Abstract
In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations.
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1 Introduction
Gronwall-Bellman inequality [1, 2] and Bihari inequality [3] provided important devices in the study of existence, uniqueness, boundedness, oscillation, stability, invariant manifolds and other qualitative properties of solutions to differential equations, integral equations and integro-differential equations. In the past few decades, a number of studies have focused on generalizations of the Gronwall-Bellman inequality. For example, in [4–10], the Gronwall-Bellman-Gamidov type integral inequalities and their generalizations were studied; in [11–14], the Gronwall-like inequalities and their deformations were investigated; in [15–18], the Volterra type iterated inequalities were discussed; in [19–24], the Volterra-Fredholm type inequalities were examined.
The Gronwall-Bellman inequality can be stated as follows.
If u and f are nonnegative continuous functions on an interval \([a,b]\), and u satisfies the following inequality:
where \(c\geq0\) is a constant. Then
In 2004, Pachpatte [6] investigated the retarded linear Volterra-Fredholm type integral inequality in two independent variables:
In 2010, Wang [14] investigated a retarded Volterra type integral inequality with two variables:
In 2014, Lu et al. [21] studied the nonlinear retarded Volterra-Fredholm type iterated integral inequality:
In 2016, Huang and Wang [23] discussed the retarded nonlinear Volterra-Fredholm type integral inequality with maxima:
Motivated by the work presented in [14, 21, 23], we establish some new retarded nonlinear Volterra-Fredholm type integral inequality with maxima in two independent variables in this paper:
and
By the amplification method, differential and integration, and the inverse function, we obtain the lower bound estimation of the unknown function. The example is given to illustrate the application of our results.
2 Main results
In what follows, R denotes the set of real numbers, \(R_{+}=[0,+\infty)\), \(I_{1}=[M,+\infty)\), \(I_{2}=[N,+\infty)\) are the given subsets of R, \(\Delta=I_{1}\times I_{2} \). \(C^{1}(\Omega,S)\) denotes the class of all continuously differentiable functions defined on set Ω with range in the set S, \(C(\Omega,S)\) denotes the class of all continuous functions defined on set Ω with range in the set S, and \(\alpha'(t)\) denotes the derived function of a function \(\alpha(t)\). For convenience, we cite some useful lemmas in the discussion of our proof as follows:
Lemma 2.1
See [25]
Let \(u(t)\), \(a(t)\), \(b(t)\) be nonnegative and continuous functions defined for \(t\in R_{+}\). Assume that \(a(t)\) is non-increasing function for \(t\in R_{+}\). If
then
From Lemma 2.1, we can get the generalization in two dimensions.
Lemma 2.2
