# Existence Theory forq-Antiperiodic Boundary Value Problems of Sequential q-Fractional Integrodifferential Equations

Authors: Ravi P. Agarwal; Bashir Ahmad; Ahmed Alsaedi; Hana Al-Hutami
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We discuss the existence and uniqueness of solutions for a new class of sequentialq-fractional integrodifferential equations with q-antiperiodic boundary conditions. Our results rely on the standard tools of fixed-point theory such as Krasnoselskii's fixed-point theorem, Leray-Schauder nonlinear alternative, and Banach's contraction principle. An illustrative example is also presented.


---

## Body

## 1. Introduction

We consider aq-antiperiodic boundary value problem of sequentialq-fractional integrodifferential equations given by
(1)
D
c
q
α
(
D
c
q
γ
+
λ
)
x
(
t
)
=
A
f
(
t
,
x
(
t
)
)
+
B
I
q
ρ
g
(
t
,
x
(
t
)
)
,
nnnnnnnnnnnnnnnn
.
nnnnnl
0
≤
t
≤
1
,
0
<
q
<
1
,
x
(
0
)
=
-
x
(
1
)
,
(
t
(
1
-
γ
)
D
q
x
(
t
)
)
|
t
=
0
=
-
D
q
x
(
1
)
,
where D
c
q
αand D
c
q
γdenote the fractionalq-derivative of the Caputo type, 0
<
α
,
γ
≤
1, I
q
ρ
(
·
)denotes Riemann-Liouville integral with0
<
ρ
<
1
,
f
,
gbeing given continuous functions,λ
∈
ℝ andA
,
Bbeing real constants.The aim of the present study is to establish some existence and uniqueness results for the problem (1) by means of Krasnoselskii’s fixed-point theorem, Leray-Schauder nonlinear alternative, and Banach’s contraction principle. Though the tools employed in this work are standard, yet their exposition in the framework of the given problem is new.Fractional calculus has developed into a popular mathematical modelling tool for many real world phenomena occurring in physical and technical sciences, see, for example, [1–4]. A fractional-order differential operator distinguishes itself from an integer-order differential operator in the sense that it is nonlocal in nature and can describe the memory and hereditary properties of some important and useful materials and processes. This feature has fascinated many researchers and several results ranging from theoretical analysis to asymptotic behavior and numerical methods for fractional differential equations have been established. For some recent work on the topic, see [5–12] and references therein.The mathematical modeling of linear control systems, concerning the controllability of systems consisting of a set of well-defined interconnected objects, is based on the linear systems of divided difference functional equations. The controllability in mathematical control theory studies the concepts such as controllability of the state, controllability of the output, controllability at the origin, and complete controllability. Theq-difference equations play a key role in the control theory as these equations are always completely controllable and appear in theq-optimal control problem [13]. The variationalq-calculus is known as a generalization of the continuous variational calculus due to the presence of an extra-parameterqwhose nature may be physical or economical. The study of theq-uniform lattice rely on theq-Euler equations. In other words, it suffices to solve theq-Euler-Lagrange equation for finding the extremum of the functional involved instead of solving the Euler-Lagrange equation [14]. One can find more details in a series of papers [15–21].The subject of fractionalq-difference (q-fractional) equations is regarded as fractional analogue ofq-difference equations and has recently gained a considerable attention. For examples and details, we refer the reader to the works [22–33] and references therein while some earlier work on the subject can be found in [34–36]. The present work is motivated by recent interest in the study of fractional-order differential equations.


## 2. Preliminaries on Fractionalq-Calculus

Let us describe the notations and terminology forq-fractional calculus [35].For a real parameterq
∈
ℝ
+
∖
{
1
}, aq-real number denoted by[
a
]
qis defined by
(2)
[
a
]
q
=
1
-
q
a
1
-
q
,
a
∈
ℝ
.Theq-analogue of the Pochhammer symbol (q-shifted factorial) is defined as
(3)
(
a
;
q
)
0
=
1
,
(
a
;
q
)
k
=
∏
i
=
0
k
-
1
‍
(
1
-
a
q
i
)
,
k
∈
ℕ
∪
{
∞
}
.Theq-analogue of the exponent (
x
-
y
)
kis
(4)
(
x
-
y
)
(
0
)
=
1
,
(
x
-
y
)
(
k
)
=
∏
j
=
0
k
-
1
‍
(
x
-
y
q
j
)
,
n
n
n
n
n
n
n
n
n
n
n
n
.
n
n
n
n
.
n
.
n
n
n
n
n
k
∈
ℕ
,
x
,
y
∈
ℝ
.Theq-gamma functionΓ
q
(
y
)is defined as
(5)
Γ
q
(
y
)
=
(
1
-
q
)
(
y
-
1
)
(
1
-
q
)
y
-
1
,
where y
∈
ℝ
∖
{
0
,
-
1
,
-
2
,
…
}. Observe that Γ
q
(
y
+
1
)
=
[
y
]
q
Γ
q
(
y
).Definition 1 (see [35]).
