# A Generalized Nonlinear Volterra-Fredholm Type Integral Inequality and Its Application

Authors: Limian Zhao; Shanhe Wu; Wu-Sheng Wang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We establish a new nonlinear retarded Volterra-Fredholm type integral inequality. The upper bounds of the embedded unknown functions are estimated explicitly by using the theory of inequality and analytic techniques. Moreover, an application of our result to the retarded Volterra-Fredholm integral equations for estimation is given.


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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, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equation. A lot of its generalizations in various cases can be found from the literature (e.g., [3–7]). During the past few years, some investigators have established a lot of useful and interesting integral inequalities in order to achieve various goals; see [8–18] and the references cited therein.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].In 2004, Pachpatte [9] has discussed the linear Volterra-Fredholm type integral inequality with retardation:
(3)u(t)≤k+∫α(t0)α(t)a(t,s)[∫α(t0)sc(s,τ)u(τ)dτf(s)u(s)+∫α(t0)sc(s,τ)u(τ)dτ]ds+∫α(t0)α(T)b(t,s)u(s)ds,∀t∈I.
In 2011, Abdeldaim and yakout [17] studied a new integral inequality of Gronwall-Bellman-Pachpatte type:
(4)u(t)≤u0+∫α(t0)tf(s)u(s)×[u(s)+∫α(t0)s‍h(τ)×[∫α(t0)τ‍u(τ)+∫α(t0)τ‍g(ξ)u(ξ)dξ]dτ]ds.In this paper, on the basis of [9, 17], we discuss a new retarded nonlinear Volterra-Fredholm type integral inequality:
(5)u(t)≤k+∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1h2(t2)f1(t1)ϕ1(u(t1))+∫α(t0)t1h2(t2)×[∫α(t0)tn-2hn-1(tn-1)f2(t2)ϕ2(u(t2))+⋯+∫α(t0)tn-2hn-1(tn-1)×[∫α(t0)tn-2hn-1(tn-1)fn-1(tn-1)ϕn-1(u(tn-1))+∫α(t0)tn-1‍hn(tn)ϕn×(u(tn))dtn∫α(t0)tn-2hn-1(tn-1)]dtn-1⋯]dt2f1(t1)ϕ1(u(t1))+∫α(t0)t1h2(t2)]dt1+∫α(t0)α(T)‍h1(t1)×[∫α(t0)t1‍h2(t2)f1(t1)ϕ1(u(t1))+∫α(t0)t1‍h2(t2)×[∫α(t0)t1‍h2(t2)f2(t2)ϕ2(u(t2))+⋯+∫α(t0)tn-2hn-1(tn-1)×[∫α(t0)t1‍h2(t2)fn-1(tn-1)ϕn-1(u(tn-1))+∫α(t0)tn-1‍hn(tn)ϕn×(u(tn))dtn∫α(t0)t1‍h2(t2)]dtn-1⋯]dt2f1(t1)ϕ1(u(t1))+∫α(t0)t1‍h2(t2)]dt1,
where k is a constant. The upper bound estimation of the unknown function is given by integral inequality technique, such as change of variable, amplification method, differential and integration, inverse function, and the dialectical relationship between constants and variables. Furthermore, we apply our result to retarded nonlinear Volterra-Fredholm type equations for estimation.


## 2. Main Result

Throughout this paper,R denotes the set of real numbers, R+=[0,+∞), I=[t0,T], 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).We give the following notations used to simplify the details of presentation.We technically define a sequence of functions{wi(u)} by ϕi(u) in (5), which can be defined recursively by
(6)w1(u)≔max⁡τ∈[0,u]{ϕ1(τ)},wi+1(u)≔max⁡τ∈[0,u]{ϕi+1(τ)wi(τ)}wi(u),i=1,…,n.
