Ulam stability for nonlinear implicit differential equations with Hilfer-Katugampola fractional derivative and impulses

1.   Introduction

Numerous authors have been interested in fractional differential equations throughout the years ^{[1,4,5,10,12,13,14,16,18,19,20,28]}. Several natural events are known to be modeled using fractional differential equations, which provides for a better description of the true state of the problem as compared to the problem modeled using differential equations of integer order ^[8,20,24,27].

Fractional calculus has played an essential role in several domains during the last two decades, including mechanics, chemistry, economics, biology, control theory, and signal and image processing. Furthermore, it has been discovered that fractional order models may accurately capture the dynamical behavior of many complex systems. Such models are appealing not just to engineers and physicists, but also to mathematicians. For further details and applications ^{[2,3,17,20,21,23,29,31,32,33,34,35]}.

Several researchers have recently explored impulsive differential equations given their considerable applicability in diverse domains of science and technology. For a detailed study, see for instance ^[9,11,25].

Motivated by the aforementioned works and the paper ^[26], in this paper, we consider the following impulsive problem:

where $^\rho D_{{\alpha}^{+}}^{{\zeta_1},{\zeta_2}},^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}$ are the Hilfer-Katugampola fractional derivative of order ${\zeta_1} \in(0,1)$ and type ${\zeta_2} \in [0,1]$ and Katugampola fractional integral of order $1-{\xi},({\xi}={\zeta_1}+{\zeta_2}-{\zeta_1}{\zeta_2})$ respectively, ${\psi} :{\Theta}\times {{\mathbb R}} \times {{\mathbb R}} \to {{\mathbb R}} ,$ and $0<{\alpha}={\vartheta}_{0}<{\vartheta}_{1}<\dots <{\vartheta}_{{\varsigma}}<{\vartheta}_{{\varsigma}+1}={\mu}$, $\Delta ^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}{\mathfrak{p}}|_{{\vartheta}={\vartheta}_{\nu}}= \ ^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}{\mathfrak{p}}({\vartheta}_{{\nu}}^{+})-^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}{\mathfrak{p}}({\vartheta}_{{\nu}}^{-})$,where $^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}{\mathfrak{p}}({\vartheta}_{{\nu}}^{+})=\underset{{\kappa}\rightarrow 0^+}{\lim }^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}{\mathfrak{p}}({\vartheta}_{\nu}+{\kappa})$ and $^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}{\mathfrak{p}}({\vartheta}_{{\nu}}^{-})=\underset{{\kappa}\rightarrow 0^-}{\lim }^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}{\mathfrak{p}}({\vartheta}_{{\nu}}+{\kappa})$ represent the right and left limits of $^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}{\mathfrak{p}}({\vartheta})$ at ${\vartheta}={\vartheta}_{{\nu}}$.

The following is the structure of paper. Section 2 presents certain notations and revisits several notions and auxiliary results. Section 3 presents two results for the problems (1.1)–(1.3) by employing suitable fixed point theorems. In Section 4, the Ulam-Hyers stability for the problems (1.1)–(1.3) is given. Finally, we give an example to illustrate the applicability of our theoretical results.

2.   Preliminaries

Let $0<{\alpha}<{\mu}, {\Theta}=[{\alpha},{\mu}]$ and $C({\Theta},{{\mathbb R}})$ denotes a Banach space composed of all continuous functions from ${\Theta}$ into ${{\mathbb R}}$ with the norm

Consider the weighted spaces

with the norms

and

Consider the weighted Banach space of piecewise continuous functions defined by

with the norm

For ${\xi}=1,$ we obtain the space

with the norm

Now, we consider the weighted spaces:

with the norms

By $X_{{\tilde{\alpha}}}^{p}({\alpha},{\bar{\alpha}}), \ ({\tilde{\alpha}}\in {{\mathbb R}},\ 1\leq p \leq \infty)$, we denote the space of those complex-valued Lebesgue measurable functions ${\psi}$ on $[{\alpha},{\bar{\alpha}}]$ for which $\|{\psi}\|_{X_{{\tilde{\alpha}}}^{p}}<\infty,$ with the norm

Definition 2.1. ^[26] Let ${\zeta_1}\in{{\mathbb R}}_{+},{\tilde{\alpha}}\in{{\mathbb R}}$ and ${\varkappa}\in X_{{\tilde{\alpha}}}^{p}({\alpha},{\bar{\alpha}}).$ The Katugampola fractional integral of order ${\zeta_1}$ is given by