Letfbe a function defined on[
0,1
]
.The fractional q-integral of the Riemann-Liouville type of orderβ
≥
0is(
I
q
0
f
)
(
t
)
=
f
(
t
)and
(6)
I
q
β
f
(
t
)
≔
∫
0
t
‍
(
t
-
q
s
)
(
β
-
1
)
Γ
q
(
β
)
f
(
s
)
d
q
s
=
t
β
(
1
-
q
)
β
∑
k
=
0
∞
‍
q
k
(
q
β
;
q
)
k
(
q
;
q
)
k
f
(
t
q
k
)
,
n
n
n
n
n
n
n
n
n
n
n
n
n
n
.
n
β
>
0
,
t
∈
[
0,1
]
.Observe that the aboveq-integral reduces to the following one forβ
=
1.
(7)
I
q
f
(
t
)
∶
=
∫
0
t
‍
f
(
s
)
d
q
s
=
t
(
1
-
q
)
∑
k
=
0
∞
‍
q
k
f
(
t
q
k
)
.
Further details ofq-integrals and fractionalq-integrals can be found respectively in Section  1.3 and Section  4.2 of the text [35].Remark 2.
The semigroup property holds forq-fractional integration (Proposition  4.3 [35]):
(8)
I
q
γ
I
q
β
f
(
t
)
=
I
q
β
+
γ
f
(
t
)
;
γ
,
β
∈
ℝ
+
.
Further, it has been shown in Lemma  6 of [37] that
(9)
I
q
β
(
x
)
(
υ
)
=
Γ
q
(
υ
+
1
)
Γ
q
(
β
+
υ
+
1
)
(
x
)
(
β
+
υ
)
,
0
<
x
<
a
,
β
∈
ℝ
+
,
υ
∈
(
-
1
,
∞
)
.Before giving the definition of fractionalq-derivative, we recall the concept ofq-derivative.Letfbe a real valued function defined on aq-geometric setA(|
q
|
≠
1). Then theq-derivative of a function fis defined as
(10)
D
q
f
(
t
)
=
f
(
t
)
-
f
(
q
t
)
t
-
q
t
,
t
∈
A
∖
{
0
}
.
For0
∈
A, theq-derivative at zero is defined for|
q
|
<
1by
(11)
D
q
f
(
0
)
=
lim
⁡
n
→
∞
f
(
t
q
n
)
-
f
(
0
)
t
q
n
,
t
∈
A
∖
{
0
}
.
Provided that the limit exists and does not depend ont.Furthermore,(12)
D
q
0
f
=
f
,
D
q
n
f
=
D
q
(
D
q
n
-
1
f
)
,
n
=
1,2
,
3
,
…
.Definition 3 (see [35]).
The Caputo fractionalq-derivative of orderβ
>
0is defined by
(13)
c
D
q
β
f
(
t
)
=
I
q
⌈
β
⌉
-
β
D
q
⌈
β
⌉
f
(
t
)
,
where ⌈
β
⌉is the smallest integer greater than or equal toβ
.Next we enlist some properties involving Riemann-Liouvilleq-fractional integral and Caputo fractionalq-derivative (Theorem  5.2 [35]):
(14)
I
q
β
D
c
q
β
f
(
t
)
=
f
(
t
)
-
∑
k
=
0
⌈
β
⌉
-
1
‍
t
k
Γ
q
(
k
+
1
)
(
D
q
k
f
)
(
0
+
)
,
.
nnnnnnnnnnnnnnnnnnnnnn
∀
t
∈
(
0
,
a
]
,
β
>
0
;
c
D
q
β
I
q
β
f
(
t
)
=
f
(
t
)
,
∀
t
∈
(
0
,
a
]
,
β
>
0
.Now we establish a lemma that plays a key role in the sequel.Lemma 4.
For a givenh
∈
C
(
[
0,1
]
,
ℝ
), the boundary value problem
(15)
D
c
q
α
(
D
q
γ
c
+
λ
)
x
(
t
)
=
h
(
t
)
,
0
≤
t
≤
1
,
0
<
q
<
1
,
x
(
0
)
=
-
x
(
1
)
,
(
t
(
1
-
γ
)
D
q
x
(
t
)
)
|
t
=
0
=
-
D
q
x
(
1
)
is equivalent to theq-integral equation
(16)
x
(
t
)
=
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
(
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
h
(
m
)
d
q
m
-
λ
x
(
u
)
)
d
q
u
+
(
1
-
2
t
γ
)
4
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
(
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
h
(
m
)
d
q
m
-
λ
x
(
u
)
)
d
q
u
-
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
(
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
h
(
m
)
d
q
m
-
λ
x
(
u
)
)
d
q
u
.Proof.