Obviously, for all j>i, the function wj(u)/wi(u) is increasing and the sequence {wi(u)} consists of nondecreasing nonnegative functions and satisfies wi(u)≥ϕi(u), i=1,…,n. Moreover,
(7)wi∝wi+1,i=1,2,…,n-1,
as defined in [4] for comparison of monotonicity of functions, because the ratios wi+1(u)/wi(u), i=1,…,n-1, are all nondecreasing.For given constantui>0, we define functions
(8)W1(u,u1)=∫u1u‍dsw1(s),(9)Wi(u,ui)=∫uiu‍wi-1(W1-1(⋯Wi-1-1(s)⋯))dswi(W1-1(⋯Wi-1-1(s)⋯)),i=2,…,n,
which are strictly increasing. When there is no confusion, we simply let Wi(u) denote Wi(u,ui) and Wi-1 denote its inverse.We define functions{Hi(t)}(i=1,2,…,n):
(10)H1(t)=∫α(t0)α(t)‍h1(t1)f1(t1)dt1,H2(t)=∫α(t0)α(t)‍h1(t1)[∫α(t0)t1‍h2(t2)f2(t2)dt2]dt1,⋮Hn-1(t)=∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1‍h2(t2)×[⋯[∫α(t0)tn-2‍hn-1(tn-1)fn-1(tn-1)dtn-1]⋯∫α(t0)tn-2‍]dt2]dt1,Hn(t)=∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1‍h2(t2)×[⋯[∫α(t0)tn-2‍hn-1(tn-1)×[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯∫α(t0)tn-2‍]dt2]dt1.We define function(11)G(u)=Wn{Wn-1{⋯{W2{W1(2u-k)}}⋯}}-Wn{Wn-1{⋯{W2{W1(u)+H1(T)}+H2(T)}⋯}+Hn-1(T)}-Hn(T),∀u>k.Theorem 1.
Suppose thathn(t),fi(t),hi(t)∈C(I,R+),(i=1,…,n-1),α∈C1(I,I) is nondecreasing with α(t)≤t and α(t0)=t0 on I; all ϕi are continuous functions with ϕi(u)>0(i=1,…,n) for u>0, Wi(+∞)=+∞, i=1,2,…,n. Suppose that the function G(u) is increasing and G(u)=0 has a solution c for u>k. If u(t) satisfies (5), then
(12)u(t)≤W1-1×{W2-1{⋯{Wn-1{Wn{Wn-1{⋯{W2{W1(c)+H1(t)}+H2(t)}⋯}+Hn-1(t)}+Hn(t)}W2-1{⋯{Wn-1{Wn{Wn-1{⋯{W2{W1(c)+H1(t)}}⋯}},∀t∈I,
where Wi-1(i=1,2,…,n) are inverse functions ofWi, respectively.Proof.
From (5) and (6), we have
(13)u(t)≤k+∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1h2(t2)f1(t1)w1(u(t1))+∫α(t0)t1h2(t2)×[∫α(t0)t1h2(t2)f2(t2)w2(u(t2))+⋯+∫α(t0)tn-2hn-1(tn-1)×[∫α(t0)t1h2(t2)fn-1(tn-1)wn-1(u(tn-1))+∫α(t0)tn-1‍hn(tn)wn×(u(tn))dtn∫α(t0)tn-2hn-1(tn-1)f2(t2)w2(u(t2))]dtn-1⋯]dt2f1(t1)w1(u(t1))+∫α(t0)t1h2(t2)]dt1+∫α(t0)α(T)‍h1(t1)×[∫α(t0)t1‍h2(t2)f1(t1)w1(u(t1))+∫α(t0)t1‍h2(t2)×[∫α(t0)t1‍h2(t2)f2(t2)w2(u(t2))+⋯+∫α(t0)tn-2hn-1(tn-1)×[∫α(t0)t1‍h2(t2)fn-1(tn-1)wn-1(u(tn-1))+∫α(t0)tn-1‍hn(tn)wn×(u(tn))dtn∫α(t0)t1‍h2(t2)]dtn-1⋯]dt2f1(t1)w1(u(t1))+∫α(t0)t1‍h2(t2)]dt1,
for all t∈I. Let z1(t) denote the function on the right-hand side of (13), which is a positive and nondecreasing function on I. Then (13) is equivalent to
(14)u(t)≤z1(t),∀t∈I,(15)z1(t0)=k+∫α(t0)α(T)‍h1(t1)×[∫α(t0)α(T)‍h1(t1)f1(t1)w1(u(t1))+∫α(t0)t1‍h2(t2)×[∫α(t0)α(T)‍h1(t1)f2(t2)w2(u(t2))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[∫α(t0)tn-2‍fn-1(tn-1)wn-1(u(tn-1))+∫α(t0)tn-1‍hn(tn)wn×(u(tn))dtn∫α(t0)α(T)‍h1(t1)fn-1(tn-1)wn-1(u(tn-1))]dtn-1⋯]dt2f1(t1)w1(u(t1))+∫α(t0)t1‍h2(t2)]dt1.