Definition 2.2. ^[26] Let ${\zeta_1}\in{{\mathbb R}}_{+}\setminus \mathbb{N}$ and $\rho>0.$ The Katugampola fractional derivative $^{\rho}D_{{\alpha}^{+}}^{{\zeta_1}}$ of order ${\zeta_1}$ is given by

where ${\beta}=[{\zeta_1}]+1$ and $\delta_{\rho}^{{\beta}}=\left({\vartheta}^{1-\rho} \frac {d}{d{\vartheta}}\right)^{{\beta}}.$

Theorem 2.3. ^[26] Let ${\zeta_1} >0, {\zeta_2} >0, 1\leq p \leq \infty, 0<{\alpha}<{\bar{\alpha}}<\infty$ and $\rho,{\tilde{\alpha}} \in {{\mathbb R}}, \rho\geq {\tilde{\alpha}}.$ Then, for ${\varkappa}\in X_{{\tilde{\alpha}}}^{p}({\alpha},{\bar{\alpha}})$ the semigroup property is valid, i.e.

Lemma 2.4. ^[22] Let ${\zeta_1} >0,$ and $0\leq {\xi} <1.$ Then, $^{\rho}I_{{\alpha}^{+}}^{{\zeta_1}}$ is bounded from $C_{{\xi},\rho}({\Theta})$ into $C_{{\xi},\rho}({\Theta}).$

Lemma 2.5. ^[7] Let ${\vartheta}>{\alpha}.$ Then, for ${\zeta_1} \geq 0$ and ${\zeta_2}>0,$ we have

Lemma 2.6. ^[22] Let ${\zeta_1}>0, 0\leq{\xi}<1$ and ${\varkappa}\in C_{{\xi}}[{\alpha},{\bar{\alpha}}].$ Then,

Lemma 2.7. ^[22] Let $0<{\zeta_1}<1, 0\leq{\xi}<1.$ If ${\varkappa}\in C_{{\xi},\rho}[{\alpha},{\bar{\alpha}}]$ and $^{\rho}I_{{\alpha}^{+}}^{1-{\zeta_1}}{\varkappa}\in C_{{\xi},\rho}^{1}[{\alpha},{\bar{\alpha}}],$ then

Definition 2.8. ^[22] Let the order ${\zeta_1}$ and the type ${\zeta_2}$ satisfy ${\beta}-1<{\zeta_1}<{\beta}$ and $0\leq{\zeta_2}\leq1,$ with ${\beta}\in \mathbb{N}.$ The Hilfer-Katugampola fractional derivative of a function ${\varkappa}\in C_{1-{\xi},\rho}[{\alpha},{\bar{\alpha}}],$ is defined by

In this paper we consider the case ${\beta}=1$ since $0<{\zeta_1}<1.$

Property 2.9. ^[22] The operator $^{\rho}D^{{\zeta_1},{\zeta_2}}_{{\alpha}^{+}}$ can be written as

Definition 2.10. We assume that the parameters ${\zeta_1},{\zeta_2},{\xi}$ satisfy

Then, we can define the spaces

and

Since $^{\rho}D_{{\alpha}^{+}}^{{\zeta_1},{\zeta_2}}{\mathfrak{p}}=\ ^{\rho}I_{{\alpha}^{+}}^{{\xi}(1-{\zeta_1})} \ ^{\rho}D_{{\alpha}^{+}}^{{\xi}}{\mathfrak{p}},$ by Lemma 2.4, we get

and

Lemma 2.11. ^[22] Let $0<{\zeta_1}<1, 0\leq{\zeta_2}\leq1$ and ${\xi}={\zeta_1}+{\zeta_2}-{\zeta_1}{\zeta_2}.$ If ${\mathfrak{p}}\in C_{1-{\xi},\rho}^{{\xi}}({\Theta}),$ then

and

Lemma 2.12. ^[22] Let $0<{\zeta_1}<1, 0\leq{\zeta_2}\leq1$ and ${\xi}={\zeta_1}+{\zeta_2}-{\zeta_1}{\zeta_2}.$ If $\varphi \in C_{1-{\xi}}[{\alpha},{\bar{\alpha}}]$ and $^{\rho}I_{{\alpha}^{+}}^{1-{\zeta_2}(1-{\zeta_1})}\varphi \in C_{1-{\xi},\rho}^{1}[{\alpha},{\bar{\alpha}}],$ then $^{\rho}D_{{\alpha}^{+}}^{{\zeta_1},{\zeta_2}} \ ^{\rho}I_{{\alpha}^{+}}^{{\zeta_1}}\varphi$ exist on $({\alpha},{\bar{\alpha}}]$ and

