Abstract
In this manuscript, we examine the existence and the Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces. The results are based on fixed point theorems of Darbo and Mönch associated with the technique of measure of noncompactness. We provide some examples to indicate the applicability of our results.
Similar content being viewed by others
1 Introduction
As it is known very well, the roots of fixed point theory go to the method of successive approximations (or Picard’s iterative method) that is used to solve certain differential equations. Roughly speaking, Banach derived the fixed point theorem from the method of successive approximations. In the last decades fixed point theory has been enormously and independently from the differential equations. But, recently, fixed point results turn to be the tools for the solutions of the differential equation. In this paper, we shall involve two interesting fixed point theorems (Darbo’s fixed point theorem and Mönch’s fixed point theorem) in the setting of “measure of noncompactness” to solve the boundary value problem for nonlinear implicit fractional differential equations with instantaneous impulses.
Differential equations of fractional order have been recently proved to be a powerful tool to study many phenomena in various fields of science and engineering such as electrochemistry, electromagnetics, viscoelasticity, finance, and so on. In the literature, it is very common to propose a solution for fractional differential equations by involving different kinds of fractional derivatives, see e.g. [1–10, 12, 13, 21, 22, 36]. On the other hand, there a few results that deal with the boundary value problems for fractional differential equations. The aim of the present paper is to underline the importance of the theory of impulsive differential equations. Further, by the help of these observations, we aim to understand several phenomena that are not clarified by the non-impulsive equations (see e.g. [15, 16, 18, 32]).
In 1940, Ulam [34, 35] raised the following problem of the stability of the functional equation (of group homomorphisms): “Under what conditions does it exist an additive mapping near an approximately additive mapping?”
Let \(G_{1}\) be a group, and let \(G_{2}\) be a metric group with a metric \(d(\cdot ,\cdot )\). Given any \(\epsilon >0\), does there exist \(\delta >0\) such that if a function \(h: G_{1} \rightarrow G_{2}\) satisfies the inequality \(d(h(xy),h(x)h(y))<\delta \) for all \(x,y \in G_{1}\), then there exists a homomorphism \(H: G_{1} \rightarrow G_{2}\) with \(d(h(x), H(x))<\epsilon \) for all \(x \in G_{1}\)?
A partial answer was given by Hyers [20] in 1941, and between 1982 and 1998 Rassias [28, 29] established the Hyers–Ulam stability of linear and nonlinear mappings. Subsequently, many works have been published in order to generalize Hyers results in various directions, see for example [24, 25, 30, 31, 33].
Inspired by the papers mentioned above, we examine the existence results to the boundary value problem with nonlinear implicit generalized Hilfer-type fractional differential equation with instantaneous impulses:
where
-
\({}^{\rho }D_{t_{k}^{+}}^{\alpha , \beta }\) is the generalized Hilfer fractional derivative of order \(\alpha \in (0,1)\) and type \(\beta \in [0,1]\) and \(\rho >0\);
-
\({}^{\rho }{\mathcal{J}}_{t_{k}^{+}}^{1-\gamma }\) is the generalized Hilfer fractional integral of order \(1-\gamma \), \((\gamma =\alpha +\beta -\alpha \beta )\);
-
\(c_{1}\), \(c_{2}\) are reals with \(c_{1}+c_{2}\neq 0 \), \(J_{k}:=(t_{k},t_{k+1}]\); \(k=0,\ldots ,m\), \(a=t_{0}< t_{1}<\cdots <t_{m}<t_{m+1}=b<\infty \);
-
\(u(t_{k}^{+})=\lim_{\epsilon \rightarrow 0^{+}} u(t_{k}+\epsilon )\) and \(u(t_{k}^{-})=\lim_{\epsilon \rightarrow 0^{-}} u(t_{k}+\epsilon )\) represent the right-hand and left-hand limits of \(u(t)\) at \(t=t_{k}\), \(c_{3}\in E\);
-
\(f : (a,b]\times E \times E \to E \) is a given function over a Banach space \((E,\|\cdot \|)\);
-
\(\varpi _{k}:E \to E\); \(k=1,\ldots ,m\), are given continuous functions.