It is well known that the solution ofq-fractional equation in (15) can be written as
(17)
x
(
t
)
=
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
(
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
h
(
m
)
d
q
m
-
λ
x
(
u
)
)
d
q
u
-
t
γ
Γ
q
(
γ
+
1
)
c
0
-
c
1
.
Differentiating (17), we obtain
(18)
D
q
x
(
t
)
=
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
(
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
h
(
m
)
d
q
m
-
λ
x
(
u
)
)
d
q
u
-
[
γ
]
q
t
γ
-
1
Γ
q
(
γ
+
1
)
c
0
.
Using the boundary conditions (15) in (17) and (18) and solving the resulting expressions forc
0and c
1, we get
(19)
c
0
=
Γ
q
(
γ
+
1
)
2
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
(
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
h
(
m
)
d
q
m
-
λ
x
(
u
)
)
d
q
u
,
c
1
=
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
(
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
h
(
m
)
d
q
m
-
λ
x
(
u
)
)
d
q
u
-
1
4
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
(
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
h
(
m
)
d
q
m
-
λ
x
(
u
)
)
d
q
u
.
Substituting the values ofc
0andc
1in (17) yields the solution (16). The converse follows in a straightforward manner. This completes the proof.Let𝒞
=
C
(
[
0,1
]
,
ℝ
) denote the Banach space of all continuous functions from[
0,1
]into ℝ endowed with the usual norm defined by∥
x
∥
=
sup
⁡
{
|
x
(
t
)
|
,
t
∈
[
0,1
]
}.In view of Lemma4, we define an operator 𝒰
:
𝒞
→
𝒞 as
(20)
(
𝒰
x
)
(
t
)
=
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
(
A
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
n
×
f
(
m
,
x
(
m
)
)
d
q
m
+
B
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
g
(
m
,
x
(
m
)
)
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
20000
-
λ
x
(
u
)
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
)
d
q
u
+
(
1
-
2
t
γ
)
4
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
(
A
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
n
l
l
×
f
(
m
,
x
(
m
)
)
d
q
m
+
B
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
g
(
m
,
x
(
m
)
)
d
q
m
-
λ
x
(
u
)
)
d
q
u
-
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
(
A
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
f
(
m
,
x
(
m
)
)
d
q
m
+
B
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
g
(
m
,
x
(
m
)
)
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
-
λ
x
(
u
)
)
d
q
u
.Observe that the problem (1) has solutions only if the operator equationx
=
𝒰
xhas fixed points.


## 3. Main Results

For the forthcoming analysis, the following conditions are assumed.(
A
1
)
f
,
g
:
[
0,1
]
×
ℝ
→
ℝ are continuous functions such that|
f
(
t
,
x
)
-
f
(
t
,
y
)
|
≤
L
1
|
x
-
y
|and|
g
(
t
,
x
)
-
g
(
t
,
y
)
|
≤
L
2
|
x
-
y
|, for all t
∈
[
0,1
]
,
L
1
,
L
2
>
0
,
x
,
y
∈
ℝ.
(
A
2
)
There existζ
1
,
ζ
2
∈
C
(
[
0,1
]
,
ℝ
+
)with|
f
(
t
,
x
)
|
≤
ζ
1
(
t
)
,
|
g
(
t
,
x
)
|
≤
ζ
2
(
t
), for all (
t
,
x
)
∈
[
0,1
]
×
ℝ, where∥
ζ
i
∥
=
sup
⁡
t
∈
[
0,1
]
|
ζ
i
(
t
)
|
,
i
=
1,2. For computational convenience, we set
(21)
δ
1
=
3
2
Γ
q
(
α
+
γ
+
1
)
+
1
4
[
γ
]
q
Γ
q
(
α
+
γ
)
,
δ
2
=
3
2
Γ
q
(
α
+
ρ
+
γ
+
1
)
+
1
4
[
γ
]
q
Γ
q
(
α
+
ρ
+
γ
)
,
δ
3
=
3
2
Γ
q
(
γ
+
1
)
+
1
4
Γ
q
(
γ
+
1
)
,
(22)
Λ
=
L
[
|
A
|
(
1
4
[
γ
]
q
Γ
q
(
α
+
γ
)
+
1
2
Γ
q
(
α
+
γ
+
1
)
)
+
|
B
|
(
1
4
[
γ
]
q
Γ
q
(
α
+
ρ
+
γ
)
+
1
2
Γ
q
(
α
+
ρ
+
γ
+
1
)
)
]
+
|
λ
|
[
1
4
Γ
q
(
γ
+
1
)
+
1
2
Γ
q
(
γ
+
1
)
]
.Our first existence result is based on Krasnoselskii’s fixed point theorem.Lemma 5 (see, Krasnoselskii [38]).
