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Generalized nonlinear weakly singular retarded integral inequalities with maxima and their applications
Journal of Inequalities and Applications volume 2018, Article number: 294 (2018)
Abstract
This paper deals with a generalized nonlinear weakly singular retarded Wendroff-type integral inequality with maxima of an unknown function of two variables. The key is that a technique of monotonization without separability and monotonicity of given functions is used for estimating the boundedness of unknown functions. Then our outcomes can be helpful to weaken conditions for some known results. By applying our results, the uniqueness of solutions for some singular integral equation with maxima may be proven.
1 Introduction
The Gronwall inequality [1] holds a vital place in studying qualitative properties of the solutions of integral equations and differential equations. Some linear and nonlinear generalizations (e.g. [2–11]) of the Gronwall inequality have been extensively discussed. With further study of fractional differential equations, integral inequalities with weakly singular kernels have attracted more and more attention (see [12–20]). In [14], a new method was presented to analyze the nonlinear singular integral inequalities of Henry type:
In 2008, Cheung et al. [20] solved the nonlinear weakly singular inequality
On the other hand, since differential equations with maxima of the unknown function [21–26] can be applied in control theory, some significant results for integral inequalities containing the maxima of the unknown function [22, 27–30] have been obtained. The integral inequality with maxima
where f, g, and ψ are nonnegative continuous functions and \(a(x,y)>0\) is a nondecreasing continuous function, was discussed in [22].
Combining (1.2) with (1.3), we will consider the integral inequality with maxima
where a, g, \(\omega_{j}\), \(f_{j}\), \(b_{j}\), and \(c_{j}\) are nonnegative continuous functions, \(b_{j}\) and \(c_{j}\) are increasing functions and belong to \(C^{1}\), \(b_{*}(x_{0}):=\min_{1\le j\le{m}}b_{j}(x_{0})\), \(h>0\) is a constant. Specially, the monotonicity of a, \(\omega_{j}\), \(\mu_{j}\), \(f_{j}\), and g is not required. Further, \(\omega_{j}\)’s are used to construct a sequence of stronger monotonized functions. Then the obtained result is applied for considering the uniqueness of solutions to a boundary value problem of an integral equation with maxima.
2 Main result
Let \(\mathbb {R}:=(-\infty, +\infty)\), \(\mathbb {R}_{+}:=[0,\infty)\), \(\Delta:=[x_{0},x_{1})\times[y_{0}, y_{1})\) and \(\Xi:= [b_{*}(x_{0})-h,x_{0}]\times[y_{0}, y_{1})\). Define \(\Phi_{1}, \Phi_{2}: B\subset\mathbb {R} \rightarrow \mathbb {R}\setminus\{0\}\). As in [4], if \(\Phi_{1}/\Phi _{2}\) is nondecreasing on B, then \(\Phi_{1}\varpropto\Phi_{2}\). Considering inequality (1.4), we make the following assumptions for all \(j=1,\ldots,m\):
- (A1):
-
\(b_{j}\in C^{1}([x_{0},x_{1}),\mathbb {R}_{+})\) and \(c_{j}\in C^{1}([y_{0},y_{1}), [y_{0},y_{1}))\) are nondecreasing such that \(b_{j}(x)\leq x\) and \(c_{j}(y)\le y\), and \(c_{j}(y_{0})=y_{0}\);
- (A2):
-
\(a\in C(\Delta,\mathbb {R}_{+}) \), \(f_{j}\in C(\Delta\times [b_{*}(x_{0}),x_{1})\times[y_{0},y_{1}), \mathbb{R}_{+})\), \(\omega_{j},\mu_{j}\in C(\mathbb{R}_{+},\mathbb {R}_{+}) \) with \(\omega_{j}(t)>0\), \(\mu _{j}(t)>0\) for \(t>0\);
- (A3):
-
\(g, \varphi\in C(\mathbb {R}_{+},\mathbb{R}_{+})\) and \(\psi\in C(\Xi, \mathbb {R}_{+})\), and φ is strictly increasing such that \(\lim_{t\rightarrow\infty}\varphi(t)=\infty\);
- (A4):
-
\(\alpha_{j}, \bar{\alpha}_{j}\in(0,1]\), \(\beta_{j},\bar{\beta}_{j}\in(0,1)\), \(\gamma_{j}>1-\frac{1}{p}\), \(\bar{\gamma}_{j}>1-\frac{1}{p}\) such that \(\frac{1}{p}+\alpha_{j}(\beta_{j}-1)+\gamma_{j}-1\ge0\), \(\frac {1}{p}+\bar{\alpha}_{j}(\bar{\beta}_{j}-1)+\bar{\gamma}_{j}-1\ge0\), \(p(\beta_{j}-1)+1>0\), \(p(\bar{\beta}_{j}-1)+1>0\), \(p>1\).
