0 Advances in Difference Equations

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On a weakly singular quadratic integral equations of Volterra type in Banach algebras

Xiulan Yu1, Chun Zhu2 and JinRong Wang2,3*

"Correspondence: wjr9668@126.com

2Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China

31 ndustriall nternet ofThings Engineering Research Center of the Higher Education Institutions of Guizhou Province and Schoolof Mathematics and Computer Science, Guizhou NormalCollege, Guiyang, Guizhou 550018, P.R. China Fulllist of author information is available at the end of the article

Abstract

In this paper, we present existence and uniqueness theorems of nonnegative, asymptotically stable, and ultimately nondecreasing solutions for weakly singular quadratic integral equations of Volterra type in Banach algebras. The concept of the measure of noncompactness and a fixed point theorem due to Darbo acting in a Banach algebra are the main tools in carrying out our proof An effective numerical example is given to illustrate our theory results. MSC: 45G05; 47H30

Keywords: weakly singular quadratic integral equations; Banach algebras; measure of noncompactness; fixed point theorem

1 Introduction

Quadratic integral equations with nonsingular kernels are associated with epidemic models [1-5]. Recently, quadratic integral equations with singular kernels have received a lot of attention because of their useful applications in describing numerous events and problems of the real world. A simple type of quadratic integral equation involving singular kernels in Banach algebras on an unbounded interval can be written as

x(t) = (Vix)(t)(V2x)(t), t e R+ := [0, to), (1)

(Vix)(t)= mi(t)+fi(t,x(t)) i (t - s)-aiui(t,s,x(s)) ds, Jo

ringer

where ai e (0,1), mi, fi, and ui are functions satisfying certain conditions for i = 1,2. As regards (1), Banas and Dudek [6] apply the techniques of a certain measure of noncompactness to study the existence of theories in the Banach algebra BC(R+).

As pointed out in [6], one would have to explore some sophisticated tools to study quadratic integral equations involving singular kernels in Banach algebras due to such type equations have rather complicated form. More in particular, we have the important condition (m) (see Definition 2.2) related to the operator Vi, i = 1,2, which plays an essential role in the application of the technique of measures of noncompactness in a certain Banach algebras setting.

It seems interesting to ask what happens if the singular (t - s)-ai is replaced by a certain function of, say, (tpi - spi)a'-lsYi where ai, ¡3i, and yi are pre-fixed numbers for i = 1,2.

©2014 Yu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly cited.

In this paper we continue the work in [6] and apply the explored tools and develops the techniques to study the existence of solutions to (1) with

in BC(R+), where r(-) is the gamma function, ai e (0,1), fa > 0, and Yi e R are pre-fixed numbers for i = 1,2.

Let us notice that weakly singular kernels appears in the second term of (2) are corresponding to the so-called Erdelyi-Kober type fractional integrals [7] of the function h which is given by

In the past two decades, differential and integral equations involving fractional integral operators draws a great application in nonlinear oscillations of earthquakes, many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic model. One can also find more details of such fractional equations in physics, viscoelasticity, electrochemistry, and porous media in [8-13] and in the important monographs [7,14-18].

In the present paper, we apply a certain measure of noncompactness introduced by Ba-nas and Dudek [6] and develop some useful methods in Wang et al. [19] to derive new existence, asymptotically stable, and ultimately nondecreasing of nonnegative solutions to (1) with (2) under the restriction on the parameters 1 > ai > 0, fa > 0, Yi > fa(1 - ai) - 1. Moreover, we also give the uniqueness result by requiring gi to satisfy the Lipschitz continuous condition and addressing ai = a2, fa = fa2, yi = Y2. As an application, we give an effective numerical example to illustrate our theoretical results.

Comparing with the corresponding existence results in Olaru [3, 4] and Brestovanska and Medved [5], we discuss a quadratic integral equation with a class of special singular kernels by using a different method, a fixed point theorem due to Darbo acting in a Banach algebra via the technique of the measure of noncompactness. As a result, we obtain a new and interesting existence result.

Comparing with the corresponding existence result in Banas and Dudek [6], there are at least three different points: (i) the singular kernels in our equation is more general; (ii) we not only present the existence result but also derive the uniqueness result; (iii) we give an effective numerical example and draw a curve of the unique nonnegative, asymptotically stable and ultimately nondecreasing solution to support our theoretical results.

2 Mathematical preliminary

Let E be a Banach space with the norm || • || and the zero element 0. Denote by B(x, r) the closed ball centered at x and with radius r. The symbol Br stands for the ball B(0, r). If X c E we use X, ConvX to denote the closure and convex closure of X, respectively. The symbol diamX denote the diameter of a bounded set X and ||X|| denotes the norm of X, that is, ||X|| = sup{||x||: x e X}. Moreover, we denote by ME the family of all nonempty and bounded subsets of E and by NE its subfamily consisting of all relatively compact sets.

We collect the following definition of a measure of noncompactness.