LetYbe a closed, convex, bounded, and nonempty subset of a Banach spaceX
.LetQ
1
,
Q
2be the operators such that (i)Q
1
x
+
Q
2
y
∈
Ywheneverx
,
y
∈
Y; (ii)Q
1is compact and continuous; and (iii)Q
2is a contraction mapping. Then there existsz
∈
Ysuch thatz
=
Q
1
z
+
Q
2
z
.Theorem 6.
Letf
,
g
:
[
0,1
]
×
ℝ
→
ℝ be continuous functions satisfying (
A
1
)
–
(
A
2
).FurthermoreΛ
<
1, whereΛis given by (22) andL
=
max
⁡
{
L
1
,
L
2
}
.Then the problem (1) has at least one solution on[
0,1
].Proof.
Consider the setB
σ
=
{
x
∈
𝒞
:
∥
x
∥
≤
σ
}, whereσis given by
(23)
σ
≥
|
A
|
∥
ζ
1
∥
δ
1
+
|
B
|
∥
ζ
2
∥
δ
2
1
-
|
λ
|
δ
3
,
1
-
|
λ
|
δ
3
>
0
.
Define operators𝒰
1and𝒰
2onB
σas
(24)
(
𝒰
1
x
)
(
t
)
=
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
(
A
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
f
(
m
,
x
(
m
)
)
d
q
m
nnnnn
+
B
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
n
n
n
n
n
n
n
n
n
n
×
g
(
m
,
x
(
m
)
)
d
q
m
n
n
n
n
n
n
n
n
n
l
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
-
λ
x
(
u
)
)
d
q
u
,
t
∈
[
0,1
]
,
(
𝒰
2
x
)
(
t
)
=
(
1
-
2
t
γ
)
4
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
nnnn
×
(
A
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
n
n
n
n
n
n
n
×
f
(
m
,
x
(
m
)
)
d
q
m
nnnnnn
+
B
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
n
n
n
n
n
n
l
n
n
n
n
×
g
(
m
,
x
(
m
)
)
d
q
m
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
nnnnnn
-
λ
x
(
u
)
)
d
q
u
-
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
(
A
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
f
(
m
,
x
(
m
)
)
d
q
m
+
B
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
g
(
m
,
x
(
m
)
)
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
-
λ
x
(
u
)
)
d
q
u
,
t
∈
[
0,1
]
.
Forx
,
y
∈
B
σ, we find that
(25)
∥
𝒰
1
x
+
𝒰
2
y
∥
≤
|
A
|
∥
ζ
1
∥
δ
1
+
|
B
|
∥
ζ
2
∥
δ
2
+
|
λ
|
σ
δ
3
≤
σ
.
Thus,𝒰
1
x
+
𝒰
2
y
∈
B
σ
.Continuity offandgimply that the operator𝒰
1 is continuous. Also,𝒰
1is uniformly bounded onB
σas
(26)
∥
𝒰
1
x
∥
≤
|
A
|
∥
ζ
1
∥
Γ
q
(
α
+
γ
+
1
)
+
|
B
|
∥
ζ
2
∥
Γ
q
(
α
+
ρ
+
γ
+
1
)
+
|
λ
|
σ
Γ
q
(
γ
+
1
)
.
Now, we prove the compactness of the operator𝒰
1
.In view of (
A
1
), we define
(27)
sup
⁡
(
t
,
x
)
∈
[
0,1
]
×
B
σ
|
f
(
t
,
x
)
|
=
f
¯
,
sup
⁡
(
t
,
x
)
∈
[
0,1
]
×
B
σ
|
g
(
t
,
x
)
|
=
g
¯
.
Consequently, fort
1
,
t
2
∈
[
0,1
], we have
(28)
∥
(
𝒰
1
x
)
(
t
2
)
-
(
𝒰
1
x
)
(
t
1
)
∥
≤
∫
0
t
1
‍
(
t
2
-
q
u
)
(
γ
-
1
)
-
(
t
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
f
¯
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
+
|
B
|
g
¯
×
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
d
q
m
+
|
λ
|
σ
]
d
q
u
+
∫
t
1
t
2
‍
(
t
2
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
f
¯
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
+
|
B
|
g
¯
×
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
d
q
m
+
|
λ
|
σ
]
d
q
u
,
which is independent ofxand tends to zero ast
2
→
t
1. Thus,𝒰
1is relatively compact onB
σ. Hence, by the Arzelá-Ascoli Theorem,𝒰
1is compact onB
σ
.Now, we shall show that𝒰
2is a contraction.
From(
A
1
)and forx
,
y
∈
B
σ, we have
(29)
∥
𝒰
2
x
-
𝒰
2
y
∥
≤
sup
⁡
t
∈
[
0,1
]
{
1
4
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
n
n
.
