# Global Asymptotic Stability for Nonlinear Functional Integral Equation of Mixed Type

Authors: Zhinan Xia
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The existence results of global asymptotic stability of the solution are proved for
functional integral equation of mixed type. The measure of noncompactness and the fixed-point
theorem of Darbo are the main tools in carrying out our proof. Furthermore, some examples are
given to show the efficiency and usefulness of the main findings.


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## Body

## 1. Introduction

It is well known that functional integral equation of various types creates an important subject of numerous mathematical investigations and constitutes a significant branch of nonlinear analysis. It has great applications in physics, engineering, economics, and biologyed modelling problems connect with real world. With the help of several tools of functional analysis, topology, and fixed-point theory, many authors have made important contributions to this theory [1–5].Letℝ denote the real line and ℝ+=[0,+∞). In this paper, we investigate the nonlinear functional integral equation of mixed type, namely,
(1)x(t)=F(∫0∞‍t,f(t,x(α(t))),∫0β(t)‍u(t,s,x(γ(s)))dsg(t,s),∫0∞‍v(t,s,x(δ(s)))ds),t∈ℝ+.
It is worthwhile mentioning that (1) contains numerous functions and integral and functional integral equations encountered in nonlinear analysis. For example, the classical Volterra integral equation, the famous Chandrasekhar integral equation, the Urysohn integral equation, and the Volterra-Stieltjes integral equation are the special cases of (1).The goal of this paper is to investigate the existence and asymptotic behavior of solutions for (1). Using the technique associated with a suitable measure of noncompactness and the fixed-point theorem of Darbo, we show that (1) has at least one solution under rather general and convenient assumptions. We also obtain some asymptotic characterization of the solutions of (1). The results of the present paper generalize several ones obtained previously in the papers [1, 3, 6–8] and references therein.


## 2. Preliminaries

In this section, we introduce some notations, definitions, and preliminary facts which are used in this paper. Letx be a real function defined on [a,b], and let ⋁abx denotes the variation of x on [a,b]; if ⋁abx is finite, we say that x is of bounded variation. Let g(t,s):[a,b]×[c,d]→ℝ be a function, and let ⋁t=pqg(t,s) indicate the variation of the function t→g(t,s) on [p,q]⊂[a,b], where s is arbitrarily fixed in [c,d]. Similarly, define ⋁s=pqg(t,s). If x and φ are two real functions defined on [a,b], define the Stieltjes integral ∫ab‍x(t)dφ(t) and say that x is Stieltjes integrable on [a,b] with respect to φ.Lemma 1 (see [9]).
Ifx is Stieltjes integrable on [a,b] with respect to φ of bounded variation; then
(2)|∫ab‍x(t)dφ(t)|≤∫ab‍|x(t)|d(⋁atφ).Lemma 2 (see [9]).
Letx1, x2 be Stieltjes integrable functions on [a,b] with respect to a nondecreasing function φ such that x1(t)≤x2(t) for t∈[a,b], then
(3)∫ab‍x1(t)dφ(t)≤∫ab‍x2(t)dφ(t).We also consider Stieltjes integrals∫ab‍x(s)dsg(t,s), where g:[a,b]×[a,b]→ℝ and ds indicates the integration with respect to s. The details concerning the integral of this type will be given later.Let(E,∥·∥) be a real Banach space with zero element θ. Denote by B(x,r) the closed ball centered at x and with radius r, Br stands for the ball B(θ,r). If X is a nonempty subset of E, we denote by X¯ and ConvX the closure and the closed convex hull of X, respectively. The family of all nonempty and bounded subsets of E is denoted by 𝔐E and its subfamily consisting of all relatively compact sets is denoted by 𝔑E.Definition 3 (see [10]).
A functionμ:𝔐E→ℝ+:=[0,∞) is said to be a measure of noncompactness in E if it satisfies the following conditions. (1)
The familykerμ={X∈𝔐E:μ(X)=0} is nonempty and kerμ⊂𝔑E.(2)
X⊂Y⇒μ(X)≤μ(Y).(3)
μ(X¯)=μ(ConvX)=μ(X).(4)
μ(λX+(1-λ)Y)≤λμ(X)+(1-λ)μ(Y) for λ∈[0,1].(5)
If{Xn}n is a sequence of closed sets from 𝔐E such that Xn+1⊂Xn(n=1,2,…) and if limn→∞μ(Xn)=0, then the set X∞=⋂n=1∞Xn is nonempty.
The familykerμ in (1) is called the kernel of μ. Note that X∞ from (5) is a member of kerμ.
Consider the Banach spaceBC(ℝ+) of all real functions defined, bounded and continuous on ℝ+, and equipped with the maximum norm ∥x∥=sup{|x(t)|:t≥0}. Fix a nonempty bounded subset X of BC(ℝ+) and T>0, and for x∈X, ε>0, define
(4)ωT(x,ε):=sup{|x(t)-x(s)|:t,s∈[0,T],|t-s|≤ε}.
