ and a positive real constant $\mathfrak{C}_{\chi} > 0$ such that

Here we give notation for the constants

Theorem 4.2. Assume that $\alpha_{k} \in (1, 2]$, $\beta_{k} \in [0, 1]$, $\rho_{k} \in {R}^{+}$, $\gamma_{k} = (\beta_{k}(2\rho_{k}-\alpha_{k})+\alpha_{k})/\rho_{k}$, $\psi_{k} \in {C}({J}, {R})$ where $\psi_{k}^{\prime} > 0$, $k = 1, 2, \ldots, m$ and $f \in {C}({J}\times{R}^{3}, {R})$. If the assumptions $({H}_{1})$ and $({H}_{2})$ and the inequality (3.8) hold, then the considered problem (1.4) is UHR stable with respect to $(\delta, \chi)$ on ${J}$.

Proof. Assume that $z \in {E}$ is any solution of (4.3) and $u \in {E}$ is a solution of the considered problem (1.4). By the same argument as in Theorem 4.1, it follows that

Under $({C}_{1})$ of Remark 4.4 and $({H}_{1})$, $({H}_{2})$ and $({U}_{1})$, we see that

It follows that, $\Vert z - u \Vert_{{PC}} \leq \mathfrak{C}_{{F}, \chi_{{F}}} \; \epsilon \; (\delta + \chi(t))$, where

Therefore, the considered problem (1.4) is UHR stable with respect to $(\delta, \chi)$ in ${E}$.

Corollary 4.2. By taking $\epsilon = 1$ and $\chi(0) = 0$ in Theorem 4.2, we obtain the considered problem (1.4) is GUHR stable.

5.   Applications

Example 5.1. Consider the following impulsive problem of the form:

From the considered problem (5.1), we set $\alpha_k = (2k+8)/7$, $\beta_k = (3-k)/4$, $\rho_k = (3k+46)/50$, $\psi_k(t) = 1/(k+2) + \sin((k+2)t/((k+3)t-k+5))$, $\sigma_k = (3k+2)/(5-k)$, $\nu_k = (2k+3)/8$, $t_k = k/2$, $k = 0, 1, 2$, $T = 3/2$, $\mu_i = (4i+3)/(12-2i)$, $\eta_i = (2i+2)/5$, $\lambda_l = (2l+2)/(7-2l)$, $\theta_l = (2l+3)/4$, $\xi_l = (3l+2)/6$, $i = 0, 1, 2$, $l = 0, 1, 2$ and ${A} = e$. Thanks to the given data, we can compute that $\Lambda \approx 1.319519900$, $\Omega_1 \approx 0.226529808$, $\Omega_2 \approx 0.656891205$, $\Omega_3 \approx 1.166135348$, $\Omega_4 \approx 0.688903756$, $\Omega_5 \approx 0.228769110$, $\Omega_6 \approx 6.890783193$, $\Omega_7 \approx 5.410714286$, $\Omega_8 \approx 0.341866237$, and $\Omega_9 \approx 0.707853736$. The following functions will be considered for theoretical confirmation:

For any $u_i$, $v_i$, $w_i \in {R}$, $i = 1, 2$, and $t \in [0, 3/2]$, it follows that

It is easy to see that the conditions $({H}_1)$ and $({H}_2)$ are fulfilled under ${L}_1 = 3/10$, ${L}_2 = 2/5$, ${L}_3 = 3/7$, ${I}_1 = 1/4$ and, ${I}_2 = 2/35$. Then we have $\Delta_{1} \approx 0.449865758$ and $\Delta_{2} \approx 0.278219606$, which implies that $\Delta_{1}+\Delta_{2} \approx 0.728085364 < 1$. Since all the conditions of Theorem 3.1 are satisfied, the considered problem (5.1) has a unique solution on $[0, 3/2]$. Furthermore, thanks of (4.10), we get

Therefore, the considered problem (5.1) is UH stable on $[0, T]$. By setting $\chi(\epsilon) = \mathfrak{C}_{{F}} \epsilon$ via $\chi(0) = 0$, we obtain from Corollary 4.1 that the considered problem (5.1) is GUH stable on $[0, 3/2]$. Moreover, if we put $\chi(t) = \Psi_{\psi_{k}}^{\frac{3}{\rho_{k}}}(t, t_{k})$ into $({U}_{1})$, we have