Definition 2.1 (see [20]) A mapping ¡x : ME ^ R+ is said to be a measure of noncompactness in E if it satisfies the following conditions:

(i) The family ker ¡x = {X e ME: ¡(X) = 0} is nonempty and ker x с NE where ker ¡x denotes the kernel of the measure of noncompactness ¡.

(ii) X с Y ¡(X) < ¡(Y).

(iii) ¡(X) = x(X).

(iv) ¡(ConvX) = x(X).

(v) ¡(XX +(1 - X)Y) < Xx(X) + (1 - X)x(Y) for X e [0,1].

(vi) If (Xn) is a sequence of closed sets from ME such that X„+1 с Xn (n = 1,2,...) and if lim„^TO ¡(Xn) = 0, then the intersection XTO = P|Xn is nonempty.

We see that the intersection set XTO from (vi) belongs to ker ¡. In fact, since ¡(XTO) < ¡(Xn) for every n, we have ¡(XTO) = 0.

We assume that the space E has the structure of Banach algebra. In such a case we write xy in order to denote the product of elements x,y e E. Similarly, we denote XY = {xy: x e X,y e Y}.

Now, we recall a useful concept of the condition (m).

Definition 2.2 (see [21]) One says that the measure of noncompactness ¡x defined on the Banach algebra E satisfies condition (m):

if ¡(XY) < ||X||x(Y) + || Y ||x(X) for arbitrary sets X, Y e Me.

In what follows we recall some measures of noncompactness in the Banach algebra BC(R+) consisting of all real functions defined, continuous, and bounded on R+. The algebra BC(R+) is endowed with the usual supremum norm ||x|| = sup{|x(t)| : t e R+} for x e BC(R+). In fact, the present measures of noncompactness were considered in detail in [21].

We assume that X is an arbitrarily fixed nonempty and bounded subset of the Banach algebra BC(R+), that is, X e MBC(R+). Choose arbitrarily e > 0 and T >0. Forx e X denote by wT (x, e) the modulus of continuity of the function x on the interval [t0, T], i.e., wT (x, e) = sup{|x(t) — x(s)|: t,5 e [0, T], |t -s| < e}.

Let (X, e) = sup{«T(x, e) : x e X}, a>T(X) = lime^0 о/(X, e) and wg°(X) = limT^TO (X). Define a(X) = limT^TO{supxeX{sup{|x(t) — x(s)|: t,s > T}}}. Finally, we set ¡a(X)=«0°(X)+fl(X).

It was shown [21] that the function ¡a is the measure of noncompactness in the algebra BC(R+) as introduced in [21]. The kernel ker ¡a of this measure contains all sets X e Mbc(r+) such that functions belonging to X are locally equicontinuous on R+ and have finite limits at infinity. Moreover, all functions from the set X tend to their limits with the same rate. Further, it was also proved that the measure of noncompactness ¡a satisfies condition (m) in [21].

For our problem, we consider another measure of noncompactness. In order to define this measure, similarly as above, fix a set X e MBC(R+) and a number t e R+. Denote by X(t) the cross-section of the set X at the point t, that is, X(t) = {x(t): x e X}. Denote by diamX(t) the diameter of X(t). Further, for a fixed T >0 and x e X denote by dT(x) the so-called modulus of decrease of the function x on the interval [T, то), which is defined by the formula dT(x) = sup{|x(t) — x(s)| — [x(t) — x(s)]: T < s < t}. We denote dT(X) = sup{dT(x): x e X}, dTO(X) = limT^TO dT (X).

In a similar way one can define the modulus of increase of function x and the set X (see [21]).

Finally, let us define the set quantity id in the following way:

ld(X) = «¡f (X) + dTO(X) + lim sup diamX(t). (3)

Linking the facts established in [21, 22], id is the measure of noncompactness in the algebra BC(R+). The kernel ker id of this measure consists of all sets X e MBC(R+) such that functions belonging to X are locally equicontinuous on R+ and the thickness of the bundle X(t) formed by functions from X tends to zero at infinity. Moreover, all functions from X are ultimately nondecreasing on R+ (see [23]). The measure l d has also an additional property.

Lemma 2.3 ([6, Theorem 6]) The measure of noncompactness id defined by (3) satisfies condition (m) on the family of all nonempty and bounded subsets X of Banach algebra BC(R+) such that functions belonging to X are nonnegative on R+.

The measure of noncompactness id defined by (3) allows us to characterize solutions of considered operator equations in terms of the concept of asymptotic stability.

To formulate precisely that concept (see [24]) assume that fi is a nonempty subset of the Banach algebra BC(R+) and F: fi — BC(R+) is an operator. Consider the operator equation

x(t) = (Fx)(t), t e R+, (4)

where x e fi.

Definition 2.4 ([6, Definition 7]) One says that solutions of (4) are asymptotically stable if there exists a ball B(x0, r) in BC(R+) such that B(x0, r) n fi = 0, and for each e >0 there exists T >0 such that |x(t)-y(t) | < e for all solutions x,y e B(x0, r) n fi of (4) and for t > T.

