Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 971540, 18 pages doi:10.1155/2010/971540 Research Article Existence and Uniqueness of Positive Solutions for Discrete Fourth-Order Lidstone Problem with a Parameter Yanbin Sang, 1, 2 Zhongli Wei, 2, 3 and Wei Dong 4 1 Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China 2 School of Mathematics, Shandong University, Jinan, Shandong 250100, China 3 Department of Mathematics, Shandong Jianzhu University, Jinan, Shandong 250101, China 4 Department of Mathematics, Hebei University of Engineering, Handan, Hebei 056021, China Correspondence should be addressed to Yanbin Sang, sangyanbin@126.com Received 9 January 2010; Revised 23 March 2010; Accepted 26 March 2010 Academic Editor: A. Pankov Copyright q 2010 Yanbin Sang et al. 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. This work presents sufficient conditions for the existence and uniqueness of positive solutions for a discrete fourth-order beam equation under Lidstone boundary conditions with a parameter; the iterative sequences yielding approximate solutions are also given. The main tool used is monotone iterative technique. 1. Introduction In this paper, we are interested in the existence, uniqueness, and iteration of positive solutions for the following nonlinear discrete fourth-order beam equation under Lidstone boundary conditions with explicit parameter β given by Δ 4 y  t − 2  − βΔ 2 y  t − 1   h  t   f 1  y  t    f 2  y  t   ,t∈  a  1,b− 1  Z , 1.1 y  a   0 Δ 2 y  a − 1  ,y  b   0 Δ 2 y  b − 1  , 1.2 where Δ is the usual forward difference operator given by Δytyt  1 − yt, Δ n yt Δ n−1 Δyt, c, d Z : {c, c  1, ,d− 1,d},andβ>0 is a real parameter. In recent years, the theory of nonlinear difference equations has been widely applied to many fields such as economics, neural network, ecology, and cybernetics, for details, see 2 Advances in Difference Equations 1–7 and references therein. Especially, there was much attention focused on the existence and multiplicity of positive solutions of fourth-order problem, for example, 8–10,andin particular the discrete problem with Lidstone boundary conditions 11–17. However, very little work has been done on the uniqueness and iteration of positive solutions of discrete fourth-order equation under Lidstone boundary conditions. We would like to mention some results of Anderson and Minh ´ os 11 and He and Su 12, which motivated us to consider the BVP 1.1 and 1.2. In 11, Anderson and Minh ´ os studied the following nonlinear discrete fourth-order equation with explicit parameters β and λ given by Δ 4 y  t − 2  − βΔ 2 y  t − 1   λf  t, y  t   ,t∈  a  1,b− 1  Z , 1.3 with Lidstone boundary conditions 1.2, where β>0andλ>0 are real