First of all, we monotonize some given functions
f,
\(\omega_{i}\), and
a in system (
1.6) of integral inequalities. Let
$$\begin{aligned} \tilde{f}(t):=\max_{\tau\in[0, t] } \bigl\{ f(\tau) \bigr\} ,\quad t\ge0, \qquad\tilde{a}(t):=\max_{\tau\in[t_{0}, t] } \bigl\{ a(\tau) \bigr\} , \quad t\ge t_{0} . \end{aligned}$$
(2.7)
From (
2.3) we see that the function
\(W_{i}\) is strictly increasing and therefore its inverse
\(W_{i}^{-1}\) is well defined, continuous, and increasing in its domain. The sequence
\(\{\tilde{\omega}_{i}(t)\}\), defined by
\(\omega_{i}(s)\), consists of nondecreasing nonnegative functions on
\(\mathbb{R}_{+} \) and satisfies
$$ \begin{aligned} &\omega_{i}(t) \le \tilde{ \omega}_{i}(t),\quad i=1,2,\ldots ,m, \\ &\omega_{i}(t) \le \hat{\omega}_{i}(t),\quad i=m+1,\ldots ,m+n, \\ &\hat{\omega}_{i} \bigl(\tilde{f}(t) \bigr)\le \tilde{ \omega}_{i}(t),\quad i=m+1,\ldots,m+n. \end{aligned} $$
(2.8)
Moreover,
$$\begin{aligned} \tilde{\omega}_{i}\varpropto\tilde{\omega}_{i+1},\quad i=1,2, \ldots,m+n, \end{aligned}$$
(2.9)
because the ratios
\({\tilde{\omega}_{i+1}(t)}/{\tilde{\omega}_{i}(t)}\),
\(i=1,2,\ldots,m+n\), are all nondecreasing. Furthermore, let
$$\begin{aligned} \hat{g}_{i}(t,s) :=\max_{\iota\in[t_{0}, t ]}g_{i}( \iota,s), \end{aligned}$$
(2.10)
which is nondecreasing in
t for each fixed
s and satisfies
\(\hat {g}_{i}(t,s)\geq g_{i}(t,s)\geq0\) for all
\(i=1,2,\ldots,m+n\). We note that
\(\tilde{a}(t)\ge a(t)\) and
\(\hat{g}_{i}(t,s)\ge f_{i}(t,s)\) and they are continuous and nondecreasing in
t. From the monotonicity of
\(\tilde{f}(t)\) we obtain the inequality
$$\begin{aligned} \max_{\xi\in[c_{i}(s)-h,c_{i}(s)]} f \bigl(u(\xi) \bigr) \le& \max _{\xi\in[c_{i}(s)-h,c_{i}(s)]} \tilde{f} \bigl(u(\xi) \bigr) \\ \le& \tilde{f} \Bigl(\max_{\xi\in[c_{i}(s)-h,c_{i}(s)]} u(\xi) \Bigr), \quad\forall s \in\bigl[b^{*}(t_{0}), t_{1}\bigr). \end{aligned}$$
(2.11)
From (
1.6), (
2.8), (
2.11), and the definition of
\(\hat{g}_{i}(t,s)\), we obtain
$$ \begin{aligned} &\varphi \bigl(u(t) \bigr) \leq \tilde{a}(t)+\sum _{i=1}^{m} \int_{b_{i}(t_{0})}^{b_{i}(t)} \bigl(t^{\alpha_{i}}-s^{\alpha_{i}} \bigr)^{k_{i}(\beta_{i}-1)}s^{q_{i}({\gamma }_{i}-1)}\hat{g}_{i}(t,s)\tilde{ \omega}_{i} \bigl(u(s) \bigr)\,ds \\ &\hphantom{\varphi (u(t) ) \leq}{} +\sum_{j=m+1}^{m+n} \int_{b_{j}(t_{0})}^{b_{j}(t)} \bigl(t^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{k_{j}(\beta_{j}-1)}s^{q_{j}(\gamma_{j}-1)}\hat {g}_{j}(t,s) \\ &\hphantom{\varphi (u(t) ) \leq}{} \times\hat{\omega}_{j} \Bigl(\tilde{f} \Bigl(\max _{\xi\in [c_{j}(s)-h,c_{j}(s)]}u(\xi) \Bigr) \Bigr)\,ds \\ &\hphantom{\varphi (u(t) )}\leq \tilde{a}(t)+\sum_{i=1}^{m} \int_{b_{i}(t_{0})}^{b_{i}(t)} \bigl(t^{\alpha_{i}}-s^{\alpha_{i}} \bigr)^{k_{i}(\beta_{i}-1)}s^{q_{i}({\gamma }_{i}-1)}\hat{g}_{i}(t,s)\tilde{ \omega_{i}} \bigl(u(s) \bigr)\,ds \\ &\hphantom{\varphi (u(t) ) \leq}{} +\sum_{j=m+1}^{m+n} \int_{b_{j}(t_{0})}^{b_{j}(t)} \bigl(t^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{k_{j}(\beta_{j}-1)}s^{q_{j}(\gamma_{j}-1)}\hat {g}_{j}(t,s) \\ &\hphantom{\varphi (u(t) ) \leq}{}\times\tilde{\omega}_{j} \Bigl(\max _{\xi\in[c_{j}(s)-h,c_{j}(s)]}u(\xi) \Bigr)\,ds,\quad t\in\bigl[b_{j}(t_{0}),t_{1}\bigr), \\ &u(t) \leq \psi(t),\quad t\in \bigl[b^{*}(t_{0})-h,t_{0} \bigr]. \end{aligned} $$
(2.12)
Let
\(\frac{1}{p}+\frac{1}{q}=1\),
\(p>1\), then
\(q>0\). Since
\(pq_{i}(\gamma_{i}-1)+1>0\),
\(pk_{i}(\beta_{i}-1)+1>0\), and
\(\frac {1}{p}+k_{i}\alpha_{i}(\beta_{i}-1)+q_{i}(\gamma_{i}-1)\ge0\) for
\(i=1,\ldots ,m+n\). By Lemma
2.1, Hölder’s inequality, and (
2.12) we get for
\(t\in[t_{0}, t_{1})\)
$$\begin{aligned} \varphi \bigl(u(t) \bigr) \leq& \tilde{a}(t)+\sum _{i=1}^{m} \biggl( \int _{b_{i}(t_{0})}^{b_{i}(t)} \bigl(t^{\alpha_{i}}-s^{\alpha_{i}} \bigr)^{pk_{i}(\beta _{i}-1)}s^{pq_{i}(\gamma_{i}-1)}\,ds \biggr)^{\frac{1}{p}} \biggl( \int_{b_{i}(t_{0})}^{b_{i}(t)} \hat{g}^{q}_{i}(t,s) \tilde{\omega}^{q}_{i} \bigl(u(s) \bigr)\,ds \biggr)^{\frac{1}{q}} \\ &{} +\sum_{j=m+1}^{m+n} \biggl( \int_{b_{j}(t_{0})}^{b_{j}(t)} \bigl(t^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{pk_{j}(\beta_{j}-1)}s^{pq_{j}(\gamma _{j}-1)}\,ds \biggr)^{\frac{1}{p}} \\ &{}\times \biggl( \int_{b_{j}(t_{0})}^{b_{j}(t)} \hat{g}^{q}_{j}(t,s) \tilde{\omega}^{q}_{j} \Bigl(\max_{\xi\in [c_{j}(s)-h,c_{j}(s)]}u( \xi) \Bigr)\,ds \biggr)^{\frac{1}{q}} \\ \le& \tilde{a}(t)+\sum_{i=1}^{m} \biggl( \int_{0}^{t} \bigl(t^{\alpha _{i}}-s^{\alpha_{i}} \bigr)^{pk_{i}(\beta_{i}-1)}s^{pq_{i}((\gamma_{i}-1))}\,ds \biggr)^{\frac{1}{p}} \biggl( \int_{b_{i}(t_{0})}^{b_{i}(t)} \hat{g}^{q}_{i}(t,s) \tilde{\omega}^{q}_{i} \bigl(u(s) \bigr)\,ds \biggr)^{\frac{1}{q}} \\ &{} +\sum_{j=m+1}^{m+n} \biggl( \int_{0}^{t} \bigl(t^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{pk_{j}(\beta_{j}-1)}s^{pq_{j}(\gamma _{j}-1)}\,ds \biggr)^{\frac{1}{p}} \\ &{}\times \biggl( \int_{b_{j}(t_{0})}^{b_{j}(t)} \hat{g}^{q}_{j}(t,s) \tilde{\omega}^{q}_{j} \Bigl(\max_{\xi\in[c_{j}(s)-h,c_{j}(s)]}u( \xi) \Bigr)\,ds \biggr)^{\frac{1}{q}} \\ \le& \tilde{a}(t)+\sum_{i=1}^{m} d_{i}(t) \biggl( \int_{b_{i}(t_{0})}^{b_{i}(t)} \hat{g}^{q}_{i}(t,s) \tilde{\omega}_{i}^{q} \bigl(u(s) \bigr)\,ds \biggr)^{\frac{1}{q}} \\ &{} +\sum_{j=m+1}^{m+n} d_{j}(t) \biggl( \int_{b_{j}(t_{0})}^{b_{j}(t)}\hat {g}^{q}_{j}(t,s) \tilde{\omega}_{j}^{q} \Bigl(\max_{\xi\in[c_{j}(s)-h,c_{j}(s)]}u( \xi) \Bigr)\,ds \biggr)^{\frac{1}{q}}, \end{aligned}$$
(2.13)
where we use
\(0\le b_{i}(t)\le t\) and the definition of
\(d_{i}(t)\).
