# Solvability of  Nonlinear Integral Equations of Volterra Type

Authors: Zeqing Liu; Sunhong Lee; Shin Min Kang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper deals with the existence of continuous bounded solutions for a rather general nonlinear integral equation of Volterra type and discusses also the existence and asymptotic stability of continuous bounded solutions for another nonlinear integral equation of Volterra type. The main tools used in the proofs are some techniques in analysis and the Darbo fixed point theorem via measures of noncompactness. The results obtained in this paper extend and improve essentially some known results in the recent literature. Two nontrivial examples that explain the generalizations and applications of our results are also included.


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## Body

## 1. Introduction

It is well known that the theory of nonlinear integral equations and inclusions has become important in some mathematical models of real processes and phenomena studied in mathematical physics, elasticity, engineering, biology, queuing theory economics, and so on (see, [1–3]). In the last decade, the existence, asymptotical stability, and global asymptotical stability of solutions for various Volterra integral equations have received much attention, see, for instance, [1, 4–22] and the references therein.In this paper, we are interested in the following nonlinear integral equations of Volterra type:(1.1)x(t)=f(t,x(a(t)),(Hx)(b(t)),∫0α(t)‍u(t,s,x(c(s)))ds),∀t∈ℝ+,(1.2)x(t)=h(t,x(t),∫0α(t)‍u(t,s,x(c(s)))ds),∀t∈ℝ+,
where the functions f,h,u,a,b,c,α and the operator H appearing in (1.1) are given while x=x(t) is an unknown function.To the best of our knowledge, the papers dealing with (1.1) and (1.2) are few. But some special cases of (1.1) and (1.2) have been investigated by a lot of authors. For example, Arias et al. [4] studied the existence, uniqueness, and attractive behaviour of solutions for the nonlinear Volterra integral equation with nonconvolution kernels
(1.3)x(t)=∫0t‍k(t,s)g(x(s))ds,∀t∈ℝ+.Using the monotone iterative technique, Constantin [13] got a sufficient condition which ensures the existence of positive solutions of the nonlinear integral equation
(1.4)x(t)=L(t)+∫0t‍[M(t,s)x(s)+K(t,s)g(x(s))]ds,∀t∈ℝ+.Roberts [21] examined the below nonlinear Volterra integral equation
(1.5)x(t)=∫0t‍k(t-s)G((x(s),s))ds,∀t∈ℝ+,
which arose from certain models of a diffusive medium that can experience explosive behavior; utilizing the Darbo fixed point theorem and the measure of noncompactness in [7], Banaś and Dhage [6], Banaś et al. [8], Banaś and Rzepka [9, 10], Hu and Yan [16] and Liu and Kang [19] investigated the existence and/or asymptotic stability and/or global asymptotic stability of solutions for the below class of integral equations of Volterra type:
(1.6)x(t)=(Tx)(t)∫0t‍u(t,s,x(s))ds,∀t∈ℝ+,x(t)=f(t,x(t))+∫0t‍u(t,s,x(s))ds,∀t∈ℝ+,x(t)=f(t,x(t))∫0t‍u(t,s,x(s))ds,∀t∈ℝ+,x(t)=g(t,x(t))+x(t)∫0t‍u(t,s,x(s))ds,∀t∈ℝ+,x(t)=f(t,x(t))+g(t,x(t))∫0t‍u(t,s,x(s))ds,∀t∈ℝ+,x(t)=f(t,x(α(t)))+∫0β(t)‍g(t,s,x(γ(s)))ds,∀t∈ℝ+,
respectively. By means of the Schauder fixed point theorem and the measure of noncompactness in [7], Banaś and Rzepka [11] studied the existence of solutions for the below nonlinear quadratic Volterra integral equation:
(1.7)x(t)=p(t)+f(t,x(t))∫0t‍v(t,s,x(s))ds,∀t∈ℝ+.Banaś and Chlebowicz [5] got the solvability of the following functional integral equation
(1.8)x(t)=f1(t,∫0t‍k(t,s)f2(s,x(s))ds),∀t∈ℝ+
in the space of Lebesgue integrable functions on ℝ+. El-Sayed [15] studied a differential equation of neutral type with deviated argument, which is equivalent to the functional-integral equation
