Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 60, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
INTEGRO-DIFFERENTIAL EQUATIONS OF FRACTIONAL
ORDER WITH NONLOCAL FRACTIONAL BOUNDARY
CONDITIONS ASSOCIATED WITH FINANCIAL ASSET MODEL
BASHIR AHMAD, SOTIRIS K. NTOUYAS
Abstract. In this article, we discuss the existence of solutions for a boundaryvalue problem of integro-differential equations of fractional order with nonlocal
fractional boundary conditions by means of some standard tools of fixed point
theory. Our problem describes a more general form of fractional stochastic
dynamic model for financial asset. An illustrative example is also presented.
1. Formulation and basic result
Fractional calculus, regarded as a branch of mathematical analysis dealing with
derivatives and integrals of arbitrary order, has been extensively developed and applied to a variety of problems appearing in sciences and engineering. It is worthwhile
to mention that this branch of mathematics has played a crucial role in exploring
various characteristics of engineering materials such as viscoelastic polymers, foams,
gels, and animal tissues, and their engineering and scientific applications. For a recent detailed survey of the activities involving fractional calculus, we refer a recent
paper by Machado, Kiryakova and Mainardi [16]. Some recent work on the topic
can be found in [1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 17] and references therein.
The underlying dynamics of equity prices following a jump process or a Levy
process provide a basis for modeling of financial assets. The CGMY, KoBoL and
FMLS are examples of some interesting financial models involving the dynamics of
stock prices. In [8], it is shown that the prices of financial derivatives are expressible
in terms of fractional derivative.
In [15], the author described the dynamics of a financial asset by the fractional
stochastic differential equation of order µ (representing the dynamical memory
effects in the market stochastic evolution) with fractional boundary conditions. In
the present paper, we study a more general model associated with financial asset.
2000 Mathematics Subject Classification. 34A08, 34B10, 34B15.
Key words and phrases. Fractional differential equations; integral boundary conditions;
existence; fixed point theorems; financial asset.
c
2013
Texas State University - San Marcos.
Submitted December 6, 2012. Published February 26, 2013.
1
2
B. AHMAD, S. K. NTOUYAS
EJDE-2013/60
Precisely, we consider the following problem:
−Dα x(t) = Af (t, x(t)) + BI β g(t, x(t)),
Dδ x(0) = 0,
(n − 1) < α ≤ n, t ∈ [0, 1],
Z η
Dδ x(s)ds,
Dδ+1 x(0) = 0, . . . , Dδ+(n−2) x(0) = 0, Dδ x(1) =
0
(1.1)
where 0 < δ ≤ 1, α − δ > n, 0 < β < 1, 0 < η < 1, D(·) denotes the RiemannLioville fractional derivative of order (·), f, g are given continuous function, and
A, B are real constants.
We remark that the problem (1.1) also arises in real estate asset securitization
modeling [18].
By the substitution x(t) = I δ y(t) = D−α y(t), the problem (1.1) takes the form
−Dα−δ y(t) = Af (t, I δ y(t)) + BI β g(t, I δ y(t)),
y(0) = 0,
y 0 (0) = 0, . . . , y (n−2) (0) = 0,
t ∈ [0, 1],
Z η
y(1) =
y(s)ds.
(1.2)
0
Lemma 1.1. For any h ∈ C(0, 1) ∩ L(0, 1), the unique solution of the linear fractional boundary-value problem
−Dα−δ y(t) = h(t),
y(0) = 0,
t ∈ [0, 1],
y 0 (0) = 0, . . . , y (n−2) (0) = 0,
Z
y(1) =
η
y(s)ds,
(1.3)
0
is
y(t) = −I α−δ h(t) +
(α − δ)tα−δ−1 α−δ
I
h(1) − I α−δ+1 h(η) ,
α−δ
α−δ−η
where I (·) (·) denotes Riemann-Liouville integral.
Proof. It is well known that the solutions of fractional differential equation in (1.1)
can be written as
y(t) = −I α−δ h(t) + c1 tα−δ−1 + c2 tα−δ−2 + c3 tα−δ−3 + · · · + cn tα−δ−n ,
(1.4)
where c1 , c2 , . . . , cn ∈ R are arbitrary constants [12]. Using the given boundary
conditions, we find that c2 = 0, c3 = 0, . . . , cn = 0 and
α−δ
α−δ
α−δ+1
c1 =
I
h(1)
−
I
h(η)
.
