Bulletin of the
Korean Mathematical Society BKMS

In this paper, we consider the second-order nonlinear differential equation with the nonlocal boundary conditions. We first reformulate this boundary value problem as a fixed point problem for a Fredholm integral equation operator, and then present a result on the existence and uniqueness of the solution by using the contraction mapping theorem. Furthermore, we establish a sufficient condition on the functions $\mu$ and $h_{i}$, $i=1,2$ that guarantee a unique solution for this nonlocal problem in a Hilbert space. Also, accurate analytic solutions in series forms for this boundary value problems are obtained by the Adomian decomposition method (ADM).

Keywords: second-order equation, nonlocal boundary conditions, existence and uniqueness solution, Adomian decomposition method

MSC numbers: Primary 34B10