(1.9)x(t)=f(t,∫0H(t)‍x(s)ds,x(h(t))),∀t∈ℝ+
by the technique linking measures of noncompactness with the classical Schauder fixed point principle. Using an improvement of the Krasnosel’skii type fixed point theorem, Taoudi [22] discussed the existence of integrable solutions of a generalized functional-integral equation
(1.10)x(t)=g(t,x(t))+f1(t,∫0t‍k(t,s)f2(s,x(s))ds),∀t∈ℝ+.Dhage [14] used the classical hybrid fixed point theorem to establish the uniform local asymptotic stability of solutions for the nonlinear quadratic functional integral equation of mixed type
(1.11)x(t)=f(t,x(α(t)))(q(t)+∫0β(t)‍u(t,s,x(γ(s)))ds),∀t∈ℝ+.The purpose of this paper is to prove the existence of continuous bounded solutions for (1.1) and to discuss the existence and asymptotic stability of continuous bounded solutions for (1.2). The main tool used in our considerations is the technique of measures of noncompactness [7] and the famous fixed point theorem of Darbo [23]. The results presented in this paper extend proper the corresponding results in [6, 9, 10, 15, 16, 19]. Two nontrivial examples which show the importance and the applicability of our results are also included.This paper is organized as follows. In the second section, we recall some definitions and preliminary results and prove a few lemmas, which will be used in our investigations. In the third section, we state and prove our main results involving the existence and asymptotic stability of solutions for (1.1) and (1.2). In the final section, we construct two nontrivial examples for explaining our results, from which one can see that the results obtained in this paper extend proper several ones obtained earlier in a lot of papers.


## 2. Preliminaries

In this section, we give a collection of auxiliary facts which will be needed further on. Letℝ=(-∞,∞) and ℝ+=[0,∞). Assume that (E,∥·∥) is an infinite dimensional Banach space with zero element θ and Br stands for the closed ball centered at θ and with radius r. Let B(E) denote the family of all nonempty bounded subsets of E.Definition 2.1.
LetD be a nonempty bounded closed convex subset of the space E. A operator f:D→E is said to be a Darbo operator if it is continuous and satisfies that μ(fA)≤kμ(A) for each nonempty subset A of D, where k∈[0,1) is a constant and μ is a measure of noncompactness on B(E).The Darbo fixed point theorem is as follows.Lemma 2.2 (see [23]).
LetD be a nonempty bounded closed convex subset of the space E and let f:D→D be a Darbo operator. Then f has at least one fixed point in D.LetBC(ℝ+) denote the Banach space of all bounded and continuous functions x:ℝ+→ℝ equipped with the standard norm
(2.1)∥x∥=sup{|x(t)|:t∈ℝ+}.For any nonempty bounded subsetX of BC(ℝ+),x∈X,t∈ℝ+,T>0 and ε≥0, define
(2.2)ωT(x,ε)=sup{|x(p)-x(q)|:p,q∈[0,T]with|p-q|≤ε},ωT(X,ε)=sup{ωT(x,ε):x∈X},ω0T(X)=limε→0ωT(X,ε),ω0(X)=limT→+∞ω0T(X),X(t)={x(t):x∈X},diamX(t)=sup{|x(t)-y(t)|:x,y∈X},μ(X)=ω0(X)+limsupt→+∞diamX(t).It can be shown that the mappingμ is a measure of noncompactness in the space BC(ℝ+) [4].Definition 2.3.
Solutions of an integral equation are said to beasymptotically stable if there exists a ball Br in the space BC(ℝ+) such that for any ε>0, there exists T>0 with
(2.3)|x(t)-y(t)|≤ε
for all solutions x(t),y(t)∈Br of the integral equation and any t≥T.
It is clear that the concept of asymptotic stability of solutions is equivalent to the concept of uniform local attractivity [9].Lemma 2.4.
Letφ and ψ:ℝ+→ℝ+ be functions with
(2.4)limt→+∞ψ(t)=+∞,limsupt→+∞φ(t)<+∞.
Then(2.5)limsupt→+∞φ(ψ(t))=limsupt→+∞φ(t).Proof.