For those \(\omega_{j}\)’s, \(\mu_{j}\)’s given in (A4), define \(\tilde {\omega}_{j}(t)\) inductively by
where \(\hat{\omega}_{j}(t):=\max_{\tau\in[0, t] }\{\bar {\omega}_{j}(\tau)\}\), \(\hat{\mu}_{j}(t):=\max_{\tau\in[0, t] }\{\bar{\mu}_{j}(\tau)\}\), \(\tilde{g}(t):=\max_{\tau\in[0, t] }\{g(\tau)\}\), \(\bar{\omega}_{j}(t):=\omega_{j}(t)+\varepsilon_{j}\), \(\bar{\mu }_{j}(t):=\mu_{j}(t)+\varepsilon_{j}\) for \(t\ge0\), \(\epsilon_{j}:= \varepsilon\) if \(\omega_{j}(0)=0\) or \(:=0\) if \(\omega_{j}(0)\neq0\) for all \(j=1,2,\ldots,m\), where \(\varepsilon>0\) is an arbitrarily given constant.
Lemma 1
([16])
Let α, β, γ, and p be positive constants. Then
where \(\theta:=p[\alpha(\beta-1)+\gamma-1]+1\), \(B(\xi,\eta)=\int ^{1}_{0}s^{\xi-1}(1-s)^{\eta-1}\,ds\) (\(\operatorname{Re} \xi>0\), \(\operatorname{Re} \eta>0\)) is the beta function.
Lemma 2
Suppose that
-
(C1)
\(b_{j}\in C^{1}([x_{0},x_{1}),\mathbb {R}_{+})\) and \(c_{j}\in C^{1}([y_{0},y_{1}), [y_{0},y_{1}))\) are nondecreasing with \(b_{j}(x)\leq x\) on \([x_{0},x_{1})\), \(c_{j}(y)\le y\) on \([y_{0},y_{1})\) and \(c_{j}(y_{0})=y_{0}\) for all \(j=1,\ldots,m\);
-
(C2)
\(\psi\in C(\Xi,\mathbb {R}_{+})\), \(g_{j}\in C(\Delta\times\mathbb {R}^{2}_{+},\mathbb {R}_{+})\) are nondecreasing functions in x and y for all \(j=1,\ldots,m\);
-
(C3)
\(h_{j}, \bar{h}_{j}\in C(\mathbb {R}_{+},\mathbb{R}_{+})\) (\(j=1,\ldots,m\)) are all nondecreasing with \(h_{j}(t)>0\), \(\bar{h}_{j}(t)>0\) for \(t>0\), and \(h_{j}\bar{h}_{j}\propto h_{j+1}\bar{h}_{j+1}\) (\(j=1,\ldots,m-1\));
-
(C4)
\(b\in C(\Delta, \mathbb{R}_{+})\), \(b_{x}, b_{y}\in(\Delta, \mathbb{R})\), and \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\psi(s,t)\le b(x_{0},t)\) for all \(t\in [y_{0},y_{1})\).
If \(u\in C([b_{*}(x_{0})-h,x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\) satisfies the integral inequality
then
for all \((x,y)\in[x_{0}, X_{1}^{*}]\times[y_{0},Y_{1}^{*}]\), where \(H_{j}^{-1}\) is the inverse of the function
\(t_{j}\) is a given constant, and \(\eta_{j}\) is defined by
for \(j=1,\ldots,m-1\), and \(x_{0}\le X_{1}^{*}< x_{1}\), \(y_{0}\le Y_{1}^{*}< y_{1}\) are chosen such that
for \(j=1,\ldots,m\).
Proof
Let b be positive on Δ. It means that \(\eta_{1}(x,y)\) is positive on Δ. Under such a circumstance, \(\eta_{1}\) is nondecreasing on Δ and \(\eta_{1}(x,y)>0\),
Concerning (2.8), we consider the auxiliary inequality
where \(x_{0}\leq\xi\le X^{*}_{1}\) and \(y_{0}\leq\eta\le Y^{*}_{1}\) are chosen arbitrarily. Having (2.9) we claim
for \(x_{0}\le x \le\min\{\xi, X^{*}_{2}\}\), \(y_{0}\le y \le\min\{\eta, Y^{*}_{2}\}\), where \(\tilde{\eta}_{j}(\xi,\eta,x,y)\) is defined inductively by \(\tilde {\eta}_{1}(\xi,\eta,x,y):=\eta_{1}(x,y)\) and
for \(j=2,\ldots, m\), and \(X^{*}_{2}\in[x_{0},x_{1})\), \(Y^{*}_{2}\in[y_{0},y_{1})\) are chosen such that
for \(j=1,2,\ldots,m\). Note that \(X^{*}_{2}\ge X^{*}_{1}\) and \(Y^{*}_{2}\ge Y^{*}_{1}\). In fact, both \(\tilde{\eta}_{j}(\xi,\eta,x,y)\) and \(g_{j}(\xi,\eta,x,y)\) are nondecreasing in ξ and η. Thus \(X^{*}_{2}\), \(Y^{*}_{2}\) satisfying (2.11) will get smaller as ξ, η are chosen larger.
