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Some weakly singular Volterra integral inequalities with maxima in two variables
Journal of Inequalities and Applications volume 2023, Article number: 36 (2023)
Abstract
In this paper, we establish some new weakly singular Volterra type integral inequalities that include the maxima of the unknown function of two variables. We also use the results to research the boundedness of solutions to retarded nonlinear Volterra type integral equations.
1 Introduction
Alongside mathematics development, inequalities have played increasingly important roles in theory and applications. Gronwall–Bellman inequality and Bihari inequality are highly prominent inequalities [1–3], which provided important tools to study the qualitative properties of differential equations, integral equations, and integro-differential equations such as existence, uniqueness, oscillation, stability, boundedness, invariant manifolds, and other properties. Over the past few decades, various researchers have worked on related issues, and a lot of research results have been obtained, including differential system, difference system, time-scale system [4–31]. In [7–12, 29], the Volterra–Fredholm type inequalities were examined. There have also been some results for integral inequalities containing the maxima of the unknown functions [12, 23–29]. In recent years, with the rising of fractional order calculation, the study on weak singular inequalities has become a hot topic [13–16, 31].
In 1997, Medved [4] discussed the following Henry type integral inequalities:
In 2008, Ma and Pec̆airé [13] investigated some new explicit bounds for weakly singular integral inequalities
In 2010, Wang and Zheng [31] investigated the nonlinear weakly singular integral inequalities with two variables
In 2013, Yan [27] investigated the nonlinear Gronwall–Bellman type integral inequalities with maxima of two variables
In 2014, Thiramanus et al. [28] investigated the Henry–Gronwall integral inequalities with maxima
In 2015, Yan [23] investigated some new weakly singular Volterra integral inequalities with maxima
In 2017, Xu and Ma [29] investigated some new retarded nonlinear Volterra–Fredholm type integral inequalities with maxima in two variables
In this paper, we are concerned with the following weakly singular Volterra integral inequalities with maxima in two variables:
where \(\Omega =[x_{0},x_{1})\times [y_{0},y_{1}), \Omega _{0}=[\alpha _{\ast}(x_{0})-h,x_{0})\times [\beta _{\ast}(y_{0})-k,y_{1}) \cup [x_{0},x_{1})\times [\beta _{\ast}(y_{0})-k,y_{0})\).
2 Main results
In this section, we consider the integral inequality (1.9) with \(x_{0}< x_{1}\) and \(y_{0}< y_{1}\). First, we give the following conditions:
\((A_{1})\) \(b_{i}(x):[x_{0},x_{1})\rightarrow [0,\infty )\ (i=1,2,\ldots ,m+n)\) and \(c_{i}(y):[y_{0},y_{1})\rightarrow [y_{0},y_{1})\ (i=1,2,\ldots ,m+n)\) are differentiable continuously and nondecreasing such that \(b_{i}(x)\leq x\) on \([x_{0},x_{1})\), \(c_{i}(y)\leq y\) on \([y_{0},y_{1})\);
\((A_{2})\) All \(g_{i}\ (i=1,2,\ldots ,m+n)\) are continuous nonnegative functions on \(\Omega \times \Omega _{0}\);
\((A_{3})\) \(f,\varphi :R_{+}\rightarrow R_{+}, \psi :\Omega _{0}\rightarrow R_{+}\) are continuous functions and φ is a strictly increasing function, \(\lim_{t\rightarrow \infty}\varphi (t)=+\infty \);
\((A_{4})\) All \(\omega _{i}:R_{+}\rightarrow R_{+}\ (i=1,2,\ldots ,m+n)\) are continuous functions;
\((A_{5})\) \(a(x,y)\) is a continuous nonnegative function on Ω;
\((A_{6})\) \(k_{i},v_{i}\in [0,1], \alpha _{i}\in (0,1], \beta _{i}\in (0,1), pk_{i}(\beta _{i}-1)+1>0, pv_{i}(\gamma _{i}-1)+1>0\) such that \(\frac{1}{p}+k_{i}\alpha _{i}(\beta _{i}-1)+v_{i}(\gamma _{i}-1)\geq 0 \ (p>1, i=1,2,\ldots ,m+n), h,k \) are positive constants;
\((A_{7})\) \(\alpha _{\ast}(x_{0}):=\min \{\min_{1\leq i\leq m}b_{i}(x_{0}), \min_{m+1\leq j\leq m+n}(b_{j}(x_{0}))\}\), \(\beta _{\ast}(y_{0}):=\min \{\min_{1\leq i\leq m}c_{i}(y_{0}), \min_{m+1 \leq j\leq m+n}(c_{j}(y_{0}))\}\);
\((A_{8})\) \(\max_{(t,s)\in \Omega _{0}}\psi (t,s)\leq \varphi ^{-1}((1+m+n)^{1- \frac{1}{q}}a(x_{0},y))\) and \(u\in C(\Omega _{0},R_{+})\).
