# Generalized Nonlinear Volterra-Fredholm Type Integral Inequality with Two Variables

Authors: Yusong Lu; Wu-Sheng Wang; Xiaoliang Zhou; Yong Huang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We establish a class of new nonlinear retarded Volterra-Fredholm type integral inequalities, with two variables, where known functionw in integral functions in Q.-H. Ma and J. Pečarić, 2008 is changed into the functions w1,w2. By adopting novel analysis techniques, such as change of variable, amplification method, differential and integration, inverse function, and the dialectical relationship between constants and variables, the upper bounds of the embedded unknown functions are estimated. The derived results can be applied in the study of solutions of ordinary differential equations and integral equations.


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## Body

## 1. Introduction

Gronwall-Bellman inequality [1, 2] is an important tool in the study of existence, uniqueness, boundedness, oscillation, stability, and other qualitative properties of solutions of differential equations and integral equation. There can be found a lot of its generalizations in various cases from the literature (e.g., [3–6]).Gronwall-Bellman inequality [1, 2] can be stated as follows. If u and f are nonnegative continuous functions on an interval [a,b] satisfying
(1)u(t)≤c+∫at‍f(s)u(s)ds,t∈[a,b],
for some constant c≥0, then
(2)u(t)≤cexp⁡(∫at‍f(s)ds),t∈[a,b].
During the past few years, some investigators have established a lot of useful and interesting integral inequalities in order to achieve various goals; see [7–25] and the references cited therein.In 2004, Pachpatte [8] has established the linear Volterra-Fredholm type integral inequality with retardation
(3)u(t)≤k+∫α(t0)α(t)‍a(t,s)×[f(s)u(s)+∫α(t0)s‍c(s,τ)u(τ)dτ]dsvvvvvv+∫α(t0)α(T)‍b(t,s)u(s)ds,∀t∈I.
In 2005, Agarwal et al. [9] investigated the inequality
(4)u(t)≤a(t)+∑i=1n‍∫bi(t0)bi(t)‍gi(t,s)wi(u(s))ds,t0≤t<t1.
In 2006, Cheung [10] studied the inequality
(5)up(x,y)≤a+pp-q×∫b1(x0)b1(x)‍∫c1(y0)c1(y)‍g1(s,t)uq(s,t)dtds+pp-q∫b2(x0)b2(x)‍∫c2(y0)c2(y)‍g2(s,t)uq(s,t)×ψ(u(s,t))dtds.
In 2008, Ma and Pečarić [13] have discussed the following useful nonlinear Volterra-Fredholm type integral inequality with retardation
(6)u(x,y)‍≤k+∫α(x0)α(x)∫β(y0)β(y)h1(s,t)‍u(x,y)≤k+vvvv×[∫β(y0)t‍h2f(s,t)w(u(s,t))u(x,y)≤k+vvvvvvv‍+∫α(x0)s∫β(y0)th2(τ,ξ)w‍vvvvvvvvvvvvvvvvvvvvvvvvvvv×(u(τ,ξ))dτdξ∫β(y0)t‍h2]≤k+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s,t)u(x,y)≤+∫α(x0)α(M)×[∫β(y0)t‍h2f(s,t)w(u(s,t))u(x,y)≤k+∫α(x0)α(M)vvvv+∫α(x0)s‍∫β(y0)t‍h2(τ,ξ)wvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv×(u(τ,ξ))dτdξ∫β(y0)t‍h2].In 2011, Abdeldaim and Yakout [20] studied a new integral inequality of Gronwall-Bellman-Pachpatte type
(7)u(t)≤u0+∫α(t0)tf(s)u(s)111111×[u(s)+∫α(t0)s‍h(τ)1111111111111111×[u(τ)+∫α(t0)τ‍g(ξ)1111111111111111111111111111×u(ξ)dξ∫α(t0)τ‍]dτ∫α(t0)τ‍]ds.In this paper, on the basis of [13, 20], we discuss a new retarded nonlinear Volterra-Fredholm type integral inequality(8)u(x,y)≤k+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)w(u(s1,t1))×[∫β(y0)t2‍f1(s1,t1)w1(u(s1,t1))+∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)×[∫β(y0)t2‍f2(s2,t2)w2(u(s2,t2))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)w3(u(s3,t3))dt3ds3]dt2ds2]dt1ds1+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)w(u(s1,t1))×[∫β(y0)t2‍f1(s1,t1)w1(u(s1,t1))+∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)×[∫β(y0)t2‍f2(s2,t2)w2(u(s2,t2))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)w3(u(s3,t3))dt3ds3]dt2ds2]dt1ds1,wherek is a constant, where the known function w in integral functions in [13] is replaced to the functions w1,w2. The upper bound estimation of the unknown function is given by integral inequality technique. Furthermore, we apply our result to retarded nonlinear Volterra-Fredholm type equations for estimation.


