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Some new finite difference inequalities arising in the theory of difference equations
Advances in Difference Equations volume 2011, Article number: 21 (2011)
Abstract
In this work, some new finite difference inequalities in two independent variables are established, which can be used in the study of qualitative as well as quantitative properties of solutions of certain difference equations. The established results extend some existing results in the literature.
MSC 2010: 26D15
1. Introduction
Finite difference inequalities in one or two independent variables which provide explicit bounds play a fundamental role in the study of boundedness, uniqueness, and continuous dependence on initial data of solutions of difference equations. Many difference inequalities have been established (for example, see [1–11] and the references therein). In the research of difference inequalities, generalization of known inequalities has been paid much attention by many authors. Here we list some recent results in the literature.
In [[12], Theorems 2.62.8], Pachpatte presents the following six discrete inequalities, based on which some new bounds on unknown functions are established.
Preprint submitted to Advances in Difference Equations June 16, 2011
where u, a, b, c are nonnegative functions defined on m ∈ ℕ_{0}, n ∈ ℕ_{0}, and L : ℕ_{0} × ℕ_{0} × ℝ_{+} → ℝ_{+} satisfies 0 ≤ L(m, n, u)  L(m, n, v) ≤ M(m, n, v)(u  v) for u ≥ v ≥ 0, where M : ℕ_{0} × ℕ_{0} × ℝ_{+} → ℝ_{+}.
Recently, in [[13], Theorems 16], Meng and Li present the following inequalities with more general forms.
where p ≥ 1 is a constant, u, a, b, c, e are nonnegative functions defined on m ∈ ℕ_{0}, n ∈ ℕ_{0}, and L is defined the same as in (a 5)a(6).
As one can see, (b 1)(b 2) are generalizations of (a 1)(a 2), while (b 4)(b 6) are generalizations of (a 4)(a 6).
More recently, Meng and Ji [[14], Theorems 3, 4, 7, 8] extended (b 1)(b 4) to the following inequalities.
where p, q, r are constants with p ≥ q, p ≥ r, p ≠ 0, and u, a, b, c, d, e are nonnegative functions defined on m ∈ ℕ_{0}, n ∈ ℕ_{0}.
The presented inequalities above have proved to be very useful in the study of quantitative as well as qualitative properties of solutions of certain difference equations.
Motivated by the work mentioned above, in this paper, we will establish some more generalized finite difference inequalities, which provide new bounds for unknown functions lying in these inequalities. We will illustrate the usefulness of the established results by applying them to study the boundedness, uniqueness, and continuous dependence on initial data of solutions of certain difference equations.
Throughout this paper, ℝ denotes the set of real numbers and ℝ_{+} = [0, ∞), and ℤ denotes the set of integers, while ℕ_{0} denotes the set of nonnegative integers. I := [m _{0}, ∞] ∩ ℤ and are two fixed lattices of integral points in ℝ, where m _{0}, n _{0} ∈ ℤ. Let . We denote the set of all ℝvalued functions on Ω by ℘(Ω), and denote the set of all ℝ_{+}valued functions on Ω by ℘_{+}(Ω). The partial difference operators Δ_{1} and Δ_{2} on u ∈ ℘(Ω) are defined as Δ_{1} u(m, n) = u(m +1, n)  u(m, n), Δ_{2} u(m, n) = u(m, n + 1)  u(m, n).
2. Main results
Lemma 2.1. [[15]] Assume that a ≥ 0, p ≥ q ≥ 0, and p ≠ 0, then for any K > 0
Lemma 2.2. Let u(m, n), a(m, n), b(m, n) are nonnegative functions defined on Ω with a(m, n) not equivalent to zero.

(1)
Assume that a(m, n) is nondecreasing in the first variable. If
for (m, n) ∈ Ω, then

(2)
Assume that a(m, n) is decreasing in the first variable. If
for (m, n) ∈ Ω, then
Remark 1. Lemma 2.2 is a direct variation of [[12], Lemma 2.5].
Theorem 2.1. Suppose u, a, b, f, g, h, w ∈ ℘_{+} (Ω), and b, f, g, h, w are nondecreasing in the first variable, while decreasing in the second variable. α : I → I is nondecreasing with α (m) ≤ m for ∀m ∈ I, while is nondecreasing with β(n) ≥ n for . p, q, r, l are constants with p ≥ q, p ≥ r, p ≥ l, p ≠ 0.
If for (m, n) ∈ Ω, u(m, n) satisfies the following inequality
then we have
provided H(m.n) > 0, where K > 0 is a constant, and
Proof. Let .
Then we have
Furthermore, if given (X, Y ) ∈ Ω, and (m, n) ∈ ([m _{0}, X]×[Y, ∞]) ∩ Ω, then using (4) and Lemma 2.1 we have
where H, , , are defined in (3).