2 Preliminaries
In this section, we recall and recollect the basic notion, notations together with some fundamental results that will be necessary in the main results. Throughout the paper, \((E,\|\cdot \|)\) represents a Banach space. Set \(J=[a,b]\) where \(0< a< b\). The letter C is reserved to represent the Banach space which consists of all continuous functions \(u:J\to E\) where the norm is
In what follows, we pay attention to the weighted spaces of continuous functions
where \(0\leq \gamma <1\),
with the norms
and
Consider the Banach space
Also, we consider the weighted space
and
equipped with the norm
The letter \(L^{1}(J)\) indicates the space of Bochner-integrable functions \(f: J\longrightarrow E\) with the norm
Definition 2.1
([23])
Let \(\alpha \in {\mathbb{R}}_{+}\) and \(g\in L^{1}(J)\). The generalized Hilfer fractional integral of order α is
with \(\Gamma (\alpha )=\int _{0}^{\infty }t^{\alpha -1}e^{-t}\,dt\), \(\alpha >0\) (the Euler gamma function)
Definition 2.2
([23] Generalized Hilfer fractional derivative)
Set \(\rho >0\) and \(\alpha \in {\mathbb{R}}_{+}\setminus \mathbb{N}\). The generalized Hilfer fractional derivative \({}^{\rho }D_{a^{+}}^{\alpha }\) of order α is defined by
where \(n=[\alpha ]+1\) and \(\delta _{\rho }^{n}= (t^{1-\rho }\frac{d}{dt} )^{n}\).
Theorem 2.3
([23])
Let \(\alpha >0\), \(\beta >0\), \(1\leq \rho \leq \infty \), \(0< a< b<\infty \). Then, for \(g\in L^{1}(J)\), we have
Lemma 2.4
Let \(\alpha >0\) and \(0\leq \gamma <1\). Then \({}^{\rho }{\mathcal{J}}_{a^{+}}^{\alpha }\) lies between \(PC_{\gamma ,\rho }(J)\) and \(PC_{\gamma ,\rho }(J)\).
Lemma 2.5
([27])
Suppose that \(0\leq \gamma <1\), \(0< a< b<\infty \), \(\alpha >0 \), and \(u \in PC_{\gamma ,\rho }(J)\). If \(\alpha > 1-\gamma \), then \({}^{\rho }{\mathcal{J}}_{a^{+}}^{\alpha }u\) is continuous on J and
Lemma 2.6
([11])
Let \(t>a\). Then, for \(\alpha \geq 0\) and \(\beta >0\), we have
Lemma 2.7
([27])
Suppose that \(g\in PC_{\gamma }[a,b]\) with \(0\leq \gamma <1\) and \(\alpha >0\). Then we have
Lemma 2.8
([27])
Let \(0<\alpha <1\), \(0\leq \gamma <1\). If \(g\in PC_{\gamma ,\rho }[a,b]\) and \({}^{\rho }{\mathcal{J}}_{a^{+}}^{1-\alpha }g\in PC_{\gamma ,\rho }^{1}[a,b]\), then
Definition 2.9
([27])
For a function \(g\in PC_{\gamma ,\rho }[a,b]\) with \(\rho >0\), the generalized Hilfer-type fractional derivative is defined by
where type β with \(0\leq \beta \leq 1\), and order α with \(n-1<\alpha <n\) for \(n\in \mathbb{N}\).
Since \(0<\alpha <1\), we shall focus only on the case \(n=1\).
Property 2.10
([27])
The operator \({}^{\rho }D^{\alpha ,\beta }_{a^{+}}\) can be written as
Property 2.11
([27])
The fractional derivative \({}^{\rho }D^{\alpha ,\beta }_{\alpha ^{+}}\) is an interpolator of the following fractional derivatives: Hilfer \((\rho \rightarrow 1)\), Hilfer–Hadamard \((\rho \rightarrow 0^{+})\), generalized \((\beta =0)\), Caputo-type \((\beta =1)\), Riemann–Liouville \((\beta =0, \rho \rightarrow 1)\), Hadamard \((\beta =0, \rho \rightarrow 0^{+})\), Caputo \((\beta =1, \rho \rightarrow 1)\), Caputo–Hadamard \((\beta =1, \rho \rightarrow 0^{+})\), Liouville \((\beta =0, \rho \rightarrow 1, a=0)\), and Weyl \((\beta =0,\rho \rightarrow 1, a=-\infty )\).
Let \(\gamma =\alpha +\beta -\alpha \beta \), where \(0<\alpha \), \(\beta ,\gamma <1 \). Construct the space
and
where \(k=0,\ldots ,m\).
