We treat an initial boundary value problem for a nonlinear wave equation u t t − u x x + K | u | α u + λ | u t | β u t = f ( x , t ) in the domain 0 < x < 1 , 0 < t < T . The boundary condition at the boundary point x = 0 of the domain for a solution u involves a time convolution term of the boundary value of u at x = 0 , whereas the boundary condition at the other boundary point is of the form u x ( 1 , t ) + K 1 u ( 1 , t ) + λ 1 u t ( 1 , t ) = 0 with K 1 and λ 1 given nonnegative constants. We prove existence of a unique solution of such a problem in classical Sobolev spaces. The proof is based on a Galerkin-type approximation, various energy estimates, and compactness arguments. In the case of α = β = 0 , the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution ( u , P ) of this problem up to order N + 1 in two small parameters K , λ .