Since \(\max_{s\in[b^{*}(x_{0})-h,x_{0}]}\psi(s,t)\le b(x_{0},t)\) and \(b(x_{0},t)\le \eta_{1}(x_{0},t)\le\eta_{1}(x,t)\), we obtain
First, (2.10) holds for \(m=1\). In fact,(2.9) for \(m=1\) is written as
where
\(z_{1}(x,y)\) is a nondecreasing function on \([x_{0}, \xi]\times[y_{0},\eta]\). Then
for all \((x,y)\in[x_{0},\xi]\times[y_{0},\eta] \). We have \(0< h_{1}(u(s,t))\bar{h}_{1}(u(s,t))\le h_{1}(z_{1}(s,t))\bar{h}_{1}(z_{1}(s,t)) \le h_{1}(z_{1}(x,y))\bar{h}_{1}(z_{1}(x,y)) \) by (C3) and (2.13) \(s\le b_{1}(x)\le x\), \(t\le c_{1}(y)\le y\) and both \(z_{1}\) and \(h_{1}\tilde{h}_{1}\) are nondecreasing. Thus
Integrating inequality (2.15) from \(x_{0}\) to x, from (2.4) we get
for all \((x,y)\in[x_{0},\xi]\times[y_{0},\eta]\). From (2.14), (2.16), and the monotonicity of \(H^{-1}_{1}\), we have
for \(x_{0}\le x\le\xi< X^{*}_{2}\), \(Y_{0}\le y \le\eta< Y^{*}_{2}\), implying that (2.7) is true for \(m=1\).
Assume that (2.10) holds for \(m=k\). Consider
Let
Then \(z_{2}\) is a nondecreasing function on \([x_{0}, x]\times[y_{0},\eta]\). By (2.19) and the definition of \(z_{2}\), it follows that
Since \(h_{j}\bar{h}_{j}\) is nondecreasing and \(z_{2}(x,y)>0\), \(b'_{j}(x)\ge 0\), and \(b_{j}(x)\le x\), we have
for all \((x,y)\in[x_{0},X_{1}^{*}]\times[y_{0},Y_{1}^{*}]\), where \(\tilde {h}_{j+1}(u):=h_{j+1}(u)/h_{1}(u)\), \(\hat{h}_{j+1}(u):=\bar{h}_{j+1}(u)/\bar{h}_{1}(u)\), \(j=1,\ldots,k\). Integrating the above inequality from \(x_{0}\) to x, we can obtain
for all \((x,y)\in[x_{0},X]\times[y_{0},Y]\). Let
Then inequality (2.21) can be rewritten as
the same form as (2.9) for \(m=k\). By (C3), each \((\bar {h}_{j+1}\circ H_{1}^{-1})(\tilde{h}_{j+1}\circ H_{1}^{-1})\) (\(j=1,\ldots,k\)) is a nonnegative continuous and increasing function on \(\mathbb{R}_{+}\) and positive on \((0,+\infty)\). Moreover, \((\tilde{h}_{j}\circ H_{1}^{-1})\propto(\hat{h}_{j+1}\circ H_{1}^{-1})\) for all \(j=2,\ldots, k\). By the inductive assumption, we have
for \(x_{0}\le x\le\min\{\xi, X_{3}^{*}\}\), \(y_{0}\le y\le\min\{\eta, Y_{3}^{*}\}\), where
\(\tilde{t}_{j+1}=H_{1}(t_{j+1})\), \(\bar{H}^{-1}_{j+1}\) is the inverse of \(\bar{H}_{j+1}\), \(j=1,\ldots, k\),
\(j=1,\ldots,k-1\), and \(X^{*}_{3}\), \(Y^{*}_{3}\) are chosen such that
Note that
Then, from (2.20), (2.24), and (2.28), we get
for \(x_{0}\le x\le\min\{X, X_{3}^{*}\}\), \(y_{0}\le y\le\min\{Y, Y_{3}^{*}\} \). Let \(\tilde{\varrho}_{j}(x,y)=H^{-1}_{1}(\varrho_{j}(x,y))\). Then
Moreover, with the assumption that \(\tilde{\varrho}_{k}(x,y)=\tilde {\eta}_{k+1}(\xi,\eta,x,y)\), we get
This proves that
Therefore, (2.27) becomes
which implies that \(X^{*}_{2}=X^{*}_{3}\), \(\xi\le X^{*}_{3}\), \(Y^{*}_{2}=Y^{*}_{3}\), \(\eta\le Y^{*}_{3}\). From (2.29) we obtain
for \(x_{0}\le x\le \min\{X,X_{2}^{*}\}\), \(y_{0}\le y\le \min\{Y,Y_{2}^{*}\}\). This proves (2.10) by induction.
Taking \(x=\xi,\eta\), \(y=\xi,\eta\) in (2.10), we have
for \(x_{0}\le\xi\le X^{*}_{1}\), \(y_{0}\le\eta\le Y^{*}_{1}\), since \(x^{*}_{2}\ge X^{*}_{1}\), \(Y^{*}_{2}\ge Y^{*}_{1}\) and \(\tilde{\eta}_{m}(\xi,\eta,\xi,\eta)= \eta_{m}(\xi,\eta)\). Since ξ, η are arbitrary, replacing ξ and η with x and y, respectively, we have
for all \((x,y)\in[x_{0}, X^{*}_{1}]\times[y_{0},Y^{*}_{1}]\).