Let
(5)ωT(X,ε)=sup{ωT(x,ε):x∈X},ω0T(X)=limε→0ωT(X,ε),ω0(X)=limT→∞ω0T(X).
Moreover, for a fixed t∈ℝ+, define
(6)X(t)={x(t):x∈X},diamX(t)=sup{|x(t)-y(t)|:x,y∈X}.
Finally, define the function μ on 𝔐BC(ℝ+) by
(7)μ(X)=ω0(X)+limsupt→∞diamX(t).
It can be shown that μ is a measure of noncompactness in BC(ℝ+) [10].Remark 4.
ker
μ contains nonempty and bounded sets X such that functions from X are locally equicontinuous on ℝ+ and tend to zero at infinity uniformly with respect to X.AssumeΩ is a nonempty subset of BC(ℝ+) and Q is an operator defined on Ω with values in BC(ℝ+). Consider the operator equation
(8)x(t)=(Qx)(t),t∈ℝ+.Definition 5 (see [8]).
The solutionx(t) of (8) is said to be locally attractive if there exists a ball B(x0,r) in BC(ℝ+) such that for arbitrary solutions x=x(t) and y=y(t) of (8) belonging to B(x0,r)∩Ω such that
(9)limt→∞(x(t)-y(t))=0.Definition 6 (see [3]).
The solutionx=x(t) of (8) is said to be globally attractive, if (9) holds for each solutions y=y(t) of (8). If (9) is satisfied uniformly with respect to Ω, we say that solutions of (8) are globally asymptotically stable (or uniformly globally attractive).Theorem 7 (Darbo fixed-point theorem [10]).
LetΩ be a nonempty bounded closed and convex subset of Banach space E and let Q:Ω→Ω be a continuous mapping such that μ(QX)≤kμ(X) for any nonempty subset of X of Ω, where k∈[0,1) is a constant. Then Q has a fixed-point in Ω.From Theorem7, it can be shown that the set fix Q of fixed-points of Q is a member of kerμ.


## 3. Main Results

In this section, we will investigate the existence and asymptotic behavior of solutions for (1) in BC(ℝ+). Assume that the following conditions are satisfied. (H1)
α,β,γ,δ:ℝ+→ℝ+ are continuous functions.(H2)
g:ℝ+×ℝ+→ℝ satisfies the following conditions.
(H2.a)
For allt1,t2∈ℝ+ such that t1<t2, the function s→g(t2,s)-g(t1,s) is nondecreasing on ℝ+.(H2.b)
The functions→g(0,s) is nondecreasing on ℝ+.(H2.c)
The functionss→g(t,s) and t→g(t,s) are continuous on ℝ+ for each fixed t∈ℝ+ or s∈ℝ+, respectively.(H3)
u:ℝ+×ℝ+×ℝ→ℝ is a continuous function and there exist a continuous function k(t,s):ℝ+×ℝ+→ℝ+ and a continuous, nondecreasing function φ:ℝ+→ℝ+ such that
(10)|u(t,s,x)|≤k(t,s)φ(|x|),∀t,s∈ℝ+,x∈ℝ.(H4)
v:ℝ+×ℝ+×ℝ→ℝ is a continuous function, v(t,s,0)=0 and there exist a continuous function p(t,s):ℝ+×ℝ+→ℝ+ and a continuous, nondecreasing functions ψ:ℝ+→ℝ+ such that ψ(0)=0 and
(11)|v(t,s,x)-v(t,s,y)|≤p(t,s)ψ(|x-y|),∀t,s∈ℝ+,x,y∈ℝ.(H5)
F:ℝ+×ℝ×ℝ×ℝ→ℝ is a continuous function and there exist a constant M>0 and continuous functions m1(t),m2(t):ℝ+→ℝ+ such that
(12)|F(t,x1,x2,x3)-F(t,y1,y2,y3)|≤M|x1-y1|+m1(t)|x2-y2|+m2(t)|x3-y3|,
for all t∈ℝ+, x1,x2,x3,y1,y2,y3∈ℝ. Moreover, the function t→F(t,0,0,0) is bounded with F0=supt≥0|F(t,0,0,0)|.(H6)
f:ℝ+×ℝ→ℝ is a continuous function and there exists a constant λ>0 such that
(13)|f(t,x)-f(t,y)|≤λ|x-y|,∀x,y∈ℝ,t∈ℝ+.
Moreover, f(t,0)∈BC(ℝ+) and f0=supt≥0|f(t,0)|.(H7)
limt→∞m1(t)∫0β(t)‍k(t,s)dsg(t,s)=0, limt→∞m2(t)∫0∞‍p(t,s)ds=0.(H8)
Consider(14)limT→∞{sup{∫T∞‍p(t,s)ds:t∈ℝ+}}=0.(H9)
There exists a positive solutionr0 of the inequality
(15)Mλr+Mf0+M1φ(r)+M2ψ(r)+F0≤r
such that Mλ<1, where
(16)M1=sup{m1(t)∫0β(t)‍k(t,s)dsg(t,s):t≥0},M2=sup{m2(t)∫0∞‍p(t,s)ds:t≥0}.Remark 8.