## 2. Main Result

Throughout this paper,I1=[x0,M] and I2=[y0,N] are the given subsets of real numbers R, R+=[0,+∞), Δ=I1×I2, C1(M,S) denotes the class of continuously differentiable functions defined on set M with range in the set S, C(M,S) denotes the class of continuous functions defined on set M with range in the set S, and α′(t) denotes the derived function of a function α′(t).Theorem 1.
Suppose thatk>0 is a constant; functions u(x,y),h3(x,y),fi(x,y),hi(x,y)∈C(Δ,R+), (i=1,2), both α∈C1(I1,I1) and β∈C1(I2,I2) are nondecreasing with α(x)≤x on I1, β(y)≤y on I2, w,w1,w2/w1,w3/w1,w3/w2∈C(R+,R+) are nondecreasing functions with wi(u)>0 (i=1,2,3) for u>0,
(9)W1(z)=∫cz‍dsw(s)w1(s),c>0,z∈(0,+∞),W1(+∞)=+∞,(10)W2(z)=∫cz‍w1(W1-1(s))dsw2(W1-1(s)),c>0,z∈(0,+∞),W2(+∞)=+∞,(11)W3(z)=∫cz‍w2(W1-1(W2-1(s)))dsw3(W1-1(W2-1(s))),c>0,z∈(0,+∞),W3(+∞)=+∞,(12)H(u)=W3{W2{W1(2u-k)}}-W3×{W2{∫α(x0)α(M)‍W1(u)1111111111+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)f1(s1,t1)dt1ds1}+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)1111111111111×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)1111111111111111111111×f2(s2,t2)dt2ds2∫β(y0)t1‍]dt1ds1}-∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)111111111111×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)×[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1
is increasing, and H(u)=0 has a solution c for u>k. If u(x,y) satisfies (8), then(13)u(x,y)≤W1-1{×[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}}W2-1{×[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}W3-1{W3{W2{W1(c)+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)f1(s1,t1)dt1ds1}+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)f2(s2,t2)dt2ds2]dt1ds1∫β(y0)β(y)‍}+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)×[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}}},∀(x,y)∈Δ,where Wi-1 (i=1,2,3) are inverse functions of Wi, respectively.Proof.
