1 Introduction
The well-known inequalities, such as the Gronwall type inequality, the Gronwall-Bellman type inequality, the Henry-Gronwall type inequality, the Henry-Bihari type inequality and their variants in retarded form played important roles in the research of quantitative analysis of the solutions to differential and integral equations, as well as in the modeling of engineering and science problems. Recently, with the development of fractional differential equations, integral inequalities with weakly singular kernels have drawn more attention [1‐11]. In 1981, Henry [12] proposed a method to find solutions and proved some results concerning linear integral inequalities with a weakly singular kernel. In 1997, Medved̆ [3] presented a new method to solve integral inequalities of Henry-Gronwall type and their Bihari version, then he got the explicit bounds with a quite simple formula, similar to the classic Gronwall-Bellman inequalities. Furthermore, he also obtained global solutions of the semilinear evolutions in [4].
In 2011, Ye and Gao [5] presented the integral inequalities of Henry-Gronwall type,
$$ \textstyle\begin{cases} u(t) \leq a(t)+\int_{ t_{0}}^{t}(t-s)^{\beta -1}[b(s)u(s)+c(s)u(s-r)]\,ds, \quad t\in[t_{0}, T), \\ u(t)\leq\varphi(t), \quad t\in[t_{0}-r, t_{0}). \end{cases} $$
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In 2013, Shao and Meng [6] established Gronwall-Bellman type inequalities with a weakly singular integral
$$ \textstyle\begin{cases} u(t) \leq a(t)+\int_{ t_{0}}^{t}(t-s)^{\beta-1}b(s)u(s)\,ds+\int_{ t_{0}}^{t}(t-s)^{\beta-1}p(s)u^{\gamma}(s-r)\,ds, \quad t\in[t_{0}, T), \\ u(t)\leq\varphi(t), \quad t\in[t_{0}-r, t_{0}). \end{cases} $$
In [7], Feng and Meng studied the following Gronwall-Bellman type inequalities:
$$ u^{p}(x) \leq a(x)+ \int_{0}^{x}b(t)u^{q}(t)\,dt+ \frac{1}{\Gamma(\alpha )}h(x) \int_{0}^{x}(x-t)^{\alpha-1}L \bigl(t, u(t) \bigr) \,dt. $$
In 2008, by using the modified Medved̆ method, Ma and Pečarić [8] studied the inequality
$$ u^{p}(t) \leq a(t)+b(t) \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta -1}s^{\gamma-1}f(s) u^{q}(s)\,ds,\quad t\in R_{+}. $$
The aim of this paper was to give explicit bounds to some new nonlinear Henry-Gronwall type retarded integral inequalities with weakly singular integral kernel of the form
and Gronwall-Bellman type integral inequalities with nonlinear weakly singular integral kernel of the form
which can be used as handy and effective tools in the study of the delay fractional differential equations. We also give some examples to illustrate applications of our results.
$$ \textstyle\begin{cases} u^{p}(t) \leq a(t)+\int_{t_{0}}^{t}(t-s)^{\beta-1}b(s) u^{q}(s)\,ds+\int_{t_{0}}^{t}(t-s)^{\beta-1}c(s) u^{l}(s-r)\,ds, \quad t\in I, \\ u(t)\leq\varphi(t), \quad t\in[t_{0}-r, t_{0}), \end{cases} $$
$$ u^{p}(t) \leq a(t)+b(t) \int_{0}^{t}(t-s)^{\beta-1}c(s) u^{m}(s)\,ds+d(t) \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta -1}s^{\gamma-1}f(s) u^{q}(s)\,ds, $$
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2 Preliminary knowledge
In the following, R denotes the set of real numbers, N denotes the set of integer numbers, \(R_{+}= [0, +\infty)\), \(I=[t_{0}, T)\subset R_{+}\), \(T<\infty\). For convenience, we give some lemmas which will be used in the proof of the main results.
Lemma 2.1
(Jensen’s inequality)
Let
\(n\in N\), \(a_{1}, a_{2}, \ldots, a_{n}\)
be nonnegative real numbers. Then, for
\(r>1\),
$$ \Biggl(\sum_{i=1}^{n}a_{i} \Biggr)^{r}\leq n^{r-1}\sum_{i=1}^{n}a_{i}^{r} . $$
Lemma 2.2
(see [13])
Let
\(a\geq0\), \(p\geq q\geq0\), \(p\neq0\), then
$$ a^{\frac{q}{p}}\leq \frac{q}{p}K^{\frac{q-p}{p}}a+\frac {p-q}{p}K^{\frac{q}{p}} \quad \textit{for any } K> 0 . $$
Lemma 2.3
(see [14])
Let
α, β, γ, and
p
be positive constants, then
Here
\(B(m,n)=\int^{1}_{0}s^{m-1}(1-s)^{n-1}\,ds\) (\(m > 0\), \(n > 0\)) is the
B-function, \(\theta=p[\alpha(\beta-1)+\gamma-1]+1\).
$$ \int^{t}_{0} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{p(\beta-1)}s^{p(\gamma -1)}\,ds=\frac{t^{\theta}}{\alpha}B \biggl[ \frac{p(\gamma -1)+1}{\alpha}, p(\beta-1)+1 \biggr], \quad t\in R_{+}. $$
Definition 2.1
(see [15])
Let \([x, y, z]\) be an ordered parameter group of nonnegative real numbers. The group is said to belong to the first class distribution and denoted by \([x, y, z] \in I\) if the conditions \(x \in(0, 1]\), \(y \in(\frac {1}{2}, 1) \), and \(z \geq\frac{3}{2}-y\) are satisfied; the group is said to belong to the second class distribution and denoted by \([x, y, z]\in\mathit{II}\) if the conditions \(x\in(0, 1]\), \(y \in(0, \frac{1}{2}]\), and \(z > \frac {1-2y^{2}}{1-y^{2}}\) are satisfied.
Lemma 2.4
(see [15])
Suppose that the positive constants
α, β, γ, \(p_{1}\), and
\(p_{2}\)
satisfy the conditions
(a)
if
\([\alpha, \beta, \gamma]\in I\), \(p_{1}=\frac {1}{\beta}\);
(b)
if
\([\alpha, \beta, \gamma]\in \mathit{II}\), \(p_{2}=\frac {1+4\beta}{1+3\beta}\), then
and
are valid for
\(i=1, 2\).
$$ B \biggl[\frac{p_{i}(\gamma-1)+1}{\alpha}, p_{i}(\beta-1)+1 \biggr]\in(0, \infty) $$
$$ \theta_{i}=p_{i} \bigl[\alpha(\beta-1)+\gamma-1 \bigr]+1 \geq0 $$
Lemma 2.5
([13])
Let
\(a(t), b(t)\in C(I, R_{+})\), \(p\geq q>0\). If
\(u(t)\in C(I, R_{+})\), and
then
Here
$$ u^{p}(t) \leq a(t)+ \int_{t_{0}}^{t}b(s) u^{q}(s)\,ds, $$
$$ u(t) \leq \biggl\{ a(t)+ \int_{t_{0}}^{t}A_{1}(s)e^{\int _{s}^{t}B_{1}(\tau)\,d\tau}\,ds \biggr\} ^{\frac{1}{p}}, \quad t\in I. $$
(2.1)
$$\begin{aligned}& A_{1}(t)=b(t) \biggl[\frac{q}{p}K^{\frac{q-p}{p}}a(t)+ \frac {p-q}{p}K^{\frac{q}{p}} \biggr],\qquad B_{1}(t)= \frac{q}{p}K^{\frac {q-p}{p}}b(t). \end{aligned}$$
Lemma 2.6
Let
\(a(t), b(t), c(t)\in C(I, R_{+})\), \(p\geq q>0\), \(p\geq l>0\), \(\varphi(t)\in C([t_{0}-r, t_{0}], R_{+})\), \(a(t_{0})=\varphi(t_{0})\). If
\(u(t)\in C(I, R_{+})\), and
Then, for
\(t\in[t_{0}, t_{0}+r)\),
Here
\(r\in R_{+}\),
$$ \textstyle\begin{cases} u^{p}(t) \leq a(t)+\int_{t_{0}}^{t}b(s) u^{q}(s)\,ds+\int _{t_{0}}^{t}c(s) u^{l}(s-r)\,ds, \quad t\in I, \\ u(t)\leq\varphi(t), \quad t\in[t_{0}-r, t_{0}). \end{cases} $$
(2.2)
$$ u(t) \leq \biggl\{ a(t)+ \int_{t_{0}}^{t}c(s)\varphi^{l}(s-r)\,ds+ \int _{t_{0}}^{t}A_{2}(s)e^{\int_{s}^{t}B_{2}(\tau)\,d\tau} \,ds \biggr\} ^{\frac{1}{p}}. $$
(2.3)
$$\begin{aligned} \begin{aligned} &A_{2}(t)=\frac{q}{p}K^{\frac{q-p}{p}}b(t) \biggl(a(t)+ \int ^{t}_{t_{0}}c(s)\varphi^{l}(s-r)\,ds \biggr)+\frac{p-q}{p}K^{\frac{q}{p}}, \\ &B_{2}(t)=B_{1}(t)=\frac{q}{p}K^{\frac{q-p}{p}}b(t). \end{aligned} \end{aligned}$$
For
\(t\in[t_{0}+r, T)\),
where
$$ u(t) \leq \biggl\{ a(t)+ \int_{t_{0}}^{t}A_{3}(s)e^{\int _{s}^{t}B_{3}(\tau)\,d\tau} \,ds \biggr\} ^{\frac{1}{p}}, $$
(2.4)
$$\begin{aligned}& A_{3}(t)=\frac{q}{p}K^{\frac{q-p}{p}}a(t)b(t)+ \frac {p-q}{p}K^{\frac{q}{p}}b(t)+\frac{l}{p}K^{\frac {l-p}{p}}a(t-r)c(t)+ \frac{p-l}{p}K^{\frac{l}{p}}c(t), \\& B_{3}(t)=\frac{q}{p}K^{\frac {q-p}{p}}b(t)+ \frac{l}{p}K^{\frac{l-p}{p}}c(t). \end{aligned}$$
Proof
For \(t\in[t_{0}, t_{0}+r)\), we get
then from Lemma 2.5, the inequality (2.3) holds.