Differentiating z1(t) with respect to t, using (14), we have
(16)z1′(t)=α′(t)h1(α(t))×[∫α(t0)α(t)‍h2(t2)f1(α(t))w1(u(α(t)))+∫α(t0)α(t)‍h2(t2)×[∫α(t0)α(t)‍h2(t2)f2(t2)w2(u(t2))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[∫α(t0)α(t)‍h2(t2)fn-1(tn-1)wn-1(u(tn-1))+∫α(t0)tn-1‍hn(tn)wn×(u(tn))dtn∫α(t0)α(t)‍h2(t2)]dtn-1⋯]dt2f1(α(t))w1(u(α(t)))+∫α(t0)α(t)‍h2(t2)]≤α′(t)h1(α(t))×[∫α(t0)α(t)‍h2(t2)f1(α(t))w1(z1(α(t)))+∫α(t0)α(t)‍h2(t2)×[∫α(t0)α(t)‍h2(t2)f2(t2)w2(z1(t2))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[∫α(t0)α(t)‍h2(t2)fn-1(tn-1)wn-1(z1(tn-1))+∫α(t0)tn-1‍hn(tn)wn×(z1(tn))dtn∫α(t0)α(t)‍h2(t2)]dtn-1⋯]dt2],∀t∈I,
by the monotonicity of w1 and z1 and the property of α. From (16), we have
(17)z1′(t)w1(z1(t))≤α′(t)h1(α(t))×[∫α(t0)α(t)‍h2(t2)f1(α(t))+∫α(t0)α(t)‍h2(t2)×[f2(t2)w2(z1(t2))w1(z1(t2))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[fn-1(tn-1)wn-1(z1(tn-1))w1(z1(tn-1))+∫α(t0)tn-1‍hn(tn)×wn(z1(tn))w1(z1(tn))dtn]dtn-1⋯]dt2],∀t∈I.
Integrating both sides of the above inequality from t0 to t, we obtain
(18)W1(z1(t))≤W1(z1(t0))+∫α(t0)α(t)‍h1(s)f1(s)ds+∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1‍h2(t2)×[f2(t2)w2(z1(t2))w1(z1(t2))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[fn-1(tn-1)wn-1(z1(tn-1))w1(z1(tn-1))+∫α(t0)tn-1‍hn(tn)×wn(z1(tn))w1(z1(tn))dtn]dtn-1⋯]dt2]dt1≤W1(z1(t0))+∫α(t0)α(T1)‍h1(s)f1(s)ds+∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1‍h2(t2)×[f2(t2)w2(z1(t2))w1(z1(t2))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[fn-1(tn-1)wn-1(z1(tn-1))w1(z1(tn-1))+∫α(t0)tn-1‍hn(tn)×wn(z1(tn))w1(z1(tn))dtn]dtn-1⋯]dt2]dt1,
for t0≤t≤T1≤T; T1 is chosen arbitrarily, where W1 is defined by (8).
Letz2(t) denote the function on the right-hand side of (18), which is a positive and nondecreasing function on [t0,T1]. Then (18) is equivalent to
(19)z1(t)≤W1-1(z2(t)),∀t∈[t0,T1],(20)z2(t0)=W1(z1(t0))+∫α(t0)α(T1)h1(s)f1(s)ds.