Since \({}^{\rho }D_{t_{k}^{+}}^{\alpha ,\beta }u= {}^{\rho }{\mathcal{J}}_{t_{k}^{+}}^{ \gamma (1-\alpha )} {}^{\rho }D_{t_{k}^{+}}^{\gamma }u\), it follows from Lemma 2.4 that
Lemma 2.12
([27])
Let \(\gamma =\alpha +\beta -\alpha \beta \) with \(0<\alpha <1\), \(0\leq \beta \leq 1\). If \(u\in PC_{\gamma ,\rho }^{\gamma }(J)\), then
and
Definition 2.13
([14])
Let X be a Banach space, and let \(\Omega _{X}\) be the family of bounded subsets of X. The Kuratowski measure of noncompactness is the map \(\mu : \Omega _{X} \longrightarrow [ 0,\infty )\) defined by
where \(M\in \Omega _{X}\).
For all \(M,M_{1},M_{2} \in \Omega _{X}\), the map μ satisfies the following properties:
-
M is relatively compact (\(\mu (M)=0 \Leftrightarrow \overline{M} \) is compact).
-
\(\mu (M)=\mu (\overline{M})\).
-
\(M_{1}\subset M_{2} \Rightarrow \mu (M_{1})\leq \mu (M_{2})\).
-
\(\mu (M_{1}+M_{2})\leq \mu (M_{1})+\mu (M_{2})\).
-
\(\mu (cM)=|c|\mu ({M})\), \(c \in {\mathbb{R}}\).
-
\(\mu (conv M)=\mu (M)\).
Lemma 2.14
([19])
Let \(D\subset PC_{\gamma ,\rho }(J)\) be a bounded and equicontinuous set, then
(i) the function \(t\rightarrow \mu (D(t))\) is continuous on \((a,b]\), and
(ii) \(\mu (\int ^{b}_{a}u(s)\,ds : u \in D )\leq \int ^{b}_{a}\mu (D(s))\,ds\), where
Lemma 2.15
(Theorem 4.1, [27])
Let \(f:(a,b]\times E\rightarrow E\) be a function such that \(f(\cdot ,u(\cdot ), {}^{\rho }D_{t_{k}^{+}}^{\alpha ,\beta }u(\cdot )) \in C_{\gamma ,\rho }(J)\) for any \(u\in PC_{\gamma ,\rho }(J)\). Then \(u\in PC_{\gamma ,\rho }^{\gamma }(J)\) is a solution of the differential equation
if and only if u satisfies the following Volterra integral equation:
where \(\gamma =\alpha +\beta -\alpha \beta \).
Theorem 2.16
(Mönch’s theorem [26])
Suppose that D is a closed, bounded, and convex subset of a Banach space X such that \(0 \in D\). Let T be a continuous mapping of D into itself. If the implication
holds for every subset V of D, then T has a fixed point.
Theorem 2.17
(Darbo’s theorem [17])
Suppose that D is a nonempty, closed, bounded, and convex subset of a Banach space X. Let T be a continuous mapping of D into itself such that, for any nonempty subset C of D,
where \(0\leq k<1\), and μ is the Kuratowski measure of noncompactness. Then T has a fixed point in D.
3 The existence of solutions
We start this section by stating the following linear fractional differential equation:
where \(0<\alpha <1\), \(0\leq \beta \leq 1\), \(\rho >0\), with the conditions
and
where \(\gamma =\alpha +\beta -\alpha \beta \), \(c_{3}\in E\), \(c_{1},c_{2}\in {\mathbb{R}}\) with \(c_{1}+c_{2}\neq 0\) and \(\xi _{1}=\frac{c_{2}}{c_{1}+c_{2}}\), \(\xi _{2}= \frac{c_{3}}{c_{1}+c_{2}}\). The following theorem shows that problem (6)–(8) has a unique solution given by
Theorem 3.1
Let \(\gamma =\alpha +\beta -\alpha \beta \), where \(0<\alpha <1\) and \(0\leq \beta \leq 1\). If \(\psi :(a,b] \rightarrow E\) is a function such that \(\psi (\cdot ) \in C_{\gamma ,\rho }(J)\), then \(u\in PC_{\gamma ,\rho }^{\gamma }(J)\) satisfies problem (6)–(8) if and only if it satisfies (9).
Proof
Assume that u satisfies (6)–(8). If \(t \in J_{0}\), then
Lemma 2.15 implies we have the solution that can be written as
If \(t \in J_{1}\), then Lemma 2.15 implies
If \(t \in J_{2}\), then Lemma 2.15 implies
Repeating the process in this way, the solution \(u(t)\) for \(t \in J_{k}\), \(k=1,\ldots ,m\), can be written as
Applying \({}^{\rho }{\mathcal{J}}_{t_{m}^{+}}^{1-\gamma }\) on both sides of (11), using Lemma 2.6, and taking \(t=b\), we obtain
Multiplying both sides of (12) by \(c_{2}\) and using condition (8), we obtain
which implies that
Substituting (13) into (11) and (10), we obtain (9).