There are many examples of functiong(t,s) which verify (H2), such as(i)
g(t,s)=s,(ii)
g(t,s)=sln(t+1),(iii)
g(t,s)={tln((t+s)/t),fort>0,s≥00,ln((t+s)/t)fort=0,s≥0,(iv)
g(t,s)=∫0t‍(∫0s‍p(z,y)dy)dz, where p:ℝ+×ℝ+→ℝ+ is a bounded and integrable function. One can see [2] for more details.In what follows, we provide some properties of the functiong(t,s).Lemma 9 (see [9]).
Under assumptions(H2.a) and (H2.b), the function s→g(t,s) is nondecreasing on ℝ+ for any fixed t∈ℝ+.Lemma 10 (see [9]).
Assume that the functiong satisfies the assumption (H2.a); then for arbitrarily fixed s1,s2∈ℝ+ such that s1<s2, the function t→g(t,s2)-g(t,s1) is nondecreasing on ℝ+.Remark 11.
Observe that ifr0>0 satisfies (15), then
(17)Mλ≤1-Mf0r0-F0r0-M1φ(r0)r0-M2ψ(r0)r0.
Thus, if one of the terms of Mf0, F0, M1, and M2 dose not vanish, then Mλ<1 is automatically satisfied.For the sake of convenience and simplicity, we will use the following notations in the paper. For fixedT>0, let
(18)αT=sup{α(t):t∈[0,T]},βT=sup{β(t):t∈[0,T]},m1T=inf{m1(t):t∈[0,T]},m2T=inf{m2(t):t∈[0,T]},m1T=sup{m1(t):t∈[0,T]},kT=sup{k(t,s):t∈[0,T],s∈[0,βT]}.The main result of this section is the following theorem.Theorem 12.
Under assumptions(H1)–(H9), (1) has at least one solution x∈BC(ℝ+). Moreover, solutions of (1) are globally asymptotically stable.Proof.
Consider the operatorℱ defined on BC(ℝ+) by
(19)(ℱx)(t)=F(∫0∞‍t,f(t,x(α(t))),∫0β(t)‍u(t,s,x(γ(s)))dsg(t,s),∫0∞‍v(t,s,x(δ(s)))ds),t∈ℝ+.
Taking into account (H2), (H3), and Lemma 9, we deduce that ℱ is well defined on BC(ℝ+).
First, we show thatℱ maps BC(ℝ+) into itself. It is clear that ℱx is continuous on ℝ+ for each x∈BC(ℝ+). By (H3)–(H6) and Lemma 1, for t∈ℝ+, one has
(20)|(ℱx)(t)|≤|∫0∞‍F(∫0∞‍t,f(t,x(α(t))),∫0β(t)u(t,s,x(γ(s)))dsg(t,s),∫0∞v(t,s,x(δ(s)))ds)|≤|∫0∞‍F(∫0∞‍t,f(t,x(α(t))),∫0β(t)‍u(t,s,x(γ(s)))dsg(t,s),∫0∞‍v(t,s,x(δ(s)))ds)-F(t,0,0,0)|+F(t,0,0,0)≤M|f(t,x(α(t)))|+m1(t)×|∫0β(t)‍u(t,s,x(γ(s)))dsg(t,s)|+m2(t)|∫0∞‍v(t,s,x(δ(s)))ds|+|F(t,0,0,0)|≤Mλ|x(α(t))|+M|f(t,0)|+m1(t)×∫0β(t)‍|u(t,s,x(γ(s)))|ds(⋁p=0sg(t,p))+m2(t)∫0∞‍|v(t,s,x(δ(s)))|ds+|F(t,0,0,0)|≤Mλ∥x∥+M|f(t,0)|+m1(t)×∫0β(t)‍k(t,s)φ(|x(γ(s))|)dsg(t,s)+m2(t)∫0∞‍p(t,s)ψ(|x(δ(s))|)ds+|F(t,0,0,0)|≤Mλ∥x∥+Mf0+φ(∥x∥)m1(t)×∫0β(t)‍k(t,s)dsg(t,s)+ψ(∥x∥)m2(t)∫0∞‍p(t,s)ds+F0≤Mλ∥x∥+Mf0+M1φ(∥x∥)+M2ψ(∥x∥)+F0,
By (H9), it is clear that ℱ(Br0)⊂Br0.
Next, we show thatℱ is continuous on Br0. Let ε>0 be given, and since (H7) holds, there exists T>0, such that m1(t)∫0β(t)‍k(t,s)dsg(t,s)<ε and m2(t)∫0∞‍p(t,s)ds<ε for t≥T. Consider the following two cases.