$$ u^{p}(t) \leq a(t)+ \int_{t_{0}}^{t}c(s) \varphi^{l}(s-r)\,ds+ \int_{t_{0}}^{t}b(s) u^{q}(s)\,ds, $$
For \(t\in[t_{0}+r, T)\), let \(z(t)=\int_{t_{0}}^{t}b(s) u^{q}(s)\,ds+\int_{t_{0}}^{t}c(s) u^{l}(s-r)\,ds\), then \(z(t_{0})=0\), \(z(t)\) is nondecreasing, and
Then from Lemma 2.2 we obtain, for any \(K>0\),
and we have \(z(t)\leq \int_{t_{0}}^{t}A_{3}(s)e^{\int_{s}^{t}B_{3}(\tau)\,d\tau}\,ds\). Then the inequality (2.4) holds. □
$$ u^{p}(t) \leq a(t)+z(t). $$
$$\begin{aligned} z'(t) =&b(t) u^{q}(t)+c(t) u^{l}(t-r) \\ \leq& b(t) \biggl[\frac{q}{p}K^{\frac{q-p}{p}} \bigl(a(t)+z(t) \bigr)+ \frac{p-q}{p}K^{\frac{q}{p}} \biggr] \\ &{}+c(t) \biggl[\frac{l}{p}K^{\frac{l-p}{p}} \bigl(a(t-r)+z(t-r) \bigr)+ \frac {p-l}{p}K^{\frac{l}{p}} \biggr] \\ \leq& A_{3}(t)+B_{3}(t)z(t), \end{aligned}$$
Lemma 2.7
([16])
Let
\(a(t)\), \(b(t, s)\), \(b'(t,s)\)
be continuous and nonnegative. If
\(u(t)\in C(I, R_{+})\)
and
then
$$ u(t) \leq a(t)+ \int_{t_{0}}^{t}b(t, s)u(s)\,ds, $$
$$ u(t) \leq a(t)+e^{\int_{t_{0}}^{t}b(t, s)\,ds} \int_{t_{0}}^{t}e^{-\int _{t_{0}}^{\tau} b(\tau, s)\,ds} \biggl[a(\tau)b(\tau, \tau)+ \int_{t_{0}}^{\tau }b'_{\tau}(\tau, s)a(s)\,ds \biggr]\,d\tau. $$
(2.5)
3 Main results
We are now to deal with a certain class of Henry-Gronwall type retarded inequalities with weak integral kernels.
Theorem 3.1
Let
\(a(t), b(t), c(t)\in C(I, R_{+})\), \(p\geq q>0\), \(p\geq l>0\), \(\varphi(t)\in C([t_{0}-r, t_{0}], R_{+})\), \(a(t_{0})=\varphi(t_{0})\), \(u(t)\in C(I, R_{+})\). Suppose
where
\(\beta> 0\).
$$ \textstyle\begin{cases} u^{p}(t) \leq a(t)+\int_{t_{0}}^{t}(t-s)^{\beta-1}b(s) u^{q}(s)\,ds+\int_{t_{0}}^{t}(t-s)^{\beta-1}c(s) u^{l}(s-r)\,ds, \quad t\in I, \\ u(t)\leq\varphi(t), \quad t\in[t_{0}-r, t_{0}), \end{cases} $$
(3.1)
(i) Suppose that
\(\beta>\frac{1}{2}\), then
Here
$$\begin{aligned} \textstyle\begin{cases} u(t) \leq \{a_{1}(t)+\int_{t_{0}}^{t}c_{1}(s)\varphi ^{l}(s-r)\,ds+\int_{t_{0}}^{t}A_{4}(s)e^{\int_{s}^{t}B_{4}(\tau)\,d\tau }\,ds \}^{\frac{1}{p}}, \quad t\in[t_{0}, t_{0}+r), \\ u(t)\leq \{a_{1}(t)+\int_{t_{0}}^{t}A_{5}(s)e^{\int_{s}^{t}B_{5}(\tau )\,d\tau}\,ds \}^{\frac{1}{p}}, \quad t\in[t_{0}+r, T). \end{cases}\displaystyle \end{aligned}$$
(3.2)
$$\begin{aligned}& a_{1}(t)=3a^{2}(t)e^{-2t} , \qquad b_{1}(t)=\frac{6\Gamma(2\beta-1)}{4^{\beta}}b^{2}(t)e^{2t(\frac{q}{p}-1)} , \\& c_{1}(t)=\frac{6\Gamma(2\beta-1)}{4^{\beta}}c^{2}(s)e^{2t(\frac{l}{p}-1)} , \\& A_{4}(t)=\frac{q}{p}K^{\frac{q-p}{p}}b_{1}(t) \biggl[a_{1}(t)+ \int_{t_{0}}^{t}c_{1}(s) \varphi^{l}(s-r)\,ds \biggr]+\frac{p-q}{p}K^{\frac{q}{p}} , \\& B_{4}(t)=\frac{q}{p}K^{\frac {q-p}{p}}b_{1}(t) , \\& A_{5}(t)= \biggl[\frac{q}{p}K^{\frac{q-p}{p}}a_{1}(t)+ \frac {p-q}{p}K^{\frac{q}{p}} \biggr]b_{1}(t)+ \biggl[ \frac{l}{p}K^{\frac {l-p}{p}}a_{1}(t-r) +\frac{p-l}{p}K^{\frac{l}{p}} \biggr]c_{1}(t) , \\& B_{5}(t)=\frac{q}{p}K^{\frac{q-p}{p}}b_{1}(t)+ \frac{l}{p}K^{\frac {l-p}{p}}c_{1}(t) . \end{aligned}$$
(ii) If
\(0<\beta\leq\frac{1}{2}\), let
\(q_{1}=\frac{1+\beta }{\beta}\), \(p_{1}=1+\beta\), then
Here
$$ \textstyle\begin{cases} u(t) \leq \{a_{2}(t)+\int_{t_{0}}^{t}c_{2}(s)\varphi ^{l}(s-r)\,ds+\int_{t_{0}}^{t}A_{6}(s)e^{\int_{s}^{t}B_{6}(\tau)\,d\tau }\,ds \}^{\frac{1}{p}}, \quad t\in[t_{0}, t_{0}+r), \\ u(t)\leq \{a_{2}(t)+\int_{t_{0}}^{t}A_{7}(s)e^{\int _{s}^{t}B_{7}(\tau)\,d\tau}\,ds \}^{\frac{1}{p}},\quad t\in [t_{0}+r, T). \end{cases} $$
(3.3)
$$\begin{aligned}& a_{2}(t)=3^{q_{1}-1}a^{q_{1}}(t)e^{-q_{1}t} , \qquad b_{2}(t)=3^{q_{1}-1} \biggl(\frac{\Gamma(1-(1-\beta )p_{1})}{p_{1}^{1-(1-\beta)p_{1}}} \biggr)^{\frac{q_{1}}{p_{1}}}b^{q_{1}}(t)e^{(-q_{1}+\frac {q_{1}q}{p})t} , \\& c_{2}(t)=3^{q_{1}-1} \biggl(\frac{\Gamma(1-(1-\beta )p_{1})}{p_{1}^{1-(1-\beta)p_{1}}} \biggr)^{\frac{q_{1}}{p_{1}}} c^{q_{1}}(t)e^{(-q_{1}+\frac{q_{1}l}{p})t} , \\& A_{6}(t)=\frac {q}{p}K^{\frac{q-p}{p}}b_{2}(t) \biggl(a_{2}(t)+ \int _{t_{0}}^{t}c_{2}(s) \varphi^{l}(s-r)\,ds \biggr)+\frac{p-q}{p}K^{\frac {q}{p}} , \\& B_{6}(t)=\frac{q}{p}K^{\frac{q-p}{p}}b_{2}(t), \\& A_{7}(t)= \biggl[\frac{q}{p}K^{\frac{q-p}{p}}a_{2}(t)+ \frac {p-q}{p}K^{\frac{q}{p}} \biggr]b_{2}(t)+ \biggl[ \frac{l}{p}K^{\frac {l-p}{p}}a_{2}(t-r) +\frac{p-l}{p}K^{\frac{l}{p}} \biggr]c_{2}(t) , \\& B_{7}(t)=\frac{q}{p}K^{\frac{q-p}{p}}b_{2}(t)+ \frac{l}{p}K^{\frac {l-p}{p}}c_{2}(t) . \end{aligned}$$
