# A Multiple Iterated Integral Inequality and Applications

Authors: Zongyi Hou; Wu-Sheng Wang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We establish new multiple iterated Volterra-Fredholm type integral inequalities, where the composite functionw(u(s)) of the unknown function u with nonlinear function w  in integral functions in [Ma, QH, Pečarić, J: Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities. Nonlinear Anal. 69  (2008) 393–407] is changed into the composite functions w1(u(s)),w2(u(s)),…,wn(u(s)) of the unknown function u with different nonlinear functions w1,w2,…,wn, respectively. By adopting novel analysis techniques, the upper bounds of the embedded unknown functions are estimated explicitly. 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

The well-known Gronwall-Bellman inequality [1, 2] is the following or can be equivalently regarded as the following:
(1)u(t)≤c+∫atf(s)u(s)ds,t∈[a,a+T],
where c≥0 is a constant, f is a given nonnegative continuous function, and u is the unknown function. It is often used to estimate solutions of differential equations. In 1956 Bihari [3] discussed
(2)u(t)≤c+∫0t‍f(s)w(u(s))ds.
In 1990 Pinto [4] investigated
(3)u(t)≤c(t)+∑i=1n‍∫at‍gi(s)wi(u(s))ds.
Replacing the upper limit t of the integral with a function b(t) in (2), in 2000 Lipovan [5] improved Bihari’s results by investigating the following so-called retarded Gronwall-like inequalities:
(4)u(t)≤c+∫b(t0)b(t)‍f(s)w(u(s))ds,(5)u(t)≤c+∫t0t‍f(s)w(u(s))ds+∫b(t0)b(t)‍g(s)w(u(s))ds.
In 2005 Agarwal et al. [6] generally discussed
(6)u(t)≤c(t)+∑i=1n‍∫bi(t0)bi(t)‍gi(t,s)wi(u(s))ds,t0≤t<t1.
As required in estimation for solutions, invariant sets, and stability, many generalized versions of the Gronwall-Bellman inequality were given with an invariant decomposition [7–9], a singular kernel [10, 11], and maxima [12, 13]. More results about integral inequalities of single variable and multivariables can be found, for example, the books [14, 15].In order to investigate the behavior of solutions of a linear Volterra-Fredholm type integral equation, a form of integral inequalities which contains multiple integrals of the unknown,(7)u(t)≤c+∫α(t0)α(T)‍b(t,s)u(s)ds+∫α(t0)α(t)‍a(t,s)[f(s)u(s)+∫α(t0)s‍c(s,τ)u(τ)dτ]ds,SSSSSSSSSSSSSSssssSSSSSSSSSSSSSSSSSSSS∀t∈[t0,T],
called linear Volterra-Fredholm type integral inequality with retardation, is discussed by Pachpatte [16] in 2004.In 2008 Ma and Pec˘aric´ [17] discussed more generally the following inequality:
(8)u(t)≤c+∫α(t0)α(t)‍σ1(s)ssssssss×[f(s)w(u(s))+∫α(t0)s‍σ2(τ)w(u(τ))dτ]ds+∫α(t0)α(T)‍σ1(s)ssssss×[f(s)w(u(s))+∫α(t0)s‍σ2(τ)w(u(τ))dτ]ds,ssssssssssssssssssssssssssssssssssssssssssssssssssss∀t∈I,
where I=[t0,T]. In 2011 Abdeldaim and Yakout [18] investigated the following:
(9)u(t)≤c+∫α(t0)tf(s)u(s)×[u(s)+∫α(t0)s‍h(τ)[u(τ)+∫α(t0)τ‍g(ξ)u(ξ)dξ]dτ]ds.
In 2013 Wang et al. [19] studied a new integral inequality of Gronwall-Bellman-Pachpatte type
(10)φ(u(t))≤c(t)+∫α(t0)α(t)‍f1(t,s)w1(u(s))ds+∫α(t0)α(t)‍f1(t,s)w1(u(s))sssssss×(∫t0s‍f2(s,τ)w2(u(τ))dτ)ds+∫α(t0)α(t)‍f1(t,s)w1(u(s))sssssss×(∫t0s‍f2(s,τ)w2(u(τ))ssssssssssss×(∫t0τ‍f3(τ,ξ)w3(u(ξ))dξ)dτ)ds.
In this paper, on the basis of [17, 18], we discuss a new multiple iterated Volterra-Fredholm type integral inequality
(11)u(t)≤c+∫α(t0)α(t)‍h1(t1)ssssssssssss×[∫α(t0)t1‍f1(t1)φ1(u(t1))+∫α(t0)t1‍h2(t2)sssssssssssssssss×[∫α(t0)t1‍f2(t2)φ2(u(t2))+⋯sssssssssssssssssssss+∫α(t0)tn-2‍hn-1(tn-1)sssssssssssssssssssssssss×[∫α(t0)tn-1‍fn-1(tn-1)φn-1(u(tn-1))sssssssssssssssssssssssssssss+∫α(t0)tn-1‍hn(tn)φn(u(tn))dtn]ssssssssssssssssssssssssssssssssssss×dtn-1⋯∫α(t0)tn-1‍]dt2]dt1+∫α(t0)α(T)‍h1(t1)[∫α(t0)tn-1‍f1(t1)φ1(u(t1))+∫α(t0)t1‍h2(t2)sssssssssssssssssss×[∫α(t0)tn-1‍f2(t2)φ2(u(t2))+⋯sssssssssssssssssssssss+∫α(t0)tn-2‍hn-1(tn-1)ssssssssssssssssssssssssssss×[∫α(t0)tn-2‍fn-1(tn-1)φn-1(u(tn-1))sssssssssssssssssssssssssssssssss+∫α(t0)tn-1‍hn(tn)φn(u(tn))ssssssssssssssssssssssssssssssssssss×dtn∫α(t0)tn-1‍]dtn-1⋯]dt2]dt1.
Using monotonization of some functions, we simplify the above multicomposition in an operator form. The unknown function will be estimated by known functions. Furthermore, we apply our result to retarded nonlinear Volterra-Fredholm type equations for estimation of solutions.