Let \(b(x,y)=0\) for some \((x,y)\in\Delta\). Let \(\eta_{1,\epsilon }(x,y):=r_{1}(x,y)+\epsilon\) for any \(\epsilon>0\). Then \(\eta_{1,\epsilon}(x,y)>0\). Using the same arguments as above, where \(\eta_{1}(x,y)\) is replaced with \(\eta_{1,\epsilon}(x,y)\), we get
for \(x_{0}\le x\le X^{*}_{1}\), \(y_{0}\le Y^{*}_{1}\). Then consider the continuity of \(\eta_{i,\epsilon}\) in ϵ and the continuity of \(H_{j}\) and \(H_{j}^{-1}\) for \(j=1,\ldots, m\), and let \(\epsilon\rightarrow0^{+}\). Then we obtain (2.7). This completes the proof. □
Theorem 2.1
Suppose that (A1)–(A4) hold. \(\max_{s\in [b_{*}(x_{0})-h,x_{0}]}\psi(s,y)\leq\varphi^{-1}( (1+m)^{1-1/q}a(x_{0}, y))\) for \(y\in[y_{0},y_{1})\) and \(u\in C([b_{*}(x_{0})-h,x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\) are satisfied (1.4). Then, for all \((x,y)\in[x_{0}, X_{1})\times[y_{0},Y_{1})\), we have
where \(W_{j}^{-1}\)is the inverse of the function
In (2.36) and (2.37), \(t_{j}\) is a given constant, \(\frac {1}{p}+\frac{1}{q}=1\), \(\tilde{\omega}_{j}\) (\(j=1,2,\ldots,m\)) are defined by (2.1),
\(X_{1}\in[x_{0}, x_{1})\), \(Y_{1}\in[y_{0}, y_{1})\) are chosen such that
for \(j=1,\ldots,m\).
Proof
Above all, we monotonize functions \(f_{j}\), \(\omega_{j}\), \(\mu_{j}\), g, and a in (1.4). Let
which is increasing in x and y. The sequence \(\{\tilde{\omega}_{j}\}\), defined by \(\omega_{j}(s)\) and \(\mu_{j}(s)\) in (2.1), consists of nonnegative and nondecreasing functions on \(\mathbb{R}_{+} \) and satisfies
Moreover, because the ratios \({\tilde{\omega}_{j+1}}/{\tilde{\omega }_{j}}\) (\(j=1,\ldots,m-1\)) are all nondecreasing, we have
Let
which are increasing in x and y and satisfy \(\tilde {f}_{j}(x,y,s,t)\geq f_{j}(x,y,s,t)\geq0\) for \(j=1,2,\ldots,m\). Since g̃ is nondecreasing, we obtain
for all \((s,y)\in[b_{*}(x_{0}), x_{1})\times[y_{0},y_{1})\). From (1.4), (2.42), (2.45), and the definition of \(\hat{f}_{j}\), we can obtain
Let \(\frac{1}{p}+\frac{1}{q}=1\), \(p>1\), then \(q>0\). By Lemma 1, Hölder’s inequality, (A4) and (2.46), we obtain, for all \((x,y)\in[x_{0},x_{1})\times[y_{0}, y_{1})\),
where \(0\le b_{j}(t)\le t \), \(0\le c_{j}(t)\le t\), \(M_{j}\), \(\bar {M}_{j}\), \(\theta_{j}\), and \(\bar{\theta}_{j}\) are given by (2.40) for \(j=1,\ldots,m\).
By Jensen’s inequality and (2.47), we get, for all \((x,y)\in \Delta\),
Then, from (2.38), \(r_{1}\) is increasing on Δ. Then, by the definition of \(r_{1}\) and \(\tilde{f}_{j}\), from (2.48) we have
According to (2.49), we consider the inequalities
where \(x_{0}\leq X\le X_{1}\) and \(y_{0}\leq Y\le Y_{1}\) are chosen arbitrarily.
Since \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\psi(s,y)\leq\varphi ^{-1}((1+m)^{1-1/q}a(x_{0},y)) \) for \(y\in[y_{0},y_{1})\), \(a(x_{0},y)\le\hat{ a}(x_{0},y)\), we have \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\psi(s,y)\leq\varphi ^{-1}(r^{1/q}_{1}(X,Y))\), \(y\in[y_{0},Y]\). Define a function \(z(x,y): [b_{*}(x_{0})-h, X)\times[y_{0},Y)\rightarrow \mathbb {R}_{+}\) by
Clearly, \(z(x,y)\) is increasing in x. By the definition of \(z(x,y)\) and (2.50), we have
Since \(\varphi(t)\) is strictly increasing and \(z(x,y)\) is nondecreasing, from (2.51) we get, for \((s,y)\in[b_{*}(x_{0}), X]\times[y_{0},Y]\),
From the definition of \(z(x,y)\), (2.42), (2.51), and (2.52), it follows that
where \(\vartheta_{j}(t)\equiv1\), \(t\ge0\).
