Bohner We establish a general form of sum-difference inequality in two variables, which includes both more than two distinct nonlinear sums without an assumption of monotonicity and a non
Trang 1Volume 2009, Article ID 486895, 17 pages
doi:10.1155/2009/486895
Research Article
An Extension to Nonlinear Sum-Difference
Inequality and Applications
Wu-Sheng Wang1 and Xiaoliang Zhou2
1 Department of Mathematics, Hechi University, Yizhou, Guangxi 546300, China
2 Department of Mathematics, Guangdong Ocean University, Zhanjiang 524088, China
Received 31 March 2009; Revised 31 March 2009; Accepted 17 May 2009
Recommended by Martin J Bohner
We establish a general form of sum-difference inequality in two variables, which includes both more than two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums We employ a technique of monotonization and use a property of stronger monotonicity to give an estimate for the unknown function Our result enables us to solve those discrete inequalities considered in the work of W.-S Cheung2006 Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence
the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
Gronwall-Bellman inequality 1, 2 is a fundamental tool in the study of existence, uniqueness, boundedness, stability, invariant manifolds and other qualitative properties
of solutions of differential equations and integral equation There are a lot of papers investigating them such as 3 15 Along with the development of the theory of integral inequalities and the theory of difference equations, more attentions are paid to some discrete versions of Bellman-Gronwall type inequalitiese.g., 16–18 Starting from the basic form
u n ≤ an n−1
s0
discussed in19, an interesting direction is to consider the inequality
u2n ≤ P2u20 2n−1
s0
Trang 2
a discrete version of Dafermos’ inequality20, where α, P, Q are nonnegative constants and
u, g are nonnegative functions defined on {1, 2, , T} and {1, 2, , T − 1}, respectively Pang
and Agarwal21 proved for 1.2 that un ≤ 1 α n Pu0 n−1
s0Qg s for all 0 ≤ n ≤ T.
Another form of sum-difference inequality
u2n ≤ c2 2n−1
s0
was estimated by Pachpatte22 as un ≤ Ω−1Ωcn−1
s0f2sn−1
s0f1s, where Ωu :
u
u0 ds/w s Recently, Pachpatte 23,24 discussed the inequalities of two variables
u m, n ≤ c m−1
s0
n−1
t0
u s, ta s, t log us, t bs, tg log us, t ,
u m, n ≤ c m−1
s0
n−1
t0
f1s, tgus, t m−1
s0
n−1
t0
s−1
σ0
t−1
τ0
κ s, t, σ, τguσ, τ
m−1
s0
n−1
t0
⎛
⎝s−1
σ0
t−1
τ0
⎛
⎝σ−1
ξ0
τ−1
η0
h
s, t, σ, τ, ξ, η g
u
ξ, η
⎞
⎠
⎞
⎠,
1.4
where g is nondecreasing In25 another form of inequality of two variables
u2m, n ≤ c2 m−1
s m0
n−1
t n0
a s, tus, t m−1
s m0
n−1
t n0
b s, tus, twus, t 1.5
was discussed Later, this result was generalized in26 to the inequality
u p m, n ≤ c m−1
s m0
n−1
t n0
d s, tu q s, t m−1
s m0
n−1
t n0
e s, tu q s, twus, t, 1.6
where c, p, and q are all constants, c ≥ 0, p > q > 0, and d, e are both nonnegative real-valued
functions defined on a lattice inZ2
, and w is a continuous nondecreasing function satisfying
w u > 0 for all u > 0.
