A note on a class of problems for a higher-order fully nonlinear equation under one-sided Nagumo-type condition

The purpose of this work is to establish existence and location results for the higher-order fully nonlinear differential equation u ( n ) ( t ) = f ( t , u ( t ) , u ′ ( t ) , … , u ( n − 1 ) ( t ) ) , n ≥ 2 , with the boundary conditions u ( i ) ( a ) = A i , for i = 0 , … , n − 3 , u ( n − 1 ) ( a ) = B , u ( n − 1 ) ( b ) = C or u ( i ) ( a ) = A i , for i = 0 , … , n − 3 , c 1 u ( n − 2 ) ( a ) − c 2 u ( n − 1 ) ( a ) = B , c 3 u ( n − 2 ) ( b ) + c 4 u ( n − 1 ) ( b ) = C , with A i , B , C ∈ R , for i = 0 , … , n − 3 , and c 1 , c 2 , c 3 , c 4 real positive constants. It is assumed that f : [ a , b ] × R n − 1 → R is a continuous function satisfying one-sided Nagumo-type conditions which allows an asymmetric unbounded behaviour on the nonlinearity. The arguments are based on the Leray–Schauder topological degree and lower and upper solutions method.