Báo cáo hoa học: "Research Article An Extension to Nonlinear Sum-Difference Inequality and Applications"

Hindawi Publishing Corporation Advances in Difference Equations Volume 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 Department of Mathematics, Hechi University, Yizhou, Guangxi 546300, China Department of Mathematics, Guangdong Ocean University, Zhanjiang 524088, China Correspondence should be addressed to Xiaoliang Zhou, zjhdzxl@yahoo.com.cn 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 Cheung 2006 Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence Copyright q 2009 W.-S Wang and X Zhou 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 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 inequalities e.g., 16–18 Starting from the basic form un ≤ an  n−1  fsus, 1.1 s0 discussed in 19, an interesting direction is to consider the inequality n−1    αu2 s  Qgsus , u2 n ≤ P u2 0  s0 1.2 Advances in Difference Equations a discrete version of Dafermos’ inequality 20, 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 Agarwal 21 proved for 1.2 that un ≤ 1  αn P u0  n−1 s0 Qgs for all ≤ n ≤ T Another form of sum-difference inequality n−1    u2 n ≤ c2  f1 suswus  f2 sus 1.3 s0  n−1 estimated by Pachpatte 22 as un ≤ Ω−1 Ωc  n−1 s0 f2 s  s0 f1 s, where Ωu : was u ds/ws Recently, Pachpatte 23, 24 discussed the inequalities of two variables u0 um, n ≤ c  m−1 n−1    us, t as, t log us, t  bs, tg log us, t , s0 t0 um, n ≤ c  m−1 n−1  f1 s, tgus, t  s0 t0 s0 t0 κs, t, σ, τguσ, τ σ0 τ0 1.4 ⎞⎞ ⎛ τ−1 s−1  σ−1  t−1   ⎝ ⎝ h s, t, σ, τ, ξ, η g u ξ, η ⎠⎠, ⎛  s−1  t−1  m−1 n−1  m−1 n−1  s0 t0 σ0 τ0 ξ0 η0 where g is nondecreasing In 25 another form of inequality of two variables u2 m, n ≤ c2  m−1 n−1   as, tus, t  sm0 tn0 m−1 n−1   bs, tus, twus, t 1.5 sm0 tn0 was discussed Later, this result was generalized in 26 to the inequality up m, n ≤ c  m−1 n−1   ds, tuq s, t  sm0 tn0 m−1 n−1   es, tuq s, twus, t, 1.6 sm0 tn0 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 in Z2 , and w is a continuous nondecreasing function satisfying wu > for all u > In this paper we establish a more general form of sum-difference inequality with positive integers m, n, ψum, n ≤ am, n  k m−1 n−1    fi m, n, s, tϕi us, t, 1.7 i1 sm0 tn0 where k ≥ In 1.7 we replace the constant c, the functions up , ds, t, es, t, uq and uq wu in 1.6 with a function am, n, more general functions ψu, f1 m, n, s, t, f2 m, n, s, t, ϕ1 u and ϕ2 u, respectively Moreover, we consider more than two nonlinear terms and not require the monotonicity of every ϕi i  1, 2, , k We employ a technique of Advances in Difference Equations monotonization to construct a sequence of functions which possesses stronger monotonicity than the previous one Unlike the work in 26 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 