Case
1.For t≥T, x,y∈Br0 with ∥x-y∥≤ε,
(21)|(ℱx)(t)-(ℱy)(t)|≤|F(∫0β(t)‍t,f(t,x(α(t))),∫0β(t)‍u(t,s,x(γ(s)))dsg(t,s),∫0∞‍v(t,s,x(δ(s)))ds)-F(∫0β(t)‍t,f(t,y(α(t))),∫0β(t)‍u(t,s,y(γ(s)))dsg(t,s),∫0∞‍v(t,s,y(δ(s)))ds)|≤M|f(t,x(α(t)))-f(t,y(α(t)))|+m1(t)×|∫0β(t)‍[u(t,s,x(γ(s)))-u(t,s,y(γ(s)))]dsg(t,s)∫0β(t)‍|+m2(t)|∫0∞‍[v(t,s,x(δ(s)))-v(t,s,y(δ(s)))]ds|≤Mλ|x(α(t))-y(α(t))|+m1(t)×∫0β(t)‍|u(t,s,x(γ(s)))-u(t,s,y(γ(s)))|ds×(⋁p=0sg(t,p))+m2(t)×∫0∞‍|v(t,s,x(δ(s)))-v(t,s,y(δ(s)))|ds≤Mλ∥x-y∥+m1(t)×∫0β(t)‍|u(t,s,x(γ(s)))-u(t,s,y(γ(s)))|dsg(t,s)+m2(t)∫0∞‍|v(t,s,x(δ(s)))|ds+m2(t)∫0∞‍|v(t,s,y(δ(s)))|ds≤Mλ∥x-y∥+m1(t)×∫0β(t)‍k(t,s)φ(|x(γ(s))|)dsg(t,s)+m1(t)∫0β(t)‍k(t,s)φ(|y(γ(s))|)dsg(t,s)+m2(t)∫0∞‍p(t,s)ψ(|x(δ(s))|)ds+m2(t)∫0∞‍p(t,s)ψ(|y(δ(s))|)ds≤Mλ∥x-y∥+φ(∥x∥)m1(t)×∫0β(t)‍k(t,s)dsg(t,s)+φ(∥y∥)m1(t)×∫0β(t)‍k(t,s)dsg(t,s)+ψ(∥x∥)m2(t)∫0∞‍p(t,s)ds+ψ(∥y∥)m2(t)∫0∞‍p(t,s)ds≤Mλ∥x-y∥+2φ(r0)m1(t)×∫0β(t)‍k(t,s)dsg(t,s)+2ψ(r0)m2(t)∫0∞‍p(t,s)ds≤Mλε+2φ(r0)ε+2ψ(r0)ε.
Case
2. For t∈[0,T], x,y∈Br0 with ∥x-y∥≤ε, by Lemmas 9 and 10, one has(22)|(ℱx)(t)-(ℱy)(t)|≤M|f(t,x(α(t)))-f(t,y(α(t)))|+m1(t)×|∫0β(t)‍[u(t,s,x(γ(s)))-u(t,s,y(γ(s)))]dsg(t,s)∫0β(t)‍|+m2(t)×|∫0∞‍[v(t,s,x(δ(s)))-v(t,s,y(δ(s)))]ds|≤Mλ|x(α(t))-y(α(t))|+m1(t)×∫0β(t)‍|u(t,s,x(γ(s)))-u(t,s,y(γ(s)))|ds×(⋁p=0sg(t,p))+m2(t)×∫0∞‍|v(t,s,x(δ(s)))-v(t,s,y(δ(s)))|ds≤Mλ∥x-y∥+m1(t)×∫0β(t)‍|u(t,s,x(γ(s)))-u(t,s,y(γ(s)))|dsg(t,s)+m2(t)∫0∞‍p(t,s)ψ(|x(δ(s))-y(δ(s))|)ds≤Mλε+m1Tω(u,ε)∫0β(t)‍dsg(t,s)+ψ(∥x-y∥)m2(t)∫0∞‍p(t,s)ds≤Mλε+m1Tω(u,ε)×(g(t,β(t))-g(t,0))+M2ψ(ε)≤Mλε+m1Tω(u,ε)×(g(T,β(t))-g(T,0))+M2ψ(ε)≤Mλε+m1Tω(u,ε)×(g(T,βT)-g(T,0))+M2ψ(ε),
where
(23)ω(u,ε)=sup{|u(t,s,x)-u(t,s,y)|:t∈[0,T],s∈[0,βT],x,y∈[-r0,r0],∥x-y∥≤ε}.
By (21) and (22),
(24)|(ℱx)(t)-(ℱy)(t)|≤max{(g(T,βT)-g(T,0))Mλε+2φ(r0)ε+2ψ(r0)ε,Mλε+m1Tω(u,ε)×(g(T,βT)-g(T,0))+M2ψ(ε)}.
Note that ω(u,ε)→0, ψ(ε)→0 as ε→0, and ℱ is continuous on Br0 by (24).