Differentiating z2(t) with respect to t, using (19), we have(21)z2′(t)=α′(t)h1(α(t))×[wn(W1-1(z2(tn)))w1(W1-1(z2(tn)))∫α(t0)α(t)‍h2(t2)×[f2(t2)w2(z1(t2))w1(z1(t2))+∫α(t0)t2‍h3(t3)×[f3(t3)w3(z1(t2))w1(z1(t2))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[fn-1(tn-1)wn-1(z1(tn-1))w1(z1(tn-1))+∫α(t0)tn-1‍hn(tn)×wn(z1(tn))w1(z1(tn))dtn]dtn-1⋯]dt3]dt2]≤α′(t)h1(α(t))×[∫α(t0)α(t)‍h2(t2)×[f2(t2)w2(W1-1(z2(t2)))w1(W1-1(z2(t2)))+∫α(t0)t2‍h3(t3)×[f3(t3)w3(W1-1(z2(t3)))w1(W1-1(z2(t3)))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[fn-1(tn-1)wn-1(W1-1(z2(tn-1)))w1(W1-1(z2(tn-1)))+∫α(t0)tn-1‍hn(tn)×wn(W1-1(z2(tn)))w1(W1-1(z2(tn)))dtn]dtn-1⋯]dt3]dt2],∀t∈[t0,T1],by the monotonicity of wi/w1(i=1,2,…,n) and the property of α. From (21), we have
(22)z2′(t)w1(W1-1(z2(t)))w2(W1-1(z2(t)))≤α′(t)h1(α(t))×∫α(t0)α(t)‍h2(t2)f2(t2)dt2+α′(t)h1(α(t))×∫α(t0)α(t)‍h2(t2)×[∫α(t0)t2‍h3(t3)×[f3(t3)w3(W1-1(z2(t2)))w2(W1-1(z2(t2)))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[fn-1(tn-1)wn-1(W1-1(z2(tn-1)))w2(W1-1(z2(tn-1)))+∫α(t0)tn-1‍hn(tn)×wn(W1-1(z2(tn)))w2(W1-1(z2(tn)))dtn]dtn-1⋯]dt3]dt2,
for all t∈[t0,T1]. From (22), we have(23)W2(z2(t))≤W2(z2(t0))+∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1‍h2(t2)f2(t2)dt2]dt1+∫α(t0)α(t)‍h1(t1)×[w3(W1-1(z2(t2)))w2(W1-1(z2(t2)))∫α(t0)t1‍h2(t2)×[∫α(t0)t2‍h3(t3)×[f3(t3)w3(W1-1(z2(t2)))w2(W1-1(z2(t2)))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[fn-1(tn-1)wn-1(W1-1(z2(tn-1)))w2(W1-1(z2(tn-1)))+∫α(t0)tn-1‍hn(tn)×wn(W1-1(z2(tn)))w2(W1-1(z2(tn)))dtn]dtn-1⋯wn(W1-1(z2(tn)))w2(W1-1(z2(tn)))]dt3]dt2]dt1≤W2(z2(t0))+∫α(t0)α(T1)‍h1(t1)×[∫α(t0)t1‍h2(t2)f2(t2)dt2]dt1+∫α(t0)α(t)‍h1(t1)×[w3(W1-1(z2(t2)))w2(W1-1(z2(t2)))∫α(t0)t1‍h2(t2)×[w3(W1-1(z2(t2)))w2(W1-1(z2(t2)))∫α(t0)t2‍h3(t3)×[f3(t3)w3(W1-1(z2(t2)))w2(W1-1(z2(t2)))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[fn-1(tn-1)wn-1(W1-1(z2(tn-1)))w2(W1-1(z2(tn-1)))+∫α(t0)tn-1‍hn(tn)×wn(W1-1(z2(tn)))w2(W1-1(z2(tn)))dtn]×dtn-1⋯wn(W1-1(z2(tn)))w2(W1-1(z2(tn)))]dt3]dt2]dt1,for all t∈[t0,T1], where W2 is defined by (9). Repeating the same derivation as in (19), (23), and so on, we obtain
(24)Wn-2(zn-2(t))≤Wn-2(zn-2(t0))+∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1‍h2(t2)⋯[∫α(t0)tn-3‍hn-2(tn-2)fn-2(tn-2)dtn-2]⋯]dt1+∫α(t0)α(t)‍h1(t1)×[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))∫α(t0)t1‍h2(t2)×[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))⋯[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))∫α(t0)tn-2‍hn-1(tn-1)×[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))fn-1(tn-1)×wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))+∫α(t0)tn-1‍hn(tn)×wn(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))×dtnwn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))]dtn-1]⋯]dt2]dt1≤Wn-2(zn-2(t0))+∫α(t0)α(T1)‍h1(t1)×[∫α(t0)t1‍h2(t2)⋯[∫α(t0)tn-3‍hn-2(tn-2)fn-2(tn-2)dtn-2]⋯]dt1+∫α(t0)α(t)‍h1(t1)×[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))∫α(t0)t1‍h2(t2)×[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))⋯[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))∫α(t0)tn-2‍hn-1(tn-1)×[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))fn-1(tn-1)×wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))+∫α(t0)tn-1‍hn(tn)×wn(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))×dtnwn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))]dtn-1]⋯]dt2]dt1,
for all t∈[t0,T1], where Wn-2 is defined by (9).
