Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 605169, 9 pages
http://dx.doi.org/10.1155/2013/605169
Research Article
Sequential Derivatives of Nonlinear 𝑞-Difference Equations with
Three-Point 𝑞-Integral Boundary Conditions
Nittaya Pongarm, Suphawat Asawasamrit, and Jessada Tariboon
Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok,
Bangkok 10800, Thailand
Correspondence should be addressed to Jessada Tariboon; jessadat@kmutnb.ac.th
Received 31 October 2012; Accepted 24 January 2013
Academic Editor: Hak-Keung Lam
Copyright © 2013 Nittaya Pongarm et al. 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.
This paper studies sufficient conditions for the existence of solutions to the problem of sequential derivatives of nonlinear qdifference equations with three-point q-integral boundary conditions. Our results are concerned with several quantum numbers of
derivatives and integrals. By using Banach’s contraction mapping, Krasnoselskii’s fixed-point theorem, and Leray-Schauder degree
theory, some new existence results are obtained. Two examples illustrate our results.
1. Introduction
The study of 𝑞-calculus or quantum calculus was initiated by
the pioneer works of Jackson [1], Carmichael [2], Mason [3],
Adams [4], Trjitzinsky [5], and so forth. Since then, in the last
few decades, this subject has evolved into a multidisciplinary
research area with many applications; for example, see [6–14].
For some recent works, we refer the reader to [15–21] and
references therein. However, the theory of boundary value
problems for nonlinear 𝑞-difference equations is still in the
beginning stages and it needs to be explored further.
In [22], Ahmad investigated the existence of solutions
for a nonlinear boundary value problem of third-order 𝑞difference equation:
𝐷𝑞3 𝑢 (𝑡) = 𝑓 (𝑡, 𝑢 (𝑡)) ,
𝑢 (0) = 0,
𝐷𝑞 𝑢 (0) = 0,
𝐷𝑞2 𝑢 (𝑡) = 𝑓 (𝑡, 𝑢 (𝑡)) ,
𝑢 (0) = 𝜂𝑢 (𝑇) ,
𝑡 ∈ [0, 𝑇] ,
𝐷𝑞 𝑢 (0) = 𝜂𝐷𝑞 𝑢 (𝑇) .
(1)
Using Leray-Schauder degree theory and standard fixedpoint theorems, some existence results were obtained. Moreover, he showed that if 𝑞 → 1, then his results corresponded
to the classical results. Ahmad et al. [23] studied a boundary
(2)
They proved the existence and uniqueness theorems of the
problem (2) using the Leray-Schauder nonlinear alternative
and some standard fixed-point theorems. For some very
recent results on nonlocal boundary value problems of
nonlinear 𝑞-difference equations and inclusions, see [24–26].
In this paper, we discuss the existence of solutions for the
following nonlinear 𝑞-difference equation with three-point
integral boundary condition:
𝐷𝑞 (𝐷𝑝 + 𝜆) 𝑥 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡)) ,
0 ≤ 𝑡 ≤ 1,
𝑢 (1) = 0.
value problem of a nonlinear second-order 𝑞-difference
equation with nonseparated boundary conditions
𝑥 (0) = 0,
𝜂
𝑡 ∈ [0, 𝑇] ,
𝛽 ∫ 𝑥 (𝑠) 𝑑𝑟 𝑠 = 𝑥 (𝑇) ,
(3)
0
where 0 < 𝑝, 𝑞, 𝑟 < 1, 𝑓 ∈ 𝐶([0, 𝑇] × R, R), 𝛽 ≠ 𝑇(1 + 𝑟)/𝜂2 ,
𝜂 ∈ (0, 𝑇) is a fixed point, and 𝜆 is a given constant.
The aim of this paper is to prove some existence and
uniqueness results for the boundary value problem (3). Our
results are based on Banach’s contraction mapping, Krasnoselskii’s fixed-point theorem, and Leray-Schauder degree
2
Journal of Applied Mathematics
theory. Since the problem (3) has different values of the
quantum numbers of the 𝑞-derivative and the 𝑞-integral, the
existence results of such problem are also new.
Lemma 1. Let 𝑇(1 + 𝑟) ≠ 𝛽𝜂2 , 0 < 𝑝, 𝑞, 𝑟 < 1, and let 𝜆
be a constant. Then for any ℎ ∈ 𝐶[0, 𝑇], the boundary value
problem
2. Preliminaries
𝐷𝑞 (𝐷𝑝 + 𝜆) 𝑥 (𝑡) = ℎ (𝑡) ,
Let us recall some basic concepts of quantum calculus [15].
For 0 < 𝑞 < 1, we define the 𝑞-derivative of a real-valued
function 𝑓 as
𝐷𝑞 𝑓 (𝑡) =
𝑓 (𝑡) − 𝑓 (𝑞𝑡)
,
(1 − 𝑞) 𝑡
𝐷𝑞 𝑓 (0) = lim 𝐷𝑞 𝑓 (𝑡) . (4)
𝜂
𝛽 ∫ 𝑥 (𝑠) 𝑑𝑟 𝑠 = 𝑥 (𝑇) ,
𝑥 (0) = 0,
0
0 < 𝜂 < 𝑇,
is equivalent to the integral equation
𝑛 ∈ N. (5)
𝑡
𝑠
𝑡
0
0
0
+
The 𝑞-integral of a function 𝑓 defined on the interval [0, 𝑇] is
given by
𝛽 (1 + 𝑟) 𝑡
𝑇 (1 + 𝑟) − 𝛽𝜂2
𝜂
V
𝑠
0
0
0
× ∫ ∫ (∫ ℎ (𝑢) 𝑑𝑞 𝑢 − 𝜆𝑥 (𝑠)) 𝑑𝑝 𝑠𝑑𝑟 V
∞
𝑎
𝑛=0
∫ 𝑓 (𝑠) 𝑑𝑞 𝑠 := ∑ (1 − 𝑞) 𝑞
(14)
𝑥 (𝑡) = ∫ ∫ ℎ (𝑢) 𝑑𝑞 𝑢𝑑𝑝 𝑠 − 𝜆 ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠
𝐷𝑞𝑛 𝑓 (𝑡) = 𝐷𝑞 𝐷𝑞𝑛−1 𝑓 (𝑡) ,
𝑡
(13)
𝑡→0
The higher-order 𝑞-derivatives are given by
𝐷𝑞0 𝑓 (𝑡) = 𝑓 (𝑡) ,
𝑡 ∈ [0, 𝑇] ,
𝑛
(6)
× [𝑡𝑓 (𝑡𝑞𝑛 ) − 𝑎𝑓 (𝑞𝑛 𝑎)] ,
𝑡 ∈ [0, 𝑇] ,
and for 𝑎 = 0, we denote
𝑡
∞
0
𝑛=0
𝐼𝑞 𝑓 (𝑡) = ∫ 𝑓 (𝑠) 𝑑𝑞 𝑠 = ∑ 𝑡 (1 − 𝑞) 𝑞𝑛 𝑓 (𝑡𝑞𝑛 ) ,
(7)
𝑏
𝑎
𝑎
0
0
𝑇
𝜆 (1 + 𝑟) 𝑡
𝑥 (𝑠) 𝑑𝑝 𝑠
∫
𝑇 (1 + 𝑟) − 𝛽𝜂2 0
−
𝑇 𝑠
(1 + 𝑟) 𝑡
∫
∫ ℎ (𝑢) 𝑑𝑞 𝑢𝑑𝑝 𝑠.
