Volume 2010, Article ID 127093, 11 pagesdoi:10.1155/2010/127093 Research Article On the Existence of Locally Attractive Solutions of a Nonlinear Quadratic Volterra Integral Equation of F
Trang 1Volume 2010, Article ID 127093, 11 pages
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 abbas77@yahoo.com
Received 19 May 2010; Accepted 25 November 2010
Academic Editor: Mouffak Benchohra
Copyrightq 2010 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 fractionalarbitrary order 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:
x t f t, xt
q t 1 Γα
t
0
g t, s, xs
t − s1−α ds
for all t∈Ê and α ∈ 0, 1, in the space of real functions defined, continuous, and bounded
on an unbounded interval
Trang 2It 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 papers 8,9
2 Notations, Definitions, and Auxiliary Facts
Denote by L1a, b the space of Lebesgue integrable functions on the interval a, b, which
is equipped with the standard norm Let x ∈ L1a, b and let α > 0 be a fixed number The Riemann-Liouville fractional integral of order α of the function xt is defined by the formula:
I α x t 1
Γα
t
0
x s
t − s1−αds, t ∈ a, b, 2.1 whereΓα denotes the gamma function.
It may be shown that the fractional integral operator, I α transforms the space L1a, b
into itself and has some other propertiessee 10–12
Let X BCÊ be the space of continuous and bounded real-valued functions onÊ
and letΩ be a subset of X Let P : X → X be an operator and consider the following operator equation in X, namely,
for all t ∈ Ê Below we give different characterizations of the solutions for the operator equation2.2 onÊ We need the following definitions in the sequel
Definition 2.1 We say that solutions of2.2 are locally attractive if there exists an x0∈ BCÊ
and an r > 0 such that for all solutions x xt and y yt of 2.2 belonging to B r x0 ∩ Ω
we have that:
lim
t→ ∞
Px − Py ≤ kx − y for all x, y ∈ X The constant k is called the Lipschitz constant of P on
X.
called compact if for any bounded subset S of X, P S is a relatively compact subset of X If
P is continuous and compact, then it is called completely continuous on X.
We seek the solutions of1.1 in the space BCÊ of continuous and bounded real-valued functions defined onÊ Define a standard supremum norm · and a multiplication
“·” in BCÊ by
x sup{|xt| : t ∈ }, xy
t xtyt, t ∈ . 2.4
Trang 3Clearly, BCÊ becomes a Banach space with respect to the above norm and the
multiplication in it By L1Ê we denote the space of Lebesgue integrable functions onÊ
with the norm · L1defined by
x L1
∞
0
We employ a hybrid fixed point theorem of Dhage14 for proving the existence result
Theorem 2.4 Dhage 14 Let S be a closed-convex and bounded subset of the Banach space X and
let F, G : S → S be two operators satisfying:
a F is Lipschitz with the Lipschitz constant k,
b G is completely continuous,
c FxGx ∈ S for all x ∈ S, and
d Mk < 1 where M GS sup{Gx : x ∈ S}.
Then the operator equation
has a solution and the set of all solutions is compact in S.
3 Existence Result
We consider the following set of hypotheses in the sequel
H1 The function f : Ê ×Ê → Ê is continuous, and there exists a bounded function
l :Ê → Ê with bound L satisfying
f t, x − f
for all t∈Ê and x, y∈Ê
H2 The function f1:Ê → Êdefined by f1 |ft, 0| is bounded with
f0 sup f1t : t ∈Ê
H3 The function q :Ê → Ê is continuous and limt→ ∞q t 0.