Proof
(i) Using the Cauchy-Schwarz inequality by (3.1), we get
Using Lemma 2.1 for \(n=3\), \(r=2\), we obtain
$$\begin{aligned} u^{p}(t) \leq& a(t)+ \int_{t_{0}}^{t}(t-s)^{\beta-1}e^{s}e^{-s}b(s) u^{q}(s)\,ds+ \int_{t_{0}}^{t}(t-s)^{\beta-1}e^{s}e^{-s}c(s) u^{l}(s-r)\,ds \\ \leq& a(t)+ \biggl( \int_{t_{0}}^{t}(t-s)^{2\beta-2}e^{2s} \,ds \biggr)^{\frac{1}{2}} \biggl( \int_{t_{0}}^{t}b^{2}(s)e^{-2s} u^{2q}(s)\,ds \biggr)^{\frac{1}{2}} \\ &{}+ \biggl( \int_{t_{0}}^{t}(t-s)^{2\beta-2}e^{2s} \,ds \biggr)^{\frac {1}{2}} \biggl( \int_{t_{0}}^{t}c^{2}(s)e^{-2s} u^{2l}(s-r)\,ds \biggr)^{\frac{1}{2}} \\ \leq&a(t)+ \biggl(\frac{2e^{2t}\Gamma(2\beta-1)}{4^{\beta}} \biggr)^{\frac{1}{2}} \\ &{}\cdot\biggl[ \biggl( \int_{t_{0}}^{t}b^{2}(s)e^{-2s} u^{2q}(s)\,ds \biggr)^{\frac{1}{2}}+ \biggl( \int_{t_{0}}^{t}c^{2}(s)e^{-2s} u^{2l}(s-r)\,ds \biggr)^{\frac {1}{2}} \biggr] . \end{aligned}$$
$$ u^{2p}(t)\leq3a^{2}(t)+3\frac{2e^{2t}\Gamma(2\beta-1)}{4^{\beta }} \biggl( \int_{t_{0}}^{t}b^{2}(s)e^{-2s} u^{2q}(s)\,ds+ \int _{t_{0}}^{t}c^{2}(s)e^{-2s} u^{2l}(s-r)\,ds \biggr). $$
Let \(v(t)=e^{-\frac{2}{p}t}u^{2}(t)\), then we get
Using Lemma 2.6, we get (3.2).
$$\begin{aligned} v^{p}(t) \leq&3a^{2}(t)e^{-2t}+\frac{6\Gamma(2\beta-1)}{4^{\beta }} \\ &{}\cdot \biggl( \int_{t_{0}}^{t}b^{2}(s)e^{2s(\frac{q}{p}-1)} v^{q}(s)\,ds+ \int_{t_{0}}^{t}c^{2}(s)e^{2s(\frac{l}{p}-1)} v^{l}(s-r)\,ds \biggr). \end{aligned}$$
(ii) By the hypothesis, we get \(\frac{1}{p_{1}}+\frac {1}{q_{1}}=1\), using the Hölder inequality by (3.1), we obtain
$$\begin{aligned} u^{p}(t) \leq& a(t)+ \biggl( \int_{t_{0}}^{t}(t-s)^{p_{1}(\beta -1)}e^{p_{1}s} \,ds \biggr)^{\frac{1}{p_{1}}} \biggl( \int _{t_{0}}^{t}b^{q_{1}}(s)e^{-q_{1}s} u^{q_{1}q}(s)\,ds \biggr)^{\frac {1}{q_{1}}} \\ &{}+ \biggl( \int_{t_{0}}^{t}(t-s)^{p_{1}(\beta-1)}e^{p_{1}s} \,ds \biggr)^{\frac{1}{p_{1}}} \biggl( \int_{t_{0}}^{t}e^{-q_{1}s}c^{q_{1}}(s) u^{q_{1}l}(s-r)\,ds \biggr)^{\frac{1}{q_{1}}} \\ =&a(t)+ \biggl(\frac{e^{p_{1}t}\Gamma(1-(1-\beta )p_{1})}{p_{1}^{1-(1-\beta)p_{1}}} \biggr)^{\frac{1}{p_{1}}} \\ &{}\cdot \biggl[ \biggl( \int_{t_{0}}^{t}b^{q_{1}}(s)e^{-q_{1}s} u^{q_{1}q}(s)\,ds \biggr)^{\frac{1}{q_{1}}}+ \biggl( \int_{t_{0}}^{t}c^{q_{1}}(s)e^{-q_{1}s} u^{q_{1}l}(s-r)\,ds \biggr)^{\frac{1}{q_{1}}} \biggr] . \end{aligned}$$
Using Lemma 2.1 for \(n=3\), \(r=q_{1}\), we have
Let \(v(t)=e^{-\frac{q_{1}}{p}t}u^{q_{1}}(t)\), then we get
$$\begin{aligned} u^{pq_{1}}(t) \leq&3^{q_{1}-1} a^{q_{1}}(t)+3^{q_{1}-1} \biggl(\frac {e^{p_{1}t}\Gamma(1-(1-\beta)p_{1})}{p_{1}^{1-(1-\beta)p_{1}}} \biggr)^{\frac{q_{1}}{p_{1}}} \\ &{}\cdot \biggl[ \int_{t_{0}}^{t}b^{q_{1}}(s)e^{-q_{1}s} u^{q_{1}q}(s)\,ds+ \int_{t_{0}}^{t}c^{q_{1}}(s)e^{-q_{1}s} u^{q_{1}l}(s-r)\,ds \biggr] . \end{aligned}$$
$$\begin{aligned} v^{p}(t) \leq&3^{q_{1}-1} a^{q_{1}}(t)e^{-q_{1}t}+3^{q_{1}-1} \biggl(\frac {\Gamma(1-(1-\beta)p_{1})}{ p_{1}^{1-(1-\beta)p_{1}}} \biggr)^{\frac{q_{1}}{p_{1}}} \\ &{}\cdot \biggl[ \int_{t_{0}}^{t}b^{q_{1}}(s)e^{(-q_{1}+\frac {q_{1}q}{p})s} v^{q}(s)\,ds+ \int_{t_{0}}^{t}c^{q_{1}}(s)e^{(-q_{1}+\frac{q_{1}l}{p})s} v^{l}(s-r)\,ds \biggr]. \end{aligned}$$
Remark 3.1
Theorem 3.2
Suppose that
\(u(t)\), \(a(t)\), \(b(t)\), \(c(t)\), \(d(t)\), \(f(t)\)
are nonnegative continuous functions for
\(t\in R_{+}\). Let
p, m, q
be constants with
\(p\geq m>0\), \(p\geq q >0\). If
then for
\([\alpha, \beta, \gamma]\in I\),
where
for
\([\alpha, \beta, \gamma]\in \mathit{II}\),
where
$$ u^{p}(t) \leq a(t)+b(t) \int_{0}^{t}(t-s)^{\beta-1}c(s) u^{m}(s)\,ds+d(t) \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta -1}s^{\gamma-1}f(s) u^{q}(s)\,ds, $$
(3.4)