inequalities 1.6 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 Main Result Throughout this paper, let R  −∞, ∞, 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 ⊂ N20 , and let Λs,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 Δ1 gm, n  gm  1, n − gm, n and Δ2 gm, n  gm, n  1 − gm, n Obviously, the linear difference equation Δxm  bm with the initial condition xm0   has the solution m0 −1 m−1 sm0 bs In the sequel, for convenience, we complementarily define that sm0 bs  We give the following basic assumptions for the inequality 1.7 H1  ψ is a strictly increasing continuous function on R satisfying that ψ∞  ∞ and ψu > for all u > H2  All ϕi i  1, 2, , k are continuous and positive functions on R H3  am, n ≥ on Λ H4  All fi i  1, 2, , k are nonnegative functions on Λ × Λ With given functions ϕ1 , ϕ2 , and ψ, we technically consider a sequence of functions wi s, which can be calculated recursively by   w1 s : max ϕ1 τ , τ∈0,s  wi1 s : max τ∈0,s 2.1  ϕi1 τ wi s, wi τ i  1, , k − For given constants ui > and variable u > 0, we define Wi u, ui  : u dx −1 , w ψ x ui i i  1, 2, , k 2.2 Obviously, Wi is strictly increasing in u > and therefore the inverses Wi−1 are well defined, continuous, and increasing Let fi m, n, s, t : max τ,ξ∈m0 ,m×n0 ,n fi τ, ξ, s, t, 2.3 which is nondecreasing in m and n for each fixed s and t and satisfies fi x, y, t, s ≥ fi x, y, t, s ≥ for all i  1, , k 4 Advances in Difference Equations Theorem 2.1 Suppose that H1 –H4  hold and um, n is a nonnegative function on Λ satisfying 1.7 Then, for m, n ∈ ΛM1 ,N1  , a sublattice in Λ,  um, n ≤ ψ −1  Wk−1 m−1 n−1   Wk Υk m, n   fk m, n, s, t , 2.4 sm0 tn0 where Υk m, n is determined recursively by Υ1 m, n : am0 , n0   m−1  |as  1, n0  − as, n0 |  sm0 Υi1 m, n : |am, t  1 − am, t|, tn0  Wi−1 n−1  Wi Υi m, n  m−1 n−1   2.5  fi m, n, s, t , i  1, , k − 1, sm0 tn0 and M1 , N1  ∈ Λ is arbitrarily given on the boundary of the lattice   dx fi m, n, s, t ≤ −1 , i  1, 2, , k m, n ∈ Λ : Wi Υi m, n  ui wi ψ x sm0 tn0 2.6 U : ∞ m−1 n−1   Remark 2.2 As explained in 3, Remark 2, since different choices of ui in Wi i  1, 2, , k not affect our results, we simply let Wi u denote Wi u, ui  when there is no confusion For  i u  u dx/wi ψ −1 x Obviously, W i u  Wi u  W i ui   ui , let W positive constants vi / vi i ui  It follows that  −1 v  W −1 v − W and W i i   −1 W i i Υi m, n  W m−1 n−1    fi m, n, s, t  sm0 tn0  Wi−1 Wi Υi m, n  m−1 n−1    fi m, n, s, t , sm0 tn0 2.7 i , i  1, 2, , k that is, we obtain the same expression in 2.4 if we replace Wi with W  Moreover, by replacing Wi with Wi , the condition in the definition of U in Theorem 2.1 reads i Υi M1 , N1   W M −1 N −1   fi m, n, s, t ≤ sm0 tn0 ∞ dx , −1 vi wi ψ x i  1, 2, , k, 2.8 the left-hand side of which