α − δ − η α−δ
Substituting these values in (1.1) yields
y(t) = −I α−δ h(t) +
This completes the proof.
(α − δ)tα−δ−1 α−δ
α−δ+1
I
h(1)
−
I
h(η)
.
α − δ − η α−δ
Thus, the solution of the linear variant of the problem (1.1) can be written as
x(t) = I δ y(t)
h
i
(α − δ)tα−δ−1 α−δ
α−δ+1
I
h(1)
−
I
h(η)
= I δ − I α−δ h(t) +
α − δ − η α−δ
Z t (t − s)δ−1
(α − δ) α−δ
α
α−δ+1
= −I h(t) +
I
h(1) − I
h(η)
sα−δ−1 ds
α − δ − η α−δ
Γ(δ)
0
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NONLOCAL FRACTIONAL BOUNDARY-VALUE PROBLEMS
3
(α − δ) α−δ
α−δ+1
= −I α h(t) +
I
h(1)
−
I
h(η)
×
α − δ − η α−δ
n tα−1 Z 1
o
×
(1 − ν)δ−1 ν α−δ−1 dν ,
Γ(δ) 0
where we have used the substitution s = νt in the integral of the last term. Using
the relation for Beta function B(·, ·):
Z 1
Γ(α)Γ(β + 1)
,
B(β + 1, α) =
(1 − u)α−1 uβ du =
Γ(α + β + 1)
0
we obtain
x(t) = −I α h(t) +
Γ(α − δ + 1)tα−1 α−δ
α−δ+1
I
h(1)
−
I
h(η)
.
(α − δ − η α−δ )Γ(α)
(1.5)
The solution of the original nonlinear problem (1.1) can be obtained by replacing
h with the right hand side of the fractional equation of (1.1) in (1.5).
Let C = C([0, 1], R) denote the Banach space of all continuous functions from
[0, 1] → R endowed with the norm defined by kxk = sup{|x(t)|, t ∈ [0, 1]}.
In relation to problem (1.1), we define an operator U : C → C as
(Ux)(t)
Z
= −A
t
Z t
(t − s)α−1
(t − s)α+β−1
f (s, x(s))ds − B
g(s, x(s))ds
Γ(α)
Γ(α + β)
0
0
Z 1
h Z 1 (1 − s)α−δ−1
(1 − s)α−δ+β−1
f (s, x(s))ds + B
g(s, x(s))ds
+ Qtα−1 A
Γ(α − δ)
Γ(α − δ + β)
0
0
Z η
Z η
i
(η − s)α−δ
(η − s)α−δ+β
−A
f (s, x(s))ds − B
g(s, x(s))ds ,
0 Γ(α − δ + 1)
0 Γ(α − δ + β + 1)
where
Q=
Γ(α − δ + 1)
,
(α − δ − η α−δ )Γ(α)
α 6= δ + η α−δ .
For the sake of convenience, we set
n h
tα
η α−δ+1 i
1
Ω = sup |A|
+ |Q|tα−1
+
Γ(α + 1)
Γ(α − δ + 1) Γ(α − δ + 2)
t∈[0,1]
h
io
α+β
t
1
η α−δ+β+1
+ |B|
+ |Q|tα−1
+
.
Γ(α + β + 1)
Γ(α − δ + β + 1) Γ(α − δ + β + 2)
(1.6)
1.1. Existence results via Banach’s fixed point theorem.
Theorem 1.2. Assume that f, g : [0, 1]×R → R are continuous functions satisfying
the condition:
(A1) |f (t, x) − f (t, y)| ≤ L1 |x − y|, |g(t, x) − g(t, y)| ≤ L2 |x − y|, for all t ∈ [0, 1],
L1 , L2 > 0, x, y ∈ R.
Then the boundary-value problem (1.1) has a unique solution if L < 1/Ω, where
L = max{L1 , L2 } and Ω is given by (1.6).
