The space \(\mathscr{W}=C([0, 1],\mathbb {R})=\{\upsilon (t):\upsilon \in C([0, 1]) \}\) is a Banach space with respect to the norm defined by
$$ \|\upsilon \|_{\mathscr{W}}=\max_{t\in [0, 1]}\bigl\{ \bigl\vert \upsilon (t) \bigr\vert : t \in [0, 1]\bigr\} .$$
(2)
Definition 2.1
([1])
The noninteger order integral of a function \(\mathrm{g}\in L^{1}([a,b],\mathbb {R}^{+})\) of order \(\varsigma \in \mathbb {R}^{+}\) is defined by
$$ \mathcal{I}^{\varsigma }_{a} \mathrm{g}(t)= \int _{a}^{t} \frac{(t-z)^{\varsigma -1}}{\Gamma (\varsigma )} \mathrm{g}(z)\,dz, $$
(3)
where Γ is the gamma function.
Definition 2.2
([1])
For a function g given on the interval \([a,b]\), the Caputo fractional order derivative of υ is defined by
$$ {}^{C}D^{\varsigma }\mathrm{g}(t)=\frac{1}{\Gamma (\ell -\varsigma )} \int _{a}^{t} (t-z)^{\ell -\varsigma -1} \mathrm{g}^{(\ell )}(z)\,dz, $$
(4)
where \(\ell =[\varsigma ]+1\).
Lemma 2.3
([39])
For \(\varsigma >0\), the given result holds:
$$ \mathcal{I}^{\varsigma } \bigl({}^{C}D^{\varsigma } \mathrm{g}(t) \bigr)= \mathrm{g}(t)-\sum_{\jmath =0}^{\ell -1} \frac{\mathrm{g}^{(\jmath )}(0)}{\jmath !}t^{\jmath },\quad \textit{where } \ell =[\varsigma ]+1.$$
We construct the following three inequalities for studying Hyers–Ulam stability of problem (1). Let \(x\in C([0, 1], \mathbb {R}_{+})\) be a nondecreasing function, \(\xi \geq 0\), \(\psi \in \mathscr{W}\), such that for
,
, the following sets of inequalities are satisfied:
where ς and m are the same as defined in problem (1).
Definition 2.4
([40])
If for \(\epsilon >0\) there exists a constant \(C_{\mathrm{g}}>0\) such that, for any solution \(\psi \in \mathscr{W}\) of inequality (5), there is a unique solution \(\upsilon \in \mathscr{W}\) of system (1) that satisfies
$$ \bigl\vert \psi (t)-\upsilon (t) \bigr\vert \leq C_{\mathrm{g}}\epsilon ,\quad t\in I, $$
then problem (1) is Hyers–Ulam stable.
Definition 2.5
If for \(\epsilon >0\) and a set of positive real numbers \(\mathbb {R}^{+}\) there exists \(x\in C(\mathbb {R}^{+},\mathbb {R}^{+})\) with \(x(0)=0\) such that, for any solution \(\psi \in \mathscr{W}\) of inequality (6), there exist \(\epsilon >0\) and a unique solution \(\upsilon \in \mathscr{W}\) of problem (1) that satisfies
$$ \bigl\vert \psi (t)-\upsilon (t) \bigr\vert \leq x(\epsilon ),\quad t\in I, $$
then problem (1) is generalized Hyers–Ulam stable.
Definition 2.6
([40])
If for \(\epsilon >0\) there exists a real number \(C_{\mathrm{g}}>0\) such that, for any solution \(\psi \in \mathscr{W}\) of inequality (7), there is a unique solution \(\upsilon \in \mathscr{W}\) of problem (1) that satisfies
$$ \bigl\vert \psi (t)-\upsilon (t) \bigr\vert \leq C_{\mathrm{g}}\epsilon \bigl(\xi +x(t)\bigr), \quad t \in I, $$
then problem (1) is Hyers–Ulam–Rassias stable with respect to \((\xi ,x)\).