Let \(v(t):=\varphi^{-1}(t^{1/q})\), which is a continuous and increasing function on \(\mathbb {R}_{+}\). Thus \(\tilde{\omega }^{q}_{j}(h(t))\) (\(j=1,\ldots, m\)) are continuous and increasing on \(\mathbb {R}_{+}\) and satisfy \(\tilde{\omega}_{j}(v(t))>0 \) for \(t>0\). Moreover, since \(\tilde {\omega}_{j}(t)\propto\tilde{\omega}_{j+1}(t)\), \(\tilde{\omega}^{q}_{j+1}(v(t))/\tilde{\omega}^{q}_{j}(v(t))\) are continuous and increasing on \(\mathbb {R}_{+}\) and positive on \((0,\infty)\), then \((\tilde{\omega}_{j}\circ v)\vartheta_{j}\propto (\tilde{\omega}_{j+1}\circ v)\vartheta_{j+1}\) for \(j=1,2,\ldots,m-1\).
Applying Lemma 2 to specified \(g_{j}(x,y,s,t)=\tilde{f}_{j}(X,Y,s,t)\), \(h_{j}(t)=\tilde{\omega}^{q}_{j}(\varphi^{-1}(t^{1/q}))\), \(\bar{h}_{j}(t)=\vartheta_{j}(t)\equiv1\) (\(j=1,2,\ldots,m\)), and (2.53), we obtain
for \(x_{0}\le x \le\min\{X, X_{2}\}\), \(y_{0}\le y \le\min\{Y, Y_{2}\}\), where \(\tilde{r}_{j}\) is defined inductively by \(\tilde{r}_{1}(X,Y, x,y):=\gamma _{1}(X,Y)\) and
for \(j=2,\ldots, m\), and \(\bar{X}_{1}\), \(\bar{Y}_{1}\) are chosen such that
for \(j=1,\ldots,m\).
Note that \(X_{2}\ge X_{1}\) and \(Y_{2}\ge Y_{1}\). In fact, both \(\tilde {r}_{j}(X,Y,x,y)\) and \(\tilde{f}_{j}(X,Y,x,y)\) are increasing in X and Y. Thus \(X_{2}\), \(Y_{2}\) satisfying (2.55) get smaller as X, Y are chosen larger.
According to (2.51) and (2.54),
for \(x_{0}\le x \le\min\{X, X_{2}\}\), \(y_{0}\le y \le\min\{Y, Y_{2}\}\).
Taking \(x=X\), \(y=Y\) in (2.56), we have
for \(x_{0}\le X\le X_{1}\), \(y_{0}\le Y\le Y_{1}\). It is easy to verify \(\tilde {r}_{m}(X,Y,X,Y)= r_{m}(X,Y)\). Thus, (2.57) can be written as
Since X, Y are arbitrary, replacing Y and X with y and x, respectively, we have
for all \((x,y)\in[x_{0}, X^{*}_{1}]\times[y_{0},Y^{*}_{1}]\).
This completes the proof. □
Theorem 2.2
We make the following assumptions:
- (S1):
-
\(c(x,y)\in C(\Delta, \mathbb{R}_{+}) \) and \(b_{j}\in C^{1}([x_{0},x_{1}),\mathbb {R}_{+})\), and \(c_{j}\in C^{1}([y_{0},y_{1}), [y_{0},y_{1}))\) are nondecreasing with \(b_{j}(x)\leq x\) on \([x_{0},x_{1})\) and \(c_{j}(y)\le y\) on \([y_{0},y_{1})\), and \(c_{j}(y_{0})=y_{0}\) for \(j=1,\ldots,m\);
- (S2):
-
\(\hat{\psi}\in C(\Xi,\mathbb {R}_{+})\), \(\hat{g}_{j}\in C(\Delta\times[b_{*}(x_{0}),x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\) (\(j=1,2,\ldots, m\));
- (S3):
-
\(\phi_{j}, \hat{\phi}_{j}\in C(\mathbb {R}_{+},\mathbb{R}_{+})\) (\(j=1,\ldots ,m\)) are all nondecreasing with \(\{\phi_{j},\hat{\phi}_{j}\}(t)>0\) for \(t>0\), and \(\phi_{j}\hat{\phi}_{j}\propto\phi_{j+1}\hat{\phi}_{j+1}\) (\(j=1,\ldots,m-1\));
- (S4):
-
\(k\ge1\), \(\alpha_{j}, \bar{\alpha}_{j}\in(0,1]\), \(\beta_{j},\bar{\beta }_{j}\in(0,1)\), \(\gamma_{j}>1-\frac{1}{p}\), \(\bar{\gamma}_{j}>1-\frac{1}{p}\) such that \(\frac{1}{p}+\alpha_{j}(\beta_{j}-1)+\gamma_{j}-1\ge0\), \(\frac {1}{p}+\bar{\alpha}_{j}(\bar{\beta}_{j}-1)+\bar{\gamma}_{j}-1\ge0\), \(p(\beta_{j}-1)+1>0\), \(p(\bar{\beta}_{j}-1)+1>0\), \(p>1\) for all \(j=1,\ldots,m\).
If \(u\in C([b_{*}(x_{0})-h,x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\) satisfies the integral inequality
where \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\hat{\psi}(s,y)\leq( (1+m)^{1-1/q}c(x_{0},y))^{1/k}\) for all \(y\in[y_{0},y_{1})\).