Let the right side of (5) be v(m, n). Then
and
Considering v(m, n) ≥ v(m, n + 1), we have
Setting n = t in (7), and a summary with respect to t from n to r  1 yields
Letting r → ∞ in (8), using v(m, ∞) = H(X, Y ) we obtain
which is followed by
Setting m = s in (9), and a multiple with respect to s from m _{0} to m  1 yields
Considering v(m _{0}, n) = H(X, Y ), and then combining (4), (6) and (10) we obtain
Setting m = X, n = Y in (11), and considering (X, Y ) ∈ Ω is selected arbitrarily, then after substituting X, Y with m, n we obtain the desired inequality.
Remark 2. If we take Ω = ℕ_{0} × ℕ_{0}, w(m, n) ≡ 0, α (m) = m, β(n) = n, and omit the conditions "b, f, g, h, w are nondecreasing in the first variable, while decreasing in the second variable" in Theorem 2.1, which is unnecessary for the proof since α(m) = m, β(n) = n, then Theorem 2.1 reduces to [[14], Theorem 3]. Furthermore, if g(m, n) ≡ 0, q = 1, p ≥ 1, then Theorem 2.1 reduces to [[13], Theorem 1].
Following a similar process as the proof of Theorem 2.1, we have the following three theorems.
Theorem 2.2. Suppose u, a, b, f, g, h, w ∈ ℘_{ + } (Ω), and b, f, g, h, w are decreasing both in the first variable and the second variable. α : I → I is nondecreasing with α(m) ≥ m for ∀m ∈ I, while is nondecreasing with β(n) ≥ n for . p, q, r, l are defined as in Theorem 2.1. If for (m, n) ∈ Ω, u(m, n) satisfies the following inequality
then we have
provided H(m.n) > 0, where , , are defined as in Theorem 2.1, and
Remark 3. If we take Ω = ℕ_{0} × ℕ_{0}, w(m, n) ≡ 0, α(m) = m, β(n) = n, and omit the conditions "b, f, g, h, w are decreasing both in the first variable and the second variable" in Theorem 2.2, which are unnecessary for the proof since α(m) = m, β(n) = n, then Theorem 2.2 reduces to [[14], Theorem 4]. Furthermore, if g(m, n) ≡ 0, q = 1, p ≥ 1, then Theorem 2.2 reduces to [[13], Theorem 2].
Theorem 2.3. Suppose u, a, b, f, g, h, w ∈ ℘_{ + } (Ω), and b, f, g, h, w are nondecreasing both in the first variable and the second variable. α : I → I is nondecreasing with α(m) ≤ m for ∀m ∈ I, while is nondecreasing with β(n) ≤ n for . p, q, r, l are defined as in Theorem 2.1. If for (m, n) ∈ Ω, u(m, n) satisfies the following inequality
then we have
provided H(m.n) > 0, where , , are defined as in Theorem 2.1, and
Theorem 2.4. Suppose u, a, b, f, g, h, w ∈ ℘_{+} (Ω), and b, f, g, h, w are decreasing in the first variable, while nondecreasing in the second variable. α : I → I is nondecreasing with α(m) ≥ m for ∀m ∈ I, while is nondecreasing with β(n) ≤ n for . p, q, r, l are defined as in Theorem 2.1. If for (m, n) ∈ Ω, u(m, n) satisfies the following inequality
then we have
provided H(m.n) > 0, where , , are defined as in Theorem 2.1, and
Next we will study the following difference inequality:
where u, a, b, f, g, h, w ∈ ℘_{+}(Ω) with a(m, n) not equivalent to zero, and f, g, h, w are nondecreasing in the first variable, while decreasing in the second variable, a is nondecreasing in the first variable, and b is decreasing in the second variable, α, β, p, q, r, l are defined as in Theorem 2.1.
Theorem 2.5. If for (m, n) ∈ Ω, u(m, n) satisfies (12), then we have
provided , where K > 0 is a constant, and
Proof: Denote , and v(m, n) = a(m, n) + z(m, n). Then v(m, n) is nondecreasing in the first variable, and
By Lemma 2.2 we obtain
where J(m, n) is defined in (14). Furthermore, using Lemma 2.1 we have
where , , , are defined in (14).
Obviously, , , are nondecreasing in the first variable, while decreasing in the second variable. Following in a same manner as the proof of Theorem 2.1 we obtain
Combining (16) and (18) we obtain the desired result.
Remark 4. If we take Ω = ℕ_{0} × ℕ_{0}, w(m, n) ≡ 0, α(m) = m, β(n) = n, and omit the conditions "f, g, h, w are nondecreasing in the first variable, while decreasing in the second variable" and "b is decreasing in the second variable" in Theorem 2.5, then Theorem 2.5 reduces to [[14], Theorem 7]. Furthermore, if g(m, n) ≡ 0, q = 1, p ≥ 1, then Theorem 2.5 reduces to [[13], Theorem 3].