Reciprocally, applying \({}^{\rho }{\mathcal{J}}_{t_{k}^{+}}^{1-\gamma }\) on both sides of (9) and using Lemma 2.6 and Theorem 2.3, we get
Next, taking the limit \(t\rightarrow a^{+}\) of (14) and using Lemma 2.5, with \(1-\gamma <1-\gamma +\alpha \), we obtain
Now, taking \(t=b\) in (14), we get
From (15) and (16), we find that
which shows that the boundary condition \(c_{1} ({}^{\rho }{\mathcal{J}}_{a^{+}}^{1-\gamma }u )(a^{+})+c_{2} ({}^{\rho }{\mathcal{J}}_{t_{m}^{+}}^{1-\gamma }u )(b)=c_{3}\) is satisfied. Next, we apply operator \({}^{\rho }D_{t_{k}^{+}}^{\gamma }\) on both sides of (9), where \(k=0,\ldots ,m\). Then, from Lemma 2.6 and Lemma 2.12, we obtain
Since \(u\in PC_{\gamma ,\rho }^{\gamma }(J) \) and by definition of \(PC_{\gamma ,\rho }^{\gamma }(J)\), we have \({}^{\rho }D_{t_{k}^{+}}^{\gamma }u \in PC_{\gamma ,\rho }(J)\), then (17) implies that
As \(\psi (\cdot ) \in C_{\gamma ,\rho }(J)\) and from Lemma 2.4, it follows
Regarding the definition of the space \(PC_{\gamma ,\rho }^{n}(J)\), together with (18), (19), we derive that
Applying operator \({}^{\rho }{\mathcal{J}}_{t_{k}^{+}}^{\beta (1-\alpha )}\) on both sides of (17) and using Lemma 2.8, Lemma 2.5, and Property 2.10, we have
that is, (6) holds.
Also, we can easily show that
This completes the proof. □
As a consequence of Theorem 3.1, we have the following result.
Lemma 3.2
Let \(\gamma =\alpha +\beta -\alpha \beta \), where \(0\leq \beta \leq 1 \) and \(0<\alpha <1 \). Suppose that \(f:(a,b]\times E\times E\rightarrow E\) is a function so that \(f(\cdot ,u(\cdot ),w(\cdot ))\in C_{\gamma ,\rho }(J)\) for each \(w,u \in PC_{\gamma ,\rho }(J)\).
If \(u\in PC_{\gamma ,\rho }^{\gamma }(J) \), then u satisfies problem (1)–(3) iff u is the fixed point of \(\Psi :PC_{\gamma ,\rho }(J)\rightarrow PC_{\gamma ,\rho }(J)\) defined by
where \(h:(a,b]\rightarrow {\mathbb{R}}\) is defined as
Evidently, \(h\in C_{\gamma ,\rho }(J)\). Also, by Lemma 2.4, \(\Psi u\in PC_{\gamma ,\rho }(J)\).
In the sequel, we shall use the following hypotheses efficiently:
- (Ax1):
-
The mapping \(t \mapsto f(t,u,w) \) is measurable on \((a,b]\) for each \(u,w \in E\), and the functions \(u \mapsto f(t,u,w) \) and \(w \mapsto f(t,u,w) \) are continuous on E for a.e. \(t \in (a,b] \), and
$$ f\bigl(\cdot ,u(\cdot ),w(\cdot )\bigr)\in PC^{\beta (1-\alpha )}_{\gamma ,\rho } \quad \text{for any }u, w \in PC_{\gamma ,\rho }(J). $$ - (Ax2):
-
There exists a continuous function \(p :J\longrightarrow [0,\infty )\) such that
$$\begin{aligned}& \bigl\Vert f(t,u,w) \bigr\Vert \leq p(t)\quad \mbox{for a.e. }t \in (a,b] \mbox{ and for each }u,w \in E. \end{aligned}$$ - (Ax3):
-
For each bounded set \(B\subset E\) and for each \(t \in (a,b] \), we have
$$ \mu \bigl(f\bigl(t,B,\bigl({}^{\rho }D_{a^{+}}^{\alpha ,\beta }B\bigr) \bigr)\bigr)\leq \biggl( \frac{t^{\rho }-t_{k}^{\rho }}{\rho } \biggr)^{1-\gamma }p(t) \mu (B), $$with \({}^{\rho }D_{a^{+}}^{\alpha ,\beta }B=\{{}^{\rho }D_{a^{+}}^{\alpha ,\beta }w:w \in B\}\) and \(k=1,\ldots ,m\).
