Directory UMM :Journals:Journal_of_mathematics:EJQTDE:
Electronic Journal of Qualitative Theory of Differential Equations
2011, No. 22, 1-15; http://www.math.u-szeged.hu/ejqtde/
A FOUR-POINT NONLOCAL INTEGRAL BOUNDARY
VALUE PROBLEM FOR FRACTIONAL DIFFERENTIAL
EQUATIONS OF ARBITRARY ORDER
Bashir Ahmada , Sotiris K. Ntouyasb
a
Department of Mathematics, Faculty of Science, King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, SAUDI ARABIA
e-mail: bashir− [email protected]
b
Department of Mathematics, University of Ioannina
451 10 Ioannina, GREECE
e-mail: [email protected]
Abstract
This paper studies a nonlinear fractional differential equation of an arbitrary
order with four-point nonlocal integral boundary conditions. Some existence results are obtained by applying standard fixed point theorems and Leray-Schauder
degree theory. The involvement of nonlocal parameters in four-point integral
boundary conditions of the problem makes the present work distinguished from
the available literature on four-point integral boundary value problems which
mainly deals with the four-point boundary conditions restrictions on the solution or gradient of the solution of the problem. These integral conditions may be
regarded as strip conditions involving segments of arbitrary length of the given
interval. Some illustrative examples are presented.
Key words and phrases: Fractional differential equations; four-point integral boundary conditions; existence; Fixed point theorem; Leray-Schauder degree.
AMS (MOS) Subject Classifications: 26A33, 34A12, 34A40.
1
Introduction
Boundary value problems for nonlinear fractional differential equations have recently
been studied by several researchers. Fractional derivatives provide an excellent tool
for the description of memory and hereditary properties of various materials and processes. These characteristics of the fractional derivatives make the fractional-order
models more realistic and practical than the classical integer-order models. As a matter of fact, fractional differential equations arise in many engineering and scientific
disciplines such as physics, chemistry, biology, economics, control theory, signal and
image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc. [17, 18, 19, 20]. Some recent work on boundary value problems of
fractional order can be found in [1, 2, 3, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 22] and the
EJQTDE, 2011 No. 22, p. 1
references therein.
In this paper, we consider a boundary value problem of nonlinear fractional differential equations of an arbitrary order with four-point integral boundary conditions
given by
c q
D x(t) = f (t, x(t)), 0 < t < 1, m − 1 < q ≤ m,
Z ξ
x(0) = α
x(s)ds, x′ (0) = 0, x′′ (0) = 0, . . . , x(m−2) (0) = 0,
(1.1)
0
Z
η
x(1) = β
x(s)ds, 0 < ξ, η < 1,
0
where c D q denotes the Caputo fractional derivative of order q, f : [0, 1] × X → X
is continuous and α, β ∈ R. Here, (X, k · k) is a Banach space and C = C([0, 1], X)
denotes the Banach space of all continuous functions from [0, 1] → X endowed with a
topology of uniform convergence with the norm denoted by k · k.
Integral boundary conditions have various applications in applied fields such as
blood flow problems, chemical engineering, thermoelasticity, underground water flow,
population dynamics, etc. For a detailed description of the integral boundary conditions, we refer the reader to the papers [4, 5] and references therein. It has been
observed that the limits of integration in the integral part of the boundary conditions
are usually taken to be fixed, for instance, from 0 to 1 in case the independent variable belongs to the interval [0, 1]. In the present study, we have introduced a nonlocal
type of integral boundary conditions with limits of integration involving the parameters
0 < ξ, η < 1. It is imperative to note that the available literature on nonlocal boundary
conditions is confined to the nonlocal parameters involvement in the solution or gradient of the solution of the problem. The present work is motivated by a recent article
[10], in which some existence results were obtained for nonlinear fractional differential
equations with three-point nonlocal integral boundary conditions.
2
Preliminaries
Let us recall some basic definitions of fractional calculus [17, 18, 20].
Definition 2.1 For a function g : [0, ∞) → R, the Caputo derivative of fractional
order q is defined as
Z t
1
c q
(t − s)n−q−1g (n) (s)ds, n − 1 < q < n, n = [q] + 1,
D g(t) =
Γ(n − q) 0
where [q] denotes the integer part of the real number q.
EJQTDE, 2011 No. 22, p. 2
Definition 2.2 The Riemann-Liouville fractional integral of order q is defined as
1
I g(t) =
Γ(q)
q
Z
0
t
g(s)
ds, q > 0,
(t − s)1−q
provided the integral exists.
Lemma 2.1 ([17]) For q > 0, the general solution of the fractional differential equation
D q x(t) = 0 is given by
c
x(t) = c0 + c1 t + c2 t2 + . . . + cn−1 tn−1 ,
where ci ∈ R, i = 0, 1, 2, . . . , n − 1 (n = [q] + 1).
In view of Lemma 2.1, it follows that
I q c D q x(t) = x(t) + c0 + c1 t + c2 t2 + ... + cn−1 tn−1 ,
(2.1)
for some ci ∈ R, i = 0, 1, 2, ..., n − 1 (n = [q] + 1).
Lemma 2.2 For a given σ ∈ C[0, 1], the unique solution of the boundary value problem
c q
D x(t) = σ(t), 0 < t < 1, m − 1 < q ≤ m,
Z ξ
x(0) = α
x(s)ds, x′ (0) = 0, x′′ (0) = 0, . . . , x(m−2) (0) = 0,
(2.2)
0
Z η
x(1) = β
x(s)ds, 0 < ξ, η < 1,
0
is given by
x(t) =
−
−
−
t
Z ξ Z s
(t − s)q−1
(s − k)q−1
α h
m
(βη − m)
σ(s)ds +
σ(k)dk ds
Γ(q)
m∆
Γ(q)
0
0
0
Z
Z η Z s
i
1
(1 − s)q−1
(s − k)q−1
m
m
σ(k)dk ds + ξ
σ(s)ds
βξ
Γ(q)
Γ(q)
0
0
0
Z ξ Z s
tm−1 h
(s − k)q−1
α(βη − 1)
σ(k)dk ds
∆
Γ(q)
0
0
Z 1
Z η Z s
i
q−1
(1 − s)q−1
(s − k)
σ(k)dk ds + (αξ − 1)
σ(s)ds .
β(αξ − 1)
Γ(q)
Γ(q)
0
0
0
Z
where
∆=
αξ m (βη − 1) − (αξ − 1)(βη m − m)
6= 0.
m
(2.3)
EJQTDE, 2011 No. 22, p. 3
Proof. It is well known [17] that the general solution of the fractional differential
equation in (2.1) can be written as
x(t) =
Z
0
t
(t − s)q−1
σ(s)ds − c0 − c1 t − c2 t2 − . . . − cm−1 tm−1 ,
Γ(q)
(2.4)
where c0 , c1 , c2 , . . . , cm−1 are arbitrary constants. Applying the boundary conditions
for the problem (2.2), we find that c1 = 0, ..., cm−2 = 0,
c0
Z ξ Z s
α h
(s − k)q−1
m
= −
(βη − m)
σ(k)dk ds
m∆
Γ(q)
0
0
Z 1
Z η Z s
i
(1 − s)q−1
(s − k)q−1
m
m
σ(k)dk ds + ξ
σ(s)ds
− βξ
Γ(q)
Γ(q)
0
0
0
and
cm−1
Z ξ Z s
1h
(s − k)q−1
=
α(βη − 1)
σ(k)dk ds
∆
Γ(q)
0
0
Z 1
Z η Z s
i
(1 − s)q−1
(s − k)q−1
σ(k)dk ds + (αξ − 1)
σ(s)ds ,
− β(αξ − 1)
Γ(q)
Γ(q)
0
0
0
where ∆ is given by (2.3). Substituting the values of c0 , c1 , . . . , cm−1 in (2.4), we obtain
x(t) =
−
−
−
t
Z ξ Z s
(t − s)q−1
(s − k)q−1
α h
m
(βη − m)
σ(s)ds +
σ(k)dk ds
Γ(q)
m∆
Γ(q)
0
0
0
Z
Z η Z s
i
1
(1 − s)q−1
(s − k)q−1
m
m
σ(k)dk ds + ξ
σ(s)ds
βξ
Γ(q)
Γ(q)
0
0
0
Z ξ Z s
(s − k)q−1
tm−1 h
α(βη − 1)
σ(k)dk ds
∆
Γ(q)
0
0
Z η Z s
Z 1
i
q−1
(s − k)
(1 − s)q−1
β(αξ − 1)
σ(k)dk ds + (αξ − 1)
σ(s)ds .