Then, we have

By applying (4.14), one has

Then, by Theorem 4.2, the considered problem (5.1) is UHR stable on $[0, 3/2]$. Finally, if we set $\chi(\epsilon) = \mathfrak{C}_{{F}, \chi_{{F}}} \epsilon$ via $\chi(0) = 0$ and $\epsilon = 1$ in Corollary 4.2, we obtain that the considered problem (5.1) is GUHR stable with respect to $(\delta, \chi)$ on $[0, 3/2]$. In addition, we will present the graphical relations between $\Delta_{1} + \Delta_{2}$, $\alpha_{k}$, and $\beta_{k} \in [0, 1]$ for $k = 0, 1, 2$ in Figure 1, while Table 1 shows the relationship between $\alpha_{k}$, $\beta_{k}$, $\Lambda$, $\Omega_{i}$, $i = 1, 2, \ldots, 6$, and $\Delta_{1} + \Delta_{2} < 1$.

Example 5.2. Consider the following impulsive problem of the form:

From the considered problem (5.1), we set $\alpha_k = (e^{k-1}+2)/(e^{k-1}+1)$, $\beta_k = (3-k)/4$, $\rho_k = (3k+46)/50$, $\psi_k(t) = (t^{t^2-k+2})/(t+2k+10)$, $\sigma_k = (3k+2)/(5-k)$, $\nu_k = (2k+3)/8$, $t_k = k/2$, $k = 0, 1, 2$, $T = 3/2$, $\mu_i = (4i+3)/(12-2i)$, $\eta_i = (2i+2)/5$, $\lambda_l = (2l+2)/(7-2l)$, $\theta_l = (2l+3)/4$, $\xi_l = (3l+2)/6$, $i = 0, 1, 2$, $l = 0, 1$ and ${A} = e$. Thanks to the given data, we can compute that $\Lambda \approx 0.812170396$, $\Omega_1 \approx 0.167309417$, $\Omega_2 \approx 0.079568573$, $\Omega_3 \approx 1.027705947$, $\Omega_4 \approx 0.923594556$, $\Omega_5 \approx 0.229753595$, $\Omega_6 \approx 14.887728830$. The following functions will be considered for theoretical confirmation:

For any $u$, $v$, $w \in {R}$, and $t \in [0, 3/2]$, it follows that

By applying $({H}_{3})$–$({H}_{5})$ with $\Theta(u) = 2|u|+1$, we obtain that ${M}_1 = 0.296890807$, ${M}_2 = 0.178134484$, $g_{1}^{*} = 0.273381295$, $g_{2}^{*} = 0.249373747$, $g_{3}^{*} = 0.750430599$, $\Xi_1 = 1/5$, $\Xi_2 = 1/10$, and

By using (3.30) and $({H}_{6})$, we have that

Then,

which yields $\mathfrak{R}_{2} > 4.041270259$. Since all the conditions of Theorem 3.2 are satisfied, the considered problem (5.2) has a solution on $[0, 3/2]$.

Example 5.3. Consider the following the impulsive $(\rho_{k}, \psi_{k})$-${HFP}$-${IDE}$-${MIBC}$s of the form:

Form the considered problem (5.3), we set $\alpha_k \in \{(2\pi+k-2)/4, 1+\sqrt{(k+1)/(k+6)}, (e^{k-1}+2)/(e^{k-1}+1), \ln(k+4)\}$, $\beta_k = (4-k)/5$, $\rho_k = (k+18)/20$, $\psi_k(t) \in \{1/(k+2) + \sin((k+2)t/((k+3)t-k+5))$, $(k+4)/2-\arccos((t^2+kt-2)/10), (t^{t^2-k+3})/(t+2k+8)$, $2-(\ln[(k+2)t+3k+3])/(\ln\left[(k+1)t+2k+2\right])\}$, $t_k = k/2$, $k = 0, 1, 2, 3$, $T = 2$, $\phi_{k}(u(t_{k})) = -(k+1)/(k+2)$, $\phi_{k}^{*}(u(t_{k})) = (-1)^k (3/2)$, $k = 1, 2, 3$, $\mu_i = (4i+3)/(12-2i)$, $\eta_i = (2i+2)/5$, $\lambda_l = (2l+2)/(7-2l)$, $\theta_l = (2l+3)/4$, $\xi_l = (3l+2)/6$, $i = 0, 1, 2, 3$, $l = 0, 1, 2, 3$, and ${A} = e$. Thanks to the given data, we can compute that $\Lambda \approx 1.3195199$. By using Lemma 2.6 with $f(t, u(t), {_{\rho_{k}}^{}}{I}_{t_{k}}^{\sigma_{k}; \psi_{k}} u(t), {_{\rho_{k}}^{}}{I}_{t_{k}}^{\nu_{k}; \psi_{k}} u(t)) = 2$, the solution of the considered problem (5.3) can be written as

Hence, the solution of the considered problem (5.3) is divided into three cases.