- (Ax4):
-
The functions \(\varpi _{k}:E\longrightarrow E\) are continuous and there exists \(\eta ^{*}>0\) such that
$$ \bigl\Vert \varpi _{k}(u) \bigr\Vert \leq \eta ^{*} \Vert u \Vert \quad \text{for each } u \in E, k=1, \ldots ,m. $$ - (Ax5):
-
For any bounded set \(B\subset E\) and for any \(t \in (a,b] \), we find
$$ \mu \bigl(\varpi _{k}(B)\bigr)\leq \eta ^{*} \biggl( \frac{t^{\rho }-t_{k}^{\rho }}{\rho } \biggr)^{1-\gamma }\mu (B),\quad k=1, \ldots ,m. $$Set \(p^{*}=\sup_{t\in J}p(t)\).
We are now in a position to investigate the existence result for problem (1)–(3) based on the fixed point theorem of Mönch.
Theorem 3.3
Assume that (Ax1)–(Ax5) hold. If
then problem (1)–(3) has at least one solution in \(PC_{\gamma ,\rho }^{\gamma }(J)\subset PC_{\gamma ,\rho }^{\alpha , \beta }(J)\).
Proof
Consider the operator \(\Psi :PC_{\gamma ,\rho }(J)\rightarrow PC_{\gamma ,\rho }(J)\) defined in (20) and the ball \(B_{R}:=B(0,R)=\{w \in PC_{\gamma ,\rho }(J) : \|w\|_{PC_{\gamma ,\rho }} \leq R \}\).
For any \(u \in B_{R}\) and any \(t \in (a,b]\), we find
By Lemma 2.6, we have
Hence, for all \(u \in PC_{\gamma ,\rho }(J)\) and each \(t \in (a,b]\), we get
Hence, we conclude that Ψ transforms the ball \(B_{R}\) into itself. Now, we prove that the operator \(\Psi :B_{R}\rightarrow B_{R}\) fulfills all conditions of Theorem 2.16. For the sake of transparency, we shall divide the proof in four steps.
Step 1: We shall prove that the operator \(\Psi :B_{R}\rightarrow B_{R}\) is continuous.
Suppose that the sequence \(\{u_{n}\}\) converges to u in \(PC_{\gamma ,\rho }(J)\).
Then we get
for each \(t\in (a,b]\), where \(h_{n},h\in C_{\gamma ,\rho }(J)\) in a way that
Since \(u_{n}\rightarrow u\), then we get \(h_{n}(t)\rightarrow h(t)\) as \(n\rightarrow \infty \) for each \(t\in (a,b]\). So we find that
by the Lebesgue dominated convergence theorem.
Step 2: We shall indicate that \(\Psi (B_{R})\) is equicontinuous and bounded.
Indeed, \(\Psi (B_{R})\) is bounded since \(\Psi (B_{R})\subset B_{R}\) and \(B_{R}\) is bounded. Next, let \(\epsilon _{1},\epsilon _{2} \in J\), \(\epsilon _{1}<\epsilon _{2}\), and let \(u\in B_{R}\). Then
Consequently, we conclude that \(\Psi (B_{R})\) is bounded and equicontinuous.
Step 3: The implication (4) of Theorem 2.16 holds.
Now let D be an equicontinuous subset of \(B_{R}\) such that \(D\subset \overline{\Psi (D)}\cup \{0\}\), therefore the function \(t \longrightarrow d(t)=\mu (D(t))\) is continuous on J. Regarding the properties of the measure μ, together with (Ax3) and (Ax5), for each \(t \in (a,b]\), we find
Thus
From (21), we get \(\|d\|_{PC_{\gamma ,\rho }}=0\), that is, \(d(t)=\mu (D(t))=0\), for each \(t\in J_{k}\), \(k=0,\ldots ,m\), and then \(D(t)\) is relatively compact in E. In view of the Ascoli–Arzela theorem, D is relatively compact in \(B_{R}\). Applying now Theorem 2.16, we conclude that Ψ has a fixed point \(u^{*}\in PC_{\gamma ,\rho }(J)\), which is a solution of problem (1)–(3).
Step 4: We show that such a fixed point \(u^{*}\in PC_{\gamma ,\rho }(J)\) lies in \(PC^{\gamma }_{\gamma ,\rho }(J)\).