Let \(u(x,y)\), \(a(x,y)\), \(b(x,y)\) be nonnegative and continuous functions defined for \((x,y)\in\Delta\). Assume that \(a(x,y)\) is a non-increasing function in the first variable. If
then
Lemma 2.3
See [26]
Assume that \(a\geq0\), \(p\geq q\geq 0\), and \(p\neq0\). Then, for any \(K>0\),
Theorem 2.1
Suppose that the following conditions hold:
-
(i)
\(\psi\in C(R_{+},R_{+})\) is an increasing function and \(\psi(u)>0\), \(\forall u>0\), \(\psi(\infty)=\infty\). \(\psi^{-1}\) is the inverse function of ψ. \(\varphi_{1}\), \(\varphi_{2}\), \(\varphi_{2}/\varphi_{1}\in C(R_{+},R_{+})\) are increasing functions with \(\varphi_{i}(u)>0\) (\(i=1,2\)) for \(u>0\). \(\psi^{-1}\), \(\varphi_{i}\) (\(i=1,2\)) are sub-multiplicative and sub-additive, that is,
$$\begin{gathered} \psi^{-1}(\alpha\beta)\leq \psi^{-1}( \alpha)\psi^{-1}(\beta),\qquad \psi^{-1}(\alpha+\beta)\leq \psi^{-1}(\alpha)+\psi^{-1}(\beta), \\ \varphi_{i}(\alpha\beta)\leq \varphi_{i}(\alpha) \varphi_{i}(\beta),\qquad \varphi_{i}(\alpha+\beta)\leq \varphi_{i}(\alpha)+\varphi_{i}(\beta), \quad \alpha,\beta \in R_{+}; \end{gathered} $$ -
(ii)
\(k(x,y), a(x,y)\in C(\Delta,R_{+})\) and \(k(x,y)\) is non-increasing in the first variable;
-
(iii)
\(b_{i}(s,t,x,y), c_{i}(s,t,x,y), d_{j}(s,t,x,y), e_{j}(s,t,x,y)\in C(\Delta^{2},R_{+})\) for \(i=1,2,\ldots,l_{1}\); \(j=1,2,\ldots,l_{2}\); \(b_{i}\), \(c_{i}\), \(d_{j}\), \(e_{j}\) are all non-increasing functions in the last two variables;
-
(iv)
\(\alpha, \alpha_{i}, \alpha_{j}\in C(I_{1},I_{1})\), \(\beta_{i}, \beta_{j}\in C(I_{2},I_{2})\) are non-decreasing functions with \(\alpha(x), \alpha_{i}(x), \alpha_{j}(x)\geq x\), \(\beta_{i}(y), \beta_{j}(y)\geq y\) (\(i=1,2,\ldots,l_{1}\); \(j=1,2,\ldots,l_{2}\)). \(h\geq1\) is a constant;
-
(v)
the function \(u\in C(\Delta,R_{+})\) satisfies the inequality
$$\begin{aligned} \psi \bigl(u(x,y) \bigr)&\leq k(x,y)+ \int_{\alpha(x)}^{\infty}a(s,y)\psi \bigl(u(s,y) \bigr)\,ds \\ &\quad {} +\sum _{i=1}^{l_{1}} \int_{\alpha_{i}(x)}^{\infty} \int_{\beta_{i}(y)}^{\infty} \biggl[b_{i}(s,t,x,y) \varphi_{1} \bigl(u(s,t) \bigr) \\ &\quad{}+ \int_{s}^{\infty} \int_{t}^{\infty}c_{i}(\xi,\eta,x,y) \varphi_{2} \Bigl(\max_{\sigma\in [\xi,h\xi]} u(\sigma,\eta) \Bigr)\,d \xi \,d \eta \biggr]\,ds\,dt \\ &\quad {} +\sum_{j=1}^{l_{2}} \int_{\alpha_{j}(M)}^{\infty} \int_{\beta_{j}(N)}^{\infty} \biggl[d_{j}(s,t,x,y)\psi \bigl(u(s,t) \bigr) \\ &\quad {}+ \int_{s}^{\infty} \int_{t}^{\infty}e_{j}(\xi,\eta,x,y)\psi \Bigl( \max_{\sigma\in [\xi,h\xi]} u(\sigma,\eta) \Bigr)\,d\xi \,d\eta \biggr] \,ds\,dt, \\ &\quad (x,y)\in\Delta, \end{aligned}$$(2.1)
then we have
where
on condition that \(W_{1}(+\infty)=+\infty\), \(W_{2}(+\infty)=+\infty\), and \(G(u)\) is a strictly increasing function on \(R_{+}\). We have
Proof
Let
Obviously, \(z(x,y)\) is non-increasing in each of the variables. From (2.1), we have
Applying Lemma 2.2, we obtain
i.e.