Now, and letX be a nonempty set of Br0 and x∈X, fix arbitrarily T>0 and ε>0, let t1,t2∈[0,T] such that t1<t2 and t2-t1≤ε,
(25)|(ℱx)(t2)-(ℱx)(t1)|≤|F(∫0β(t2)‍t2,f(t2,x(α(t2))),∫0β(t2)‍u(t2,s,x(γ(s)))dsg(t2,s),∫0∞‍v(t2,s,x(δ(s)))ds)-F(∫0∞‍t1,f(t1,x(α(t1))),∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t1,s),∫0∞‍v(t1,s,x(δ(s)))ds)|≤|F(∫0β(t2)‍t2,f(t2,x(α(t2))),∫0β(t2)‍u(t2,s,x(γ(s)))dsg(t2,s),∫0∞‍v(t2,s,x(δ(s)))ds)-F(∫0∞‍t2,f(t1,x(α(t1))),∫0β(t2)‍u(t2,s,x(γ(s)))dsg(t2,s),∫0∞‍v(t2,s,x(δ(s)))ds)|+|∫0β(t1)‍F(∫0β(t2)‍t2,f(t1,x(α(t1))),∫0β(t2)‍u(t2,s,x(γ(s)))dsg(t2,s),∫0∞‍v(t2,s,x(δ(s)))ds)-F(∫0β(t2)‍t2,f(t1,x(α(t1))),∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t1,s),∫0∞‍v(t2,s,x(δ(s)))ds)|+|∫0β(t1)‍F(∫0β(t2)‍t2,f(t1,x(α(t1))),∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t1,s),∫0∞‍v(t2,s,x(δ(s)))ds)-F(∫0β(t2)‍t2,f(t1,x(α(t1))),∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t1,s),∫0∞‍v(t1,s,x(δ(s)))ds)|+|F(∫0β(t2)‍t2,f(t1,x(α(t1))),∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t1,s),∫0∞‍v(t1,s,x(δ(s)))ds)-F(∫0β(t2)‍t1,f(t1,x(α(t1))),∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t1,s),∫0∞‍v(t1,s,x(δ(s)))ds)|≤M|f(t2,x(α(t2)))-f(t1,x(α(t1)))|+m1T|∫0β(t2)‍u(t2,s,x(γ(s)))dsg(t2,s)-∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t1,s)|+m2T|∫0∞‍v(t2,s,x(δ(s)))ds-∫0∞‍v(t1,s,x(δ(s)))ds|+ωr0T(F,ε):=Mϑ0+m1Tϑ1+m2Tϑ2+ωr0T(F,ε),
where
(26)ϑ0=|f(t2,x(α(t2)))-f(t1,x(α(t1)))|,ϑ1=|∫0β(t2)‍u(t2,s,x(γ(s)))dsg(t2,s)-∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t1,s)|,ϑ2=|∫0∞‍v(t2,s,x(δ(s)))ds-∫0∞‍v(t1,s,x(δ(s)))ds|,ωr0T(F,ε)=sup{[-M2ψ(r0)m2T,M2ψ(r0)m2T]|F(t2,x,y,z)-F(t1,x,y,z)|:t1,t2∈[0,T],|t2-t1|≤ε,x∈[-λr0-f0,λr0+f0],y∈[-φ(r0)M1m1T,φ(r0)M1m1T],z∈[-M2ψ(r0)m2T,M2ψ(r0)m2T]}.
Moreover,
(27)ϑ0=|f(t2,x(α(t2)))-f(t1,x(α(t1)))|≤|f(t2,x(α(t2)))-f(t2,x(α(t1)))|+|f(t2,x(α(t1)))-f(t1,x(α(t1)))|≤λ|x(α(t2))-x(α(t1))|+|f(t2,x(α(t1)))-f(t1,x(α(t1)))|≤λωT(x,ωT(α,ε))+ωr0T(f,ε),
where
(28)ωT(x,ωT(α,ε))=sup{|x(θ2)-x(θ1)|:θ1,θ2∈[0,αT],|θ2-θ1|<ωT(α,ε)},ωr0T(f,ε)=sup{|f(t2,x)-f(t1,x)|:t1,t2∈[0,T],|t2-t1|≤ε,x∈[-r0,r0]}.