×
|
f
(
m
,
x
(
m
)
)
-
f
(
m
,
y
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
|
g
(
m
,
x
(
m
)
)
-
g
(
m
,
y
(
m
)
)
|
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
|
x
(
u
)
-
y
(
u
)
|
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
|
f
(
m
,
x
(
m
)
)
-
f
(
m
,
y
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
|
g
(
m
,
x
(
m
)
)
-
g
(
m
,
y
(
m
)
)
|
d
q
m
+
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
|
λ
|
|
x
(
u
)
-
y
(
u
)
|
]
d
q
u
}
≤
sup
⁡
t
∈
[
0,1
]
{
1
4
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
L
1
×
|
x
(
m
)
-
y
(
m
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
n
n
.
×
L
2
|
x
(
m
)
-
y
(
m
)
|
d
q
m
+
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
|
λ
|
|
x
(
u
)
-
y
(
u
)
|
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
n
.
l
×
L
1
|
x
(
m
)
-
y
(
m
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
L
2
|
x
(
m
)
-
y
(
m
)
|
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
|
x
(
u
)
-
y
(
u
)
|
]
d
q
u
}
≤
[
L
[
|
A
|
(
1
4
[
γ
]
q
Γ
q
(
α
+
γ
)
+
1
2
Γ
q
(
α
+
γ
+
1
)
)
+
|
B
|
(
1
4
[
γ
]
q
Γ
q
(
α
+
ρ
+
γ
)
+
1
2
Γ
q
(
α
+
ρ
+
γ
+
1
)
)
]
+
|
λ
|
[
1
4
[
γ
]
q
Γ
q
(
γ
)
+
1
2
Γ
q
(
γ
+
1
)
]
]
∥
x
-
y
∥
=
Λ
∥
x
-
y
∥
,
where we have used (22). In view of the assumptionΛ
<
1, the operator𝒰
2is a contraction. Thus, all the conditions of Lemma 5 are satisfied. Hence, by the conclusion of Lemma 5, the problem (1) has at least one solution on[
0,1
].Our next result is based on Leray-Schauder nonlinear alternative.Lemma 7 (nonlinear alternative for single valued maps, see [39]).
LetEbe a Banach space,Ca closed, convex subset of E
,
Wan open subset ofC, and0
∈
W
.Suppose that 𝒰
:
W
¯
→
Cis a continuous, compact (
i
.
e
.
,
𝒰
(
W
¯
)is a relatively compact subset ofC) map. Then either (i)
𝒰 has a fixed point in W
¯, or
(ii)
there is ax
∈
∂
W(the boundary ofWinC) andκ
∈
(
0,1
)withx
=
κ
𝒰
(
x
)
.Theorem 8.
Letf
,
g
:
[
0,1
]
×
ℝ
→
ℝ be continuous functions and the following assumptions hold: (
A
3
)
there exist functionsν
1
,
ν
2
∈
C
(
[
0,1
]
,
ℝ
+
), and nondecreasing functionsϑ
1
,
ϑ
2
:
ℝ
+
→
ℝ
+such that|
f
(
t
,
x
)
|
≤
ν
1
(
t
)
ϑ
1
(
∥
x
∥
)
,
|
g
(
t
,
x
)
|
≤
ν
2
(
t
)
ϑ
2
(
∥
x
∥
)
, for all (
t
,
x
)
∈
[
0,1
]
×
ℝ
;
(
A
4
)
there exists a constantω
>
0such that
(30)
ω
>
|
A
|
∥
ν
1
∥
ϑ
1
(
ω
)
δ
1
+
|
B
|
∥
ν
2
∥
ϑ
2
(
ω
)
δ
2
1
-
|
λ
|
δ
3
,
1
-
|
λ
|
δ
3
>
0
.
Then the boundary value problem (1) has at least one solution on[
0,1
]
.Proof.
Consider the operator𝒰
:
𝒞
→
𝒞 defined by (20). The proof consists of several steps. (i)
It is easy to show that𝒰 is continuous.
(ii)
𝒰 maps bounded sets into bounded sets in C
(
[
0,1
]
×
ℝ
).