𝑇 (1 + 𝑟) − 𝛽𝜂2 0 0
Proof. For 𝑡 ∈ [0, 𝑇], 𝑞-integrating (13) from 0 to 𝑡, we obtain
provided the series converges. If 𝑎 ∈ [0, 𝑇] and 𝑓 is defined
on the interval [0, 𝑇], then
𝑏
+
t
(𝐷𝑝 + 𝜆) 𝑥 (𝑡) = ∫ ℎ (𝑠) 𝑑𝑞 𝑠 + 𝑐1 .
0
∫ 𝑓 (𝑠) 𝑑𝑞 𝑠 = ∫ 𝑓 (𝑠) 𝑑𝑞 𝑠 − ∫ 𝑓 (𝑠) 𝑑𝑞 𝑠.
(15)
(16)
(8)
Equation (16) can be written as
Similarly, we have
𝐼𝑞0 𝑓 (𝑡) = 𝑓 (𝑡) ,
𝐼𝑞𝑛 𝑓 (𝑡) = 𝐼𝑞 𝐼𝑞𝑛−1 𝑓 (𝑡) ,
𝑛 ∈ N.
𝑡
𝐷𝑝 𝑥 (𝑡) = ∫ ℎ (𝑠) 𝑑𝑞 𝑠 − 𝜆𝑥 (𝑡) + 𝑐1 .
(9)
0
(17)
Observe that
𝐷𝑞 𝐼𝑞 𝑓 (𝑡) = 𝑓 (𝑡) ,
(10)
For 𝑡 ∈ [0, 𝑇], 𝑝-integrating (17) from 0 to 𝑡, we have
and if 𝑓 is continuous at 𝑡 = 0, then
𝐼𝑞 𝐷𝑞 𝑓 (𝑡) = 𝑓 (𝑡) − 𝑓 (0) .
(11)
𝑡
𝑡
𝑡
∫ 𝑓 (𝑠) 𝐷𝑞 𝑔 (𝑠) 𝑑𝑞 𝑠 = [𝑓 (𝑠) 𝑔 (𝑠)]0 − ∫ 𝐷𝑞 𝑓 (𝑠) 𝑔 (𝑞𝑠) 𝑑𝑞 𝑠.
0
0
(12)
In the limit 𝑞 → 1, the 𝑞-calculus corresponds to the classical
calculus. The above results are also true for quantum numbers
𝑝, 𝑟 such that 0 < 𝑝 < 1 and 0 < 𝑟 < 1.
𝑠
0
0
(18)
𝑡
− 𝜆 ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠 + 𝑐1 𝑡 + 𝑐2 .
In 𝑞-calculus, the product rule and integration by parts
formula are
𝐷𝑞 (𝑔ℎ) (𝑡) = (𝐷𝑞 𝑔 (𝑡)) ℎ (𝑡) + 𝑔 (𝑞𝑡) 𝐷𝑞 ℎ (𝑡) ,
𝑡
𝑥 (𝑡) = ∫ ∫ ℎ (𝑢) 𝑑𝑞 𝑢𝑑𝑝 𝑠
0
From the first condition of (14), it follows that 𝑐2 = 0. For
𝑡 ∈ [0, 𝑇], 𝑟-integrating equation (18) from 0 to 𝑡, we get
𝑡
𝑡
V
𝑠
0
0
0
0
∫ 𝑥 (V) 𝑑𝑟 V = ∫ ∫ ∫ ℎ (𝑢) 𝑑𝑞 𝑢𝑑𝑝 𝑠𝑑𝑟 V
𝑡2
− 𝜆 ∫ ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠𝑑𝑟 V + 𝑐1
.
1+𝑟
0 0
𝑡
V
(19)
Journal of Applied Mathematics
3
problem. In view of Lemma 1, for 𝑡 ∈ [0, 𝑇], 𝑥 ∈ C, we define
the operator 𝐴 : C → C as
The second boundary condition (14) implies that
𝜂
𝛽 ∫ 𝑥 (V) 𝑑𝑟 V
(𝐴𝑥) (𝑡)
0
𝜂
V
𝑠
0
0
0
𝜂
V
0
0
− 𝛽𝜆 ∫ ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠𝑑𝑟 V + 𝑐1
𝛽𝜂2
1+𝑟
V
𝑠
𝑇
𝑠
𝑇
0
0
0
𝑡
0
0
0
(20)
Therefore,
𝜂 V
𝑠
𝛽 (1 + 𝑟)
𝑐1 =
(∫
ℎ (𝑢) 𝑑𝑞 𝑢 − 𝜆𝑥 (𝑠)) 𝑑𝑝 𝑠𝑑𝑟 V
∫
∫
𝑇 (1 + 𝑟) − 𝛽𝜂2 0 0 0
+
𝑇
𝜆 (1 + 𝑟)
∫ 𝑥 (𝑠) 𝑑𝑝 𝑠
2
𝑇 (1 + 𝑟) − 𝛽𝜂 0
−
𝑇 𝑠
1+𝑟
∫
∫ ℎ (𝑢) 𝑑𝑞 𝑢𝑑𝑝 𝑠.
𝑇 (1 + 𝑟) − 𝛽𝜂2 0 0
V
𝑠
0
0
0
+
𝑇
𝜆 (1 + 𝑟) 𝑡
∫ 𝑥 (𝑠) 𝑑𝑝 𝑠
2
𝑇 (1 + 𝑟) − 𝛽𝜂 0
−
𝑇 𝑠
(1 + 𝑟) 𝑡
∫
∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢𝑑𝑝 𝑠.
𝑇 (1 + 𝑟) − 𝛽𝜂2 0 0
(24)
Note that the problem (3) has solutions if and only if the
operator equation 𝐴𝑥 = 𝑥 has fixed points.
Our first result is based on Banach’s fixed-point theorem.
(𝐻1 ) |𝑓(𝑡, 𝑥)−𝑓(𝑡, 𝑦)| ≤ 𝐿|𝑥−𝑦|, for all 𝑡 ∈ [0, 𝑇], 𝑥, 𝑦 ∈ R;
Substituting the values of 𝑐1 and 𝑐2 in (18), we obtain (15). This
completes the proof.