4
B. AHMAD, S. K. NTOUYAS
EJDE-2013/60
Proof. Let us define M = max{M1 , M2 }, where M1 , M2 are finite numbers given by
ΩM
supt∈[0,1] |f (t, 0)| = M1 , supt∈[0,1] |g(t, 0)| = M2 . Selecting r ≥ 1−LΩ
, we show that
UBr ⊂ Br , where Br = {x ∈ C : kxk ≤ r}. Using that |f (s, x(s)) ≤ |f (s, x(s)) −
f (s, 0)|+|f (s, 0)| ≤ L1 r+M1 , |g(s, x(s))| ≤ |g(s, x(s))−g(s, 0)|+|g(s, 0)| ≤ L2 r+M2
for x ∈ Br and (1.6), it can easily be shown that
k(Ux)k
1
η α−δ+1 i
tα
+ |Q|tα−1
+
Γ(α + 1)
Γ(α − δ + 1) Γ(α − δ + 2)
t∈[0,1]
h
io
α+β
η α−δ+β+1
t
1
+ |B|
+ |Q|tα−1
+
Γ(α + β + 1)
Γ(α − δ + β + 1) Γ(α − δ + β + 2)
= (Lr + M )Ω ≤ r,
≤ (Lr + M ) sup
n h
|A|
which implies that UBr ⊂ Br . Now, for x, y ∈ C we obtain
kUx − Uyk
n Z
≤ sup |A|
t∈[0,1]
0
t
(t − s)α−1
|f (s, x(s)) − f (s, y(s))|ds
Γ(α)
t
(t − s)α+β−1
|g(s, x(s)) − g(s, y(s))|ds
Γ(α + β)
0
h Z 1 (1 − s)α−δ−1
α−1
+ |Q|t
|A|
|f (s, x(s)) − f (s, y(s))|ds
Γ(α − δ)
0
Z 1
(1 − s)α−δ+β−1
+ |B|
|g(s, x(s)) − g(s, y(s))|ds
Γ(α − δ + β)
0
Z η
(η − s)α−δ
+ |A|
|f (s, x(s)) − f (s, y(s))|ds
0 Γ(α − δ + 1)
Z η
io
(η − s)α−δ+β
+ |B|
|g(s, x(s)) − g(s, y(s))|ds
0 Γ(α − δ + β + 1)
n h
tα
η α−δ+1 i
1
≤ L sup |A|
+ |Q|tα−1
+
Γ(α + 1)
Γ(α − δ + 1) Γ(α − δ + 2)
t∈[0,1]
h
io
α+β
t
η α−δ+β+1
1
+ |B|
+ |Q|tα−1
+
Γ(α + β + 1)
Γ(α − δ + β + 1) Γ(α − δ + β + 2)
× kx − yk
Z
+ |B|
= LΩkx − yk.
By the given assumption, L < 1/Ω. Therefore U is a contraction. Thus, the
conclusion of the theorem follows by the contraction mapping principle (Banach
fixed point theorem).
Now we present another variant of existence-uniqueness result. This result is
based on the Hölder’s inequality.
Theorem 1.3. Suppose that the continuous functions f and g satisfy the following
assumptions:
(H1) |f (t, x) − f (t, y)| ≤ m(t)|x − y|, |g(t, x) − g(t, y)| ≤ n(t)|x − y|, for t ∈ [0, 1],
1
x, y ∈ R, and m, n ∈ L γ ([0, 1], R+ ), γ ∈ (0, α − δ − n).
EJDE-2013/60
NONLOCAL FRACTIONAL BOUNDARY-VALUE PROBLEMS
5
(H2) |A|kmkZ1 + |B|knkZ2 < 1, where
1 1 − γ 1−γ
|Q| 1 − γ 1−γ
Z1 =
+
Γ(α) α − γ
Γ(α − δ) α − δ − γ
1−γ
|Q|
1−γ
+
η α−δ+1−γ ,
Γ(α − δ + 1) α − δ + 1 − γ
1 − γ 1−γ
1−γ
1
1−γ
|Q|
+
Γ(α + β) α + β − γ
Γ(α − δ + β) α − δ + β − γ
1−γ
1−γ
|Q|
η α−δ+β+1−γ ,
+
Γ(α − δ + β + 1) α − δ + β + 1 − γ
γ
R1
1
and kµk = 0 |µ(s)| γ ds , µ = m, n. Then the boundary value problem (1.1) has
a unique solution.
