A Generalized Sum-Difference Inequality and Applications to Partial Difference Equations

We establish a general form of sum-difference inequality in two variables, which includes both 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 by Cheung and Ren 2006 . Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.


Introduction
Being an important tool in the study of differential equations and integral equation, various generalizations of Gronwall inequality 1, 2 and their applications have attracted great interests of many mathematicians see 3-5 .Some recent works can be found, for example, in 6-9 and some references therein.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 Gronwall-type inequalities see, e.g., 10-12 for some early works .Found in 13 , the unknown function u in the fundamental form of sum-difference inequality can be estimated by u n ≤ a n n−1 s 0 1 f s .Pang and Agarwal 14 considered the inequality where α, P , and Q are nonnegative constants and u and g are nonnegative functions defined on {1, 2, . . ., T} and {1, 2, . . ., T − 1}, and they estimated that u n ≤ 1 α n Pu 0 n−1 s 0 Qg s , for all 0 ≤ n ≤ T .Another form of sum-difference inequality, where c is a constant, f 1 and f 2 are both real-valued nonnegative functions defined on N 0 {0, 1, 2, . ..}, and w is a continuous nondecreasing function defined on u 0 , ∞ such that w u > 0 on u 0 , ∞ and w u 0 0, for a real constant u 0 , was estimated by Pachpatte 15 as , where Ω u : u u 0 ds/w s .Recently, discretization see 16, 17 was also made for Ou-Yang's inequality 18 .In 16 , the inequality of two variables, was discussed.Later, this result was generalized in 17 to the inequality where c ≥ 0 and p > q > 0 are all constant, a and b are both nonnegative real-valued functions defined on a lattice in Z 2 , 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 for nonnegative integers m, n.In 1.6 , we replace the constant c in 1.5 with a function a m, n and replace the functions u p , u q , u q w u in 1.5 with the more general form of functions ψ u , ϕ 1 u , ϕ 2 u , respectively.Moreover, we do not require the monotonicity of ϕ 1 and ϕ 2 .We employ a technique of monotonization and use a property of stronger monotonicity to overcome the difficulty from nonmonotonicity so as to give an estimate for the unknown function u.Our result enables us to solve the discrete inequality 1.5 and other inequalities considered in 17 .Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.

Main result
Throughout this paper, let R denote the set of all real numbers, R 0, ∞ , and For functions w m , z m, n , m, n ∈ N 0 , their first-order differences are defined by Δw Wu-Sheng Wang 3 Our basic assumptions for inequality 1.6 are given in the following.
H 1 ψ is a strictly increasing continuous function on R satisfying that ψ u > 0, for all u > 0.
H 2 All ϕ i i 1, 2 are continuous functions on R and positive on 0, ∞ .With given functions ϕ 1 , ϕ 2 , and ψ, we define where u i > 0 i 1, 2 are given constants.Sometimes we simply let W i u denote W i u, u i when there is no confusion.Obviously, W 1 and W 2 are both strictly increasing in u > 0 and therefore the inverses and m 1 , n 1 ∈ Λ is arbitrarily given on the boundary of the lattice Remark 2.2.Different choices of u i in W i i 1, 2 do not affect our results.For positive con- that is, we obtain the same expression in 2.5 if we replace W i with W i .Moreover, by replacing W i with W i , the condition in the definition of U in our theorem reads the left-hand side of which is equal to t n 0 f i s, t and the right-hand side of which equals Comparison between both sides implies that 2.8 is equivalent to the condition given in the definition of U in our theorem with m, n m 1 , n 1 .
Remark 2.3.If we choose ψ u u p , ϕ 1 u u q , ϕ 2 u u q w u , f 1 s, t a s, t , and f 2 s, t b s, t with p > q > 0 in 1.6 and restrict a m, n to be a constant c, then we can apply Theorem 2.1 to inequality 1.5 as discussed in 17 .