Γ(q)
Γ(q)
0
0
0
Z
This completes the proof.
✷
EJQTDE, 2011 No. 22, p. 4
In view of Lemma 2.2, we define an operator F : C → C by
Z t
(t − s)q−1
f (s, x(s))ds
(Fx)(t) =
Γ(q)
0
Z ξ Z s
(s − k)q−1
α h
m
(βη − m)
f (k, x(k))dk ds
+
m∆ Z
Γ(q)
0
0
η Z s
q−1
(s
−
k)
f (k, x(k))dk ds
−βξ m
Γ(q)
0
i
R 0
q−1
m 1 (1−s)
+ξ 0 Γ(q) f (s, x(s))ds
Z ξ Z s
tm−1 h
(s − k)q−1
α(βη − 1)
−
f (k, x(k))dk ds
∆
Γ(q)
0
Z η Z 0s
(s − k)q−1
f (k, x(k))dk ds
−β(αξ − 1)
Γ(q)
0
Z 10
i
(1 − s)q−1
+(αξ − 1)
f (s, x(s))ds , t ∈ [0, 1].
Γ(q)
0
(2.5)
For convenience, let us set
n
|α| h |βη m − m|ξ q+1 + (|β|η q+1 + (q + 1))ξ m i
tq
ϑ = max
+
t∈[0,1] Γ(q + 1)
m|∆|
Γ(q + 2)
q+1
q+1
m−1 h
|α(βη − 1)|ξ
+ |αξ − 1|(|β|η
+ (q + 1)) io
t
+
|∆|
Γ(q + 2)
1
ϑ1 + ϑ2
=
1+
,
(2.6)
Γ(q + 1)
m|∆|(q + 1)
where
ϑ1 = |α|(|βη m − m| + m|βη − 1|)ξ q+1, ϑ2 = (|α|ξ m + m|αξ − 1|)(|β|η q+1 + (q + 1)).
Observe that the problem (1.1) has solutions if the operator equation Fx = x has
fixed points.
For the forthcoming analysis, we need the following assumption:
(H) Assume that f : [0, 1] × X → X is a jointly continuous function and maps
bounded subsets of [0, 1] × X into relatively compact subsets of X.
Furthermore, we need the following fixed point theorem to prove the existence of
solutions for the problem at hand.
Theorem 2.1 [21] Let X be a Banach space. Assume that Ω is an open bounded subset
of X with θ ∈ Ω and let T : Ω → X be a completely continuous operator such that
kT uk ≤ kuk,
∀u ∈ ∂Ω.
Then T has a fixed point in Ω.
EJQTDE, 2011 No. 22, p. 5
3
Existence results in Banach space
Lemma 3.1 The operator F : C → C is completely continuous.
Proof. Clearly, continuity of the operator F follows from the continuity of f. Let
Ω ⊂ C be bounded. Then, ∀x ∈ Ω, by the assumption (H), there exists L1 > 0 such
that |f (t, x)| ≤ L1 . Thus, we have
Z t
(t − s)q−1
|f (s, x(s))|ds
|(F)(t)| ≤
Γ(q)
0
Z ξ Z s
(s − k)q−1
|α| h m
|βη − m|
|f (k, x(k))|dk ds
+
m|∆|
Γ(q)
0
0
Z η Z s
(s − k)q−1
m
+|β|ξ
|f (k, x(k))|dk ds
Γ(q)
0
0
Z 1
i
(1 − s)q−1
m
|f (s, x(s))|ds
+ξ
Γ(q)
0
Z ξ Z s
m−1 h
(s − k)q−1
|t
|
|α(βη − 1)|
|f (k, x(k))|dk ds
+
|∆|
Γ(q)
0
0
Z η Z s
(s − k)q−1
+|β(αξ − 1)|
|f (k, x(k))|dk ds
(3.1)
Γ(q)
0
0
Z 1
i
(1 − s)q−1
|f (s, x(s))|ds
+|αξ − 1|
Γ(q)
0
Z ξ Z s
n Z t (t − s)q−1
|α| h m
(s − k)q−1
≤ L1
|βη − m|
ds +
dk ds
Γ(q)
m|∆|
Γ(q)
0
0
0
Z 1
Z η Z s
(1 − s)q−1 i
(s − k)q−1
m
m
dk ds + ξ
ds
+|β|ξ
Γ(q)
Γ(q)
0
0
0
Z ξ Z s
|tm−1 | h
(s − k)q−1
+
|α(βη − 1)|
dk ds
|∆|
Γ(q)
0
0
Z 1
Z η Z s
(1 − s)q−1 io
(s − k)q−1
dk ds + |αξ − 1|
ds
+|β(αξ − 1)|
Γ(q)
Γ(q)
0
0
0
ϑ1 + ϑ2
L1
1+
= L2 ,
≤
Γ(q + 1)
m|∆|(q + 1)
which implies that k(Fx)k ≤ L2 . Furthermore,
Z t
(t − s)q−2
′
|f (s, x(s))|ds
|(Fx) (t)| =
0 Γ(q − 1)
|(m − 1)tm−2 | h
+
|α(βη − 1)|
|∆|
Z ξ Z
0
0
s
(s − k)q−1
|f (k, x(k))|dk ds
Γ(q)
EJQTDE, 2011 No. 22, p. 6
+|β(αξ − 1)|
Z
η
0
+|αξ − 1|
Z
1
0
≤ L1
nZ
t
0
Z
s
0
(s − k)q−1
|f (k, x(k))|dk ds
Γ(q)
i
(1 − s)q−1
|f (s, x(s))|ds
Γ(q)
(t − s)q−2
ds
Γ(q − 1)
(3.2)
|(m − 1)tm−2 | h
|α(βη − 1)|
+
|∆|
+|β(αξ − 1)|
Z
η
0
≤ L1
Z
0
s
Z ξ Z
0
0
s
(s − k)q−1
dk ds
Γ(q)
(s − k)q−1
dk ds + |αξ − 1|
Γ(q)
Z
0
1
(1 − s)q−1 io
ds
Γ(q)
n 1
|m − 1| h |α(βη − 1)|ξ q+1 + |β(αξ − 1)|η q+1 |(αξ − 1)| io
+
+
Γ(q)
|∆|
Γ(q + 2)
Γ(q + 1)
= L3 .
Hence, for t1 , t2 ∈ [0, 1], we have
|(Fx)(t2 ) − (Fx)(t1 )| ≤
Z
t2
|(Fx)′ (s)|ds ≤ L3 (t2 − t1 ).
t1
This implies that F is equicontinuous on [0, 1]. Thus, by the Arzela-Ascoli theorem, we
have that F(Ω)(t) is relatively compact in X for every t, and so the operator F : C → C
is completely continuous.
f (t, x)
= 0, where the limit is uniform
x→0
x
with respect to t. Then the problem (1.1) has at least one solution.
Theorem 3.1 Assume that (H) holds and lim
f (t, x)
= 0, therefore there exists a constant r > 0 such that
x→0
x
|f (t, x)| ≤ δ|x| for 0 < |x| < r, where δ > 0 is such that
Proof. Since lim
ϑδ ≤ 1,
(3.3)
where ϑ is given by (2.6). Define Ω1 = {x ∈ C | kxk < r} and take x ∈ C such that
kxk = r, that is, x ∈ ∂Ω1 . As before, it can be shown that F is completely continuous
EJQTDE, 2011 No. 22, p. 7
and
|α| h |βη m − m|ξ q+1 + (|β|η q+1 + (q + 1))ξ m i
tq
+
t∈[0,1] Γ(q + 1)
m|∆|
Γ(q + 2)
tm−1 h |α(βη − 1)|ξ q+1 + |αξ − 1|(|β|η q+1 + (q + 1)) io
+
δkxk
|∆|
Γ(q + 2)
|Fx(t)| ≤ max
n
(3.4)
= ϑδkxk,
which, in view of (3.3), yields kFxk ≤ kxk, x ∈ ∂Ω1 . Therefore, by Theorem 2.1, the
operator F has at least one fixed point, which in turn implies that the problem (1.1)
has at least one solution.
