A global result for discrete φ-Laplacian eigenvalue problems

where T >  is a given positive integer, u(k) = u(k + ) – u(k), λ is a positive parameter, p : [,T]Z → (,∞) and g : R+ → (,∞) is continuous. We assume that (A) φ : R→ R is an odd and strictly increasing homeomorphism; (A) limu→∞ g(u) φ(u) = ∞. The function φ(u) covers two important cases: φ(u) = u and φ(u) = |u|p–u (p > ). If φ(u) = u, then problem () is the classical second order difference Dirichlet boundary value problem. For the case that φ(u) = |u|p–u, problem () is the well-known discrete p-Laplacian problem. The two cases have been widely studied. To name a few, see [–] and the references therein. Problem () can be viewed as the discrete analogue of the following differential φLaplacian problem { (φ(u′))′ + λp(t)g(u(t)) = ,  < t < , u() = u() = , ()

Problem () can be viewed as the discrete analogue of the following differential φ-Laplacian problem (φ(u )) + λp(t)g(u(t)) = ,  < t < , which rises from the study of radial solutions for p-Laplacian equations (φ(u) = |u| p- u) on an annular domain (see [], and references therein). Recently, the differential φ-Laplacian problems have been widely studied in many different papers. We refer the readers to [ , the existence and uniqueness of solutions were discussed for mixed and Dirichlet boundary value problems by the fixed point theory of contraction mapping. To the best of our knowledge, there are no results on the existence and multiplicity of positive solutions for difference φ-Laplacian problems. Therefore, the purpose of this paper is to establish a global result of positive solutions of (). We state our main result as follows.
Theorem . Let (A) and (A) hold. Then there exists λ * >  such that problem () has at least two positive solutions for λ ∈ (, λ * ), at least one positive solution for λ = λ * and no solution for λ > λ * .
The result is motivated mainly by the ideas in [, ], in which some global results of positive solutions were established for boundary value problems of p-Laplacian differential systems and φ-Laplacian differential systems, respectively.
Generally, in order to make priori estimations on possible positive solutions of φ-Laplacian problems, the function φ satisfies not only condition (A), but also other additional conditions. For example, Wang [] used the following condition.
In our discussion for the single discrete problem (), we only assume that φ satisfies condition (A). In addition, in the discussion of nonlinear differential systems in [, ], monotonicity conditions were imposed on nonlinear terms. In this paper g does not have to satisfy monotonicity conditions. The remaining part of this paper is organized as follows. In Section , we show some lemmas for the later use. In Section , we show the proof of Theorem .. Our proofs are mainly based on the upper and lower solutions technique arguments and the fixed-point index theory for cones.

Some lemmas
First, we introduce an existence result of solutions based on lower and upper solutions method for discrete φ-Laplacian boundary value problems, which has been proved by Cabada [].
Consider the following boundary value problem where http://www.advancesindifferenceequations.com/content/2013/1/264 (B) φ : R → R is an odd and strictly increasing function; In a same way, we define the upper solution of () by reversing the above inequalities.

Lemma . []
Let (B) and (B) hold. Assume that there exist α and β, respectively lower and upper solutions of () such that α ≤ β. Then problem () has at least one solution u with α ≤ u ≤ β.
A function u of integer variable is said to be concave if  u(k -) ≤ . This is the same as saying that the first difference u(k -) is non-increasing. If  u(k -) < , then u is said to be strictly concave.
We have by the monotonicity of It is easy to see that u = u C[,T+] R . By the concavity ofũ(t) on [, T + ] R , we have that for any δ ∈ (, T+  ) R , which implies the expected results. http://www.advancesindifferenceequations.com/content/2013/1/264 Lemma . Let (B) hold and u satisfy the following difference inequality Lemmas . and . yield the following result.