Letlimsupt→+∞φ(t)=A. It follows that for each ε>0, there exists T>0 such that
(2.6)φ(t)<A+ε,∀t≥T.
Equation (2.4) means that there exists C>0 satisfying
(2.7)ψ(t)≥T,∀t≥C.
Using (2.6) and (2.7), we infer that
(2.8)φ(ψ(t))<A+ε,∀t≥C,
that is, (2.5) holds. This completes the proof.Lemma 2.5.
Leta:ℝ+→ℝ+ be a differential function. If for each T>0, there exists a positive number a-T satisfying
(2.9)0≤a′(t)≤a-T,∀t∈[0,T],
then
(2.10)ωT(x∘a,ε)≤ωa(T)(x,a-Tε),∀(x,ε)∈BC(R+)×(0,+∞).Proof.
LetT>0. It is clear that (2.9) yields that the function a is nondecreasing in [0,T] and for any t,s∈[0,T], there exists ξ∈(0,T) satisfying
(2.11)|a(t)-a(s)|=a′(ξ)|t-s|≤a-T|t-s|
by the mean value theorem. Notice that (2.9) means that a(t)∈[a(0),a(T)]⊆[0,a(T)] for each t∈[0,T], which together with (2.11) gives that
(2.12)ωT(x∘a,ε)=sup{|x(a(t))-x(a(s))|:t,s∈[0,T],|t-s|≤ε}≤sup{|x(p)-x(q)|:p,q∈[a(0),a(T)],|p-q|≤a-Tε}≤ωa(T)(x,a-Tε),∀(x,ε)∈BC(R+)×(0,+∞),
which yields that (2.10) holds. This completes the proof.Lemma 2.6.
Letφ:ℝ+→ℝ+ be a function with limt→+∞φ(t)=+∞ and X be a nonempty bounded subset of BC(ℝ+). Then
(2.13)ω0(X)=limT→+∞ω0φ(T)(X).Proof.
SinceX is a nonempty bounded subset of BC(ℝ+), it follows that ω0(X)=limT→+∞ω0T(X). That is, for given ε>0, there exists M>0 satisfying
(2.14)|ω0T(X)-ω0(X)|<ε,∀T>M.
It follows fromlimt→+∞φ(t)=+∞ that there exists L>0 satisfying
(2.15)φ(T)>M,∀T>L.
By means of (2.14) and (2.15), we get that
(2.16)|ω0φ(T)(X)-ω0(X)|<ε,∀T>L,
which yields (2.13). This completes the proof.


## 3.  Main Results

Now we formulate the assumptions under which (1.1) will be investigated.(H1)
f:ℝ+×ℝ3→ℝ is continuous with f(t,0,0,0)∈BC(ℝ+) and f-=sup{|f(t,0,0,0)|: t∈ℝ+};(H2)
a,b,c,α:ℝ+→ℝ+ satisfy that a and b have nonnegative and bounded derivative in the interval [0,T] for each T>0,c and α are continuous and α is nondecreasing and
(3.1)limt→+∞a(t)=limt→+∞b(t)=+∞;(H3)
u:ℝ+2×ℝ→ℝ is continuous;(H4)
there exist five positive constantsr,M,M0,M1, and M2 and four continuous functions m1, m2,m3,g:ℝ+→ℝ+ such that g is nondecreasing and
(3.2)|f(t,v,w,z)-f(t,p,q,y)|≤m1(t)|v-p|+m2(t)|w-q|+m3(t)|z-y|,∀t∈ℝ+,v,p∈[-r,r],w,q∈[-g(r),g(r)],z,y∈[-M,M],(3.3)limt→+∞sup{m3(t)∫0α(t)‍|u(t,s,v(c(s)))-u(t,s,w(c(s)))|ds:v,w∈Br}=0,(3.4)sup{m3(t)∫0α(t)‍|u(t,s,v(c(s)))|ds:t∈ℝ+,v∈Br}≤M0,(3.5)sup{∫0α(t)‍|u(t,s,v(c(s)))|ds:t∈ℝ+,v∈Br}≤M,(3.6)sup{mi(t):t∈ℝ+}≤Mi,i∈{1,2},(3.7)M1r+M2g(r)+M0+f-≤r,M1+QM2<1;(H5)
H:BC(ℝ+)→BC(R+) satisfies that H:Br→BC(ℝ+) is a Darbo operator with respect to the measure of noncompactness of μ with a constant Q and
(3.8)|(Hx)(b(t))|≤g(|x(b(t))|),∀(x,t)∈Br×ℝ+.Theorem 3.1.
