Two Positive Solutions of Third-Order BVP with Integral Boundary Condition and Sign-Changing Green's Function

Abstract EN

We are concerned with the following third-order boundary value problem with integral boundary condition: u ′ ′ ′ ( t ) = f ( t , u ( t ) ) , t ∈ [ 0,1 ] , u ′ ( 0 ) = u ( 1 ) = 0 , u ′ ′ ( η ) + ∫ α β u ( t ) d t = 0 , where 1 / 2 < α ≤ β ≤ 1 , α + β ≤ 4 / 3 , and η ∈ ( 1 / 2 , α ] .

Although the corresponding Green's function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions on f by using the two-fixed-point theorem due to Avery and Henderson.

An example is also included to illustrate the main results obtained.