Letzn-1(t) denote the function on the right-hand side of (24), which is a positive and nondecreasing function on [t0,T1]. Then (24) is equivalent to
(25)zn-2(t)≤Wn-2-1(zn-1(t)),∀t∈[t0,T1],(26)zn-1(t0)=Wn-2(zn-2(t0))+∫α(t0)α(T1)‍h1(t1)×[∫α(t0)t1‍h2(t2)⋯×[∫α(t0)tn-3‍hn-2(tn-2)fn-2(tn-2)dtn-2]⋯]dt1.
Differentiating zn-1(t) with respect to t, we have
(27)zn-1′(t)=α′(t)h1(α(t))×[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))∫α(t0)α(t)‍h2(t2)×[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))⋯[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))∫α(t0)tn-2‍hn-1(tn-1)×[wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))fn-1(tn-1)×wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))+∫α(t0)tn-1‍hn(tn)×wn(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))×dtnwn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))]dtn-1]⋯]dt2],
for all t∈[t0,T1]. From (27), using (25), we have
(28)zn-1′(t)wn-2(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn-1)))⋯)))≤α′(t)h1(α(t))×[wn(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))∫α(t0)α(t)‍h2(t2)×[wn(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))⋯[wn(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))∫α(t0)tn-2‍hn-1(tn-1)×[wn(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))fn-1(tn-1)+∫α(t0)tn-1‍hn(tn)×wn(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))×dtnwn(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-3-1(zn-2(tn)))⋯)))]dtn-1]⋯]dt2]≤α′(t)h1(α(t))×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)α(t)‍h2(t2)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))⋯[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)tn-2‍hn-1(tn-1)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))fn-1(tn-1)+∫α(t0)tn-1‍hn(tn)×wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))×dtnwn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))]dtn-1]⋯]dt2],
for all t∈[t0,T1], by the monotonicity of zn-1,W1-1,…,Wn-2-1 and wn-2/wn-1 and the property of α. Integrating both sides of the above inequality from t0 to t, we obtain
(29)Wn-1(zn-1(t))≤Wn-1(zn-1(t0))+∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1‍h2(t2)×[⋯[∫α(t0)tn-2‍hn-1(tn-1)fn-1(tn-1)dtn-1]⋯wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))]dt2]dt1+∫α(t0)α(t)‍h1(t1)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)t1‍h2(t2)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))⋯[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)tn-2‍hn-1(tn-1)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)tn-1‍hn(tn)×wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))×dtnwn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))]dtn-1]⋯]dt2]dt1≤Wn-1(zn-1(t0))+∫α(t0)α(T1)‍h1(t1)×[∫α(t0)t1‍h2(t2)×[∫α(t0)tn-2‍hn-1(tn-1)⋯[∫α(t0)tn-2‍hn-1(tn-1)fn-1(tn-1)dtn-1]⋯wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))]dt2]dt1+∫α(t0)α(t)‍h1(t1)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)t1‍h2(t2)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))⋯[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)tn-2‍hn-1(tn-1)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)tn-1‍hn(tn)×wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))×dtnwn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))]dtn-1]⋯]dt2]dt1,
for all t∈[t0,T1], where Wn-1 is defined by (9). Let zn(t) denote the function on the right-hand side of (29), which is a positive and nondecreasing function on [t0,T1]. Then (29) is equivalent to
(30)zn-1(t)≤Wn-1-1(zn(t)),∀t∈[t0,T1],(31)zn(t0)=Wn-1(zn-1(t0))+∫α(t0)α(T1)‍h1(t1)×[∫α(t0)t1‍h2(t2)×[∫α(t0)t1‍⋯[∫α(t0)tn-2‍hn-1(tn-1)×fn-1(tn-1)dtn-1∫α(t0)tn-2‍]⋯]dt2]dt1.
