Existence of solution for a nonlinear fifth-order three-point boundary value problem

In this paper, we explore the existence of nontrivial solution for the fifth-order three-point boundary value problem of the form u ( 5 ) ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , with boundary conditions u ( 0 ) = 0 , u ‘ ( 0 ) = u ” ( 0 ) = u ” ′ ( 0 ) = 0 , u ( 1 ) = α u ( η ) , where 0 < η < 1 , α ∈ R , α η 4 ≠ 1 , f ∈ C ( [ 0 , 1 ] × R , R ) . Under certain growth conditions on the non-linearity f and using Leray-Schauder nonlinear alternative, we prove the existence of at least one solution of the posed problem. Furthermore, the obtained results are further illustrated by mean of some examples.