In this paper we establish a more general form of sum-difference inequality with
positive integers m, n,
ψ um, n ≤ am, n k
i1
m−1
s m0
n−1
t n0
f i m, n, s, tϕ i us, t, 1.7
where k≥ 2 In 1.7 we replace the constant c, the functions u p , ds, t, es, t, u q and u q w u
in 1.6 with a function am, n, more general functions ψu, f1m, n, s, t, f2m, n, s, t,
ϕ1u and ϕ2u, respectively Moreover, we consider more than two nonlinear terms and
do not require the monotonicity of every ϕ i i 1, 2, , k We employ a technique of
Trang 3monotonization to construct a sequence of functions which possesses stronger monotonicity than the previous one Unlike the work in26 for two sum terms, the maximal regions of
validity for our estimate of the unknown function u are decided by boundaries of more than
two planar regions Thus we have to consider the inclusion of those regions and find common regions We demonstrate that inequalities1.6 and other inequalities considered in 26 can also be solved with our result Furthermore, we apply our result to boundary value problems
of a partial difference equation for boundedness, uniqueness, and continuous dependence
2 Main Result
Throughout this paper, letR −∞, ∞, R 0, ∞, and N0 {0, 1, 2, }, m0, n0∈ N0, X, Y ∈
N0∪ {∞} are given nonnegative integers For any integers s < t, let diss, t {j : s ≤ j ≤
t, j ∈ N0}, I dism0, X , and J disn0, Y Define Λ I × J ⊂ N2
0, and letΛs,t denote the sublattice dism0, s × disn0, t in Λ for any s, t ∈ Λ.
For functions gm, n, m, n ∈ N0, their first-order differences are defined by
Δ1g m, n gm 1, n − gm, n and Δ2g m, n gm, n 1 − gm, n Obviously, the
linear difference equation Δxm bm with the initial condition xm0 0 has the solution
m−1
s m0b s In the sequel, for convenience, we complementarily define thatm0 −1
s m0b s 0.
We give the following basic assumptions for the inequality1.7
H1 ψ is a strictly increasing continuous function on Rsatisfying that ψ∞ ∞ and
ψ u > 0 for all u > 0.
H2 All ϕ i i 1, 2, , k are continuous and positive functions on R
H3 am, n ≥ 0 on Λ.
H4 All f i i 1, 2, , k are nonnegative functions on Λ × Λ.
With given functions ϕ1, ϕ2, and ψ, we technically consider a sequence of functions
w i s, which can be calculated recursively by
w1s : max
τ ∈0,s
ϕ1τ,
w i1s : max
τ ∈0,s
ϕ
i1τ
w i τ
w i s, i 1, , k − 1.
2.1
For given constants u i > 0 and variable u > 0, we define
W i u, u i :
u
u i
dx
w i
Obviously, W i is strictly increasing in u > 0 and therefore the inverses W i−1are well defined, continuous, and increasing Let
f i m, n, s, t : max
τ,ξ∈m0,m ×n0,nf i τ, ξ, s, t, 2.3
which is nondecreasing in m and n for each fixed s and t and satisfies f i x, y, t, s ≥
f i x, y, t, s ≥ 0 for all i 1, , k.
Trang 4Theorem 2.1 Suppose that H1–H4 hold and um, n is a nonnegative function on Λ satisfying
1.7 Then, for m, n ∈ Λ M1,N1 , a sublattice in Λ,
u m, n ≤ ψ−1
W k−1
W kΥk m, n m−1
s m0
n−1
t n0
f k m, n, s, t
whereΥk m, n is determined recursively by
Υ1m, n : am0, n0 m−1
s m0
|as 1, n0 − as, n0| n−1
t n0
|am, t 1 − am, t|,
Υi1m, n : W−1
i
W iΥi m, n m−1
s m0
n−1
t n0
f i m, n, s, t
, i 1, , k − 1,
2.5
and M1, N1 ∈ Λ is arbitrarily given on the boundary of the lattice
U :
m, n ∈ Λ : W iΥi m, n m−1
s m0
n−1
t n0
f i m, n, s, t ≤
∞
u i
dx
w i
ψ−1x
.
