Positive solutions of m-point boundary value problems for higher order ordinary differential equations

• Published in: Nonlinear analysis Vol. 66; no. 7; pp. 1573 - 1586
• Main Authors: Guo, Yanping, Tian, Jiwei
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.04.2007; Elsevier
• Subjects: 34B10; 34B15; Exact sciences and technology; Fixed point theorem in cones; Green’s function; Higher order ordinary differential equation; m-point boundary value problem; Mathematical analysis; Mathematics; Ordinary differential equations; Positive solution; Sciences and techniques of general use; Fixed point theorem in cones; 34B15; Green’s function; 34B10; Positive solution; Higher order ordinary differential equation; m-point boundary value problem; Cone; Differential equation; Nonlinear analysis; 34B15 Higher order ordinary differential equation; Green's function; Green function; Partial differential equation; Numerical analysis; Boundary value problem; Fixed point theorem; MR subject classification: 34B10
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2006.02.010

In this paper, we study m-point boundary value problems for higher order ordinary differential equation { y ( 2 n ) ( t ) = f ( t , y ( t ) , y ″ ( t ) , … , y ( 2 ( n − 1 ) ) ( t ) ) , 0 ≤ t ≤ 1 , y ( 2 i ) ( 0 ) = 0 , y ( 2 i ) ( 1 ) = ∑ j = 1 m − 2 k i j y ( 2 i ) ( ξ j ) , 0 ≤ i ≤ n − 1 , where f is allowed to change sign, and 0 = ξ 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < ξ m − 1 = 1 . We show sufficient conditions for the existence of at least two positive solutions by applying a new fixed point theorem in cones and the associated Green’s function. In particular, the second positive solutions for the above problem is not concave.