Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 RESEARCH Open Access Some new finite difference inequalities arising in the theory of difference equations Qinghua Feng1,2*, Fanwei Meng2 and Yaoming Zhang1 * Correspondence: fqhua@sina.com School of Science, Shandong University of Technology, Zhangzhou Road 12, Zibo, Shandong, 255049, China Full list of author information is available at the end of the article 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 Keywords: Finite difference inequalities, Difference equations, Explicit bounds, Qualitative analysis, Quantitative analysis 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 2.6-2.8], Pachpatte presents the following six discrete inequalities, based on which some new bounds on unknown functions are established m−1 ∞ (a1 ) : u(m, n) ≤ a(m, n) + b(m, n) c(s, t)u(s, t), s=0 t=n+1 Preprint submitted to Advances in Difference Equations June 16, 2011 ∞ ∞ (a2 ) : u(m, n) ≤ a(m, n) + b(m, n) c(s, t)u(s, t), s=m+1 t=n+1 ∞ m−1 (a3 ) : u(m, n) ≤ a(m, n) + ∞ b(s, n)u(s, n) + s=0 c(s, t)u(s, t), s=m+1 t=n+1 ∞ (a4 ) : u(m, n) ≤ a(m, n) + ∞ ∞ b(s, n)u(s, n) + s=m+1 c(s, t)u(s, t), s=m+1 t=n+1 © 2011 Feng et al; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page of 17 m−1 ∞ m−1 (a5 ) : u(m, n) ≤ a(m, n) + b(s, n)u(s, n) + s=0 L(s, t, u(s, t)), s=0 t=n+1 ∞ ∞ (a6 ) : u(m, n) ≤ a(m, n) + ∞ b(s, n)u(s, n) + s=m+1 L(s, t, u(s, t)), s=m+1 t=n+1 where u, a, b, c are nonnegative functions defined on m Ỵ N0, n Ỵ N0, and L : N0 ì N0 ì + đ + satisfies ≤ L(m, n, u) - L(m, n, v) ≤ M(m, n, v)(u - v) for u ≥ v ≥ 0, where M : N0 ì N0 ì + đ ℝ+ Recently, in [[13], Theorems 1-6], Meng and Li present the following inequalities with more general forms m−1 ∞ (b1) : up (m, n) ≤ a(m, n) + b(m, n) [c(s, t)u(s, t) + e(s, t)], s=0 t=n+1 ∞ ∞ (b2) : up (m, n) ≤ a(m, n) + b(m, n) [c(s, t)u(s, t) + e(s, t)], s=m+1 t=n+1 m−1 (b3) : up (m, n) ≤ a(m, n) + ∞ m−1 b(s, n)up (s, n) + s=0 [c(s, t)u(s, t) + e(s, t)], s=0 t=n+1 ∞ ∞ (b4) : up (m, n) ≤ a(m, n) + ∞ b(s, n)up (s, n) + s=m+1 [c(s, t)u(s, t) + e(s, t)] s=m+1 t=n+1 m−1 ∞ m−1 (b5 ) : up (m, n) ≤ a(m, n) + b(s, n)up (s, n) + s=0 L(s, t, u(s, t)), s=0 t=n+1 ∞ ∞ (b6 ) : up (m, n) ≤ a(m, n) + ∞ b(s, n)up (s, n) + s=m+1 L(s, t, u(s, t)), s=m+1 t=n+1 where p ≥ is a constant, u, a, b, c, e are nonnegative functions defined on m Ỵ N0, n Ỵ N0, and L is defined the same as in (a5)-a(6) As one can see, (b1)-(b2) are generalizations of (a1)-(a2), while (b4)-(b6) are generalizations of (a4)-(a6) More recently, Meng and Ji [[14], Theorems 3, 4, 7, 8] extended (b1)-(b4) to the following inequalities m−1 ∞ (c1) : up (m, n) ≤ a(m, n)+b(m, n) [c(s, t)uq (s, t) + d(s, t)ur (s, t) + e(s, t)], s=0 t=n+1 ∞ ∞ (c2) : up (m, n) ≤ a(m, n)+b(m, n) [c(s, t)uq (s, t) + d(s, t)ur (s, t) + e(s, t)]], s=m+1 t=n+1 m−1 (c3) : up (m, n) ≤ a(m, n)+ m−1 ∞ b(s, n)up (s, n)+ s=0 [c(s, t)uq (s, t) + d(s, t)ur (s, t) + e(s, t)], s=0 t=n+1 Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page of 17 ∞ ∞ (c4) : up (m, n) ≤ a(m, n)+ ∞ b(s, n)up (s, n)+ s=m+1 [c(s, t)uq (s, t) + d(s, t)ur (s, t) + e(s, t)] s=m+1 t=n+1 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 Ỵ N0, n Ỵ N0 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 N0 denotes the set of nonnegative integers I := [m0, ∞] ∩ ℤ and I := [n0 , ∞] Z are two fixed lattices of integral points in ℝ, where m0, n0 ẻ Let := I ì I Z2 We denote the set of all ℝ-valued functions on Ω by ℘(Ω), and denote the set of all ℝ+-valued functions on Ω by ℘+(Ω) The partial difference operators Δ1 and Δ2 on u Ỵ ℘(Ω) are defined as Δ1 u(m, n) = u(m +1, n) - u(m, n), Δ2u(m, n) = u (m, n + 1) - u(m, n) Main results Lemma 2.1 [[15]] Assume that a ≥ 0, p ≥ q ≥ 0, and p ≠ 0, then for any K >0 q ap ≤ q q−p p−q q K p a+ Kp p p 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 m−1 u(m, n) ≤ a(m, n) + b(s, n)u(s, n) s=m0 for (m, n) Ỵ Ω, then m−1 u(m, n) ≤ a(m, n) [1 + b(s, n)] s=m0 (2) Assume that a(m, n) is decreasing in the first variable If ∞ u(m, n) ≤ a(m, n) + b(s, n)u(s, n) s=m+1 for (m, n) Ỵ Ω, then ∞ u(m, n) ≤ a(m, n) [1 + b(s, n)] s=m+1 Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page of 17 Remark Lemma 2.2 is a direct variation of [[12], Lemma 2.5] Theorem 2.1 Suppose u, a, b, f, g, h, w Ỵ ℘+ (Ω), and b, f, g, h, w are nondecreasing in the first variable, while decreasing in the second variable a : I ® I is nondecreasing with a (m) ≤ m for ∀m Ỵ I, while β : I → I is nondecreasing with b(n) ≥ n for ∀n ∈ I p, q, r, l are constants with p ≥ q, p ≥ r, p ≥ l, p ≠ If for (m, n) Ỵ Ω, u(m, n) satisfies the following inequality α(m)−1 ∞ up (m, n) ≤ a(m, n)+b(m, n) s ∞ w(ξ , η)ul (ξ , η)], [f (s, t)uq (s, t) + g(s, t)ur (s, t) + h(s, t) + ξ =0 η=t s=α(m0 ) t=β(n)+1 (1) then we have ∞ m−1 u(m, n) ≤ {a(m, n) + b(m, n)H(m, n) {1+ s=m0 q q−p r r−p [f (s, t) K p + g(s, t) K p + p p [α(s + 1) − α(s)][β(t) − β(t − 1)] t=n+1 s ∞ ξ =0 η=t (2) l l−p w(ξ , η) K p ]}} p p provided H(m.n) >0, where K > is a constant, and ⎧ α(m)−1 ∞ r−p r ⎪ q q−p p−q q p−r p ⎪ r ⎪ {f (s, t)[ K p a(s, t) + K p ] + g(s, t)[ p K p a(s, t) + K ] ⎪ H(m, n) = ⎪ ⎪ p p p ⎪ ⎪ s=α(m0 ) t=β(n)+1 ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s ∞ +h(s, t) + ξ =0 η=t l l l−p p−l p w(ξ , η)[ K p a(ξ , η) + K ]}, p p (3) f = f (m, n)b(m, n), g = g(m, n)b(m, n), w = w(m, n)b(m, n) Proof Let α(m)−1 ∞ s ∞ [f (s, t)uq (s, t) + g(s, t)ur (s, t) + h(s, t) + z(m, n) = ξ =0 η=t s=α(m0 ) t=β(n)+1 f (ξ , η)ul (ξ , η)] Then we have (4) u(m, n) ≤ [a(m, n) + b(m, n)z(m, n)] p Furthermore, if given (X, Y ) ẻ , and (m, n) ẻ ([m0, X]ì[Y, ∞]) ∩ Ω, then using (4) and Lemma 2.1 we have α(m)−1 ∞ q z(m, n) ≤ r {f (s, t)[a(s, t) + b(s, t)z(s, t)] p + g(s, t)[a(s, t) + b(s, t)z(s, t)] p s=α(m0 ) t=β(n)+1 ∞ s l w(ξ , η)[a(ξ , η) + b(ξ , η)z(ξ , η)] p } + h(s, t) + ξ =0 η=t α(m)−1 ∞ q q−p p−q q {f (s, t)[ K p (a(s, t) + b(s, t)z(s, t)) + Kp] p p ≤ s=α(m0 ) t=β(n)+1 r r r−p p−r p + g(s, t)[ K p (a(s, t) + b(s, t)z(s, t)) + K ] p p s ∞ + h(s, t) + ξ =0 η=t α(m)−1 l l l−p p−l p w(ξ , η)[ K p (a(ξ , η) + b(ξ , η)z(ξ , η)) + K ]} p p ∞ = H(m, n) + s=α(m0 ) t=β(n)+1 s ∞ + ξ =0 η=t q q−p r r−p {f (s, t)b(s, t) K p z(s, t) + g(s, t)b(s, t) K p z(s, t) p p l l−p w(ξ , η)b(ξ , η) K p z(ξ , η)} p α(m)−1 ≤ H(X, Y) + ∞ q q−p r r−p {f (s, t) K p z(s, t) + g(s, t) K p z(s, t) p p s=α(m0 ) t=β(n)+1 s ∞ + ξ =0 η=t l l−p w(ξ , η) K p z(ξ , η)}, p (5) Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page of 17 where H, f , g, w are defined in (3) Let the right side of (5) be v(m, n) Then z(m, n) ≤ v(m, n), (6) and [v(m + 1, n) − v(m, n)] − [v(m + 1, n + 1) − v(m, n + 1)] α(m+1)−1 β(n+1) = s=α(m) t=β(n)+1 q q−p r r−p [f (s, t) K p z(s, t) + g(s, t) K p z(s, t) + p p α(m+1)−1 β(n+1) ≤ s=α(m) t=β(n)+1 ≤ [f (s, s=α(m) ξ =0 η=t q q−p r r−p [f (s, t) K p v(s, t) + g(s, t) K p v(s, t) + p p α(m+1)−1 β(n+1) t=β(n)+1 q q−p t) K p p + g(s, r r−p t) K p p s ∞ + ξ =0 η=t ∞ s s l l−p w(ξ , η) K p z(ξ , η)] p ∞ ξ =0 η=t l l−p w(ξ , η) K p v(ξ , η)] p l l−p w(ξ , η) K p ]v(s, t) p q q−p ≤ [α(m + 1) − α(m)][β(n + 1) − β(n)][f (α(m + 1) − 1, β(n) + 1) K p p r r−p +g(α(m + 1) − 1, β(n) + 1) K p + p α(m+1)−1 ∞ ξ =0 η=β(n)+1 l l−p w(ξ , η) K p ]v(α(m + 1) − 1, β(n) + 1) p q q−p r r−p ≤ [α(m + 1) − α(m)][β(n + 1) − β(n)][f (m, n + 1) K p + g(m, n + 1) K p p p m + ∞ l l−p w(ξ , η) K p ]v(m, n + 1) p ξ =0 η=n+1 Considering v(m, n) ≥ v(m, n + 1), we have v(m + 1, n) − v(m, n) v(m + 1, n + 1) − v(m, n + 1) − v(m, n) v(m, n + 1) q q−p r r−p ≤ [α(m + 1) − α(m)][β(n + 1) − β(n)][f (m, n + 1) K p + g(m, n + 1) K p + p p ∞ m l l−p w(ξ , η) K p ] p ξ =0 η=n+1 (7) Setting n = t in (7), and a summary with respect to t from n to r - yields v(m + 1, n) − v(m, n) v(m + 1, r) − v(m, r) − v(m, n) v(m, r) r ≤ q q−p r r−p [α(m + 1) − α(m)][β(t) − β(t − 1)][f (m, t) K p + g(m, t) K p + p p t=n+1 ∞ m l l−p w(ξ , η) K p ] p ξ =0 η=t (8) Letting r ® ∞ in (8), using v(m, ∞) = H(X, Y ) we obtain v(m + 1, n) − v(m, n) v(m, n) ∞ ≤ q q−p r r−p [α(m + 1) − α(m)][β(t) − β(t − 1)][f (m, t) K p + g(m, t) K p + p p t=n+1 ∞ m ξ =0 η=t l l−p w(ξ , η) K p ], p which is followed