Under Assumptions(H1)–(H5), (1.1) has at least one solution x=x(t)∈Br.Proof.
Letx∈Br and define
(3.9)(Fx)(t)=f(t,x(a(t)),(Hx)(b(t)),∫0α(t)‍u(t,s,x(c(s)))ds),∀t∈ℝ+.
It follows from (3.9) and Assumptions (H1)–(H5) that Fx is continuous on ℝ+ and that
(3.10)|(Fx)(t)|≤|f(t,x(a(t)),(Hx)(b(t)),∫0α(t)‍u(t,s,x(c(s)))ds)-f(t,0,0,0)|+|f(t,0,0,0)|≤m1(t)|x(a(t))|+m2(t)|(Hx)(b(t))|+m3(t)|∫0α(t)‍u(t,s,x(c(s)))ds|+f-≤M1r+M2g(r)+M0+f-≤r,∀t∈ℝ+,
which means that Fx is bounded on ℝ+ and F(Br)⊆Br.
We now prove that(3.11)μ(FX)≤(M1+M2Q)μ(X),∀X⊆Br.
LetX be a nonempty subset of Br. Using (3.2), (3.6), and (3.9), we conclude that
(3.12)|(Fx)(t)-(Fy)(t)|=|f(t,x(a(t)),(Hx)(b(t)),∫0α(t)‍u(t,s,x(c(s)))ds)-f(t,y(a(t)),(Hy)(b(t)),∫0α(t)‍u(t,s,y(c(s)))ds)|≤m1(t)|x(a(t))-y(a(t))|+m2(t)|(Hx)(b(t))-(Hy)(b(t))|+m3(t)∫0α(t)‍|u(t,s,x(c(s)))-u(t,s,y(c(s)))|ds≤M1|x(a(t))-y(a(t))|+M2|(Hx)(b(t))-(Hy)(b(t))|+sup{m3(t)∫0α(t)‍|u(t,s,w(c(s)))-u(t,s,z(c(s)))|ds:w,z∈Br},∀x,y∈X,t∈ℝ+,
which yields that
(3.13)diam(FX)(t)≤M1diamX(a(t))+M2diam(HX)(b(t))+sup{m3(t)∫0α(t)‍|u(t,s,w(c(s)))-u(t,s,z(c(s)))|ds:w,z∈Br},∀t∈ℝ+,
which together with (3.3), Assumption (H2) and Lemma 2.4 ensures that
(3.14)limsupt→+∞diam(FX)(t)≤M1limsupt→+∞diamX(a(t))+M2limsupt→+∞diam(HX)(b(t))+limsupt→+∞sup{m3(t)∫0α(t)‍|u(t,s,w(c(s)))-u(t,s,z(c(s)))|ds:w,z∈Br}=M1limsupt→+∞diamX(t)+M2limsupt→+∞diam(HX)(t),
that is,
(3.15)limsupt→+∞diam(FX)(t)≤M1limsupt→+∞diamX(t)+M2limsupt→+∞diam(HX)(t).
For eachT>0 and ε>0, put
(3.16)M3T=sup{m3(p):p∈[0,T]},urT=sup{|u(p,q,v)|:p∈[0,T],q∈[0,α(T)],v∈[-r,r]},ωrT(u,ε)=sup{|u(p,τ,v)-u(q,τ,v)|:p,q∈[0,T],|p-q|≤ε,τ∈[0,α(T)],v∈[-r,r]},ωT(u,ε,r)=sup{|u(p,q,v)-u(p,q,w)|:p∈[0,T],q∈[0,α(T)],v,w∈[-r,r],|v-w|≤ε},ωrT(f,ε,g(r))=sup{|f(p,v,w,z)-f(q,v,w,z)|:p,q∈[0,T],|p-q|≤ε,v∈[-r,r],w∈[-g(r),g(r)],z∈[-M,M]}.