Then
for all \((x,y)\in[x_{0}, X_{2})\times[y_{0},Y_{2})\), where \(G_{j}^{-1}\)is the inverse of the function
In (2.61) and (2.62), \(t_{j}>0\) is a given constant, \(\frac{1}{p}+\frac{1}{q}=1\), \(e_{j}(x,y)\) is defined recursively by
\(X_{2}\in[x_{0}, x_{1})\), \(Y_{2}\in[y_{0}, y_{1})\) are chosen such that
for \(j=1,2,\ldots,m\).
Proof
Let
which are increasing in x and y and satisfy \(\bar {g}_{j}(x,y,s,t)\geq g_{j}(x,y,s,t)\geq0\) for \(j=1,2,\ldots,m\). From (2.60), (2.67), and the definition of \(\tilde {g}_{j}\), we obtain
Let \(\frac{1}{p}+\frac{1}{q}=1\), \(p>1\), then \(q>0\). By Lemma 1, Hölder’s inequality, (S4), and (2.68), we obtain, for all \((x,y)\in\Delta\),
where \(0\le b_{j}(t)\le t \), \(0\le c_{j}(t)\le t\), \(M_{j}\), \(\bar {M}_{j}\), \(\theta_{j}\), and \(\bar{\theta}_{j}\) are given by (2.65) for \(j=1,\ldots,m\).
By Jensen’s inequality and (2.69), we get, for all \((x,y)\in \Delta\),
By the definition of \(e_{1}\) and \(\tilde{g}_{j}\), from (2.70) we obtain
Concerning (2.71), we consider the auxiliary inequalities
where \(x_{0}\leq X\le X_{2}\) and \(y_{0}\leq Y\le Y_{2}\) are chosen arbitrarily.
Since \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\hat{\psi}(s,y)\leq ((1+m)^{1-1/q}a(x_{0},y))^{\frac{1}{k}}\) for \(y\in[y_{0},y_{1})\), \(a(x_{0},y)\le\hat{ c}(x_{0},y)\), we have \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\psi(s,y)\leq (e^{1/q}_{1}(X,Y))^{\frac{1}{k}}\), \(y\in[y_{0},Y]\). Define a function \(z(x,y): [b_{*}(x_{0})-h, X)\times[y_{0},Y)\rightarrow \mathbb {R}_{+}\) by
Clearly, \(z(x,y)\) is increasing in x. From (2.72) and the definition of z, we have
Then, noting that z is increasing, from (2.51) we get for \((s,y)\in[b_{*}(x_{0}), X]\times[y_{0},Y]\)
From (2.42), (2.73), (2.74), and the definition of z, we have
Let \(v(t):=t^{1/(kq)}\), which is a continuous and increasing function on \(\mathbb {R}_{+}\). Thus \(\phi^{q}_{j}(v(t)) \) and \(\hat {\phi}^{q}_{j}(v(t))\) (\(j=1,\ldots, m\)) are continuous and increasing on \(\mathbb {R}_{+}\) and positive on \((0,\infty)\). Moreover, since \(\phi_{j}\hat{\phi}_{j}\propto\phi_{j+1}\hat{\phi}_{j+1}\), we have \((\phi_{j+1}\circ v)^{q}(\hat{\phi}_{j+1}\circ v)^{q} \propto(\phi _{j}\circ v)^{q}(\hat{\phi}_{j}\circ v)^{q}\) (\(j=1,\ldots,m-1\)). Taking \(g_{j}(x,y,s,t)=\tilde{g}_{j}(X,Y,s,t)\) and \(h_{j}(t)=\phi^{q}_{j}(v(t))\), \(\bar{h}_{j}(t)=\hat{\phi}^{q}_{j}(v(t))\), \(j=1,2,\ldots,m\), in Lemma 2 and (2.75),we obtain
for \(x_{0}\le x \le\min\{X, X^{*}_{2}\}\), \(y_{0}\le y \le\min\{Y, Y^{*}_{2}\}\), where \(\tilde{e}_{j}(X,Y,x,y)\) is defined inductively by \(\tilde {e}_{1}(X,Y,x,y):=e_{1}(X,Y)\) and
for \(j=2,\ldots, m\), and \(X^{*}_{2}\), \(Y^{*}_{2}\) are chosen such that
for \(j=1,\ldots,m\).
Note that \(X^{*}_{2}=X_{2}\) and \(Y^{*}_{2}=Y_{2}\). It follows from (2.73) and (2.76) that
for \(x_{0}\le x \le\min\{X, X^{*}_{2}\}\), \(y_{0}\le y \le\min\{Y, Y^{*}_{2}\}\).