$$\begin{aligned} u(t) \leq& \biggl\{ a(t)+ \biggl[\bar{a}_{1}(t)+e^{\int_{0}^{t}\bar {b}_{1}(t,s)\,ds}\int_{0}^{t}e^{-\int_{0}^{\tau}\bar{b}_{1}(\tau ,s)\,ds} \\ &{}\cdot \biggl( \bar{a}_{1}(s) \bar{b}_{1}(\tau,\tau)+ \int_{0}^{\tau}\bar{b}'_{1\tau}( \tau ,s)\bar{a}_{1}(s)\,ds \biggr)\,d\tau \biggr]^{1-\beta} \biggr\} ^{\frac{1}{p}}, \end{aligned}$$
(3.5)
$$\begin{aligned}& \bar{a}_{1}(t) = 3^{\frac{\beta}{1-\beta}} \bigl[b(t)e^{t}M_{1}^{\beta}A^{1-\beta}_{1}(t)+d(t) \bigl(N_{1} t^{\theta_{1}} \bigr)^{\beta}B_{1}^{1-\beta}(t) \bigr]^{\frac{1}{1-\beta}}, \\& \bar{b}_{1}(t,s) = 3^{\frac{\beta}{1-\beta}} \biggl[ \biggl( \frac {m}{p}K^{\frac{m-p}{p}}b(t)e^{t}M_{1}^{\beta} \biggr)^{\frac {1}{1-\beta}} e^{-\frac{1}{1-\beta}s} c^{\frac{1}{1-\beta}}(s) \\& \hphantom{\bar{b}_{1}(t,s) ={}}{}+ \biggl( \frac{q}{p}K^{\frac{q-p}{p}}d(t) \bigl(N_{1}t^{\theta_{1}} \bigr)^{\beta} \biggr)^{\frac{1}{1-\beta}}f^{\frac {1}{1-\beta}}(s) \biggr], \\& M_{1} = \frac{\Gamma (1-(1-\beta)\frac{1}{\beta} )}{ (\frac{1}{\beta} )^{1-(1-\beta)\frac{1}{\beta}}}= \frac{\Gamma(2-\frac{1}{\beta})}{ \beta^{\frac{1}{\beta}-2}}, \\& N_{1} = \frac{1}{\alpha}\cdot B \biggl(\frac{\frac{\gamma-1}{\beta }+1}{\alpha}, \frac{\beta-1}{\beta}+1 \biggr)=\frac{1}{\alpha}\cdot B \biggl(\frac{\beta+\gamma-1}{\alpha}, \frac{2\beta-1}{\beta} \biggr), \\& \theta_{1} = \frac{1}{\beta} \bigl[\alpha(\beta-1)+\gamma-1 \bigr]+1, \\& A_{1}(t) = \int_{0}^{t}e^{-\frac{1}{1-\beta}s}c^{\frac{1}{1-\beta }}(s) \biggl(\frac{m}{p}K^{\frac{m-p}{p}}a(s)+\frac{p-m}{p} K^{\frac{m}{p}} \biggr)^{\frac{1}{1-\beta}}\,ds, \\& B_{1}(t) = \int_{0}^{t}f^{\frac{1}{1-\beta}}(s) \biggl( \frac {q}{p}K^{\frac{q-p}{p}}a(s)+\frac{p-q}{p} K^{\frac{q}{p}} \biggr)^{\frac{1}{1-\beta}}\,ds; \end{aligned}$$
$$\begin{aligned} u(t) \leq& \biggl\{ a(t)+ \biggl[\bar{a}_{2}(t)+e^{\int_{0}^{t}\bar {b}_{2}(t,s)\,ds} \int_{0}^{t}e^{-\int_{0}^{\tau}\bar{b}_{2}(\tau ,s)\,ds} \\ &{}\cdot\biggl( \bar{a}_{2}(s) \bar{b}_{2}(\tau,\tau)+ \int_{0}^{\tau}\bar{b}'_{2\tau}( \tau ,s)\bar{a}_{2}(s)\,ds \biggr)\,d\tau \biggr]^{\frac{ \beta}{1+4\beta }} \biggr\} ^{\frac{1}{p}}, \end{aligned}$$
(3.6)
$$\begin{aligned}& \bar{a}_{2}(t) = 3^{\frac{1+3\beta}{\beta}} \bigl[b(t)e^{t}M_{2}^{\frac{1+3 \beta}{1+4\beta}}A^{\frac{\beta}{1+4\beta }}_{2}(t)+d(t) \bigl(N_{2} t^{\theta_{2}} \bigr)^{\frac{1+3\beta}{1+4 \beta}}B_{2}^{\frac{\beta }{1+4\beta}}(t) \bigr]^{\frac{1+4\beta}{\beta}}, \\& \bar{b}_{2}(t,s)= 3^{\frac{1+3\beta}{\beta}} \biggl[ \biggl( \frac {m}{p}K^{\frac{m-p}{p}}b(t)e^{t}M_{2} ^{\frac{1+3\beta}{1+4\beta}} \biggr)^{\frac{1+4\beta}{\beta}} e^{-\frac{1+4\beta}{\beta}s} c^{\frac{1+4\beta}{\beta}}(s) \\& \hphantom{\bar{b}_{2}(t,s)={}}{}+ \biggl(\frac{q}{p}K^{\frac {q-p}{p}}d(t) \bigl(N_{2}t^{\theta_{2}} \bigr)^{\frac{1+3\beta}{1+4 \beta}} \biggr) ^{\frac{1+4\beta}{\beta}}f^{\frac{1+4\beta}{\beta}}(s) \biggr], \\& M_{2} = \frac{\Gamma (1-(1-\beta)\frac{1+4\beta}{1+3\beta } )}{ (\frac{1+4\beta}{1+3\beta} )^{1-(1-\beta )\frac{1+4\beta}{1+3\beta}}} =\frac{\Gamma (\frac{4\beta^{2}}{1+3\beta} )}{ (\frac{1+4\beta}{1+3\beta} )^{\frac{4\beta^{2}}{1+3\beta }}}, \\& N_{2} = \frac{1}{\alpha}\cdot B \biggl(\frac{\frac{1+4\beta}{1+3\beta}(\gamma-1)+1}{\alpha}, \frac{1+4\beta}{1+3\beta}(\beta-1)+1 \biggr) \\& \hphantom{N_{2}}=\frac{1}{\alpha }\cdot B \biggl( \frac{\gamma-\beta+4\beta\gamma}{\alpha(1+3\beta)},\frac {4\beta^{2}}{1+3\beta} \biggr), \\& \theta_{2} = \frac{1+4\beta}{1+3\beta} \bigl[\alpha(\beta-1)+\gamma-1 \bigr]+1, \\& A_{2}(t) = \int_{0}^{t}e^{-\frac{1+4\beta}{\beta}s}c^{\frac {1+4\beta}{\beta}}(s) \biggl(\frac{m}{p}K^{\frac{m-p}{p}}a(s)+\frac{p-m}{p} K^{\frac{m}{p}} \biggr)^{\frac{1+4\beta}{\beta}}\,ds, \\& B_{2}(t) = \int_{0}^{t}f^{\frac{1+4\beta}{\beta}}(s) \biggl( \frac {q}{p}K^{\frac{q-p}{p}}a(s)+\frac{p-q}{p} K^{\frac{q}{p}} \biggr)^{\frac{1+4\beta}{\beta}}\,ds. \end{aligned}$$
Proof
Define
then
By Lemma 2.2, from (3.7), for any \(K>0\), we have
Here
Now we try to estimate \(A(t)\), \(B(t)\), and \(C(t) \).