For those \(\omega _{i}\) given in \((A_{4})\), we can define \(\tilde{\omega}_{i}(t)\) \((i=1,2,\ldots ,m+n, t>0)\) by
for \(i=1,2,\ldots ,m-1\) and
for \(j=m+1,\ldots ,m+n-1\), where
\(\varepsilon _{i}>0\) are very small constants.
Remark 1
If f and \(\omega _{i}(u)\ (i=1,2,\ldots ,m)\) given in \((A_{3})\) and \((A_{4})\) are nondecreasing and continuous functions and satisfy
then we define functions \(\tilde{\omega}_{i}(u):=\omega _{i}(u)\ (i=1,\ldots ,m)\), \(\tilde{\omega}_{j}(u):=\omega _{i}(f(u))\ (j=m+1,\ldots ,m+n)\).
To prove our results, we need the following lemmas.
Lemma 1
([27])
Suppose that \((B_{1})\)–\((B_{5})\) hold:
\((B_{1})\) \(\alpha _{i}(x):[x_{0},x_{1})\rightarrow [x_{0},x_{1})\ (i=1,2, \ldots ,m+n)\) and \(\beta _{i}(y):[y_{0},y_{1})\rightarrow [y_{0},y_{1})\ (i=1,2, \ldots ,m+n)\) are nondecreasing such that \(\alpha _{i}(x)\leq x\) on \([x_{0},x_{1})\), \(\beta _{i}(y)\leq y\) on \([y_{0},y_{1})\) and \(\beta _{i}(y_{0})=y_{0}\);
\((B_{2})\) All \(f_{i}\ (i=1,2,\ldots ,m+n)\) are continuous nonnegative functions on \(\Lambda \times [\alpha _{\ast}(x_{0}),x_{1})\times [y_{0},y_{1})\);
\((B_{3})\) \(g,\varphi :R_{+}\rightarrow R_{+}, \psi :[\alpha _{\ast}(x_{0})-h,x_{1}) \rightarrow R_{+}\) are continuous and φ is strictly increasing such that \(\lim_{t\rightarrow \infty}\varphi (t)=+\infty \);
\((B_{4})\) All \(\omega _{i}\ (i=1,2,\ldots ,m+n)\) are continuous on \(R_{+}\) and positive on \((0,+\infty )\);
\((B_{5})\) \(a(x,y)\) is a continuous and nonnegative function on Λ.
Thereinto, \(\Lambda :=[x_{0},x_{1}]\times [y_{0},y_{1}], \Omega :=[\alpha _{ \ast}(x_{0}),x_{0})\times [y_{0},y_{1})\), and \(x_{0}< x_{1}, y_{0}< y_{1}\) in \(R_{+}:=[0,\infty )\). \(\max_{s\in [\alpha _{\ast}(x_{0})-h,x_{0}]}\psi (s,y)\leq \varphi ^{-1}(a(x_{0},y))\) for all \(y\in [y_{0},y_{1})\) and \(u\in C(\Omega ,R_{+})\) satisfies the system of inequalities as follows:
Then
for all \((x,y)\in [x_{0},X_{1})\times [y_{0},Y_{1})\), where
\(u_{i}>0\) are given constants, \(\tilde{\omega}_{i}\) are defined in (2.1) and (2.2), and \(r_{i}(x,y)\) are defined recursively by
for \(i=1,2,\ldots ,m+n\), and \(X_{1}\in [x_{0},x_{1}), Y_{1}\in [y_{0},y_{1})\) are chosen such that
for \(i=1,2,\ldots ,m+n\).