Letz1(x,y) denote the function on the right-hand side of (8), which is positive and nondecreasing in each of the variables (x,y)∈Δ. From (8), we have
(14)u(x,y)≤z1(x,y),∀(x,y)∈Δ,(15)z1(x0,y)=k+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)w(u(s1,t1))×[∫α(x0)s1‍f1(s1,t1)w1(u(s1,t1))+∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)×[∫α(x0)s1‍f2(s2,t2)w2(u(s2,t2))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)×w3(u(s3,t3))dt3ds3∫α(x0)s1‍]dt2ds2∫α(x0)s1‍]dt1ds1.Differentiating z1(x,y) with respect to x, using (14), we have
(16)∂∂xz1(x,y)=α′(x)∫β(y0)β(y)‍h1(α(x),t1)w(u(α(x),t1))×[∫β(y0)t2‍f1(α(x),t1)w1(u(α(x),t1))+∫α(x0)α(x)‍∫β(y0)t1‍h2(s2,t2)×[∫β(y0)t2‍f2(s2,t2)w2(u(s2,t2))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)1111111111111111111111111111111111111×w3(u(s3,t3))dt3ds3∫β(y0)t2‍]dt2ds2]dt1≤α′(x)∫β(y0)β(y)‍h1(α(x),t1)w(z1(α(x),t1))×[∫β(y0)t2‍f1(α(x),t1)w1(z1(α(x),t1))+∫α(x0)α(x)‍∫β(y0)t1‍h2(s2,t2)×[∫β(y0)t2‍f2(s2,t2)w2(z1(s2,t2))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt111111111111111111111111111111111111×w3(z1(s3,t3))dt3ds3∫β(y0)t2‍]dt2ds2]
by the monotonicity of w, w1, w2, w3, and z1 and the property of α,β. From (16), we have
(17)(∂/∂x)z1(x,y)w(z1(x,y))w1(z1(x,y))≤α′(x)∫β(y0)β(y)‍h1(α(x),t1)×[∫β(y0)t2‍f1(α(x),t1)+∫α(x0)α(x)‍∫β(y0)t1‍h2(s2,t2)×[f2(s2,t2)w2(z1(s2,t2))w1(z1(s2,t2))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)111111111111111111111111111111111×w3(z1(s3,t3))w1(z1(s3,t3))dt3ds3]dt2ds2]dt1.
Integrating both sides of the above inequality from x0 to x, we obtain
(18)W1(z1(x,y))≤W1(z1(x0,y))+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)×[w3(z1(s3,t3))w1(z1(s3,t3))f1(s1,t1)+∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)×[f2(s2,t2)w2(z1(s2,t2))w1(z1(s2,t2))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)×w3(z1(s3,t3))w1(z1(s3,t3))dt3ds3]dt2ds2]dt1ds1≤W1(z1(x0,y))+∫α(x0)α(X)‍∫β(y0)β(Y)‍h1(s1,t1)f1(s1,t1)dt1ds1+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)×[f2(s2,t2)w2(z1(s2,t2))w1(z1(s2,t))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)11111111111111111111111111111111111111111111×w3(z1(s3,t3))w1(z1(s3,t3))dt3ds3]dt2ds2]dt1ds1,
for all (x,y)∈[x0,X]×[y0,Y], X∈I1, Y∈I2, and X,Y are chosen arbitrarily, where W1 is defined by (9).
Letz2(x,y) denote the function on the right-hand side of (18), which is positive and nondecreasing in each of the variables (x,y)∈[x0,X]×[y0,Y]. From (18), we have
(19)z1(x,y)≤W1-1(z2(x,y)),∀(x,y)∈[x0,X]×[y0,Y],(20)z2(x0,y)=W1(z1(x0,y))+∫α(x0)α(X)‍∫β(y0)β(Y)‍h1(s1,t1)f1(s1,t1)dt1ds1.