LetT>0,ε>0,x∈X and t,s∈[0,T] with |t-s|≤ε. It follows from (H2) that there exist a-T and b-T satisfying
(3.17)0≤a′(t)≤a-T,0≤b′(t)≤b-T,∀t∈[0,T].
In light of (3.2), (3.6), (3.9), (3.16), (3.17), and Lemma 2.5, we get that
(3.18)|(Fx)(t)-(Fx)(s)|≤|f(t,x(a(t)),(Hx)(b(t)),∫0α(t)‍u(t,τ,x(c(τ)))dτ)-f(t,x(a(s)),(Hx)(b(s)),∫0α(s)‍u(s,τ,x(c(τ)))dτ)|+|f(t,x(a(s)),(Hx)(b(s)),∫0α(s)‍u(s,τ,x(c(τ)))dτ)-f(s,x(a(s)),(Hx)(b(s)),∫0α(s)‍u(s,τ,x(c(τ)))dτ)|≤m1(t)|x(a(t))-x(a(s))|+m2(t)|(Hx)(b(t))-(Hx)(b(s))|+m3(t)[|∫α(s)α(t)‍|u(t,τ,x(c(τ)))|dτ|+∫0α(s)‍|u(t,τ,x(c(τ)))-u(s,τ,x(c(τ)))|dτ]+sup{|f(p,v,w,z)-f(q,v,w,z)|:p,q∈[0,T],|p-q|≤ε,v∈[-r,r],w∈[-g(r),g(r)],z∈[-M,M]}≤M1ωT(x∘a,ε)+M2ωT((Hx)∘b,ε)+M3T|α(t)-α(s)|sup{|u(p,τ,v)|:p∈[0,T],τ∈[0,α(T)],v∈[-r,r]}+M3Tα(T)sup{|u(p,τ,v)-u(q,τ,v)|:p,q∈[0,T],|p-q|≤ε,τ∈[0,α(T)],v∈[-r,r]}+ωrT(f,ε,g(r))≤M1ωa(T)(x,a-Tε)+M2ωb(T)(Hx,b-Tε)+M3TωT(α,ε)urT+M3Tα(T)ωrT(u,ε)+ωrT(f,ε,g(r)),
which implies that
(3.19)ωT(Fx,ε)≤M1ωa(T)(x,a-Tε)+M2ωb(T)(Hx,b-Tε)+M3TωT(α,ε)urT+M3Tα(T)ωrT(u,ε)+ωrT(f,ε,g(r)),∀T>0,ε>0,x∈X.
Notice that Assumptions(H1)–(H3) imply that the functions α=α(t),f=f(t,p,q,v) and u=u(t,y,z) are uniformly continuous on the sets [0,T],  [0,T]×[-r,r]×[-g(r),g(r)]×[-M,M] and [0,T]× [0,α(T)]×[-r,r], respectively. It follows that
(3.20)limε→0ωT(α,ε)=limε→0ωrT(u,ε)=limε→0ωrT(f,ε,g(r))=0.
In terms of (3.19) and (3.20), we have
(3.21)ω0T(FX)≤M1ω0a(T)(X)+M2ω0b(T)(HX),
lettingT→+∞ in the above inequality, by Assumption (H2) and Lemma 2.6, we infer that
(3.22)ω0(FX)≤M1ω0(X)+M2ω0(HX).
By means of (3.15), (3.22), and Assumption (H5), we conclude immediately that
(3.23)μ(FX)=ω0(FX)+limsupt→+∞diam(FX)(t)≤M1ω0(X)+M2ω0(HX)+M1limsupt→+∞diamX(t)+M2limsupt→+∞diam(HX)(t)=M1μ(X)+M2(ω0(HX)+limsupt→+∞diam(HX)(t))≤(M1+M2Q)μ(X),
that is, (3.11) holds.
