Existence and Multiplicity of Positive Solutions of a Nonlinear Discrete Fourth-Order Boundary Value Problem

and Applied Analysis 3 Ma and Xu 15 also applied the fixed point theorem in cones to obtain some results on the existence of generalized positive solutions. It is the purpose of this paper to show some new results on the existence and multiplicity of generalized positive solutions of 1.4 , 1.8 by Dancer’s global bifurcation theorem. To wit, we get the following. Theorem 1.3. Let h : 2 → 0,∞ , f ∈ C , 0,∞ , and lim s→ 0 f s s f0 ∈ 0,∞ , lim s→∞ f s s f∞ ∞. 1.9 Assume that there exists B ∈ 0, ∞ such that f is nondecreasing on 0, B . Then i 1.4 , 1.8 have at least one generalized positive solution if 0 < λ < λ1/f0; ii 1.4 , 1.8 have at least two generalized positive solutions if λ1 f0 < λ < sup s∈ 0,B s γ ∗f s , 1.10 where γ ∗ maxt∈ 1 ∑T s 2 K t, s h s , K t, s is defined as 2.3 and λ1 is the first eigenvalue of Δ4u t − 2 λh t u t , t ∈ 2, u 1 u T 1 Δ2u 0 Δ2u T 0. 1.11 The “dual” of Theorem 1.3 is as follows. Theorem 1.4. Let h : 2 → 0,∞ , f ∈ C , 0,∞ , and lim s→ 0 f s s f0 ∈ 0,∞ , lim s→∞ f s s f∞ 0. 1.12 Assume that there exists B ∈ 0, ∞ such that f is nondecreasing on 0, B . Then i 1.4 , 1.8 have at least a generalized positive solution provided λ > inf s∈ 0,c1B s c1γ∗f s , 1.13 where γ∗ mint∈ 2 ∑T s 2 K t, s h s ; 4 Abstract and Applied Analysis ii 1.4 , 1.8 have at least two generalized positive solutions provided inf s∈ 0,c1B s c1γ∗f s < λ < λ1 f0 . 1.14 The rest of the paper is organized as follows: in Section 2, we present the form of the Green’s function of 1.4 , 1.8 and its properties, and we enunciate the Dancer’s global bifurcation theorem 16, Corollary 15.2 . In Section 3, we use the Dancer’s bifurcation theorem to prove Theorems 1.3 and 1.4 and in Section 4, we finish the paper presenting a couple of illustrative examples. Remark 1.5. For other results on the existence and multiplicity of positive solutions and nodal solutions for fourth-order boundary value problems based on bifurcation techniques, see 17– 21 . 2. Preliminaries and Dancer’s Global Bifurcation Theorem Lemma 2.1. Let h : 2 → . Then the linear boundary value problem Δ4u t − 2 h t , t ∈ 2, u 1 u T 1 Δ2u 0 Δ2u T 0 2.1