H4 The function g :Ê ×Ê ×Ê → Êis continuous Moreover, there exist a function
m :Ê → Ê being continuous onÊ and a function h :Ê → Ê being continuous
onÊ with h0 0 and such that
g t, s, x − g
for all t, s∈ such that s ≤ t and for all x, y ∈
Trang 4For further purposes let us define the function g1:Ê → Ê by putting
g1t max g t, s, 0 : 0≤ s ≤ t
Obviously the function g1is continuous onÊ
In what follows we will assume additionally that the following conditions are satisfied
H5 The functions a, b :Ê → Ê defined by the formulas
a t mtt α , b t g1tt α , 3.5 are bounded onÊ and vanish at infinity, that is, limt→ ∞a t lim t→ ∞b t 0.
and K2 > 0 such that:
K1 sup q t : t ∈Ê
, K2 sup
a thr bt
Γα 1 : t, r ∈Ê . 3.6
Theorem 3.2 Assume that the hypotheses H1–H5 hold Furthermore, if LK1 K2 < 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 of1.1 are locally attractive onÊ .
radius r, where r f0K1 K2/1 − LK1 K2 > 0.
Let us define two operators F and G on B r0 by
Fx t ft, xt,
Gx t qt 1
Γα
t
0
g t, s, xs
for all t∈Ê
According to the hypothesis H1, the operator F is well defined and the function
function Gx is continuous and bounded in view of hypothesis H4 Therefore F and G define the operators F, G : B r 0 → X We will show that F and G satisfy the requirements of
Theorem 2.4on B r0
The operator F is a Lipschitz operator on B r 0 In fact, let x, y ∈ B r0 be arbitrary Then by hypothesisH1, we get
Fx t − Fyt ft,xt− ft,yt ≤ lt xt− yt ≤ Lx − y, 3.8
Trang 5for all t∈Ê Taking the supremum over t,
for all x, y ∈ B r 0 This shows that F is a Lipschitz on B r 0 with the Lipschitz constant L Next, we show that G is a continuous and compact operator on B r0 First we show
that G is continuous on B r 0 To do this, let us fix arbitrary > 0 and take x, y ∈ B r0 such thatx − y ≤ Then we get
Gxt −Gyt ≤ 1
Γα
t
0
g t, s, xs − g
t, s, y s
t − s1−α ds
≤ 1
Γα
t
0
m th x s − ys
t − s1−α ds
≤ m tt α
Γα 1 h r
≤ a t
Γα 1 h r.
3.10
Since hr is continuous on Ê , then it is bounded onÊ , and there exists a nonnegative
constant, say h∗, such that h∗ sup{hr : r > 0} Hence, in view of hypothesis H5, we infer that there exists T > 0 such that at ≤ Γα 1/h∗for t > T Thus, for t > T we derive
that
Furthermore, let us assume that t ∈ 0, T Then, evaluating similarly to the above we obtain
the following estimate:
Gxt − Gyt ≤ 1
Γα
t
0
g t, s, xs − g
t, s, y s
t − s1−α ds≤
T α Γα 1 ω T r
g,
, 3.12
where ω T
r g, 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 ω T
r g, → 0 as → 0 Hence, from the above-established facts we conclude that the operator G maps the ball B r0 continuously into itself
Trang 6Now, we show that G is compact on B r0 It is enough to show that every sequence
{Gx n } in GB r 0 has a Cauchy subsequence In view of hypotheses H3 and H4, we
infer that:
|Gx n t| ≤ q t 1
Γα
t
0
g t, s, x n s
t − s1−α ds
≤ q t 1
Γα
t
0
g t, s, x n s − gt, s, 0
t − s1−α ds
1
Γα
t
0
g t, s, 0
t − s1−α ds
≤ q t 1 Γαt
0
m th|x n s|
t − s1−α ds 1
Γα
t
0
g1t
t − s1−αds
≤ q t mtt α
Γα 1 h r
g1tt α Γα 1
≤ q t athr bt
Γα 1
≤ K1 K2,
3.13
for all t ∈Ê Taking the supremum over t, we obtain Gx n ≤ K1 K2 for all n ∈ Æ This shows that {Gx n } is a uniformly bounded sequence in GB r0 We show that it is also
equicontinuous Let > 0 be given Since lim t→ ∞q t 0, there is constant T > 0 such that
|qt| < /2 for all t ≥ T.