✷
Theorem 3.2 Assume that f : [0, 1] × X → X is a jointly continuous function and
satisfies the condition
kf (t, x) − f (t, y)k ≤ Lkx − yk, ∀t ∈ [0, 1], L > 0, x, y ∈ X,
with L < 1/ϑ, where ϑ is given by (2.6). Then the boundary value problem (1.1) has a
unique solution.
Proof. Setting supt∈[0,1] |f (t, 0)| = M and choosing r ≥
Br , where Br = {x ∈ C : kxk ≤ r}. For x ∈ Br , we have:
ϑM
, we show that FBr ⊂
1 − Lϑ
t
(t − s)q−1
|f (s, x(s))|ds
Γ(q)
t∈[0,1]
0
Z ξ Z s
|α| n m
(s − k)q−1
|βη − m|
+
|f (k, x(k))|dk ds
m|∆|
Γ(q)
0
0
Z η Z s
q−1
(s − k)
+|β|ξ m
|f (k, x(k))|dk ds
Γ(q)
0
0
Z 1
o
q−1
(1 − s)
m
|f (s, x(s))|ds
+ξ
Γ(q)
0
Z ξ Z s
|tm−1 | n
(s − k)q−1
+
|α(βη − 1)|
|f (k, x(k))|dk ds
|∆|
Γ(q)
0
Z η Z s 0
q−1
(s − k)
|f (k, x(k))|dk ds
+|β(αξ − 1)|
Γ(q)
0
0
Z 1
oi
(1 − s)q−1
+|αξ − 1|
|f (s, x(s))|ds
Γ(q)
0
h Z t (t − s)q−1
(|f (s, x(s)) − f (s, 0)| + |f (s, 0)|)ds
≤ sup
Γ(q)
t∈[0,1]
0
k(F)(t)k ≤
sup
hZ
EJQTDE, 2011 No. 22, p. 8
Z ξ Z s
|α| n m
+
|βη − m|
(s − k)q−1 (|f (k, x(k)) − f (k, 0)|
m|∆|
0
0
+|f (k, 0)|)dk ds
Z η Z s
m
+|β|ξ
(s − k)q−1 (|f (k, x(k)) − f (k, 0)| + |f (k, 0)|)dk ds
0
Z 10
o
+ξ m
(1 − s)q−1 (|f (s, x(s)) − f (s, 0)| + |f (s, 0)|)ds
0
Z ξ Z s
m−1 n
|t
|
|α(βη − 1)|
(s − k)q−1 (|f (k, x(k)) − f (k, 0)|
+
|∆|
0
0
+|f (k, 0)|)dk ds
Z η Z s
+|β(αξ − 1)|
(s − k)q−1 (|f (k, x(k)) − f (k, 0)|
0
0
+kf (k, 0)k)dk ds
Z 1
oi
+|(αξ − 1)|
(1 − s)q−1 (|f (s, x(s)) − f (s, 0)| + |f (s, 0)|)ds
0
Z ξ Z s
h Z t (t − s)q−1
(s − k)q−1
|α| n m
≤ (Lr + M) sup
|βη − m|
ds +
dk ds
Γ(q)
m|∆|
Γ(q)
t∈[0,1]
0
0
0
+|β|ξ
m
Z
0
η
Z
s
0
(s − k)q−1
dk ds + ξ m
Γ(q)
|tm−1 | n
+
|α(βη − 1)|
|∆|
η
Z ξ Z
0
0
s
Z
0
1
(1 − s)q−1 o
ds
Γ(q)
(s − k)q−1
dk ds
Γ(q)
s
(s − k)q−1
+|β(αξ − 1)|
dk ds + |αξ − 1|
Γ(q)
0
0
ϑ1 + ϑ2
(Lr + M)
1+
= (Lr + M)ϑ ≤ r.
≤
Γ(q + 1)
m|∆|(q + 1)
Z
Z
Z
0
1
(1 − s)q−1 oi
ds
Γ(q)
Now, for x, y ∈ C and for each t ∈ [0, 1], we obtain
k(Fx)(t) − (Fy)(t)k
h Z t (t − s)q−1
|f (s, x(s)) − f (s, y(s))|ds
≤ sup
Γ(q)
0
t∈[0,1]
Z ξ Z s
(s − k)q−1
|α| n m
|βη − m|
+
|f (k, x(k)) − f (k, y(k))|dk ds
m|∆|
Γ(q)
0
0
Z η Z s
q−1
(s − k)
+|β|ξ m
|f (k, x(k)) − f (k, y(k))|dk ds
Γ(q)
0
0
EJQTDE, 2011 No. 22, p. 9
1
o
(1 − s)q−1
|f (s, x(s)) − f (s, y(s))|ds
Γ(q)
0
Z ξ Z s
n
m−1
(s − k)q−1
|t
|
|α(βη − 1)|
|f (k, x(k)) − f (k, y(k))|dk ds
+
|∆|
Γ(q)
0
0
Z η Z s
q−1
(s − k)
+|β(αξ − 1)|
|f (k, x(k)) − f (k, y(k))|dk ds
Γ(q)
0
0
Z 1
oi
q−1
(1 − s)
+|αξ − 1|
|f (s, x(s)) − f (s, y(s))|ds
Γ(q)
0
Z ξ Z s
h Z t (t − s)q−1
(s − k)q−1
|α| n m
|βη − m|
ds +
dk ds
≤ Lkx − yk sup
Γ(q)
m|∆|
Γ(q)
0
0
0
t∈[0,1]
+ξ
m
Z
+|β|ξ
m
Z
0
+
η
Z
s
0
(s − k)q−1
dk ds + ξ m
Γ(q)
|tm−1 | n
|α(βη − 1)|
|∆|
η
Z
ξ
0
Z
s
0
Z
0
1
(1 − s)q−1 o
ds
Γ(q)
(s − k)q−1
dk ds
Γ(q)
s
(s − k)q−1
dk ds + |(αξ − 1)|
+|β(αξ − 1)|
Γ(q)
0
0
L
ϑ1 + ϑ2
≤
1+
kx − yk
Γ(q + 1)
m|∆|(q + 1)
= Lϑkx − yk,
Z
Z
Z
1
0
(1 − s)q−1 oi
ds
Γ(q)
where ϑ is given by (2.6). Observe that L depends only on the parameters involved
in the problem. As L < 1/ϑ, therefore F is a contraction. Thus, the conclusion of the
theorem follows by the contraction mapping principle (Banach fixed point theorem).✷
Our next existence result is based on Leray-Schauder degree theory.
Theorem 3.3 Suppose that (H) holds. Furthermore, it is assumed that there exist
1
constants 0 ≤ κ < , where θ is given by (2.6) and M > 0 such that |f (t, x)| ≤ κ|x|+M
θ
for all t ∈ [0, 1], x ∈ X. Then the boundary value problem (1.1) has at least one solution.
Proof. Consider a fixed point problem
x = Fx,
(3.5)
where F is defined by (2.5). In view of the fixed point problem (3.5), we just need to
prove the existence of at least one solution x ∈ C satisfying (3.5). Define a suitable
ball BR with radius R > 0 as
BR = {x ∈ C : kxk < R} ,
EJQTDE, 2011 No. 22, p. 10
where R will be fixed later. Then, it is sufficient to show that F : B R → C satisfies
x 6= λFx, ∀ x ∈ ∂BR and ∀ λ ∈ [0, 1].
(3.6)
Let us set
H(λ, x) = λFx, x ∈ X, λ ∈ [0, 1].