Next we prove thatF is continuous on the ball Br. Let x∈Br and {xn}n≥1⊂Br with limn→∞xn=x. It follows from (3.3) that for given ε>0, there exists a positive constant T such that
(3.24)sup{m3(t)∫0α(t)‍|u(t,s,v(c(s)))-u(t,s,w(c(s)))|ds:v,w∈Br}<ε3,∀t>T.
Sinceu=u(t,s,v) is uniformly continuous in [0,T]×[0,α(T)]×[-r,r], it follows from (3.16) that there exists δ0>0 satisfying
(3.25)ωT(u,δ,r)<ε1+3M3Tα(T),∀δ∈(0,δ0).
By Assumption(H5) and limn→∞xn=x, we know that there exists a positive integer N such that
(3.26)(1+M1)∥xn-x∥+M2∥Hxn-Hx∥<12min{ε,δ0},∀n>N.
In view of (3.2), (3.6), (3.9), (3.24)–(3.26), and Assumption (H2), we gain that for any n>N and t∈ℝ+(3.27)|(Fxn)(t)-(Fx)(t)|=|f(t,xn(a(t)),(Hxn)(b(t)),∫0α(t)‍u(t,s,xn(c(s)))ds)-f(t,x(a(t)),(Hx)(b(t)),∫0α(t)‍u(t,s,x(c(s)))ds)|≤m1(t)|xn(a(t))-x(a(t))|+m2(t)|(Hxn)(b(t))-(Hx)(b(t))|+m3(t)∫0α(t)‍|u(t,s,xn(c(s)))-u(t,s,x(c(s)))|ds≤M1∥xn-x∥+M2∥Hxn-Hx∥+max{{∑1211}supτ>Tsup{m3(τ)∫0α(τ)‍|u(τ,s,z(c(s)))-u(τ,s,y(c(s)))|ds:z,y∈Br},supτ∈[0,T]sup{m3(τ)∫0α(τ)‍|u(τ,s,z(c(s)))-u(τ,s,y(c(s)))|ds:z,y∈Br,∥z-y∥≤δ02{∫0α(τ)‍}}{∑1211}}<ε2+max{ε3,supτ∈[0,T]{m3(τ)∫0α(τ)‍ωT(u,δ02,r)ds}}<ε2+max{ε3,M3Tα(T)ε1+3M3Tα(T)}=5ε6,
which yields that
(3.28)∥Fxn-Fx∥<ε,∀n>N,
that is, F is continuous at each point x∈Br.
Thus Lemma2.2 ensures that F has at least one fixed point x=x(t)∈Br. Hence (1.1) has at least one solution x=x(t)∈Br. This completes the proof.Now we discuss (1.2) under below hypotheses:(H6)
h:ℝ+×ℝ2→ℝ is continuous with h(t,0,0)∈BC(ℝ+) and h-=sup{|h(t,0,0)|:t∈ℝ+};(H7)
c,α:ℝ+→ℝ+ are continuous and α is nondecreasing;(H8)
there exist two continuous functionsm1,m3:ℝ+→ℝ+ and four positive constants r,M,M0 and M1 satisfying (3.3)–(3.5),
(3.29)|h(t,v,w)-h(t,p,q)|≤m1(t)|v-p|+m3(t)|w-q|,∀t∈ℝ+,v,p∈[-r,r],w,q∈[-M,M],(3.30)sup{m1(t):t∈ℝ+}≤M1,(3.31)M0+h-≤r(1-M1).Theorem 3.2.
Under Assumptions(H3) and (H6)–(H8), (1.2) has at least one solution x=x(t)∈Br. Moreover, solutions of (1.2) are asymptotically stable.Proof.
As in the proof of Theorem3.1, we conclude that (1.2) possesses at least one solution in Br.
Now we claim that solutions of (1.2) are asymptotically stable. Note that r,M0, and M1 are positive numbers and h-≥0, it follows from (3.31) that M1<1. In terms of (3.3), we infer that for given ε>0, there exists T>0 such that
(3.32)sup{m3(t)∫0α(t)‍|u(t,s,v(c(s)))-u(t,s,w(c(s)))|ds:v,w∈Br}<ε(1-M1),∀t≥T.