is equal to i ui   Wi Υi M1 , N1   W M −1 N −1   sm0 tn0 fi m, n, s, t, 2.9 Advances in Difference Equations and the right-hand side of which equals  ui dx −1  vi wi ψ x ∞ dx i ui   −1  W ui wi ψ x ∞ dx −1 ui wi ψ x 2.10 The comparison between the both sides implies that 2.8 is equivalent to the condition given in the definition of U in Theorem 2.1 with m, n  M1 , N1  Remark 2.3 If we choose k  2, ψu  up , ϕ1 u  uq , ϕ2 u  uq wu with p > q > 0, f1 m, n, s, t  ds, t and f2 m, n, s, t  es, t and restrict am, n to be a constant c in 1.7, then we can apply Theorem 2.1 to inequality 1.6 discussed in 26 Proof of Theorem First of all, we monotonize some given functions ϕi , fi in the sums Obviously, the sequence wi s defined by ϕi i  1, , k in 2.1 consists of nondecreasing nonnegative functions and satisfies wi s ≥ ϕi s, for i  1, , k Moreover, wi ∝ wi1 , i  1, 2, , k − 1, 3.1 as defined in 27 for comparison of monotonicity of functions wi s i  1, , k, because every ratio wi1 s/wi s is nondecreasing By the definitions of functions wi , fi , ψ, and Υ1 , from 1.7 we get  um, n ≤ ψ −1 Υ1 m, n  k m−1 n−1      fi m, n, s, twi us, t , ∀m, n ∈ Λ 3.2 i1 sm0 tn0 Then, we discuss the case that am, n > for all m, n ∈ Λ Because Υ1 satisfies m−1  Υ1 m, n  am0 , n0   |as  1, n0  − as, n0 |  sm0 n−1  |am, t  1 − am, t| tn0 3.3 ≥ am, n, it is positive and nondecreasing on Λ We consider the auxiliary inequality to 3.2, for all m, n ∈ ΛM,N ,  um, n ≤ ψ −1 Υ1 M, N  k m−1 n−1    i1 sm0 tn0   fi M, N, s, twi us, t , 3.4 Advances in Difference Equations where M ∈ dism0 , M1  and N ∈ disn0 , N1  are chosen arbitrarily, and claim that, for m, n ∈ Λmin{M2 ,M},min{N2 ,N} , a sublattice in ΛM1 ,N1  ,  um, n ≤ ψ −1  Wk−1 Wk n−1  m−1    k M, N, m, n  fk M, N, s, t Υ   , 3.5 sm0 tn0  k M, N, m, n is determined recursively by where Υ  M, N, m, n : Υ1 M, N, Υ   n−1  m−1    −1    Υi1 M, N, m, n : Wi Wi Υi M, N, m, n  fi M, N, s, t , 3.6 sm0 tn0 i  1, 2, , k − 1, and M2 , N2  ∈ ΛM1 ,N1  is arbitrarily chosen on the boundary of the lattice  U1 : n−1  m−1     i M, N, m, n  fi M, N, s, t m, n ∈ Λ : Wi Υ sm0 tn0  dx ≤ −1 , i  1, 2, , k ui wi ψ x ∞ 3.7 We note that M2 , N2 can be chosen appropriately such that M2 M, N  M1 , N2 M, N  N1 , ∀M, N ∈ ΛM1 ,N1  3.8 In fact, from the fact of M1 , N1  being on the boundary of the lattice U, we see that  M  −1 N −1   i M1 , N1 , M1 , N1   Wi Υ fi M1 , N1 , s, t sm0 tn0  Wi Υi M1 , N1   ≤ ∞ M −1 N −1   fi M1 , N1 , s, t 3.9 sm0 tn0 dx , −1 ui wi ψ x i  1, 2, , k 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 prove 3.5 n−1   For k  1, let zm, n  m−1 sm0 tn0 f1 M, N, s, tw1 us, t Then z is nonnegative and nondecreasing in each variable on ΛM,N From 3.4 we observe that um, n ≤ ψ −1 Υ1 M, N  zm, n, ∀m, n ∈ ΛM N 3.10 Advances in Difference Equations