Regarding the fact that \(u^{*} \) is the only fixed point of the mapping Ψ at \(PC_{\gamma ,\rho }(J)\), for any \(t\in J_{k}\), with \(k=0,\ldots ,m\), we have
where \(h\in C_{\gamma ,\rho }(J)\) in a way that
Applying \({}^{\rho }D^{\gamma }_{t_{k}^{+}}\) to both sides and by Lemma 2.6 and Lemma 2.12, we have
On account of \(\gamma \geq \alpha \), and (Ax1), it lies in \(PC_{\gamma ,\rho }(J)\). Consequently, \({}^{\rho }D^{\gamma }_{t_{k}^{+}}u^{*} \in PC_{\gamma ,\rho }(J) \), which implies that \(u^{*}\in PC^{\gamma }_{\gamma ,\rho }(J) \). Regarding Theorem 3.3, as an outcome of Step 1 to Step 4, we deduce that problem (1)–(3) has at least one solution in \(PC^{\gamma }_{\gamma ,\rho }(J)\). □
Our second existence result for problem (1)–(3) is based on Darbo’s fixed point theorem.
Theorem 3.4
Assume that (Ax1)–(Ax5) and (21) hold. Then problem (1)–(3) has at least one solution in \(PC_{\gamma ,\rho }^{\gamma }(J)\subset PC_{\gamma ,\rho }^{\alpha , \beta }(J)\).
Proof
Consider the operator Ψ defined in (20). We know that \(\Psi :B_{R}\longrightarrow B_{R}\) is bounded and continuous and that \(\Psi (B_{R})\) is equicontinuous, we need to prove that the operator Ψ is a \(\mathfrak{L}\)-contraction.
Let \(D\subset B_{R}\) and \(t \in J\). Then we have
By Lemma, we have
Therefore
So, by (21), the operator Ψ is a \(\mathfrak{L}\)-contraction. □
As a consequence of Theorem 2.17 and using Step 4 of the last result, we deduce that Ψ possesses a fixed point. In particular, this fixed point forms a solution for problem (1)–(3).
Example 3.5
Consider the Banach space
with the norm
We examine the following impulsive boundary value problem of the generalized Hilfer FDE:
where \(J_{k}=(t_{k},t_{k+1}]\), \(t_{k}=1+\frac{k}{5}\) for \(k=0,\ldots ,9\), \(m=9\), \(a=t_{0}=1\), and \(b=t_{10}=3\).
Set
We have
where \(\alpha =\gamma =\frac{1}{2}\), \(\beta =0\), \(\rho =1\), and \(k=0,\ldots ,9\). Clearly, the continuous function \(f \in PC^{0}_{\frac{1}{2},1}([1,2])\).
As a result, condition (Ax1) is fulfilled.
For each \(u,w\in E\) and \(t\in (1,3] \),
Hence condition (Ax2) is satisfied with \(p^{*}=\frac{7}{213}\).
And let
Let \(u\in E\). Then we have
and so condition (Ax4) is satisfied with \(\eta ^{*}=\frac{1}{40}\).
The condition (21) of Theorem 3.3 is satisfied for
Then problem (22)–(24) has at least one solution in \(PC_{\frac{1}{2},1}^{\frac{1}{2}}([1,3])\subset PC_{\frac{1}{2},1}^{ \frac{1}{2},0}([1,3])\).
4 Ulam-type stability
Now, we consider the Ulam stability for problem (1)–(3). Let \(u\in PC_{\gamma ,\rho }(J)\), \(\epsilon >0\), \(\tau >0\), and \(\vartheta :(a,b]\longrightarrow [0,\infty )\) be a continuous function. We consider the following inequality:
Definition 4.1
Problem (1)–(3) is Ulam–Hyers–Rassias (U-H-R) stable with respect to \((\vartheta ,\tau )\) if there exists a real number \(a_{f,m,\vartheta }>0\) such that, for each \(\epsilon >0\) and for each solution \(u \in PC_{\gamma ,\rho }(J)\) of inequality (25), there exists a solution \(w \in PC_{\gamma ,\rho }(J)\) of (1)–(3) with
Remark 4.2
A function \(u \in PC_{\gamma ,\rho }(J)\) is a solution of inequality (25) if and only if there exist \(\sigma \in PC_{\gamma ,\rho }(J)\) and a sequence \(\sigma _{k}, k=0,\ldots ,m\), such that
-
1
\(\|\sigma (t)\|\leq \epsilon \vartheta (t)\) and \(\|\sigma _{k}\|\leq \epsilon \tau \), \(t \in J_{k}\), \(k=1,\ldots ,m\);
-
2
\(({}^{\rho }D_{t_{k}^{+}}^{\alpha ,\beta }u )(t)=f (t, u(t), ({}^{\rho }D_{t_{k}^{+}}^{\alpha ,\beta }u )(t) )+ \sigma (t)\), \(t \in J_{k}\), \(k=0,\ldots ,m\);
-
3
\(({}^{\rho }{\mathcal{J}}_{t_{k}^{+}}^{1-\gamma }u )(t_{k}^{+})= ({}^{\rho }{\mathcal{J}}_{t_{k-1}^{+}}^{1-\gamma }u )(t_{k}^{-})+ \varpi _{k}(u(t_{k}^{-}))+\sigma _{k}\), \(k=1,\ldots ,m\).