Z2 =
Proof. For x, y ∈ R and for each t ∈ [0, 1], by Hölder inequality, we have
kUx − Uyk
n Z
≤ sup |A|
t∈[0,1]
0
t
t
(t − s)α−1
m(s)|x(s) − y(s)|ds
Γ(α)
(t − s)α+β−1
n(s)|x(s) − y(s)|ds
Γ(α + β)
0
h Z 1 (1 − s)α−δ−1
+ |Q| |A|
m(s)|x(s) − y(s)|ds
Γ(α − δ)
0
Z 1
(1 − s)α−δ+β−1
+ |B|
n(s)|x(s) − y(s)|ds
Γ(α − δ + β)
0
Z η
(η − s)α−δ
+ |A|
m(s)|x(s) − y(s)|ds
0 Γ(α − δ + 1)
Z η
io
(η − s)α−δ+β
+ |B|
n(s)|x(s) − y(s)|ds
0 Γ(α − δ + β + 1)
n |A|kmk 1 − γ 1−γ
|B|knk 1 − γ 1−γ α+β−γ
≤ sup
tα−γ +
t
Γ(α) α − γ
Γ(α + β) α + β − γ
t∈[0,1]
h |A|kmk 1 − γ 1−γ
1−γ
|B|knk 1−γ
+ |Q|
+
Γ(α − δ) α − δ − γ
Γ(α − δ + β) α − δ + β − γ
1−γ
|A|kmk 1−γ
+
η α−δ+1−γ
Γ(α − δ + 1) α − δ + 1 − γ
1−γ
io
|B|knk
1−γ
+
η α−δ+β+1−γ kx − yk
Γ(α − δ + β + 1) α − δ + β + 1 − γ
h 1 1 − γ 1−γ
|Q| 1 − γ 1−γ
≤ |A|kmk
+
Γ(α) α − γ
Γ(α − δ) α − δ − γ
1−γ i
|Q|
1−γ
+
kx − yk
Γ(α − δ + 1) α − δ + 1 − γ
h
1 − γ 1−γ
1−γ
1
|Q|
1−γ
+
+ |B|knk
Γ(α + β) α + β − γ
Γ(α − δ + β) α − δ + β − γ
Z
+ |B|
6
B. AHMAD, S. K. NTOUYAS
EJDE-2013/60
i
1−γ
|Q|
1−γ
η α−δ+β+1−γ kx − yk
Γ(α − δ + β + 1) α − δ + β + 1 − γ
= [|A|kmkZ1 + |B|knkZ2 ]kx − yk.
+
In view of condition (H2), it follows that U is a contraction mapping. Hence,
Banach’s fixed point theorem applies and U has a unique fixed point which is the
unique solution of problem (1.1). This completes the proof.
1.2. Existence result via Leray-Schauder Alternative.
Lemma 1.4 (Nonlinear alternative for single valued maps [11]). Let E be a Banach
space, C a closed, convex subset of E, U an open subset of C and 0 ∈ U . Suppose
that F : U → C is a continuous, compact (that is, F (U ) is a relatively compact
subset of C) map. Then either
(i) F has a fixed point in U , or
(ii) there is a u ∈ ∂U (the boundary of U in C) and λ ∈ (0, 1) with u = λF (u).
Theorem 1.5. Assume that f, g : [0, 1]×R → R are continuous functions. Assume
that:
(A3) There exist functions p1 , p2 ∈ L1 ([0, 1], R+ ), and nondecreasing functions
ψ1 , ψ2 : R+ → R+ such that
|f (t, x)| ≤ p1 (t)ψ1 (kxk),
|g(t, x)| ≤ p2 (t)ψ2 (kxk),
for all (t, x) ∈ [0, 1] × R.
(A4) There exists a constant M > 0 such that
M
> 1,
|A|Λ1 ψ1 (M )kp1 kL1 + |B|Λ1 ψ2 (M )kp2 kL1
where
1
|Q|
|Q|
+
+
,
Γ(α + 1) Γ(α − δ + 1) Γ(α − δ + 2)
|Q|
|Q|
1
+
+
.
Λ2 =
Γ(α + β + 1) Γ(α − δ + β + 1) Γ(α − δ + β + 2)
Λ1 =
Then the boundary-value problem (1.1) has at least one solution on [0, 1].
Proof. Consider the operator U : C → C with x = Ux, where
(Ux)(t)
Z
= −A
t
Z t
(t − s)α−1
(t − s)α+β−1
f (s, x(s))ds − B
g(s, x(s))ds
Γ(α)
Γ(α + β)
0
0
Z 1
h Z 1 (1 − s)α−δ−1
(1 − s)α−δ+β−1
α−1
+ Qt
A
f (s, x(s))ds + B
g(s, x(s))ds
Γ(α − δ)
Γ(α − δ + β)
0
0
Z
Z η
η
i
(η − s)α−δ
(η − s)α−δ+β
−A
f (s, x(s))ds − B
g(s, x(s))ds .