Differentiating zn(t) with respect to t, using (30), we have
(32)zn′(t)=α′(t)h1(α(t))×∫α(t0)α(t)‍h2(t2)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))⋯[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)tn-2‍hn-1(tn-1)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)tn-1‍hn(tn)×wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))×dtnwn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))]dtn-1]⋯]dt2≤α′(t)h1(α(t))×∫α(t0)α(t)‍h2(t2)×[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))⋯[wn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))∫α(t0)tn-2‍hn-1(tn-1)×[wn(W1-1(W2-1(⋯(Wn-1-1(zn(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-1-1(zn(tn)))⋯)))∫α(t0)tn-1‍hn(tn)×wn(W1-1(W2-1(⋯(Wn-1-1(zn(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-1-1(zn(tn)))⋯)))×dtnwn(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn)))⋯)))]dtn-1]⋯]dt2,
for all t∈[t0,T1]. From (32), we have
(33)zn′(t)wn-1(W1-1(W2-1(⋯(Wn-1-1(zn(tn)))⋯)))wn(W1-1(W2-1(⋯(Wn-1-1(zn(tn)))⋯)))=α′(t)h1(α(t))×∫α(t0)α(t)‍h2(t2)×[⋯[∫α(t0)tn-2‍hn-1(tn-1)×[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]]dt2,
for all t∈[t0,T1]. Integrating both sides of the above inequality from t0 to t, we obtain
(34)Wn(zn(t))-Wn(zn(t0))≤∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1‍h2(t2)×[⋯[∫α(t0)tn-2‍hn-1(tn-1)×[∫α(t0)tn-1‍hn(tn)dtn∫α(t0)tn-2‍]dtn-1∫α(t0)tn-2‍]⋯[∫α(t0)t1‍h2(t2)]dt2]dt1,
for all t∈[t0,T1]. From (19), (25), (30), and (34), we have
(35)z1(t)≤W1-1(W2-1(⋯(Wn-1-1(zn(t)))⋯))≤W1-1{W2-1{⋯{Wn-1{[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1Wn(zn(t0))+∫α(t0)α(t)‍h1(t1)×[∫α(t0)t1‍h2(t2)[⋯[∫α(t0)tn-2‍hn-1(tn-1)×[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1}}⋯}}
for all t∈[t0,T1]. Substituting (20), (26), and (31) into (35), we have
(36)z1(t)≤W1-1{[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1}}⋯}W2-1×{[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1}}⋯⋯{[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1}Wn-1{Wn{∫α(t0)tn-1‍Wn-1×{Wn-2{⋯{W2{[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1W1(z1(t0))+∫α(t0)α(T1)‍h1(t1)f1(t1)dt1}+∫α(t0)α(T1)‍h1(t1)[∫α(t0)t1h2(t2)f2(t2)dt2]dt1}⋯}+∫α(t0)α(T1)‍h1(t1)[∫α(t0)t1‍h2(t2)[∫α(t0)t1‍⋯[∫α(t0)tn-3‍hn-2(tn-2)fn-2(tn-2)dtn-2]⋯]dt2]dt1}+∫α(t0)α(T1)‍h1(t1)[∫α(t0)t1‍h2(t2)[∫α(t0)t1‍⋯[∫α(t0)tn-2‍hn-1(tn-1)fn-1(tn-1)dtn-1]⋯]dt2]dt1}+∫α(t0)α(t)‍h1(t1)[∫α(t0)t1‍h2(t2)[⋯[∫α(t0)tn-2‍hn-1(tn-1)×[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1}}⋯}},∀t∈[t0,T1].