For the forthcoming analysis, let C = 𝐶([0, 𝑇], R) denote
the Banach space of all continuous functions from [0, 𝑇] to
R endowed with the norm defined by ‖𝑥‖ = sup{|𝑥(𝑡)|, 𝑡 ∈
[0, 𝑇]}.
In the following, for the sake of convenience, we set
1
1+𝑝
(𝐻2 ) Λ := (Φ + 𝐿Ω) < 1,
where 𝐿 is a Lipschitz constant, Ω and Φ are defined by (22)
and (23), respectively.
Then, the boundary value problem (3) has a unique
solution.
Proof. Assume that sup𝑡∈[0,𝑇] |𝑓(𝑡, 0)| = 𝑀0 ; we choose a
constant
𝑅≥
𝑀0 Ω
.
1−Λ
(22)
󵄨󵄨 𝑡 𝑠
𝑡
󵄨
= sup 󵄨󵄨󵄨∫ ∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢𝑑𝑝 𝑠 − 𝜆 ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠
0
𝑡∈[0,𝑇] 󵄨󵄨 0 0
+
(23)
3. Main Results
Now, we are in the position to establish the main results. We
transform the boundary value problem (3) into a fixed-point
(25)
Now, we will show that 𝐴𝐵𝑅 ⊂ 𝐵𝑅 , where 𝐵𝑅 = {𝑥 ∈ C :
‖𝑥‖ ≤ 𝑅}. For any 𝑥 ∈ 𝐵𝑅 , we have
‖(𝐴𝑥)‖
𝑇3 (1 + 𝑟)
+ 󵄨󵄨
󵄨) ,
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 |𝜆| 𝑇𝜂2
|𝜆| 𝑇2 (1 + 𝑟)
Φ = |𝜆| 𝑇 + 󵄨󵄨 󵄨 󵄨
+
󵄨
󵄨
󵄨.
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 󵄨󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
𝜂
Theorem 2. Assume that 𝑓 : [0, 𝑇] × R → R is a jointly
continuous function satisfying the conditions
(21)
󵄨󵄨 󵄨󵄨 3
󵄨󵄨𝛽󵄨󵄨 𝑇𝜂 (1 + 𝑟)
× (𝑇2 + 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 (1 + 𝑟 + 𝑟2 )
𝛽 (1 + 𝑟) 𝑡
𝑇 (1 + 𝑟) − 𝛽𝜂2
× ∫ ∫ (∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢 − 𝜆𝑥 (𝑠)) 𝑑𝑝 𝑠𝑑𝑟 V
= ∫ ∫ ℎ (𝑢) 𝑑𝑞 𝑢𝑑𝑝 𝑠 − 𝜆 ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠 + 𝑐1 𝑇.
Ω=
𝑠
+
𝛽𝜂2
= 𝛽 ∫ ∫ (∫ ℎ (𝑢) 𝑑𝑞 𝑢 − 𝜆𝑥 (𝑠)) 𝑑𝑝 𝑠𝑑𝑟 V + 𝑐1
1+𝑟
0 0
0
𝜂
𝑡
= ∫ ∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢𝑑𝑝 𝑠 − 𝜆 ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠
= 𝛽 ∫ ∫ ∫ ℎ (𝑢) 𝑑𝑞 𝑢𝑑𝑝 𝑠𝑑𝑟 V
𝛽 (1 + 𝑟) 𝑡
𝑇 (1 + 𝑟) − 𝛽𝜂2
𝜂
V
𝑠
0
0
0
× ∫ ∫ (∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢 − 𝜆𝑥 (𝑠)) 𝑑𝑝 𝑠𝑑𝑟 V
+
𝑇
𝜆 (1 + 𝑟) 𝑡
𝑥 (𝑠) 𝑑𝑝 𝑠
∫
𝑇 (1 + 𝑟) − 𝛽𝜂2 0
−
󵄨󵄨
𝑇 𝑠
(1 + 𝑟) 𝑡
󵄨󵄨
𝑓
𝑥
𝑑
𝑢𝑑
𝑠
∫
∫
(𝑢,
(𝑢))
𝑞
𝑝 󵄨󵄨󵄨
𝑇 (1 + 𝑟) − 𝛽𝜂2 0 0
󵄨󵄨
4
Journal of Applied Mathematics
𝑡
𝑠
|𝜆| ‖𝑥‖ (1 + 𝑟) 𝑡 𝑇
+ 󵄨󵄨
󵄨 ∫ 𝑑𝑝 𝑠
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 0
󵄨
󵄨 󵄨
󵄨
≤ sup {∫ ∫ (󵄨󵄨󵄨𝑓 (𝑢, 𝑥 (𝑢)) − 𝑓 (𝑢, 0)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑓 (𝑢, 0)󵄨󵄨󵄨) 𝑑𝑞 𝑢𝑑𝑝 𝑠
0 0
𝑡∈[0,𝑇]
𝑡
𝑇 𝑠
(1 + 𝑟) 𝑡
(𝐿
+
𝑀
+ 󵄨󵄨
)
∫
∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠}
‖𝑥‖
󵄨
0
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
0 0
+ |𝜆| ∫ |𝑥 (𝑠)| 𝑑𝑝 𝑠
0
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 (1 + 𝑟) 𝑡
+ 󵄨󵄨 󵄨 󵄨
󵄨
𝑇
󵄨󵄨 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
𝜂
V
= sup {(𝐿 ‖𝑥‖ + 𝑀0 )
𝑡∈[0,𝑇]
𝑠
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 (1 + 𝑟) 𝑡
+ 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
󵄨
󵄨 󵄨
󵄨
× (∫ ∫ ∫ (󵄨󵄨󵄨𝑓 (𝑢, 𝑥 (𝑢)) − 𝑓 (𝑢, 0)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑓 (𝑢, 0)󵄨󵄨󵄨)
0 0 0
× 𝑑𝑞 𝑢𝑑𝑝 𝑠𝑑𝑟 V
𝜂
V
0
0
× ((𝐿 ‖𝑥‖+𝑀0 )
+ |𝜆| ∫ ∫ |𝑥 (𝑠)| 𝑑𝑝 𝑠𝑑𝑟 V)
(1 + 𝑟) 𝑡
+ 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
𝑇
𝑠
󵄨
󵄨
× ∫ ∫ (󵄨󵄨󵄨𝑓 (𝑢, 𝑥 (𝑢)) − 𝑓 (𝑢, 0)󵄨󵄨󵄨
0 0
𝑇2
(1 + 𝑟) 𝑡
+ 󵄨󵄨
(𝐿
+
𝑀
)
}
‖𝑥‖
󵄨
0
1+𝑝
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 |𝜆| 𝜂2 𝑡
|𝜆| 𝑇 (1 + 𝑟) 𝑡
+
)
= sup {‖𝑥‖ (|𝜆| 𝑡+ 󵄨󵄨 󵄨 󵄨
2 󵄨󵄨 󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨
𝑇
+
𝑟)−𝛽𝜂
