Sign-changing solutions to discrete fourth-order Neumann boundary value problems

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1 Yang Advances in Difference Equations 2013, 2013:10 R E S E A R C H Open Access Sign-changing solutions to discrete fourth-order Neumann boundary value problems Jingping Yang * * Correspondence: fuj09@lzu.cn Gansu Institute of Political Science and Law, Lanzhou, , P.R. China Abstract By using topological degree theory and fixed point index theory, we consider a discrete fourth-order Neumann boundary value problem. We provide sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions. Keywords: topological degree theory; fixed point index theory; sign-changing solutions; positive solutions; negative solutions 1 Introduction In this paper, we consider the existence of sign-changing solutions to the following discrete nonlinear fourth-order boundary value problem (BVP: 4 u(t 2 α 2 u(t 1+βu(t=f ( t, u(t, t [2, T] Z, (1.1 u(1 = u(t= 3 u(0 = 3 u(t 1=0, (1.2 where denotes the forward difference operator defined by u(t=u(t +1 u(t, T >2 is an integer, f :[2,T] Z R R is continuous, and α, β are real parameters and satisfy α 2 4β, and α α 2 4β > 8sin 2 π 2(T 1. Let a, b be two integers with a < b.weemploy[a, b] Z to denote the discrete interval given by {a, a +1,...,b}. The theory of nonlinear difference equations has been widely used to study discrete models in many fields such as computer science, economics, neural network, ecology, cybernetics, etc. In recent years, a great deal of work has been done in the study of the existence and multiplicity of solutions for a discrete boundary value problem. For the background and recent results, we refer the reader to [1 5] and the references therein. We may think of BVP (1.1, (1.2 as a discrete analogue of the fourth-order boundary value problem u (4 (t αu (2 (t+βu(t=λf ( t, u(t, t (0, 1, (1.3 u (0 = u (1 = u 3 (0 = u 3 (1 = 0. ( Yang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Yang Advances in Difference Equations 2013, 2013:10 Page 2 of 11 The special case of BVP (1.3, (1.4 has been studied by many authors using various approaches; for example, see [6, 7]. However, it seems that there is no similar result in the literature on the existence of sign-changing solutions, positive solutions, and negative solutions for BVP (1.1, (1.2. Motivated by [8], our purpose is to apply some basic theorems in topological degree theory and fixed point index theory to establish some conditions for the nonlinear function f, which are able to guarantee the existence of sign-changing solutions, positive solutions, and negative solutions for the above discrete boundary value problem. The organization of this paper is as follows. In Section 2, we state some notations and preliminary knowledge about the topological degree theory and fixed point index theory. In Section 3, wepresent thespectrumofsecond-ordereigenvalue problems. In Section4, we give the expression of Green s function of second-order Neumann problems and consider the eigenvalue problem of fourth-order BVPs. In Section 5, by computing the topological degree and the fixed point index, we discuss the existence of multiple sign-changing solutions to BVP (1.1, ( Preliminaries As we have mentioned, we will use the theory of the