Following a almost same process as the proof of Theorem 2.5, we have the following two theorems.
Theorem 2.6. Suppose u, a, b, f, g, h, w ∈ ℘_{+} (Ω) with a(m, n) not equivalent to zero, and f, g, h, w are decreasing both in the first variable and the second variable, a is decreasing in the first variable, and b is decreasing in the second variable, α, β are defined as in Theorem 2.2, and p, q, r l are defined as in Theorem 2.1. If for (m, n) ∈ Ω, u(m, n) satisfies the following inequality
then we have
provided , where
Theorem 2.7. Suppose u, a, b, f, g, h, w ∈ ℘_{+} (Ω) with a(m, n) not equivalent to zero, and f, g, h, w are decreasing both in the first variable and the second variable, a is nondecreasing in the first variable, and b is decreasing in the second variable, α, β are defined as in Theorem 2.2, and p, q, r, l are defined as in Theorem 2.1. If for (m, n) ∈ Ω, u(m, n) satisfies the following inequality
then we have
provided , where K > 0 is a constant, and
and , , , J(m, n) are defined as in Theorem 2.5.
Remark 5. If we take Ω = ℕ_{0} × ℕ_{0}, w(m, n) ≡ 0, α(m) = m, β(n) = n, and omit the conditions "f, g, h, w are decreasing both in the first variable and the second variable" and "b is decreasing in the second variable" in Theorem 2.6, then Theorem 2.6 reduces to [[14], Theorem 8]. Furthermore, if g(m, n) ≡ 0, q = 1, p ≥ 1, then Theorem 2.6 reduces to [[13], Theorem 4].
Remark 6. If we take Ω = ℕ_{0} × ℕ_{0}, w(m, n) ≡ 0, α(m) = m, β(n) = n, and omit the conditions "f, g, h, w are decreasing both in the first variable and the second variable" and "b is decreasing in the second variable" in Theorem 2.7, then Theorem 2.7 reduces to [[12], Theorem 2.7(q1)].
In the following, we will study the difference inequality with the following form
where u, a, b, w ∈ ℘_{+}(Ω) with a(m, n) not equivalent to zero, and w is nondecreasing in the first variable, while decreasing in the second variable, a is nondecreasing in the first variable, and b is decreasing in the second variable, α, β are defined as in Theorem 2.1, L : Ω × ℝ_{+} → ℝ_{+} satisfies 0 ≤ L(m, n, u)  L(m, n, v) ≤ M(m, n, v)(u  v) for u ≥ v ≥ 0, where M : Ω × ℝ_{+} → ℝ_{+}. p, l are defined as in Theorem 2.1 with p ≥ 1.
Theorem 2.8. If for (m, n) ∈ Ω, u(m, n) satisfies (19), then
provided that , and is nondecreasing in the first variable and decreasing in the second variable, where K > 0 is a constant, and
Proof: Denote , and v(m, n) = a(m, n) + z(m, n). Then v(m, n) is nondecreasing in the first variable, and
By Lemma 2.2 we obtain
where J(m, n) is defined in (21). Furthermore,
where , , are defined in (21).
Then following in a same manner as the proof of Theorem 2.1 we obtain
The desired inequality can be deduced by the combination of (23) and (25).
Theorem 2.9. Suppose u, a, b, w ∈ ℘_{+}(Ω) with a(m, n) not equivalent to zero, and w is decreasing both in the first variable and the second variable, a is decreasing in the first variable, and b is decreasing in the second variable, α, β are defined as in Theorem 2.2, and L is defined as in Theorem 2.8. p, l are defined as in Theorem 2.1 with p ≥ 1. If for (m, n) ∈ Ω, u(m, n) satisfies the following inequality
then
provided that , and is decreasing both in the first variable and the second variable, where
The proof for Theorem 2.8 is similar to Theorem 2.7, and we omit it here.
Remark 7. If we take Ω = ℕ_{0} × ℕ_{0}, w(m, n) ≡ 0, α(m) = m, β(n) = n, and omit the conditions "w is nondecreasing in the first variable, while decreasing in the second variable", " is nondecreasing in the first variable and decreasing in the second variable", and "b is decreasing in the second variable" in Theorem 2.8, then Theorem 2.8 reduces to [[13], Theorem 5].
Remark 8. If we take Ω = ℕ_{0} × ℕ_{0}, w(m, n) ≡ 0, α(m) = m, β(n) = n, and omit the conditions "w is decreasing both in the first variable and the second variable", " is decreasing both in the first variable and the second variable" and "b is decreasing in the second variable" in Theorem 2.9, then Theorem 2.9 reduces to [[13], Theorem 6].