$$ v(t)=b(t) \int_{0}^{t}(t-s)^{\beta-1}c(s) u^{m}(s)\,ds+d(t) \int _{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta-1}s^{\gamma-1}f(s) u^{q}(s)\,ds, $$
(3.7)
$$ u^{p}(t)\leq a(t)+v(t)\quad \mbox{or} \quad u(t)\leq \bigl\{ a(t)+v(t) \bigr\} ^{\frac{1}{p}}. $$
(3.8)
$$\begin{aligned} v(t) \leq& b(t) \int_{0}^{t}(t-s)^{\beta-1}c(s) \biggl[ \frac {m}{p}K^{\frac{m-p}{p}} \bigl(a(s)+v(s) \bigr)+\frac{p-m}{p}K^{\frac {m}{p}} \biggr] \,ds \\ &{}+d(t) \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta-1}s^{\gamma -1}f(s) \biggl[\frac{q}{p}K^{\frac{q-p}{p}} \bigl(a(s)+v(s) \bigr)+\frac {p-q}{p}K^{\frac{q}{p}} \biggr]\,ds \\ =&A(t)+\frac{m}{p}K^{\frac{m-p}{p}}b(t) \int_{0}^{t}(t-s)^{\beta -1}c(s) v(s)\,ds \\ &{}+ \frac{q}{p}K^{\frac{q-p}{p}}d(t) \int _{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta-1}s^{\gamma -1}f(s)v(s)\,ds \\ =&A(t)+\frac{m}{p}K^{\frac{m-p}{p}}b(t)\cdot B(t)+\frac{q}{p}K^{\frac{q-p}{p}}d(t) \cdot C(t). \end{aligned}$$
(3.9)
$$\begin{aligned}& A(t) = b(t) \int_{0}^{t}(t-s)^{\beta-1}c(s) \biggl[ \frac {m}{p}K^{\frac{m-p}{p}}a(s)+\frac{p-m}{p}K^{\frac{m}{p}} \biggr] \,ds \\& \hphantom{A(t) ={}}{} + d(t) \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta-1}f(s)s^{\gamma -1} \biggl[\frac{q}{p}K^{\frac{q-p}{p}}a(s)+ \frac{p-q}{p}K^{\frac{q}{p}} \biggr]\,ds, \\& B(t) = \int_{0}^{t}(t-s)^{\beta-1}c(s) v(s)\,ds, \\& C(t) = \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta-1}s^{\gamma-1}f(s)v(s)\,ds. \end{aligned}$$
If \([\alpha,\beta,\gamma]\in I\), let \(p_{1}=\frac{1}{\beta}\), \(q_{1}=\frac{1}{1-\beta}\); if \([\alpha,\beta,\gamma]\in\mathit{II}\), let \(p_{2}=\frac{1+4\beta}{1+3\beta}\), \(q_{2}=\frac{1+4\beta}{\beta}\). Then \(\frac{1}{p_{i}}+\frac{1}{q_{i}}=1\) for \(i=1, 2\). Using Hölder’s inequality, Lemma 2.3, and Lemma 2.4 we get
Here
$$\begin{aligned} A(t) \leq& b(t) \biggl[ \int_{0}^{t}(t-s)^{p_{i}(\beta -1)}e^{p_{i}s} \,ds \biggr]^{\frac{1}{p_{i}}} \\ &{}\cdot\biggl[ \int_{0}^{t} e^{-q_{i}s}c^{q_{i}}(s) \biggl( \frac{m}{p}K^{\frac {m-p}{p}}a(s)+\frac{p-m}{p}K^{\frac{m}{p}} \biggr)^{q_{i}} \,ds \biggr]^{\frac{1}{q_{i}}} \\ &{}+d(t) \biggl[ \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{p_{i}(\beta -1)}s^{p_{i}(\gamma-1)}\,ds \biggr]^{\frac{1}{p_{i}}} \\ &{}\cdot \biggl[ \int_{0}^{t}f^{q_{i}}(s) \biggl( \frac{q}{p}K^{\frac {q-p}{p}}a(s)+\frac{p-q}{p}K^{\frac{q}{p}} \biggr)^{q_{i}}\,ds \biggr]^{\frac{1}{q_{i}}} \\ =&b(t) \biggl(\frac{e^{p_{i}t}\Gamma(1-(1-\beta )p_{i})}{p_{i}^{1-(1-\beta)p_{i}}} \biggr) ^{\frac{1}{p_{i}}} \\ &{}\cdot \biggl[ \int_{0}^{t} e^{-q_{i}s}c^{q_{i}}(s) \biggl( \frac{m}{p}K^{\frac {m-p}{p}}a(s)+\frac{p-m}{p}K^{\frac{m}{p}} \biggr)^{q_{i}} \,ds \biggr]^{\frac{1}{q_{i}}} \\ &{}+d(t) \biggl[\frac{t^{\theta_{i}}}{\alpha}B \biggl(\frac {p_{i}(\gamma-1)+1}{\alpha}, p_{i}(\beta-1)+1 \biggr) \biggr]^{\frac {1}{p_{i}}} \\ &{}\cdot\biggl[ \int_{0}^{t}f^{q_{i}}(s) \biggl( \frac{q}{p}K^{\frac {q-p}{p}}a(s)+\frac{p-q}{p}K^{\frac{q}{p}} \biggr)^{q_{i}}\,ds \biggr]^{\frac{1}{q_{i}}} \\ =& b(t)e^{t}M_{i}^{\frac{1}{p_{i}}}A_{i}^{\frac {1}{q_{i}}}(t)+d(t) \bigl(N_{i}t^{\theta_{i}} \bigr)^{\frac {1}{p_{i}}}B_{i}^{\frac{1}{q_{i}}}(t). \end{aligned}$$
(3.10)
$$\begin{aligned}& \theta_{i} = p_{i} \bigl[\alpha(\beta-1)+\gamma-1 \bigr]+1, \\& M_{i} = \frac{\Gamma(1-(1-\beta)p_{i})}{p_{i}^{1-(1-\beta)p_{i}}},\qquad N_{i}= \frac{1}{\alpha}B \biggl(\frac{p_{i}(\gamma -1)+1}{\alpha}, p_{i}(\beta-1)+1 \biggr), \\& A_{i}(t) = \int_{0}^{t} e^{-q_{i}s}c^{q_{i}}(s) \biggl( \frac {m}{p}K^{\frac{m-p}{p}}a(s)+\frac{p-m}{p}K^{\frac{m}{p}} \biggr)^{q_{i}} \,ds, \\& B_{i}(t) = \int_{0}^{t}f^{q_{i}}(s) \biggl( \frac{q}{p}K^{\frac {q-p}{p}}a(s)+\frac{p-q}{p}K^{\frac{q}{p}} \biggr)^{q_{i}}\,ds , \quad i=1, 2. \end{aligned}$$
Similarly, we can get
and
From (3.9), (3.11), and (3.12) we get
Using Lemma 2.1 and (3.10) we can get
Here
$$\begin{aligned} B(t) \leq& \biggl[ \int_{0}^{t}(t-s)^{p_{i}(\beta-1)}e^{p_{i}s} \,ds \biggr]^{\frac{1}{p_{i}}} \biggl[ \int_{0}^{t}e^{-q_{i}s} c^{q_{i}}(s)v^{q_{i}}(s) \,ds \biggr]^{\frac{1}{q_{i}}} \\ =&e^{t}M_{i}^{\frac{1}{p_{i}}} \biggl[ \int_{0}^{t}e^{-q_{i}s} c^{q_{i}}(s)v^{q_{i}}(s) \,ds \biggr]^{\frac{1}{q_{i}}} \end{aligned}$$
(3.11)
$$\begin{aligned} C(t) \leq& \biggl[ \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{p_{i}(\beta -1)}s^{p_{i}(\gamma-1)}\,ds \biggr]^{\frac{1}{p_{i}}} \biggl[ \int _{0}^{t} f^{q_{i}}(s)v^{q_{i}}(s) \,ds \biggr]^{\frac{1}{q_{i}}} \\ =& \bigl(N_{i}t^{\theta_{i}} \bigr)^{\frac{1}{p_{i}}} \biggl[ \int_{0}^{t} f^{q_{i}}(s)v^{q_{i}}(s) \,ds \biggr]^{\frac{1}{q_{i}}}. \end{aligned}$$
(3.12)