Now, we recall fixed point theorems for operators acting in a Banach algebra and satisfying some conditions expressed with the help of the measure of noncompactness.

Definition 2.5 ([20]) Let fi be a nonempty subset of a Banach space E, and let F: fi — E be a continuous operator which transforms bounded subsets of fi onto bounded ones. One says that F satisfies the Darbo condition with a constant k with respect to the measure of noncompactness i if i(FX) < k|(X) for each X e ME such that X c fi.If k < 1, then F is called a contraction with respect to l .

Lemma 2.6 ([6, Lemma 4]) Let E be a Banach algebra and assume that i is a measure of noncompactness on E satisfying condition (m). Assume that fi is nonempty, bounded, closed, and convex subset of the Banach algebra E, and the operators P and T transform continuously the set fi into E in such a way thatP(fi) and T(fi) are bounded. Moreover, one assumes that the operator F = P ■ T transforms fi into itself. If the operators P and T satisfy on the set fi the Darbo condition with respect to the measure of noncompactness i with the constants k1 andk2, respectively, then the operator F satisfies on fi the Darbo condition

with the constant ||P(fi) k + || T(fi) ||k1. Particularly, if ||P(fi) ||/<2 + || T(fi) ||k1 < 1, then F is a contraction with respect to the measure of noncompactness ¡x and has at least one fixed point in the set fi.

Remark 2.7 It can be shown [20] that the set Fix F of all fixed points of the operator F on the set fi isa member of the kernel ker ¡x.

In what follows we recall facts concerning the superposition operator which are drawn from [25]. In order to define this operator assume that J c R but J = 0 and f: R+ x J ^ R is a given function. Denote Xj by all the functions acting from R+ into J. For any x e Xj, a function F is defined by

(Fx)(t)=f (t,x(t)), t e R. (5)

Then the operator F defined in (5) is called the superposition operator generated byf.

The following result presents a useful property of the superposition operator which is considered in the Banach space B(R+) consisting of all real functions defined and bounded on R+.

Lemma 2.8 (see [23]) Assume that the following hypotheses are satisfied. (CI) The function f is continuous on the set R+ x J.

(C2) The function 1i—► f (t, u) is ultimately nondecreasing uniformly with respect to u belonging to bounded subintervals of J, that is,

lim {sup{ f (t, u) -f (s, u) | - f (t, u) -f (s, u)]: t > s > T, u e J1 =0

for any bounded subinterval J1 c J. (C3) For any fixed t e R+ the function u i—► f (t, u) is nondecreasing on J. (C4) The function u i—► f (t, u) satisfies a Lipschitz condition; that is, there exists a constant k >0 such that

f (t, u) -f (t, v)| < k\u - v|

for all t e R+ and all u, v e J. Then the inequality

dTO(Fx) < kdTO(x)

holds for any function x e B(R+), where k is the Lipschitz constantfrom assumption (iv).

The following basic equality will be used in the sequel.

Lemma 2.9 (see [26]) Let a, fa, y , andp be constants such that a > 0,p(Y -1) +1 > 0, and pfa -1) + 1 > 0. Then

£ (ta - sa )p(fa-1)sp(Y -1) ds = a ®(p(Y -a1) + 1, pfa - 1) + 1), t e R+,

B(f,n)= f s?-1(1-s)n-1 ds (Re(f)>0,Re(n)>0)

is the well-known Beta function and 0 = p[a* -1) + y -1] + 1. 3 Main results

In order to derive the existence theorem of nonnegative, asymptotically stable and ultimately nondecreasing solution, we consider (1) with (2) under the following assumptions:

(HI) The function mi, i = 1,2 is nonnegative, bounded, continuous, and ultimately nondecreasing.

(H2) The functionf: R+ x R+ — R+ satisfies the conditions (C1)-(C3) of Lemma 2.8 for i = 1,2.

(H3) The functions f, i = 1,2 satisfy the Lipschitz condition with respect to the second variable; that is, there exists a constant ki such that

f(t, x)-f (t, y) | < ki |x - y|, t e R+, i = 1,2.

(H4) ui : R+ x R+ x R — R is a continuous function such that ui: R+ x R+ x R+ — R+ i = 1,2.

(H5) There exists a continuous and nondecreasing function Gi: R+ — R+ and a bounded and continuous function gi: R+ x R+ — R+ such that ui(t,s,x) =gi(t,s)Gi(|x|) for t,s e R+, x e R, and i = 1,2.

(H6) The function t — f0 (t* - s*)ai-1sYigi(t, s) ds is bounded on R+ and

lim / (t* -sA)ai-1sYigi(t,s)ds = 0, i = 1,2.

t—^J 0

(H7) The function gi, i = 1,2, satisfies r t2

rl—m^J su^0 { K t2 - sft) ai-1sYigi(t2, s) - - sA) ai-1sYigi(t1, s)|

- [(t2ft - sP>)ai-1sY'gi(t2,s) - (4i - s*)ai-1sYigi(t1,s)]} ds: t2 > t1 > ^ J =0.