By Lemma 1, one has
(29)ϑ1=|∫0β(t2)‍u(t2,s,x(γ(s)))dsg(t2,s)-∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t1,s)|≤|∫0β(t2)‍u(t2,s,x(γ(s)))dsg(t2,s)-∫0β(t1)‍u(t2,s,x(γ(s)))dsg(t2,s)|+|∫0β(t1)‍u(t2,s,x(γ(s)))dsg(t2,s)-∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t2,s)|+|∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t2,s)-∫0β(t1)‍u(t1,s,x(γ(s)))dsg(t1,s)|≤∫β(t1)β(t2)‍|u(t2,s,x(γ(s)))|ds×(⋁p=β(t1)sg(t2,p))+∫0β(t1)‍|u(t2,s,x(γ(s)))-u(t1,s,x(γ(s)))|ds×(⋁p=0sg(t2,p))+∫0β(t1)‍|u(t1,s,x(γ(s)))|ds×(⋁p=0s[g(t2,p)-g(t1,p)])≤∫β(t1)β(t2)‍|u(t2,s,x(γ(s)))|dsg(t2,s)+∫0β(t1)‍|u(t2,s,x(γ(s)))-u(t1,s,x(γ(s)))|dsg(t2,s)+∫0β(t1)‍|u(t1,s,x(γ(s)))|ds×(g(t2,s)-g(t1,s))≤∫β(t1)β(t2)‍k(t2,s)φ(|x(γ(s))|)dsg(t2,s)+∫0β(t1)‍ωr0T(u,ε)dsg(t2,s)+∫0β(t1)‍k(t1,s)φ(|x(γ(s))|)ds×(g(t2,s)-g(t1,s))≤kTφ(r0)∫β(t1)β(t2)‍dsg(t2,s)+ωr0T(u,ε)∫0β(t1)‍dsg(t2,s)+kTφ(r0)∫0β(t1)‍ds(g(t2,s)-g(t1,s))=kTφ(r0)[g(t2,β(t2))-g(t2,β(t1))]+ωr0T(u,ε)[g(t2,β(t1))-g(t2,0)]+kTφ(r0)([g(t2,β(t1))-g(t1,β(t1))]-[g(t2,0)-g(t1,0)][g(t2,β(t1))-g(t1,β(t1))])≤kTφ(r0)[g(T,β(t2))-g(T,β(t1))]+ωr0T(u,ε)[g(T,β(t1))-g(T,0)]+kTφ(r0)([g(t2,βT)-g(t1,βT)]+|g(t2,0)-g(t1,0)|[g(t2,βT)-g(t1,βT)])≤kTφ(r0)ωT(g(T,·),ωT(β,ε))+ωr0T(u,ε)[g(T,βT)-g(T,0)]+kTφ(r0)[ωT(g(·,βT),ε)+ωT(g(·,0),ε)],
where
(30)ωT(g(T,·),ωT(β,ε))=sup{|g(T,θ2)-g(T,θ1)|:θ1,θ2∈[0,βT],|θ2-θ1|<ωT(β,ε)},ωr0T(u,ε)=sup{[0,βT]|u(t2,s,x)-u(t1,s,x)|:t1,t2∈[0,T],s∈[0,βT],|t2-t1|≤ε,x∈[-r0,r0]}.
By (H4), one has
(31)ϑ2=|∫0∞‍v(t2,s,x(δ(s)))ds-∫0∞‍v(t1,s,x(δ(s)))ds|≤∫0∞‍|v(t2,s,x(δ(s)))ds-v(t1,s,x(δ(s)))|ds≤∫0T‍|v(t2,s,x(δ(s)))ds-v(t1,s,x(δ(s)))|ds+∫T∞‍|v(t2,s,x(δ(s)))ds-v(t1,s,x(δ(s)))|ds≤Tωr0T(v,ε)+∫T∞‍|v(t2,s,x(δ(s)))|ds+∫T∞‍|v(t1,s,x(δ(s)))|ds≤Tωr0T(v,ε)+∫T∞‍p(t2,s)ψ(|x(δ(s))|)ds+∫T∞‍p(t1,s)ψ(|x(δ(s))|)ds≤Tωr0T(v,ε)+2sup{∫T∞‍p(t,s)ds:t∈ℝ+}·ψ(r0),
where
(32)ωr0T(v,ε)=sup{|v(t2,s,x)-v(t1,s,x)|:t1,t2,∈[0,T],s∈[0,T],|t2-t1|≤ε,x∈[-r0,r0]}.
By (25), (27), (29), and (31), one has
(33)ωT(ℱx,ε)≤MλωT(x,ωT(α,ε))+Mωr0T(f,ε)+m1TkTφ(r0)ωT(g(T,·),ωT(β,ε))+m1Tωr0T(u,ε)[g(T,βT)-g(T,0)]+m1TkTφ(r0)[ωT(g(·,βT),ε)+ωT(g(·,0),ε)]+m2TTωr0T(v,ε)+2m2Tψ(r0)×sup{∫T∞‍p(t,s)ds:t∈ℝ+}+ωr0T(F,ε),
and then
(34)ωT(ℱX,ε)≤MλωT(X,ωT(α,ε))+Mωr0T(f,ε)+m1TkTφ(r0)ωT×(g(T,·),ωT(β,ε))+m1Tωr0T(u,ε)[g(T,βT)-g(T,0)]+m1TkTφ(r0)×[ωT(g(·,βT),ε)+ωT(g(·,0),ε)]+m2TTωr0T(v,ε)+2m2Tψ(r0)×sup{∫T∞‍p(t,s)ds:t∈ℝ+}+ωr0T(F,ε).
