Existence and uniqueness of solutions for a fourth-order boundary value problem

Abstract

In this paper, we study the fourth-order two-point boundary value problem x ⁗ ( t ) − f ( t , x ( t ) , x ′ ( t ) , x ″ ( t ) , x ‴ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , x ( 0 ) = x ′ ( 1 ) = 0 , a x ″ ( 0 ) − b x ‴ ( 0 ) = 0 , c x ″ ( 1 ) + d x ‴ ( 1 ) = 0 . By means of lower and upper solution method, growth conditions on the nonlinear term f which guarantee the existence of solutions for the above boundary value problem are given. In particular, we obtain the uniqueness of the solution by imposing a monotone condition of the term f .