Proof of theorem
First of all, we monotonize some given functions ϕ i in the sums.Obviously, w 1 s and w 2 s , defined by ϕ 1 and ϕ 2 in 2.1 and 2.2 , are nondecreasing and nonnegative functions and satisfy w i s ≥ ϕ i s , i 1, 2.Moreover, we can check that the ratio w 2 s /w 1 s is also nondecreasing.Therefore, from 1.6 we get 3.1 We first discuss in the case that a m, n > 0, for all m, n ∈ Λ.It means that Υ 1 m, n > 0, for all m, n ∈ Λ.In such a circumstance, Υ 1 is positive and nondecreasing on Λ and satisfies

3.2
Because ψ is strictly increasing, from 3.1 we have From the properties of f i and w i , we see that z is nonnegative and nondecreasing in each variable on Λ.Since Υ 1 is nondecreasing, for arbitrarily fixed pair of integers K, L ∈ Λ m 1 ,n 1 , we observe from 3.3 that Moreover, we note that w i is nondecreasing and satisfies w i u > 0, for u > 0 i 1, 2 , and that Υ 1 K, L z m, n > 0. It implies by 3.5 that where On the other hand, by the mean-value theorem for integrals, for arbitrarily given m, n , m by the monotonicity of w 1 and ψ.It follows from 3.6 and 3.8 that 3.9 Keep n fixed and substitute m with s in 3.9 .Then, taking the sum on both sides of 3.9 over s m 0 , m 0 1, m 0 2, . . ., m − 1, we get

3.10
Advances in Difference Equations for all m, n ∈ Λ K,L , where we note from the definition of z m, n in 3.3 and the remark about m 0 −1 s m 0 in the second paragraph of Section 2 that z m 0 , n 0. For convenience, let Then, 3.10 can be rewritten as for all m, n ∈ Λ K,L , where we note that σ K, L ≥ σ m, n , for all m, n ∈ Λ K,L .Let g m, n denote the function on the right-hand side of 3.13 , which is obviously a positive function and nondecreasing in each variable.Since the composition θ ψ −1 W −1 1 u is also nondecreasing in u, by 3.13 , that is the fact that Ξ m, n ≤ g m, n , we have

3.14
In order to estimate the left-hand side of 3.14 further, we consider the following integral:

3.15
where we note the definitions of W 1 , W 2 , and θ in 2.3 , 2.4 , and 3.7 .Applying the meanvalue theorem to 3.15 , we see that for arbitrarily given m, n , m 1, n ∈ Λ K,L , there exists η in the open interval g m, n , g m 1, n such that .

3.16
Thus, it follows from 3.14 , 3.15 , and 3.16 that for all m, n ∈ Λ K,L .Furthermore, using the same procedure as done for 3.9 , we keep n fixed and setting m s in 3.17 .Then, summing up both sides of 3.17 over s m 0 , m 0 1, m 0 2, . . ., m − 1, we get for all m, n ∈ Λ K,L , where we note the fact that g m 0 , n σ K, L and the definition of σ in 3.12 .By the monotonicity of W 1 and ψ, the fact that Ξ m, n ≤ g m, n , given in 3.13 , and inequality 3.18 , we obtain from 3.5 that

3.19
for all m, n ∈ Λ K,L , where we note the definitions of Ξ in 3.11 and g just after 3.13 .This result also implies the particular case that

3.20
For the arbitrary choice of K, L ∈ Λ m 1 ,n 1 , it also implies that 2.5 holds for all m, n ∈ Λ m 1 ,n 1 .
The remainder case is that a m, n 0, for some m, n ∈ Λ.Let where ε > 0 is an arbitrary small number.Obviously, Υ 1,ε m, n > 0, for all m, n ∈ Λ.Using the same arguments as above, where Letting ε → 0 , we obtain 2.5 because of continuity of Υ i,ε in ε and continuity of W i and W −1 i , for i 1, 2. This completes the proof.Remark that m 1 and n 1 lie on the boundary of the lattice U.In particular, 2.5 is true for all m, n ∈ Λ when all w i s i 1, 2 satisfy ∞, so we may take m 1 M, n 1 N.