By the uniform continuity of the functions f, g, β, u, v, and F on compact set, one has
(35)ωr0T(f,ε)⟶0,ωr0T(u,ε)⟶0,ωr0T(v,ε)⟶0,ωr0T(F,ε)⟶0,asε⟶0,ωT(g(T,·),ωT(β,ε))⟶0,ωT(g(·,βT),ε)⟶0,ωT(g(·,0),ε)⟶0,asε⟶0,
so
(36)ω0T(ℱX)≤Mλω0T(X)+2m2Tψ(r0)×sup{∫T∞‍p(t,s)ds:t∈ℝ+},
and then, by (H8),
(37)ω0(ℱX)≤Mλω0(X).
On the other hand, take arbitrary x,y∈X and fix t∈ℝ+; one has
(38)|(ℱx)(t)-(ℱy)(t)|≤M|f(t,x(α(t)))-f(t,y(α(t)))|+m1(t)×|∫0β(t)‍[u(t,s,x(γ(s)))-u(t,s,y(γ(s)))]dsg(t,s)|+m2(t)|∫0∞‍[v(t,s,x(δ(s)))-v(t,s,y(δ(s)))]ds|≤Mλ|x(α(t))-y(α(t))|+m1(t)×∫0β(t)‍|u(t,s,x(γ(s)))-u(t,s,y(γ(s)))|ds×(⋁p=0sg(t,p))+m2(t)∫0∞‍|v(t,s,x(δ(s)))-v(t,s,y(δ(s)))|ds≤MλdiamX(t)+m1(t)×∫0β(t)‍k(t,s)φ(|x(γ(s))|)dsg(t,s)+m1(t)∫0β(t)‍k(t,s)φ(|y(γ(s))|)dsg(t,s)+m2(t)∫0∞‍p(t,s)ψ(|x(δ(s))|)ds+m2(t)∫0∞‍p(t,s)ψ(|y(δ(s))|)ds≤MλdiamX(t)+2φ(r0)m1(t)×∫0β(t)‍k(t,s)dsg(t,s)+2ψ(r0)m2(t)∫0∞‍p(t,s)ds,
and hence
(39)diam(ℱX)(t)≤MλdiamX(t)+2φ(r0)m1(t)×∫0β(t)‍k(t,s)dsg(t,s)+2ψ(r0)m2(t)∫0∞‍p(t,s)ds.
By (H7), one has
(40)limsupt→∞diam(ℱX)(t)≤Mλlimsupt→∞diamX(t).
By (37) and (40), one has
(41)μ(ℱX)≤Mλμ(X),
where μ is the measure of noncompactness defined in (7). Hence, in view of Theorem 7, we conclude that ℱ has at least one fixed-point x∈Br0, which is a solution of (1). Moreover, taking into account the description of sets belonging to kerμ (see Remark 4), we deduce that the solutions of (1) are globally asymptotically stable. This completes the proof.Remark 13.
Equation (1) is quite general and includes some nonlinear integral equations studied earlier by various authors as special cases. For example, we have the following. (i)
Iff(t,x)=x, g(t,s)=s, α(t)=β(t)=γ(t)=δ(t)=t, (1) reduces to the nonlinear functional integral equation which studied by [7], that is,
(42)x(t)=F(t,x(t),∫0t‍u(t,s,x(s))ds,∫0∞‍v(t,s,x(s))ds),t∈ℝ+.(ii)
IfF(t,x,y,z)=q(t)+xy, α(t)=β(t)=γ(t)=δ(t)=t, v(t,s,x)=0, (1) reduces to the nonlinear quadratic integral equation of Volterra-Stieltjes type which is studied by the author of [2], that is,
(43)x(t)=q(t)+f(t,x(t))×∫0t‍u(t,s,x(s))dsg(t,s),t∈ℝ+.(iii)
IfF(t,x,y,z)=q(t)+xz, g(t,s)=s, α(t)=β(t)=γ(t)=δ(t)=t, u(t,s,x)=0, (1) reduces to the quadratic Urysohn integral equation which is studied by the author of [22], that is,
(44)x(t)=q(t)+f(t,x(t))×∫0∞‍v(t,s,x(s))ds,t∈ℝ+.For more detail, one can see Table 1.Table 1
Some of papers studied earlier as special cases, whereq, ψ, φ, k, h are suitable continuous functions.
F
(
t
,
x
,
y
,
z
)
f
(
t
,
x
)
g
(
t
,
s
)
α
(
t
), β(t), γ(t), δ(t)
u
(
t
,
s
,
x
)
v
(
t
,
s
,
x
)
Reference
x
s
[1]
x
s
t
[7]
x
s
t
0
[8, 11]
x
s
0
[12]
s
0
[13]
x
+
y
s
t
0
[6]
s
0
[3]
x
+
ψ
(
y
)
s
0
[14]
q
(
t
)
+
y
0
s
0
[15]
q
(
t
)
+
z
0
s
t
0
[16]
0
s
0
[17]
x
y
s
t
0
[8, 18]
q
(
t
)
+
x
y
s
0
[19]
t
0
[2]
q
(
t
)
+
x
z
s
t
0
k
(
t
,
s
)
h
(
s
,
x
)
[20, 21]
s
t
0
[22]
x
+
φ
(
t
,
z
)
s
0
[23]
x
(
q
(
t
)
+
y
)
s
0
[4]
q
(
t
)
+
x
ψ
(
y
)
s
0
[24]


## 4. Examples

In this section, we present some examples to illustrate our main result.Example 14.