Remark 3.1 For some certain k, v,a, b,c,d e R, we can choosegi(t,s) = kev(t+s) orgi(t,s) = (at+bs+c)d to guarantee the assumptions (H6) and (H7) hold. For more details, one can see the example in Section 4.

For brevity, we set

Fi = sup| |fi(t,0) |: t e R+}, gi = sup{gi(t,s): t,s e R+},

Gi = sup{ TO) f01(t* - s*r>gi(t,s) ds: t e R+J, F = max{F1,F2}, k = max{k1, k2}, m = maxj ||m1y, ||m2^.

(H8) There exists a solution r0 > 0 of the following inequality:

[m + kG1rG1(r) + FG1G1(r)] x [m + kG2rG2(r) + FG2G2(r)] < r (6)

such that

L := mk(G1G1(r0) + G2G2(r0)) + 2//FGlGl(ro)G2G2(ro)

+ 2k2r0G1G1(r0)G2G2(r0) < 1. (7)

Consider the operators on the space BC(R+) by defining

(Fix)(t) =f (t, x(t)),

1 t ( ) ( )

(Uix)(t) = (tfai - """VWt,s,x(s)) ds, i = 1,2.

T(ai^0

As a result, we obtain

(Vix)(t)=mi(t) + (Fix)(t)(Uix)(t), t e R+, i = 1,2. (8)

In order to achieve our goal, we present the following results.

Lemma 3.2 The operators Ui and Fi, i = 1,2 transform continuously the set fi c BC(R+) into BC(R+) in such a way that Ui(fi) and Fi(fi) are positive and bounded.

Proof For any x e fi, keeping in mind of our assumptions (H1)-(H5) one can derive the fact that Uix is nonnegative on R+, i = 1,2. Moreover, for t e R+, linking (2) with the imposed assumptions, we derive

(Vix)(t) < mi(t) + kix(t)+f(t,0) ft(tfa - sfa)ai-1sYiui(t,s,x(s)) ds T(ai) J0

< mi(t) + [kix(t)^^(:,°)]Gi(|x|) f - fi )-1sYigi(t, s) ds T(ai) Jq

< ||mi| + kiGi||x||Gi(||x||) + FGiGi(|x|), i = 1,2, (9)

which shows that Vix, i = 1,2, is bounded on R+.

Next, keeping in mind of the properties of the superposition operator in [27] and (H2) we find that Fix is continuous and bounded on R+, i = 1,2.

To show that Vix is continuous on R+, i = 1,2, it is sufficient to show that Uix is continuous on R+, i = 1,2.

Fix T >0 and e > 0, we choose any t1, t2 e [0, T] such that \t2 - t1 \< e. Without loss of generality we assume that t1 < t2. Since Yi > fa (1 - ai) -1 and ai > 0 for i = 1,2, we can take Zi > 1 such that ZiYi > Zifai(1 - ai) -1 and Zi(ai -1) + 1 > 0, i = 1,2. Set Z* := ^^ (i = 1,2). By

Lemma 2.9 and the Holder inequality, we have

|(U,x)(t2)-(Uix)(t1)|

■ i 2 (t* - s*)ai-1sYiu(t2,s,x(s)) ds 02

- tOt Jt2 (4' - s*)ai-1sYiui(t1,s,x(s)) ds

1 t (t* - s*)ai-1sYiui(t1,s,x(s)) ds 02

T(ai) |1

T(ai) J0 1 /•t1

1 1 (t* - s*)<H-1sYiu(t1,s,x(s)) ds

f1 (t* - s*)ai-1sYiui(t1,s,x(s)) ds 02

i 1 (t* - s*)ai-1sYiui(t1,s,x(s)) ds

< (t* - s*;)ai-1sYi |u^t2,s,x(s)) - u^t1,s,x(s)) | ds

+ f2 (t* - s* )ai-1sYiui(t1, s, x(s)) ds t1 2

f1 u(t1, s,x(s))sYi [(t* - *y-1 - (t* - s*)ai-1] ds J0

, N it2 (t* - s*^V* ds + ^^^ it2 (t2* - s*f"1 sY' ds T(ai) h 12 ' T(ai) A1 V2 '

Gi(|x|)gi ft1 + T(ai)

*iT(ai)

T(ai) Jt1 1 /• t1 + T^

(ui, e) /"t2

T [(t*i - s*)ai-1- (t2*i - s*)ai-1]sYi-ds

T^-D+y'VYL+i,ai\ +

V *i / T(ai)

(t*i - s*)Z(ai-1]sZYids +

Gi(||x||)gi

ai) 11

* - s*)ai-1 sK ds

£ {* - s*^V* d^ + £2 {* - s*^ ds

*iT(ai)

T*i(a;-1)+Yi-+1W Yi + 1,

2Gi(|x|)gi ^ J»^«*1 »(^ai)

+ Gi(||x||)gi [J?;(a/-1)+Y/+1 - tfii(ai-1)+Yi+1]»( Y + 1 a

T(ai) L2 1 J V *i ' '

0)ln(ui,e)T*M-1)+Yi+1 + Gi(|x|)gib)T(h e)

*iT(ai) T(ai) ( , \

2Gi(|x|)gi ^

B ^ ai

«¡^(ui,e) = sup{|ui(t2,s,x)-ui(t1,s,x)1: t1, t2,s e [0, T],

\t2-11 \ < e,x e [-d, d]}, (1)

and h(t) := t^"^^1 is continuous on [0, T].