by ∞ m v(m + 1, n) q q−p r r−p [α(m + 1) − α(m)][β(t)−β(t−1)][f (m, t) K p +g(m, t) K p + ≤ {1+ v(m, n) p p t=n+1 ξ =0 ∞ η=t l l−p w(ξ , n) K p ]} p (9) Setting m = s in (9), and a multiple with respect to s from m0 to m - yields m−1 ∞ v(m, n) q q−p r r−p {1 + [α(s + 1) − α(s)][β(t) − β(t − 1)][f (s, t) K p + g(s, t) K p + ≤ v(m0 , n) s=m p p t=n+1 s ∞ ξ =0 η=t l l−p w(ξ , η) K p ]} p (10) Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page of 17 Considering v(m0, n) = H(X, Y ), and then combining (4), (6) and (10) we obtain ∞ m−1 u(m, n) ≤ {a(m, n) + b(m, n)H(X, Y) {1 + s=m0 [α(s + 1) − α(s)][β(t) − β(t − 1)] t=n+1 q q−p r r−p [f (s, t) K p + g(s, t) K p + p p s ∞ ξ =0 η=t (11) l l−p w(ξ , η) K p ]}} p p Setting m = X, n = Y in (11), and considering (X, Y ) Ỵ Ω is selected arbitrarily, then after substituting X, Y with m, n we obtain the desired inequality Remark If we take Ω = N0 × N0, w(m, n) ≡ 0, a (m) = m, b(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 a(m) = m, b (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 Ỵ ℘+ (Ω), and b, f, g, h, w are decreasing both in the first variable and the second variable a : I ® I is nondecreasing with a(m) ≥ m for ∀m Ỵ I, while β : I → I is nondecreasing with b(n) ≥ n for ∀n ∈ I p, q, r, l are defined as in Theorem 2.1 If for (m, n) Ỵ Ω, u(m, n) satisfies the following inequality ∞ ∞ ∞ up (m, n) ≤ a(m, n)+b(m, n) ∞ w(ξ , η)ul (ξ , η)], [f (s, t)uq (s, t) + g(s, t)ur (s, t) + h(s, t) + ξ =s η=t s=α(m)+1 t=β(n)+1 then we have ∞ u(m, n) ≤ {a(m, n) + b(m, n)H(m, n) ∞ {1 + q q−p r r−p [f (s, t) K p + g(s, t) K p + p p [α(s + 1) − α(s)][β(t) − β(t − 1)] t=n+1 s=α(m)+1 ∞ ∞ ξ =s η=t l l−p w(ξ , η) K p ]}} p p provided H(m.n) >0, where f , g, w are defined as in Theorem 2.1, and ∞ ∞ H(m, n) = s=α(m)+1 t=β(n)+1 r q q−p p−q q r r−p p−r p {f (s, t)[ K p a(s, t) + K p ] + g(s, t)[ K p a(s, t) + K ] p p p p ∞ ∞ +h(s, t) + ξ =s η=t l l l−p p−l p w(ξ , η)[ K p a(ξ , η) + K ]} p p Remark If we take Ω = N0 × N0, w(m, n) ≡ 0, a(m) = m, b(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 a(m) = m, b(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 Ỵ ℘+ (Ω), and b, f, g, h, w are nondecreasing both in the first variable and the second variable a : I ® I is nondecreasing with a(m) ≤ m for ∀m Ỵ I, while β : I → I is nondecreasing with b(n) ≤ n for ∀n ∈ I p, q, r, l are defined as in Theorem 2.1 If for (m, n) Î Ω, u(m, n) satisfies the following inequality Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page of 17 α(m)−1 β(n)−1 up (m, n) ≤ a(m, n)+b(m, n) s t w(ξ , η)ul (ξ , η)], [f (s, t)uq (s, t) + g(s, t)ur (s, t) + h(s, t) + ξ =0 η=0 s=α(m0 )t =β(n0 ) then we have m−1 u(m, n) ≤ {a(m, n) + b(m, n)H(m, n) n−1 {1 + s=m0 [α(s + 1) − α(s)][β(t) − β(t − 1)] t=n0 q q−p r r−p [f (s, t) K p + g(s, t) K p + p p s ξ =0 l l−p p p ]}} w(ξ , η) K p η=0 t provided H(m.n) >0, where f , g, w are defined as in Theorem 2.1, and α(m)−1 β(n)−1 H(m, n) = s=α(m0 ) t=β(n0 ) q {f (s, t)[ K p q−p q r−p r p a(s, t) + p − q K p ] + g(s, t)[ r K p a(s, t) + p − r K p ] p p p l l l−p p−l p p a(ξ , η) + w(ξ , η)[ K K ]} p p η=0 s t + h(s, t) + ξ =0 Theorem 2.4 Suppose u, a, b, f, g, h, w Ỵ ℘+ (Ω), and b, f, g, h, w are decreasing in the first variable, while nondecreasing in the second variable a : I ® I is nondecreasing with a(m) ≥ m for ∀m Ỵ I, while β : I → I is nondecreasing with b(n) ≤ n for ∀n ∈ I p, q, r, l are defined as in Theorem 2.1 If for (m, n) Ỵ Ω, u(m, n) satisfies the following inequality ∞ β(n)−1 up (m, n) ≤ a(m, n)+b(m, n) ∞ t w(ξ , η)ul (ξ , η)], [f (s, t)uq (s, t) + g(s, t)ur (s, t) + h(s, t) + ξ =s η=0 s=α(m)+1 t=β(n0 ) then we have ∞ u(m, n) ≤ {a(m, n) + b(m, n)H(m, n) n−1 {1 + q q−p r r−p [f (s, t) K p + g(s, t) K p + p p [α(s + 1) − α(s)][β(t) − β(t − 1)] t=n0 s=α(m)+1 ∞ ξ =s t l l−p w(ξ , η) K p ]}} p p η=0 provided H(m.n) >0, where f , g, w are defined as in Theorem 2.1, and ∞ β(n)−1 H(m, n) = s=α(m)+1 t=β(n0 ) r q q−p p−q q r r−p p−r p {f (s, t)[ K p a(s, t) + K p ] + g(s, t)[ K p a(s, t) + K ] p p p p ∞ t l l