Definition 2.13. ^[20] A two-parameter Mittag-Leffler function $E_{{\zeta_1},{\zeta_2}}({\mathfrak{p}}), {\zeta_1},{\zeta_2},{\mathfrak{p}} \in {{\mathbb R}}$ with ${\zeta_1}>0$ and ${\zeta_2}>0,$ is defined by

If ${\zeta_2}=1,$ we obtain:

Lemma 2.14. ^[6] Let ${\zeta_1} >0, {\mathfrak{p}}({\vartheta}), {\varpi_1}({\vartheta})$ be nonnegative functions and ${\varpi_2}({\vartheta})$ be nonnegative and nondecreasing function for ${\vartheta} \in [{\vartheta}_{0},{\mu}), {\mu}>0, {\varpi_2}({\vartheta})\leq {\theta}$ where ${\theta}$ is a constant. If

Then

Corollary 2.15. ^[6] Assume that the requirements of Lemma 2.14 are met, and that ${\varpi_1}({\vartheta})$ is a nondecreasing function for ${\vartheta} \in [{\vartheta}_{0},{\mu}).$ Then

Definition 2.16. The Eq $(1.1)$ is Ulam-Hyers stable if there exists a real number $c_{{\psi},{\varsigma}}>0$ such that for each $\kappa >0$ and for each solution ${\mathfrak{w}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ of the inequality

there exists a solution ${\mathfrak{p}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ of Eq $(1.1)$ with

Definition 2.17. The Eq $(1.1)$ is generalized Ulam-Hyers stable if there exists ${\varpi} _{{\psi},{\varsigma}}\in PC_{1-{\xi}}({{\mathbb R}}_{+},{{\mathbb R}}_{+}) ,$ ${\varpi} _{{\psi},{\varsigma}}(0) =0,$ such that for each solution ${\mathfrak{w}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ of the inequality $(2.1),$ there exists a solution ${\mathfrak{p}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ of the Eq $(1.1)$ with

Definition 2.18. The Eq $(1.1)$ is Ulam-Hyers-Rassias stable with respect to $(\omega,{\varpi})$ if there exists a real number $c_{{\psi},{\varsigma},\omega}>0$ such that for each $\kappa >0$ and for each solution ${\mathfrak{w}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ of the inequality

there exists a solution ${\mathfrak{p}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ of Eq $(1.1)$ with

Definition 2.19. The Eq $(1.1)$ is generalized Ulam-Hyers-Rassias stable with respect to $(\omega,{\varpi})$ if there exists a real number $c_{{\psi},{\varsigma},\omega }>0$ such that for each solution ${\mathfrak{w}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ of the inequality

there exists a solution ${\mathfrak{p}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ of Eq $(1.1)$ with

Remark 2.20. A function ${\mathfrak{w}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ is a solution of the inequality (2.2) if and only if there is $\eta\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ and a sequence $\eta_{{\nu}}, {\nu}=1,\ldots,{\varsigma}$, where

i) $|\eta({\vartheta})|\leq\kappa \omega({\vartheta}), {\vartheta}\in ({\vartheta}_{{\nu}}, {\vartheta}_{{\nu}+1}], {\nu}=1,\ldots,{\varsigma}$ and $|\eta_{{\nu}}|\leq\kappa{\varpi}, {\nu}=1,\ldots,{\varsigma};$

ii) $^{\rho}D^{{\zeta_1},{\zeta_2}}_{{\alpha}^{+}}{\mathfrak{w}}({\vartheta})={\psi}({\vartheta},{\mathfrak{w}}({\vartheta}),^{\rho}D^{{\zeta_1},{\zeta_2}}_{{\alpha}^{+}}{\mathfrak{w}}({\vartheta}))+\eta({\vartheta}), {\vartheta}\in ({\vartheta}_{{\nu}}, {\vartheta}_{{\nu}+1}], {\nu}=1,\ldots,{\varsigma}$;

iii) $\Delta ^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}{\mathfrak{w}}|_{{\vartheta}={\vartheta}_{\nu}}=\chi_{{\nu}}+\eta_{{\nu}}, {\nu}=1,\ldots,{\varsigma}.$

Theorem 2.21. ^[30] Let ${\mathfrak{D}}\subset PC({\Theta},{{\mathbb R}}).$ ${\mathfrak{D}}$ is relatively compact (i.e $\overline{{\mathfrak{D}}}$ is compact) if:

(1) ${\mathfrak{D}}$ is uniformly bounded i.e, there exists ${\bar{\theta}}>0$ where