(1
󵄨
󵄨
󵄨
󵄨󵄨
𝑡∈[0,𝑇]
󵄨
󵄨 󵄨
+
󵄨
󵄨
+ 󵄨󵄨󵄨𝑓 (𝑢, 0)󵄨󵄨󵄨) 𝑑𝑞 𝑢𝑑𝑝 𝑠}
𝑡
󵄨
󵄨
≤ sup {∫ ∫ (𝐿 |𝑥 (𝑢)| + 󵄨󵄨󵄨𝑓 (𝑢, 0)󵄨󵄨󵄨) 𝑑𝑞 𝑢𝑑𝑝 𝑠
0 0
𝑡∈[0,𝑇]
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 (1 + 𝑟) 𝑡
+ |𝜆| ∫ |𝑥 (𝑠)| 𝑑𝑝 𝑠 + 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
0
𝜂
V
𝑠
0
0
0
󵄨
󵄨
× (∫ ∫ ∫ (𝐿 |𝑥 (𝑢)| + 󵄨󵄨󵄨𝑓 (𝑢, 0)󵄨󵄨󵄨) 𝑑𝑞 𝑢𝑑𝑝 𝑠𝑑𝑟 V
𝜂
V
0
0
+ |𝜆| ∫ ∫ |𝑥 (𝑠)| 𝑑𝑝 𝑠𝑑𝑟 V)
𝑇
|𝜆| (1 + 𝑟) 𝑡
+ 󵄨󵄨
∫
|𝑥 (𝑠)| 𝑑𝑝 𝑠
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 0
(1 + 𝑟) 𝑡
+ 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
𝑇
𝑠
0
0
𝑠
0
0
≤ sup {(𝐿 ‖𝑥‖ + 𝑀0 ) ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠
𝑡∈[0,𝑇]
𝑇2 (1 + 𝑟) 𝑡
+ 󵄨󵄨
󵄨 )}
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 |𝜆| 𝑇𝜂2
|𝜆| 𝑇2 (1 + 𝑟)
≤ 𝑅 (|𝜆| 𝑇 + 󵄨󵄨 󵄨 󵄨
󵄨󵄨 + 󵄨󵄨
󵄨)
2
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂 󵄨󵄨 󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
+
(𝐿𝑅 + 𝑀0 )
1+𝑝
󵄨󵄨 󵄨󵄨 3
󵄨󵄨𝛽󵄨󵄨 𝑇𝜂 (1 + 𝑟)
× (𝑇2 + 󵄨󵄨
󵄨
𝑇
+
𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 (1 + 𝑟 + 𝑟2 )
(1
󵄨󵄨
𝑇3 (1 + 𝑟)
+ 󵄨󵄨
󵄨)
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
󵄨
󵄨
× ∫ ∫ (𝐿 |𝑥 (𝑢)| + 󵄨󵄨󵄨𝑓 (𝑢, 0)󵄨󵄨󵄨) 𝑑𝑞 𝑢𝑑𝑝 𝑠}
𝑡
(𝐿 ‖𝑥‖ + 𝑀0 )
1+𝑝
󵄨󵄨 󵄨󵄨
󵄨󵄨𝛽󵄨󵄨 (1 + 𝑟) 𝜂3 𝑡
× (𝑡2 + 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 (1 + 𝑟 + 𝑟2 )
𝑠
𝑡
𝜂2
(1−𝑟) 𝜂3
+|𝜆|
)
‖𝑥‖
1+𝑟
(1 + 𝑝) (1 − 𝑟3 )
|𝜆| ‖𝑥‖ (1 + 𝑟) 𝑡
+ 󵄨󵄨
󵄨𝑇
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
𝑇
|𝜆| (1 + 𝑟) 𝑡
+ 󵄨󵄨
󵄨 ∫ |𝑥 (𝑠)| 𝑑𝑝 𝑠
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 0
𝑡2
+ |𝜆| ‖𝑥‖ 𝑡
1+𝑝
󵄨󵄨 󵄨󵄨
𝑡
󵄨𝛽󵄨 (1 + 𝑟) 𝑡
+ |𝜆| ‖𝑥‖ ∫ 𝑑𝑝 𝑠 + 󵄨󵄨 󵄨 󵄨
󵄨
𝑇
󵄨󵄨 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
0
𝜂
V
𝑠
0
0
0
× ((𝐿 ‖𝑥‖ + 𝑀0 ) ∫ ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠𝑑𝑟 V
𝜂
V
0
0
+ |𝜆| ‖𝑥‖ ∫ ∫ 𝑑𝑝 𝑠𝑑𝑟 V)
= 𝑅Φ + (𝐿𝑅 + 𝑀) Ω ≤ 𝑅.
(26)
Next, we will show that 𝐴 is a contraction. For any 𝑥, 𝑦 ∈
C and for each 𝑡 ∈ [0, 𝑇], we have
󵄩󵄩
󵄩
󵄩󵄩(𝐴𝑥) − (𝐴𝑦)󵄩󵄩󵄩
󵄨
󵄨
= sup 󵄨󵄨󵄨(𝐴𝑥) (𝑡) − (𝐴𝑦) (𝑡)󵄨󵄨󵄨
𝑡∈[0,𝑇]
󵄨󵄨 𝑡 𝑠
󵄨
= sup 󵄨󵄨󵄨∫ ∫ (𝑓 (𝑢, 𝑥 (𝑢)) − 𝑓 (𝑢, 𝑦 (𝑢))) 𝑑𝑞 𝑢𝑑𝑝 𝑠
𝑡∈[0,𝑇] 󵄨󵄨 0 0
Journal of Applied Mathematics
5
𝑡
− 𝜆 ∫ (𝑥 (𝑠) − 𝑦 (𝑠)) 𝑑𝑝 𝑠
0
+
𝛽 (1 + 𝑟) 𝑡
𝑇 (1 + 𝑟) − 𝛽𝜂2
𝜂
V
𝑠
0
0
0
× ∫ ∫ (∫ (𝑓 (𝑢, 𝑥 (𝑢)) − 𝑓 (𝑢, 𝑦 (𝑢))) 𝑑𝑞 𝑢
−𝜆 (𝑥 (𝑠) − 𝑦 (𝑠))) 𝑑𝑝 𝑠𝑑𝑟 V
+
𝑇
𝜆 (1 + 𝑟) 𝑡
∫ (𝑥 (𝑠) − 𝑦 (𝑠)) 𝑑𝑞 𝑠
2
𝑇 (1 + 𝑟) − 𝛽𝜂 0
(1 + 𝑟) 𝑡
− 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
󵄨󵄨
𝑇 𝑠
󵄨
× ∫ ∫ (𝑓 (𝑢, 𝑥 (𝑢)) − 𝑓 (𝑢, 𝑦 (𝑢))) 𝑑𝑞 𝑢𝑑𝑝 𝑠󵄨󵄨󵄨󵄨
󵄨󵄨
0 0
𝑡
𝑠
󵄩
󵄩
≤ sup {𝐿 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠
0 0
𝑡∈[0,𝑇]
𝑡
2
(1 + 𝑟) 𝑡
󵄩󵄩 𝑇
󵄩󵄩
𝑥
−
𝑦
𝐿
+ 󵄨󵄨
󵄩
󵄩
󵄨
󵄩 1 + 𝑝}
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 󵄩
󵄨󵄨󵄨𝛽󵄨󵄨󵄨 |𝜆| 𝑇𝜂2
󵄩
󵄩
≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 (|𝜆| 𝑇 + 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
|𝜆| 𝑇2 (1 + 𝑟)
+ 󵄨󵄨
󵄨)
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
󵄩
󵄩
󵄨󵄨 󵄨󵄨 3
𝐿 󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄨󵄨𝛽󵄨󵄨 𝜂 𝑇 (1 + 𝑟)
+ 󵄩
(𝑇2 + 󵄨󵄨
󵄨
1+𝑝
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 (1 + 𝑟 + 𝑟2 )
𝑇3 (1 + 𝑟)
+ 󵄨󵄨
󵄨)
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
󵄩
󵄩
= (Φ + 𝐿Ω) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩
󵄩
≤ Λ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
(27)
Since Λ < 1, 𝐴 is a contraction. Thus, the conclusion of the
theorem follows by Banach’s contraction mapping principle.