Moreover, we note that w1 is nondecreasing and satisfies w1 u > for u > and that Υ1 M, N  zm, n > From 3.10 we have n−1  Δ1 Υ1 M, N  zm, n tn0 f1 M, N, m, tw1 um, t  −1 w1 ψ Υ1 M, N  zm, n w1 ψ −1 Υ1 M, N  zm, n ≤ n−1  3.11 f1 M, N, m, t tn0 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 ∈ ΛM N there exists ξ in the open interval Υ1 M, N  zm, n, Υ1 M, N  zm  1, n such that W1 Υ1 M, N  zm  1, n − W1 Υ1 M, N  zm, n  Υ1 M,Nzm1,n du  −1 w ψ u Υ1 M,Nzm,n  Δ1 Υ1 M, N  zm, n w1 ψ −1 ξ ≤ Δ1 Υ1 M, N  zm, n w1 ψ −1 Υ1 M, N  zm, n 3.12 It follows from 3.11 and 3.12 that W1 Υ1 M, N  zm  1, n − W1 Υ1 M, N  zm, n ≤ n−1  f1 M, N, m, t 3.13 tn0 Substituting m with s and summing both sides of 3.13 from s  m0 to m − 1, we get, for all m, n ∈ ΛM N , W1 Υ1 M, N  zm, n ≤ W1 Υ1 M, N  m−1 n−1   f1 M, N, s, t 3.14 sm0 tn0 We note from the definition of zm, n in 3.2 and the definition of zm0 , n  By the monotonicity of W −1 and 3.10 we obtain  um, n ≤ ψ −1 W1−1 W1 Υ1 M, N  that is, 3.5 is true for k  m−1 n−1   sm0 tn0 m0 −1 sm0 in Section that  f1 M, N, s, t , ∀m, n ∈ ΛM N , 3.15 Advances in Difference Equations Next, we make the inductive assumption that 3.5 is true for k  l Consider  um, n ≤ ψ −1 Υ1 M, N  l1 m−1 n−1      fi M, N, s, twi us, t , 3.16 i1 sm0 tn0 l1 m−1 n−1  for all m, n ∈ ΛM N Let ym, n  sm0 tn0 fi M, N, s, twi us, t, which is i1 nonnegative and nondecreasing in each variable on ΛM,N Then 3.16 is equivalent to um, n ≤ ψ −1 Υ1 M, N  ym, n , ∀m, n ∈ ΛM N 3.17 Since wi is nondecreasing and satisfies wi u > for u > i  1, 2, , l  1 and Υ1 K, L  ym, n > 0, from 3.17 we obtain, for all m, n ∈ ΛM N , n−1  Δ1 Υ1 M, N  ym, n tn0 f1 M, N, m, tw1 um, t  −1 w1 ψ Υ1 M, N  ym, n w1 ψ −1 Υ1 M, N  ym, n l1 n−1  tn fi M, N, m, twi um, t i2  −10 w1 ψ Υ1 M, N  ym, n ≤ n−1  f1 M, N, m, t  tn0 n−1 l   fi1 M, N, m, tφi1 um, t, i1 tn0 3.18 where φi u : wi u , w1 u i  2, 3, , l  3.19 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 ∈ ΛM,N there exists ξ in the open interval Υ1 M, N  ym, n, Υ1 M, N  ym  1, n such that W1 Υ1 M, N  ym  1, n − W1 Υ1 M, N  ym, n  Υ1 M,Nym1,n du −1 w ψ u Υ1 M,Nym,n Δ1 Υ1 M, N  ym, n  w1 ψ −1 ξ Δ1 Υ1 M, N  ym, n ≤ w1 ψ −1 Υ1 M, N  ym, n  3.20 Advances in Difference Equations Therefore, it follows from 3.18 and 3.20 that W1 Υ1 M, N  ym  1, n − W1 Υ1 M, N  ym, n ≤ n−1  f1 M, N, m, t  tn0 n−1 l   3.21 fi1 M, N, m, tφi1 um, t i1 tn0 substituting m with s in 3.21 and summing both sides of 3.21 from s  m0 to m − 1, we get, for all m, n ∈ ΛM,N , W1 Υ1 M, N  ym, n − W1 Υ1 M, N ≤ m−1 n−1   f1 M, N, s, t  sm0 tn0 l m−1 n−1    fi1 M, N, s, tφi1 us, t, 3.22 i1 sm0 tn0 where we note that ym0 , n  For convenience, let ψΞm, n : W1 Υ1 M, N  ym, n , θM, N, m, n : W1 Υ1 M, N  m−1 n−1   3.23 f1 M, N, s, t sm0 tn0 From 3.17 and 3.22 we can get  Ξm, n ≤ ψ −1 θM, N, M, N  