(2) ${\mathfrak{D}}$ is equicontinuous on $({\vartheta}_{{\nu}},{\vartheta}_{{\nu}+1}]$ i.e, for every $\kappa >0,$ there exists $\delta >0$ such that for each ${\mathfrak{p}},\overline{{\mathfrak{p}}}\in ({\vartheta}_{{\nu}},{\vartheta}_{{\nu}+1}],$ $\left\vert {\mathfrak{p}}-\overline{{\mathfrak{p}}}\right\vert \leq \delta$ implies $\left\vert {\psi}({\mathfrak{p}})-{\psi}(\overline{{\mathfrak{p}}})\right\vert \leq \kappa$, for every ${\psi}\in {\mathfrak{D}}.$

With appropriate modifications, the preceding theorem may be extended to the weighted space $PC_{1-{\xi}}({\Theta},{{\mathbb R}})$.

Theorem 2.22. ^[26] (${PC_{1-{\xi}}}$ type Arzela-Ascoli theorem). Let ${\mathfrak{D}}\subset PC_{1-{\xi}}({\Theta},{{\mathbb R}}).$ ${\mathfrak{D}}$ is relatively compact (i.e $\overline{{\mathfrak{D}}}$ is compact) if:

(1) ${\mathfrak{D}}$ is uniformly bounded i.e, there exists ${\bar{\theta}}>0$ such that

(2) ${\mathfrak{D}}$ is equicontinuous on $({\vartheta}_{{\nu}},{\vartheta}_{{\nu}+1}]$ i.e, for every $\kappa >0,$ there exists $\delta >0$ such that for each ${\mathfrak{p}},\overline{{\mathfrak{p}}}\in ({\vartheta}_{{\nu}},{\vartheta}_{{\nu}+1}],$ $\left\vert {\mathfrak{p}}-\overline{{\mathfrak{p}}}\right\vert \leq \delta$ implies $\left\vert {\psi}({\mathfrak{p}})-{\psi}(\overline{{\mathfrak{p}}})\right\vert \leq \kappa$, for every ${\psi}\in {\mathfrak{D}}.$

3.   Existence of solutions

In this section, we study the existence of solution for the problems (1.1)–(1.3). We employ the following lemma to establish our main results.

Lemma 3.1. Let $0<{\zeta_1}<1,$ and $0\leq{\zeta_2}\leq1, {\xi}={\zeta_1}+{\zeta_2}-{\zeta_1}{\zeta_2}$ and ${\gamma}:{\Theta}\rightarrow {{\mathbb R}}$ be continuous function. For any ${\bar{\alpha}}\in {\Theta}$, a function ${\mathfrak{p}}\in C_{1-{\xi},\rho}^{{\xi}} ({\Theta},{{\mathbb R}})$ is a solution of the equation:

if and only if, ${\mathfrak{p}}$ is solution of the Hilfer-Katugampola fractional differential equation

Proof. $(\Rightarrow )$ Let ${\mathfrak{p}} \in C_{1-{\xi},\rho}^{{\xi}}({\Theta})$ satisfying $(3.1).$ We demonstrate that ${\mathfrak{p}}$ also verifies the fractional differential equation $(3.2).$ Applying $^\rho D_{{\alpha}^{+}}^{{\zeta_1},{\zeta_2}}$ on both sides of the Eq $(3.1),$ we get

From the Lemma 2.4 and by the definition of the space $C_{1-{\xi},\rho}^{1}({\Theta}),$ we get

By Lemma 2.12 and Lemma 2.5 we obtain

$(\Leftarrow)$ Let ${\mathfrak{p}} \in C_{1-{\xi},\rho}^{{\xi}}({\Theta})$ by a solution of the fractional differential equation $(3.2).$ We prove that ${\mathfrak{p}}$ is also a solution of $(3.1).$ Then

By Lemma 2.7, with ${\zeta_1}={\xi},$ we obtain

where ${\vartheta}\in ({\alpha},{\mu}].$ By hypothesis, ${\mathfrak{p}}\in C_{1-{\xi},\rho}^{{\xi}}({\Theta}),$ using Lemma 2.11, we have

which implies that

By applying Lemma 2.5 we get

By replacing $(3.4)$ and $(3.5)$ in $(3.3)$ we have

with ${\vartheta}\in({\alpha},{\bar{\alpha}}],$ that is ${\mathfrak{p}}(\cdot)$ satisfies $(3.1).$ This completes the proof of the Lemma.