where \(A(x,y)\) is defined in (2.6), and \(A(x,y)\) is non-increasing in the first variable. So we have
By (2.11), (2.14), (2.15), and condition (i), we deduce
where \(B(M,N)\) is defined in (2.7), and \(C(M,N)\) is defined as follows:
\(\forall X\in I_{1}\), \(Y\in I_{2}\), for all \((x,y)\in[X,\infty)\times[Y,\infty)\), we have
Let \(z_{1}(x,y)\) denote the function on the right-hand side of (2.18), which is positive and non-increasing in each of the variables \((x,y)\in[X,\infty)\times[Y,\infty)\). From (2.18), we have
Differentiating \(z_{1}(x,y)\) with respect to x, we have
By the monotonicity of \(\varphi_{1}\), \(\varphi_{2}\), \(z_{1}\) and the property of \(\alpha_{i}\), \(\beta_{i}\), from (2.21), we get
Replacing x with s, and integrating it from x to ∞, we obtain
i.e.
where \(E(X,Y)\) is defined in (2.9). Let \(z_{2}(x,y)\) denote the function on the right-hand side of (2.24), which is positive and non-increasing in each of the variables \((x,y)\in[X,\infty)\times[Y,\infty)\). From (2.24), we have
Differentiating \(z_{2}(x,y)\) with respect to x, we have
By the monotonicity of \(\varphi_{2}/\varphi_{1}\) and \(z_{2}\), from (2.27), we obtain
Replace x with s, and integrating it from x to ∞, we get
where
Obviously, \(F(x,y,x,y)=F(x,y)\), which is defined in (2.10). From (2.19), (2.20), (2.25), (2.26) and (2.29), we have
Since X and Y are chosen arbitrarily, we have
By the definition of \(C(M,N)\) and (2.19), we get
or
where \(D(M,N)\) is defined in (2.8). By (2.5) and the hypothesis of G, we obtain
Combining (2.31), (2.34) and (2.14), we get the desired result. □
Corollary 2.1
Let the functions k, a, α, \(b_{i}\), \(c_{i}\), \(\alpha_{i}\), \(\beta_{i}\) (\(i=1,2,\ldots,l_{1}\)), \(d_{j}\), \(e_{j}\), \(\alpha_{j}\), \(\beta_{j}\) (\(j=1,2,\ldots,l_{2}\)) and u be defined as in Theorem 2.1, p is a positive constant and \(p\geq1\). If the function \(u(x,y)\) satisfies the inequality,
then: (i) if \(p>1\), we have
where
on condition that \(G_{1}(u)\) is a strictly increasing function on \(R_{+}\).
(ii) If \(p=1\), we have
where
and
Proof
Inequality (2.35) followed by letting \(\psi(u(x,y))=u^{p}(x,y)\), \(\varphi_{1}(u(x,y))=\varphi_{2}(u(x,y))=u(x,y)\) in Theorem 2.1. Then \(\psi^{-1}(u(x,y))=u^{\frac{1}{p}}(x,y)\) and \((u+v)^{\frac{1}{p}}\leq u^{\frac{1}{p}}+v^{\frac{1}{p}}\), \((uv)^{\frac{1}{p}}= u^{\frac{1}{p}}v^{\frac{1}{p}}\).
If \(p>1\), we have
Applying Theorem 2.1, we can easily get (2.36).
If \(p=1\), we have
Obviously, \(G_{2}(u)\) is a strictly increasing function on \(R_{+}\), \(G_{2}^{-1}(u)\) is the inverse of \(G_{2}(u)\), we get
where \(\overline{B}(M,N)\) is defined in (2.42). Applying Theorem 2.1, we can easily get (2.40). Details are omitted here. □
Theorem 2.2
Suppose that the following conditions hold:
-
(i)
(ii)-(iv) of Theorem 2.1 are satisfied;
-
(ii)
\(q_{i}\), \(r_{i}\) are nonnegative constants with \(p\geq q_{i}\), \(p\geq r_{i}\), \(i=1,2,\ldots,l_{1}\), and \(\varepsilon_{j}\), \(\delta_{j}\) are nonnegative constants with \(p\geq\varepsilon_{j}\), \(p\geq\delta_{j}\), \(j=1,2,\ldots,l_{2}\).