By Lemma
2.2 and (
2.13), we get for
\(t\in[t_{0}, t_{1})\)
$$\begin{aligned} \varphi^{q} \bigl(u(t) \bigr) \le& (1+m+n)^{q-1} \Biggl[\tilde{a}^{q}(t)+\sum _{i=1}^{m} d^{q}_{i}(t) \int _{b_{i}(t_{0})}^{b_{i}(t)} \hat{g}^{q}_{i}(t,s) \tilde{\omega}^{q}_{i} \bigl(u(s) \bigr)\,ds \\ &{} + \sum_{j=m+1}^{m+n} d^{q}_{j}(t) \int_{b_{j}(t_{0})}^{b_{j}(t)}\hat {g}^{q}_{j}(t,s) \tilde{\omega}^{q}_{j} \Bigl(\max_{\xi\in [c_{j}(s)-h,c_{j}(s)]}u( \xi) \Bigr)\,ds \Biggr]. \end{aligned}$$
(2.14)
Then from (
2.4), we see that
\(\hat{r}_{1}(t)\) is nondecreasing on
\([t_{0},t_{1})\). By the definition of
\(\tilde{g}_{i}(t,s)\) and
\(\hat{r}_{1}(t)\), and (
2.14), we have
$$ \begin{aligned} &\varphi^{q} \bigl(u(t) \bigr) \le \hat{r}_{1}(t)+\sum_{i=1}^{m} \int_{b_{i}(t_{0})}^{b_{i}(t)} \tilde{g}_{i}(t,s)\tilde{ \omega}^{q}_{i} \bigl(u(s) \bigr)\,ds \\ &\hphantom{\varphi^{q} (u(t) ) \le}{} +\sum_{j=m+1}^{m+n} \int_{b_{j}(t_{0})}^{b_{j}(t)}\tilde {g}_{j}(t,s)\tilde{ \omega}^{q}_{j} \Bigl(\max_{\xi\in [c_{j}(s)-h,c_{j}(s)]}u(\xi) \Bigr)\,ds,\quad t\in[t_{0}, t_{1}), \\ &u(t) \le \psi (t),\quad t\in \bigl[b^{*}(t_{0})-h,t_{0} \bigr]. \end{aligned} $$
(2.15)
Consider the auxiliary system of inequalities with (
2.15)
$$\begin{aligned} \varphi^{q} \bigl(u(t) \bigr) \le& \hat{r}_{1}(\sigma)+\sum_{i=1}^{m} \int_{b_{i}(t_{0})}^{b_{i}(t)} \tilde{g}_{i}(\sigma,s) \tilde{\omega}^{q}_{i} \bigl(u(s) \bigr)\,ds \\ &{} +\sum_{j=m+1}^{m+n} \int_{b_{j}(t_{0})}^{b_{j}(t)}\tilde{g}_{j}(\sigma ,s) \tilde{\omega}^{q}_{j} \Bigl(\max_{\xi\in[c_{j}(s)-h,c_{j}(s)]}u( \xi ) \Bigr)\,ds, \end{aligned}$$
(2.16)
for all
\(t\in[t_{0},\sigma]\), where
σ is chosen arbitrarily such that
\(t_{0}\leq\sigma\leq T_{1}\).
