Existence and uniqueness of solutions of a fourth-order boundary value problem with non-homogeneous boundary conditions

Abstract

Let m ≥ 2 and a , b , c > 0 . We consider the existence and uniqueness of solutions for the fourth order iterative boundary value problem, x ( 4 ) ( t ) = − f ( t , x ( t ) , x [ 2 ] ( t ) , … , x [ m ] ( t ) ) , − a ≤ t ≤ a where x [ 2 ] ( t ) = x ( x ( t ) ) and for j = 3 , … , m , x [ j ] ( t ) = x ( x [ j − 1 ] ( t ) ) , with solutions satisfying one of the following sets of conjugate boundary conditions: x ( − a ) = − a , x ′ ( − a ) = b , x ″ ( − a ) = c , x ( a ) = a , x ( − a ) = − a , x ( a ) = a , x ′ ( a ) = b , x ″ ( a ) = c . The main tool used is the Schauder fixed point theorem.