If \((x,y)\in\Delta\), \(u(x,y)\) satisfies the following inequality:
then we have
where
Proof
Let
Obviously, \(z(x,y)\) is non-increasing in every variable. From (2.44) and (2.49), we have
By Lemma 2.2, we obtain
where \(A(x,y)\) is defined in (2.6). Then we get
By Lemma 2.3, we have
Combining (2.53) and (2.49), we have
where \(B_{1}(M,N)\) is defined in (2.46), \(C_{1}(M,N)\) is defined as follows:
\(\forall X\in I_{1}\), \(\forall Y\in I_{2}\), for all \((x,y)\in[X,\infty)\times[Y,\infty)\), we have
Let \(z_{1}(x,y)\) denote the function on the right-hand side of (2.56), which is positive and non-increasing in each of the variables \((x,y)\in[X,\infty)\times[Y,\infty)\). From (2.56), we have
Differentiating \(z_{1}(x,y)\) with respect to x, we have
Dividing both sides of (2.59) by \(z_{1}(x,y)\), noticing that \(z_{1}(x,y)\) is non-increasing in each variable, we have
Replace x with s, and integrate it from x to ∞, we get
where
It is obvious that \(F_{1}(x,y,x,y)=F_{1}(x,y)\), which is defined in (2.47). From (2.57), (2.58) and (2.61), we get
Due to the fact that X, Y are chosen arbitrarily, we have
By the definition of \(C_{1}(M,N)\), we have
where \(D_{1}(M,N)\) is defined in (2.48). Then, according to \(D_{1}(M,N)<1\), we have
From (2.64) and (2.66), we get
Combining (2.52) and (2.67), we obtain the desired result. □
Remark 2.1
If \(q_{i}=r_{i}=1\) (\(i=1,2,\ldots,l_{1}\)), \(\varepsilon_{j}=\delta_{j}=p\) (\(j=1,2,\ldots,l_{2}\)), the inequality (2.44) becomes (2.35), but the proof of Theorem 2.2 is different from that of Corollary 2.1.
Corollary 2.2
Let k, a, α, \(\alpha_{i}\), \(\beta_{i}\), \(b_{i}\), \(c_{i}\) (\(i=1,2,\ldots,l_{1}\)), \(\alpha_{j}\), \(\beta_{j}\), \(d_{j}\), \(e_{j}\) (\(i=1,2,\ldots,l_{2}\)) be defined as in Theorem 2.1, then q, r are nonnegative constants with \(0\leq q\leq 2\), \(0\leq r\leq 2\). For \((x,y)\in\Delta\), \(u(x,y)\) satisfies the following inequality:
then we have
where
Proof
Inequality (2.68) follows by inequality (2.44) with \(p=2\), \(q_{i}=q\), \(r_{i}=r\) (\(i=1,2,\ldots,l_{1}\)), \(\varepsilon_{j}=\delta_{j}=1\) (\(j=1,2,\ldots,l_{2}\)). Then, applying Theorem 2.2, we can easily get (2.69). Details are omitted here. □
Remark 2.2
As one can see, the established results above mainly deal with Volterra-Fredholm type integral inequalities with maxima in two variables. And they are different from the results presented in [14, 21, 23]. In Theorem 2.1, in the case of one variable, if we take \(k(x,y)=k\), \(a(x,y)=0\), \(l_{1}=l_{2}=1\), \(b_{1}(s,t,x,y)=d_{1}(s,t,x,y)=h_{1}(s)\), \(c_{1}(\xi,\eta,x,y)=e_{1}(\xi,\eta,x,y)=h_{2}(\xi)\), \(\psi(u)=\varphi_{1}(u)\), \(\psi(u)=\varphi_{2}(u)\) in the second iterated integral, orderly, we will get the inequality that is similar to inequality (1.5). If the above conditions are satisfied in two dimensions and \(\varphi_{2}(\max_{\sigma\in[\xi,h\xi]}u(\sigma,\eta))=\varphi_{2}(u(\xi,\eta))\), we get analogs of the inequality (1.4). And if we take \(l_{1}=2\), \(l_{2}=0\), \(b_{i}(s,t,x,y)=f_{i}(s,t)\), \(c_{i}(\xi,\eta,x,y)=0\) in Theorem 2.1, inequality (2.1) reduces to (1.3).