The estimate (10) implies that Ui is uniformly continuous on [0, T] x [0, T] x [-||x||, ||x||]. Thus, we known that Uix, i = 1,2 is continuous on R+, which guarantees that Vi, i = 1,2, transforms the set fi into fi due to (10). The proof is done. □

Lemma 3.3 There exists a r0 > 0 such that W = V1V2 transforms fir0 into fir0 where

firo = {x e BC(R+): 0 < x(t) < r0, t e R+}, (2)

Vi(firo)| < m + kiGroGi(ro) + FGGi(ro), i = 1,2.

Proof Note that (9) and (6), there exists a r0 > 0 such that W = V1V2 transforms firo into fir0 and (13) holds. □

Lemma 3.4 The operator W = V1V2 is a contraction with respect to the measure of non-compactness ¡d with the constantLgiven in (7).

Proof Choose a nonempty subset X of the set firo and choose any fixed T >0 and e > 0. Then, for x e X and for t1, t2 e [0, T] such that \t2 -11\ < e and t2 > t1, we have

|(Vix)(t2) - (Vix)(t1)| < «T(mi,e) + |(Fix)fe) - (Fix)(t1)||(Uix)(t2)|

+ |(Fix)(t1)||(Uix)(t2) - (Uix)(t1)|. (14)

In a similar way we obtain

|(Fix)(t2) - (Fix)(t1)| < ki|x(t2) -x(t1)| + ^(t2,x(ft)) -fi{t1,x(t1))|

< ki«T(x,e)+oT^tfi,e), (15)

wTd (fi, e) = sup{ f(t2, x)-fi(t1, x)| : t1, t2 e [0, T],\t2 - t1 \ < e, x e [-d, d}). Moreover,

|(Uix)(t2)| < P№ -sfai)ai-1sYigi(t,s)ds < Gi(||x||)Gi,

1 (ai) Jo

|(Fix)(t1)| < ki|x(t1)| + ^(t1,0)| < kiro + Fi, for any t1, t2 e R+.

Further, linking (14), (15) with the above fact, we arrive at

|(Vix)(t2)-(Vix)(t1)|

< «T(mi,e) + [ki«T(x,e)+«^(/i,e)]Gi(||x||)Gi + [kiro + Fi]

'<*{Ui'e)Tß,a-i)+v,+i + GQW(h e)

ftr(a) r(a) v ' \

+ 2G im)gt fa Z j (16)

1(ai) V fa J'

Clearly, ^^iTx! (fi, e) and «Tc| (ui, e) tend to zero as e ^ 0 since the functions fi and ui are uniformly continuous on the set [0, T] x [-||x||, ||x||] and [0, T] x [0, T] x [-||x||, ||x||], respectively. Hence we have

«TT(ViX) <fGiG^ro«(X) (7)

«o°(VX) <ZiGiGi(ro)«000(X). (18)

Next, choose any x,y e X and t e R+. Keeping in mind our assumptions, we obtain |(Vix)(t)-(Vy)(t)|

< f(t,x(t))-f:(t,y(t))\ ft(tfa. -^rViui(t,s,x(s))ds

r(ai) Jo

■ / (tfai -sfai)ai-1sYi|u^t,s,x(s)) - ui(t,s,y(s)) | ds Jo

fi(t>y(t)) I - (.ß, ß^a,-1

r(«i) ./0

^w-yi^M) f(tß, -5ß,^(t,,)^

r(«i) J o

WfVft (tßi - 5ßi ^^(t, s)[G,(x(s)) - G,(y(s))] ds 1 (ai) Jo

ki\x(t)-y(t)\Gi(ro) I ■ uß. ß.,a,-1 n

■ / (tßi - sß0ai-1 sy'gi(t,s) ds

+ (kro+rf2G(ro) it(tß, - ß)ai lsVigi(t,s) ds. (19)

r(a;) Jo

Now using (H6), we derive

lim sup diam(y,X)(t) = o. (2o)

In what follows, we prove that y is continuous on ^ro.

Fix e > o and take x,y e ^ro such that \\x - y|| < e. In view of (2o) we know that we may find a number T > o such that for any t > T we get

|(y,x)(t) - (y,y)(t) I < e.