Lemma 2
([14])
\(\alpha ,\beta ,\gamma \), and p are positive constants. Then
therein \(B[\xi ,\eta ]=\int ^{1}_{0}s^{\xi -1}(1-s)^{\eta -1}\,ds\ (\mathrm{Re}\xi >0,\mathrm{Re} \eta >0)\) and \(\theta =p[\alpha (\beta -1)+\gamma -1]+1\).
Theorem 2.1
Suppose that \((A_{1})\)–\((A_{8})\) hold, \((x,y)\in \Omega \cup \Omega _{0}\), \(u(x,y)\) satisfies the integral inequalities (1.9). Then we have
for all \((x,y)\in [x_{0},X_{1})\times [y_{0},Y_{1})\), where \(W_{i}^{-1}\) are the inverse of the functions
\(u_{i}>0\) are given constants, \(r_{i}(t)\) are defined by
and
\(\frac{1}{p}+\frac{1}{q}=1, p>1\), \(q>0\). \(pv_{i}(\gamma _{i}-1)+1>0, pk_{i}(\beta _{i}-1)+1>0\) and \(\frac{1}{p}+k_{i}\alpha _{i}(\beta _{i}-1)+v_{i}(\gamma _{i}-1)\geq 0\) for \(i=1,2,\ldots,m+n\). \(X_{1}\in [x_{0},x_{1}), Y_{1}\in [y_{0},y_{1})\) are the largest numbers such that
\(i=1,2,\ldots ,m+n\).
Proof
First of all, for those \(f, a(x,y)\) given in \((A_{3})\) and \((A_{5})\), we define \(\tilde{a}(x,y)\) by (2.6) and
By \((A_{4})\) and Remark 1, the functions \(W_{i}\) are strictly increasing. Therefore we know that \(W_{i}^{-1}\) are continuous and increasing functions in their domain. The sequence \(\{\tilde{\omega}_{i}(t)\}\) defined by \(\omega _{i}(t)\) is nondecreasing nonnegative functions on \(R_{+}\) and satisfies
Since the ratios \(\frac{\tilde{\omega}_{i+1}(t)}{\tilde{\omega}_{i}(t)}\ (i=1,2, \ldots ,m+n)\) are all nondecreasing, we have \(\tilde{\omega}_{i}(t)\propto \tilde{\omega}_{i+1}(t)\ (i=1,2,\ldots ,m+n)\).
Furthermore, \(\hat{g}_{i}(x,y,t,s)\) defined by (2.9) are nondecreasing in \(x, y\) for each fixed \(t, s\) and satisfy \(\hat{g}_{i}(x,y,t,s)\geq g_{i}(x,y,t,s)\geq 0\) for all \(i=1,2,\ldots ,m+n\). We have \(\tilde{a}(x,y)\geq a(x,y)\) and \(\hat{g}_{i}(x,y,t,s)\geq g_{i}(x,y,t,s)\), and they are continuous and nondecreasing in \(t, s\). From the monotonicity of \(\tilde{f}(u)\), we obtain the inequality
for \((x,y)\in \Omega \cup \Omega _{0}\).
From (1.9), (2.6), (2.9), (2.12), (2.13), and (2.14), we obtain
Let \(\frac{1}{p}+\frac{1}{q}=1, p>1\), then \(q>0\). Since \(pv_{i}(\gamma _{i}-1)+1>0, pk_{i}(\beta _{i}-1)+1>0\) and \(\frac{1}{p}+k_{i}\alpha _{i}(\beta _{i}-1)+v_{i}(\gamma _{i}-1)\geq 0\) for \(i=1,2,\ldots,m+n\). By Lemma 2 and Holder’s inequality, we get
for \((x,y)\in \Omega \), where \(M_{i}\) and \(\theta _{i}\) are defined by (2.10) and (2.11), \(i=1,2,\ldots ,m+n\).