Then, by the Arzel´a-Ascoli Theorem, hλ (x) = x − H(λ, x) = x − λFx is completely
continuous. If (3.6) is true, then the following Leray-Schauder degrees are well defined
and by the homotopy invariance of topological degree, it follows that
deg(hλ , BR , 0) = deg(I − λF, BR , 0) = deg(h1 , BR , 0)
= deg(h0 , BR , 0) = deg(I, BR , 0) = 1 6= 0, 0 ∈ Br ,
where I denotes the identity operator. By the nonzero property of Leray-Schauder
degree, h1 (t) = x − λFx = 0 for at least one x ∈ BR . In order to prove (3.6), we
assume that x = λFx for some λ ∈ [0, 1] and for all t ∈ [0, 1] so that
Z t
(t − s)q−1
|f (s, x(s))|ds
|x(t)| = |λFx(t)| ≤
Γ(q)
0
Z ξ Z s
(s − k)q−1
|α| h m
|βη − m|
|f (k, x(k))|dk ds
+
m|∆|
Γ(q)
0
0
Z ηZ s
q−1
(s − k)
+|β|ξ m
|f (k, x(k))|dk ds
Γ(q)
0
0
Z 1
i
q−1
(1 − s)
m
|f (s, x(s))|ds
+ξ
Γ(q)
0
Z ξ Z s
|tm−1 | h
(s − k)q−1
+
|α(βη − 1)|
|f (k, x(k))|dk ds
|∆|
Γ(q)
0
Z η Z s0
q−1
(s − k)
|f (k, x(k))|dm ds
+|β(αξ − 1)|
Γ(q)
0
0
Z 1
i
q−1
(1 − s)
|f (s, x(s))|ds
+|αξ − 1|
Γ(q)
0
Z ξ Z s
n Z t (t − s)q−1
|α| h m
(s − k)q−1
≤ (κ|x| + M)
|βη − m|
ds +
dk ds
Γ(q)
m|∆|
Γ(q)
0
0
0
+|β|ξ
m
Z
η
0
s
Z
0
(s − k)q−1
dk ds + ξ m
Γ(q)
|tm−1 | h
+
|α(βη − 1)|
|∆|
+|β(αξ − 1)|
Z
0
η
Z
Z ξ Z
0
0
s
s
0
Z
0
1
(1 − s)q−1 i
ds
Γ(q)
(s − k)q−1
dk ds
Γ(q)
(s − k)q−1
dk ds + |(αξ − 1)|
Γ(q)
Z
0
1
(1 − s)q−1 io
ds
Γ(q)
EJQTDE, 2011 No. 22, p. 11
≤
ϑ1 + ϑ2
(κ|x| + M)
1+
= (κ|x| + M) ϑ,
Γ(q + 1)
m|∆|(q + 1)
which, on taking norm and solving for kxk, yields
kxk ≤
Letting R =
4
Mϑ
.
1 − κϑ
Mϑ
+ 1, (3.6) holds. This completes the proof.
1 − κϑ
✷
Examples
Let us define
X = {x = (x1 , x2 , ..., xp , ...) : xp → 0}
with the norm kxk = supp |xp |.
Example 4.1 Consider the problem
1
C q
D xp (t) = (121 + x3p (t)) 2 + 2(t + 1)(xp − sin xp (t)) − 11, 0 < t < 1,
Z
ξ
xp (0) = α
xp (s)ds, x′p (0) = 0, x′′p (0) = 0, . . . , xp(m−2) (0) = 0,
0Z η
xp (1) = β
xp (s)ds, 0 < ξ, η < 1,
(4.1)
0
where m − 1 < q ≤ m, m ≥ 2.
It can easily be verified that all the assumptions of Theorem 3.1 hold. Consequently,
the conclusion of Theorem 3.1 implies that the problem (4.1) has at least one solution.
Example 4.2 Consider the following four-point integral fractional boundary value problem
|xp |
512
c 13/2
, t ∈ [0, 1],
D xp (t) =
2
p(t + 2) (1 + |xp |)
Z
Z 3/4
(4.2)
1 1/4
′
(v)
xp (0) =
xp (s)ds, xp (0) = 0, . . . , xp (0) = 0, xp (1) =
xp (s)ds.
2 0
0
Here, q = 13/2, m = 7, α = 1/2, β = 1, ξ = 1/4, η = 3/4, and fp (t, x) =
512
|xp |
, f = (f1 , f2 , ..., fp , ...). As |fp (t, x) − fp (t, y)| ≤ 128|xp − yp |, there2
p(t + 2) (1 + |xp |)
fore, kf (t, x) − f (t, y)k ≤ 128kx − yk with L = 128. Further,
ϑ1 +ϑ2
L
Lϑ = Γ(q+1) 1 + m|∆|(q+1) = 128 × (1.087591 × 10−03 ) = 0.139212 < 1.
Thus, by the conclusion of Theorem 3.2, the boundary value problem (4.2) has a unique
solution on [0, 1].
EJQTDE, 2011 No. 22, p. 12
Example 4.3 Consider the following boundary value problem
1
|xp |
c 13/2
sin(2πxp ) +
, t ∈ [0, 1],
D xp (t) =
(4π)
1 + |xp |
Z 1/4
Z
′
(v)
xp (0) = 2
xp (s)ds, xp (0) = 0, . . . , xp (0) = 0, xp (1) = 3
0
(4.3)
3/4
xp (s)ds.
0
Here, q = 13/2, α = 2, β = 3, ξ = 1/3, η = 2/3, and
1
|xp | 1
≤ |xp | + 1.
sin(2πxp ) +
fp (t, x) =
(4π)
1 + |xp | 2
So kf (t, x)k ≤ 21 kxk + 1. Clearly M = 1, κ = 1/2, ϑ = 1.081553 × 10−03 , and κ < 1/ϑ.
Thus, all the conditions of Theorem 3.3 are satisfied and consequently the problem
(4.3) has at least one solution.
Acknowledgement. The authors thank the referee and the editor for their useful
comments.
References
[1] R.P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for fractional
differential equations, Georgian Math. J. 16 (2009), 401-411.
[2] R.P. Agarwal, B. Ahmad, Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal., to appear.
[3] B. Ahmad, J.J. Nieto, Existence of solutions for nonlocal boundary value problems
of higher order nonlinear fractional differential equations, Abstr. Appl. Anal. 2009,
Art. ID 494720, 9 pp.
[4] B. Ahmad, J.J. Nieto, Existence results for nonlinear boundary value problems of
fractional integrodifferential equations with integral boundary conditions, Bound.
Value Probl. 2009, Art. ID 708576, 11 pp.
[5] B. Ahmad, A. Alsaedi, B. Alghamdi, Analytic approximation of solutions of the
forced Duffing equation with integral boundary conditions, Nonlinear Anal. Real
World Appl. 9 (2008), 1727-1740.
[6] B. Ahmad, J.J. Nieto, Existence results for a coupled system of nonlinear fractional
differential equations with three-point boundary conditions, Comput. Math. Appl.
58 (2009) 1838-1843.
EJQTDE, 2011 No. 22, p. 13
[7] B. Ahmad, S. Sivasundaram, On four-point nonlocal boundary value problems of
nonlinear integro-differential equations of fractional order, Appl. Math. Comput.
217 (2010), 480-487.
[8] B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl. Math. Lett. 23 (2010), 390-394.
[9] B. Ahmad, Existence of solutions for fractional differential equations of order
q ∈ (2, 3] with anti-periodic boundary conditions, J. Appl. Math. Comput. 34
(2010), 385-391.
[10] B. Ahmad, S.K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv.
Difference Equ. 2011, Art. ID 107384, 11 pp.
[11] Z. Bai, On positive solutions of a nonlocal fractional boundary value problem,
Nonlinear Anal. 72 (2010), 916-924.
[12] M. Benchohra, S. Hamani, Nonlinear boundary value problems for differential
inclusions with Caputo fractional derivative, Topol. Methods Nonlinear Anal.
32(2008), 115-130.
[13] M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential
equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009)
2391-2396.
[14] R.P Agarwal, M. Benchohra and S. Hamani, A survey on existence result for
boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. 109 (2010), 973-1033.
[15] Y.-K. Chang, J.J. Nieto, Some new existence results for fractional differential
inclusions with boundary conditions, Math. Comput. Model. 49 (2009), 605-609.
[16] S. Hamani, M. Benchohra, John R. Graef, Existence results for boundary value
problems with nonlinear fractional inclusions and integral conditions, Electronic
J. Differential Equations, Vol. 2010(2010), No. 20, pp. 1-16.
[17] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional
Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science
B.V., Amsterdam, 2006.
[18] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[19] J. Sabatier, O.P. Agrawal, J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering,
Springer, Dordrecht, 2007.
EJQTDE, 2011 No. 22, p. 14
[20] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives,
Theory and Applications, Gordon and Breach, Yverdon, 1993.
[21] D.R. Smart, Fixed Point Theorems. Cambridge University Press, 1980.
[22] S. Zhang, Positive solutions to singular boundary value problem for nonlinear
fractional differential equation, Comput. Math. Appl. 59 (2010), 1300-1309.