## 2. Preliminaries

Throughout this paper, letR denote the set of real numbers, R+=[0,+∞) and I=[t0,T]. For k=0,1, let Ck(M,S) denote the class of kth order continuously differentiable functions defined on the set M and ranged in the set S. For simplicity, we use the product ∏i=1nfi and ∏i=n1fi to present the composition f1∘f2∘⋯∘fn and fn∘fn-1∘⋯∘f1.


### 2.1. Monotonization

First, we monotonize thoseφis in inequality (11). Define
(12)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,
recursively. One can prove that (P1)
eachwi is a nondecreasing nonnegative continuous function,(P2)
φi(u)≤wi(u), i=1,…,n,(P3)
wi+1 hasstronger monotonicity than wi, denoted by wi∝wi+1, i=1,2,…,n-1; that is, by the definition given in [4, 6], the ratios wi+1(u)/wi(u), i=1,…,n-1, are all nondecreasing.Thus, the sequence{φi} can be replaced by a larger but monotonous one {wi} in (11). For a given constant ui>0, define functions Wi(u,u1), i=1,2,…,n, recursively by
(13)W1(u,u1):=∫u1u‍dsw1(s),(14)Wi(u,ui):=∫uiu‍wi-1(∏l=1i-1Wl-1(s))dswi(∏l=1i-1Wl-1(s)),i=2,…,n,
where we use Wl and Wl-1 to denote Wl(u,ul) and its inverse when there is no confusion. Clearly, they are all strictly increasing.For given positive constantsci, i=1,2,…,n, define W^i(u) by
(15)W^i(u):=Wi(u)+ci,∀u∈R+,i=1,2,…,n,
where Wi(i=1,2,…,n) are defined by (13) and (14), respectively.Lemma 1.
Suppose thatφi(i=1,2,…,n) are nonnegative and integrable on [t0,T]. Then Wi(i=1,2,…,n) are increasing and continuous differentiable functions, and
(16)ddξ(∏i=n2‍Wi∘W1(ξ))=1wn(ξ),(17)ddξ(∏i=n2‍W^i∘W^1(ξ))=wn-1(∏i=1n-1Wi-1∘(∏i=n-12W^i∘W^1(ξ)))wn(∏i=1n-1Wi-1∘(∏i=n-12W^i∘W^1(ξ)))⋯w1(W1-1∘W^1(ξ))w2(W1-1∘W^1(ξ))1w1(ξ),∀ξ≥k(T).Proof.
By the definition,Wi(s)(i=1,2,…,n) are increasing and continuous differentiable functions. From (13) and (14), we have
(18)ddξ(∏i=n2‍Wi∘W1(ξ))=Wn′(∏i=n-11‍Wi(ξ)⋯)⋯W2′(W1(ξ))W1′(ξ)=wn-1(∏i=1n-1Wi-1(∏i=n-11Wi(ξ)))wn(∏i=1n-1(Wi-1(∏i=n-11Wi(ξ))))⋯w1(W1-1(W1(ξ)))w2(W1-1(W1(ξ)))1w1(ξ)=1wn(ξ).
Moreover,
(19)ddξ(∏i=n2‍W^i∘W^1(ξ))=d(Wn(Wn-1(⋯(W2(W1(ξ)+c1)+c2)⋯)+cn-1)+cn)dξ=Wn′(Wn-1(⋯(W2(W1(ξ)+c1)+c2)⋯)+cn-1)⋯W2′(W1(ξ)+c1)W1′(ξ)=wn-1(∏i=1n-1Wi-1(sssssssss+c2)⋯)+cn-1∏i=1n-1Wi-1(Wn-1))Wn-1(⋯(W2(W1(ξ)+c1)sssssssssssssssssssssssssssss+c2)⋯)+cn-1∏i=1n-1Wi-1(Wn-1)))×(wn(∏i=1n-1Wi-1(Wn-1(⋯(W2(W1(ξ)+c1)+c2)⋯)sssssssssssssssssssssssssssssss+cn-1)∏i=1n-1Wi-1))-1⋯w1(W1-1(W1(ξ)+c1))w2(W1-1(W1(ξ)+c1))1w1(ξ).
By the definitions of W^i(u), we have the relation (17).


