doi:10.1155/2010/127093
Research Article
On the Existence of Locally Attractive
Solutions of a Nonlinear Quadratic Volterra Integral Equation of Fractional Order
Mohamed I. Abbas
Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt
Correspondence should be addressed to Mohamed I. Abbas,m i [email protected] Received 19 May 2010; Accepted 25 November 2010
Academic Editor: Mouffak Benchohra
Copyrightq2010 Mohamed I. Abbas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The authors employs a hybrid fixed point theorem involving the multiplication of two operators for proving an existence result of locally attractive solutions of a nonlinear quadratic Volterra integral equation of fractionalarbitraryorder. Investigations will be carried out in the Banach space of real functions which are defined, continuous, and bounded on the real half axisÊ.
1. Introduction
The theory of differential and integral equations of fractional order has recently received a lot of attention and now constitutes a significant branch of nonlinear analysis. Numerous research papers and monographs have appeared devoted to differential and integral equations of fractional ordercf., e.g.,1–6. These papers contain various types of existence results for equations of fractional order.
In this paper, we study the existence of locally attractive solutions of the following nonlinear quadratic Volterra integral equation of fractional order:
xt
ft, xt
qt 1
Γα t
0
gt, s, xs t−s1−α ds
, 1.1
for allt∈Ê andα∈0,1, in the space of real functions defined, continuous, and bounded on an unbounded interval.
It is worthwhile mentioning that up to now integral equations of fractional order have only been studied in the space of real functions defined on a bounded interval. The result obtained in this paper generalizes several ones obtained earlier by many authors.
In fact, our result in this paper is motivated by the extension of the work of Hu and Yan7. Also, We proceed and generalize the results obtained in the papers8,9.
2. Notations, Definitions, and Auxiliary Facts
Denote byL1a, bthe space of Lebesgue integrable functions on the interval a, b, which is equipped with the standard norm. Letx ∈L1a, band letα > 0 be a fixed number. The Riemann-Liouville fractional integral of orderαof the functionxtis defined by the formula:
Iαxt 1 Γα
t
0
xs
t−s1−αds, t∈a, b, 2.1 whereΓαdenotes the gamma function.
It may be shown that the fractional integral operator,Iαtransforms the spaceL1a, b into itself and has some other propertiessee10–12.
LetX BCÊbe the space of continuous and bounded real-valued functions onÊ and letΩbe a subset ofX. LetP :X → Xbe an operator and consider the following operator equation inX, namely,
xt P xt, 2.2
for all t ∈ Ê. Below we give different characterizations of the solutions for the operator equation2.2onÊ. We need the following definitions in the sequel.
Definition 2.1. We say that solutions of2.2are locally attractive if there exists anx0∈BCÊ and anr >0 such that for all solutionsxxtandyytof2.2belonging toBrx0∩Ω we have that:
t→ ∞lim
xt−yt
0. 2.3
Definition 2.2. An operatorP :X → Xis called Lipschitz if there exists a constantksuch that P x−P y ≤kx−yfor allx, y∈X. The constantkis called the Lipschitz constant ofP on X.
Definition 2.3Dugundji and Granas13. An operatorP on a Banach spaceXinto itself is called compact if for any bounded subsetSofX,PSis a relatively compact subset ofX. If Pis continuous and compact, then it is called completely continuous onX.
We seek the solutions of1.1in the space BCÊof continuous and bounded real- valued functions defined onÊ. Define a standard supremum norm · and a multiplication
“·” in BCÊby
xsup{|xt|:t∈Ê}, xy
t xtyt, t∈Ê. 2.4
Clearly, BCÊ becomes a Banach space with respect to the above norm and the multiplication in it. ByL1Êwe denote the space of Lebesgue integrable functions onÊ with the norm · L1defined by
xL1 ∞
0
|xt|dt. 2.5
We employ a hybrid fixed point theorem of Dhage14for proving the existence result.
Theorem 2.4Dhage14. LetSbe a closed-convex and bounded subset of the Banach spaceXand letF, G:S → Sbe two operators satisfying:
aFis Lipschitz with the Lipschitz constantk, bGis completely continuous,
cFxGx∈Sfor allx∈S, and
dMk <1 whereMGSsup{Gx:x∈S}.
Then the operator equation
FxGxx 2.6
has a solution and the set of all solutions is compact inS.
3. Existence Result
We consider the following set of hypotheses in the sequel.
H1The functionf : Ê ×Ê → Ê is continuous, and there exists a bounded function l:Ê → Ê with boundLsatisfying
ft, x−f
t, y ≤lt x−y 3.1
for allt∈Ê andx, y∈Ê.