Since T1 is chosen arbitrarily, we have
(37)z1(t)≤W1-1{[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1}}⋯}W2-1×{[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1}}⋯⋯{[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1}Wn-1{∫α(t0)tn-1‍Wn{∫α(t0)t1‍Wn-1×{Wn-2{⋯{W2{[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1W1(z1(t0))+∫α(t0)α(t)‍h1(t1)f1(t1)dt1}+∫α(t0)α(t)‍h1(t1)[∫α(t0)t1h2(t2)f2(t2)dt2]dt1}⋯}+∫α(t0)α(t)‍h1(t1)[∫α(t0)t1‍h2(t2)[∫α(t0)t1‍⋯[∫α(t0)tn-3‍hn-2(tn-2)fn-2(tn-2)dtn-2]⋯]dt2]dt1}+∫α(t0)α(t)‍h1(t1)[∫α(t0)t1‍h2(t2)[∫α(t0)t1‍⋯[∫α(t0)tn-2‍hn-1(tn-1)fn-1(tn-1)dtn-1]⋯]dt2]dt1}+∫α(t0)α(t)‍h1(t1)[∫α(t0)t1‍h2(t2)[⋯[∫α(t0)tn-2‍hn-1(tn-1)×[∫α(t0)tn-1‍hn(tn)dtn]dtn-1]⋯]dt2]dt1}}⋯}}=W1-1{W2-1×{⋯{Wn-1{Wn{Wn-1{⋯{W2×{W1(z1(t0))+H1(t)}+H2(t)}⋯}+Hn-1(t)}+Hn(t)}Wn-1{Wn{Wn-1{⋯{W2{W1(z1(t0))+H1(t)}+H2(t)}⋯}}⋯⋯{Wn-1{Wn{Wn-1{⋯{W2{W1(z1(t0))+H1(t)}+H2(t)}⋯}}{⋯{Wn-1{Wn{Wn-1{⋯{W2{W1(z1(t0))+H1(t)}+H2(t)}⋯}},∀t∈[t0,T].
By the definition ofz1 and (15), we have
(38)2z1(t0)-k=k+2∫α(t0)α(T)‍h1(t1)×[∫α(t0)tn-1‍hn(tn)f1(t1)w1(u(t1))+∫α(t0)t1‍h2(t2)×[∫α(t0)tn-1‍hn(tn)f2(t2)w2(u(t2))+⋯+∫α(t0)tn-2‍hn-1(tn-1)×[∫α(t0)tn-1‍hn(tn)fn-1(tn-1)wn-1(u(tn-1))+∫α(t0)tn-1‍hn(tn)×wn(u(tn))dtn∫α(t0)tn-1‍]dtn-1⋯∫α(t0)tn-1‍]dt2]dt1=z1(T).
From (37) and (38), we have
(39)2z1(t0)-k≤W1-1{W2-1{⋯{Wn-1{Wn{Wn-1{⋯{W2{W1(z1(t0))+H1(T)}+H2(T)}⋯}+Hn-1(T)}+Hn(T)}W2-1}⋯}}
or
(40)Wn{Wn-1{⋯{W2{W1(2z1(t0)-k)}}⋯}}-Wn{Wn-1{⋯{W2{W1(z1(t0))+H1(T)}+H2(T)}⋯}+Hn-1(T)}-Hn(T)≤0.
By the definition of G, the assumption of Theorem 1, and (40), we observe that
(41)G(z1(t0))≤0=G(c).
Since H2 is increasing, from the last inequality and (14), we have the desired estimation (12).
We define the following functions:(42)H1(t)=∫α(t0)α(t)h1(t1)f1(t1)dt1,H2(t)=∫α(t0)α(t)h1(t1)[∫α(t0)t1‍h2(t2)dt2]dt1,E(u)=W2{W1(2u-k)}-W2{W1(u)+H1(T)}-H2(T),
for all u>k, where Wi,i=1,2 are defined by (8) and (9), respectively.Corollary 2.
Letn=2, f1(t), f2(t), hi(t), ϕi, Wi, i=1,2,α be as in Theorem 1. Suppose that the function E(u) is increasing and E(u)=0 has a solution c for u>k. If u(t) satisfies (5), then
(43)u(t)≤W1-1{W2-1{W2{W1(c)+H1(t)}+H2(t)}},∀t∈I,
where Wi-1(i=1,2) are inverse functions ofWi, respectively.