By Jensen’s inequality and (2.16), we get for \((x,y)\in \Omega \)
By (2.5), (2.8), and (2.17), we have
Concerning (2.18), we consider the auxiliary system of inequalities
for all \((x,y)\in [x_{0},X)\times [y_{0},Y)\), where X and Y are chosen arbitrarily such that \(x_{0}\leq X\leq X_{1}, y_{0}\leq Y\leq Y_{1}\).
Since
we get
Now we can define the function
Obviously, \(z(x,y)\) is nondecreasing.
From (2.21) and (2.22) we have
Let \(e(z):=\varphi ^{-1}(z^{\frac{1}{q}})\) and \(e(z)\) is a continuous and nondecreasing function on \(R_{+}\). Thus, \(\tilde{w}_{i}(e(z))\) is continuous and nondecreasing on \(R_{+}\ (i=1,2,\ldots ,m+n)\), \(\tilde{w}_{i}(e(z))>0\) for \(z>0\).
Since \(\tilde{w}_{i}(z)\propto \tilde{w}_{i+1}(z)\), we get that \(\frac{\tilde{w}_{i+1}(e(z))}{\tilde{w}_{i}(e(z))}\) is also continuous and nondecreasing on \(R_{+}\). So we obtain \(\tilde{w}_{i}^{q}(e(z))\propto \tilde{w}_{i+1}^{q}(e(z)), i=1,2,3, \ldots ,m+n-1\). By (2.23), we let \(\varphi (u(x,y))=z(x,y), a(x,y)=r_{1}(X,Y), f_{i}(x,y,s,t)= \tilde{g}_{i}(X,Y,t,s), \omega _{i}(u(s,t))=\tilde{\omega}_{i}(e(z))\), applying Lemma 1, we have
for all \(x_{0}\leq x\leq \min \{X,X_{2}\}\) and \(y_{0}\leq y\leq \min \{Y,Y_{2}\}\), where \(W_{m+n}\) is defined in (2.4),
\(X_{2}\leq x_{1}, Y_{2}\leq y_{1}\) are the largest numbers such that
\(i=1,2,\ldots ,m+n\).
It follows from (2.21) and (2.24) that we have
for all \(x_{0}\leq x\leq \min \{X,X_{2}\}\) and \(y_{0}\leq y\leq \min \{Y,Y_{2}\}\).
Let \(x=X, y=Y, X_{2}=X_{1}, Y_{2}=Y_{1}\), we have
for all \(x_{0}\leq X\leq X_{1}\) and \(y_{0}\leq Y\leq Y_{1}\).
It is easy to obtain \(r_{m+n}(X,Y,X,Y)=r_{m+n}(X,Y)\). So (2.26) can be restated as
Because \(X,Y\) are arbitrary, we can replace X and Y with x and y. Thus we get
for all \((x,y)\in [x_{0},X_{1})\times [y_{0},Y_{1})\).
The proof is complete. □
Corollary 2.1
Suppose that \((A_{1})\)–\((A_{8})\) hold if \(u(x,y)\) satisfy the following inequality:
where \(u\in C(\Omega \cup \Omega _{0},R_{+})\) and \(c\geq 0\) is a constant. Then
for all \((x,y)\in [x_{0},X_{1})\times [y_{0},Y_{1})\), where \(\bar{r}_{i}(x,y)\) is defined by \(\bar{r}_{1}(x,y):=\varphi ^{q}(M)\) and
\(\frac{1}{p}+\frac{1}{q}=1, p>1, q>0\). \(pv_{i}(\gamma _{i}-1)+1>0, pk_{i}(\beta _{i}-1)+1>0\) and \(\frac{1}{p}+k_{i}\alpha _{i}(\beta _{i}-1)+v_{i}(\gamma _{i}-1)\geq 0\) for \(i=1,2,\ldots,m+n. X_{1}< x_{1}\), \(Y_{1}< y_{1}\) are the largest numbers such that
\(W_{i}\) is defined in (2.4) and \(W_{i}^{-1}\) is the inverse of \(W_{i}\). \(\tilde{\omega}_{i}\) is defined in (2.1) and (2.2). \(\tilde{g}_{i}\) is defined in (2.7).