Leray-Schauder degree and the fixed point index in a cone to prove our main existence results. Let us collect some results that will be used below. One can refer to [9 12] for more details. Lemma 2.1 (see [9, 10] Let E be a Banach space and X EbeaconeinE. Assume that is a bounded open subset of E. Suppose that A : X X is a completely continuous operator. If there exists x 0 X\{θ} such that x Ax μx 0, x X, μ 0, (2.1 then the fixed point index i(a, X, X=0. Lemma 2.2 (see [9, 10] Let E be a Banach space and let X EbeaconeinE. Assume that is a bounded open subset of E, θ. Suppose that A : X Xisacompletely continuous operator. If Ax μx, x X, μ 1, (2.2 then the fixed point index i(a, X, X=1. Lemma 2.3 (see [11] Let E be a Banach space, let be a bounded open subset of E, θ, and A : E be completely continuous. Suppose that Ax x, Ax x, x. (2.3 Then deg(i A,, θ=1. Lemma 2.4 (see [12] Let A be a completely continuous operator which is defined on a Banach space E. Let x 0 E be a fixed point of A and assume that A is defined in a neighborhood of x 0 and Fréchet differentiable at x 0. If 1 is not an eigenvalue of the linear operator

3 Yang Advances in Difference Equations 2013, 2013:10 Page 3 of 11 A (x 0, then x 0 is an isolated singular point of the completely continuous vector field I A, and for small enough r >0, deg ( I A, B(x 0, r, θ =( 1 k, (2.4 where k is the sum of algebraic multiplicities of real eigenvalues of A (x 0 in (1, +. Lemma 2.5 (see [12] Let A be a completely continuous operator which is defined on a Banach space E. Assume that 1 is not an eigenvalue of the asymptotic derivative. Then the completely continuous vector field I A is nonsingular on spheres S ρ = {x E : x = ρ} of sufficiently large radius ρ and deg ( I A, B(θ, ρ, θ =( 1 k, (2.5 where k is the sum of algebraic multiplicities of real eigenvalues of A ( in (1, +. Lemma 2.6 (see [12] Let X be a solid cone of a Banach space E (X is nonempty, let be a relatively bounded open subset of X, and let A : X X be a completely continuous operator. If any fixed point of A in is an interior point of X, there exists an open subset O of E(O such that deg(i A, O, θ=i(a,, X. (2.6 Now, we will consider the space E = { u :[2,T] Z R, u(1 = u(t= 3 u(0 = 3 u(t 1=0 } (2.7 equipped with the norm u = max t [2,T]Z u(t. Clearly, E is a (T 1-dimensional Banach space. Choose the cone P E defined by P = { u E u(t 0, t [2, T] Z }. (2.8 Obviously, the interior of P is P = {u E u(t >0,t [2, T] Z }.Foreachu, v E, we write u v if u(t v(t fort [2, T] Z.Asolutionu of BVP (1.1, (1.2 issaidtobea positive solution (a negative solution, resp. if u P\{θ} (u ( P\{θ}, resp.. A solution u of BVP (1.1, (1.2 is said to be a sign-changing solution if u / P ( P. 3 Spectrum of second-order eigenvalue problems Consider the second-order discrete linear eigenvalue problems 2 y(t 1+λy(t=0, t [1, T] Z, (3.1 y(1 = y(t=0. (3.2 Lemma 3.1 (see [13]Theorem1 The eigenvalues of (3.1, (3.2 can be given by λ k =2cos kπ T +1 2,