H2The functionf1:Ê → Êdefined byf1|ft,0|is bounded with f0sup
f1t:t∈Ê
. 3.2
H3The functionq:Ê → Ê is continuous and limt→ ∞qt 0.
H4The functiong :Ê ×Ê ×Ê → Êis continuous. Moreover, there exist a function m:Ê → Êbeing continuous onÊand a functionh:Ê → Êbeing continuous onÊwithh0 0 and such that
gt, s, x−g
t, s, y ≤mth x−y 3.3
for allt, s∈Ê such thats≤tand for allx, y∈Ê.
For further purposes let us define the functiong1:Ê → Êby putting g1t max gt, s,0 : 0≤s≤t
. 3.4
Obviously the functiong1is continuous onÊ.
In what follows we will assume additionally that the following conditions are satisfied.
H5The functionsa, b:Ê → Ê defined by the formulas
at mttα, bt g1ttα, 3.5
are bounded onÊ and vanish at infinity, that is, limt→ ∞at limt→ ∞bt 0.
Remark 3.1. Note that if the hypothesesH3andH5hold, then there exist constantsK1>0 andK2 >0 such that:
K1sup
qt:t∈Ê
, K2sup
athr bt
Γα1 :t, r∈Ê
. 3.6
Theorem 3.2. Assume that the hypothesesH1–H5hold. Furthermore, ifLK1K2<1, where K1 and K2 are defined in Remark 3.1, then 1.1 has at least one solution in the space BCÊ. Moreover, solutions of1.1are locally attractive onÊ.
Proof. SetX BCÊ,Ê. Consider the closed ball Br0 inX centered at origin 0 and of radiusr, wherer f0K1K2/1−LK1K2>0.
Let us define two operatorsFandGonBr0by Fxt ft, xt,
Gxt qt 1
Γα t
0
gt, s, xs
t−s1−α ds, 3.7
for allt∈Ê.
According to the hypothesis H1, the operator F is well defined and the function Fxis continuous and bounded on Ê. Also, since the function qis continuous on Ê, the functionGxis continuous and bounded in view of hypothesisH4. ThereforeFandGdefine the operatorsF, G : Br0 → X. We will show that F and G satisfy the requirements of Theorem 2.4onBr0.
The operatorF is a Lipschitz operator onBr0. In fact, letx, y ∈ Br0be arbitrary.
Then by hypothesisH1, we get
Fxt−Fyt ft, xt−f
t, yt ≤lt xt−yt ≤Lx−y, 3.8
for allt∈Ê. Taking the supremum overt,
Fx−Fy≤Lx−y, 3.9
for allx, y∈Br0. This shows thatFis a Lipschitz onBr0with the Lipschitz constantL.
Next, we show thatGis a continuous and compact operator onBr0. First we show thatGis continuous onBr0. To do this, let us fix arbitrary > 0 and takex, y ∈Br0such thatx−y ≤. Then we get
Gxt− Gy
t ≤ 1 Γα
t
0
gt, s, xs−g
t, s, ys t−s1−α ds
≤ 1 Γα
t
0
mth xs−ys t−s1−α ds
≤ mttα Γα1hr
≤ at Γα1hr.
3.10
Sincehr is continuous on Ê, then it is bounded onÊ, and there exists a nonnegative constant, sayh∗, such thath∗ sup{hr : r > 0}. Hence, in view of hypothesisH5, we infer that there existsT >0 such thatat≤Γα1/h∗fort > T. Thus, fort > Twe derive that
Gxt− Gy
t ≤. 3.11 Furthermore, let us assume thatt∈0, T. Then, evaluating similarly to the above we obtain the following estimate:
Gxt− Gy
t ≤ 1 Γα
t
0
gt, s, xs−g
t, s, ys
t−s1−α ds≤ Tα Γα1ωTr
g,
, 3.12
whereωTrg, sup{|gt, s, x−gt, s, y|:t, s∈0, T,x, y∈−r, r,|x−y| ≤}.
Therefore, from the uniform continuity of the function gt, s, x on the set 0, T× 0, T × −r, r we derive that ωrTg, → 0 as → 0. Hence, from the above- established facts we conclude that the operator G maps the ball Br0 continuously into itself.