Lemma 3.2. Let ${\xi}={\zeta_1}+{\zeta_2}-{\zeta_1}{\zeta_2},$ where $0<{\zeta_1}<1$ and $0\leq {\zeta_2} \leq1.$ Let $\Psi:({\alpha},{\mu}] \rightarrow {{\mathbb R}}$ is a continuous function. A function ${\mathfrak{p}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta})$ is a solution of the fractional integral equation

where ${\nu}=1,\dots,{\varsigma}$, if and only if ${\mathfrak{p}}$ is a solution of the problem:

Proof. Assume that ${\mathfrak{p}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta},{{\mathbb R}})$ satisfies the problems $(3.7)$–$(3.9).$

If ${\vartheta}\in({\alpha},{\vartheta}_{1}]$ then

Then the problem $(3.10)$ is equivalent to the following fractional integral ^[22].

Now, if ${\vartheta}\in({\vartheta}_{1},{\vartheta}_{2}]$ then

By Lemma 3.1, we have

Now, from $(3.11),$ we have

This gives

Using $(3.13)$ in $(3.12),$ we obtain

Next, if ${\vartheta}\in ({\vartheta}_{2},{\vartheta}_{3}]$ then

Again, by Lemma 3.1, we have

From $(3.14),$ we have

which gives

Using $(3.16)$ in $(3.15),$ we get

Continuing the above process, we obtain

Conversely, let ${\mathfrak{p}} \in PC_{1-{\xi},\rho}^{{\xi}}({\Theta},{{\mathbb R}})$ satisfies the fractional integral equation $(3.11).$ Then, for ${\vartheta}\in ({\alpha},{\vartheta}_{1}],$ we have

Applying $^{\rho}D_{{\alpha}^{+}}^{{\zeta_1},{\zeta_2}}$ on both sides, we get

From Lemma 2.4 and by the definition of the space $C_{1-\lambda,\rho}^{1}({\Theta}),$ we can get

Using Lemma 2.12 and Lemma 2.5, we obtain

Now, for ${\vartheta}\in ({\vartheta}_{{\nu}};{\vartheta}_{{\nu}+1}],$ we have

Applying the operator $^{\rho}D_{{\alpha}^{+}}^{{\zeta_1},{\zeta_2}}$ on both sides, we get

From $(3.17)$ and using Lemma 2.12 and Lemma 2.5, we obtain

Thus, ${\mathfrak{p}}$ satisfies $(3.7).$ Next, we demonstrate that ${\mathfrak{p}}$ also verify $(3.8)$ and $(3.9).$ Applying the operator $^{\rho}I_{{\alpha}^{+}}^{1-{\xi}}$ on both sides of Eq $(3.11),$ we get

By Lemma 2.5 we get

which implies that

Since ${\mathfrak{p}}$ satisfies $(3.7)$ we have

Using Lemma 2.11 we have

From the definition of $C_{1-{\xi},\rho}^{{\xi}}({\Theta}),$ Lemma 2.4 and using Definition 2.2, we have

Thus

By Lemma 2.7, with ${\zeta_1}={\xi}$ we can write

By Lemma 2.5 we have

which is the condition $(3.9).$

Further, from Eq $(3.6),$ for ${\vartheta}\in ({\vartheta}_{{\nu}},{\vartheta}_{{\nu}+1}],$ we have

and for ${\vartheta}\in ({\vartheta}_{{\nu}-1},{\vartheta}_{{\nu}}],$ we have

Therefore, from $(3.18)$ and $(3.19),$ we obtain

which condition $(3.8).$ We have proved that ${\mathfrak{p}}$ satisfies the problems $(3.7)$–$(3.9).$

As a consequence of Lemma 3.2, we have the following lemma.

Lemma 3.3. Let ${\xi}={\zeta_1}+{\zeta_2}-{\zeta_1}{\zeta_2}$ where $0<{\zeta_1}<1$ and $0\leq{\zeta_2}\leq1,$ let ${\psi}:({\alpha},{\mu}]\times{{\mathbb R}}\times{{\mathbb R}}\rightarrow {{\mathbb R}}$ be a continuous function where ${\psi}(\cdot,{\mathfrak{p}}(\cdot),{\mathfrak{x}}(\cdot))\in PC_{1-{\xi},\rho}({\Theta})$ for any ${\mathfrak{p}},{\mathfrak{x}} \in PC_{1-{\xi},\rho}({\Theta}).$ If ${\mathfrak{p}}\in PC_{1-{\xi},\rho}^{{\xi}}({\Theta}),$ then ${\mathfrak{p}}$ verifies $(1.1)$–$(1.3)$ if and only if ${\mathfrak{p}}$ is the fixed point of the operator ${\mathcal S}:PC_{1-{\xi},\rho}({\Theta})\rightarrow PC_{1-{\xi},\rho}({\Theta})$ given by

where ${\varkappa}:(0,{\mu}]\rightarrow {{\mathbb R}}$ be a function verifying the functional equation