(Received February 8, 2011)
EJQTDE, 2011 No. 22, p. 15
2011, No. 22, 1-15; http://www.math.u-szeged.hu/ejqtde/
A FOUR-POINT NONLOCAL INTEGRAL BOUNDARY
VALUE PROBLEM FOR FRACTIONAL DIFFERENTIAL
EQUATIONS OF ARBITRARY ORDER
Bashir Ahmada , Sotiris K. Ntouyasb
a
Department of Mathematics, Faculty of Science, King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, SAUDI ARABIA
e-mail: bashir− [email protected]
b
Department of Mathematics, University of Ioannina
451 10 Ioannina, GREECE
e-mail: [email protected]
Abstract
This paper studies a nonlinear fractional differential equation of an arbitrary
order with four-point nonlocal integral boundary conditions. Some existence results are obtained by applying standard fixed point theorems and Leray-Schauder
degree theory. The involvement of nonlocal parameters in four-point integral
boundary conditions of the problem makes the present work distinguished from
the available literature on four-point integral boundary value problems which
mainly deals with the four-point boundary conditions restrictions on the solution or gradient of the solution of the problem. These integral conditions may be
regarded as strip conditions involving segments of arbitrary length of the given
interval. Some illustrative examples are presented.
Key words and phrases: Fractional differential equations; four-point integral boundary conditions; existence; Fixed point theorem; Leray-Schauder degree.
AMS (MOS) Subject Classifications: 26A33, 34A12, 34A40.
1
Introduction
Boundary value problems for nonlinear fractional differential equations have recently
been studied by several researchers. Fractional derivatives provide an excellent tool
for the description of memory and hereditary properties of various materials and processes. These characteristics of the fractional derivatives make the fractional-order
models more realistic and practical than the classical integer-order models. As a matter of fact, fractional differential equations arise in many engineering and scientific
disciplines such as physics, chemistry, biology, economics, control theory, signal and
image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc. [17, 18, 19, 20]. Some recent work on boundary value problems of
fractional order can be found in [1, 2, 3, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 22] and the
EJQTDE, 2011 No. 22, p. 1
references therein.
In this paper, we consider a boundary value problem of nonlinear fractional differential equations of an arbitrary order with four-point integral boundary conditions
given by
c q
D x(t) = f (t, x(t)), 0 < t < 1, m − 1 < q ≤ m,
Z ξ
x(0) = α
x(s)ds, x′ (0) = 0, x′′ (0) = 0, . . . , x(m−2) (0) = 0,
(1.1)
0
Z
η
x(1) = β
x(s)ds, 0 < ξ, η < 1,
0
where c D q denotes the Caputo fractional derivative of order q, f : [0, 1] × X → X
is continuous and α, β ∈ R. Here, (X, k · k) is a Banach space and C = C([0, 1], X)
denotes the Banach space of all continuous functions from [0, 1] → X endowed with a
topology of uniform convergence with the norm denoted by k · k.
Integral boundary conditions have various applications in applied fields such as
blood flow problems, chemical engineering, thermoelasticity, underground water flow,
population dynamics, etc. For a detailed description of the integral boundary conditions, we refer the reader to the papers [4, 5] and references therein. It has been
observed that the limits of integration in the integral part of the boundary conditions
are usually taken to be fixed, for instance, from 0 to 1 in case the independent variable belongs to the interval [0, 1]. In the present study, we have introduced a nonlocal
type of integral boundary conditions with limits of integration involving the parameters
0 < ξ, η < 1. It is imperative to note that the available literature on nonlocal boundary
conditions is confined to the nonlocal parameters involvement in the solution or gradient of the solution of the problem. The present work is motivated by a recent article
[10], in which some existence results were obtained for nonlinear fractional differential
equations with three-point nonlocal integral boundary conditions.
2
Preliminaries
Let us recall some basic definitions of fractional calculus [17, 18, 20].
Definition 2.1 For a function g : [0, ∞) → R, the Caputo derivative of fractional
order q is defined as
Z t
1
c q
(t − s)n−q−1g (n) (s)ds, n − 1 < q < n, n = [q] + 1,
D g(t) =
Γ(n − q) 0
where [q] denotes the integer part of the real number q.
EJQTDE, 2011 No. 22, p. 2
Definition 2.2 The Riemann-Liouville fractional integral of order q is defined as
1
I g(t) =
Γ(q)
q
Z
0
t
g(s)
ds, q > 0,
(t − s)1−q
provided the integral exists.
Lemma 2.1 ([17]) For q > 0, the general solution of the fractional differential equation
D q x(t) = 0 is given by
c
x(t) = c0 + c1 t + c2 t2 + . . . + cn−1 tn−1 ,
where ci ∈ R, i = 0, 1, 2, . . . , n − 1 (n = [q] + 1).
In view of Lemma 2.1, it follows that
I q c D q x(t) = x(t) + c0 + c1 t + c2 t2 + ... + cn−1 tn−1 ,
(2.1)
for some ci ∈ R, i = 0, 1, 2, ..., n − 1 (n = [q] + 1).
Lemma 2.2 For a given σ ∈ C[0, 1], the unique solution of the boundary value problem
c q
D x(t) = σ(t), 0 < t < 1, m − 1 < q ≤ m,
Z ξ
x(0) = α
x(s)ds, x′ (0) = 0, x′′ (0) = 0, . . . , x(m−2) (0) = 0,
(2.2)
0
Z η
x(1) = β
x(s)ds, 0 < ξ, η < 1,
0
is given by
x(t) =
−
−
−
t
Z ξ Z s
(t − s)q−1
(s − k)q−1
α h
m
(βη − m)
σ(s)ds +
σ(k)dk ds
Γ(q)
m∆
Γ(q)
0
0
0
Z
Z η Z s
i
1
(1 − s)q−1
(s − k)q−1
m
m
σ(k)dk ds + ξ
σ(s)ds
βξ
Γ(q)
Γ(q)
0
0
0
Z ξ Z s
tm−1 h
(s − k)q−1
α(βη − 1)
σ(k)dk ds
∆
Γ(q)
0
0
Z 1
Z η Z s
i
q−1
(1 − s)q−1
(s − k)
σ(k)dk ds + (αξ − 1)
σ(s)ds .
β(αξ − 1)
Γ(q)
Γ(q)
0
0
0
Z
where
∆=
αξ m (βη − 1) − (αξ − 1)(βη m − m)
6= 0.
m
(2.3)
EJQTDE, 2011 No. 22, p. 3
Proof. It is well known [17] that the general solution of the fractional differential
equation in (2.1) can be written as
x(t) =
Z
0
t
(t − s)q−1
σ(s)ds − c0 − c1 t − c2 t2 − . . . − cm−1 tm−1 ,
Γ(q)
(2.4)
where c0 , c1 , c2 , . . . , cm−1 are arbitrary constants. Applying the boundary conditions
for the problem (2.2), we find that c1 = 0, ..., cm−2 = 0,
c0
Z ξ Z s
α h
(s − k)q−1
m
= −
(βη − m)
σ(k)dk ds
m∆
Γ(q)
0
0
Z 1
Z η Z s
i
(1 − s)q−1
(s − k)q−1
m
m
σ(k)dk ds + ξ
σ(s)ds
− βξ
Γ(q)
Γ(q)
0
0
0
and
cm−1
Z ξ Z s
1h
(s − k)q−1
=
α(βη − 1)
σ(k)dk ds
∆
Γ(q)
0
0
Z 1
Z η Z s
i
(1 − s)q−1
(s − k)q−1
σ(k)dk ds + (αξ − 1)
σ(s)ds ,
− β(αξ − 1)
Γ(q)
Γ(q)
0
0
0
where ∆ is given by (2.3). Substituting the values of c0 , c1 , . . . , cm−1 in (2.4), we obtain
x(t) =
−
−
−
t
Z ξ Z s
(t − s)q−1
(s − k)q−1
α h
m
(βη − m)
σ(s)ds +
σ(k)dk ds
Γ(q)
m∆
Γ(q)
0
0
0
Z
Z η Z s
i
1
(1 − s)q−1
(s − k)q−1
m
m
σ(k)dk ds + ξ
σ(s)ds
βξ
Γ(q)
Γ(q)
0
0
0
Z ξ Z s
(s − k)q−1
tm−1 h
α(βη − 1)
σ(k)dk ds
∆
Γ(q)
0
0
Z η Z s
Z 1
i
q−1
(s − k)
(1 − s)q−1
β(αξ − 1)
σ(k)dk ds + (αξ − 1)
σ(s)ds .