Differentiating z2(x,y) with respect to x, by the monotonicity of z2, W1-1, w2/w1, and w3/w1, the property of α,β, and (19), we have(21)∂∂xz2(x,y)=α′(x)∫β(y0)β(y)‍h1(α(x),t1)×[∫α(x0)α(x)‍∫β(y0)t1‍h2(s2,t2)×[f2(s2,t2)w2(z1(s2,t2))w1(z1(s2,t2))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)1111111111111111111111111111111111×w3(z1(s3,t3))w1(z1(s3,t3))dt3ds3]dt2ds2]dt1≤α′(x)∫β(y0)β(y)‍h1(α(x),t1)×[w2(W1-1(z2(s2,t2)))w1(W1-1(z2(s2,t1)))∫α(x0)α(x)‍∫β(y0)t1‍h2(s2,t2)×[f2(s2,t2)w2(W1-1(z2(s2,t2)))w1(W1-1(z2(s2,t1)))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)11111111111111111111111111111111111×w3(W1-1(z2(s3,t3)))w1(W1-1(z2(s3,t3)))dt3ds3]dt2ds2]dt1
for all (x,y)∈[x0,X]×[y0,Y]. From (21), we have
(22)w1(W1-1(z2(x,y)))(∂/∂x)z2(x,y)w2(W1-1(z2(x,y)))≤α′(x)∫β(y0)β(y)‍h1(α(x),t1)×[∫α(x0)α(x)‍∫β(y0)t1‍h2(s2,t2)×[∫β(y0)t2‍f2(s2,t2)+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)1111111111111111111111111111111×w3(W1-1(z2(s3,t3)))w2(W1-1(z2(s3,t3)))dt3ds3]dt2ds2]dt1,
for all(x,y)∈[x0,X]×[y0,Y]. From (22), we have(23)W2(z2(x,y))≤W2(z2(x0,y))+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)×[∫β(y0)t1‍f2(s2,t2)+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)×w3(W1-1(z2(s3,t3)))w2(W1-1(z2(s3,t3)))dt3ds3]dt2ds2]dt1ds1≤W2(z2(x0,y))+∫α(x0)α(X)‍∫β(y0)β(Y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)f2(s2,t2)dt2ds2]dt1ds1+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)×[w3(W1-1(z2(s3,t3)))w2(W1-1(z2(s3,t3)))∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)×[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)w3(W1-1(z2(s3,t3)))w2(W1-1(z2(s3,t3)))dt3ds3]dt2ds2]dt1ds1,
for all(x,y)∈[x0,X]×[y0,Y], where W2 is defined by (10). Let z3(x,y) denote the function on the right-hand side of (23), which is positive and nondecreasing in each of the variables (x,y)∈[x0,X]×[y0,Y]. Then (23) is equivalent to
(24)z2(x,y)≤W2-1(z3(x,y)),∀(x,y)∈[x0,X]×[y0,Y],(25)z3(x0,y)=W2(z2(x0,y))+∫α(x0)α(X)‍∫β(y0)β(Y)‍h1(s1,t1)×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)vvvvvvvvvvvvvvv∫α(x0)s1‍v×f2(s2,t2)dt2ds2∫β(y0)t1‍]dt1ds1.
Differentiatingz3(x,y) with respect to x, using (24), we have(26)∂∂xz3(x,y)=α′(x)∫β(y0)β(y)‍h1(α(x),t1)×[∫α(x0)α(x)‍∫β(y0)t1‍h2(s2,t2)×[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)w3(W1-1(z2(s3,t3)))w2(W1-1(z2(s3,t3)))dt3ds3]dt2ds2]dt1≤α′(x)∫β(y0)β(y)‍h1(α(x),t1)×[∫α(x0)α(x)‍∫β(y0)t1‍h2(s2,t2)×[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)w3(W1-1(W2-1(z3(s3,t3))))w2(W1-1(W2-1(z3(s3,t3))))dt3ds3]dt2ds2]dt1,
for all(x,y)∈[x0,X]×[y0,Y]. From (26), using the monotonicity of z3, W1-1, W2-1, and w3/w2 and the property of α,β, we have(27)w2(W1-1(W2-1(z3(x,y))))(∂/∂x)z3(x,y)w3(W1-1(W2-1(z3(x,y))))≤α′(x)∫β(y0)β(y)‍h1(α(x),t1)vvvvvvvvvv×[∫α(x0)α(x)‍∫β(y0)t1‍h2(s2,t2)vvvvvvvvvvvvvvvvvvv×[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3∫α(x0)α(x)‍]dt2ds2∫α(x0)α(x)‍]dt1,∫α(x0)α(x)‍
for all(x,y)∈[x0,X]×[y0,Y]. Integrating both sides of the above inequality from x0 to x, we obtain(28)W3(z3(x,y))≤W3(z3(x0,y))+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1,∀(x,y)∈[x0,X]×[y0,Y],
whereW3 is defined by (11).
