Existence of Positive Solutions to Singular φ-Laplacian Nonlocal Boundary Value Problems when φ is a Sup-multiplicative-like Function

Abstract

In this paper, using a fixed point index theorem on a cone, we present some existence results for one or multiple positive solutions to φ -Laplacian nonlocal boundary value problems when φ is a sup-multiplicative-like function and the nonlinearity may not satisfy the L 1 -Carath e ´ odory condition.

Existence and Multiplicity Results for Nonlocal Boundary Value Problems with Strong Singularity

“…The following lemmas (Lemmas 1-3) can be proved by the similar arguments in [4] (Section 2) and [39] (Section 2). For the sake of completeness, we give the proofs of them.…”

“…is sup-multiplicative-like, where c k ≥ 0 and p k ∈ (1, ∞) for 1 ≤ k ≤ n and c 1 c n > 0 for some n ∈ N (see, e.g., [2,4]). Lee and Xu ( [5,6]) generalized the condition (A1) to the one with ψ 2 is a function not requiring that ψ 2 (0) = 0 and studied the existence of positive solutions to singularly weighted nonlinear systems.…”

“…For more general ϕ which does not satisfy (A1), Kaufmann and Milne [21] considered BVP (1) and ( 2) with α1 = α2 = 0, 0 ≤ h ∈ L 1 (0, 1) with h ≡ 0 and f = f (u) ∈ C(R + , R + ) and showed the existence of a positive solution for all λ > 0 under the assumptions on the nonlinearity f which induce the sublinear nonlinearity provided ϕ(s) = |s| p−1 s with p > 1. Recently, for an odd increasing homeomorphism ϕ satisfying (A1), Kim and Jeong [4] studied various existence results for positive solutions to BVP (1) and ( 2) with λ = 1. For other interesting results, we refer the reader to and the references therein.…”

Multiplicity of Positive Solutions to Nonlocal Boundary Value Problems with Strong Singularity

“…For any g ∈ H ϕ and any σ satisfying (14), T(g) is monotone increasing on [0, σ) and monotone decreasing on (σ, 1]. We notice that σ = σ(g) is not necessarily unique, but T(g) is independent of the choice of σ satisfying (14) (see [10], [Remark 2]).…”

“…Goodrich [9] studied perturbed Volterra integral operator equations and, as an application, established the existence of at least one positive solution to the p-Laplacian differential equation with nonlocal boundary conditions. Jeong and Kim [10] obtained sufficient conditions on the nonlinearity f for the existence of multiple positive solutions to problem (1)-( 2) with λ = 1. For the nonlinearity f = f (t, s) satisfying f (t, 0) ≡ 0, Kim [11] showed the existence, nonexistence and multiplicity of positive solutions to problem (1)-( 2) by investigating the shape of the unbounded solution continuum.…”