0 Γ(α − δ + 1)
0 Γ(α − δ + β + 1)
We show that F maps bounded sets into bounded sets in C([0, 1], R). For a positive
number r, let Br = {x ∈ C([0, 1], R) : kxk ≤ r} be a bounded set in C([0, 1], R).
Then
|(Ux)(t)|
EJDE-2013/60
NONLOCAL FRACTIONAL BOUNDARY-VALUE PROBLEMS
7
Z t
(t − s)α−1
(t − s)α+β−1
p1 (s)ψ1 (kxk)ds + |B|
p2 (s)ψ2 (kxk)ds
Γ(α)
Γ(α + β)
0
0
h Z 1 (1 − s)α−δ−1
α−1
|A|
+ |Q|t
p1 (s)ψ1 (kxk)ds
Γ(α − δ)
0
Z 1
(1 − s)α−δ+β−1
+ |B|
p2 (s)ψ2 (kxk)ds
Γ(α − δ + β)
0
Z η
(η − s)α−δ
p1 (s)ψ1 (kxk)ds
+ |A|
0 Γ(α − δ + 1)
Z η
i
(η − s)α−δ+β
+ |B|
p2 (s)ψ2 (kxk)ds
0 Γ(α − δ + β + 1)
o
n
|Q|
|Q|
1
+
+
≤ |A|ψ1 (r)kp1 kL1
Γ(α + 1) Γ(α − δ + 1) Γ(α − δ + 2)
n
o
1
|Q|
|Q|
+ |B|ψ2 (r)kp2 kL1
+
+
.
Γ(α + β + 1) Γ(α − δ + β + 1) Γ(α − δ + β + 2)
Z
t
≤ |A|
Consequently
kUxk
o
1
|Q|
|Q|
+
+
Γ(α + 1) Γ(α − δ + 1) Γ(α − δ + 2)
n
o
1
|Q|
|Q|
+ |B|ψ2 (r)kp2 kL1
+
+
.
Γ(α + β + 1) Γ(α − δ + β + 1) Γ(α − δ + β + 2)
≤ |A|ψ1 (r)kp1 kL1
n
Next we show that F maps bounded sets into equicontinuous sets of C([0, 1], R).
Let t1 , t2 ∈ [0, 1] with t1 < t2 and x ∈ Br , where Br is a bounded set of C([0, 1], R).
Then we obtain
k(Ux)(t2 ) − (Ux)(t1 )k
|A| Z t1
≤
[(t2 − s)α−1 − (t1 − s)α−1 ]f (s, x(s))ds
Γ(α) 0
Z t2
|A|
+
(t2 − s)α−1 f (s, x(s))ds
Γ(α) t1
Z t1
|B|
+
[(t2 − s)α+β−1 − (t1 − s)α+β−1 ]g(s, x(s))ds
Γ(α + β) 0
Z t2
|B|
+
(t2 − s)α+β−1 g(s, x(s))ds
Γ(α + β) t1
h Z 1 (1 − s)α−δ−1
+ |Q|[(t2 )α−1 − (t1 )α−1 ] |A|
|f (s, x(s))|ds
Γ(α − δ)
0
Z 1
(1 − s)α−δ+β−1
|g(s, x(s))|ds
+ |B|
Γ(α − δ + β)
0
Z η
Z η
i
(η − s)α−δ
(η − s)α−δ+β
+ |A|
|f (s, x(s))|ds + |B|
|g(s, x(s))|ds Γ(α
−
δ
+
1)
Γ(α
−
δ
+
β
+
1)
0
0
|A| Z t1
[(t2 − s)α−1 − (t1 − s)α−1 ]p1 (s)ψ1 (r)ds
≤
Γ(α) 0
8
B. AHMAD, S. K. NTOUYAS
+
|A|
Γ(α)
Z
t2
(t2 − s)α−1 p1 (s)ψ1 (r)ds
t1
|B|
+
Γ(α + β)
Z
|B|
Γ(α + β)
Z
+
t1
[(t2 − s)α+β−1 − (t1 − s)α+β−1 ]p2 (s)ψ2 (r)ds
0
t2
(t2 − s)α+β−1 p2 (s)ψ2 (r)ds
t1
h Z
+ |Q|[(t2 )α−1 − (t1 )α−1 ] |A|
1
0
1
Z
+ |B|
Z
EJDE-2013/60
0
η
+ |A|
0
(1 − s)α−δ−1
p1 (s)ψ1 (r)ds
Γ(α − δ)
(1 − s)α−δ+β−1
p2 (s)ψ2 (r)ds
Γ(α − δ + β)
Z η
i
(η − s)α−δ+β
(η − s)α−δ
p1 (s)ψ1 (r)ds + |B|
p2 (s)ψ2 (r)ds .
