Eigenvalue problem for p-Laplacian three-point boundary value problems on time scales

Abstract

Let T be a time scale such that 0 , T ∈ T , β , γ ⩾ 0 and 0 < η < ρ ( T ) . We consider the following p-Laplacian three-point boundary problem on time scales ( φ p ( u Δ ( t ) ) ) ∇ + λ h ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , T ) , u ( 0 ) − β u Δ ( 0 ) = γ u Δ ( η ) , u Δ ( T ) = 0 , where p > 1 , λ > 0 , h ∈ C ld ( ( 0 , T ) , [ 0 , ∞ ) ) and f ∈ C ( [ 0 , ∞ ) , ( 0 , ∞ ) ) . Some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. In doing so the usual restriction that f 0 = lim u → 0 + f ( u ) φ p ( u ) and f ∞ = lim u → ∞ f ( u ) φ p ( u ) exist is removed. An example is also given to illustrate the main results.