3 Applications in the integral equation
In this section, we apply our results in Theorem 2.1 and Theorem 2.2 to study the retarded Volterra-Fredholm type integral equations with maxima in two variables. Some results on the boundedness of their solutions are presented, which demonstrate that our results can be used to investigate the qualitative properties of solutions of some integral equations.
Example
We consider the retarded Volterra-Fredholm type integral equation of the form
Suppose that the following conditions hold:
-
(i)
\(g_{1}(x,y)\), \(g_{2}(x,y)\), \(v(x,y)\in C(\Delta,R)\);
-
(ii)
\(x+\rho(x)\), \(x+\rho_{i}(x)\), \(x+\rho_{j}(x)\in C^{1}(I_{1},I_{1})\) and \(y+\gamma_{i}(y)\), \(y+\gamma_{j}(y)\in C^{1}(I_{2},I_{2})\) are strictly increasing with
$$\begin{gathered} \rho(M)=\rho_{i}(M)=\rho_{j}(M)=0,\qquad \gamma_{i}(N)=\gamma_{j}(N)=0, \\ \rho(x)\geq 0,\qquad \rho_{i}(x)\geq 0,\qquad \rho_{j}(x) \geq 0\quad \mbox{for }x\geq M, \\ \gamma_{i}(y)\geq 0,\qquad \gamma_{j}(y)\geq 0\quad \mbox{for }y\geq N, \\ \rho'(x)>-1,\qquad \rho_{i}'(x)>-1,\qquad \rho_{j}'(x)>-1,\\ \gamma_{i}'(y)>-1, \qquad \gamma_{j}'(y)>-1\quad (i=1,2, \ldots,l_{1}; j=1,2,\ldots,l_{2}); \end{gathered} $$ -
(iii)
\(F_{1i}, G_{1j}\in C(\Delta^{2}\times R^{2},R)\), \(F_{2i}, G_{2j}\in C(\Delta^{2}\times R,R)\) (\(i=1,2,\ldots,l_{1}\); \(j=1,2,\ldots,l_{2}\)).
Let \(\alpha(x)=x+\rho(x)\), \(\alpha_{i}(x)=x+\rho_{i}(x)\), \(\alpha_{j}(x)=x+\rho_{j}(x)\), \(\beta_{i}(y)=y+\gamma_{i}(y)\), \(\beta_{j}(y)=y+\gamma_{j}(y)\). Then α, \(\alpha_{i}\), \(\alpha_{j}\), \(\beta_{i}\), \(\beta_{j}\) satisfy the condition (iv) of Theorem 2.1.
Theorem 3.1
In Eq. (3.1), suppose that the following conditions hold:
where k, a, \(b_{i}\), \(c_{i}\), \(d_{j}\), \(e_{j}\), \(\varphi_{1}\), \(\varphi_{2}\) are defined in Theorem 2.1. Assume that the function \(G_{3}(u)=W_{2} (W_{1}(\frac{u}{D_{3}(M,N)}) )-W_{2} (W_{1}(B_{3}(M,N)+u)+E_{3}(M,N) )\) is increasing. Then we have the following estimate:
where
\(W_{1}\), \(W_{2}\) are defined in Theorem 2.1.
