Positive Solutions for a Second-Order p-Laplacian Boundary Value Problem with Impulsive Effects and Two Parameters

and Applied Analysis 3 The rest of the paper is organized as follows: in Section 2, we state the main results of problem (7). In Section 3, we provide some preliminary results, and the proofs of the main results together with several technical lemmas are given in Section 4. The final section of the paper contains an example to illustrate the theoretical results. 2. Main Results In this section, we state the main results, including existence and multiplicity results of positive solutions for problem (7). For convenience, we introduce the following notations: f 0 = lim sup u→0 max t∈J f (t, u) φp (u) , f ∞ = lim sup u→∞ max t∈J f (t, u) φp (u) , f0 = lim inf u→0 min t∈J f (t, u) φp (u) , f∞ = lim inf u→∞ min t∈J f (t, u) φp (u) , I 0 (k) = lim sup u→0 max t∈J Ik (t, u) u , I ∞ (k) = lim sup u→∞ max t∈J Ik (t, u) u , I0 (k) = lim inf u→0 min t∈J Ik (t, u) u , I∞ (k) = lim inf u→∞ min t∈J Ik (t, u) u , J = [0, 1] , k = 1, 2, . . . , n. (10) Moreover, we choose four numbers r, r1, r2, and R satisfying 0 < r < r1 < δr2 < r2 < R < +∞, (11) where δ is defined in (23). Theorem 1. Assume that (H1)–(H4) hold and f∞, f, I∞(k), and I(k) (k = 1, 2, . . . , n) are positive constants. Then, (i) there exist λ0 > 0 and μ0 > 0 such that, for any λ > λ0 and μ > μ0, problem (7) has a positive solution u with