### 2.2. Simplification with Operators

Lethi,fi be positive continuous functions in (11), where i=1,2,…,n, fn(t)≡1. Define Hi:C([t0,T])→C([t0,T]) by
(20)Hiv(t):=∫α(t0)t‍hi(s)v(s)ds,i=1,2,…,n,
and define Fi[u]:C([t0,T])→C([t0,T]) by
(21)Fi[u]v(t):=fi(t)wi(u(t))+v(t),i=1,2,…,n.Having defined those operators, we can enlarge inequality (11) by (12) in the simpler form
(22)u(t)≤k(t)+{∏i=1n-1‍(Hi∘Fi[u])∘Hn∘Fn[u]0}(α(t))+{∏i=1n-1‍(Hi∘Fi[u])∘Hn∘Fn[u]0}(α(T)),ssssssssssssssssssssssssssssssssssssssss∀t∈[t0,T],
where 0 denotes a zero function.


### 2.3.W-Function

DefineWi(u)(α(T)) by
(23)Wi(u)(α(T)):=Wi(u)+(∏j=1i-1‍Hj)∘(Hifi(t))(α(T)),sssssssssssssssssssss∀u∈R+,i=1,2,…,n,
where Wi and Hi are defined by (14) and (20), respectively. Since Wi(i=1,2,…,n) are continuous functions, Wi(i=1,2,…,n) are also continuous functions.Define a function(24)W~(ξ):=∏i=n2‍Wi∘W1(2ξ-c)-∏i=n2‍Wi∘W1(ξ)(α(T)),∀ξ>c.Lemma 2.
Suppose thatφi(u)(i=1,…,n) are all continuous such that φi(u)>0 for u>0, Wi(i=1,2,…,n) defined by (13) and (14) satisfy Wi(+∞)=+∞. Suppose that hi,fi, and φi(i=1,…,n) satisfy
(25)1wn(2ξ-c)-V(ξ)≥0,∀ξ>c,
where
(26)V(ξ):=wn-1(∏i=1n-1Wi-1∘(∏i=n-12Wi∘W1(ξ)(α(T))))wn(∏i=1n-1Wi-1∘(∏i=n-12W~i∘W1(ξ)(α(T))))⋯w1(W1-1∘W1(ξ)(α(T)))w2(W1-1∘W1(ξ)(α(T)))1w1(ξ),
and Wi(i=1,2,…,n) are defined by (23). Then, W~(ξ) is nondecreasing, and W~(ξ)=0 has a solution ξ=ξ0 with ξ0>c.Proof.
Using Lemma1 and condition (25), we have
(27)W′(ξ)=1wn(2ξ-c)-V(ξ)≥0.
Thus W~(ξ) is a nondecreasing function. Since W~(c)<0 and lim⁡ξ→∞W~(ξ)=+∞, we see that W~(ξ)=0 has a solution ξ=ξ0 with ξ0>c.