For a positive numberr
¯, letB
r
¯
=
{
x
∈
𝒞
:
∥
x
∥
≤
r
¯
} be a bounded set in C
(
[
0,1
]
×
ℝ
) andx
∈
B
r
¯. Then, we have
(31)
∥
(
𝒰
x
)
∥
≤
sup
⁡
t
∈
[
0,1
]
{
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
|
f
(
m
,
x
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
|
g
(
m
,
x
(
m
)
)
|
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
|
x
(
u
)
|
]
d
q
u
+
1
4
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
|
f
(
m
,
x
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
|
g
(
m
,
x
(
m
)
)
|
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
|
x
(
u
)
|
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
|
f
(
m
,
x
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
|
g
(
m
,
x
(
m
)
)
|
d
q
m
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
+
|
λ
|
|
x
(
u
)
|
]
d
q
u
}
≤
sup
⁡
t
∈
[
0,1
]
{
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
ν
1
(
m
)
ϑ
1
(
∥
x
∥
)
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
ν
2
(
m
)
ϑ
2
(
∥
x
∥
)
d
q
m
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
+
|
λ
|
|
x
(
u
)
|
]
d
q
u
+
1
4
[
γ
]
q
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
ν
1
(
m
)
ϑ
1
(
∥
x
∥
)
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
ν
2
(
m
)
ϑ
2
(
∥
x
∥
)
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
|
x
(
u
)
|
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
ν
1
(
m
)
ϑ
1
(
∥
x
∥
)
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
ν
2
(
m
)
ϑ
2
(
∥
x
∥
)
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
|
x
(
u
)
|
]
d
q
u
}
≤
|
A
|
∥
ν
1
∥
ϑ
1
(
∥
x
∥
)
×
{
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
[
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
]
d
q
u
+
1
4
[
γ
]
q
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
]
d
q
u
}
+
|
B
|
∥
ν
2
∥
ϑ
2
(
∥
x
∥
)
×
{
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
[
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
]
d
q
u
+
1
4
[
γ
]
q
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
nn
.
×
[
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
]
d
q
u
}
+
|
λ
|
∥
x
∥
{
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
d
q
u
+
1
4
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
d
q
u
}
≤
|
A
|
∥
ν
1
∥
ϑ
1
(
r
¯
)
δ
1
+
|
B
|
∥
ν
2
∥
ϑ
2
(
r
¯
)
δ
2
+
|
λ
|
r
¯
δ
3
≤
r
¯
.
This shows that𝒰
x
∈
B
r
¯.(iii)
𝒰 maps bounded sets into equicontinuous sets of C
(
[
0,1
]
×
ℝ
).
Lett
1
,
t
2
∈
[
0,1
] with t
1
<
t
2 and x
∈
B
r
¯, where B
r
¯is a bounded set of C
(
[
0,1
]
,
ℝ
). Then, we obtain
(32)
∥
(
𝒰
x
)
(
t
2
)
-
(
𝒰
x
)
(
t
1
)
∥
≤
|
∫
0
t
1
‍
(
t
2
-
q
u
)
(
γ
-
1
)
-
(
t
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
ν
1
(
m
)
ϑ
1
(
r
¯
)
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
ν
2
(
m
)
ϑ
2
(
r
¯
)
d
q
m
+
|
λ
|
r
¯
]
d
q
u
+
∫
t
1
t
2
‍
(
t
2
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
ν
1
(
m
)
ϑ
1
(
r
¯
)
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
ν
2
(
m
)
ϑ
2
(
r
¯
)
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
r
¯
]
d
q
u
|
+
(
t
2
γ
-
t
1
γ
)
2
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
ν
1
(
m
)
ϑ
1
(
r
¯
)
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
ν
2
(
m
)
ϑ
2
(
r
¯
)
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
r
¯
]
d
q
u
.
Obviously the right-hand side of the above inequality tends to zero independently ofx
∈
B
r
¯ as t
2
-
t
1
→
0
.Therefore, it follows by the Arzelá-Ascoli theorem that 𝒰
:
𝒞
→
𝒞 is completely continuous.(iv)
Letxbe a solution of the given problem such thatx
=
κ
𝒰
xforκ
∈
(
0,1
)
.Then, fort
∈
[
0,1
], it follows by the procedure used to establish (ii) that
(33)
|
x
(
t
)
|
=
|
κ
(
𝒰
x
)
(
t
)
|
≤
|
A
|
∥
ν
1
∥
ϑ
1
(
∥
x
∥
)
δ
1
+
|
B
|
∥
ν
2
∥
ϑ
2
(
∥
x
∥
)
δ
2
+
|
λ
|
∥
x
∥
δ
3
.
Consequently, we have(34)
∥
x
∥
≤
|
A
|
∥
ν
1
∥
ϑ
1
(
∥
x
∥
)
δ
1
+
|
B
|
∥
ν
2
∥
ϑ
2
(
∥
x
∥
)
δ
2
1
-
|
λ
|
δ
3
.
In view of(
A
4
), there existsωsuch that∥
x
∥
≠
ω. Let us set
(35)
W
=
{
x
∈
𝒞
:
∥
x
∥
<
ω
}
.
Note that the operator𝒰
:
W
¯
→
C
(
[
0,1
]
,
ℝ
)is continuous and completely continuous. From the choice ofW, there is nox
∈
∂
Wsuch thatx
=
κ
𝒰
(
x
)for someκ
∈
(
0,1
). In consequence, by the nonlinear alternative of Leray-Schauder type (Lemma 7), we deduce that 𝒰has a fixed pointx
∈
W
¯which is a solution of the problem (1). This completes the proof.Finally we show the existence of a unique solution of the given problem by applying Banach’s contraction mapping principle (Banach fixed-point theorem).Theorem 9.