Letz=z(t),y=y(t) be two arbitrarily solutions of (1.2) in Br. According to (3.29)–(3.32), we deduce that
(3.33)|z(t)-y(t)|=|h(t,z(t),∫0α(t)‍u(t,τ,z(c(τ)))dτ)-h(t,y(t),∫0α(t)‍u(t,τ,y(c(τ)))dτ)|≤m1(t)|z(t)-y(t)|+m3(t)∫0α(t)‍|u(t,τ,z(c(τ)))-u(t,τ,y(c(τ)))|dτ≤M1|z(t)-y(t)|+sup{m3(t)∫0α(t)‍|u(t,τ,v(c(τ)))-u(t,τ,w(c(τ)))|dτ:v,w∈Br}<M1|z(t)-y(t)|+ε(1-M1),∀t≥T,
which means that
(3.34)|z(t)-y(t)|<ε
whenever z,y are solutions of (1.2) in Br and t≥T. Hence solutions of (1.2) are asymptotically stable. This completes the proof.Remark 3.3.
Theorems3.1 and 3.2 generalize Theorem  3.1 in [6], Theorem  2 in [9], Theorem  3 in [10], Theorem  1 in [15], Theorem  2 in [16], and Theorem  3.1 in [19]. Examples 4.1 and 4.2 in the fourth section show that Theorems 3.1 and 3.2 substantially extend the corresponding results in [6, 9, 10, 15, 16, 19].


## 4.  Examples

In this section, we construct two nontrivial examples to support our results.Example 4.1.
Consider the following nonlinear integral equation of Volterra type:(4.1)x(t)=110+ln3(1+t3)+tx2(1+3t2)200(1+t)+1103+20t+3|x(t3)|+19+1+t(∫0t2‍s2x(4s)sin(1+t3s5-(t-s)4x3(4s))1+t12+|stx3(4s)-3s3+7(t-2s)2|x2(4s)ds)3,∀t∈ℝ+.
Put
(4.2)f(t,v,w,z)=110+ln3(1+t3)+tv2200(1+t)+1103+20t+|w|+z39+1+t,u(t,s,p)=s2psin(1+t3s5-(t-s)4p3)1+t12+|stp3-3s3+7(t-2s)2|p2,a(t)=1+3t2,b(t)=t3,c(t)=4t,α(t)=t2,∀t,s∈ℝ+,v,w,z,p∈ℝ.
Let(4.3)r∈[9-512,9+512],M=3,M0=9r10,M1=r100,M2=1300,f-=110,Q=3,m1(t)=rt100(1+t),m2(t)=1(103+20t)2,m3(t)=3M29+1+t,(Hy)(t)=3y(t),g(t)=3t,∀t∈ℝ+,y∈BC(ℝ+).
It is easy to verify that (3.6) and Assumptions (H1)–(H3) and (H5) are satisfied. Notice that
(4.4)M1r+M2g(r)+M0+f-≤r⟺r∈[9-512,9+512],|f(t,v,w,z)-f(t,p,q,y)|≤t200(1+t)|v2-p2|+|1103+20t+|w|-1103+20t+|q||+19+1+t|z3-y3|≤m1(t)|v-p|+m2(t)|w-q|+m3(t)|z-y|,∀t∈ℝ+,v,p∈[-r,r],w,q∈[-g(r),g(r)],z,y∈[-M,M],sup{m3(t)∫0α(t)‍|u(t,s,v(c(s)))-u(t,s,w(c(s)))|ds:v,w∈Br}=sup{3M29+1+t∫0t2‍|s2v(4s)sin(1+t3s5-(t-s)4v3(4s))1+t12+|stv3(4s)-3s3+7(t-2s)2|v2(4s)-s2w(4s)sin(1+t3s5-(t-s)4w3(4s))1+t12+|stw3(4s)-3s3+7(t-2s)2|w2(4s)|ds:v,w∈Br}≤3M29+1+t∫0t2‍2s2(1+t12)2[r(1+t12)+r3(r3t3+3r6+7(t+2t2)2)]ds≤2M2t6(9+1+t)(1+t12)2{r(1+t12)+r3[r3t3+3r6+7(t+2t2)2]}→0ast→+∞,sup