## 2.1. Monotonization

First, we monotonize thoseφis in inequality (11). Define
(12)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,
recursively. One can prove that (P1)
eachwi is a nondecreasing nonnegative continuous function,(P2)
φi(u)≤wi(u), i=1,…,n,(P3)
wi+1 hasstronger monotonicity than wi, denoted by wi∝wi+1, i=1,2,…,n-1; that is, by the definition given in [4, 6], the ratios wi+1(u)/wi(u), i=1,…,n-1, are all nondecreasing.Thus, the sequence{φi} can be replaced by a larger but monotonous one {wi} in (11). For a given constant ui>0, define functions Wi(u,u1), i=1,2,…,n, recursively by
(13)W1(u,u1):=∫u1u‍dsw1(s),(14)Wi(u,ui):=∫uiu‍wi-1(∏l=1i-1Wl-1(s))dswi(∏l=1i-1Wl-1(s)),i=2,…,n,
where we use Wl and Wl-1 to denote Wl(u,ul) and its inverse when there is no confusion. Clearly, they are all strictly increasing.For given positive constantsci, i=1,2,…,n, define W^i(u) by
(15)W^i(u):=Wi(u)+ci,∀u∈R+,i=1,2,…,n,
where Wi(i=1,2,…,n) are defined by (13) and (14), respectively.Lemma 1.
Suppose thatφi(i=1,2,…,n) are nonnegative and integrable on [t0,T]. Then Wi(i=1,2,…,n) are increasing and continuous differentiable functions, and
(16)ddξ(∏i=n2‍Wi∘W1(ξ))=1wn(ξ),(17)ddξ(∏i=n2‍W^i∘W^1(ξ))=wn-1(∏i=1n-1Wi-1∘(∏i=n-12W^i∘W^1(ξ)))wn(∏i=1n-1Wi-1∘(∏i=n-12W^i∘W^1(ξ)))⋯w1(W1-1∘W^1(ξ))w2(W1-1∘W^1(ξ))1w1(ξ),∀ξ≥k(T).Proof.
By the definition,Wi(s)(i=1,2,…,n) are increasing and continuous differentiable functions. From (13) and (14), we have
(18)ddξ(∏i=n2‍Wi∘W1(ξ))=Wn′(∏i=n-11‍Wi(ξ)⋯)⋯W2′(W1(ξ))W1′(ξ)=wn-1(∏i=1n-1Wi-1(∏i=n-11Wi(ξ)))wn(∏i=1n-1(Wi-1(∏i=n-11Wi(ξ))))⋯w1(W1-1(W1(ξ)))w2(W1-1(W1(ξ)))1w1(ξ)=1wn(ξ).
Moreover,
(19)ddξ(∏i=n2‍W^i∘W^1(ξ))=d(Wn(Wn-1(⋯(W2(W1(ξ)+c1)+c2)⋯)+cn-1)+cn)dξ=Wn′(Wn-1(⋯(W2(W1(ξ)+c1)+c2)⋯)+cn-1)⋯W2′(W1(ξ)+c1)W1′(ξ)=wn-1(∏i=1n-1Wi-1(sssssssss+c2)⋯)+cn-1∏i=1n-1Wi-1(Wn-1))Wn-1(⋯(W2(W1(ξ)+c1)sssssssssssssssssssssssssssss+c2)⋯)+cn-1∏i=1n-1Wi-1(Wn-1)))×(wn(∏i=1n-1Wi-1(Wn-1(⋯(W2(W1(ξ)+c1)+c2)⋯)sssssssssssssssssssssssssssssss+cn-1)∏i=1n-1Wi-1))-1⋯w1(W1-1(W1(ξ)+c1))w2(W1-1(W1(ξ)+c1))1w1(ξ).
By the definitions of W^i(u), we have the relation (17).


## 2.2. Simplification with Operators

Lethi,fi be positive continuous functions in (11), where i=1,2,…,n, fn(t)≡1. Define Hi:C([t0,T])→C([t0,T]) by
(20)Hiv(t):=∫α(t0)t‍hi(s)v(s)ds,i=1,2,…,n,
and define Fi[u]:C([t0,T])→C([t0,T]) by
(21)Fi[u]v(t):=fi(t)wi(u(t))+v(t),i=1,2,…,n.Having defined those operators, we can enlarge inequality (11) by (12) in the simpler form
(22)u(t)≤k(t)+{∏i=1n-1‍(Hi∘Fi[u])∘Hn∘Fn[u]0}(α(t))+{∏i=1n-1‍(Hi∘Fi[u])∘Hn∘Fn[u]0}(α(T)),ssssssssssssssssssssssssssssssssssssssss∀t∈[t0,T],
where 0 denotes a zero function.