parameters. The authors obtained the following result. Theorem 1.1 see 11. Assume that the following condition is satisfied A 1  ft, ygtwy,whereg : a  1,b − 1 Z → 0, ∞ with  b−1 za1 gz > 0, w : 0, ∞ → 0, ∞ is continuous and nondecreasing, and there exists θ ∈ 0, 1 such that wκy ≥ κ θ wy for κ ∈ 0, 1 and y ∈ 0, ∞, then, for any λ ∈ 0, ∞,theBVP1.3 and 1.2 has a unique positive solution y λ . Furthermore, such a solution y λ satisfies the following properties: i lim λ → 0  y λ   0 and lim λ →∞ y λ   ∞; ii y λ is nondecreasing in λ; iii y λ is continuous in λ,thatis,ifλ → λ 0 ,theny λ − y λ 0 →0. Very recently, in 12, He and Su investigated the existence, multiplicity, and nonexistence of nontrivial solutions to the following discrete nonlinear fourth-order boundary value problem Δ 4 u  t − 2   ηΔ 2 u  t − 1  − ξu  t   λf  t, u  t  ,t∈ Z  a  1,b 1  , u  a   0 Δ 2 u  a − 1  ,u  b  2   0 Δ 2 u  b  1  , 1.4 where Δ denotes the forward difference operator defined by Δutut  1 − ut, Δ n ut ΔΔ n−1 ut, Za  1,b 1 is the discrete interval given by {a  1,a 2, ,b 1} with a and b a<b integers, η, ξ, λ are real parameters and satisfy η<8sin 2 π 2  b − a  2  ,η 2 4ξ ≥ 0,ξ4η sin 2 π 2  b − a  2  < 16 sin 4 π 2  b − a  2  ,λ>0. 1.5 For the function f, the authors imposed the following assumption: B 1  ft, xgthx, where g : Za  1,b  1 → 0, ∞ with  b1 ta1 gt > 0, h : R → 0, ∞ is continuous and nondecreasing, and there exists θ ∈ 0, 1 such that hμx ≥ μ θ hx for μ ∈ 0, 1 and x ∈ 0, ∞. Advances in Difference Equations 3 Their main result is the following theorem. Theorem 1.2 see 12 . Assume that B 1  holds. Then for any λ ∈ 0, ∞,theBVP1.4 has a unique positive solution u λ . Furthermore, such a solution u λ satisfies the properties (i)–(iii) stated in Theorem 1.1. The aim of this work is to relax the assumptions A 1  and B 1  on the nonlinear term, without demanding the existence of upper and lower solutions, we present conditions for the BVP 1.1 and 1.2 to have a unique solution and then study the convergence of the iterative sequence. The ideas come from Zhai et al. 18, 19 and Liang 20. Let B denote the Banach space of real-valued functions on a − 1,b  1 Z ,withthe supremum norm   y    sup t∈a−1,b1 Z   y  t    . 1.6 Throughout this paper, we need the following hypotheses: H 1  f i : 0, ∞ → 0, ∞ are continuous and f i y > 0fory>0 i  1, 2; H 2  h : a  1,b− 1 Z → 0, ∞ with  b−1 za1 hz > 0; H 3  f 1 : 0, ∞ → 0, ∞ is nondecreasing, f 2 : 0, ∞ → 0, ∞ is nonincreasing, and there exist ϕτ,ψτ on interval a1,b−1 Z with ϕ : a1,b−1 Z → 0, 1,for all e 0 ∈ 0, 1, there exists τ 0 ∈ a1,b−1 Z such that ϕτ 0 e 0 ,andψτ >ϕτ, for all τ ∈ a  1,b− 1 Z which satisfy f 1  ϕ  τ  y  ≥ ψ  τ  f 1  y  ,f 2  1 ϕ  τ  y  ≥ ψ  τ  f 2  y  , ∀τ ∈  a  1,b− 1  Z ,y≥ 0. 