Lemma . Let (B) hold. Then each solution u of () is strictly concave and u(k)
Consider the following boundary value problem where h(k) >  for all k ∈ [, T] Z and φ satisfies (A). It is easy to check that u ∈ E is a solution of () if and only if where C  satisfies u(T + ) = , i.e., Since φ - is a homeomorphism from R onto itself, the solution C  of () is unique. Let Then () can be rewritten as follows Lemma . Let (A) hold. Assume that u solves () and u = u(k * ). Then Equivalently, where C  is the unique solution of the following equation Proof Suppose that the conclusion is not true. Then there exists a sequence {λ n } ⊂ and u n corresponding positive solutions of () at λ n such that u n → ∞ as n → ∞. Let λ = inf{λ n }, p = min k∈[,T] Z p(k) and =  λp . Since  / ∈ , we know λ > . By (A), there exists R >  such that g(u) > φ(u) for all u > R. By the concavity of u n and Lemma ., we know that holds for n sufficiently large. It follows that g u n (k) > φ u n (k) , k ∈ [, T] Z . http://www.advancesindifferenceequations.com/content/2013/1/264 Let u n = u n (k * ). If k * > , then by Lemma ., we have a contradiction that where C (n)  satisfies u n (T + ) = . If k * = , then we also have a contradiction that where C (n)  satisfies u n () = . The proof is completed.
Lemma . Let (B) hold. Then there exists λ * >  such that () has a positive solution at λ * .
Proof Take α ≡ . Then α is a lower solution of (). Let β solve the following boundary value problem Therefore, β is an upper solution of () at λ * . By Lemma ., () has a positive solution u at λ * , and the proof is done.

Lemma . Let (A) and (A) hold. Then is bounded.
Proof Suppose, on the contrary, that there exists a sequence {λ n } ⊂ and u n corresponding positive solutions of () at λ n such that λ n → ∞ as n → ∞. By (A), there exists >  such that for all u ≥ , g(u) ≥ φ(u). Since λ n → ∞ as n → ∞, there exists δ >  satisfying δp >  such that λ n > δ holds for n sufficiently large. Let u n = u n (k * ). Similar to the arguments in the proof of Lemma ., if k * > , we have the following contradiction and if k * = , we also have a contradiction that Lemma . Let (A) and (A) hold. Then there exists λ * >  such that = (, λ * ]. Proof By Lemmas .-., is a nonempty bounded interval open at the left with the left endpoint  / ∈ . We only need to show that it is closed at the right. Let sup = λ * . Then there exists a sequence {λ n } ⊂ satisfying λ n < λ n+ such that λ n → λ * as n → ∞. Since {λ n } is bounded, Lemma . implies that there exists a constant ℵ such that for all n and all possible positive solutions u n of () at λ n , u n ≤ ℵ. It follows that {u n } has a convergent subsequence, say again {u n }, which converges to u * . Since (λ n , u n ) solves problem (), we know that (φ( u n (k -))) + λ n p(k)g(u n (k)) = , k ∈ [, T] Z , u n () = u n (T + ) = .
In the succeeding arguments, we need the following well-known fixed point index theorem on cones. For proof and details, see Guo [].

Lemma . Let E be a Banach space, let P be a cone in E, and let be a bounded open
set in E with  ∈ . Let T : P ∩¯ → P be a complete continuous operator satisfying that Then i(T, P ∩ , P) = . http://www.advancesindifferenceequations.com/content/2013/1/264 3 The proof of Theorem 1.1 By Lemma ., there exists λ * >  such that problem () has at least one positive solution for λ ∈ (, λ * ], and no solution for λ > λ * . So we only need to show the existence of the second positive solution for λ ∈ (, λ * ). Let u * be the positive solution of () at λ * , and λ :  < λ < λ * be given. Choose a constant M > such that g(u * (k)) > M for k ∈ [, T] Z . Since g is uniformly continuous on [, u * + ], there exits a sufficiently small number >  such that for all u ∈ E : uu * ≤ , Then is bounded and open in E. Consider the following problem (φ( u(k -))) + λp(k)g * (u(k)) = , k ∈ [, T] Z , u() = u(T + ) = , where Then g * : R → R is continuous and bounded function. We show that all solutions u of () It follows that u(k  -) ≥ u * (k  -) and u * (k  ) ≥ u(k  ). Thus, On the other hand, ≤ g u * (k  ) http://www.advancesindifferenceequations.com/content/2013/1/264 and () implies that which contradicts (). Thus, if u is a solution of (), then u ∈ . Define T * λ the same as T λ replacing g by g * . Then T * λ : K → K is continuous. Since g * is bounded, there exists R  >  such that T * λ u < R  for all u ∈ K . Thus, we can choose R sufficiently large such that R > R  and ⊂ B R . Here B R = {u ∈ E : u < R}. Then by Lemma ., we have i(T * λ , B R ∩K, K) = . Since all positive fixed points of T * λ are contained in , and T λ u = u is equivalent to T * λ u = u on ∩ K , one has ()