Taking \(x=X\), \(y=Y\) in (2.56), we have
for \(x_{0}\le X\le X_{2}\), \(y_{0}\le Y\le Y_{2}\). It is easy to verify \(\tilde {e}_{m}(X,Y,X,Y)= e_{m}(X,Y)\). Thus, (2.57) can be written as
Since \(X,Y\) are arbitrary, replacing X and Y with x and y, respectively, we get
for all \((x,y)\in[x_{0}, X_{2}]\times[y_{0},Y_{2}]\). This completes the proof. □
Corollary 2.3
Let the following conditions be fulfilled:
- (B1):
-
all \(b_{j}\in C^{1}([x_{0},x_{1}),\mathbb {R}_{+})\) and \(c_{j}\in C^{1}([y_{0},y_{1}), [y_{0},y_{1}))\) are nondecreasing with \(b_{j}(x)\leq x\) on \([x_{0},x_{1})\), \(c_{j}(y)\le y\) on \([y_{0},y_{1})\), and \(c_{j}(y_{0})=y_{0}\) for all \(j=1,\ldots,m\);
- (B2):
-
\(a\in C(\Delta, \mathbb{R}_{+})\) and \(\hat{\psi}\in C(\Xi,\mathbb {R}_{+})\), \(\varphi_{1} \in C(\mathbb {R}_{+},\mathbb {R}_{+})\), and \(\varphi_{1}\) is strictly increasing such that \(\lim_{t\rightarrow\infty}\varphi(t)=\infty\),and \(f_{j}\in C(\Delta\times[b_{*}(x_{0}),x_{1})\times[y_{0},y_{1}),\mathbb{R}_{+})\) for all \(j=1,\ldots, m\);
- (B3):
-
all \(\psi_{j}\) (\(j=1,\ldots,m\)) are continuous and increasing functions on \(\mathbb {R}_{+}\) and positive on \((0,+\infty)\) such that \(\psi_{1}\propto\psi_{2}\propto\ldots\propto\psi_{m}\);
- (B4):
-
\(\alpha_{j}, \bar{\alpha}_{j}\in(0,1]\), \(\beta_{j},\bar{\beta}_{j}\in(0,1)\), \(\gamma_{j}>1-\frac{1}{p}\), \(\bar{\gamma}_{j}>1-\frac{1}{p}\) such that \(\frac{1}{p}+\alpha_{j}(\beta_{j}-1)+\gamma_{j}-1\ge0\), \(\frac {1}{p}+\bar{\alpha}_{j}(\bar{\beta}_{j}-1)+\bar{\gamma}_{j}-1\ge0\), \(p(\beta_{j}-1)+1>0\), \(p(\bar{\beta}_{j}-1)+1>0\), \(p>1\), \(j=1,2,\ldots,m\);
- (B5):
-
\(u\in C([b_{*}(x_{0})-h,x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\) satisfies the integral inequality
$$ \begin{aligned} &\varphi_{1}\bigl(u(x,y) \bigr) \leq a(x,y)+\sum_{j=1}^{M} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} \times t^{\bar{\gamma}_{j}-1} f_{j}(x,y,s,t) \psi_{j}\bigl(u(s,t)\bigr)\,dt\,ds \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} +\sum_{j=M+1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1} \bigl(y^{\bar{\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} \times t^{\bar{\gamma}_{j}-1}f_{j}(x,y,s,t) \psi_{j} \Bigl(\max_{\tilde {\eta}\in[s-h,s]}u(\tilde{\eta},t) \Bigr) \,dt\,ds, \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} (x,y)\in[x_{0},x_{1})\times [y_{0}, y_{1}), \\ &u(x,y) \leq \hat{\psi }(x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}), \end{aligned} $$(2.82)where \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\hat{\psi}(s,y)\leq \varphi_{1}^{-1}( (1+m)^{1-1/q}a(x_{0},y))\) for all \(y\in[y_{0},y_{1})\).
Then
for all \((x,y)\in[x_{0}, X_{2})\times[y_{0},Y_{2})\), where \(G_{j}^{-1}\) is the inverse of the function
\(t_{j}\) is a given constant, \(r_{j}(x,y)\) is defined recursively by
\(M_{j}:=\alpha_{j}^{-1}B(\frac{p(\gamma_{j}-1)+1}{\alpha_{j}}, p(\beta_{j}-1)+1)\), \(\bar{M}_{j}:=\bar{\alpha}_{j}^{-1}B(\frac{p(\bar {\gamma}_{j}-1)+1}{\bar{\alpha}_{j}}, p(\beta_{j}-1)+1)\), \(\theta_{j}:=p(\alpha_{j}(\beta_{j}-1)+\gamma _{j}-1)+1\), \(\bar{\theta}_{j}:=p(\bar{\alpha}_{j}(\bar{\beta }_{j}-1)+\bar{\gamma}_{j}-1)+1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(X_{2}\in[x_{0}, x_{1})\), \(Y_{2}\in[y_{0}, y_{1})\) are chosen such that
for \(j=1,2,\ldots,m\).