Definition 2.7
([40])
If there exists a constant \(C_{\mathrm{g}}>0\) such that, for any solution \(\psi \in \mathscr{W}\) of inequality (6), there is a unique solution \(\upsilon \in \mathscr{W}\) of problem (1) that satisfies
$$ \bigl\vert \psi (t)-\upsilon (t) \bigr\vert \leq C_{\mathrm{g}}\bigl(\xi +x(t)\bigr),\quad t\in I, $$
then problem (1) is generalized Hyers–Ulam–Rassias stable with respect to \((\xi ,x)\).
Remark 1
The function \(\psi \in \mathscr{W}\) is a solution of inequality (5) if there exist a function \(y\in \mathscr{W}\) and a sequence
,
, which depends on ψ such that
-
(i)
\(|y(t)|\leq \epsilon \),
, \(t\in I\),
-
(ii)
\({}_{0}^{C}D_{t}^{\varsigma }\psi (t)=\mathrm{g}(t, \psi (t), \psi (mt), {}_{0}^{C}D_{t}^{\varsigma }{\psi (t)})+y(t)\),
-
(iii)
,
-
(iv)
.
Remark 2
The function \(\psi \in \mathscr{W}\) is a solution of inequality (6) if there exist a function \(y\in \mathscr{W}\) and a sequence
,
, which depends on ψ such that
-
(i)
\(|y(t)|\leq x(t)\),
, \(t\in I\),
-
(ii)
\({}_{0}^{C}D_{t}^{\varsigma }\psi (t)=\mathrm{g}(t, \psi (t), \psi (mt), {}_{0}^{C}D_{t}^{\varsigma }{\psi (t)})+y(t)\),
-
(iii)
,
-
(iv)
.
Remark 3
The function \(\psi \in \mathscr{W}\) is a solution of inequality (6) if there exist a function \(y\in \mathscr{W}\) and a sequence
,
, which depends on ψ such that
-
(i)
\(|y(t)|\leq \epsilon x(t)\),
, \(t\in I\),
-
(ii)
\({}_{0}^{C}D_{t}^{\varsigma }\psi (t)=\mathrm{g}(t, \psi (t), \psi (mt), {}_{0}^{C}D_{t}^{\varsigma }{\psi (t)})+y(t)\),
-
(iii)
,
-
(iv)
.
We give the proof of the following lemma, which provides a base for obtaining a solution to problem (1).
Lemma 2.8
Let \(\varsigma \in (1, 2]\), \(\alpha :[0, 1]\rightarrow \mathbb {R}\) be a continuous function, then the function \(\upsilon \in \mathscr{W}\) is the solution to the following problem:
if and only if υ satisfies the following integral equation:
where
Proof
Assume that, for \(t\in [0, t_{1}]\), υ is a solution of (8). Then, by Lemma 2.3, there exist \(a_{1},a_{2}\in \mathbb {R}\) such that
$$ \upsilon (t) = {}_{0}I_{t}^{\varsigma } \alpha (t)-a_{1}-a_{2}t= \frac{1}{\Gamma (\varsigma )} \int _{0}^{t}(t-z)^{\varsigma -1}\alpha (z)\,dz-a_{1}-a_{2}t,$$
(10)
which also yields
$$ \upsilon '(t) = \frac{1}{\Gamma (\varsigma -1)} \int _{0}^{t}(t-z)^{ \varsigma -2}\alpha (z)\,dz-a_{2}. $$
(11)
Let, for \(t\in (t_{1}, t_{2}]\), us have \(d_{1},d_{2}\in \mathbb {R}\) with