## 2.3.W-Function

DefineWi(u)(α(T)) by
(23)Wi(u)(α(T)):=Wi(u)+(∏j=1i-1‍Hj)∘(Hifi(t))(α(T)),sssssssssssssssssssss∀u∈R+,i=1,2,…,n,
where Wi and Hi are defined by (14) and (20), respectively. Since Wi(i=1,2,…,n) are continuous functions, Wi(i=1,2,…,n) are also continuous functions.Define a function(24)W~(ξ):=∏i=n2‍Wi∘W1(2ξ-c)-∏i=n2‍Wi∘W1(ξ)(α(T)),∀ξ>c.Lemma 2.
Suppose thatφi(u)(i=1,…,n) are all continuous such that φi(u)>0 for u>0, Wi(i=1,2,…,n) defined by (13) and (14) satisfy Wi(+∞)=+∞. Suppose that hi,fi, and φi(i=1,…,n) satisfy
(25)1wn(2ξ-c)-V(ξ)≥0,∀ξ>c,
where
(26)V(ξ):=wn-1(∏i=1n-1Wi-1∘(∏i=n-12Wi∘W1(ξ)(α(T))))wn(∏i=1n-1Wi-1∘(∏i=n-12W~i∘W1(ξ)(α(T))))⋯w1(W1-1∘W1(ξ)(α(T)))w2(W1-1∘W1(ξ)(α(T)))1w1(ξ),
and Wi(i=1,2,…,n) are defined by (23). Then, W~(ξ) is nondecreasing, and W~(ξ)=0 has a solution ξ=ξ0 with ξ0>c.Proof.
Using Lemma1 and condition (25), we have
(27)W′(ξ)=1wn(2ξ-c)-V(ξ)≥0.
Thus W~(ξ) is a nondecreasing function. Since W~(c)<0 and lim⁡ξ→∞W~(ξ)=+∞, we see that W~(ξ)=0 has a solution ξ=ξ0 with ξ0>c.