4 Yang Advances in Difference Equations 2013, 2013:10 Page 4 of 11 and the corresponding eigenfunction is if k =0,then ϕ 0 (t =1.If k {1,2,...,T 2}, we have (i if k T 1 3, then ϕ 1 k(t= cos 3kπ (ii if k = T 1 3, then ϕ k(t=sin πt 3. 2(T 1 cos kπ (2t 3, (2T 1 4 Green s function and eigenvalue problems to fourth-order BVPs In this section, we construct Green s function associated with BVP (1.1, (1.2. Let r 1, r 2 be roots of the polynomial P(r=r 2 αr + β,namely, r 1 = α + α 2 4β 2 Then we have, r 2 = α α 2 4β. 2 4 u(t 2 α 2 u(t 1+βu(t= ( 2 L + r 1 ( 2 L + r 2 u(t = ( 2 L + r 2 ( 2 L + r 1 u(t, where u =(u(0, u(1,...,u(t +2,Lu = u(t 1,t [2, T] Z. Under the basic assumption on α, β,itiseasytoseethatr 1 r 2 > 4sin 2 π 2(T 1. Consider the two initial value problems: { 2 u(t 1+r i u(t=0, t [2, T] Z, (4.1 u(1 = 0, u(1 = 1, { 2 v(t 1+r i v(t=0, t [2, T] Z, (4.2 v(t=0, v(t=1. By the direct computing, we get (4.1 hasa unique solution u(t= r 2 1 r 1 (r 2 r 1 rt r 1 r 2 (r 2 r 1 rt 2, and (4.2hasauniquesolution Let v(t= r 2 1 r T 1 (r 2 r 1 rt r 1 r T 2 (r 2 r 1 rt 2. ρ := (r 1 1(r 2 1(r2 T 1 r1 T 1 r1 T 1 r2 T 1. Lemma 4.1 Let h :[1,T] Z R and i {1, 2} be fixed. Then the problem { 2 u(t 1+r i u(t=h(t, t [2, T] Z, u(1 = u(t=0 (4.3 has a unique solution T u(t= G i (t, sh(s, t [1, T +1] Z,

5 Yang Advances in Difference Equations 2013, 2013:10 Page 5 of 11 where G i (t, s is given by G i (t, s= 1 ρ { u(tv(s, 1 s t T +1, u(sv(t, 1 t s T +1. Remark 4.1 Green s function G i (t, s defined by Lemma 4.1 is positive on [1, T] Z [1, T] Z. Define operators K, f, A : E E,respectively,by T T (Ku(t= G 2 (t, kg 1 (k, su(s, u E, t [1, T] Z ; (4.4 k=2 (fu(t=f ( t, u(t, u E, t [1, T] Z ; A = Kf. (4.5 Now, from Lemma 4.1, itiseasytoseethatbvp(1.1, (1.2 hasasolutionu = u(t ifand only if u is a fixed point of the operator A. It follows from the continuity of f that A : E E is completely continuous. Lemma 4.2 Let h :[2,T] Z R. Then the linear discrete fourth-order boundary value problem { 4 u(t 2 α 2 u(t 1+βu(t=h(t, t [2, T] Z, u(1 = u(t= 3 u(0 = 3 u(t 1=0 has a unique solution and u(t= = ( T T G 1 (t, kg 2 (k, sh(s k=2 ( T T G 2 (t, kg 1 (k, sh(s, t [2, T] Z, k=2 (4.6 u(0 = u(3, u(1 = u(2, u(t=u(t +1, u(t 1=u(T +2. Proof The conclusion is obvious, so we omit it. We will use the following assumptions. (H1 α, β are real parameters and satisfy α 2 4β, and α α 2 4β > 8sin 2 π 2(T 1. (H2 f :[2,T] Z R R is continuous, for any t [2, T] Z, f (t,0=0; for any t [2, T] Z and x R, xf (t, x 0.

6 Yang Advances in Difference Equations 2013, 2013:10 Page 6 of 11 (H3 There exists an even number k 0 [0, T 2] Z such that 1 λ 2 k 0 < β 0 < 1 λ 2 k 0 +1, (4.7 f (t,x where lim x 0 = β x 0 uniformly for t [2, T] Z, λ k is defined in Lemma 3.1 and λ T 1. (H4 Thereexistsanevennumberk 1 [0, T 2] Z such that 1 λ 2 < β < 1 k 1 λ 2, (4.8 k 1 +1 f (t,x where lim x = β x uniformly for t [2, T] Z,andλ 0, λ 1,...,λ T 1 are given in the condition (H3. (H5 There exists a constant M >0such that for any (t, x [2, T] Z [ M, M], f (t, x < ω 1 M, (4.9 where ω = max t [2,T]Z T T k=2 G 2(t, kg 1 (k, s. Lemma 4.3 Suppose that (H2 holds and u P\{θ} is a solution of BVP (1.1, (1.2. Then u P. Proof According to (H2 and the positivity of Green s function defined in Lemma 4.1,we can easily get the desired conclusion. Remark 4.2 Similarly to Lemma 4.3,wealsoknowthatif(H2holdsandu ( P\{θ} is asolutionofbvp(1.1, (1.2, then u ( P. Lemma 4.4 Suppose that (H2-(H4 hold. Then the operator A is Fréchet differentiable at θ and, where the operator A is defined by (4.5. Moreover, A (θ=β 0 KandA ( = β K. Proof By (H3, for any ɛ >0,thereexistsδ >0suchthat