This completes the proof.
󵄩
󵄩
+ |𝜆| 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ∫ 𝑑𝑝 𝑠
Our second result is based on the following Krasnoselskii’s fixed-point theorem [27].
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 (1 + 𝑟) 𝑡
+ 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
Theorem 3. Let 𝐾 be a bounded closed convex and nonempty
subset of a Banach space 𝑋. Let 𝐴, 𝐵 be operators such that
0
𝜂
V
0
0
𝑠
󵄩
󵄩
× (𝐿 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ∫ ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠𝑑𝑟 V
0
𝜂
V
0
0
󵄩
󵄩
+ |𝜆| 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ∫ ∫ 𝑑𝑝 𝑠𝑑𝑟 V)
+
|𝜆| (1 + 𝑟) 𝑡 󵄩󵄩
󵄩
󵄩𝑥 − 𝑦󵄩󵄩󵄩
𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄩
𝑇
(1 + 𝑟) 𝑡
× ∫ 𝑑𝑞 𝑠 + 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
0
𝑇
𝑠
0
0
󵄩
󵄩
×𝐿 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠}
2
󵄩 𝑡
󵄩
󵄩
󵄩
= sup {𝐿 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
+ |𝜆| 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 𝑡
1
+
𝑝
𝑡∈[0,𝑇]
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 (1 + 𝑟) 𝑡
+ 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
(1 − 𝑟) 𝜂3
󵄩
󵄩
× (𝐿 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
(1 + 𝑝) (1 − 𝑟3 )
2
󵄩 𝜂
󵄩
+ |𝜆| 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
)
1+𝑟
(1 + 𝑟) 𝑡
󵄩
󵄩
+ |𝜆| 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 󵄨󵄨
󵄨𝑇
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
(i) 𝐴𝑥 + 𝐵𝑦 ∈ 𝐾 whenever 𝑥, 𝑦 ∈ 𝐾;
(ii) 𝐴 is compact and continuous;
(iii) 𝐵 is a contraction mapping.
Then, there exists 𝑧 ∈ 𝐾 such that 𝑧 = 𝐴𝑧 + 𝐵𝑧.
Theorem 4. Assume that (𝐻1 ) and (𝐻2 ) hold with
(𝐻3 ) |𝑓(𝑡, 𝑥)| ≤ 𝜇(𝑡), for all (𝑡, 𝑥) ∈ [0, 𝑇] × R, with 𝜇 ∈
𝐿1 ([0, 𝑇], R+ ).
If
󵄨󵄨 󵄨󵄨
󵄨󵄨𝛽󵄨󵄨 |𝜆| 𝑇𝜂2
|𝜆| 𝑇2 (1 + 𝑟)
󵄨󵄨
󵄨󵄨 + 󵄨󵄨
󵄨
2
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂 󵄨󵄨 󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
+
󵄨󵄨 󵄨󵄨 3
󵄨󵄨𝛽󵄨󵄨 𝜂 𝑇 (1 + 𝑟)
1
( 󵄨󵄨
1 + 𝑝 󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨󵄨 (1 + 𝑟 + 𝑟2 )
(28)
𝑇3 (1 + 𝑟)
+ 󵄨󵄨
󵄨 ) < 1,
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
then the boundary value problem (3) has at least one solution
on [0, 𝑇].
Proof. Setting max𝑡∈[0,𝑇] |𝜇(𝑡)| = ||𝜇|| and choosing a constant
󵄩󵄩 󵄩󵄩
󵄩𝜇󵄩 Ω
𝑅≥ 󵄩 󵄩 ,
1−Φ
(29)
where Ω and Φ are given by (22) and (23), respectively, we
consider that 𝐵𝑅 = {𝑥 ∈ C : ||𝑥|| ≤ 𝑅}.
6
Journal of Applied Mathematics
In view of Lemma 1, we define the operators 𝐹1 and 𝐹2 on
the set 𝐵𝑅 as
𝑡
𝑠
𝑡
0
0
0
(𝐹1 𝑥) (𝑡) = ∫ ∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢𝑑𝑝 𝑠 − 𝜆 ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠,
(𝐹2 𝑥) (𝑡) =
𝛽 (1 + 𝑟) 𝑡
𝑇 (1 + 𝑟) − 𝛽𝜂2
𝜂
V
𝑠
0
0
0
× ∫ ∫ (∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢 − 𝜆𝑥 (𝑠)) 𝑑𝑝 𝑠𝑑𝑟 V
+
−
𝑇
𝜆 (1 + 𝑟) 𝑡
𝑥 (𝑠) 𝑑𝑝 𝑠
∫
𝑇 (1 + 𝑟) − 𝛽𝜂2 0
𝑇 𝑠
(1 + 𝑟) 𝑡
∫
∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢𝑑𝑝 𝑠,
𝑇 (1 + 𝑟) − 𝛽𝜂2 0 0
(30)
󵄩󵄩
󵄩
󵄩󵄩(𝐹1 𝑥) + (𝐹2 𝑦)󵄩󵄩󵄩
𝑠
𝑡
0
0
0
󵄩 󵄩
≤ 󵄩󵄩󵄩𝜇󵄩󵄩󵄩 ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠 + |𝜆| ‖𝑥‖ ∫ 𝑑𝑝 𝑠
V
𝑠
𝜂
𝑇
|𝜆| (1 + 𝑟) 𝑡 󵄩󵄩 󵄩󵄩
+ 󵄨󵄨
󵄨 󵄩𝑦󵄩 ∫ 𝑑𝑝 𝑠
𝑇
󵄨󵄨 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 󵄩 󵄩 0
0
0
We define a ball 𝐵𝑅 ⊂ C, with a constant radius 𝑅 > 0, given
by
𝑡∈[0,𝑇]
󵄨󵄨
𝑡2
󵄨
+𝜆 ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠󵄨󵄨󵄨
󵄨󵄨
0
(34)
Then, it is sufficient to show that 𝐴 : 𝐵𝑅 → C satisfies
𝑥 ≠ 𝜃𝐴𝑥,
(31)
󵄨󵄨
󵄨
󵄨󵄨𝐹1 𝑥 (𝑡1 ) − 𝐹2 𝑥 (𝑡2 )󵄨󵄨󵄨
󵄨󵄨 𝑡1 𝑠
𝑡1
󵄨󵄨
≤
sup
󵄨󵄨∫ ∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢𝑑𝑝 𝑠 − 𝜆 ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠
0
0
(𝑡,𝑥)∈[0,𝑇]×𝐵𝑅 󵄨󵄨 0
− ∫ ∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢𝑑𝑝 𝑠
(33)
𝐵𝑅 = {𝑥 ∈ C : max |𝑥 (𝑡)| < 𝑅} .