It is obvious that ${\varkappa}\in PC_{1-{\xi},\rho}({\Theta}).$ Also, by Lemma $2.4,$ ${\mathcal S}{\mathfrak{p}}\in PC_{1-{\xi},\rho}({\Theta}).$

Assume that the function ${\psi}:({\alpha},{\mu}]\times{{\mathbb R}}\times{{\mathbb R}}\rightarrow {{\mathbb R}}$ is continuous and verifies the following:

$(H1)$ ${\psi}(\cdot,{\mathfrak{x}}(\cdot),{\mathfrak{y}}(\cdot))\in PC^{{\zeta_2}(1-{\zeta_1})}_{1-{\xi},\rho}$ for any ${\mathfrak{x}}, {\mathfrak{y}} \in PC_{1-{\xi},\rho}({\Theta}).$

(H2) There exist constants ${\theta_1} >0$ and $0<{\theta_2}<1$ such that

for any ${\mathfrak{x}}, {\mathfrak{y}}, \bar{{\mathfrak{x}}},\bar{{\mathfrak{y}}} \in PC_{1-{\xi},\rho}({\Theta})$ and ${\vartheta}\in ({\alpha},{\mu}]$.

We can now declare and demonstrate our existence result for problems (1.1)–(1.3) based on Banach's fixed point ^[15].

Theorem 3.4. If $(H1)$ and $(H2)$ are met, and

then the problems (1.1)–(1.3) has unique solution in $PC_{1-{\xi},\rho}^{{\xi}}({\Theta})\subset PC_{1-{\xi},\rho}^{{\zeta_1},{\zeta_2}}({\Theta})$.

Proof. The proof will be presented in two segments.

Step 1: We demonstrate that ${\mathcal S}$ defined in $(3.20)$ has a unique fixed point ${\mathfrak{p}}^*$ in $PC_{1-{\xi},\rho}({\Theta}).$ Let ${\mathfrak{p}},{\mathfrak{x}} \in PC_{1-{\xi},\rho}({\Theta})$ and ${\vartheta}\in ({\alpha},{\mu}],$ then, we have

where ${\varkappa},{\gamma}\in PC_{1-{\xi},\rho}({\Theta})$ such that

By (H2), we have

Then,

Therefore, for each ${\vartheta}\in ({\alpha},{\mu}]$

By Lemma $2.5,$ we have

hence

which implies that

By $(3.21),$ the operator ${\mathcal S}$ is a contraction. Hence, by Banach's contraction principle, ${\mathcal S}$ has a unique fixed point ${\mathfrak{p}}^* \in PC_{1-{\xi},\rho}({\Theta})$.

Step 2: We show that such a fixed point ${\mathfrak{p}}^*\in PC_{1-{\xi},\rho}({\Theta})$ is actually in $PC^{{\xi}}_{1-{\xi},\rho}({\Theta}).$

Since ${\mathfrak{p}}^*$ is the unique fixed point of operator ${\mathcal S}$ in $PC_{1-{\xi},\rho}({\Theta})$, then, for each ${\vartheta}\in ({\alpha},{\mu}],$ we have

Applying $^{\rho}D^{{\xi}}_{{\alpha}^{+}}$ to both sides and by Lemma $2.5,$ and Lemma $2.11,$ we have

Since ${\xi}\geq{\zeta_1},$ by (H1), the right hand side is in $PC_{1-{\xi},\rho}({\Theta})$ and thus $^{\rho}D^{{\xi}}_{{\alpha}^{+}}{\mathfrak{p}}^* \in PC_{1-{\xi},\rho}({\Theta})$ which implies that ${\mathfrak{p}}^*\in PC^{{\xi}}_{1-{\xi},\rho}({\Theta}).$ As a consequence of Steps 1 and 2 together with Lemma $3.3,$ we can conclude that the problems $(1.1)$–$(1.3)$ has a unique solution in $PC^{{\xi}}_{1-{\xi},\rho}({\Theta}).$

Our second result is based on Krasnoselskii fixed point theorem ^[15].

Theorem 3.5. Assume $(H1)$ and,

(H3) There exist constants $0<{\theta_1}< \frac {(1-{\theta_2})\Gamma({\zeta_1}+{\xi})}{2\Gamma({\xi})}\left(\frac{{\mu}^{\rho}-{\alpha}^{\rho}}{\rho}\right)^{-{\zeta_1}}$ and $0<{\theta_2}<1$ such that

for any ${\mathfrak{x}}, {\mathfrak{y}}, \bar{{\mathfrak{x}}},\bar{{\mathfrak{y}}} \in PC_{1-{\xi},\rho}({\Theta})$ and ${\vartheta}\in ({\alpha},{\mu}]$.

