Singular Three-point Boundary Value Problem Research Articles

Positive solutions for singular second order three-point boundary value problems

In this paper, the singular three-point boundary value problem { u ″ ( t ) + h ( t ) f ( t , u ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = α u ( η ) , where η ∈ ( 0 , 1 ) , 0 < α < 1 / η , is considered under some conditions concerning the first eigenvalue corresponding to the relevant linear operator. h ( t ) is allowed to be singular at t = 0 , 1 and f may be singular at u = 0 . The existence of a positive solution is obtained by a fixed index point.

Positive solutions of singular three-point boundary value problems for the One-dimensional p-Laplacian

Abstract-We use the fixed-point index theory to establish the existence of at least one or two positive solutions for the singular three-point boundary value problems ( φ p ( y ′ ) ) ′ + a ( t ) f ( y ( t ) ) = 0 , 0 < t < 1 , y ′ ( 0 ) = 0 , y ( 1 ) = β y ( η ) , where φ p ( s ) = | s | p −2 s , p 2, 0 ⪯ β < 1, 0 < η < 1, ƒ C ([O,+∞), [0,+∞)), a ∈ C ((0,1), [0, +∞ )), and a ( t ) is allowed to have a singularity at the endpoints of (0, 1). Applications of our results are provided to yield positive radial solutions of some partial differential equations boundary value problems on an annulus. As an application, we also give some examples to demonstrate our results.

A second-order singular three-point boundary value problem

The existence of a positive solution is obtained for the second-order three-point boundary value problem γ″ + ƒ (χ, γ) = 0, 0 < χ ≤ 1, y(0) = 0, γ(p) −γ(1) = 0, where 0 < p < 1 is fixed and where (χ, γ) is singular at χ = 0, = 0, and possibly at γ = ∞. The method applies a fixed-point theorem for mappings that are decreasing with respect to a cone.

Positive solutions to a singular second order three-point boundary value problem

A fixed point theorem is used to study a singular second order three-point boundary value problem. The problem is more general. Combining the method of constructing Green functions with operators defined piecewise, the existence resultof positive solutions to a singular second order three-point boundary value problem is established. The nonlinearity can be allowed to change sign.