Notice that
\(\max_{s\in[b^{*}(t_{0})-h,t_{0}]}\psi(s)\leq\varphi^{-1}(\hat {r}^{1/q}_{1}(\sigma)) \) because
\(\max_{s\in[J(t_{0})-h,t_{0}]}\psi(s)\le\varphi^{-1} ((1+m+n)^{\frac{p-1}{p}}a^{\frac{1}{q}}(t_{0}))\le\varphi^{-1}(\hat {r}_{1}(\sigma))\). Define a function
\(z(t): [B^{*}(t_{0})-h, \sigma]\rightarrow\mathbb{R}_{+}\) such that
$$\begin{aligned} z(t)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \hat{r}_{1}(\sigma)+\sum_{i=1}^{m} \int_{b_{i}(t_{0})}^{b_{i}(t)} \tilde{g}_{i}(\sigma,s)\tilde{\omega}_{i}^{q}(u(s))\,ds\\ \quad{}+\sum_{j=m+1}^{m+n} \int_{b_{j}(t_{0})}^{b_{j}(t)} \tilde{g}_{j}(\sigma,s)\tilde{\omega}_{j}^{q} (\max_{\xi\in [c_{j}(s)-h,c_{j}(s)]}u(\xi) )\,ds,& t\in[t_{0},\sigma],\\ \hat{r}_{1}(\sigma),& t\in[b^{*}(t_{0})-h, t_{0}]. \end{array}\displaystyle \right . \end{aligned}$$
Clearly,
\(z(t)\) is nondecreasing. By (
2.16) and the definition of
\(z(t)\) we have
$$ u(t)\leq \varphi^{-1} \bigl(z^{\frac{1}{q}}(t) \bigr), \quad t \in \bigl[b^{*}(t_{0})-h, \sigma \bigr]. $$
(2.17)
Since
\(z(t)\) is nondecreasing, from (
2.17) we obtain
$$\begin{aligned} \max_{\xi\in[c_{j}(s)-h,c_{j}(s)]} u(\xi) \le& \max _{\xi\in[c_{j}(s)-h,c_{j}(s)]} \varphi^{-1} \bigl(z^{\frac{1}{q}}(\xi) \bigr) \\ \le& \varphi^{-1} \bigl(z^{\frac{1}{q}} \bigl(c_{j}(s) \bigr) \bigr)\le\varphi ^{-1} \bigl(z^{\frac{1}{q}}(s) \bigr),\quad s\in \bigl[b_{j}(t_{0}), b_{j}(\sigma) \bigr]. \end{aligned}$$
(2.18)
It follows from (
2.17), (
2.18), and the definition of
\(z(t)\) that
$$\begin{aligned} z(t) \leq& \hat{r}_{1}(\sigma)+\sum _{i=1}^{m} \int _{b_{i}(t_{0})}^{b_{i}(t)} \tilde{g}_{i}(\sigma,s) \tilde{\omega}^{q}_{i} \bigl(\varphi^{-1} \bigl(z^{\frac {1}{q}}(s) \bigr) \bigr)\,ds \\ &{} +\sum_{j=m+1}^{m+n} \int_{b_{j}(t_{0})}^{b_{j}(t)} \tilde{g}_{j}(\sigma,s) \tilde{\omega}^{q}_{j} \bigl(\varphi^{-1} \bigl(z^{\frac {1}{q}}(s) \bigr) \bigr)\,ds,\quad t\in[t_{0},\sigma]. \end{aligned}$$
(2.19)
In order to demonstrate the basic condition of monotonicity, let
\(e(t):=\varphi^{-1}(t^{\frac{1}{q}})\), which is clearly a continuous and nondecreasing function on
\(\mathbb{R}_{+}\). Thus, for each
i,
\(\tilde{\omega}_{i}(e(t))\) is continuous and nondecreasing on
\(\mathbb{R}_{+}\) and
\(\tilde{\omega}_{i}(e(t))>0 \) for
\(t>0\). Moreover, since
\(\tilde{\omega }_{i}(t)\propto\tilde{\omega}_{i+1}(t)\), we see that the ratio
\(\tilde{\omega}_{i+1}(b(t))/\tilde{\omega }_{i}(e(t))\) is also a continuous and nondecreasing function on