Γ(q)
Γ(q)
0
0
0
Z
This completes the proof.
✷
EJQTDE, 2011 No. 22, p. 4
In view of Lemma 2.2, we define an operator F : C → C by
Z t
(t − s)q−1
f (s, x(s))ds
(Fx)(t) =
Γ(q)
0
Z ξ Z s
(s − k)q−1
α h
m
(βη − m)
f (k, x(k))dk ds
+
m∆ Z
Γ(q)
0
0
η Z s
q−1
(s
−
k)
f (k, x(k))dk ds
−βξ m
Γ(q)
0
i
R 0
q−1
m 1 (1−s)
+ξ 0 Γ(q) f (s, x(s))ds
Z ξ Z s
tm−1 h
(s − k)q−1
α(βη − 1)
−
f (k, x(k))dk ds
∆
Γ(q)
0
Z η Z 0s
(s − k)q−1
f (k, x(k))dk ds
−β(αξ − 1)
Γ(q)
0
Z 10
i
(1 − s)q−1
+(αξ − 1)
f (s, x(s))ds , t ∈ [0, 1].
Γ(q)
0
(2.5)
For convenience, let us set
n
|α| h |βη m − m|ξ q+1 + (|β|η q+1 + (q + 1))ξ m i
tq
ϑ = max
+
t∈[0,1] Γ(q + 1)
m|∆|
Γ(q + 2)
q+1
q+1
m−1 h
|α(βη − 1)|ξ
+ |αξ − 1|(|β|η
+ (q + 1)) io
t
+
|∆|
Γ(q + 2)
1
ϑ1 + ϑ2
=
1+
,
(2.6)
Γ(q + 1)
m|∆|(q + 1)
where
ϑ1 = |α|(|βη m − m| + m|βη − 1|)ξ q+1, ϑ2 = (|α|ξ m + m|αξ − 1|)(|β|η q+1 + (q + 1)).
Observe that the problem (1.1) has solutions if the operator equation Fx = x has
fixed points.
For the forthcoming analysis, we need the following assumption:
(H) Assume that f : [0, 1] × X → X is a jointly continuous function and maps
bounded subsets of [0, 1] × X into relatively compact subsets of X.
Furthermore, we need the following fixed point theorem to prove the existence of
solutions for the problem at hand.
Theorem 2.1 [21] Let X be a Banach space. Assume that Ω is an open bounded subset
of X with θ ∈ Ω and let T : Ω → X be a completely continuous operator such that
kT uk ≤ kuk,
∀u ∈ ∂Ω.
Then T has a fixed point in Ω.
EJQTDE, 2011 No. 22, p. 5
3
Existence results in Banach space
Lemma 3.1 The operator F : C → C is completely continuous.
Proof. Clearly, continuity of the operator F follows from the continuity of f. Let
Ω ⊂ C be bounded. Then, ∀x ∈ Ω, by the assumption (H), there exists L1 > 0 such
that |f (t, x)| ≤ L1 . Thus, we have
Z t
(t − s)q−1
|f (s, x(s))|ds
|(F)(t)| ≤
Γ(q)
0
Z ξ Z s
(s − k)q−1
|α| h m
|βη − m|
|f (k, x(k))|dk ds
+
m|∆|
Γ(q)
0
0
Z η Z s
(s − k)q−1
m
+|β|ξ
|f (k, x(k))|dk ds
Γ(q)
0
0
Z 1
i
(1 − s)q−1
m
|f (s, x(s))|ds
+ξ
Γ(q)
0
Z ξ Z s
m−1 h
(s − k)q−1
|t
|
|α(βη − 1)|
|f (k, x(k))|dk ds
+
|∆|
Γ(q)
0
0
Z η Z s
(s − k)q−1
+|β(αξ − 1)|
|f (k, x(k))|dk ds
(3.1)
Γ(q)
0
0
Z 1
i
(1 − s)q−1
|f (s, x(s))|ds
+|αξ − 1|
Γ(q)
0
Z ξ Z s
n Z t (t − s)q−1
|α| h m
(s − k)q−1
≤ L1
|βη − m|
ds +
dk ds
Γ(q)
m|∆|
Γ(q)
0
0
0
Z 1
Z η Z s
(1 − s)q−1 i
(s − k)q−1
m
m
dk ds + ξ
ds
+|β|ξ
Γ(q)
Γ(q)
0
0
0
Z ξ Z s
|tm−1 | h
(s − k)q−1
+
|α(βη − 1)|
dk ds
|∆|
Γ(q)
0
0
Z 1
Z η Z s
(1 − s)q−1 io
(s − k)q−1
dk ds + |αξ − 1|
ds
+|β(αξ − 1)|
Γ(q)
Γ(q)
0
0
0
ϑ1 + ϑ2
L1
1+
= L2 ,
≤
Γ(q + 1)
m|∆|(q + 1)
which implies that k(Fx)k ≤ L2 . Furthermore,
Z t
(t − s)q−2
′
|f (s, x(s))|ds
|(Fx) (t)| =
0 Γ(q − 1)
|(m − 1)tm−2 | h
+
|α(βη − 1)|
|∆|
Z ξ Z
0
0
s
(s − k)q−1
|f (k, x(k))|dk ds
Γ(q)
EJQTDE, 2011 No. 22, p. 6
+|β(αξ − 1)|
Z
η
0
+|αξ − 1|
Z
1
0
≤ L1
nZ
t
0
Z
s
0
(s − k)q−1
|f (k, x(k))|dk ds
Γ(q)
i
(1 − s)q−1
|f (s, x(s))|ds
Γ(q)
(t − s)q−2
ds
Γ(q − 1)
(3.2)
|(m − 1)tm−2 | h
|α(βη − 1)|
+
|∆|
+|β(αξ − 1)|
Z
η
0
≤ L1
Z
0
s
Z ξ Z
0
0
s
(s − k)q−1
dk ds
Γ(q)
(s − k)q−1
dk ds + |αξ − 1|
Γ(q)
Z
0
1
(1 − s)q−1 io
ds
Γ(q)
n 1
|m − 1| h |α(βη − 1)|ξ q+1 + |β(αξ − 1)|η q+1 |(αξ − 1)| io
+
+
Γ(q)
|∆|
Γ(q + 2)
Γ(q + 1)
= L3 .
Hence, for t1 , t2 ∈ [0, 1], we have
|(Fx)(t2 ) − (Fx)(t1 )| ≤
Z
t2
|(Fx)′ (s)|ds ≤ L3 (t2 − t1 ).
t1
This implies that F is equicontinuous on [0, 1]. Thus, by the Arzela-Ascoli theorem, we
have that F(Ω)(t) is relatively compact in X for every t, and so the operator F : C → C
is completely continuous.
f (t, x)
= 0, where the limit is uniform
x→0
x
with respect to t. Then the problem (1.1) has at least one solution.
Theorem 3.1 Assume that (H) holds and lim
f (t, x)
= 0, therefore there exists a constant r > 0 such that
x→0
x
|f (t, x)| ≤ δ|x| for 0 < |x| < r, where δ > 0 is such that
Proof. Since lim
ϑδ ≤ 1,
(3.3)
where ϑ is given by (2.6). Define Ω1 = {x ∈ C | kxk < r} and take x ∈ C such that
kxk = r, that is, x ∈ ∂Ω1 . As before, it can be shown that F is completely continuous
EJQTDE, 2011 No. 22, p. 7
and
|α| h |βη m − m|ξ q+1 + (|β|η q+1 + (q + 1))ξ m i
tq
+
t∈[0,1] Γ(q + 1)
m|∆|
Γ(q + 2)
tm−1 h |α(βη − 1)|ξ q+1 + |αξ − 1|(|β|η q+1 + (q + 1)) io
+
δkxk
|∆|
Γ(q + 2)
|Fx(t)| ≤ max
n
(3.4)
= ϑδkxk,
which, in view of (3.3), yields kFxk ≤ kxk, x ∈ ∂Ω1 . Therefore, by Theorem 2.1, the
operator F has at least one fixed point, which in turn implies that the problem (1.1)
has at least one solution.