Let t1, t2∈Ê be arbitrary If t1, t2∈ 0, T, then we have
|Gx n t2 − Gx n t1|
≤ q t2 − qt1 1
Γα
t2
0
g t2, s, x n s
t2− s1−α ds−
t1
0
g t1, s, x n s
t1− s1−α ds
≤ q t2 − qt1 1
Γα
t1
0
g t2, s, x n s
t2− s1−α ds
t2
t1
g t2, s, x n s
t2− s1−α ds
t1
0
g t1, s, x n s
t1− s1−α ds
≤ q t2 − qt1
1
Γα
t1
0
g t2, s, x n s
t2− s1−α −g t1, s, x n s
t2− s1−α
ds
t1
0
g t1, s, x n s
t2− s1−α −
g t1, s, x n s
t1− s1−α
ds
t2
t
g t2, s, x n s
t2− s1−α ds
Trang 7
≤ q t2 − qt1
1
Γα
t1
0
g t2, s, x n s − gt1, s, x n s
t2− s1−α ds
t1
0
g t1, s, x n s 1
t2− s1−α − 1
t1− s1−α
t2
t1
g t2, s, x n s
t2− s1−α ds
≤ q t2 − qt1
1
Γα
t1
0
g t2, s, x n s − gt1, s, x n s 1
t2− s1−αds
t1
0
g t1, s, x n s − gt1, s, 0 gt1, s, 0 1
t2− s1−α − 1
t1− s1−α
ds
t2
t1
g t2, s, x n s − gt2, s, 0 gt2, s, 0
≤ q t2 − qt1
1
Γα
t1
0
g t2, s, x n s − gt1, s, x n s 1
t2− s1−αds
t1
0
m t1h|x n s| g1t1
1
t2− s1−α − 1
t1− s1−α
ds
t2
t1
m t2h|x n s| g1t2
t2− s1−α ds
≤ q t2 − qt1 1
Γα
t1
0
g t2, s, x n s − gt1, s, x n s 1
t2− s1−αds
m t1hr g1t1
Γα 1
t α1− t α
2 t2− t1α
m t2hr g1t2
Γα 1 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 |Gx n t2 − Gx n t1| → 0 as t1 → t2
If t1, t2≥ T, then we have
|Gx n t2 − Gx n t1| ≤ q t2 − qt1 1Γα t2
0
g t2, s, x n s
t2− s1−α ds−
t1
0
g t1, s, x n s
t1− s1−α ds
≤ q t1 qt2 1Γα t2
0
g t2, s, x n s
t2− s1−α ds−
t1
0
g t1, s, x n s
t1− s1−α ds
< ,
3.15
as t1 → t2
Trang 8Similarly, if t1, t2∈Ê with t1< T < t2, then we have
|Gx n t2 − Gx n t1| ≤ |Gx n t2 − Gx n T| |Gx n T − Gx n t1|. 3.16
Note that if t1 → t2, then T → t2and t1 → T Therefore from the above obtained estimates,
it follows that:
|Gx n t2 − Gx n T| −→ 0, |Gx n T − Gx n t1| −→ 0, as t1−→ t2. 3.17
As a result,|Gx n t2 − Gx n T| → 0 as t1 → t2 Hence{Gx n} is an equicontinuous sequence
of functions in X Now an application of the Arzel´a-Ascoli theorem yields that {Gx n} 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{Gx n } is Cauchy sequence in X Now |Gx n t − Gxt| → 0 as n → ∞ for all t ∈ 0, T Then for given > 0 there exists an n0 ∈Æ such that for m, n ≥ n0, then we have
|Gx m t − Gx n t| 1
Γα
t
0
g t, s, x m s − gt, s, x n t
t − s1−α ds
≤ 1
Γα
t
0
g t, s, x m s − gt, s, x n t
t − s1−α ds
≤ 1
Γα
t
0
m th|x m s − x n s|
t − s1−α ds
≤ m tt α h r
Γα 1
≤ a th∗
Γα 1
< .