l−p p−l p w(ξ , η)[ K p a(ξ , η) + K ]} p p η=0 + h(s, t) + ξ =s Next we will study the following difference inequality: m−1 up (m, n) ≤ a(m, n) + b(s, n)up (s, n)+ s=m0 α(m)−1 ∞ s s=α(m0 ) t=β(n)+1 (12) ∞ w(ξ , η)ul (ξ , η)], [f (s, t)uq (s, t) + g(s, t)ur (s, t) + h(s, t) + ξ =0 η=t where u, a, b, f, g, h, w Ỵ ℘+(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, a, b, p, q, r, l are defined as in Theorem 2.1 Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page of 17 Theorem 2.5 If for (m, n) Ỵ Ω, u(m, n) satisfies (12), then we have ∞ m−1 u(m, n) ≤{{a(m, n) + H(m, n) {1 + [α(s + 1) − α(s)][β(t) − β(t − 1)] s=m0 q q−p [f (s, t) K p + g(s, p t=n+1 s r−p r t) K p + p ∞ ξ =0 η=t (13) l l−p w(ξ , η) K p ]}}J(m, n)} p p provided H(m.n) > 0, where K >0 is a constant, and ⎧ α(m)−1 ∞ ⎪ q q−p p−q q ⎪ ⎪ ⎪ H(m, n) = {f (s, t)[ K p a(s, t)J(s, t) + Kp] ⎪ ⎪ p p ⎪ ⎪ s=α(m0 ) t=β(n)+1 ⎪ ⎪ ⎪ ⎪ r ⎪ r r−p p−r p ⎪ ⎪ +g(s, t)[ K p a(s, t))J(s, t) + K ] ⎪ ⎪ ⎪ p p ⎪ ⎪ ⎨ s ∞ l−p l l p−l p (14) w(ξ , η)[ K p a(ξ , η)J(ξ , η) + K ]}, +h(s, t) + ⎪ ⎪ p p ⎪ ξ =0 η=t ⎪ ⎪ ⎪ ⎪ q r l ⎪ ⎪ ⎪ f (m, n) = f (m, n)J p (m, n), g(m, n) = g(m, n)J p (m, n), w(m, n) = w(m, n)J p (m, n), ⎪ ⎪ ⎪ ⎪ ⎪ m−1 ⎪ ⎪ ⎪ ⎪ J(m, n) = ⎪ [1 + b(s, n)] ⎩ s=m0 Proof: Denote α(m)−1 ∞ z(m, n) = [f (s, t)uq (s, t) + g(s, t)ur (s, t) + h(s, t) + s=α(m0 ) t=β(n)+1 s ∞ ξ =0 η=t w(ξ , η)ul (ξ , η)], and v(m, n) = a(m, n) + z(m, n) Then v(m, n) is nondecreasing in the first variable, and m−1 up (m, n) ≤ v(m, n) + b(s, n)up (s, n) (15) s=m0 By Lemma 2.2 we obtain m−1 up (m, n) ≤ v(m, n) [1 + b(s, n)] = (a(m, n) + z(m, n))J(m, n), (16) s=m0 where J(m, n) is defined in (14) Furthermore, using Lemma 2.1 we have α(m)−1 ∞ q z(m, n) ≤ r {f (s, t)[(a(s, t) + z(s, t))J(s, t)] p + g(s, t)[(a(s, t) + z(s, t))J(s, t)] p s=α(m0 ) t=β(n)+1 s ∞ l w(ξ , η)[(a(ξ , η) + z(ξ , η))J(ξ , η)] p } + h(s, t) + ξ =0 η=t α(m)−1 ∞ ≤ s=α(m0 ) t=β(n)+1 q q q−p p−q q {f (s, t)J p (s, t)[ K p (a(s, t) + z(s, t)) + Kp] p p r r r r−p p−r p + g(s, t)J p (s, t)[ K p (a(s, t) + z(s, t)) + K ] p p s ∞ + h(s, t) + ξ =0 η=t α(m)−1 l l l l−p p−l p w(ξ , η)J p (ξ , η)[ K p (a(ξ , η) + z(ξ , η)) + K ]} p p ∞ = H(m, n) + s=α(m0 ) t=β(n)+1 s ∞ + h(s, t) + ξ =0 η=t r−p q q−p r {f (s, t) K p z(s, t) + g(s, t)K p z(s, t) p p l l−p w(ξ , η)[ K p z(ξ , η)}, p where H, f , g , w are defined in (14) (17) Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page of 17 Obviously, f , g , w 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 ∞ m−1 z(m, n) ≤H(m, n) {1 + s=m0 [α(s + 1) − α(s)][β(t) − β(t − 1)] t=n+1 q q−p r r−p [f (s, t) K p + g(s, t) K p + p p s ∞ ξ =0 η=t l l−p w(ξ , η) K p ]} p (18) Combining (16) and (18) we obtain the desired result Remark If we take Ω = N0 × N0, w(m, n) ≡ 0, a(m) = m, b(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 Ỵ ℘+ (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, a, b are defined as in Theorem 2.2, and p, q, r l are defined as in Theorem 2.1 If for (m, n) Ỵ Ω, u(m, n) satisfies the following inequality ∞ up (m, n) ≤ a(m, n) + b(s, n)up (s, n)+ s=m+1 ∞ ∞ ∞ ∞ w(ξ , η)ul (ξ , η)], [f (s, t)uq (s, t) + g(s, t)ur (s, t) + h(s, t) + ξ =s η=t s=α(m)+1 t=β(n)+1 then we have ∞ u(m, n) ≤{{a(m, n) + H(m, n) ∞ {1 + [α(s + 1) − α(s)][β(t) − β(t − 1)] s=m+1 q q−p [f (s, t) K p + g(s, p t=n+1 ∞ r−p r t) K p + p ∞ ξ =s η=t l l−p w(ξ , η) K p ]}}J(m, n)} p p provided H(m.n) > 0, where ⎧ ∞ ∞ q q−p p−q q ⎪ ⎪ H(m, n) = {f (s, t)[ K p a(s, t)J(s, t) + Kp] ⎪ ⎪ ⎪ p p ⎪ ⎪ s=α(m)+1 t=β(n)+1 ⎪ ⎪ ⎪ ⎪ r ⎪ p−r p r r−p ⎪ ⎪ +g(s, t)[ K p a(s, t))J(s, t) + K ] ⎪ ⎪ p p ⎪ ⎪ ⎪ ⎨ ∞ ∞ l l l−p p−l p w(ξ , η)[ K p a(ξ , η)J(ξ , η) + +h(s, t) + K ]}, ⎪ ⎪ p p ⎪ ξ =s η=t ⎪ ⎪ ⎪ ⎪ q r l ⎪ ⎪ ⎪ f (m, n) = f (m, n)J p (m, n), g(m, n) = g(m, n)J p (m, n), w(m, n) = w(m, n)J p (m, n), ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ J(m, n) = ⎪ [1 + b(s, n)] ⎩ s=m+1 Theorem 2.7 Suppose u, a, b, f, g, h, w Ỵ ℘+ (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 Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page 10 of 17 is nondecreasing in the first