{m3(t)∫0α(t)‍|u(t,s,v(c(s)))|ds:t∈ℝ+,v∈Br}=sup{3M29+1+t∫0t2‍|s2v(4s)sin(1+t3s5-(t-s)4v3(4s))1+t12+|stv3(4s)-3s3+7(t-2s)2|v2(4s)|ds:t∈ℝ+,v∈Br}≤sup{3M210∫0t‍s2ds1+t12:t∈ℝ+}=rM220<M0,sup{∫0α(t)‍|u(t,s,v(c(s)))|ds:t∈ℝ+,v∈Br}=sup{3M29+1+t∫0t2‍|s2v(4s)sin(1+t3s5-(t-s)4v3(4s))1+t12+|stv3(4s)-3s3+7(t-2s)2|v2(4s)|ds:t∈ℝ+,v∈Br}≤sup{3M210∫0t‍s2ds1+t12:t∈ℝ+}=r6≤M,
that is, (3.2)–(3.5) and (3.7) hold. Hence all Assumptions of Theorem 3.1 are fulfilled. Consequently, Theorem 3.1 ensures that (4.1) has at least one solution x=x(t)∈Br. However Theorem  3.1 in [6], Theorem  2 in [9],Theorem  3 in [10], Theorem  1 in [15], Theorem  2 in [16], and Theorem  3.1 in [19] are unapplicable for (4.1).Example 4.2.
Consider the following nonlinear integral equation of Volterra type:(4.5)x(t)=3+sin4(1+t2)16+x2(t)8+t2+11+t(∫0t‍sx3(1+s2)1+t2+scos2(1+t3s7x2(1+s2))ds)2,∀t∈ℝ+.
Put(4.6)h(t,v,w)=3+sin4(1+t2)16+v28+t2+w21+t,u(t,s,p)=sp31+t2+scos2(1+t3s7p2),α(t)=t,c(t)=1+t2,m1(t)=18+t2,m3(t)=41+t,∀t,s∈ℝ+,v,w,p∈ℝ,r=12,M=2,M0=M1=18,h-=14.
It is easy to verify that (3.30), (3.31) and Assumptions (H6) and (H7) are satisfied. Notice that
(4.7)|h(t,v,w)-h(t,p,q)|≤18+t2|v2-p2|+11+t|w2-q2|≤m1(t)|v-p|+m3(t)|z-y|,∀t∈ℝ+,v,p∈[-r,r],w,q∈[-M,M],sup{m3(t)∫0α(t)‍|u(t,s,v(c(s)))-u(t,s,w(c(s)))|ds:x,y∈Br}=sup{41+t∫0t‍|sv3(1+s2)1+t2+scos2(1+t3s7v2(1+s2))-sw3(1+s2)1+t2+scos2(1+t3s7w2(1+s2))ds|ds:v,w∈Br}≤4r3t(1+t+t2)(1+t)(1+t2)2→0ast→+∞,sup{m3(t)∫0α(t)‍|u(t,s,v(c(s)))|ds:t∈ℝ+,v∈Br}=sup{41+t∫0t‍|sv3(1+s2)1+t2+scos2(1+t3s7v2(1+s2))ds|:t∈ℝ+,v∈Br}≤sup{4r31+t∫0t‍s1+t2ds:t∈ℝ+}≤sup{t4(1+t)(1+t2):t∈ℝ+}≤18=M1,sup{∫0α(t)‍|u(t,s,v(c(s)))|ds:t∈ℝ+,v∈Br}=sup{∫0t‍|sv3(1+s2)1+t2+scos2(1+t3s7v2(1+s2))ds|:t∈ℝ+,v∈Br}≤sup{r3∫0t‍s1+t2ds:t∈ℝ+}=132<M,
which yield (3.3)–(3.5). That is, all Assumptions of Theorem 3.2 are fulfilled. Therefore, Theorem 3.2 guarantees that (4.5) has at least one solution x=x(t)∈Br. Moreover, solutions of (4.5) are asymptotically stable. But Theorem  2 in [9], Theorem  3 in [10], Theorem  1 in [15], Theorem  2 in [16], and Theorem  3.1 in [19] are invalid for (4.5).



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*Source: 932019-2012-12-05.xml*