Proof
Applying Theorem 2.1 to specified \(\omega_{j}(u)\equiv\psi _{j}(u)\) (\(j=1,\ldots,M\)), \(\mu_{j}(u)\equiv1\) (\(j=1,\ldots,M\)), \(\omega_{j}(u)\equiv1\) (\(j=M+1,\ldots,m\)), \(\mu_{j}(u)\equiv\psi_{j}(u)\) (\(j=M+1,\ldots,m\)), \(f_{j}(x,y,s,t)=\check{f}_{j}(x,y,s,t)\), \(g(t)=t\), from (2.82) we obtain estimate (2.83). The proof is complete. □
3 Applications
Consider a nonlinear weakly singular integral equation with maxima
where \(F\in C(\Delta\times\mathbb {R}^{4},\mathbb {R})\), h is a positive constant, \(\psi\in C([x_{0}-h,x_{0}]\times[y_{0},y_{1}),\mathbb{R})\), \(a\in C(\Delta, \mathbb {R})\), \(\theta_{j}\in(0,1)\), and \(p(\gamma_{j}-1)+1>0\) such that \(\frac{1}{p}+\theta_{j}+\gamma_{j}-2\ge0\) and \(p(\theta _{j}-1)+1>0\), \(p>1\), \(j=1,2\).
The following result gives an estimate for its solutions.
Corollary 3.1
Suppose that functions F in (3.1) satisfy
where \(h_{j}\in C([x_{0},x_{1})\times[y_{0},y_{1})\times\mathbb {R}^{2},\mathbb {R}_{+})\), and \(h_{j}(x,y,s,t)\) is nondecreasing in x and y for each fixed s and t, and \(\mu_{j}\in C(\mathbb {R}_{+},(0,\infty))\) (\(j=1,2\)) such that \(\mu_{1}\propto\mu_{2}\), \(\max_{s\in[x_{0}-h,x_{0}]}\psi(s,y)\le3^{1-1/q}|a(x_{0}, y)|\) for all \(y\in[y_{0}, y_{1})\).
Then any solution \(z(x,y)\) of (3.1) has the estimate
for all \((x,y)\in[x_{0},X_{1})\times[y_{0},Y_{1})\), where
\(M_{j}:=B(p(\gamma_{j}-1)+1, p(\theta_{j}-1)+1)\) (\(j=1,2\)), \(\delta _{j}:=p(\theta_{j}+\gamma_{j}-2)+1\), \(j=1,2\), \(\frac{1}{p}+\frac{1}{q}=1\), and constants \(u_{1}\), \(u_{2}\) are given arbitrarily, \(X_{1}\in[x_{0}, x_{1})\), \(Y_{1}\in[y_{0}, y_{1})\) are chosen such that
Proof
From (3.1) we obtain
Set \(v(x,y)=|z(x,y)|\) for all \((x,y)\in[x_{0}-h,x_{1})\times[y_{0},y_{1})\). From (3.4) we get
Applying Corollary 2.3 to the specified \(M=1\), \(m=2\), \(\varphi_{1} (u)=u\), \(f_{j}(x,y,s,t)=h_{j}(x,y,s,t)\), \(b_{j}(t)=t\), \(c_{j}(t)=t\), \(\alpha_{j}=\bar{\alpha}_{j}=1\), \(g(t)=t\), we obtain (3.3) from (3.5). □
Corollary 3.2
Suppose that functions F and ψ in (3.1) satisfy
for all \((x,y)\in\Delta\) and \(s_{j},t_{j}\in\mathbb {R}\) (\(i =1,2\)), where \(h_{j}\in C(\Delta,\mathbb {R}_{+})\). Then system (3.1) has at most one solution on Δ.
Proof
Assume that equation (3.1) has two solutions \(u(x,y)\), \(v(x,y)\). By the equivalent integral equation (3.1), we have
for all \((x,y)\in[x_{0},x_{1})\times[y_{0},y_{1})\). Since \(u(x, y)\) is a continuous function, it implies that, for any fixed \(t \in[y_{0}, y]\) and \(s \in[x_{0}, x]\), there exists \(\tau\in [s-h, s]\) such that \(\max_{ \tilde{\eta}\in[s-h,s]}u( \tilde{\eta },t) = u(\tau,t)\) holds. Now we suppose \(\max_{ \tilde{\eta }\in[s-h,s]}u( \tilde{\eta},t)\ge\max_{ \tilde{\eta}\in [s-h,s]}v( \tilde{\eta},t)\) and have
It follows from (3.7) and (3.8) that
Let
From (3.7) we obtain
Let \(\varepsilon>0\) be an arbitrary number. Then from (3.10) we have
Applying Corollary 2.3 to specified \(N=1\), \(m=2\), \(\varphi _{1}(u)=u\), \(g(t)=t\), \(b_{j}(t)=c_{j}(t)=t\), \(f_{j}(x,y,s,t)=h_{2}(s,t)\), \(j=12\), \(a(x,y)=\epsilon\), from (3.11) we obtain, for all \((x,y)\in\Delta\),
where \(\frac{1}{p}+\frac{1}{q}=1\), \(M_{j}\) and \(\delta_{j}\) (\(j=1,2\)) are defined as in Corollary 3.1. Letting \(\varepsilon\rightarrow 0\), we obtain the uniqueness of the solution of equation (3.1). The uniqueness is proved. □
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This research was supported by the National Natural Science Foundation of China (No. 11461058).
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Yan, Y., Zhou, D. & Zhao, J. Generalized nonlinear weakly singular retarded integral inequalities with maxima and their applications. J Inequal Appl 2018, 294 (2018). https://doi.org/10.1186/s13660-018-1885-6
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DOI: https://doi.org/10.1186/s13660-018-1885-6