$$\begin{aligned} v(t) \leq& A(t)+\frac{m}{p}K^{\frac{m-p}{p}}b(t)e^{t}M_{i}^{\frac {1}{p_{i}}} \biggl[ \int_{0}^{t} e^{-q_{i}s} c^{q_{i}}(s)v^{q_{i}}(s) \,ds \biggr]^{\frac{1}{q_{i}}} \\ &{}+\frac{q}{p}K^{\frac{q-p}{p}}d(t) \bigl(N_{i}t^{\theta_{i}} \bigr)^{\frac{1}{p_{i}}} \biggl[ \int_{0}^{t} f^{q_{i}}(s)v^{q_{i}}(s) \,ds \biggr]^{\frac{1}{q_{i}}}. \end{aligned}$$
$$\begin{aligned} v^{q_{i}}(t) \leq& 3^{q_{i}-1} A^{q_{i}}(t)+3^{q_{i}-1} \biggl(\frac {m}{p}K^{\frac{m-p}{p}}b(t)e^{t}M_{i}^{\frac{1}{p_{i}}} \biggr)^{q_{i}} \int_{0}^{t} e^{-q_{i}s} c^{q_{i}}(s)v^{q_{i}}(s) \,ds \\ &{}+3^{q_{i}-1} \biggl(\frac{q}{p}K^{\frac{q-p}{p}}d(t) \bigl(N_{i}t^{\theta _{i}} \bigr)^{\frac{1}{p_{i}}} \biggr)^{q_{i}} \int_{0}^{t} f^{q_{i}}(s)v^{q_{i}}(s) \,ds \\ =&3^{q_{i}-1} A^{q_{i}}(t)+e_{i}(t) \int_{0}^{t}e^{-q_{i}s} c^{q_{i}}(s)v^{q_{i}}(s) \,ds+g_{i}(t) \int_{0}^{t} f^{q_{i}}(s)v^{q_{i}}(s) \,ds \\ \leq&3^{q_{i}-1} \bigl[b(t)e^{t}M_{i}^{\frac{1}{p_{i}}}A_{i}^{\frac {1}{q_{i}}}(t) +d(t) \bigl(N_{i}t^{\theta_{i}} \bigr)^{\frac{1}{p_{i}}}B_{i}^{\frac {1}{q_{i}}}(t) \bigr]^{q_{i}} \\ &{}+ \int_{0}^{t} \bigl[e_{i}(t)e^{-q_{i}s}c^{q_{i}}(s)+g_{i}(t) f^{q_{i}}(s) \bigr]v^{q_{i}}(s)\,ds. \end{aligned}$$
(3.13)
$$ e_{i}(t)=3^{q_{i}-1} \biggl(\frac{m}{p}K^{\frac {m-p}{p}}b(t)e^{t}M_{i}^{\frac{1}{p_{i}}} \biggr)^{q_{i}},\qquad g_{i}(t)=3^{q_{i}-1} \biggl( \frac{q}{p}K^{\frac {q-p}{p}}d(t) \bigl(N_{i}t^{\theta_{i}} \bigr)^{\frac{1}{p_{i}}} \biggr)^{q_{i}}. $$
By Lemma 2.7 we have
Here
$$ v^{q_{i}}(t) \leq \bar{a}(t)+e^{\int_{0}^{t}\bar{b}(t,s)\,ds} \int _{0}^{t}e^{-\int_{0}^{\tau}\bar{b}(\tau,s)\,ds} \biggl[\bar{a}(s) \bar {b}(\tau, \tau)+ \int_{0}^{\tau}\bar{b}'_{\tau}( \tau,s)\bar {a}(s)\,ds \biggr]\,d\tau. $$
(3.14)
$$\begin{aligned}& \bar{a}_{i}(t)=3^{q_{i}-1} \bigl[b(t)e^{t}M_{i}^{\frac {1}{p_{i}}}A_{i}^{\frac{1}{q_{i}}}(t)+d(t) \bigl(N_{i}t^{\theta _{i}} \bigr)^{\frac{1}{p_{i}}}B_{i}^{\frac{1}{q_{i}}}(t) \bigr]^{q_{i}}, \\& \bar{b}_{i}(t, s)=e_{i}(t)e^{-q_{i}s}c^{q_{i}}(s)+g_{i}(t)f^{q_{i}}(s). \end{aligned}$$
Combining (3.8) and (3.14) we get
Considering the two situations for \(i=1, 2\) and using the parameters α, β, and γ to \(p_{i}\), \(q_{i}\) and \(\theta_{i}\) in (3.15), we can get (3.4) and (3.5), respectively. This completes the proof of Theorem 3.2. □
$$\begin{aligned} u(t) \leq& \biggl\{ a(t)+ \biggl[\bar{a}_{i}(t)+e^{\int_{0}^{t}\bar{b}_{i}(t,s)\,ds} \int_{0}^{t} e^{-\int_{0}^{\tau}\bar{b}_{i}(\tau,s)\,ds} \\ &{}\cdot\biggl( \bar{a}_{i}(s)\bar {b}_{i}(\tau, \tau)+ \int_{0}^{\tau}\bar{b}'_{i\tau}( \tau, s)\bar{a}_{i}(s)\,ds \biggr)\,d\tau \biggr]^{\frac{1}{q_{i}}} \biggr\} ^{\frac{1}{p}}. \end{aligned}$$
(3.15)
Corollary 3.1
Suppose that the conditions of Theorem
3.2
hold, if
\(b(t)=0\), then
and we deduce the following results.
$$ u^{p}(t) \leq a(t)+d(t) \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta -1}s^{\gamma-1}f(s) u^{q}(s)\,ds $$
For
\([\alpha, \beta, \gamma]\in I\),
where
\(N_{1}\), \(B_{1}(t)\)
are the same as in Theorem
3.2.
$$\begin{aligned} u(t) \leq& \biggl\{ a(t)+ \biggl[\bar{a}_{3}(t)+e^{\int_{0}^{t}\bar {b}_{3}(t,s)\,ds} \int_{0}^{t}e^{\int_{0}^{\tau}\bar{b}_{3}(\tau ,s)\,ds} \\ &{}\cdot\biggl( \bar{a}_{3}(s) \bar{b}_{3}(\tau,\tau)+ \int_{0}^{\tau}\bar{b}'_{3\tau}( \tau ,s)\bar{a}_{3}(s)\,ds \biggr)\,d\tau \biggr]^{1-\beta} \biggr\} ^{\frac{1}{p}}, \end{aligned}$$
(3.16)
$$\begin{aligned}& \bar{a}_{3}(t) = \bigl(3^{\beta}d(t)N_{1}^{\beta}t^{(\alpha +1)(\beta-1)+\gamma} \bigr)^{\frac{1}{1-\beta}}B_{1}(t), \\& \bar{b}_{3}(t,s) = 3^{\frac{\beta}{1-\beta}} \biggl(\frac {q}{p}K^{\frac{q-p}{p}}d(t)N_{1}^{\beta}t^{(\alpha+1)(\beta -1)+\gamma} f(s) \biggr)^{\frac{1}{1-\beta}}, \end{aligned}$$
For
\([\alpha, \beta, \gamma]\in \mathit{II}\),
where
\(N_{2}\), \(B_{2}(t)\)
are the same as in Theorem
3.2.
$$\begin{aligned} u(t) \leq& \biggl\{ a(t)+ \biggl[\bar{a}_{4}(t)+e^{\int_{0}^{t}\bar {b}_{4}(t,s)\,ds} \int_{0}^{t}e^{\int_{0}^{\tau}\bar{b}_{4}(\tau ,s)\,ds} \\ &{}\cdot\biggl( \bar{a}_{4}(s) \bar{b}_{4}(\tau,\tau)+ \int_{0}^{\tau}\bar{b}'_{4\tau}( \tau ,s)\bar{a}_{4}(s)\,ds \biggr)\,d\tau \biggr]^{\frac{ \beta}{1+4\beta }} \biggr\} ^{\frac{1}{p}}, \end{aligned}$$
(3.17)
$$\begin{aligned}& \bar{a}_{4}(t) = 3^{\frac{1+3\beta}{\beta}} \bigl[d(t)N_{2}^{\frac {1+3\beta}{1+4\beta}} t^{\frac{ [\alpha(\beta-1)+\gamma ](1+3\beta)+\beta}{1+3\beta }} \bigr]^{\frac{1+4\beta}{\beta}}B_{2}(t), \\& \bar{b}_{4}(t,s) = 3^{\frac{1+3\beta}{\beta}} \biggl(\frac {q}{p}K^{\frac{q-p}{p}}d(t)N_{2}^{\frac{1+3\beta}{1+4\beta}} t^{\frac{[\alpha(\beta-1)+\gamma](1+3\beta)+\beta}{1+3\beta }}f(s) \biggr)^{\frac{1+4\beta}{\beta}}, \end{aligned}$$
Corollary 3.2
Suppose that the conditions of Theorem
3.2
hold, \(\alpha=\gamma=1\). If
then for
\([\alpha, \beta, \gamma]\in I\),
where
\(M_{1}\), \(A_{1}(t)\), \(B_{1}(t)\)
are the same as in Theorem
3.2.
