Positive solutions of a discrete second-order boundary value problems with fully nonlinear term

In this paper, we mainly consider a kind of discrete second-order boundary value problem with fully nonlinear term. By using the fixed-point index theory, we obtain some existence results of positive solutions of this kind of problems. Instead of the upper and lower limits condition on f, we may only impose some weaker conditions on f.

In the past few years, boundary value problems for difference equations have been deduced from different disciplines, such as the computer sciences, economics, mechanical engineering and control systems and so on; see, for instance, [1,6,23,24]. Therefore, many scholars studied the discrete boundary value problems, including the linear discrete problems and nonlinear discrete problems [2-5, 8, 10-17, 19-21, 26, 28-30, 33-35]. In 1999, by using the upper and lower solution method, Agarwal and O'Regan [2] studied the existence of solutions and nonnegative solutions for the following discrete problem: Thereafter, many authors focused on the existence of solutions and positive solutions of (1.2). In particular, since Zhou et al. [26] introduced the variation method to solve the discrete boundary condition, several excellent existence results of discrete boundary value problems have been obtained by using this method; see, for instance, [8,11,26,33,35] and the references therein. For example, by using the variation method, Bonanno et al. [8] studied the existence of multiple positive solutions of (1.2). Meanwhile, as a very important method, the bifurcation technique has also been introduced to discuss the discrete problem as (1.2). For example, by using the bifurcation technique, Gao et al. [12] studied the continuum of the positive and negative solutions of the boundary value problem (1.2) and they also obtained the existence of positive solutions and negative solutions of (1.2). Meanwhile, Ma et al. [27][28][29] and Gao [10] also used the same method to consider different discrete boundary value problems. Finally, another important method used to discuss the positive solutions of the discrete boundary value problems should be noted: fixed-point theory in cones. In fact, since Merdivenci [31] introduced the fixed-point theory in cones to consider the positive solutions of the two-point discrete boundary value problems as (1.2), lots of interesting and excellent results have been obtained. For example, by using the fixed-point theory in cones, Wong and Agarwal [34] considered the existence results of positive solutions for a boundary value problems of a higher-order difference equation, Ma and Raffoul [30] considered the existence of positive solutions of the discrete threepoint boundary value problems in 2004. Later, Henderson and Luca [19][20][21], Agarwal and Luca [2] considered the existence of positive solutions of the discrete multi-point systems.
However, it is noted that most of the above results focus on the problems as (1.2) which does not contain the damping term u in the nonlinear term f . As we know, the damping phenomenon exists widely in the real world. Therefore, it is interesting to consider such a problem which has the damping term in the nonlinear term; see, for instance, [7,22,32]. In [7], Anderson et al. considered the existence of the solutions of this kind of problems by using Schaefer's theorem. In [22,32], the method of lower and upper solutions are used to consider the existence of solutions a kind of discrete problems with the fully nonlinear term. Therefore, inspired by the above the results, we try our best to consider the existence of positive solutions of the discrete boundary value problem (1.1), which has a damping term u in the nonlinear term. Our main tools here are also some fixed-point theories in a cone, called the fixed-point index theories, we only briefly list them in Sect. 3 and we can find them in the references [9,18] for more details. Furthermore, in the present paper, the superlinear and the sublinear conditions on the nonlinear term f at 0 and ∞ do not hold as the limitation form, but some weaker conditions hold at 0 and ∞; see Remarks 3.1 and 3.2. Finally, it is noted that the continuous problems with fully nonlinear terms have been studied by [25].
The rest of the present paper is organized as follows: In Sect. 2, we give some preliminaries, including the work space, the properties of the Green's function and the spectral results of the linear eigenvalue problems. In Sect. 3, we give our main results and prove them.

Preliminaries
At first, let us introduce our work space. Let Then j is an isomorphism from E to Y . Furthermore, define then P is a cone in E. Now, let us consider the following linear boundary value problems: Then the following results hold.

where G(t, s) is the Green's function defined as
Then continuing to sum the above equation from s = 1 to s = t -1, we obtain Combining this with the boundary condition u(T + 1) = 0, we get

Lemma 2.2 The Green's function G(t, s) satisfies the following properties:
Proof The properties (i)-(iv) are obvious. We only prove (v) here. In fact, Therefore, (v) holds.