l m−1 n−1    i1 sm0 tn0     −1 −1  , fi1 M, N, s, tφi1 ψ W1 ψΞm, n 3.24 the same form as 3.4 for k  l, for all m, n ∈ ΛM,N , where we note that θM, N, M, N ≥ θM, N, m, n for all m, n ∈ ΛM,N We are ready to use the inductive assumption for 3.24 In order to demonstrate the basic condition of monotonicity, let hs  ψ −1 W1−1 ψs, obviously which is a continuous and nondecreasing function on R Thus each φi hs is continuous and nondecreasing on R and satisfies φi hs > for s > Moreover,   φi1 hs wi1 hs ϕi1 τ   max , φi hs wi hs wi τ τ∈0,hs 3.25 which is also continuous nondecreasing on R and positive on R This implies that φi hs ∝ φi1 hs, for i  2, , l Therefore, the inductive assumption for 3.5 can be used to 3.24 and we obtain, for all m, n ∈ Λmin{M,M3 },min{N,N3 } ,  Ξm, n ≤ ψ −1 Φ−1 l1  Φl1 θl1 M, N, m, n  m−1 n−1   sm0 tn0  fl1 M, N, s, t , 3.26 10 Advances in Difference Equations u where Φi u : ui  ds/φi hs, u > 0, u  ψ −1 W1 u, Φ−1 i is the inverse of Φi for i  2, 3, , l  1, θl1 M, N, m, n is determined recursively by θ1 M, N, m, n : θM, N, M, N,  θi1 M, N, m, n : Φ−1 i m−1 n−1   Φi θi M, N, m, n   fi M, N, s, t , i  1, 2, , l, 3.27 sm0 tn0 and M3 , N3 are functions of M, N such that M3 M, N, N3 M, N ∈ ΛM1 ,N1  lie on the boundary of the lattice  U2 : m, n ∈ Λ : Φi θi M, N, m, n  m−1 n−1   fi M, N, s, t sm0 tn0  ds , i  2, 3, , l  , ≤ ui  φi hs  ∞ 3.28 where ∞ denotes either limu → ∞ u if it converges or ∞ Note that u ds W1−1 ψs ui  θ u w1 ψ −1 W1−1 ψs ds  −1 −1 W1 ψs ui  wi ψ Φi u  ψ −1  W −1 ψu 3.29 dx −1 w ψ x i ui    Wi W1−1 ψu , i  2, 3, , l   Thus, from 3.17, 3.23, and 3.27, 3.26 can be equivalently written as   um, n ≤ ψ −1 W1−1 ψΞm, n     −1 ≤ ψ −1 Wl1 Wl1 W1−1 ψθl1 M, N, m, n  m−1 n−1    fl1 M, N, s, t , 3.30 ∀m, n ∈ Λmin{M,M3 },min{N,N3 } sm0 tn0  i M, N, m, n, defined We further claim that the term W1−1 ψθi M, N, m, n is the same as Υ in 3.6, i  1, 2, , l  For convenience, let θi M, N, m, n  W1−1 ψθi M, N, m, n  M, N, m, n Obviously, it is that θ1 M, N, m, n  Υ Advances in Difference Equations 11 The remainder case is that am, n  for some m, n ∈ Λ Let Υ1,ε m, n  Υ1 m, n  ε, 3.31 where ε > is an arbitrary small number Obviously, Υ1,ε m, n > for all m, n ∈ Λ Using the same arguments as above and replacing Υ1 m, n with Υ1,ε m, n, we get  um, n ≤ ψ −1 W2−1  W2 W1−1 W1 Υ1, m, n  m−1 n−1   f1 s, t sm0 tn0  m−1 n−1   sm0 tn0  f2 s, t 3.32 for all m, n ∈ Λm1 ,n1  Considering continuities of Wi and Wi−1 for i  1, as well as of Υi,ε in ε and letting ε → 0 , we obtain 2.4 This completes the proof We remark that m1 , n1 lie on the boundary of the lattice U In particular, 2.4 is true ∞ for all m, n ∈ Λ when every wi i  1, 2 satisfies ui dx/wi ψ −1 x  ∞ Therefore, we may take m1  M, n1  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: Δ1 Δ2 ψzm, n  