Positive solutions of the singular eigenvalue problem for a higher-order differential equation on time scales

In this paper, we are concerned with the singular eigenvalue problem for 2 n th-order differential equations { ( − 1 ) n y △ 2 n ( t ) = μ h ( t ) f ( t , y ( t ) ) , t ∈ [ a , b ] , y △ 2 i ( a ) − β i + 1 y △ 2 i + 1 ( a ) = α i + 1 y △ 2 i ( η ) , γ i + 1 y △ 2 i ( η ) = y △ 2 i ( b ) , 0 ≤ i ≤ n − 1 , where μ is a positive parameter, η ∈ ( a , b ) , n ≥ 1 , β i ≥ 0 , 1 < γ i < b − a + β i η − a + β i , 0 ≤ α i < b − γ i η + ( γ i − 1 ) ( a − β i ) b − η , i = 1 , 2 , ⋯ , n . The nonlinearities h : ( a , b ) → [ 0 , + ∞ ) and f : [ a , b ] × ( 0 , + ∞ ) → [ 0 , + ∞ ) are continuous; h may have singularity at t = a and/or t = b and f has singularity at y = 0 . Using the fixed point index theorem and the first eigenvalue of the positive linear operator obtained from the Krein–Rutman theorem, we investigate the existence of positive solutions of the eigenvalue problem and obtain the interval of parameter μ .