Singular discrete and continuous mixed boundary value problems

Abstract

For each n ∈ N , n ≥ 2 , we prove the existence of a solution ( u 0 , … , u n ) ∈ R n + 1 of the singular discrete problem 1 h 2 Δ 2 u k − 1 + f ( t k , u k ) = 0 , k = 1 , … , n − 1 , Δ u 0 = 0 , u n = 0 , where u k > 0 for k = 0 , … , n − 1 . Here T ∈ ( 0 , ∞ ) , h = T n , t k = h k , f ( t , x ) : [ 0 , T ] × ( 0 , ∞ ) → R is continuous and has a singularity at x = 0 . We prove that for n → ∞ the sequence of solutions of the above discrete problems converges to a solution y of the corresponding continuous boundary value problem y ″ ( t ) + f ( t , y ( t ) ) = 0 , y ′ ( 0 ) = 0 , y ( T ) = 0 .