Applications to a difference equation
In this section, we apply our result to the following boundary value problem simply called BVP for the partial difference equation: where Λ : I × J is defined as in the beginning of Section 2, ψ ∈ C 0 R, R is a strictly increasing odd function satisfying ψ u > 0, for u > 0, 2 satisfying ϕ i u > 0, for u > 0, and functions f : I → R and g : J → R satisfy f m 0 g n 0 0. Obviously, 4.1 is a generalization of the BVP problem considered in 17, Section 3 .So the results of 17 cannot be applied immediately.In what follows we first apply our main result to discuss boundedness of solutions of 4.1 .
, where m 1 , n 1 are given as in Theorem 2.1 and

4.4
Proof.Clearly, the difference equation of BVP 4.1 is equivalent to It follows that for u 1 , u 2 ∈ R and m, n ∈ Λ : I × J, where I m 0 , M ∩ N 0 , J n 0 , N ∩ N 0 as assumed in the beginning of Section 2 with natural numbers M and N, h 1 , h 2 are both nonnegative functions defined on the lattice Λ, ϕ 1 , ϕ 2 ∈ C 0 R , R are both nondecreasing with the nondecreasing ratio ϕ 2 /ϕ 1 such that ϕ i 0 0, ϕ i u > 0, for all u > 0 and 1 0 ds/ϕ i s ∞, for i 1, 2, and ψ ∈ C 0 R, R is a strictly increasing odd function satisfying ψ u > 0, for u > 0.Then, BVP 4.1 has at most one solution on Λ.
Proof.Assume that both z m, n and z m, n are solutions of BVP 4.1 .From the equivalent form 4.5 of 4.1 , we have since m < M, n < N. Thus, by 4.10 Thus, we conclude from 2.5 that |ψ z m, n −ψ z m, n | ≤ 0, implying that z m, n z m, n , for all m, n ∈ Λ since ψ is strictly increasing.It proves the uniqueness.Remark 4.3.If h 1 ≡ 0 or h 2 ≡ 0 in 4.8 , the conclusion of Corollary 4.2 also can be obtained.Finally, we discuss the continuous dependence of solutions of BVP 4.1 on the given functions F, f, and g.Consider a variation of BVP 4.1 Our requirement on the small difference, F − F in Corollary 4.4, is stronger than the condition iii in 17, Theorem 3.3 , but ours may be easier to check because one has to verify the inequality in his condition iii for each solution z m, n of BVP 4.14 .

m w m 1 −
w m , Δ 1 w m, n w m 1, n − w m, n , and Δ 2 z m, n z m, n 1 − z m, n .Obviously, the linear difference equation Δx m b m with the initial condition x m 0 0 has the solution m−1 s m 0 b s .For convenience, in the sequel we complementarily define that m 0 −1 s m 0 b s 0.

H 3 H 4
a m, n ≥ 0 on Λ.All f i i 1, 2 are nonnegative functions on Λ.

Corollary 4 . 1 .
All solutions z m, n of BVP 4.1 have the estimate 2 are well defined, continuous, and increasing.
< ∞, 4.7 for all m, n ∈ Λ m 1 ,n 1 , then every solution z m, n of BVP 4.1 is bounded on Λ m 1 ,n 1 .Next, we discuss the uniqueness of solutions for BVP 4.1 .
ϕ 1 ψ z s, t − ψ z s, t R , and f : I → R, g : J → R are functions satisfying f m 0 Let F be a function as assumed in the beginning of Section 4 and satisfy 4.2 and 4.8 on the same lattice Λ as assumed in Corollary 4.2.Suppose that the three differences By Corollary 4.2, the solution z m, n is unique.By the continuity and the strict monotonicity of ψ, we suppose that an inequality of the form 1.6 .Applying Theorem 2.1 to 4.17 , we obtain ∈ Λ m 1 ,n 1 , where m 1 , n 1 are given as in Theorem 2.1, By 4.10 we see that Υ i m, n → 0 i 1, 2 as → 0. It follows from 4.18 that lim →0 |ψ z m, n − ψ z m, n | 0, and hence z m, n depends continuously on F, f, and g since ψ is strictly increasing.
s,t,u ∈Λ×R F s, t, u − F s, t, u 4.15 are all sufficiently small.Then, solution z m, n of BVP 4.14 is sufficiently close to the solution z m, n of BVP 4.1 .Proof.