Proof
By applying the conditions (3.2) to (3.1), we have
for \((x,y)\in\Delta\), where \(M_{1}\), \(M_{1i}\), \(M_{2i}\) (\(i=1,2,\ldots,l_{1}\)), \(M_{1j}\), \(M_{2j}\) (\(j=1,2,\ldots,l_{2}\)) are defined in (3.9). Applying the results of Theorem 2.1 to (3.10) with \(\psi(u)=u\), \(a(s,y)=M_{1}a(\alpha^{-1}(s),y)\), \(b_{i}(s,t,x,y)=M_{1i}M_{2i}b_{i}(\alpha_{i}^{-1}(s),\beta_{i}^{-1}(t),x,y)\), \(c_{i}(\xi,\eta,x,y)=M_{1i}^{2}M_{2i}^{2}c_{i}(\alpha_{i}^{-1}(\xi),\beta_{i}^{-1}(\eta),x,y)\), \(d_{j}(s,t,x,y)=M_{1j}M_{2j}d_{j}(\alpha_{j}^{-1}(s),\beta_{j}^{-1}(t),x,y)\), \(e_{j}(\xi,\eta,x,y)=M_{1j}^{2}M_{2j}^{2}e_{j}(\alpha_{j}^{-1}(\xi),\beta_{j}^{-1}(\eta),x,y)\), we obtain the desired estimation (3.3). □
Theorem 3.2
In equation (3.1), suppose that the following conditions hold:
where p, \(q_{i}\), \(r_{i}\), \(\varepsilon_{j}\), \(\delta_{j}\), \(b_{i}\), \(c_{i}\), \(d_{j}\), \(e_{j}\) (\(i=1,2,\ldots,l_{1}\); \(j=1,2,\ldots,l_{2}\)) are defined as in Theorem 2.2. Then we have the following estimate:
where
Proof
Applying the conditions of (3.11) to (3.1), we have
for \((x,y)\in\Delta\), where \(M_{1}\), \(M_{1i}\), \(M_{2i}\) (\(i=1,2,\ldots,l_{1}\)), \(M_{1j}\), \(M_{2j}\) (\(j=1,2,\ldots,l_{2}\)) are defined in (3.9). Applying the results of Theorem 2.2 to (3.16) with \(a(s,y)=M_{1}a(\alpha^{-1}(s),y)\), \(b_{i}(s,t,x,y)=M_{1i}M_{2i}b_{i}(\alpha_{i}^{-1}(s),\beta_{i}^{-1}(t),x,y)\), \(c_{i}(\xi,\eta,x,y)=M_{1i}^{2}M_{2i}^{2}c_{i}(\alpha_{i}^{-1}(\xi),\beta_{i}^{-1}(\eta),x,y)\), \(d_{j}(s,t,x,y)=M_{1j}M_{2j}d_{j}(\alpha_{j}^{-1}(s),\beta_{j}^{-1}(t),x,y)\), \(e_{j}(\xi,\eta,x,y)=M_{1j}^{2}M_{2j}^{2}e_{j}(\alpha_{j}^{-1}(\xi),\beta_{j}^{-1}(\eta),x,y)\), we obtain the desired estimation (3.12). □
4 Conclusion
In this paper, we established several new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables in Theorem 2.1 and Theorem 2.2, and gave their specific cases in Corollary 2.1 and Corollary 2.2, respectively, which can be used in the analysis of the qualitative properties to solutions of integral equations with maxima. In Theorem 3.1 and Theorem 3.2, we also presented the applications to research the boundedness of solutions of retarded nonlinear Volterra-Fredholm type integral equations.
Using our method, one can further study the integral inequality with more dimensions.
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Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions and comments, which helped to improve the quality of the paper.
This research is supported by National Science Foundation of China (11671227).
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RX proved parts of the results in Section 2 and participated in Section 3 - Applications. XM carried out the generalized weakly singular integral inequalities and completed part of the proof. All authors read and approved the final manuscript.
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Xu, R., Ma, X. Some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two variables and their applications. J Inequal Appl 2017, 187 (2017). https://doi.org/10.1186/s13660-017-1460-6
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DOI: https://doi.org/10.1186/s13660-017-1460-6