1.7 2. Two Lemmas To prove the main results in this paper, we will employ two lemmas. These lemmas are based on the linear discrete fourth-order equation Δ 4 y  t − 2  − βΔ 2 y  t − 1   u  t  ,t∈  a  1,b− 1  Z , 2.1 with Lidstone boundary conditions 1.2. Lemma 2.1 see 11. Let u : a  1,b− 1 Z → R be a function. Then the nonhomogeneous discrete fourth-order Lidstone boundary value problem 2.1, 1.2 has solution y  t   b  sa b−1  za1 G 2  t, s  G 1  s, z  u  z  ,t∈  a − 1,b 1  Z , 2.2 4 Advances in Difference Equations where G 2 t, s given by G 2  t, s   1   1, 0    b, a  ⎧ ⎨ ⎩   t, a    b, s  : t ≤ s,   s, a    b, t  : s ≤ t,  t, s  ∈  a − 1,b 1  Z ×  a, b  Z 2.3 with t, sμ t−s − μ s−t for μ β  2   ββ  4/2, is the Green’s function for the second-order discrete boundary value problem −  Δ 2 y  t − 1  − βy  t    0,t∈  a, b  Z , y  a   0  y  b  , 2.4 and G 1 s, z given by G 1  s, z   1 b − a ⎧ ⎨ ⎩  s − a  b − z  : s ≤ z,  z − a  b − s  : z ≤ s,  s, z  ∈  a, b  Z ×  a  1,b− 1  Z 2.5 is the Green’s function for the second-order discrete boundary value problem −Δ 2 x  s − 1   0,s∈  a  1,b− 1  Z , x  a   0  x  b  . 2.6 Lemma 2.2 see 11. Let m :   1, 0    b, a  1   b − a   2  b, a  ,M:  b − a   2  b/2,a/2  4  1, 0    b, a  . 2.7 Then, for t, s, z ∈ a  1,b− 1 3 Z , one has m ≤ G 2  t, s  G 1  s, z  ≤ M. 2.8 3. Main Results Theorem 3.1. Assume that H 1 –H 3  hold. Then, the BVP 1.1 and 1.2 has a unique solution y ∗ t in D,where D   y ∈ B | y  a   0  y  b  ,y  t  > 0,t∈  a  1,b− 1  Z  . 3.1 Advances in Difference Equations 5 Moreover, for any x 0 ,y 0 ∈ D, constructing successively the sequences x n1  t   b  sa b−1  za1 G 2  t, s  G 1  s, z  h  z   f 1  x n  z   f 2  y n  z   , t ∈  a − 1,b 1  Z ,n 0, 1, 2, , y n1  t   b  sa b−1  za1 G 2  t, s  G 1  s, z  h  z   f 1  y n  z    f 2  x n  z   , t ∈  a − 1,b 1  Z ,n 0, 1, 2, , 3.2 One has x n t,y n t converge uniformly to y ∗ t in a − 1,b 1 Z . Proof. First, we show that the BVP 1.1 and 1.2 has a solution. It is easy to see that the BVP 1.1 and 1.2 has a solution y  yt if and only if y is a fixed point of the operator equation A  y 1 ,y 2   t   b  sa b−1  za1 G 2  t, s  G 1  s, z  h  z   f 1  y 1  z    f 2  y 2  z   ,t∈  a − 1,b 1  Z . 3.3 In view of H 3  and 3.3, Ay 1 ,y 2  is nondecreasing in y 1 and nonincreasing in y 2 . Moreover, for any τ ∈ a  1,b− 1 Z , we have A  ϕ  τ  y 1 , 1 ϕ  τ  y 2   t   b−1  sa1 b−1  za1 G 2  t, s  G 1  s, z  h  z   f 1  ϕ  τ  y 1  z    f 2  1 ϕ  τ  y 2  z   ≥ ψ  τ  b−1  sa1 b−1  za1 G 2  t, s  G 1  s, z  h  z   f 1  y 1  z    f 2  y 2  z    ψ  τ  A  y 1 ,y 2   t  3.4 for t ∈ a, b Z and y 1 ,y 2 ∈ D. Let L   b − a − 1  b−1  za1 h  z  , 3.5 6 Advances in Difference Equations condition H 2  implies L>0. Since