Positive Solutions of Singular Initial-Boundary Value Problems to Second-Order Functional Differential Equations. (26th August 2008) - Jin, Fengfei; Yan, Baoqiang
Abstract : Positive solutions to the singular initial-boundary value problemsx _ _ = − f ( t,   x t ),   0 < t < 1,   x 0 = 0,   x ( 1 ) = 0, are obtained by applying the Schauder fixed-point theorem, wherex t ( u ) = x ( t + u )   ( 0 ≤ t ≤ 1 ) on[ − r, 0 ] andf ( ⋅, ⋅ ) : ( 0, 1 ) × ( C + \ { 0 } ) → R +   ( C + = { x ∈ C ( [ − r, 0 ], R ),   x ( t ) ≥ 0,   ∀ t ∈ [ − r, 0 ] } ) may be singular atφ ( u ) = 0   ( − r ≤ u ≤ 0 ) andt = 0 . As an application, an example is given to demonstrate our result.