## 3. Main Result

The following theorem shows that the unknown function is estimated by the given known functions.Theorem 3.
Letc be a positive constant. Suppose that hn(t), fi(t), and hi(t)∈C(I,R+), i=1,…,n-1. Suppose that hi, fi, hi, and φi(i=1,…,n) satisfy the assumption of Lemma 2. Suppose that α∈C1(I,I) is nondecreasing such that α(t)≤t on I. Then the unknown u(t) in (22) is estimated as
(28)u(t)≤∏i=1n‍Wi-1{∏i=n2‍Wi∘W1(ξ0)(α(t))},∀t∈[t0,T].Remark 4.
As explained in Remark 2 in [6], different choices of ui in the definitions (13)-(14) of Wi do not affect our results (28).Proof.
For convenience, we cite some definitions in the discussion of our proof as follows: for each fixed positive continuous functionu, define Fij[u]:C([t0,T])→C([t0,T]) by
(29)Fij[u]v(t):=fi(t)wi(W1-1(⋯(Wj-1-1(u(t)))⋯))wj(W1-1(⋯(Wj-1-1(u(t)))⋯))+v(t),
where j<i, i=2,…,n, j=1,2,…,n-1.
From (22), we have
(30)u(t)≤c+{(∏i=1n-1‍Hi∘Fi[u])∘Hn∘Fn[u]0}(α(t))+{(∏i=1n-1‍Hi∘Fi[u])∘Hn∘Fn[u]0}(α(T)),sssssssssssssssssssssssssssssssssssssssss∀t∈[t0,T1].
Define a function z1(t) by the function on the right-hand side of (30). Then, z1(t) is a positive and nondecreasing function on [t0,T]. Using (30), we have
(31)u(t)≤z1(t),∀t∈[t0,T],(32)z1(t0)=c+{(∏i=1n-1‍Hi∘Fi[u])∘Hn∘Fn[u]0}(α(T)).
Differentiating z1(t) with respect to t, using (31) we have
(33)z1′(t)=α′(t)h1(α(t))×[∏i=2n-1‍f1(α(t))w1(u(α(t)))ssss+[(∏i=2n-1‍Hi∘Fi[u])∘Hn∘Fn[u]0](α(t))]≤α′(t)h1(α(t))×[∏i=2n-1‍f1(α(t))w1(z1(α(t)))ssss+[(∏i=2n-1‍Hi∘Fi[z1])∘Hn∘Fn[z1]0](α(t))],
for all t∈[t0,T]. From (33), we have
(34)z1′(t)w1(z1(t))≤α′(t)h1(α(t))×[∏i=2n-1‍f1(α(t))+[(∏i=2n-1‍Hi∘Fi1[z1])ssssssssssssssssssss∘Hn∘Fn1[z1]0∏i=2n-1‍](α(t))],
for all t∈[t0,T]. Integrating both sides of the above inequality from t0 to t, we have
(35)W1(z1(t))≤W1(z1(t0))+H1(f1(s))(α(t))+[H1∘(∏i=2n-1‍Hi∘Fi1[z1])∘Hn∘Fn1[z1]0](α(t))≤W1(z1(t0))+H1(f1(s))(α(T))+[H1∘(∏i=2n-1‍Hi∘Fi1[z1])∘Hn∘Fn1[z1]0](α(t)),
for t∈[t0,T].
Letz2(t) denote the function on the right-hand side of (35); we can see that z2(t) is a positive and nondecreasing function on [t0,T]. From (35), we obtain
(36)z1(t)≤W1-1(z2(t)),∀t∈[t0,T],(37)z2(t0)=W1(z1(t0))+(H1f1(s))(α(T)).
Differentiating z2(t) with respect to t, using (36) we obtain
(38)z2′(t)=α′(t)h1(α(t))×[(∏i=3n-1‍Hi∘Fi1[z1])∫α(t0)α(t)‍h2(t2)[(∏i=3n-1‍Hi∘Fi1[z1])f2(t2)w2(z1(t2))w1(z1(t2))ssssssssssssssssssssssss+[(∏i=3n-1‍Hi∘Fi1[z1])sssssssssssssssssssssssss∘Hn∘Fn1[z1]0(∏i=3n-1‍Hi∘Fi1[z1])]]dt2(∏i=3n-1‍Hi∘Fi1[z1])]≤α′(t)h1(α(t))[∫α(t0)α(t)‍h2(t2)sssssssssssssssssssss×[(∏i=3n-1‍Hi∘Fi1[W1-1∘z2])f2(t2)w2(W1-1(z2(t2)))w1(W1-1(z2(t2)))ssssssssssssssssssssssssss+[(∏i=3n-1‍Hi∘Fi1[W1-1∘z2])∘Hnssssssssssssssssssssssssssssss∘Fn1[W1-1∘z2]0(∏i=3n-1‍Hi∘Fi1[W1-1∘z2])](∏i=3n-1‍Hi∘Fi1[W1-1∘z2])]dt2(∏i=3n-1‍Hi∘Fi1[W1-1∘z2])],ssssssssssssssssssssssssssssssssssssssssssssssss∀t∈[t0,T].
From (38), we have
(39)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))[H2∘(∏i=3n-1‍Hi∘Fi2[z2])ssssssssssssssssssssssss∘Hn∘Fn2[z2]0∏i=3n-1‍](α(t)),
for all t∈[t0,T]. From (39), we have
(40)W2(z2(t))≤W2(z2(t0))+[H1∘(H2f2(s))](α(t))+[H1∘H2∘(∏i=3n-1‍Hi∘Fi2[z2])sssssss∘Hn∘Fn2[z2]0∏i=3n-1‍](α(t))≤W2(z2(t0))+[H1∘(H2f2(s))](α(T))+[H1∘H2∘(∏i=3n-1‍Hi∘Fi2[z2])ssssss∘Hn∘Fn2[z2]0∏i=3n-1‍](α(t)),
for all t∈[t0,T]. Proceeding with the same derivation as in (36) to (40) and so on, we obtain
(41)Wn-2(zn-2(t))≤Wn-2(zn-2(t0))+[(∏i=1n-2‍Hi)fn-2(s)](α(t))+[(∏i=1n-2‍Hi)∘Hn-1∘F(n-1)(n-2)[zn-2]ssssssss∘Hn∘Fn(n-2)[zn-2]0∏i=1n-2‍](α(t))≤Wn-2(zn-2(t0))+[(∏i=1n-2‍Hi)fn-2(s)](α(T))+[(∏i=1n-2‍Hi)∘Hn-1∘F(n-1)(n-2)[zn-2]sssssssss∘Hn∘Fn(n-2)[zn-2]0∏i=1n-2‍](α(t)),