f i y > 0fory>0 i  1, 2,byLemma 2.2, we have A  L, L   b−1  sa1 b−1  za1 G 2  t, s  G 1  s, z  h  z   f 1  L   f 2  L   ≥ m  f 1  L   f 2  L   b−1  sa1 b−1  za1 h  z   m  f 1  L   f 2  L   L 3.6 for m in 2.1 and L in 3.5. Moreover, we obtain A  L, L  ≤ M  f 1  L   f 2  L   L 3.7 for M in 2.1. Thus m  f 1  L   f 2  L   L ≤ A  L, L  ≤ M  f 1  L   f 2  L   L. 3.8 Therefore, we can choose a sufficiently small number e 1 ∈ 0, 1 such that e 1 L ≤ A  L, L  ≤ L e 1 , 3.9 which together with H 3  implies that there exists τ 1 ∈ a  1,b− 1 Z such that ϕτ 1 e 1 ,so ϕ  τ 1  L ≤ A  L, L  ≤ L ϕ  τ 1  . 3.10 Since ψτ 1 /ϕτ 1  > 1, we can take a sufficiently large positive integer k such that  ψτ 1  ϕτ 1   k ≥ 1 ϕ  τ 1  . 3.11 It is clear that  ϕτ 1  ψτ 1   k ≤ ϕ  τ 1  . 3.12 Advances in Difference Equations 7 We define u 0  t   ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −  ϕ  τ 1   k L: t  a − 1,b 1, 0: t  a, b,  ϕ  τ 1   k L: t ∈  a  1,b− 1  Z , v 0  t   ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ − L  ϕ  τ 1   k : t  a − 1,b 1, 0: t  a, b, L  ϕ  τ 1   k : t ∈  a  1,b− 1  Z . 3.13 Evidently, for t ∈ a, b Z , u 0 ≤ v 0 . Take any λ ∈ 0, ϕτ 1  2k , then λ ∈ 0, 1 and u 0 ≥ λv 0 . By the mixed monotonicity of A, we have Au 0 ,v 0  ≤ Av 0 ,u 0 . In addition, combining H 3  with 3.10 and 3.11,weget A  u 0 ,v 0   A   ϕ  τ 1   k L, 1  ϕ  τ 1   k L   A  ϕ  τ 1   ϕ  τ 1   k−1 L, 1 ϕ  τ 1   ϕ  τ 1   k−1 L  ≥ ψ  τ 1  A   ϕ  τ 1   k−1 L, 1  ϕ  τ 1   k−1 L  ≥··· ≥  ψ  τ 1   k A  L, L  ≥  ψ  τ 1   k ϕ  τ 1  L ≥  ϕ  τ 1   k L  u 0 . 3.14 From H 3 , we have A  y 1 ,y 2   A  ϕ  s  y 1 ϕ  s  , 1 ϕ  s  ϕ  s  y 2  ≥ ψ  s  A  y 1 ϕ  s  ,ϕ  s  y 2  , ∀s ∈  a  1,b− 1  Z ,y 1 ,y 2 ≥ 0, 3.15 and hence A  y 1 ϕ  s  ,ϕ  s  y 2  ≤ 1 ψ  s  A  y 1 ,y 2  , ∀s ∈  a  1,b− 1  Z ,y 1 ,y 2 ≥ 0. 3.16 8 Advances in Difference Equations Thus, we have A  v 0 ,u 0   A  L  ϕ  τ 1   k ,  ϕ  τ 1   k L   A  L ϕ  τ 1   ϕ  τ 1   k−1 ,ϕ  τ 1   ϕ  τ 1   k−1 L  ≤ 1 ψ  τ 1  A  L  ϕ  τ 1   k−1 ,  ϕ  τ 1   k−1 L  ≤··· ≤ 1  ψ  τ 1   k A  L, L  ≤ 1  ψ  τ 1   k L ϕ  τ 1  . 3.17 In accordance with 3.12, we can see that A  v 0 ,u 0  ≤ L  ϕ  τ 1   k  v 0 . 3.18 Construct successively the sequences u n  A  u n−1 ,v n−1  ,v n  A  v n−1 ,u n−1  ,n 1, 2, 3.19 By the mixed monotonicity of A, we have u 1  Au 0 ,v 0  ≤ Av 0 ,u 0 v 1 . By induction, we obtain u n ≤ v n ,n 1, 2, It follows from 3.14, 3.18, and the mixed monotonicity of A that u 0 ≤ u 1 ≤···≤ u n ≤···≤v n ≤···≤v 1 ≤ v 0 . 