Global structure of sign-changing solutions for discrete Dirichlet problems

Abstract Let T > 1 T\gt 1 be an integer, T ≔ [ 1 , T ] Z = { 1 , 2 , … , T } , T ˆ ≔ { 0 , 1 , … , T + 1 } {\mathbb{T}}:= {{[}1,T]}_{{\mathbb{Z}}}=\{1,2,\ldots ,T\},\hspace{.0em}\hat{{\mathbb{T}}}:= \{0,1,\ldots ,T+1\} . In this article, we are concerned with the global structure of the set of sign-changing solutions of the discrete second-order boundary value problem { Δ 2 u ( x − 1 ) + λ h ( x ) f ( u ( x ) ) = 0 , x ∈ T , u ( 0 ) = u ( T + 1 ) = 0 , \left\{\begin{array}{l}{\mathrm{\Delta}}^{2}u(x-1)+\lambda h(x)f(u(x))=0,\hspace{1em}x\in {\mathbb{T}},\\ u(0)=u(T+1)=0,\end{array}\right. where λ > 0 \lambda \gt 0 is a parameter, f ∈ C ( ℝ , ℝ ) f\in C({\mathbb{R}},{\mathbb{R}}) satisfies f ( 0 ) = 0 , s f ( s ) > 0 f(0)=0,\hspace{.1em}sf(s)\gt 0 for all s ≠ 0 s\ne 0 and h : T ˆ → [ 0 , + ∞ ) h:\hat{{\mathbb{T}}}\to {[}0,+\infty ) . By using the directions of a bifurcation, we obtain existence and multiplicity of sign-changing solutions of the above problem for λ \lambda lying in various intervals in ℝ {\mathbb{R}} . Moreover, we point out that these solutions change their sign exactly k − 1 k-1 times, where k ∈ { 1 , 2 , … , T } k\in \{1,2,\ldots ,T\} .

. By using the directions of a bifurcation, we obtain existence and multiplicity of sign-changing solutions of the above problem for λ lying in various intervals in . Moreover, we point out that these solutions change their sign exactly − k 1 times, where ∈ { … } k

Introduction
In this article, we are concerned with the discrete second-order boundary value problem In assumption (A2) and throughout the article, we use the following well-known conceptions of a generalized zero and a simple generalized zero at ∈ t . , then y has a generalized zero at ∈ t 0 .

Definition 1.2. A function
→ y:ˆis said to have a simple generalized zero at ∈ t 0 provided that one of the following conditions is satisfied: It is well-known that the eigenvalue problem has a finite sequence of simple eigenvalues and the eigenfunction ϕ k corresponding to μ k has exactly − k 1 simple generalized zeros in (see [1], and for other spectral results of related problems we refer to [2]). Let be the generalized zeros of ϕ k .
In the special case ( ) ≡ h x 1, the mth eigenvalue and eigenfunction corresponding to it are characterized by be the generalized zeros of ( ) 1,2 , , 1 2 , such that In order to state our main result, we first recall some standard notations to describe the properties of sign- Our main result is the following.
Existence and multiplicity of positive solutions for discrete boundary value problems have been extensively studied by several authors. We refer to Agarwal et al. [4,5], Cheng and Yen [6], Rachunkova and Tisdell [7], Ma [8], Rodriguez [9] and references therein. However, the methods in [4][5][6][7][8][9] are analytic techniques and various fixed point theorems, which cannot be used to prove the existence of S k ν solutions.
By using the global bifurcation theory [10,11], Davidson and Rynne [12] and Luo and Ma [13] studied second-order Dirichlet boundary value problems on measure chain. It is well-known that measure chain d enotes an arbitrary closed subset of real numbers , and we can regard the equations in [12,13] as difference equations when = . They showed that unbounded continua of nontrivial solutions prescribed nodal properties emanated from the trivial branch at the eigenvalues of the linearization problems. However, they only showed that for each fixed integer ≥ k 1 and ∈ {+ −} ν , , there exists at least one nodal solution. . By using Mathematica 9.0, the distance between two consecutive generalized zeros of ( ) ψ t m2 was found to be 20, which means = c 20 ⁎ . Thus, it is easy to check that = ĉ 10 by the definition of ĉ.
The rest of the article is organized as follows. In Section 2, we show the existence of bifurcation from some eigenvalues for the corresponding problem according to the standard argument and the rightward direction of bifurcation. In Section 3, the change in direction of bifurcation is given. The final section is devoted to show an a priori bound of solutions for (1.1) and complete the proof of Theorem 1.3.

Rightward bifurcation
We study the global behavior from the trivial branch with the rightward direction of bifurcation under suitable assumptions on h and f.
Since u ≢ 0 on ˆ, we may assume that On the other hand, it follows from equation (1.1) and However, by (2.2) and the fact ( ) = u t 0 0 , we get which contradicts (2.1). □ Next following the similar arguments in Gao and Ma [14], with obvious changes, we may get the following.

4)
Proof. According to discrete Rolle's theorem (see [1]), there exists . Then by a direct computation, it is easy to see that Dividing both sides of (2.7) by ‖ ‖ u n , we get

Direction turn of bifurcation
In this section, we show that the connected components grow to the left at some point under (A5) condition.
Lemma 3.1. Assume that (A2) holds for some ∈ k . Let u be a S k ν solution of (1.1). Then there exist two generalized zeros α u and β u , such that u t for some t α β , .
u u 0 0 Moreover, , Proof. It is an immediate consequence of the fact that u is concave down in ˆ. □ Lemma 3.2. Assume that (A1) and (A5) hold. Let σ be as in Lemma 3.1 and u be a S k ν solution of (1.1) with Proof. Let u be a S k ν solution of (1.1). It follows from Lemma 3.1 that We note that u is a solution of Suppose on the contrary that ≥ / λ μ f k 0 . Then for ∈ x J u , we have from (A5) that

  
Combine this with a generalized version of the Sturm-type comparison theorem [3]. We deduce that u has at least one generalized zero on J u . This contradicts the fact that ( ) > u x 0 on J u . □ 4 Second turn and proof of Theorem 1.3 The main ingredient of this section is an a priori estimate and finally we shall give a proof of Theorem 1.3.
Assume that (A1) and (A4) hold. Let (A2) be satisfied for some ∈ k . Let u be an S k ν solution of (1.1).
Then, there exists a constant > C 0 independent of u such that for Proof. Let be the generalized zeros of u. Then, by (A2) we may get