From (19), (24), and (28), we have(29)z1(x,y)≤W1-1(z2(x,y))≤W1-1(W2-1(z3(x,y)))≤W1-1{+∫α(t0)α(t)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}}W2-1{+∫α(t0)α(t)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}W3-1{[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]W3(z3(x0,y))+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}}},∀(x,y)∈[x0,X]×[y0,Y].Substituting (20) and (25) into (29), we have(30)z1(x,y)≤W1-1(z2(x,y))≤W1-1(W2-1(z3(x,y)))≤W1-1{+∫α(t0)α(t)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}}W2-1{+∫α(t0)α(t)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}W3-1{W3{W2{W1(z1(x0,y))+∫α(x0)α(X)‍∫β(y0)β(Y)‍h1(s1,t1)f1(s1,t1)dt1ds1}+∫α(x0)α(X)‍∫β(y0)β(Y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)f2(s2,t2)dt2ds2]dt1ds1}+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}}},∀(x,y)∈[x0,X]×[y0,Y]. Since X,Y are chosen arbitrarily, we have(31)z1(x,y)≤W1-1×{+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}}W2-1{+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}}W3-1{+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}W3{W2{W1(z1(x0,y))+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)f1(s1,t1)dt1ds1}+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)f2(s2,t2)dt2ds2]dt1ds1}+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}}},∀(x,y)∈Δ.
By the definition ofz1 and (15), we have(32)2z1(x0,y)-k=k+2∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)w(u(s1,t1))×[∫β(y0)t1‍f1(s1,t1)w1(u(s1,t1))+∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)×[f2(s2,t2)w2(u(s2,t2))+∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)w3(u(s3,t3))dt3ds3]dt2ds2]dt1ds1=z1(M,N).From (31) and (32), we have(33)2z1(x0,y)-k≤W1-1×{W2-1{W3-1{W3{W2{W1(z1(x0,y))+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)f1(s1,t1)dt1ds1}+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)f2(s2,t2)dt2ds2]dt1ds1}+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1}}}or(34)W3{W2{W1(2z1(x0,y)-k)}}≤W3{W2{W1(z1(x0,y))+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)f1(s1,t1)dt1ds1}+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)f2(s2,t2)dt2ds2]dt1ds1}+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)[∫α(x0)s2‍∫β(y0)t2‍h3(s3,t3)dt3ds3]dt2ds2]dt1ds1.By the definition ofH, the assumption of Theorem 1, and (34), we observe that
(35)H(z1(x0,y))≤0=H(c).
Since H is increasing, from (14), (31), and (35), we have the desired estimation (13).We consider a special case of Theorem1. If u(x,y) satisfies nonlinear Volterra-Fredholm type integral inequality with retardation,(36)u(x,y)≤k+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)w(u(s1,t1))×[f1(s1,t1)w1(u(s1,t1))+∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)w2(u(s2,t2))dt2ds2]dt1ds1+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)w(u(s1,t1))×[f1(s1,t1)w1(u(s1,t1))+∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)w2(u(s2,t2))dt2ds2]dt1ds1,jjjjjjj∀(x,y)∈Δ.Corollary 2.
Let functionsu(x,y), f1(x,y), h1(x,y), h2(x,y), α, β, w, w1, w2/w1, W1, and  W2 and constant k be as in Theorem 1. Suppose that
(37)H(u)=W2{W1(2u-k)}-W2{W1(u)+∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)f1(s1,t1)dt1ds1}-∫α(x0)α(M)‍∫β(y0)β(N)‍h1(s1,t1)×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)dt2ds2]dt1ds1
is increasing and H(u)=0 has a solution c for u>k. If u(x,y) satisfies (36), then
(38)u(x,y)≤W1-1{W2-1{W2{∫α(x0)α(x)‍W1(c)11111111111111111+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)vvvvvvvvvvvvvvvvvvvvvvvvvv×f1(s1,t1)dt1ds1W1(c)+∫α(x0)α(x)‍∫β(y0)β(y)‍}+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)vvvvvvvvvvvvvvvvvvvvvv×[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)dt2ds2]dt1ds1W1(c)+∫α(x0)α(x)‍∫β(y0)β(y)‍}W1(c)+∫α(x0)α(x)‍∫β(y0)β(y)‍}.
for all(x,y)∈Δ, where Wi-1 (i=1,2) are inverse functions of Wi, respectively.Remark 3.