3.20 Note that u 0 ≥ λv 0 , so we can get u n t ≥ u 0 t ≥ λv 0 t ≥ λv n t,t∈ a, b Z ,n 1, 2, Let λ n  sup { λ>0 | u n  t  ≥ λv n  t  ,t∈  a, b  Z } ,n 1, 2, 3.21 Thus, we have u n  t  ≥ λ n v n  t  ,t∈  a, b  Z ,n 1, 2, , 3.22 and then u n1  t  ≥ u n  t  ≥ λ n v n  t  ≥ λ n v n1  t  ,t∈  a, b  Z ,n 1, 2, 3.23 Therefore, λ n1 ≥ λ n , that is, {λ n } is increasing with {λ n }⊂0, 1.Set  λ  lim n →∞ λ n . We can show that  λ  1. In fact, if 0 <  λ<1, by H 3 , there exists τ 2 ∈ a1,b−1 Z such that ϕτ 2   λ. Consider the following two cases. Advances in Difference Equations 9 i There exists an integer N such that λ N   λ. In this case, we have λ n   λ for all n ≥ N holds. Hence, for n ≥ N, it follows from 3.4 and the mixed monotonicity of A that u n1  A  u n ,v n  ≥ A   λv n , 1  λ u n   A  ϕ  τ 2  v n , 1 ϕ  τ 2  u n  ≥ ψ  τ 2  A  v n ,u n   ψ  τ 2  v n1 . 3.24 By the definition of λ n , we have λ n1   λ ≥ ψ  τ 2  >ϕ  τ 2    λ. 3.25 This is a contradiction. ii For all integer n, λ n <  λ. In this case, we have 0 <λ n /  λ<1. In accordance with H 3 , there exists θ n ∈ a  1,b− 1 Z such that ϕθ n λ n /  λ. Hence, combining 3.4 with the mixed monotonicity of A, we have u n1  A  u n ,v n  ≥ A  λ n v n , 1 λ n u n   A ⎛ ⎜ ⎝ λ n  λ  λv n , u n  λ n /  λ   λ ⎞ ⎟ ⎠  A  ϕ  θ n  ϕ  τ 2  v n , u n ϕ  θ n  ϕ  τ 2   ≥ ψ  θ n  A  ϕ  τ 2  v n , u n ϕ  τ 2   ≥ ψ  θ n  ψ  τ 2  A  v n ,u n   ψ  θ n  ψ  τ 2  v n1 . 3.26 By the definition of λ n , we have λ n1 ≥ ψ  θ n  ψ  τ 2  >ϕ  θ n  ψ  τ 2   λ n  λ ψ  τ 2  . 3.27 Let n →∞, we have  λ ≥   λ/  λψτ 2  >   λ/  λϕτ 2 ϕτ 2   λ, and this is also a contradiction. Hence, lim n →∞ λ n  1. Thus, combining 3.20 with 3.22, we have 0 ≤ u nl  t  − u n  t  ≤ v n  t  − u n  t  ≤ v n  t  − λ n v n  t    1 − λ n  v n  t  ≤  1 − λ n  v 0  t  3.28 for t ∈ a, b Z , where l is a nonnegative integer. Thus,  u nl − u n  ≤  v n − u n  ≤  1 − λ n  v 0 . 3.29 Therefore, there exists a function y ∗ ∈ D such that lim n →∞ u n  t   lim n →∞ v n  t   y ∗  t  for t ∈  a − 1,b 1  Z . 3.30 10 Advances in Difference Equations By the mixed monotonicity of A and 3.20, we have u n1  t   A  u n  t  ,v n  t  ≤ A  y ∗  t  ,y ∗  t   ≤ A  v n  t  ,u n  t   v n1  t  . 3.31 Let n →∞and we get Ay ∗ t,y ∗ t  y ∗ t, t ∈ a − 1,b 1 Z .Thatis,y ∗ is a nontrivial solution of the BVP 1.1 and 1.2. Next, we show the uniqueness of solutions of the BVP 1.1 and 1.2. Assume, to the contrary, that there exist two nontrivial solutions y 1 and y 2 of the BVP 1.1 and 1.2 such that Ay 1 t,y 1 t  y 1 t and Ay 2 t,y 2 t  y 2 t for t ∈ a − 1,b 1 Z . According to 3.9, we can know that there exists 0 <η≤ 1 such that ηy 2 t ≤ y 1 t ≤ 1/ηy 2 t for t ∈ a, b Z . Let η 0  sup  0 <η≤ 1 | ηy 2 ≤ y 1 ≤ 1 η y 2  . 3.32 Then 0 <η 0 ≤ 1andη 0 y 2 t ≤ y 1 t ≤ 1/η 0 y 2 t for t ∈ a, b Z . We now show that η 0  1. In fact, if 0 <η 0 < 1, then, in view of H 3 , there exists τ ∈ a  1,b− 1 Z such that ϕτη 0 . Furthermore, we have y 1  A  y 1 ,y 1  ≥ A  η 0 y 2 , 1 η 0 y 2   A  ϕ  τ  y 2 , 1 ϕ  τ  y 2  ≥ ψ  τ  A  y 2 ,y 2   ψ  τ  y 2 , 3.33 y 1  A  y 1 ,y 1  ≤ A  y 2 η 0 ,η 0 y 2   A  y 2 ϕ  τ  ,ϕ  τ  y 2  ≤ 1 ψ  τ  A  y 2 ,y 2   1 ψ  τ  y 2 . 