Case Ⅰ. If we set $\alpha_{k} \in \{ \pi/2 + (k-2)/4$, $1 + \sqrt{(k+1)/(k+6)}$, $(e^{k-1}+2)/(e^{k-1}+1)$, $\ln(k+4)\}$ and $\psi_{k}(t) = 1/(k+2) + \sin(((k+2)t)/((k+3)t+(5-k)))$ for $k = 0, 1, 2, 3$, then the solution of the considered problem (5.3) is displayed in Figure 2.

Case Ⅱ. If we set $\alpha_{k} = \pi/2 + (k-2)/4$ and $\psi_{k}(t) \in \{ 1/(k+2) + \sin(((k+2)t)/((k+3)t-k+5)), (k+4)/(2) - \arccos((t^2+kt-2)/(10)), (t^{t^2-k+3})/(t+2k+8), 2 - (\ln[(k+2)t+3k+3])/(\ln[(k+1)t+2k+2]) \}$ for $k = 0, 1, 2, 3$, then the solution of the considered problem (5.3) is displayed in Figure 3.

Case Ⅲ. If we set $\alpha_{k} \in \{ \pi/2 + (k-2)/4$, $1 + \sqrt{(k+1)/(k+6)}$, $(e^{k-1}+2)/(e^{k-1}+1)$, $\ln(k+4)\}$ and $\psi_{k}(t) \in \{ 1/(k+2) + \sin(((k+2)t)/((k+3)t-k+5)), (k+4)/(2) - \arccos((t^2+kt-2)/(10)), (t^{t^2-k+3})/(t+2k+8), 2 - (\ln[(k+2)t+3k+3])/(\ln[(k+1)t+2k+2]) \}$ for $k = 0, 1, 2, 3$, then the solution of the considered problem (5.3) is displayed in Figure 4.

6.   Conclusions

In this paper, we have investigated existence theory and stability results for a class of nonlinear impulsive boundary value problem of fractional integro-differential equations supplemented with mixed nonlocal multi-point and multi-term integral boundary conditions in the context of the $(\rho_{k}, \psi_{k})$-Hilfer fractional derivatve. Firstly, the solution to the linear variant impulsive considered problem was introduced in terms of a Volterra integral equation. The uniqueness result was proved by using Banach's fixed point theorem, while the existence result was established by means of a fixed point theorem due to O'Regan. In addition, a variety of Ulam's stability such as UH, GUH, UHR and GUHR stability were studied by applying nonlinear functional analysis technique. Finally, three examples illustrating the results are also provided to confirm the correctness of the theoretical results. The novelty of our results is not only finding a distinctive qualitative theory for this problem within the given frame but also addressing some new, interesting exceptional cases for various values of the parameters related to the considered problem. For example,

$(i)$ If we set $\lambda_{l} = 0$ for all $l = 0, 1, \ldots, n$, then the considered problem (1.4) reduces to ${BVP}$ for nonlinear impulsive $(\rho_{k}, \psi_{k})$-Hilfer-FIDEs under nonlocal multi-point boundary conditions: $u(0) = 0$, $\sum_{i = 0}^{m}\mu_{i} u(\eta_{i}) = {A}$.

$(ii)$ If we set $\mu_{i} = 0$ for all $i = 0, 1, \ldots, m$, then the considered problem (1.4) reduces to ${BVP}$ for nonlinear impulsive $(\rho_{k}, \psi_{k})$-Hilfer-FIDEs under nonlocal multi-term integral boundary conditions: $u(0) = 0,\sum\nolimits_{l = 0}^n {{\lambda _l}{_{{\rho _l}}} I}_{{t_l}}^{{\theta _l};{\psi _l}}u({\xi _l}) = A$.

This research would provide a significant contribution to the literature on the qualitative theory, which might involve the growth of the idea introduced in this field as well as the possibility for further generalizations in a wide range of exclusive outputs for applications and theories. One proposal is that future studies explore the existence and uniqueness of solutions for additional forms of nonlinear differential-integral equations in the setting of other fractional operators with varied boundary conditions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

W. Sudsutad would like to thank you for supporting this paper through Ramkhamhaeng University. C. Thaiprayoon and J. Kongson would like to extend their appreciation to Burapha University.

Conflict of interest

The authors declare no conflict of interest.