Therefore, (𝐹1 𝑥) + (𝐹2 y) ∈ 𝐵𝑅 . The condition (28) implies
that 𝐹2 is a contraction mapping. Next, we will show that
𝐹1 is compact and continuous. Continuity of 𝑓 coupled
with the assumption (𝐻3 ) implies that the operator 𝐹1
is continuous and uniformly bounded on 𝐵𝑅 . We define
sup(𝑡,𝑥)∈[0,𝑇]×𝐵𝑅 |𝑓(𝑡, 𝑥)| = 𝑓max < ∞. For 𝑡1 , 𝑡2 ∈ [0, 𝑇] with
𝑡2 < 𝑡1 and 𝑥 ∈ 𝐵𝑅 , we have
𝑠
𝑥 = 𝐴𝑥.
𝑠
(1 + 𝑟) 𝑡
󵄩 󵄩
+ 󵄨󵄨
󵄨 󵄩󵄩𝜇󵄩󵄩 ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠,
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 󵄩 󵄩 0 0
󵄩 󵄩
≤ 𝑅Φ + 󵄩󵄩󵄩𝜇󵄩󵄩󵄩 Ω ≤ 𝑅.
𝑡2
As the third result, we prove the existence of solutions of
(3) by using Leray-Schauder degree theory.
V
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩𝜇󵄩󵄩󵄩 ∫ ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠𝑑𝑟 V + |𝜆| 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ∫ ∫ 𝑑𝑝 𝑠𝑑𝑟 V)
0 0 0
0 0
𝑇
Actually, as 𝑡1 − 𝑡2 → 0, the right-hand side of the above
inequality tends to zero. So 𝐹1 is relatively compact on 𝐵𝑅 .
Hence, by the Arzelá-Ascoli theorem, 𝐹1 is compact on 𝐵𝑅 .
Therefore, all the assumptions of Theorem 5 are satisfied and
the conclusion of Theorem 5 implies that the three-point
integral boundary value problem (3) has at least one solution
on [0, 𝑇]. This completes the proof.
Proof. Let us define an operator 𝐴 : C → C as in (24). We
will prove that there exists at least one solution 𝑥 ∈ C of the
fixed-point equation
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 (1 + 𝑟) 𝑡
+ 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
𝜂
sup
Theorem 5. Let 𝑓 : [0, 𝑇] × R → R. Assume that there exist
constants 0 ≤ 𝜅 < (1−Φ)Ω−1 , where Ω and Φ are given by (22)
and (23), respectively, and 𝑀 > 0 such that |𝑓(𝑡, 𝑥)| ≤ 𝜅|𝑥|+𝑀
for all 𝑡 ∈ [0, 𝑇], 𝑥 ∈ C. Then, the boundary value problem (3)
has at least one solution.
for 𝑥, 𝑦 ∈ 𝐵𝑅 . By computing directly, we have
𝑡
󵄨󵄨
󵄨󵄨 𝑡1 𝑠
𝑡1
󵄨
󵄨󵄨
󵄨󵄨∫ ∫ 𝑓 (𝑢, 𝑥 (𝑢)) 𝑑𝑞 𝑢𝑑𝑝 𝑠 − 𝜆 ∫ 𝑥 (𝑠) 𝑑𝑝 𝑠󵄨󵄨󵄨
󵄨󵄨
󵄨
𝑡2
(𝑡,𝑥)∈[0,𝑇]×𝐵𝑅 󵄨󵄨 𝑡2 0
󵄨
󵄨󵄨 2 2 󵄨󵄨
󵄨󵄨𝑡1 − 𝑡2 󵄨󵄨
󵄨 + |𝜆| (𝑡 − 𝑡 ) 𝑅.
≤ 𝑓max 󵄨
1
2
1+𝑝
(32)
=
∀𝑥 ∈ 𝜕𝐵𝑅 , ∀𝜃 ∈ [0, 1] .
(35)
Now, we set
𝐻 (𝜃, 𝑥) = 𝜃𝐴𝑥,
𝑥 ∈ C, 𝜃 ∈ [0, 1] .
(36)
Then, by the Arzelá-Ascoli theorem, we get that ℎ𝜃 (𝑥) =
𝑥 − 𝐻(𝜃, 𝑥) = 𝑥 − 𝜃𝐴𝑥 is completely continuous. If (35) holds,
then the following Leray-Schauder degrees are well defined.