f (t, x β 0 x < ɛ x for any 0 < x < δ, t [2, T] Z. Hence, noticing that f (t, 0 = 0 for any t [2, T] Z,wehave Au Aθ β 0 Ku = K(fu β 0 u K max t [2,T] Z f ( t, u(t β0 u(t < ɛ K u (4.10 for any u E with 0 < u < δ,where K = max t [2,T]Z T T k=2 G 2(t, k G 1 (k, s.consequently, Au Aθ β 0 Ku lim =0. (4.11 u 0 u This means that the nonlinear operator A is Fréchet differentiable at θ and A (θ=β 0 K. By (H4, for any ɛ >0,thereexistsW >0suchthat f (t, x β x < ɛ x for any x > W, t [2, T] Z.Letc = max (t,x [2,T]Z [ W,W] f (t, x β x.bythecontinuityoff (t, xwith

7 Yang Advances in Difference Equations 2013, 2013:10 Page 7 of 11 respect to x, wehavec <+. Then,forany(t, x [2, T] Z R, f (t, x β x < ɛ x + c. Thus Au β Ku K max t [2,T] Z f (t, u β u(t < K ( ɛ u + c (4.12 for any u E.Consequently, Au Aθ β Ku lim = 0, (4.13 u u which implies that operator A is Fréchet differentiable at and A ( =β K.Theproof is completed. Lemma 4.5 Let M be given in the condition (H5. Suppose that (H1-(H4 hold. Then A(P P, A( P ( P. Moreover, onehasthefollowing. (i There exists an r 0 (0, M such that for any 0<r r 0, i ( A, P B(θ, r, P =0, i ( A,( P B(θ, r, P =0. (4.14 (ii There exists an R 0 > M such that for any R R 0, i ( A, P B(θ, R, P =0, i ( A,( P B(θ, R, P = 0. (4.15 Proof By (H2 and the fact that G i (t, s is positive on [2, T] Z [2, T] Z,wegetthatfor any t [2, T] Z, f (t, P P, f (t, P P, andk(p P, K( P P. ThenA(P P and A( P P. We only need to prove conclusion (i. The proof of conclusion (ii is similar and will be omitted here. Let γ 0 = inf u =1 u β 0 Ku. The condition (H3 yields γ 0 > 0. It follows from (4.12 that there exists r 0 [0, M]suchthat Au β Ku < 1 2 γ 0 u, (4.16 where 0 < u r 0. Setting H(s, u=sau +(1 sβ 0 Ku,thenH :[0,1] E E is completely continuous. For any s [0, 1] and 0 < u r 0,weobtainthat u H(s, u u β 0 Ku s Au β 0 Ku γ 0 u 1 2 γ 0 u >0. (4.17 According to the homotopy invariance of the fixed point index, for any 0 < r r 0,wehave i ( A, P B(θ, r, P = i ( β 0 K, P B(θ, r, P, (4.18 i ( A, P B(θ, r, P = i ( β 0 K, P B(θ, r, P. (4.19 Let ϕ 0 (t=1.thenkϕ 0 = λ 2 0 ϕ 0 and ϕ 0 P (see Lemma 3.1 and the proof of Lemma 4.2. We claim u β 0 Ku σϕ 0, u P B(θ, r, σ 0. (4.20

8 Yang Advances in Difference Equations 2013, 2013:10 Page 8 of 11 Indeed, we assume that there exist u 1 P B(θ, randσ 1 0suchthatu 1 β 0 Ku 1 = σ 1 ϕ 0. Obviously, u 1 = β 0 Ku 1 + σ 1 ϕ 0 σ 1 ϕ 0.Sinceβ 0 λ 2 k, k =1,2,...,T 2,thenσ 1 >0.Set σ max = sup{σ : u 1 σϕ 0 }.Itisclearthatσ 1 σ max < and u 1 σ max ϕ 0.Then u 1 = β 0 Ku 1 + σ 1 ϕ 0 β 0 Kσ max ϕ 0 + ϕ 0 σ 1 = ( β 0 λ 2 0 σ max + σ 1 ϕ0. (4.21 Since β 0 λ 2 0 >1,thenβ 0λ 2 0 σ max + σ 1 > σ max, which contradicts the definition of σ max.this proves (4.21. It follows from Lemma 2.1 and (4.20that i ( β 0 K, P B(θ, r, P =0. (4.22 Similarly to (4.22, we know also that i ( β 0 K, P B(θ, r, P =0. (4.23 By (4.18, (4.19, (4.22, and (4.23, we conclude i ( A, P B(θ, r, P =0, i ( A, P B(θ, r, P =0. ( Main results Now, with the aid of the lemmas in Section 2, we are in a position to state and prove our main results. Theorem 5.1 Assume that the conditions (H1-(H5 hold. Then BVP (1.1, (1.2 has at least two sign-changing solutions. Moreover, BVP (1.1, (1.2 has at least two positive solutions and two negative solutions. Proof Since G(t, s is positive on [2, T] Z [1, T] Z, by (H5, we have for any u E with u = M, Au(t T T = G 2 (t, kg 1 (k, sf ( t, u(t T T G 2 (t, kg 1 (k, s f ( t, u(t This gives k=2 < ω 1 M T k=2 T G 2 (t, kg 1 (k, s M, t [2, T] Z. (5.1 k=2 Au < M = u. (5.2 By (5.2 and Lemmas 2.2 and 2.3,wehave deg ( I A, B(θ, M, θ =1, (5.3 i ( A, P B(θ, M, P =1, (5.4 i ( A, P B(θ, M, P = 1. (5.5

