Some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two variables and their applications

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.


Introduction
Gronwall-Bellman inequality [, ] and Bihari inequality [] 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 [-], the Gronwall-Bellman-Gamidov type integral inequalities and their generalizations were studied; in [-], the Gronwall-like inequalities and their deformations were investigated; in [-], the Volterra type iterated inequalities were discussed; in [-], 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: In , Lu et al. [] studied the nonlinear retarded Volterra-Fredholm type iterated integral inequality: (  .  ) In , Huang and Wang [] discussed the retarded nonlinear Volterra-Fredholm type integral inequality with maxima: (.) Motivated by the work presented in [, , ], we establish some new retarded nonlinear Volterra-Fredholm type integral inequality with maxima in two independent variables in this paper: 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.

Main results
In what follows, R denotes the set of real numbers, R + = [, +∞), I  = [M, +∞), I  = [N, +∞) are the given subsets of R, = I  × I  . C  ( , S) denotes the class of all continuously differentiable functions defined on set with range in the set S, C( , S) denotes the class of all continuous functions defined on set with range in the set S, and α (t) denotes the derived function of a function α(t). For convenience, we cite some useful lemmas in the discussion of our proof as follows: , a(t), b(t) be nonnegative and continuous functions defined for t ∈ R + . Assume that a(t) is non-increasing function for t ∈ R + . If Theorem . Suppose that the following conditions hold: are sub-multiplicative and sub-additive, that is, (ii) k(x, y), a(x, y) ∈ C( , 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) ∈ C(  , R + ) for i = , , . . . , l  ; j = , , . . . , l  ; b i , c i , d j , e j are all non-increasing functions in the last two variables; a(s, y)ψ u(s, y) ds on condition that W  (+∞) = +∞, W  (+∞) = +∞, and G(u) is a strictly increasing function on R + . We have By (.), (.), (.), and condition (i), we deduce

x, y) k(s, t)A(s, t) + A(s, t)z(s, t)
where B(M, N) is defined in (.), and C(M, N) is defined as follows: Let z  (x, y) denote the function on the right-hand side of (.), which is positive and non- By the monotonicity of ϕ  , ϕ  , z  and the property of α i , β i , from (.), we get Replacing x with s, and integrating it from x to ∞, we obtain where E(X, Y ) is defined in (.). Let z  (x, y) denote the function on the right-hand side of (.), which is positive and non-increasing in each of the variables ( Differentiating z  (x, y) with respect to x, we have By the monotonicity of ϕ  /ϕ  and z  , from (.), we obtain Replace x with s, and integrating it from x to ∞, we get Obviously, F(x, y, x, y) = F(x, y), which is defined in (.). From (.), (.), (.), (.) and (.), we have (.) Since X and Y are chosen arbitrarily, we have u p (σ , η)) dξ dη ds dt, (x, y) ∈ , (.)

s, t, M, N)k(s, t)A(s, t)
on condition that G  (u) is a strictly increasing function on R + .
If p = , we have Obviously, G  (u) is a strictly increasing function on R + , G -  (u) is the inverse of G  (u), we get (ii) q i , r i are nonnegative constants with p ≥ q i , p ≥ r i , i = , , . . . , l  , and ε j , δ j are nonnegative constants with p ≥ ε j , p ≥ δ j , j = , , . . . , l  .
If (x, y) ∈ , u(x, y) satisfies the following inequality: Obviously, z(x, y) is non-increasing in every variable. From (.) and (.), we have a(s, y)u p (s, y) ds.
(  .   ) By Lemma ., we obtain where A(x, y) is defined in (.). Then we get By Lemma ., we have Combining (.) and (.), we have N) is defined as follows: Let z  (x, y) denote the function on the right-hand side of (.), which is positive and nonincreasing in each of the variables ( Differentiating z  (x, y) with respect to x, we have where u(x, y) satisfies the following inequality: a(s, y)u  (s, y) ds Proof Inequality (.) follows by inequality (.) with p = , q i = q, r i = r (i = , , . . . , l  ), ε j = δ j =  (j = , , . . . , l  ). Then, applying Theorem ., we can easily get (.). Details are omitted here.

Applications in the integral equation
In this section, we apply our results in Theorem . and Theorem . 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.