On the other hand, for any t e [0, T], we have

|(Vx)(t)-(Viy)(t)|

fi(t, x(t))-fi(t, y(t))| f (* * )ai-1

fi(t, y(t))

T(ai) 0

M a' Yi

i (t*' - s*)ai-1sYiui(t,s,x(s)) ds

f (t* -s*')a'-1sY'^u(t,s,x(s)) - ui(t,s,y(s)) | ds

ki|x(t)-y(t)!Gi(||x||) f - s* )a'-1sYigi(t, s) ds T(ai) 0

MMfe«^ r T - ^ ^^

^TTT1 + 'iTn T*'M»( T' a) e),

T(ai) PT(ai) \ *i J 0

(a.) *T(ai)

where we denoted

«^(u.,e) = sup{|u.(t,s,x)-u.(t,s,y)|: t,s e [0, T],x,y e [-r0,^0], |x -y| < e}. By using the uniform continuity of the function u. on [0, T] x [0, T] x [r0, r0], we derive ^T0(ui,e) as e — 0.

This shows that we can find T >0 such that wTQ (ue) is sufficiently small for t > T and i = 1,2.

Now, we take any fixed T >0 and choose t1, t1 such that t2 > t1 > T. Then, for any x e X, we obtain

|(Vx)(t2) - (Vix)(ti)| - [(Vix)(t2) - (Vix)(ti)] < dT(m.) + dT(Fix)(Vix)(t2) + (F.x)(ti)| |(Uix)(t2) - (Uix)(ti)| - [(Uix)(t2) - (Uix)(ti)]}. (22)

Note that

|(Uix)(t2) - (Uix)(ti)| - [(U.x)(t2) - (Uix)(ti)]

i 2 (4' - s*)a'-1sY'gi(t2, s)G(x(s)) ds

P (t* - s*)a'-1sY'gi(ti,s)Gi(x(s)) ds

P (t*' - s*)a'-1sY'gi(ti,s)Gi(x(s)) ds

P (t* - s*)a'-1sY'gi(t2,s)Gi(x(s)) ds

P (t*' - s*)a'-1sY'gi(ti,s)Gi(x(s)) ds

T(ai) 1

- rO) £2^ -sPiyi-1syg^s)Gi(x(s)) ds

- radii-sft)ai-1sYigi(i2's)Gi(x(s))

- (tf -sf)ai-1sYigi(ti,s)G(x(s))\ds

- f12 [(tf - r'-V'gife,s)Gi(x(s)) jo

- (tf - sf)ai-1sKgi(ti,s)Gi(x(s))] dsj

- ll1 f1 \ (tfi - sfi r-v^, s)-(tf - ^-v^, s)\

; (ai) Jo

- [(t2fi -sf)ai-1sYigi(t2,s) - (tf -sf)ai-1sYigi(ti,s)]} ds. (23) By (Hi), (H5), and (H7) and (22), we obtain

dm(Vix) - dm(Fix)Gi(ro)Gi, i = 1,2.

Hence, in view of Lemma 2.8, we derive

dm(Vix) - kiGiGi(ro)dK,(x), i = 1,2.

Linking (23), (15), (18), and (22), we obtain

ld(ViX) - kiGiGi(ro)xd(X), i = 1,2.

It comes from (13) and (7) we obtain

L := |Vi(^ro)|k2 + |V2(^ro)|ki<1,

which implies that W is a contraction with respect to the measure of noncompactness id with the constant 0 < L <1. This completes the proof. □

Now we are ready to state the main result in this paper.

Theorem 3.5 Let the assumptions (H1)-(H8) be satisfied. Then (1) with (2) has at least one solutionx = x(-) in the space BC(R+). Moreover, thissolution is nonnegative, asymptotically stable, and ultimately nondecreasing.

Proof By Lemmas 3.2-3.4, one can see that all the assumptions in Lemma 2.6 are satisfied. Thus, one can infer that the operator W has at least one fixed point x e Due to Remark 2.7 we know that x is nonnegative on R+, asymptotically stable, and ultimately nondecreasing. The proof is done. □

To end this section, we establish some sufficient conditions to derive the uniqueness of solution.

Theorem 3.6 Let the assumptions of Theorem 3.5 be satisfied. There exists a positive constant hi such that

\u(t,s,x) - u(t,s,y)\< hi|x-y|, i = 1,2 (24)

for any t e R+ and all x, y e where ^ro is defined in (12). Then for

ai= a2 e (0,1), ft = ^2 e (0,+to), yi = Y2 e (ft(1- ai)-1,+ro),

(1) with (2) has a unique nonnegative, asymptotically stable, and ultimately nondecreasing solution.

Proof Suppose that y be another nonnegative and nondecreasing solution of (1). Then y satisfies the following integral equation:

y(t) = (Viy)(t)(V2y)(t), t e R+.

