Two Positive Solutions of Third-Order BVP with Integral Boundary Condition and Sign-Changing Green's Function. (19th May 2015) - Niu, Bing-Wei; Sun, Jian-Ping; Ren, Qiu-Yan
Abstract : 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, where1 / 2 < α ≤ β ≤ 1,     α + β ≤ 4 / 3, andη ∈ ( 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 onf by using the two-fixed-point theorem due to Avery and Henderson. An example is also included to illustrate the main results obtained.