9 Yang Advances in Difference Equations 2013, 2013:10 Page 9 of 11 From (H3 and Lemma 3.1, one has that the eigenvalues of the operator A (θ=β 0 K which are larger than 1 are β 0 λ 2 0, β 0λ 2 1,...,β 0λ 2 k 0. (5.6 From (H4 and Lemma 3.1, one has that the eigenvalues of the operator A ( =β K which are larger than 1 are β λ 1 0, β λ 2 1,...,β λ 2 k 1. (5.7 It follows from Lemmas 2.4 and 2.5 that there exist 0 < r 1 < r 0 and R 1 > R 0 such that deg ( I A, B(θ, r 1, θ =( 1 k 0 =1, (5.8 deg ( I A, B(θ, R 1, θ =( 1 k 1 =1, (5.9 where r 0 and R 0 are given in Lemma 4.5. Owing to Lemma 4.3,onehas i ( A, P B(θ, r 1, P =0, (5.10 i ( A, P B(θ, r 1, P =0, (5.11 i ( A, P B(θ, R 1, P =0, (5.12 i ( A, P B(θ, R 1, P = 0. (5.13 According to the additivity of the fixed point index, by (5.4, (5.10, and (5.12, we have i ( A, P ( B(θ, M\B(θ, r 1, P = i ( A, P B(θ, M, P i ( A, P B(θ, r 1, P =1 0=1, (5.14 i ( A, P ( B(θ, R 1 \B(θ, M, P = i ( A, P B(θ, R 1, P i ( A, P B(θ, M, P =0 1= 1. (5.15 Hence, the nonlinear operator A has at least two fixed points u 1 P (B(θ, M\B(θ, r 1 and u 2 P (B(θ, R 1 \B(θ, M, respectively. Then, u 1 and u 2 are positive solutions of BVP (1.1, (1.2. Using again the additivity of the fixed point index, by (5.5, (5.11, and (5.13, we get i ( A, P ( B(θ, M\B(θ, r 1, P =1 0=1, (5.16 i ( A, P ( B(θ, R 1 \B(θ, M, P =0 1= 1. (5.17 Hence, the nonlinear operator A has at least two fixed points u 3 P (B(θ, M\B(θ, r 1 and u 4 P (B(θ, R 1 \B(θ, M, respectively. Then, u 3 and u 4 are negative solutions of BVP (1.1, (1.2. Let Ɣ 1 = { u P ( B(θ, M\B(θ, r 1 : Au = u }, Ɣ 2 = { u P ( B(θ, R 1 \B(θ, M : Au = u },

10 Yang Advances in Difference Equations 2013, 2013:10 Page 10 of 11 Ɣ 3 = { u P ( B(θ, M\B(θ, r 1 : Au = u }, Ɣ 4 = { u P ( B(θ, R 1 \B(θ, M : Au = u }. It follows from Lemmas 2.6, 4.3,Remark4.2,and(5.14-(5.17thatthereexistopensubsets O 1, O 2, O 3,andO 4 of E such that Ɣ 1 O 1 P ( B(θ, M\B(θ, r 1, Ɣ 1 O 1 P ( B(θ, R 1 \B(θ, M, (5.18 Ɣ 3 O 3 P ( B(θ, M\B(θ, r 1, Ɣ 4 O 4 P ( B(θ, R 1 \B(θ, M, (5.19 deg(i A, O 1, θ=1, (5.20 deg(i A, O 2, θ= 1, (5.21 deg(i A, O 3, θ=1, (5.22 deg(i A, O 4, θ= 1. (5.23 By (5.3, (5.20, (5.22, (5.8, and the additivity of the Leray-Schauder degree, we get deg ( I A, B(θ, M\ ( O 1 O 3 B(θ, r 1, θ = = 2, (5.24 which implies that the nonlinear operator A has at least one fixed point u 5 B(θ, M\(O 1 O 3 B(θ, r 1. Similarly, by (5.9, (5.21, (5.23, and (5.3, we get deg ( I A, B(θ, R 1 \ ( O 2 O 4 B(θ, M, θ = =2, (5.25 which implies that the nonlinear operator A has at least one fixed point u 6 B(θ, R 1 \(O 2 O 4 B(θ, M. Then, u 5 and u 6 are two distinct sign-changing solutions of BVP (1.1, (1.2. Thus, the proof of Theorem 5.1 is finished. Theorem 5.2 Assume that the conditions (H1-(H5 hold, and that f (t, x= f (t, x for t [2, T] Z and x R. Then BVP (1.1, (1.2 has at least four sign-changing solutions. Moreover, BVP (1.1, (1.2 has at least two positive solutions and two negative