In Corollary2, when w1(x,y)≡w2(x,y) on Δ, W2(u)=u-c, and W2-1(u)=u+c, (38) is equivalent to(39)u(x,y)≤W1-1×{W1(c)+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)f1(s1,t1)dt1ds1+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)[∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)dt2ds2]dt1ds1}=W1-1{W1(c)+∫α(x0)α(x)‍∫β(y0)β(y)‍h1(s1,t1)[f1(s1,t1)+∫α(x0)s1‍∫β(y0)t1‍h2(s2,t2)dt2ds2]dt1ds1},∀(x,y)∈Δ.
Corollary2 reduces to Theorem 3.1′ in [13].


## 3. Application

In this section, we apply our result in Theorem1 to study the retarded Volterra-Fredholm integral equations with two variables, which demonstrates that our result is useful to investigate the qualitative properties of solutions of some retarded Volterra-Fredholm integral equations with two variables.We consider the retarded Volterra-Fredholm integral equation of the form(40)v(x,y)=g(x,y)+∫x0x‍∫y0y‍F1{∫y0t1‍s1,t1,v(s1-d(s1),t1-e(t1)),×∫x0s1‍∫y0t1‍F2[s2,t2,v(s2-d(s2),t2-e(t2))]dt2ds2}dt1ds1+∫x0M‍∫y0N‍F1{s1,t1,v(s1-d(s1),t1-e(t1)),∫x0s1‍∫y0t1‍F2[s2,t2,v(s2-d(s2),t2-e(t2))]dt2ds2}dt1ds1for all(x,y)∈[x0,M]×[y0,N], where g,v∈C(Δ,R), x-d(x)∈C1(I1,I1), and y-e(y)∈C1(I2,I2) are strictly increasing with d(x0)=0, e(y0)=0, d(x)≥0, e(y)≥0, d′(x)<1, e′(y)<1, and F1,F2∈C(Δ×R,R). Let α(x)=x-d(x), β(y)=y-e(y); then α(x),β(y) satisfy the condition in Theorem 1 and are invertible functions.The following corollary gives the bound on the solution of (40).Corollary 4.
Letk=max⁡(x,y)∈Δ{|g(x,y)|}. Suppose that F1,F2 in (40) satisfy the conditions
(41)|F1(s,t,x,y)|≤h1(s,t)w(|x|)[f1(s,t)w1(|x|)+|y|],|F2(s,t,x)|≤h2(s,t)w2(|x|),
where f1(s,t), h1(s,t), h2(s,t), w(s), w1(s), and w2(s) are as in Theorem 1. Assume that the function(42)H*(u)=W2{W1(2u-k)}-W2{W1(u)+∫α(x0)α(M)‍∫β(y0)β(N)‍Eh1(α-1(s1),β-1(t1))f1(α-1(s1),β-1(t1))dt1ds1}-∫α(x0)α(M)‍∫β(y0)β(N)‍Eh1(α-1(s1),β-1(t1))[∫α(x0)s1‍∫β(y0)t1‍Eh2(α-1(s2),β-1(t2))dt2ds2]dt1ds1
is increasing andH*(u)=0 has a solution c for u>k. If v(x,y) is a solution of (40) on Δ, then(43)|v(x,y)|≤W1-1×{+∫α(x0)α(x)‍∫β(y0)β(y)‍Eh1(α-1(s1),β-1(t1))[∫α(x0)s1‍∫β(y0)t1‍Eh2(α-1(s2),β-1(t2))dt2ds2]dt1ds1}W2-1{W2{W1(c)+∫α(x0)α(x)‍∫β(y0)β(y)‍Eh1(α-1(s1),β-1(t1))f1(α-1(s1),β-1(t1))dt1ds1}+∫α(x0)α(x)‍∫β(y0)β(y)‍Eh1(α-1(s1),β-1(t1))[∫α(x0)s1‍∫β(y0)t1‍Eh2(α-1(s2),β-1(t2))dt2ds2]dt1ds1}},
whereE=max⁡x∈I1(1/α′(α-1(x)))max⁡y∈I2(1/β′(β-1(y)))<∞)<∞ and W1, W2, W1-1, and W2-1 are as in Theorem 1.Proof.