Γ(α − δ + 1)
Γ(α
−
δ
+
β
+
1)
0
Obviously the right hand side of the above inequality tends to zero independently
of x ∈ Br as t2 − t1 → 0. As U satisfies the above assumptions, therefore it follows
by the Arzelá-Ascoli theorem that U : C([0, 1], R) → C([0, 1], R) is completely
continuous.
The result will follow from the Leray-Schauder nonlinear alternative (Lemma
1.4) once we have proved the boundendness of the set of all solutions to equations
x = λUx for λ ∈ [0, 1].
Let x be a solution. Then, for t ∈ [0, 1], and using the computations in proving
that U is bounded, we have
|x(t)|
o
1
|Q|
|Q|
+
+
Γ(α + 1) Γ(α − δ + 1) Γ(α − δ + 2)
n
o
1
|Q|
|Q|
+ |B|ψ2 (kxk)kp2 kL1
+
+
.
Γ(α + β + 1) Γ(α − δ + β + 1) Γ(α − δ + β + 2)
= |λ(Ux)(t)| ≤ |A|ψ1 (kxk)kp1 kL1
n
Consequently,
kxk
≤ 1.
|A|Λ1 ψ1 (kxk)kp1 kL1 + |B|Λ1 ψ2 (kxk)kp2 kL1
In view of (A4), there exists M such that kxk =
6 M . Let us set
U = {x ∈ C([0, 1], X) : kxk < M }.
Note that the operator U : U → C([0, 1], R) is continuous and completely continuous. From the choice of U , there is no x ∈ ∂U such that x = λU(x) for
some λ ∈ (0, 1). Consequently, by the nonlinear alternative of Leray-Schauder type
(Lemma 1.4), we deduce that U has a fixed point x ∈ U which is a solution of the
problem (1.1). This completes the proof.
Example. Consider a boundary-value problem of integro-differential equations of
fractional order with nonlocal fractional boundary conditions given by
−D5/2 x(t) = Af (t, x(t)) + BI β g(t, x(t)),
D1/4 x(0) = 0,
D5/4 x(0) = 0,
D1/4 x(1) =
t ∈ [0, 1],
Z
0
η
D1/4 x(s)ds,
(1.7)
EJDE-2013/60
NONLOCAL FRACTIONAL BOUNDARY-VALUE PROBLEMS
where n = 3, A = B = 1, β = 3/4, η = 2/3, f (t, x) =
1
−1
x + sin2 t. With the given data, we find that
2 tan
Q=
3|x|(2+|x|)
8(1+|x|)
9
+ 4t, g(t, x) =
Γ(α − δ + 1)
= 1.037485,
(α − δ − η α−δ )Γ(α)
and
1
η α−δ+1 i
tα
+ |Q|tα−1
+
Γ(α + 1)
Γ(α − δ + 1) Γ(α − δ + 2)
t∈[0,1]
h
io
α+β
t
1
η α−δ+β+1
+ |B|
+ |Q|tα−1
+
Γ(α + β + 1)
Γ(α − δ + β + 1) Γ(α − δ + β + 2)
= 1.043555,
Ω = sup
n h
|A|
and L1 = 3/4, L2 = 1/2 as |f (t, x) − f (t, y)| ≤ 34 |x − y|, |g(t, x) − g(t, y)| ≤ 12 |x − y|.
Clearly L = max{L1 , L2 } = 3/4 and L < 1/Ω. Thus all the assumptions of
Theorem 1.2 are satisfied. Hence, by the conclusion of Theorem 1.2, the problem
(1.7) has a unique solution.
Acknowledgments. The authors gratefully acknowledge the editor for his constructive comments.
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Bashir Ahmad
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box
80203, Jeddah 21589, Saudi Arabia
E-mail address: bashirahmad qau@yahoo.com
Sotiris K. Ntouyas
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
E-mail address: sntouyas@uoi.gr