## 3. Application

In this section, we apply our result in Theorem1 to investigate the retarded Volterra-Fredholm integral equations:
(44)x(t)=x0+∫t0t‍F1{s,x(s-γ(s)),∫t0s‍F2[τ,x(τ-γ(τ))]dτ}ds+∫t0T‍F1{s,x(s-γ(s)),∫t0s‍F2[τ,x(τ-γ(τ))]dτ}ds,
for t∈I, where x∈C(I,R),γ∈C1(I,I) is nondecreasing with t-γ(t)≥t0,γ(t0)=0,γ′(t)<1,F1∈C(I×R2,R),F2∈C(I×R,R). Let β(t)=t-γ(t); then β(t)∈C1(I,I),β(t)≤t. Since β′(t)=1-γ′(t)>0, β(t) is an increasing and invertible function.The following theorem gives the bound on the solution of (44).Theorem 3.
Suppose thatF1,F2 in (44) satisfy the conditions
(45)|F1(s,x,y)|≤h1(s)[f1(s)w1(|x|)+|y|],|F2(s,x)|≤h2(s)w2(|x|),
where f1(s),h1(s),h2(s),w1(s) and w2(s) are as in Theorem 1; let M=max⁡t∈I(1/β′(β-1(t)))<∞. Assume that the function
(46)H3(u)=W2[W1(2u-k)]-W2[W1(u)+∫β(t0)β(T)‍h1(s)f1(s)ds]-∫β(t0)β(T)‍h1(s)[∫β(t0)s‍h2(τ)f2(τ)dτ]ds
is increasing and H3(t)=0 has a solution c for u>k. If x(t) is a solution of (44), then(47)|x(t)|≤W1-1{W2-1[W2[W1(c)+∫β(t0)β(t)‍Mh1(β-1(s))f1(β-1(s))ds]+∫β(t0)β(t)‍Mh1(β-1(s))[∫β(t0)s‍Mh2(β-1(τ))dτ]ds]},∀t∈I,where W1,W2,W1-1, and W2-1 are as in Theorem 1.Proof.
Using the condition (45), we have
(48)|x(t)|≤|x0|+∫t0t‍h1(s)[∫t0s‍f1(s)w1(|x(s-γ(s))|)+∫t0s‍h2(τ)×w2(|x(τ-γ(τ))|)dτ∫β(t0)β(s)‍]ds+∫t0T‍h1(s)[∫t0s‍f1(s)w1(|x(s-γ(s))|)+∫t0s‍h2(τ)w2(|x(τ-γ(τ))|)dτ]ds=|x0|+∫t0t‍h1(s)[∫t0s‍f1(s)w1(|x(β(s))|)+∫t0s‍h2(τ)w2(|x(β(τ))|)dτ]ds+∫t0T‍h1(s)[∫t0s‍f1(s)w1(|x(β(s))|)+∫t0s‍h2(τ)w2(|x(β(τ))|)dτ]ds≤|x0|+∫t0t‍h1(s)[∫β(t0)β(s)‍f1(s)w1(|x(β(s))|)+∫β(t0)β(s)‍Mh2(β-1(τ))×w2(|x(τ)|)dτ∫β(t0)β(s)‍]ds+∫t0T‍h1(s)[∫β(t0)β(s)‍f1(s)w1(|x(β(s))|)+∫β(t0)β(s)‍Mh2(β-1(τ))×w2(|x(τ)|)dτ∫β(t0)β(s)‍]ds≤|x0|+∫β(t0)β(t)‍Mh1(β-1(s))×[∫β(t0)s‍f1(β-1(s))w1(|x(s)|)+∫β(t0)s‍Mh2(β-1(τ))×w2(|x(τ)|)dτ∫β(t0)β(s)‍]ds+∫β(t0)β(T)‍Mh1(β-1(s))×[∫β(t0)s‍f1(β-1(s))w1(|x(s)|)+∫β(t0)s‍Mh2(β-1(τ))×w2(|x(τ)|)dτ∫β(t0)β(s)‍]ds,
for t∈I, where several changes of variables are made. Applying the result of Theorem 1 to the last inequality, we obtain the desired estimation (47).



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*Source: 865136-2014-05-21.xml*