Using the conditions (41), we have(44)|v(x,y)|≤|g(x,y)|+∫x0x‍∫y0y‍h1(s1,t1)w(|v(s1-d(s1),t1-e(t1))|)×[∫y0t1‍f1(s1,t1)w1(|v(s1-d(s1),t1-e(t1))|)+∫x0s1‍∫y0t1‍h2(s2,t2)w2(|v(s2-d(s2),t2-e(t2))|)dt2ds2]dt1ds1+∫x0M‍∫y0N‍h1(s1,t1)w(|v(s1-d(s1),t1-e(t1))|)×[∫y0t1‍f1(s1,t1)w1(|v(s1-d(s1),t1-e(t1))|)+∫x0s1‍∫y0t1‍h2(s2,t2)w2(|v(s2-d(s2),t2-e(t2))|)dt2ds2]dt1ds1≤k+∫x0x‍∫y0y‍h1(s1,t1)w(|v(α(s1),β(t1))|)×[∫y0t1‍f1(s1,t1)w1(|v(α(s1),β(t1))|)+∫x0s1‍∫y0t1‍h2(s2,t2)w2(|v(α(s2),β(t2))|)dt2ds2]dt1ds1+∫x0M‍∫y0N‍h1(s1,t1)w(|v(α(s1),β(t1))|)×[∫y0t1‍f1(s1,t1)w1(|v(α(s1),β(t1))|)+∫x0s1‍∫y0t1‍h2(s2,t2)w2(|v(α(s2),β(t2))|)dt2ds2]dt1ds1,
for all(x,y)∈Δ. From (44), using change of variables, we obtain
(45)≤k+∫x0x‍∫y0y‍h1(s1,t1)w(|v(α(s1),β(t1))|)×[1α′(α-1(s2))f1(s1,t1)w1(|v(α(s1),β(t1))|)vvvvvvvvvvvv+∫α(x0)α(s1)‍∫β(y0)β(t1)‍1α′(α-1(s2))vvvvvvvvvvvvvvvvvvvvvv×1β′(β-1(t2))h2vvvvvvvvvvvvvvvvvvvvvv×(α-1(s2),β-1(t2))w2vvvvvvvvvvvvvvvvvvvvvv×(|v(s2,t2)|)dt2ds21α′(α-1(s2))]dt1ds1k+∫x0M‍∫y0N‍h1(s1,t1)w(|v(α(s1),β(t1))|)×[1α′(α-1(s2))1β′(β-1(t2))f1(s1,t1)w1(|v(α(s1),β(t1))|)+∫α(x0)α(s1)‍∫β(y0)β(t1)‍1α′(α-1(s2))1β′(β-1(t2))h2vvvvvvvvvvvvvvvvvvvv×(α-1(s2),β-1(t2))w2vvvvvvvvvvvvvvvvvvvv×(|v(s2,t2)|)dt2ds21α′(α-1(s2))1β′(β-1(t2))]dt1ds1≤k+∫α(x0)α(x)‍∫β(y0)β(y)‍Eh1(α-1(s1),β-1(t1))w(|v(s1,t1)|)×[∫β(y0)t1‍f1(α-1(s1),β-1(t1))w1(|v(s1,t1)|)+∫α(x0)s1‍∫β(y0)t1‍Eh2(α-1(s2),β-1(t2))w2vvvvvvvvvvvvvvvvvvvvvv×(|v(s2,t2)|)dt2ds21α′(α-1(s2))]dt1ds1+∫α(x0)α(M)‍∫β(y0)β(N)‍Eh1(α-1(s1),β-1(t1))w(|v(s1,t1)|)×[∫β(y0)t1‍f1(α-1(s1),β-1(t1))w1(|v(s1,t1)|)+∫α(x0)s1‍∫β(y0)t1‍Eh2(α-1(s2),β-1(t2))w2vvvvvvvvvvvvvvvvvvvvvv×(|v(s2,t2)|)dt2ds2vvvvvvvvvvvvvvvvvvvvvv×1α′(α-1(s2))]dt1ds1,
for all(x,y)∈Δ. Applying the result of Theorem 1 to the inequality (45), we obtain the desired estimation (39).



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*Source: 359280-2014-04-24.xml*