2.6
Remark 2.2 As explained in3, Remark 2, since different choices of ui in W i i 1, 2, , k
do not affect our results, we simply let Wi u denote W i u, u i when there is no confusion For
positive constants v i / u i, let W i u u
v i dx/w i ψ−1x Obviously, W i u W i u W i u i and W i−1v W−1
i v − W i u i It follows that
W i−1
W iΥi m, n m−1
s m0
n−1
t n0
f i m, n, s, t
W−1
i
W iΥi m, n m−1
s m0
n−1
t n0
f i m, n, s, t
,
2.7
that is, we obtain the same expression in 2.4 if we replace W i with W i , i 1, 2, , k Moreover, by replacing W iwith W i , the condition in the definition of U inTheorem 2.1reads
W iΥi M1, N1 M1−1
s m0
N1 −1
t n0
f i m, n, s, t ≤
∞
v i
dx
w i
the left-hand side of which is equal to
W i u i W iΥi M1, N1 M1−1
s m
N1 −1
t n
Trang 5
and the right-hand side of which equals
u i
v i
dx
w i
ψ−1x
∞
u i
dx
w i
ψ−1x W i u i
∞
u i
dx
w i
The comparison between the both sides implies that2.8 is equivalent to the condition given
in the definition of U inTheorem 2.1withm, n M1, N1
Remark 2.3 If we choose k 2, ψu u p , ϕ1u u q , ϕ2u u q w u with p > q > 0,
f1m, n, s, t ds, t and f2m, n, s, t es, t and restrict am, n to be a constant c in 1.7, then we can applyTheorem 2.1to inequality1.6 discussed in 26
3 Proof of Theorem
First of all, we monotonize some given functions ϕ i , f i in the sums Obviously, the sequence
w i s defined by ϕ i i 1, , k in 2.1 consists of nondecreasing nonnegative functions and
satisfies w i s ≥ ϕ i s, for i 1, , k Moreover,
as defined in27 for comparison of monotonicity of functions w i s i 1, , k, because every ratio w i1s/w i s is nondecreasing By the definitions of functions w i , f i , ψ, andΥ1, from1.7 we get
u m, n ≤ ψ−1
Υ1m, n k
i1
m−1
s m0
n−1
t n0
f i m, n, s, tw i us, t
Then, we discuss the case that am, n > 0 for all m, n ∈ Λ Because Υ1satisfies
Υ1m, n am0, n0 m−1
s m0
|as 1, n0 − as, n0| n−1
t n0
|am, t 1 − am, t|
≥ am, n,
3.3
it is positive and nondecreasing onΛ We consider the auxiliary inequality to 3.2, for all
m, n ∈ Λ M,N,
u m, n ≤ ψ−1
Υ1M, N k
i1
m−1
s m
n−1
t n
f i M, N, s, tw i us, t
Trang 6
where M ∈ dism0, M1 and N ∈ disn0, N1 are chosen arbitrarily, and claim that, for
m, n ∈ Λ min{M2,M },min{N2,N}, a sublattice inΛM1,N1 ,
u m, n ≤ ψ−1
W k−1
W k
Υk M, N, m, n m−1
s m0
n−1
t n0
f k M, N, s, t
where Υk M, N, m, n is determined recursively by
Υ1M, N, m, n : Υ1M, N,
Υi1M, N, m, n : W−1
i
W i
Υi M, N, m, n m−1
s m0
n−1
t n0
f i M, N, s, t
,
3.6
i 1, 2, , k − 1, and M2, N2 ∈ ΛM1,N1 is arbitrarily chosen on the boundary of the lattice
U1:
m, n ∈ Λ : W i
Υi M, N, m, n m−1
s m0
n−1
t n0
f i M, N, s, t
≤
∞
u i
dx
w i
ψ−1x
.
3.7
We note that M2, N2can be chosen appropriately such that
M2M, N M1, N2M, N N1, ∀M, N ∈ Λ M1,N1 . 3.8
In fact, from the fact ofM1, N1 being on the boundary of the lattice U, we see that
W i
Υi M1, N1, M1, N1 M1−1
s m0
N1 −1
t n0
f i M1, N1, s, t
W iΥi M1, N1 M1−1
s m0
N1 −1
t n0
f i M1, N1, s, t
≤
∞
u i
dx
w i
ψ−1x
3.9
Thus, it means that we can take M2 M1, N2 N1 Moreover, M min{M2, M }, N
min{N2, N}
In the following, we will use mathematical induction to prove3.5
For k 1, let zm, n m−1
s m0
n−1
t n0f1M, N, s, tw1us, t Then z is nonnegative and nondecreasing in each variable onΛM,N From3.4 we observe that
u m, n ≤ ψ−1Υ1M, N zm, n, ∀m, n ∈ Λ M N 3.10
Trang 7Moreover, we note that w1 is nondecreasing and satisfies w1u > 0 for u > 0 and that
Υ1M, N zm, n > 0 From 3.10 we have
Δ1Υ1M, N zm, n
w1
ψ−1Υ1M, N zm, n
n−1
t n0f1M, N, m, tw1um, t
w1
ψ−1Υ1M, N zm, n
≤n−1
t n0
f1M, N, m, t.