Consider the nonlinear functional integral equation of Volterra-Stieltjes type(45)x(t)=f(t,x(α(t)))+∫0β(t)‍u(t,s,x(γ(s)))dsg(t,s),t∈ℝ+.
Equation (45) is a special case of (1), where F(t,x,y,z)=x+y, v(t,s,x)=0, δ(t)=0. Since F(t,x,y,z)=x+y, (H5) is satisfied with M=m1(t)=1, m2(t)=0, and F0=0. By Theorem 12, one has the following.Theorem 15.
Assume that(H1)–(H3), (H6), (H7), and (H9) hold; then (45) has at least one solution in BC(ℝ+) which is globally asymptotically stable.Remark 16.
Wheng(t,s)=s, (45) is investigated by [3], and if g(t,s)=s, α(t)=β(t)=γ(t)=t, it is investigated by the author of [6], see Table 1 for more details. Theorem 15 improves, and generalizes the results of [3, 6].Remark 17.
IfF(t,x,y,z)=q(t)+xy, the following quadratic nonlinear functional integral equation of Volterra-Stieltjes type:
(46)x(t)=q(t)+f(t,x(α(t)))×∫0β(t)‍u(t,s,x(γ(s)))dsg(t,s),t∈ℝ+.
When α(t)=β(t)=γ(t)=t in (46), it is investigated by the authors of [2]. For (46), by Theorem 12, one has a similar result. Here we omit it.Example 18.
Consider the quadratic Urysohn functional integral equation(47)x(t)=q(t)+f(t,x(α(t)))×∫0∞‍v(t,s,x(δ(s)))dst∈ℝ+.
Equation (47) is a special case of (1), where F(t,x,y,z)=q(t)+xz, u(t,s,x)=0, and β(t)=γ(t)=0. By Theorem 12, one has the following theorem.Theorem 19.
Assume that(H1) and (H4)–(H9) hold, and (47) has at least one solution in BC(ℝ+) which is globally asymptotically stable.Finally, we provide an example of concrete nonlinear functional integral equation of (1) which satisfies all the assumptions of Theorem 12.Example 20.
Consider the functional integral equation(48)x(t)=132e-t2+t2+232(t2+1)x(2t)+11+t2∫0t2‍x2(s)1+t4+s2ds+14sin2t∫0∞‍x(s)e-setds,t∈ℝ+.
Observe that (48) is a special case of (1), where F(t,x,y,z)=x+(1/(1+t2))y+zsin2t, f(t,x)=(1/32)e-t2+((t2+2)/(32(t2+1)))x, u(t,s,x)=x2/(1+t4+s2), v(t,s,x)=(1/4)xe-set, g(t,s)=s, α(t)=2t, β(t)=t2, and γ(t)=δ(t)=t. Then
(49)|u(t,s,x)|≤11+t4+s2x2,|v(t,s,x)-v(t,s,y)|≤14e-set|x-y|,
so k(t,s)=1/(1+t4+s2), p(t,s)=(1/4)e-set, φ(x)=x2, and ψ(x)=x, and then (H1)–(H4) hold.
ByF(t,x,y,z)=x+(1/(1+t2))y+zsin2t and f(t,x)=(1/32)e-t2+((t2+2)/(32(t2+1)))x, it is easy to see that (H5) and (H6) are satisfied with M=1, m1(t)=1/(1+t2), m2(t)=sin2t, λ=1/16, and F0=0, f0=1/32. Since
(50)11+t4+s2≤11+s2,∫0∞‍p(t,s)ds=14e-t,
which implies that
(51)limt→∞∫0β(t)‍k(t,s)dsg(t,s)≤limt→∞∫0t2‍11+s2ds=π2,limt→∞∫0∞‍p(t,s)ds=0,
so (H7) holds. It can be easily seen that p(t,s) satisfies (H8). Moreover,
(52)M1=sup{11+t2∫0t2‍11+t4+s2ds:t≥0}≤π2,M2=sup{14sin2t∫0∞‍e-setds:t≥0}=sup{14e-tsin2t:t≥0}≤14,
then the inequality from (H9) has the following form:
(53)116r+132+π2r2+14r≤r.
Obviously, r0=1/8 is a positive solution of this inequality for which Mλ<1, and thus (H9) holds. By Theorem 12, (48) has at least one solution in BC(ℝ+) which is globally asymptotically stable.



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*Source: 193602-2013-10-01.xml*