3.34 In 3.34, we used the relation formula 3.16. Since ψ τ >ϕτη 0 , this contradicts the definition of η 0 . Hence η 0  1. Therefore, the BVP 1.1 and 1.2 has a unique solution. Finally, we show that “moreover” part of the theorem. For any initial x 0 ,y 0 ∈ D,in accordance with 3.9, we can choose a sufficiently small number e 2 ∈ 0, 1 such that e 2 L ≤ x 0 ≤ 1 e 2 L, e 2 L ≤ y 0 ≤ 1 e 2 L. 3.35 It follows from H 3  that there exists τ 3 ∈ a  1,b− 1 Z such that ϕτ 3 e 2 , and hence ϕ  τ 3  L ≤ x 0 ≤ L ϕ  τ 3  ,ϕ  τ 3  L ≤ y 0 ≤ L ϕ  τ 3  . 3.36 Thus, we can choose a sufficiently large positive integer k such that  ψτ 3  ϕτ 3   k ≥ 1 ϕ  τ 3  . 3.37 Define u 0   ϕ  τ 3   k L, v 0  L  ϕ  τ 3   k . 3.38 [...]... 15 P J Y Wong and R P Agarwal, “Results and estimates on multiple solutions of Lidstone boundary value problems,” Acta Mathematica Hungarica, vol 86, no 1-2, pp 137–168, 2000 16 P J Y Wong and R P Agarwal, “Characterization of eigenvalues for difference equations subject to Lidstone conditions,” Japan Journal of Industrial and Applied Mathematics, vol 19, no 1, pp 1–18, 2002 17 P J Y Wong and L Xie, “Three... solutions of fourth-order nonlinear c difference equations,” Lithuanian Mathematical Journal, vol 49, no 1, pp 71–92, 2009 11 D R Anderson and F Minhos, A discrete fourth-order Lidstone problem with parameters,” Applied ´ Mathematics and Computation, vol 214, no 2, pp 523–533, 2009 12 T He and Y Su, “On discrete fourth-order boundary value problems with three parameters,” Journal of Computational and. .. Computational and Applied Mathematics, vol 233, no 10, pp 2506–2520, 2010 13 R P Agarwal and D O’Regan, Lidstone continuous and discrete boundary value problems,” Memoirs on Differential Equations and Mathematical Physics, vol 19, pp 107–125, 2000 14 P J Y Wong and R P Agarwal, “Multiple solutions of difference and partial difference equations with Lidstone conditions,” Mathematical and Computer Modelling,... “Multiple positive solutions of singular and nonsingular discrete problems via variational methods,” Nonlinear Analysis: Theory, Methods & Applications, vol 58, no 1-2, pp 69–73, 2004 4 V Lakshmikantham and D Trigiante, Theory of Difference Equations: Numerical Methods and Applications, vol 181, Academic Press, Boston, Mass, USA, 1988 5 W G Kelley and A C Peterson, Difference Equations: An Introduction with Applications,... Zhang, L Kong, Y Sun, and X Deng, Existence of positive solutions for BVPs of fourth-order difference equations,” Applied Mathematics and Computation, vol 131, no 2-3, pp 583–591, 2002 9 Z He and J Yu, “On the