$$\begin{aligned}& \upsilon (t) = \frac{1}{\Gamma (\varsigma )} \int _{t_{1}}^{t}(t-z)^{ \varsigma -1}\alpha (z)\,dz-d_{1}-d_{2}(t-t_{1}), \\& \upsilon '(t) = \frac{1}{\Gamma (\varsigma -1)} \int _{t_{1}}^{t}(t-z)^{ \varsigma -2}\alpha (z)\,dz-d_{2}. \end{aligned}$$
(12)
This leads us to
$$\begin{aligned}& \upsilon \bigl(t_{1}^{-}\bigr) = \frac{1}{\Gamma (\varsigma )} \int _{t_{0}}^{t_{1}}(t_{1}-z)^{ \varsigma -1} \alpha (z)\,dz-a_{1}-a_{2}t_{1},\qquad \upsilon \bigl(t_{1}^{+}\bigr)=-d_{1}, \\& \upsilon '\bigl(t_{1}^{-}\bigr) = \frac{1}{\Gamma (\varsigma -1)} \int _{0}^{t_{1}}(t_{1}-z)^{ \varsigma -2} \alpha (z)\,dz-a_{2},\qquad \upsilon '\bigl(t_{1}^{+} \bigr)=-d_{2}. \end{aligned}$$
Corresponding to impulsive conditions, we have
$$ \Delta \upsilon (t_{1})=\upsilon \bigl(t_{1}^{+} \bigr)-\upsilon \bigl(t_{1}^{-}\bigr)= \mathcal{F}_{1}\bigl(\upsilon (t_{1})\bigr) \quad \text{and}\quad \Delta \upsilon '(t_{1})= \upsilon '\bigl(t_{1}^{+}\bigr)-\upsilon '\bigl(t_{1}^{-}\bigr)=\bar{ \mathcal{F}}_{1}\bigl( \upsilon (t_{1})\bigr), $$
one has
$$\begin{aligned}& -d_{1} = \frac{1}{\Gamma (\varsigma )} \int _{t_{0}}^{t_{1}}(t_{1}-z)^{ \varsigma -1} \alpha (z)\,dz-a_{1}-a_{2}t_{1}+ \mathcal{F}_{1}\bigl(\upsilon (t_{1})\bigr), \\& -d_{2} = \frac{1}{\Gamma (\varsigma -1)} \int _{0}^{t_{1}}(t_{1}-z)^{ \varsigma -2} \alpha (z)\,dz-a_{2}+\bar{\mathcal{F}}_{1}\bigl(\upsilon (t_{1})\bigr). \end{aligned}$$
Thus (12) implies
$$\begin{aligned} \upsilon (t) =&\frac{1}{\Gamma (\varsigma )} \int _{t_{1}}^{t}(t-z)^{ \varsigma -1}\alpha (z)\,dz+ \frac{1}{\Gamma (\varsigma )} \int _{0}^{t_{1}}(t_{1}-z)^{ \varsigma -1} \alpha (z)\,dz \\ &{}+\frac{t-t_{1}}{\Gamma (\varsigma -1)} \int _{0}^{t_{1}}(t_{1}-z)^{ \varsigma -2} \alpha (z)\,dz+\mathcal{F}_{\jmath }\bigl(\upsilon (t_{1}) \bigr)+(t-t_{1}) \bar{\mathcal{F}}_{1}\bigl(\upsilon (t_{1})\bigr) \\ &{}-a_{1}-a_{2}t, \quad t\in (t_{1}, t_{2}]. \end{aligned}$$
Similarly, for
, one has
which by differentiation gives the result
Using the given boundary conditions in (10), (11), we obtain
$$ -pa_{1}-qa_{2}= \int _{0}^{1}h_{1}\bigl(\upsilon (z) \bigr)\,dz $$
(15)
and
Thus, in view of \(p\upsilon (1)+q\upsilon '(1)=\int _{0}^{1}h_{2}(\upsilon (z))\,dz\) and the result (15), we get the following values for \(-a_{1}\) and \(-a_{2}\):
Putting these values for \(-a_{1}\) and \(-a_{2}\) in (10) and (13), we get (9). □