Note that

\x(t)-y(t)\ < \(V1x)(t)(V2x)(t)-(V1y)(t)(V2x)(t)\ + \(V1y)(t)(V2x)(t) - (V1y)(t)(V2y)(t)\ < \(V2x)(t)\\(V1x)(t)-(V1y)(t)\ + \(V1y)(t)\\(V2x)(t)-(V2y)(t)\, (25)

\(Vix)(t)-(Viy)(t)\

„ fi(t,x(t))-fi(t,y(t))|

+ fe^ f ti -)"'-1sn\ui(t,s,x(s)) - ui(t,s,y(s)) \ ds I (ai) Jo

< ^i|x(t)-y(t);Gi(yxy) ft ^ - s- rv^t, s) ds r(«i) Jo

/ (tPi -sPi)ai-1syim(t,s,x(s)) ds

f t^ -rVi|x(s)-y(s)|ds T(ai) Jo

x(t)-y(t)\

(ai) kiGiGi(r 0)

~ r(oii)

+ hi(kiro+ F) f* ^ -s?iyi-1sy'\x(s)-y(s)\ds, i = 1,2. (26)

T(«i) Jo

Using (26) and (9) in (25), we obtain \x(t) - y(t) \ < L\x(t) - y(t) \

+ 2h(kro+ F )[m + kGroG + FGG] ft ^ - s,1 r> \x(s) -y(s)\ ds

T(«1) Jo

for any t e R+ and all x,y e where L is given in (7), G = max{G1, G2}, G = max{G1(r0), G2(ro)}, and h = max{hi, h2}. Note that (7), we have

(1- L)\x(t)-y(t)\

2h(kro + F)[m + kGroG + FGG]

- 2h(kr0+ F)[fkGr0G + FGG] <1 f (tfi - sfi)Z^Vn ds

t ( ) \ \

/ (tfi-sfi)ai-1sYi \x(s)-y(s)\ds

j*(tfi -sfi)Z

x \x(s)-y(s)\Z* ds 2h(kro + F)[m + kGro G + FGG]

where Z and Z * are defined in the proof of Lemma 3.2. In view of (7), we can rewrite the above inequality to

z(t) - c(t) i z(s) ds - (i + c(t)) i z(s) ds, (7)

where z(t) := |x(t) -y(t)|Z* and

2Z*hZ* (kro + F)Z* [m + kGroG + FGG]z*

C(t): =

(1- L)Z * T(ai)Z *

x( b(^ z(«1-1) + ^ r

-. Z(a-1) + 1ll

From (27), we get

z(t) ^ 1 + c(t) -

i (1 + c(s))

_1 + c(s)_

and the Gronwall inequality implies j4j|) = 0, so z(t) = 0. This completes the proof. □ 4 An example

Motivated by Example 11 in [6], we treat a numerical example to illustrate the main results. Consider the following quadratic fractional integral equation:

x(t) = (Vix)(t)(V2x)(t), (28)

2t I arctan(t2 + x(t)) ft. i i.i -+ ^- i ^

5t + 1 T(f)

(Vi x)(t) =-+ ^--pp—— 12- s ip se sx (t) ds, (29)

1 - e-2t 2 ln(x(t) + 1) f tx4(t)(t2 - 52)-5s Tf) J0 (t + s + 2)3

(y2x)(t) = —— +2 :2;—- ——— ds, (30)

for t e R+.

Clearly, a1 = a2 = 3, f1 = ft = *, Y1 = Y2 = 1, and the functions in (1) have the form m1 = 52+1, m2 = ,/1(t,x) = 3 arctan(t2 + x(t)),f2(t,x) = 2 ln(x(t) + 1), M1(t,s,x) = e-t-sx2, u2(t,s, x)= (¡+¡+23.

In what follows, we check that the above functions will satisfy all the assumptions of Theorem 3.5.

Step 1, the function mi, i = 1,2 is nonnegative, bounded, and continuous on R+. Since m1 and m2 are increasing on R+, they must be ultimately nondecreasing on R+. Meanwhile, ||m1N = § and ||m2 H = 5. Thus, (H1) holds.

Step 2, fi, i = 1,2 transform continuously the set R+ x R+ into R+. Moreover,/1 is non-decreasing with respect to both variables and satisfies the Lipschitz condition with the constant k1 = 5. Similarly, the functionf2 = f2(t,x) is increasing with respect to x and satisfies the Lipschitz condition with the constant k2 = 2. Also, F1 = 3, F2 = 0. Thus,/ and /2 satisfy (H2) and (H3).

Step 3, ui(t,s,x) is continuous on R+ x R+ x R and transforms R+ x R+ x R+. Meanwhile, ui(t,s,x) = gi(t,s)Gi(|x|), i = 1,2, where g1(t,s) = e-t-s, G:(x) = x2, g2(t,s) = (t+s+2)3, and G2(x) = x4. It is easily seen that (H4) and (H5) are satisfied for u1 and u2.

Step 4, one has

^ (t2- s2)-3se-t-sds < e-t^ (t2- s2)-3sds = 2B^4,2^e-tt11,

/' T^ds < ^f' {l»- s 2)-» sds = .

Jo (t + s + 2) (t + 2) Jo (t + 2)

lim / (tfi - sft)ai-1sYi'gi(t, s) ds = 0, i = 1,2.