Fm, n, zm, n, m, n ∈ Λ, 4.1 zm, n0   fm, zm0 , n  gn, m, n ∈ Λ, where Λ : I × J is defined as in the beginning of Section 2, ψ ∈ C0 R, R is strictly increasing odd function satisfying ψu > for u > 0, F : Λ × R → R satisfies |Fm, n, u| ≤ h1 m, nϕ1 |u|  h2 m, nϕ2 |u|, 4.2 for given functions h1 , h2 : Λ → R and ϕi ∈ C0 R , R  i  1, 2 satisfying ϕi u > for u > 0, and functions f : I → R and g : J → R satisfy that fm0   gn0   Obviously, 4.1 is a generalization of the BVP problem considered by 26, Section 3, and the theorems of 26 are not able to solve it In the following we first apply our main result to the discussion of boundedness of 4.1 12 Advances in Difference Equations Corollary 4.1 All solutions zm, n of BVP 4.1 have the following estimation for all m, n ∈ Λm1 ,n1   |zm, n| ≤ ψ −1  W2−1 W2 Υ2 m, n  m−1 n−1    h2 s, t , 4.3 sm0 tn0 where m1 , n1 are given as in Theorem 2.1 and W2 u  u dx   ,   −1  −1  maxτ∈0,x ϕ1 ψ −1 τ maxτ∈0,x ϕ2 ψ τ /maxτ1 ∈0,τ ϕ1 ψ τ1  u dx  −1  , max τ∈0,x ϕ1 ψ τ   m−1 n−1   −1 Υ2 m, n  W1 W1 Υ1 m, n  h1 t, s , W1 u  sm0 tn0 Υ1 m, n ≤ m−1  n−1     ψ fs  1 − ψ fs   ψ gt  1 − ψ gt  sm0 tn0 4.4 Proof Clearly, the difference equation of BVP 4.1 is equivalent to n−1   m−1 ψzm, n  ψ fm  ψ gn  Fs, t, zs, t 4.5 sm0 tn0 It follows, by 4.2, that n−1       m−1 ψzm, n ≤ ψ fm  ψ gn   h1 s, tϕ1 |zs, t| sm0 tn0  m−1 n−1   4.6 h2 s, tϕ2 |zs, t| sm0 tn0 Let am, n  |ψfm  ψgn| Since |ψzm, n|  ψ|zm, n|, 4.6 is of the form 1.6 Applying our Theorem 2.1 to inequality 4.6, we obtain the estimate of zm, n as given in this corollary Corollary 4.1 gives a condition of boundedness for solutions Concretely, if Υ1 m, n < ∞, m−1 n−1   sm0 tn0 h1 s, t < ∞, m−1 n−1   h2 s, t < ∞ sm0 tn0 for all m, n ∈ Λm1 ,n1  , then every solution zm, n of BVP 4.1 is bounded on Λm1 ,n1  4.7 Advances in Difference Equations 13 Next, we discuss the uniqueness of solutions for BVP 4.1 Corollary 4.2 Suppose additionally that     |Fm, n, u1  − Fm, n, u2 | ≤ h1 m, nϕ1 ψu1  − ψu2   h2 m, nϕ2 ψu1  − ψu2  4.8 for u1 , u2 ∈ R and m, n ∈ Λ : I × J, where I  m0 , M ∩ N0 , J  n0 , N ∩ N0 as assumed in the beginning of Section with natural numbers M and N, h1 , h2 are both nonnegative functions defined on the lattice Λ, ϕ1 , ϕ2 ∈ C0 R , R  are both nondecreasing with the nondecreasing ratio ϕ2 /ϕ1 such 1 that ϕi 0  0, ϕi u > for all u > and ds/ϕi s  ∞ for i  1, and ψ ∈ C0 R, R is strictly increasing odd function satisfying ψu > for u > Then BVP 4.1 has at most one solution on Λ Proof Assume that both zm, n and zm, n are solutions of BVP 4.1 From the equivalent form 4.5 of 4.1 we have n−1    m−1    ψzm, n − ψ zm, n ≤ zs, t h1 s, tϕ1 ψzs, t − ψ sm0 tn0  m−1 n−1     zs, t h2 s, tϕ2 ψzs, t − ψ 4.9 sm0 tn0 for all m, n ∈ Λ, which is an inequality of the form 1.7, where am, n ≡ Applying our