3.11
On the other hand, by the Mean Value Theorem for integral and by the monotonicity of w1 and ψ, for arbitrarily given m, n, m 1, n ∈ Λ M N there exists ξ in the open interval
Υ1M, N zm, n, Υ1M, N zm 1, n such that
W1Υ1M, N zm 1, n − W1Υ1M, N zm, n
Υ1M,Nzm1,n
Υ 1M,Nzm,n
du
w1
ψ−1u
Δ1Υ1M, N zm, n
w1
ψ−1ξ
≤ Δ1Υ1M, N zm, n
w1
ψ−1Υ1M, N zm, n
3.12
It follows from3.11 and 3.12 that
W1Υ1M, N zm 1, n − W1Υ1M, N zm, n ≤ n−1
t n0
f1M, N, m, t. 3.13
Substituting m with s and summing both sides of3.13 from s m0to m− 1, we get, for all
m, n ∈ Λ M N,
W1Υ1M, N zm, n ≤ W1Υ1M, N m−1
s m0
n−1
t n0
f1M, N, s, t. 3.14
We note from the definition of zm, n in 3.2 and the definition ofm0 −1
s m0 inSection 2that
z m0, n 0 By the monotonicity of W−1and3.10 we obtain
u m, n ≤ ψ−1
W1−1
W1Υ1M, N m−1
s m0
n−1
t n0
f1M, N, s, t
, ∀m, n ∈ Λ M N , 3.15
that is,3.5 is true for k 1.
Trang 8Next, we make the inductive assumption that3.5 is true for k l Consider
u m, n ≤ ψ−1
Υ1M, N l1
i1
m−1
s m0
n−1
t n0
f i M, N, s, tw i us, t
for all m, n ∈ Λ M N Let ym, n l1
i1m−1
s m0
n−1
t n0 fi M, N, s, tw i us, t, which is nonnegative and nondecreasing in each variable onΛM,N Then3.16 is equivalent to
u m, n ≤ ψ−1
Υ1M, N ym, n , ∀m, n ∈ Λ M N 3.17
Since w i is nondecreasing and satisfies w i u > 0 for u > 0 i 1, 2, , l 1 and Υ1K, L
y m, n > 0, from 3.17 we obtain, for all m, n ∈ Λ M N,
Δ1
Υ1M, N ym, n
w1
ψ−1
Υ1M, N ym, n
n−1
t n0 f1M, N, m, tw1um, t
w1
ψ−1
Υ1M, N ym, n
l1
i2n−1
t n0 fi M, N, m, tw i um, t
w1
ψ−1
Υ1M, N ym, n
≤n−1
t n0
f1M, N, m, t l
i1
n−1
t n0
f i1M, N, m, tφ i1um, t,
3.18 where
φ i u : w i u
On the other hand, by the Mean Value Theorem for integrals and by the monotonicity of
w1 and ψ, for arbitrarily given m, n, m 1, n ∈ Λ M,N there exists ξ in the open interval
Υ1M, N ym, n, Υ1M, N ym 1, n such that
W1
Υ1M, N ym 1, n − W1
Υ1M, N ym, n
Υ1M,Nym1,n
Υ 1M,Nym,n
du
w1
ψ−1u
Δ1
Υ1M, N ym, n
w1
ψ−1ξ
Υ1M, N ym, n
w1
ψ−1
Υ1M, N ym, n
3.20
Trang 9Therefore, it follows from3.18 and 3.20 that
W1
Υ1M, N ym 1, n − W1
Υ1M, N ym, n
≤n−1
t n0
f1M, N, m, t l
i1
n−1
t n0
f i1M, N, m, tφ i1um, t. 3.21
substituting m with s in3.21 and summing both sides of 3.21 from s m0 to m− 1, we get, for allm, n ∈ Λ M,N,
