Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence

Abstract In this paper, we study second-order nonlinear discrete Robin boundary value problem with parameter dependence. Applying invariant sets of descending flow and variational methods, we establish some new sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions of the system when the parameter belongs to appropriate intervals. In addition, an example is given to illustrate our results.


Introduction
Throughout this paper, let N, Z and R denote the sets of all nature numbers, integers and real numbers, respectively. We consider the following second-order nonlinear di erence equation with Robin boundary value problem (BVP for short) where T ≥ is a given integer, [ , T] =∶ { , , ⋯, T}, parameter α > , f ∶ [ , T] × R → R is continuous in the second variable, ∆ denotes the forward di erence operator de ned by ∆x(k) = x(k + ) − x(k), ∆ x(k) = ∆(∆x(k)). Discrete nonlinear equations with parameter dependence play an important role in describing many physical problems, such as nonlinear elasticity theory or mechanics and engineering topics [ , ]. In recent years, some authors also contributed to the study of (1) and obtained some interesting results. For example, when α = , Jiang and Zhou [ ] employed strongly monotone operator and critical point theory to establish the existence of nontrivial positive solutions. By virtue of variational methods and critical point theory, Guo and Song [ ] investigated the existence of positive solutions. Zhang  View from above reasons and motivated by [ ], the purpose of this paper is to apply invariant sets of descending ow and variational techniques to get some su cient conditions for the existence of signchanging solutions, negative solutions and positive solutions to (1).
In the following, we rst consider the linear eigenvalue problem corresponding to (1) Let λ k be eigenvalues of (2) and {z k } T k= be the corresponding eigenvectors of {λ k } T k= , then In this paper, we focus on the following assumptions: where F(k, u) = ∫ u f (k, s)ds.
Our results read as follows: The remainder of this paper is organized as follows. After introducing some notations and preliminary results in Section , we complete the proof of Theorem 1.1 and give an example to illustrate our result in Section .

Variational structure and preliminary results
Given for all x, y ∈ G, then the induced norm ⋅ m is Let H be the T-dimensional Hilbert space equipped with the usual inner product (⋅, ⋅) and norm ⋅ . It is easy to see that G is isomorphic to H, ⋅ m and ⋅ are equivalent. Denote x + = max{x, }, x − = min{x, }. Then for any x ∈ H, ⟨⋅, ⋅⟩ m ≥ . De ne functional I ∶ H → R as For any x = (x( ), x( ), ⋯, x(T)) ⊺ ∈ H, I(x) can be rewritten as here α τ is the transpose of the vector α on H, Remark 2.1. In fact, many existing results are applicable. Namely, one can apply numerous results for the variational formulation (4), see [ − ].
where h ∶ [ , T] → R. It is not hard to see that (5) and the system of linear algebra equations (A + mI)x = h are equivalent, then the unique solution of (5) can be expressed by On the other side, we have Lemma 2.2. The unique solution of (5) is here G m (k, s) can be written as Proof. First study the homogeneous equation of (5) then the corresponding characteristic equation of (7) is p −( + m)p + = . Consider m > (m = is trivial), then ( + m) − > , which means we have Then two independent solutions of (7) can be expressed by x (k) = p k and x (k) = p k . Therefore, the general solution of (5) is x(k) = a (k)p k + a (k)p k . The next step is to determine coe cients a (k) and a (k). Using the method of variation of constant, we get the general solution of (5) as From initial conditions, we nd a ( ) = −a ( ) and which means the proof of Lemma 2.2 is completed.
Proof. For any x, y ∈ H, using the mean value theorem, it follows As f is continuous in x, we nd  In this paper, we will analyse the properties of the ow, pay close attention to the direction and the destination to which the ow goes, and seek the limit along the ow. We are interested in those points in H across which the ow does not go to in nity and work for seeking such points in H. If we have such a point, then the ow curve crossing it goes ultimately to a critical point. It seems that one would obtain many critical points if he or she is given many such points. However, even if there may be many such points, we cannot get more than one critical point in general since the di erent ow curves may ultimately go to the same critical point. In order to get more critical points, we will de ne the concept of invariant set of descending ow and then we will divide the whole space H into several invariant subsets of descending ow. In this way, we can get more than one critical point.

Proof of main result
Let convex cones Λ = {x ∈ H ∶ x ≥ } and −Λ = {x ∈ H ∶ x ≤ }. The distance respecting to ⋅ m in H is written by dist m . For arbitrary ε > , we denote contains only sign-changing functions.

Lemma 3.1. Suppose one of the following condition holds.
(i) r = +∞ or (ii) r < +∞ is not an eigenvalue of (2), here r is de ned by (J ).
Then the functional I de ned by (3) satis es (PS) condition for all α ∈ , +∞ .
Proof. (i) Assume r = +∞. Let {x n } ⊂ H be a (PS) sequence. Since H is a nite dimensional space, we only need to show {x n } is bounded. If r = +∞, choosing a constant γ > , for all Then (ii) suppose r < +∞ is not an eigenvalue of (2). We are now ready to prove that {x n } is bounded. Arguing by contradiction, we suppose there is a subsequence of {x n } with ρ n = x n → +∞ as n → ∞ and for each k ∈ [ , T], either {x n (k)} is bounded or x n (k) → +∞. Put y n = x n ρ n . Clearly, y n = .
Then there have a subsequence of {y n } and y ∈ H satisfying that y n → y as n → ∞.
For I ′ (x n ) ρ n → as n → ∞, we have y − K ry → . In view of Lemma 2.5, we nd that r is an eigenvalue of matrix A, which contradicts the assumption. So {x n } is bounded and the proof is nished.
We claim {x n } is bounded. Actually, if {x n } is unbounded, it possesses a subsequence of {x n } and some k ∈ [ , T] satisfying x n (k ) → +∞ as n → ∞. According to (J )(i), we get and there is a constant M > such that uf (k, u) − F(k, u) ≤ M for any k ∈ [ , T] and u ∈ R. Therefore, which contradicts (10). So our claim is proved and I satis es the (C) condition. Finally, assume (J )(ii) hold. In a similar way as above, we nd that I satis es (C) condition. Then Lemma 3.2 is veri ed. Lemma 3.3. If (J ) and (J ) hold, there exist m ≥ and ε > such that for < ε < ε , we have: ε is a nontrivial critical point of I and A m (∂B − ε ) ⊂ B − ε , then x is a negative solution of (1); (ii) if x ∈ B + ε is a nontrivial critical point of I and A m (∂B + ε ) ⊂ B + ε , then x is a positive solution of (1).
(ii) can be discussed similarly, we only need to change y + to y − to prove (ii). For simplicity, we omit its proof. (2)Assume r ∈ λ α , +∞ . For u ∈ H , x = ε z + ε z . In general, we can suppose (z , z ) = .