$$ u^{p}(t) \leq a(t)+b(t) \int_{0}^{t}(t-s)^{\beta-1}c(s) u^{m}(s)\,ds+d(t) \int_{0}^{t}(t-s)^{\beta-1}f(s) u^{q}(s)\,ds, $$
$$\begin{aligned} u(t) \leq& \biggl\{ a(t)+ \biggl[\bar{a}_{5}(t)+e^{\int_{0}^{t}\bar {b}_{1}(t,s)\,ds} \int_{0}^{t}e^{\int_{0}^{\tau}\bar{b}_{1}(\tau ,s)\,ds} \\ &{}\cdot \biggl( \bar{a}_{5}(s) \bar{b}_{1}(\tau,\tau)+ \int_{0}^{\tau}\bar{b}'_{5\tau}( \tau ,s)\bar{a}_{5}(s)\,ds \biggr)\,d\tau \biggr]^{1-\beta} \biggr\} ^{\frac{1}{p}}, \end{aligned}$$
(3.18)
$$\begin{aligned}& \bar{a}_{5}(t) = 3^{\frac{\beta}{1-\beta}} \bigl[b(t)e^{t}M_{1}^{\beta}A_{1}^{1-\beta}(t) +d(t) \bigl(\bar{N}_{1}t^{\bar{\theta}_{1}} \bigr)^{\beta}B_{1}^{1-\beta }(t) \bigr]^{\frac{1}{1-\beta}}, \\& \bar{b}_{5}(t,s) = 3^{\frac{\beta}{1-\beta}} \biggl[ \biggl( \frac {m}{p}K^{\frac{m-p}{p}}b(t)M_{1}^{\beta} \biggr)^{\frac{1}{1-\beta }}e^{\frac{1}{\beta-1}s} c^{\frac{1}{\beta-1}}(s)+ \biggl( \frac{q}{p}K^{\frac {q-p}{p}}d(t) \bigl(N_{1}t^{\bar{\theta}_{1}} \bigr)^{\beta} \biggr)^{\frac{1}{1-\beta}}f^{\frac{1}{1-\beta}}(s) \biggr], \\& \bar{N}_{1} = B \biggl(1, \frac{2\beta-1}{\beta} \biggr), \qquad \bar{ \theta}_{1}=\frac{1}{\beta}(\beta-1)+1=\frac{2\beta -1}{\beta}, \end{aligned}$$
For
\([\alpha, \beta, \gamma]\in \mathit{II}\),
where
\(M_{2}\), \(A_{2}(t)\), \(B_{2}(t)\)
are the same as in Theorem
3.2.
$$\begin{aligned} u(t) \leq& \biggl\{ a(t)+ \biggl[\bar{a}_{6}(t)+e^{\int_{0}^{t}\bar {b}_{6}(t,s)\,ds} \int_{0}^{t}e^{\int_{0}^{\tau}\bar{b}_{6}(\tau ,s)\,ds} \\ &{}\cdot\biggl( \bar{a}_{6}(s) \bar{b}_{6}(\tau,\tau)+ \int_{0}^{\tau}\bar{b}'_{6\tau}( \tau ,s)\bar{a}_{6}(s)\,ds \biggr)\,d\tau \biggr]^{\frac{ \beta}{1+4\beta }} \biggr\} ^{\frac{1}{p}}, \end{aligned}$$
(3.19)
$$\begin{aligned}& \bar{a}_{6}(t) = 3^{\frac{1+3\beta}{\beta}} \bigl[b(t)e^{t}M_{2}^{\frac{1+3 \beta}{1+4\beta}}A_{2}^{\frac{\beta }{1+4\beta}}(t)+d(t) \bigl(\bar{N}_{2} t^{\bar{\theta}_{2}} \bigr)^{\frac{1+3\beta}{1+4\beta}}B_{2}^{\frac{\beta}{1+4\beta}}(t) \bigr]^{\frac{1+4\beta}{\beta}}, \\& \bar{b}_{6}(t,s) = 3^{\frac{1+3\beta}{\beta}} \biggl[ \biggl( \frac {m}{p}K^{\frac{m-p}{p}}b(t)e^{t} M_{2}^{\frac{1+3 \beta}{1+4\beta}} \biggr)^{\frac{1+4\beta}{\beta}} e^{-\frac{1+4\beta}{\beta}s} c^{\frac{1+4\beta}{\beta}}(s) \\& \hphantom{\bar{b}_{6}(t,s) ={}}{}+ \biggl(\frac{q}{p}K^{\frac {q-p}{p}}d(t) \bigl(\bar{N}_{2}t^{\bar{\theta}_{2}} \bigr)^{\frac {1+3\beta}{1+4\beta}} \biggr) ^{\frac{1+4\beta}{\beta}}f^{\frac{1+4\beta}{\beta}}(s) \biggr], \\& \bar{N}_{2} = B \biggl(1, \frac{4\beta^{2}}{1+3\beta} \biggr), \\& \bar{\theta}_{2} = \frac{1+4\beta}{1+3\beta}(\beta-1)+1=\frac {4\beta^{2}}{1+3\beta}, \end{aligned}$$
4 Applications
(1) Consider the delay fractional differential equations with initial condition
Here \({}^{\mathrm{C}}_{t_{0}}D_{t}^{\beta}\) is the Caputo fractional derivative of order β (\(m-1\leq\beta< m\)), \(r\in R_{+}\) is a real constant, \(f(t, y, z)\in C(I\times R^{2}, R)\), φ is a given continuously differentiable function on \([t_{0}-r, t_{0}]\) up to order m (\(m=-[-\beta]\)). In this case, we denote \(\varphi^{k}(t_{0})=b_{k}\), \(k=0,1,2,\ldots, m-1\).
$$ \textstyle\begin{cases} {}^{\mathrm{C}}_{t_{0}}D_{t}^{\beta}x^{p}(t)=f(t, x(t), x(t-r)), \quad t\in I=[t_{0}, T), \\ {}^{\mathrm{C}}_{t_{0}}D_{t}^{\beta} x(t_{0})=b_{k}, \quad k=0,1,2,\ldots,m-1, \\ x(t)=\varphi(t), \quad t\in I=[t_{0}-r, t_{0}). \end{cases} $$
(4.1)
In [17], the initial value problem (4.1) is equivalent to the Volterra fractional integral equation
$$ \textstyle\begin{cases} x^{p}(t)=\sum_{k=0}^{m-1}\frac{b_{k}}{k!}(t-t_{0})^{k}+\frac {1}{\Gamma(\beta)}\int_{t_{0}}^{t}(t-s)^{\beta-1}f(s, x(s), x(s-r))\,ds, \quad t\in I, \\ x(t)=\varphi(t), \quad t\in I=[t_{0}-r, t_{0}). \end{cases} $$
(4.2)
The next theorem deals with the estimates of the solution of (4.2).
Theorem 4.1
Suppose that
\(|f(t,y,z)|\leq b(t)|y|^{q}+c(t)|z|^{l}\), \(t\in I\), \(m, p, q, l\in R_{+}\), \(p\geq q\), \(p\geq l\), \(b(t), c(t)\in C(I, R_{+})\). If
\(x(t)\)
is the solution of initial problem (4.2), then the following estimations hold.