Now, we show thatGis compact onBr0. It is enough to show that every sequence {Gxn} inGBr0has a Cauchy subsequence. In view of hypothesesH3and H4, we infer that:
|Gxnt| ≤ qt 1 Γα
t
0
gt, s, xns t−s1−α ds
≤ qt 1 Γα
t
0
gt, s, xns−gt, s,0
t−s1−α ds 1 Γα
t
0
gt, s,0 t−s1−αds
≤ qt 1 Γα
t
0
mth|xns|
t−s1−α ds 1 Γα
t
0
g1t t−s1−αds
≤ qt mttα
Γα1hr g1ttα Γα1
≤ qt athr bt Γα1
≤K1K2,
3.13
for allt ∈Ê. Taking the supremum overt, we obtainGxn ≤ K1K2 for alln∈ Æ. This shows that {Gxn} is a uniformly bounded sequence in GBr0. We show that it is also equicontinuous. Let > 0 be given. Since limt→ ∞qt 0, there is constantT >0 such that
|qt|< /2 for allt≥T.
Lett1, t2∈Ê be arbitrary. Ift1, t2∈0, T, then we have
|Gxnt2−Gxnt1|
≤ qt2−qt1 1 Γα
t2
0
gt2, s, xns t2−s1−α ds−
t1
0
gt1, s, xns t1−s1−α ds
≤ qt2−qt1 1 Γα
t1
0
gt2, s, xns t2−s1−α ds
t2
t1
gt2, s, xns t2−s1−α ds
t1
0
gt1, s, xns t1−s1−α ds
≤ qt2−qt1 1
Γα t1
0
gt2, s, xns
t2−s1−α −gt1, s, xns t2−s1−α
ds
t1
0
gt1, s, xns
t2−s1−α − gt1, s, xns t1−s1−α
ds t2
t1
gt2, s, xns t2−s1−α ds
≤ qt2−qt1 1
Γα t1
0
gt2, s, xns−gt1, s, xns t2−s1−α ds
t1
0
gt1, s, xns 1
t2−s1−α − 1 t1−s1−α
ds
t2
t1
gt2, s, xns t2−s1−α ds
≤ qt2−qt1 1
Γα t1
0
gt2, s, xns−gt1, s, xns 1 t2−s1−αds
t1
0
gt1, s, xns−gt1, s,0 gt1, s,0 1
t2−s1−α − 1 t1−s1−α
ds
t2
t1
gt2, s, xns−gt2, s,0 gt2, s,0 t2−s1−α ds
≤ qt2−qt1 1
Γα t1
0
gt2, s, xns−gt1, s, xns 1 t2−s1−αds
t1
0
mt1h|xns| g1t1 1
t2−s1−α − 1 t1−s1−α
ds
t2
t1
mt2h|xns| g1t2 t2−s1−α ds
≤ qt2−qt1 1 Γα
t1
0
gt2, s, xns−gt1, s, xns 1 t2−s1−αds mt1hr g1t1
Γα1
tα1−tα2 t2−t1α
mt2hr g1t2
Γα1 t2−t1α.
3.14 From the uniform continuity of the function qt on 0, T and the function g in 0, T× 0, T×−r, r, we get|Gxnt2−Gxnt1| → 0 ast1 → t2.
Ift1, t2≥T, then we have
|Gxnt2−Gxnt1| ≤ qt2−qt1 1 Γα
t2
0
gt2, s, xns t2−s1−α ds−
t1
0
gt1, s, xns t1−s1−α ds
≤ qt1 qt2 1 Γα
t2
0
gt2, s, xns t2−s1−α ds−
t1
0
gt1, s, xns t1−s1−α ds
< ,
3.15 ast1 → t2.
Similarly, ift1, t2∈Ê witht1< T < t2, then we have
|Gxnt2−Gxnt1| ≤ |Gxnt2−GxnT||GxnT−Gxnt1|. 3.16
Note that ift1 → t2, thenT → t2andt1 → T. Therefore from the above obtained estimates, it follows that:
|Gxnt2−GxnT| −→0, |GxnT−Gxnt1| −→0, ast1−→t2. 3.17
As a result,|Gxnt2−GxnT| → 0 ast1 → t2. Hence{Gxn}is an equicontinuous sequence of functions in X. Now an application of the Arzel´a-Ascoli theorem yields that{Gxn}has a uniformly convergent subsequence on the compact subset 0, T of Ê. Without loss of generality, call the subsequence of the sequence itself.
We show that{Gxn}is Cauchy sequence inX. Now|Gxnt−Gxt| → 0 asn → ∞ for allt∈0, T. Then for given >0 there exists ann0 ∈Æ such that form, n≥n0, then we have
|Gxmt−Gxnt| 1 Γα
t
0
gt, s, xms−gt, s, xnt t−s1−α ds
≤ 1 Γα
t
0
gt, s, xms−gt, s, xnt t−s1−α ds
≤ 1 Γα
t
0
mth|xms−xns|
t−s1−α ds
≤ mttαhr Γα1
≤ ath∗ Γα1
< .