3. Applications
In this section, we will present some applications for the established results above, and show they are useful in the study of boundedness, uniqueness, continuous dependence of solutions of certain difference equations.
Example 1. Consider the following difference equation
with the initial condition
where p ≥ 1 is an odd number, u ∈ ℘ (Ω), F _{1}, F _{2} : Ω × ℝ → ℝ.
Theorem 3.1. Suppose u(m, n) is a solution of (26) and (27). If f(m) + g(n)  C ≤ σ, F _{1}(m, n, u) ≤ f _{1}(m, n)u, and F _{2}(m, n, u) ≤ f _{2}(m, n)u, where f _{1}, f _{2} ∈ ℘_{+}(Ω), then we have
where K > 0 is a constant, and
Proof. The equivalent form of (26) and (27) is denoted by
Then we have
We note that it is unnecessary for f _{1}, f _{2} being nondecreasing or decreasing since α(m) = m, β(n) = n here, and a suitable application of Theorem 2.1 to (31) yields the desired result.
The following theorem deals with the uniqueness of solutions of (26) and (27).
Theorem 3.2. Suppose F _{ i }(m, n, u)  F _{ i }(m, n, v) ≤ f _{ i }(m, n)u ^{p}  v ^{p} , i = 1, 2, where f _{ i } ∈ ℘_{+}(Ω), i = 1, 2, then (26) and (27) has at most one solution.
Proof. Suppose u _{1}(m, n), u _{2}(m, n) are two solutions of (26) and (27). Then
Treat as one variable, and a suitable application of Theorem 2.1 to (32) yields , which implies . Since p is an odd number, then we have u _{1}(m, n) ≡ u _{2}(m, n), and the proof is complete.
The following theorem deals with the continuous dependence of the solution of (26) and (27) on the functions F _{1}, F _{2} and the initial value f (m), g(n).
Theorem 3.3. Assume , i = 1,2, where f _{ i } ∈ ℘_{+}(Ω), i = 1, 2, , where ε > 0 is a constant, and furthermore, assume , is the solution of the following difference equation
with the initial condition
where , : Ω × ℝ → ℝ, then
provided that G(m, n) ≤ K, where
Proof. The equivalent form of (33) and (34) is denoted by
Then from (30) and (36) we have
Then a suitable application of Theorem 2.1 to (37) yields the desired result.
Example 2. Consider the following difference equation
where u, a, b ∈ ℘(Ω) with a(m, n) not equivalent to zero, p ≥ 1 is an odd number, F _{1}, F _{2} : Ω × ℝ → ℝ.
Theorem 3.4. Suppose u(m, n) is a solution of (38). If F _{1}(m, n, u) ≤ L(m, n, u), F _{2}(m, n, u) ≤ w(m, n)u ^{l}, where L is defined as in Theorem 2.8, and w ∈ ℘_{+}(Ω), l ≥ 0, p ≥ l, then we have
where
Proof. From (38) we have
Then a suitable application of Theorem 2.9 (with α(m) = m, β(n) = n) to (41) yields the desired result.
Similar to Theorems 3.2 and 3.3, we also have the following two theorems dealing with the uniqueness and continuous dependence of the solution of (38) on the functions a, b, F _{1}, F _{2}.
Theorem 3.5. Suppose F _{ i }(m, n, u)  F _{ i }(m, n, v) ≤ f _{ i }(m, n)u ^{p}  v ^{p} , i = 1, 2, where f _{ i } ∈ ℘_{+}(Ω), i = 1, 2, then (38) has at most one solution.
Theorem 3.6. Assume , i = 1, 2, where f _{ i } ∈ ℘_{+}(Ω), i = 1, 2, , and furthermore, assume , is the solution of the following difference equation
where , : Ω × ℝ → ℝ, then
provided that , where
and
The proof for Theorems 3.53.6 is similar to Theorems 3.23.3, in which Theorem 2.6 is used. Due to the limited space, we omit it here.
4. Conclusions
In this paper, some new finite difference inequalities in two independent variables are established, which can be used as a handy tool in the study of boundedness, uniqueness, continuous dependence on initial data of solutions of certain difference equations. The established inequalities generalize some existing results in the literature.
6. Authors'contributions
QF carried out the main part of this article. All authors read and approved the final manuscript.
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7. Acknowledgements
This work is supported by National Natural Science Foundation of China (Grant No 10571110), Natural Science Foundation of Shandong Province (ZR2009AM011 and ZR2010AZ003) (China) and Specialized Research Fund for the Doctoral Program of Higher Education (20103705110003)(China). The authors thank the referees very much for their careful comments and valuable suggestions on this paper.
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5. Competing interests
The authors declare that they have no competing interests.
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Keywords
 Finite difference inequalities
 Difference equations
 Explicit bounds
 Qualitative analysis
 Quantitative analysis