Then the problems $(1.1)$–$(1.3)$ has at least one solution.

Proof. Consider the set

where

where ${\psi}^{*}=\sup\limits_{{\vartheta}\in {\Theta}}|{\psi}({\vartheta},0,0)|.$

We define the operators ${{\mathcal S}_1}$ and ${{\mathcal S}_2}$ on $B_{\varepsilon^*}$ by

Then $(3.20)$ can be written as

Step 1: We demonstrate that ${{\mathcal S}_1}{\mathfrak{p}}+{{\mathcal S}_2}{\mathfrak{x}} \in B_{\varepsilon^*}$ for any ${\mathfrak{p}},{\mathfrak{x}} \in B_{\varepsilon^*}.$ For operator ${{\mathcal S}_1},$ multiplying both sides of $(3.22)$ by $\left( \frac {{\vartheta}^{\rho}-{\alpha}^{\rho}}{\rho}\right)^{1-{\xi}},$ we have

then

This gives

By (H3), we have for each ${\vartheta} \in ({\alpha},{\mu}],$

Multiplying both sides of the above inequality by $\left( \frac {{\vartheta}^{\rho}-{\alpha}^{\rho}}{\rho}\right)^{1-{\xi}},$ we get

Then, for each ${\vartheta}\in({\alpha},{\mu}],$ we have

Thus, (3.23) and Lemma $2.5,$ implies

Therefore

Thus

Linking $(3.24)$ and $(3.26)$ for every ${\mathfrak{p}},{\mathfrak{x}} \in B_{\varepsilon^*}$ we obtain

Since

and

we have

which infers that ${{\mathcal S}_1}{\mathfrak{p}}+{{\mathcal S}_2}{\mathfrak{x}} \in B_{\varepsilon^*}.$

Step 2: Clearly ${{\mathcal S}_1}$ is a contraction.

Step 3: ${{\mathcal S}_2}$ is compact and continuous.

The continuity of ${{\mathcal S}_2}$ follows from the continuity of ${\psi}.$ Next we prove that ${{\mathcal S}_2}$ is uniformly bounded on $B_{\varepsilon^*}.$ Let any ${\mathfrak{x}}\in B_{\varepsilon^*}.$ Then by $(3.26)$ we have

This means that ${{\mathcal S}_2}$ is uniformly bounded on $B_{\varepsilon^*}.$ Next, we show that ${{\mathcal S}_2}B_{\varepsilon^*}$ is equicontinuous. Let any ${\mathfrak{x}}\in B_{\varepsilon^*}$ and $0<{\alpha}<\tau_{1}<\tau_{2}\leq {\mu}.$ Then

by using $(3.25)$ we have

Note that

This shows that ${{\mathcal S}_2}$ is equicontinuous on ${\Theta}.$ Therefore ${{\mathcal S}_2}$ is relatively compact on $B_{\varepsilon^*}.$ By $PC_{1-{\xi}};$ type Arzela-Ascoli theorem ${{\mathcal S}_2}$ is compact on $B_{\varepsilon^*}.$

By Krasnoselskii's fixed point theorem, ${\mathcal S}$ has at least a fixed point ${\mathfrak{p}}^* \in C_{1-{\xi},\rho}({\Theta})$ and by the same way of the proof of Theorem $3.4,$ ${\mathfrak{p}}^* \in C^{{\xi}}_{1-{\xi},\rho}({\Theta}).$ Using Lemma 3.3, we conclude that the problems $(1.1)$–$(1.3)$ has at least one solution in the space $C^{{\xi}}_{1-{\xi},\rho}({\Theta}).$

4.   Ulam-Hyers-Rassias stability

In what follows, we give the following result on Ulam-Hyers-Rassias stability.

Theorem 4.1. Assume that (H1), (H2), (3.21) hold and,

(H4) There exists a nondecreasing function $\omega \in PC_{{\xi}-1}({\Theta})$ and there exists $\lambda _{\omega }>0$ such that for any ${\vartheta}\in ({\alpha},{\mu}]:$

Then, the Eq $(1.1)$ is Ulam-Hyers-Rassias stable with respect to $(\omega,{\varpi})$.

