Several Existence Theorems of Multiple Positive Solutions of Nonlinear m-Point BVP for an Increasing Homeomorphism and Homomorphism on Time Scales. (7th December 2009) - Han, Wei; Kang, Shugui
Abstract : By using fixed point theorems in cones, the existence of multiple positive solutions is considered for nonlinearm -point boundary value problem for the following second-order boundary value problem on time scales( ϕ ( u Δ ) ) ∇ + a ( t ) f ( t, u ( t ) ) = 0, t ∈ ( 0, T ), ϕ ( u Δ ( 0 ) ) = ∑ i = 1 m − 2 a i ϕ ( u Δ ( ξ i ) ), u ( T ) = ∑ i = 1 m − 2 b i u ( ξ i ), whereϕ : R → R is an increasing homeomorphism and homomorphism andϕ ( 0 ) = 0 . Some new results are obtained for the existence of twin or an arbitrary odd number of positive solutions of the above problem by applying Avery-Henderson and Leggett-Williams fixed point theorems, respectively. In particular, our criteria generalize and improve some known results by Ma and Castaneda (2001). We must point out for readers that there is only thep -Laplacian case for increasing homeomorphism and homomorphism. As an application, one example to demonstrate our results is given.