✷
Theorem 3.2 Assume that f : [0, 1] × X → X is a jointly continuous function and
satisfies the condition
kf (t, x) − f (t, y)k ≤ Lkx − yk, ∀t ∈ [0, 1], L > 0, x, y ∈ X,
with L < 1/ϑ, where ϑ is given by (2.6). Then the boundary value problem (1.1) has a
unique solution.
Proof. Setting supt∈[0,1] |f (t, 0)| = M and choosing r ≥
Br , where Br = {x ∈ C : kxk ≤ r}. For x ∈ Br , we have:
ϑM
, we show that FBr ⊂
1 − Lϑ
t
(t − s)q−1
|f (s, x(s))|ds
Γ(q)
t∈[0,1]
0
Z ξ Z s
|α| n m
(s − k)q−1
|βη − m|
+
|f (k, x(k))|dk ds
m|∆|
Γ(q)
0
0
Z η Z s
q−1
(s − k)
+|β|ξ m
|f (k, x(k))|dk ds
Γ(q)
0
0
Z 1
o
q−1
(1 − s)
m
|f (s, x(s))|ds
+ξ
Γ(q)
0
Z ξ Z s
|tm−1 | n
(s − k)q−1
+
|α(βη − 1)|
|f (k, x(k))|dk ds
|∆|
Γ(q)
0
Z η Z s 0
q−1
(s − k)
|f (k, x(k))|dk ds
+|β(αξ − 1)|
Γ(q)
0
0
Z 1
oi
(1 − s)q−1
+|αξ − 1|
|f (s, x(s))|ds
Γ(q)
0
h Z t (t − s)q−1
(|f (s, x(s)) − f (s, 0)| + |f (s, 0)|)ds
≤ sup
Γ(q)
t∈[0,1]
0
k(F)(t)k ≤
sup
hZ
EJQTDE, 2011 No. 22, p. 8
Z ξ Z s
|α| n m
+
|βη − m|
(s − k)q−1 (|f (k, x(k)) − f (k, 0)|
m|∆|
0
0
+|f (k, 0)|)dk ds
Z η Z s
m
+|β|ξ
(s − k)q−1 (|f (k, x(k)) − f (k, 0)| + |f (k, 0)|)dk ds
0
Z 10
o
+ξ m
(1 − s)q−1 (|f (s, x(s)) − f (s, 0)| + |f (s, 0)|)ds
0
Z ξ Z s
m−1 n
|t
|
|α(βη − 1)|
(s − k)q−1 (|f (k, x(k)) − f (k, 0)|
+
|∆|
0
0
+|f (k, 0)|)dk ds
Z η Z s
+|β(αξ − 1)|
(s − k)q−1 (|f (k, x(k)) − f (k, 0)|
0
0
+kf (k, 0)k)dk ds
Z 1
oi
+|(αξ − 1)|
(1 − s)q−1 (|f (s, x(s)) − f (s, 0)| + |f (s, 0)|)ds
0
Z ξ Z s
h Z t (t − s)q−1
(s − k)q−1
|α| n m
≤ (Lr + M) sup
|βη − m|
ds +
dk ds
Γ(q)
m|∆|
Γ(q)
t∈[0,1]
0
0
0
+|β|ξ
m
Z
0
η
Z
s
0
(s − k)q−1
dk ds + ξ m
Γ(q)
|tm−1 | n
+
|α(βη − 1)|
|∆|
η
Z ξ Z
0
0
s
Z
0
1
(1 − s)q−1 o
ds
Γ(q)
(s − k)q−1
dk ds
Γ(q)
s
(s − k)q−1
+|β(αξ − 1)|
dk ds + |αξ − 1|
Γ(q)
0
0
ϑ1 + ϑ2
(Lr + M)
1+
= (Lr + M)ϑ ≤ r.
≤
Γ(q + 1)
m|∆|(q + 1)
Z
Z
Z
0
1
(1 − s)q−1 oi
ds
Γ(q)
Now, for x, y ∈ C and for each t ∈ [0, 1], we obtain
k(Fx)(t) − (Fy)(t)k
h Z t (t − s)q−1
|f (s, x(s)) − f (s, y(s))|ds
≤ sup
Γ(q)
0
t∈[0,1]
Z ξ Z s
(s − k)q−1
|α| n m
|βη − m|
+
|f (k, x(k)) − f (k, y(k))|dk ds
m|∆|
Γ(q)
0
0
Z η Z s
q−1
(s − k)
+|β|ξ m
|f (k, x(k)) − f (k, y(k))|dk ds
Γ(q)
0
0
EJQTDE, 2011 No. 22, p. 9
1
o
(1 − s)q−1
|f (s, x(s)) − f (s, y(s))|ds
Γ(q)
0
Z ξ Z s
n
m−1
(s − k)q−1
|t
|
|α(βη − 1)|
|f (k, x(k)) − f (k, y(k))|dk ds
+
|∆|
Γ(q)
0
0
Z η Z s
q−1
(s − k)
+|β(αξ − 1)|
|f (k, x(k)) − f (k, y(k))|dk ds
Γ(q)
0
0
Z 1
oi
q−1
(1 − s)
+|αξ − 1|
|f (s, x(s)) − f (s, y(s))|ds
Γ(q)
0
Z ξ Z s
h Z t (t − s)q−1
(s − k)q−1
|α| n m
|βη − m|
ds +
dk ds
≤ Lkx − yk sup
Γ(q)
m|∆|
Γ(q)
0
0
0
t∈[0,1]
+ξ
m
Z
+|β|ξ
m
Z
0
+
η
Z
s
0
(s − k)q−1
dk ds + ξ m
Γ(q)
|tm−1 | n
|α(βη − 1)|
|∆|
η
Z
ξ
0
Z
s
0
Z
0
1
(1 − s)q−1 o
ds
Γ(q)
(s − k)q−1
dk ds
Γ(q)
s
(s − k)q−1
dk ds + |(αξ − 1)|
+|β(αξ − 1)|
Γ(q)
0
0
L
ϑ1 + ϑ2
≤
1+
kx − yk
Γ(q + 1)
m|∆|(q + 1)
= Lϑkx − yk,
Z
Z
Z
1
0
(1 − s)q−1 oi
ds
Γ(q)
where ϑ is given by (2.6). Observe that L depends only on the parameters involved
in the problem. As L < 1/ϑ, therefore F is a contraction. Thus, the conclusion of the
theorem follows by the contraction mapping principle (Banach fixed point theorem).✷
Our next existence result is based on Leray-Schauder degree theory.
Theorem 3.3 Suppose that (H) holds. Furthermore, it is assumed that there exist
1
constants 0 ≤ κ < , where θ is given by (2.6) and M > 0 such that |f (t, x)| ≤ κ|x|+M
θ
for all t ∈ [0, 1], x ∈ X. Then the boundary value problem (1.1) has at least one solution.
Proof. Consider a fixed point problem
x = Fx,
(3.5)
where F is defined by (2.5). In view of the fixed point problem (3.5), we just need to
prove the existence of at least one solution x ∈ C satisfying (3.5). Define a suitable
ball BR with radius R > 0 as
BR = {x ∈ C : kxk < R} ,
EJQTDE, 2011 No. 22, p. 10
where R will be fixed later. Then, it is sufficient to show that F : B R → C satisfies
x 6= λFx, ∀ x ∈ ∂BR and ∀ λ ∈ [0, 1].
(3.6)
Let us set
H(λ, x) = λFx, x ∈ X, λ ∈ [0, 1].