From the homotopy invariance of topological degree, it
follows that
deg (ℎ𝜃 , 𝐵𝑅 , 0) = deg (𝐼 − 𝜃𝐴, 𝐵𝑅 , 0)
= deg (ℎ, 𝐵𝑅 , 0)
(37)
= deg (ℎ0 , 𝐵𝑅 , 0)
= deg (𝐼, 𝐵𝑅 , 0) = 1 ≠ 0,
0 ∈ 𝐵𝑅 ,
where 𝐼 denotes the unit operator. By the nonzero property
of Leray-Schauder degree, ℎ1 (𝑥) = 𝑥 − 𝐴𝑥 = 0 for at least one
Journal of Applied Mathematics
7
𝑥 ∈ 𝐵𝑅 . Let us assume that 𝑥 = 𝜃𝐴𝑥 for some 𝜃 ∈ [0, 1]. Then,
for all 𝑡 ∈ [0, 𝑇], we obtain
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 |𝜆| 𝑇𝜂2
|𝜆| 𝑇2 (1 + 𝑟)
= |𝑥| (|𝜆| 𝑇 + 󵄨󵄨 󵄨 󵄨
󵄨󵄨 + 󵄨󵄨
󵄨)
2
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂 󵄨󵄨 󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
|𝑥 (𝑡)| = |𝜃 (𝐴𝑥) (𝑡)|
𝑡
+
𝑠
󵄨
󵄨
≤ ∫ ∫ 󵄨󵄨󵄨𝑓 (𝑢, 𝑥 (𝑢))󵄨󵄨󵄨 𝑑𝑞 𝑢𝑑𝑝 𝑠
0 0
󵄨󵄨 󵄨󵄨 3
(𝜅 |𝑥| + 𝑀)
󵄨󵄨𝛽󵄨󵄨 𝑇𝜂 (1 + 𝑟)
(𝑇2 + 󵄨󵄨
󵄨
1+𝑝
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 (1 + 𝑟 + 𝑟2 )
𝑇3 (1 + 𝑟)
+ 󵄨󵄨
󵄨)
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
𝑡
+ |𝜆| ∫ |𝑥 (𝑠)| 𝑑𝑝 𝑠
0
= |𝑥| Φ + (𝜅 |𝑥| + 𝑀) Ω.
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 (1 + 𝑟) 𝑡
+ 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
𝜂
V
(38)
𝑠
󵄨
󵄨
× ∫ ∫ (∫ 󵄨󵄨󵄨𝑓 (𝑢, 𝑥 (𝑢))󵄨󵄨󵄨 𝑑𝑞 𝑢 + |𝜆𝑥 (𝑠)|) 𝑑𝑝 𝑠𝑑𝑟 V
0 0
0
Taking norm sup𝑡∈[0,𝑇] |𝑥(𝑡)| = ‖𝑥‖ and solving for ‖𝑥‖, this
yields
|𝜆| (1 + 𝑟) 𝑡
+ 󵄨󵄨
󵄨
𝑇
󵄨󵄨 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
‖𝑥‖ ≤
(39)
Let 𝑅 = 𝑀Ω/(1−(Φ+𝜅Ω))+1, then (35) holds. This completes
the proof.
𝑇
(1 + 𝑟) 𝑡
× ∫ |𝑥 (𝑠)| 𝑑𝑝 𝑠 + 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
0
𝑇
𝑀Ω
.
1 − (Φ + 𝜅Ω)
𝑠
󵄨
󵄨
× ∫ ∫ 󵄨󵄨󵄨𝑓 (𝑢, 𝑥 (𝑢))󵄨󵄨󵄨 𝑑𝑞 𝑢𝑑𝑝 𝑠
0 0
4. Examples
𝑡
𝑠
𝑡
0
0
0
≤ (𝜅 |𝑥| + 𝑀) ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠 + |𝜆| |𝑥| ∫ 𝑑𝑝 𝑠
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 (1 + 𝑟) 𝑡
+ 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
Example 6. Consider the following nonlinear 𝑞-difference
equation with boundary value problem:
𝜂
V
𝑠
0
0
0
× ((𝜅 |𝑥| + 𝑀) ∫ ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠𝑑𝑟 V
𝜂
V
0
0
+ |𝜆| |𝑥| ∫ ∫ 𝑑𝑝 𝑠𝑑𝑟 V)
|𝜆| |𝑥| (1 + 𝑟) 𝑡
+ 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
𝑇
(1 + 𝑟) 𝑡
× ∫ 𝑑𝑝 𝑠 + 󵄨󵄨
󵄨 (𝜅 |𝑥| + 𝑀)
𝑇
+ 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
(1
󵄨
0
󵄨
𝑇
𝑠
0
0
× ∫ ∫ 𝑑𝑞 𝑢𝑑𝑝 𝑠
≤ (𝜅 |𝑥| + 𝑀)
2
1
|𝑥|
⋅
,
𝐷1/2 (𝐷1/3 − ) 𝑥 (𝑡) =
7
(𝑡 + 2)2 |𝑥| + 1
𝑥 (0) = 0,
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 (1 + 𝑟) 𝑇
+ 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
𝜂2
(1 − 𝑟) 𝜂3
+
)
|𝜆|
|𝑥|
1+𝑟
(1 + 𝑝) (1 − 𝑟3 )
|𝜆| |𝑥| (1 + 𝑟) 𝑇2
+ 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
(1 + 𝑟) 𝑇3
+ (𝜅 |𝑥| + 𝑀) 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 (1 + 𝑝)
𝑥 (1) +
𝑡 ∈ [0, 1] ,
2 3/4
∫ 𝑥 (𝑠) 𝑑1/4 𝑠 = 0.
3 0
(40)
Set 𝑞 = 1/2, 𝑝 = 1/3, 𝑟 = 1/4, 𝑇 = 1, 𝜆 = −2/7, 𝜂 = 3/4,
𝛽 = −2/3, and 𝑓(𝑡, 𝑥) = (1/(𝑡 + 2)2 )(‖𝑥‖/(1 + ‖𝑥‖)). Since
|𝑓(𝑡, 𝑥) − 𝑓(𝑡, 𝑦)| ≤ (1/4)‖𝑥 − 𝑦‖, then, (𝐻1 ) and (𝐻2 ) are
satisfied with 𝑇(1 + 𝑟) − 𝛽𝜂2 = 13/8 ≠ 0,
Ω=
𝑇2
+ |𝜆| |𝑥| 𝑇
1+𝑝
× ((𝜅 |𝑥| + 𝑀)
In this section, we give two examples to illustrate our results.
󵄨󵄨 󵄨󵄨 3
󵄨󵄨𝛽󵄨󵄨 𝑇𝜂 (1 + 𝑟)
1
(𝑇2 + 󵄨󵄨
󵄨
1+𝑝
𝑇
+
𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 (1 + 𝑟 + 𝑟2 )
(1
󵄨󵄨
132
𝑇3 (1 + 𝑟)
+ 󵄨󵄨
󵄨 ) = 91 ,
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 |𝜆| 𝑇𝜂2
Φ = |𝜆| 𝑇 + 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
(41)
4
|𝜆| 𝑇2 (1 + 𝑟)
+ 󵄨󵄨
󵄨= ,
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 7
𝐿 = 1/4. Hence, Λ =: Φ + 𝐿Ω = 85/91 < 1. Therefore, by
Theorem 2, the boundary value problem (40) has a unique
solution on [0, 1].
8
Journal of Applied Mathematics
Example 7. Consider the following nonlinear 𝑞-difference
equation with boundary value problem:
1
sin (5𝜋𝑥)
|𝑥|
,
𝐷2/3 (𝐷4/5 + ) 𝑥 (𝑡) =
+
9
25𝜋
|𝑥| + 1
𝑥 (0) = 0,
3
𝑡 ∈ [0, ] ,
2
1
3
1
𝑥 ( ) = ∫ 𝑥 (𝑠) 𝑑1/2 𝑠.