Suppose that the assumption(
A
1
) holds and
(36)
Λ
¯
=
(
L
Λ
1
+
|
λ
|
δ
3
)
<
1
,
Λ
1
=
|
A
|
δ
1
+
|
B
|
δ
2
,
whereδ
1
,
δ
2, andδ
3are given by (21) andL
=
max
⁡
{
L
1
,
L
2
}. Then the boundary value problem (1) has a unique solution.Proof.
FixM
=
max
⁡
{
M
1
,
M
2
}, where M
1
,
M
2are finite numbers given byM
1
=
sup
⁡
t
∈
[
0,1
]
|
f
(
t
,
0
)
|
,
M
2
=
sup
⁡
t
∈
[
0,1
]
|
g
(
t
,
0
)
|. Selecting σ
≥
M
Λ
1
/
(
1
-
Λ
¯
), we show that 𝒰
B
σ
⊂
B
σ, where B
σ
=
{
x
∈
𝒞
:
∥
x
∥
≤
σ
}. For x
∈
B
σ, we have
(37)
∥
(
𝒰
x
)
∥
≤
sup
⁡
t
∈
[
0,1
]
{
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
|
f
(
m
,
x
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
|
g
(
m
,
x
(
m
)
)
|
d
q
m
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
+
|
λ
|
|
x
(
u
)
|
]
d
q
u
+
1
4
[
γ
]
q
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
|
f
(
m
,
x
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
|
g
(
m
,
x
(
m
)
)
|
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
|
x
(
u
)
|
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
|
f
(
m
,
x
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
|
g
(
m
,
x
(
m
)
)
|
d
q
m
+
|
λ
|
|
x
(
u
)
|
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
]
d
q
u
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
}
≤
sup
⁡
t
∈
[
0,1
]
{
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
(
|
f
(
m
,
x
(
m
)
)
-
f
(
m
,
0
)
|
+
|
f
(
m
,
0
)
|
)
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
(
|
g
(
m
,
x
(
m
)
)
-
g
(
m
,
0
)
|
+
|
g
(
m
,
0
)
|
)
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
|
x
(
u
)
|
]
d
q
u
+
1
4
[
γ
]
q
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
(
|
f
(
m
,
x
(
m
)
)
-
f
(
m
,
0
)
|
+
|
f
(
m
,
0
)
|
)
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
(
|
g
(
m
,
x
(
m
)
)
-
g
(
m
,
0
)
|
+
|
g
(
m
,
0
)
|
)
d
q
m
+
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
|
λ
|
|
x
(
u
)
|
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
(
|
f
(
m
,
x
(
m
)
)
-
f
(
m
,
0
)
|
+
|
f
(
m
,
0
)
|
)
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
(
|
g
(
m
,
x
(
m
)
)
-
g
(
m
,
0
)
|
+
|
g
(
m
,
0
)
|
)
d
q
m
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
+
|
λ
|
|
x
(
u
)
|
]
d
q
u
}
≤
|
A
|
(
L
1
σ
+
M
1
)
×
sup
⁡
t
∈
[
0,1
]
{
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
]
d
q
u
+
1
4
[
γ
]
q
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
d
q
m
]
d
q
u
}
+
|
B
|
(
L
2
σ
+
M
2
)
×
sup
⁡
t
∈
[
0,1
]
{
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
d
q
m
]
d
q
u
+
1
4
[
γ
]
q
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
d
q
m
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
d
q
m
]
d
q
u
}
+
|
λ
|
σ
sup
⁡
t
∈
[
0,1
]
{
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
d
q
u
+
1
4
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
d
q
u
}
≤
M
Λ
1
+
Λ
¯
σ
≤
σ
This shows that𝒰
B
σ
⊂
B
σ
.For x
,
y
∈
ℝ, we obtain
(38)
∥
𝒰
x
-
𝒰
y
∥
≤
sup
⁡
t
∈
[
0,1
]
{
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
|
f
(
m
,
x
(
m
)
)
-
f
(
m
,
y
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
|
g
(
m
,
x
(
m
)
)
-
g
(
m
,
y
(
m
)
)
|
d
q
m
+
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
|
λ
|
|
x
(
u
)
-
y
(
u
)
|
]
d
q
u
+
1
4
[
γ
]
q
×
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
2
)
Γ
q
(
γ
-
1
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
|
f
(
m
,
x
(
m
)
)
-
f
(
m
,
y
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
|
g
(
m
,
x
(
m
)
)
-
g
(
m
,
y
(
m
)
)
|
d
q
m
+
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
|
λ
|
|
x
(
u
)
-
y
(
u
)
|
]
d
q
u
+
1
2
∫
0
1
‍
(
1
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
×
[
|
A
|
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
×
|
f
(
m
,
x
(
m
)
)
-
f
(
m
,
y
(
m
)
)
|
d
q
m
+
|
B
|
∫
0
u
‍
(
u
-
q
m
)
(
α
+
ρ
-
1
)
Γ
q
(
α
+
ρ
)
×
|
g
(
m
,
x
(
m
)
)
-
g
(
m
,
y
(
m
)
)
|
d
q
m
∫
0
t
‍
(
t
-
q
u
)
(
γ
-
1
)
Γ
q
(
γ
)
+
∫
0
u
‍
(
u
-
q
m
)
(
α
-
1
)
Γ
q
(
α
)
|
λ
|
|
x
(
u
)
-
y
(
u
)
|
]
d
q
u
}
≤
Λ
¯
∥
x
-
y
∥
.