3.18
This shows that{Gx n } ⊂ GB r 0 ⊂ X is Cauchy Since X is complete, then {Gx n} converges
to a point in X As GB r 0 is closed, {Gx n } converges to a point in GB r0 Hence,
G B r 0 is relatively compact and consequently G is a continuous and compact operator on
B r0
Trang 9Next, we show that FxGx ∈ B r 0 for all x ∈ B r 0 Let x ∈ B r0 be arbitrary, then
|FxtGxt| ≤ |Fxt||Gxt|
≤ f t, xt q t 1
Γα
t
0
g t, s, xs
t − s1−α ds
≤ f t, xt − ft, 0 ft,0
q t 1 Γαt
0
g t, s, xs − gt, s, 0 gt,s,0
≤l t|xt| f1t
q t 1 Γαt
0
m th|xt| g1t
t − s1−α ds
≤L x f0
qt mtt α h r g1tt α
Γα 1
≤L x f0
qt athr bt
Γα 1
≤L x f0
K1 K2
≤ LK1 K2x f0K1 K2
f0K1 K2
1− LK1 K2
r,
3.19
for all t ∈Ê Taking the supremum over t, we obtain FxGx ≤ r for all x ∈ B r0 Hence hypothesisc ofTheorem 2.4holds
Also we have
M GB r0
sup{Gx : x ∈ B r0}
sup
sup
t≥0
q t 1 Γαt
0
g t, s, xs
t − s1−α ds
: x ∈ B r0
≤ sup
t≥0
q t sup
t≥0
a thr bt
Γα 1
≤ K1 K2,
3.20
and therefore Mk LK1 K2 < 1 Now we applyTheorem 2.4to conclude that1.1 has
a solution on
Trang 10Finally, we show the local attractivity of the solutions for1.1 Let x and y be any two
solutions of1.1 in B r0 defined onÊ , then we get
x t − yt ≤ f t, xt
q t 1 Γα
t
0
g t, s, xs
t − s1−α ds
f
t, y t
q t 1 Γα
t
0
g
t, s, y s
t − s1−α ds
≤ f t, xt q t 1 Γαt
0
g t, s, xs
t − s1−α ds
f
t, y t q t 1 Γαt
0
g
t, s, y s
t − s1−α ds
≤ 2Lr f0 q t athr bt Γα 1 ,
3.21
for all t∈Ê Since limt→ ∞q t 0, lim t→ ∞a t 0 and lim t→ ∞b t 0, for > 0, there are real numbers T > 0, T > 0 and T > 0 such that |qt| < for t ≥ T , at < h∗/ Γα 1 for all t ≥ T and bt < /Γα 1 for all t ≥ T If we choose T∗ max{T , T , T }, then from the above inequality it follows that|xt − yt| ≤ ∗for t ≥ T∗, where ∗ 6Lr f0 > 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:
x t t t2x tte −t2/2 1
Γ2/3
t
0
x 2/3 se −3ts 1/10t 8/3 1
t − s 1/3 ds
, 4.1
where t∈Ê
Observe that the above equation is a special case of1.1 Indeed, if we put α 2/3
and
f t, x t t2x,
q t te −t2/2 ,
g t, s, x x 2/3 se −3ts 1
10t 8/3 1.
4.2
Then we can easily check that the assumptions of Theorem 3.2 are satisfied In fact, we
have that the function f t, x is continuous and satisfies assumption H1, where lt t2
... ExampleIn 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... is Cauchy Since X is complete, then {Gx n} converges
to a point in X As GB r 0 is closed, {Gx n } converges to a point in GB r0...
According to the hypothesis H1, the operator F is well defined and the function
function Gx is continuous and bounded in view of hypothesis H4 Therefore F and G define the