A four-point boundary value problem with singular ϕ-Laplacian

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 21; no. 2; pp. 1 - 16
• Main Authors: Chinní, Antonia, Di Bella, Beatrice, Jebelean, Petru, Precup, Radu
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2019; Springer Nature B.V
• Subjects: Analysis; Boundary conditions; Boundary value problems; Fixed points (mathematics); Localization; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Singular; positive solutions; Krasnoselskii’s fixed point theorem; Laplace equations; Primary 34L30; Schauder’s fixed point theorem; Secondary 34B10; four-point boundary value problems; 34B18
• ISSN: 1661-7738; 1661-7746; 2730-5422
• DOI: 10.1007/s11784-019-0703-1

We prove that the four-point boundary value problem - ϕ ( u ′ ) ′ = f ( t , u , u ′ ) , u ( 0 ) = α u ( ξ ) , u ( T ) = β u ( η ) , where f : [ 0 , T ] × R 2 → R is continuous, α , β ∈ [ 0 , 1 ) , 0 < ξ < η < T , and ϕ : ( - a , a ) → R ( 0 < a < ∞ ) is an increasing homeomorphism, which is always solvable. When instead of f is some g : [ 0 , T ] × [ 0 , ∞ ) → [ 0 , ∞ ) , we obtain existence, localization, and multiplicity of positive solutions. Our approach relies on Schauder and Krasnoselskii’s fixed point theorems, combined with a Harnack-type inequality.