Proof. Let ${\mathfrak{w}}\in PC_{1-{\xi},\rho}({\Theta})$ be a solution of the inequality $(2.2).$ Denote by ${\mathfrak{p}}$ the unique solution of the problem:

Using Lemma 3.3, we obtain

where ${\varkappa}:(0,{\mu}]\rightarrow {{\mathbb R}}$ be a function satisfying the functional equation

Since ${\mathfrak{w}}$ solution of the inequality $(2.2)$ and by Remark $2.20$, we have

Clearly, the solution of the problems (4.1) and (4.2) is given by

where ${\gamma}:(0,{\mu}]\rightarrow {{\mathbb R}}$ be a function satisfying the functional equation

If ${\vartheta}\in ({\alpha},{\vartheta}_{1}],$ it follows that

where ${\varkappa},{\gamma}\in C_{1-{\xi},\rho}({\Theta})$ such that

By (H2), we have

Then,

Therefore, for each ${\vartheta}\in ({\alpha},{\vartheta}_{1}]$

Applying Corollary 2.15, we get

where

If ${\vartheta} \in ({\vartheta}_{\nu},{\vartheta}_{{\nu}+1}],\ {\nu}=1, \ldots,{\varsigma},$ then we have

where ${\varkappa},{\gamma}\in C_{1-{\xi},\rho}({\Theta})$ such that

By (H2), we have

Then,

Therefore, for each ${\vartheta} \in ({\vartheta}_{\nu},{\vartheta}_{{\nu}+1}],\ {\nu}=1, \ldots,{\varsigma},$

Applying Corollary 2.15, we get

where

Thus, the Eq $(1.1)$ is Ulam-Hyers-Rassias stable with respect to $(\omega,{\varpi})$. The proof is complete.

The following theorem gives Ulam-Hyers stable result.

Theorem 4.2. Assume that (H1), (H2) and $(3.21)$ hold. Then, the Eq $(1.1)$ is Ulam-Hyers stable.

Proof. Let ${\mathfrak{w}}\in C_{1-{\xi},\rho}({\Theta})$ be a solution of the inequality $(2.1).$ Denote by ${\mathfrak{p}}$ the unique solution of the problem:

By the same way of the proof of Theorem 4.1, we can easily show that:

If ${\vartheta}\in ({\alpha},{\vartheta}_{1}],$ it follows that

Applying Lemma $2.14,$ we get

If ${\vartheta} \in ({\vartheta}_{\nu},{\vartheta}_{{\nu}+1}],\ {\nu}=1, \ldots,{\varsigma},$ then we have

Applying Lemma 2.14, we get

which completes the proof of the theorem. Moreover, if we set ${\varpi}_{{\psi},{\varsigma}}(\kappa)=(b_{1}+b_{2})\kappa; {\varpi}_{{\psi},{\varsigma}}(0)=0$, then, the Eq $(1.1)$ is generalized Ulam-Hyers stable.

5.   An example

Consider the following initial value problem with impulse

where $J_{0}=\left(1, \frac{3}{2}\right],$ $J_{1}=\left(\frac{3}{2},2\right].$

Set

We have

with ${\xi}={\zeta_1}=\rho=\frac{1}{2}$ and ${\zeta_2}=0.$ Clearly, the function ${\psi} \in PC_{\frac{1}{2},\frac{1}{2}}([1,2]).$

Hence condition (H1) is satisfied.

For each ${\mathfrak{x}},\bar{{\mathfrak{x}}},{\mathfrak{y}},\bar{{\mathfrak{y}}}\in{{\mathbb R}}$ and ${\vartheta}\in(1,2]$:

Hence condition $(H{2})$ is satisfied with ${\theta_1}={\theta_2}= \frac {1}{108e}.$

The condition

is satisfied with with ${\alpha}=1$ and ${\mu}=2.$ It follows from Theorem 3.4 that the problems (5.1)–(5.3) has a unique solution in the space $PC^{\frac{1}{2}}_{\frac{1}{2},\frac{1}{2}}([1,2]).$ Moreover, Theorem 4.2, implies that the Eq $(1.1)$ is Ulam-Hyers stable.

6.   Conclusions

We have investigated the existence, uniqueness and stability of solutions for a class of nonlinear impulsive Hilfer-Katugampola problems. Our reasoning is founded on the Banach contraction principle and Krasnoselskii’s fixed point theorem. In addition, an example is provided to demonstrate the effectiveness of the main results. We plan to consider for for a futur study the same problem in infinite dimensional Banach space and make us of Darbo and Monch's fixed point theorems associated with the notion of measure of noncompactness.

Acknowledgements

The research of J. J. Nieto was partially supported by the AEI of Spain under Grant MTM2016-75140-P and co-financed by European Community fund FEDER. The work of Y. Zhou was supported by the National Natural Science Foundation of China (12071396).

Conflict of interest

The authors declare that there is no conflict of interest.