\(\mathbb{R}_{+}\) and satisfies
\(\tilde{\omega}_{i}(e(t))>0\) for
\(t>0\), implying that
\(\tilde{\omega}^{q}_{i}(e(t))\varpropto\tilde{\omega}^{q}_{i+1}(e(t))\),
\(i=2,\ldots, m+n-1\). Applying Lemma
2.3 to the case that
\(f_{i}(t,s)=\tilde{g}_{i}(\sigma ,s)\),
\(a(t)=\hat{r}_{1}(\sigma)\), and
\(\omega_{i}(t)=\tilde{\omega}^{q}_{i}(\varphi^{-1}(t^{\frac{1}{q}}))\),
\(i=1,2,\ldots,m+n\), from (
2.19) we obtain
$$\begin{aligned} z(t)\leq W_{m+n}^{-1} \biggl(W_{m+n} \bigl( \hat{r}_{m+n}(\sigma,t) \bigr)+ \int _{b_{m+n}(t_{0})}^{b_{m+n}(t)}\tilde{g}_{m+n}(\sigma,s) \,ds \biggr) \end{aligned}$$
(2.20)
for all
\(t_{0}\le t \le\min\{\sigma,T_{1}\}\), where
$$\begin{aligned} &\tilde{r}_{1}(\sigma,t) := \hat{r}_{1}( \sigma ), \\ &\tilde{r}_{i+1}(\sigma,t) := W_{i}^{-1} \biggl(W_{i} \bigl(\tilde {r}_{i}(\sigma,t) \bigr) + \int_{b_{i}(t_{0})}^{b_{i}(t)}\tilde{g}_{i}(\sigma,s)\,ds \biggr),\quad i=1,2,\ldots,m+n-1, \end{aligned}$$
(2.21)
and
\(T_{1}< t_{1}\) is the largest number such that
$$\begin{aligned} W_{i} \bigl(\tilde{r}_{i}(\sigma,T_{1}) \bigr)+ \int_{b_{i}(t_{0})}^{b_{i}(T_{1})}\tilde {g}_{i}(\sigma,s) \,ds \leq \int_{u_{i}}^{\infty}\frac{dz}{\tilde{\omega }^{q}_{i}(\varphi^{-1}(z^{\frac{1}{q}}))} \end{aligned}$$
(2.22)
for
\(i=1,2,3,\ldots,m+n\). Notice that
\(T\leq T_{1}\). In fact,
\(W_{i}\) is strictly increasing by (
2.3), so its inverse
\(W_{i}^{-1}\) is continuous and increasing in its corresponding domain by (
2.3). It follows from (
2.21) and the definition of
\(\tilde{g}_{i}(\sigma,s)\) that
\(\tilde{r}_{i}(\sigma,t)\) and
\(\tilde{g}_{i}(\sigma,s)\) are nondecreasing in
σ. Thus,
\(T_{1}\) satisfying (
2.22) gets smaller as
σ is chosen larger. In particular,
\(T_{1}\) satisfies the same equation (
2.6) as
T when
\(\sigma=T\). It follows from (
2.17) and (
2.20) that
$$\begin{aligned} u(t)\leq\varphi^{-1} \biggl( \biggl(W_{m+n}^{-1} \biggl(W_{m+n} \bigl(\tilde {r}_{m+n}(\sigma,t) \bigr)+ \int_{b_{m+n}(t_{0})}^{b_{m+n}(t)}\tilde {g}_{m+n}(\sigma,s) \,ds \biggr) \biggr)^{1/q} \biggr) . \end{aligned}$$
(2.23)
Taking
\(t=\sigma\) in (
2.23), we have
$$\begin{aligned} u(\sigma)\leq\varphi^{-1} \biggl( \biggl( W_{m+n}^{-1} \biggl(W_{m+n} \bigl( \tilde{r}_{m+n}(\sigma,\sigma) \bigr)+ \int _{b_{m+n}(t_{0})}^{b_{m+n}(\sigma)}\tilde{g}_{m+n}(\sigma,s) \,ds \biggr) \biggr)^{1/q} \biggr) \end{aligned}$$
(2.24)
for
\(0\le\sigma\le T\). It is easy to verify
\(\tilde{r}_{i}(\sigma,\sigma)=\hat{r}_{i}(\sigma)\).