Theorem 2.1 with the choice that u1  u2  1, we obtain an estimate of the difference |ψzm, n − ψ zm, n| in the form 2.4, where Υ1 m, n ≡ because am, n ≡ Furthermore, by the definition of Wi we see that lim Wi u  −∞, u→0 lim Wi−1 u  0, u → −∞ i  1, 4.10 It follows that W1 Υ1 m, n  m−1 n−1   h1 s, t  −∞, 4.11 sm0 tn0 since m < M, n < N Thus, by 4.10,  Υ2 m, n  W1−1 W1 Υ1 m, n  m−1 n−1   sm0 tn0  h1 s, t  4.12 14 Advances in Difference Equations Similarly, we get W2 Υ2 m, n  m−1 n−1 sm0 tn0 h2 s, t  −∞ and therefore  W2−1 W2 Υ2 m, n  m−1 n−1    h2 s, t  4.13 sm0 tn0 Thus we conclude from 2.4 that |ψzm, n − ψ zm, n| ≤ 0, implying that zm, n  zm, n for all m, n ∈ Λ since ψ is strictly increasing It proves the uniqueness Remark 4.3 If h1 ≡ or h2 ≡ in 4.8, the conclusion of the 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  Δ1 Δ2 ψzm, n  Fm, n, zm, n, m, n ∈ Λ,  zm, n0   fm, m, n ∈ Λ, zm0 , n  g n, 4.14 where ψ ∈ C0 R, R is strictly increasing odd function satisfying ψu > for u > 0, F ∈    g n0   C0 Λ × R, R, and f : I → R, g : J → R are functions satisfying fm Corollary 4.4 Let F be a function as assumed in the beginning of Section and satisfy 4.2 and 4.8 on the same lattice Λ as assumed in Corollary 4.2 Suppose that the three differences     maxf − f , m∈I   max g − g , n∈J max s,t,u∈Λ×R     Fs, t, u − Fs, t, u 4.15 are all sufficiently small Then solution zm, n of BVP 4.14 is sufficiently close to the solution zm, n of BVP 4.1 Proof By Corollary 4.2, the solution zm, n is unique By the continuity and the strict monotonicity of ψ, we suppose that       maxψ fm − ψ fm  < , m∈I max   maxψ g n − ψ gn  < , n∈J     Fs, t, u − Fs, t, u < , s,t,u∈I×J×R 4.16 Advances in Difference Equations 15 where  > is a small number By the equivalent difference equation 4.5 and the inequality 4.8 we get        ψ zm, n − ψzm, n  ≤ ψ fm − ψ fm  ψ g n − ψ gn      Fs, t, zs, t − Fs, t, zs, t m−1 n−1    sm0 tn0 ≤ 2     Fs, t, zs, t − Fs, t, zs, t m−1 n−1    sm0 tn0  m−1 n−1   |Fs, t, zs, t − Fs, t, zs, t| 4.17 sm0 tn0 ≤ {2  m1 − m0 n1 − n0 }  m−1 n−1     zs, t − ψzs, t h1 s, tϕ1 ψ sm0 tn0  m−1 n−1     zs, t − ψzs, t , h2 s, tϕ2 ψ sm0 tn0 that is an inequality of the form 1.7 Applying Theorem 2.1 to 4.17, we obtain, for all m, n ∈ Λm1 ,n1  , that   m−1 n−1     ψ zm, n − ψzm, n  ≤ W −1 W2 Υ2 m, n  h2 s, t , 4.18 sm0 tn0 where m1 , n1 are given as in Theorem 2.1,  Υ2 m, n  W1−1 W1 Υ1 m, n  m−1 n−1    h1 t, s , sm0 tn0 4.19 Υ1 m, n  {2  m1 − m0 n1 − n0 } By 4.10 we see that Υi m, n → i  1, 2 as  → It follows from 4.18 that zm, n − ψzm, n|  and hence zm, n depends continuously on F, f, and lim → |ψ g Remark 4.5 Our requirement of the small difference F − F in Corollary 4.4 is stronger than the condition iii in 26, Theorem 3.3, but 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