Forward-backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators

In this paper, we construct a forward-backward splitting algorithm for approximating a zero of the sum of an alpha-inverse strongly monotone operator and a maximal monotone operator. The strong convergence theorem is then proved under mild conditions. Then, we add a nonexpansive mapping in the algorithm and prove that the generated sequence converges strongly to a common element of a fixed points set of a nonexpansive mapping and zero points set of the sum of monotone operators. We apply our main result both to equilibrium problems and convex programming.