Proof
By (2.29) and the definition of M, we have
We choose \(a(x,y)=(1+m+n)^{\frac{1}{q-1}}\varphi (M)\), then (2.33) can be converted to (2.30) by Theorem 2.1.
The proof is complete. □
3 Applications
In this section, we apply the results to study the boundedness of the solutions for an integral equation with the maxima.
Example 1
Consider the system of integral equations with maxima.
\(\psi \in C([\hat{\alpha}(x_{0})-h,x_{0})\times [\hat{\beta}(y_{0})-k,y_{0}),R)\), \(x_{0}\geq 0, y_{0}\geq 0, h>0\).
Suppose that \((C_{1})\) \(|f_{i}(x,y,t,s,u)|\leq g_{i}(x,y,t,s)\omega _{i}(|u|)\), \(\omega _{i}\) are continuous positive and nondecreasing functions on \(R_{+}\ (i=1,2)\), \(\omega _{1}\propto \omega _{2}\), \(g_{i}(x,y,t,s)\) is nondecreasing in \((x,y)\) for each fixed \((t,s)\), \(\beta _{i}\in (0,1), \gamma _{i}>1-\frac{1}{p}, \frac{1}{p}+\beta _{i}+ \gamma _{i}-2\geq 0\ (p>1, i=1,2)\);
\((C_{2})\) \(\bar{\alpha}(x),\hat{\alpha}(x)\in C^{1}([x_{0},\infty ),R_{+}), \bar{\beta}(y),\hat{\beta}(y)\in C^{1}([y_{0},\infty ),R_{+}), \bar{\alpha}(x),\hat{\alpha}(x),\bar{\beta}(y),\hat{\beta}(y)\) are nondecreasing, \(\hat{\alpha}(x)\leq x, \bar{\alpha}(x)\leq x, 0<\hat{\alpha}(x)- \bar{\alpha}(x)\leq h\) for \(x\geq x_{0}\) and \(\bar{\beta}(y)\leq y, \hat{\beta}(y)\leq y, 0<\hat{\beta}(y)- \bar{\beta}(y)\leq k\) for \(y\geq y_{0}\);
\((C_{3})\) \(a(x,y)\) is continuous on \([x_{0},\infty )\times [y_{0},\infty )\);
\((C_{4})\) \(\max_{(\eta ,\xi )\in [\bar{\alpha}(t),\hat{\alpha}(t)]\times [ \bar{\beta}(s),\hat{\beta}(s)]}|\psi (\eta ,\xi )|\leq 3^{1- \frac{1}{q}}|a(x,y)|\).
Then we give an estimate for the solutions of (3.1).
Theorem 3.1
Suppose that \((C_{1})\)–\((C_{4})\) hold, then from (3.1) we have
for \((x,y)\in [x_{0},X_{1}]\times [y_{0},Y_{1}]\), where
\(\tilde{a}(x,y)\) is a continuous and nondecreasing function on \([x_{0},\infty )\times [y_{0},\infty )\).
\(W_{i}^{-1}\) are the inverse of the functions
\(X_{1},Y_{1}\) are the largest numbers such that
Proof
From (3.1) and \((C_{1})\), we have
Let \(z(x,y)=|u(x,y)|\) for \((x,y)\in [\hat{\alpha}(x_{0})-h,\infty )\times [\hat{\beta}(y_{0})-k, \infty )\), then we have
\(z(x,y)=|\psi (x,y)|, (x,y)\in [\hat{\alpha}(x_{0})-h,x_{0})\times [ \hat{\beta}(y_{0})-k,y_{0})\).
From condition \((C_{2})\), we have
By (3.4), we have
\(z(x,y)=|\psi (x,y)|, (x,y)\in [\hat{\alpha}(x_{0})-h,x_{0})\times [ \hat{\beta}(y_{0})-k,y_{0})\).