existence of positive solutions of fourth-order difference equations,” Applied Mathematics and Computation, vol 161, no 1, pp 139–148, 2005 10 J V Manojlovi´ , “Classification and existence of positive. .. ZR2009AM004 , and the Youth Science Foundation of Shanxi Province 2009021001-2 References 1 R P Agarwal, Difference Equations and Inequalities, vol 155, Marcel Dekker, New York, NY, USA, 1992 2 R P Agarwal, D O’Regan, and P J Y Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999 3 R P Agarwal, K Perera, and D O’Regan,... symmetric solutions of Lidstone boundary value problems for difference and partial difference equations,” Computers & Mathematics with Applications, vol 45, no 6–9, pp 1445–1460, 2003 18 Advances in Difference Equations 18 C.-B Zhai and X.-M Cao, “Fixed point theorems for τ-ϕ-concave operators and applications,” Computers and Mathematics with Applications, vol 59, no 1, pp 532–538, 2010 19 C B Zhai, W X Wang,... Introduction with Applications, Academic Press, Boston, Mass, USA, 1991 6 J Yu and Z Guo, “On boundary value problems for a discrete generalized Emden-Fowler equation,” Journal of Differential Equations, vol 231, no 1, pp 18–31, 2006 7 D B Wang and W Guan, “Three positive solutions of boundary value problems for p-Laplacian difference equations,” Computers & Mathematics with Applications, vol 55, no 9,... Mathematics with Applications, vol 59, no 1, pp 532–538, 2010 19 C B Zhai, W X Wang, and L L Zhang, “Generalizations for a class of concave and convex operators,” Acta Mathematica Sinica, vol 51, no 3, pp 529–540, 2008 Chinese 20 Z D Liang, Existence and uniqueness of fixed points for mixed monotone operators,” Journal of Dezhou University, vol 24, no 4, pp 1–6, 2008 Chinese ... τ5 A L 3.60 L ϕ τ5 An application of 3.56 yields v1 ≤ 1 ϕ τ5 k L v0 3.61 Therefore, we obtain u0 ≤ u1 ≤ v1 ≤ v0 For t a − 1, b theorem 3.62 1, the proof is similar and hence omitted This completes the proof of the 16 Advances in Difference Equations Remark 3.4 In Theorem 3.1, the more general conditions are imposed on the nonlinear term than Theorem 1.1 In particular, in Theorem 3.3, ψ τ, y contains . China 2 School of Mathematics, Shandong University, Jinan, Shandong 250100, China 3 Department of Mathematics, Shandong Jianzhu University, Jinan, Shandong 250101, China 4 Department of Mathematics,. boundary value problems with three parameters,” Journal of Computational and Applied Mathematics, vol. 233, no. 10, pp. 2506–2520, 2010. 13 R. P. Agarwal and D. O’Regan, Lidstone continuous and. multiple solutions of Lidstone boundary value problems,” Acta Mathematica Hungarica, vol. 86, no. 1-2, pp. 137–168, 2000. 16 P. J. Y. Wong and R. P. Agarwal, “Characterization of eigenvalues for