for all t∈[t0,T], where Wn-2 is defined by (14).
Define a functionzn-1(t) by the function on the right-hand side of (41). Then zn-1(t) is a positive and nondecreasing function on [t0,T]. From (41), we get
(42)zn-2(t)≤Wn-2-1(zn-1(t)),∀t∈[t0,T],(43)zn-1(t0)=Wn-2(zn-2(t0))+[(∏i=1n-2‍Hi)fn-2(s)](α(T)).
Differentiating zn-1(t) with respect to t, we have
(44)zn-1′(t)=α′(t)h1(α(t))×[(∏i=2n-1‍Hi)∘F(n-1)(n-2)[zn-2]sssssss∘Hn∘Fn(n-2)[zn-2]0∏i=2n-1‍](α(t))≤α′(t)h1(α(t))×[(∏i=2n-1‍Hi)∘F(n-1)(n-2)[Wn-2-1∘zn-1]ssssss∘Hn∘Fn(n-2)[Wn-2-1∘zn-1]0∏i=2n-1‍](α(t)),
for all t∈[t0,T]. Then (44) is equivalent to
(45)zn-1′(t)wn-2(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn-1)))⋯)))wn-1(W1-1(W2-1(⋯(Wn-2-1(zn-1(tn-1)))⋯)))≤α′(t)h1(α(t))[(∏i=2n-1‍Hi)fn-1(s)](α(t))+α′(t)h1(α(t))×[(∏i=2n-1‍Hi)∘Hn∘Fn(n-1)[zn-1]0∏i=2n-1‍](α(t)),
for all t∈[t0,T]. Integrating both sides of (45) from t0 to t, we have
(46)Wn-1(zn-1(t))≤Wn-1(zn-1(t0))+[(∏i=1n-1‍Hi)fn-1(s)](α(t))+[(∏i=1n-1‍Hi)∘Hn∘Fn(n-1)[zn-1]0](α(t))≤Wn-1(zn-1(t0))+[(∏i=1n-1‍Hi)fn-1(s)](α(T))+[(∏i=1n-1‍Hi)∘Hn∘Fn(n-1)[zn-1]0](α(t)),
for all t∈[t0,T].
Define a functionzn(t) by the function on the right-hand side of (46); then, zn(t) is a positive and nondecreasing function on [t0,T]. From (46), we have
(47)zn-1(t)≤Wn-1-1(zn(t)),∀t∈[t0,T],(48)zn(t0)=Wn-1(zn-1(t0))+[(∏i=1n-1‍Hi)fn-1(s)](α(T)).
Differentiating zn(t) with respect to t, using (47) we have
(49)zn′(t)=α′(t)h1(α(t))[(∏i=2n‍Hi)∘Fn(n-1)[zn-1]0](α(t))≤α′(t)h1(α(t))×[(∏i=2n‍Hi)∘Fn(n-1)[Wn-1-1∘zn]0](α(t)),
for all t∈[t0,T]. From (49), we have
(50)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))[(∏i=2n‍Hi)1](α(t)),
for all t∈[t0,T], where 1 denote the constant function v(t)≡1. Integrating both sides of the above inequality from t0 to t, we obtain
(51)Wn(zn(t))-Wn(zn(t0))≤[(∏i=1n‍Hi)1](α(t)),
for all t∈[t0,T]. From (36), (42), (47), and (51), we have
(52)z1(t)≤W1-1(W2-1(⋯(Wn-1-1(zn(t)))⋯))≤∏i=1n‍Wi-1{Wn(zn(t0))+[(∏i=1n‍Hi)1](α(t))},
for all t∈[t0,T]. Substituting (37), (43), and (48) into (52), we have
(53)z1(t)≤∏i=1n‍Wi-1{Wn(∏i=n-12‍Wi∘W1(z1(t0))(α(T)))ssssssssssssss+[(∏i=1n‍Hi)1](α(t))},
for all t∈[t0,T]. Since T is chosen arbitrarily, we have
(54)z1(t)≤∏i=1n‍Wi-1{Wn(∏i=n2‍Wi∘W1(z1(t0))(α(t)))},
for all t∈[t0,T]. By the definition of z1 and (32), we have
(55)2z1(t0)-c=c+2{(∏i=1n-1‍Hi∘Fi[u])∘Hn∘Fn[u]0}(α(T))=z1(T).
From (54) and (55), we have
(56)2z1(t0)-c≤∏i=1n‍Wi-1{∏i=n2‍Wi∘W1(z1(t0))(α(T))},
or
(57)∏i=n2‍Wi∘W1(2ξ-c)-∏i=n2‍Wi∘W1(ξ)(α(T))≤0.
By the definition of W(ξ), the assumption of Theorem 3, and (57), we observe that
(58)W(z1(t0))≤0=W(ξ0).
By Lemma 2, W(ξ) is increasing. From the last inequality and (31) we have the desired estimation (28).We define the following functions:(59)E(u)=W2{W1(2u-c)}-W2{W1(u)+(H1f1(t))(α(T))}-H1∘(H21)(α(T)),
for all u>c, where Wi, i=1,2 are defined by (13) and (14), respectively.Example 5.
Letn=2, f1(t), hi(t), φi, Wi, i=1, 2, α be as in Theorem 3; c is a positive constant. Suppose that the function E(u) is increasing and E(u)=0 has a solution c for u>c. If u(t) satisfies (22), then
(60)u(t)≤W1-1{W2{W1(c)+(H1f1(t))(α(t))}W2-1{W2{W1(c)+(H1f1(t))(α(t))}W2{W1(c)+(H1f1(t))(α(t))}sssssssssssssssss-H1∘(H21)(α(t))W2{W1(c)+(H1f1(t))(α(t))}}W2{W1(c)+(H1f1(t))(α(t))}W2-1{W2{W1(c)+(H1f1(t))(α(t))}W2{W1(c)+(H1f1(t))(α(t))}},∀t∈[t0,T],
where Wi-1(i=1,2) are inverse functions of Wi, respectively.Remark 6.
Ifw1=w2 in Example 5, then the result in Example 5 will yield the conclusion that appeared in Theorem 2.1′ in [17]. Since if w1=w2, then W2(u)=u-u2, W2-1(u)=u+u2, from (60), we have u(t)≤W1-1{W1(c)+H1(t)+H2(t)} for all t∈[t0,T].