(i) Suppose that
\(\beta>\frac{1}{2}\), then
Here
$$ \textstyle\begin{cases} |x(t)| \leq \{g_{1}(t)+\int_{t_{0}}^{t}k_{1}(s)\varphi ^{l}(s-r)\,ds+\int_{t_{0}}^{t}F_{1}(s)e^{\int_{s}^{t}G_{1}(\tau)\,d\tau }\,ds \}^{\frac{1}{p}}, \quad t\in[t_{0}, t_{0}+r), \\ |x(t)|\leq \{g_{1}(t)+\int_{t_{0}}^{t}F_{2}(s)e^{\int_{s}^{t}G_{2}(\tau )\,d\tau}\,ds \}^{\frac{1}{p}} \quad t\in[t_{0}+r, T). \end{cases} $$
(4.3)
$$\begin{aligned}& g_{1}(t)=3e^{-2t} \Biggl[\sum_{k=0}^{m-1} \frac {|b_{k}|}{k!}(t-t_{0})^{k} \Biggr]^{2}, \qquad h_{1}(t)=\frac{6\Gamma(2\beta-1)}{4^{\beta}}e^{2t(\frac {q}{p}-1)} \biggl[ \frac{b(t)}{\Gamma(\beta)} \biggr]^{2}, \\& k_{1}(t)=\frac{6\Gamma(2\beta-1)}{4^{\beta}} \biggl[\frac {c(t)}{\Gamma(\beta)} \biggr]^{2}e^{2t(\frac{l}{p}-1)}, \\& F_{1}(t)=\frac{q}{p}K^{\frac{q-p}{p}}h_{1}(t) \biggl[g_{1}(t)+ \int _{t_{0}}^{t}k_{1}(s) \varphi^{l}(s-r)\,ds \biggr]+\frac{p-q}{p}K^{\frac{q}{p}},\qquad G_{1}(t)=\frac{q}{p}K^{\frac{q-p}{p}}h_{1}(t), \\& F_{2}(t)= \biggl[\frac{q}{p}K^{\frac{q-p}{p}}g_{1}(t)+ \frac {p-q}{p}K^{\frac{q}{p}} \biggr]h_{1}(t)+ \biggl[ \frac{l}{p}K^{\frac {l-p}{p}}g_{1}(t-r) +\frac{p-l}{p}K^{\frac{l}{p}} \biggr]k_{1}(t), \\& G_{2}(t)=\frac{q}{p}K^{\frac{q-p}{p}}h_{1}(t)+ \frac{l}{p}K^{\frac {l-p}{p}}k_{1}(t) . \end{aligned}$$
(ii) If
\(0<\beta\leq\frac{1}{2}\), then
Here
$$ \textstyle\begin{cases} |x(t)| \leq \{g_{2}(t)+\int_{t_{0}}^{t}k_{2}(s)\varphi ^{l}(s-r)\,ds+\int_{t_{0}}^{t}F_{3}(s) e^{\int_{s}^{t}G_{3}(\tau)\,d\tau}\,ds \}^{\frac{1}{p}}, \quad t\in[t_{0}, t_{0}+r), \\ |x(t)|\leq \{g_{2}(t)+\int_{t_{0}}^{t}F_{4}(s)e^{\int_{s}^{t}G_{4}(\tau )\,d\tau}\,ds \}^{\frac{1}{p}}, \quad t\in[t_{0}+r, T). \end{cases} $$
(4.4)
$$\begin{aligned}& g_{2}(t)=3^{\frac{1}{\beta}}e^{-\frac{1+\beta}{\beta}t} \Biggl[\sum _{k=0}^{m-1}\frac{|b_{k}|}{k!}(t-t_{0})^{k} \Biggr]^{\frac {1+\beta}{\beta}}, \\& h_{2}(t)=3^{\frac{1}{\beta}} \biggl(\frac{\Gamma(\beta ^{2})}{(1+\beta)^{\beta^{2}}} \biggr)^{\frac{1}{\beta}} e^{ [-\frac{1+\beta}{\beta}+\frac{q}{p}(1+\beta) ]t} \biggl[\frac{b(t)}{\Gamma(\beta)} \biggr]^{\frac{1+\beta}{\beta }}, \\& k_{2}(t)=3^{\frac{1}{\beta}} \biggl(\frac{\Gamma(\beta ^{2})}{(1+\beta)^{\beta^{2}}} \biggr)^{\frac{1}{\beta}} e^{ [-\frac{1+\beta}{\beta}+\frac{q}{p}(1+\beta) ]t} \biggl[\frac{c(t)}{\Gamma(\beta)} \biggr]^{\frac{1+\beta}{\beta }}, \\& F_{3}(t)=\frac{q}{p}K^{\frac{q-p}{p}}h_{2}(t) \biggl[g_{2}(t)+ \int _{t_{0}}^{t}k_{2}(s) \varphi^{l}(s-r)\,ds \biggr]+\frac{p-q}{p}K^{\frac {q}{p}}, \qquad G_{3}(t)=\frac{q}{p}K^{\frac{q-p}{p}}h_{2}(t), \\& F_{4}(t)= \biggl[\frac{q}{p}K^{\frac{q-p}{p}}g_{2}(t)+ \frac {p-q}{p}K^{\frac{q}{p}} \biggr]h_{2}(t)+ \biggl[ \frac{l}{p}K^{\frac {l-p}{p}}g_{2}(t-r) +\frac{p-l}{p}K^{\frac{l}{p}} \biggr]k_{2}(t), \\& G_{4}(t)=\frac{q}{p}K^{\frac{q-p}{p}}h_{2}(t)+ \frac{l}{p}K^{\frac {l-p}{p}}k_{2}(t) . \end{aligned}$$
Proof
By (4.2), we derive that
Using Theorem 3.1, we get the desired conclusion. This proves the results (4.3) and (4.4). □
$$ \textstyle\begin{cases} |x(t)|^{p}\leq\sum_{k=0}^{m-1}\frac{|b_{k}|}{k!}(t-t_{0})^{k}+\frac {1}{\Gamma(\beta)}\int_{t_{0}}^{t}(t-s)^{\beta-1}b(s)|x(s)|^{q}\,ds \\ \hphantom{|x(t)|^{p}\leq{}}{}+\frac{1}{\Gamma(\beta)}\int_{t_{0}}^{t}(t-s)^{\beta-1}c(s)|x(s-r)|^{l}\,ds,\quad t\in I, \\ |x(t)|=|\varphi(t)|, \quad t\in[t_{0}-r, t_{0}). \end{cases} $$
(4.5)
(2) In this section, based on the definition of Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral, we will study the boundedness of a certain FDE with R-L fractional operator and E-K fractional operator. The definitions of two fractional operators are given below.
Definition 4.1
([17])
The R-L fractional integral and fractional derivative of order α of the function \(f(x)\in C(R_{+}, R)\) are given by
provided that the right side is point-wise defined on \(R_{+}\).
$$\begin{aligned}& I^{\alpha}f(x)=\frac{1}{\Gamma(\alpha)} \int_{0}^{x}(x-t)^{\alpha-1}f(t)\,dt, \quad \alpha>0, \\& D^{\alpha}f(x)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dx} \int _{0}^{x}(x-t)^{-\alpha}f(t)\,dt,\quad 0< \alpha< 1, \end{aligned}$$
Definition 4.2
The E-K fractional integral of continuous function \(f(x)\in C(R_{+}, R)\) is defined by
provided that the right side is point-wise defined on \(R_{+}\).
$$ I^{\gamma,\delta}_{\beta}f(x)=\frac{x^{-\beta(\gamma+\delta )}}{\Gamma(\delta)} \int_{0}^{x} \bigl(x^{\beta}-t^{\beta} \bigr)^{\delta -1}t^{\beta\gamma}f(t)d \bigl(t^{\beta} \bigr),\quad \delta, \gamma, \beta\in R_{+}, $$
Consider the following Volterra type integral equations:
$$ u^{p}(t)-\frac{b(t)}{\Gamma(\beta)} \int_{0}^{t}(t-s)^{\beta -1}c(s)u^{m}(s) \,ds-\frac{d(t)}{\Gamma(\beta)} \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta-1}s^{\gamma-1}f(s)u^{q}(s)\,ds=a(t). $$
(4.6)
Theorem 4.2
Let
\(u(t), a(t), b(t), c(t), d(t), f(t)\in C[0,\infty)\), p, m, q, α, β, γ
be the same as in Theorem
3.2. Then, for any
\(K>0\), every solution of equation (4.6) has the bounds and the same modality as (3.5), (3.6), in which we have
\(|u(t)|\), \(|a(t)|\), \(\frac{|b(t)|}{\Gamma (\beta)}\), \(|c(t)|\), \(\frac{|d(t)|}{\Gamma(\beta)}\), \(|f(t)|\)
instead of
\(u(t)\), \(a(t)\), \(b(t)\), \(c(t)\), \(d(t)\), and
\(f(t)\).
Proof
From (4.6) we have
Then by the difference in the selection of \(a(t)\), \(b(t)\), \(c(t)\), \(d(t)\), \(f(t)\), p, m, q, α, β, γ, we get the desired results. □
$$\begin{aligned} \bigl\vert u(t) \bigr\vert ^{p} \leq& \bigl\vert a(t) \bigr\vert + \frac{\vert b(t)\vert }{\Gamma(\beta)} \int_{0}^{t}(t-s)^{\beta -1} \bigl\vert c(s) \bigr\vert \bigl\vert u(s) \bigr\vert ^{m}\,ds \\ &{}-\frac{\vert d(t)\vert }{\Gamma(\beta)} \int_{0}^{t} \bigl(t^{\alpha}-s^{\alpha} \bigr)^{\beta-1}s^{\gamma-1} \bigl\vert f(s) \bigr\vert \bigl\vert u(s) \bigr\vert ^{q}\,ds. \end{aligned}$$
Acknowledgements
This research is supported by National Science Foundation of China (11171178 and 11271225).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
RX carried out the generalized weakly singular integral inequalities and completed the corresponding proof. FM participated in Section 4 - Applications. All authors read and approved the final manuscript.