variable, and b is decreasing in the second variable, a, b are defined as in Theorem 2.2, and p, q, r, l are defined as in Theorem 2.1 If for (m, n) Ỵ Ω, u(m, n) satisfies the following inequality m−1 up (m, n) ≤ a(m, n) + b(s, n)up (s, n)+ s=m0 ∞ ∞ ∞ ∞ w(ξ , η)ul (ξ , η)], [f (s, t)uq (s, t) + g(s, t)ur (s, t) + h(s, t) + ξ =s η=t s=α(m)+1 t=β(n)+1 then we have ∞ u(m, n) ≤ {{a(m, n) + H(m, n) ∞ {1 + s=m+1 q q−p r r−p [f (s, t) K p + g(s, t) K p + p p [α(s + 1) − α(s)][β(t) − β(t − 1)] t=n+1 ∞ ∞ ξ =s η=t l l−p w(ξ , η) K p ]}}J(m, n)} p p provided H(m.n) > 0, where K >0 is a constant, and ∞ ∞ H(m, n) = s=α(m)+1 t=β(n)+1 q q−p p−q q {f (s, t)[ K p a(s, t)J(s, t) + Kp] p p r p−r p r r−p + g(s, t)[ K p a(s, t))J(s, t) + K ] p p ∞ ∞ + h(s, t) + ξ =s η=t l l l−p p−l p w(ξ , η)[ K p a(ξ , η)J(ξ , η) + K ]}, p p and f (m, n), g(m, n), w(m, n), J(m, n) are defined as in Theorem 2.5 Remark If we take Ω = N0 × N0, w(m, n) ≡ 0, a(m) = m, b(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 If we take Ω = N0 × N0, w(m, n) ≡ 0, a(m) = m, b(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 α(m)−1 m−1 up (m, n) ≤ a(m, n)+ ∞ b(s, n)up (s, n)+ s=m0 s=α(m0 ) t=β(n)+1 s ∞ w(ξ , η)ul (ξ , η)], [L(s, t, u(s, t)) + (19) ξ =0 η=t where u, a, b, w Î ℘ + (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, a, b are defined as in Theorem 2.1, L : ì + đ + satisfies L(m, n, u) - L(m, n, v) ≤ M(m, n, v)(u - v) for u ≥ v ≥ 0, where M : ì + đ + p, l are defined as in Theorem 2.1 with p ≥ Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page 11 of 17 Theorem 2.8 If for (m, n) Ỵ Ω, u(m, n) satisfies (19), then ∞ m−1 u(m, n) ≤{{a(m, n) + H(m, n) 1−p [f (s, t) K p + p s {1 + s=m0 ∞ ξ =0 η=t [α(s + 1) − α(s)][β(t) − β(t − 1)] t=n+1 (20) l l−p w(ξ , η) K p ]}}J(m, n)} p , p provided that H(m.n) > 0, and f (m, n) is nondecreasing in the first variable and decreasing in the second variable, where K >0 is a constant, and ⎧ α(m)−1 ∞ ⎪ 1−p p−1 ⎪ ⎪ {L(s, t, J p (s, t)( K p a(s, t) + K p )) ⎪ H(m, n) = ⎪ ⎪ p p ⎪ ⎪ s=α(m0 ) t=β(n)+1 ⎪ ⎪ ⎪ s ∞ ⎪ l l ⎪ l l−p p−l p ⎪ ⎪ ⎪ w(ξ , η)J p (ξ , η)[ K p a(ξ , η) + + K ]}, ⎨ p p ξ =0 η=t ⎪ 1−p 1−p ⎪ 1 ⎪ ⎪ f (m, n) = M(m, n, J p (m, n)( K p a(m, n) + p − K p ))J p (m, n) K p , ⎪ ⎪ ⎪ p p p ⎪ ⎪ ⎪ ⎪ m−1 ⎪ l ⎪ ⎪ ⎪ w(m, n) = w(m, n)J p (m, n), J(m, n) = ⎪ [1 + b(s, n)] ⎩ (21) s=m0 α(m)−1 Proof: Denote z(m, n) = ∞ ∞ s [L(s, t, u(s, t)) + s=α(m0 ) t=β(n)+1 ξ =0 η=t w(ξ , η)ul (ξ , η)], and v(m, n) = a(m, n) + z(m, n) Then v(m, n) is nondecreasing in the first variable, and m−1 up (m, n) ≤ v(m, n) + b(s, n)up (s, n) (22) s=m0 By Lemma 2.2 we obtain m−1 up (m, n) ≤ v(m, n) [1 + b(s, n)] = (a(m, n) + z(m, n))J(m, n), (23) s=m0 where J(m, n) is defined in (21) Furthermore, α(m)−1 ∞ z(m, n) ≤ s ∞ {L(s, t, ((a(s, t) + z(s, t))J(s, t)) p ) + s=α(m0 ) t=β(n)+1 α(m)−1 ∞ l w(ξ , η)((a(ξ , η) + z(ξ , η))J(ξ , η)) p } ξ =0 η=t 1 1−p p−1 {L(s, t, J p (s, t)( K p (a(s, t) + z(s, t)) + K p )) p p ≤ s=α(m0 ) t=β(n)+1 s ∞ + ξ =0 η=t α(m)−1 l l l l−p p−l p w(ξ , η)J p (ξ , η)( K p (a(ξ , η) + z(ξ , η)) + K )} p p ∞ 1 1−p p−1 p {L(s, t, J (s, t)( K p (a(s, t) + z(s, t)) + K p )) p p = s=α(m0 ) t=β(n)+1 1 1−p p−1 1 K p )) + L(s, t, J p (s, t)( K − L(s, t, J p (s, t)( K p a(s, t) + p p p s ∞ + ξ =0 η=t α(m)−1 1−p p l l l 1−p p−l p p w(ξ , η)J (ξ , η)[ K p (a(ξ , η) + z(ξ , η)) + K ]} p p ∞ ≤ 1 1−p p−1 1 1−p {M(s, t, J p (s, t)( K p a(s, t) + K p ))J p (s, t) K p z(s, t) p p p s=α(m0 ) t=β(n)+1 1 1−p p−1 + L(s, t, J p (s, t)( K p a(s, t) + K p )) p p s ∞ + ξ =0 η=t a(s, t) + l l l−p p−l p w(ξ , η)J p (ξ , η)[ K p (a(ξ , η) + z(ξ , η)) + K ]} p p α(m)−1 ∞ s ∞ {f (s, t)z(s, t) + = H(m, n) + s=α(m0 ) t=β(n)+1 ξ =0 η=t l l−p w(ξ , η) K p z(ξ , η)}, p p−1 K p )) p (24) Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page 12 of 17 where H, f , w are defined in (21) Then following in a same manner as the proof of Theorem 2.1 we obtain ∞ m−1 z(m, n) ≤ H(m, n) {1 + s=m0 ∞ s 1−p [α(s + 1) − α(s)][β(t)−β(t−1)][f (s, t) K p + p t=n+1 ξ =0 η=t l l−p w(ξ , η) K p ]} p (25) The desired inequality can