In this project we successfully study iterative algorithm, which is based on the modified Mann iterative method and the gradient-projection algorithm for solving a convex minimization problems and fixed point problems in a real Hilbert space. We give the numerical example to support the algorithm for solving a common solution of convex minimization problem and fixed point problem. We also recommend authors to refer this project work, for further extension of the results to more general class of mappings called, contractive type and generalized nonexpansive mappings and to more general problems called equilibrium and split feasibility problems, and application type problems.