Now we are concerned with the Ulam–Hyers–Rassias stability of our problem (1)–(3).
Theorem 4.3
Assume that in addition to (Ax1)–(Ax5) and (21), the following hypotheses hold:
- (Ax6):
-
There exist a nondecreasing function \(\vartheta \in PC_{\gamma ,\rho }(J)\) and \(\lambda _{\vartheta }>0\) such that, for each \(t\in (a,b]\), we have
$$ \bigl({}^{\rho }{\mathcal{J}}_{a^{+}}^{\alpha }\vartheta \bigr) (t)\leq \lambda _{\vartheta }\vartheta (t). $$ - (Ax7):
-
There exists a continuous function \(\chi :[a,b]\longrightarrow [0,\infty )\) such that, for each \(t\in J_{k}\); \(k=0,\ldots ,m\), we have
$$ p(t)\leq \chi (t)\vartheta (t). $$
Then equation (1) is U-H-R stable with respect to \((\vartheta ,\tau )\).
Set \(\chi ^{*}=\sup_{t\in [a,b]}\chi (t)\).
Proof
Consider the operator Ψ defined in (20). Let \(u\in PC_{\gamma ,\rho }(J)\) be a solution of inequality (25), and let us assume that w is the unique solution of the problem
By Theorem 3.1 and Lemma 3.2, we obtain for each \(t\in (a,b]\)
where \(h:(a,b]\rightarrow E \) is a function satisfying the functional equation
Since u is a solution of inequality (25), by Remark 4.2, we have
Clearly, the solution of (26) is given by
where \(g:(a,b]\rightarrow E \) is a function satisfying the functional equation
We have, for each \(t \in J_{k}\), \(k=0,\ldots ,m\),
Hence, for each \(t\in (a,b]\), we have
Thus,
where \(a_{\vartheta }=(1+\frac{2\chi ^{*}}{\epsilon })\lambda _{\vartheta }\). Hence, equation (1) is U-H-R stable with respect to \((\vartheta ,\tau )\). □
Example 4.4
Consider the Banach space
with the norm
and let the following impulsive anti-periodic boundary value problem hold:
where \(J_{0}=(1,2]\), \(J_{1}=(2,e]\), \(t_{1}=2\), \(m=1\), \(a=t_{0}=1\), and \(b=t_{2}=e\).
Set
We have
with \(\gamma =\alpha =\frac{1}{2}\), \(\rho =\frac{1}{2}\), \(\beta =0\), and \(k\in \{0,1\}\). Clearly, the continuous function \(f \in PC^{0}_{\frac{1}{2},\frac{1}{2}}([1,e])\).
Hence condition (Ax1) is satisfied.
For each \(u,w\in E\) and \(t\in (1,e] \),
Hence condition (Ax2) is satisfied with
and
And let
Let \(u\in E\). Then we have
and so condition (Ax4) is satisfied with \(\eta ^{*}=\frac{1}{77e^{4-e}+2}\).
The condition (21) of Theorem 3.3 is satisfied for
Then problem (27)–(29) has at least one solution in \(PC_{\frac{1}{2},\frac{1}{2}}^{\frac{1}{2}}([1,e])\subset PC_{ \frac{1}{2},\frac{1}{2}}^{\frac{1}{2},0}([1,e])\). Also, hypothesis (Ax6) is satisfied with \(\tau =1\), \(\vartheta (t)=e^{3}\), and \(\lambda _{\vartheta }=3\). Indeed, for each \(t\in (1,e]\), we get
Let the function \(\chi :[1,e]\longrightarrow [0,\infty )\) be defined by
then, for each \(t\in (1,e]\), we have
with \(\chi ^{*}=p^{*}e^{-3}\). Hence, condition (Ax7) is satisfied. Consequently, Theorem 4.3 implies that equation (27) is U-H-R stable.
Availability of data and materials
No data were used to support this study. It is not applicable for our paper.
