On a Three-Point Boundary Value Problem for Third Order Differential Equations with Singularities in Phase Variables

Abstract

Abstract The singular boundary value problem (ϕ(𝑥″))′ = 𝑓(𝑡, 𝑥, 𝑥′, 𝑥″), 𝑥(0) = 𝑥(𝑇1) = 𝑥(𝑇) = 0 is considered. Here 0 < 𝑇1 < 𝑇, ϕ is an increasing homeomorphism from ℝ onto ℝ, positive 𝑓 satisfies the local Carathéodory conditions on [0, 𝑇] × (ℝ \ {0})3 and 𝑓 may be singular at the value 0 of all its phase variables. The conditions guaranteeing the solvability of the above problem are presented. The proofs are based on regularization and sequential techniques and in limit processes a combination of the Fatou theorem and the Lebesgue dominated convergence theorem is used.