On the number of positive solutions of systems of nonlinear dynamic equations on time scales

In this paper we consider the following n-dimensional second-order nonlinear system on time scales u Δ Δ ( t ) + λ a ( t ) f ( u σ ( t ) ) = 0 , t ∈ [ a , b ] T with the Sturm–Liouville boundary conditions α u ( a ) - β u Δ ( a ) = 0 , γ u ( σ ( b ) ) + δ u Δ ( σ ( b ) ) = 0 , where u = ( u 1 , … , u n ) , α = diag [ α 1 , … , α n ] , β = diag [ β 1 , … , β n ] , γ = diag [ γ 1 , … , γ n ] , δ = diag [ δ 1 , … , δ n ] . Let f 0 = ∑ i = 1 n lim ∥ u ∥ → 0 f i ( u ) / ∥ u ∥ and f ∞ = ∑ i = 1 n lim ∥ u ∥ → ∞ f i ( u ) / ∥ u ∥ . Define i 0 = number of zeros in the set { f 0 , f ∞ } and i ∞ = number of infinities in the set { f 0 , f ∞ } . By using fixed point index theory, we show that:(i) = 1 or 2, then there exist λ 0 > 0 such that the system has i 0 positive solution(s) for λ > λ 0 ; = 1 or 2, then there exist λ 0 > 0 such that the system has i 0 positive solution(s) for 0 < λ < λ 0 ; = 0 or i ∞ = 0 , then the system has no positive solution for sufficiently large or small λ > 0 , respectively.