On positive solutions for some second-order three-point boundary value problems with convection term

• Published in: Journal of inequalities and applications 2019-03, Vol.2019 (1), p.1-11, Article 72
• Main Authors: Wei, Yongfang, Bai, Zhanbing, Sun, Sujing
• Format: Article
• Language: English
• Subjects: Analysis; Applications of Mathematics; Boundary value problems; Continuity (mathematics); Convection; Dependence; Fixed point theorem; Fixed points (mathematics); Mathematics; Mathematics and Statistics; Positive solution; Three-point boundary value problem
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-019-2029-3

In this paper, a fixed point theorem in a cone and some inequalities of the associated Green’s function are applied to obtain the existence of positive solutions of second-order three-point boundary value problem with dependence on the first-order derivative x ″ ( t ) + f ( t , x ( t ) , x ′ ( t ) ) = 0 , 0 < t < 1 , x ( 0 ) = 0 , x ( 1 ) = μ x ( η ) , where f : [ 0 , 1 ] × [ 0 , ∞ ) × R → [ 0 , ∞ ) is a continuous function, μ > 0 , η ∈ ( 0 , 1 ) , μ η < 1 . The interesting point is that the nonlinear term is dependent on the convection term.