W1
Υ1M, N ym, n − W1Υ1M, N
≤ m−1
s m0
n−1
t n0
f1M, N, s, t l
i1
m−1
s m0
n−1
t n0
f i1M, N, s, tφ i1us, t, 3.22
where we note that ym0, n 0 For convenience, let
ψ Ξm, n : W1
Υ1M, N ym, n ,
θ M, N, m, n : W1Υ1M, N m−1
s m0
n−1
t n0
From3.17 and 3.22 we can get
Ξm, n ≤ ψ−1
θ M, N, M, N l
i1
m−1
s m0
n−1
t n0
f i1M, N, s, tφ i1
ψ−1
W1−1
ψ Ξm, n
,
3.24
the same form as3.4 for k l, for all m, n ∈ Λ M,N , where we note that θM, N, M, N ≥
θ M, N, m, n for all m, n ∈ Λ M,N We are ready to use the inductive assumption for3.24
In order to demonstrate the basic condition of monotonicity, let hs ψ−1W−1
1 ψs,
obviously which is a continuous and nondecreasing function onR Thus each φ i hs is
continuous and nondecreasing onRand satisfies φ i hs > 0 for s > 0 Moreover,
φ i1hs
φ i hs
w i1hs
w i hs maxτ ∈0,hs
ϕ
i1τ
w i τ
which is also continuous nondecreasing onRand positive onR This implies that φ i hs ∝
φ i1hs, for i 2, , l Therefore, the inductive assumption for 3.5 can be used to 3.24 and we obtain, for allm, n ∈ Λ min{M,M3},min{N,N3 },
Ξm, n ≤ ψ−1
Φ−1
l1
Φl1θ l1M, N, m, n m−1
s m
n−1
t n
f l1M, N, s, t
Trang 10
whereΦi u : u
u ids/φ i hs, u > 0, u ψ−1W1u, Φ−1
i is the inverse ofΦi for
i 2, 3, , l 1, θ l1M, N, m, n is determined recursively by
θ1M, N, m, n : θM, N, M, N,
θ i1M, N, m, n : Φ−1
i
Φi θ i M, N, m, n m−1
s m0
n−1
t n0
f i M, N, s, t
, i 1, 2, , l, 3.27
and M3, N3are functions ofM, N such that M3M, N, N3M, N ∈ Λ M1,N1 lie on the boundary of the lattice
U2 :
m, n ∈ Λ : Φ i θ i M, N, m, n m−1
s m0
n−1
t n0
f i M, N, s, t
≤
∞
u i
ds
φ i hs , i 2, 3, , l 1
,
3.28
where ∞ denotes either limu→ ∞ u if it converges or ∞ Note that
Φi u
u
u i
ds
θ
ψ−1
W1−1
ψ s
u
u i
w1
ψ−1
W1−1
ψ s ds
w i
ψ−1
W1−1
ψ s
W−1
1 ψu
u i
dx
w i
ψ−1x
W i
W1−1
ψ u , i 2, 3, , l 1.
3.29
Thus, from3.17, 3.23, and 3.27, 3.26 can be equivalently written as
u m, n ≤ ψ−1
W1−1
ψ Ξm, n
≤ ψ−1
W l−11
W l1
W1−1
ψ θ l1M, N, m, n
m−1
s m0
n−1
t n0
f l1M, N, s, t
, ∀m, n ∈ Λ min{M,M3},min{N,N3 }.
3.30
We further claim that the term W1−1ψθ i M, N, m, n is the same as Υ i M, N, m, n, defined
in 3.6, i 1, 2, , l 1 For convenience, let θ i M, N, m, n W−1
1 ψθ i M, N, m, n.
Obviously, it is that θ1M, N, m, n Υ1M, N, m, n.