¡^J 0

So we find that (H6) is satisfied. Step 5, for 0 < T < ti < t2, we get

f {|(f -sf1 pV^fe,s) - (tf1 -sf1 rWfe,s)| 0

- [(tf1 - sf1 ^V g1(t2, s) - (if1 - sf1 )a1-1sY1g1(t1, s)]} ds

t2 {|(t22 -s2)-3se-t2-s - (t12 -s2)-3se-t1-s|

- [(t22 -s2)-3se-t2-s - (t? -s2)-3se-t1-s]} ds

= 2 i¡2 [(t2 -s2)-3se-t1-s - (t22 - s2)-3se-t2-s] ds 0 1 2

r12 1 , 1 r ¡2 1, 1

< 2e-tW (t2-s2)-3sds-2e-2t2 / (t22 -s2)-3sds 0 1 0 2

• ft2 1 1 _ 1 3sds + 2e-ti / (t?-si)-5

ri i 1 1

(?- s 2 P

2\ 11 11 />t2 1 1-13 1 4, -j [e-tit16 - e-2t2tf] + 4e-ti J (ti -sips2 ds 1

4,3) e2t21221

4, 2 [e-ti ti1 - e-2t2126 ] - 12e-ti tf (12 - tf)

2 \ 11 11

4, - j [e-ti tf - e-2t21!6 ] — 0 as t2 > ti — to.

Similarly, we obtain f t2

- sf2) a2 ^s^2 g2(t2, s) -tf2 - sf2) a2 !sY2 g2(ti, s)\ - [(tf2 - sf2 ) a2 -1sY2 g2 (t2, s) - (tf2 - sf2 )a2-1 sY2g2(ti, s)]} ds

1 1 1 1 1 1

(t22 - s 2) 3 s (ti - s 2) 3 s

(t2 + s + 2)3 (ti + s + 2)3

(t22 - s 1) 1 s (ti2 - s2) 1 s' _ (t2+ s + 2)3 - (ti+ s + 2)3 _

(ti2 - s2)-3s (t22 - s2)-3s' (ti + s + 2)3 - (t2 + s + 2)3

(ti + 2)3 1, 3)ti

p 2 1 1 1 (tf- s 2 )-3

(2t2 + 2)3

ft2 1 1 1 (tf- s 2 P

, 3)l2

Z112 1 1 1

(ti1- s2 P

(ti + 2)3 (2t2 + 2)3 (ti + 2)3 J ti

, 3)tir

,|)t216 U^t1- tf)2

(ti + 2)3 (2t2 + 2)3

(ti + 2)3

— 0 as t2 > t1 —^ to.

Thus, we see that (H7) is satisfied.

Step 6, using the above facts we find m = max{ym1y, ||m2||} = 2, k = max{k1, k2} = 2, F = max{F 1,F2} = 3 and Gi = 0.3962, G2 = 0.0489. Thus, the first inequality in (H8) reduces to

2 2 — o n — / 2 2 — n — „;

1 + 1 Gir3 +1^jU + aG2r5 + 3^ -r.

One can verify that r0 = 1 is a solution of the above inequality such that it satisfies also the second inequality in (H8)

2 2 _ _ 4n__/2\2__

2 x 2(Girg + G2r4) + —GiG2r6 + (jJ GiG2r7 = 0.1544 < 1.

As a result, all the assumptions in Theorem 3.5 are satisfied. Moreover, (24) in Theorem 3.6 is also satisfied. Thus, (28) with V1, V2 in (29) and (30) has a unique solution x e ^ where ^ = {x e BC(R+): 0 - x(t) - 1 for t e R+}, which is asymptotically stable and ultimately nondecreasing.

The unique asymptotically stable and ultimately nondecreasing solution of (28) with (29) and (30) is displayed in Figure 1.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

This work was carried out in collaboration between allauthors. JRW raises these interesting problems in this research. JRW, ZC and XLY proved the theorems, interpreted the results and wrote the article. Allauthors defined the research theme, read and approved the manuscript.

Author details

'College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030031, P.R. China. 2Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China. 3IndustrialInternet ofThings Engineering Research Center of the Higher Education Institutions of Guizhou Province and Schoolof Mathematics and Computer Science, Guizhou NormalCollege, Guiyang, Guizhou 550018, P.R. China.

Acknowledgements

The authors thank the referees for their carefulreading of the manuscript and insightfulcomments, which helped to improve the quality of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to the improvement of the presentation of the paper. This work is supported by Science and Technology Program of Guiyang (No. ZhuKeHeTong[2013101]10-6), Key Support Subject (Applied Mathematics), Key project on the reforms of teaching contents and course system of Guizhou NormalCollege, Doctor Project of Guizhou NormalCollege (13BS010) and Guizhou Province Education Planning Project (2013A062).

Received: 16 March 2014 Accepted: 15 April 2014 Published: 06 May 2014

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10.1186/1687-1847-2014-130

Cite this article as: Yu et al.: On a weakly singular quadratic integral equations of Volterra type in Banach algebras.

Advances in Difference Equations 2014, 2014:130

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