solutions. Proof It follows from the proof of Theorem 5.1 that BVP (1.1, (1.2 has at least six different nontrivial solutions u i (i =1,2,...,6satisfying u 1, u 2 P, u 3, u 4 P, u 5, u 6 / P ( P, r 1 < u 5 < M < u 6 < R 1. (5.26 By the condition that f (t, x = f (t, x fort [2, T] Z and x R, weknowthat u 5 and u 6 are also solutions of BVP (1.1, (1.2. Let u 7 = u 5, u 8 = u 6,thenu i (i =1,2,...,8are different nontrivial solutions of BVP (1.1, (1.2.The proofis completed. By the method used in the proof of Theorems 5.1 and 5.2, we can prove the following corollaries.

11 Yang Advances in Difference Equations 2013, 2013:10 Page 11 of 11 Corollary 5.3 Assume that the conditions (H1-(H3 and (H5 or (H1, (H2, (H4, and (H5 hold. Then BVP (1.1, (1.2 has at least one sign-changing solution. Moreover, BVP (1.1, (1.2 has at least one positive solution and one negative solution. Corollary 5.4 Assume that the conditions (H1-(H3 and (H5 or (H1, (H2, (H4, and (H5 hold, and that f (t, x= f (t, x for t [2, T] Z and x R. Then BVP (1.1, (1.2 has at least two sign-changing solutions. Moreover, BVP (1.1, (1.2 has at least one positive solution and one negative solution. Competing interests The author declares that they have no competing interests. Acknowledgements The author is very grateful to the anonymous referees for their valuable suggestions. Received: 20 October 2012 Accepted: 27 December 2012 Published: 14 January 2013 References 1. Agarwal, RP: Difference Equations and Inequalities: Theory, Methods, and Applications, 2nd edn. Dekker, New York ( Agarwal, RP, O Regan, D, Wong, PJY: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht ( Gao, CH, Dai, GW, Ma, RY: Existence of positive solutions to discrete second-order boundary value problems with indefinite weight. Adv. Differ. Equ. 2012, 3 ( A12 (34B1839A23 4. Wong, PJY, Xie, L: Three symmetric solutions of Lidstone boundary value problems for difference and partial difference equations. Comput. Math. Appl. 45, ( Yu, JS, Guo, ZM: On boundary value problems for a discrete generalized Emden-Fowler equation. J. Differ. Equ. 231, ( Yang, Y, Zhang, JH: Infinitely many mountain pass solutions on a kind of fourth-order Neumann boundary value problem. Appl. Math. Comput. 213, ( Yang, Y, Zhang, JH: Nontrivial solutions on a kind of fourth-order Neumann boundary value problems. Appl. Math. Comput. 218, ( He, TS, Yang, W, Yang, FJ: Sign-changing solutions for discrete second-order three-point boundary value problems. Discrete Dyn. Nat. Soc. 2010, Article ID (2010. doi: /2010/ Guo, DJ, Sun, JX, Liu, ZL: Functional Methods for Nonlinear Differential Equations. Shandong Science and Technology Press, Jinan ( Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics in Science and Engineering, vol. 5. Academic Press, Boston ( Guo, DJ: Nonlinear Functional Analysis, 2nd edn. Shandong Science and Technology Press, Jinan ( Krasnoselskiĭ, MA, Zabreĭko, PP: Geometrical Methods of Nonlinear Analysis. Grundlehren der Mathematischen Wissenschaften, vol Springer, Berlin ( Agarwal, RP, Bohner, M, Wong, PJY: Sturm-Liouville eigenvalue problems on time scales. Appl. Math. Comput. 99, (1999 doi: / Cite this article as: Yang: Sign-changing solutions to discrete fourth-order Neumann boundary value problems. Advances in Difference Equations :10.