## 4. Application

In this section, we apply our result in Theorem3 to investigate the retarded Volterra-Fredholm integral equations
(61)x(t)=x0+∫t0t‍F1{s,x(s-d(s)),∫t0s‍F2[τ,x(τ-d(τ))]dτ}ds+∫t0T‍F1{s,x(s-d(s)),∫t0s‍F2[τ,x(τ-d(τ))]dτ}ds,
for t∈[t0,T], where x∈C(I,R), d∈C1(I,I) is nondecreasing with t-d(t)≥t0, d(t0)=0, d′(t)<1, F1∈C(I×R2,R), and F2∈C(I×R,R). Let β(t)=t-d(t); then β(t)∈C1(I,I), β(t)≤t. Since β′(t)=1-d′(t)>0, β(t) is an increasing and invertible function.The following corollary gives the bound on the solution of (61).Corollary 7.
Suppose that theF1, F2 in (61) satisfy the conditions
(62)|F1(s,x,y)|≤h1(s)[f1(s)w1(|x|)+|y|],(63)|F2(s,x)|≤h2(s)w2(|x|),
where f1(s), h1(s), h2(s), w1(s), and w2(s) are as in Theorem 3; let M=max⁡t∈[t0,T](1/β′(β-1(t)))<∞. Assume that the function
(64)E*(u)=W2[W1(2u-|x0|)]-W2[W1(u)+∫β(t0)β(T)‍Mh1(s)f1(s)ds]-∫β(t0)β(T)‍Mh1(s)×[∫β(t0)s‍Mh2(τ)dτ]ds
is increasing and E*(u)=0 has a solution c for u>ξ0. If x(t) is a solution of (61), then
(65)|x(t)|≤W1-1×{W2-1[W2[∫β(t0)β(t)‍W1(c)sssssssssssssssss+∫β(t0)β(t)‍Mh1(β-1(s))f1(β-1(s))ds]sssssssssssss+∫β(t0)β(t)‍Mh1(β-1(s))sssssssssssssss×[∫β(t0)s‍Mh2(β-1(τ))dτ]ds]},sssssssssssssssssssssssssssssssssssssssss∀t∈[t0,T],
where W1, W2, W1-1, and W2-1 are as in Theorem 3.Proof.
Using the conditions (62)-(63) we have
(66)|x(t)|≤|x0|+∫t0t‍h1(s)[∫t0s‍f1(s)w1(|x(s-d(s))|)ssssssssssssss+∫t0s‍h2(τ)w2(|x(τ-d(τ))|)dτ]ds+∫t0T‍h1(s)[∫t0s‍f1(s)w1(|x(s-d(s))|)sssssssssssssss+∫t0s‍h2(τ)w2(|x(τ-d(τ))|)dτ]ds=|x0|+∫t0t‍h1(s)[∫t0s‍f1(s)w1(|x(β(s))|)ssssssssssssssssssss+∫t0s‍h2(τ)w2(|x(β(τ))|)dτ]ds+∫t0T‍h1(s)[∫t0s‍f1(s)w1(|x(β(s))|)sssssssssssssss+∫t0s‍h2(τ)w2(|x(β(τ))|)dτ]ds≤|x0|+∫t0t‍h1(s)[∫t0s‍f1(s)w1(|x(β(s))|)sssssssssssssss+∫β(t0)β(s)‍Mh2(β-1(τ))w2(|x(τ)|)dτ]ds+∫t0T‍h1(s)[∫t0s‍f1(s)w1(|x(β(s))|)sssssssssssssss+∫β(t0)β(s)‍Mh2(β-1(τ))w2(|x(τ)|)dτ]ds≤|x0|+∫β(t0)β(t)‍Mh1(β-1(s))×[∫β(t0)s‍f1(β-1(s))w1(|x(s)|)sssss+∫β(t0)s‍Mh2(β-1(τ))w2(|x(τ)|)dτ]ds+∫β(t0)β(T)‍Mh1(β-1(s))sssss×[∫β(t0)β(T)‍f1(β-1(s))w1(|x(s)|)ssssssss+∫β(t0)s‍Mh2(β-1(τ))w2(|x(τ)|)dτ]ds,
for t∈[t0,T], where several changes of variables are made. Applying the result of Theorem 3 to the last inequality, we obtain the desired estimation (65).



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*Source: 327943-2014-07-22.xml*