3.18
This shows that{Gxn} ⊂GBr0⊂Xis Cauchy. SinceXis complete, then{Gxn}converges to a point in X. As GBr0 is closed, {Gxn} converges to a point in GBr0. Hence, GBr0is relatively compact and consequentlyGis a continuous and compact operator on Br0.
Next, we show thatFxGx∈Br0for allx∈Br0. Letx∈Br0be arbitrary, then
|FxtGxt| ≤ |Fxt||Gxt|
≤ ft, xt
qt 1 Γα
t
0
gt, s, xs t−s1−α ds
≤ ft, xt−ft,0 ft,0
qt 1 Γα
t
0
gt, s, xs−gt, s,0 gt, s,0 t−s1−α ds
≤
lt|xt|f1t
qt 1 Γα
t
0
mth|xt| g1t t−s1−α ds
≤
Lxf0
qt mttαhr g1ttα Γα1
≤
Lxf0
qt athr bt
Γα1
≤
Lxf0
K1K2
≤LK1K2xf0K1K2 f0K1K2
1−LK1K2 r,
3.19
for allt ∈Ê. Taking the supremum overt, we obtainFxGx ≤r for allx ∈Br0. Hence hypothesiscofTheorem 2.4holds.
Also we have
MGBr0
sup{Gx:x∈Br0}
sup
sup
t≥0
qt 1 Γα
t
0
gt, s, xs t−s1−α ds
:x∈Br0
≤sup
t≥0
qt sup
t≥0
athr bt
Γα1
≤K1K2,
3.20
and thereforeMk LK1K2 <1. Now we applyTheorem 2.4to conclude that1.1has a solution onÊ
Finally, we show the local attractivity of the solutions for1.1. Letxandybe any two solutions of1.1inBr0defined onÊ, then we get
xt−yt ≤
ft, xt
qt 1
Γα t
0
gt, s, xs t−s1−α ds
f
t, yt
qt 1
Γα t
0
g
t, s, ys t−s1−α ds
≤ ft, xt
qt 1 Γα
t
0
gt, s, xs t−s1−α ds
f
t, yt
qt 1 Γα
t
0
g
t, s, ys t−s1−α ds
≤2
Lrf0 qt athr bt Γα1
,
3.21
for allt∈Ê. Since limt→ ∞qt 0, limt→ ∞at 0 and limt→ ∞bt 0, for > 0, there are real numbersT>0,T >0 andT >0 such that|qt|< fort≥T,at< h∗/Γα1for allt≥Tandbt< /Γα1for allt≥T. If we chooseT∗max{T, T, T}, then from the above inequality it follows that|xt−yt| ≤∗fort≥T∗, where∗ 6Lrf0 >0. This completes the proof.
4. An Example
In this section we provide an example illustrating the main existence result contained in Theorem 3.2.
Example 4.1. Consider the following quadratic Volterra integral equation of fractional order:
xt
tt2xt
te−t2/2 1 Γ2/3
t
0
x2/3se−3ts1/
10t8/31 t−s1/3 ds
, 4.1
wheret∈Ê.
Observe that the above equation is a special case of1.1. Indeed, if we putα 2/3 and
ft, x tt2x, qt te−t2/2, gt, s, x x2/3se−3ts 1
10t8/31.
4.2
Then we can easily check that the assumptions of Theorem 3.2 are satisfied. In fact, we have that the functionft, xis continuous and satisfies assumptionH1, wherelt t2
and ft,0 ft,0 t f1 as in assumption H2. We have that the function qt is continuous and it is easily seen thatqt → 0 ast → ∞, thus assumptionH3is satisfied.
Next, let us notice that the functiongt, s, xsatisfies assumptionH4, wheremt e−3t, hr r2/3andgt, s,0 1/10t8/31. Thusg1 gt, s,0. To check that assumptionH5 is satisfied let us observe that the functionsa, bappearing in that assumption take the form:
at t2/3e−3t, bt t2/3
10t8/31. 4.3
Thus it is easily seen thatat, bt → 0 ast → ∞. Finally, let us note that inRemark 3.1 there are two constants K1, K2 > 0 such thatLK1K2 < 1. It is also easy to check that K1 q1 e−1/2 0.60653. . .,K2 e−3 0.1/0.8856 0.16913. . . and L 1. Then LK1K2 0.77566. . . < 1. Hence, taking into account thatΓ5/3 > 0.8856cf.4, all the assumptions ofTheorem 3.2are satisfied and4.1has a solution in the space BCÊ. Moreover, solutions of4.1are uniformly locally attractive in the sense ofDefinition 2.1.
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