Existence of positive solutions for second order multi-point boundary value problems

Abstract

We study the second order nonlinear differential equation: (E) y′′ +h(t) f (y) = 0, 0 < t < 1, h(t) ∈ L(0,1), f (y) ∈C(R,R+) subject to multi-point boundary conditions: (i) y(0) = m ∑ i=1 αiy(ξi), y(1) = m ∑ i=1 βiy(ξi), (ii) y′(0) = m ∑ i=1 αiy′(ξi), y(1) = m ∑ i=1 βiy(ξi), (iii) y(0) = m ∑ i=1 αiy(ξi), y′(1) = m ∑ i=1 βiy′(ξi), where 0 < ξ1 < · · · < ξm < 1,αi 0,βi 0, i = 1,2, · · · ,m . We also assume that h(t) is nonnegative and can be singular at t = 0 or t = 1 or both and α1 + · · ·+αm < 1 , β1 + · · ·+βm < 1 . We prove existence theorems for positive solutions of (E) (i), (E) (ii) and (E) (iii) when the limit f (y)/y as y → 0 and y→ ∞ do not necessarily exist. Our results extend recent results of Zhang and Sun [18] for the boundary value problem (E)(i) when αi ≡ 0 for all i = 1, . . . ,m . Mathematics subject classification (2010): 34B10, 34B15.