By condition \((C_{4})\), we can obtain
Compare with (1.9), we let \(\psi (u(x,y))=z(x,y), a(x,y)=\tilde{a}(x,y), m=n=1, b_{i}(x)=x\ (i=1,2), c_{i}(y)=y\ (i=1,2), \alpha _{i}=1\ (i=1,2), k_{i},v_{i}=1\ (i=1,2)\), applying Theorem 2.1, from (3.5) we obtain (3.2).
Then the proof is complete. □
Example 2
Consider the system of integral inequations with maxima:
where
\(h, k\) are constants, \(\psi (x,y)\in C(\Omega _{0},R)\).
Theorem 3.2
\(u(x,y)\) satisfies the integral inequalities (3.6), \((x,y)\in \Omega \cup \Omega _{0}\), we let \(p=\frac{3}{2}, q=3\).
Then we have
for all \((x,y)\in [0,1)\times [0,1)\).
Proof
Compare with (1.9), from (3.6), we let \(\varphi (u)=u, m=n=1, b_{i}(x)=x\ (i=1,2), c_{i}(y)=y\ (i=1,2), \alpha _{i}=1\ (i=1,2), v_{i}=0\ (i=1,2), k_{i}=1\ (i=1,2), \beta _{i}=\frac{1}{2}\ (i=1,2), x_{0}=0,\ y_{0}=0, a(x,y)=xy\).
Setting \(p=\frac{3}{2}, q=3\), we get
From the definition
we have
Applying Theorem 2.1, we obtain
We have \(\tilde{\omega}_{i}(u)=\omega _{i}(u)=u, i=1,2\). So we obtain
Then we have (3.8) holds for all \((x,y)\in [0,1)\times [0,1)\).
So the proof is complete. □
More generally, consider the system of differential equations with maxima:
where \(\lambda (0<\lambda <1), x_{0}\geq 0, y_{0}\geq 0, h,k>0\) are constants, \(\psi \in C(\Omega _{0},R), F\in C(\Omega \times R^{2},R), f\in C([x_{0},x_{1}),R), g\in C([y_{0},y_{1}),R), f(x_{0})=g(y_{0})\).
System (3.9) is more generalized than system (3.6). By Theorem 2.1, we can estimate solutions for the nonlinear equation. By analogy with the equation considered in Sect. 3, Corollary 3.2 of [23], we can prove that system (3.9) has at most one solution on Ω.
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References
Gronwall, T.H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20(4), 292–296 (1919)
Bellman, R.: The stability of solutions of linear differential equations. Duke Math. J. 10, 643–647 (1943)
Bihari, I.: A generalization of a lemma of Bellman and its applications to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hung. 7, 81–94 (1956)
Medved, M.: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. Appl. 214, 349–366 (1997)
Agarwal, R.P., Ryoo, C.S., Kim, Y.H.: New integral inequalities for iterated integrals with applications. J. Inequal. Appl. 2007, Article ID 024385 (2007)
Abdeldaim, A.: On some new integral inequalities of Gronwall–Bellman–Pachpatte type. Appl. Math. Comput. 217(20), 7887–7899 (2011)
Meng, F.W., Shao, J.: Some new Volterra–Fredholm type dynamic integral inequalities on time scales. Appl. Math. Comput. 223(3), 444–451 (2013)
Ma, Q.H., Pec̆arié, J.: Estimates on solutions of some new nonlinear retarded Volterra–Fredholm type integral inequalities. Nonlinear Anal., Theory Methods Appl. 69(2), 393–407 (2008)
Hou, Z.Y., Wang, W.S.: A class of nonlinear retarded Volterra–Fredholm type integral inequality and its application. Math. Pract. Theory 44, 21 (2014)
Lu, Y.S., Wang, W.S., Zhou, X.L., Hang, Y.: Generalized nonlinear Volterra–Fredholm type integral inequality with two variables. J. Appl. Math. 2014, Article ID 359280 (2014)
Hou, Z.Y., Wang, W.S.: A class of nonlinear Volterra–Fredholm type integral inequality with variable lower limit and its application. J. Southwest China Norm. Univ. 41, 2 (2016)
Huang, C.M., Wang, W.S.: A class of nonlinear Volterra–Fredholm type integral inequality with maxima. J. Sichuan Normal Univ. 39, 3 (2016)