be deduced by the combination of (23) and (25) Theorem 2.9 Suppose u, a, b, w Ỵ ℘+(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, a, b 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 ≥ If for (m, n) Ỵ Ω, u(m, n) satisfies the following inequality ∞ up (m, n) ≤ a(m, n)+ ∞ s=m+1 ∞ ∞ b(s, n)up (s, n)+ ∞ w(ξ , η)ul (ξ , η)], [L(s, t, u(s, t)) + ξ =s η=t s=α(m)+1 t=β(n)+1 then ∞ u(m, n) ≤{{a(m, n) + H(m, n) 1−p [f (s, t) K p + p s=m+1 ∞ ∞ ξ =s η=t ∞ {1 + [α(s + 1) − α(s)][β(t) − β(t − 1)] t=n+1 l l−p w(ξ , η) K p ]}}J(m, n)} p , p provided that H(m.n) > 0, and f (m, n) is decreasing both in the first variable and the second variable, where ⎧ ∞ ∞ 1 1−p p−1 ⎪ ⎪ H(m, n) = ⎪ {L(s, t, J p (s, t)[ K p a(s, t) + K p ]) ⎪ ⎪ p p ⎪ ⎪ s=α(m)+1 t=β(n)+1 ⎪ ⎪ ⎪ ⎪ ∞ ∞ l l ⎪ ⎪ l l−p p−l p ⎪ ⎪ + w(ξ , η)J p (ξ , η)[ K p a(ξ , η) + K ]}, ⎪ ⎨ p p ξ =s η=t ⎪ ⎪ 1−p 1−p 1 ⎪ ⎪ ⎪ f (m, n) = M(m, n, J p (m, n)( K p a(m, n) + p − K p ))J p (m, n) K p , ⎪ ⎪ ⎪ p p p ⎪ ⎪ ⎪ ∞ ⎪ l ⎪ ⎪ ⎪ w(m, n) = w(m, n)J p (m, n), J(m, n) = ⎪ [1 + b(s, n)] ⎩ s=m+1 The proof for Theorem 2.8 is similar to Theorem 2.7, and we omit it here Remark If we take Ω = N0 × N0, w(m, n) ≡ 0, a(m) = m, b(n) = n, and omit the conditions “w is nondecreasing in the first variable, while decreasing in the second variable”, “ f (m, n) 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 If we take Ω = N0 × N0, w(m, n) ≡ 0, a(m) = m, b(n) = n, and omit the conditions “w is decreasing both in the first variable and the second variable”, “ f (m, n) 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] 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 Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page 13 of 17 Example Consider the following difference equation ∞ m − F2 (ξ , η, u(ξ , η)), p 12 u (m, n) = F1 (m, n + 1, u(m, n + 1)) + (26) ξ =m0 η=n+1 with the initial condition up (m, ∞) = f (m), up (m0 , n) = g(n), f (m0 ) = g(∞) = C, (27) where p ≥ is an odd number, u Ỵ ℘ (Ω), F1, F2 : ì đ Theorem 3.1 Suppose u(m, n) is a solution of (26) and (27) If |f(m) + g(n) - C| ≤ s, |F1(m, n, u)| ≤ f1(m, n)|u|, and |F2(m, n, u)| ≤ f2(m, n)|u|, where f1, f2 Ỵ ℘+(Ω), then we have ∞ m−1 |u(m, n)| ≤ {σ + H(m, n) {1+ s=m0 1−p [f1 (s, t) K p + p t=n+1 ∞ s 1−p f2 (ξ , η) K p ]}} p , (28) p ξ =0 η=t where K >0 is a constant, and m−1 H(m, n) = s=m0 ∞ s 1−p p−1 {f1 (s, t)[ K p σ + K p ]+ p p t=n+1 ξ =0 ∞ η=t 1−p p−1 f2 (ξ , η)[ K p σ + K p ]} (29) p p Proof The equivalent form of (26) and (27) is denoted by m−1 ∞ ∞ s up (m, n) = f (m) + g(n) − C + F2 (ξ , η, u(ξ , η))] (30) [F1 (s, t, u(s, t)) + ξ =0 η=t s=m0 t=n+1 Then we have m−1 ∞ |F1 (s, t, u(s, t)) + F2 (ξ , η, u(ξ , η))| ξ =0 η=t s=m0 t=n+1 m−1 ∞ s ∞ |F1 (s, t, u(s, t))| + ≤ |f (m) + g(n) − C| + |F2 (ξ , η, u(ξ , η))| (31) ξ =0 η=t s=m0 t=n+1 m−1 ∞ ≤σ + ∞ s |u(m, n)|p ≤ |f (m) + g(n) − C| + ∞ s f2 (ξ , η)|u(ξ , η)| f1 (s, t)|u(s, t)| + ξ =0 η=t s=m0 t=n+1 We note that it is unnecessary for f1, f2 being nondecreasing or decreasing since a(m) = m, b(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 |Fi(m, n, u) - Fi(m, n, v)| ≤ fi(m, n)|up - vp|, i = 1, 2, where fi Î ℘+(Ω), i = 1, 2, then (26) and (27) has at most one solution Proof Suppose u1(m, n), u2(m, n) are two solutions of (26) and (27) Then p p |u1 (m, n) − u2 (m, n)| m−1 ∞ s = | ∞ F1 (s, t, u1 (s, t)) − F1 (s, t, u2 (s, t)) + m−1 ∞ ≤ [F2 (ξ , η, u1 (ξ , η)) − F2 (ξ , η, u2 (ξ , η))]| ξ =0 η=t s=m0 t=n+1 s ∞ |F1 (s, t, u1 (s, t)) − F1 (s, t, u2 (s, t))| + ∞ ≤ s p p ∞ |f1 (s, t)|u1 (s, t) − u2 (s, t)| + s=m0 t=n+1 (32) ξ =0 η=t s=m0 t=n+1 m−1 |F2 (ξ , η, u1 (ξ , η)) − F2 (ξ , η, u2 (ξ , η))| p p |f2 (ξ , η)|u1 (ξ , η) − u2 (ξ , η)| ξ =0 η=t Treat |up (m, n) − up (m, n)| as one variable, and a suitable application of Theorem 2.1 to (32) yields |up (m, n) − up (m, n)| ≤ 0, which implies up (m, n) ≡ up (m, n) 2 Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page 14 of 17 Since p is an odd number, then we have u (m, n) ≡ u (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 F1, F2 and the initial value f (m), g(n) Theorem 3.3 Assume |Fi (m, n, u1 ) − Fi (s, t, u2 )| ≤ fi (s, t)|up − up |, i = 1,2, where f i Ỵ ℘ + (Ω), i = 1, 2, |f (m) − f (m) + g(n) − g(n) ≤ ε, where ε >0 is a constant, and ∞ m−1 furthermore, assume ∞ s {|F1 (s, t, u(s, t))−F1 (s, t, u(s, t))|+ , | F2 (ξ , η, u(ξ , η))−F2 (ξ , η, u(ξ , η))|} ≤ ε ξ =0 η=t s=m0 t=n+1 u ∈ ℘( ) is the solution of the following difference equation ∞ m − p 12 u (m, F2 (ξ , η, u(ξ , η)), n) = F1 (m, n + 1, u(m, n + 1)) + (33) ξ =0 η=n+1 with the initial condition up (m, ∞) = f (m), up (m0 , n) = g(n), f (m0 ) = g(∞) = C, (34) where F1, F2: Ω × ℝ ® ℝ, then (35) |up (m, n) − up (m, n)| ≤ (2ε) p K, provided that G(m, n) ≤ K, where m−1 ∞ ∞ s G(m, n) = {1+{ ξ =0 η=t s=m0 t=n=1 ∞ m−1 f2 (ξ , η)]} [f1 (s, t)+ {1+ s=m0 s ∞ f2 (ξ , η)]}} p [f1 (s, t) + ξ =0 η=t t=n=1 Proof The equivalent form of (33) and (34) is denoted by m−1 ∞ s up (m, n) = f (m) + g(n) − C + ∞ F2 (ξ , η, u(ξ , η))] (36) [F1 (s, t, u(s, t)) + ξ =0 η=t s=m0 t=n+1 Then from (30) and (36) we have |up (m, n) − up (m, n)| m−1 ∞ = |f (m) + g(n) − C + ∞ s F2 (ξ , η, u(ξ , η))] [F1 (s, t, u(s, t)) + ξ =0 η=t s=m0 t=n+1 m−1 ∞ −f (m) − g(n) + C − ∞ s [f (s, t, u(s, t)) − f (ξ , η, u(ξ , η))]| ξ =0 η=t s=m0 t=n+1 m−1 ∞ ≤ |f (m) − f (m) + g(n) − g(n)| + {|F1 (s, t, u(s, t)) − f (s, t, u(s, t))| s=m0 t=n+1 s ∞ |F2 (ξ , η, u(ξ , η)) − f (ξ , η, u(ξ , η))|} + (37) ξ =0 η=t m−1 ∞ ≤ε+ {|F1 (s, t, u(s, t)) − F1 (s, t, u(s, t))| + |F1 (s, t, u(s, t)) − f (s, t, u(s, t))| s=m0 t=n+1 s ∞ |F2 (ξ , η, u(ξ , η)) − F2 (ξ , η, u(ξ , η))| + |F2 (ξ , η, u(ξ , η)) − f (ξ , η, u(ξ , η))|} + ξ =0 η=t m−1 ∞ s ∞ {|F1 (s, t, u(s, t)) − F1 (s, t, u(s, t))| + ≤ 2ε + m−1 ∞ ≤ 2ε + s ∞ {f1 (s, t)|up (s, t) − up (s, t)| + s=m0 t=n+1 |F2 (ξ , η, u(ξ , η)) − F2 (ξ , η, u(ξ , η))|} ξ =0 η=t s=m0 t=n+1 f2 (ξ , η)|up (ξ , η) − up (ξ , η)|} ξ =0 η=t Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page 15 of 17 Then a suitable application of Theorem 2.1 to (37) yields the desired result Example Consider the following difference equation ∞ ∞ up (m, n) = a(m, n)+ s=m+1 ∞ ∞ b(s, n)up (s, n)+ ∞ F2 (ξ , η, u(ξ , η))], [F1 (s, t, u(s, t)) + ξ =s η=t s=m+1 t=n+1 (38) where u, a, b Ỵ ℘(Ω) with a(m, n) not equivalent to zero, p ≥ is an odd number, F1, F2 : Ω × ℝ ® ℝ Theorem 3.4 Suppose u(m, n) is a solution of (38) If |F1(m, n, u)| ≤ L(m, n, u), |F2 (m, n, u)| ≤ w(m, n)|u|l, where L is defined as in Theorem 2.8, and w Ỵ ℘+(Ω), l ≥ 0, p ≥ l, then we have ∞ ∞ |u(m, n)| ≤ {{|a(m, n)| + H(m, n) {1 + s=m+1 1−p [f (s, t) K p + p t=n+1 ∞ ∞ ξ =s η=t l l−p w(ξ , η) K p ]}J(m, n)} p , p (39) where ⎧ ∞ ∞ ⎪ 1−p p−1 ⎪ ⎪ H(m, n) = {L(s, t, J p (s, t)[ K p a(s, t) + K p ]) ⎪ ⎪ ⎪ p p ⎪ s=m+1 t=n+1 ⎪ ⎪ ⎪ ⎪ l ∞ ∞ ⎪ l ⎪ l l−p p−l p ⎪ p (ξ , η)[ K p a(ξ , η) + ⎪ + w(ξ , η)J K ]}, ⎪ ⎨ p p ξ =s η=t (40) ⎪ ⎪ ⎪ 1−p p−1 1 1−p ⎪ f (m, n) = M(m, n, J p (m, n)( K p a(m, n) + ⎪ K p ))J p (m, n) K p , ⎪ ⎪ p p p ⎪ ⎪ ⎪ ⎪ ∞ ⎪ l ⎪ ⎪ w(m, n) = w(m, n)J p (m, n), J(m, n) = ⎪ [1 + |b(s, n)|] ⎪ ⎩ s=m+1 Proof From (38) we have ∞ ∞ |u(m, n)|p ≤ |a(m, n)| + ∞ | b(s, n)||u(s, n)|p + s=m+1 ∞ ∞ ∞ | b(s, n)||u(s, n)|p + s=m+1 ∞ |F2 (ξ , η, u(ξ , η))|] ξ =s η=t s=m+1 t=n+1 ∞ ≤ |a(m, n)| + ∞ [|F1 (s, t, u(s, t))| + ∞ s=m+1 t=n+1 (41) w(ξ , η)|u(ξ , η)|l ] [L(s, t, u(s, t)) + ξ =s η=t Then a suitable application of Theorem 2.9 (with a(m) = m, b(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, F1, F2 Theorem 3.5 Suppose |Fi(m, n, u) - Fi(m, n, v)| ≤ fi(m, n)|up - vp|, i = 1, 2, where fi Ỵ ℘+(Ω), i = 1, 2, then (38) has at most one solution Theorem 3.6 Assume |Fi (m, n, u1 ) − Fi (s, t, u2 )| ≤ fi (s, t)|up − up |, i = 1, 2, where f i Ỵ ℘ + (Ω), i = 1, 2, |f (m) − f (m) + g(n) − g(n) ≤ ε, and furthermore, assume u ∈ ℘( ), u ∈ ℘( ) is the solution of the following difference equation ∞ up (m, n) = a(m, n)+ ∞ ∞ b(s, n)up (s, n)+ s=m+1 s=m+1 t=n+1 where F1, F2: ì đ , then |up (m, n) − up (m, n)| ≤ (2ε) p K, ∞ ∞ F2 (ξ , η, u(ξ , η))], [F1 (s, t, u(s, t)) + ξ =s η=t (42) Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 Page 16 of 17 provided that G(m, n) ≤ K , where ∞ ∞ ∞ G(m, n) = {{1 + { ∞ f2 (ξ , η)J(ξ , η)]}× [f1 (s, t)J(s, t) + ξ =s η=t s=m+1 t=n+1 ∞ ∞ {1 + s=m+1 ∞ ∞ t=n+1 f2 (ξ , η)]}}J(m, n)} p , [f1 (s, t) + ξ =s η=t and ∞ [1 + |b(s, n)|] f1 (m, n) = f1 (m, n)J(m, n), f2 (m, n) = w(m, n)J(m, n), J(m, n) = s=m+1 The proof for Theorems 3.5-3.6 is similar to Theorems 3.2-3.3, in which Theorem 2.6 is used Due to the limited space, we omit it here 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 Competing interests The authors declare that they have no competing interests Authors’contributions QF carried out the main part of this article All authors read and approved the final manuscript 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 Author details School of Science, Shandong University of Technology, Zhangzhou Road 12, Zibo, Shandong, 255049, China 2School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, China Received: 28 February 2011 Accepted: 15 July 2011 Published: 15 July 2011 References Zhao XQ, Zhao QX, Meng FW: On some new nonlinear discrete inequalities and their applications J Ineq Pure Appl Math 2006, 7:1-9, Article 52 Pachpatte BG: Inequalities applicable in the theory of finite differential equations J Math Anal Appl 1998, 222:438-459 Ma QH: N-independent-variable discrete inequalities of Gronwall-Ou-Iang type Ann Differen Equations 2000, 16:813-820 Pachpatte BG: On some new inequalities related to a certain inequality arising in the theory of differential equations J Math Anal Appl 2000, 251:736-751 Cheung WS, Ma QH, Pečarić J: Some discrete nonlinear inequalities and applications to difference equations Acta Math Scientia 2008, 28(B):417-430 Deng SF: Nonlinear discrete inequalities with two variables and their applications Appl Math Comput 2010, 217:2217-2225 Cheung WS, Ren JL: Discrete nonlinear inequalities and applications to boundary value problems J Math Anal Appl 2006, 319:708-724 Ma QH, Cheung WS: Some new nonlinear difference inequalities and their applications J Comput Appl Math 2007, 202:339-351 Feng et al Advances in Difference Equations 2011, 2011:21 http://www.advancesindifferenceequations.com/content/2011/1/21 10 11 12 13 14 15 Ma QH: Estimates on some power nonlinear Volterra-Fredholm type discrete inequalities and their applications J Comput Appl Math 2010, 233:2170-2180 Ma QH: Some new nonlinear Volterra-Fredholm-type discrete inequalities and their applications J Comput Appl Math 2008, 216:451-466 Pang PYH, Agarwal RP: On an integral inequality and discrete analogue J Math Anal Appl 1995, 194:569-577 Pachpatte BG: On some fundamental integral inequalities and their discrete analogues J Ineq Pure Appl Math 2001, 2:1-13, Article 15 Meng FW, Li WN: On some new nonlinear discrete inequalities and their applications J Comput Appl Math 2003, 158:407-417 Meng FW, Ji DH: On some new nonlinear discrete inequalities and their applications J Comput Appl Math 2007, 208:425-433 Jiang FC, Meng FW: Explicit bounds on some new nonlinear integral inequality with delay J Comput Appl Math 2007, 205:479-486 doi:10.1186/1687-1847-2011-21 Cite this article as: Feng et al.: Some new finite difference inequalities arising in the theory of difference equations Advances in Difference Equations 2011 2011:21 Submit your manuscript to a journal and benefit from: Convenient online submission Rigorous peer review Immediate publication on acceptance Open access: articles freely available online High visibility within the field Retaining the copyright to your article Submit your next manuscript at springeropen.com Page 17 of 17 ... doi:10.1186/1687-1847-2011-21 Cite this article as: Feng et al.: Some new finite difference inequalities arising in the theory of difference equations Advances in Difference Equations 2011 2011:21 Submit your manuscript... decreasing in the second variable”, “ f (m, n) 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... variable, a is decreasing in the first variable, and b is decreasing in the second variable, a, b 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