Then, by the Arzel´a-Ascoli Theorem, hλ (x) = x − H(λ, x) = x − λFx is completely
continuous. If (3.6) is true, then the following Leray-Schauder degrees are well defined
and by the homotopy invariance of topological degree, it follows that
deg(hλ , BR , 0) = deg(I − λF, BR , 0) = deg(h1 , BR , 0)
= deg(h0 , BR , 0) = deg(I, BR , 0) = 1 6= 0, 0 ∈ Br ,
where I denotes the identity operator. By the nonzero property of Leray-Schauder
degree, h1 (t) = x − λFx = 0 for at least one x ∈ BR . In order to prove (3.6), we
assume that x = λFx for some λ ∈ [0, 1] and for all t ∈ [0, 1] so that
Z t
(t − s)q−1
|f (s, x(s))|ds
|x(t)| = |λFx(t)| ≤
Γ(q)
0
Z ξ Z s
(s − k)q−1
|α| h m
|βη − m|
|f (k, x(k))|dk ds
+
m|∆|
Γ(q)
0
0
Z ηZ s
q−1
(s − k)
+|β|ξ m
|f (k, x(k))|dk ds
Γ(q)
0
0
Z 1
i
q−1
(1 − s)
m
|f (s, x(s))|ds
+ξ
Γ(q)
0
Z ξ Z s
|tm−1 | h
(s − k)q−1
+
|α(βη − 1)|
|f (k, x(k))|dk ds
|∆|
Γ(q)
0
Z η Z s0
q−1
(s − k)
|f (k, x(k))|dm ds
+|β(αξ − 1)|
Γ(q)
0
0
Z 1
i
q−1
(1 − s)
|f (s, x(s))|ds
+|αξ − 1|
Γ(q)
0
Z ξ Z s
n Z t (t − s)q−1
|α| h m
(s − k)q−1
≤ (κ|x| + M)
|βη − m|
ds +
dk ds
Γ(q)
m|∆|
Γ(q)
0
0
0
+|β|ξ
m
Z
η
0
s
Z
0
(s − k)q−1
dk ds + ξ m
Γ(q)
|tm−1 | h
+
|α(βη − 1)|
|∆|
+|β(αξ − 1)|
Z
0
η
Z
Z ξ Z
0
0
s
s
0
Z
0
1
(1 − s)q−1 i
ds
Γ(q)
(s − k)q−1
dk ds
Γ(q)
(s − k)q−1
dk ds + |(αξ − 1)|
Γ(q)
Z
0
1
(1 − s)q−1 io
ds
Γ(q)
EJQTDE, 2011 No. 22, p. 11
≤
ϑ1 + ϑ2
(κ|x| + M)
1+
= (κ|x| + M) ϑ,
Γ(q + 1)
m|∆|(q + 1)
which, on taking norm and solving for kxk, yields
kxk ≤
Letting R =
4
Mϑ
.
1 − κϑ
Mϑ
+ 1, (3.6) holds. This completes the proof.
1 − κϑ
✷
Examples
Let us define
X = {x = (x1 , x2 , ..., xp , ...) : xp → 0}
with the norm kxk = supp |xp |.
Example 4.1 Consider the problem
1
C q
D xp (t) = (121 + x3p (t)) 2 + 2(t + 1)(xp − sin xp (t)) − 11, 0 < t < 1,
Z
ξ
xp (0) = α
xp (s)ds, x′p (0) = 0, x′′p (0) = 0, . . . , xp(m−2) (0) = 0,
0Z η
xp (1) = β
xp (s)ds, 0 < ξ, η < 1,
(4.1)
0
where m − 1 < q ≤ m, m ≥ 2.
It can easily be verified that all the assumptions of Theorem 3.1 hold. Consequently,
the conclusion of Theorem 3.1 implies that the problem (4.1) has at least one solution.
Example 4.2 Consider the following four-point integral fractional boundary value problem
|xp |
512
c 13/2
, t ∈ [0, 1],
D xp (t) =
2
p(t + 2) (1 + |xp |)
Z
Z 3/4
(4.2)
1 1/4
′
(v)
xp (0) =
xp (s)ds, xp (0) = 0, . . . , xp (0) = 0, xp (1) =
xp (s)ds.
2 0
0
Here, q = 13/2, m = 7, α = 1/2, β = 1, ξ = 1/4, η = 3/4, and fp (t, x) =
512
|xp |
, f = (f1 , f2 , ..., fp , ...). As |fp (t, x) − fp (t, y)| ≤ 128|xp − yp |, there2
p(t + 2) (1 + |xp |)
fore, kf (t, x) − f (t, y)k ≤ 128kx − yk with L = 128. Further,
ϑ1 +ϑ2
L
Lϑ = Γ(q+1) 1 + m|∆|(q+1) = 128 × (1.087591 × 10−03 ) = 0.139212 < 1.
Thus, by the conclusion of Theorem 3.2, the boundary value problem (4.2) has a unique
solution on [0, 1].
EJQTDE, 2011 No. 22, p. 12
Example 4.3 Consider the following boundary value problem
1
|xp |
c 13/2
sin(2πxp ) +
, t ∈ [0, 1],
D xp (t) =
(4π)
1 + |xp |
Z 1/4
Z
′
(v)
xp (0) = 2
xp (s)ds, xp (0) = 0, . . . , xp (0) = 0, xp (1) = 3
0
(4.3)
3/4
xp (s)ds.
0
Here, q = 13/2, α = 2, β = 3, ξ = 1/3, η = 2/3, and
1
|xp | 1
≤ |xp | + 1.
sin(2πxp ) +
fp (t, x) =
(4π)
1 + |xp | 2
So kf (t, x)k ≤ 21 kxk + 1. Clearly M = 1, κ = 1/2, ϑ = 1.081553 × 10−03 , and κ < 1/ϑ.
Thus, all the conditions of Theorem 3.3 are satisfied and consequently the problem
(4.3) has at least one solution.
Acknowledgement. The authors thank the referee and the editor for their useful
comments.
References
[1] R.P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for fractional
differential equations, Georgian Math. J. 16 (2009), 401-411.
[2] R.P. Agarwal, B. Ahmad, Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal., to appear.
[3] B. Ahmad, J.J. Nieto, Existence of solutions for nonlocal boundary value problems
of higher order nonlinear fractional differential equations, Abstr. Appl. Anal. 2009,
Art. ID 494720, 9 pp.
[4] B. Ahmad, J.J. Nieto, Existence results for nonlinear boundary value problems of
fractional integrodifferential equations with integral boundary conditions, Bound.
Value Probl. 2009, Art. ID 708576, 11 pp.
[5] B. Ahmad, A. Alsaedi, B. Alghamdi, Analytic approximation of solutions of the
forced Duffing equation with integral boundary conditions, Nonlinear Anal. Real
World Appl. 9 (2008), 1727-1740.
[6] B. Ahmad, J.J. Nieto, Existence results for a coupled system of nonlinear fractional
differential equations with three-point boundary conditions, Comput. Math. Appl.
58 (2009) 1838-1843.
EJQTDE, 2011 No. 22, p. 13
[7] B. Ahmad, S. Sivasundaram, On four-point nonlocal boundary value problems of
nonlinear integro-differential equations of fractional order, Appl. Math. Comput.
217 (2010), 480-487.
[8] B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl. Math. Lett. 23 (2010), 390-394.
[9] B. Ahmad, Existence of solutions for fractional differential equations of order
q ∈ (2, 3] with anti-periodic boundary conditions, J. Appl. Math. Comput. 34
(2010), 385-391.
[10] B. Ahmad, S.K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv.
Difference Equ. 2011, Art. ID 107384, 11 pp.
[11] Z. Bai, On positive solutions of a nonlocal fractional boundary value problem,
Nonlinear Anal. 72 (2010), 916-924.
[12] M. Benchohra, S. Hamani, Nonlinear boundary value problems for differential
inclusions with Caputo fractional derivative, Topol. Methods Nonlinear Anal.
32(2008), 115-130.
[13] M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential
equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009)
2391-2396.
[14] R.P Agarwal, M. Benchohra and S. Hamani, A survey on existence result for
boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. 109 (2010), 973-1033.
[15] Y.-K. Chang, J.J. Nieto, Some new existence results for fractional differential
inclusions with boundary conditions, Math. Comput. Model. 49 (2009), 605-609.
[16] S. Hamani, M. Benchohra, John R. Graef, Existence results for boundary value
problems with nonlinear fractional inclusions and integral conditions, Electronic
J. Differential Equations, Vol. 2010(2010), No. 20, pp. 1-16.
[17] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional
Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science
B.V., Amsterdam, 2006.
[18] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[19] J. Sabatier, O.P. Agrawal, J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering,
Springer, Dordrecht, 2007.
EJQTDE, 2011 No. 22, p. 14
[20] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives,
Theory and Applications, Gordon and Breach, Yverdon, 1993.
[21] D.R. Smart, Fixed Point Theorems. Cambridge University Press, 1980.
[22] S. Zhang, Positive solutions to singular boundary value problem for nonlinear
fractional differential equation, Comput. Math. Appl. 59 (2010), 1300-1309.
(Received February 8, 2011)
EJQTDE, 2011 No. 22, p. 15