Ma, Q.H., Pec̆arié, J.: Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations. J. Math. Anal. Appl. 341, 894–905 (2008)
Ma, Q.H., Yang, E.H.: Estimates on solutions of some weakly singular Volterra integral inequalities. Acta Math. Appl. Sin. 25(3), 505–515 (2002)
Xu, R., Meng, F.W.: Some new weakly singular integral inequalities and their applications to fractional differential equations. J. Inequal. Appl. 2016, Article ID 78 (2016)
Xu, R.: Some new nonlinear weakly singular integral inequalities and their applications. J. Math. Inequal. 11(4), 1007–1018 (2017)
Wang, T.L., Xu, R.: Some integral inequalities in two independent variables on time scales. J. Math. Inequal. 6(1), 107–118 (2012)
Wang, T.L., Xu, R.: Bounds for some new integral inequalities with delay on time scales. J. Math. Inequal. 6(1), 1–12 (2012)
Xu, R., Meng, F.W., Song, C.H.: On some integral inequalities on time scales and their applications. J. Inequal. Appl. 2010), Article ID 464976 (2010)
Du, L.W., Xu, R.: Some new Pachpatte type inequalities on time scales and their applications. J. Math. Inequal. 6(2), 229–240 (2012)
Wan, L.L., Xu, R.: Some generalized integral inequalities and there applications. J. Math. Inequal. 7(3), 495–511 (2013)
Xu, R., Zhang, Y.: Generalized Gronwall fractional summation inequalities and their applications. J. Inequal. Appl. 2015, 242 (2015)
Yan, Y.: On some new weakly singular Volterra integral inequality with maxima and their applications. J. Inequal. Appl. 369, 16 (2015)
Hristova, S., Stafanova, K.: Linear integral inequalities involving maxima of the unknown scalar functions. Funkc. Ekvacioj 53, 381–394 (2010)
Henderson, J., Hristova, S.: Nonlinear integral inequalities involving maxima of unknown scalar functions. Math. Comput. Model. 53, 871–882 (2011)
Bohner, M., Hristova, S., Stefanova, K.: Nonlinear integral inequalities involving maxima of the unknown scalar functions. Math. Inequal. Appl. 12, 811–825 (2012)
Yan, Y.: Nonlinear Gronwall–Bellman type integral inequalities with maxima. Math. Inequal. Appl. 16, 911–928 (2013)
Thiramanus, P., Tariboon, J., Ntouyas, S.: Henry–Gronwall integral inequalities with maxima and their applications to fractional differential equations. Abstr. Appl. Anal. 2014, Article ID 276316 (2014)
Xu, R., Ma, X.T.: Some new retarded nonlinear Volterra–Fredholm type integral inequality with maxima in two variables and their applications. J. Inequal. Appl. 187, 25 (2017)
Wang, W.S.: A generalized retarded Gronwall-like inequality in two variables and applications to BVP. Appl. Math. Comput. 191(1), 144–154 (2007)
Hong, W., Kelong, Z.: Some nonlinear weakly singular integral inequalities with two variables and applications. J. Inequal. Appl. 2010, Article ID 345701 (2010)
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YS carried out the bounds of solutions to retarded nonlinear Volterra type integral equations and completed the corresponding proof. RX participated in the proof of the theorem and examples. All authors read and approved the final manuscript.
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Sun, Y., Xu, R. Some weakly singular Volterra integral inequalities with maxima in two variables. J Inequal Appl 2023, 36 (2023). https://doi.org/10.1186/s13660-023-02939-9
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DOI: https://doi.org/10.1186/s13660-023-02939-9
MSC
- 39A12
- 26A33
Keywords
- Volterra type
- Integral inequalities
- Maxima
- Two variables
- Integral equations