Positive solutions of nonlinear multi-point boundary value problems

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 22; no. 5; pp. 1387 - 1402
• Main Author: Dogan, Abdulkadir
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.11.2018; Springer Nature B.V
• Subjects: Boundary value problems; Calculus of Variations and Optimal Control; Optimization; Econometrics; Fixed points (mathematics); Fourier Analysis; Mathematics; Mathematics and Statistics; Nonlinear differential equations; Operator Theory; Potential Theory; Nonlinear boundary value problems; 34B15; Fixed point theorem; Positive solutions; Differential equation; 34B18
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-018-0583-4

This paper deals with the existence of positive solutions of nonlinear differential equation u ″ ( t ) + a ( t ) f ( u ( t ) ) = 0 , 0 < t < 1 , subject to the boundary conditions u ( 0 ) = ∑ i = 1 m - 2 a i u ( ξ i ) , u ′ ( 1 ) = ∑ i = 1 m - 2 b i u ′ ( ξ i ) , where ξ i ∈ ( 0 , 1 ) with 0 < ξ 1 < ξ 2 < ⋯ < ξ m - 2 < 1 , and a i , b i satisfy    a i , b i ∈ [ 0 , ∞ ) , 0 < ∑ i = 1 m - 2 a i < 1 , and ∑ i = 1 m - 2 b i < 1 . By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality inf 0 ≤ t ≤ 1 u ( t ) ≥ γ ‖ u ‖ ∞ .