Multiple Positive Solutions for Singular Quasilinear Multipoint BVPs with the First-Order Derivative. (17th April 2008) - Jiang, Weihua; Wang, Bin; Guo, Yanping
Abstract : The existence of at least three positive solutions for differential equation( ϕ p ( u ‣ ( t ) ) ) ‣ + g ( t ) f ( t, u ( t ), u ‣ ( t ) ) = 0, under one of the following boundary conditions:u ( 0 ) = ∑ i = 1 m − 2 a i u ( ξ i ), φ p ( u ′ ( 1 ) ) = ∑ i = 1 m − 2 b i φ p ( u ′ ( ξ i ) ) orφ p ( u ′ ( 0 ) ) = ∑ i = 1 m − 2 a i φ p ( u ′ ( ξ i ) ), u ( 1 ) = ∑ i = 1 m − 2 b i u ( ξ i ) is obtained by using the H. Amann fixed point theorem, whereφ p ( s ) = | s | p − 2 s, p > 1, 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1, a i > 0, b i > 0, 0 < ∑ i = 1 m − 2 a i < 1, 0 < ∑ i = 1 m − 2 b i < 1 . The interesting thing is thatg ( t ) may be singular at any point of [0, 1] andf may be noncontinuous.