2
4 0
(42)
Set 𝑞 = 2/3, 𝑝 = 4/5, 𝑟 = 1/2, 𝑇 = 3/2, 𝜆 = 1/9, 𝜂 = 1,
and 𝛽 = 1/4. Here, |𝑓(𝑡, 𝑥)| = | sin(5𝜋𝑥)/25𝜋+|𝑥|/(1+|𝑥|)| ≤
(|𝑥|/5) + 1. So, 𝑀 = 1, 𝑇(1 + 𝑟) − 𝛽𝜂2 = 2 ≠ 0, and
Ω=
1
1+𝑝
󵄨󵄨 󵄨󵄨
󵄨𝛽󵄨 |𝜆| 𝑇𝜂2
Φ = |𝜆| 𝑇 + 󵄨󵄨 󵄨 󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨
[9] R. Floreanini and L. Vinet, “Automorphisms of the q-oscillator
algebra and basic orthogonal polynomials,” Physics Letters A,
vol. 180, no. 6, pp. 393–401, 1993.
[10] R. Floreanini and L. Vinet, “Symmetries of the q-difference heat
equation,” Letters in Mathematical Physics, vol. 32, no. 1, pp. 37–
44, 1994.
[11] R. Floreanini and L. Vinet, “q-gamma and q-beta functions in
quantum algebra representation theory,” Journal of Computational and Applied Mathematics, vol. 68, no. 1-2, pp. 57–68, 1996.
[13] G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 35
of Encyclopedia of Mathematics and Its Applications, Cambridge
University Press, Cambridge, UK, 1990.
(43)
3
|𝜆| 𝑇2 (1 + 𝑟)
+ 󵄨󵄨
󵄨= ,
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 8
𝜅=
[8] R. J. Finkelstein, “q-deformation of the Lorentz group,” Journal
of Mathematical Physics, vol. 37, no. 2, pp. 953–964, 1996.
[12] P. G. O. Freund and A. V. Zabrodin, “The spectral problem
for the q-Knizhnik-Zamolodchikov equation and continuous qJacobi polynomials,” Communications in Mathematical Physics,
vol. 173, no. 1, pp. 17–42, 1995.
󵄨󵄨 󵄨󵄨 3
󵄨󵄨𝛽󵄨󵄨 𝑇𝜂 (1 + 𝑟)
× (𝑇2 + 󵄨󵄨
󵄨
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂2 󵄨󵄨󵄨 (1 + 𝑟 + 𝑟2 )
615
𝑇3 (1 + 𝑟)
+ 󵄨󵄨
󵄨󵄨 ) = 224 ,
2
󵄨󵄨𝑇 (1 + 𝑟) − 𝛽𝜂 󵄨󵄨
[7] R. J. Finkelstein, “q-field theory,” Letters in Mathematical
Physics, vol. 34, no. 2, pp. 169–176, 1995.
28
1
< (1 − Φ) Ω−1 =
.
5
123
Hence, by Theorem 5, the boundary value problem (42) has
at least one solution on [0, 3/2].
Acknowledgments
The authors would like to thank the reviewers for their valuable comments and suggestions on the paper. This research
is supported by King Mongkut’s University of Technology
North Bangkok, Thailand.
References
[1] F. H. Jackson, “On q-difference equations,” American Journal of
Mathematics, vol. 32, no. 4, pp. 305–314, 1910.
[2] R. D. Carmichael, “The general theory of linear q-difference
equations,” American Journal of Mathematics, vol. 34, no. 2, pp.
147–168, 1912.
[3] T. E. Mason, “On properties of the solutions of linear qdifference equations with entire function coefficients,” American Journal of Mathematics, vol. 37, no. 4, pp. 439–444, 1915.
[4] C. R. Adams, “On the linear ordinary q-difference equation,”
American Mathematical Series II, vol. 30, pp. 195–205, 1929.
[5] W. J. Trjitzinsky, “Analytic theory of linear q-differece equations,” Acta Mathematica, vol. 61, no. 1, pp. 1–38, 1933.
[6] T. Ernst, “A new notation for q-calculus and a new q-Taylor formula,” U.U.D.M. Report, Department of Mathematics, Uppsala
University, 1999.
[14] G.-N. Han and J. Zeng, “On a q-sequence that generalizes the median Genocchi numbers,” Annales des Sciences
Mathématiques du Québec, vol. 23, no. 1, pp. 63–72, 1999.
[15] V. Kac and P. Cheung, Quantum Calculus, Universitext,
Springer, New York, NY, USA, 2002.
[16] G. Bangerezako, “Variational q-calculus,” Journal of Mathematical Analysis and Applications, vol. 289, no. 2, pp. 650–665, 2004.
[17] A. Dobrogowska and A. Odzijewicz, “Second order q-difference
equations solvable by factorization method,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 319–346,
2006.
[18] G. Gasper and M. Rahman, “Some systems of multivariable
orthogonal q-Racah polynomials,” The Ramanujan Journal, vol.
13, no. 1–3, pp. 389–405, 2007.
[19] M. E. H. Ismail and P. Simeonov, “q-difference operators for
orthogonal polynomials,” Journal of Computational and Applied
Mathematics, vol. 233, no. 3, pp. 749–761, 2009.
[20] M. Bohner and G. Sh. Guseinov, “The h-Laplace and q-Laplace
transforms,” Journal of Mathematical Analysis and Applications,
vol. 365, no. 1, pp. 75–92, 2010.
[21] M. El-Shahed and H. A. Hassan, “Positive solutions of qdifference equation,” Proceedings of the American Mathematical
Society, vol. 138, no. 5, pp. 1733–1738, 2010.
[22] B. Ahmad, “Boundary-value problems for nonlinear thirdorder q-difference equations,” Electronic Journal of Differential
Equations, vol. 94, pp. 1–7, 2011.
[23] B. Ahmad, A. Alsaedi, and S. K. Ntouyas, “A study of
second-order q-difference equations with boundary conditions,” Advances in Difference Equations, vol. 2012, article 35,
2012.
[24] B. Ahmad, S. K. Ntouyas, and I. K. Purnaras, “Existence results
for nonlinear q-difference equations with nonlocal boundary
conditions,” Communications on Applied Nonlinear Analysis,
vol. 19, no. 3, pp. 59–72, 2012.
[25] B. Ahmad and J. J. Nieto, “On nonlocal boundary value
problems of nonlinear q-difference equations,” Advances in
Difference Equations, vol. 2012, article 81, 2012.
Journal of Applied Mathematics
[26] B. Ahmad and S. K. Ntouyas, “Boundary value problems for qdifference inclusions,” Abstract and Applied Analysis, vol. 2011,
Article ID 292860, 15 pages, 2011.
[27] M. A. Krasnoselskii, “Two remarks on the method of successive
approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, no. 1,
pp. 123–127, 1955.
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