SinceΛ
¯
∈
(
0,1
)by the given assumption, therefore 𝒰 is a contraction. Hence, it follows by Banach’s contraction principle that the problem (1) has a unique solution.Example 10.
Consider aq-fractional integrodifferential equation withq-antiperiodic boundary conditions given by
(39)
D
c
q
α
(
D
q
γ
c
+
1
15
)
x
(
t
)
=
1
5
f
(
t
,
x
(
t
)
)
+
1
8
I
q
ρ
g
(
t
,
x
(
t
)
)
,
nnnnnnnnnnnnnn
.
0
<
t
<
1
,
0
<
q
<
1
,
x
(
0
)
=
-
x
(
1
)
,
(
t
(
1
-
γ
)
D
q
x
(
t
)
)
|
t
=
0
=
-
D
q
x
(
1
)
,
where α
=
γ
=
ρ
=
q
=
1
/
2
,
λ
=
1
/
15, A
=
1
/
5
,
B
=
1
/
8
,
f
(
t
,
x
)
=
(
t
/
20
π
)
sin
2
π
x+(
t
+
1
)
(
1
+
x
)
4
/
(
1
+
(
1
+
x
)
4
), g
(
t
,
x
)
=
(
1
/
8
π
)
tan
-
1
x
+
1
/
16
.With the given data,δ
1
≃
1.92678
,
δ
2
≃
1.72332
,
δ
3
≃
1.90037
,and
(40)
|
f
(
t
,
x
)
|
≤
(
t
+
1
)
(
∥
x
∥
10
+
1
)
,
|
g
(
t
,
x
)
|
≤
1
8
.
Clearly,ν
1
(
t
)
=
t
+
1
,
ϑ
1
(
∥
x
∥
)
=
∥
x
∥
/
10
+
1
,
ν
2
(
t
)
=
1
/
8
,
ϑ
1
(
∥
x
∥
)
=
1, and the condition (
A
4
)implies that ω
>
1.00176. Thus all the assumptions of Theorem 8 are satisfied. Hence, the conclusion of Theorem 8 applies to the problem (39).Example 11.
Consider the followingq-fractionalq-antiperiodic boundary value problem:
(41)
D
c
q
α
(
D
q
γ
c
+
1
8
)
x
(
t
)
=
1
2
f
(
t
,
x
(
t
)
)
+
1
4
I
q
ρ
g
(
t
,
x
(
t
)
)
,
nnnnnnnnnnn
0
<
t
<
1
,
0
<
q
<
1
,
x
(
0
)
=
-
x
(
1
)
,
(
t
(
1
-
γ
)
D
q
x
(
t
)
)
|
t
=
0
=
-
D
q
x
(
1
)
,
where α
=
γ
=
ρ
=
q
=
A
=
1
/
2, λ
=
1
/
8
,
B
=
1
/
4
,
f
(
t
,
x
)
=
(
1
/
(
t
2
+
2
)
)
(
|
x
|
/
(
1
+
|
x
|
)
)
+
sin
2
t
,
g
(
t
,
x
)
=
(
1
/
4
)
tan
-
1
x
+
cos
⁡
2
t
+
t
3
+
5
.With the given data, it is found thatL
1
=
1
/
2
,
L
2
=
1
/
4as |
f
(
t
,
x
)
-
f
(
t
,
y
)
|
≤
(
1
/
2
)
|
x
-
y
|, |
g
(
t
,
x
)
-
g
(
t
,
y
)
|
≤
(
1
/
4
)
|
x
-
y
|
.Clearly L
=
max
⁡
{
L
1
,
L
2
}
=
1
/
2. Moreover, δ
1
≃
1.92678
,
δ
2
≃
1.72332
,and δ
3
≃
1.90037. Using the given values, it is found thatΛ
¯
≃
0.934655
<
1. Thus all the assumptions of Theorem 9 are satisfied. Hence, by the conclusion of Theorem 9, there exists a unique solution for the problem (41).



---

*Source: 207547-2014-04-30.xml*