Lemma 2.3
Let u ∈ P be a solution of (2.1), (2.2). Then u satisfies the following properties: Therefore, Furthermore, by the property (v) of G(t, s), we know that (ii) By direct calculation, we know that Then, for t ∈ [1, T] Z , we get Combining this with the fact that u(0) = u(T + 1) = 0, we see that the assertion (ii) holds.

Main results
In this section, we try our best to find the nontrivial positive solution of the problem (1.1). Let Then K is a positive cone in E. Define an operator A : K → E by

s)f s, u(s), u(s) .
Since f : [1, T] Z × R + × R → R + is a continuous function, it is not difficult to see that A : K → K is a completely continuous mapping. Now, it suffices to find the nontrivial positive fixed-point of A. To get it, let us recall some basic concepts and lemmas on the fixed-point theory in a cone; see [9,18]. Let E be a Banach space, K ⊂ E is a closed convex cone. Suppose that D is a bounded open subset of E with boundary ∂D, and K ∩ D = ∅. Then the following lemmas hold.

Lemma 3.1 Let D be a bounded open subset of E with θ ∈ D, and
The first main result is as follows.
hold. Then the boundary value problem (1.1) has at least one positive solution in K .
Proof Let r 1 ∈ (0, δ) small enough, where δ is the positive constant introduced by (H1). Then, by Lemma 3.1, we try to prove that, for any u ∈ K ∩ ∂ r 1 and 0 < μ ≤ 1, Suppose to the contrary that there exists u 0 ∈ K ∩ ∂ r 1 and 0 < μ 0 ≤ 1 such that μ 0 Au 0 = u 0 . This implies that u 0 is a positive solution of the problem Therefore, combining this with the fact we get Combining this with Lemma 2.3 (ii) and (iii), we obtain This contradicts the assumption a + 2b < 1 T 2 . Therefore, (3.1) holds. By Lemma 3.1, we get Now, let L 0 = max{|f (t, u, v) -cu| : (t, u, v) ∈ [1, T] Z × R + × R, |u| + |v| ≤ H} + 1. Then, the condition (H2) implies that Define a operator A 1 : K → E by Then A 1 : K → K is a completely continuous operator. Now, let r 2 > δ, we show that To get it, by Lemma 3.2, we only need to show that where ϕ 1 (t) = sin π t T+1 / sin π t T+1 E is the eigenfunction of the linear eigenvalue problem (2.4), which corresponds to the first eigenvalue λ 1 = 2 -2 cos π T+1 . Then ϕ 1 E = 1 and ϕ 1 (t) > 0 on [1, T] Z . Suppose to the contrary that there exist u 1 ∈ K ∩ ∂ r 2 and τ 1 ≥ 0 such that u 1 -A 1 u 1 = τ 1 ϕ 1 . Combining this with the definition of A 1 , we know that u 1 is a solution of the problem Therefore, by (H2), we get Multiplying this inequality by ϕ 1 (t) and summing from s = 1 to s = t, we get Now, if T t=1 u 1 (t)ϕ 1 (t) = 0, then we get λ 1 ≥ c. In fact, by Lemma 2.3 (i), for t ∈ [0, T + 1] Z , we get This implies that (1t)Au + tA 1 u = u, u ∈ K ∩ ∂ r 2 , 0 ≤ t ≤ 1. (3.5) Suppose to the contrary that there exist u 2 ∈ K ∩ ∂ r 2 and 0 ≤ t 0 ≤ 1 such that Therefore, by the definition of A and A 2 , we know that u 2 is a solution of the problem Therefore, Multiplying both sides of this inequality by ϕ 1 (t) and summing from s = 1 to s = T, we get Furthermore, by Lemma 2.3 (i), we know that Combining this with (3.6), we obtain . (3.7) Let .
(H4) there exist three positive constants a > 0, b > 0 and H > 0 with a + 2b < 1 T 2 , such that hold. Then the boundary value problem (1.1) has at least one positive solution in K .