References
Abbas, S., Benchohra, M., Graef, J.R., Henderson, J.: Implicit Differential and Integral Equations: Existence and Stability. de Gruyter, London (2018)
Abbas, S., Benchohra, M., Henderson, J., Lazreg, J.E.: Measure of noncompactness and impulsive Hadamard fractional implicit differential equations in Banach spaces. Math. Eng. Sci. Aerosp. 8(3), 1–19 (2017)
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Advanced Fractional Differential and Integral Equations. Nova Science Publishers, New York (2014)
Abdeljawad, T., Agarwal, R.P., Karapinar, E., Kumari, P.S.: Solutions of he nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space. Symmetry 11, 686 (2019)
Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6652
Afshari, H., Kalantari, S., Baleanu, D.: Solution of fractional differential equations via \(\alpha -\psi \)-Geraghty type mappings. Adv. Differ. Equ. 2018, 347 (2018)
Afshari, H., Kalantari, S., Karapinar, E.: Solution of fractional differential equations via coupled FP. Electron. J. Differ. Equ. 2015(286), 1 (2015)
Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J.: Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities. Springer, Cham (2017)
Ahmad, B., Ntouyas, S.K.: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 15, 362–382 (2012)
Almeida, R., Malinowska, A.B., Odzijewicz, T.: Fractional differential equations with dependence on the Caputo–Katugampola derivative. J. Comput. Nonlinear Dyn. 11(6), 1–11 (2016)
Alqahtani, B., Aydi, H., Karapinar, E., Rakocevic, V.: A solution for Volterra fractional integral equations by hybrid contractions. Mathematics 7, 694 (2019)
Baleanu, D., Güvenç, Z.B., Machado, J.A.T.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, New York (2010)
Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Dekker, New York (1980)
Benchohra, M., Bouriah, S., Graef, J.R.: Boundary value problems for nonlinear implicit Caputo–Hadamard-type fractional differential equations with impulses. Mediterr. J. Math. 14, 206 (2017)
Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York (2006)
Goebel, K.: Concise Course on Fixed Point Theorems. Yokohama Publishers, Japan (2002)
Graef, J.R., Henderson, J., Ouahab, A.: Impulsive Differential Inclusions. A Fixed Point Approch. de Gruyter, Berlin (2013)
Guo, D.J., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht (1996)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)
Karapinar, E., Abdeljawad, T., Jarad, F.: Applying new fixed point theorems on fractional and ordinary differential equations. Adv. Differ. Equ. 2019, 421 (2019)
Karapınar, E., Fulga, A., Rashid, M., Shahid, L., Aydi, H.: Large contractions on quasi-metric spaces with an application to nonlinear fractional differential-equations. Mathematics 7, 444 (2019)
Katugampola, U.: A new approach to a generalized fractional integral. Appl. Math. Comput. 218, 860–865 (2011)
Kucche, K.D., Mali, A.D.: Initial time difference quasilinearization method for fractional differential equations involving generalized Hilfer fractional derivative. Comput. Appl. Math. 39(31), 1–33 (2020)
Mali, A.D., Kucche, K.D.: Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations. Math. Methods Appl. Sci. 43(15), 8608–8631 (2020)
Mönch, H.: BVP for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980)
Oliveira, D.S., Capelas de Oliveira, E.: Hilfer–Katugampola fractional derivatives. Comput. Appl. Math. 37(3), 3672–3690 (2018)
Rassias, J.M.: Functional Equations, Difference Inequalities and Ulam Stability Notions (F.U.N.). Nova Science Publishers, New York (2010)
Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Rassias, Th.M., Brzdek, J.: Functional Equations in Mathematical Analysis, vol. 86. Springer, New York (2012)
Rus, I.A.: Ulam stabilities of ordinary differential equations in a Banach space. Carpath. J. Math. 26, 103–107 (2010)
Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)
Sousa, J.V., Kucche, K.D., Oliveira, E.C.: On the Ulam–Hyers stabilities of the solutions of Ψ-Hilfer fractional differential equation with abstract Volterra operator. Comput. Appl. Math. 37(3), 3672–3690 (2018)
Ulam, S.M.: Problems in Modern Mathematics. Wiley, New York (1940). Chap. 6
Ulam, S.M.: A Collection of Mathematical Problems. Interscience, New York (1960)
Zhou, Y., Wang, J.-R., Zhang, L.: Basic Theory of Fractional Differential Equations, 2nd edn. World Scientific, Hackensack (2017)
Acknowledgements
The authors are grateful to the referees for their helpful remarks and suggestions.
Funding
We declare that funding is not applicable for our paper.
Author information
Authors and Affiliations
Contributions
The authors contributed equally to the manuscript and read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Salim, A., Benchohra, M., Karapınar, E. et al. Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations. Adv Differ Equ 2020, 601 (2020). https://doi.org/10.1186/s13662-020-03063-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-020-03063-4